University of Amsterdam - UvA...Scienti c abstract Background: Sagittarius A* (Sgr A*) is the...

University of Amsterdam

MSc Physics and Astronomy

Track: Astronomy & Astrophysics

Master Thesis

The impact of radiation on the dynamics and

appearance of the supermassive black hole in

the Galactic center

“Have you seen Interstellar?”

by

David Jacob van Eijnatten

10645462

July 29, 2018

60 ECTS

2017 - 2018

Supervisor:prof. dr. S.B. MarkoffDaily supervisor:Koushik Chatterjee, MSc

Examiners:prof. dr. S.B. Markoff

prof. dr. M.B.M. van der Klis

Anton Pannekoek Institute for Astronomy

Scientific abstract

Background: Sagittarius A* (Sgr A*) is the supermassive black hole of 4.6×106M in thecenter of the Milky Way. It is accreting at a very low rate of 10−9 − 10−7M/yr. Flaresare observed in the near-infrared and X-ray frequencies. The origin of the flares is theaccretion flow near the black hole, but its physical mechanism is unknown. Recently, theEvent Horizon Telescope imaged the flow around the black hole with a resolution that ison the order of the event horizon of the black hole. This work focuses on discovering theorigin of these flares and the appearance of Sagittarius A* on these event horizon scalesthrough radiative cooling.Method: We use the general relativistic magnetohydrodynamic simulation code HAMRR in2D combined with semi-analytical cooling functions for optically thin flows. The electronsare assumed thermal and the temperature is some fraction < 1 of the proton temperature.We combine this with the ray-tracing code BHOSS to generate images and spectra. Thegas starts out in a disk with a radius of 50 RG (RG = GM

c2) around the black hole and is

allowed to be accreted for a minimum of one day.Results: We find that even an accretion disk around a black hole with an accretion rateas low as Sgr A* is dynamically heavily affected by radiative cooling. The temperaturecan drop an order of magnitude compared to a non-cooled disk. Its spectral appearanceis also affected. The synchrotron self-absorption frequency shifts, the flux decreases anda large portion of the high energy tail disappears from the synchrotron spectrum whencooling is incorporated. We find a best-fit accretion rate of M = 1 × 10−8M/yr at aninclination of 85 deg and a temperature ratio of 0.3. Unfortunately we were not ableto statistically investigate flares as the MRI is not resolved anymore after a few days ofsimulation time.Conclusion: At all accretion rates the dynamics of an accretion disk are affected by theloss of energy through radiation. When analyzing the results from the Event HorizonTelescope it is important to account for radiative losses. The accretion rate we find ishigher than was found previously with other studies, although we are consistent whenwe lower our resolution. Flaring studies using GRMHD simulations are difficult withouthigh-resolution 3D simulation with strongly magnetized disk where the MRI functionsfor tens of days.

Popular abstract

Black holes are some of the most fascinating objects in the cosmos. They can grow tomasses millions or billions times heavier than the sun, they can impede the growth ofgalaxies and sling out matter to distances trillion times their diameter. Our Milky Wayalso contains one of these supermassive black holes, called Sagittarius A*. Its mass isabout five million times the mass of the sun and we can observe it sucking up (“accreting”)small amounts of gas that is swirling around in its neighbourhood. We do not fullyunderstand what we see, however: almost every day we see “flares”, burst of radiationthat last a short time. To investigate this and to observe the black hole “shadow”, theshape the black hole imprints on the surrounding gas, astronomers are taking a pictureof Sagittarius A* with a resolution that is about the diameter of the black hole with theEvent Horizon Telescope. To understand what we will see in this picture and to explainthe flaring behaviour of Sagittarius A* we use computer simulations to replicate theinteraction between the black hole and its surrounding gas. This gas behaves like a fluidcontaining a magnetic field, so the simulated physics is the laws of fluid dynamics andgeneral relativity. To this, I added optically thin radiation. Radiation allows energy toescape in the form of light. We find that this energy loss has a significant impact on howthe gas behaves and on what the picture from the Event Horizon Telescope will look. Wealso found that that although our simulations were useful to determine the appearance ofthe accretion process, it was not possible to simulate this process long enough to gatherenough statistics to say anything about the flares.

Samenvatting

Zwarte gaten behoren tot de meest fascinerende objecten in het heelal. Ze kunnen miljoe-nen tot miljarden keer zo zwaar zijn als de zon, ze bepalen de formatie en groei vansterrenstelsels en ze kunnen enorme hoeveelheden materiaal over lengten afschieten diebiljarden mall zo lang zijn als hun diameter. Ons melkwegstelsel heeft ook een superzwaarzwart gat, genaamd Sagittarius A*. Het weegt 4,6 miljoen keer zo veel als de zon en hetis geobserveerd dat het kleine hoeveelheden gas opzuigt (“accreteert”). Hoewel we ditweten, begrijpen we niet alles wat we zien: elke dag zien we flikkeringen van korte duurin het geobserveerde licht. Om deze flikkeringen te onderzoeken en om de “schaduw”van het zwarte gat te zien, nemen astronomen een foto van het zwarte gat met eenresolutie hoger dan de diameter van het zwarte gat met de Event Horizon Telescope.De “schaduw” is de vertekening van het gas door het zwarte gat. Om deze foto en deflikkeringen te begrijpen draaien wij computer simulaties van de de relevante natuurkunderond het zwarte gat. Het gas gedraagt zich als een gemagnetiseerde vloeistof en aldussimuleren wij de vloeistofwetten en zwaartekracht. We hebben diverse stralingsprocessentoegevoegd, waardoor energie kan ontsnappen aan het systeem in de vorm van licht. Wehebben ondervonden dat dit energieverlies een significant effect heeft op het gedrag vanhet gas en op het uiterlijk van accretieproces dat wij op aarde kunnen observeren. Hoewelonze simulaties nuttig waren om dit uiterlijk te bepalen, was het lastig de origine van deflikkeringen te bepalen omdat onze simulaties het verzamelen van voldoende statistiekenniet toelieten.

Contents

1 Introduction 61.1 Supermassive black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Sagittarius A* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 Flares, spectra and accretion flows . . . . . . . . . . . . . . . . . . . . . . 111.4 The fundamental plane of black hole accretion and state transitions . . . 141.5 The Event Horizon Telescope . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Introduction to simulations and research question . . . . . . . . . . . . . 18

2 Theory 212.1 General relativity and the Kerr metric . . . . . . . . . . . . . . . . . . . 212.2 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.2 Magneto-rotational instability . . . . . . . . . . . . . . . . . . . . 23

2.3 General relativistic magnetohydrodynamics . . . . . . . . . . . . . . . . . 252.3.1 Conserved fluid quantities . . . . . . . . . . . . . . . . . . . . . . 252.3.2 The energy-momentum tensor . . . . . . . . . . . . . . . . . . . . 252.3.3 Conserved electromagnetic quantities . . . . . . . . . . . . . . . . 27

2.4 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4.1 The implications of low optical depth . . . . . . . . . . . . . . . . 302.4.2 Approximations for Sagittarius A* . . . . . . . . . . . . . . . . . 30

3 Numerical methods 363.1 Finite volume methods and Godunov solvers . . . . . . . . . . . . . . . . 363.2 The ∇ · ~B = 0 condition . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.3 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.4 Density floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.5 Unit scaling and accretion rates . . . . . . . . . . . . . . . . . . . . . . . 383.6 Timestep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.7 Scale height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.8 Initial conditions and other parameters . . . . . . . . . . . . . . . . . . . 41

4 Results 43

5 Discussion 51

6 Conclusion 536.1 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Acknowledgements 55

A Ray-tracing results 63

1

List of Figures

1.1 The M-σ relation. Black hole mass is correlated with the bulge velocitydispersion in galaxies through a powerlaw. This implies that black holegrowth and galaxy formation are related. Taken from Gillessen et al. (2009a). 7

1.2 Image of galaxy cluster MS 0735.6+7421. In orange, optical data fromthe Hubble Space Telescope is shown, indicating the location of the stars.In purple, radio data from the Very Large Array is shown, indicating thelocation of the jet. In blue, X-ray data from the Chandra X-ray observatoryis shown. The cavities in the in the X-ray emitting gas coincide with thejet, implying that the jet “blew bubbles” in the surrounding medium. Thisevacuation of gas could halt galaxy growth and is called AGN feedback.Credit: Chandra archive. . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3 Combined radio image from the Very Large Array and Green Bank Tele-scope. In the center of the image the Galactic center, Sagittarius A (SgrA) is visible. Some supernova remnants (SNRs) and non-thermal radiofilaments (NRFs) are also visible. Figure taken from Yusef-Zadeh et al.(2004). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4 The orbit of S2 around Sgr A* as monitored by the Very Large Telescope(VLT) And Keck Observatory. S2 is on a very elliptical orbit (e = 0.88)with a pericentre of 120 AU. The best-fit mass of the central body is3.7± 0.15× 106 M Credit: The MPE-IR Galactic Center Team. . . . . 10

1.5 Ray-traced images of a simulation of Sgr A*. Left: 1.3 mm image at anangle of 60 deg accreting at 4.2 × 10−8 M/yr. Right: scatter-broadenedversion of the left image. The color scale is normalized intensity. In bothimages the black hole shadow is easily visible as the dark area surroundedby radiation. Credit: Moscibrodzka et al. (2014) . . . . . . . . . . . . . . 11

1.6 The three canonical accretion disk models. On the left the scale heightH/R and optical depth τ are given. In the middle a schematic drawingis given of the system, where the arrows refer to an outflow. On theright the luminosity is given as a fraction of the Eddington luminosity, thetheoretical luminosity at which no more matter can be accreted. Credit:Alexander Tchekhovskoy. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.7 A typical multiwavelength spectra of Sgr A*. The dot-dashed line is thethermal synchrotron and SSC emission, the short-dashed lines in the non-thermal synchrotron emission, the dotted line is the total synchrotron andSSC emission, the long-dashed line is the bremsstrahlung emission and thesolid line is the total quiescent emission. The upper bowtie is a strongX-ray flare. The data was not taken simultaneous but over a period of 5years. Taken from Yuan et al. (2003). . . . . . . . . . . . . . . . . . . . . 14

2

1.8 Three hardness-intensity diagrams for different compact objects from Fender(2010). Hardness is defined as the ratio of fluxes in two X-ray bands, wherehigh hardness (A in the diagram) indicates a nonthermal spectrum and lowhardness (B in the diagram) indicates a thermal spectrum. The sourcestravel counterclockwise around the diagram and spend most of their timein the lower right corner, where they are faintest. For neutron stars andblack holes hardness is used on the x-axis while for white dwarfs a definitionanalogous to the DFL is used: powerlaw luminosity over total luminosity. 15

1.9 The DFLD from Kording et al. (2006). This diagram is similar to the HIDof stellar mass compact objects. Powerlaw luminosity over total luminosityis plotted versus total luminosity. R is the radio loudness, the ratio of radioover optical luminosity. The data is binned and only bins with more than10 counts are shown. The gap between the LLAGN on the lower rightand the quasars on the top is caused by various selection effects in theused survey. As you can see, LLAGN statistically show a similar patternas XRBs even with the quiescent state in the lower right being the mostheavily populated. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.10 The fundamental plane of black hole accretion. There is a powerlaw rela-tion between radio and X-ray luminosity for all low-luminosity black holesources when corrected for mass. The only exception is Sgr A*, even itsmost powerful flares shown here can not boost its X-ray emission to theto sufficient heights. The light green datapoints show other black holesources. Taken from Markoff (2005). . . . . . . . . . . . . . . . . . . . . 17

2.1 A schematic of the MRI. Two particles in a differentially rotating flowconnected by a spring exercise forces on each other. Angular momentumis transferred from the inner particle to the outer particle. The spring inthis case is a magnetic field line. Credit: Nick Murphy. . . . . . . . . . . 24

4.1 The initial setup of the simulation. The torus is plotted in density, withthe black hole in the middle. The black lines in the disk give the poloidalmagnetic field loops. The gray lines give contours of the grid. Not all cellsare plotted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 The initial temperature profile of the simulation. These are temperaturestypical for an ADAF, going up to > 1011 K. The black dotted lines giveH/R = 1. The inner regions of the ADAF also agree with this. . . . . . . 44

4.3 The accretion rate for three different accretion rates with Te/Ti=0.3 witha resolution of 10242. Their average values are very close to the intendedaccretion rate. They also remain constant over the whole simulation domain. 45

4.4 Angle-averaged, radial profiles of density, magnetic field strength and tem-perature for three different accretion rates. The solid lines display thecooled simulations M9T3HR, M8T3HR and M7T3HR. The dashed linesdisplay the non-cooled simulation HR scaled to the same density as thethree cooled simulations. The density and magnetic field scale with theaccretion rates as expected. The profiles are clearly affected by the cooling.We do not expect the temperature to explicitly scale with accretion rate,although the temperature is severely lowered by radiative cooling. . . . . 46

3

4.5 A plot of the angle-dependent density profile for M9T3HR, M8T3HR andM7T3HR. As the accretion rate rises, the flow starts to collapse and theequatorial density rises with two orders of magnitude. . . . . . . . . . . . 47

4.6 A plot of the angle-dependent distribution of the positron to proton ratio zfor M9T3HR, M8T3HR and M7T3HR. This ratio times two tells you howmuch leptons there are due to pair-production compared to the leptonsfrom ionized gas. The collapsed high density, low temperature areas of thedisk have a reduced value for z. . . . . . . . . . . . . . . . . . . . . . . . 48

4.7 The angle-dependent distribution of the synchrotron self-absorption fre-quency for M9T3HR, M8T3HR and M7T3HR. The self-absorption fre-quency rises with the accretion rate, making the optically thick inner flowappear increasingly larger. . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.8 Three different time series. On top, the total spatially integrated coolingrate in a 10RG sphere around the black hole is given in arbitrary units. Inthe middle, the accretion rate over the event horizon is given in arbitraryunits. Below, the average ratio of the MRI wavelength over the cell sizeis given separately for each dimension. The horizontal line is drawn at aratio of unity where the MRI wavelength drops below the resolution of thesimulation. The φ-component of the MRI can never be resolved due to theaxisymmetric nature of these simulations. . . . . . . . . . . . . . . . . . . 50

A.1 The 230 GHz images and radio spectra at i = 5 deg, i = 45 deg and i = 85deg for M9T3HR. Observations taken from Falcke et al. (1998); Schodelet al. (2003); Bower et al. (2015). . . . . . . . . . . . . . . . . . . . . . . 64



A.4 The 230 GHz image at i = 45 deg and spectra of M9T3HR compared toHR for which we assumed M = 1× 10−9 M/yr. Observations taken fromFalcke et al. (1998); Schodel et al. (2003); Bower et al. (2015). . . . . . . 67



4

List of Tables

2.1 The radiative and pair-producing processes in a non-magnetized plasma.Taken from Svensson (1982a). . . . . . . . . . . . . . . . . . . . . . . . . 31

4.1 The simulations discussed in this work. We ran 2D simulations at threedifferent accretion rates within the observational constraints of Sgr A*.We run each of these for two different temperature ratios. at a low andhigh resolution of 5122 and 10242, respectively. We ran one very highresolution purposes for a convergence test. For each resolution we also rana noncooled test to benchmark the cooling effects against. The namingworks as follows: M is followed by the negative logarithm of the accretionrate, T is followed by ten times Te/Ti. This is followed by the resolution.LR refers to low resolution (5122), HR refers to higher resolution (10242)and VHR refers to very high resolution (2600x1300). . . . . . . . . . . . 43

5

CHAPTER 1

Introduction

1.1 Supermassive black holes

When Albert Einstein presented the theory of general relativity in 1916 (Einstein, 1916),one of the first solutions found to the Einstein equations was the vacuum solution near astatic, neutral and spherical massive object, like a star or planet. However, the solutionalso seemed to indicate that any object that was sufficiently dense as to have a radiusr < rS = GM

c2, where rS is called the Schwartzschild radius, G is the gravitational con-

stant, M is the mass of the object and c is the speed of light, would collapse to a pointor singularity and nothing inside rS could ever leave this sphere, not even light. What ismore, it is possible for some core collapse supernovae from massive stars to achieve thisdensity. Black holes are now observed beyond a reasonable doubt across a huge rangeof masses: from a couple to ten times the mass of the sun (stellar-mass black holes) asdetected in binary systems to millions or billions of solar masses (supermassive blackholes or SMBHs) as detected in the cores of some galaxies. The formation paths of thestellar-mass black holes are clear: direct formation from supernovae, collapse of accret-ing neutron stars or binary neutron star mergers with a final core mass exceeding theTolman-Oppenheimer-Volkoff limit, the theoretical maximum mass a neutron star canhave (Tolman, 1939; Oppenheimer & Volkoff, 1939). The formation paths of supermas-sive black holes are less well understood (Volonteri, 2010). Most galaxies are thought toto contain their very own SMBH and these galaxies also seem to be heavily influencedby their central black hole even though the black hole contains only a fraction of thetotal galaxy mass. This influence is most clearly expressed in the form of the M − σrelation, the strong correlation between black hole mass and stellar velocity dispersionin the bulge (Ferrarese & Merritt, 2000; Gebhardt et al., 2000), see Fig. 1.1. The mostwidely accepted explanation uses accretion powered outflows as the main influence on theneighbouring regions (Silk & Rees, 1998; King, 2003). More recently it was discoveredthat galaxy merger models could also reproduce the M − σ relation without any interac-tions of the black holes with the surrounding galaxies (Hirschmann et al., 2010).Large scale cosmological simulations of galaxy formation tend to overpredict the amountof massive galaxies that form (Bower et al., 2006b). The only known mechanism thatcan quench galaxy growth on such scales are large scale jets produced by SMBHs.

6

Figure 1.1: The M-σ relation. Black hole mass is correlated with the bulge velocitydispersion in galaxies through a powerlaw. This implies that black hole growth andgalaxy formation are related. Taken from Gillessen et al. (2009a).

Some supermassive black holes are “active”: they are accreting significant amounts ofgas and occasionally driving an outflow. These sources are called “active galactic nuclei”(AGN) and produce some of the most impressive images in astronomy, see Fig. 1.2.Clearly these outflows must have a huge influence on their environment by heating andpushing surrounding gas. Other non-active or “quiescent” galaxies show a history ofoutflow activity (Lintott et al., 2009; Comerford et al., 2017). This suggests that SMBHscan switch on and off, a behaviour more clearly seen in X-ray binaries (XRBs), whichundergo this switch in more human time scales of a few to tens of years (Falcke et al.,2004; Remillard & McClintock, 2006). The influence of SMBH on their environment isusually called AGN feedback. This feedback can have a large impact on star formationrates in the galactic bulge, galaxy formation and galaxy evolution (Fabian, 1994; Di Mat-teo et al., 2005; Croton et al., 2006; Fontanot et al., 2006; Hardcastle et al., 2007; Fabian,2012). By pumping a huge amount of energy in their environment, AGN self-regulatetheir growth. Both low-density interstellar matter (ISM) is evacuated from the bulgeand young stars commonly associated with winds do not form. This is observed as aproportionality between galaxy mass and SMBH mass (Magorrian et al., 1998).

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Figure 1.2: Image of galaxy cluster MS 0735.6+7421. In orange, optical data from theHubble Space Telescope is shown, indicating the location of the stars. In purple, radiodata from the Very Large Array is shown, indicating the location of the jet. In blue,X-ray data from the Chandra X-ray observatory is shown. The cavities in the in theX-ray emitting gas coincide with the jet, implying that the jet “blew bubbles” in thesurrounding medium. This evacuation of gas could halt galaxy growth and is called AGNfeedback. Credit: Chandra archive.

1.2 Sagittarius A*

The Milky Way also has a supermassive black hole at its center. Because the Milky Wayis the only galaxy that we can not look at from a distance, the story starts with theactual determination of its center.Before the radio astronomy boom following the end of the second world war, the onlyway to determine the center of the Milky Way was through optical observations of stellar

8

motions (Goss & McGee, 1996). The center had already appeared in one of Karl Jansky’sradio observations, but was not identified as the Galactic center (Jansky, 1933). It couldbe inferred that our host Galaxy was a spiral galaxy with its center somewhere in thedirection of Scorpius and Sagittarius. Because we can only look towards the center of theMilky Way through its disk, optical observations are almost impossible because of a 25magnitude extinction (Genzel et al., 2010).

Figure 1.3: Combined radio image from the Very Large Array and Green Bank Telescope.In the center of the image the Galactic center, Sagittarius A (Sgr A) is visible. Somesupernova remnants (SNRs) and non-thermal radio filaments (NRFs) are also visible.Figure taken from Yusef-Zadeh et al. (2004).

In the fifties radio emission from Sagittarius was quickly identified as the Galactic centerand the source was dubbed Sagittarius A (Sgr A) (Piddington & Minnett, 1951), see Fig.1.3. A few years after the idea was proposed that a supermassive black could be presentin the center of every galaxy by Lynden-Bell (1969) and Lynden-Bell & Rees (1971) acompact radio source was found inside Sgr A and was identified with our own SMBH.Both the radio source and the SMBH are referred to as Sagittarius A* or Sgr A* (Brown,1982). In later studies, the size of Sgr A* was cut down to < 1 A.U. (Ghez et al., 2003;Shen et al., 2005; Bower et al., 2006a; Doeleman et al., 2008). This indicates a density> 1021 M/pc3. The only stable system on Galactic time scales with such a high densityis a black hole.

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Figure 1.4: The orbit of S2 around Sgr A* as monitored by the Very Large Telescope(VLT) And Keck Observatory. S2 is on a very elliptical orbit (e = 0.88) with a pericentreof 120 AU. The best-fit mass of the central body is 3.7 ± 0.15 × 106 M Credit: TheMPE-IR Galactic Center Team.

The most convincing evidence for the presence of a black hole at the center of our Galaxycomes from monitoring stellar orbits at the Galactic center. In Gillessen et al. (2009a,b)the orbits of 28 stars are presented over a period of 21 years. One of those stars, S2, hascompleted a full orbit since the monitoring began in 1995. From the orbit of S2 a mass forSgr A* can be derived, 3.7± 0.15× 106 M, and a minimum size, 120 A.U., since S2 didnot collide with Sgr A*. Using the periastron approach of S2, the gravitational redshift ofits light due to Sgr A* could actually be measured (GRAVITY Collaboration et al., 2018).

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Figure 1.5: Ray-traced images of a simulation of Sgr A*. Left: 1.3 mm image at an angleof 60 deg accreting at 4.2 × 10−8 M/yr. Right: scatter-broadened version of the leftimage. The color scale is normalized intensity. In both images the black hole shadowis easily visible as the dark area surrounded by radiation. Credit: Moscibrodzka et al.(2014)

1.3 Flares, spectra and accretion flows

We know that Sgr A* has daily flaring activity in the near-infrared (NIR) and X-rays(Baganoff et al., 2001; Genzel et al., 2003; Ghez et al., 2004). Determining the origin andbehaviour of these flares is imperative for understanding and interpreting the data thatwill be gathered by the Event Horizon Project, see Sec. 1.5. Many different explanationshave been put forward for the cause of the increased emission. Due to observationalconstraints and the short timescales the possible flaring mechanisms are actually quiterestricted. All possible physical mechanisms proposed below have in common that theyoccur in the inner tens of RG in either an inflow or outflow. Markoff et al. (2001) sug-gested jet activity, Yusef-Zadeh et al. (2006a) suggested an expanding plasmon from theVan der Laan model and Zubovas et al. (2012) suggested tidal distruption of asteroidsor exoplanets. Some explanations have to do with the accretion flow, from orbiting hotspots (Broderick & Loeb, 2005, 2006) and wave instabilities (Tagger & Melia, 2006) tononthermal electrons produced by reconnection or turbulence (Yusef-Zadeh et al., 2004;Dodds-Eden et al., 2009).The accretion flow around Sgr A* is thought to be advection dominated (ADAF; Narayanet al., 1998; Yuan et al., 2003). This is a low accretion rate version of an accretion disk.Accretion disks are produced when gas surrounding a massive object becomes gravitation-ally bound while retaining some angular momentum. The gas starts to spiral around themassive objects and a disk-like object forms, usually just referred to as an accretion disk.Since the disk is not rigid, the gravitational field causes a differential rotation. This in

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turn causes viscous dissipation of energy and gas slowly moves through the disk towardsthe central source. The importance of viscosity is usually measured by the Reynolds num-ber Re, the ratio of inertial forces to viscous forces. If Re >> 1, inertial forces dominateand the disk is stationary. If Re << 1, viscous forces dominate and the disk moves in-wards. From purely hydrodynamic considerations we obtain Re ∼ 1014, meaning the diskis completely stable (Frank et al., 1992). We do however see accreting accretion disks, sosomething else must cause viscous dissipation. This unknown effect was parametrized bythe α-prescription by Shakura & Sunyaev (1973). This is simply the phenomenologicalformula for the kinmatic viscosity ν = αcsH. Later the magneto-rotational instability(MRI) was discovered by Balbus & Hawley (1991), see Sect. 2.2.2. The appearance ofaccretion disks at different accretion rates is wildly different. Like stated before, one ofthe low accretion rate versions is referred to as an ADAF, a rather puffy disk with scaleheights (H/R, where H is the height of the flow at radius R) of unity where the energyreleased by viscous dissipation can not easily be radiated away because of a decouplingbetween the hot protons and the radiating electrons and as such these flows tend to beextremely hot (up to 1010− 1011 K) and underluminous (Narayan & Yi, 1994). The termadvection dominated refers to the fact that most thermal energy is carried over the eventhorizon or away in an outflow and thereby “cools” the accretion flow instead of radia-tion. As the accretion rate increases, the higher density allows particles to radiate moreefficiently and the disk collapses into a razor-thin disk or α-disk (Shakura & Sunyaev,1973). These emit blackbody radiation and are therefore much brighter than ADAFseven when normalized with accretion rate. At higher accretion rates still, the escapetime of a photon becomes longer than the accretion time scale of the parcel of gas thattrapped it. This leads to a radiatively inefficient disk again often referred to as a slimdisk (Begelman & Meier, 1982; Abramowicz et al., 1988). For a schematic visualizationof these three accretion states, see Fig. 1.6.

Figure 1.6: The three canonical accretion disk models. On the left the scale height H/Rand optical depth τ are given. In the middle a schematic drawing is given of the system,where the arrows refer to an outflow. On the right the luminosity is given as a fractionof the Eddington luminosity, the theoretical luminosity at which no more matter can beaccreted. Credit: Alexander Tchekhovskoy.

Getting back to the flares, their appearance is a well-studied subject. X-ray flares appearas discrete “events”, with processes different to the steady state. The X-ray flares arestrong with an amplification of a few to hundreds of times the quiescent flux and the

12

quiescent flux is very constant (Baganoff et al., 2003). The NIR flares are less separatefrom their background. A quasi-periodic oscillation or simply statistical outliers fromBrownian noise could also be an explanation (Meyer et al., 2008; Do et al., 2009). Thisis contradictory to the fact that every X-ray flare seems associated with an NIR flare, al-though not every NIR flare can be associated with an X-ray flare (Hornstein et al., 2007).Due to the larger spatial resolution of X-ray instruments the thermal bremsstrahlungemitted around the Bondi radius could be obscuring the smaller X-ray flares (Witzelet al., 2012, 2018). NIR flares occur on average 4 times a day, while X-ray flares occurtwice a day, with strong X-ray flares occurring daily (Baganoff et al., 2003; Eckart et al.,2006; Porquet et al., 2008).In 2012, a campaign was launched with the Chandra X-ray observatory to observe Sgr A*for 3 megaseconds or ∼ 35 days. This campaign was called the X-ray Visionary Project(XVP) (PIs: Baganoff, Markoff, Nowak). During the 3 megaseconds, 39 new flares wereobserved, effectively doubling the statistics from the decade prior. Because of the largenumber of flares, a statistical study could be done and many distribution of durations,fluences and luminosities were calculated, see Neilsen et al. (2013). It was also foundthat all of the variable emission is about 10 % of the total emission and was attributableto the inner ∼ 20 RG (Neilsen et al., 2015). In theory, one could try to replicate theseresults using global simulation to find the origins of the flares.A typical multiwavelength spectrum is given in Fig. 1.7. The spectrum has a strongturnover at 1012 Hz called the submillimeter peak. It can be explained by thermal syn-chrotron radiation with a self-absorpion frequency of 1012 Hz (Yuan et al., 2003). Between109 and 1010 Hz there is an excess attributable to synchrotron radiation from nonthermalelectrons. The radio excess could also be explained by an outflow (Falcke & Biermann,1996), and the variability profile of the spectrum is strong evidence for this (Bower et al.,2015). The thermal synchrotron radiation is a necessary presence, because otherwise theNIR upper limits would be violated. The quiescent X-ray emission is produced at largescales out to the Bondi radius, while the flared X-rays have to come from close to theblack hole (Wang et al., 2013; Neilsen et al., 2015).

13

8 10 12 14 16 18 20

log[ν(Hz)]

30

31

32

33

34

35

36

37

log

[νL

ν(e

rg s

−1)]

Sgr A*

Figure 1.7: A typical multiwavelength spectra of Sgr A*. The dot-dashed line is the ther-mal synchrotron and SSC emission, the short-dashed lines in the nonthermal synchrotronemission, the dotted line is the total synchrotron and SSC emission, the long-dashed lineis the bremsstrahlung emission and the solid line is the total quiescent emission. Theupper bowtie is a strong X-ray flare. The data was not taken simultaneous but over aperiod of 5 years. Taken from Yuan et al. (2003).

1.4 The fundamental plane of black hole accretion and state transi-tions

Some stellar mass compact objects like X-ray binaries (XRBs) go through different accre-tion phases on time scales of months, years or decades. These phases are mostly clearlyseen in the hardness-intensity diagram (HID), see Fig. 1.8. X-ray hardness is a measureof how non-thermal that X-ray spectrum is. High hardness means a spectrum that isdominated by a typical powerlaw from a corona or jet base (Markoff et al., 2005), usuallyaccompanied by radio emission from a jet. Low hardness typically means a soft spectrumor a spectrum dominated by a thermal disk component close to the innermost stable cir-cular orbit (ISCO). These compact objects travel counterclockwise along the hysteresiscurve of the HID, first increasing in flux while the hardness stays approximately constant,then travelling towards a soft spectrum where the disk dominates and the jet turns off.As the source gets fainter, it goes towards a quiescent hard spectrum. These sources havea very different formation path (supernovae versus mergers and accretion for XRBs) andaccrete very differently (Roche lobe overflow in XRBs versus interstellar gas and tidaldisruption events for SMBHs) than SMBHs. But there are some indications that theirsupermassive counterparts go through state transitions also. Time scales close to the

14

black hole should scale directly with the black hole mass (see Chap. 2) so that statetransitions can never be observed directly since there is at least a factor of a few millionbetween masses of Galactic and supermassive black holes. However, the HID of SMBHscould be inferred statistically by plotting the current sample. There is another problem:in standard disk theory, the peak temperature of the disc scales as follows: T ∝ M−1/4

(Shakura & Sunyaev, 1973). Where the disk shines in the X-rays for Galactic black holes,it shines in the optical or UV for SMBHs. In Kording et al. (2006) the HID is replacedby the disk fraction luminosity diagram or DFLD, see Fig. 1.9. Hardness is replacedby the powerlaw luminosity over the total luminosity and the intensity is replaced bythe total luminosity. The used sample from Kording et al. (2006) is very similar to theconventional HIDs from XRBs. The same models that are used to fit hard-state XRBscan also be used to fit low-luminosity AGN (LLAGN), see for example Connors et al.(2017).

Figure 1.8: Three hardness-intensity diagrams for different compact objects from Fender(2010). Hardness is defined as the ratio of fluxes in two X-ray bands, where high hardness(A in the diagram) indicates a nonthermal spectrum and low hardness (B in the diagram)indicates a thermal spectrum. The sources travel counterclockwise around the diagramand spend most of their time in the lower right corner, where they are faintest. Forneutron stars and black holes hardness is used on the x-axis while for white dwarfs adefinition analogous to the DFL is used: powerlaw luminosity over total luminosity.

15

Figure 1.9: The DFLD from Kording et al. (2006). This diagram is similar to the HIDof stellar mass compact objects. Powerlaw luminosity over total luminosity is plottedversus total luminosity. R is the radio loudness, the ratio of radio over optical luminosity.The data is binned and only bins with more than 10 counts are shown. The gap betweenthe LLAGN on the lower right and the quasars on the top is caused by various selectioneffects in the used survey. As you can see, LLAGN statistically show a similar pattern asXRBs even with the quiescent state in the lower right being the most heavily populated.

Another indication of a similarity between light and heavy black holes is the fundamentalplane of black hole accretion (Merloni et al., 2003; Falcke et al., 2004), see Fig. 1.4.Galactic black holes in the low-hard state and their supermassive analogs show the samecorrelation between their radio and X-ray luminosity when corrected for mass. This is avery strong correlation, suggesting that the physics of sub-Eddington accretion flows andjet launching are similar across the mass scale. Once normalized for mass, the radiativeprocesses at the inner accretion flow (producing X-rays) and the radiative processes in thecore of the jet (producing the bulk of the radio emission) seem to be linked in the sameway for black holes of all masses. Theoretically, this can be supported by the fact that jetlaunching physics might be scale-invariant, more specifically its equations can be scaled byRG = GM

c2(Falcke & Biermann, 1995; Heinz & Sunyaev, 2003; Markoff et al., 2003), where

RG is the gravitational radius, see Sect. 3.5 for a further explanation of scale-invariance.Although the jet launching plasma physics that dominates the dynamics might be scale-invariant, the associated radiative processes are not, hence the mass correction in thefundamental plane of black hole accretion. There is however one anomalous source, SgrA*. Our very own SMBH lies below the correlation most of the time, while strong X-rayflares boost it within a few standard deviations of the fundamental plane of black holeaccretion (Markoff, 2005). The statistics suggest that flares could be observed that putSgr A* exactly on the fundamental plane of black hole accretion, at which point Sgr A*might start to follow it, for example by launching radiatively stronger jets. It is unlikelythat Sgr A* is the one LLAGN that has a fundamentally different accretion/ejectionmechanism. It might be that Sgr A* is in some intermediate, quiescent phase. There issome evidence to suggest this is that case. The Soltan relation (Soltan, 1982), a method

16

of estimating SMBH mass that works very well for its simplicity, assumes that all massin SMBHs is accreted in gas. At the current accretion rate of Sgr A* it would take manyHubble times to get up to its measured mass. By looking at reflected hard X-rays fromSgr B2 (see Fig. 1.3) and other molecular clouds one can infer an X-ray luminosity 106

times brighter than a hundred years ago for Sgr A* (Revnivtsev et al., 2004; Ponti et al.,2010; Clavel et al., 2013). Lastly, a few years ago the Fermi bubbles were discovered,two γ-ray emitting teardrop-shaped bubbles on either side of the galaxy, extending for8-10 kpc, but originating from within a 100 pc zone near Sgr A* (Su et al., 2010). Manytheories about the formation of these bubbles involve recent jet or general outflow activityof Sgr A* (Zubovas et al., 2011; Guo & Mathews, 2011; Cheng et al., 2011).

Figure 1.10: The fundamental plane of black hole accretion. There is a powerlaw relationbetween radio and X-ray luminosity for all low-luminosity black hole sources when cor-rected for mass. The only exception is Sgr A*, even its most powerful flares shown herecan not boost its X-ray emission to the to sufficient heights. The light green datapointsshow other black hole sources. Taken from Markoff (2005).

1.5 The Event Horizon Telescope

Out of all the discovered SMBHs, our own is by far the closest by a factor of 103 (Fal-cke et al., 2000). Even when taking into account that SMBH masses can go above1010 and a black hole’s radius scales linearly with its mass, Sgr A* also subtends the

17

biggest angular size on the sky, with the nearby LLAGN M87 a close second. Its angu-lar size is also many times larger than those of Galactic stellar-mass black holes, mak-ing Sgr A* the perfect candidate for black hole imaging. The angular size is givenby ∠ = RH/D ≈ 0.2 nanoarcsec MBH

M

kpcD

, giving microarcsecond scales for Sgr A* butnanoarcsecond scales for all other Galactic black holes and most other extragalactic blackholes (Goddi et al., 2017). The Event Horizon Telescope (EHT) (Doeleman et al., 2008,2009; Fish et al., 2011) is going to actually resolve the black hole by looking at theaccreting gas surrounding Sgr A*. From simulations one expects to see the black hole“shadow”, an emissionless patch on the event horizon (Falcke et al., 2000), see Fig. 1.5.The shape of this shadow can be used to test alternative theories of gravity in the stronglimit. This “shadow” is thought to have an angular size of ∼ 50 microarcseconds. Mostof the observations will be done at 230 GHz or 1.3 mm, where the angular resolution is∼ 22 microacrseconds. At longer wavelengths, line-of-sight scattering would dominateany measurement and the gas becomes optically thick, potentially obscuring the shadow(Bower et al., 2004). Shorter wavelengths are technically very difficult, as the accuracyof the measurements needs to be a small fraction of the wavelength. The next phase ofEHT observation can hopefully be done at 0.8 mm.The results of these observations will need to be checked against theoretical predictions.The usual way this is done is by simulating the relevant fluid dynamics in general rela-tivity, then calculating the radiation properties this would produce. This profile is thenfed through a scattering screen calculating to account for the ionized matter between usand Sgr A*. Finally, using VLBI simulations the actual image produced by EHT canbe predicted. The effect of the radiation profile calculation and scattering screen can beseen in Fig. 1.5. Other things can also be calculated in a similar manner, like the spec-trum, lightcurve or polarization profile. The latter can be used to distinguish differentmagnetic field line configurations and as such provides another method to distinguish thejet from the disk. All of these different diagnostics can be fit to the eventual EHT datato determine which processes are exactly at play.

1.6 Introduction to simulations and research question

There are multiple ways to make predictions about these types of flows. One way is theanalytical, the pen-and-paper approach, as in Shakura & Sunyaev (1973); Narayan &Yi (1994). In this method either explicit equations are derived or implicit equations aresolved by hand in order to make predictions. Inevitably, one has to make many simplifica-tions to be able to get a solvable system. Another is semi-analytical, where the amount ofsimplifications are smaller because equations are solved by the computer. This may soundsimilar to a simulation, however, it is useful to distinguish the two. In semi-analytics yousolve for a small number (∼ 10) of physically distinct parameters. In a simulation ornumerical physics, you solve for any number of physically similar parameters, for exam-ple discretized in either space or time. By doing this, very little simplification of yourequations is required. The results are however much harder to interpret because you nowhave millions to billions of values that need to be made sense of.For some problems, analytic methods are adequate as can be seen by the huge amount ofphysics already done before the advent of the computer. For the specific type of problemwe will be addressing in this thesis, the behaviour of gas in the proximity of a black hole,the equations are highly non-linear and are best described by simulations. The differ-ential equations are strictly local in space (only involves directly surrounding points) so

18

we can chop up the space around the block hole in cells and solve the equations betweenneighbouring cells. After the equations are solved, they are evolved in time and thisprocess is repeated.In the limit of infinite spatial and temporal resolution, this allows an accurate repro-duction of reality as long as the relevant physics is taken into account. However, thisinfinite resolution is never achieved due to limited machine precision and computationalspeed. In order to maintain stable solution and speed up the simulation other numericalmethods are used, see Chap. 3. Including the relevant physics is not always straight-forward either. For the simulations in this thesis, the equations we will be solving arethe ideal general relativistic magnetohydrodynamics (GRMHD) equations. These equa-tions describe how gas moves in a curved spacetime. The assumptions of these equationsare valid for astrophysical plasmas, see Chap. 2. In these equations a lot of physics isexcluded, like resistivity, radiation or particle acceleration. Inclusion of these effects isobviously important for the observational signal but even when worrying only about thelarge-scale dynamics, these effects can be important, even for ADAFs where the verylittle radiation is produced (Dibi et al., 2012). The effects of radiation are also nonlinearin the simulations: a change in dynamics caused by the loss of internal energy throughradiation can change the radiation profile.Radiation is a manifestly non-local effect. Depending on the medium, photons can travelgreat lengths before interacting. In two limits, however, radiation can be treated as alocal effect: for plasmas so tenuous that any produced radiation escapes the system im-mediately (optically thin) or plasmas so dense that all radiation interacts many timesinside a single cell (optically thick), allowing again for the possibility of the local trans-port of a radiation field.As stated before, the inclusion of radiation also allows us to compare simulation resultswith observations. In astronomy as a whole, the only way to “observe” in the most generalsense is to detect either radiation, cosmic rays (highly energetic massive particles), neu-trinos and gravitational waves. Because of the large effective spatial resolution of particleexperiements, it is difficult to locate the source of neutrinos, cosmic rays and gravitationalwaves, which leaves radiation as the main source of observational constraints for specificsources.Ray tracing, a technique employed in the making of animated pictures by for examplePixar, allows for the generation of images of 3D objects. This can also be applied toGRMHD simulations (see Fig. 1.5) and can also be used to generate the temporal andspectral signature, see e.g. Dibi et al. (2012); Drappeau et al. (2013).In this thesis, I willll be combining the GRMHD equations with an optically thin radia-tion field to determine the accretion dynamics and hopefully the origins of the flares ofSgr A*. This is done with the GRMHD code HAMR (Liska et al., 2018) a state-of-the-art,highly parallel, GPU-accelerated code that allows us to run bigger and longer simulationsthan ever before. The accretion flow around Sgr A* has been investigated before withGRMHD simulations. In most studies, radiation is not taking into account in the energyconservation equation (Moscibrodzka et al., 2009; Dexter et al., 2009, 2010). If we canobserve these systems, some energy must be lost from these systems. This could be anegligible amount, but this might not always be the case as we will see in Chap. 4. I willtry to show that subtracting the radiative energy from the total energy is important forthe dynamics and observational appearance of Sgr A*. One study did self-consistentlycool their simulations, but with a different code (COSMOS++ Anninos et al., 2005), with-out pair-production and at a much lower resolution, which determines the duration of

19

steady accretion and the amplification of the magnetic field by the MRI (Dibi et al., 2012;Drappeau et al., 2013). This all makes a direct comparison to our results Our highestresolution run is a factor of 50 higher than theirs and we can run simulations for a muchlonger period. In (Drappeau et al., 2013) no images were generated, which we do. Thisall combined will allow us to study the dynamics, observational signature and flaringbehaviour of Sgr A* better than ever before.

20

CHAPTER 2

Theory

2.1 General relativity and the Kerr metric

According to general relativity, the metric of a non-charged, spinning black hole is theKerr metric (Kerr, 1963). There is observational evidence for a nonzero spin in bothstellar-mass and supermassive black holes (Fabian et al., 1989; Zhang et al., 1997; Bren-neman & Reynolds, 2006). Nonzero spin is achieved by black hole mergers and the accre-tion of gas with angular momentum of the same sign. The Kerr metric in Boyer-Lindquistcoordinates is (Boyer & Lindquist, 1967):

ds2 = −(

1−2r

Σ

)dt2+

Σ

∆dr2+Σdθ2+

(r2+a2+

2ra2

Σsin θ

)sin θ2dφ2−4ra sin θ2

Σdφdt (2.1)

where (t, r, θ, φ) are the standard spherical coordinates. Σ = r2+a2 cos θ2, ∆ = r2−2r+a2

and a is the dimensionless spin parameter, which ranges from 0 to 1. When consideringaccretion flows, it is common to let it range from -1 to 1 as to allow for prograde andretrograde disks or photon orbits, but for the metric itself is symmetric in this range.Spacetime itself actually rotates around a spinning black hole. Imagine a test particle atinfinity, where gµν = ηµν , with no angular momentum with respect to the black hole. Asthe particle approaches, spacetime is no longer flat and the test particle’s reference framestarts to rotate with Ω = − gφt

gφφ= 2rac

Σ(r2+a2)+2ra2 sin θ2. This effect is called frame dragging

and means that everything that approaches the black hole will experience a “force” alongthe rotation of the black hole (Misner et al., 1973).A few surfaces can be distinguished from this metric. When ∆ → 0, grr → ∞. This isalso known as the ”event horizon”, located at rH = 1+

√1− a2 (we pick the higher-valued

solution because this reduces to the Schwarzschild solution when a → 0). Nothing canleave the volume the horizon encompasses, thus providing a natural boundary conditionfor the radial coordinate. Another interesting surface exists outside the event horizon.When gtt switches sign, timelike paths can only stay timelike if they rotate in the samedirection as the black hole with a minimum velocity of Ω. Any retrograde flow is no longerpossible. This surface is referred to as the “ergosphere”, located at rE = 1+

√1− a2 cos θ

(we choose the highest-valued solution because the lower valued solution lies within theevent horizon).For the stability of the simulations and to make use of the aforementioned natural bound-ary condition, it would be useful to find a coordinate system in which the event horizonis no longer a coordinate singularity so we can integrate across it. As it turns out, this ispossible with two coordinate transformations, following Meier (2012):

21

dt = dt′ − 4r

∆dr (2.2)

and

dφ = dφ′ − 2a

∆dr (2.3)

These transformations give the line element

ds2 = −(1− 4r

Σ2)dt′2 + (1 +

4r

Σ2)dr2

+Σ2dθ2 + (Σ2 + a2(1 +2r

Σ)) sin θ2dφ′2

+4r

Σ2drdt− 4ar sin θ2

Σ2dφ′dt

−2a(1 +2r

Σ2) sin θ2drdφ′

(2.4)

which only has the curvature singularity at Σ = 0, at the cost of 3 extra nonzero indepen-dent components. Notice that energy and angular momentum in the φ−direction are stillconserved. This is the metric is called Kerr-Schild and used inside HAMR. For numericalreasons usually a simple (diagonal) transformation is made from (t′, r, θ, φ′) to (x0, x1,x2, x3) where t = x0, r = ex1 , θ = πx2 + 1

2(1−h) sin 2πx2 and φ = x3. Collimation of the

grid toward the equatorial plan can be set by h = [0, 1].

2.2 Magnetohydrodynamics

Plasmas appear throughout the astrophysics. From the low-density gas in the ISM or thesolar wind to the dense plasmas of stellar interiors or the universe before recombination.Because plasmas are made up of charged particles, they can sustain magnetic fields andas such behave differently than ordinary gases. When the scales of neutral gases aremuch longer than typical microscopic scales of their constituent molecules or atoms, wecan describe the gas by the macroscopic theory of fluid dynamics. So too, for plasmas,where the macroscopic theory is called magnetohydrodynamics or MHD. The equationsof MHD are similar to those of fluid dynamics and because of this the theories share mostassumptions.

2.2.1 Assumptions

The assumptions of hydrodynamics

There is only one basic assumption that hydrodynamics needs in its most general form, therest of the assumptions are only needed to simplify the equations. This basic assumptionis the continuum assumption: molecular properties are averaged on scales much smallerthat characteristic length scales for the fluid. This is usually expressed as the Knudsennumber Kn = λ

Lwhere λ is the mean free path of the molecules and L is some character-

istic length scale. The same can be done for time scales.Another assumption we will use is that of a perfect fluid, where we ignore all viscous,shear and conduction effects. We can do this if the particles can be approximated aspoint-like: the size of the particles is small compared to the mean separation (n−1/3

where n is the particle density).

22

The assumptions of MHD

Similar to the continuum assumption of fluid dynamics above, MHD has the quasi-neutrality assumption. Even though the plasma is made up of charge particles, eachvolume element is assumed neutral. This is because any net charge produces such a largeelectrostatic force that it quickly disappears. These net charges get removed on time

scales of 1ωp

where ωp =√

4πnee2

meis the plasma frequency, ne is the electron density, me is

the electron mass and e is the electron charge.In most astrophysical scenarios, the plasma has a very high conductivity. MHD withinfinite conductivity is called ideal MHD. The conductivity is σ = e2ne

meνcwhere νc is the

collisional frequency.

2.2.2 Magneto-rotational instability

An important consequence of ideal MHD is Alven’s theorem or the flux freezing theorem.This theorem says that magnetic field lines are “frozen” in the plasma: the plasma andmagnetic flux move together. We start by deriving Ohm’s law for plasmas. A covariantversion of a part of the derivation below is given in chapter 9.2 of Meier (2012). Ohm’slaw is not clearly applicable to plasmas, especially without external field. To derive thegeneralized Ohm’s law, we start by writing down the momentum equation for a multi-component fluid. We do not derive this expression rigorously, but the usual shape ofa Lagrangian derivative on the LHS and force terms on the RHS should instill someconfidence.

msnsd~usdt

= ens( ~E + ~us × ~B)−∇~P + ~Fc (2.5)

where the subscript s denotes the particle species, ~P is the pressure, ~us is the velocity, ~Eis the electric field, ~B is the magnetic field and ~Fc denotes the force-vector holding thecollisional terms with the other species in the fluid. Summing the plugged-in equations

for ions and electrons and noting that me << mi and that ~ui = ~u+ me~jmien

and ~ue = ~u− ~jen

for the center-of-mass velocity ~u = mi~ui+me~uemi+me

gets us to

me

ne2

∂~j

∂t= ~E + ~u× ~B − 1

ne(~j × ~B)− 1

ne∇Pe −

1

σ~j (2.6)

where ~j is the current. The only density that appears is n since quasi-neutrality impliesthat ni = ne. The term on the left is known as the electron inertia for obvious reasons.The first and second term on the right give the Lorentz force. The third term is knownas the Hall effect. The fourth term is the electron pressure and the fifth term is aparametrization of the collisional force. As we will be performing global simulations, wewill assume our fluid varies weakly on typical plasma time and distance scales. Usingthese assumptions, we can neglect the inertial, Hall and pressure term. Equation 2.6 nowreduces to

~E + ~u× ~B =1

σ~j (2.7)

In the limit of perfect conductivity (σ →∞) we get the ideal Ohm’s law

~E = −~u× ~B (2.8)

23

For completely ionizated plasma νc = 2.91 × 10−6ne ln ΛT−3/2 s−1 where ln Λ ≈ 10 isthe Coulomb algorithm. For typical values for our accretion flow this gives σ ≈ 1027,justifying our infinite conductivity assumption. Now we can start to derive the fluxfreezing theorem. The magnetic flux Ψ through a curve C spanned by a surface S isgiven by

Ψ =

∫S

~B · dS (2.9)

We take the time derivative

dΨ

dt=

∫S

d ~B

dt· dS +

∫C

~B · ~u× dl (2.10)

The first term on the right we can rewrite using one of Maxwell’s law ∂ ~B∂t

= −∇× ~E andthe second term we can rewrite Stokes’ theorem, which gives

dΨ

dt= −

∫S

∇× ~E · dS −∫S

∇× (~u× ~B) · dS = −∫S

∇× ( ~E + ~u× ~B) · dS = 0 (2.11)

where we use the ideal Ohm’s law in the last step. So the flux through any curve movingwith the plasma is conserved.In a differentially rotating accretion flow, when two particles at the same radius, linked bya magnetic field line, get radially perturbed, the inner particle feels a decelerating forceand the outer particle feels an accelerating force caused by magnetic stresses, see Fig.2.2.2. This transport angular momentum to the outer particle and causes turbulence.This effect is called the magnetorotational instability and is thought to be the maindriver of accretion.

Figure 2.1: A schematic of the MRI. Two particles in a differentially rotating flow con-nected by a spring exercise forces on each other. Angular momentum is transferred fromthe inner particle to the outer particle. The spring in this case is a magnetic field line.Credit: Nick Murphy.

24

2.3 General relativistic magnetohydrodynamics

2.3.1 Conserved fluid quantities

For a semi-rigorous derivation of the GRMHD equations as they are solved in GRMHDcodes, we follow closely Gammie et al. (2003). Where warranted, we give justificationsof assumptions and extra steps.The conservation of particle number for a single-species fluid is given by

∇µJµ = 0 (2.12)

where Jµ = ρuµ is the density current, where ρ is the rest-mass density and uµ is theplasma velocity. After a choice of coordinates this becomes

∂

∂xµJµ + ΓλλµJ

µ =∂

∂xµ(√−gJµ) = 0 (2.13)

where Γµνλ is the Christoffel symbol. This is true because Γλλµ = 12gδν∂µgδν = ∂µ ln g =

1√−g∂µ√−g.

The conservation of stress-energy (Bianchi identities) is given by

∇µTµν = 0 (2.14)

where ∇µ is the covariant derivative associated with the spacetime metric gµν and T µν isthe stress-energy tensor. Mathematically this can be shown to be true since the covariantdivergence of the Einstein tensor Gµν disappears and these two quantities can be relatedthrough the Einstein equation:

Gµν = 8πT µν (2.15)

∇µGµν = 0 (2.16)

Due to the symmetry of T µν if we choose a coordinate system we can write equation 2.14as

∂

∂xµT µν + ΓµµλT

νλ + ΓνµλTµλ =

1√−g

∂

∂xµ(√−gT µν) + ΓνµλT

µλ = 0 (2.17)

for the same reason given at equation 2.13.

2.3.2 The energy-momentum tensor

Any symmetric (0,2) tensor (in this case Tµν) can be decomposed into two scalar fields,one vector field and one tensor field with respect to any time-like vector field. In thiscase that timelike vector field uµ. We obtain the decomposition by projecting Tµν alonguµ and the projection tensor hµν = uµuν + gµν (so that hµνu

ν = 0).

Tµν = Tκλuκuλuµuν − 2hκµTκλu

λuν +1

3hκλTκλhµν + (hκµh

λνT(κλ) −

1

3hκλTκλhµν) (2.18)

We now have scalar fields Tκλuκuλ and 1

3hκλTκλ, vector field −2hκµTκλu

λ and tracelesssymmetric tensor field (hκµh

λνT(κλ) − 1

3hκλTκλhµν).

The usual physical interpretation of T µν for a fluid is the flux of the µth component ofthe energy-momentum vector through a hypersurface perpendicular to eν . This meansthat

25

• For µ = ν = 0, we propagate energy density through time, so T 00 = ε, where ε isthe energy density.

• For µ = 0, ν = i, we propagate energy density through space, so T 0i = T i0 = qi,where qi is the momentum density vector. Alternatively, for µ = i, ν = 0 wepropagate momentum through time, leading to the same conclusion.

• For µ = ν = i, we propagate momentum along parallel to its direction, so T ii = pi,where pi is the pressure density vector.

• For µ = i, ν = j, i 6= j, we propagate momentum perpendicular to itself, so T i 6=j =σij, where σij is the symmetric traceless shear stress density tensor.

Using these physical interpretations, we can now identify ε = Tκλuκuλ, qµ = −2hκµTκλu

λ,p = 1

3hκλTκλ and σµν = (hκµh

λνT(κλ) − 1

3hκλTκλhµν), resulting in

Tµν = εuµuν + qµuν + phµν + σµν (2.19)

Now we go to the “perfect fluid” approximation, where we ignore all viscous, shear andconduction effects such that qµ = 0 and σµν = 0 and truncate the moment equations atsecond order, reducing our energy-momentum tensor to

Tµν = εuµuν + phµν = (ρ+ u+ p)uµuν + pgµν (2.20)

where ρ is the rest-mass energy and u is the internal energy density. The usual wayequation 2.20 was constructed is by noting that for a fluid not to be affected by anisotropic(.eg. viscous) effects, its energy-momentum tensor needs to be diagonal where T 11 =T 22 = T 33 or

T = diag(ε, p, p, p) (2.21)

If we find a tensorial equation that reduces to this expression in our rest-frame we aredone. A relatively easy guess is

T µν = (ε+ p)uµuν + pηµν (2.22)

where ηµν = −1,1,1,1 is the Minkowski metric. Because uµ = (−1, 0, 0, 0) in the rest-frame of the fluid, equation 2.22 reduces to equation 2.21. Generalizing the metric bringsus back to equation 2.20.Equations 2.17 and 2.13 give us 5 equalities for 6 unknowns (uµ, ρ, u and p). We canclose this system of equations by introducing an equation of state of the form

p = p(ρ, u) (2.23)

The most commonly used equation of state is that of an ideal gas

p = (γ − 1)u (2.24)

where γ is the adiabatic index. For a non-relativistic gas, γ = 5/3 and for an ultra-relativistic gas, γ = 4/3.

26

2.3.3 Conserved electromagnetic quantities

Until now, we have ignored the “M”-part in “GRMHD”. In this Sect. we will explorethe effects on coupling the equations we have already established with those of Maxwell.To remind everyone of their undergraduate physics, the homogeneous Maxwell equationsare

∇ · ~B = 0 (2.25)

∇× ~E − ∂ ~B

∂t= 0 (2.26)

and the inhomogeneous Maxwell equations are

∇ · ~E = ρ (2.27)

∇× ~B − ∂ ~E

∂t= ~J (2.28)

These can be reduced to two equations

∂µFµν = Jµ (2.29)

and

∂[µFνλ] = 0 (2.30)

which can also be written as

∂µGµν = 0 (2.31)

where we introduced the Faraday tensor F µν and its Hodge dual Gµν = 12εµνκλFκλ. The

contravariant representation of F µν takes the form

F µν =

0 −Ex −Ey −EzEx 0 −Bz By

Ey Bz 0 −Bx

Ez −By Bx 0

(2.32)

Because of the homogeneous Maxwell equations 2.25 and 2.26 we can write

~B = ∇× ~A = 0 (2.33)

~E = −∇φ− ∂ ~A

∂t(2.34)

~A and φ are not uniquely defined so we can choose our gauge for which we will choosethe Lorentz gauge

∂φ

∂t+∇ ~A = 0 (2.35)

Grouping our new scalar and vector potential into a 4-vector we get

27

Aµ = (φ, ~A) (2.36)

and it is straightforward to show that

F µν = ∂µAν − ∂νAµ (2.37)

Analogous to the fluid energy-momentum tensor of equation 2.20 we can construct theelectromagnetic energy-momentum tensor T µνEM using less-than-rigorous arguments basedon undergraduate physics.

• For µ = ν = 0, we propagate energy density through time, so T 00 = ε, where ε isthe energy density. From electromagnetism we know ε = E2 +B2.

• For µ = 0, ν = i, we propagate energy density through space, so T 0i = T i0 =qi, where qi is the momentum density vector. Alternatively, for µ = i, ν = 0we propagate momentum through time, leading to the same conclusion. Fromelectromagnetism we know this as the Poynting flux, or qi = ~E × ~B.

• For µ = i, ν = j, we need a tensor whose divergence gives the force vector and wecan identify this with the Maxwell stress tensor σij =

δij2

(E2 +B2) +EiEj +BiBj.

In Minkowski space the arguments lead to the following tensor

T µν =

E2 +B2 EyBz − EzBy ExBz − EzBx ExBy − EyBx

EyBz − EzBy −σxx −σxy σxzEzBx − ExBz −σyx −σyy σyzExBy − EyBx −σzx −σzy σzz

(2.38)

A more rigorous derivation can be done by employing the Lorentz 4-force density fµ. Asthis is the time-derivative of pµ, the spatial entries are given by the Lorentz force density~f = ρ ~E + ~J × ~B. The temporal entry is given by the time derivative of the energy or~f · ~u. This means we can write the 4-force density as

fµ = FµνJν (2.39)

Using equation 2.29 this becomes

fµ = Fµν∂λFλν = ∂ν(F

νλFµλ)− F νλ∂νFλµ (2.40)

The last term in equation 2.40 can also be written as

F νλ∂νFλµ =1

2F νλ(∂νFλµ + ∂µFνλ) = −1

2F νλ∂µFλν =

1

4∂µ(F νλFνλ) (2.41)

where in the first step we used the anti-symmetry of F µν , in the second step we usedequation 2.30 and in the third step we used the anti-symmetry again. Plugging this resultinto equation 2.40 gives

fµ = Fµν∂λFλν = ∂ν(F

νλFµλ)−1

4∂µ(F νλFνλ) = −∂νT νµEM (2.42)

where we recover our electromagnetic energy-momentum tensor again, this time formu-lated in a general spacetime as

28

T µνEM = F µλF νλ −

1

4gµνFλκF

λκ (2.43)

With very laborious checking one can verify that this is equivalent to 2.38. You maynotice that the covariant version of equation 2.42 is given by

fµ = ∇νTµνEM (2.44)

but do not we also have equation 2.14? Peace is restored to the universe by realizingT µνEM is only a part of the energy-momentum tensor, because the existence of ρ and ~Jcontained in fµ means we need matter, in our case given by T µνF . Equation 2.14 is thusa kind of local version of Newton’s third law or conservation of momentum. This is thefirst time we see the deep relation between the fluid and the electromagnetic field, therelation that is so important in GRMHD.We can rewrite equation 2.8 in the fluid frame to a form compatible with our otherequations

uµFµν = 0 (2.45)

one of the most important results for ideal magnetohydrodynamics.Because of the ideal MHD condition it is useful to define the magnetic 4-vector

bµ = εµνκλuνFλκ (2.46)

which one can check reduces to b0 = 0, bi = ~B in the rest-frame of the fluid. Note thatequation 2.45 implies that bµu

µ = 0. We can use this to invert equation 2.46, giving

F µν = εµνκλuκbλ (2.47)

and its dual

Gµν = bµuν − bνuµ (2.48)

Using this last equation together with 2.31 and the identity used in equation 2.13 we getthe following equations

∂i(√−gBi) = 0

√−g∂tBi + ∂i[

√−g(bjui − bibj)] = 0

(2.49)

where Bi is implicitly defined as bt = Biuµgiµ, bi = Bi+btui

ut. The first equation is the

general relativistic equivalent of ∇ · ~B = 0 and the second equation is the inductionequation. Using equation 2.47 we can reduce the electromagnetic energy-momentumtensor (equation 2.43) to

T µνEM = b2uµuν +1

2b2gµν − bµbν (2.50)

Our total energy-momentum tensor, to conclude is

T µνMHD = T µνF + T µνEM = (ρ+ u+ p+ b2)uµuν + (p+1

2b2)gµν − bµbν (2.51)

Now we have established our basic equations (equation 2.13, 2.17, 2.49 and 2.24) and ourstress-energy tensor T µνMHD = T µνF +T µνEM . How these equations are solved, we will discussin chapter 3.

29

2.4 Radiation

In order to cool the simulation, we need to subtract the radiated energy from the internalenergy of the simulation. In exchange for the luxury of a low optical depth everywhere (seeSect. 2.4.1) we have to deal with non-blackbody radiation, see Tab. 2.1 for a summaryof radiative and pair-producing processes in non-magnetized plasmas, to which we onlyneed to add synchrotron radiation. Luckily, in Sgr A* we can ignore bremsstrahlung atradii < 103 RG (Narayan et al., 1998; Moscibrodzka et al., 2009), we do however takebremsstrahlung into account for the cooling so our simulations can be used at some laterpoint when larger simulation sizes might be considered. For the spectra and images we donot consider bremsstrahlung. The two radiative types we need to worry about for that aresynchrotron radiation and Compton scattering. Compton scattering will be only used forthe synchrotron radiation field and will be described by a correction factor η describingthe energy amplification per synchrotron photon by scattering. The total cooling rate isgiven by

q−total = ηq−sync + q−brems (2.52)

This is substracted from the source term belonging to the energy equation. Before wediscuss the produced radiation, we first need to explore a key factor in our simulations:low optical depth.

2.4.1 The implications of low optical depth

In an ADAF, we can estimate the optical depth for Thomson scattering by τe = σTneHwhere σT is the Thomson cross-section and H ∼ R is the scale-height of the flow (Esinet al., 1996). H typically lies in the range 1010 to 1013 for our grid. In our highest-accretion rate simulation, ne,max ≈ 1012, providing an upper limit of τe < 0.06 butusually τe << 10−3. For other radiation types the optical depth is similarly low forthe frequencies that cool the most. Thus, we can assume that any radiation producedleaves the simulation without interacting with the fluid, meaning that radiation onlyremoves energy. Low optical depth also means that the only pair-producing interactionthat matters is ee↔ eee+e− while electron-positron pair annihilation still dominates thereverse reaction (Svensson, 1982a).

2.4.2 Approximations for Sagittarius A*

We assume the electrons get thermalized at some fraction of the proton temperature

Te = fTTp (2.53)

fT = 0.1, 0.3, 1 (2.54)

where Te is the electron temperature, Tp is the proton temperature and fT is the ratioas in Moscibrodzka et al. (2009); Esin et al. (1996); Dibi et al. (2012); Fragile & Meier(2009). This is our main assumption, it allows for cheap analytic cooling functions thatare imperative for GRMHD codes. It has been shown to agree with more sophisticatedmethods that track electron distributions (Dibi et al., 2012). This is an approximationof a weak coupling between protons and electrons. The protons heat up through viscousdissipation and in turn heat up the electrons, while the electrons cool through radiativeprocesses and in turn cool the protons. There will always be some coupling between the

30

Two-body interaction Radiative variant Pair-producing variant

Møller & Bhaba scat.ee→ ee

Bremsstrahlungee↔ eeγ

ee↔ eee+e−

Compton scat.γe→ γe

Double Compton scat.γe↔ γeγ

γe↔ ee+e−

Pair annihilatione+e− → γγ

Three quantum annihilatione+e− ↔ γγγ

−

Photon-photon pair productionγγ → e+e−

Radiative pair productionγγ ↔ e+e−γ

−

Coulomb interactionep→ ep

Bremsstrahlungep↔ epγ

ep↔ epe+e−

γp↔ pe+e−

Table 2.1: The radiative and pair-producing processes in a non-magnetized plasma. Takenfrom Svensson (1982a).

different species of charged particles through Coulomb interactions for example, althoughthermalization is doubtful when the mean free path of an electrons is too large, althouighthis is likeluy not the case in Sgr A* (Sharma et al., 2007).

Particle-particle pair equillibria

Simple electron-positron pair annihilation dominates since all other pair annihilationprocesses are 3+-body interactions. For a thermal plasma, this is given by (Svensson,1982b)

n+,ann = πcr2enen+g(θ) (2.55)

where

g(θ) = (1 + [2θ2

ln 1.12θ + 1.3])−1 (2.56)

For pair production, there are multiple processes to consider, see Table 2.1. The threequantum annihilation cross section is strictly smaller than the pair annihilation crosssection so we ignore the former. Except for very small fT , the protons are much slowerso any process involving protons will be much smaller than the same process involvingelectrons. Lastly, for low optical depths, we can neglect all processes involving photons.This leaves us with electron-electron pair production. For a thermal plasma, this is (Esinet al., 1996)

n+,ee = cr2en

2e

2× 10−4θ3/2e/θ(1 + 0.015θ) θ ≤ 1

(112/27π)α2f ln θ3(1 + 0.058/θ)−1 θ > 1

(2.57)

We assume a pair-equillibrium on GRMHD time scales since runaway pair productioncan only exist on very short time scales in optically thin flows (Svensson, 1982b). Usingthe positron-electron ratio calculated from equations 2.57 and 2.55 we can calculate all

31

other particle densities

n+

ne

=1

πg(θ)

2× 10−4θ3/2e−2/θ(1 + 0.015θ) θ ≤ 1

(112/27π)α2f ln θ3(1 + 0.058/θ)−1 θ > 1

(2.58)

z =n+

np

=n+

ne − n+

=

n+

ne

1− n+

ne

(2.59)

ne = np(µ+ z) (2.60)

n+ = npz (2.61)

n± = ne + n+ (2.62)

while np is calculated directly from the plasma density ρ (np = ρµmH

), where mH is themass of an hydrogen atom and µ = 1.69 is the mean molecular weight. This functionblows up when Te > 2.4× 1011 so we limit the temperature going into this calculation tothat value. This will limit the pair production very close to the horizon and thus decreasethe cooling.

Thermal bremsstrahlung

Bremsstrahlung is radiation produced charged particles accelerated by Coulomb interac-tions. Per Coulomb interaction, the total radiated energy is

E =4

3

e6

m2c3

1

b3v(2.63)

where b is the impact parameter. E ∝ m−2 such that lepton bremsstrahlung dominatesproton bremsstrahlung. Following Esin et al. (1996) we adopt the electron-ion, electron-electron and electron-positron cooling rates from Svensson (1982a); Stepney & Guilbert(1983):

q−ei = 1.48× 10−22npn± ergs cm−3 s−1 (2.64)

q−ee =

2.56× 10−22(n2

e + n2+)θ1.5(1 + 1.1θ + θ2 − 1.25θ2.5) ergs cm−3 s−1 θ ≤ 1

3.42× 10−22(n2e + n2

+)θ[ln 1.123θ + 1.28] ergs cm−3 s−1 θ > 1(2.65)

q−−+ =

3.43× 10−22nen+(θ0.5 − 1.7θ2) ergs cm−3 s−1 θ ≤ 1

6.84× 10−22nen+θ[ln 1.123θ + 1.24] ergs cm−3 s−1 θ > 1(2.66)

The total bremsstrahlung cooling rate is then

q−brems = q−ee + q−ei + q−−+ (2.67)

Thermal synchrotron

Synchrotron is radiation produced by the acceleration of charged particles in magneticfields. The appearance of the radiation depends on the angle between the magnetic fieldline and the path of the electron. For an isotropic distribution of electron incidences, asmight exist in an accretion disk, the power produced per particle is given by

32

P =4

9

e4

m2c3B2γ2β2 (2.68)

where γ is the Lorentz factor and β = v/c. Because of the strong dependence on massand velocity, proton synchrotron is negligible to electron synchrotron radiation. Goingforward, synchrotron refers to sychrotron produced by leptons only.We need to derive a cooling rate from a description that allows us to create images andspectra as well. The emission and absorption coefficient are respectively given by (deKool et al., 1989; Rybicki & Lightman, 1979)

jν,sync =1

4π

∫ ∞1

p(ν, γ)nγdγ (2.69)

αν,sync = − 1

8πmν2

∫ ∞1

p(ν, γ)γ2 ∂

∂γ

nγγ2dγ (2.70)

where

nγ = n±γ2β

θK2(1/θ)e−

γθ (2.71)

for a thermal distribution of electrons,

p(ν, γ) =31/2e3Bν

mc2νssa

∫ ∞ν/νssa

K5/3(x)dx (2.72)

and

νssa =3eB

4πmcγ2 (2.73)

Again, for the cooling function in the simulation we need to integrate over both frequencyand electron energy. Pacholczyk (1970) has found a the following equation for the emissioncoefficient for a thermal distribution of electrons

jν,sync,thermal = 4.43× 1030 4πn±K2(1/θ)

I( xM

sinφ

)erg/cm3/s (2.74)

where xM = 2ν3ν0θ2

, ν0 = eB2πmec

and I( xMsinφ

) is some tabulated function whose angle-averaged

version can be approximated by the following fitted function (Mahadevan et al., 1996)

I ′(xM) =4.0505

x1/6M

(1 +

0.4

x1/4M

+0.5316

x1/2M

)e−1.8899x

1/3M (2.75)

The spectrum radiated from a volume of gas should always be less than or equal to ablack body spectrum emitted on the surface. We use this fact to estimate the synchrotronself-absorption frequency, νssa. The volume we consider is an annulus. This gives us thefollowing expression

2H(2πR∆R)jν,sync,thermal = 2(2πR∆R)2πν2

ssa

c2kT (2.76)

or

jνssa,sync,thermal = 2πν2

ssa

c2

kT

H(2.77)

33

which we numerically solve for νssa. Below this frequency, the emission coefficient isthe Rayleigh-Jeans limit. To find the total synchrotron cooling rate q−sync we integratethe Rayleigh-Jeans law up to νssa and equation 2.74 from νssa which can be done usingsymbolic maths software resulting in

q−sync =2πR2

2HπR2

∫ νssa

0

dν2πν2

c2kT +

∫ ∞νssa

dνjνssa,sync,thermal (2.78)

=2πkTν3

ssa

3Hc2+ (6.76× 10−28)

n±

K2(1/θ)a1/61

(2.79)[1

a11/24

Γ(11

2, a4ν

1/3ssa

(+

a2

a19/44

+a2

a19/44

Γ(19

4, a4ν

1/3ssa

)(2.80)

+a3

a44

(a34νssa + 3a2

4ν2/3ssa + 6a4ν

1/3ssa + 6)e−a4ν

1/3ssa

](2.81)

where a1 = 23ν0θ2

, a2 = 0.4

a1/41

, a3 = 0.5316

a1/21

, a4 = 1.8899a1/31 and Γ(a, x) =

∫∞xdt ta−1e−t is

the upper incomplete gamma function. In the code we solve for νssa with Brent’s methodin log space.

Thermal comptonization

Inverse Compton scattering of photons by electrons where the electron where producedby synchrotron radiation of the same populations of electron is called synchrotron self-Compton or SSC. The SSC emissivity is described by an integral over the approximatesynchrotron radiation field Isync = jν,sync

αν,sync(1−e−αν,syncR) as (Chiaberge & Ghisellini, 1999;

Drappeau et al., 2013)

jν,SSC =σT4

∫ νmax0

νmin0

dν0

ν0

∫ γ2

γ1

dγ

γ2β2n(γ)f(ν0, ν)

ν

ν0

Is(ν0) (2.82)

where νmin0 and νmax

0 are the lowest and highest emitted synchrotron frequencies. Theintegration limits are given by

γ1 = max[(ν

4ν0

)1/2, γmin] (2.83)

and

γ2 = min[(3mec

2

4hν0

)1/2, γmax] (2.84)

and the spectrum produced by a single electron and photon is given by

f(ν0, ν) =

(1 + β) ν

ν0− (1− β) 1−β

1+β≤ ν

ν0≤ 1

(1 + β)− νν0

(1− β) 1 ≤ νν0≤ 1+β

1−β

0 otherwise

(2.85)

To avoid having to do the expensive numerical calculation of equation 2.82 we calculatethe contribution of SSC to the cooling using the Compton enhancement factor η, theaverage factor of energy increase due to SSC:

34

η(ν) = es(A−1)[1− P (jm + 1, As)] + ηmaxP (jm + 1, s) (2.86)

where jm = ln ηmax/ lnA, A = 1 + 4θ + 16θ2, s = τes + τ 2es, ηmax = 3kT

hνand P (a, x) =

1Γ(a,0)

∫ x0dt ta−1e−t is the normalized lower incomplete gamma function. The derivation

of equation 2.86 can be found in Esin et al. (1996). Since the emission coefficient of syn-chrotron radiation is strongly peaked at νssa we can approximate the frequency-integratedη by ∫ ∞

0

dν η(ν) = η(νssa) (2.87)

The total cooling rate is then given by

q−sync,SSC = η(νssa)q−sync (2.88)

35

CHAPTER 3

Numerical methods

Now that we have all the formulae we need, it is time to solve them. HAMR is a grid basedcode: the simulated volume is chopped into cells inside which the fluid quantities havea constant value. The quantities are then evolved through time with discrete timesteps.There are multiple ways to do these calculations. We will be using a finite volume method.

3.1 Finite volume methods and Godunov solvers

We start with a hyperbolic system of equations written in conservative form as

d~U

dt+∇ · ~F (~U) = ~S (3.1)

where ~U are the conserved variables, ~F (~U) are the fluxes and ~S are the source terms.We will ignore the source terms for now as they complicate the following derivationunnecessarily. We can integrate this using the divergence theorem to

d

dt

∫~UdV +

∫~F · ~ndS = 0 (3.2)

Now, for convenience we switch to 1D (higher dimensions are straightforward). If wemake the integral over some finite volume, let us say [xi−1/2, xi+1/2]

d

dt

∫ xi+1/2

xi−1/2

~Udx+1

∆x(~F (~U)− ~F (~U)) = 0 (3.3)

and then integrate over some time interval∫ xi+1/2

xi−1/2

~Udx−∫ xi+1/2

xi−1/2

~Udx+1

∆x(

∫ tn+1

tn

~F (~U)dt−∫ tn+1

tn

~F (~U))dt = 0 (3.4)

Now we identify

~Uni =

1

∆x

∫ xi+1/2

xi−1/2

~Udx (3.5)

and

Fi±1/2 =1

∆t

∫ tn+1

tn

~F (~U)dt (3.6)

36

to give

~Un+1i = ~Un

i +∆t

∆x(~Fi−1/2 − ~Fi+1/2) (3.7)

This is the finite-volume method. Godunov (1959) realized that this leads to a series ofRiemann problems. A Riemann problem is an initial value problem with a conservationequation where the initial data is constant except for one discontinuity, in this case thecell edge. For every dimension a cell has, it has two Riemann problems. The solution ofa Riemann solution involves waves and the timestep can be chosen such that the wavesfrom the two boundaries can not interact, see Sect. 3.6.Before we use the wave solution of the Riemann problem to calculate the flux, we needto know what Un

i±1/2 are. One can assume that Uni is constant across a cell, giving

Uni±1/2 =

Uni±1+Uni2

. There are also other possibilities to interpolate Uni to the cell edges,

like monotized central (MC) or piecewise parabolic method (PPM), resulting in a higherorder spatial accuracy (van Leer, 1977; Colella & Woodward, 1984).There is no known explicit solution to the (GR)MHD Riemann problems. Iterative pro-cedures to find the full solution can be expensive and unnecessary (Giacomazzo & Rez-zolla, 2006). That is why most MHD codes use approximate solvers like the Roe andHLL solvers (Roe, 1981; Harten et al., 1983). These methods work by approximatingsome or all wavespeeds and then calculating how much flows from one cell to the next.Roe solvers work by solving the characteristic equations to calculate the wavespeeds ex-actly, while HLL solvers approximate the fastest wavespeed, which for GRMHD is thefast magnetosonic speed. Time evolution can be done by the canonical forward Euler orfourth order Runge-Kutta, an explicit fourth-order method. In HAMR a leapfrog methodis deployed, an implicit second-order method.So what are ~U , ~F and ~S for GRMHD? For this we go back to our governing equations ofchapter 2 and we see that

~U =√−g(ρut, T tt , T

ti , Bi) (3.8)

~F =√−g(ρui, T it , T

ii , b

jui − biuj) (3.9)

~S =√−g(0, T κλΓλtκ, T

κλΓλiκ, 0) (3.10)

3.2 The ∇ · ~B = 0 condition

In Sect. 2.3 we saw that we do not need the no monopoles constraint ∇ · B = 0 toevolve our simulation. If one makes sure that this constraint is satisfied within the initialconditions it will always remain satisfied in the limit of infinite precision and resolution.Since we are not in this limit this constraint can get increasingly more violated as thesimulation evolves and even relatively small violations of this∇·B = 0 can affect dynamics(Gardiner & Stone, 2005). There are multiple ways to make sure this condition stayssatisfied through time. One method involves removing the divergence of B each timestep,another method involves adding an evolutionary equation while the third designs thedifference equations in such as way as to explicitly satisfy the no monopoles constraint(Toth, 2000; Gammie et al., 2003; Gardiner & Stone, 2005). This last method is calledconstrained transport and involves defining the magnetic field not at the cell centers but

37

at the cell faces. This better captures the essence of the magnetic field as it is a surface-integrated conserved quantity instead of a volume-integrated conserved quantity as forexample density.

3.3 Boundary conditions

Every grid code needs a domain of simulation. For two coordinates, we want the fluid atthe most extreme values of the coordinates to stay within the grid. For 3D, φ = [0, 2π],the grid is periodic: matter flowing out at φ = 2π is flowing in at φ = 0 with itsvelocity vector conserved and vice versa. For 2D, the simulation is only one cell thick,but the boundary condition is still periodic. For 3D, θ = [0, π] and the polar axis istransmissive: matter that flows to the polar axis gets transported around the pole (soin the φ-direction) with a factor of π. For 2D, θ = [0, π], matter flowing out at θ = πis also flowing in at θ = π with the θ-component of its velocity vector reversed in sign,also known as a reflective boundary condition. The formation of jets keeps the regionaround the polar axis evacuated so the relevant physics for this work is not affected bythis boundary condition. For the r-coordinate we have both in 2D and 3D at the innerboundary a natural boundary: the event horizon. Here we use an outflow boundarycondition. However, we do place a few cells inside the horizon for numerical stability.The boundary condition is located at 0.85rH . The outer boundary condition is also anoutflow boundary condition and is the only one that is clearly non physical. As such,it is placed far away from the torus as not to interact with the accretion process atRout = 1000RG.

3.4 Density floors

Every cell in the simulation start out with a nonzero value for the magnetic field strength,density and internal energy, called a floor value. In the case that one of these densitiesdrops below its floor value there is a risk that the inversion error for some quantity islarger than that density. Because of this, these floors are constantly enforced. Anytime afloor is enforced, the conservation laws are broken, for example by adding mass. The onlyreal area where this has an effect is inside the jet, where, because of the ideal assumption,mass cannot penetrate the field lines and so the jet becomes “empty” (McKinney, 2006).For the current study we focus on the disk and its observational signatures so we do notworry about the jet dynamics. The radiation produced in the jet might have a significantimpact on our results, but a lot still needs to happen to reproduce realistic jets, so wepostpone this to later research. We do not expect the density floors in the jet to affectthe disk since mass added by the density floors cannot cross over to the disk due to thejet’s strongly magnetized nature (Blandford & Znajek, 1977), although the jet collimationcould be affected and this the jet might exert a different pressure on the disk and wind.

3.5 Unit scaling and accretion rates

In the code, distances are scaled by the gravitational radius RG and time is scaled by thelight-crossing time tc = RG/c

L =GM

c2(3.11)

38

T =GM

c3(3.12)

so that both L and T scale with black hole mass. In the code natural units are used forsimplicity and to keep the calculations close to unity.

G = c = 1 (3.13)

We do not have such an explicit unit for mass in the simulation. The maximum densityin the initial torus is always ρmax = 1. This is a result of GRMHD simulations beingscale-free: you can choose any mass and accretion rate and use those to determine themass scale. In the following we use the subscript c to denote any values in code units.The code accretion rate Mc at some radius can be calculated using the following formula:

Mc =

∫dθdφ

√−gρcurc (3.14)

To compare this to the known accretion rate of some black hole system we need to scaleby the appropriate units. Velocity is easy to scale by a factor of c. The determinant ofa spherical metric needs to absorb the two factors of [D]2 that are needed to transformthe angle infinitesimals dθ2 and dφ2 into lengths. Thus

√−g is scaled by R2

G. We wantto determine the density scaling so ρc is scaled by ρ

ρc. This gives

M =

∫dθdφ

√−gρcurc

ρ

ρc

(GMc2

)2

c = Mcρ

ρc

(GMc2

)2

c (3.15)

We know M = 4.6×106M, M = 10−9,−8,−7M/yr and Mc we can calculate easily fromthe simulation. Assuming adding radiative cooling will not change the accretion rate tooseverely, we can then calculate the density scaling

ρ

ρc= 8.44 · 10−15,−16,−17 g

cm3(3.16)

and all other scalings:

[Bi] =[M ]

12

[T ][D]12

=

√g

s√

cm= 1G→ Bi = Bi

c

√ρ

ρcc (3.17)

[u] =[M]

[D][T]2=

g

s2cm→ u = uc

ρ

ρcc2 (3.18)

[p] =[M]

[D][T]2=

g

s2cm→ p = pc

ρ

ρcc2 (3.19)

3.6 Timestep

To correctly solve a system of equations using a Godunov scheme, we need to makesure that the different waves can not interact within a cell. To guarantee this, we firstdetermine the maximum flow speed in or out of a particular cell.

ci,max = max(0, ci,+,L, ci,+,R)

ci,min = −min(0, ci,−,L, ci,−,R)

ci,top = max(ci,min, ci,max)

(3.20)

39

where ci,+,L and ci,+,R are the left and right inflowing wave speeds, ci,−,L and ci,−,R arethe left and right outflowing wave speeds in cell i. Therefore the maximum flow speedis given by ci,top. To get the timestep, we have to use half the cell size and the Courantfactor:

dti,1 = Cdxi,12ci,top

(3.21)

where C is the Courant factor and dx is the cell width in this particular dimension. Weuse C = 0.9. We repeat this procedure in all three dimensions and calculate the finaltime step:

dti,MHD =1

dti,1+

1

dti,2+

1

dti,3(3.22)

In this research we are not only solving the fluid equations but also our set of radiativeequations so we need to make sure the timestep also properly resolves the radiative effects.We tried this by forcing the cooling to be some fraction of the internal energy:

dti,cool = fuiλi

(3.23)

where f ∼ 0.9. This ensures that u will always stay positive. However, in the event ofrunaway cooling, the timestep will become so small that simulation grinds to a halt. Ourfinal timestep is

dt = min(dti,MHD, dti,cool) (3.24)

We found that the simulation could not efficiently integrate through the start of accretionusing this method, sometimes slowing down by a factor of 103 − 104 For many RG/c.Not only is this not efficient use of computation time, but taking some many timestepsincreases inversion errors. We decided to use the same cooling subcycle as in Dibi et al.(2012), letting the cooling reduce the internal energy by a maximum of 50% four timesevery MHD timestep for a total of ∼ 94%.This is combined with hierarchical timestepping where past some radii the timestep inter-mittently increased by some power of two. This is both done to speed up the simulationand to keep errors down in outer regions of the simulation. To clarify, imagine the outerregions to be stationary on time scales of dt. Any calculation of of the conserved orprimitive values will just build up truncation or inversion errors, respectively.

3.7 Scale height

All values needed for the semi-analytical calculation of the radiation are locally defined,except for one: the scale height. Analytically the scale height is defined as the aspectratio of the disk, or H/R where H is the height of the disk in the z-direction. H/R isusually chosen to be a fixed value of order unity. In simulations this height is not clearlydefined and no physically justified definition is possible that is constant along the radialdirection in the z = 0 plane. As HAMR solves the GRMHD PDEs using a finite-differencingscheme, global properties of the accretion flow are not straightforward to calculate. Scaleheights are defined as the distance over which quantities drop off by a factor of 1

e. If one

assumes these quantities have a exponential decay in some or all direction(s), the scaleheight is given by

40

Q = Q0e− ~xHQ

∇Q =Q0

HQ

e− ~xHQ

HQ =Q

|∇Q|

(3.25)

In Fragile & Meier (2009) Q = T 4 is used as this reduces to the correct scale height forradiation-dominated thin disks as P = 4σ

3cT 4. However, we need a definition that is robust

and works well in both collapsed areas in the inner disk and puffy areas in the outer diskas well as any filaments or other structures within the flow. We find Q = T 4 successfullycaptures these properties.

3.8 Initial conditions and other parameters

As stated in Sect. 3.5, we use M = 4.6× 106 M for Sgr A* (Ghez et al., 2008; Gillessenet al., 2009a). For the accretion rate, we use three values within the observational con-straints of Faraday rotation at dotM = 10−9,−8,−7 M/yr giving a density scaling ofρc = 8.44 × 10−15,−16,−17 g

cm3 (Bower et al., 2005; Marrone et al., 2006). Most of our

runs are done with TiTe

= 10/3, but we also consider TiTe

= 10. For our control runs withoutcooling, we do not need to choose an accretion rate or a temperature ratio, so we canbenchmark all combinations of those two parameters against one control run. We use amean molecular weight µ = 1.69 and an adiabatic index Γ = 5/3.We evolve our disk for 5000RG/c = 12hr or about 20 orbital periods at the outer radiusof the disk.The initial disk is the equillibrium torus of (Fishbone & Moncrief, 1976) with its innerradius rin = 6RG and its pressure maximum at rmax = 12RG. The disk is seeded with aweak magnetic field given by the vector potential

Aφ =

Cρ, if ρ

ρmax− 0.2 > 0

0, otherwise

where C is a normalization constant, chosen such that βp,max = 100. This gives us apurely poloidal (only r, θ dependencies) magnetic field. There are different ways to cal-culate βp,max, we use βp,max = (Γ− 1)2umax

B2max

.We use the HLL flux with PPM reconstruction. All simulations in this work are done inaxisymmetry, so a 2D grid containing 3D vector quantities. As explained in Sect. 3.3,the inner radial boundary is located at 0.85 rH and the outer radial boundary is locatedat 1000RG = 45 A.U.. Any given resolution is always in MKS coordinates.Our density floors are set such that ρ > max(pB/20, pG/150, 10−6r−2) and u > max(pB/750, 10−7r−2)to make sure the simulation does not fail due to inaccuracies caused by values that aretoo small.

For the ray-tracing, we consider three different inclination angles: 5, 45, and 85 deg. Thefield of view for the calculation of the spectra and images is [−20, 20]RG = [−105.7, 105.7]µas.The resolution of the images is 5122 px. As mentioned in chapter 2, currently we onlyconsider synchrotron radiation for the images and spectra. This is to make sure we canfit the synchrotron self-absorption peak and confirm the previous results of McKinney

41

& Blandford (2009); Dibi et al. (2012). In later work we can consider different typesof radiation and possibly larger disks to do variability studies in the NIR and X-rays.Currently this can only be done in the radio, where the variability is smallest.

42

CHAPTER 4

Results

Name Te/Ti Cooling M(M/yr) Dimension ResolutionM9T3LR 0.3 Yes 1× 10−9 2D 5122

M9T3HR 0.3 Yes 1× 10−9 2D 10242

M9T1HR 0.1 Yes 1× 10−9 2D 10242

M8T3LR 0.3 Yes 1× 10−8 2D 5122

M8T3HR 0.3 Yes 1× 10−8 2D 10242

M8T3HR 0.1 Yes 1× 10−8 2D 10242

M7T3LR 0.3 Yes 1× 10−7 2D 5122

M7T3HR 0.3 Yes 1× 10−7 2D 10242

M7T1HR 0.1 Yes 1× 10−7 2D 10242

M7T3VHR 0.3 Yes 1× 10−7 2D 2600× 1300LR - No - 2D 5122

HR - No - 2D 10242

Table 4.1: The simulations discussed in this work. We ran 2D simulations at threedifferent accretion rates within the observational constraints of Sgr A*. We run each ofthese for two different temperature ratios. at a low and high resolution of 5122 and 10242,respectively. We ran one very high resolution purposes for a convergence test. For eachresolution we also ran a noncooled test to benchmark the cooling effects against. Thenaming works as follows: M is followed by the negative logarithm of the accretion rate,T is followed by ten times Te/Ti. This is followed by the resolution. LR refers to lowresolution (5122), HR refers to higher resolution (10242) and VHR refers to very highresolution (2600x1300).

We have studied different combinations of the parameters Te/Ti and M , the models arelisted in Table 4.1. We ran all cases at two different resolutions to to check for conver-gence, with one very high resolution case for the highest accretion rate case as we expectthe most deviation there from previous studies of non-cooled ADAFs.In Fig. 4.1 the initial density profile, magnetic field lines and some grid lines are shown asexplained in 3.8. In Fig. 4.2 the initial temperature is shown together with the H/R = 1lines. A large portion of the initial torus is above 1011 K and the scale height is close tounity, typical attributes of an ADAF. It will remain an ADAF if the flow stays radiativelyinefficient.

43

Figure 4.1: The initial setup of the simulation. The torus is plotted in density, with theblack hole in the middle. The black lines in the disk give the poloidal magnetic fieldloops. The gray lines give contours of the grid. Not all cells are plotted.

Figure 4.2: The initial temperature profile of the simulation. These are temperaturestypical for an ADAF, going up to > 1011 K. The black dotted lines give H/R = 1. Theinner regions of the ADAF also agree with this.

44

Once accretion starts, the simulations quickly achieve their desired accretion rate, seeFig. 4.3. Even though the radiation clearly affects the dynamics as we will see later, nocorrection of the density scaling was needed between the simulations and all simulationran with the density scaling calculated from the non-cooled runs. This was true for allresolutions. The standard deviations of the accretion rates are approximately the same,implying no large changes in the variability of the accretion rate. The accretion rate wasour main test of convergence and the average accretion rates of M7T3LR, M7T3HR andM7T3VHR lie within 10 % of each other, so we conclude the simulations are converged.Due to the sensitivity of the radiation functions to small changes in density, temperatureand magnetic field strength we concluded that cooling rates were a bad convergence test.

Figure 4.3: The accretion rate for three different accretion rates with Te/Ti=0.3 with aresolution of 10242. Their average values are very close to the intended accretion rate.They also remain constant over the whole simulation domain.

In the radial profiles we get when we time- and angle-average our simulations, we seesome large differences, see Fig. 4.4. For ρ and B2 we expect them to scale directly withthe accretion rate. However, we see some significant changes apart from this. There isan increasing excess of mass density at small radii with increasing accretion rate and viceversa for outer radii. As mass moves vertically towards the midplane, mass at larger radiiexperiences an inward pressure force. Because of the increasing loss of internal energy,this effect gets stronger with larger accretion rates. The density profile for the highest

45

accretion rate is significantly less smooth than the others, implying an increase in turbu-lence. The magnetic field radial profile are pretty similar, except, again, for the highestaccretion rate case. This is because of an increased MRI due to the added turbulenceby the cooling. The radial profiles with the most striking differences are those of thetemperatures. There is no explicit scaling of the temperature with accretion rate so anychanges seen here are directly due to cooling. A clear trend exists between accretion rateand temperature. The highest accretion rate case is at least a full order of magnitudecooler than the non-cooled case. This has large consequences for the cooling functionsand pair-production rates.

Figure 4.4: Angle-averaged, radial profiles of density, magnetic field strength and tem-perature for three different accretion rates. The solid lines display the cooled simulationsM9T3HR, M8T3HR and M7T3HR. The dashed lines display the non-cooled simulationHR scaled to the same density as the three cooled simulations. The density and mag-netic field scale with the accretion rates as expected. The profiles are clearly affected bythe cooling. We do not expect the temperature to explicitly scale with accretion rate,although the temperature is severely lowered by radiative cooling.

In Fig. 4.5 the effect of cooling can be seen on the angle-dependent mass density. As theaccretion rate increases, a clear collapse of the disk in the inner ∼ 10 RG can be seen.For the high accretion rate case a very thin disk forms, increasing the optical depth andsynchrotron self-absorption frequency. This collapse is similar to the collapse shown in

46

Dibi et al. (2012). This is still far away from the thin disk solution of Shakura & Sunyaev(1973) where the temperature is lower by a factor of 103 and a density higher by a factor> 106. However, this might be a good model for the start of a state transition in XRBsor AGN. By steadily increasing the accretion rate on state-transition timescale, a changein the spectra might be seen that is similar to state transitions. Because of the longtimescales of AGN state transitions, this can only be investigated in XRBs. This is notdone in this work.

Figure 4.5: A plot of the angle-dependent density profile for M9T3HR, M8T3HR andM7T3HR. As the accretion rate rises, the flow starts to collapse and the equatorialdensity rises with two orders of magnitude.

A large value of z (the ratio of the positron to proton number density) occurs in lowdensity, high temperature areas. The collapse around the equatorial plane as seen in Fig.4.5 lowers the value of z. In the area around the jet, sometimes called the jet sheath orwind, is the lowest density area we consider. As can be seen in Fig. 4.6, this is exactlywhere the highest value of z is achieved. Closed field loops, likely recent reconnectionsites, are associated with dissipation of a large amount of magnetic energy as thermalenergy. These local hotspots clearly experience higher pair production rates.

47

Figure 4.6: A plot of the angle-dependent distribution of the positron to proton ratio zfor M9T3HR, M8T3HR and M7T3HR. This ratio times two tells you how much leptonsthere are due to pair-production compared to the leptons from ionized gas. The collapsedhigh density, low temperature areas of the disk have a reduced value for z.

As the accretion rate increases, the synchrotron self-absorption frequency increases. As areference, the EHT will image Sgr A* at 230 GHz or 2.3×1011 Hz. In Fig. 4.7 you can seethe spatial dependence of νssa. For M9T3HR, the νEHT >> νssa everywhere, resulting inan optically thin flow that reveals structure of both the flow and the black hole shadow.For M8T3HR, νEHT ∼ νssa, revealing even better structure of both flow and shadow. ForM7T3HR however, νEHT < νssa, giving a very well defined flow but almost completelyobscuring the black hole shadow for high inclination.

Figure 4.7: The angle-dependent distribution of the synchrotron self-absorption frequencyfor M9T3HR, M8T3HR and M7T3HR. The self-absorption frequency rises with the ac-cretion rate, making the optically thick inner flow appear increasingly larger.

48

This can be seen in figures A.1, A.2 and A.3 in App. A. Also spectra are shown. Theimages were made by Ziri Younsi at ITP, Goethe University using his ray-tracing codeBHOSS (Younsi et al., in prep.). The images are linearly scaled between no intensity andthe maximum intensity in the images. The spectra are normalized physically: normal-ization is not a free parameter. In general, as the inclination increases, the spectra gaina more pronounced high energy tail due to Doppler shifting. Also beaming and lightbending seem to more than compensate for the decreased visible surface of the disk interms of normalization. This effect on the normalization is small compared to the effectof increasing the accretion rate. Clearly, the best fit to the 230 GHz and surroundingdata points has an accretion rate between 10−9 and 10−8M/yr and a high inclinationof ∼ 45 − 85 deg. Fitting the spectra to the data is difficult, since the cooling demandsthat M is set to a fixed dimensionful value according to in code units. If cooling wouldaffect the dynamics, the scaling between these two would not have to be linear. In orderto fit the data points at lower and higher energies, synchrotron emission from the outflowwould need to be included.We saw already some dynamical differences in some important values like density andtemperature in figures 4.4 and 4.5. We can also clearly see this for the images and spectrain figures A.4, A.5 and A.6 in App. A. There the cooled simulations are compared againstnon-cooled simulations scaled to the same accretion rate. In Fig. A.4 M9T3HR is shown.The high-energy tail is significantly decreased by the cooling, reducing the contrast inthe image. Again, this does not mean the cooled simulation would appear brighter, onthe contrary, the 230 GHz flux is slightly lower in this case compared to the non-cooledsimulation. In Fig. A.5 M8T3HR is shown and we can see that the cool case is less brightand again the high-energy tail is reduced. Here the contrast is higher because of a smallshift in the self-absorption frequency which lies pretty close to 230 GHz. In Fig. A.6M7T3HR is shown and almost the entirety of the flow is optically thick since νssa >> 230GHz. This simulation radiates/cools the most because the density and magnetic fieldscale with the accretion rate. As such, the spectrum decreases in brightness by an orderof magnitude when the simulation is cooled. Remarkably, the high-energy tail does notseem to disappear, probably caused by the growth of a highly radiating thin disk, seeFig. 4.5. It might also be only true for this particular timeslice due to the occurrenceof a magnetic reconnection event. Sgr A* has X-ray flares on a daily basis so in orderto gather proper statistics we would need to run simulations for at least tens of daysor ∼ 1 × 105 RG. We extended M8T3LR as this seemed to produce the spectra thatmost closely matched observations, see Fig. A.2. Unfortunately, the MRI could not besustained for more than ∼ 4 days. After this time, the wavelength of the MRI insta-bility drops below the cell size for most cells. In order to sustain the MRI, it needs tobe resolved by a couple of cells. When the MRI dies out, the magnetic field dissipatesaway without being amplified by the MRI. This severely affects the synchrotron and SSCcooling rates. The decreased cooling rates cause a less turbulent flow which results in avery low and steady accretion rate, see Fig. 4.8. Of the 4 days of statistics, one day isneeded for accretion to properly start, so we would expect between 3 and 6 flares. Evenduring this time, the simulation is clearly not in some kind of steady state. We concludethat these 2D simulations are not adequate to investigate flaring behaviour.

49

Figure 4.8: Three different time series. On top, the total spatially integrated coolingrate in a 10 RG sphere around the black hole is given in arbitrary units. In the middle,the accretion rate over the event horizon is given in arbitrary units. Below, the averageratio of the MRI wavelength over the cell size is given separately for each dimension. Thehorizontal line is drawn at a ratio of unity where the MRI wavelength drops below theresolution of the simulation. The φ-component of the MRI can never be resolved due tothe axisymmetric nature of these simulations.

The simulation with Te/Ti = 0.1 behaved exactly as expected from a cooler flow, theresults were less affected by cooling and showed less interesting results because of this.Due to time constraints we could not generate spectra and images for these simulationsso it is not known if any of these would provide a better fit to the spectrum of Sgr A*.As such all results involving these simulations are omitted.

50

CHAPTER 5

Discussion

The results discussed in Chap. 4 can be compared to previous obtained results withsimilar methods. In Moscibrodzka et al. (2009); Dexter et al. (2009, 2010); Drappeauet al. (2013) spectral fitting of GRMHD simulations to Sgr A* was also done, where onlythe latter had included self-consistent cooling. In Moscibrodzka et al. (2009) a best-fitsolution of Te/Ti = 0.3, M = 1.86 × 10−9 M/yr, i = 85 deg is found with very similarinitial conditions. In Drappeau et al. (2013) Te/Ti = 0.3, M = 2.56×10−9M/yr, i = 85deg is found at a similar spin (a = 0.9), but with a larger disk and four poloidal fieldloops instead of one. In Dexter et al. (2009, 2010) equally low accretion rates are foundin 3D but with a large scatter, such that they agree with our results. From this we canconclude that our results of a best-fit with simulation M8T3HR are inconsistent withprevious work.These inconsistency might be caused by a few factors. Firstly, all other studies have anaccretion rate that dies out after a few orbits while our simulations in contrast maintaina steady accretion rate up to a the simulated time of 5000 RG/c or 20 orbits, see Fig. 4.3.This difference is most likely due to the MRI being properly resolved in our simulationsup to that time. When there is a rapid decrease in the accretion rate, the time-averagedspectra do not necessarily produce the right result since we know Sgr A* accretion rateis not rapidly decreasing. The simulations from previous work were all run on a lowerresolution by at least a factor 4. Secondly, Moscibrodzka et al. (2009) uses the same diskand magnetic field but has a grid that only extends to 40 RG, cutting off the disk whileour grid extends to 1000 RG. They also employ a monotonic central slope limiter (seeSec. 3.1), which is more diffusive (Colella & Woodward, 1984), and they employ differentdensity floors. Drappeau et al. (2013) had a much stronger magnetic field with β = 10and and employed a different initial field configuration. They also found using β = 50resulted in a density that is twice as high and a temperature that’s twice as low thanthe β = 10 case. So far, we have only ray-traced one timeslice. It is possible our best-fitaccretion rate will end up lower when we time-average over some period since we see fromFig. 4.3 that the accretion rate is highly variable.We replicated the setup from Moscibrodzka et al. (2009), including the same collimatedgrid, density floors, slope limiter, absorptivities and emissivities. Time-averaging thesame period 1500 − 2000 RG, we found we need a higher flux by a factor of 2.5. Thiscould be due to the stochastic nature of the MRI, as the simulation has not achieved asteady state yet. Compared to HR we need a higher flux by a factor of 6.5. A strictlybetter resolution, slope limiter and grid setup thus increases the accretion rate needed tofit the 3.4 Jy flux at 230 GHz. To validate GRMHD results for the EHT predictions, a

51

code comparison project was undertaking comparing all major GRMHD codes, such asHAMR, HAMR3D (Porth et al., in prep.) a scatter larger than a factor 2.5 was found betweendifferent GRMHD codes. So maybe our results are as consistent as could be expected.All of the above factors could cause the inconsistencies. Except for the fact that we onlyuse one timeslice to produce the spectra, all differences should speak in favor for ourresults. The spectra in these different studies are difficult to compare to ours since we donot include SSC and they do. This causes the high energy tail of the thermal electronsto be obscured by first order scatterings.The collapse of the disk however happens very similar to that of Dibi et al. (2012). Sincetheir torus starts at a larger radius, the radial profiles are quantitatively different, butbehave qualitatively similar under the influence of cooling. As mentioned in Chap. 4, thisquick collapse to a much thinner disk by assuming reasonable accretion rates for ADAFsmight give insight into state transitions for black holes accross the mass scale. By slowlyincreasing the accretion rate during the simulation and subsequently generating spectra,the thermalization of the X-ray spectrum might be seen. The jet might even disappear fora thin disk, although we do not yet calculate radiation from this part of the simulation.The two main assumptions in our cooling functions are optical thinness and the thermal-ization of electrons. We checked that the assumption of optical thinness holds by justsimply calculating the optical depth at each cell. For M7, the maximum optical depthreached is ∼ 0.1, while on average τ << 0.01. The more precarious and harder to verifyassumption is that of thermalization of electrons on a time scale shorter than the coolingtime scale. There is some theoretical evidence for magnetic and viscous instabilities thatcan heat electrons faster than Coulomb interactions (Begelman & Chiueh, 1988; Sharmaet al., 2007). We also know however that turbulence and magnetic reconnection can accel-erate electrons into a powerlaw (Sironi & Spitkovsky, 2014). We certainly see evidence forturbulence through the MRI and we also spotted some (numerical) reconnection events.How we should compare the thermal and non-thermal contributions is not obvious. Wedo know that the majority of the electrons should be thermal from observations (Falcke& Biermann, 1996; Yuan et al., 2003). It still could be that the small fraction of non-thermal electrons is important for the cooling, we have not estimated this as of yet.As we have shown cooling is important even for an underluminous source like Sgr A*,even for the lowest accretion rates possible according to observations. Although the dy-namics might not be significantly affected for these low accretion rates, the image andspectrum are, see Fig. 4.5 and A.4. When the dynamics are significantly affected, asthey are for M = 10−8,−7, obviously cooling has to be included.

52

CHAPTER 6

Conclusion

The structure and self-absorption profile of an accretion profile is very important infor-mation for analysis of the eventual EHT results. There are many degeneracies that canarise, for example variations in the scattering screen can resemble a change in inclination.It is therefore important that the observation-independent physics resembles reality asclosely as possible. Cooling GRMHD simulations, as is done in this work, is strictly betterthan the alternative and high-resolution studies are important as can be seen from thedeviations from previous results. Flows at the same accretion rate have a significantlydifferent appearance when cooled and even their dynamics can change as the accretionrate rises. We find a best fit-by-eye at M = 1 × 10−8 M/yr, i = 45 − 85 deg withTe/Ti = 0.3.We have seen the start of the formation of a thin disk as the accretion rate increases. Thisis part of standard accretion disk theory. We have seen that it happens on a very shorttime scale. For Sgr A*, this happens within a day. Increasing the accretion rate wouldsurely collapse the disk even further and this can be done until the assumption of opticalthinness breaks down. It is also likely that Te/Ti rises as thermal coupling between theions and electrons gets stronger. Investigating state transitions like this is easier in X-raybinaries since data during a whole cycle is available for several sources.Explaining Sgr A*’s position on the fundamental plane of black hole accretion or thestatistics from the XVP is unfortunately not possible using the simulation in this work.We have seen that the MRI dies out within a few days, causing the synchrotron fluxto drastically decrease. Any timeseries analysis would require very long, high resolu-tion 3D runs, possibly with stronger and larger magnetic field in order to keep the MRIcontinually going.

6.1 Future prospects

If one assumes that the flaring mechanics of Sgr A* are well-represented in ideal GRMHDsimulations, then in order to gather 60 days of data as in the Chandra X-ray Visionaryproject we need to run a simulation for 2.4 × 105 RG/c. This can however only be donein 3D since in 2D the MRI wavelength drops below the resolution and the MRI shuts offat about 5× 104 RG also due to Cowling’s anti-dynamo theorem (Cowling, 1933). Thismeans only about a fifth of the statistics could be gathered. But in 3D, if one wants tokeep the same resolution in r and θ for our lowest resolution simulations, we need also512 cells in φ making the simulation a factor 512 more expensive and the data size 512times as large. And the simulations need to run ∼ 50 times longer, at a higher cadence

53

as to make sure to catch all flares and recover their origin. The cadence for all simulationin this research is 20RG/c. On the machine used to do all runs in this works, simulationone day of Sgr A* at a resolution of 10242 takes about a day of real time. Simulating60 days in 3D is therefore an ambitious project and not allow for any parameter studies.This makes it very important to use other metrics than flaring statistics to make sure wereplicate Sgr A* inner accretion flow as closely as possible. These other metrics includethe spectral fitting and image comparison with the models from this research. Flaring inthe disk would most likely be a result of turbulence or numerical reconnection, where thelatter effect might not be a good representation of its physical equivalent and also has aresolution dependency. This is also something that needs to be investigated.At the moment, only synchrotron radiation is included for the spectra. In order to repro-duce also the X-rays, which flare the most, we need to include also bremsstrahlung andinverse Compton scattering for the spectra. In the radio a excess is seen towards higherwavelengths. By giving the thermal distribution of electrons a small powerlaw tail (1%of electrons) we could most likely reproduce the whole radio spectrum, although mostlikely the excess is caused by jet emission (Brinkerink et al., 2015). If the X-ray flaresare due to enhanced SSC we also need a partly non-thermal electron distribution (Yuanet al., 2002).One can also keep track of the electron distribution. This however breaks the thermalassumption and necessitates the use of not only cooling terms, but also electron-ion cou-pling terms that energize the electrons. Not to mention that keeping track of electrondistributions at each cell is a very big memory demand.Another extension of the current work would be to not only consider the disk but also thejet as radiation source as in Moscibrodzka et al. (2014). In that study, as in Moscibrodzkaet al. (2009), internal energy is not removed at run-time. It needs to be investigated ifthis would affect the dynamics as we saw for the disk. In the case where it could affectthe dynamics we need to investigate if this does not constantly trigger the (non-physical)density floors because the jet is mainly empty and the internal energy is low. One fixfor this would be resistive GRMHD, where the jet could actually be mass-loaded fromthe disk and wind. As an added benefit this would result in a more physical source ofreconnection in the disk.My focus for this work was Sgr A* and as such the assumption of optical thinness wasjustified. When considering other occurrences of ADAFs, for example state transitionsin XRBs, we probably have to deal with different regimes. Currently the radiation isonly included as a sink in the energy evolution equation. For flows that are opticallythicker radiative energy diffusion needs to be included and for dense winds even momen-tum transport due to the radiation field. One can do this by taking moments of theradiation transfer equation and introducing some closure relation analogous to how thefluid equations are derived and solving them on the same grid as the GRMHD equations.This is both more difficult and more numerically expensive than our approach. The mostcommon closure relations, flux-limited diffusion and M1 both have severe flaws (Turner& Stone, 2001; Gonzalez et al., 2007).

54

Acknowledgements

First of all I would like to thank my daily supervisor, Koushik Chatterjee. I had themost interaction with him and he helped my with many theoretical and practical issues,especially when I felt stuck. Secondly, I would like to thank Matthew Liska for helpingme with some problems that probably no one else could help me with. I also highlyappreciated the help I received from Ziri Younsi. The images and spectra he generatedfor me really tied my thesis together.Many thanks to prof. dr. Sera Markoff, my main supervisor and first examiner. Wehave been working together now for almost 2.5 years, since the start of my bachelor’sproject. Before that, I never considered astronomy as a master’s or career choice. Itis been educational and challenging the whole time. I appreciate the faith that I couldhandle important and relevant scientific research.Also a thank you to my second examiner prof. dr. Michiel van der Klis for selflesslyreading through my thesis.I would also like to thank and congratulate all of my colleagues from C4.105, my fellowmaster’s students, Emma, Martijn, Frank, Filipe, Floor, Jorrit, Kriek and David. The last2 years were a blast and although I do not remember receiving any help, I do rememberlaughing a little bit too loud more than I would like to admit.All other members of the jetset research group, Dimitris, Tom, Matteo, Atul, Tobi andFe, many thanks for the insightful group meetings. In particular the latter two for usefulfeedback on my thesis.Outside of the university, I would like to thank my friends and office mates Johan andAbel for patiently waiting for me to finish school, although Abel recently gave up andretired. I look forward to working with Johan for the next few years. I also enjoyed allthe refreshing drinks with my other office mates Laurens and Puk.Lastly, I would like to thank my parents Joris and Marielle and my brother Bram forkeeping my on the path of education when other things might have been more tempting.More generally, I would like to thank them for giving my the childhood and educationthat allows me to do what I do and succeed.

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APPENDIX A

Ray-tracing results

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Figure A.1: The 230 GHz images and radio spectra at i = 5 deg, i = 45 deg and i = 85deg for M9T3HR. Observations taken from Falcke et al. (1998); Schodel et al. (2003);Bower et al. (2015).

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Figure A.4: The 230 GHz image at i = 45 deg and spectra of M9T3HR compared toHR for which we assumed M = 1 × 10−9 M/yr. Observations taken from Falcke et al.(1998); Schodel et al. (2003); Bower et al. (2015).

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