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ARENBERG DOCTORAL SCHOOL
Faculty of Science
Relativistic jet dynamics from
X-ray binaries to ActiveGalactic Nuclei
Rémi Monceau-Baroux
Dissertation presented in partial
fulfillment of the requirements for the
degree of Doctor in Mathematics
October 2014
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Relativistic jet dynamics from X-ray binaries to
Active Galactic Nuclei
Rémi MONCEAU-BAROUX
Examination committee:Prof. Giovanni Lapenta, chairProf. Rony
Keppens, supervisorDr. Zakaria Meliani, co-supervisor
(LUTh, Observatoire de Paris)Prof. Hans Van WinckelProf. Paul
GibbonProf. Tom Van DoorsselaereProf. Alexandre Marcowith
(LUPM, Université Montpellier II)
Dr. Oliver Porth(Department of Applied Mathematics, The
University of Leeds)
Dissertation presented in partialfulfillment of the requirements
forthe degree of Doctorin Mathematics
October 2014
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© 2013 KU Leuven – Faculty of ScienceUitgegeven in eigen
beheer, Rémi Monceau-Baroux, Celestijnenlaan 200B, bus 2400, B-3001
Heverlee (Belgium)
Alle rechten voorbehouden. Niets uit deze uitgave mag worden
vermenigvuldigd en/of openbaar gemaakt wordendoor middel van druk,
fotokopie, microfilm, elektronisch of op welke andere wijze ook
zonder voorafgaandeschriftelijke toestemming van de uitgever.
All rights reserved. No part of the publication may be
reproduced in any form by print, photoprint, microfilm,electronic
or any other means without written permission from the
publisher.
ISBN 978-90-8649-759-1D/2014/10705/67
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Preface
What does it mean to be a physicist? I think the answer is
simple, to askyourself: how does the universe work? Even the direct
world around us hasendless wonders if you start asking yourself
this question. I remember one of the first times I asked it
myself: if gravity bring objects down, how comes thewater get out
of the sink on the second floor?
This fascination brought me through my whole childhood and
teenage years.It was cultivated by some fantastic if not slightly
’surprising’ teachers. I recallentering a classroom one day and
finding my Physics teacher in a wheelchairwith a bowling ball in
his arm and asking me to come and stand in front of
him. That was the best test of the principal of action-reaction
I can remember.Or another teacher in high-school who made a point
of not only talking aboutPhysics and Chemistry, but also
demonstrating it. As a teenager, to see a canof coke implode under
the pressure of the atmosphere because the water vaporinside goes
violently back into liquid state can be tagged as ’cool’. I thank
allmy teachers past and future.
My path kept going through university to finally end up at the
start of mythesis. And what I though to be bad luck finally brought
me to the door of aman without whom I’m not sure this manuscript
would exist: Rony Keppens.
If I had to use one word to describe him, it would be the word
’fair’. When I didgood, I knew it. When I messed up, I knew it too.
He is one of those physicistthat we are scared of and hope for
during our talk: he is always interested andwill try to understand
what you are talking about whatever the topic is and hewill ask
meaningful questions. He will push you in the understanding of
yourown subject and make you grow for it. Will thank you be enough?
If Rony wasmy mentor, Zakaria Meliani and Oliver Porth were my
guides in the new landof simulations. I bow to both of you for your
patience towards my questions.
I know some people can do only with science. Personally I need
to share it,
discuss it with friends. And friends I made many in Leuven. From
work:
i
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ii PREFACE
Alkis, Marie-Elenna, Franscesco, Jorge, Mikael, and so on
(beware the pandainvasion!). And Jan, my dear Jan, my nemesis Jan?
I’m not too sure. . .Anyway, I will miss you and your rolling eyes.
I also made friend in town during
my few tries at mastering Dutch: Becky (so English), Simon (so
Irish!), Mikael,Stefanie and so on. Speaking about discussing
Physics, sometimes taking someliberty with it: I need to thank the
“Orsay gang”: Clément, Walter, Vincent,Théo, Julien, etc. We are
not working in the same field, but we will probablybe walking
together for a while still!
If friend are important, family is important too. Among it some
role models:given the number of doctors in my close relatives, how
could I not become oneof them when my Christmas presents where
ranging from ’Particle dictionary’to ’Feynman introductory courses
to physic’. And I have to do that one: See
Daniel, I did it! And as family goes, I need to say that you, my
dear Laura,make me feel more complete than I have been for many
years. Your supportthrough the grumpiest phase of my Phd has a lot
to do with me not collapsingunder the pressure.
Finally I would like to thank the members of my comity that I
didn’t mentionyet for their examination of my work: Dr Marcowith,
Dr Lapenta, Dr vanWinckel, Dr Gibbon and Dr van Doorsselaere. I
would like to thank you allnot only for the formal validation of my
thesis, but also for all the scientificdiscussions which helped me
improve my current and future work.
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Abstract
In this thesis I study the dynamics of relativistic plasma jets
in the contextof astrophysics. I consider jets with low to medium
Lorentz factors associatedwith Active Galactic Nuclei and X-Ray
Binaries. I want to highlight how thedifferent parameters of the
jet, and of its surroundings, affect its propagation. Imore
specifically study the propagation of the jet far away from its
launchingregion where the jet is supposed to be dominated by
kinetic energy with a weakmagnetic field. For this reason I use a
special relativistic hydrodynamical model.To solve my equations I
use the code MPI-AMRVAC with a self designed userfile. This file is
written in a general way which allows to study a large varietyof
relativistic jets, using an appropriate parametrization based on
the observed
jet kinetics.
After a general introduction on relativistic jets and their
context, I will presentthe tools used and developed in the course
of my Phd. I will present the codeMPI-AMRVAC and the basic
numerical algorithms it employs.
In my first series of simulations, I will place myself in the
context of jetsassociated with Active Galactic Nuclei with mildly
relativistic Lorentz factorfor powerful jet sources. I will study
the impact of the opening angle of the
jet on its propagation and compare my results with the
more commonly used
cylindrical topology. I will also study the impact of the medium
by varying theinertia ratio between jet and medium, and by varying
the stratification of themedium.
In my second series of simulations, I will study the specific
case of the relativistic jet associated with SS433. This is a
well-studied X-ray binary with a precessing jet. I will study
the impact of the Lorentz factor on the energy exchange
between jet and medium, and the resulting consequences for the
jet propagation. I willalso compare my simulations with data from
observations. Therefore, I willrealize synthetic radio maps of my
simulations and confront them directly with
those obtained by the VLA instrument.
iii
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iv ABSTRACT
In my simulation for my third project, I will study a
discrepancy known fromobservations of SS433 between the large scale
of the jet, where it interacts witha supernova remnant called W50,
and the smaller scale where it is directly
observed. I will show that by means of dynamical processes
specific to precessingrelativistic jets, that it is possible to
recollimate a precessing jet and transformit into a continuous
conical jet. This resolves the discrepancy mentioned.
I will finally conclude on what I learned during this work and
highlight the thightlink between analytic work, advanced numerical
approaches and observationalworks.
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Beknopte samenvatting
In deze doctoraatsthesis bestudeer ik de dynamica van
relativistische plasma jetsin een astrofysische context. Ik
beschouw hierbij jets met lage tot middelgroteLorentzfactoren; deze
worden geassocieerd met het centrum van actievesterrenstelsels
(AGN’s of Active Galactic Nuclei ), en
met röntgendubbelsterren.Ik wil aan het licht brengen hoe de
verschillende parameters van de jet enzijn omgeving, de propagatie
ervan beïnvloeden. In het bijzonder bestudeer ikde propagatie van de
jet op grote afstand van zijn ontstaanspositie, waar,zo wordt
verondersteld, de kinetische energie de bepalende factor is voorde
dynamica. Het magnetische veld is er zwak. Omwille van deze
redengebruik ik een hydrodynamisch model met inclusie van speciale
relativiteit.
Om de vergelijkingen op te lossen gebruik ik de MPI-AMRVAC code
met eenzelfgeschreven user file. Dit bestand is zeer algemeen
opgesteld, zodat een groteverscheidenheid aan relativistische jets
ermee beschreven kan worden via eengepaste parametrisatie,
gebaseerd op de geobserveerde mechanica van de jets.
Na een algemene inleiding over relativistische jets en hun
context, zal ik demethoden beschrijven die ik tijdens mijn doctoraat
heb gebruikt en ontwikkeld.Ik zal de MPI-AMRVAC code presenteren en
de numerieke algoritmen die eringebruikt worden uitleggen.
In een eerste reeks simulaties plaats ik me in de context van
jets geassocieerd metAGN’s met zwak relativistische Lorentzfactoren
voor krachtige bronnen van jets.Ik zal de impact van de
openingshoek van de jet op zijn propagatiekenmerkenbestuderen en mijn
resultaten vergelijken met die bekomen via de vaak
gebruiktecylindrische topologie. Ik zal ook het effect van het
medium bestuderen door deinertieverhouding tussen jet en medium te
laten variëren, alsook de stratificatievan het medium zelf.
In een tweede reeks van simulaties, zal ik het bijzondere geval
bekijken van derelativistische jet geassocieerd met SS433. Dit is
een röntgendubbelster met
een jet die een precessie beweging ondergaat, die reeds
uitvoerig bestudeerd
v
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vi BEKNOPTE SAMENVATTING
werd. Ik zal de impact van de Lorentzfactor bestuderen op de
uitwisseling vanenergie tussen jet en medium, en de gevolgen voor
de propagatie van de jet. Ikzal ook mijn simulaties vergelijken met
observaties. Hiervoor zal ik synthetische
radio-observaties genereren van mijn simulaties en deze direct
vergelijken metdie van VLA.
In simulaties voor een derde project zal ik de schijnbare
tegenstrijdigheidonderzoeken, bekend van observaties van SS433,
tussen het gedrag van de jet opgrote schaal, waar hij interageert
met een supernovarestant genaamd W50, enhet gedrag op kleinere
schaal dat we direct kunnen waarnemen. Ik zal aantonendat door
middel van dynamische processen, typisch voor jets in precessie,
hetmogelijk is om een dergelijke jet te hercollimeren en ze te
transformeren tot eencontinue, kegelvormige jet. Dit lost de zonet
vermelde discrepantie op.
Uiteindelijk zal ik samenvatten wat ik tijdens dit onderzoek
geleerd heb ende nauwe banden tussen analytisch werk, gevorderde
numerieke methoden enobservaties in de verf zetten.
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Abbreviations
AGN Active Galactic NucleiAMR Adaptive Mesh Refinement
BH Black Hole
FR-I Faranoff-Riley type IFR-II Faranoff-Riley type II
G Gauss (unit)GR General Relativity
GRB Gamma Ray BurstGRMHD General Relativistic
Magneto-hydrodynamic
HD Hydrodynamic (model)HLL Harten-Lax-van LeerHLLC
Harten-Lax-van Leer-Contact
IGM Inter Galactic MediumISCO Innermost Stable Circular
Orbit
ISM Inter Stellar Medium
KH Kelvin-Helmholtz (instability)
MHD Magneto-hydrodynamic (model)MPI-AMRVAC Message
Passing Interface Adaptive Mesh
Refinement Versatile Advection Code
pc parsec (unit)PeV Peta electron Volt
vii
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viii ABBREVIATIONS
RT Rayleigh-Taylor (instability)
SFR Star Formation RegionSR Special RelativitySRMHD Special
Relativistic Magneto-hydrodynamic
TVD Total Variation DiminishingTVDLF Total Variation Diminishing
Lax-Friedrichs
VLA Very Large ArrayVLBA Very Long Baseline Array
XRB X-Ray Binary
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List of Symbols
Ṁ BH Black hole accretion rate
Lab frame energy
(a) Radiative efficiency
0 Vaccum permittivity
η Inertia ratio
η Density ratio
Γ Polytropic index
γ Lorentz factor
Γe Electron power law
Γeff Effective polytropic index
Λ Plasma parameter
λD Debye length
λfree Mean free path
λLarmor Larmor radius
µ Frequency
ρ Density (in g.cm−3)
τ Energy density
B Magnetic field
S Momentum density
ix
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x LIST OF SYMBOLS
vth Thermal velocity
v Velocity
a Spin of the black hole
BP Poloidal component of B
c Speed of light
D Dopler factor
e Elementary charge
eint Specific internal energy
E spin Spin energy
eth Specific thermal energy
F (U ) Flux
G Gravitational constant
h Specific enthalpy
I Intensity
j Synchrotron emission
kB Constant of Boltzmann
L Luminosity
Lacc Luminisoty of the jet from accretion potential
Lmagnetic Luminisoty of the jet from spin/magnetic field
extraction
lnΛ Coulomb logarithm
m Mass
M Solar mass
m p Proton mass
M BH Mass of the black hole
M red Reducible mass (of BH)
n Number density (in cm−3)
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LIST OF SYMBOLS xi
p pressure
rin ISCO radius
S phys Physical source term
T Temperature
tcoll Collision time
U Variable
V Volume
Z Charge number
zc King atmosphere core radius
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Contents
Abstract iii
Contents xiii
List of Figures xvii
List of Tables xxv
1 Introduction 1
1.1 On the presence of jets in astrophysics . . . . . . .
. . . . . . . . 1
1.2 Discovery . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 1
1.3 Jet formation . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 5
1.4 Propagation and main features . . . . . . . . . . . .
. . . . . . 7
1.5 Interaction of jets with their surroundings . . . . .
. . . . . . . 10
1.6 Where we stand . . . . . . . . . . . . . . . . . . . .
. . . . . . . 12
2 Models and Tools 15
2.1 Special relativistic hydrodynamics . . . . . . . . .
. . . . . . . 15
2.2 Code - MPI-AMRVAC . . . . . . . . . . . . . . . . . .
. . . . . 20
2.2.1 MPI-AMRVAC structure . . . . . . . . . . . . . . .
. . 20
2.2.2 Finite Volume Model . . . . . . . . . . . . . . . . .
. . . 20
xiii
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xiv CONTENTS
2.2.3 Spatio temporal discretization . . . . . . . . . . .
. . . . . 21
2.2.4 AMR and MPI . . . . . . . . . . . . . . . . . . . .
. . . 27
2.3 Observational instruments . . . . . . . . . . . . . .
. . . . . . . 29
2.3.1 Chandra . . . . . . . . . . . . . . . . . . . . . . .
. . . . 30
2.3.2 VLA . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 31
2.3.3 VLBA . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 33
2.4 Model for radio mapping . . . . . . . . . . . . . . . .
. . . . . . 34
3 Effect of angular opening on the dynamics of relativistic
hydro jets 37
3.1 Introduction . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 37
3.1.1 Code setup . . . . . . . . . . . . . . . . . . . .
. . . . . 39
3.1.2 Initial conditions . . . . . . . . . . . . . . . .
. . . . . . 39
3.2 Results -parametric study . . . . . . . . . . . . . .
. . . . . . . 42
3.2.1 Dynamics - general description . . . . . . . . . .
. . . . 42
3.2.2 Dynamics - overall propagation distances . . . . .
. . . 46
3.2.3 Dynamics study - volumes and radial expansion . . . .
. . 51
3.2.4 Temporal evolution of the energy content . . . . .
. . . 53
3.3 Dynamical details in conical relativistic HD jets . . .
. . . . . . 56
3.3.1 Internal structure of the jet . . . . . . . . . . .
. . . . . 56
3.3.2 Dynamic formation of a layered structure . . . . .
. . . 59
3.4 Conclusion . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 61
4 Relativistic 3D precessing jet simulations for the X-ray
binary SS433 67
4.1 Introduction . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 67
4.1.1 Code setup . . . . . . . . . . . . . . . . . . . .
. . . . . 70
4.1.2 Initial conditions . . . . . . . . . . . . . . . .
. . . . . . 70
4.1.3 Data analysis . . . . . . . . . . . . . . . . . . .
. . . . . 73
4.2 Results . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 74
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CONTENTS xv
4.2.1 General dynamics . . . . . . . . . . . . . . . . .
. . . . 74
4.2.2 Quantification of dynamics . . . . . . . . . . . .
. . . . 78
4.2.3 Radio mapping . . . . . . . . . . . . . . . . . . .
. . . . 85
4.3 Conclusion . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 87
5 Spatial evolution of the precessing SS433 jet 91
5.1 Introduction . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 91
5.2 Code setup . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 92
5.3 Data analysis . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 93
5.4 Spatial evolution . . . . . . . . . . . . . . . . . .
. . . . . . . . 94
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 104
6 Conclusions and Prospectives 105
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 105
6.2 Prospectives . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 107
6.2.1 VLBA observations: a movie of SS433 . . . . . . . .
. . 108
7 Nederlandstalige samenvatting 113
A Appendix 117
A.1 Parfile . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 117
A.2 User file . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 119
Bibliography 135
Curriculum vitae 145
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List of Figures
1.1 Left: AGN jet (Credit: Pearson Education, Inc., Upper
SaddleRiver, New Jersey); Right: XRB jet (Credit: CXC/M.Weiss)
. 2
1.2 Left: M87 observation with space telescope Hubble,
AGN jet.The jet is about 5000 light year or 1.5 kpc; Right:
Firstobservation of superluminal material in GRS1915+105, an
XRB
jet, from [Mirabel and Rodriguez, 1994]. The scale
given is of about 1”, for a estimate distance of the object of
12.5 kpc. . . . 2
1.3 VLBI and VLA observation of AGN jet 3C120 showing the
multi-
scale appearance of the same jet. Credit Harris et al.
2005[Harris et al., 2005] . . . . . . . . . . . . . . . . .
. . . . . . . . 4
1.4 The magnetic flux trapping effect. Left panel:
Structure of themagnetosphere in the model where the disk is
terminated at thelast stable orbit. Right panel: Structure of the
magnetospherewith included accretion flow in the plunging region
(light shadowof grey ). Notice that in this case the magnetic flux
threadingthe black hole is higher. Credit: [Boettcher et al.,
2012] . . . . 6
1.5 Sketch of the launching region and early propagation.
Thestructure is matched to the emission type (adapted
from[Marscher, 2005]). The poloidal field geometry from the
originalsketch was switched to a helical one to reflect the recent
consensus. 8
1.6 Sketch of the general structure of the jet while
propagating in amedium. . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 9
xvii
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xviii LIST OF FIGURES
1.7 The ‘Swordy spectrum’ of the Pierre Auger
Observatory, updatedby William F. Hanlon from the University of
Utah. The particleflux as function of their energy displays the
cosmic ray spectrum
measured on and close to earth, with the famous knee and
ankle.At the same time, the plot indicates the typical energy that
canbe reached by particle accelerators. . . . . . . . . . .
. . . . . . . 11
2.1 We consider a volume V with
associated quantities U n. Acrosseach edge of this
volume we consider the fluxes of the quantitiesF (U n),
here indicated to be given in a Cartesian x-y sense. . . .
21
2.2 Top: Logarithm of density for three simulations at
same physicaltime for schemes HLLC (left), HLLCD (middle) and
TVDLF(right). Plotting the full domain in the (x,y) plane.
Bottom:Horizontal cut at y = 0.8 of the logarithm
of the density throughthe domain for HLLC (continuous), HLLCD
(cross) and TVDLF(dash). The horizontal axis is the x distance and
the vertical axisis log(ρ). . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 26
2.3 For the hypothetical Cartesian grid exploiting 4×3
grid blocksat level l = 1, the left panel shows the global grid
indiceswhile the tree representation is given at the right panel.
From[Keppens et al., 2012] . . . . . . . . . . . . . . . . .
. . . . . . 28
2.4 Example of dynamic AMR following the propagation of
arelativistic jet into a 2D domain. . . . . . . . . . . . .
. . . . . 29
2.5 Tycho supernovae remnant in X-ray. Credit: X-ray:
NASA/CX-C/Rutgers/K.Eriksen et al . . . . . . . . . . . . . .
. . . . . . . 32
2.6 M87 galaxy and associated jet observed with VLA.
Credit:NRAO/AUI/NSF . . . . . . . . . . . . . . . . . . . . .
. . . . . 34
3.1 Geometry of the jet as used in the simulation: the
jet entersthe domain at z0 = 0.5 pc from the
source with a width settingrb as calculated assuming a first
angle of 5°. A flaring-like caseassumes a fixed angle θ,
which is added to a reference cylindrical
jet, thereby introducing a ‘virtual source location’. The
actualdomain of the simulation is comprised between z0
and zout forthe axial direction and r
= 0 and rout for the radial direction.The
velocity field is initiated in a finite region up to z1,
accordingto the formulae given in the text. . . . . . . . .
. . . . . . . . . 42
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LIST OF FIGURES xix
3.2 Example of our method to instantaneously identify jet
regionsfor a jet opening angle of 10° at time t=20 (case E).
We showthe masks locating the jet (left) mixing region (center) and
the
shocked ISM (right). We identify the jet beam by
γ ≥ 2, findingthe mixing region where |(∇ ×
v)φ| > 1 without the jet beam,and the cocoon
where τ = τ t=0 with beam and
mixing regionexcluded. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 44
3.3 Comparison of the density distribution at time t=20
for anopening angle of 0° (left - case C) and 10°
(right - Case E). Fulldomain: 10x19.5 pc. A wider opening angle
increases the widthof the jet beam that stays collimated. The
maximal reach at agiven t of the jet beam is inversely
proportional to the opening
angle. The radial reach of the shock surrounding the jet
remainsof the same order. . . . . . . . . . . . . . . . . .
. . . . . . . . 45
3.4 Comparison of the proper density distribution at time
t=20 foran opening angle of 5° (left - case D) and 10°
(right - Case E).Full domain: 10x19.5 pc. Conclusions from visual
investigationagree qualitatively with fig. 3.3. . . . .
. . . . . . . . . . . . . . 46
3.5 Density profile along the symmetry axis of the jet
for cases C,D and E and for the unperturbed atmosphere. All cases
show
an increased density at the head of the jet beam. The maximalz
reach is inversely proportional to the opening angle. Case
C(cylindrical case) keeps a near constant density. Cases D and
Eshow a series of nodes with increased density. The position
of the nodes depends on the opening angle: the wider the
openingangle, the longer the internodal distance. The jet density
incase C becomes higher than the ISM density from about
z ≈ 12,whereas for the other two cases, it remains below
the ISM densityprofile throughout the domain. . . . . . . .
. . . . . . . . . . . 47
3.6 Matter swept up by the jet beam for different cases.
For thesame density profile (cases C, D and E, or cases, F and H)
wider jets sweep away more matter. We can see that the density
profileclearly matters. . . . . . . . . . . . . . . . . . .
. . . . . . . . 48
3.7 Volume fraction of the mixing region up to a time
t=20 fordifferent cases. The volume depends both on the opening
angleand the properties of the ISM density profile. The
volumeincreases for a smaller opening angle although the radial
reachof this region diminishes. The maximal z reach
of the jet is
therefore dominant over the radial expansion of the mixing
region. 49
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xx LIST OF FIGURES
3.8 Radial expansion of the shocked region over time for
differentcases. Despite initial differences, the radial width of
the jet-influenced region does not vary much for a varying opening
angle. 50
3.9 Position of the bow shock marking the head of the jet
over timefor different cases. The density ratio and the opening
angle areboth relevant for the propagation of this bow shock. The
widerthe opening angle, the more it decelerates. The higher the
densityratio η, the slower the shock front deceleration.
. . . . . . . . . 51
3.10 Fraction of the domain volume occupied by the
shocked ISMregion up to a time t=20 for different cases. This
volume fractionof the shocked region both depends on the opening
angle and onthe density profile of the ISM. We know from Fig.
3.8 that allsimulations have comparable radial reach,
therefore we expectthe volume of the cases to be ordered as in Fig.
3.9. This is thecase within the same series of simulations
(cases C, D and E, andcases F and H). . . . . . . . . . . .
. . . . . . . . . . . . . . . . 52
3.11 Fraction of the total domain energy excess present in
the mixingregion, which is dominated by instabilities, for the
different cases. 56
3.12 Excess energy ratio in the shocked ISM region for different
cases. 57
3.13 Total excess energy ratio in the mixing region and
shocked ISMfor different cases. By comparison with Figs.
3.11 and 3.12 weconclude that the transfer is
dominated by heating from theshocks in the shocked ISM region.
. . . . . . . . . . . . . . . . 58
3.14 Thermal term of the total energy transfer. . . . . .
. . . . . . . 59
3.15 Left: Comparison of the effective polytropic index
for cases C(left) and E (right) at time 20 of the simulation. This
figurereveals more clearly a series of secondary shocks that
propagate
in the shocked ISM region and also reheat it. Right:
Comparisonof the effective polytropic index for cases D (left) and
E (right)at time 20 of the simulation. Full domain: 10x19.5 pc.
. . . . . 60
3.16 Details of two nodes along the axis. Left is the
effective polytropicindex, right is the density. Internal structure
of the jet beamappears with a finite opening angle: a static shock,
in the labframe, forms along the path of the jet, which can be the
site of Fermi-type-II acceleration of particles. The size of
the image is7 × 8 pc, starting from the bottom of the domain,
with the twonodes at 3.5 and 7 pc along the symmetry axis. .
. . . . . . . . . 61
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LIST OF FIGURES xxi
3.17 Presence of a dynamically formed layer surrounding
the beam of the jet. Left is the effective polytropic index,
right is the curl of the velocity. This is another zoom on the
inlet (as in Fig. 3.16)
for case E, now showing the (scaled) vorticity at the right.
. . . 623.18 Radial cuts of the density through the
beam at distances z=2, 5
and 10 for an opening angle of 0° (top - case C) and 10°
(bottom- case E), both with a King atmosphere density
profile. The cutsreveal the position of the jet beam and of a layer
between the
jet beam and the shocked ISM. For the cylindrical case
(left)the layer is near constant in the symmetry axis direction
andhas a width comparable to the radius of the beam. The
openingangle case (bottom) shows a more complicated structure of
this
dynamically formed layer, with two peaks in the curl of the
velocity. 63
3.19 Radial cuts of the curl of the velocity through the
beam atdistances z=2, 5 and 10 for an opening angle of 0°
(top - caseC) and 10° (bottom - case E), both with a King
atmospheredensity profile. Same as fig. 3.18 . . . .
. . . . . . . . . . . . . 64
3.20 Radial cuts of the axial velocity through the beam at
distancesz=2, 5 and 10 for an opening angle of 0° (top -
case C) and10° (bottom - case E), both with a King
atmosphere density
profile. Same as fig. 3.18 . . . . . . . . . . . . .
. . . . . . . . . 65
4.1 Geometry used for the inner boundary jet-injection
prescription. 72
4.2 Filter for the region decomposition of the domain.
Case A, timet = 3.26 years. From left to right we
identify in a cross-sectionalplane the unaffected ISM, the shocked
ISM, the jet cocoon, andthe jet beam. . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 74
4.3 Global view of three cases at times t = 2
(6.53 years) for case A
and B and time t = 0.25 (0.82 year) for case D.
Density is shownon a 2D cut. The jet beam is drawn with the jet
filter fromsection 4.1.3. The pressure is quantified on
the jet beam surface.Left: case A γ = 1.036;
right: case B γ = 1.87; bottom: case
Dγ = 1.87, no precession. The whole domain is shown
0.2x0.1x0.1pc. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 75
4.4 Left: Radial speed profiles for various times. Right:
farthestradial reach of SISM, cocoon, and jet regions. Top: case
Aγ = 1.036; middle: case B γ = 1.87;
bottom: case D γ = 1.87,
no precession. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 79
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xxii LIST OF FIGURES
4.5 2D cut of the domain along the (z,x) plane, mapping
of the localsound speed cs at time t =
3.26 years. Top: case A; bottom:case B. . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 80
4.6 Energy transfer evolution as function of time. Top
left: case Aγ = 1.036; top right: case B
γ = 1.87; bottom: case D γ =
1.87,no precession. . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 81
4.7 Internal structure of the non-precessing case D.
Internal structureappears with standing recollimation shocks at
0.02 pc, 0.047 and0.07 pc. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 81
4.8 Top: density plot of one jet of case D at
time t = 0.25. Bottom:pressure plot of one jet of case D
at time t = 0.25. Formation
of a structured beam for the jet by pushing material from theISM
by the head of the jet, as by a piston. Because of
adiabaticexpansion, the density in that outer layer is low, whereas
densityand pressure build up in front of the head of the jet. Note
theformation of standing recollimation shocks along the path of
the
jet. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 84
4.9 Left to right: radio map from simulations cases A, B,
and C.Units are given in parsec, the object is estimated to be at
adistance of 5.5 kpc. All graphs overplot the kinematic model
with parameters corresponding to the case. Right: VLA imageof
the microquasar SS433 in the constellation Aquila,
adaptedfrom [Roberts et al., 2008], units are given in
arcsecond, with thekinematic model for SS433 overplotted. Both
simulated radiomapand observations take contours with steps of
factors of
√ 2. . . 87
4.10 Radio maps from the simulation Case D. Because the
jet is notrelated to SS433, we use a different angle to the line of
sight toenhance details. Left: angle to the line of sight 85◦.
Right: angleto the line of sight 78◦. Distance units are given in
parsec. Theobject is estimated to be at a distance of 5.5 kpc.
. . . . . . . . 88
5.1 Global view of the large simulation of SS433 zoomed
to the scalereached at t = 2 (6.5 years),
i.e. O(0.1 pc). Volume is renderedusing the tracer to
locate the jet. The volume is colored with thepressure. The 2D cut
is showing the proper density. . . . . . . 94
5.2 In the same format as Fig. 4.6 energy
transfer from jet to cocoonand SISM regions, as a function of time.
. . . . . . . . . . . . . 95
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LIST OF FIGURES xxiii
5.3 Top: radial speed profiles. Bottom: farthest radial
reach of SISM,cocoon and jet regions. As in Fig.
4.4, but for the enlarged caseA, run on a much extended
domain. . . . . . . . . . . . . . . . 97
5.4 Collapsed emission images of SS433 at time (from left
to right,top to bottom) t = 6.5 years,
t = 10.5 years, t = 15.5
yearsand t = 29.25 years. Logarithm of the
intensity from eqn. 5.1.Scale is indicated and the same for
all panels. . . . . . . . . . . 98
5.5 Top: Maximal distance R max (z) from the precession
axisreached by the jet material, at various times. Midle: Distance
R
j (z) from the precession axis of the jet beam axis, at
the sametimes. Bottom: Fit of R j (z) for t = 29.25 years. The dash
linerepresent a precessing angle of 20
◦. The two continuous line fit a
ax+b function between the transition region (z = 0.07pc) andthe
position where the jet get deflected (z = 0.17pc). The curveis fit
for the negative z with 9.8◦ and the positive z with
11.6◦
precessing angles. . . . . . . . . . . . . . . . . . . .
. . . . . . . 100
5.6 Horizontal cut along the (X, Z )
plane (precession axis along theZ axis) of SS433 jet
simulation at time (from left to right, topto bottom)
t = 6.5 years, t = 10.5 years,
t = 15.5 years andt = 29.25
years. Plot of the Lorentz factor. The scale is as
indicated and shared for all panels. . . . . . . . . . .
. . . . . . . 1015.7 Cuts along the (X, Y )
plane (precession axis along the Z axis)
of SS433 jet simulation at distance z = −0.15
pc,z = −0.07 pcand z =
−0.02 pc at time t = 29.25
years. Top: Lorentzfactor. Bottom: Material coming from the jet as
located withthe weighted tracer. Scale is indicated and shared for
all panels. 102
5.8 Radio map of the spatial evolution of SS433 in VLA
conditions.The reconstruction beam is equivalent to 0.005 pc. We
showa dynamical range of 40 contours with incremental levels of 2on
the left and a radiomap in 128 colors on the right revealingfurther
details. Unit are in pc. . . . . . . . . . . . . . . . . . .
103
6.1 Detail of the SS433 movie: brightening of radio
elements. Fromthe NRAO website (see text). . . . . . . . . .
. . . . . . . . . . 109
6.2 4 panels of the radio movie made from our simulation.
Eachpanel is separated by 1.9 days. Distances are in units
of 1015 cm.112
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List of Tables
1.1 Parameters for different jets mentioned in this work.
. . . . . . 5
2.1 Typical parameters for jet associated astrophysical
objects andtheir environment. Typical values for parameters are
extractedfrom the literature as indicated. Due to uncertainty
errors inobservations and the variety of objects in the same class,
thoseparameters can vary up to 2 orders of magnitude. . . . .
. . . . 18
2.2 Typical computational aspects for simulations in this work.
. . 30
2.3 Resolution of the VLA instrument for the different channels
. . 32
2.4 Resolution of the VLBA instrument for the different channels
. 33
3.1 General parameters for all 2.5D simulations studying
the effectof the opening angle. . . . . . . . . . . . . . .
. . . . . . . . . . 40
3.2 Parameters distinguishing the 2.5D simulations
studying theeffect of the opening angle. . . . . . . . . . .
. . . . . . . . . . 40
4.1 General parameters for all 3D simulations studying
the modelof SS433. See section 4.1.2 for references and a
more detaileddescription. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 71
4.2 Parameters for the 3D simulations studying the model
of SS433,which vary from case to case. . . . . . . . . . . .
. . . . . . . . . 71
5.1 General parameters for the 3D simulation studying the
spatial
evolution of SS433. . . . . . . . . . . . . . . . . . . .
. . . . . . 92
xxv
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xxvi LIST OF TABLES
5.2 Normalization of quantities for the 3D simulation
studying thespatial evolution of SS433. . . . . . . . . . .
. . . . . . . . . . . 92
6.1 General parameters for the simulations at the VLBA
scale forSS433. . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 110
6.2 Normalisation of quantities for the simulations. . .
. . . . . . . 110
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Chapter 1
Introduction
1.1 On the presence of jets in astrophysics
Jets are a common occurrence in astrophysics. They can be found
associatedwith a great variety of objects of various mass and
energy, from slow jets inproto-star systems to high-energy beams
associated with pulsars and Gamma
Ray Bursts (GRB) or Active Galactic Nuclei (AGN). Especially the
latter jetshave such high speed and energy that their study
requires the use of generalrelativity (GR) or at least special
relativity (SR). Although the recipe variesfrom one case to the
other, the sources of those jets mostly rely on the magneticfield,
the pressure distribution and the rotational momentum of the
mediumthey originate from. Our study focuses on mildly relativistic
jets with Lorentzfactor γ between 1 and 10. Two
very similar jets fall in this category: the jetsassociated with
AGN and x-ray binaries (XRB).
Both AGN and XRB jets originate from the vicinity of a massive
object likea black hole. They are thought to be a scaled version of
each other. Howeverthe origin of the matter feeding their accretion
disk gives rise to jet differences.AGN jets are fed by their host
galaxy whereas the XRB jets are fed by a massivecompanion (see fig.
1.1).
1.2 Discovery
The history of relativistic jet observations is strongly linked
with the evolutionof observational means. At first those cosmic
radio sources were not resolved.
1
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2 INTRODUCTION
Figure 1.1: Left: AGN jet (Credit: Pearson Education, Inc.,
Upper SaddleRiver, New Jersey); Right: XRB jet (Credit:
CXC/M.Weiss)
Figure 1.2: Left: M87 observation with space telescope Hubble,
AGN jet. The jet is about 5000 light year or 1.5 kpc; Right:
First observation of superluminalmaterial in GRS1915+105, an XRB
jet, from [Mirabel and Rodriguez, 1994].The scale given is of
about 1”, for a estimate distance of the object of 12.5 kpc.
1
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DISCOVERY 3
Therefore these discrete objects were commonly referred to as
radio stars asit was known that the Sun emits radio waves. The
first observation of arelativistic jet was reported in 1918 with
the Crossley Reflector (an optical
ground telescope). In a list of observations appeared «a Curious
straight ray lies in a gap in the nebulosity in p.a.
20 ◦, apparently connected with the nucleus by a thin
line of matter. The ray is brightest at its inner end which is 11”
from
the nucleus » (see [Curtis, 1918]). This object will later
be known as M87, of the Virgo cluster and is to date the more
famous and observed example of anAGN jet and one of the most famous
images of Hubble (see fig. 1.2). After thesecond world war,
techniques developed quickly, like radio interferometry,
andincreased resolution revealed extended emissions with central
radio cores linkedwith by a “ faint wisp of jet ”
(see [Schmidt, 1963]). Ground arrays like the VLAor the VLBA
(respectively operational in 1980 and 1993) started to
becomecommon. The first one, with an angular resolution of 1
milliarcsecond is usedfor parsec scale observations. The latter is
more commonly used for observationon larger scale. With this
technique, a whole new world opened and lead to thefirst
observation of a XRB jet within the galaxy ([Mirabel and Rodriguez,
1994],fig. 1.2). Quickly this object was described as a
scaled-down version of AGN
jets. As radio sources from AGN were called Quasars, XRB
radio sources wereclassified as µ-quasars. Since then many
other jets have been observed forboth AGN and XRB, like in Hercules
A or Cygnus A (see [Krause, 2005] and[Boettcher et al.,
2012]). In the following years observations multiplied and
increased resolution allowed to break them down into
observational categories.For AGN jets it was the classification of
Faranoff-Riley type I (FR-I) and type II(FR-II) [Fanaroff and
Riley, 1974] with the first being brighter near his sourcecore
like 3C 296 [Laing et al., 2006]. The latter, seemingly more
powerful FR-IItype, was brighter near its head like in 3C 216
[Fejes et al., 1992].
Nowadays, relativistic jets are a common research theme in
published material,and quite a few reviews are available both on
AGN jets [Boettcher et al., 2012],[Homan, 2012] and [Perucho,
2014] and XRB jets [Miller-Jones et al., 2006],[Fender,
2001] and [Miller-Jones et al., 2011]. In the past
decade, reviews
tend to present both AGN and XRB jets together, showing their
similarities([Livio, 2000], [Fender, 2010]). We observe every
year an increasing number of publications for this field. The
main reason is the wide variety of wavelengthsin which they can be
observed and their multi-scale nature. Firstly, althoughsimilar in
nature, the scales of XRB and AGN jets are quite different
(fewparsecs (pc) for the first and up to mega-parsecs (Mpc) for the
latter) see table1.1. This results also in the time scale of XRB
jets to be of human proportion,where the evolution of AGN jets
takes place in terms of millions of years, makingthe study of the
first to be of prime interest in the study of the second. Butthe
difference of size is not only present between AGN and XRB jets: a
unique
jet is already multi-scale in nature (see [Harris
et al., 2005], fig. 1.3). This
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4 INTRODUCTION
Figure 1.3: VLBI and VLA observation of AGN jet 3C120 showing
the multi-scale appearance of the same jet. Credit Harris et al.
2005 [Harris et al., 2005]
variety of scale can also mean that different physical
descriptions will be needed.The large scale will use models like
MHD or HD fluid description, as we willdiscuss later, where
particle acceleration, like the type II Fermi acceleration,may
require kinetic models or a statistical treatment.
Despite all those observations, a great deal of information is
still missing onrelativistic jets. The emission mechanisms are not
fully understood and some
properties are not easily accessible from observations. A good
example is theremaining lack of consensus on the exact composition
of those jets: protondominated (mixture of electron and proton with
the kinetic energy mostly carriedby the proton), lepton dominated
(mostly pairs of electrons and positrons) orother [Kundt and
Gopal-Krishna, 1980]. These missing keys are the foundationof the
multi disciplinarity of the field: researchers are forced to use a
combinationof observations, models and simulations to shed some
light on the missing pieces.
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JET FORMATION 5
Host type Lkin γ b sizeM87 AGN
2 1043erg.s−1 6-40 30 kpc3C120 AGN N/A 4.1 150 kpc
GRS1915+105 XRB 1038
erg.s−1
11.7 N/ASS433 XRB 1039erg.s−1 1.04 50 pc
Host referenceM87 [Allen et al., 2006], [Biretta et al.,
2002] and
[Boettcher et al., 2012]3C120 [Cohen et al.,
2007] and [Harris et al., 2005]GRS1915+105 [Harmon
et al., 1997] and
[Miller-Jones et al., 2006]SS433 [Brinkmann et al., 2005]
Table 1.1: Parameters for different jets mentioned in this
work.
1.3 Jet formation
The first question which comes to mind about relativistic jets,
is the questionof their origin. Three requirements have to be met:
launching, acceleration andcollimation. To allow for the launch of
a relativistic jet, we first require a high
energy source. In the vicinity of black holes we have access to
both potentialenergy released from accreting matter (eq.
1.1) or the rotational energy of ablack hole (eq. 1.2). For
the latter it was shown how this spin energy could beextracted to
power the jet (see [Blandford and Znajek, 1977] and eq 1.3).
Bothenergy sources are enough on their own to power either an AGN
jet or a XRB
jet given the respective mass of the compact object
involved.
The maximal luminosity of the jet reachable by extracting energy
from theaccretion potential is expressed as:
Lacc = 12
GM BH Ṁ rin
= (a) Ṁ c2 = 5.7 1046(a) ˙M BH
108M yr−1[erg.s−1] (1.1)
where the radiative efficiency (a) parametrizes the
position of rin, also calledISCO (Innermost Stable
Circular Orbit) in terms of gravitational radius, Gis the
gravitational constant, M BH the mass of the
central black hole, Ṁ theaccretion rate of the
black hole and c the light speed (see [Krolik,
1998]).
The available spin energy is express as:
E spin = M redc2 = M BH (1−
((1 + 1− a2)/2)1/2)c2 (1.2)
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6 INTRODUCTION
Figure 1.4: The magnetic flux trapping effect. Left panel:
Structure of themagnetosphere in the model where the disk is
terminated at the last stable orbit.Right panel: Structure of the
magnetosphere with included accretion flow in theplunging region
(light shadow of grey ). Notice that in this case the magneticflux
threading the black hole is higher. Credit: [Boettcher et
al., 2012]
Where M red is the reducible mass of the black
hole and a the spin parameter.For a maximally spinning
black hole (i.e a = 1): E spin =
0.29M BH c
2 =5.2 1061M BH /108M [erg]. Resulting in
a maximal luminosity of the jetexpressed as:
Lmagnetic 1042( BP 3KGauss
)2( M BH 108M
)2a2 [erg.s−1] (1.3)
where BP is the poloidal component of the
magnetic field, scaled to 3Kilo Gauss. The details of this
extraction mechanism can be found in[Blandford and Znajek,
1977].
The presence of a magnetic field, both in the source region and
the propagationarea, was quickly noticed due to the consensus on
the synchrotron process to bethe main mechanism of
emission [Meisenheimer and Roeser, 1986]. Neverthelessthe
origin, strength and geometry of that field is still unknown. A
possiblescenario would be a dynamo process arising from the
rotation of the accretion
disk, or an accretion of the surrounding magnetic field. In all
cases, modelsexplaining how to form a jet from an accretion disk
appeared early in thefield ([Blandford and Payne, 1982] and
[Blandford and Znajek, 1977]). But thelaunching mechanism is still
studied [Markoff, 2006] as it seems really difficultfor the wind of
the disk to be collimated to observed opening angles, allowingfor
large scale propagation [Bogovalov and Tsinganos, 2005].
From the late 20th century onwards, simulations started being
commonlyused to test models and to find new ideas. Two constraints
were faced: therelativistic treatment for HD and MHD, and the
computational cost of large 2D
and 3D simulations. Nevertheless SRMHD and GRMHD simulations
startedflourishing first in 2D (see [Koide et al., 1998]
and [Komissarov, 2005]) and
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PROPAGATION AND MAIN FEATURES 7
then in 3D ([McKinney and Blandford, 2009] and [Fendt
et al., 2014]). Thosesimulations were successful in obtaining
highly relativistic jets from the vicinityof black holes. But
problems remain, like the density loading of the jet. The
magnetic configuration needed in the model for jet formation has
its field linesanchored in the black hole. With most of the matter
coming from the disk andbeing in a ’frozen-in’ context, it is
difficult to feed matter to the inner part of the beam (see
figure 1.4). Another problem resides in the fact that
withoutan external collimating agent, those jets have wide opening
angles and lowassociated Lorentz factor. We know from observations
that relativistic jets inthe context of AGN and XRB jets, have
Lorentz factors notably higher than1 and up to 30 [Kellermann
et al., 2004]. Any mechanism (magnetic, thermal,centrifugal,
radiative. . .) used to accelerate the jet has to be able to
bringit up to such velocity. The Lorentz factor is a crucial factor
to most studiesof relativistic jet propagation (see [Marti et al.,
1997], [Cohen et al., 2007] and[Lister et al., 2009]) since it
determines the overall energy and jet morphology.What accelerates
the jet, also plays a role in its collimation: the magnetic
fieldcan pinch the jet together when helical, and the pressure from
a disk wind mayhelp to confine the central jet as well.
1.4 Propagation and main features
Collimation is difficult in the launching region. For instance
the Blandford-Znajek mechanism (extraction of energy from the
spinning black hole) doesnot give rise to any collimation of the
jet [Blandford and Znajek, 1977]. Butobservations show that
jets are staying collimated for many orders of magnitudecompared to
the size of their source. A crucial non-answered question
istherefore the mechanism by which the jet is kept collimated. The
magneticfield could help with that collimation depending on its
geometry. But manystudies show that on the contrary, a magnetic
field would tend to make the jetunstable through
magnetohydrodynamical phenomena like the kink instability.
These instabilities could explain the particular shape of some
jets with a helicalpattern while they are non-precessing [Perucho,
2013]. Two other argumentsare therefore also invoked for
re-collimation of the jet. The first one is theinteraction with the
medium. As the jet propagates in the Inter StellarMedium (ISM) or
Inter Galactic Medium (IGM), shocks are formed at itssurface of
interaction with the outer material. Those shocks then
propagatetoward the jet axis pinching it and re-collimating it.
This gives birth to re-collimation nodes along the path of the beam
which are expected to be theseat of particles acceleration. This
process could increase locally the beam
emission and explain the radio blobs (or knots) well known from
observations([Blandford and Rees, 1974], [Scheuer,
1973] and fig. 1.5) and studied since then
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8 INTRODUCTION
Figure 1.5: Sketch of the launching region and early
propagation. The structure
is matched to the emission type (adapted from [Marscher, 2005]).
The poloidalfield geometry from the original sketch was switched to
a helical one to reflectthe recent consensus.
(see [Komissarov and Falle, 1997], [Cohen et al., 2007]
and [Lister et al., 2009]).Radio blobs could on the other hand
be explained by blobs of matter beingejected sporadically by the
jet or its source. These blobs could be comingfrom nonlinearly
developing instabilities in the jet or be molecular cloudsbeing
caught in the path of the beam [Sahayanathan and Misra, 2005].
Thesecond argument for re-collimation uses the relativistic nature
of the jet andsimply considers the inertia of the jet beam which
keeps it collimated withthe opening angle it has leaving the
launching region. Observations indicatethat this angle is always
finite. Even so, in recent years, most simulationsuse cylindrical
jets for their injection model. Sometime the opening angle
isdiscarded by using a wider width of the jet, or using a
precession of the jet[O’Neill et al., 2005]. The argument is that
the re-collimation process happenstoo fast after leaving the
injection region such that it does not play a role
in further propagation. Few models actually implemented an
opening angle
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PROPAGATION AND MAIN FEATURES 9
Figure 1.6: Sketch of the general structure of the jet while
propagating in amedium.
([Meliani et al., 2008], [Krause et al., 2013] and
chapter 3). The propagation of the jet is linked with
other features studied in both observations and simulations.Among
the most spectacular are the bubbles inflated by the jet, also
calledthe cocoon ([Falle, 1991], [Alexander, 2002] and fig.
1.6). This cocoon madeof hot matter is often an indirect way to
measure properties of the jet. Itsemission flux, coupled with an
estimated age of the jet and predictions on theemission mechanism,
can for example determine the jet kinetic luminosity. Thiscocoon is
also what remains of the jet after it turns off. An example of
sucha remnant is close at hand with the recent discovery of
emission from whatappears to be a bubble inflated by a jet coming
from our own galaxy nucleus[Su and Finkbeiner, 2012]. A
simulation [Ruszkowski et al., 2013] was made toreproduce this
feature using the MHD equations coupled with a cosmic
rayspopulation on a 10 by 20 Kpc domain. It shows how this
particular jet couldhave formed the characteristic morphological
features.
As for the launching region, observations are only providing
part of the picture.Models are constructed to try to explain it and
simulations take the place of the in-situ experiments. The
study of propagation follows a similar pattern andstarts with 2D
axisymmetry ([O’Neill et al., 2005] and [Meliani et al.,
2008]).
But the fundamental flaw in axisymmetry is that it prevents the
jet to haveany non-axisymmetric behavior. Studies show that a
propagating jet is non-
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10 INTRODUCTION
axisymmetric [Perucho, 2013]. For this reason, simulations are
more andmore realized in a 3D setting, using schemes like Adaptive
Mesh Refinement(AMR) and massively parallel codes to cope with the
size of calculations
([Walg et al., 2013],[Cielo et al., 2014] and [Guan et
al., 2014]). 3D simulationsalso allowed for dramatically different
dynamics, like the precession of the jet.One of the most famous
observations of a precessing jet, is the jet associatedwith SS433
in our galaxy. Low resolution and non-relativistic simulation
onthis XRB object were made as early as 2000 ([Muller and
Brinkmann, 2000]and [Brinkmann et al., 2005]). In our work we
extend the study to a specialrelativistic treatment at high
resolution ([Monceau-Baroux et al., 2014]
andchapters 3, 4 and 5).
1.5 Interaction of jets with their surroundings
The interest in AGN and XRB jet is also relevant due to their
role in shapingtheir host galaxy. Depending on the parameters used
for these jets, they canprevent the formation of stars or, on the
contrary, increase it. A study by Gaibleret al. 2012 [Gaibler
et al., 2012] shows how the jet produces a redistribution
of the Star Formation Rate in its host galaxy: near the jet
source, in the centerof the galaxy, the SFR is reduced because of
the disruption caused by the
jet. In the rest of the galaxy on the contrary, the
additional thermal energyproduced by the jet interacting with the
ISM causes an increase of the SFR. Thisresults in a star population
completely different with or without the jet. Jetsalso carry away
rangular momentum, matter and kinetic and thermal
energy([Schawinski et al., 2006], [Sutherland and Bicknell, 2007],
[Elbaz et al., 2009]and [Gaibler et al., 2012]). The energy present
in the jet, and the presenceof both shocks and magnetic field, make
them to be a strong candidate foracceleration of particles to the
peta-electronvolt (PeV). This could be the sourceof the far right
part of the cosmic ray power spectrum observed on earth (fig.1.7).
Recent studies use simulations to see how efficient this
acceleration could
be ([Meli and Biermann, 2013] and [Asano et al.,
2014]). In particular, the firstuses a Monte Carlo technique
following relativistic electrons evolving along
staticre-collimation shocks of an AGN jet. Using in turn sub- or
super-sonic shocks,they obtain a good understanding of the
efficiency of such acceleration andproduce electron population
spectra comparable with observations. On a similaraspect, an idea
would be to have particles accelerated in the magnetosphereof stars
and planets crossing the path of the jet beam. [Araudo et al.,
2013]shows how a bow shock would form when an early stage star
interacted with anAGN jet. [Bosch-Ramon et al.,
2012] simply drops a molecular cloud or an old
red giant in the path of the jet and follows how the material is
accelerated andcould be the site of particle accelerations.
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INTERACTION OF JETS WITH THEIR SURROUNDINGS 11
Figure 1.7: The ‘Swordy spectrum’ of the Pierre Auger
Observatory, updatedby William F. Hanlon from the University of
Utah. The particle flux as functionof their energy displays the
cosmic ray spectrum measured on and close toearth, with the famous
knee and ankle. At the same time, the plot indicatesthe typical
energy that can be reached by particle accelerators.
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12 INTRODUCTION
Finally one of the most dynamic fields of the study on
relativistic jets, is theirrole in cosmological models. Since a
long time, there exists a discrepancybetween observed temperatures
of galaxy clusters and the ones predicted from
models. The first are higher than the latter. Cosmology needs to
feedback energyaccreted in objects like galaxies. It is shown that
AGN jets have powerful enoughluminosity to reheat the cosmological
models to the observed temperatures([Silk and Rees, 1998]
and [Zanni et al., 2005]). This feedback is now part
of most of cosmological models. But the way relativistic jets
share their energywith their surroundings is not yet completely
understood. Many mechanismsare thought to play a part: conversion
from kinetic to thermal energy at thehead of the jet; shocks
created by the jet and propagating outward; mixing of material
through instabilities of Kelvin-Helmholtz (KH) and
Rayleigh-Taylor(RT) type. Recent works, including our chapter
3, try to give more insighton this energy transfer
([Monceau-Baroux et al., 2012] and [Cielo et al.,
2014]).Some recent studies even show how XRB jets could contribute
to this feedback[Fragos et al., 2013].
1.6 Where we stand
In my own work I will consider the jet out of its formations
regions. I will follow
relativistic jets propagation and their interactions with their
surroundings. I willfollow two axis of study. The first axis is the
complementarity in the study of relativistic jets of analytic,
simulation and observation. The second axis of studycompares two
classes of relativistic jets: AGN related jets, and XRB related
jet. In chapter 2, I will introduce the tools which
are necessary throughout mywhole study and highlights the second
axis: I will describe the models and codesused in my different
simulations as well as the observational instruments neededto
obtain the data with which my own results were compared. The second
axiswill become more evident as we move to chapter 3 were
I used parameter typicalfor AGN jets of the FR-II classification.
Through a parametric study we will
try to have a global picture of the early time propagation of an
AGN jet. Iwill study the impact of the parameters both from the jet
and its surroundings,that affect their common interaction. In
chapter 4 I will be able to comparethat first case with a
jet typical of XRB associated jet. In the same chapter, Iwill also
study the effect of the precession of the jet on its interaction
with itssurroundings. I will move further and take parameters
typical for the SS433XRB jets. This jet is a precessing jet well
observed on the parsec scales. In thissame chapter, by mean of
reconstructed radio-maps, I will directly compare mysimulations
with observation. One of the consequences of this study is to
realize
the presence of a discrepancy between the observations of SS433
at parsec scales,and around the 50 parsec scale where it interacts
with the supernova remnant
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WHERE WE STAND 13
in W50. I will offer a solution to this discrepancy in
chapter 5 where I will showthe drastic change in
morphology of the jet as it propagates and interacts withthe
medium.
By mean of state of the art simulations, I explore active topic
and place myself in the growing field that strongly linked
together observations and simulations.I develop analytical tools
often missing from many publications on relativistic
jets which often rely on generic morphology to draw their
conclusion. Morespecifically I develops ’on the fly’ algorithm
allowing me to locate the differentregions of the jet and its
medium as we will discuss further in chapter
3 and 4.
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Chapter 2
Models and Tools
2.1 Special relativistic hydrodynamics
Many simulations for relativistic jets use a fluid model, it
being special orgeneral relativistic (magneto) hydrodynamic. Such
model can only be used if the scale of the system (L) is large
by comparison with the scale (λ) of theindividual dynamics of the
(charged) particles,
L λ. (2.1)
In the context of astrophysical objects we can consider
different λ. If we wereto consider a purely hydrodynamic
fluid, the relevant λ would be the mean freepath. For
ion-ion coulomb interaction, this expresses as:
λfree = 1.25 1010 ( T 109 )2
ρ1010 lnΛ
[cm], (2.2)
with T the temperature in
Kelvin, ρ the density in cm−3 and lnΛ the
Coulomblogarithm equal to 20 for ionized hydrogen. This also gives
a constraint on thetime scale. The thermal velocity writes as
vth = kBT
m , (2.3)
15
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16 MODELS AND TOOLS
with kB the Boltzmann constant and m the
mass of the particle. We can thendefine a collision timescale, i.e
the typical timespan in between consecutivecollisions, as
tcoll = λfree
vth. (2.4)
We can use a macroscopic model when the scale of the system is
large withrespect to the mean free path and its timescale has to be
large with respect tothis collision time. Looking at
table 2.1, we obtain that the mean free path is of the
order of our jet radius. This alone cannot justify the use of the
fluid model.
In reality, jets undoubtedly contain charged particles, and
plasma aspects must
be accounted for. The Larmor radius, or gyro-radius, is the
radius of the circularmotion of a charged particle in the presence
of a uniform magnetic field of strength B, and this
length-scale is set by
λLarmor = mv⊥ZeB
, (2.5)
with Z the charge number (Z = 1
for proton), e the elementary charge andv⊥
the velocity component perpendicular to the direction of the
magnetic
field. As the magnetic field is expected to be turbulent, the
latter v⊥ velocityis approximated as being equal to
the thermal velocity as seen in equation2.3. Looking at
table 2.1 we see that this λ is many
orders of magnitudesmaller than the typical scale of our problem.
As in many studies like in[Blandford and Rees, 1974], this is the
argument that will allow us to considerfluid-like behavior, even if
the magnetic field is dynamically weak. In the restof this thesis,
we consider that this weak magnetic fluid (observations tendto show
a weak magnetic field of a few µG up to a few G for AGN jets
andµ-quasar) is advected with the fluid resulting in the variation
of the pressurethrough adiabatic compression and expansion to
result in a similar variation of
the intensity of the magnetic field strengh. This will allow us
to use the pressureas a proxy of the magnetic field for emission
mapping purposes in section 2.4.
Finally we ensure neutrality by calculating the Debye
length:
λD =
0kBT
ne2 , (2.6)
with 0 the void permittivity and n the
particle density. We see that the Debye
length remains much smaller than our system size as seen in
table 2.1. Finallywe see that the plasma parameter Λ
expressed as
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SPECIAL RELATIVISTIC HYDRODYNAMICS 17
Λ = ρλ3D, (2.7)
is always large, ensuring sufficient particles in the Debye
sphere for fluid modeling.Again using table 2.1 we
calculate that Λ is higher than 8 108 for all
medium,ensuring that we can consider a non magnetized plasma, since
the fluid strengthsare very weak.
Our model does not take into account full general relativity.
Sufficiently far awayfrom the compact object giving rise to the
jet, we are not subject to its stronggravitational field. We use a
flat Minkowski metric, as the jet reaches velocitiesclose to the
speed of light. Its Lorentz factor γ =
1√
1−v2/c2 is significantly
higher than one as seen in table 2.1. For this reason we
will make use of a
special relativistic fluid model.
From these considerations the model we use is therefore special
relativistichydrodynamics expressing the conservation of different
relevant quantities in 4dimensional space time. Using a lab frame
equipped with a Minkowski metricwe can use a familiar space-time
split. The first equation is the conservation of particle
number
∂ργ ∂t
+ ∇ · (ργv) = 0 , (2.8)
where γ is the Lorentz factor, ρ is
the proper rest frame density and v is
thethree-velocity of the fluid in the chosen Lorentzian lab frame.
The secondequation is the conservation of momentum, written as
∂ S
∂t + ∇ · (
Sv) +
∇ p = 0 , (2.9)
where p is the pressure in the fluid frame and
S is the momentum densitydefined as
S = hγ 2ρv . (2.10)
where h is the specific enthalpy. The third equation
expresses the conservationof energy using the energy density
variable τ
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18 MODELS AND TOOLS
Object Masssource(M )
Schwarzschildradius (cm)
Size jet(pc =3 1018 cm)
γ Lorentzfactor
AGN 107−8 [1] ∼ 3 1013 ∼ 106 [2] 2−
40 [3]µ-quasar 4− 16 [4] ∼ 12− 48 105 ∼ 1
[2]
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CODE - MPI-AMRVAC 19
∂τ
∂t + ∇ · ( S − ργv) = 0 ,
(2.11)
where the energy density τ in the lab frame
with the rest mass contributionalready subtracted, is related to
the proper density and pressure as follows
τ = ρhγ 2 − p− ργ . (2.12)
This gives us a system of equations with variables ρ,
S , τ and p. This
callsfor a closure relation usually of the form p =
p(ρ, τ ). In our case we have a
relativistic fluid and we can use the Mathews approximation to
the Synge gasequation as our closure relation (see [Blumenthal and
Mathews, 1976]), and itwrites as
p =
Γ− 1
2
ρ
eintm p
− m peint
. (2.13)
where Γ is a parameter, taken as the
non-relativistic value of the polytropic
index of the gas 53 , eint =
m p + eth is the specific internal
energy includingthe rest (proton) mass m p and the
specific thermal energy eth. Therefore, therelativistic
specific enthalpy h used before is defined as
h = 1
2
(Γ + 1)
eintm p
− (Γ− 1) m peint
, (2.14)
This is equivalent to using a spatially varying, effective
polytropic index Γeff =
Γ− Γ−12 (1− m2
p
e2int
) that varies between its classical value
of 5/3 and its relativistic
value of 4/3. The speed of light is taken equal to
unity in our model andtherefore in all equations above.
2.2 Code - MPI-AMRVAC
MPI-AMRVAC is a finite volume code designed to solve for sets of
partial
differential equations that relate to a number of physical
systems. Its corefunction aims to solve a set of equations in a
conservative form :
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20 MODELS AND TOOLS
∂ tU +∇ · F (U )
= S phys(U, ∂ tU, ∂ i∂ jU,x,t)
(2.15)where U is the set of (conserved)
variables and F (U ) the corresponding
fluxes.
When the right-hand-side source terms S phys
are absent, these clearly describeconservation laws for
U . In the special relativistic hydro (SRHD) case, we
have
U = (ργ, S, τ ).
MPI-AMRVAC is designed to solve conservative equations and the
correspondingconserved variables U =
(ργ, S, τ ). It is nevertheless possible, given the
usageof the correct code parameter, to set the initial conditions
of your problem intoprimitive variables u = (ρ, v, p).
The code will therefore transform the variablesbefore going through
the computational cycle. Particular care was taken inMPI-AMRVAC
that this conversion does not result in a loss of precision.
2.2.1 MPI-AMRVAC structure
The modular architecture of MPI-AMRVAC allows to easily adapt to
a widerange of problems. Pre-existing physics modules exist for HD,
MHD, SRHD.New physics modules can be created for any set of
equations that can be writtenin the form of equation 2.15. It
is possible to choose between a wide variety of both time and
space discretization. After the solver and physics module are
defined, the user can write a user file. This file implements
the initial condition,or acts on the evolving quantities during
simulation time to for example definean inner boundary. Post
processing or analysis during runtime of the data canbe added. The
code has fully automated adaptive mesh refinement
capabilities,while it is also possible to in addition define manual
behavior of the refinementprocess of the grid as seen later on.
2.2.2 Finite Volume Model
MPI-AMRVAC typically uses a finite volume model although
recently alsoconservative high order finite differencing solvers
have become available[Keppens and Porth, 2014]. The finite volume
method considers a finite volumeV for which we define
quantities U . For each edge of that volume, the
codecalculates a flux F of those quantities going
through the said edge A (see fig.2.1). Such fluxes
F (U ) are calculated by using a variety of
solvers we brieflydescribe later on. Knowing both quantities
U and fluxes F (U ) the code
thenadvances in time using a selection of multi-stage time stepping
schemes (e.g.Runge Kutta) and updates the quantities in the control
volume.
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CODE - MPI-AMRVAC 21
Figure 2.1: We consider a volume V with
associated quantities U n. Acrosseach edge of this
volume we consider the fluxes of the quantities
F (U n), hereindicated to be given in a Cartesian
x-y sense.
2.2.3 Spatio temporal discretization
The basic ingredients for the discretization are a method to
calculate the fluxesat the edges of the volume, and a solver to
advance in time. All spatial solvers for
fluxes work on the same principle: we use a cell-face
reconstruction employinga finite sized stencil of quantities on
both sides normal to the edge. Since we dothis for obtaining both a
left- and a right-faced value, we end up with a systemwith two
continuous states. Such states separated by a discontinuity result
in aso called Riemann problem. Analytic methods exist to calculate
fluxes resultingfrom the known temporal evolution of this
discontinuity.
In MPI-AMRVAC two time discretizations are mainly used. The
first one is asimple two step predictor-corrector scheme and
computes as:
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22 MODELS AND TOOLS
U n+ 1
2x =U
nx +
∆t
2 S (U nx )
− ∆t2∆V x
N D1
(Ai+1/2F (U ni+1/2)−Ai−1/2F (U ni−1/2)),
U n+1x =U nx + ∆tS (U
n+ 12
x )
− ∆t∆V x
N D
1(Ai+1/2F (U
n+1/2i+1/2 )−Ai−1/2F (U
n+1/2i−1/2 )),
(2.16)
with U x the component
of U in grid point x,
S the source term including boththe physical
source term from equation 2.15 and the geometrical source
term(null in Cartesian geometry), ∆V x the volume
of the cell at position x which isbound in the
N D dimensions by edge centered surfaces indicated
by A.
The second one is a three step scheme and this writes as:
U n+1x
=U nx
+ ∆tS (U nx
)
− ∆t∆V x
N D1
(Ai+1/2F (U ni+1/2)−Ai−1/2F (U ni−1/2)),
U n+1/2x =3
4U nx +
1
4U n+1x +
∆t
4 S (U nx )
−
∆t
4∆V x
N D
1
(Ai+1/2F (U n+1i+1/2)
−Ai−1/2F (U
n+1i
−1/2),
U n+1x =1
3U nx +
2
3U n+1/2x +
2∆t
3 S (U nx )
− 2∆t3∆V x
N D1
(Ai+1/2F (U n+1/2i+1/2
)−Ai−1/2F (U
n+1/2i−1/2 ),
(2.17)
To compute the fluxes MPI-AMRVAC offers a variety of schemes
among
which the TVDLF method (Total Variation Diminishing
Lax-Friedrichs, see[Tóth and Odstrčil, 1996]) is a robust scheme.
It has the advantage of not using
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CODE - MPI-AMRVAC 23
a Riemann solver and is therefore quite fast for heavy 3D
simulations. It doesnot produce spurious oscillations at shock
front. It has the downside to be morediffusive than other schemes
like the HLLC or TVD schemes. The physical flux
finally writes as
F LF i+1/2 = 1
2[F (U Li+1/2) + F (U
Ri+1/2)− LF cmaxi+1/2(U Ri+1/2 − U Li+1/2)],
(2.18)
with
cmaxi+1/2 ≡
cmaxU Ri+1/2 + U
Li+1/2
2 , (2.19)
the fastest characteristic speed cmax in the system at
hand. LF is a usercontrolled parameter to reduce
dissipation, we can use LF = 1 or less.
The HLL method (Harten-Lax-van Leer) uses further approximations
than theTVDLF. The physical fluxes compute as
F HLLi+1/2 = λ
+
F F (U
L
i+1/2)− λ−F F (U R
i+1/2) +
HLL
λ−F λ+
F (U
R
i+1/2 − U L
i+1/2)λ+F − λ−F
,
(2.20)
where λ±F are the fastest wave speeds in both
directions and HLL is again a
user controlled parameter against dissipation.
When λ+F = −λ−F , this reducesto the previous
TVDLF scheme. The HLL scheme can be further improved byintroducing
an intermediate state at the contact discontinuity as developed
in[Mignone and Bodo, 2006] resulting in the HLLC scheme.
A full derivation of the different schemes can be found in
chapter 19 of [Goedbloed et al., 2010]. An adapted scheme was
used for our 2D simulations:to resolve fine structure the scheme
uses HLLC for the whole domain. Wherethat scheme would induce
important spurious oscillations, the scheme thenswitches to the
more robust TVDLF (see [Meliani et al., 2008]).
One important point to keep all explicit schemes numerically
stable is for thewaves used in the flux calculation to not go
further than one cell. This resultsin the so-called CFL condition
which expresses as:
|∆t∆x
cmax|
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24 MODELS AND TOOLS
This sets an important condition on the maximal value of the
time step ∆t.One has to be careful that when increasing the
number of levels allowed in thesimulation, we also reduce the cell
size. By doing so it will lower the value of
the time step by reducing the spatial step. This results in an
increased numberof time steps, which increases the wall-clock time
needed to reach the samecomputational time.
Cell interface states U L,Ri+1/2 are computed
from the cell center values as:
U Li+1/2 = U i + ∆U i/2,
U Ri+1/2 = U i+1−
∆U i+1/2,(2.22)
where ∆U i is a limited slope. Different
flavors of this limiter are available inMPI-AMRVAC among which we
mention three here in full. Most robust butdiffusive is the minmod
flavor:
∆U i = ∆U imax[0, min(1, ri)]. (2.23)
Up to third order accuracy can be reached with the Koren limiter
[Koren, 1993],
given by
∆U i = ∆U imax[0, min(2ri, 2 + ri
3 , 2)] for U L,
∆U i = ∆U imax[0, min(2ri, 1 + 2ri
3 , 2)] for U R,
(2.24)
and the cada3 limiter[Cada and Torrilhon, 2009], given by
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CODE - MPI-AMRVAC 25
∆U i =sgn(∆U i)max[0,
min[2|∆U i|+ sgn(∆U i)∆U i−1
3 ],
max[−αsgn(∆U i)∆U i−1,
min[β sgn(∆U i)∆U i−1,
2|∆U i|+ sgn(∆U i)∆U i−13
, γ |∆U i|]]]] for U L,
∆U i+1 =sgn(∆U i+1)max[0,
min[2sgn(∆U i+1)∆U i + |∆U i+1|
3 ],
max[−
α|∆U i
|, min[β
|∆U i+1
|, 2sgn(∆U i+1)∆U i + |∆U i+1|
3 ,
γ sgn(∆U i+1)∆U i]]]] for U R,
(2.25)
where free parameters α = 0.5, β = 2
and γ = 1.6 are typically taken.
A demonstration of the use of these different schemes can be
seen in fig. 2.2 usinga three step time advance and
’cada3’ limiter. We use a simple 2D cartesiansimulation of a
relativistic continuous and cylindrical jet (SRHD equations).The
set up is similar to the one which will be used in chapter 3.
The resolution of 60x120 for a domain of [-0.5 0.5]x[0
1.5] is low on purpose to highlight the effectof the different
schemes. Only one level of refinement is turned on resulting
inturning off the AMR scheme (see following section). We can see
how the HLLCand HLLCD schemes reveal more features where the
diffusivity of the TVLDFsmooths the sub-structure. This simulation
took 3264 s for HLLC, 3156 s forHLLCD and
3114 s for TVLDF. We can conclude that while it
conserves theshocks and secondary features as well as the HLLC
scheme, the hybrid HLLCDdoes represent a gain in computational
time. Nevertheless the differences areonly visible in the logarithm
of the quantities and the TVDLF scheme willstill be accurate enough
for the 3D simulations. All schemes are second order
accurate on smooth solutions, when combined with a suitable
limiting procedure.The choice of the limiter, together with the
time integrator method, can evenlead to higher, up to third order
accurate behavior. Their validations and theirimplementations in
MPI-AMRVAC are described in [Keppens et al., 2012] and[Keppens and
Porth, 2014].
For our simulations described in the following chapter, the
different schemesused can be found in table 2.2.
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26 MODELS AND TOOLS
Figure 2.2: Top: Logarithm of density for three simulations at
same physicaltime for schemes HLLC (left), HLLCD (middle) and TVDLF
(right). Plottingthe full domain in the (x,y) plane. Bottom:
Horizontal cut at y = 0.8 of thelogarithm of the
density through the domain for HLLC (continuous), HLLCD(cross) and
TVDLF (dash). The horizontal axis is the x distance and the
verticalaxis is log(ρ).
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CODE - MPI-AMRVAC 27
2.2.4 AMR and MPI
MPI-AMRVAC uses a scheme called Adaptive Mesh Refinement (AMR).
This
allows to have a dynamic adjustment of the resolution of the
domain. Thedomain is divided in a number of blocks set by the user
on the first level. Eachblock is in turn divided into a number of
cells also defined by the user. Thissets up the base resolution on
our domain. For example a domain of 4x3 blockswith 12x12 cells per
block results in a base resolution on the first level of 48x36.The
most evident way to increase the resolution is to augment the
number of blocks in the domain, and/or the number of cells per
block. But applying thisto the whole domain greatly increases the
computational cost, and refines thedomain even in parts where
nothing happens.
The AMR scheme in MPI-AMRVAC allows to set automatic as well as
enforcedconditions to locate parts of the domain where a higher
resolution is allowed. Insuch part of the domain, the block on the
first level is divided into a number of blocks on a level
equal to 2D with D the number of dimensions. Blocks on
thesecond level can again be divided into blocks on an extra level
of resolution andso on as seen on figure 2.3. Therefore for
each extra level of blocks the effectiveresolution is multiplied by
a factor 2 in each dimension without enforcing amaximum resolution
on the whole domain. A set of indices allow to easily locateany
block during execution time as seen on the left panel of figure
2.3. Note
that those indices are the position of the block for its given
level. For examplethe block noted (5,3) is indeed the fifth block
horizontally and the third blockvertically if the domain was fully
on level 2. This combined with the knowledgeof the level of the
block allows to easily locate it.
The user can enforce resolution in parts of the domain (in our
cases we enforcemaximum resolution around the injection region of
the relativistic jet) or bydynamically evolving quantities and
discrete evaluations of their spatio-temporalvariation over the
domain to follow the dynamics as seen on fig. 2.4. If in
anyblock the code finds a cell with a local relative error such
as
E Relx,iw > (2.26)
with the user defined threshold, then the refinement
is triggered for that block.MPI-AMRVAC offers three different
estimations of the local error resultingfrom the variation of the
quantities over the domain. The first method usesa Richardson-based
estimator predicting the quantities at the later time
tn+1xusing the value at the same location at a time
tn−1x and t
nx . The second method
uses only the latter two sets of quantities. Finally the third
method that will
be used in our work is a Löhner type estimator. It expresses the
local relativeerror at position x and for quantity
u as:
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28 MODELS AND TOOLS
Figure 2.3: For the hypothetical Cartesian grid exploiting 4×3
grid blocksat level l = 1, the left panel shows the global grid
indices while the tree
representation is given at the right panel. From [Keppens
et al., 2012]
E Relx,U =
N U
max(DU , ), (2.27)
with numerator
N U =i1
i2
[∆i1(∆i2δ U )]2, (2.28)
and denominator
DU =i1
i2
[S i1(|∆i1δ U |)
+ f wave,1S i2(S i1 |δ U |)]2,
(2.29)
with δ U the local gradient of the
U variable, ∆i the discrete central
differencesintroduced per dimension i and
S i are sum operators. Finally is
theerror threshold set by the user. f wave is a
parameter usually of the order10−2. A more complete description of
this error estimator can be found in[Keppens et al., 2012]. The
code is also able to trigger refinement both onconservative or
primitive variables.
The domain is therefore divided into a number of blocks on
different levels.MPI-AMRVAC then uses a space filling curve through
the block hierarchy toorder the blocks, and depends on Message
Passing Interface (MPI) to distribute