Penultimate Year Group Project Report

University of BirminghamSchool of Physics and Astronomy

Asteroseismology & Finding Planets Group Studies

Investigating the Relationship Between the Characteristics of an ExtrasolarSystem and the Properties of its Host Star

Final Report

Jeremy Galiszewski (Project Leader) (JG), Alexander Willett (Editor-in-Chief) (AW), MarkGilbert (Sub-Group Leader) (MG), Peter Steele (Sub-Group Leader) (PS), Matthew Harrison(Sub-Group Editor) (MH), Louis Pollock (Sub-Group Editor) (LP), Fergus Cowie (FC), Asher

Ezekiel (AE), Oliver Hall (OH), Morgan Lamb (ML), Chester Lewis (CL), Gareth Miller (GM),Edward Murton (EM), Laura Scott (LS)

March 26, 2015

Abstract (JG)

We have carried out an investigation into how the characteristics of an extrasolar system correspond withthose of its host star. This has been done using luminosity-time and frequency-power data of 39 stars observedby NASA’s Kepler mission. From this data, along with additional values of temperature and metallicity, wehave calculated the mass, radius, age and core hydrogen fraction of each of the stars in question using methodsof direct power spectrum analysis and asteroseismic diagrams. In the data set provided to us, the calculatedstellar mass range was 0.91 ± 0.17 M� to 2.32 ± 1.05 M�, the stellar radii ranged from 0.83 ± 0.10 R� to4.53± 0.55 R� and the age range was 0.52± 0.13 Gyrs to 8.06± 2.03 Gyrs.

In addition to the results from asteroseismology, 30 exoplanets have been detected in the luminosity-timegraphs, from which the orbital parameters for each planet could be calculated. This was achieved via thecreation of a transit detection code with accuracy determined by a separate phase folding code. A transitfitting code was used to fit a model to the data and output more accurate values, taking into account stellarlimb darkening. Comparisons of this model to an independent numerical model revealed that limb-darkeningcoefficients produced by the fit were inaccurate, decreasing the accuracy on the planetary radii as a result.With this in mind, the calculated range of planetary radii was 1.407 ± 0.180R⊕ to 1.364 ± 0.162RJup andthe calculated range for orbital periods was 2.204735299± 0.000000139days to 179.4304524± 0.00213173days.The computational methods used in obtaining these period values justify the precision in the errors. For starswith more accurate asteroseismic results, additional planetary characteristics such as mass, orbital eccentricity,composition and equilibrium temperature were estimated or constrained.

Contents

1 Introduction 11.1 Introduction to the Report (AW & MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Aims of the Report (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction to Asteroseismology (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2.1 Discovery and Development (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 MOST, CoRoT, and Kepler (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 The Current State and Future of Asteroseismology (MH) . . . . . . . . . . . . . . . . . . 2

1.3 Introduction to Planet-Detection (LP & AW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.1 Transit Method (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.2 Astrometric Wobble and Doppler Wobble Method (LP) . . . . . . . . . . . . . . . . . . . 31.3.3 Transit Timing Variation (TTV) Method (AE) . . . . . . . . . . . . . . . . . . . . . . . . 3

2 The Kepler and PLATO Missions 42.1 The Kepler Mission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Mission Design (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1.2 Spacecraft Design (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 The PLATO Mission (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Asteroseismology Theory 63.1 Asteroseismic Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 The Description of Oscillations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.1.2 The Origin of Oscillations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.2 Analysis of Asteroseismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 Frequency-Power Spectra (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Data Visualisation (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2.3 Spectrum Noise (MG & LS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2.4 Scaling Relations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2.5 Asteroseismic Diagrams (MH, EM & MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Results from Asteroseismic Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.1 Stellar Properties (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3.2 Stellar Evolutionary Theory (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4 Planet-Finding Theory 164.1 Planet Formation (AE & EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.2 Orbital Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4.2.1 Kepler’s Laws (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.2.2 Two- and Three-Body Problem (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.3 Planet-Finding Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.1 Transit Method (LP) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3.2 Transit Timing Variations (TTV) (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.3 Other Planet Detection Methods (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.4 Stellar Limb Darkening (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5 Mass Constraining Methods (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.6 Auxiliary Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.6.1 Stellar Variability (ML) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.6.2 The Effect of Starspots on Transits (CL) . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.6.3 Hill Spheres (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6.4 Roche Limit (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.6.5 Three-Day Pileup (EM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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4.6.6 Exomoon Detection (AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6.7 Circumstellar Habitable Zone (EM & MG) . . . . . . . . . . . . . . . . . . . . . . . . . . 35

5 Asteroseismology Method 385.1 Initial Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.1.1 Plotting Raw Data (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.2 Filtering and Removing Noise (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.1.3 Isolating Regions of Interest (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5.2 Manually Measuring Stellar Oscillation Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 395.2.1 Measuring Large and Small Frequency Spacing (JG & ML) . . . . . . . . . . . . . . . . . 39

5.3 Computationally Measuring Stellar Oscillation Properties . . . . . . . . . . . . . . . . . . . . . . 395.3.1 Measuring Large and Small Frequency Spacing (LS) . . . . . . . . . . . . . . . . . . . . . 395.3.2 Measuring Frequency of Maximum Power (LS & GM) . . . . . . . . . . . . . . . . . . . . 40

5.4 Obtaining Asteroseismic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4.1 Using Scaling Relations (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.4.2 Using Asteroseismic Diagrams (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

6 Planet-Finding Method 446.1 Manual Methods (LP, CL & AE) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446.2 Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

6.2.1 Introduction (FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2.2 Transit Detection Code (FC & AW) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 456.2.3 Phase Folding Code (FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 466.2.4 Transit Fitting Code (OH & FC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.2.5 Modelling for Stellar Limb Darkening (CL & OH) . . . . . . . . . . . . . . . . . . . . . . 49

6.3 Processing of Results (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

7 Results and Discussion 557.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.1.1 Key to Results Tables (MH & PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.1.2 Nomenclature (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

7.2 Full Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557.2.1 Stellar, Orbital, and Planetary Results (PS, AW, JG & ML) . . . . . . . . . . . . . . . . 557.2.2 Omissions (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.3 Results from Asteroseismic Diagrams (GM) . . . . . . . . . . . . . . . . . . . . . . . . . . 757.2.4 Comparison Between Scaling Relation and Asteroseismic Diagram Values (GM) . . . . . . 80

7.3 General Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 827.3.1 Relationship Between Stellar Mass and System Age (GM) . . . . . . . . . . . . . . . . . . 827.3.2 Relationship Between Stellar Mass and Luminosity (GM) . . . . . . . . . . . . . . . . . . 827.3.3 Relationship Between Stellar Mass and Radius (MH) . . . . . . . . . . . . . . . . . . . . . 837.3.4 Relationship Between Stellar Metallicity and Planet Composition (MH) . . . . . . . . . . 837.3.5 Relationship Between Stellar Radius and Planet Radius (JG) . . . . . . . . . . . . . . . . 837.3.6 Frequency of Orbital Period for Discovered Exoplanets (PS) . . . . . . . . . . . . . . . . . 847.3.7 The Mass-Radius Relation for Discovered Exoplanets (PS) . . . . . . . . . . . . . . . . . 847.3.8 Evidence for Tidal Circularisation (PS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

7.4 Comparison of Stellar Limb-Darkening Coefficients (CL & OH) . . . . . . . . . . . . . . . . . . . 85

8 Conclusions 908.1 General Conclusions (JG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.2 Stellar Properties (MG) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3 Exoplanet Detection Methods and Properties (LP, FC & PS) . . . . . . . . . . . . . . . . . . . . 918.4 Relationships Between Characteristics of the Extrasolar Systems and the Properties of their Host

Star (MH) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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Chapter 1

Introduction

1.1 Introduction to the Report(AW & MH)

Asteroseismology - the study of oscillations in stars- and the hunt for extrasolar planets are both rela-tively recent fields, having their beginnings as seriousresearch areas within the last half century. They arehighly interconnected, with similar technical require-ments and using similar data, which has resulted incombined space-borne missions such as CoRoT and Ke-pler.

A key area of interest with respect to exoplanetsis whether or not they would be able to support lifeas we know it. This means special attention is givento finding Earth-like planets orbiting Sun-like stars, inan orbital region known as the ‘Habitable Zone’, wherewater can exist in liquid state. However, the study ofexoplanets goes beyond the simple detection of theirexistence, as it involves the analysis of the physicaland dynamical processes of the systems and can givean insight into their formation. This requires detailedknowledge of the physical characteristics of both theplanets and their host star. It is here that both aster-oseismology and the different planet-detection meth-ods come into play. Asteroseismology provides de-tails about the host star, while various planet-detectionmethods can provide information about the planets rel-ative to their host star. Together, these two fields ofstudy can build a detailed picture of the properties ofthe star-planet systems. With enough systems doc-umented, apparent trends and relationships betweenstellar properties and planetary properties can be iden-tified, which can then increase our understanding of allplanetary systems, including our own, and their forma-tion.

In this project, we research and investigate a selec-tion of the methods available to astrophysicists in theseareas, and then go on to apply some of these methodsto Kepler data that we were given on 39 stars (exclud-ing GSIC 28 which was found to be a duplicate of GSIC1, and GSIC 37, a duplicate of GSIC 11) with a rangeof mass, temperature, metallicity, age, and number ofdetectable planets, in an attempt to derive as muchinformation about those stars and planets as possiblefrom the data we have. We then characterise the starsand planets, and attempt to discern trends betweenproperties of each of the host stars and their planet orplanets.

1.1.1 Aims of the Report (JG)

This report first of all aims to provide a sufficientbackground to provide the reader with a context intowhich the rest of the work can be placed. This comesin the form of a short history of asteroseismology andplanet-finding, an overview of the general methodsused in both fields, and a discussion of the work carriedout by the Kepler mission and the planned strategy ofthe PLATO mission.

The next aim is to provide a thorough descriptionof all the theory used by the members of the project incompiling this report. This is required by the readerto understand the rest of the report as all sections willrefer back to the theory chapters. The theory chapterswill contain all equations used later on in the reportand explain their origin, culminating in a useful com-pilation of information given in a wide range of astero-seismology and planet-finding papers published by theastrophysics community in recent years.

Once the theory has been discussed the report willtake the reader step-by-step through the methods usedin obtaining useful results from the data given. Therewere both manual and computational methods of dataanalysis used in this project and members of each teamwill explain their own methods and how they compareto those used by their counterparts.

The final aim of the report is to outline the resultsin a thorough and informative manner. An overall de-scription of each star and detectable planet is given,aiming to be consistent with the accepted practicesused in announcing newly discovered planets. Resultsfrom a variety of methods will also be displayed. Thissection will then discuss in more detail the initial re-sults by analysing trends between the stellar and plan-etary properties.

1.2 Introduction toAsteroseismology (MH)

Asteroseismology (from the Greek aster, seismos,logia meaning star-earthquake-study) as a field has itsbeginnings in the study of our own star, the Sun, knownas helioseismology. This field of research, pioneered inBirmingham, involves measuring the complex oscilla-tions that manifest on the surface of the Sun due tostanding waves set up by various internal mechanisms,and then analysing the oscillation spectra to obtain

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information on the internal structure. Asteroseismol-ogy applies these methods to other observable stars,with the principal goals of advancing theories of stellarstructure and evolution. Clearly, given the proximityof the Sun, it can be observed in much greater detailthan other stars, but the principles remain the samein many types of stars. We can treat the Sun as anyother star for comparison by viewing it in integratedlight as we do the others, rather than with the detailedimaging possible.

1.2.1 Discovery and Development (MH)

In the 1960s solar oscillation modes were detectedfor the first time. Leighton, Noyes and Simon (1962)used a Doppler shift in spectral lines technique to ob-serve radial velocity variations, in time and with po-sition on the disk, to identify oscillations with a fiveminute period [1]. In the early 1970s these oscilla-tions were explained [2] and in the following years anddecades the theory was developed and eventually ex-tended to other stars in the 1990s - probably the firstcertain detection of individual modes in another starwas by Kjeldsen et al. in 1995 [3]. Initially performedwith ground-based telescopes, asteroseismic observa-tions suffered from the usual problems of ground-basedastronomy, such as rotation of the Earth, atmosphere-related scintillation, and inclement weather.

The launch of the first dedicated space-based as-teroseismology instrument, the Canadian-run MOST(Microvariability and Oscillations of STars or Micro-variabilite et Oscillations STellaires) marked the be-ginning of much better asteroseismic data. Followingon from MOST, the French-led CoRoT (Convection,Rotation, and planetary Transits or Convection, Rota-tion et Transits planetaires) and NASA’s Kepler space-based missions permitted an explosion in the amount ofdata and the number of stars being investigated astero-seismically. Kepler in particular, with its dual, comple-mentary asteroseismology and exoplanet-hunting mis-sions, made a huge contribution: asteroseismic obser-vations often provide the best way to constrain theglobal properties of exoplanets by constraining the stel-lar properties relative to which exoplanet properties aremeasured using transit methods.

1.2.2 MOST, CoRoT, and Kepler (MH)

Launched in mid-2003, MOST was capable of ob-serving one star at a time for up to 60 days [4]. Tar-geting solar-like and also metal-poor subdwarf stars,MOST was able to obtain impressive data relative toits size (approximately 50kg and the size of a suitcase),thanks to its attitude control system and set of reac-tion wheels, on stars of apparent magnitude V = 6 andbrighter. As of June 2007, MOST had conducted 64campaigns, collecting data for more than 850 stars [5].

Beginning observations at the end of 2006, the Eu-ropean Space Agency’s CoRoT mission was extendedtwice and eventually lasted 2137 days [6], almost sixyears. CoRoT was able to observe thousands of stars

simultaneously, for up to 150 days without interrup-tion. CoRoT obtained asteroseismic data on severalhundred stars before the failure of several componentsin late 2012 [7].

NASA’s Kepler telescope launched in 2009 and iscapable of observing around 150,000 stars simultane-ously in long cadence (29.4 minutes of exposure andreadout time) and just over 500 stars simultaneouslyin short cadence (58.9 seconds of exposure and readouttime) [8] - for explanations of the cadences, see Chapter2. Kepler is responsible for the detection and confirma-tion of more than a thousand exoplanets, which can becharacterised thanks to combining transit data withasteroseismically-derived stellar properties. The Ke-pler Asteroseismic Science Consortium (KASC), withmore than 500 members around the world, is dedicatedto maintaining the Kepler Asteroseismic Science Oper-ations Centre (KASOC) database which holds all thephotometry data collected by the telescope [9].

1.2.3 The Current State and Future ofAsteroseismology (MH)

The CoRoT and Kepler missions in particular havegenerated a wealth of data on hundreds of thousandsof stars, much of which remain to be exploited. Asidefrom the simple global stellar properties that can be de-rived from asteroseismic data, novel methods are con-stantly being devised to interpret the data in order to,for example, detect binary systems that cannot be re-solved with photometry [10], measure stellar differen-tial rotation [11], or investigate the excitation of modesby gravitational waves [12], to name but a few.

Not only do we still have a lot to gain fromanalysing the data produced by those missions, butwith the launch of NASA’s TESS (Transiting Exo-planet Survey Satellite) expected in 2017, and ESA’sPLATO planned for 2024, there will be no shortageof new and improved data. This promises both im-proved accuracy for currently known stellar properties- and therefore exoplanet properties - and a large in-crease in the number of stars which we can analyseasteroseismically for exoplanet purposes or to betterour understanding of stellar evolution.

1.3 Introduction to PlanetDetection (LP & AW)

The term extrasolar planet, or exoplanet, refersto a planet orbiting a star other than our own Sun.The existence of these has been theorised for centuries,beginning with Giordano Bruno, an Italian Catholicmonk and supporter of heliocentricity, controversiallyasserting the existence of “countless suns and count-less Earths all rotating around their suns” in 1584 [13].However, it hasn’t been until the last 23 years that thefield has yielded any results, although there had pre-viously been a couple of unconfirmed detections. Forexample, in 1988 a suspected planet was detected out-side of the Solar System [14], however due to the lowprecision of measurements, this could not be confirmed

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until 2003 [15]. A more significant development wasmade in 1992 when Dr. Aleksander Wolszczan pub-lished his findings of the first confirmed exoplanets, or-biting a pulsar, PSR B1257+12 [16]. The planets werediscovered by studying the variations in the pulsationsof the pulsar, brought about by the gravitational ef-fects of the mass of each planet. It wasn’t until 1995that the first exoplanet orbiting a main-sequence (MS)star, 51 Pegasi, was discovered by Michael Mayor andDidier Queloz of the University of Geneva [17]. It wasthis discovery that has now led to the discovery of over1800 confirmed exoplanets within the last twenty years,with projects such as CoRoT and Kepler working onthis task [18].

The detection of extrasolar planets is very chal-lenging due to the low apparent size and brightnessof distant stars. With exoplanets only a fraction of thesize of these stars and with a negligible flux, very highprecision telescopes and spectroscopes are required tolocate these planets. It is generally more straight-forward to detect large, massive planets than small,Earth-like planets, as such planets have greater gravi-tational influences on their host stars. It is for this rea-son that the majority of exoplanets located to date ex-hibit Jupiter-like properties. There are many methodsthat can be used to detect exoplanets, the most com-mon of which are the transit method and the Dopplerwobble method. The Doppler method was generallyused to discover the first extrasolar planets, howeverin recent years the most successful means of discoveryfor new planets has been via the transit method. TheKepler mission has been instrumental in this, with thevast majority of these transit detections having beenachieved using data from the Kepler spacecraft. TheDoppler wobble method and transit method are de-scribed below, along with the less practical astrometricmethod.

1.3.1 Transit Method (LP)

The transit method takes into account any peri-odic changes in the flux of a distant star that mayindicate the presence of a transiting planet [19]. Whenan exoplanet moves between its host star and an ob-server on Earth, a small fraction of the flux from thestar is blocked by the planet. Dips in flux from a starare common however, and many considerations needto be made before confirming the presence of a planet;these are discussed in Chapter 4. The transit methodprovides information on the period of the orbit of theplanet as well as the ratio of the planetary radius to thestellar radius. Phenomena such as stellar limb darken-ing and varying impact parameters add complicationsto calculations that determine planetary characteris-tics; these are also discussed later in the report.

1.3.2 Astrometric Wobble and DopplerWobble Method (LP)

The astrometric wobble method considers the rel-ative position of the stars in the night sky, with the

positions of stars changing due to the gravitational at-traction of orbiting bodies. When a planet orbits astar, the gravitational pull of the planet can move thestar to such a degree that its motion can be detected.The stellar motion is largest and therefore most easilydetectable with a massive planet in orbit with a largesemi-major axis. The mass of the star is also an im-portant factor in the motion, with less massive starswobbling to a larger degree. To detect planets usingthis method, data regarding the positions of stars withtime is required. No extrasolar planets have yet beenconfirmed using this method, as movements of distantstars due to orbiting planets are too small to detectfrom Earth.

The Doppler method, on the other hand, consid-ers the changing wavelength of light detected from adistant star that indicates the presence of a compan-ion planet. A distant star moving towards and awayfrom us will give a sinusoidally varying spectrum ofshifted wavelengths, from which the period of orbit ofits companion planet can be extrapolated. Over 300extrasolar planets have been found to date using thismethod with the vast majority lying within 3 AU oftheir host star [20]. On top of this, the majority of thelocated planets possess masses close to that of Jupiter,with the masses increasing as the distance from the starincreases. The method only works if the planet andstar are moving towards and away from us. The radialvelocity can be found using this method (the velocitytowards or away from the observer). From this a min-imum mass for the planet can be found - inclination ofthe system will reduce the radial velocity component,so we would see an effect on the star as though from aless massive planet. If the spectral data of the planetare known, this can be used to find the planet’s actualmass. The Doppler wobble of distant stars has beendetected by such facilities as PRIMA [21] - an instru-ment used in the VLT (Very Large Telescope) whichdetects Jupiter-like planets.

1.3.3 Transit Timing Variation (TTV)Method (AE)

Transit Timing Variation (TTV) is a technique thatlooks for changes in the timings of transits for a planetdetected by the transit method to infer the presence ofanother planet.

When using the transit method to detect planets,the transits should be strictly periodic, that is the timebetween each transit should be exactly the same. Ifthe time of each transit is measured accurately and avariation in transit timing can be seen, this suggeststhe presence of another body, usually a planet, grav-itationally perturbing the orbit of the planet aroundthe star. By measuring the variation in transit timingand observing it as a function of time, it is possible todetermine properties of the perturbing body such asmass and orbital radius [22]. Natural satellites orbit-ing exoplanets (or exomoons) can also cause TTVs andthis is also discussed.

3

Chapter 2

The Kepler and PLATO Missions

2.1 The Kepler Mission

2.1.1 Mission Design (PS)

Launched in March 2009, the Kepler spacecraft wasdesigned to observe one specific region of The MilkyWay in optical wavelengths, with the aim of detectingextrasolar planets via the transit method. The datafrom the initial data collection period has been used toconfirm the existence of over 1000 extrasolar planets todate, with the possibility of many more detections tocome [23].

The spacecraft orbits the Sun in an ‘Earth-trailing’orbit of period 372.5 days, appearing to slowly recedefrom the planet. In order to prevent light from theSun saturating the telescope, the spacecraft executesa 90◦ roll four times each year in order to orient theSunshade correctly.

The Kepler spacecraft was planned to be stabilisedby three reaction wheels (with a fourth wheel in re-serve). These wheels rotated the spacecraft in threedimensions to both ensure that the telescope aper-ture continued observing the intended area, and also tomaximise the area of the solar panels in direct sunlight.Unfortunately, by May 2013 two of these wheels hadfailed, jeopardising future data collection [24]. In re-sponse to this, a second mission plan, K2, was devised.The spacecraft will be oriented along the ecliptic planeof the solar system, with stellar radiation pressure thenused to balance the spacecraft in lieu of a third reac-tion wheel, so that several regions on the plane of theecliptic can be studied [25].

2.1.2 Spacecraft Design (PS)

The Kepler spacecraft comprises a large photome-ter, along with non-measurement-oriented systems totransmit collected data to Earth, control spacecraft at-titude and regulate spacecraft power and temperature.The photometer is the instrument from which all thedata used in this report ultimately stems, and thus it isappropriate to discuss its design in greater detail here.

Figure 2.1.1 – Basic structure of the Kepler spacecraft,showing several important systems [26].

Photometer

The telescope for the Kepler spacecraft is a Schmidtcamera of aperture diameter 0.95m, which provides a101deg2 field of view. To prevent saturation of pix-els by contaminant light from the Sun, a large shadeprotects the aperture of the photometer.

The photometer itself comprises a total of 4250x25mm CCDs, each with 2200x1024 pixels. TheCCDs are mounted on a slightly curved base, as theprimary mirror produces a curved focal plane. As thetelescope’s field of vision is fixed, each star in Kepler ’sview corresponds to a certain pixel or group of pixels,the data from which are read out at a constant interval

4

of 6 seconds in order to prevent pixel saturation. Onlypixels corresponding to stars of interest in the studyare used. The data from these pixels is then integratedinto a cadence, which comprises 270 readings, mean-ing that each cadence records a total amount of lightabsorbed from each star over a period of 0.49 hours,known as a ‘long cadence’. For stars of particular in-terest, ‘short cadence’ data can also be recorded, witha length of 1 minute [27].

In addition to the 42 ‘science’ CCDs, 4 furtherCCDs in the corners of the array are used to monitor40 ‘guide’ stars, in order to ensure that the instrumentis stable and the field of view remains the same. Datafrom these CCDs has shown the spacecraft to be drift-ing only a few millipixels from its intended positionover each 3 month period of observation [28].

Figure 2.1.2 – Cross-section of the Kepler photometer,showing focusing of input light via the primary mirror ontothe CCD array [26].

2.2 The PLATO Mission (EM)

PLAnetary Transits and Oscillations of stars isa mission planned by ESA that will be the spiritualsuccessor to the NASA Kepler mission. PLATO isplanned for launch in 2024 when it will be launchedup to the L2 Lagrange point [29], an orbit that con-stantly shields the spacecraft from much of the sun-light. The mission plans for 6 years of observationsfrom the shadow of the Earth where the 34 indepen-dent telescopes and cameras will cover ∼ 2250 squaredegrees of the sky per pointing. Two of the cam-eras are short 2.5 second cadence cameras looking atthe brighter stars with the remaining thirty-two long25 second cadence cameras recording data from starsthat have visual magnitudes greater than eight. [30]The mission will be providing vital data to continueprogress made by the now limited capabilities of theKepler mission and is planned to last for 6 years andobserve over 1 million stars [29]

Photometer

Each camera on the PLATO spacecraft is made upof an array of four 4510x4510 pixels each of side 18μm.Different regions within each pointing are monitoredby different numbers of cameras. The centre is ob-served by 32 cameras, 24 are used in the observationof another region with 16 and 8 for a further two.

Figure 2.2.1 – Field of view of the PLATO mission dividedinto sections corresponding to number of telescopes moni-toring each region [31]

For every two cameras there is one Data Process-ing Unit which completes the necessary calculationsand modifications on the data before it is sent to theInstrument Control Unit, which sends the data to theground in order to be analysed.

5

Chapter 3

Asteroseismology Theory

3.1 Asteroseismic Oscillations

3.1.1 The Description of Oscillations(MH)

Before discussing the mechanics of the oscillations,it is useful to introduce the notation used to describethem. The oscillations within a star cause the surfaceto expand and contract periodically, thus a star thathas only radial oscillations would simply expand andcontract uniformly and symmetrically. However, theintroduction of non-radial oscillations causes sphericalsymmetry to be broken. Parts of the surface expandwhile other areas contract, with node lines betweenthese areas where there is neither expansion nor con-traction. This results in a huge variety of different sur-face patterns of expansion and contraction dependenton the combinations of radial and non-radial oscillationmodes.

Figure 3.1.1 – Depiction of the oscillations for several valuesof l and m. The dark grey regions are expanding, the lightgrey regions contracting, and the white lines represent nodelines. The equator and pole are marked [32].

The radial components are described by the radialovertone number n (sometimes written in the literaturewith k), the number of nodes along the radius. Thespherical components (corresponding to θ in spheri-cal polar coordinates) are described by the angular orspherical degree l , the number of node lines parallelto the equator between the two poles, and finally theazimuthal components are described by the azimuthalorder, m (corresponding to φ), the number of node

lines that pass through the pole.The angular degree l and azimuthal order m also

define the spherical harmonic functions that describethe small-amplitude displacements [19]:

Yml (θ,φ) = (–1)mcl,mPm

l cos(θ)exp(imφ) (3.1)

where θ and φ are the latitude and longitude anglesrespectively, cl ,m are normalisation constants and Pm

lare Legendre polynomials. Figure 3.1.1 shows a visuali-sation of the manifestation of these spherical harmonicsfor some low-degree modes.

An oscillation with angular degree l = 0 is a radialone which passes along a diameter through the centre ofthe star. These oscillations therefore have the longestsurface wavelength and the lowest frequency for anyparticular radial overtone n.

3.1.2 The Origin of Oscillations (MH)

Oscillations in a star can be generated by severalmechanisms, among them the kappa-mechanism whichis the mechanism for strongly varying classical pul-sators like Cepheids. This mechanism results from lay-ers of high opacity (due to ionised helium) acting likea valve and blocking radiation from escaping. Thiscauses a build up of pressure which eventually resultsin a rapid expansion of the star as the energy is re-leased. The star can then contract and heat up again,and the process repeats. These types of pulsation candisplay visual amplitudes of up to two magnitudes, andperiods of days to weeks [33]. The important class ofoscillations for this project is solar-like oscillators, sonamed because this is how our Sun oscillates. Solar-likeoscillations are typically on the order of several partsper million in amplitude, with frequencies of minutesto hours. There exist two main types of oscillations: p-mode acoustic oscillations for which pressure gradientsprovide the restoring force, and g-mode buoyancy oscil-lations which are damped by gravity. The frequency re-gions of the two types can be defined with reference totwo characteristic frequencies: the Brunt-Vaisala andthe Lamb frequency, NBV and Ll respectively [32]:

N2BV = g

(1

Γ1

dln(P)

dr–

dln(ρ)

dr

)(3.2)

L2l =

l(l + 1)c2

r2(3.3)

6

where g is the gravitational field strength, Γ1 is thefirst adiabatic exponent, P is the pressure, ρ is thedensity, l is the angular degree, c is the local speed ofsound, and r is the radius. Oscillations with frequencyhigher than both the Lamb and the Brunt-Vaisala fre-quency are p-modes; oscillations with frequency lessthan both the characteristic frequencies are g-modes.As a star evolves, the g-mode frequencies increase andthe p-mode frequencies decrease. This can result ininteractions between the two modes, producing oscilla-tions that combine both g- and p-behaviour, known asmixed-modes.

P-mode Oscillations

In stars which have near-surface convection zones(CZs), i.e. generally those stars with mass less than1.1M� and temperature less than 7500K [19], turbu-lence in the outermost part of the convective regionsrandomly excites acoustic oscillations. These soundwaves can travel through the rest of the star, and pen-etrate to different depths depending on the angle atwhich they are propagated from the CZ. Due to theincrease of temperature going inwards towards the cen-tre of the star, wavefronts are refracted away from thecentre back towards the surface, and because, abovethe surface, the solar atmosphere is much less dense,waves cannot be sustained and are reflected back in,thus oscillations are trapped within the star and travelaround beneath the surface. There is therefore a fun-damental maximum frequency of the oscillations, theacoustic cut-off frequency, νac . The frequency is de-fined by the speed of sound in that medium and thescale height, H [3]:

νac ∝c

2H∝ gT

–1/2eff (3.4)

If the frequency of a wave is higher than this, i.e. itsperiod is short enough, it can be sustained in the lowerdensity and will escape from the stellar interior into theatmosphere. Therefore only oscillations of frequencieslower than νac reflect back in and become trapped andgenerate the standing waves which cause the surfaceexpansion and contraction. Figure 3.1.2 shows how awave that propagates deeper into the star before turn-ing back up towards the surface will reach the surfacefewer times on its trip around the star, so will exhibit alonger wavelength on the surface and will have a lowerfrequency (for the same radial order), denoted by alower angular degree, l .

P-mode oscillations are observed to follow theasymptotic relation, established by Tassoul (1980) [34],which is valid for high values of the radial overtone [35]:

νn,l '(

n +l

2+ ε

)Δν0 – l(l + 1)D0 (3.5)

where ε is related to physics of the surface layers andD0 depends on the sound speed gradient near the coreof the star.

The capabilities of the telescopes used to detect theoscillations in other stars by their changes in bright-ness limit the number of degrees we can observe. Since

the amplitude of the oscillations decreases with increas-ing l , the highest degree oscillation measured in otherstars is l = 3 , for stars with particularly good obser-vational data and high signal to noise ratios (SNR).This is less problematic than it may seem, as theselow-degree modes are those that penetrate most deeplyinto the star and hence are most useful for probing stel-lar structure, as well as being used to determine globalstellar properties. The radial overtones of the frequen-cies measured are typically in the range n = 10 – 30 .

Figure 3.1.2 – Oscillations of varying angular degree, l . Thelower the angular degree, the deeper into the star the oscil-lation penetrates, down to l = 0 , the radial mode passingthrough the centre [36].

G-mode Oscillations

In the central regions of solar-like stars, which arestable against convection, the displacements of pock-ets of gas upwards and their subsequent sinking due togravity result in oscillatory motions around their po-sition of equilibrium. This motion in turn regularlydisplaces adjacent material perpendicular to the ra-dial direction, and hence g-mode oscillations are al-ways non-radial. They are also restricted to the ra-diative central regions of the star; they cannot crossthe convective boundary as in the convectively unsta-ble zones the oscillatory motion of pockets of gas doesnot occur, and they therefore manifest only with verysmall amplitudes at the surface of the star, makingthem usually very difficult to detect. This is unfor-tunate, as g-mode oscillations, being very sensitive toconditions in the core of the star, are very useful forinvestigating stellar evolution. G-modes are much eas-ier to detect when their frequencies increase and theyundergo avoided crossing interactions with p-modes,i.e. in evolved stars. In these cases, mixed-mode os-cillations allow g-mode properties to be brought to thesurface via their interactions with p-modes. Thus g-modes are more often detected in stars evolved off themain sequence (MS).

7

F-mode Oscillations

F-mode, or fundamental mode oscillations, are theoscillations with radial order n = 0 . These oscillationsbehave similarly to p-mode oscillations, but are tech-nically distinct. At high angular degrees l > 20 , wherethey penetrate only very slightly into the star from thesurface, they can be approximated as surface gravitywaves - like ripples on the surface of a large body ofwater - with frequency given by [36]:

ω =√

g0kh (3.6)

where g0 is the equilibrium value of the surface gravityand kh is the horizontal wavenumber given by [19]:

kh =

√l(l + 1)

R(3.7)

where l is the angular degree and R is the stellar radius.

Mixed-mode Oscillations

After some evolution, typically into the subgiantand red-giant phases, when the core has contractedand the envelope expanded, g-mode frequencies are in-creased and p-mode frequencies are decreased due tothe decrease in mean density, eventually to the extentthat the frequency regimes of the two types of oscilla-tion overlap. When a g-mode and p-mode of similarfrequency and the same angular degree (except l = 0for which there are no g-modes) approach, they caninteract and undergo an ‘avoided crossing’, resultingin a change in the frequencies called mode-bumpingwhich causes the oscillations to no longer follow theasymptotic relation. This phenomenon can make thefrequency-power spectrum harder to interpret as themode peaks become more irregular, but the interac-tion can be witnessed in an Echelle diagram as thelower end of the l = 1 ridge fragments due to the ef-fect on the large spacing (see Section 3.2.2). Avoidedcrossings of l = 2 modes are much weaker.

3.2 Analysis of AsteroseismicData

3.2.1 Frequency-Power Spectra (MH)

The observational data obtained from telescopessuch as CoRoT and Kepler are used to generate plots ofthe frequencies detected. The Doppler shifts or inten-sity changes in emitted light are measured for the inte-grated light from the disk, and the variation with timeis analysed by a Fourier Transform [32]. The result-ing frequency-power spectrum (FPS), shown for one ofthe stars from our dataset (GSIC 0) in Figure 3.2.1,when smoothed and filtered to account for noise (seeSection 3.2.3), shows the relative power of the oscilla-tions as a function of frequency and from these plotscan be extracted several useful quantities. The regionof interest, where the amplitudes of the oscillations onthe surface are large enough to be measured above the

noise (either intrinsic to the star or instrumental), lieswithin a region bounded by two frequency limits. Themaximum frequency is the acoustic cut-off frequencymentioned above. The lower limit is the frequency ofthe fundamental oscillation, where n = l = 0 .

Figure 3.2.1 – Part of the raw frequency-power spectrum forGSIC 0 from our set of stellar data. Peaks of l = 0 , 1 , 2 canbe discerned (and are labelled) even in the raw spectrum.

First, the visible frequencies can be fitted with aGaussian envelope in order to find the frequency forwhich the oscillation power is greatest (not just themode with the strongest amplitude), νmax . This fre-quency is observed to scale with the acoustic cut-offfrequency [3]:

νmax ∝ νac ∝ gT–1/2eff (3.8)

such that for solar-like stars the ratio νmax/νac is thesame.

Next, the ‘frequency separations (or spacings)’ canbe measured. The large spacing, Δν, is the differencebetween the frequencies of two oscillation modes of thesame angular degree and consecutive radial overtone:

Δνn,l = νn,l – νn–1,l (3.9)

The large spacing corresponds to the inverse of thetime taken for sound to travel the diameter of the star.The small separation, δν, is given by the differencein frequency of oscillation modes of consecutive radialovertone and odd or even consecutive angular degree:

δνn,l = νn,l – νn–1,l+2 (3.10)

The large spacing can be combined with the fre-quency for maximum power and the effective tempera-ture of the star (measured independently by photome-try) to estimate, to a reasonable degree of accuracy,global properties like mass and radius, from whichmean density and surface gravity can be obtained (seeSection 3.2.4). The large and small spacings can beused together to determine stellar properties such asmass, hydrogen concentration, and age; the methodsare described in Section 3.2.5.

As implied above, one can also use the large spacingto calculate the acoustic radius of the star, that is thetime taken for sound waves to travel the radius of the

8

star, given by [37]:

T0 =

∫ R

0

1

csdr ≈ 1

2〈Δνn,l〉(3.11)

where T0 is the acoustic radius and cs is the speed ofsound. This quantity can be useful as it can be usedto estimate the depth of the base of the sub-surfaceconvection zone. At the base of the convective zonethere is a sharp change in structure. If the region ofthe change, known as a glitch, lies within the cavityprobed by an oscillation mode, a periodic variation isintroduced into the frequency, a component of which isthe acoustic depth of the glitch from the surface. Thesinusoidal variation can be fitted to find the acousticdepth - which is proportional to the period of the vari-ation, which can be compared to the acoustic radiusof the star in order to find an absolute value for theradius of the base of the convection zone [37], if this isthe feature causing the glitch. In Solar-type stars, thebase of the convection zone lies at an acoustic depth ofapproximately T0/2 or deeper [37].

Also from the FPS, by examining the oscillationpeaks in fine detail, it is possible to find evidence ofa star’s rotation, from which can be determined theangle of inclination of the rotation axis [38]. This isuseful information, providing a likely value of the incli-nation of the plane of any planets in the system - spin-orbit misalignment is possible but uncommon. When astar has significant, but not too fast, rotation, and hasan inclination, i , which means its rotation axis is notparallel to the line of sight (which would be i = 0 ◦),oscillation frequencies are shifted: higher when the os-cillation moves with the rotation and lower when itmoves against. Thus peaks of degree l are split intoeither triplet or doublet peaks of degree and order l ,|m| ≤ l . The relative power of these peaks can be usedto calculate the angle of inclination [38]. The relativepower of the central peak to the split peaks, εl ,m , is re-lated to the inclination, as shown in Figure 3.2.2, anddepends on the angular degree [38]:

ε1,0(i) = cos2(i) (3.12)

ε1,±1(i) =1

2sin2(i) (3.13)

ε2,0(i) =1

4(3cos2(i) – 1)2 (3.14)

ε2,±1(i) =3

8sin2(2i) (3.15)

ε2,±2(i) =3

8sin4(i) (3.16)

where ∑m

εl,m(i) = 1 (3.17)

There are some limitations on the usability of thistechnique. This method works best for angles of in-clination less than 30◦, approximately 15% of stars.

At least a useful estimate value can be obtained fromspectra with an SNR down to 20. If the rotation axisis parallel to the line of sight, no splitting is seen, andit is not possible to distinguish the cases of no incli-nation and no rotation without other information. Forthe splitting to be resolvable, the angular velocity, Ω,must be greater than twice the line width of the peak.Thus, as the line-width increases with frequency [38],lower frequency modes offer the best chance for thisanalysis. Ω can be obtained approximately from [38]:

νn,l,m = νn,l,0 + mΩ (3.18)

Figure 3.2.2 – Rotational splitting of frequency peaks fortwo different inclinations. The greater inclination has splitpeaks with a higher relative power [38].

3.2.2 Data Visualisation (MH)

Figure 3.2.3 – The frequency-power spectrum for GSIC 40.This is an evolved star, and with the p-mode frequenciesspikes labelled, g-modes can be seen between them.

9

Figure 3.2.4 – Some examples of Echelle diagrams. Theupper panel (MS star 16 Cyg A) shows very clean ridgesof each angular degree, even faintly l = 3 ; the lower panel(star KIC6442183, a subgiant) shows evidence of an avoidedcrossing [37].

The Echelle diagram provides a useful method oflooking at a frequency-power spectrum’s information.The diagram is created by dividing the FPS into slicesof width Δν and essentially superposing them on topof each other. On the horizontal axis is frequency mod-ulo the large spacing, ν%Δν, and on the vertical axis isfrequency. The frequencies of the oscillation modes inthe slices is plotted, and thus vertical lines are visible,each corresponding to a value of l , because the largespacing is approximately the same for each l . In realitythe ridges are almost always curved, concave towardsincreasing ν%Δν. This is an artifact of the inaccu-racy of the simplified asymptotic relation which doesn’ttake into account the change of the large separationwith frequency: the large frequency is usually mea-sured around νmax , so for higher frequencies the sepa-ration is greater and for lower frequencies it is smaller.An Echelle diagram can be used to identify avoidedcrossings and mixed-mode oscillations, as shown in thelower panel in Figure 3.2.4: the fragmented lower endof the l = 1 ridge is evidence of the bumped frequenciesand subsequently affected large spacing. This could beemployed in helping to identify modes in irregular spec-tra. For example, without the labels on Figure 3.2.3which comes from an evolved star, the spectrum wouldlook very irregular due to the presence of the g-modes.An Echelle diagram could be used to find the verti-

Figure 3.2.5 – Granulation as it appears on the surface ofthe Sun. Hot, rising plasma will appear brighter than thecooler, sinking intergranular lanes between cells [39]

cal ridges corresponding to p-modes and thus identifythose peaks.

3.2.3 Spectrum Noise (MG & LS)

Granulation (MG)

Stellar granulation is a highly significant contribu-tor of noise in the FPS. Granules are cells of convectionthat appear on surface of stars, such as the Sun, witheffective temperatures lower than about 7500K thathave an outer convective zone. The typical diameterof these cells is 1000-1500km. They arise from convec-tion currents which transport hot plasma from belowthe surface upward, with the intergranular lanes in be-tween cells providing the return flow of cooler, henceless luminous, plasma. Importantly, the lifetimes ofthe granules are similar to the periods of the p-modeoscillations so attempting to model and reduce contri-bution of granulation to the noise in the FPS wouldgreatly benefit analysis.

The effect of granulation in the FPS can be seenpredominately at the lower frequencies (≤1 mHz)where the noise can be seen to increase as the fre-quency decreases. Mesogranulation and supergranula-tion – groups of convective cells of increasing diameterand lifetime – along with granulation create the layerof noise as described above. On the stellar surface thevertical granule velocity (ranging up to the order ofkms–1) far exceeds that of the oscillations we wish tostudy and, in the FPS of evolved stars, the frequencyof p-modes can have been reduced low enough to over-lap with the granulation noise, making it difficult toanalyse these oscillations (especially at low signal tonoise ratios). One distinguishing property of granula-tion that can be exploited is its coherence as discussed

10

by Elsworth and Thompson (2004) [40]. Oscillationscan rise above the granulation background providedthat both the variations are observed for long enoughand the mode lifetimes are long enough. Modellingthe signature of this phenomenon should be possibleand, if achieved, would greatly improve the ability toanalyse stars where the granulation noise in the FPSsignificantly affects detection of the oscillations. Oneobstacle faced when trying to build such a model is theunpredictable nature of granulation from one star toanother. A successful attempt to model solar granula-tion was made by Samadi et al. (2013) [41] with theaim of extending their methods to stars being observedby CoRoT and Kepler. They produced a simple, 1Dtheoretical model which reproduced the observed solargranulation spectrum, as well as a more complex 3D ra-diative hydrodynamical model which saw granulationin disk-integrated intensity (as is seen in our FPS) rep-resentative of the surface layers of F-type dwarfs andred giant stars. These models, however, are non-trivialand beyond the scope of this investigation, as muchof the focus is on analysis of the FPS and creation ofasteroseismic diagrams detailed later in Section 3.2.5.

Reducing the Noise (LS)

If a star’s modes of oscillation are to be found mea-sures must be taken to account for the noise in theFPS. One way of doing this would be to smooth the rawdata using a boxcar filter, as was done in this project.This can be done by convolving the data with an ar-ray whose size is known as the window size and whoseelements are all one, or equivalently by calculating amoving average. The effect of the boxcar filter is tosmooth the data, with a larger window size producingmore severe smoothing. The boxcar is most effectiveat dealing with white noise as the filter has a constantamplitude.

However, the total noise in the FPS is not justwhite. The low-frequency areas of the FPS have morepowerful noise than the high-frequency areas due togranulation. Applying the boxcar filter to an FPSwill not remove the increase in power seen in the low-frequency areas. Although the stars studied in thisproject did not require anything more than the boxcarfilter (see Section 5.1.2), it may not be adequate forother stars. In more evolved stars for example, wherethe modes appear at lower frequencies, the signal maybe well within the region of granulation noise.

Modelling the noise and then subtracting the modelaway from the data is one way to deal with thefrequency-dependance of the noise. In order to do thisthe noise due to each component of stellar activity mustbe summed. The power of an individual component ofthe stellar activity noise can be written as [42]:

P(ν) = η(ν)2 4ζσ2τ

1 + (2πντ)c (3.19)

where ν is the frequency, η is a damping factor arisingfrom the discrete nature of the data, ζ is a normalisa-tion constant that depends on the stellar activity pro-

cess, σ is the granulation RMS velocity, τ is the char-acteristic timescale of the noise-causing process and cis an exponent equal to 4.

The form of equation 3.19 is based on the modelproposed by Harvey (1985) [43] that estimates the ef-fects of granulation as being a pulse in power that sud-denly appears and then exponentially decays. In Har-vey’s model the value of c is 2 (making the noise powera Lorentzian function). Corrections made by Kallingeret al. (2014) [42] when better data became availablemodified the value of c to be 4. This is consistentwith a power pulse which symmetrically and exponen-tially rises and decays with a characteristic timescale,τ. The rise and decay of the power suggested by thismodel is representative of the fact that the granules ona star’s surface appear and disappear with those sametimescales. Granulation, mesogranulation and super-granulation all have different values of τ.

3.2.4 Scaling Relations (MH)

Once the FPS has been filtered, smoothed, and theuseful quantities extracted, we can use those quanti-ties to obtain estimates of the global properties of thestar. For solar-like stars, i.e. those that have near-surface convection zones, if we assume that for themain-sequence (MS) and red-giant (RG) stages the os-cillation properties scale well against a solar reference,we can derive relations to estimate properties of thesestars just from the FPS and an independently mea-sured temperature. Kjeldsen and Bedding (1995) de-rive relations for the large spacing and the frequency ofmaximum power [3]. The large spacing, Δν, is approx-imately equal to the inverse of the time for sound tocross the star, which is related to the temperature andthe density and hence the mass and radius. The fre-quency for maximum oscillation power, νmax , can alsobe scaled via the cut-off frequency, νac , which defines adynamical timescale and hence is related to the surfacegravity and temperature. From these relations, scalinglaws can be derived for these two quantities [3]:

νmax

νmax,�=

(M

M�

)(R

R�

)–2( Teff

Teff,�

)–1/2

(3.20)

Δν0

Δν0,�=

(M

M�

)1/2( R

R�

)–3/2

(3.21)

These relations can be easily rearranged to giveequivalent reverse scaling relations to find estimatesof the mass and radius, and hence density and surfacegravity of any star for which the large spacing, fre-quency for maximum oscillations power, and effectivetemperature are known [37]:

R

R�'(νmax

νmax,�

)( 〈Δνn,l〉〈Δνn,l〉�

)–2(Teff

Teff,�

)1/2

(3.22)

M

M�'(νmax

νmax,�

)3( 〈Δνn,l〉〈Δνn,l〉�

)–4(Teff

Teff,�

)3/2

(3.23)

11

These scaling relations will give errors on the orderof one to ten percent [37], of which the error on masswill be larger. If the goal is to constrain the massand radius of extrasolar planets around a star analysedin this way, these large errors render these values oflimited use. Other methods must therefore be usedto retrieve accurate deductions of the value of theseproperties.

3.2.5 Asteroseismic Diagrams(MH, EM & MG)

Attempting to use a Hertzsprung-Russell diagramto match an observed star to an evolutionary track isdifficult due to the degeneracy of the HR diagram: cer-tain regions of the diagram have many tracks, of widelyvarying properties such as mass and age, very close to-gether such that only a small error on the temperatureand luminosity of the star can result in a large num-ber of possibilities. Hence, so-called ‘asteroseismic HRdiagrams’ (AHRDs) are used instead. These are plotsof asteroseismic quantities which result in much moredistinct evolutionary tracks and a much lower degree ofdegeneracy, particularly for solar-like stars on the MS.Stellar modelling is used to generate a large number ofmodel stars with a range of masses, radii, metallicities,and other initial parameters, including relevant stel-lar physics, whose evolution is simulated. From thesemodel stars, simulated oscillations are generated usingthe asymptotic relation in Equation (3.5), in which D0

is related to the small frequency spacing by:

δνn,l = (4l + 6)D0 (3.24)

The spectra are analysed and the asteroseismicquantities are calculated and plotted against each otheron an AHRD. The small and large spacing are usedparticularly because they are sensitive to the evolutionand mass of stars. The large spacing is proportional tothe square root of the mean density of the star, whichincreases with stellar mass and decreases slightly withevolution. D0 is sensitive to the sound speed gradient.With evolution, the sound speed decreases towards thecentre as the proportion of helium and therefore themean molecular weight, μ, rises, and so D0 decreaseswith evolution. The small spacing δν therefore alsodecreases. Thus on this grid of stellar models, evolu-tionary tracks as lines of constant mass, hydrogen con-tent (isopleths), and age (isochrones) can be drawn,and a real star whose FPS has been analysed can belocated on this plot to give a very good estimate of itsglobal properties. Figure 3.2.6 shows an asteroseismicdiagram with the small separation plotted against thelarge separation, to give a grid with lines of constantmass and hydrogen content.

Figure 3.2.6 – An asteroseismic HR diagram (or JCD orC-D diagram) of the small separation plotted against thelarge separation for MS stars. The solid lines are lines ofconstant mass, and the dotted lines are lines of constanthydrogen content [44].

This is the most common form of AHRD, as thehydrogen fraction is a well-defined quantity, whereasage is more subject to other parameters. The age canbe approximately obtained, assuming correct models,from the hydrogen fraction [19]:

T =10 – 14.3X

M/M�x109years (3.25)

where T is the age of the star, Xc is the hydrogenfraction and M is the mass of the star.

It can also be useful to plot a ratio of the small andlarge separations, rl ,l+2 :

r0,2(n) =δν0,2(n)

Δν1(n)(3.26)

or

r1,3(n) =δν1,3(n)

Δν0(n + 1)(3.27)

against the large spacing, as this quantity is not sen-sitive to surface effects or the structure of the outerlayers of a star, which can vary strongly between mod-els [45]. It should be noted that Chaplin et al. (2005)found evidence that for the Sun as a star, the frequencyratio changes with the activity level [46], and so variessinusoidally on a timescale of decades. This timescaleis much longer than the timebase of our data, so shouldbe insignificant, assuming our stars have similar activ-ity periods, if they have activity. White et al. (2011)also noted that for the l = 1 modes there is departurefrom the asymptotic relation for more evolved stars,making Δν0 more reliable, and that when there are noavoided crossings Δν0 ' Δν1, so there is little impacton the diagram from changing the definition of the ra-tio [44]. Hence the most useful form is:

r =δν0,2

Δν0(3.28)

Another important building block in AHRDs ismetallicity. Metallicity defines the fraction of heavy

12

elements, or metals, in a star. In this context a metalis any element heavier than helium. Stellar hydrogen,helium and metal fractions – X, Y and Z, respectively(where X + Y + Z = 1) – are used to describe the ma-terial content inside stars and play a significant rolein how both actual stars, and the stellar models usedto build the AHRDs, evolve. Metallicity values areoften presented as the logarithm of the ratio betweenconcentrations of iron (Fe) and hydrogen (H) observednear the stellar surface, calibrated with the Sun (Z� =0.017).[

Fe

H

]= log10

(NFe

NH

)∗

– log10

(NFe

NH

)�

(3.29)

where NFe and NH are the respective number densitiesof iron and hydrogen near the surface of the stars. Withreference to HR diagrams, a relatively low metallicity,solar-like, Zero Age Main Sequence (ZAMS) star willappear bluer, i.e. of higher temperature, compared to astar of higher metallicity, due to decreased opacity andline blanketing which increases its effective tempera-ture. The luminosity will vary very little with metal-licity, therefore it can be inferred that the radius of alow metallicity star will be lower than that of a highmetallicity star of the same mass to maintain similarluminosities (see Equation 4.84). Having a reduced ra-dius will increase the overall density of a low metallicitystar so the value of the large spacing would be expectedto increase, and vice versa.

A small change in the metallicity value of the starsused to model the grid in the AHRDs will have a greatimpact on position of the grid itself, as can be seen inFigures 3.3.1 and 3.3.2 in Section 3.3.2.

It is an apparent fact for any AHRD, such asthe those illustrated, that metal-poor stars have theirtracks moved up and to the left – with increased smallspacing and decreased large spacing – with the oppo-site for metal-rich stars. This effect (at least for thelarge spacing) is contrary to what was previously ex-pected. It is, however, very important to state how theparameters on which the large spacing is dependent,previously outlined in the text, are only approxima-tions and it is likely that there are many other subtledependencies which will affect it. When stars evolveoff from the MS the evolutionary tracks of stars of dif-ferent masses appear to converge as both the small andlarge spacings decrease in value over time. It is oftenthe case that the effect of metallicity on the positionof the tracks becomes less important as the differencebecomes similar to the error on the small spacing (SeeFigure 3.3.1).

3.3 Results from AsteroseismicData

3.3.1 Stellar Properties (MH)

The accuracy of the scaling relations quoted aboveis, as previously mentioned, not very good. They can,however, be improved by a second-order asymptotic

treatment that takes into account more precise varia-tion in the frequencies. For example, the large spacingincreases with frequency, hence the positive curvatureof the ridges on Echelle diagrams calculated using aconstant Δν. The second order asymptotic relation is[35]:

νn,l =

(n +

l

2+ ε

)Δν – [l(l + 1)d0 + d1]

Δν2

νn,l(3.30)

where d0 is related to the gradient of the speed of soundand d1 is a correction related to surface boundary con-ditions. Following this relation, we can obtain an equa-tion to modify the observed large spacing, Δνobs , to getan asymptotic large spacing, Δνas :

Δνas = (1 + ζ)Δνobs (3.31)

where ζ = 0.57/nmax for MS stars, nmax = νmax/Δνobs ,and ζ = 0.038 for RG stars. ζ is a parameter calculatedas a measure of the offset between the observed andasymptotically-derived values.

The scaling relations can then be rewritten:

R

R�'(νmax

νref

)( 〈Δνn,l〉〈Δνn,l〉ref

)–2(Teff

Teff,�

)1/2

(3.32)

M

M�'(νmax

νref

)3( 〈Δνn,l〉〈Δνn,l〉ref

)–4(Teff

Teff,�

)3/2

(3.33)

where νref and 〈Δνn,l 〉ref are reference values differ-ent to the Solar values, derived from a set of modelstars. Mosser et al. (2013) give the values 3104μHzand 138.8μHz respectively [35]. These forms of thescaling relations have uncertainties of 4% and 8% forradius and mass respectively. Mosser et al. suggestthat these relations are appropriate for stars of massless than 1.3M� and temperatures in the range 5000-6500K. Additionally, estimates of the radius and massof stars that were calculated using first-order asymp-totic relation-derived scaling relations could be alteredusing:

Ras ' (1 – 2ζ)Robs (3.34)

Mas ' (1 – 4ζ)Mobs (3.35)

In addition, White et al. (2011) find that the rela-tion between mean stellar density and the large spacinghas a temperature dependence, by observing deviationfrom the scaling relations. They deduce that an im-proved form of the relation should be [44]:

ρ

ρ�=

(Δν

Δν�

)2

· (f (Teff))–2 (3.36)

where

f (Teff) = –4.29

(Teff

104K

)2

+ 4.84

(Teff

104K

)– 0.35

(3.37)

13

for stars with temperature between 4700K and 6700Kand mass above ∼ 1.2M�. They found that this ad-justment gives an error on the density scaling relationof approximately 1%.

Asteroseismic diagrams, on the other hand, shouldoffer a much more accurate value for the age and themass of a star, provided appropriate models are used.However, because the large spacing is less sensitiveto mass for stars above 1.4M� [47], the evolutionarytracks become more degenerate above this value, re-ducing the accuracy of this method for mass. Similarly,as stars evolve off the MS, there is a higher degeneracyin the lines of constant hydrogen content as the smallspacing changes less with evolution up the red-giantbranch.

Asteroseismic diagram grids are particularlystrongly dependent on metallicity. For example, Lebre-ton and Montalban found that halving the metallicityfrom Z = 0.02 to 0.01 caused a 15 to 30% change inthe age value obtained [48]. In addition, they can alsobe affected significantly by the inclusion or alterationof the overshooting parameter, αov [48], which mea-sures the carrying over of material from the convectivezone into the convectively-stable zone beneath it dueto inertia while sinking, and by changes to the initialhydrogen content [47]. Convective core overshootingcan also result in the intake of more nuclear fuel tothe core, affecting the evolution of the star. A changefrom 0.0 to 0.2 of this parameter can cause the linesof constant hydrogen content to stretch apart by up to50% their original spacing [48]. Figure 3.3.2 displaysthese effects. Changing the initial hydrogen and he-lium content has a smaller but still noticeable effect onthe mass result.

In general this method, assuming the correct inputphysics and parameters to the models used, should beable to give stellar ages with an uncertainty of less than10% and mass with an uncertainty of several percent.

3.3.2 Stellar Evolutionary Theory (MH)

So far we have a reasonably poor understanding ofthe physics of the surface layer of stars, with models, aspreviously noted, often differing strongly on this part ofthe star. We have a much better understanding of thedeeper layers and cores of stars. As discussed in Section3.2.5 above, the small separation δν is very sensitiveto the structure of the stellar core and hence to theevolutionary state of the star. This makes the smallseparation a valuable diagnostic tool for obtaining thehydrogen fraction and therefore estimating the age, asfeatured in Figure 3.2.6 and with Equation (3.25).

The study of stellar surface layers using asteroseis-mology could be greatly augmented with improved ob-servational equipment and techniques. Modes of higherangular degree penetrate less deeply into the star andhence would be more useful for probing the surfacestructure. The ability to resolve the surface of starsand measure smaller amplitudes with less noise wouldpermit the measurement of modes of higher angulardegree, which with current techniques is not possible

due to their small amplitudes and the self-cancellationin low-resolution integrated-light observations.

We can also use asteroseismic results to test cur-rent stellar models. We can measure some stars’ globalproperties independently, for example in eclipsing bi-nary systems whose mass can be measured with pho-tometry and spectroscopy [49], or bright stars whoseradius can be measured by ground-based interferome-try or parallax techniques which require the distanceand magnitude to be known as well as the tempera-ture [50]. Then, asteroseismic results that differ fromthe known values can reveal discrepancies in the modelused. Equivalently, asteroseismic results that agreewell with independently established properties will giveconfidence to the scaling relations or AHRD modelsused.

14

Figure 3.3.1 – Small against large spacing AHRD with evolutionary tracks for stars that are metal-poor (Z0 = 0.011,blue) and metal-rich (Z0 = 0.028, red). Isochrones are not shown in this example[44]

Figure 3.3.2 – Asteroseismic diagrams showing the effect of changing the metallicity or overshooting parameters in themodels used [48]. On the left, metallicity changes from Z = 0.02 (black lines) to 0.01 (light grey lines). On the right,the overshooting parameter is changed from 0.0 (black lines) to 0.2 (grey lines).

15

Chapter 4

Planet-Finding Theory

In our study, we were provided with long and shortcadence Kepler data showing the flux changes from 40different stars over time. Such data can only be anal-ysed using the transit method and so 40 long and shortcadence light curves were produced using a Pythoncode. The Planetary group was tasked with detect-ing possible transits in these data. After detecting thetransits and analysing their shape, relevant data aboutthe various star-planet systems could be produced inconcert with the Asteroseismology group.

4.1 Planet Formation (AE & EM)

Much of what is known about the formation of plan-ets is based on the planets in our solar system. Theseplanets can be grouped into two categories. The ter-restrial planets, Mercury, Venus, Earth and Mars, arepredominantly rocky and orbit close to the sun, all witha semi-major axis of 1.52AU or less [51]. The giantplanets, Jupiter, Saturn, Uranus and Neptune, havesmall rocky cores surrounded by huge amounts of iceand gas, and are more massive than terrestrial planetswith larger orbits. Giant planets can be further dividedinto gas giants such as Jupiter and Saturn which con-tain mostly hydrogen and helium, and icy giants suchas Uranus and Neptune which contain heavier, volatileelements such as oxygen, carbon, and sulphur.

The formation of stars and planets occurs in densecollections of gas and dust particles called Giant Molec-ular Clouds (GMCs). Approximately 10 Gyrs after aGMC is formed, sections of the cloud collapse due togravitational instabilities, which results in the forma-tion of a protostar. This attracts further matter intoa disk surrounding the protostar. Disc dissipation fol-lows and the material in the disk surrounding the pro-tostar is transported inwardly towards it [52]. Thisprocess continues until the protostar reaches an age of1 Myr, at this point the protostar finishes this accu-mulation and becomes a T Tauri star [53]. As a resultof this process, planetary formation can begin.

Terrestrial planets form via the Planetesimal Ac-cretion Model which states that rocky planets formfrom dust grains that collide and combine into plan-etesimals which in turn grow in size to eventually be-come a planet [54]. As these planets form in the innersolar nebula, temperatures are so high that the onlydust particles available for accretion are those with a

high melting point such as iron and aluminium. Asthese molecules are relatively rare, there is only a smallamount of material available to the planets, thereforethey have a small size.

The currently favoured model for the formation ofgiant planets is the Core Accretion Model. As withterrestrial planets, the core of the planet forms fromsolid particles. However, it is thought that giant plan-ets form further out from the protostar, past a pointknown as the ice line. The ice line is the region arounda protostar where the energy transferred from lightand pressure in the nebula enable volatile molecules tofreeze. Since boiling point and volatility are dependenton the molecular type, the ice line varies depending onthe molecule in question. Ices, with the exception ofwater, are more dense than their previous gaseous stateand can therefore exert a stronger gravitational forceand form planetesimals with the already present dustgrains [55]. Since the ices are abundant, the core islarger than the case for terrestrial planets and so hasa greater gravitational attraction. This larger core canattract the huge amounts of gas that are present ingiant planets.

From observations of other stellar systems, planetswith a rocky core surrounded by an envelope of gassessuch as hydrogen, helium and volatiles exist. Theseare given the name gas dwarfs as they contain gasseslike the giants but have a lower mass. Buchhave etal. (2014) constrained the radii of each type of planet.They constrained terrestrial-like planets to less than1.7R⊕, gas dwarf planets to between 1.7R⊕ and 3.9R⊕and giant planets to larger than 3.9R⊕ [56].

The first planet found orbiting a Sun-like star, 51Pegasi b, was found to have a mass around half thatof Jupiter, suggesting that it was a gas giant. How-ever given a semi-major axis of 0.052AU and a highersurface temperature than Jupiter [17], 51 Pegasi b be-came the prototype of a class of planets labelled ‘hotJupiters’. Since then, many more planets with sim-ilar properties have been discovered. These planetscall into question the planetary formation theories thatwere developed by observing our solar system, in whichthere are no hot Jupiters. According to these theories,it is not possible for a planet the size of Jupiter to haveformed this close to the star. Giant planets are onlyable to grow so large due to the fact that their corecontains volatile elements that have frozen. This freez-

16

ing only occurs when the the planets orbit lies beyondthe ice line. For this reason, these planets must haveformed further out and somehow moved to a closer or-bit [57].

The most developed model to explain this phe-nomenon is migration. This describes how when plan-ets form, tidal interactions between the planet and thesurrounding protoplanetary disk result in the exchangeof angular momentum and so the orbit of the planetcan change [58]. There are two main types of migra-tion regarding planets. Type I describes the relativelyfast migration of planets with masses less than 10M⊕whereas type II describes the slower migration of plan-ets with masses between 10M⊕ and 30M⊕ [59]. As hotJupiters have a high mass, they fall under the cate-gory of type II migration. In this migration, due tothe high mass of the planet, the planet and disk inter-act so strongly that a gap in the disk actually opens upwhere the planet orbits, forming a barrier. Some gasdoes leak into the path of planetary orbit and is eitheraccreted by the planet or passes straight through. Asthe particles in the protoplanetary disk are generallymoving inwards, the planet, which is obstructing theviscous evolution of the disk, moves in the same direc-tion to a smaller orbit. This migration slows down asthe disk dissipates since there is less force exerted onthe planet enabling it to settle into a smaller orbit [60].

Another way to explain the presence of hot Jupitersis to examine the situation where a giant planet inter-acts with a smaller body in the solar system, reducingthe energy of its own orbit. This model is examinedin more detail in Subsection 4.6.5 which discusses thethree-day pileup of hot Jupiters.

It is worth mentioning that hot Jupiters presentother surprising properties. An example is that obser-vations show that some hot Jupiters orbit their hoststars in the opposite direction to the direction of ro-tation of the star. This poses a problem as all plan-etary formation theories predict that planets formedfrom the same disk of gas and dust as their parent starshould orbit in the same direction. Li et al. (2014)propose a mechanism to explain this seemingly unnat-ural observation. They describe how the orbit of theinner hot Jupiter is greatly perturbed to an extremelyelliptical orbit by an outer orbiting planet until the or-bit of the inner planet suddenly flips over. The outerplanet steadily removes angular momentum from theinner planet until, when the inner planet is essentiallyon a collision course with the star and has very littleangular momentum, a small gravitational kick appliedto the planet can flip it over and reverse the spin andorbital direction [61].

4.2 Orbital Mechanics

4.2.1 Kepler’s Laws (AE)

Celestial mechanics is a branch of astronomy thatdeals with the motion of celestial objects such as plan-ets and moons under gravity. In the early 17th century,Johannes Kepler developed three laws that describe the

motion of planets around their respective stars in thesky. A summary of the laws are:

· All planets move in elliptical orbits around theirstar which is at a focus.

· The area swept out by a planet in a given timeis always the same.

· The period of the orbit of a planet squared isproportional to the semi-major axis of the orbitcubed.

Since the planets are in elliptical orbits, the distancebetween the planet and star will vary depending on thepoint in orbit of the planet, unlike a circular orbit, asshown in Figure 4.2.1[62].

Figure 4.2.1 – Kepler’s first law describing the ellipticalorbit of a planet around the Sun [62].

As Figure 4.2.2shows, Kepler’s second law meansthat when the planet is at the perihelion of the orbit,it must be moving faster than at the aphelion so thatthe areas swept out by both positions are the same.

Figure 4.2.2 – Kepler’s second law showing the area sweptout by a planet as it moves around the star [62].

17

The mathematical representation of Kepler’s thirdlaw is:

P2 =4π2

GM∗a3 (4.1)

where P is the orbital period of a planet around astar of mass M∗, G is the gravitational constant anda is the semi-major axis of the orbit. The semi-majoraxis, defined as the mean of the perihelion and aphe-lion, must be used in this law instead of the radiusof orbit since the planet follows an elliptical orbit andso the distance between the planet and star will varyfrom a minimum at the perihelion to a maximum at theaphelion. When using the transit method to detect aplanet, it is possible to constrain the eccentricity abovea certain minimum by comparing the semi-major axiscalculated using Kepler’s third law to the distance be-tween the planet and star at the time of transit. Adetailed explanation of this can be found in Section6.3.

4.2.2 Two- and Three-Body Problem(LP)

The two and three body problems are used to deter-mine the motion of bodies that have noticeable gravita-tional influence over each other, such as orbiting plan-ets or stars. In the two body problem, the two objectsare given set masses, velocities and different startingpositions. From this information, the change in motionof the two bodies can be determined as a function oftime. The three body problem is much more complexthan the two body problem as a third object is addedto the system with its own velocity, mass and position.The two body problem yields clear and understand-able results via methods of integration, the three bodyproblem on the other hand is too complex to solve an-alytically and is yet to be completely understood.

Two-Body Problem (LP)

The most basic two body problem scenario is toconsider two masses m1 and m2 with a commonbarycentre. The vectors from each respective objectto the centre of mass are defined as r1 and r2. Withthis, the following equation can be written [63]:

m1r1 + m2r2 = 0 (4.2)

The total mass of the system (m1 plus m2 ) is equalto M and the vector displacement between the twobodies is r. The distance between the two objects islabelled d , which is equal to the sum of r1 and r2 .From this and the above equation, further identitiescan be derived:

r1 =m2

m1r2 =

m2

M – m2(d – r1) (4.3)

r1(1 +m2

M – m2) = r1(

M

M – m2) = d(

m2

M – m2) (4.4)

These equations can then be used to derive the sim-plified equations shown below:

r1 = d(m2

M) (4.5)

r2 = (m1

m2)r1 = d(

m1

M) (4.6)

The above equations can be written in vector formwith vectors r1 and r2 displayed in terms of r, thevector displacement:

r1 = –(m2

M)r (4.7)

r2 = (m1

M)r (4.8)

Differential equations of motion can be written, de-scribing the force that each body exerts on one another:

m1r1 = (Gm1m2

r2)r (4.9)

m2r2 = –(Gm1m2

r2)r (4.10)

Using substitutions between equations (4.7) and(4.8), and equations (4.9) and (4.10), the followingequation of motion can be derived:

r = –(GM

r2)r (4.11)

Here the unit vector r is included which is equal toone. These two body considerations are made with theassumption that the two bodies move in a plane withrespect to each other, the centre of mass frame. It isalso possible to derive equations that consider the totalenergy of a two body system:

E =1

2(m1r2

1 + m2r22) –

Gm1m2

r(4.12)

Similarly, the total angular momentum of a twobody system can be written:

L = m1r21θ+ m2r2

2θ (4.13)

Three-Body Problem (LP)

Although the general three body problem has noanalytical solution, an idealised, restricted three bodyproblem can be considered. This three body probleminvolves two bodies with masses m1 and m2 and athird body, m3 with a much lower mass that has anegligible gravitational effect on the other two bodies.The two more massive bodies are in orbit and bothlie along the x-axis of a Cartesian co-ordinate system(x , y , z ) at a time t = 0 . The origin of the system canbe considered as the centre of mass C and the distancebetween m1 and m2 remains a constant distance R.The distance between m1 and C is constantly r1 whilstthe distance between m2 and C is constantly r2 . Anequation for the orbital angular velocity of the system

18

can be written using M as the total mass of the system[64]:

ω2 =

GM

R3(4.14)

We can also state the following relationship:

m1

m2=

r1

r2(4.15)

For simplicity the values of R and GM are made tobe equal to 1. Using Equation (4.14) it can be clearlyseen that this also results in ω being equal to 1. Wetake the following identities:

μ1 = Gm1 (4.16)

μ2 = Gm2 (4.17)

Hence:r1 = μ2 (4.18)

r2 = 1 – r1 = μ1 (4.19)

From these equations, position vectors can be as-signed to m1 and m2 . The position vector for m1 isr1 = (x1, y1, 0) with the position vector for m2 as r2

= (x2, y2, 0), where:

r1 = μ2(–cosωt, –sinωt, 0) (4.20)

r2 = μ1(cosωt, sinωt, 0) (4.21)

At this point, the third mass can finally be consid-ered with a position vector of r = (x, y, z). Cartesianequations of motion of the particle can therefore befound using this consideration. These are displayedbelow:

x = –μ1(x – x1)

ρ31

– μ2(x – x2)

ρ32

(4.22)

y = –μ1(y – y1)

ρ31

– μ2(y – y2)

ρ32

(4.23)

z = –μ1z

ρ31

– μ2z

ρ32

(4.24)

where:

ρ21 = (x – x1)2 + (y – y1)2 + z2 (4.25)

ρ22 = (x – x2)2 + (y – y2)2 + z2 (4.26)

The light particle m3 is considered as a moving par-ticle in the rotating frame, forces on stationary bodiesin this frame can be represented by using a Roche po-tential. When two bodies orbit one another circularly,their gravitational fields interact with one another andmerge to form one field. Contoured images can be pro-duced that display areas of equal and varying gravita-tional potential. This is known as the Roche potentialnamed after Edouard Roche [65].

Figure 4.2.3 – The Contours of a Roche Potential [66]

This diagram not only shows the relative positionsof the bodies and the gravitational field, but it alsoshows the five Lagrangian points in the system. Atthese points, a stationary body would feel no overallforce. These positions are often ideal for satellites.Mathematically, the Roche potential is defined as [67]:

ΦR(r) = –Gm1

|r – r1|–

Gm2

|r – r2|–ω

2r2

2(4.27)

In the equation, the third term accounts for thecentrifugal force. The five Lagrangian points appearwhen ΔΦR = 0. An understanding of the Roche po-tential can be used to model the motion of a third, lessmassive body through systems of this type.

4.3 Planet-Finding Methods

4.3.1 Transit Method (LP)

When an object passes between a distant star andan observer on Earth, there is a drop in the amountof flux reaching the observer as a fraction of the lightfrom the distant star has been blocked. The drop influx corresponds to the area of the star blocked by thepassing object. Often, such occurrences are as a resultof a planet orbiting the star. The presence of a planetcan be confirmed by detecting such flux changes pe-riodically, with the change in flux being the same forevery transit. Since planets have a very small radiuscompared to that of stars, the change in detected fluxduring the transit of an exoplanet on a distant star isvery small. For this reason, the instruments that de-tect changes in flux of distant stars must have a highprecision.

19

Figure 4.3.1 – Transit Example [68].

Figure 4.3.2 – Impact Parameter Diagram with Basic LightCurve [69].

When recording data on a distant star, a light curveis often produced, displaying the flux of the star againsttime. The radius of a transiting planet has an effecton the transit depth of the light curve. A light curve isshown in Figure 4.3.1 with one transit displayed. Thedepth of the transit δ is displayed as a fraction of thetotal flux of the host star and it can be related to theratio of the planetary radius and stellar radius [70]:

δ =

(Rp

R∗

)2

(4.28)

So long as the star remains much greater in sizethan the planet, the above equation can be applied.These transits reveal more than just the ratio betweenplanetary and stellar radii; they also reveal the pe-riod of transit of the planet and the time that it takesfor a transit to occur. The period can be inferred bythe time between transits with the transit length mea-sured by considering the time it takes from the startof a transit to its end. Detection of many transits isrequired to ensure that these values remain constantover a substantial period of time. With these valuesknown, further information regarding exoplanets can

be inferred. The transit time and period of orbit canbe related by [70]:

τ =PR∗πa

(4.29)

Here, τ corresponds to the transit time, R∗ to thestellar radius and a to the semi major axis of theplanet. This equation is somewhat simplified, as itassumes that the planet moves in a straight line acrossthe star and does not account for any varied planetarymotion.

When a planet transits a distant star, it is rare forit to take the longest path across the star, along thestars’ radius. For this reason an impact parameter isintroduced to account for the variety of possible pathsa transiting planet can take in front of a star. Theimpact parameter is represented by the letter b, and isa measure of how off-centre a transit is. The geometryis depicted in Figure 4.3.2.

The impact parameter of a planet affects the transitduration due to the circular appearance of a star. Thiscan be related to the transit depth, transit time andtime of partial transit t . The partial transit time isthe sum of the ingress and egress time where a fractionof the planet blocks the star, but not the whole of theplanet. The ingress marks the beginning of a transit,the egress marks the end of a transit [71]:

b = 1 –τ

√δ

t(4.30)

The impact parameter can also be found by consid-ering the inclination of the system, as follows:

b =a cos(i)

R∗(4.31)

This equation can then be combined with Equation(4.29) to get a more accurate transit time:

τ =

Psin–1

(√(Rp+R∗)2–b2

a

)π

(4.32)

Many stars host exoplanets that cannot be detectedby this method as the planets do not transit the starrelative to our position on Earth. The probability ofdetecting the planet for a system with a star and oneexoplanet, using the transit method, can be quantifiedusing:

p =R∗a

(4.33)

The probability of detecting an exoplanet whenonly analysing one star is very low, however when hun-dreds of stars are analysed, the probability of findingexoplanets soon becomes very high. Equation (4.33)only takes into consideration planets with a circularorbit; for an elliptical orbit, Equation (4.34) must beapplied [72]:

p =R∗

a(1 – e2)(4.34)

As well as detecting primary transits, where aplanet passes in front of a star relative to an observer,

20

secondary transits can also be detected. A secondarytransit occurs when an exoplanet is ‘hidden’ from anobserver on Earth by its host star. Although planetsdo not emit light, they are able to reflect star lightwhich can be detected on Earth. When an exoplanetis obstructed from view by its host star, the reflectedlight is no longer able to reach us and a dip in the fluxfrom the system is noticed. The depth of a secondarytransit is significantly smaller than a primary transitand therefore these transits are much more difficult todetect. The collection of secondary transit data canbe used to determine the albedo of the exoplanet. Thealbedo is the reflection coefficient for a planet which ifknown, can be used as an indication of the nature ofthe atmosphere or surface of a planet.

The detection of a transit does not necessarily meanthat a planet is present, as there are often many ‘falsetransits’ detected. The majority of false transits areas a result of binary stars, where the smaller of thetwo stars moves in front of the larger star. This re-sults in a dip in flux of the system which usually hasa much greater transit depth than that of a typicalplanet-star transit. For this reason, the majority of bi-nary star transits can be identified easily. If the smallerof the two stars only eclipses the larger star partially,then a dip comparable to that of a planet can be de-tected. In this situation, the spectroscopic make up ofthe system must be analysed. Since stars emit lightof different wavelengths, any dips in the intensity ofparticular wavelengths of light can be monitored to in-dicate whether the passing body is a star or planet. Asplanets emit no light of their own, when a planet tran-sits a star, all emitted wavelengths from the systemwill drop in intensity.

The luminosity of a star changes due to internaloscillating standing waves that alter the size of thestar. The luminosity changes periodically and there-fore this must be considered when analysing transitdata. A change in flux due to fluctuating luminosityoccurs more gradually than the dip in flux seen when abody transits the star, however such changes do make itmore difficult to detect transits in the first place. Theluminosity of a star can change by a factor of 10–5.Although this seems small, this value is comparableto the change in flux of a Sun-like star by the transitof an Earth-like planet (8.4x10–5W). Observing starsfrom Earth produces a limit on detecting changes ofluminosity smaller than 8x10–4W [73]. The presenceof sunspots on stars can result in a dip in the flux de-tected as the stars rotate. This can be mistaken fora transit, however further analysis of such stars willreveal that as time goes on, the numbers of sunspotswill change and so there will be no periodicity to thechanges in flux.

When a planet transits a star, some of the light fromthe star can pass through the planetary atmosphere be-fore it reaches Earth. Spectroscopic examination of thelight could reveal the contents of the atmosphere witha detection of either O2 or O3 indicating a potentialfor the presence of life [74]. This information is practi-cally impossible to obtain using light reflected from an

orbiting planet that is not in transit.If an exoplanet has an approximately circular orbit,

and the radius of its host star is known, the planetaryradius, semi-major axis, orbital period and inclinationcan all be calculated. The transit method is most usefulwhen finding planets with large radii and small semi-major axes as these are the most likely types of planetsto be detected. The semi-major axis is important as itallows us to determine whether a planet is in the hab-itable zone. This describes the distance from the starat which water can collect on the surface of a planetwithout boiling or freezing. This, together with spec-troscopic observations, can determine the make up ofthe atmosphere of the planet, which is of paramountimportance when searching for habitable planets.

4.3.2 Transit Timing Variations (TTV)(AE)

In a multiple planet system, the motions of the starand its orbiting planets will not follow Kepler’s Laws,but slightly irregular orbits due to the gravitationalinfluence of the other bodies in the system. This pro-duces a slight variation in the length of the planet’sorbital period, which can be detected by the variationin time between transits. From this, the presence ofa second planet can be inferred. As of February 2015,15 planets have been discovered using TTVs [18], 14 ofwhich were detected using the Kepler spacecraft. Thefirst planet detected using this technique was Kepler-19c in 2011 [75].

Take the case looking at j transits, where0 ≤ j ≤ N and N is the total number of transits ob-served. The time of any given transit is given by:

t(j) = jP + t(0) (4.35)

where P is the period of transits and t(0 ) the timeat which the the first transit begins. If there is a pe-riodic change in this time, there is a transit timingvariation given by:

δt(j) = t(j) – jP (4.36)

Transit timing variations can be categorised intofour situations [76].

1. Non-Interacting Inner Planet

Take the case of observing transits of a planet farfrom a star. Timings of the transits will vary if the starhas formed a binary system with another planet closeto the star. As the star orbits the centre of mass of thebinary system and changes its relative position in thesky, the line of sight of the transit will be shifted peri-odically so the time between transits will change. Thiscase is represented in Figure 4.3.3 which shows how thepoint in a planet’s orbit during transit is dependent onthe position of the star as it wobbles.

21

Figure 4.3.3 – TTVs caused by an inner planet forming abinary system with the star. The yellow sphere representsthe star and the blue and green spheres represent the innerand transiting planets respectively [76].

If the star is to the right of the centre of mass ofthe binary system, as is the case for the left side partof Figure 4.3.3, the planet has to move further roundin order to transit the star. For this reason there willbe a larger time between this transit and the previousone resulting in a positive δt(j). The opposite happenswhen the star is to the left of the centre. This resultsin a sinusoidal TTV signal and it is clear to see thatthe larger the mass of the inner planet, the larger thewobble of the star, and so the TTV signal is greater.

This case assumes that the two planets are farenough away to not interact with each other. The TTVfor a given transit can be given by [77]:

δt(j) ≈ –Pt

2π

ac

at

mc

M∗sin

(2π(jPt – t(o))

Pc

)(4.37)

where Pc , Pt , ac and at are the periods and radii of theplanet close to the star causing the perturbation andthe transiting planet respectively and mc and M∗ arethe masses of the inner planet and the star. A usefulmeasure of the time variation is the root mean squaregiven by [77]:

〈δt〉RMS =1√2

Pt

2π

ac

at

mc

M∗(4.38)

as this just leaves two unknown variables, ac and mc ,assuming the period and radius of the transiting planetare known.

2. Interaction from Exterior Planet in LargeOrbit

In this situation, an inner circular orbiting systemconsisting of a star and transiting planet is perturbedby an exterior planet on a large orbit with high ec-centricity producing TTVs in the transiting planet’speriod, as is shown in the Figure 4.3.4.

Figure 4.3.4 – TTVs caused by an outer planet with a higheccentricity perturbing a binary system consisting of thestar and an inner transiting planet. The spheres representthe same bodies as for case one [76].

Agol et al. (2005) use Jacobian coordinates to de-rive a first order Legendre series approximation for theperturbing effective force on the inner binary system.This approximates the TTV signal as [77]:

δt = β(1 – e20)–3/2[f0 – n0(t – τ0) + e0 sin(f0)] (4.39)

where e0 is the eccentricity of the outer planet, n0 (t –t0 ) is the mean motion of the planet, f0 is the trueanomaly of the planet, which is defined as the angulardistance between the perihelion of the planet and theplanet itself observed from the star it orbits. β is givenby:

β =m0

2π(M∗ + mt)

P2t

P0(4.40)

The RMS of the TTV is [77]:

〈δt〉RMS =3βe0√

2(1 – e20)3/2

[1 –

3e20

16–

47e40

1296–

413e60

27648

]1/2(4.41)

This model is particularly useful for finding exteriorplanets perturbing a transiting hot Jupiter as the TTVsignal is highest when mt > m0 .

3. Interactions Between Two Planets

Consider a case where two planets with orbits thatare close to one another, but not in resonance, perturbeach other. When the two planets are at conjunction(the orbits of the planets are at their closest positions),the planets interact most strongly [78]. This results inthe orbit of the transiting planet being perturbed fromcircular to eccentric as Figure 4.3.5 shows.

22

Figure 4.3.5 – TTVs caused by two planets with initiallycircular orbits close to one another. The orbit of the tran-siting planet, shown in green, is perturbed from circular toeccentric so a TTV signal is produced [76].

Since the planets are not in resonance, the positionof conjunction will not be constant and the perturba-tions will cancel out when the angular position movesby π radians. This means that the eccentricity willgrow over half a period of circulation of the position ofconjunction. The closer the planets are to resonance,the longer it takes for the position of conjunction tocomplete a full cycle, therefore the eccentricity andhence the period of orbit are affected greatly.

There are two factors which contribute to the TTV:fluctuations in the mean motion of the planets andchanges due to the eccentricity not remaining constant.For the equations below, it is assumed we have a casewhere the planets are near but not at resonance sothere is a transit ratio j : j + 1 with a high value ofj . When eccentricity dominates, the TTV is given by[77]:

δt ≈ μpertε–1P (4.42)

where μpert is the ratio of the mass of the perturbingplanet and the star and ε is the fractional distance fromresonance:

ε = |1 – (1 + j–1)P1

P2|. (4.43)

When fluctuations in mean motion dominate, theresulting TTV for the lighter planet is [77]:

δtlight ≈ μ2ε–3P (4.44)

where μ is the mass ratio of the heavy planet and thestar. Due to conservation of energy, the TTV for theheavier planet is reduced to [77]:

δtheavy ≈mlight

mheavyμ

2ε–3P (4.45)

As the equations show, when ε > μ1/2 the eccentric-ity term dominates the mean motion fluctuation termand the opposite is true when ε < μ1/2 until ε < j1/3μ2/3

at which point the planets are in mean motion reso-nance.

If both planets in the system transit the star (whichis not improbable as planets tend to orbit on similarplanes), it will be possible to derive values for bothplanets individually. This data would then be used totest our model and show whether the observed TTVis caused by the planets observed or by other, non-observed planets.

4. Interactions of Two Planets in Mean-MotionResonance

Take the case of two planets with initially circu-lar orbits with a first order resonance. Initially, con-junctions occur at exactly the same position in spacerelative to the star. Since the interaction between theplanets is strong at the conjunctions, perturbations arecaused in the orbit of the lighter planet making it ec-centric, and so the semi-major axis and period change.These changes result in movements of the position ofconjunction and, like the previous case, when the po-sition shifts by approximately π radians, the pertur-bations start to cancel out causing the eccentricity todecrease. So over a cycle of 2π radians, the eccentrici-ties and change in period increase to a maximum beforereturning to their original values.

Assuming a large j so that j ≈ j + 1 and P1 ≈ P2

as before, the maximum transit timing variation forthe lighter planet is [77]:

δtmax,light ∼P

j(4.46)

and for the heavier planet is [77]:

δtmax,heavy ∼mlight

mheavy

P

j(4.47)

The time it takes for the conjunction to return toits original position is called the libration period andis [77]:

Plib ∼ 0.5j–1ε–1P ∼ 0.5j–4/3

μ–2/3P. (4.48)

In the context of this project, the accuracy with whichwe are able to measure the period of transits is too lowto be able to detect a TTV and hence infer the presenceof extra planets in the system with any certainty. How-ever, these equations are important as they are neces-sary in calculating properties of the planets detectedby TTVs, if more accurate periods can be obtained.

4.3.3 Other Planet Detection Methods(CL)

With the Kepler data provided being the only dataavailable for study, exoplanets could only be detectedusing above listed methods. However, there are manyother methods by which planets can be detected. Someof these methods, which were not used in this study,have been outlined in this section. It is common formore than one method to be used to determine theproperties of a exoplanet and confirm its existence.

23

Doppler Wobble (Radial Velocity) (CL)

In the presence of an orbiting planet, a star movesaround the centre of mass, or barycentre, of the two-body system (See Section 4.2.2). This motion of thestar can be seen in the periodic shift of the spectrallines from the light emitted by the star. The wave-length of the light is alternately blue and red shiftedas the star moves towards and away from the observer,respectively. For high precision measurements of thewavelength shifts, the stellar spectra observed need tocontain many absorption lines. Stars with a low num-ber of features in their spectra have a low precision.The wavelength shifts in the observed light are used todetermine the radial velocity at which the star is mov-ing with respect to the observer, allowing properties ofthe orbiting planet to be calculated [79]. The observedradial velocity of the star, Vdop, is given by [80]:

Vdop = V∗ sin (i) (4.49)

where V∗ is the radial velocity of the star caused bythe orbiting body and i is the angle of inclination ofthe system. Equation (4.49) shows that the changein radial velocity seen by the observer is dependenton the value of the inclination and the magnitude ofthe star’s movements. This means that this methodis most sensitive to planets that are orbiting with aninclination of 90◦. This equation also shows that theDoppler method is more sensitive to massive planets,as these bodies have a greater gravitational influenceover their host star. This in turn maximises the move-ment and changes in radial velocity of the star. On thecontrary, a planet orbiting a star with an inclinationangle of 0◦, will not have an effect on the radial veloc-ity of the star with respect to the observer. Therefore,any planet orbiting a star in such an orientation wouldnot be detected [81].

Simple orbital mechanics can be used to find thevelocity of the planet, by equating the forces actingupon the planet in circular motion as it orbits. Theresult is given by [82]:

Vp =

√GM∗

a(4.50)

where Vp is the radial velocity of the planet, G is theGravitational constant, M∗ is the mass of the star anda is the semi-major axis of the planetary orbit. Thevalue Vp can be calculated if the mass of the star isknown, this can be derived from asteroseismic data.The value for a is determined by using Kepler’s thirdlaw, inputting the period of oscillation of the star, P∗[62]:

a3 =GM∗4π2

· P2∗ (4.51)

The mass of the planet, Mp is then easily deducedfrom Equation (4.52), where the conservation of mo-mentum in the star-planet system is used:

Mp =M∗V∗

Vp(4.52)

A final equation for the mass of the planet can thenbe found, by combining all of the previous relations[83]:

Mp =Vdop

sin (i)

(P∗M2

∗2πG

)1/3

(4.53)

This is the equation for the mass of a planet in a cir-cular orbit. If a planet is orbiting in an eccentric orbit,Equation 4.53 becomes [83]:

Mp =Vdop

sin (i)

(P∗M2

∗2πG

)1/3√1 – e2 (4.54)

where e represents the value of the eccentricity of theorbit. Unfortunately, the Kepler spacecraft does nothave the capacity to carry out stellar spectroscopy,therefore it can not produce radial velocity measure-ments. Kepler has had great success in detecting thepossible presence of exoplanets from light curves andthe transit method. However, star systems where Ke-pler is able to detect transits are usually then sub-ject to further ground-based observations and radialvelocity measurements. The transit and radial veloc-ity methods work well together because of their similardependence on inclination. A planet that can be ob-served by the transit method must be closely aligned toa 90◦angle of inclination, meaning that the likelihoodof also being able to detect the planet by the radialvelocity method is significantly increased. The follow-up measurements are used to eliminate false detectionsthat are produced from the Kepler light curves. Thelarge majority of such false positives are found, andeliminated, by using statistical tests on the Kepler dataitself. These false-positives are caused by the commonoccurrence of astrophysical signals that mimic the sig-nals received by transiting planets [84]. If the possibleplanet passes these statistical tests, then an observa-tion using the radial velocity method is used to ruleout another common source of false-positives; stellarbinary systems. An eclipsing binary can be very dif-ficult to distinguish from a planet in the transit data.However, due to the relatively large masses of starscompared to planets, the radial velocity method candistinguish these two objects. In the presence of aneclipsing binary, the radial velocity variations are muchlarger, ≈ 1km s–1 [84].

The follow-up radial velocity observations can notonly confirm the presence of an exoplanet in a system,they also give values for key planetary properties thatcan not be obtained by Kepler alone, like the mass ofthe planet. The combination of the planetary radiusvalues obtained by the transit method and the massvalues obtained by the radial velocity methods meansthat the density of the planet can be calculated. Thisallows conclusions to be drawn on the composition ofthe planet [85].

Astrometric Wobble (CL)

Astrometric wobbles are observed by Astrometry,a branch of astronomy which takes precise measure-ments of the relative positions and movements of stars.

24

An astrometric wobble describes the apparent motionof a star on the celestial sphere which is caused bythe gravitational effects of orbiting planets about it.Astrometry is the oldest theorised method of detect-ing extrasolar planets. In the 18th century, WilliamHerschel was researching the small, visible angular dis-placement of faint stars and from this it was concludedthat these stars had an unseen companion affecting theposition of the observed star [86]. This proves that as-trometric wobbles can even be seen by eye. However,in Herschel’s case, the effects were due to a system ofbinary stars, not the motion of a star as a result of anorbiting planet. In order to detect exoplanets by us-ing astrometric methods, more accurate methods usingphotographic and telescopic techniques must be used.

The size of the wobble is dependent on the massof the star and the planet, the radius of the orbit ofthe planet about the star and the inclination of theorbital plane, relative to the observer. If the angleof inclination is i = 90 ◦, so that the orbital plane isin line with the line of sight of the observer, the mo-tion of a star due to the presence of an orbiting planetwould appear to move from side to side. If the orbitalplane were aligned however so that it is perpendicu-lar to the observers line of sight, i = 0 ◦, then the starwould display a small circular motion as the planetsgravitational force pulls the star out of position [87].

In a star-planet system, the distance from the starto the system centre of mass, r1 , is given by the equa-tion [71]:

r1 = am

M + m=

a

1 + Mm

(4.55)

where M is the mass of the star, m is the mass of theplanet and a represents the semi-major axis. From thisequation it can be stated that the motion of the star isgreatest when the barycentre of the system is furtheraway from the star. For this reason, astrometric wob-bles are larger in systems with small stars, large planetsand large orbital radii. This is because the barycentrelies further from the centre of the star meaning thatthe star must move a greater distance about this point.The angular displacement caused by the stars orbitalmotion, at distance d away from the star is given by[87]:

Δθ ≈ r1

d≈(m

M

)(a

d

)(4.56)

Here, the small angle approximation is applied andit is assumed that M � m . This confirms what waspreviously stated; as the centre of mass moves furtheraway from the star (increased r1 ) then the angular dis-placement gets larger. Equation (4.56) also illustratesthat the motion of a star is most obvious as the dis-tance to the star decreases, relative to an observer.This is because the angular displacement gets largeras the observation distance decreases. The astrometricmethod is most obvious in two-body systems where aplanet orbits a star at a large orbital radius. This isin contrast to the transit method, which is most sen-sitive when detecting planets that orbit close to thestar. For the case where the astrometric wobble effectis caused by the orbit of one planet, the mechanics of

the situation are governed by the two-body problem(See Section 4.2.2).

The problem in the advancement of Astrometry asa detection method is the precision to which telescopescan resolve an angular change in position. The Earth’satmosphere is a barrier to improving the precision ofground-based telescopes. The atmosphere degrades thequality of astronomical images collected and noise isproduced by distortions in the telescope’s structure.These distortions are caused by the weight of the struc-ture and also the thermal response of the surroundingbuilding and the telescope itself [88]. Ground-basedtelescopes also have fixed locations, resulting in a lim-ited coverage of the sky. In order to increase this cover-age, data is combined from different observatories fromaround the world. However, this process also producesuncertainties. For these reasons, ground-based tele-scopes have not been able to surpass the barrier ofposition measurements with a precision greater thanone hundredth of an arc second [88]. For this rea-son, moving telescopes to space seemed the logical so-lution. The European Space Agency’s Gaia mission(Global Astrometric Interferometer for Astrophysics),which launched in 2013, has planned to find thousandsof planets using the astrometric wobble method. Withan astrometric precision of up to 0.00001 arc seconds,Gaia will determine positions of stars out to 30,000light-years away [89]. Gaia will be able to measure astar’s position and motion 200 times more accuratelythan previous astrometric detection attempts. A goodanalogy to show the precision of Gaia is that it is able tothe measure the angle that corresponds to the lengthof an astronauts thumbnail, who is standing on themoon, from Earth [90]. This traditional method of pos-sible planet detection has been reinvigorated with theadvance of instrumentation in astrometric techniques.On Gaia itself, astrometric, photometric (transit) andspectrospcopic (radial velocity) techniques will be com-bined to produce data from which it is easy to identify,characterise and confirm planets that orbit a star [89].

4.4 Stellar Limb Darkening(CL)

Stellar limb darkening is a phenomenon that is ob-served with varying effect in all stars, including our ownsun. It is given its name simply due to the appearanceof the edges, or limbs, of the star being darker thanits centre. The consequence of this is that the specificintensity, Iλ, at a particular point on the star is de-pendent on the distance that the point is away fromthe centre of the star [91]. This is illustrated in Figure4.4.1.

25

Figure 4.4.1 – Image of the Sun showing the decreased in-tensity of light emitted at the edges [92].

The centre of the Sun has a much greater inten-sity of light compared to that of the limbs of the Sun.This gives the sun its spherical, three-dimensional ap-pearance even though it is observed in a two dimen-sions. This is partially due to the effect of the effec-tive temperature decreasing as the radial distance fromthe centre of the star increases. The intensity of radia-tion produced is largely dependent on the temperature,which can be very simply demonstrated by the Stefan-Boltzmann Law, where the intensity is proportional totemperature to the power of four [93]. However, thetemperature dependence of the radial distance alonedoes not explain the limb-darkening effects observed.It is also reliant upon the concept known as opticaldepth, which provides a dimensionless measure of thedepth we can observe into a partially transparent gas[19]. At the centre of the star disk, the observed lightrays are penetrating radially outwards from relativelydeep with in the stars photosphere. At this point thetemperature is relatively high and the intensity is at amaximum. Light emitted from the limbs of the staroriginate in the upper, cooler regions of the photo-sphere - compared to the lower depths at which theoptical depth allows at the centre. Therefore, the in-tensity of light observed at the same optical depth fur-ther from the centre is reduced due to the decrease intemperature [94].

The modelling of the effects of stellar limb dark-ening is extremely important in obtaining an accurateestimate of planetary characteristics from their transitdata [95]. The stars in our data that have been ob-served by Kepler show the limb-darkening effect. Thiseffect causes issues with the transit method; the pres-ence of limb darkening causes a change in the inten-sity. Transit detection relies upon intensity changesproduced only by the planet blocking out the star’slight. Figure 4.4.2 shows the transits of four planetsof different sizes (scaled to the radius of the Earth) or-biting with the same period. These modelled transitstake place around a star of a constant size with a uni-form intensity across the whole disc, therefore the onlysignificant source of change in intensity comes from the

light blocked out by the transiting planets.

Figure 4.4.2 – Simulated light curves for planets of varyingradii transiting a star of uniform intensity, with a radius of0.8R� [96].

The transit shapes in Figure 4.4.2 are sharply de-fined in a box shape, with an almost instantaneousdrop in intensity as the planet moves in front of thestar. The larger the planet the more light that it blocksout and the larger the drop in intensity. The pointsplotted by the model that lie between the normal in-tensity of the star and the dropped transit intensityare due to partial coverage of the planets disc as theybegin to move in front and away from the star’s disc.This is an ideal model and in practice the light curvesreceived from stars with transiting planets present donot look as perfect as these models. This is mainly dueto limb darkening causing a variation in the intensityof the light emitted and blocked out by the transitingplanets. Applying the effects of stellar limb darken-ing to the model shown in Figure 4.4.2 produces thetransits shown in Figure 4.4.3

Figure 4.4.3 – Simulated light curves for planets of varyingradii transiting a star of radius 0.8R�, with limb darkeningpresent [96].

The shapes of the transits in Figure 4.4.3 are signif-icantly different to the shapes of the transits in Figure

26

4.4.2, where limb darkening is not accounted for. Thedifference is that as the planet moves over the limbsof star, a smaller proportion of total light is absorbedcompared to when the planet transits over the centreof the star. This is because the star is brightest atthe centre. When the planet passes over the centre ofthe star the maximum drop in intensity occurs. All ofthese transits are assumed to have an impact parame-ter equal to zero, where it transits along the radius ofthe star.

To calculate the intensity due to limb darkening ata specific point upon a star, a method to mathemati-cally mark a specific point on the star, relative to thecentre, is first needed. This is done by defining thequantity, μ, as the cosine of the angle between the lineof sight of the observer and the normal to the surfaceof the star at that point. The geometry of this is il-lustrated in Figure 4.4.4 and shown mathematically bythe following equation:

μ = cos(θ) =

√R2∗ – r2

R2∗

(4.57)

As Equation (4.57) shows, μ can also be representedin terms of the radial distance from the centre r andthe stellar radius R∗.

Figure 4.4.4 – Geometry of observer and normal vectorsrepresenting the quantity μ [97].

This angular distance, μ, can be used with theknowledge of how temperature changes with radius,and how opacity and density change with radius, topredict the change in surface brightness as a functionof angular distance [94]. This is often expressed as theratio of the intensity at a point on the star over theintensity at the centre (μ = 1). The intensity ratio re-lations have only been fully determined by observationsfor a few stars, including the Sun. This means that inorder to calculate the effects of limb darkening for starswith transiting planets, the relations and coefficientsneed to be produced by theoretical models of the stel-lar atmospheres [98]. These relations and approxima-tions have evolved over many years, as more advancedmodels have been produced to analyse limb darkeningin stellar atmospheres, unlike that of the Sun. How-ever, initially the most used intensity relation was the

linear limb-darkening law, which was based upon thelimb-darkening features of the solar atmosphere [99].The temperature and physical properties of the Sunare comparable to that of that of a grey atmosphere,where limb-darkening is well approximated to be linear[98]:

Iλ(μ)

Iλ(1)= 1 – u (1 – μ) (4.58)

The value u in Equation (4.58) is known as a limb-darkening coefficient (LDC) and varies depending onthe wavelengths of light observed and ultimately thetemperature of the star. The study of other stellaratmospheres has changed our understanding of the in-tensity laws. Other, non-linear, laws have been foundto describe the limb-darkening effects to a greater accu-racy than the linear law when analysing a star outsidethe temperature range of the Sun. The quadratic law,shown in Equation (4.59), can be used for this purposeand gives a more general intensity law for stars withdifferent properties to the Sun [98]:

Iλ(μ)

Iλ(1)= 1 – a (1 – μ) – b (1 – μ)2 (4.59)

The quadratic law has two limb-darkening coeffi-cients, a and b, which need to be determined in orderto produce values for the intensity ratio. In 1970, thelogarithmic intensity law was proposed. The resultsfrom data taken showed a close fit between the ob-served relations and this theoretical approximation ofthe intensity variation in stars with the effective tem-peratures in the range 10000K < Teff < 40000K [100]:

Iλ(μ)

Iλ(1)= 1 – c (1 – μ) – fμ ln (μ) (4.60)

At this temperature range, Equation (4.60) onlyreally applies to the stars in the spectral classes Oand B, as these are the more massive and hotter stars.These types of stars are harder to produce follow upmeasurements for as radial velocity measurements areat their most precise when the stellar spectra containlarge numbers of absorption lines. O and B group starsdo not produce many features compared to MS stars.

Limb darkening can also have an effect on the shapeof the transit depending on the value of the impact pa-rameter, b. Figure 4.4.5 shows a light curve for mod-elled transits at varying impact parameters, where animpact parameter of 0.9 revealing a transit of an objectwith an almost negligible effect on the intensity of lightreceived.

27

Figure 4.4.5 – Simulated light curves for a planet of 10Earth radii, orbiting a star with uniform intensity and aradius of 0.8×Solar Radius, at various values for the impactparameter [96].

As the impact parameter value increases, the tran-sit time decreases due to the distance travelled acrossthe stellar disk shortening. This is caused because theorbital plane, with respect to the observer, moves fur-ther away from the equator of the star as b increases.The transit where b = 0.9 has a slightly lower depththan the other transits. In this case, this will mostprobably be caused by part of the planet transitingoutside of the observers view of the stellar disk, there-fore the intensity drop is reduced, as less of the stellardisks intensity is blocked. Figure 4.4.6 shows the sametransits as in Figure 4.4.5, however, the effects of limbdarkening are included and the stellar disk is no longerof uniform intensity.

Figure 4.4.6 – Simulated light curves for a planet of 10Earth radii, orbiting a star with limb darkening presentand a radius of 0.8R�, at various values for the impactparameter [96].

In the presence of limb darkening, the shorteningeffect of the transit times is still present for increasingvalues of b. However, the depth of the transits also sig-nificantly decreases as the value for b increases. Thisis caused by the decrease in intensity of the limbs of

the star. As earlier explained, an increase in the im-pact parameter results in a transit across the star at agreater height relative to its centre. This means thatthe planetary disk blocks out a smaller proportion oflight whilst transiting near the limbs and the changein intensity of the transit is less than that of a transitacross the equatorial plane.

Introducing the limb darkening phenomenon has aneffect on the transit observed, which in turn affects themeasurement of the planetary radius and the impactparameter; it is therefore crucial to correctly model forlimb darkening in order to obtain accurate values ofthese quantities.

4.5 Mass Constraining Methods(PS)

In general, the mass of transiting exoplanets can bedetermined by the use of a second detection method,such as the Doppler Method. However, by assumingcertain parameters concerning the composition of theplanet in question based on known observables, it ispossible to constrain the mass of the planet detectedvia the Transit Method to within a mass range withoutrecourse to a second detection.

By considering theoretical planet compositions,mass-radius relations can be determined which will giverough values for exoplanet masses (Figure 4.5.1). Fromtop to bottom the plotted lines on this graph showplanets composed of:

· hydrogen (cyan solid line)· a hydrogen-helium mixture with 25% helium by

mass (cyan dotted line)· water ice (blue solid line)· water planets with 75% water ice, a 22% silicate

shell, and a 3% iron core (blue dashed line)· water planets with 45% water ice, a 48.5% sili-

cate shell, and a 6.5% iron core (blue dot-dashedline)· water planets with 25% water ice, a 52.5% sil-

icate shell, and a 22.5% iron core(blue dottedline)· silicate (MgSiO3 perovskite) (red solid line)· Earth-like silicate planets with 32.5% by mass

iron cores and 67.5% silicate mantles (reddashed line)· Mercury-like silicate planets with 70% by mass

iron core and 30% silicate mantles (red dottedline)· solid iron planets (green solid line)

Several planets of known mass and radius are alsoshown for illustrative purposes.

28

Figure 4.5.1 – Theoretical mass-radius relationship for solidplanets of various compositions. Solar system planets ap-pear as blue triangles, while extrasolar planets (with signif-icant uncertainties on each value) appear in pink. [101].

By considering orbital and planetary characteris-tics, unlikely planetary compositions can be ruled out,further narrowing the mass range. For example, theequilibrium temperature of a transiting planet can bedetermined from its radius and semi major axis, if itsalbedo and the effective temperature of the star are alsoknown. If this temperature is sufficiently high that liq-uid water cannot exist on the surface of the planet,these compositions can be discarded, greatly reducingthe possible mass range.

Note that in Figure 4.5.1, radius values tail off andactually decrease at very high masses. This is dueto electron degeneracy pressure, as the Pauli Exclu-sion Principle forbids two electrons occupying the samequantum state. This adds a significant uncertainty tocalculation of the mass ranges of planets with particu-larly large radii.

Furthermore, for gaseous planets, the radius is de-fined not just by the mass of material present but ad-ditionally by its temperature, as the volume of a gasexpands with temperature. Thus, for close orbiting gasgiants (hot Jupiters) the lower mass limit must be re-duced to account for the ‘puffiness’ caused by increasedequilibrium temperature.

There is also a discrepancy in the radius calculatedby the transit method for a gas planet and the ra-dius given by the mean distance between the centre ofa planet and its ‘surface’, as the latter is defined asthe point at which the atmospheric pressure is 1 bar[102]. The ‘true’ radius of a gas giant will thereforebe slightly lower than that calculated by the transitmethod, as above the defined surface there still existsgas which absorbs light and thus reduces the measuredstellar intensity.

For close orbiting gas giants (hot Jupiters), it can

also be useful to consider the Hill Sphere and Rochelimit (see Sections 4.6.3 and 4.6.4) in order to constrainthe mass of the planet. In order to transit multipletimes, the Hill sphere of the planet at closest approachto the star must be at least as large as the volume ofthe planet. This can provide a lower mass limit forparticularly large planets, although most planets donot orbit close enough to their host stars for this to bea productive result.

This method is particularly useful in categorisingsmall planets of around the radius of the Earth, as forthis radius a gas planet is extremely unlikely, especiallyconsidering the bias towards detection of close-orbitingplanets inherent in the use of the transit method. Theplanet can therefore safely be assumed to be rocky, andthe mass can thus be constrained to within relativelynarrow bounds.

4.6 Auxiliary Theory

4.6.1 Stellar Variability (ML)

Stellar variability is the fluctuation of the apparentmagnitude of a star. This can be caused by intrinsicor extrinsic factors. Intrinsic factors (when a star’s lu-minosity actually changes) include pulsation, eruptionand novae (including supernovae). Extrinsic factors(changes in brightness due to external forces) includebinary eclipses, planetary eclipses and rotation.

Pulsation is where the star expands and contracts.MS stars typically have a constant luminosity, but gi-ants’ and supergiants’ radii may vary as they evolve,leading to a variation in their apparent magnitude[103].

Stellar eruptions include flares and coronal massejections. Flares are violent surface eruptions, result-ing in the release of energetic particles and electromag-netic waves, whereas coronal mass ejections consist ofa slow release of large spheres of gas over the courseof numerous hours. Both of these contribute to theapparent magnitude of a star when they occur [104].

Novae are nuclear explosions caused when a cer-tain amount of hydrogen has accreted from a red giantonto the surface of a white dwarf, where the two starscomprise a binary system. The white dwarf will settledown after the explosion and will begin to accrete hy-drogen again [105]. In contrast, supernovae occur whena massive star’s core collapses, releasing huge amountsof gravitational potential energy in a cataclysmic ex-plosion [106].

Binary eclipses result in a lower apparent magni-tude when either star is being eclipsed, and the high-est possible apparent magnitude when the stars areentirely uneclipsed [107].

Planetary eclipses occur when a planet orbiting thestar moves between the observer and the star, prevent-ing a fraction of the star’s light reaching the observer.This leads to a reduction in apparent magnitude, whichresults in periodic dips sometimes seen in the lightcurves of certain stars.

29

Starspots are caused by the differential rotation ofa star. This is when the different latitudes of the starrotate at different velocities. The magnetic field associ-ated with these latitudes gets wound around the star asthe equator rotates quicker than the poles. Some mag-netic field lines exit the star one side of the equatorand enter the other hemisphere. These exit and entrypoints are cooler than the surrounding environment, soappear darker [108] [109].

4.6.2 The Effect of Starspots onTransits (CL)

Starspots are a common feature of stars. They areactive regions upon a star’s surface where the magneticfield penetrates to the surface. These spots appeardarker due to these areas dispersing less energy and oc-cupying lower temperature ranges compared to the sur-rounding stellar surface [110]. Starspots are not con-tinuous, they can have lifetimes of a few hours to a fewmonths, as the magnetic fields that cause them fade,so therefore the number and position of the starspotsvary heavily with time. Figure 4.6.1 shows how thesestarspots appeared on the Sun on the 27th September2001.

Figure 4.6.1 – NASA’s image of the Sun showing the pres-ence of Sunspots on the 27th September 2001 [111]

By studying the positions of these starspots it ispossible to determine the stellar rotational period ofthe star, this is achieved by a measuring the move-ment of a spot with respect to time. The sun and itsspots have been studied for centuries and around 400years ago the first measurement for the rotational pe-riod of our Sun was achieved by directly tracking themovement of these areas of much lower intensity [112].However, for stars further away from the Sun, it is notpossible to resolve the starspots by direct imaging, sotherefore other techniques are used in order to deter-mine the stellar rotation from the starspots. Today,

there are three main methods in which to determinethe rotational period of the star. The first is usingspectroscopy to analyse the broadening of the spectrallines caused by the rotation of the star [112]. How-ever, this method does introduce a factor of sin (i) dueto the observed inclination of the Doppler signal. Thesecond method uses the principle that the stellar fluxchanges as the starspots move from the front to theback of the stars. This intensity pattern in a star’slight curve will repeat periodically, therefore giving avalue of the period of rotation [113]. This method issuperior when measuring stars with long rotational pe-riods, as the broadening of spectral lines cannot be re-solved at low rotational velocities. This method alsohas the advantage that the uncertainty factor of sin (i)is not incorporated into calculations [112].

The final method determines the rotational periodusing information of the starspots position from a tran-sit in the light curve. When a transiting planet passesover the starspot, there is a rise in intensity at thatpoint causing a small positive bump in the light curveand misshaping the symmetric bottom of the transit[113]. This effect is shown in Figure 4.6.2.

30

Figure 4.6.2 – Top panel - Planetary Transit in light curveof the Sun recorded on the 26th April 2000 (Black) andthe 29th April 2000 (Grey). Bottom panel - subtractionof the two light curves in the top panel. The arrowsand dotted lines indicate the flux variations caused by theSunspots. Data taken by the Big Bear Solar Observatory(BBSO).[112]

The abnormality and rise in intensity is due to thestarspot not being as bright as the surrounding photo-sphere, meaning that the average intensity of the starincreases when the planet is covering this darker area[113]. In order, to determine a value for the rotationalperiod of the star, the starspot needs to be detected inmultiple transits. The ability to detect the same starspot in multiple transits depends on two main factors:the obliquity of the system (angle between the orbitalaxis and stellar rotational axis) and the rate of transitcompared to the rate of stellar rotations. If the axisof the orbit of the transiting planet is aligned with theaxis of rotation of the star then subsequent transitswill be detected, as long as there is a second transitbefore the starspot has rotated around to the unob-servable side of the star. Misalignment of theses axes

could cause a starspot to appear in the first transit butnot in subsequent transits, even if the rotation of thestar was sufficiently slow (See later in this section forfurther discussion) [113]. The period of rotation canthen be easily calculated by considering the change inlongitude of the starspot on the star in the time be-tween the transits, Δt. This is depicted in Equations4.61 and 4.62:

Ps = 2πΔt

θ1 – θ2(4.61)

where θ1 and θ2 are the longitude values of the starspotat each point of measurement in each transit [112].These values can be calculated using the equation:

θi = arcsin

([a

Rs cos (lat)

]sin [2π (f i – 0.5)]

)(4.62)

where the first transit corresponds to i = 1 and the sec-ond to i = 2. ‘lat’ is the latitude of the starspot, whichcan be deduced from the inclination and the phase f i

is found from the phases of the positive and negativepeaks in the subtracted light curve (See the bottompanel of Figure 4.6.2). The calculated values of Ps

carry an uncertainty, this uncertainty can be reducedby repeating the measurements over many transits andtaking an average value for Ps [112].

In reality, there are gaps in light curve data fromspace telescopes, therefore it is not possible to simplysubtract one light curve from the other - as it was donefor the Sun in Figure 4.6.2. Instead, a transit is mod-elled by using the predetermined parameters of the starand its planet. This produces a model transit withoutstarspots that is then subtracted from the transit datawith the spots present, to obtain a subtracted lightcurve highlighting the starspots. In order for the spot-less transit to be accurate enough to produce usefulresults, accurate values of the planet radius, stellar ra-dius, inclination and orbital radius need to be inputinto a model that accounts for limb darkening [112](See Section 6.2.5 for more detail on the Stellar LimbDarkening Model).

As mentioned previously, the presence of starspotsin transits can also give information on the obliquity ofthe star-planet system. The Rossiter-Mclaughlin (RM)technique is the most commonly used technique tomeasure the obliquity of exoplanet systems. However,this spectroscopic technique proves to be challengingwhen analysing faint host stars that are typical of theexoplanet systems that Kepler has discovered. Manyof these stars are also slow rotators, which is a disad-vantage to the RM method but crucial in the transitstarspot technique [114]. Therefore, in order to ob-tain obliquities for these systems, the transit starspotmethod must be used.

The route that a transiting planet moves across theplanet is defined as the transit chord. If this transitchord lies over a starspot then the received intensitywill be temporarily brighter and a bump will be pro-duced in the light curve, as explained before. A slowlyrotating star in a perfectly aligned system will see the

31

presence of the same starspot in many recurring tran-sits. This is due to the transiting planet tracing outexactly the same transit chord with every full orbit,which is parallel and aligned with the starspots motiondue to rotation. The starspots position will appear tomove along the transit chord in subsequent transits asthe star rotates. In this case, the starspot appears inall transits until the starspot is rotated onto the hiddenside of the star [114].

In a misaligned system, the transit chord is not par-allel with the direction of the rotating starspots. Thismeans that if a starspot is covered in one transit, thespot would have moved away from the transit chordwhen it transits for the second time and won’t be de-tected [114] [115]. Therefore, the probability of therotational period of the star and the transit period be-ing related such that the starspot interacts with thetransit chord again, with in the next several transits,is extremely low [113]. Figure 4.6.3 illustrates the twoalignments discussed.

Figure 4.6.3 – A simulation of the effect of a single starspoton four consecutive transits, in an aligned system (Toppanel) and a misaligned system (Bottom panel). The ro-tation of the star is ten times slower that the orbit of theplanet [114]

These alignments can then be translated into esti-mates on the obliquity of the system, where the per-fectly aligned system has an obliquity of 0◦. Thismethod is being used to determine the reasons be-hind the high spin-orbit angles measured between hot

Jupiters and their host stars. In some cases, the di-rection of rotation of the star can be in reverse to thedirection of the orbit of the planet [116]. It is cur-rently believed that the high obliquities in hot Jupitersystems are caused by one of two theories. The first isthat the dynamical interactions, such as planet-planetscattering, tilt the orbits of the planets with respectto the stars rotation axis. The second is that the spinaxis of the star can get tilted from its original position,in line with the protoplanetary disk, by chaotic ac-cretion, torques from neighbouring stars or magneticinteractions [114]. In principle, the latter of the twotheories would be applicable to all of the exoplanetsystems [114]. By investigating the obliquities usingthe starspot transit method for the many exoplanetsystems that Kepler has discovered, the validity of thelatter theory can be deduced [115]. So far, obliquitiesof these systems have been found to be low, suggest-ing that the high spin-orbit angles in the hot Jupitersystems are caused by dynamic interactions. However,this is ongoing research, more systems need to be ob-served in order for this conclusion to hold [114][115].

4.6.3 Hill Spheres (EM)

The Hill Sphere is the region surrounding a massivebody in which the gravity of the body is dominant,within this region objects will orbit the larger body.The radius of the Hill Sphere is given by [67]:

RH =( m

3M

)1/3a (4.63)

where M is the mass of the larger body if M >> m, mis the mass of the smaller body, and a is the separationof the two bodies.

4.6.4 Roche Limit (EM)

The Roche limit is the boundary of the regionaround any massive body inside which, if a smallerbody were to orbit, tidal interactions would cause thesmaller mass to break apart. An estimate of the equa-tion for the Roche Limit is [67]:

aR = 1.44(M

m

)1/3R (4.64)

where M is the mass of the larger body if M >> m,m is the mass of the smaller body, and R is the radiusof the orbiting body. This is derived from the equa-tion of the Hill Sphere by balancing the gravitationalattraction for a particle on the surface of the smallerbody against ‘self-gravity’ of the particle. This orbitalradius is only an estimate as it is derived under the as-sumption that the small body undergoes no distortion;it simply calculates at which point the Hill Sphere isequal to the radius of the object. The more rigorouslyderived Roche Limit [67]:

aR = 2.456(M

m

)1/3R (4.65)

has a larger factor - 2.456 vs 1.44. The larger factor of2.456 arises from tidal forces acting on the planet that

32

cause it to deform. This decreases the gravitationalattraction between the object and its outer layers asthey are pulled away from the centre of mass.

4.6.5 Three-Day Pileup (EM)

When studying the distribution of periods of gasgiants around a host star there is a peak found at aperiod of ∼ 3 days with very low eccentricity orbits.Planets found in this region are known as hot Jupitersand current understanding suggests that these planetsformed past the ice line (see Section 4.1) and migratedto their current orbital radius either by interacting withthe disk from which they formed, [57] or by ejectingsmaller bodies from the solar system.

The total energy of an orbit is given by:

EOrbit = –GMm

2a(4.66)

Ejecting matter reduces the orbital energy of the largerbody as some is transformed into the kinetic energyof the smaller body. If enough mass is ejected in ashort time, for instance a small planet being expelled,the result is a highly eccentric orbit. The subsequentplanet-star separation corresponding to the periastronis important in the circularisation of the orbit. If theperiastron is close enough to the star, the planet willundergo tidal interactions with the star. This processhinders the free rotation of the planet and acts to dis-sipate the orbital energy making it more negative andhence a more circular orbit. If the periastron is tooclose to the star it will enter the Roche Lobe and betorn apart, if it is too far from the star the tidal inter-actions will not dissipate enough energy to have anytangible effect on the orbit. The angular momentumof an orbit is described mathematically as:

L = m(GM)1/2√

a(1 – e2) (4.67)

where m is the mass of the planet, M is the mass of thestar, a is the semi-major axis, and e is the eccentricity.If we assume that the eccentricity is approximately 1then the initial orbital angular momentum is:

Lini = m√

2GMa(1 – e) (4.68)

and equating this to the orbital angular momentum ofthe circular orbit gives:

acirc = 2rper (4.69)

As these planets are found with orbital periods of ∼ 3days, a periastron value of ∼ 1.5 days appears to dis-sipate enough energy to circularise the orbit withoutpulling the planet apart. This phenomenon is onlyvalid for Sun-like stars as much more massive starshave a larger Roche Lobe and can even have radii largeenough that an 3 day orbital period would be inside thestar.

4.6.6 Exomoon Detection (AE)

Exomoons are natural satellites that orbit extraso-lar planets. Like the planets themselves, they are ex-tremely difficult to see as they have such a low apparentsize and brightness. Although no exomoons have cur-rently been detected, there are theoretical methods ofdetection and the chances of discovering an exomoonare higher than before due to the increasing numberof exoplanets being discovered [117]. The two mainmethods of detection are observing the light curve fora transiting planet and looking for unexpected dips,and by looking for TTVs in the transiting planet thatcould be caused by the presence of an exomoon.

Detection by Light Curve Observation

If an exomoon is present around a transiting planet,it can be detected by looking for distortions in the lightcurve of the flux of the star when the planet transits.This is because the moon also transits the star, as isshown in Figure 4.6.4, and so will also reduce the de-tected brightness of the star. Without the exomoon,the light curve should have a smooth, symmetric dipwhen the planet transits, as shown in Figure 4.4.6 inSection 4.4.

Distortions appear when an exomoon is present be-cause what is actually being observed is a superpositionof the transits of the planet and the moon. Figure 4.6.5shows a case where the satellite is ahead of the planet(in terms of their orbit around the star) and so thereis an initial small dip caused by the moon followed bya larger dip caused by the transit of both of the bodiesbefore the dip slightly reduces in depth when the satel-lite is no longer transiting until finally the depth goesback to zero when the entire transit of the planet-moonsystem has completely finished.

Figure 4.6.4 – The two transits of the planet and moon.

33

Figure 4.6.5 – Superposition of the two transits caused bythe planet and exomoon [118].

As the relative position of the moon around theplanet will vary from transit to transit, the distortionof the planetary dip will change.

Detection by TTV

Subsection 4.3.2, describes how TTV signals can beused to detect additional planets orbiting a star. In asimilar way, this method can theoretically be used todetect the presence of satellites orbiting a transitingplanet, i.e. exomoons. The following derivations as-sume the case where there are no other planets in thestar system and only one satellite is orbiting the planet.Additionally, it can be assumed that the orbital planesof the star-planet and planet-moon systems are alignedwith an inclination very close to 90 degrees.

The planet orbits the barycentre (the centre of massof the planet-moon system), with a semi-major axis aw .Since there will normally be a difference between thetimes at which the planet is at the mid-transit point(which is the point at which TTVs are measured) andat which the barycentre is at the mid-transit point wewill expect to see transit timing variations. For a cir-cularly orbiting satellite, the TTV signal observed issimply given by the equation [119]:

δtcir =

(asMsPp

2πapMp

)cos(fm) (4.70)

where ap and as are the semi-major axis of theplanet around the star and the exomoon around thebarycentre of the planet-moon system respectively, Mp

and Ms are their masses, Pp is the orbital period of theplanet, and fm is the true anomaly of the moon which isdefined as the angular distance between the perihelionof the moon (the point at which the moon is closestto the planet) and the actual position of the moon asseen from the planet. The RMS of the TTV signal isthen [119]:

δtcir,RMS =asMsPp

apMp√

2π(4.71)

For an eccentric orbit the TTV signal has a morecomplicated form. There are now additional variables:

the eccentricity of the planet and satellite denoted byep and es respectively, and the pericentre positions de-noted by ωp and ωs. The RMS value of the TTV signalis now given by the equation [120]:

δtell,RMS =1√2

a1/2p asMsM

–1sp

[G(M∗ + Msp)]1/2ζT(es, ωs)

Υ(ep, ωp)(4.72)

where Msp is the total mass of the planet-moonsystem, and ζT and Υ are given by the equations [120]:

ζT =(1 – e2

s )1/4

es[(e2

s +cos(2ωs))(2(1–e2s )

3/2–2+3e2s )]

1/2

(4.73)and

Υ = cos

[tan–1

(–

ep cos(ωp)

1 + ep sin(ωp)

)]·

(2(1 + ep sin(ωp))

(1 – e2p)

– 1

)1/2

(4.74)

A range of as can be obtained by noting that thesemi-major axis of a satellite around a planet must liebetween the Roche limit, denoted by aR, and the Hillradius, denoted by RH .

The Roche limit, given by Equation (4.65) in Sec-tion 4.6.4 is the minimum radius where particles cancoalesce into a moon, or for an existing moon to be sta-ble and not be torn apart by the gravity of the planet itorbits [121]. The Hill radius is the maximum distanceat which the gravity of a planet dominates that of thestar so that the satellite orbits the planet, and is givenby Equation (4.63) in Section 4.6.3. An estimate of as

can be obtained by expressing it as a fraction of RH:

as = χRH (4.75)

where χ is a number between aR and 1. In fact, χcan be further constrained to have a maximum valueof approximately 1/3 [122] because the Hill sphere doesnot take into account factors such as radiation pressurethat would move the orbit of the satellite outside theHill sphere.

The period of the exomoon can theoretically be cal-culated by taking the ratio of the period of the planetaround the star and the moon around the planet. Theperiod of the moon Ps is given by Kepler’s third law:

Ps =

(4π2a3

s

GMsp

)1/2

(4.76)

and the period of the planet, Pp , is:

Pp =

(4π2a3

p

G(Msp + M∗)

)1/2

'

(4π2a3

p

GM∗

)1/2

(4.77)

Since as is a fraction of the Hill radius, it can beexpressed as:

as = χap

(Msp – Ms

3M∗

)1/3

' χap

(Msp

3M∗

)1/3

(4.78)

34

and substituting this relation into Equation (4.76)gives Ps in the form:

Ps =

(4π2χ3a3

p

3GM∗

)1/3

(4.79)

and so the ratio of Ps and Pp is:

Ps

Pp'√χ3

3. (4.80)

The issue in the method of detecting exomoons us-ing TTVs is that there are two variables that cannot beobserved: the mass of the exomoon and the distancebetween the moon and planet. The ratio of periodsequation seems to reduce this problem since a finiterange of values for the period of the moon and hencethe size of its orbit can be calculated as shown above.Knowing as means that the only unknown in the equa-tion for the RMS of a TTV due to an exomoon on aneccentric orbit is the mass of the exomoon so this canbe calculated.

In a similar way to the TTV method, observing theduration of transit over many transits and looking forsmall variations can allow the inference of the presenceof an exomoon. Under the same assumptions as theTTV method, a transit duration variation (TDV) mustbe due to the changing velocity of the planet due toits ‘wobble’ around the barycentre of the planet-moonsystem. The amplitude of the TDV signal is given by[120]:

δTDV =(ap

as

)1/2(

M2s

Msp(Msp + M∗)

)1/2τ√2

ζD(es, ωs)

Υ(ep, ωp)

(4.81)where τ is the average transit duration over many

transits and ζD(es, ωs) is given by [120]:

ζD(es, ωs) =

(1 + e2

s – e2s cos(2ωs)

1 – e2s

)2

(4.82)

It is interesting to note that the TDV signal in-creases with increasing exomoon eccentricity whereasthe TTV signal decreases.

By taking the ratio of the TDV and TTV signals,it is possible to remove Ms and hence be able to solvethe equation to find Ps , and hence as . Using the as-sumption of a circular orbit for the satellite (es = 0 ),so that ζT = ζD = 1, and cancelling equal terms, theratio of signals, denoted by η, is [120]:

η =δTDV

δTTV' 2πτ

Pp

√3

χ3/2

=2πτ

Ps(4.83)

Ps is the only unknown variable so it can be calcu-lated. as can then be calculated simply using Kepler’sthird law. Once as is known, its value can be inputback into Equation (4.76) to determine a value of Ms .So, although using either the TTV or TDV method inisolation to determine the distance of orbit and mass

of an exomoon cannot give an accurate value, usingboth signals together does enable an accurate value tobe determined. However, in the scope of this project,the accuracy of the TTV or TDV signals that can bedetected are too low to be able to make any measure-ments about an exomoon or even infer its existence,just like when using TTVs to detect planets.

4.6.7 Circumstellar Habitable Zone(EM & MG)

The Circumstellar Habitable Zone (CHZ) is a re-gion around a star where water can exist as a liquid.The CHZ is of special interest, particularly to astro-biology, as it gives a limit on where life, to our un-derstanding, can exist. If the planet can have liquidwater on its surface then the conditions such as tem-perature and pressure would at least be similar to thatof the Earth, providing the possibility of life. Deter-mining the CHZ for stars proves to be an incrediblychallenging process as many factors have to be con-sidered and extensively modelled before an accurateresult can be obtained. These factors include (but arenot limited to) atmospheric composition, density, dis-tribution, tidal locking, stellar spectral type [123].

Habitable Zone Limits (EM & MG)

There are several different methods for estimatinglimits on the inner and outer habitable zones. In in-creasing radial distance from star, these include: Re-cent Venus, Runaway Greenhouse, and Moist Green-house for the inner habitable zone (IHZ), and Maxi-mum Greenhouse and Early Mars for the outer habit-able zone (OHZ). The Moist Greenhouse limit is thepoint at which a planet’s stratosphere becomes dom-inated by water vapour [124] when surface tempera-tures are slightly higher than that of the Earth. AnH2O-dominated atmosphere becomes opaque to infra-red radiation preventing the planet from re-emittingthe radiation into space and preventing the cooling ofthe surface. A planet capable of retaining a moderateatmosphere which orbits close to its host star, withinan orbit similar to Venus for an Earth-like planet anda sun-like star, is subject to the Runaway Greenhousegas effect. The increased surface temperatures convertmore liquid water into vapour - an effective greenhousegas. The greater content of water vapour will, in turn,increase the atmosphere’s ability to contain the infra-red radiation. This cycle repeats creating a runawayeffect, raising the surface temperature to levels whichwould be unlikely to support life, or at least anythingother than extremophiles. An estimate of the innerlimit of the CHZ is determined by considering at whatpoint this runaway effect occurs [123]. The final modelfor the inner limit is the Recent Venus estimate. It con-siders that there has been no surface water on Venusfor approximately 1 billion years and what solar lu-minosity was compared to current values, essentiallyan empirical method of the Runaway Greenhouse forvalues in the solar system.

35

The Maximum Greenhouse relies on a CO2 atmo-sphere providing the greatest heating possible for aplanet, however if the concentration of CO2 is too highthe albedo of the planet will be high and much of theincident radiation will merely be reflected into space.It requires a balance between the saturation of CO2

and the albedo as a result of this atmosphere. Finallythe Early Mars limit recognises that the early Marsmaintained liquid surface water 3.8 billion years ago,much like the Recent Venus the solar luminosity can becalculated at that time and compared to current solarflux at Mars to provide an OHZ limit.

Stellar Spectral Type (EM)

The spectral type of the star has a significant ef-fect on the CHZ. O- and B-type stars have high mass,which means the star formed more rapidly than a starsuch as the Sun. This restricts the planets to form for ashorter period reducing their accretion time, lesseningthe maximum atmospheric density. They also have in-credibly high effective temperatures, due to their highmass, and emit intensely in the UV. The simple as-sumption for the requirement of liquid water falls shortin this instance as UV radiation is damaging to cells,the energy of the light being sufficient for ionisation tooccur. The planet would need shielding against suchradiation in order to be habitable; the planet needs athick atmosphere capable of absorbing the bulk of theincident UV radiation. There is a limiting factor on asufficient atmosphere, enough time has to pass duringgaseous accretion during the planetary forming phaseof the proto-stellar nebula. This requires that the stardoes not eject substantial matter from the system, i.eenter the T Tauri phase [125]. Combining this factwith the necessity for a solid surface upon which liquidwater can lie, high mass stars present several momen-tous obstacles that must be overcome. Furthermore,the planet has to be detected before it is analysed. Thelarge semi-major axis required to be in the CHZ meansthe planet is orbiting farther from the star, makinga transit is exceedingly unlikely see Equation (4.34).Even with a transit present in the light curve data, de-tecting it will be incredibly difficult as the fractionalchange in intensity will be minute as the ratio of theradii will be very small see Equation (4.28).

K-type stars have a different problem with habit-able zones. For a planet to have the correct equilib-rium temperature around a K-type star it must havean orbit that is significantly closer than that of theEarth’s. In principle detecting planets around K-typestars is easier and they are far more likely to be hab-itable than planets around giants. The habitable zoneis located in the inner region of the system, therefore,a smaller perturbation of the planet’s orbit is requiredto cause it to become sufficiently close to the host starthat it becomes tidally locked. A planet that under-goes tidal locking will have one face that remains il-luminated whilst the other is in perpetual darkness,thus an intense temperature gradient is engendered.This environment would hold the water primarily in

two forms, vapour and solid on the respective sides ofthe planet, and prevent there from being substantialamounts of liquid water present on the surface.

Habitable Zone Limit Calculation (EM)

In order to calculate the habitable zones aroundanother star, the luminosity of the star is required:

L = 4πσSR2T4eff (4.84)

where σS is the Stefan-Boltzmann constant, R is theradius of the star, and Teff is the effective temperatureof the star. Once the luminosity of the star has beencalculated, the spectral flux must be computed:

Seff = Seff� + aT∗ + bT2∗ + cT3

∗ + dT4∗ (4.85)

where the coefficients correspond to those presentedby Kopparapu et al. (2013) and are dependent on thelimit used, T∗ = Teff –5780. The stellar luminosity andthe spectral flux are combined with the solar luminosityin the following equations:

ri =

√Lstar/L�

S�,i(4.86) ro =

√Lstar/L�

S�,o(4.87)

where S�,i and S�,o are the solar fluxes calculatedat the inner habitable zone and outer habitable zonerespectively. The results of Equation (4.85) are tab-ulated in Section 7.2. These equations produce re-sults that apply to Earth-like planets in terms of mass,1M⊕ – 10M⊕.

Every planet discovered has had a semi-major axiscalculated that was used in conjunction with the limitson the CHZ to categorise the planets as either orbitingwithin the CHZ of their host star or not. Unfortunatelyof all the planets that have been found and their orbitalparameters computed, none have orbits that are per-manently place them inside the CHZ regardless of thelimiting assumption used. No quantitative errors havebeen calculated for the limits of the CHZ. The limitsthemselves are based on a number of assumptions thatdo not account for every variable that is involved in aplanet’s atmosphere such as cloud reflections regulat-ing the temperature. Each value is only an estimateof the limits and obtaining a numerical error would beinfeasible.

Mass Dependence (MG)

The mass of a planet has also been shown to ef-fect where the boundaries of its CHZ lie, especiallyfor the IHZ. The density of a terrestrial planet’s at-mosphere will increase with its mass resulting in theatmospheric column depth (scale height) of H2O beingsmaller than that of a less massive planet. This meansthat higher temperatures are required before enoughwater vapour is present in the atmosphere to dominatethe outgoing longwave radiation, hence the Runaway

36

Greenhouse limit moves inward for high-mass terres-trial planets. The OHZ is not significantly affected bychanging planetary mass as the increased albedo com-petes with the Maximum Greenhouse limit. This massdependence of HZs around MS stars was explored byKopparapu et al. (2014) [126] where planets of between0.1M⊕ and 5M⊕ were modelled using a 1D, radiative-convective, cloud-free climate model using atmospheresof H2O with N2 as a background gas. The OHZ usedCO2 instead of H2O as the main constituent of the at-mosphere. With the inclusion of 3D global atmospheremodels from a variety of different studies, the limits ofdifferent HZs could be outlined for stars with an effec-tive temperature of 2600 K – 7200 K. From the resultsof this study, shown in Figure 4.6.6, it is clear that theIHZ limit is extended inward when the mass of a ter-restrial planet is increased, and vice versa, whereas theOHZ is not greatly affected.

Figure 4.6.6 – HZ limits labelled for different planetarymasses. The x-axis is scaled using the solar constant.Rapidly rotating and tidally locked planets (assuming a 4.5Gyr tidal locking timescale) are divided with a horizontallysloped, black-dashed line. The coloured squares representtidally locked inner-edge limits around cool stars from adifferent study. [126]

The stellar fluxes used in Figure 4.6.6 are calculatedusing Equation 4.85. As discussed in Section 4.6.7, theIHZ of planets around cool stars is considerably closerthan for Sun-like stars. Overall, it has been shown thatthe inner limit of the HZ moves outward for low-massterrestrial planets – where the flux apparent is ∼10%lower than at Earth for a planetary mass of 0.1M⊕ –and moves inward for high-mass planets – where theflux apparent is ∼7% higher than at Earth. The impli-cations of this mass dependence prove to be positive forthe search for life-harbouring, habitable planets usingthe planet-finding methods outlined in this chapter. Amore massive planet with a similar density to Earthwill have a greater radius. A greater radius means theplanet will have a greater probability of being detectedby the transit method. The probability of a transit oc-

curring increases as the semi-major axis of the planetdecreases. This all points toward the fact that a ter-restrial planet is more likely to be detected if it has alarge mass – hence a large radius – and orbits close toits host star, therefore making it significant having theIHZ limit move inward with increasing planetary mass.

Exomoons and Habitable Zones (EM)

As discussed in 4.6.6, exomoons are currentlypurely theoretical but regardless are quite likely to ex-ist. Exomoons can orbit planets that are far beyondthe limits of the habitable zone, and can still containliquid water. It has recently been suggested and isquite likely that Ganymede along with Europa, moonsof Jupiter, have a liquid ocean beneath their crusts[127] [128]. The orbit of Jupiter at 5AU is beyondthe OHZ (1.7AU) [123] requiring a different source ofenergy heating the interiors of the moons. The tidalforces acting on a planet by their host star that causeorbits to circularise as discussed in 4.6.5 are the samethat heat up the moon. Flexing the whole body awayfrom its equilibrium generates heat through friction asthe body distorts and reforms, the energy transferredin this way can be increased by the presence of otherbodies. This is the case in the Jovian system wheremultiple moons interact at different times due to vary-ing orbital radii. The presence of multiple moons or-biting planets in our own solar system makes it likelythat there will be similar structures in extrasolar sys-tems.Tidal flexing is related to habitability as it can result inwarm vents on the ocean floor which are a place life canthrive, demonstrated in our very own oceans. Moonswith liquid oceans may have analogues to warm ventsand providing the necessary molecules are present, lifeis a distinct possibility. This means that despite noneof the planets discovered orbiting within the stellarCHZ, there may be moons orbiting within a moon spe-cific HZ around their host planet. When it becomespossible to observe and analyse exomoons determiningwhether or not there are sufficient tidal interactions be-tween the satellite and the planet for an ocean to existwill be truly exciting. Delving even deeper into theoret-ical astrobiology opens the possibility for life that existsin the most extreme conditions. An example being Ti-tan, the largest moon of Saturn, undergoing precipita-tion cycles akin to that of the Earth but with methaneas the molecule found in different states. Whilst com-pletely hypothetical, including moons as possible hab-itable bodies would completely change what is consid-ered the “Habitable Zone”.

37

Chapter 5

Asteroseismology Method

5.1 Initial Data Analysis

5.1.1 Plotting Raw Data (JG)

The Asteroseismology Sub-Group was providedwith frequency-power spectra for 40 stars. These camein the form of text files which were then input to MAT-LAB to obtain the spectra visually. Understandably,there was a wide range in the quality of spectra given,with the region of interest overwhelmed by noise inapproximately half of the individual data sets. Thespectra with a low signal to noise ratio required filter-ing, but those with a high signal to noise ratio couldbe used immediately by the team aiming to manuallymeasure the large and small frequency spacing.

5.1.2 Filtering and Removing Noise(GM)

In order to better identify the regions of intereston the frequency-power spectra and isolate the oscil-lation peaks, the data needed to be filtered to removeas much noise as possible. Two filtering methods weredeveloped in order to achieve this.

The first, a MATLAB function called sm3 , mainlyused the inbuilt MATLAB function filter . The filterfunction was a one-dimensional digital filter, that useda rational transfer function as a moving-average filter.The moving-average filter was represented by the dif-ference equation

y(n) =1

W(x(n) + x(n – 1) + ... + x(n – (W – 1))) (5.1)

where W refers to the window size of the filter. Whilethis produced a moving average filter for the majorityof the data, it did cause significant problems towardsthe two ends of the data set. This was due to the firstvalue being calculated as y(1) = x(1)/W, the secondas y(2) = (x(1)+x(2))/W and so on, so until there werethe same number of x values as the window size, theaveraged y value was always much smaller. With awindow size of a few hundred points this meant that theends of the filtered spectra were significantly smallerthan expected and the problem was only accentuatedwith multiple iterations of the function on the samedata set.

To alleviate this problem, a second MATLAB func-tion was written which did not rely on the inbuilt filter

function. This function, smooth, used a boxcar func-tion to average the data of a set size T with a windowsize W. Firstly the function calculated c = 1/W: thescaling amount for each value to be averaged. It thensummed the first W values and multiplied them by cto calculate the average of that window and assigned itto the first value of the filtered set. The function theniterated (M-W+1) times through the entirety of dataset, calculating the subsequent filtered values using theequation

y(n) = y(n – 1) + (x(n + W – 1) – x(n – 1)) ∗ c (5.2)

By using equation 5.2 rather than calculating theaverage for each subsequent window size, only threecalculations were needed, rather than having to calcu-late the sum of W terms for every value to be filtered.For large window sizes this greatly reduced the num-ber of calculations that needed to be completed andtherefore the time it took to complete each filteringprocess. This function could also very easily be loopedto act as a double- or even triple-boxcar filter and fur-ther smooth the data. The only disadvantage to thisfunction was that, due to the average of the windowbeing assigned to the first value in the window, at theend of the data sets a set of (W-1) points was dis-carded. Looping the function a significant number oftimes could cause the vast majority of the data to bediscarded. For a double-boxcar filter however, used toinitially filter all of the frequency-power spectra, thiswas not a problem and smooth was used with greatpreference over sm3.

5.1.3 Isolating Regions of Interest (GM)

After filtering, it was beneficial to isolate the re-gions of interest to aid in calculating Δν and δν.With several hundred thousand data points in eachfrequency-power spectra and the regions of interestgenerally only occupying around 20% of the spectrum,isolating these areas and discarding the unnecessarydata would save processing power, and therefore time,and would potentially allow easier identification ofsmaller details in the spectra. Isolating the relevantregions of the spectra was done manually from MAT-LAB figures of the filtered data. This was done byhighlighting the area of interest on the figure, which

38

MATLAB would automatically convert into a new setof data. This was done for all but three of the spectra,for which no region of interest could be found.

5.2 Manually Measuring StellarOscillation Properties

5.2.1 Measuring Large and SmallFrequency Spacing (JG & ML)

Initially, the power spectra with a high signal tonoise ratio were plotted using MATLAB. Identifyingand labelling the l = 0 peaks was an important partof the process to gain an accurate value for the largefrequency spacing. On the spectra with a high signalto noise ratio, the l = 2 peaks were prominent in closeproximity to the left of the l = 0 peaks, allowing usto distinguish between these and the otherwise verysimilar looking l = 1 peaks.

For the power spectra with a slightly lower signalto noise ratio, the l = 2 peaks were identifiable af-ter a small amount of filtering, but this was not al-ways the case. For the power spectra with no identifi-able l = 2 peaks, the slightly irregular spacing betweenl = 1 peaks had to be used to distinguish them fromthe l = 0 peaks. If the irregular spacing was not appar-ent then it made less of an issue which peaks were usedto identify the large frequency spacing, as this methodwould not be used to yield final results.

To measure the large frequency spacing, the powerspectra were first smoothed by the computational anal-ysis team, as mentioned above. This made the centreof each peak much easier to identify. MATLAB’s datacursor tool was then used to accurately determine thefrequencies of the centre of the peaks. The large fre-quency spacing was measured for several pairs of peaksand an average taken to create a more accurate value.This process was repeated for the small frequency spac-ing where possible.

This process provided the manual measurementsteam with familiarity with each power spectrum. Asthe computational analysis team were using a builtin function to identify peaks, incorrect spacing valueswere obtained for a small number of spectra. Onceall spacings had been measured computationally, re-sults were compared with the manual measurementsand incorrect results were identified. The computa-tional measurement process was then repeated withthe aid of the manual analysis team to correct thesefew discrepancies.

5.3 Computationally MeasuringStellar OscillationProperties

5.3.1 Measuring Large and SmallFrequency Spacing (LS)

The computational method for finding the fre-quency spacings involved identifying peak positionsand from there calculating the spacings between rele-vant peaks. The MATLAB iPeak function, written byThomas O’Haver, was used to find peaks in the dataand flag them depending on threshold values for peakparameters [129], namely the peak amplitude, gradi-ent, sharpness and width. These thresholds could bealtered interactively whilst simultaneously viewing aplot of the signal, allowing the important peaks to beidentified for each star. The function also allowed thefrequency position of the peaks to be exported withhigh precision.

The iPeak function worked by using the derivativeof the data looking for zero-crossings, which indicatedturning points. To help to eliminate the effects of noise,it first smoothed the derivative. The results could befurther narrowed down by imposing a minimum valueon both the slope of the derivative at the turning pointand the amplitude of the peak. The position of thepeak was then identified by fitting a portion of theunsmoothed signal at the turning point, as smooth-ing distorts the shape of the peak. The width of thesmoothing filter, the amplitude, the slope of the deriva-tive and the amount of the original peak fitted to de-termine the position were all parameters which couldbe altered as required. However, as it was simple topick out the relevant peak positions, only the ampli-tude threshold was altered to reduce particularly largenumbers of false mode detections due to noise.

After exporting the data for each power spectrumfrom MATLAB into a spreadsheet, a number of valuesfor Δν were calculated around the mode which approx-imately corresponded to νmax . Four values of Δν werecalculated where there were sufficient numbers of l = 0peaks and the mean of these values was calculated togive a single value for the frequency-power spectrum.The same process was then used to calculate δν wherethe l = 2 could be identified.

A more simplistic approach to peak identification,which was discarded in favour of iPeak, was to applya power spectral density amplitude threshold to thedata and to use data points above that threshold tofind peaks. The major failing of the threshold methodwas its inflexibility. For instance, l = 2 modes oftenmerged into the l = 0 peaks in filtered data. It wasvery difficult for such l = 2 modes to be detected us-ing the threshold method as a peak was considered tobegin when the data were above the threshold and endwhen they went under. The peak position was givenby the position of the maximum power spectral den-sity in that frequency window. Therefore, for an l = 2mode which only manifested as a small bump on the

39

side of an l = 0 mode, the threshold value needed tobe in a narrow range of values between the bottom andthe top of the l = 2 mode in order for both peaks tobe detected. This was impractical to implement as thecorrect threshold for one mode often failed for another.

Another alternative method that was considered forextracting the frequency spacings was to use the auto-correlation of the signal. The autocorrelation is foundby multiplying the signal by itself with a certain ap-plied lag in frequency and then integrating over allvalues of frequency in the signal. It can be used tofind periodicity as it peaks when the signal and itslagged version are similar to each other. The auto-correlation function of a star’s power spectral densityplotted against frequency lag will therefore show peaksfor the frequency spacings. The peak corresponding tothe large frequency spacing would be the second mainpeak from the centre of the function, with the peakfirst from the centre being the average frequency spac-ing between l = 0 and l = 1 modes. In signals wherethe l = 2 modes are visible, plotting the autocorrela-tion function would also show a smaller peak beside thelarge frequency spacing peak. The distance betweenthis smaller peak and the large frequency spacing peakwould give the small frequency spacing.

The autocorrelation method was simple to imple-ment, but noise in the signal made peaks in the func-tion much less obvious. This could be circumventedsomewhat by setting points which were below the noisethreshold to zero before autocorrelating. However, thesmall frequency spacing often became obscured as itwas not unusual for some l = 2 modes to be less pow-erful than the noise. This was another reason thatthe iPeak function was the most suitable method forextracting the frequency spacings. It did not matterif the data was moderately noisy, because iPeak stillidentified small l = 2 modes even if it also picked upspikes in the noise. For our stars, it was not difficult topick out only the relevant peaks in the exported dataso that the noise could be ignored.

5.3.2 Measuring Frequency ofMaximum Power (LS & GM)

To find νmax , the frequency of maximum power,a Gaussian curve was fitted to the signal using theMATLAB function gaussfit [130]. This function usedan iterative least-mean-squares algorithm to optimisethe position and width of a Gaussian peak that bestmatched the data. The function used the position ofthe maximum power as an initial guess for the positionof the Gaussian peak. After one hundred iterations,the final value of the centre of the Gaussian was takento be the value of νmax .

Previously, it had been attempted to find νmax byrepeatedly smoothing the data until the signal was re-duced to a broad peak. However, it was difficult toselect an appropriate window size as both small andlarge windows had separate disadvantages. Small win-dows needed many repeated runs for any significantsmoothing to be seen. Due to the powerful noise in the

low-frequency areas of the power spectrum, large win-dow sizes caused issues with skewing the signal to bemore powerful at the low-frequency end. In addition,the smoothing method was computationally expensivewhen the whole dataset was smoothed. In an effortto correct for this, the signal was cut out of the databefore smoothing. This was not suitable either as edgeeffects, either at the beginning or the end of the datadepending on the smoothing code, caused the loss ofsignificant amounts of the signal. Experiments alsoshowed that fitting a Gaussian curve to the smootheddata using gaussfit produced almost exactly the samevalue for νmax than using gaussfit on the unsmootheddata, making it an almost redundant effort.

5.4 Obtaining AsteroseismicResults

5.4.1 Using Scaling Relations(MH)

Initial results from the scaling relations were avail-able as soon as a value for νmax and Δν0 were obtainedmanually for each star. The simplicity of the scalingrelation method made it easy to create a spreadsheetthat used the relations to calculate the mass and ra-dius as soon as the manually measured νmax and largespacing were entered, along with the effective tempera-ture for each star. The uncertainties for the measuredquantities were also entered in order to calculate anerror for the mass and radius. The errors were calcu-lated by adding the uncertainties on each component ofthe scaling relation in quadrature, with an extra errorthat accounts for the intrinsic inaccuracy of the scalingrelations due to the assumptions that they require.

Once the computational methods to obtain moreaccurate values for νmax and Δν0 were developed,these values were similarly added to the spreadsheetand the scaling relations were used to calculate a sec-ond set of values for the mass and radius of each star.The errors on the values measured computationallywere significantly lower, and this was reflected in thecalculated errors for the mass and radius. These moreaccurate results were then passed to the Planet-FindingSub-Group for use in producing absolute, as opposed torelative, values for quantities including planet radius.

5.4.2 Using Asteroseismic Diagrams(GM)

The Asteroseismology Sub-Group was providedwith the oscillation information for approximately70,000 stellar models computed using the Modules forExperiments in Stellar Astrophysics (MESA) stellarevolution code. The models ranged from a mass of0.8 to 1.5 solar masses with metallicities of Z = 0.005to Z = 0.08. Each combination of mass and metal-licity had approximately 70 models throughout theirevolution, generally starting with a central hydrogenmass fraction, Xc of 0.7 and ending at around 0.05 at

40

the end of the MS. The frequency data were detailedfor modes with angular degree ranging from l = 0 – 2and l = 0 generally had radial orders from n = 0 – 30 .The number of radial orders for l = 1 and l = 2 variedsignificantly for each model.

Extracting Frequency Data

A C++ program was written to automatically ex-tract and calculate the relevant values from the modeldata. The program worked by reading the frequencydata file path and model parameters - namely stellarmass, radius, metallicity and hydrogen mass fraction- and opening the frequency data file. The programcalculated νmax for the model using the scaling rela-tion in Equation (3.8) and the model parameters, theniterated through each of the l = 0 radial modes to findthe closest mode to νmax . While the scaling relationfor νmax could lead to slightly inaccurate results for theless solar-like stars, as it was only used to find the ap-proximate area to calculate Δν and δν this would nothave much, if any, effect on the final values so was anacceptable method to use. The program first countedthe total number of radial orders for l = 0 and l = 2angular degrees, before iterating through the frequen-cies of all of the l = 0 modes to find the radial orderclosest to νmax . The program then checked that thisradial order was not greater than the highest availableradial order for the l = 2 angular degree in order to en-sure that δν could be calculated. If the selected l = 0radial mode was greater, the program decreased theradial order incrementally until an acceptable radialmode was found or the program ran out of radial or-ders for l = 0 , at which point the model was discarded.

Following this, the program selected the three ra-dial modes above and below νmax and calculated fivevalues for Δν from the difference between adjacentl = 0 modes. To calculate δν, the program calculatedthe difference between each selected l = 0 radial modeand the previous l = 2 mode, to give six values. Themean of these values was then calculated to give singu-lar values for Δν and δν and the program input thesevalues and model parameters into a .txt file for lateruse. If the program could not compute Δν and δν forsome reason - due to lack of radial orders for l = 0 orno corresponding l = 2 modes, or the calculation pro-duced an unexpected or negative result - the model wasdiscarded. Erroneous values were mainly calculated formodels nearing the end of the MS due to the rapidlychanging properties of these stars.

Plotting the Asteroseismic Diagrams

All calculated values for Δν and δν were grouped infiles corresponding to the model mass and metallicityand these were loaded into MATLAB as arrays. Themodel data were then plotted using a MATLAB func-tion, which mainly used the inbuilt MATLAB functionscatter. Using a scatter graph for the mass lines ratherthan a line graph meant that the data did not need tofirst be ordered chronologically and meant any anoma-

lous results were easily identified and could be ignored.The function iterated through the various masses foreach metallicity and plotted the small frequency spac-ing against the large frequency spacing. Only inter-vals of 0.1 solar masses were plotted so that the graphwas still legible - plotting all available masses for eachmetallicity meant that it was almost impossible to seehowΔν and δν varied as the star evolved for each mass.For masses greater than 1.2M�, the mass lines tendedto be in much closer proximity to each other and insome cases were almost overlapping. Adding ten ex-tra mass lines between these intervals would have beenunfeasible.

In order to extract data about the stellar ages, thehydrogen mass fraction of the star needed to be deter-mined from hydrogen mass fraction isopleths. Unfor-tunately the data for various masses on the same graphdid not necessarily have the same exact hydrogen massfraction values and the closest value to a desired hy-drogen mass fraction often varied by 0.01 or greater fordifferent masses of the same metallicity. In order to gettrue isopleths the values for regular isopleth intervalsneeded to be known. For the ZAMS age, this was lessimportant, however it was decided to use intervals of0.1 with values for ZAMS (around 0.7), 0.6, 0.5, 0.4,0.3, 0.2, 0.1 and 0.05. The final value was used only onmass lines less than one solar mass, or in some cases1.1 solar mass, as these were the masses for which val-ues of Δν and δν tended to be less volatile as the starreached the end of the MS.

While the isopleth separation was not a linear re-lationship, for small intervals in the hydrogen massfraction of around 0.01 it could be approximated asone. In order to calculate values for Δν and δν for theisopleths, the known values of the isopleths immedi-ately above and below the desired isopleth value wereweighted in order to give the closest match to the iso-pleth value. For example, for the Xc = 0.4 isopleth forthe 0.4M�, Z = 0.017 model set, the known values ofΔν and δν were for Xc values of 0.410321 and 0.394821.The values for Δν and δν were thus weighted 33% and67% respectively in order to calculate the correspond-ing values for an Xc value of 0.4. By calculating theisopleths using this method it meant that the isoplethline was the best representative for the respective hy-drogen mass fraction and improved the accuracy whenadding the project’s stellar data. It also allowed anyoutliers on the mass lines to be easily identified as, ifan anomalous value was used to calculate the isopleth,it would not follow the expected trend. This methodallowed these values to easily be discarded and the nextclosest known isopleth value to be used and weighteddifferently. In total, 14 asteroseismic diagrams withcomplete hydrogen mass fraction isopleths were com-pleted to be used in the next stage of the project andan example of one can be seen in 5.4.1.

While the sub-group had been given metallicityvalues for the stars, these were in the form of [Fe/H]metallicities. Asteroseismic diagrams generally use theheavy mass fraction, or Z value, so these values neededto be converted to the correct form before the stars

41

Figure 5.4.1 – The completed asteroseismic diagram for metallicity Z=0.09

could be plotted on the correct asteroseismic diagram.The total metallic abundance of the star, [M/H], is re-lated to the metallicity by [131]:[

M

H

]= log10

(Z/X

Z�/X�

)(5.3)

The total metal abundance [M/H] is a more generalexpression than the iron abundance [Fe/H] and the twoare related through a constant, A, by [131]:[

M

H

]= A ∗

[Fe

H

](5.4)

where A generally takes values between 0.9 and 1.By equating Equations (5.3) and (5.4) the relation be-tween Z and [Fe/H] can be written as:

log10

(Z/X

Z�/X�

)= A ∗

[M

H

](5.5)

Rather than spend time writing a program to com-pute the Z value from the [Fe/H] value, it was foundthat calculators to do this already existed online [132].By using this calculator, the corresponding Z value foreach star was computed. As the Z values for the stel-lar models did not correspond exactly to the Z valuesof our stars, any small inaccuracy caused by using anon-specific calculator was rendered insignificant whenplotting the stars on a non-perfect asteroseismic dia-gram.

Since the asteroseismic diagrams were plotted asΔνagainst δν, only stars where both of these values hadbeen calculated could be added. This requirement im-mediately reduced the number of stars that were avail-able to plot, as for some of the frequency-power spectra

the signal to noise ratio had not been great enough tocalculate δν. It was decided to also only use the com-putationally calculated values to plot the stars, whichreduced the total number of stars to be plotted to 22stars.

These stars had a metallicity range of Z = 0.09to 0.35 which corresponds to roughly 0.5 to 1.5 solarmetallicity. The stars were grouped by the closest as-teroseismic diagram metallicity, with all stars havingno greater than a 0.003 difference to its asteroseismicdiagram metallicity, which would not make any signif-icant difference when the stellar values were extracted.Only six asteroseismic diagrams would be needed toplot all available stars, with metallicities of Z = 0.009,0.011, 0.014, 0.017, 0.023, 0.029.

To add the stars to the diagrams, a MATLAB func-tion was written where the user could open the relevantdiagram and input the values for Δν, δν and the starnumber. The star would then be plotted in the correctplace on the diagram and would be labelled automati-cally. The function also plotted the stars as a differentcolour and marker than the mass lines and hydrogenmass fraction isopleths so they were easily identifiableon the graph. Error bars were initially added to thestar plots to make constraining the mass and age of thestar easier when it came to extracting stellar proper-ties, however these could not be identified easily enoughdue to the size of the asteroseismic diagram so were leftout of the main plots.

Extracting Stellar Properties

Due to the design of the asteroseismic diagrams, ex-tracting the mass and hydrogen mass fraction was verysimple, and was done by calculating its position relativeto the closest mass lines and isopleths. This was donemanually, and the variation in mass and hydrogen mass

42

fraction was approximated as a linear relationship be-tween adjacent values. Using this method all but twoof the values were found. One of these values was onlyslightly beyond the scope of the asteroseismic diagramto which it was added so could be extracted, albeit toa lower accuracy, however no values could be extractedfor the other due to its position on the asteroseismicdiagram.

In order to further constrain the masses and agesfor each star an attempt was made to produce smaller-scale asteroseismic diagrams for each star - showingmass lines with separations of 0.01M� and hydrogenmass fraction isopleths with separations of 0.001. Onthese asteroseismic diagrams the errors for Δν and δνfor each star were more significant and were also plot-ted with the star. However, it was found after complet-ing this for GSIC 3 that the error for the mass and hy-drogen mass fraction value was not significantly differ-ent from the values obtained by the previous method.Calculating each isopleth for the smaller separationstook significantly longer than for the greater sepa-rations and it was decided that the amount of timeneeded to do this for each star was too significantfor the small reduction in the constraint boundariesit would produce. GSIC 3 remains the only star thisfurther method was completed on as an example of itsresults.

The stellar ages were related to the hydrogen massfraction of the star - the lower the hydrogen mass frac-tion, the older the star would be. This was, however,only true for stars on the same mass line due to starswith different masses ageing at vastly different rates.The stellar ages, radii, luminosity and effective temper-ature of each model were supplied alongside the mass,metallicity and hydrogen mass fraction of each, so com-puting these values for the project’s stars was a case ofmatching the star to the closest available model. Thiswas an optimisation problem and while a search pro-gram could have been written to do this automatically,it was easy enough to go through the model data man-ually to obtain this values. Due to the sheer number ofmodels supplied, for the vast majority of the stars anexact match to the extracted hydrogen mass fractioncould be found and for the others it was never morethan a 0.01 difference.

Due to the age, radius, luminosity and effectivetemperature values being dependent on two variables,the mass and the hydrogen mass fraction, an errorin either of these values would mean a different stel-lar model would have to be used. Therefore, to con-strain them, the values for each from the models atthe maximum and minimum mass and hydrogen massfraction were calculated and the greatest difference tothe already-calculated stellar properties were used asthe error for that value.

Amalgamating the Data

In order to compare the masses, ages, radii, lu-minosities and effective temperatures of the stars andcompare the two methods used to compute the values,

it was decided to display all of the data for the stars onseveral graphs. These were created using MATLAB’sinbuilt errorbar function, which plotted a line graphof the data with the various errors for each value. Byremoving the line using MATLAB’s figure editor andadding point markers to the various points, it was veryeasy to compare the various data. Several graphs wereplotted in this way and are discussed later in this reportin Section 7.2.3.

43

Chapter 6

Planet-Finding Method

6.1 Manual Methods(LP, CL & AE)

A Python code was used to plot light curves for theshort and long cadence Kepler data separately. Theplots produced by this light curve reader code could beanalysed manually to find any dips in flux that may cor-respond to a transiting planet. The code also countedhow many data points were present in each data set(long and short cadence) to enable the user to get afeel for the size of the data set, useful for running fur-ther codes outlined in Section 6.2. Initially, a total offorty light curves were analysed using manual methodsby a team within the Planet-Finding Sub-Group. Dur-ing the analysis, any periodic dips in flux were notedwith the periodicity, transit depths and transit dura-tions analysed. The curves produced were varied, somedisplayed clear transits with little noise, therefore en-abling results to be collected with low error values.Other curves however were more difficult to analyse.With greater noise levels, values could not be collectedwith as much certainty and error values were greater.The short cadence data contained more data pointsand therefore was used for analysis in favour of thelong cadence data, as values collected from the curveswould be more accurate. On top of this, the transitswere much easier to spot on the short cadence curvesthan the long cadence curves.

Apparent recurring transits were searched for alongeach light curve. Once the shape and periodicity ofthe transits were considered constant, analysis of thetransits could begin. Transits were analysed in groupsof ten, with data for consecutive transits being ideal asvariations in period could more easily be noted. On topof this, three of the light curves revealed two differenttransit patterns, which were analysed separately. Thisindicated that multiple planets were in orbit aroundthese particular stars, rather than just one.

Fluctuations in flux were apparent throughout eachlight curve even with transit dips disregarded. Insome systems, the flux readings consistently fluctuated,whilst in others, the changes were more gradual. Onepossible reason for such flux changes are the oscillationsof the host stars. Changes in stellar luminosity occurfrequently when stars expand and contract. The ap-pearance of starspots on stars also results in changingstellar luminosity. Gaps also occur when looking at the

data, preventing us from seeing any transits or fluctu-ations in flux, during particular time windows. Thesegaps are generally as a result of scheduled movementsor rolls of the spacecraft and down-linking events [133].These factors resulted in some slight difficulties whensearching for consecutive transits manually.

The extraction of numerical data from the tran-sits was a relatively simple, yet laborious process, withall values, (including errors) checked manually. Thetransit depth was measured by recording the flux justbefore the start of a transit, and simply subtractingthis value from the lowest flux value at the bottom ofthe same transit. Any uncertainty in these values dueto noise was recorded and used to find errors in eachvalue for transit depth. The values for the error werecollected by considering the upper and lower limits offlux just outside the transit and at the bottom of eachtransit.

The transit duration was recorded using a similarmethod, with the times at which the transits begin andend recorded. The beginning of a transit was taken tobe the point at which the flux begins to fall, with theend taken to be the point at which the flux levels offafter rising back up from the transit well. Again, un-certainty is present for these times as the beginning andend of a transit is unclear at times. This is particularlytrue when limb-darkening and varying impact param-eters are considered, which give varying gradients ofslopes at which the ingress and egress occur. Valuesfor the error were collected by considering the upperand lower limits for the time at which the beginning orend of a transit could be considered.

Period values were found in a slightly more com-plex way. The times for the beginning and end of eachtransit were added together and then divided by twoto find a central point in time for each transit. Thetimes recorded between each of these average valueswere then considered as period values for the transits.Error values for the period were found by using pre-viously obtained error values for the transit duration,with the manipulation of these values used to validatethem for the central points of each transit.

After ten sets of data had been collected for each ap-parent planet, average values for period, transit depthand transit duration were taken with errors included inthe calculations. The values were then used to aid thecomputational team, giving a preliminary indication of

44

the properties of the transits of each planet.

6.2 Computational Methods

6.2.1 Introduction (FC)

Once all the transit property values had beencalculated manually, the next step was to refine andbuild upon them using computational methods. Usinga computer program to determine these quantitiesenabled higher accuracy and greater precision, whichwas necessary in order to obtain meaningful results forplanetary and stellar properties. The hand-calculatedvalues were useful as first estimates for the parametersinput into the Python codes created, but couldnot be used for final results as the uncertainty onthese values would be far too high. Several codeswere built in Python 3.4, which all worked in col-laboration to achieve the high accuracy values forplanetary and stellar properties found in the resultsin Chapter 7. The most important of these codesare described in detail in this section, explaining howthey function and why they were necessary for theobtention of results. The codes to be discussed include:

· Transit detection code· Phase folding code· Transit fitting model (uniform source and limb-

darkening versions)· Limb-darkening numerical model

Each of these codes worked in tandem with one an-other in a step by step process of obtaining increas-ingly accurate planetary property values from lightcurves. In particular, the orbital period was first esti-mated manually using a simple light curve reader code,then improved upon using the transit detection code,then further improved using the uniform source tran-sit fitting model, and finally the most accurate andprecise value possible was determined using the limb-darkening transit-fitting model. The highly accurateperiod value could then also be used to help constrainother parameters in the limb-darkening fitting modelsuch as planet radius and the semi-major axis.

6.2.2 Transit Detection Code(FC & AW)

This computer code was built with the primary pur-pose of detecting transits and acquiring more accurateplanetary orbital period values. A more accurate pe-riod would not only clarify information about the exo-planetary system, but is essential for successful phasefolded transits, and for use in the transit fitting models.

The process for detecting transits involved binningthe light curve into bins of equal width, and takingthe mean of each bin. The code initially plotted thelight curve for the input star, as mentioned previouslyfor the manual methods. This image, along with thehand-calculated values, allowed the user to gain an ideaof the most efficient bin size to use, as the bins had to

be small enough for a drop in the mean intensity tobe noticeable, but not so small that the code took toolong a time to process this simple step. The code thencalculated the difference between the mean values ofeach consecutive bin, and flagged up any significantdrop in intensity. The user also had to define whatdrop in intensity is considered ‘significant’ based uponthe hand-calculated values of the transit depth. Theseflags are shown as the red points in Figure 6.2.1. Thisalso gave a rough estimate of the number of transitsdetected within the data set looked at by the code.However, this was dependent on the quality of the ini-tial user-input values, and so this step often requiredrepeating several times until there was approximatelyone flag per transit. However, this step did not need tobe completely accurate, as the code was able to accountfor discrepancies such as missed transits, double detec-tions of a single transit and also spikes in the signalnoise.

The next step was to find the central point for eachtransit. To do this, a small range of data was takeneither side of each flagged point, and further binnedusing very small bins (ranging from 3 to 5 data points).The range either side of the flagged coordinate to beinspected was input as the hand-calculated value forthe transit width plus some additional leeway in or-der to encompass the entire transit. Subsequently themean of each bin was once again calculated, after whichthe minimum mean value in the inspected region wastaken, representing the centre of the transit. Thesepoints were then plotted onto the light curve, an ex-ample of which is shown in Figure 6.2.2, demonstratingthe effectiveness of the code to find the centre of eachtransit. Any of these points which may not precisely belocated at the centre of the transits, due to irregularlyshaped transits or particularly large spikes in noise,will be averaged out for the most part by inspecting avast number of transits for each star.

The average difference between consecutive transitcentre times was then taken in order to determine theorbital period. The differences between every other andevery third transit were also calculated, and divided by2 and 3 respectively. This was to increase the numberof values used to calculated the mean period, with thehope of reducing the standard deviation on the result.In order to account for false detections, such as noisedetections and double transits detections, as well asmissed transits, any differences between transit centretimes that were found to be too dissimilar from thehand-calculated period were discounted. This also en-sured that the period was only calculated for a singleorbiting planet at any given time, and that the cal-culated period was not influenced by the transits ofa second planet. Once all outlying values had beendiscounted, the mean orbital period with standard de-viation could be taken and output.

Due to the constraint built into the code for theperiod value, if this value were changed to reflect theapproximate period value of a second orbiting planet,then the transit detection code was able to detect thesecond planet and determine the orbital period. How-

45

Figure 6.2.1 – Flagged transits for GSIC 1 short cadence data.

Figure 6.2.2 – A single transit for GSIC 1 short cadence data.

ever, the code was only able to detect one transitingplanet at a time, a limitation requiring the light curvesto be inspected manually initially in order to deter-mine whether or not it was necessary to look for a sec-ond planet. Although, from the output period values,one could analyse this list manually and see if a signifi-cant number of similar values were present for anothervalue of the period (which is not an integer multipleof the primary transiting planet) which would implya second transiting planet. This ‘quasi-computational’approach was actually utilised successfully for a coupleof very noisy light curves such as GSIC 8 (which wasalso found to have two transiting planets) and GSIC24, where it was easier and faster to distinguish pat-terns in the output list of period values manually, thanit was to change the code to reflect these issues.

On the other hand, for the cleaner light curveswhere the noise did not obstruct the detection of tran-sits, the transit detection code could calculate orbitalperiods in days to a reliable precision of approximatelythree to four decimal places, which was found to besuperior to those calculated manually. Although some-times struggling with more noisy light curves, if sup-

plied with enough transits to calculate over, the codecould still output the periods to a higher accuracy thanthe hand-calculated values. Another example, in addi-tion to GSIC 8 and GSIC 24, where the transit detec-tion code was particularly useful was for determiningthe period of the planet from the GSIC 22 light curve.This data was also very difficult to analyse due to lotsof noise and relatively small dips in intensity duringthe transits. Furthermore, the period value calculatedfrom the manual methods approach was not accurateenough to achieve a successful phase folded light curve.However, using the transit detection code, the periodwas found accurately enough and hence could be usedin the transit fitting models to further improve on theaccuracy, as well as helping to determining other pa-rameters, discussed in Section 6.2.4.

6.2.3 Phase Folding Code (FC)

Following on from the transit detection code, atransit phase folding code was created. This was re-quired as the light curves needed to be phase foldedin order to determine whether or not the period value

46

calculated was accurate enough. For this process tobe successful, it was essential that the period was veryaccurate, as when hundreds of transit curves are over-laid on top of one another, it only takes a very smalldifference in the period for the transit to no longer bevisible in the phase fold. Figure 6.2.3 shows an exam-ple of a successful phase folded light curve (of GSIC34), output by the phase folding code.

The code requests the required GSIC number fromthe user, followed by the period value in days. This pe-riod value could either be automatically taken from thetext file output by the transit detection code, or couldbe input manually. The actual phase fold is achievedby dividing all the time coordinates by the input pe-riod value and taking the remainder. This remainderis then plotted against the flux, resulting in a graphsimilar to that shown in Figure 6.2.3. The estimatedphase value of the transit was also obtained from thisplot for each planet, which was used as one of the inputparameters for the transit fitting code.

Figure 6.2.3 – Phase folded light curve for GSIC 34. Thetight distribution of data points means an accurate periodvalue was used.

6.2.4 Transit Fitting Code(OH & FC)

To obtain more accurate results and more preciseerrors, as stated above, it was essential that a moresophisticated method was used to extract results fromthe light curves, besides studying the data manually.For this, an analytical model was fitted to the long ca-dence light curves to produce parameters for a best fit.The model used was that set forth by Mandel & Agol(2002) [134], and transcribed into Python by Crossfield[135]. This model was fitted to the data using the em-cee Python implementation of Goodman & Weare’sAffine Invariant Markov Chain Monte Carlo EnsembleSampler [136] [137].

Two versions of this code were produced: one fit-ting a uniform source model to the data, and one fittinga quadratic stellar limb darkened model to the data.An example of the resulting shape of these models, asfitted to phase folded light curves, can be seen in Fig-ures 6.2.4 and 6.2.5 respectively. The final version ofthe uniform source model code required input of sev-eral parameters: the orbital period of the planet, the

semi-major axis in units of stellar radius, the planetaryradius in units of the stellar radius (Rp/R∗) and eitheran estimation of the time of the first visible transit orthe phase of a visible phase fold, of which the latterwas used as input in almost all instances. Besides this,the limb darkened model also required first guesses ofthe inclination of the system, and two quadratic limb-darkening LDCs. The limb-darkening equation utilisedin this model, obtained from Mandel & Agol’s work onanalytical transit curves, uses a small planet assump-tion: the ratio of the planet radius to stellar radiusis 0.1 or less. This drastically reduces the complex-ity of the model and allows for more rapid computingof transit light curves whilst maintaining a reasonablemodel accuracy [134]. A comparison to an analyticallimb-darkening model and an analysis thereof can befound in Sections 6.2.5 and 7.4 respectively.

Figure 6.2.4 – A fit of the uniform source model to a phasefold of GSIC 1. As can be seen, the period is especially wellconstrained.

47

Figure 6.2.5 – A fit of the limb-darkened model to a phasefold of GSIC 1. As can be seen, the depth is especially wellconstrained. As this figure was produced on a low amountof iterations, the fit is imperfect, as can be seen by thewidth of the fit, which is too small.

After setting reasonable ranges in which to look forthe best fit, emcee then performed a random walkwithin this range attempting to match the parametersto one another and to the data by working to find abetter value of χ2, or the likelihood, which related tothe error on the flux values and the difference betweenthe model and the data. It did so using a set amountof ‘walkers’ each ‘stepping’ a set number of iterations.If, after a random step, a walker was found to wanderfurther from the best fit, it reversed its ‘direction’ sothat the next step would tend more towards an idealfit. As such, if run for long enough using a large enoughnumber of walkers and iterations, the program returnsa visualisation of the most probable results found tobe a best fit. This visualisation can be seen for thelimb-darkened fit of GSIC 35b in Figure 6.2.6. Ascan be seen, there are seven 1D histograms, one foreach parameter, which hold the results with the high-est probability of a perfect fit. The mean value of thesehistograms is then taken to be the final result, withthe error in this result being the standard deviationon these values as calculated by Python. The plotsfound at an intersection between two different parame-ters represent the 2D probability density of correlationbetween the two parameters, which in an ideal case iscircular in shape, as can be seen in the majority ofplots in Figure 6.2.6. These serve as a clear indica-tion of how successful the code has been at finding ahigh probability of similar results (as can be seen inthis figure). This particular figure contains plots forseven parameters, whilst the plots output by the uni-form source transit fitting model would only containfive parameters, as this simpler model does not requirethe input of values of the inclination or LDCs. Assuch, one can imagine that the limb darkened fittingcode would take significantly longer to run, as was in-

deed the case. It is also important to note that the caseof GSIC 35b returned very successful results, and thatnot all resulting histograms were Gaussian in shape northe probability densities circular. While an increasednumber of iterations or walkers for the emcee func-tion naturally increases the accuracy of the results andthe visualisation thereof, it also significantly increasesthe time required to run the code. As such, the finalresults are limited, to a degree, by the time and com-puting power available during this project. The finalversion of both programs output a low resolution plotof the phase fold to gauge the success of the program, aplot of the 1D and 2D probability distributions as de-scribed above, and the resulting values and standarddeviations in ASCII format.

As close initial estimates for the exact parameters,most importantly the period, were essential to the ac-curacy of the final results and the limiting of the code’srun time, the faster uniform source model was run us-ing the estimated period values determined manually,after which the results from this model were used as theinput parameters for the limb darkened model. Ideally,the uniform source model would have used the periodvalues determined by the transit detection code for allthe stars, instead of the manual estimates for most,as this method would have constrained the region thatemcee had to analyse further and thus sped up thefitting process. A better initial estimate would resultin a more accurate result from the uniform source fit,which in turn would ensure a decreased run time of thelimb darkened model. This in turn would have enabledmore iterations and walkers to be used to increase theaccuracy of the final parameter values. However, thetransit detection code itself took a long time to com-pute the period, especially using short cadence data(which was preferable in order to achieve a better valuefor the period from the larger number of data points),so the transit detection code was only utilised for de-tection of the period of planetary orbits found in someof the more noisy light curves, including planets GSIC8b, 22b and 24b where the manual estimates were notaccurate enough to achieve a clear phase folded transit,as mentioned previously.

On the other hand, the uniform source model wasrun on all the stars with confirmed planets and avail-able parameter values with emcee using 50 walkersand 500 iterations. In addition, the limb-darkenedmodel was subsequently run using the output from theuniform source model, producing clear phase folds andaccurate values, including errors. This was initiallyperformed by members of the research group with em-cee using 100 walkers and 1000 iterations. This proveda taxing and time consuming process, due to the limi-tations set forth by the computers available. As such,the data were run simultaneously on a more sophisti-cated system with emcee using 200 walkers and 400iterations. Values were compared between both thelatter data set and those that were run beforehand onthe group’s computers, of which the most accurate re-sults with corresponding errors were selected. In thecase of GSIC 8b and GSIC 40b, the results from both

48

Figure 6.2.6 – A collection of the histogram and probability density plots of values for parameters entered into emcee forthe limb-darkened model, for GSIC 35b, after running on 200 walkers and 400 iterations. As can be seen, the histogramsare mostly Gaussian in shape, and the 2D probability density plots appear to be mostly spherical in shape.

data sets appeared to be unsatisfactory, therefore thecode was run again for these planets with emcee using200 walkers and 2000 iterations. Unfortunately, due totime constraints, this level of accuracy was unobtain-able for all discovered planets.

6.2.5 Modelling for Stellar LimbDarkening (CL & OH)

To further study the effects of stellar limb dark-ening on the transit method, and to ensure the accu-racy of the limb-darkened transit fitting model used,a numerical limb-darkening model was constructed inPython, allowing use of both linear and quadratic limb-darkening models as a function of the effective temper-

ature.

Finding Limb-Darkening Coefficients (CL)

As mentioned previously and as is clear from the useof limb darkening in the transit fitting, the modellingof the effects from the phenomenon of limb darkening isextremely important in obtaining an accurate estimateof the planets characteristics from its transit data [95].Limb-darkening coefficients (LDC) have only been di-rectly deduced for relatively very few stars apart fromthe Sun. Due to this, when producing light curve syn-thesis programs, the coefficients are usually interpo-lated from tables of theoretical values calculated fromatmosphere models [98].

49

In order to compute the specific intensity relationsdiscussed in 4.4 for a modelled star, the LDCs mustfirst be deduced for each varying equation. To make thetransit model comparable to the data available in thisstudy, the LDCs need to be computed as a function ofeffective temperature. The effective temperature, Teff ,is obtained from the Kepler data and can be easilyinput for each star.

Table 6.1 – Kepler Stellar limb-darkening Coefficients [138].

Teff Linear Quadratic Quadraticu a b

4000 0.6888 0.5079 0.22394250 0.7215 0.6408 0.09994500 0.7163 0.6483 0.08424750 0.6977 0.6036 0.11645000 0.6779 0.5528 0.15485250 0.6550 0.4984 0.19395500 0.6307 0.4451 0.22975750 0.6074 0.3985 0.25866000 0.5842 0.3539 0.28516250 0.5640 0.3198 0.30236500 0.5459 0.2901 0.31676750 0.5312 0.2672 0.32677000 0.5191 0.2478 0.33587250 0.5085 0.2308 0.34377500 0.5003 0.2165 0.3512

Table 6.1 was formulated from results from D.K.Sing’s research on the LDCs for data taken from Ke-pler and CoRot. The values of LDCs: u, a and b wereobtained as these apply to the linear and quadratic in-tensity laws used in this study. Values for c and f werenot included in this table as none of the stars in thisstudy have effective temperatures greater than 6500K;the logarithmic law applies to stars with effective tem-peratures between 10,000K and 40,000K. The valuesfor each LDC were plotted on a graph as a function ofthe effective temperature, to deduce a relationship foreach. This is shown in Figure 6.2.7.

Figure 6.2.7 – Fitted relations for limb-darkening coeffi-cients for Kepler as a function of effective temperature,Teff . Values are from Table 6.1, fit with Equation (6.1).

A Weibull function was deduced to be the best fitfor the data. This equation was of the form:

H = k

(m – 1

m

) 1–mm

(Teff – x0

l+

(m – 1

m

) 1m

)m–1

·

(Teff – x0

l+

(m – 1

m

) 1m

)m–1

·

exp

[Teff – x0

l+

(m – 1

m

) 1m

]m

+m – 1

m+ y0 (6.1)

where H is the values of the LDC. The fit to each setof data was performed by the dynamic fit wizard inSigmaPlot, by varying 5 parameters: y0, x0, k, l andm. The non-linear regression statistical measure R-squared was used to analysis the fit to the data. Thisproduced R-squared values of 0.9997±0.0015, 0.9999±0.0022 and 0.9993 ± 0.0030 to the fits of u, a and brespectively. As the best fit value of R-squared is 1this shows that the data fits very well to the fittedrelation (Equation 6.1). The values for the parametersthat form the LDC-Teff relations for u, a and b fromEquation 6.1 are included in Tables 6.2, 6.3 and 6.4respectively.

Table 6.2 – Fit parameters for fit to coefficient u from Fig-ure 6.2.7.

Fit Parameter Value

k 0.2656± 0.0063l 1749.7597± 45.6889m 1.2544± 0.0264x0 4327.5242± 10.6594y0 0.4555± 0.0059

Table 6.3 – Fit parameters for fit to coefficient a from Fig-ure 6.2.7.

Fit Parameter Value

k 0.4706± 0.0043l 1393.0336± 16.8337m 1.2766± 0.0116x0 4365.3571± 6.9795y0 0.1800± 0.0040

Table 6.4 – Fit parameters for fit to coefficient b from Figure6.2.7.

Fit Parameter Value

k –0.2679± 0.0033l 1153.4187± 19.2744m 1.3530± 0.0213x0 4400.0749± 12.5456y0 0.3548± 0.0029

These tables, along with the Weibull function inEquation 6.1, form the three equations that relate the

50

LDCs to the effective temperature of the star. Thiscompletes the linear and quadratic intensity relations,so they can be used to model the limb-darkening effectson any of the stars included in the data.

Stellar Limb Darkening Model (OH)

Using these LDCs and Equation (6.1), a modelcould be constructed which depended closely on theeffective temperature. Using Python, a code was writ-ten to take various input parameters and output alight curve displaying a transit as calculated using themodel. In part inspired by a similar approach takenby Addison, Durrance and Schwieterman (2010) [139],both the star and the planet were visualised as arraysof pixels, or ‘boxes’. To ensure that the ratio betweenthe planetary and stellar radii was maintained, an ar-ray of zeroes of 10 by 10 pixels was created to repre-sent the planet. Using the input planetary radius, thesize which a box represents in reality could be found,allowing the array representing the star to be scaledaccordingly. To allow the model to incorporate the ef-fects of egress and ingress, the total array was enlargedby 20 boxes in both width and height. The very centreof the stellar array was taken to have a radius value ofzero.

Simply using Pythagoras’ theorem, the radius valueoutward from the centre of each box in the array wascalculated. Using Equations (4.58) and (4.59) de-scribed in Section 4.4, the intensity of each box rel-ative to the intensity at the centre of the star could becalculated as a function of the radius assigned to eachbox through the variable μ as given in Equation (4.57).To ensure the star did not overrun its boundaries, anybox with a radius value that exceeded the stellar ra-dius was set to zero. For simplicity, the intensity at thecentre of the star was chosen to be 1, ensuring that theindividual intensities of boxes were between the val-ues of 0 and 1. The total intensity of the star-planetsystem was then taken to be the sum of the intensityvalues of all the boxes in the stellar array. A clarifyingvisualisation of the model can be seen in Figure 6.2.8.

Figure 6.2.8 – A contour plot of the stellar array of in-tensities, including a schematic representation of a planettransiting the star (not to scale) and the pixel grid. Notehow the effects of limb darkening are clearly visible. On theFigure, b represents the impact parameter and p representsthe distance to the planet, both relative to the centre of thestar.

The planet was then shifted along the array onebox at a time, whilst its position was recorded rel-ative to the centre of the star. For every step, theboxes which were found to be ‘covered’ by the plane-tary array were set to be zero, thus removing a num-ber of intensity values from the total during transit.The sum of the intensity values was then taken andrecorded. The stellar array was then reset for the nextstep, and this process repeated. As is to be expected,the planetary array would ‘cover’ boxes with a higherrelative intensity the closer it got to the centre of thestar, thus recreating the effects of limb darkening thatappear clearly on transit light curves.The recorded in-tensity values were then plotted against the positionof the centre of the planetary array on the stellar diskas a function of the stellar radius. An example of aresulting set of light curves can be seen in Figure 6.2.9.As it depends on the individual intensity values of theboxes on the stellar array, the total intensity differsdepending on whether the first or second order equa-tion is used, despite all other input parameters beingthe same. As such, the intensity values in Figure 6.2.9have been normalised using the differences between themean intensity of each light curve.

51

Figure 6.2.9 – A comparison between light curves output bythe stellar limb-darkening model, using the parameters ofstar GSIC 22 and planet GSIC 22b. The difference betweenthe shapes of the first (in blue) and second (in red) ordermodels is clear.

As can be seen from Figure 6.2.9, the shape outputby the model appears to be very smooth, and remi-niscent of the shape of an actual transit (such as thatshown in Figure 6.2.2 for GSIC 1), which is encourag-ing for the accuracy of this numerical model. However,as the array sizes are still fixed to a relatively low, finitevalue, the resulting model will still include an intrin-sic error on its accuracy. It is worth noting that it islikely that the second order model provides a better fitin this case, as GSIC 22 has an effective temperaturesignificantly higher than the Sun, making the secondorder model more applicable [100]. Furthermore, thisdifference between the models as seen in Figure 6.2.9has been observed to change depending on the inputparameters, indicating that the effect described by thedifferent models indeed varies significantly. The finalversion of this model allowed for user input of the pe-riod, semi-major axis, effective temperature, planetaryradius, stellar radius, inclination of the system, andwhich stellar limb-darkening model (first or second or-der) to be used. As such, the model could be very accu-rately characterised for different stars. To ensure thatit was as accurate as possible, the code also allowed thereading in of results received from the limb-darkenedtransit fitting code to act as the relevant parameters.

To test the accuracy of the model used in the transitfitting code, both the first and second order modelswere plotted over a phase fold of the quadratic limb-darkening model as given by Mandel & Agol [134] fora given star, using the same parameters. To ensurethat all three models overlapped, the x-axis value forthe transit fitting model was divided by the period torange between values of 0 and 1 for the phase. Thiswas done by ensuring that the phase fold produced atransit at exactly a phase of 0.5, by defining the dataset to start at the centre of a secondary transit. Thex-axes of the numerical models were then accordinglytranscribed from the impact parameter to the centre

of the stellar array to the phase, as

if(b ≤ 0) : φ = arccos

(b

a

)· 0.25(π

2

) + 0.25

if(b > 0) : φ = arcsin

(b

a

)· 0.25(π

2

) + 0.5

(6.2)

where a is the semi-major axis, b is the impact pa-rameter, φ is the phase and P is the period. Besidesthis, the LDCs produced by the transit fitting code andthose for the second order model produced by Equa-tion (6.1) were compared, as they represent constantsfor the same limb-darkening equation (namely Equa-tion (4.59)).The results and a discussion of these com-parisons can be found in Section 7.4.

6.3 Processing of Results (PS)

Using these various computational methods, accu-rate values for the period of a planet’s orbit (P), theratio of the planet’s radius to the star (Rp/R∗) and theinclination (i) were obtained. These results were thenused in conjunction with data collected by the Aster-oseismology group in order to characterise planets asfully as possible.

As this study only used the transit method to de-tect planets, values such as the mass of a planet andthe eccentricity of its orbit could not be calculated an-alytically, but novel methods were devised in order toobtain as much information about the detected plan-ets as possible. The methods used to calculate eachparameter will now be explained.

Planetary Radius

This value is easily obtained using the ratio of radiiprovided by the transit fitting limb-darkened modelcode, and the radius of the host star, R∗. As thereis a systematic uncertainty in the method used to cal-culate R∗ (see Section 3.2.4) this results in a ratherlarge error on this result of around 12%, almost all ofwhich is due to the uncertainty on the calculation ofstellar radius. The accuracy of this value is also depen-dent on the quality of the limb darkening coefficientsused by the program, which affect the measurement oftransit depth.

Semi-Major Axis of Orbit

Again, this value is easy calculable using Kepler’sthird law (Equation (4.51)), using the period P of theplanet (provided by the transit fitting limb-darkeningmodel) and the mass of the host star M∗, which can beobtained with the use of asteroseismic scaling relations.Although the systematic uncertainty on M∗ often givesan error of 20%. As this value is raised to the 1/3 powerthis effect is reduced, resulting in an uncertainty on aof around 6%. As planetary transits occur with highregularity, the uncertainty on the value of P providedby the code is very low, often being known to one partin a million or better.

52

Transit Duration

This value, τ, was measured manually from a phasefolded light curve, which was produced using the peri-odicity P provided by the code. As noted, this valueis often exceptionally precise, resulting in almost allcases in a clearly phase folded curve comprised usu-ally of several hundred transits, in which the transitwas clearly defined. The width of the transit was thenmeasured from the start of ingress to the end of egress,and suitable errors assigned based on the clarity of thetransit phase fold. For some planets, the value of Pwas not quite sufficient for a clear phase fold, and sofor these planets τ was not measured. In some othercases the depth of transit was too small to be measur-able manually, and thus measurement of τ was againnot possible. As the clarity of phase fold varied signifi-cantly, the uncertainty of τ also varied between 2% and26%, but the vast majority of measurements producedvalues with uncertainties of around 6%.

Star-Planet Separation at Point of Transit

This value, s, is obtained from a modified versionof Equation (4.29), which provides a value for a inthe case of a circular orbit. When eccentricities otherthan 0 are considered, the equation is more properlyrendered as:

τ =PR∗πs

(6.3)

In the above equation, s is the distance between thestar and the planet at the moment of transit, ratherthan the semi-major axis of the orbit. As this valueis dependent on R∗, which as noted had a large sys-tematic uncertainty, we find a similar uncertainty on sof around 15%, which is also a product of the manualmethod used to measure τ.

Impact Parameter

This value, b, can be obtained by similarly rear-ranging Equation (4.31). Once again, however, it mustbe remembered that this equation is valid only for acircular orbit, and so substituting a for s is thereforenecessary to obtain a more accurate value. This givesa final equation of:

b =s cos(i)

R∗(6.4)

Although the value of i is obtained computationallyto a precision of a few tenths of a degree, this equationis also dependent on s and R∗, which as previouslydiscussed have large systematic errors associated withthem. The uncertainty on b is thus very large, and inseveral cases renders the value statistically meaning-less.

Eccentricity

The true eccentricity of a planet’s orbit is not an-alytically obtainable by the use of the transit method,and is generally found by Doppler wobble methods.

However, as we had calculated both the semi-majoraxis of the planet’s orbit a, and the separation betweenthe planet and the star at transit s, by comparing thesevalues we were able to constrain the eccentricity some-what in most cases.

In the case that s was greater than a, s was treatedas the minimum value for the apoastron of the orbit,and thus eccentricity was constrained to:

emin =s – a

a(6.5)

and in the case where s was less than a, s was treatedas the maximum value for the periastron, giving aneccentricity constraint of:

emin =a – s

a(6.6)

In reality, since it is unlikely that the transit occursat the periastron or apoastron, the eccentricity maywell be significantly larger than this value, but con-sidering the lack of data which forbids an analyticallycalculated value this constraint will have to suffice forthis study. Due to the relatively high error on the mea-surement of s, an uncertainty on the minimum eccen-tricity was found which often exceeded the magnitudeof the result, but nonetheless this method was usableto detect non-circular orbits in a few cases.

Equilibrium Temperature

For this result, each planet was treated as a blackbody, heated only by radiation from its host star.Other sources of heat, such as radiation in the planet,and other effects such as the insulating effect of atmo-spheres, were not considered.

The planets were all treated as having an emissivityof 1, and as reflecting a fraction of incident light givenby A, the albedo. Based on the radius of the planet,among other considerations, an estimated albedo of 0.3was assigned to rocky planets, and 0.5 for gas giants.For Gas dwarfs, a range of values from 0.3-0.5 wereused. These values were chosen after considering theplanetary albedos of solar system planets [140] [102].The equilibrium temperature was then calculated asthe point at which the incident radiation energy ab-sorbed by the planet is equal to that radiated from itssurface, according to the equation:

Teq = Teff(1 – A)1/4

√R∗2a

(6.7)

In the above equation a circular orbit of semi-majoraxis a around a star of radius R∗ and effective temper-ature Teff is assumed.

As this equation provides only a very rough valuefor the equilibrium temperature, and this only at thesemi major axis, the value it provides is quoted hereas a range including what we consider to be a sensiblerange of temperatures, which to some extent allows forvariation in albedos and atmospheric insulation. Thisis in part to emphasise that this value is given merelyas a ‘ballpark figure’, in order that composition andthe habitable zone may be meaningfully discussed withrespect to each planet.

53

Planetary Mass and Composition

The mass of a planet is not directly obtainable byanalytical methods from the transit method. Neverthe-less, we are able to at least constrain the mass of theplanets we detected by reference to the other resultsfrom our data.

By using analytical models for planet composition,a relation between the mass and the radius can be de-rived (see Section 4.5), and thus from our relativelyprecise results for planet radius, a reasonable estimatefor the mass can be obtained. The smaller the planet,the more tightly the mass can be constrained, andby using other results, like the equilibrium tempera-ture, we can further discount certain possibilities. Theranges obtained vary significantly, but we are confidentthat we have devised a reasonable method of determin-ing mass ranges which can allow us to characterise theplanet by its likely composition.

54

Chapter 7

Results and Discussion

7.1 Introduction

In this section we will briefly discuss each star andall detected planets. As the previous sections show,many different methods were adopted to calculate awide range of information. The tables display everyparameter obtained by this project.

7.1.1 Key to Results Tables (MH & PS)

Stellar Properties

In the tables of stellar properties, the parametersare as follows:

· Δν0, the large spacing, computational value· νmax , the maximum power frequency, computa-

tional value· δν0,2, the small spacing, computational value

unless specified as manual· Teff , the effective temperature, provided· [Fe/H], the metallicity, provided· Xc , the hydrogen mass fraction, asteroseismic

diagram value· M∗, the stellar mass, computational scaling re-

lation value using Equation (3.23)· R∗, the stellar radius, computational scaling re-

lation value using Equation (3.22)· the age of the star, stellar model comparison

valueThe inner and outer Habitable Zone limits, IHZ andOHZ respectively, are calculated with the followingmethods (see Section 4.6.7):

· RG - Runaway Greenhouse· MaG - Maximum Greenhouse

Planet Orbital Properties

In the tables of planet orbital properties, the pa-rameters are as follows:

· P , the period of the orbit, computational value· a, the semi-major axis of the orbit, calculated

using Equation (4.51)· τ, the duration of the transit, measured from a

transit phase fold· s, the separation of the star and planet at the

point of transit, calculated using Equation (6.3)· e, the eccentricity of the orbit, constrained using

Equations (6.5) and (6.6)

· i , the inclination of the orbit, computationalvalue· b, the impact parameter of the orbit, calculated

using Equation (6.4)

Planet Properties

In the tables of planet characteristics, the parame-ters are as follows:

· R, the radius of the planet, calculated from theradius ratio (computational) and R∗· the rough composition of the planet, deduced

by reference to other properties· A, the rough albedo of the planet, suggested

based on composition· Teq , the equilibrium temperature of the planet,

calculated using Equation (6.7)· M , the mass of the planet, constrained by ref-

erence to other properties

7.1.2 Nomenclature (PS)

In keeping with conventional exoplanet nomencla-ture, our detected planets have been named after thestars around which they orbit, which have been identi-fied to us with GSIC numbers. Thus, a planet aroundstar GSIC 3 will be identified as GSIC 3b (GSIC 3atechnically refers to the star, but for simplicity starswill not be assigned letters). Planets in multiple planetsystems will be assigned letters continuing through thealphabet in order of increasing period. In actualitythese are usually assigned by date of discovery, but asall have been detected as a result of the same studythey must be sorted by another method.

7.2 Full Results

It should be noted that the results for stars GSIC 2,GSIC 11, GSIC 28 and GSIC 37 have been omitted forvarious reasons. These are explained in Section 7.2.2located at the end of the main results.

7.2.1 Stellar, Orbital and PlanetaryResults (PS, AW, JG & ML)

GSIC 0

The power spectrum from GSIC 0 was very easy toread. The signal to noise ratio was very high so only a

55

small amount of filtering was required, resulting in lowerrors on the calculated mass and radius of the star.The high signal to noise ratio also resulted in the visi-bility of the l = 2 peaks, meaning the small frequencyspacing, and hence the age of the star, could be esti-mated more accurately. The maximum frequency foroscillations was calculated by the computational teamwithout any trouble. This can be assumed for everystar in this report unless otherwise stated.

Table 7.1 – Stellar Parameters for GSIC 0

Parameter Value Error

Δν0 (μHz) 116.819 0.058νmax (mHz) 2.41467 0.00145δν0,2 (μHz) 6.35 0.31

Teff (K) 5766 60[Fe/H] 0.052 0.021

Xc 0.13 0.02M∗ (M�) 0.937 0.165R∗ (R�) 1.098 0.127

Age (Gyrs) 4.77 1.29RG IHZ (AU) ∼ 1.07 -

MaG OHZ (AU) ∼ 1.85 -

From the mass, radius and effective temperaturevalues given in Table 7.1, GSIC 0 can be deduced tobe very similar to the Sun. This gives a habitable zoneclose to 1AU, but unfortunately there were no planetsdetected orbiting this particular star. This star has anage of 4.77Gyrs compared to the Sun’s age of 4.6Gyrs.The only distinguishing factors between this star andthe Sun are its metallicity and hydrogen fraction, whichfor the Sun are 0 and 0.74 respectively [141].

GSIC 1

The power spectrum from GSIC 1 was more dif-ficult to read due to the lower signal to noise ratio.Once filtered, however, the l = 0 and l = 1 peaks wereeasily identifiable, making the large frequency spacingcalculations more simple. The small frequency spacingcould also be calculated after filtering.

Table 7.2 – Stellar Parameters for GSIC 1

Parameter Value Error

Δν0 (μHz) 59.469 0.229νmax (mHz) 1.08099 0.00106

δν0,2 (μHz) (BE) 3.52 0.7Teff (K) 6350 50[Fe/H] 0.26 0.08

Xc 0.14 0.02M∗ (M�) 1.446 0.260R∗ (R�) 1.990 0.236

Age (Gyrs) 1.82 0.226RG IHZ (AU) ∼ 2.28 -

MaG OHZ (AU) ∼ 3.88 -

It can be seen from Table 7.2 that this star is sig-

nificantly larger and hotter than the Sun, creating ahabitable zone centred roughly 3AU from the star. Asthis star is more massive than the Sun it will progressthrough the MS faster. This is shown by a low hydro-gen fraction, suggesting that GSIC 1 is near the end ofthe MS, and the age is approximately half that of theSun. There was a single planet detected orbiting GSIC1.

GSIC 1b

Table 7.3 – Orbital Parameters for GSIC 1b

Parameter Value Error

P (days) 2.204735299 1.39E – 07a (AU) 0.037489 0.000245τ (days) 0.1427763 5.70E – 06s (AU) 0.045505 0.000396

e > 0.214 0.161i (◦) 89.587 0.215

b 0.4005 0.0318

Table 7.4 – Planetary Characteristics for GSIC 1b

Parameter Value

R (R⊕) 15.380± 1.826Composition Gas giant

A 0.5Teq (K) 1800-2000M (M⊕) 350-1000

This is an archetypal example of a hot Jupiterplanet, orbiting very close to its host star. It is likelythat this planet formed much further from its star andthen migrated inwards through the ejection of otherplanets. This would likely result in an eccentric or-bit, which may be under the process of circularisationby tidal forces, which will act particularly strongly onthe planet at the periastron, the distance of closest ap-proach to its host star.

As the radius of this planet is so high, it is partic-ularly difficult to constrain the mass, as factors suchas electron degeneracy become relevant. However, thishigh radius value also clearly indicates a gaseous com-position (other compositions could not produce plan-ets of this radius) which in turn leads us to an albedocomparable to that of Jupiter. The equilibrium tem-perature of this planet is very high due to proximity tothe star, which is expected to artificially inflate the at-mosphere in a ‘puffy’ effect, which also influenced ourchoice of mass constraints.

In comparison to the properties of Jupiter, thisplanet has a radius of 1.364 ± 0.162RJ and a mass of1.101-3.145MJ.

It is worth noting that as the transits for this planetare particularly deep, and as the star undergoes al-most negligible background variations in flux over themeasuring time, the errors obtained on these data areamong the most precise in our study.

56

GSIC 3

For GSIC 3, the power spectrum required filtering,but only by a small amount. After filtering there werevery faint l = 2 peaks present, allowing the identifica-tion of the l = 0 peaks and the calculation of the smallfrequency spacing.

Table 7.5 – Stellar Parameters for GSIC 3

Parameter Value Error

Δν0 (μHz) 145.534 0.039νmax (mHz) 3.31667 0.00167δν0,2 (μHz) 8.89 0.2

Teff (K) 5669 75[Fe/H] –0.18 0.1

Xc 0.28 0.02M∗ (M�) 0.982 0.177R∗ (R�) 0.963 0.112

Age (Gyrs) 6.26 1.64RG IHZ (AU) ∼ 0.92 -

MaG OHZ (AU) ∼ 1.58 -

The mass, radius and temperature values in Ta-ble 7.5 suggest another star very similar to the Sun.It’s worth noting the slightly higher than average valueof the maximum frequency for oscillations, suggestingthat GSIC 3 is a star currently towards the start of itsMS lifetime. This is in contrast to the star’s age andhydrogen fraction calculated from the asteroseismic di-agrams, which would put a star of this mass well intoits MS lifetime.

GSIC 3b

Table 7.6 – Orbital Parameters for GSIC 3b

Parameter Value Error

P (days) 4.72674034 1.10E – 06a (AU) 0.05479 0.00329τ (days) 0.1188 0.01s (AU) 0.05676 0.00817

e > 0.036 0.161i (◦) 89.301 0.419

b 0.1545 0.0970

Table 7.7 – Planetary Characteristics for GSIC 3b

Parameter Value

R (R⊕) 1.427± 0.167Composition Rocky

A 0.3Teq (K) 900-1100M (M⊕) 2-4

This planet’s small radius and extremely close orbitindicate a rocky ‘Super-Earth’ planet. This small ra-dius value, coupled with the high equilibrium tempera-ture which forbids surface water, allows us to discount

this as a possible composition and thus gives us a verysmall mass range . It is to be noted here that the cal-culated minimum eccentricity is quite imprecise, andgiven the low period it is very likely that this planetfollows an almost circular orbit.

GSIC 4

The power spectrum for GSIC 4 was another spec-trum that was very easy to read, with a high signal tonoise ratio and a visible small frequency spacing be-fore filtering. This made the l = 0 peaks, and hencethe large frequency spacing, easily detectable.

Table 7.8 – Stellar Parameters for GSIC 4

Parameter Value Error

Δν0 (μHz) 60.677 0.537νmax (mHz) 1.07660 0.00116δν0,2 (μHz) 3.98 1.12

Teff (K) 6305 50[Fe/H] –0.03 0.1

Xc 0.19 0.02M∗ (M�) 1.304 0.258R∗ (R�) 1.897 0.243

Age (Gyrs) 1.54 0.160RG IHZ (AU) ∼ 2.15 -

MaG OHZ (AU) ∼ 3.66 -

Table 7.8 shows that GSIC 4 is visibly larger, hot-ter, more massive and younger than the Sun. Thestar’s age in comparison to its mass places it towardsthe end of the MS, an assumption that agrees with thelow hydrogen fraction. There were no planets detectedin orbit around this star.

GSIC 5

GSIC 5 had a power spectrum with a large amountof noise, enough for the l = 2 peaks to be undetectableby computational analysis. Therefore there was no agecalculation from the asteroseismic diagrams. The restof the table can, however, be used to generate an ageestimate.

Table 7.9 – Stellar Parameters for GSIC 5

Parameter Value Error

Δν0 (μHz) 128.525 0.683νmax (mHz) 2.91945 0.00174δν0,2 (μHz) - -

Teff (K) 6134 91[Fe/H] –0.24 0.1

Xc - -M∗ (M�) 1.240 0.237R∗ (R�) 1.131 0.139

Age (Gyrs) - -RG IHZ (AU) ∼ 1.22 -

MaG OHZ (AU) ∼ 2.09 -

57

The slightly higher than solar mass and radius val-ues given in Table 7.9 correspond to the increased ef-fective temperature. The value for the maximum fre-quency of oscillations is similar to that of the Sun, indi-cating that GSIC 5 is at a similar point in its evolution.Putting this with the mass value suggests that GSIC 5is younger than the Sun by a small amount.

GSIC 5b

Table 7.10 – Orbital Parameters for GSIC 5b

Parameter Value Error

P (days) 11.87291881 5.79E – 06a (AU) 0.10941 0.00696τ (days) 0.09959 0.01s (AU) 0.1996 0.0317

e > 0.825 0.312i (◦) 88.722 0.019

b 0.846 0.170

Table 7.11 – Planetary Characteristics for GSIC 5b

Parameter Value

R (R⊕) 2.055± 0.253Composition Gas dwarf/ rocky

A 0.3-0.5Teq (K) 750-900M (M⊕) 3-15

The radius of this planet is above, but still close tothe theoretical boundary between solid rocky planetsand those which have collected a significant envelopeof gas. As it is impossible from our data to determinethe extent of the planet’s atmosphere, both possibili-ties have been accounted for here, and this is reflectedin the large mass range, the upper reaches of whichdescribe a rocky planet, and the lower end of whichwould characterise a composition and density similarto that of Neptune.

Also notable is the planet’s high eccentricity. Al-though our calculation is imprecise, this still indicatesa planet which is surprisingly eccentric, especially con-sidering its close proximity to the host star. This willalso give rise to greater variation in temperature (andpossibly radius, depending on composition) than maybe implied by our data.

GSIC 6

The power spectrum of GSIC 6 had a slightly highersignal to noise ratio than that of GSIC 5, meaning thesmall frequency spacing could be measured and an agecan be discussed.

Table 7.12 – Stellar Parameters for GSIC 6

Parameter Value Error

Δν0 (μHz) 97.876 0.499νmax (mHz) 2.14031 0.00152δν0,2 (μHz) 8.64 0.58

Teff (K) 6270 79[Fe/H] –0.04 0.1

Xc 0.48 0.02M∗ (M�) 1.501 0.282R∗ (R�) 1.445 0.176

Age (Gyrs) 1.13 0.172RG IHZ (AU) ∼ 1.62 -

MaG OHZ (AU) ∼ 2.76 -

All values in the above table correspond to a starin the very middle of the MS. The mass of GSIC 6is greater than solar mass and it is a younger star.Just under half of the mass of GSIC 6 is composed ofhydrogen. This is the first of three stars in the dataset to have multiple planets detected in its orbit.

GSIC 6b

Table 7.13 – Orbital Parameters for GSIC 6b

Parameter Value Error

P (days) 6.23853699 1.44E – 06a (AU) 0.07593 0.00476τ (days) 0.146 0.01s (AU) 0.0915 0.0128

e > 0.204 0.185i (◦) 89.384 0.370

b 0.1462 0.0918

Table 7.14 – Planetary Characteristics for GSIC 6b

Parameter Value

R (R⊕) 2.764± 0.338Composition Gas dwarf

A 0.3-0.5Teq (K) 1100-1300M (M⊕) 2-9

This planet is likely to be a small gas planet, havinga relatively large rocky core surrounded by a thick gasenvelope. As we are unable to tell with certainty thecomposition of the envelope, an albedo range has beengiven. Being relatively small for a gas planet, the coremust take up a sizable portion of the volume, and thusa higher density has been assumed than any gas planetin our solar system, which leads us to the mass rangestated. The lower range of this mass range shows aNeptune-like composition, whereas the higher reachesdescribe a planet with a large rocky core surroundedby a gas envelope. The lower end of the range has alsobeen lowered slightly to account for the ‘puffy’ effect

58

which may inflate the atmosphere for close orbitingplanets such as this one.

The period of the orbit seems slightly inconsistentwith the eccentricity calculated. It is likely that thetrue eccentricity is lower than this value, as tidal forceswill act to circularise such low period orbits.

GSIC 6c

Table 7.15 – Orbital Parameters for GSIC 6c

Parameter Value Error

P (days) 12.7203787 1.87E – 05a (AU) 0.12210 0.00765τ (days) - -s (AU) - -

e - -i (◦) 89.491 0.037

b - -

Table 7.16 – Planetary Characteristics for GSIC 6c

Parameter Value

R (R⊕) 4.330± 0.530Composition Gas giant

A 0.5Teq (K) 800-1000M (M⊕) 10-30

This planet is larger than the other in this system,and likely to be of similar composition. This planetis likely to be less dense than its companion, as it islarger and thus more of its volume will be taken up by agaseous envelope. This planet was therefore assigned adensity similar to that of Neptune, and thus a relativelysimilar mass range was suggested, taking the inflatingeffect of temperature into account.

Unfortunately, due to a slightly imprecise value ofP , a successful phase fold was not possible for thisplanet, and thus the transit time was not measurable.This in turn meant that we were unable to calculate theseparation at transit or the eccentricity of the system,although as this is a close orbiting planet with anotherin the system, it is likely that both planets orbit atnear 0 eccentricity.

It is also notable that the periods of the two planetsare very nearly in the resonance ratio 1:2. This is theclearest example of orbital resonance seen in the data.

In comparison to the properties of Jupiter, thisplanet has a radius of 0.3841±0.0470RJ and a mass of0.0315-0.0994MJ.

GSIC 7

GSIC 7 has a slightly lower signal to noise ratio inits power spectrum than GSIC 6. Unfortunately thisdifference is great enough to make any possible l = 2peaks unreliable.

Table 7.17 – Stellar Parameters for GSIC 7

Parameter Value Error

Δν0 (μHz) 82.0.32 0.609νmax (mHz) 1.61821 0.00125δν0,2 (μHz) - -

Teff (K) 5845 88[Fe/H] 0.07 0.11

Xc - -M∗ (M�) 1.182 0.233R∗ (R�) 1.501 0.190

Age (Gyrs) - -RG IHZ (AU) ∼ 1.50 -

MaG OHZ (AU) ∼ 2.58 -

It is a shame that no small frequency spacing canbe calculated for GSIC 7, as this is one of the moreinteresting extrasolar systems in the data set and anage calculation would lead to greater discussion. How-ever, despite the lack of small frequency spacing, an agecan still be discussed through comparison with similarstars.

The value for the maximum frequency of oscilla-tions in Table 7.17 puts the star firmly on the MS andis fairly low compared to the value for more solar-likestars, implying it is more evolved. The mass of GSIC 7is similar to solar mass, suggesting a greater age thanmany of its neighbours but likely lower than the Sun.GSIC 7 is the second of three stars with multiple plan-ets detected in orbit.

GSIC 7b

Table 7.18 – Orbital Parameters for GSIC 7b

Parameter Value Error

P (days) 15.9653315 1.04E – 05a (AU) 0.13118 0.00863τ (days) 0.1854 0.01s (AU) 0.1914 0.0263

e > 0.459 0.223i (◦) 88.619 0.167

b 0.660 0.147

Table 7.19 – Planetary Characteristics for GSIC 7b

Parameter Value

R (R⊕) 3.230± 0.412Composition Gas dwarf

A 0.3-0.5Teq (K) 800-900M (M⊕) 7-15

This planet’s radius indicates a gas dwarf, and themass range and albedo have been chosen to reflect this.The close orbit is also a factor once more in the massrange, and the lower mass bound has again been re-duced to account for thermal gas expansion. This

59

planet also has a relatively eccentric orbit, which isslightly surprising given its low period, but as the otherplanet in this system has such a large period the orbitsare unlikely to affect each other.

It is quite possible that these two planets did inter-act at some point in the past, as this planet is unlikelyto have formed this close to the star, and the eccen-tricity value may be an indication that this happenedrelatively recently as tidal forces would normally beexpected to circularise this orbit.

GSIC 7c

Table 7.20 – Orbital Parameters for GSIC 7c

Parameter Value Error

P (days) 179.43045 2.13E – 03a (AU) 0.6582 0.0433τ (days) - -s (AU) - -

e - -i (◦) 89.627 0.256

b - -

Table 7.21 – Planetary Characteristics for GSIC 7c

Parameter Value

R (R⊕) 1.407± 0.180Composition Rocky

A 0.3Teq (K) 300-400M (M⊕) 0.8-4

It is quite surprising to detect a planet with thishigh a period by planetary transit, and that this planetshould prove to be rocky makes this result particularlyexciting. Even better, the equilibrium temperature ofthis planet places it within the range possible for liquidwater to exist on its surface - one of the few planets inthis study for which this is the case. This also affectsour mass constraints, as ‘water planet’ compositionsmust be included.

Due to the high period only a few transits wererecorded for this planet, and thus the value obtainedfor the period was of insufficient precision for a usablephase fold, especially considering that this planet had alow signal to noise ratio. The transit time, separation,impact parameter and eccentricity of this planet cantherefore not be calculated. This is unfortunate, asthe latter may have been interesting as it could wellhave interacted with the closer orbiting planet in thissystem.

GSIC 8

Similarly to the previous star, GSIC 8 had a powerspectrum with just enough noise to prevent the detec-tion of l = 2 peaks.

Table 7.22 – Stellar Parameters for GSIC 8

Parameter Value Error

Δν0 (μHz) 90.69 2.29νmax (mHz) 1.80987 0.00141δν0,2 (μHz) - -

Teff (K) 5952 75[Fe/H] –0.08 0.1

Xc - -M∗ (M�) 1.139 0.299R∗ (R�) 1.387 0.223

Age (Gyrs) - -RG IHZ (AU) ∼ 1.43 -

MaG OHZ (AU) ∼ 2.45 -

The values for mass, radius and effective tempera-ture given in Table 7.22 are very similar to the valuesfor GSIC 7, but with a slightly higher ratio of temper-ature to radius. This means similar assumptions canbe made about its age. This is the final star in the setwith multiple planets detected in orbit, so it is a shamethat a calculation of the age of the extrasolar systemwas not possible.

GSIC 8b

Table 7.23 – Orbital Parameters for GSIC 8b

Parameter Value Error

P (days) 6.48164 9.66E – 03a (AU) 0.07104 0.00622τ (days) 0.15177 0.04s (AU) 0.0877 0.0271

e > 0.235 0.396i (◦) 89.093 0.957

b 0.215 0.239

Table 7.24 – Planetary Characteristics for GSIC 8b

Parameter Value

R (R⊕) 2.251± 0.566Composition Gas dwarf

A 0.3-0.5Teq (K) 1000-1200M (M⊕) 2-6

This planet appears to be a close-orbiting gasdwarf, and thus again is unlikely to have formed atsuch a close orbit. It is likely that it has migratedinwards by ejecting other planets, and given the lowperiod of both planets in this system they are likely tohave interacted at some point.

The eccentricity value is likely to be near 0, astidal forces would be relatively strong at this distanceand would therefore circularise an eccentric orbit quitequickly.

60

GSIC 8c

Table 7.25 – Orbital Parameters for GSIC 8c

Parameter Value Error

P (days) 21.2227181 4.38E – 05a (AU) 0.1566 0.0137τ (days) 0.2696 0.01s (AU) 0.1617 0.0267

e > 0.032 0.193i (◦) 89.907 0.078

b 0.0406 0.0352

Table 7.26 – Planetary Characteristics for GSIC 8c

Parameter Value

R (R⊕) 2.284± 0.367Composition Gas dwarf

A 0.3-0.5Teq (K) 700-800M (M⊕) 2-6

This planet is very similar in nature to its compan-ion, and is very likely to contain a similar composition.This leads us to very similar values for each planet - theonly significant difference being the equilibrium tem-perature, which is significantly lower here - althoughstill not nearly low enough to approach habitability.

GSIC 9

The GSIC 9 power spectrum had a high signal tonoise ratio and detectable l = 2 peaks.

Table 7.27 – Stellar Parameters for GSIC 9

Parameter Value Error

Δν0 (μHz) 90.079 0.256νmax (mHz) 1.90682 0.00122δν0,2 (μHz) 8.35 0.6

Teff (K) 6169 50[Fe/H] 0.09 0.08

Xc 0.52 0.02M∗ (M�) 1.444 0.255R∗ (R�) 1.508 0.176

Age (Gyrs) 0.792 0.150RG IHZ (AU) ∼ 1.65 -

MaG OHZ (AU) ∼ 2.81 -

Table 7.27 shows that GSIC 9 is one of the youngeststars in the sample. It has a mass greater than solarmass, therefore will evolve faster. The hydrogen frac-tion puts this star towards the middle of the MS. Asingle planet has been detected orbiting GSIC 9, mak-ing this one of only two extrasolar systems in the dataset with a calculated age of less than one billion years.

GSIC 9b

Table 7.28 – Orbital Parameters for GSIC 9b

Parameter Value Error

P (days) 5.85993065 1.77E – 06a (AU) 0.07189 0.00424τ (days) 0.1637 0.01s (AU) 0.07993 0.0106

e > 0.112 0.161i (◦) 87.318 0.021

b 0.533 0.094

Table 7.29 – Planetary Characteristics for GSIC 9b

Parameter Value

R (R⊕) 2.518± 0.295Composition Gas dwarf

A 0.3-0.5Teq (K) 1100-1300M (M⊕) 3-8

In this system another close-orbiting gas giant isseen, with a period low enough such that tidal forcesare likely to have circularised the orbit. Although noother planets were detected in this system, it is likelythat they are or were present at some point, as thisplanet is likely to have migrated inwards. The massrange takes into consideration the ‘puffy’ effect of hightemperature on the radius of the planet.

GSIC 10

The power spectrum from GSIC 10 had a very lowsignal to noise ratio and as a result was nearly impossi-ble to read without lots of filtering. This resulted in ahigh uncertainty on the calculated mass and radius ofthe star. The signal to noise ratio was too low for thel = 2 peaks to be found, so the age of the star couldnot be determined.

Table 7.30 – Stellar Parameters for GSIC 10

Parameter Value Error

Δν0 (μHz) 67.328 0.228νmax (mHz) 1.27508 0.00228δν0,2 (μHz) - -

Teff (K) 5871 94[Fe/H] 0.17 0.11

Xc - -M∗ (M�) 1.284 0.242R∗ (R�) 1.761 0.212

Age (Gyrs) - -RG IHZ (AU) ∼ 1.77 -

MaG OHZ (AU) ∼ 3.05 -

From the mass, radius and effective temperaturevalues given in Table 7.30 it can be deduced that GSIC10 is larger than the Sun. Due to its large mass and

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radius, an argument could be made for this star be-ing classed as a subgiant. One orbiting planet wasdetected.

GSIC 10b

Table 7.31 – Orbital Parameters for GSIC 10b

Parameter Value Error

P (days) 13.74884265 8.78E – 06a (AU) 0.12208 0.00766τ (days) 0.2495 0.01s (AU) 0.1437 0.0183

e > 0.177 0.167i (◦) 85.263 0.900

b 1.448 0.374

Table 7.32 – Planetary Characteristics for GSIC 10b

Parameter Value

R (R⊕) 3.991± 0.506Composition Gas dwarf/gas giant

A 0.3-0.5Teq (K) 900-1000M (M⊕) 10-30

This planet occurs almost perfectly on the bound-ary between gas dwarf and gas giant classification, andthus has been estimated as having a Neptune-like massand composition. The inclination of the orbit is quitelow, resulting in a transit which may very well only‘graze’ the star - this is likely the source of the unphys-ical value obtained for the impact parameter, which islikely to be close to 1.

GSIC 12

The power spectrum from GSIC 12 was fairly sim-ple to read. The signal to noise ratio was moderatelyhigh, so not much filtering was needed. The adequatesignal to noise ratio allowed the l = 2 peaks to be seenwhich resulted in values for the small frequency spacingalong with the age of the star.

Table 7.33 – Stellar Parameters for GSIC 12

Parameter Value Error

Δν0 (μHz) 77.057 0.110νmax (mHz) 1.47346 0.00117δν0,2 (μHz) 5.29 0.21

Teff (K) 5825 75[Fe/H] 0.02 0.1

Xc 0.28 0.02M∗ (M�) 1.142 0.206R∗ (R�) 1.547 0.181

Age (Gyrs) 1.85 0.417RG IHZ (AU) ∼ 1.54 -

MaG OHZ (AU) ∼ 2.65 -

From the mass, radius and effective temperaturevalues given in Table 7.33 it can be deduced that GSIC12 is larger than the Sun, and as a result is has usedup over two-thirds of its core hydrogen despite beingless than half as old. One planet was detected orbitingGSIC 12.

GSIC 12b

Table 7.34 – Orbital Parameters for GSIC 12b

Parameter Value Error

P (days) 12.8158875 1.02E – 05a (AU) 0.11201 0.00673τ (days) 0.2655 0.01s (AU) 0.1106 0.0136

e > 0.012 0.135i (◦) 89.468 0.377

b 0.143 0.104

Table 7.35 – Planetary Characteristics for GSIC 12b

Parameter Value

R (R⊕) 2.112± 0.248Composition Gas dwarf

A 0.3-0.5Teq (K) 800-1000M (M⊕) 2.5-6

This planet is likely to be a gas dwarf, with a den-sity greater than that of Neptune due to a larger rel-ative core size. The mass radius has again been sug-gested considering the ‘puffy’ effect of high tempera-ture. Once again it is likely that this planet has mi-grated inwards and then undergone tidal circularisa-tion.

GSIC 13

The power spectrum from GSIC 13 was also quitesimple to read. The signal to noise ratio was moder-ately high, so again, not much filtering was needed tofind the large frequency spacing. However, the l = 2peaks were hard to discern without further filtering.These values were eventually obtained and the smallfrequency spacing could then be found.

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Table 7.36 – Stellar Parameters for GSIC 13

Parameter Value Error

Δν0 (μHz) 58.287 0.157νmax (mHz) 0.61878 0.00074δν0,2 (μHz) 3.78 0.34

Teff (K) 5896 75[Fe/H] –0.17 0.1

Xc - -M∗ (M�) 0.2630 0.0479R∗ (R�) 1.143 0.135

Age (Gyrs) - -RG IHZ (AU) ∼ 1.16 -

MaG OHZ (AU) ∼ 1.99 -

Due to an over-estimation of large frequency spac-ing, the mass value should be higher than shown inTable 7.36. This also had repercussions on our abil-ity to plot this star on the asteroseismic diagrams. Asthe mass was too low, the asteroseismic diagrams wereunable to be used to find the star’s age or Hydrogenfraction. Unfortunately, this error was found close tothe deadline, leaving too little time to correct it. How-ever, one planet was detected orbiting GSIC 13.

GSIC 13b

Table 7.37 – Orbital Parameters for GSIC 13b

Parameter Value Error

P (days) 105.880655 1.10E – 04a (AU) 0.2806 0.0170τ (days) - -s (AU) 0 0

e - -i (◦) 89.143 0.002

b - -

Table 7.38 – Planetary Characteristics for GSIC 13b

Parameter Value

R (R⊕) 3.722± 0.441Composition Gas dwarf/water planet

A 0.3-0.5Teq (K) 400-600M (M⊕) 10-60

The equilibrium temperature of this planet indi-cates that there is a chance that it could support liq-uid water at its surface. This in turn means that thisplanet cannot simply be classified as a gas dwarf, andthus the rather large mass range has been suggested toinclude masses of a very large rocky planet covered bywater (upper reaches of the range) in addition to themore probable gas dwarf composition.

Unfortunately the period of this transit was not suf-ficiently precise for a clear phase fold, and thus severalvalues were not calculable. As the period of this planet

is particularly large for a transiting planet, a large ec-centricity range is theoretically possible, and if indeedthe eccentricity is large the temperature at periastronwould likely have been sufficiently high to rule out thepossibility of a water-covered planet.

Due to significant uncertainty on stellar values forthis system, the characteristics of GSIC 13b are likelyto be significantly less accurate than our errors imply.

GSIC 14

GSIC 14 had a power spectrum with a very lowsignal to noise ratio, so a lot of filtering was requiredto be able to clearly separate the l = 2 peaks from thel = 0 peaks. This was successful, and as a result, themass, radius and age of the star were obtained.

Table 7.39 – Stellar Parameters for GSIC 14

Parameter Value Error

Δν0 (μHz) 88.425 0.429νmax (mHz) 1.83660 0.00153δν0,2 (μHz) 8.8 0.14

Teff (K) 6463 110[Fe/H] 0.09 0.11

Xc 0.56 0.02M∗ (M�) 1.490 0.286R∗ (R�) 1.543 0.190

Age (Gyrs) 0.515 0.134RG IHZ (AU) ∼ 1.82 -

MaG OHZ (AU) ∼ 3.09 -

From the mass, radius and effective temperaturevalues given in Table 7.39 it can be deduced that GSIC14 is larger and hotter than our Sun, and is likely to bea new MS star. The hydrogen mass fraction, however,puts this star in the middle of its MS lifetime. It is alsothe youngest star to have a planet orbiting it.

GSIC 14b

Table 7.40 – Orbital Parameters for GSIC 14b

Parameter Value Error

P (days) 18.0115861 2.66E – 05a (AU) 0.15358 0.00984τ (days) 0.2754 0.04s (AU) 0.1495 0.0285

e > 0.027 0.196i (◦) 89.569 0.335

b 0.156 0.127

63

Table 7.41 – Planetary Characteristics for GSIC 14b

Parameter Value

R (R⊕) 1.552± 0.192Composition Rocky

A 0.3Teq (K) 850-950M (M⊕) 2.5-5

This system shows a close-orbiting rocky planet,which is likely to be in a low-eccentricity orbit due tothe low value of a. It has been possible to constrainthe mass considerably, as gaseous and liquid-coveredplanet compositions have been safely discarded.

GSIC 15

GSIC 15 also had a power spectrum with a lowsignal to noise ratio, so a lot of filtering was needed tobe able to identify the l = 2 peaks separately from thel = 0 peaks. This was successful, and the mass, radiusand age of the star were obtained as a result.

Table 7.42 – Stellar Parameters for GSIC 15

Parameter Value Error

Δν0 (μHz) 68.591 0.813νmax (mHz) 1.2821 0.00128δν0,2 (μHz) 5.98 0.79

Teff (K) 6072 75[Fe/H] –0.09 0.1

Xc 0.41 0.02M∗ (M�) 1.275 0.269R∗ (R�) 1.735 0.233

Age (Gyrs) 1.03 0.0984RG IHZ (AU) ∼ 1.85 -

MaG OHZ (AU) ∼ 3.16 -

From the mass, radius and effective temperaturevalues given in Table 7.42, it can be deduced that GSIC15 is larger and hotter than our Sun and is on the mainsequence. The age and hydrogen fraction are typicalfor a star of this mass in the middle of its MS lifetime.No planets were detected orbiting GSIC 15.

GSIC 16

The power spectrum from GSIC 16 was initiallyhard to read due to its low signal to noise ratio. Afterfiltering, the l = 0 and the l = 2 peaks were locatedwith minimum effort, and the stellar properties easilycalculated.

Table 7.43 – Stellar Parameters for GSIC 16

Parameter Value Error

Δν0 (μHz) 93.253 0.369νmax (mHz) 1.93452 0.00144δν0,2 (μHz) 7.9 1.04

Teff (K) 6239 94[Fe/H] –0.14 0.1

Xc 0.42 0.02M∗ (M�) 1.335 0.251R∗ (R�) 1.436 0.174

Age (Gyrs) 1.42 0.140RG IHZ (AU) ∼ 1.60 -

MaG OHZ (AU) ∼ 2.72 -

Looking at the mass, radius and effective tempera-ture values in Table 7.43, it can be deduced that GSIC16 is a young MS star and is larger and hotter thanthe Sun. Its age and hydrogen core mass fraction putthe star in the middle of the MS. A single planet wasdetected orbiting the star.

GSIC 16b

Table 7.44 – Orbital Parameters for GSIC 16b

Parameter Value Error

P (days) 100.2829887 6.93E – 05a (AU) 0.4651 0.0291τ (days) 0.4824 0.01s (AU) 0.4419 0.0542

e > 0.050 0.131i (◦) 89.939 0.049

b 0.0708 0.0579

Table 7.45 – Planetary Characteristics for GSIC 16b

Parameter Value

R (R⊕) 2.592± 0.314Composition Gas dwarf/water planet

A 0.3-0.5Teq (K) 400-500M (M⊕) 3.5-20

This planet orbits at a relatively large distance, andit is quite surprising that it was detectable at all, es-pecially with an almost equatorial transit. The planetis also potentially cool enough for liquid water to existon the surface, which widens the mass range consider-ably, as multiple compositions must be accounted for.Were the planet hotter, the mass could be constrainedto below around 8 M⊕.

GSIC 17

The GSIC 17 power spectrum had a low signal tonoise ratio, and was therefore hard to read. Whenfiltered, the l = 0 peaks were clear to see; however the

64

l = 2 peaks could not be seen, so only the mass andradius have been calculated.

Table 7.46 – Stellar Parameters for GSIC 17

Parameter Value Error

Δν0 (μHz) 83.188 0.030νmax (mHz) 1.60261 0.00133δν0,2 (μHz) - -

Teff (K) 5699 74[Fe/H] 0.3 0.1

Xc - -M∗ (M�) 1.047 0.188R∗ (R�) 1.428 0.167

Age (Gyrs) - -RG IHZ (AU) ∼ 1.37 -

MaG OHZ (AU) ∼ 2.36 -

From the mass, radius and effective temperaturevalues given in Table 7.46, GSIC 17 was deduced to belarger than the Sun, but with approximately the samemass and effective temperature. A single planet wasdetected orbiting GSIC 17.

GSIC 17b

Table 7.47 – Orbital Parameters for GSIC 17b

Parameter Value Error

P (days) 11.52309135 4.85E – 06a (AU) 0.10136 0.00607τ (days) - -s (AU) - -

e - -i (◦) 87.849 0.281

b - -

Table 7.48 – Planetary Characteristics for GSIC 17b

Parameter Value

R (R⊕) 3.350± 0.393Composition Gas dwarf

A 0.3-0.5Teq (K) 850-950M (M⊕) 8-20

This planet is likely to be in a low-eccentricity orbitdue to tidal forces, although due to an imprecise pe-riod value it was not possible to obtain a satisfactoryphase fold for this planet, and thus τ and those valuesdependent on it could not be found.

GSIC 18

GSIC 18’s power spectrum had a low signal to noiseratio, so it needed thorough filtering to identify thel = 0 and l = 2 peaks. The large and small frequencyspacings were then found, and the mass, radius andage of the star were calculated.

Table 7.49 – Stellar Parameters for GSIC 18

Parameter Value Error

Δν0 (μHz) 178.664 0.230νmax (mHz) 4.40160 0.00165δν0,2 (μHz) 13.39 1.02

Teff (K) 5417 75[Fe/H] –0.32 0.07

Xc 0.42 0.02M∗ (M�) 0.944 0.171R∗ (R�) 0.829 0.097

Age (Gyrs) 6.60 1.39RG IHZ (AU) ∼ 0.73 -

MaG OHZ (AU) ∼ 1.27 -

The results in Table 7.49 show that GSIC 18 isslightly smaller and cooler than our Sun, as a resultit is nearly one and a half times older but remains inthe middle of its MS. GSIC 18 was found to have oneplanet orbiting it.

GSIC 18b

Table 7.50 – Orbital Parameters for GSIC 18b

Parameter Value Error

P (days) 39.79215303 3.32E – 06a (AU) 0.2238 0.0135τ (days) 0.1856 0.01s (AU) 0.2632 0.0340

e > 0.176 0.168i (◦) 89.981 0.020

b 0.0226 0.0240

Table 7.51 – Planetary Characteristics for GSIC 18b

Parameter Value

R (R⊕) 1.885± 0.221Composition Gas dwarf/ Rocky

A 0.3-0.5Teq (K) 400-500M (M⊕) 2-12

This planet is potentially small enough to be solid,but is likely to have at least some atmospheric enve-lope. The planet also orbits far enough from its hoststar to put its equilibrium temperature potentially onthe inside edge of the habitable zone, meaning watercompositions must also be considered. This leads us toa larger mass range than would have been the case fora larger planet, due to the uncertainty of composition.

GSIC 19

The power spectrum from GSIC 19 was initiallyvery hard to read due to its low signal to noise ratio.After filtering, the l = 0 peaks could be found com-putationally, although the l = 2 peaks could not be

65

located at all, therefore no result for the star’s age hasbeen logged.

Table 7.52 – Stellar Parameters for GSIC 19

Parameter Value Error

Δν0 (μHz) 61.937 1.559νmax (mHz) 1.15814 0.00123δν0,2 (μHz) - -

Teff (K) 6144 106[Fe/H] 0.13 0.1

Xc - -M∗ (M�) 1.439 0.380R∗ (R�) 1.933 0.311

Age (Gyrs) - -RG IHZ (AU) ∼ 2.10 -

MaG OHZ (AU) ∼ 3.58 -

The results in Table 7.52 show that GSIC 19 isabout eight times as voluminous as the Sun and hasone and a half times the mass. One planet was de-tected orbiting the star.

GSIC 19b

Table 7.53 – Orbital Parameters for GSIC 19b

Parameter Value Error

P (days) 42.8822 1.63E – 02a (AU) 0.2707 0.0238τ (days) 0.4002 0.01s (AU) 0.3068 0.0500

e > 0.133 0.210i (◦) 89.64 2.83

b 0.216 1.69

Table 7.54 – Planetary Characteristics for GSIC 19b

Parameter Value

R (R⊕) 1.890± 0.335Composition Gas dwarf/ Rocky

A 0.3-0.5Teq (K) 600-800M (M⊕) 2-12

This planet is again potentially rocky or gaseous,however in this case the presence of liquid water canbe discounted due to high temperature. Due to a poorfit to the inclination of the system, the value for theimpact parameter of the system has been rendered sta-tistically useless. Once again, a large mass range hasbeen suggested due to uncertainty of composition.

GSIC 20

The power spectrum for GSIC 20 required signif-icant filtering to identify the region of interest, alongwith a significant amount of analysis to identify the

l = 0 peaks. Fortunately the spacings between l = 1and l = 0 peaks were almost identical, so any errorsmade had minimal effect on the large frequency spac-ing. Eventually the small frequency spacing was alsocalculated.

Table 7.55 – Stellar Parameters for GSIC 20

Parameter Value Error

Δν0 (μHz) 61.893 0.414νmax (mHz) 1.13110 0.00097δν0,2 (μHz) 5.02 0.96

Teff (K) 5945 60[Fe/H] 0.17 0.05

Xc - -M∗ (M�) 1.279 0.244R∗ (R�) 1.860 0.231

Age (Gyrs) - -RG IHZ (AU) ∼ 1.91 -

MaG OHZ (AU) ∼ 3.28 -

Although the small frequency spacing could be cal-culated for GSIC 20, the asteroseismic diagrams failedto obtain any reliable results for the age or hydrogenfraction. This is due to the mass calculated by the as-teroseismic diagrams, 1.53M�, being slightly too largefor any valid age calculations. This mass lies just overone error above the mass calculated from the scaling re-lations. As the mass was at the top end of those used inthe asteroseismic diagrams, we could approximate theage being just below that of GSIC 4 (1.54Gyrs), as thisis the most similar star according to the asteroseismicdiagrams.

GSIC 20b

Table 7.56 – Orbital Parameters for GSIC 20b

Parameter Value Error

P (days) 4.7768795 2.18E – 05a (AU) 0.06025 0.00383τ (days) - -s (AU) - -

e - -i (◦) 89.840 0.228

b - -

Table 7.57 – Planetary Characteristics for GSIC 20b

Parameter Value

R (R⊕) 1.693± 0.211Composition Rocky

A 0.3Teq (K) 1300-1500M (M⊕) 3.5-8

This planet is below the radius threshold for a gasdwarf, but may still have some atmosphere. The tem-perature is very high due to an extremely small period

66

value, and thus the possibility of surface water can besafely discounted. In this case, the transit depth wasso small that in the phase fold obtained it was notdistinguishable from background noise manually, andthus τ and dependent values were not found. However,a low eccentricity orbit can be expected, due to tidalcirculation.

GSIC 21

The signal to noise ratio on the power spectrum ofGSIC 21 was low, so no small frequency spacing couldbe calculated.

Table 7.58 – Stellar Parameters for GSIC 21

Parameter Value Error

Δν0 (μHz) 49.91 1.93νmax (mHz) 0.95116 0.00180δν0,2 (μHz) - -

Teff (K) 5882 87[Fe/H] 0.16 0.1

Xc - -M∗ (M�) 1.771 0.560R∗ (R�) 2.393 0.449

Age (Gyrs) - -RG IHZ (AU) ∼ 2.42 -

MaG OHZ (AU) ∼ 4.15 -

This is one of the more interesting stars in terms ofevolutionary stage. Table 7.58 shows a higher than av-erage radius and mass compared to other stars in thisdata set. The high mass means that GSIC 21 will moveoff the MS sooner than the majority of other stars.The lower than average value of maximum frequencyfor oscillations also agrees with this assumption. Un-fortunately the age couldn’t be calculated, but due toits high mass and proximity to the MS, GSIC 21 islikely younger than the Sun. There was a single planetdetected in orbit around this star.

GSIC 21b

Table 7.59 – Orbital Parameters for GSIC 21b

Parameter Value Error

P (days) 46.1511576 5.69E – 05a (AU) 0.3046 0.0321τ (days) - -s (AU) - -

e - -i (◦) 88.945 0.008

b - -

Table 7.60 – Planetary Characteristics for GSIC 21b

Parameter Value

R (R⊕) 8.139± 1.527Composition Gas giant

A 0.5Teq (K) 600-800M (M⊕) 40-150

This gas giant is considered a hot Jupiter despiteorbiting comparatively far from the star by the stan-dards of this study. Unfortunately the period valueobtained for this orbit was not sufficiently precise toallow a readable phase fold, and thus values dependenton τ were not found. Of these, the eccentricity had thepotential to be interesting, as this planet is far enoughfrom the star that tidal circularisation may not haveoccurred, and considering that this planet has likelymigrated from further out in the system the orbitaleccentricity may well be high.

In comparison to the properties of Jupiter, thisplanet has a radius of 0.722 ± 0.135RJ and a mass of0.126-0.472MJ.

GSIC 22

The signal to noise ratio for GSIC 22 was highenough for the small frequency spacing to be calcu-lated after filtering.

Table 7.61 – Stellar Parameters for GSIC 22

Parameter Value Error

Δν0 (μHz) 94.025 0.688νmax (mHz) 1.96181 0.00147δν0,2 (μHz) 12.39 0.38

Teff (K) 6325 75[Fe/H] 0.01 0.1

Xc - -M∗ (M�) 1.375 0.267R∗ (R�) 1.442 0.182

Age (Gyrs) - -RG IHZ (AU) ∼ 1.64 -

MaG OHZ (AU) ∼ 2.79 -

Although the small frequency spacing was visiblefor this star, there were issues with plotting it on theasteroseismic diagrams. GSIC 22 did not fall onto anyof the mass lines, so no accurate age or hydrogen frac-tion could be calculated. This has very recently beenput down to a single error in the peak finding code usedby the computational analysis team.

An age estimate can be made, however, looking atthe mass value in Table 7.61. The mass of GSIC 22 issignificantly greater than solar mass and the maximumfrequency for oscillations places this star somewhere inthe middle of its MS lifetime. Therefore its age willbe far lower than the age of the Sun, likely by a factorof two. There was one planet detected in orbit aroundGSIC 22.

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GSIC 22b

Table 7.62 – Orbital Parameters for GSIC 22b

Parameter Value Error

P (days) 17.83367414 3.08E – 06a (AU) 0.14855 0.00962τ (days) 0.2006 0.01s (AU) 0.1898 0.0257

e > 0.278 0.192i (◦) 89.335 0.051

b 0.3285 0.0658

Table 7.63 – Planetary Characteristics for GSIC 22b

Parameter Value

R (R⊕) 2.755± 0.345Composition Gas dwarf

A 0.3-0.5Teq (K) 750-900M (M⊕) 3-8

In this system there is, once more, a hot close-orbiting gas dwarf. The mass range has once againbeen suggested considering thermal atmospheric infla-tion. This planet has a higher eccentricity than mightgenerally be expected, and may be undergoing tidalcircularisation.

GSIC 23

GSIC 23 has a power spectrum with a very highsignal to noise ratio, so all desired parameters could becalculated.

Table 7.64 – Stellar Parameters for GSIC 23

Parameter Value Error

Δν0 (μHz) 59.666 0.384νmax (mHz) 1.08708 0.00137δν0,2 (μHz) 4.63 0.54

Teff (K) 6253 85[Fe/H] –0.13 0.1

Xc 0.30 0.02M∗ (M�) 1.418 0.274R∗ (R�) 1.973 0.246

Age (Gyrs) 1.28 0.132RG IHZ (AU) ∼ 2.20 -

MaG OHZ (AU) ∼ 3.75 -

Table 7.64 shows that this is another one of thelarger stars in the data set. The high mass shows thatGSIC 23 will progress off the MS sooner than our sun,and a hydrogen fraction value of 0.3 agrees with thissuggestion. However, the age given from the asteroseis-mic diagrams is typical for a star of this mass aroundthe middle of the MS. There was a single planet de-tected in orbit around GSIC 23.

GSIC 23b

Table 7.65 – Orbital Parameters for GSIC 23b

Parameter Value ErrorP (days) 53.50521447 8.12E – 05a (AU) 0.3122 0.0201τ (days) 0.4362 0.01s (AU) 0.3583 0.0454

e > 0.148 0.163i (◦) 89.932 0.060

b 0.0464 0.0419

Table 7.66 – Planetary Characteristics for GSIC 23b

Parameter Value

R (R⊕) 2.268± 0.283Composition Gas dwarf

A 0.3-0.5Teq (K) 600-700M (M⊕) 3-7

This gas dwarf has an orbital period allowing thepotential for significant eccentricity, as tidal forces willbe weak at this separation. As it is quite small for agas dwarf, it is likely to have a dense composition.

GSIC 24

The region of interest on the power spectrum ofGSIC 24 was visible without filtering, but filtering wasrequired to find the small frequency spacing.

Table 7.67 – Stellar Parameters for GSIC 24

Parameter Value Error

Δν0 (μHz) 53.067 0.334νmax (mHz) 0.99350 0.00126δν0,2 (μHz) 5.11 1.07

Teff (K) 6174 92[Fe/H] 0.22 0.1

Xc - -M∗ (M�) 1.698 0.329R∗ (R�) 2.265 0.282

Age (Gyrs) - -RG IHZ (AU) ∼ 2.48 -

MaG OHZ (AU) ∼ 4.23 -

Although the small frequency spacing was visible,the asteroseismic diagrams weren’t able to find an ageor hydrogen fraction for this star. Similarly to GSIC20, this is due to the mass being greater than 1.5M�and the model on the asteroseismic diagrams not ac-counting for this. The large mass means that this starwill evolve off of the main sequence sooner than moreSun-like stars. The value for the maximum frequencyfor oscillations is less than 1mHz, which implies thatthis star is reaching the end of its MS lifetime, though

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its age will likely be far less than that of the Sun. Therewas one planet detected in orbit around GSIC 24.

GSIC 24b

Table 7.68 – Orbital Parameters for GSIC 24b

Parameter Value Error

P (days) 10.27783263 5.34E – 06a (AU) 0.11034 0.00713τ (days) - -s (AU) - -

e - -i (◦) 88.899 0.421

b - -

Table 7.69 – Planetary Characteristics for GSIC 24b

Parameter Value

R (R⊕) 3.140± 0.392Composition Gas dwarf

A 0.3-0.5Teq (K) 1100-1300M (M⊕) 7-15

This is a fairly large gas dwarf, with a high equi-librium temperature which will inflate its atmosphere.Due to an imprecise period value, the phase fold for thisplanet was not usable, resulting in no measurement forτ and dependent values. This planet is likely to orbitwith low eccentricity due to tidal forces however.

GSIC 25

GSIC 25 had a very standard power spectrumthat yielded the small frequency spacing after a smallamount of filtering.

Table 7.70 – Stellar Parameters for GSIC 25

Parameter Value Error

Δν0 (μHz) 153.232 0.337νmax (mHz) 3.52729 0.00183δν0,2 (μHz) 8.94 0.61

Teff (K) 5460 75[Fe/H] 0.08 0.1

Xc 0.29 0.02M∗ (M�) 0.909 0.166R∗ (R�) 0.907 0.107

Age (Gyrs) 8.06 2.03RG IHZ (AU) ∼ 0.81 -

MaG OHZ (AU) ∼ 1.41 -

Table 7.70 shows a mass and radius each within anerror of solar values, with a lower temperature. Thisstar is also the oldest in the data set for which the agecould be calculated. It is a little under twice the age ofthe Sun with a significantly lower hydrogen fraction. A

hydrogen fraction of 0.29 does, however, suggest thatGSIC 25 has a reasonable amount of time left on theMS. There were no planets detected in orbit aroundGSIC 25.

GSIC 26

GSIC 26 was another star with a low signal to noiseration on its power spectrum, but as the small spacingis visible after filtering this is not a problem.

Table 7.71 – Stellar Parameters for GSIC 26

Parameter Value Error

Δν0 (μHz) 69.940 0.493νmax (mHz) 1.27947 0.00135δν0,2 (μHz) 6.1 0.49

Teff (K) 5784 98[Fe/H] –0.11 0.11

Xc 0.42 0.02M∗ (M�) 1.090 0.216R∗ (R�) 1.625 0.206

Age (Gyrs) 1.00 0.0940RG IHZ (AU) ∼ 1.60 -

MaG OHZ (AU) ∼ 2.75 -

Table 7.71 shows another mass within one error ofsolar mass, but with a greater radius. Unfortunately,the initial mass value obtained through computationalanalysis does not agree with the value obtained inthe asteroseismic diagrams, which was 1.5M�. TheAHRD-derived mass would explain why GSIC 26 hasa hydrogen fraction of less than half at such a youngage. There were no planets detected in orbit aroundthis star.

GSIC 27

The power spectrum for GSIC 27 had a low signalto noise ratio, hence no l = 2 peaks were detected.

Table 7.72 – Stellar Parameters for GSIC 27

Parameter Value Error

Δν0 (μHz) 69.790 0.939νmax (mHz) 1.36544 0.00249δν0,2 (μHz) - -

Teff (K) 5770 75[Fe/H] 0.29 0.1

Xc - -M∗ (M�) 1.331 0.290R∗ (R�) 1.740 0.240

Age (Gyrs) - -RG IHZ (AU) ∼ 1.70 -

MaG OHZ (AU) ∼ 2.93 -

The temperature displayed in Table 7.72 is aroundthe temperature of the Sun, which is unexpected fora star of greater mass and radius. From the value ofthe maximum frequency for oscillations, it appears that

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GSIC 27 isn’t too close to evolving off of the MS andwith a large mass it can be deduced that this is a fairlyyoung star, likely around 1Gyr. There were no planetsdetected in orbit around GSIC 27.

GSIC 29

The power spectrum for GSIC 29 had a high sig-nal to noise ratio so that the l = 2 peaks were visiblewithout filtering. Although, like for all other stars inthe data, the power spectra underwent smoothing tofind accurate values for the small and large frequencyspacings.

Table 7.73 – Stellar Parameters for GSIC 29

Parameter Value Error

Δν0 (μHz) 102.915 0.276νmax (mHz) 2.11675 0.00168δν0,2 (μHz) 6.76 0.49

Teff (K) 6104 74[Fe/H] –0.2 0.1

Xc 0.15 0.02M∗ (M�) 1.141 0.207R∗ (R�) 1.276 0.151

Age (Gyrs) 3.56 0.423RG IHZ (AU) ∼ 1.37 -

MaG OHZ (AU) ∼ 2.34 -

It can be seen from Table 7.73 that the mass andradius of GSIC 29 are slightly greater than that of theSun. This star is also slightly younger than the Sun,but due to its greater mass has a much lower hydrogenfraction, as it will progress off of the MS sooner. Therewere no planets detected in orbit around GSIC 29.

GSIC 30

The power spectrum from GSIC 30 was hard toread due to the low signal to noise ratio. The spectrumwas filtered and the l = 0 peaks were found, althoughthe l = 2 peaks could not be located. This led to thecalculation of GSIC 30’s mass and radius.

Table 7.74 – Stellar Parameters for GSIC 30

Parameter Value Error

Δν0 (μHz) 107.752 0.368νmax (mHz) 2.2482 0.00124δν0,2 (μHz) - -

Teff (K) 5982 82[Fe/H] –0.02 0.1

Xc - -M∗ (M�) 1.104 0.204R∗ (R�) 1.224 0.146

Age (Gyrs) - -RG IHZ (AU) ∼ 1.27 -

MaG OHZ (AU) ∼ 2.18 -

From the mass, radius and effective temperaturevalues in Table 7.74, GSIC 30 was deduced to be a

little larger and hotter than the Sun, with one orbitingplanet detected.

GSIC 30b

Table 7.75 – Orbital Parameters for GSIC 30b

Parameter Value Error

P (days) 41.74594751 2.97E – 05a (AU) 0.2434 0.0150τ (days) 0.1918 0.01s (AU) 0.3944 0.0515

e > 0.621 0.234i (◦) 89.384 0.017

b 0.133 0.179

Table 7.76 – Planetary Characteristics for GSIC 30b

Parameter Value

R (R⊕) 2.406± 0.289Composition Gas dwarf

A 0.3-0.5Teq (K) 500-600M (M⊕) 3.5-7

This planet is another gas dwarf, with a particularlyhigh value for the eccentricity. It may very well bethe case that this planet has ejected other material tonow reside in its current orbit, which may be too farfrom the host star for tidal circularisation to have takenplace since.

GSIC 31

The power spectrum from GSIC 31 was very simpleto read due to the high signal to noise ratio. No filteringwas needed, as both the l = 0 and l = 2 peaks wereclearly visible. This led to the calculation of GSIC 31’smass, radius and age.

Table 7.77 – Stellar Parameters for GSIC 31

Parameter Value Error

Δν0 (μHz) 101.434 0.053νmax (mHz) 2.05453 0.00152δν0,2 (μHz) 5.51 0.21

Teff (K) 5793 74[Fe/H] 0.12 0.07

Xc 0.11 0.02M∗ (M�) 1.022 0.183R∗ (R�) 1.242 0.145

Age (Gyrs) 4.47 0.0543RG IHZ (AU) ∼ 1.22 -

MaG OHZ (AU) ∼ 2.11 -

From the values given in Table 7.77, it can be de-duced that GSIC 31 is similar to the Sun in terms ofage, effective temperature and mass, though it has a

70

radius about 20% larger than that of the Sun. It hasalmost finished core hydrogen-burning as it only has11% left. One planet was detected orbiting GSIC 31.

GSIC 31b

Table 7.78 – Orbital Parameters for GSIC 31b

Parameter Value Error

P (days) 5.398754123 8.34E – 07a (AU) 0.060668 0.00363τ (days) - -s (AU) - -

e - -i (◦) 89.558 0.337

b - -

Table 7.79 – Planetary Characteristics for GSIC 31b

Parameter Value

R (R⊕) 2.142± 0.250Composition Gas dwarf

A 0.3-0.5Teq (K) 1000-1200M (M⊕) 2.5-6

This gas dwarf orbits extremely close to its star,resulting in a high equilibrium temperature which willinflate it somewhat. Although a phase fold was notobtained for this planet, tidal circularisation means alow eccentricity orbit is to be expected.

GSIC 32

The GSIC 32 power spectrum was also moderatelydifficult to read. The signal to noise ratio was low,so the spectra needed a filtering to obtain l = 0 andl = 2 peaks. This led to the large and small frequencyspacing being found, and the age, radius and mass ofthe star were then acquired.

Table 7.80 – Stellar Parameters for GSIC 32

Parameter Value Error

Δν0 (μHz) 68.115 0.358νmax (mHz) 1.24896 0.00114δν0,2 (μHz) 5.41 0.22

Teff (K) 5911 66[Fe/H] –0.2 0.06

Xc 0.33 0.02M∗ (M�) 1.164 0.218R∗ (R�) 1.691 0.206

Age (Gyrs) 1.15 1.54RG IHZ (AU) ∼ 1.72 -

MaG OHZ (AU) ∼ 2.96 -

The values in Table 7.80 show that GSIC 32 is ayoung, Sun-like star with a radius about 70% bigger

than the Sun’s. Its low core hydrogen content andage also reveal that GSIC 32 is large, as more massivestars burn hydrogen and therefore evolve quicker. Oneplanet was detected orbiting the star.

GSIC 32b

Table 7.81 – Orbital Parameters for GSIC 32b

Parameter Value Error

P (days) 16.2318437 4.12E – 05a (AU) 0.13197 0.00823τ (days) - -s (AU) - -

e - -i (◦) 89.912 0.057

b - -

Table 7.82 – Planetary Characteristics for GSIC 32b

Parameter Value

R (R⊕) 2.463± 0.300Composition Gas dwarf

A 0.3-0.5Teq (K) 800-1000M (M⊕) 3.5-7

This gas dwarf’s period was not calculated to suffi-cient precision for a clear phase fold, but a low eccen-tricity can be expected due to tidal forces in this closeorbit. The mass range was chosen assuming a densityexceeding that of Neptune due to a larger fraction ofthe radius taken up by the core.

GSIC 33

In the GSIC 33 power spectrum, the peaks weresimple to find, but the spectrum was filtered to obtainmore prominent peaks, as the signal to noise ratio waspreventing the l = 2 peaks from being seen. The age,radius and mass of the star were then acquired by usingthe large and small frequency spacing.

Table 7.83 – Stellar Parameters for GSIC 33

Parameter Value Error

Δν0 (μHz) 75.888 0.733νmax (mHz) 1.48114 0.00141δν0,2 (μHz) 7.07 0.05

Teff (K) 6225 75[Fe/H] 0 0.08

Xc 0.48 0.02M∗ (M�) 1.362 0.276R∗ (R�) 1.658 0.216

Age (Gyrs) 0.798 0.0911RG IHZ (AU) ∼ 1.84 -

MaG OHZ (AU) ∼ 3.13 -

71

From the values in Table 7.83, GSIC 33 appears tobe a hot star a little larger than the Sun. It’s relativelyyoung, at less than a billion years old and appears to bein the middle of its MS lifetime from its core hydrogenfraction. No planets were detected orbiting GSIC 33.

GSIC 34

The power spectrum from GSIC 34 had a very lowsignal to noise ratio, therefore it was difficult to find thepeaks. After filtering, the l = 0 peaks could be located,but the l = 2 peaks remained hidden. This meant thatthe mass and radius values could be determined, butnot the age of the star.

Table 7.84 – Stellar Parameters for GSIC 34

Parameter Value Error

Δν0 (μHz) 74.705 0.167νmax (mHz) 1.42627 0.00108δν0,2 (μHz) - -

Teff (K) 5781 76[Fe/H] 0.09 0.1

Xc - -M∗ (M�) 1.159 0.211R∗ (R�) 1.587 0.187

Age (Gyrs) - -RG IHZ (AU) ∼ 1.56 -

MaG OHZ (AU) ∼ 2.68 -

Due to the values of radius, mass and effective tem-perature in Table 7.84, it can be deduced that GSIC 34is a little larger than the Sun and is approximately thesame temperature. One planet was detected orbitingthe star.

GSIC 34b

Table 7.85 – Orbital Parameters for GSIC 34b

Parameter Value Error

P (days) 3.213668950 8.69E – 07a (AU) 0.0448 0.00271τ (days) 0.162 0.01s (AU) 0.04664 0.00621

e > 0.042 0.152i (◦) 88.820 0.903

b 0.130 0.102

Table 7.86 – Planetary Characteristics for GSIC 34b

Parameter Value

R (R⊕) 3.998± 0.472Composition Gas dwarf/ gas giant

A 0.3-0.5Teq (K) 1350-1550M (M⊕) 8-25

This planet is approaching the size where it couldbe described as a hot Jupiter, with a period of around3 days, which is around the minimum possible. Asexpected, the eccentricity is found to likely be low, al-though this was almost definitely not the case in thepast, as the planet must have migrated inwards. It islikely that the planet is tidally locked to the star, withthe same face constantly on the day side. The atmo-sphere of this planet will be significantly inflated bytemperature, and so a relatively low mass range hasbeen suggested for this planet.

GSIC 35

The power spectrum from GSIC 35 was hard toread due to the low signal to noise ratio. Filtering thespectra resulted in the appearance of the second degreemode peaks, which in turn resulted in the values of theage, mass and radius of GSIC 35.

Table 7.87 – Stellar Parameters for GSIC 35

Parameter Value Error

Δν0 (μHz) 117.502 0.087νmax (mHz) 2.43778 0.00128δν0,2 (μHz) 5 0.54

Teff (K) 5647 74[Fe/H] –0.15 0.1

Xc 0.06 0.02M∗ (M�) 0.913 0.164R∗ (R�) 1.084 0.127

Age (Gyrs) 7.64 2.08RG IHZ (AU) ∼ 1.02 -

MaG OHZ (AU) ∼ 1.77 -

The values in Table 7.87 for mass and radius arequite similar to solar mass and radius. GSIC 35 is3 billion years older and a little bit cooler than theSun. Its hydrogen core mass fraction suggests that it isabout to leave the main sequence, though the reason-ably large maximum frequency for oscillations woulddisagree with this. One planet was detected orbitingGSIC 35.

GSIC 35b

Table 7.88 – Orbital Parameters for GSIC 35b

Parameter Value Error

P (days) 45.2942007 2.29E – 05a (AU) 0.2412 0.0145τ (days) 0.3436 0.04s (AU) 0.2116 0.0349

e > 0.123 0.154i (◦) 89.712 0.108

b 0.2111 0.0897

72

Table 7.89 – Planetary Characteristics for GSIC 35b

Parameter Value

R (R⊕) 2.311± 0.270Composition Gas dwarf

A 0.3-0.5Teq (K) 450-550M (M⊕) 2.5-6

This is a gas dwarf orbiting at a period value whichis relatively high for our study. Given the high period,the low minimum eccentricity is slightly surprising, astidal forces will not be large at this distance, althoughit is of course possible that eccentricity is larger. Therelatively low temperature constrains the mass valueslightly, as the atmosphere will not be significantly in-flated.

GSIC 36

The power spectrum from GSIC 36 had a high sig-nal to noise ratio, and as a result, both l = 0 and l = 2peaks were clearly visible when the right area of thespectrum was enlarged. This made finding the largeand small frequency spacing very simple.

Table 7.90 – Stellar Parameters for GSIC 36

Parameter Value Error

Δν0 (μHz) 103.260 0.071νmax (mHz) 2.09566 0.00125δν0,2 (μHz) 5.35 0.35

Teff (K) 5825 50[Fe/H] 0.096 0.026

Xc - -M∗ (M�) 1.017 0.176R∗ (R�) 1.225 0.140

Age (Gyrs) - -RG IHZ (AU) ∼ 1.22 -

MaG OHZ (AU) ∼ 2.09 -

The values in Table 7.90 for mass, radius and ef-fective temperature are similar to solar values. Unfor-tunately, due to a communication error at the end ofthe project, the metallicity for this star was not passedon to the computational analysis team and due to timeconstraints at the end of the project, this could not berectified. Therefore, no age or hydrogen fraction couldbe obtained. No planets were detected orbiting GSIC36.

GSIC 38

The power spectrum of GSIC 38 required no fil-tering to find the l = 0 peaks, and therefore the largefrequency spacing. However, the l = 2 peaks were toounclear after filtering to calculate the small frequencyspacing, therefore there isn’t an age value for GSIC38. Discussions led to the idea that GSIC 38’s powerspectrum may have been hard to read due to G modes

and P modes mixing, reducing the clarity of the l = 2peaks.

Table 7.91 – Stellar Parameters for GSIC 38

Parameter Value Error

Δν0 (μHz) 23.525 1.706νmax (mHz) 0.39608 0.00032δν0,2 (μHz) 0 0

Teff (K) 5470 70[Fe/H] –0.79 0.1

Xc - -M∗ (M�) 2.323 1.047R∗ (R�) 4.325 1.104

Age (Gyrs) - -RG IHZ (AU) ∼ 3.87 -

MaG OHZ (AU) ∼ 6.72 -

The mass and radius values in Table 7.91 are sig-nificantly larger than solar values, therefore GSIC 38 islikely to have left the MS and may be a red giant. Thel = 2 peaks seemed to be in the power spectra, butwere located seemingly randomly, leading to the beliefthat they were actually mixed-mode oscillations, whichare common in red giants. The lower-than-averagevalue of the maximum power frequency backs up theclaim that GSIC 38 is a strong candidate for a red gi-ant. No planets were detected orbiting the star.

GSIC 39

GSIC 39’s power spectrum also needed no filteringto find the l = 0 peaks. Once again, the l = 2 peaksweren’t where they were expected to be, even afterfiltering. As with GSIC 38, it was thought that mixed-mode oscillations may be the cause of the lack of clearl = 2 peaks.

Table 7.92 – Stellar Parameters for GSIC 39

Parameter Value Error

Δν0 (μHz) 17.457 0.060νmax (mHz) 0.24289 0.00035δν0,2 (μHz) 0 0

Teff (K) 4840 97[Fe/H] 0.2 0.16

Xc - -M∗ (M�) 1.471 0.284R∗ (R�) 4.530 0.554

Age (Gyrs) - -RG IHZ (AU) ∼ 3.28 -

MaG OHZ (AU) ∼ 5.82 -

The radius value in Table 7.92 is approximately fourand a half times larger than the radius of the Sun,leading to the belief that GSIC 39 is a red giant. Thelower effective temperature backs up this theory. Themaximum power frequency is, again, very low. Thisprovides further evidence that the star may be a redgiant. GSIC 39 also had a similar power spectrum to

73

GSIC 38, where the l = 2 peaks have been shifted dueto suspected mixed-mode oscillations. One planet wasdetected orbiting the star.

GSIC 39b

Table 7.93 – Orbital Parameters for GSIC 39b

Parameter Value Error

P (days) 21.405289 1.33E – 04a (AU) 0.1716 0.0111τ (days) 0.4623 0.04s (AU) 0.3106 0.0465

e > 0.811 0.295i (◦) 86.890 0.0577

b 0.800 0.155

Table 7.94 – Planetary Characteristics for GSIC 39b

Parameter Value

R (R⊕) 8.352± 1.022Composition Gas giant

A 0.5Teq (K) 950-1150M (M⊕) 40-120

This is a close-orbiting gas giant with a very largeeccentricity - in fact, it was calculated that the planetmay well pass within 0.5R∗ of its host. The tidal forceson the planet will be very large, and it is likely that theorbit will eventually circularise. The temperature andradius of the planet are likely to fluctuate significantlythroughout the orbit as the planet approaches the star.

In comparison to the properties of Jupiter, thisplanet has a radius of 0.7409±0.0907RJ and a mass of0.126-0.377MJ.

GSIC 40

Much like GSIC 38 and GSIC 39, GSIC 40’s powerspectrum displayed clear l = 0 peaks. However, thel = 2 peaks were actually found with ease, consideringmixed-mode oscillations slightly obscured the powerspectrum. Therefore, the mass and radius of GSIC 40could be deduced. Due to the star no longer being onthe MS, the asteroseismic diagrams could not be usedto find its age.

Table 7.95 – Stellar Parameters for GSIC 40

Parameter Value Error

Δν0 (μHz) 18.567 0.0063νmax (mHz) 0.25875 0.00348δν0,2 (μHz) 2.25 0.09

Teff (K) 4995 78[Fe/H] –0.07 0.1

Xc - -M∗ (M�) 1.446 0.302R∗ (R�) 4.323 0.549

Age (Gyrs) - -RG IHZ (AU) ∼ 0.33 -

MaG OHZ (AU) ∼ 0.59 -

The radius value in Table 7.95 indicates that GSIC40 may be a red giant. Its temperature is in the typ-ical range for red giants, and the lower-than-averagemaximum power frequency also points to the conclu-sion that GSIC 40 has left the MS. One planet wasdetected orbiting GSIC 40.

GSIC 40b

Table 7.96 – Orbital Parameters for GSIC 40b

Parameter Value Error

P (days) 52.5008334 5.14E – 05a (AU) 0.3102 0.0216τ (days) 0.6171 0.01s (AU) 0.5446 0.0697

e > 0.756 0.256i (◦) 89.819 0.141

b 0.0854 0.0684

Table 7.97 – Planetary Characteristics for GSIC 40b

Parameter Value

R (R⊕) 12.589± 1.599Composition Gas Giant

A 0.5Teq (K) 700-800M (M⊕) 350-1000

This gas giant follows a very eccentric orbit, espe-cially considering that it resides quite close to its hoststar. It is possible that the tidal forces acting on itare not sufficient to circularise this orbit. Due to thetail-off in the mass-radius relations caused by electrondegeneracy pressure, the mass range for this planet isparticularly generous, as planets above a certain ra-dius could theoretically be composed of the same massof material.

In comparison to the properties of Jupiter, thisplanet has a radius of 1.117 ± 0.142RJ and a mass of1.101-3.145MJ.

74

7.2.2 Omissions (JG)

GSIC 2

There is very little to say about this star due tothe extremely low signal to noise ratio prohibiting thedetection of the l = 0 peaks, so not even the large fre-quency spacing could be calculated. The only valuesknown are the temperature of 6343 ± 85K and metal-licity of –0.04±0.1 which were given at the start of theproject. As the temperature is higher than that of theSun, we can expect this star to have a slightly greatermass and radius.

GSIC 11

The power spectrum from GSIC 11 also had a verylow signal to noise ratio, and despite using a lot offiltering, no values could be obtained. Therefore, themass, radius and age of the star could not be calcu-lated. The only known values for this star are a tem-perature of 5046±74K and a metallicity of –0.55±0.07.The temperature here is significantly lower than thatof the Sun, suggesting a lower mass and radius.

GSIC 28

The power spectrum and light curve for this starwere found to be identical to that of GSIC 1, indicatingthat these systems are in fact the same. As GSIC 1has already been discussed above, there is no need forfurther analysis of GSIC 28.

GSIC 37

The power spectrum and light curve for this starwere found to be identical to that of GSIC 11, indicat-ing another duplicate system. As GSIC 11 has alreadybeen discussed above, there is no need for further anal-ysis of GSIC 37.

7.2.3 Results from AsteroseismicDiagrams (GM)

The stars with computational values for both Δνand δν were plotted on asteroseismic diagrams, the re-sults of which are shown on the following pages alongwith Table 7.98, which shows the complete set of dataextracted from all of the diagrams. Interestingly, GSIC22 in Figure 7.2.4 seems to be a pre-MS star as it isabove the ZAMS hydrogen mass fraction. We knowthat all of the stars investigated in this project are MSstars, which suggests that the computational value foreither Δν or δν is incorrect for this star. Indeed, byusing the value of δν calculated manually for this starit should in fact have a mass of around 1.4 M� and ahydrogen mass fraction of around 0.6, which is muchmore agreeable with expectations. The placement ofthe star with computational values means that no stel-lar properties have been extracted for this star and assuch are unavailable in Table 7.98.

Figure 7.2.8 – A more detailed asteroseismic diagram forGSIC 3. The red marker is the star position with accom-panying grey error bars.

In Figure 7.2.6 GSIC 20 and GSIC 24 are the onlytwo stars with masses outside of the scope of the dia-grams, however they are close enough that their massescan be estimated at 1.53± 0.10M� and 1.70± 0.20M�respectively. As determining the age and other stellarproperties depended on matching the star to a model,and with no model available for these masses, most ofthe stellar properties are also unavailable in Table 7.98for these two stars.

Also, while there is available data to plot GSIC 13and GSIC 36, due to a communication problem therewas no available metallicity data for these stars whenthe asteroseismic diagrams were being constructed andhence the two stars do not appear on them. Unfortu-nately, due to the time constraints of the project, thesehave not been able to have been added to the existingdiagrams and their stellar properties extracted. Thiscould be done as a future investigation.

The asteroseismic diagrams themselves are gener-ally the expected shape - matching those produced byprevious studies. As with Figure 3.2.6, the less massivestars have a much broader range for δν which gradu-ally shrinks up to a mass value of around 1M�, fromwhich point the range stays roughly the same acrossthe hydrogen mass fraction range. More massive starstend to have lower values for Δν and δν, which againagrees with expectations. The hydrogen mass fractionisopleths tend to follow the expected trend, with a fewunexpected deviations such as the ‘kink’ in the ZAMSisopleth seen around 1.3M� in Figure 7.2.4. Apartfrom these minor deviations the asteroseismic diagramsshould provide a relatively accurate tool to calculatestellar properties, presuming Δν and δν are relativelyaccurate.

Figure 7.2.8 shows a more detailed asteroseismicdiagram that was completed for GSIC 3. This allowedfor a more precise value for the mass and hydrogenmass fraction to be extracted for each star, but due totime constraints was only completed for this star.

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Figure 7.2.1 – Asteroseismic diagram of Metallicity Z=0.009 showing GSIC 18

Figure 7.2.2 – Asteroseismic diagram of Metallicity Z=0.011 showing GSIC 29 and 32

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Figure 7.2.3 – Asteroseismic diagram of Metallicity Z=0.014 showing GSIC 3, 15, 16, 23, 26 and 35

Figure 7.2.4 – Asteroseismic diagram of Metallicity Z=0.017 showing GSIC 4, 6, 12, 22 and 33

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Figure 7.2.5 – Asteroseismic diagram of Metallicity Z=0.023 showing GSIC 0, 9, 14, 25 and 31

Figure 7.2.6 – Asteroseismic diagram of Metallicity Z=0.029 showing GSIC 20 and 24

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Figure 7.2.7 – Asteroseismic diagram of Metallicity Z=0.035 showing GSIC 28

Table 7.98 – Data extracted for available stars from the asteroseismic diagrams. Stellar hydrogen mass fraction andstellar age are included in the tables in Section 7.2.1 for these stars.

GSIC Z JCD M (M�) σ(M) R (R�) σ(R) L (L�) σ(L) Teff σ(Teff)

0 0.021 0.023 1.06 0.05 1.14 0.11 1.48 0.37 5982 1173 0.013 0.014 0.92 0.05 0.93 0.06 0.88 0.23 5803 1814 0.018 0.017 1.50 0.05 1.97 0.11 6.85 1.03 6688 816 0.017 0.017 1.29 0.05 1.37 0.08 3.25 0.60 6667 1169 0.023 0.023 1.38 0.05 1.46 0.07 3.83 0.63 6724 12012 0.020 0.017 1.35 0.05 1.62 0.08 4.40 0.70 6604 7714 0.023 0.023 1.43 0.05 1.48 0.06 4.33 0.70 6873 21615 0.015 0.014 1.50 0.05 1.69 0.06 6.95 1.11 7248 17316 0.014 0.014 1.28 0.05 1.39 0.08 3.50 0.65 6721 11518 0.009 0.009 0.80 0.05 0.77 0.10 0.48 0.34 5496 41020 0.028 0.029 1.53 0.10 N/A N/A N/A N/A N/A N/A22 0.019 0.017 N/A N/A N/A N/A N/A N/A N/A N/A23 0.014 0.014 1.50 0.05 1.83 0.09 7.22 1.14 7036 12724 0.031 0.029 1.70 0.20 N/A N/A N/A N/A N/A N/A25 0.023 0.023 0.88 0.05 0.87 0.06 0.59 0.17 5431 20626 0.015 0.014 1.50 0.05 1.67 0.06 6.92 1.10 7269 17728 0.034 0.035 1.50 0.05 1.99 0.17 5.41 1.31 6276 14529 0.012 0.011 1.11 0.05 1.28 0.11 2.42 0.52 6405 9031 0.025 0.023 1.13 0.05 1.30 0.13 2.01 0.41 6047 3332 0.012 0.011 1.50 0.05 1.73 0.05 7.75 1.16 7366 19133 0.019 0.017 1.50 0.05 1.61 0.06 6.23 0.98 7214 18135 0.013 0.014 0.96 0.05 1.09 0.06 1.34 0.25 5963 124

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Figure 7.2.9 – The diagram shows the distribution of lu-minosities of the stars, as extracted from the asteroseismicdiagram data.

Stellar Luminosity

A stellar property that could not be found with theother methods was the luminosity of each of the stars.As this was given in the parameters for each model, bymatching each star with its closest model this could becalculated from the asteroseismic diagrams.

Figure 7.2.9 shows the distribution of luminositiesfor available stars. The diagram shows that the major-ity of the stars are much more luminous than the Sun,with only GSIC 3, 18 and 25 having a slightly lowerluminosity. Most of the stars are between three andeight times as the sun, which would correspond withthem being relatively solar-like, MS stars as expected.

With a non-normalised light curve for each star,or from data about the apparent luminosity of each,it would perhaps be possible to calculate the distanceeach is from our solar system using a 1/r2 relationship.Without this, however, the luminosity data serves onlyas a comparison for luminosity of the stars against theSun.

7.2.4 Comparison Between ScalingRelation and AsteroseismicDiagram Values (GM)

Stellar Mass

Figure 7.2.10 – Comparison of stellar mass values computedfrom scaling relations and extracted from asteroseismic di-agrams

As can be seen in Figure 7.2.10, the masses ex-tracted from the asteroseismic diagrams agree with thevast majority of the mass values calculated from scal-ing relations, with the exception of GSIC 26 and GSIC32. The values extracted from the diagrams also havea much smaller constraint to them than the scaling re-lation values. This is due to a relatively large error onΔν or δν not usually causing a significant change inmass and the relatively small errors on both calculatedvalues. The only stars where the masses could not beconstrained as accurately were GSIC 20 and GSIC 24,as both of these stars were slightly beyond the scale ofthe graph with masses greater than 1.5M�. However,as they were still relatively close to the available masslines, an estimate of their masses could be made byextrapolating the data.

There are a few reasons why the two methods couldhave disagreed on GSIC 26 and GSIC 32. One reasoncould be that the values derived from the scaling re-lations are incorrect, however this would suggest thateither νmax or Δν for these values have been measuredincorrectly, that these two stars were different typesof stars than which the scaling relations work for orthat the values given to us for the effective tempera-ture were incorrect for these two stars. The frequency-power spectra for these two stars had a fairly low SNRand were of a high enough quality for δν to be measuredfor them, which suggests that the measured value forΔν would have been to a higher accuracy than for otherstars and that νmax would be fairly reliable also. It isalso unlikely that these two stars alone were different

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enough for the scaling relations to be inaccurate andthe values for the effective temperature were calculatedindependent of this project so could be presumed to becorrect. This, along with the large discrepancies be-tween the model effective temperatures and given val-ues for these stars, discussed in section 7.2.4, suggeststhat either there are inaccuracies in the model used tocompute the frequency data for these stars or that theydo not fit the model that they have been matched to.Needless to say, due to the majority of the mass dataextracted from the asteroseismic diagrams agreeing tothat from the scaling relation, asteroseismic diagramscan be seen as a fairly reliable source for this stellarproperty. It would have been interesting to investigatehow the planetary properties calculated differed whenusing the asteroseismic diagram results, however dueto the time it took to calculate Δν, δν and νmax; plotthe diagrams; plot the stars and extract the data, itwas not possible to do this within the time constraintof the project.

Stellar Radius

Figure 7.2.11 – Comparison of stellar radii values computedfrom scaling relations and extracted from asteroseismic di-agrams

Figure 7.2.11 suggests that there is a very high de-gree of agreement between the stellar radii calculatedwith the two methods. Indeed, all but one of the starsin the two sets of data lie within each others constraintsand the one that does not, GSIC 23, only marginallydisagrees. The reasons for such high agreement aretwofold. Firstly, the scaling relation for radius has amuch lower power-dependency for νmax, Δν and theeffective temperature. νmax/νmax,� has a linear depen-dency as opposed to a cubic dependency, Δν is only tothe power of –2 as opposed to –4 and Teff is to thepower of 1/2 compared to 3/2. This makes it much lesssensitive to any errors in these values. Secondly, theradii values from the stellar models are also much less

dependent on less understood stellar physics, namelysurface effects, convection mixing and other free pa-rameters where published literature disagrees on theexact values, which often affects other intrinsic proper-ties of the star. This makes the radius calculation fromnearly all stellar models very reliable and they tend toagree between different stellar evolution codes, mak-ing the asteroseismic diagrams a valid alternative tocompute this property to the scaling relations. As thevalues were so similar, this probably would not havemade any significant differences to the planetary prop-erties discussed earlier in this chapter.

Effective Temperature

Figure 7.2.12 – Comparison of effective stellar temperaturevalues from given values and extracted from asteroseismicdiagrams

Unfortunately, as Figure 7.2.12 shows, there doesnot seem to be any degree of agreement between theeffective temperature extracted from data from the as-teroseismic diagrams and the values given to use to usein the project. The main reason for this is probablythe inaccuracy of stellar evolution models in modellingstellar surface physics. Due to the temperature gra-dient present in a star’s core, if the modelled surfacetemperature is wrong then this will affect the overall ef-fective temperature. From Figure 7.2.12, it seems thatthe majority of the stars have a much higher modelledeffective temperature than the known data, with onlyGSIC 3, 18, 25 and 28 agreeing to any degree withthe given values and the other temperatures being sig-nificantly higher than the actual values. Discrepancywith the given values also suggests that the luminositydata extracted from the asteroseismic diagrams mayalso disagree with values calculated using other meth-ods, but with no other data to compare the luminosityvalues to it is only possible to speculate about thisbased upon their relationship to one another.

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It is obvious that other methods are significant bet-ter and more accurate at calculating effective temper-ature of stars and until the surface physics of stars isbetter understood and modelled they are not a feasibleway of determining this property.

7.3 General Trends

To comply with the overall question asked in thetitle of the project, a group of charts have been com-piled to display the relationship between the charac-teristics of an extrasolar system and the properties ofits host star. This will include: the relationship be-tween the mass of the star and the age of the system,the relationship between the mass of the star and itsluminosity, the relationship between the mass of a starand its radius, the relationship between the metallicityof a star and the composition of its orbiting planets,the relationship between the radius of a star and theradius of its orbiting planets, a simple histogram of thenumber of planets in our data set orbiting with a par-ticular period, and finally the relationship between themass and radius of the discovered exoplanets.

7.3.1 Relationship Between StellarMass and System Age (GM)

Figure 7.3.1 – This graph shows the masses and ages of thestars, as extracted from the asteroseismic diagrams. Theblue points are the extracted values, with red error barsaccompanying them. The black line is a second-order poly-nomial that has been fitted to the points.

The age of the star has a very large dependenceon its mass and Figure 7.3.1 shows this relationshipfor the stars plotted on the asteroseismic diagrams.The second-order polynomial that has been fitted tothese stars suggests that as the stars increase in mass,they tend to decrease in age. This is in agreementwith what would be expected, especially for MS stars:smaller stars burn through their hydrogen content ata significantly slower rate than more massive stars soa less massive star has a much greater probability ofbeing older than a more massive one. More massive

stars also leave the MS at a much younger age, so find-ing a more massive star older than around two billionyears old that was not post-MS would be very unlikely.Our data correlates with this, with stars greater than1.2M� having an age between a few hundred thousandand two billion years.

For a greater range in stellar mass, you would ex-pect the relationship between the two to move awayfrom a second-order polynomial, with the age of lessmassive stars curving to reach a cut-off limit (the ageof the universe at an absolute maximum) and the age ofmore massive stars curving to either become more lin-ear or at least asymptotic to zero: there is a maximumrate at which a star could burn through its hydrogenfuel and also the consideration of a stellar mass limitto impose. However, for the small range of stars withthis data the trend line fitted to the data is acceptable.

7.3.2 Relationship Between StellarMass and Luminosity (GM)

For stars with a mass between 0.43M� and 2.0M�we would expect the luminosity to vary such that [142]

L

L�=( M

M�

)4(7.1)

Therefore, if you plotted the mass against the lu-minosity of our stars you would expect a line of bestfit to follow a fourth power-law.

Figure 7.3.2 – The diagram shows the relationship betweenthe mass and luminosity, as extracted from the asteroseis-mic diagram data. The blue circles are the positions of thestars while the red line is a line of best fit set to determineand follow a power-law relationship for the data.

Figure 7.3.2 shows such a diagram with the dataplotted and a power-law line of best fit added. Theequation to the top left of the figure shows that the

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line of best fit has the equation y = 1.235x4.099, sug-gesting that our data agrees with a fourth power-law toa high degree. This suggests that the luminosity dataextracted from the asteroseismic diagram data shouldbe reliable. However, without another method of cal-culating these values it is impossible to test the validityof this.

7.3.3 Relationship Between StellarMass and Radius (MH)

Main-sequence stars are observed to adhere to amass-radius power-law relation [143]:

R ∝ Mα (7.2)

where α = 0.8 for stars lower than solar mass andα = 0.6 for stars of mass greater than the Sun. Thisdivision is approximately the point below which a starcan have a convective envelope, the presence of whichallows energy to more easily escape the star, whichreduces radiation pressure and causes a slightly lowerradius than for a star without a convective zone, thusthe steeper relationship.

Unfortunately, as can be seen in Figure 7.3.3, ourstars do not exhibit this relationship, possibly due tothe small sample size and limited range of the vari-ables. The small number of stars all with mass justless than 1M� did not allow us to generate a fit forthat region, and a fit on the region above 1M� gavea power of 2, which is not at all close to the expectedvalue of 0.6. Despite this, it is clear that greater massstars have a larger radius, which agrees, globally if notquantitatively, with Equation (7.2).

Figure 7.3.3 – A plot of the radius against the mass of allthe stars for which we have those values.

7.3.4 Relationship Between StellarMetallicity and PlanetComposition (MH)

Figure 7.3.4 – This diagram shows the stellar metallicitiesassociated with each type of planet composition of the plan-ets detected from our data. The red circles are rocky plan-ets, the black circles are gaseous planets, and the purplecircles are those which could be either a large rocky planetor a small gas planet.

Figure 7.3.4 allows us to visualise what type ofplanet would be expected from a star with a partic-ular metallicity. Clearly, there is not a strong relation;this is perhaps because the categories of compositionare qualitative rather than quantitative. A useful mea-sure of composition might have been the mean densityof the planet, where solid rocky planets would have ahigher density than diffuse gas giants. Unfortunately,the large mass ranges for the planets, caused by nothaving a more direct method of measuring the masssuch as the Doppler wobble method, meant that get-ting a reasonable value for the mean density was im-possible.

It can, however, be noted that the average metal-licity of the stars with gas giants is higher than theaverage metallicity of the stars with rocky planets.This agrees with the findings of Fischer and Valenti(2005), who identified a subset of 850 Doppler method-detected planets. In this subset they found that theformation probability of gas giant planets rose withthe square of the number of metal atoms [144]. Theyalso determined a positive correlation between metal-licity and both the number of planets in a system andthe total mass of the planets. The three of our sys-tems that have multiple planets (GSIC 6, 7, and 8)have metallicity ranging from Z = 0.015 to 0.022, inthe middle of the range. With such a spread it is pos-sible that our sample size simply is not big enough toobserve the same trend.

7.3.5 Relationship Between StellarRadius and Planetary Radius(JG)

Due to the information given about planet forma-tion in Section 4.1, the team decided to look for a trendbetween the radius of a star and the radius of the plan-ets in its system. Section 4.1 states that stars and plan-ets each form from the same Giant Molecular Clouds

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(GMCs). A fair assumption to make is that a GMCwith the ability to form a larger protostar will alsoleave a greater amount of matter in the protoplanetarydisk, which in turn allows larger planets to form.

Figure 7.3.5 – Scatter with straight line displaying the trendbetween the radius of a star and the radius of its orbitingplanets. the uncertainty on the radius of the star has beenomitted, as the error on planetary radius is dependent onthis value and this accounts for both

In Figure 7.3.5, all planets with a known radiushave been used for the plot, giving a sample size of 28planets. A straight line was fitted to the sample andthis shows a clear positive trend. If the relationshipwere to be quantified, the gradient value of 0.1875 ±0.0585 would be used, although there is no basis for theaccuracy of this number due to the small sample sizeused to calculate it. This number is also taking intoaccount the clear outlier in the sample to the far rightof Figure 7.3.5, the gas giant in orbit around GSIC 39which is one of the proposed red giants. As most starson the MS will become larger as they evolve off theMS, the results of red giants do not relate to a trend.

7.3.6 Frequency of Orbital Period forDiscovered Exoplanets (PS)

It was decided to record the frequency of planetsdetected orbiting at various period ranges, to attemptto detect any particularly common values. For this,periods were divided into bins of width 2 days, andthe number of planets with a period in each range wascounted. The result is shown in Figure 7.3.6, and aclear peak can be seen around the 5 day mark. Thispeak can be explained by reference to the three-daypileup (4.6.5), and it is to be expected, given a highersample size, that the centroid of the peak can be seenat a slightly lower point.

Figure 7.3.6 – A histogram showing the periods of the de-tected planets, in bin sizes of 2 days. A peak around the3-6 day mark is clearly visible.

It is important to consider here that there is a heavyselection bias towards low period planets in transit de-tections. As the probability of observing a transit is in-versely proportional to the semi-major axis (see Equa-tion (4.34)), it can be can seen that by applying Ke-pler’s third law the probability of observing a transitwill scale as P–2/3 , and thus many higher period plan-ets will not be observed. On the other hand, as planetsin a system tend to orbit along a similar ‘ecliptic’ plane,if one planet in a system is observed to transit, thereis a higher probability that other planets in the samesystem will also transit the star and be detected.

This model assumes that the planets in our sampleare a representative sample of extrasolar planets. Thismay well not be the case, as our study did not detectany planets smaller than about 1.5R⊕, and so if thereare characteristic periods for small planets they willnot have been observed here.

The peak at around 5 days is significant, and clearlydemonstrates a detection of the three-day pileup in ourresults. The second peak at around 42 days may haveoccurred by chance, but it is possible that there is somereason for an increased number of detections at this dis-tance. It may be, for example, that this is the distanceat which tidal forces become too weak to significantlyaffect the orbit of a planet. A larger sample size wouldclarify whether this is the case.

7.3.7 The Mass-Radius Relation forDiscovered Exoplanets (PS)

It was decided to plot the masses and radii of thevarious planets detected by the study, in order to at-tempt to produce a means of deducing the composi-tion, mass or radius of a given planet when one or twoof these values were known. The resultant graph isshown in Figure 7.3.7, and quite clearly shows group-ing of planets of similar compositions.

It must, however, be noted that as the mass andcomposition of these planets were initially suggestedby considering composition and radius, the groupings

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shown in the graph are only useful for characteris-ing other planets provided our initial method is valid.However, the close grouping of planets described assimilar supports our choice of this method, and givesweight to the distinctions that have been used in thisstudy between planets of different composition.

7.3.8 Evidence for TidalCircularisation (PS)

It was expected that close orbiting planets wouldfollow orbits of very low eccentricity, due to the largemagnitude of the tidal forces acting upon them at thisdistance from their host stars. This may also result intidal locking.

Although we were only able to obtain a lower boundfor eccentricity due to the nature of the transit method,it is still possible to observe some influence on the ec-centricity of a planet’s orbit caused by its period.

Figure 7.3.8 – A graph displaying the minimum eccentricityand period (logarithmic scale) of the discovered planets.Note the lack of high eccentricity, low period data points.The uncertainty on period measurements is negligible, andhas thus been omitted.

This result is severely limited due to the high uncer-tainty on the calculation of minimum eccentricity, andit must be remembered that it is theoretically possiblefor all of these planets to orbit with a greater eccen-tricity than we have calculated. Nonetheless, a notice-able lack of close orbiting planets in our study haveeccentricities which are constrained above around 0.2,whereas the minimum eccentricity seems to be moreevenly distributed at higher orbital periods. From thiswe can infer that tidal effects are indeed being observedin our results.

7.4 Comparison of Stellar Limb-Darkening Coefficients(CL & OH)

As was described in Section 6.2, stellar limb darken-ing was included in the computational obtention of the

results. For the transit fitting code, discussed in Sec-tion 6.2.4, a quadratic limb-darkening transit modelby Mandel & Agol (2002) [134] was fit to the data byan emcee program. As such, best fit values for thequadratic model LDCs were output by the program,namely LD1 and LD2, as can be seen listed in Table7.99. As is described in Section 6.2.5, an independentnumerical model was made on Python to compare tothe analytical model, using Equation (6.1) to find thequadratic model LDCs as a function of effective tem-perature, namely a and b. As LD1 and LD2 representthe coefficients a and b respectively according to thesecond order limb-darkening model set forth in Equa-tion (4.59) one would expect, in an ideal case, thatthese values are equal for both models.

It is easy to see from Table 7.99, that this is notthe case, and that the values of LD1 & a and LD2 & bdiffer quite significantly for a large number of planets.Where the value for δ(a) or δ(b) is seen to be positive(shown in red), it is indicative of the difference betweenthe two corresponding values being within the error onthe fit value LD1 or LD2, which indicates a (relatively)accurate fitted value. However, for many planets this isnot the case. Despite the transit fitting code providingvery accurate results for the other output parameters,it is reasonable to assume that the best fits for theLDC values are (in some cases severely) inaccurate, ifit is assumed that the LDCs a and b resulting fromEquation (6.1) are the most accurate values available.This would indeed be the most reasonable assumptionto make, as the values of a and b (and u for the linearlimb-darkening model) were taken from literature andfit very closely to an equation (see Section 6.2.5) [138].Although LDCs are a stellar property, it is seen fromTable 7.99 that there are different values of LD1 andLD2 for GSIC 6b & c, 7b & c and 8b & c, as thetransits to which the model was fit differed for eachplanet. This gives further confirmation that the LDCsa and b are the more accurate values.

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Figure 7.3.7 – A log-log plot of the planetary mass and radius of the discovered planets, which are categorised bycomposition. Clear groups of similar planets can be seen.

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Table 7.99 – LDCs LD1 and LD2 as found by the transit fitting code, and LDCs a and b as found by Equation (6.1) forthe quadratic limb-darkening model. In red are marked those values for which a or b fall within the error on LD1 andLD2 respectively. δ is calculated by taking the difference between the corresponding LDCs and subtracting it from theerror on the analytical LDC.

GSIC LD1 σ(LD1) LD2 σ(LD2) a b δ(a) δ(b)

1b 0.281946 0.057581 0.953926 0.029235 0.307117 0.310235 0.032411 -0.6144563b 0.747188 0.039034 0.429309 0.041779 0.412344 0.249885 -0.295811 -0.1376465b 0.652608 0.131022 0.758749 0.163690 0.335458 0.295356 -0.186129 -0.2997036b 0.388625 0.036454 0.833611 0.049687 0.317087 0.305136 -0.035083 -0.4787876c 0.988516 0.064789 0.796393 0.038067 0.317087 0.305136 -0.606639 -0.4531907b 0.882470 0.087867 0.104369 0.126693 0.380669 0.269413 -0.413934 -0.0383517c 0.261469 0.074369 0.909432 0.088988 0.380669 0.269413 -0.044831 -0.5510318b 0.451898 0.126590 0.849932 0.141064 0.362937 0.279881 0.037629 -0.4289888c 0.618646 0.029096 0.967145 0.039145 0.362937 0.279881 -0.226613 -0.6481199b 0.412785 0.021592 0.978656 0.028410 0.330556 0.298013 -0.060637 -0.65223310b 0.329766 0.063529 0.881916 0.147350 0.376253 0.272053 0.017041 -0.46251312b 0.301472 0.062617 0.867344 0.116023 0.384113 0.267339 -0.020024 -0.48398213b 0.998064 0.006828 0.947266 0.007225 0.372072 0.274532 -0.619164 -0.66550814b 0.379048 0.136541 0.463178 0.249297 0.294045 0.316677 0.051538 0.10279616b 0.194638 0.064816 0.721760 0.112818 0.321116 0.303034 -0.061661 -0.30590817b 0.096899 0.096755 0.715532 0.133208 0.406729 0.253418 -0.213075 -0.32890618b 0.604716 0.006313 0.998855 0.001675 0.462686 0.216986 -0.135717 -0.78019419b 0.866771 0.349176 0.858462 0.281290 0.334045 0.296125 -0.183551 -0.28104720b 0.012750 0.012153 0.035237 0.046465 0.364062 0.279228 -0.339159 -0.19752521b 0.994807 0.012118 0.990898 0.019791 0.374406 0.273151 -0.608283 -0.69795522b 0.962142 0.022306 0.570369 0.039196 0.310167 0.308691 -0.629669 -0.22248223b 0.709106 0.019197 0.994362 0.010838 0.319285 0.303992 -0.370624 -0.67953124b 0.433532 0.149873 0.114893 0.097136 0.329866 0.298384 0.046207 -0.08635528b 0.995511 0.058440 0.083856 0.105461 0.307117 0.310235 -0.629955 -0.12091830b 0.493884 0.069008 0.948552 0.065305 0.358176 0.282630 -0.066700 -0.60061731b 0.526037 0.022445 0.673590 0.030554 0.389707 0.263945 -0.113886 -0.37909132b 0.992323 0.022641 0.974063 0.051706 0.369594 0.275993 -0.600088 -0.64636534b 0.404859 0.037072 0.808262 0.049669 0.391831 0.262647 0.024045 -0.49594635b 0.455911 0.059191 0.119908 0.086966 0.416515 0.247240 0.019795 -0.04036739b 0.930814 0.063317 0.658300 0.077747 0.587960 0.128572 -0.279537 -0.45198140b 0.653943 0.019171 0.019686 0.022956 0.554940 0.152412 -0.079832 -0.109770

Figure 7.4.1 – Comparison between Mandel & Agol’squadratic limb-darkening model [134] and the numeri-cal Python model using both linear and quadratic limb-darkening models, for GSIC 22b.

Following this, a comparison can be made betweenthe analytical limb-darkened model using LD1 & LD2and the numerical models using u, a and b, for caseswhere the LDCs are both similar and different. A com-parison of a case where the LDCs are significantly dif-ferent can be seen in Figure 7.4.1 for GSIC 22b. Itis very clear that the three models do not resembleone another very closely. Most notable are the ‘spikes’at the start and end of the transit for the analyticalmodel. These are most likely the result of the quadraticLDC dominating the equation in the calculation of thetransit, due to the values being inaccurate. This backsthe assumption that the LDCs received from the transitfitting code are indeed less accurate than those foundthrough Equation (6.1).

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Figure 7.4.2 – Comparison between Mandel & Agol’squadratic limb-darkening model [134] and the numeri-cal Python model using both linear and quadratic limb-darkening models, for GSIC 14b.

A comparison of the different models for GSIC 14bcan be seen in Figure 7.4.2, the only fit for which thevalues of a and b look to both be within the errors onthe fitted values. As can be seen, the ‘spike’ featuresare no longer present. Looking at Figures 7.4.1 and7.4.2 it is surprising how the individual relations be-tween the three separate models differ, especially con-sidering that all three use the same input parametersfor values such as the period and the semi-major axisof the orbit (namely, the final results output by thelimb darkened transit fitting code). Most striking isthe significant difference in depth between the analyti-cal model and the numerical models. As the analyticalmodel has been designed (and observed) to fit the tran-sit especially, one would assume that the depth wouldbe especially well constrained. On the other hand, asit works with an array of pixels, the numerical modeltakes its value of depth directly from a calculation ofthe area of the planetary and stellar disks. This dis-crepancy could be due to the numerical model beinga rough approximation, as the stellar array of pixelsis constrained to finite dimensions, and does not havea continuous range of intensities but rather a discretedistribution. However, this is unlikely to cause a differ-ence that significant, especially for Figure 7.4.2, whichused an array of more than 1000 by 1000 boxes tomodel the star.

It is most likely therefore that this difference indepth between models is attributable to the inaccu-rate LDCs. Note how the difference in depth in Figure7.4.1, for which the LDC values are expected to be veryinaccurate, is extremely large. Compare this to Figure7.4.2, where the LDC values are seen to be more accu-rate, and it can be seen that the difference between theanalytical and numerical model depths decreases signif-icantly. In fact, the minimum intensity at the centreof the transit for the analytical model is now found

between the first and second order models. While itcould be said that the analytical model in Figure 7.4.2appears to match the first order equation, it is worthnoting that the shape of the curve matches the shape ofthe second order model, for the majority of the transit.

Figure 7.4.3 – Comparison between Mandel & Agol’squadratic limb-darkening model [134] and the numeri-cal Python model using both linear and quadratic limb-darkening models, for GSIC 14b. Both quadratic modelsuse the same LDCs a & b.

To further confirm the suspicion that the differencein depth is due to inaccurate LDCs, Figure 7.4.2 wasrecalculated and re-plot using the LDCs a and b forboth the analytical model and the quadratic limb dark-ened model, as can be seen in Figure 7.4.3. Here it canclearly be seen that the second order model has linedup with the analytical model, most notably throughoutthe start and end of the transit. However, there is asmall discrepancy in depth between the two models atthe centre of the transit, which appears to remain rel-atively consistent throughout both models, with bothcurves resembling each other closely. This, however, isto be expected to a degree, as both models attemptto recreate an ideal representation of reality, which isdifficult to obtain, through very different means. Apossible reason for this difference, as explained before,would be the finite nature of the array used to visualisethe stellar intensity. Thus it can be seen that the valuesof LD1 and LD2 affect the shape of Mandel & Agol’smodel very significantly, and that these values of LD1and LD2 are more often than not given accurately bythe transit fitting code, as is seen in Table 7.99.

It can clearly be seen from this analysis that theLDCs for the quadratic limb-darkening equation cansignificantly vary the depth of Mandel & Agol’s modelused in the transit fitting code. It was previously as-sumed that, to fit the data, the transit fitting code var-ied the planetary radius parameter to get an accuratedepth for the fit. However, now knowing that the val-ues of LD1 and LD2 can affect the depth of the transit,it is highly possible that emcee varied these values to

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find a fit for the depth, thus possibly underestimatingthe best fit planetary radius output by the code. Thisnaturally raises concerns about the nature of the errorson the final values obtained for the planetary radii, andwhether using LDCs obtained from Equation (6.1) asinput parameters for emcee would yield more accu-rate results than allowing the LDCs to ‘roam free’ ashas been the case.

An in depth study by Espinoza & Jordan (2015)[145] further explores the manner in which allowingthe LDCs to roam free in similar transit fitting meth-ods affects the planetary radius. They conclude that,when using the quadratic limb-darkening model to fitto a light curve, it is better to allow the LDCs to varyto find a fit, as they argue that our understanding ofthe limb-darkening process is not sufficient to use fixedcoefficients without further increasing the bias on thedata. This preference for fitted values of LD1 and LD2is shared by Csizmadia et al. (2012) [146], who alsoconclude that the inconsistencies in our understandingof LDCs do not allow them to be fixed in transit fit-ting methods. Furthermore, it was found that doing sodid not allow them to obtain an accuracy in the plan-etary radius of better than 1-10%. They also mentionthat varying the LDCs as has been done in the tran-sit fitting code should provide an accuracy of less than1%, in an ideal case. However, the data used naturallyhas many imperfections, which most likely encourageemcee to vary the LDCs away from their best fit val-ues. It would be very possible that a model whichaccounted for these imperfections would allow a betterfit and more accurate values for LDCs, but this wouldrequire a more in-depth understanding of the processof stellar limb darkening, and is outside the scope ofthis project.

As an aside, it is interesting to note the differ-ences between the first and second order light curvesproduced by the numerical model. The differences indepth here most likely result from the manner in whichthe total intensity was calculated. In Figure 7.4.2 it canbe seen that the first order model appears to be tend-ing more towards a closer fit to the analytical model,as was mentioned above. This is slightly unexpected,as the temperature of GSIC 14 is 6463K, significantlyoutside the temperature range of the Sun. However,using the same LDCs, as can be seen in Figure 7.4.3,the first order model is left behind whilst the analyti-cal model lines up with the second order curve. A truegauge of the accuracy of the coefficients for first orderlimb-darkening model set forth in this report would beto compare it to the Sun, which is also outside thescope of this project.

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Chapter 8

Conclusions

8.1 General Conclusions (JG)

In this report we have introduced the fields of as-teroseismology and exoplanet detection, summarisingtheir inception, history and current state. We thenpresented an extensive explanation of the theoreticalaspects of each individual field. This included the datathat are required to first observe and then quantifythe expected results: stellar oscillations described byfrequency-power spectra in the case of asteroseismol-ogy, and planetary transits described by luminosity-time graphs in the case of exoplanet detection. Thesesections also introduced the methods that would beused to interpret and analyse the data provided.

Next, we detailed those methods, specifying pro-grams and models that were used, and sometimes cre-ated, in this project. The uses of these programs havebeen explained, along with how manual methods ofdata analysis have been used in conjunction with thecomputational methods to gain correct results. Wehave also discussed the accuracy of the results obtainedby these methods and identified where they have fallenshort in obtaining useful information.

Finally, we provided suitable presentation and dis-cussion of all results that we were able to obtainthrough the variety of methods used. These resultsincluded properties of the stars and detectable plan-ets, derived through a combination of luminosity-timedata and necessary stellar properties. We extractedand explained trends in the overall results, comparingthese where possible to what should be expected fromcurrent knowledge.

8.2 Stellar Properties (MG)

Investigating the properties of the stars as well asthe methods required has been an important endeavourin this study. A star encompasses nearly all the mat-ter in its system so it is clear that examining the starsshould provide great insight into the characteristics ofextrasolar systems which, as we have found, is certainlythe case. Having the ability to constrain the estimatesof stellar parameters such as mass, radius and age notonly improves our knowledge of the stars themselvesbut is essential in order to identify any bodies foundorbiting using current techniques, notably the transitmethod used in this project and discussed in Section4.3.1. Therefore, obtaining accurate results from as-

teroseismology was essential for the progression of therest of the project. By extension, the features of thesystems as a whole have been evaluated.

We began our investigation with the frequency-power spectra (FPS) of 41 stars, however we soon con-cluded that 2 out of the 41 were duplicates – GSIC 28of GSIC 1, and GSIC 37 of GSIC 11. We also had thevalues of effective temperature and metallicity for eachof the stars, determined independently. From the FPSwe worked to extract the key stellar oscillation valuesof Δν0, νmax and, where possible, δν0,2 which werethen used to derive information on the stars’ mass andradius, first and foremost, and age if δν0,2 was knowntoo. Both of these stages were tackled with differentmethods detailed in Chapter 5.

Once the FPS had been filtered, two different proce-dures were used to extract the values discussed above.The first was to manually analyse the FPS. This wasdone to get values quickly, which was a reasonable ap-proach for the small sample size, both for comparisonwith the coded, computational FPS analysis – whichtook much longer to develop and refine – and to pro-vide early estimates of stellar properties for use by thePlanet-Finding Sub-Group. It was difficult attemptingto find νmax manually as a function needed to be fitted.Once the FPS could be analysed much more accurately(using manually-determined values as a guide), valuesfor Δν0 and νmax were found for all stars except forGSIC 2 and GSIC 11 (and, by extension, GSIC 37).We managed to find values of δν0,2 for about half ofall stars, for which we were then able to create aster-oseismic HR diagrams (AHRDs). While it had beenhelpful in this investigation to manually decipher theFPS in the early stages, it is worth noting that weused a small sample size with only 41 FPS. With thevast collection of data by satellites such as Kepler andCoRoT many similar studies to our own may be taskedwith analysing the FPS of thousands of stars or more,so being able to develop and run efficient and accurateprograms or strings of code is paramount to keep torealistic time frames.

With the oscillation values obtained, we again usedtwo different approaches with the aim of constrainingthe parameters of each star. These values were firstused to calculate stellar radius and mass from the scal-ing relations in Equations 3.22 and 3.23, respectively.As discussed in Section 3.2.4, these equations are, forthe most part, quick and easy approximations with rea-

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sonable associated uncertainties. Using stars modelledby the MESA evolution code we were also able to findvalues for stellar parameters by building δν0,2 againstΔν0 AHRDs of differing metallicity. As is apparent,these diagrams required the values of δν0,2 for eachstar to be plotted, and as this was possible for onlyhalf of the stars the improved stellar values could notbe evaluated for all the stars, unlike the scaling relationresults which were found for nearly every star.

Comparisons of the mass and radius values ob-tained by both methods (seen in Figures 7.2.10 and7.2.11) show that using AHRDs systematically im-proves the constraints on these parameters and shouldbe used where possible. Using AHRDs also enabled usto estimate the core hydrogen fraction, Xc , and con-sequently the age of the stars – another criterion vitalfor investigating the characteristics of extrasolar sys-tems. The effective temperatures of the stars were alsore-evaluated using values from the AHRDs (see Figure7.2.12). On the whole, the temperature values givenand those calculated did not match and, while thismay indicate underlying uncertainties in the method,there are a considerable number of subtle dependenciesfrom various different values which could cause this re-sult. Unfortunately, due to the time required to buildthe different AHRDs (in Figures 7.2.2 through 7.2.7)the masses and radii used by the Planet-Finding Sub-Group to calculate their results were those evaluatedusing the scaling relations. Ideally we would have usedthe better-constrained, more accurate values from theAHRDs but we were unable to achieve this within theset time constraints.

Overall, we have analysed 39 different stars whichare on the MS and are solar-like, with a few excep-tions. These exceptions are GSIC 38, 39 and 40 whoseradii are considerably greater than their masses fromwhich we infer that they have reached the end of theirtime on the MS and have become subgiants. Thesestars could not be plotted on the AHRDs because δν0,2could not be determined due to mixed modes interfer-ing with the FPS. Ironically, the mixed modes them-selves, though preventing the stars from being agedwith the AHRDs, gave us the evidence to classify thestars as aged because the mixed modes only appearwhen p- and g-modes overlap, which occurs when asolar-like star begins to move off the MS. With thisin mind, their ages could be conjectured when takinginto consideration how the lifetime of a star on the MSchanges with mass. The stars for which δν0,2 couldnot be determined were inhibited primarily by an un-favourably low signal to noise ratio, making it difficultto find the l = 2 peaks in the FPS. Excluding GSIC13, for which we believe the oscillation values have beendetermined incorrectly which we have not been able torectify, the masses of the stars were found to range be-tween 0.91±0.17 M� and 2.32±1.05 M� with the radiiranging between 0.83±0.10 R� and 4.53±0.55 R�. Forthe stars for which the age was ascertainable, the valuesspanned from 0.52±0.13 Gyr to 8.06±2.03 Gyr. Also,when the ages were compared with the given valuesof metallicity there was no emergent correlation. This

may be because the stars in this study are of the samestellar population or perhaps the average metallicityin different parts of our galaxy increases at differentrates. The relationships between stellar mass and sys-tem age, as well as stellar mass and luminosity (Figures7.3.1 and 7.3.2, respectively) using the results from ourdata fit well with the expectations of their respectivetheories. These confirmations provide supportive evi-dence of the fact that our results are, at least for themost part, of reasonable accuracy. Therefore, amongstthe other positive results of this project, the investiga-tion into the host stars of extrasolar systems has beensuccessful.

8.3 Exoplanet DetectionMethods and Properties (LP,FC, OH & PS)

Over the course of the project a series of Pythoncomputer codes were developed which collectively cal-culated the period of an orbiting planet from raw tran-sit light curves. Orbital period values were perhapsthe most important quantity to extract to a high ac-curacy from the light curves, as many other planetaryproperties can be calculated using the period of theplanet. For this reason, a significant fraction of thecode built by the planet-finding coding team was fo-cused on calculating the period. Time is one of thequantities astronomers can measure most precisely, soit was crucial to find the period values to a precisionas high as possible.

The coding in this project used to determine planetproperties culminated in the production of two an-alytical transit model light curves to fit to the rawdata. The first model treated the star as a uniformsource, whilst the second more complex (and accu-rate) model incorporated quadratic stellar limb dark-ening. The model used was set forth by Mandel &Agol (2002) [134]. The best fit of each model withthe phase folded data was acquired using an emceerandom walk method to vary the parameters on whichthe shape of the model light curves were dependent.This was run until the closest match between the dataand the models was obtained, which subsequently out-put precise values for the model parameters. Theseparameters were the orbital period, the planet radiusand semi-major axis (both in units of stellar radius),and exclusively for the limb darkened model: the in-clination and two limb-darkening coefficients. Initialestimates for these parameters, i.e. the starting valuesfor the random walks, were mostly obtained by mea-suring transits in the data manually. However, severalstars with particularly noisy light curves were moredifficult to analyse, thus requiring a different approachdue to the first estimates needing to be relatively closeto the actual values in order for emcee to successfullyconverge. A transit detection code was used to deter-mine the period values for these stars, which acquiredhigher accuracies, enabling the transit model to fit veryclosely to the phase folded light curve data.

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The accuracy of the first estimates for the periods,as well as the period values output by the code, weretested by using them in a separate phase folding code,and observing whether a clear phase fold was produced.This allowed users to determine whether the manu-ally calculated values were accurate enough to enterinto the transit modelling code, or if they needed im-provement via the transit detection code. This processdemonstrates how the different codes worked togetherenabling us to obtain the most accurate values possiblefrom the raw light curves.

To test the accuracy of the model used and anyeffects it may have on the final results, an indepen-dent numerical model was developed using a Pythoncode. This reproduced a light curve using first andsecond order limb-darkening equations and was cus-tomisable to many properties of star-planet systems.A comparison of this model to the analytical modelby Mandel & Agol revealed that the depth of the an-alytical model was largely influenced by the value ofthe two limb-darkening coefficients, possibly underes-timating the best fit planetary radius output by thetransit fitting program. Various studies have observedthe same problem when using similar methods, butconcluded that allowing the coefficients to ‘roam free’during the fitting process yields a more accurate resultthan if they were fixed throughout [145] [146].

It can be concluded that the analytical modelused does not fully account for some perturbations ofthe Kepler data, with inaccurate values for quadraticlimb-darkening coefficients being produced as a result.Limb-darkening models with a better understandingof the underlying process must thus be used to get ac-curate fits for planetary values. Given more time, itwould have been useful to study the effects that thisimperfection in the model would have on the accuracyof the planetary radius more closely, and the manner inwhich other parameters influence the magnitude of thisuncertainty, such as the effective temperature [147].

From computational and manual analysis of Keplertransits, results were extracted regarding the proper-ties of thirty detected planets. A picture of the var-ious characteristics of the planets was built. Periodvalues were found by analysing the 39 Kepler lightcurves computationally. The majority of period val-ues were found to be less than 20 days with only threeplanets having periods greater than 100 days. Of thelow-period planets, six displayed period values lowerthan 10 days, and evidence could be seen indicating athree-day pileup. From the Kepler data, four planetswith hot Jupiter-like properties were detected, GSIC1b, GSIC 21b, GSIC 39b and GSIC 40b. The range ofperiod values extended from 2.204735299±0.000000139days to 179.4304524±0.00213173 days. The errors ob-tained have a very high precision due to the computa-tional methods used.

Every planet detected in the study had a radiusgreater than that of Earth. This may be becausemethods used in the study were not sensitive enoughto detect smaller transiting planets, due to the pres-ence of noise in the light curves. Planets with small

radii are particularly difficult to detect using the tran-sit method, as often the change in flux due to a transitis of lower magnitude than the noise in the system.Of the detected planets, eighteen were found to haveradii between 1 and 3 R⊕, with only four, the appar-ent hot Jupiters, having a radius greater than 5 R⊕.GSIC 1b was the largest detected planet, with a ra-dius of 1.364±0.162 RJ. The smallest detected planet,GSIC 7c, which slightly surprisingly also possessed thelargest period, had a radius of 1.407±0.180 R⊕. Theerrors in these results are large compared to that ofthe period, due to the uncertainty in the asteroseismicmethods used to find stellar radii.

Kepler’s third law was used to find semi-majoraxis values. All orbits were found to lie in relativeclose proximity to their host star, all with semi-majoraxes less than 0.7 AU. This meant that our datawas useful for determining the properties of planetsclose to their host stars, but unhelpful when consider-ing more distant planets. Values for semi-major axiswere collected in astronomical units and ranged from0.03749±0.00225 AU (GSIC 1b) to 0.6582±0.0433 AU(GSIC 7c). The close proximity of the planets resultedin high values for equilibrium temperature, with thecoolest planet (GSIC 7c once again) having an esti-mated temperature of around 400K and the hottestplanet (GSIC 1b) having an estimated temperature of1900K. Of the planets detected, only two were foundwith the possibility of harbouring liquid water; GSIC7c and GSIC 18b (400-500K). The latter of which is farless likely to be able to harbour life, however since theequilibrium temperature values were estimates, there isstill a slim possibility that compositions of liquid wa-ter may be present. GSIC 7c has a lower temperatureand so there is a higher chance of liquid water beingpresent. All other planets are too hot to be habitable.

Values for transit duration and star-planet separa-tion during transits were used to find the lower limitsof the eccentricities of the orbits of each planet, albeitwith a very large error margin. These values variedfrom 0.012 (GSIC 12b) to 0.824 (GSIC 5b). As onlyminimum eccentricities could be derived, values for ec-centricity closer to one were of more interest. Thedistribution of minimum eccentricities provides someevidence for tidal circularisation of low period orbits.

Analytical models describing the composition ofplanets gave an insight into the relations betweenmasses and radii of planets. Since values of radius hadbeen found to a reasonable degree of accuracy, massvalues could be estimated by considering this and otherresults. From these techniques, planetary characteris-tics could be deduced. The majority of planets exhibitgaseous features with fourteen planets likely to be gasdwarfs. A further four planets displayed features in-dicating a rocky composition with some others beingof indeterminate composition, where the data collectedhas not been sufficient to confidently categorise them.The four largest planets detected are hot Jupiters, inlow period orbits, which are likely to have formed fur-ther from their host stars and then moved inwards bymigration.

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8.4 Relationships Between Characteristics of the ExtrasolarSystems and the Properties of their Host Star (MH)

As a result of the quantity of accurate, and in somecases extremely precise, results that we obtained fromthe data we received, we have been able to examine re-lationships between many of the properties of both thestars and planets, as we hoped to be able to do fromthe beginning of the project.

Starting with the mass-age relationship of the starsfor which we were able to obtain an age, we found astrong negative correlation between the mass and ageof these stars, due to the much shorter main-sequencelifetime of higher mass stars which burn through theirhydrogen more quickly. We also observed a largerspread at the lower mass end which should be expectedbecause stars that can be old can also be young. Theseresults reinforce our confidence in the asteroseismicdiagrams we used to find the ages. Next we exam-ined the relationship between stellar mass and lumi-nosity. Our results corresponded impressively well tothe expected relationship; we obtained a power-law inwhich luminosity increases with mass to the power of4.099, when we would expect a fourth-power depen-dence. This demonstrated the potential accuracy of themethod used to obtain the luminosity from the aster-oseismic diagram data. Together, these relationshipssuggest that our asteroseismic diagram methods wereextremely well carried out.

The relationship between the stellar mass and ra-dius did not appear to agree well quantitatively withwhat could be expected. While the data showed a pos-itive correlation, which is exactly what would be ex-pected, our attempts to fit the data did not result inthe power-law relationship we expected, especially asthe diagram should have two regions either side of so-lar mass, for the lower of which we only had five stars.Similarly, the diagram plotting the stellar metallicityagainst planet composition did not reveal an obviousrelationship. Larger samples of data could have im-proved both of these, while the latter could also havebeen improved by a more quantitative measure of thecomposition of the planets and also by being able todetect more low-mass planets, though this would likely

require a different method of exoplanet-detection to thetransit method.

The stellar radius-planet radius and planet mass-planet radius diagrams show strong relationships.There is a clear trend of increasing planet radius withincreasing stellar radius. We deduced that a bigger starwould have formed from a larger cloud, which wouldhave provided more material to form bigger planets.The latter diagram demonstrates the obvious expecta-tion that more massive planets should have a largerradius, and also indicates that we were very consistentin our deductions of the planets’ composition, with thecaveat that the constraints on the mass were in refer-ence to quantities that included the other two on thediagram. This second relationship in particular givessome confidence to our method for determining planetradius and mass.

Finally, we examined the distribution of the orbitalperiods of the planets and the relation between theeccentricity and period of an orbit. In the former,we found possible evidence of the three-day pileup ofhot Jupiter planets, however it cannot be discountedthat this was due to the selection bias of the tran-sit detection method for planets that orbit close totheir star. Latterly, in the diagram of minimum eccen-tricity against period, we see that the shorter periodstars, those that take fewer than ten days to orbit theirstar, all have very low minimum values for eccentric-ity. While a maximum eccentricity would clearly havebeen useful to show that short-period stars have loweccentricities, by comparing to the spread in minimumeccentricity of stars with longer periods, we can sug-gest - though not confirm - that planets orbiting closeto their star have low eccentricity orbits.

In summary, our analysis of the data and the re-sults we obtained have allowed us to identify severalrelationships, which for the most part agree very wellwith current theories, shed light on the types of sys-tems we have observed, leaving us confident that themethods we have used throughout the project are valid.

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Thanks and Acknowledgements (JG)

We would like to thank Dr Andrea Miglio, Dr Daniel Reese, and Dr Guy Davies for their guidance, super-vision, and assistance throughout the duration of this project.

Dr Miglio provided the group with a thorough introduction to the project and gave us a good head startinto understanding the aims. His feedback on the initial worksheet was useful and advice on any unsure areasof theory ensured the accuracy of all scientific documentation produced in this project. Dr Miglio’s constantavailability meant the group could continue to move forward without delay on important issues.

Dr Reese was always on hand to provide the Asteroseismology Sub-Group advice on the practicalities ofusing the frequency-power spectra and provided the group with the MESA stellar model data. We would liketo thank him for his help in ensuring that the methods used to deal with all of the data were as efficient aspossible and his input saved a substantial amount of project time from being wasted.

Dr Davies’s advice and help was particularly useful when it came to the problems faced by the Planet-FindingSub-Group with our various Python codes. His expertise helped us make up valuable time we had lost due toour relatively slow coding progress at the outset of the project, ensuring we could obtain the high accuracy andrelevant results we set out to achieve.

Furthermore, Dr Davies contributed directly to data gathering by allowing us to use his (more powerful)computer cluster to run the transit fitting code simultaneously for all stars. This is greatly appreciated as itwould have taken group members significantly longer to obtain the same quality of results, which would likelyhave exceeded the available time.

We are also appreciative of the assistance of the University of Birmingham School of Physics and Astronomy,and in particular the Teaching Support Office for assisting us in the booking of computer rooms, without whichthe running of the project would have proceeded with significant difficulties.

Finally, I would personally like to thank all members of this project for their continued efforts from start tofinish. Everyone on the team has made an incredible contribution and as Project Leader I couldn’t make anycomplaints about a single one of them. They have truly been a pleasure to work with. Data for this projectwas obtained from the Kepler Mission, handled by the Kepler Asteroseismic Science Consortium.

94

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