1 The structure and evolution of stars Lecture 9: Computation of stellar evolutionary models.

1

The structure and evolution of stars

Lecture 9: Computation of stellar evolutionary models

2

Learning Outcomes

The student will learn• How to interpret the models of modern calculations -

(in this case the models from the Geneva theoretical stellar evolution group)

• How a realistic theoretical HRD is constructed• Understand how stars of different masses

schematically evolve• To appreciate how stellar lifetime varies with mass• How clusters are used to test models of stellar

evolution

3

Introduction and recap

Second part of course:

• Previous lectures analytical - now we will be more descriptive. Account of results for full-scale numerical calculations of the set of equations

• Numerical studies date back to 1960s (Icko Iben - momentous efforts over 30 years, often illustrated in text books)

• Results of these computations are not always anticipated or intuitively expected from fundamental principles - equations are non-linear and solutions complex

• We will concentrate on comparing the observable properties of stars (Lecture 1) and testing models by comparing to HR diagram and all its aspects

4

Example set of models - “the Geneva Group”

See handout of paper of Schaller et al. (1992): the “standard” set of stellar evolutionary models form the Geneva group. 1st line in table

NB = model number (51)

AGE = age in yrs

MASS = current mass

LOGL = log L/L

LOGTE = log Teff

X,Y,C12…NE22 = surface abundance of H,He, 12C … 22Ne (these are mass fractions)

2nd line

QCC = fraction of stellar mass within convective core

MDOT = mass loss rate:

RHOC=central density

LOGTC = log Tc

X,Y,C12…NE22 = central abundances €

log(M).

where M.

= mass loss rate in Msolyr -1

5

The Hayashi forbidden zone

The Hayashi line gives a lower limit for the Teff of stars in hydrostatic equilibrium.

First determined when evolution of protostars considered - collapsing molecular cloud to form a main-sequence star.

We will not treat it mathematically in this course:

Further reading in Böhm-Vitense, Ch. 11.2

6

Example evolution of a 5M star

H-burning in main-sequence, Xc=0 at NB=13, =100 Myr (and Yc=0.98)

Star cools and moves across HRD on thermal timescale (20 - 13= 4.6x105 yrs). From Lecture 5, the thermal timescale of the Sun is ~1015 sec or ~30Myrs

For 5M tth~2x105 yrs - similar to rapid movement timescale on HRD.

He burning begins at NB=20, ends at NB=43. Comparison of lifetimes:

H-burning 9.4 x 107 yrs

Thermal expansion 4.6 x 105 yrs

He-burning 16 x 106 yrs

tth ~GM 2

LR ~1015 M

M sol

⎛

⎝⎜⎞

⎠⎟

2Rsol

R⎛⎝⎜

⎞⎠⎟

Lsol

L⎛⎝⎜

⎞⎠⎟ s

7

Main-sequence lifetimes

Approximate main-sequence lifetimes (from Prialnik, P. 142 - note some differences with Geneva models )

Stellar clusters (from Lecture 1) large group of stars born at same time, age of cluster will show on HR-diagram as the upper end, or turn-off of the main-sequence.

We can use this as a tool (clock) for measuring age of star clusters. Stars with lifetimes less than cluster age, have left main sequence. Stars with main-sequence lifetimes longer than age, still dwell on main-sequence.

Mass M Time0.1 6 1012

0.5 7 1010

1.0 1 1010

1.25 4 109

1.5 2 109

3.0 2 108

5.0 7 107

9.0 2 107

15 1 107

25 6 106

Stars of all masses live on the main-sequence, but subsequent evolution differs enormously. We can divide the HRD into four sections, defined by mass ranges within which the evolution is similar (or related).

8

The five sections of the HRD

Note all masses approximate, boundaries overlap depending on definition.

Brown dwarfs (and planets): estimated lower stellar mass limit is 0.08 M (or 80MJup). Lower mass objects have core T too low to ignite H.

Red dwarfs: stars whose main-sequence lifetime exceeds the present age of the Universe (estimated as 1-2x1010 yr). Models yield an upper mass limit of stars that must still be on main-sequence, even if they are as old as the Universe of 0.7M

Low-mass stars: stars in the region 0.7 ≤ M ≤ 2 M . After shedding considerable amount of mass, they will end their lives as white dwarfs and possibly planetary nebulae. In Lecture 10 we will follow the evolution of a 1M star in detail.

Intermediate mass stars: stars of mass 2 ≤ M ≤ 8-10 M. Similar evolutionary paths to low-mass stars, but always at higher luminosity. Give planetary nebula and higher mass white dwarfs. Complex behaviour on the AGP branch.

High mass (or massive) stars: M >8-10 M. Distinctly different lifetimes and evolutionary paths huge variation, will study in Lecture 11.

9

Stellar evolution movies

• 5-8M evolution

QuickTime™ and aYUV420 codec decompressor

are needed to see this picture.

Frrom the “Digital Demo Room Stellar Structure and Evolution Simulator”

http://rainman.astro.uiuc.edu/ddr/stellar/index.html

10

Convection processes and uncertainties

In Schaller et al. there is some discussion on Convection Parameters (§2.5).

Mixing length theory of convection:

The description of convection which is commonly used in stellar interiors contains a free parameter called the mixing length (l). Assume that the convective elements of a characteristic size l rise or fall through a distance that is comparable with their size, before they exchange heat with their surroundings.

If it assumed that elements move adiabatically and in pressure balance with their surroundings, and that they are accelerated freely by buoyancy force.

Lconv =πr2cPρTGMr2

⎛⎝⎜

⎞⎠⎟

1/2 l2 (∇−∇ad)3/2

HP3/2

where

HP =P

−(dP dr)≡ the pressure scale height

∇=PT

dTdP

∇ad =γ −1γ

and cp is the specific heat at constant pressure

11

This expression is only useful if a value can be chosen for l. Often assumed that an appropriate value is of order a pressure scale height, and a value is defined :

The value of chosen can make a considerable difference to stellar structure, particularly in cool stars. The structure of the Sun and its Teff can be reproduced with =1.6, But nothing definite known about this value for other stars. In Schaller et al. they estimate from the average location of the red giant branch of 75 clusters, and obtained best fit for =1.6 0.1. Note that this is an empirical fit, a theory of convection is not yet developed that can predict l.

Convective Overshooting

One more important property of convection. What happens at the boundary between a convective region and non-convecitve region ? A rising convective element will still have a finite velocity as it enters the region where the convective criterion is not satisfied. This process is called convective overshooting.

This is generally not important for energy transport, but means that mixing can occur between the regions which can be significant for later evolution.

=l

HP

12

Modelling star clusters

As discussed in Lecture 1, best way to check stellar evolutionary calculations if to compare calculated and observed tracks. But can’t observe stars as they evolve - need to use star clusters.

Isochrones:

A curve which traces the properties of stars as a function of mass for a given age.

Be clear about the difference with an evolutionary track - which shows the properties of a star as a function of age for a fixed mass.

Isochrones are particularly useful for star clusters - all stars born at the same time with the same composition e.g. the Schaller et al. models. Consider stars of different masses but with the same age . Lets make a plot of Log(L/L)vs. LogTeff for an age of 1Gyr. The result is an isochrone.

Important - think about what we are looking at when we observe a cluster. We are seeing a “freeze-frame” picture at a particular age. We see how stars of different masses have evolved up to that fixed age (this is not equivalent to an evolutionary track).

13

Modelling star clusters

Meynet et al. 1993 (Astr. & Astr. Supp. Ser., 98,477)

“New dating of Galactic Open Clusters”

Using the Geneva models, they fit isochrones to real stellar clusters

14

Theoretical isochrones from Geneva models

15

Examples of young and old clusters

NGC6231 young cluster

Age~ 6Myrs

Pleiades young open cluster

Age~ 100Myrs

16

47 Tuc : globular cluster. Age= 8-10Gyrs NGC188: old open cluster .

Age= 7Gyrs

17

Summary

• We have seen examples of modern stellar evolutionary calculations (the Geneva Group)

• The main-sequence lifetimes are very dependent on initial stellar mass

• Isochrones rather than tracks for each mass. They are equivalent, but give a snapshot of the cluster at a particular age

• Excellent agreement between models, and the observed HR-diagrams

• Can be confident that we are predicting the real behaviour of these stars.

• Next two lectures will look in detail at a low-mass star, and a high mass star as case studies.