Thesis Remi

8/18/2019 Thesis Remi

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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:

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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.

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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

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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

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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

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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

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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

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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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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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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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∆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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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

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