Models of Sgr A*

Radiative Models of Sgr A* from GRMHD Simulations

Monika Mościbrodzka11affiliationmark: , Charles F. Gammie11affiliationmark: 22affiliationmark: , Joshua C. Dolence22affiliationmark: ,
Hotaka Shiokawa22affiliationmark: , Po Kin Leung22affiliationmark:
Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801 Astronomy Department, University of Illinois, 1002 West Green Street, Urbana, IL 61801 mmosc@illinois.edu
Abstract

Using flow models based on axisymmetric general relativistic magnetohydrodynamics (GRMHD) simulations, we construct radiative models for Sgr A*. Spectral energy distributions that include the effects of thermal synchrotron emission and absorption, and Compton scattering, are calculated using a Monte Carlo technique. Images are calculated using a ray-tracing scheme. All models are scaled so that the flux density is . The key model parameters are the dimensionless black hole spin , the inclination , and the ion-to-electron temperature ratio . We find that: (1) models with are inconsistent with the observed submillimeter spectral slope; (2) the X-ray flux is a strongly increasing function of ; (3) the X-ray flux is a strongly increasing function of ; (4) image size is a complicated function of , , and , but the models are generally large and at most marginally consistent with the VLBI data; (5) for models with and the event horizon is cloaked behind a synchrotron photosphere at and will not be seen by VLBI, but these models overproduce NIR and X-ray flux; (6) in all models whose SEDs are consistent with observations the event horizon is uncloaked at ; (7) the models that are most consistent with the observations have . We finish with a discussion of the limitations of our model and prospects for future improvements.

accretion, accretion disks — black hole physics — MHD — radiative transfer — Galaxy: center – (catalog Sgr A*)
slugcomment: Accepted by ApJ

1 Introduction

Long term studies of the stellar dynamics in the central parsec of our Galaxy indicate that the object in the center of the Milky Way is massive and compact and is therefore likely to be a supermassive black hole (we will use Sgr A* to refer to the radio source, the putative black hole, and the surrounding accretion flow). Recent estimates of Sgr A*’s mass and distance kpc (Ghez et al. 2008, Gillessen et al. 2008) indicate that it has the largest angular size of any known black hole ().

Sgr A* is frequently monitored at all available wavelengths: in radio since its discovery in 1974 (Balick & Brown 1974), and more recently in submillimeter, near-infrared (NIR), and X-rays. It is heavily obscured in the optical and UV ( mag). Sgr A* is a “quiescent” galactic nucleus because its bolometric luminosity in units of the Eddington luminosity is low, . The discovery of polarized emission at mm allowed the use of Faraday rotation to place a model dependent limit on the mass accretion rate at (Bower et al. 2005, Marrone et al. 2006). Submillimeter VLBI of Sgr A* shows structure at very small angular scales (Doeleman et al. 2008).

Sgr A*’s spectral energy distribution (SED) can be fit with semi-analytic quasi-spherical radiatively inefficient accretion flow (RIAF) models (e.g. Narayan et al. 1998), RIAF + outflow models (Yuan et al. 2003), and with time-dependent MHD models (e.g. Goldston et al. 2005, Ohsuga et al. 2005, Mościbrodzka et al. 2007). Other workers have modeled the VLBI and submillimeter emission (Broderick & Loeb 2005, Broderick & Loeb 2006a, Broderick & Loeb 2006b, Huang et al. 2007, Miyoshi et al. 2008, Broderick et al. 2008, Yuan et al. 2009, Dexter et al. 2009) assuming a stationary RIAF and computing emission at submillimeter wavelengths taking into account general relativistic effects.

In this work we simultaneously model the spectral energy distribution, including Compton scattering, and the VLBI data using a relativistically self-consistent approach. We assume that accretion onto Sgr A* proceeds through a geometrically thick, optically thin, two-temperature flow that we model using a general relativistic MHD (GRMHD) simulation. Black hole spin is self-consistently accounted for. We also assume that the (likely time-dependent, anisotropic, nonthermal) state of the plasma can be described by assigning a single temperature to the ions and a possibly different temperature to the electrons. Conduction is neglected.

The main goal of this work is to explore how , the inclination , and the ion-to-electron temperature ratio are constrained by the data. Our paper is organized as follows. In § 2 we review broadband observations of Sgr A*. In § 3 we outline our technique for computing the evolution of the accretion flow and the emergent radiation. In § 4 we present the results of single- and two-temperature SED computations and compare them to the observed SED. We summarize and discuss the model limitations in § 5.

2 Observations

Sgr A* has rich observational database in radio (Serabyn et al. 1997, Falcke et al. 1998, Zhao et al. 2003, An et al. 2005, Marrone et al. 2006), NIR (Davidson et al. 1992, Herbst et al. 1993, Stolovy et al. 1996, Telesco et al. 1996, Menten et al. 1997, Melia & Falcke 2001, Hornstein et al. 2002, Genzel et al. 2003, Eckart et al. 2006, Schödel et al. 2007), X-rays (Baganoff et al. 2001, Baganoff et al. 2003, Goldwurm et al. 2003, Porquet et al. 2003, Bélanger et al. 2005, Bélanger et al. 2006, Porquet et al. 2008) and even -rays (Aharonian et al. 2004, but see Aharonian et al. 2008).

In general the emission from Sgr A* in the radio band is rising with the frequency. Below the spectral slope () was found to be 111Not to be confused with the phenomenological viscosity of accretion disk theory. In this paper angular momentum transport is calculated self-consistently in a GRMHD model. (Serabyn et al. 1997, Falcke et al. 1998). Between and the spectral slope changes to (Falcke et al. 1998, An et al. 2005).

Marrone (2006) reported that the spectral slope becomes flat or declining between () and (), consistent with a transition from optically thick to optically thin radiation. He estimated a variance-weighted mean value of from four observational epochs (each epoch lasting around 2  hr, and changing from -0.46 to 0.08). The quiescent NIR counterpart of Sgr A* has been reported by Genzel et al. (2003), but it is not clear whether the “quiescent” NIR emission at the position of Sgr A* is background noise or a real detection of Sgr A* (Do et al. 2009). Thus, the measured quiescent emission in the NIR is usually interpreted as an upper limit. The quiescent luminosity at the 2-8 keV band measured with the Chandra observatory is , and the emission is extended with an intrinsic size of , consistent with the Bondi radius. The mass accretion rate at the Bondi radius deduced from X-ray observations is (Baganoff et al. 2003). Above the stationary emission Sgr A* exhibits intraday variability at all observed wavelengths (flares in submillimeter, NIR, and X-rays that often rise simultaneously).

High frequency VLBI constrains the structure of Sgr A* on angular scales comparable to . The distribution of intensities on the sky is a convolution of the (wavelength dependent) intrinsic angular structure with anisotropic interstellar broadening proportional to (Bower et al. 2006, Doeleman et al. 2008). Sgr A* has been detected by mm VLBI on baselines between Hawaii (JCMT), Arizona (SMTO), and California (CARMA) (Doeleman et al. 2008). This small number of baselines does not permit imaging of the emitting region or the “silhouette” of the black hole (Bardeen 1973, Chandrasekhar 1983, Falcke et al. 2000, Takahashi 2004), but it does constrain models of the emitting region. Using a (two-parameter) symmetric Gaussian brightness distribution model Doeleman et al. (2008) infer a full width at half maximum FWHM , or . This is very small, since the apparent diameter of the black hole is . The FWHM for a Gaussian model has also been estimated at ( mas, or , Bower et al. 2004) and at ( mas, or , Shen et al. 2005), but longer wavelength intrinsic size is more difficult to measure because scatter broadening dominates the observed image size at . VLBI observations at () and () are expected in the near future (Doeleman et al. 2009).

3 Model

Our model consists of three parts: a physical model of the accreting plasma; a numerical realization of the physical model; and a procedure for calculating the emergent radiation from the accreting plasma.

The physical model is a geometrically thick, optically thin, turbulent plasma accreting onto a rotating black hole in a statistically steady state. The angular momentum of the hole is assumed to be aligned with the angular momentum of the accreting plasma. 222Tilted, or “oblique” accretion flows, require 3D simulations; antialigned flows can be modeled using an axisymmetric simulation, but likely provide a worse fit to the data than the low-spin aligned flows considered here. The ions and electrons are assumed to have a thermal distribution function, but with a temperature ratio that may be different from one (see §2.2.1. in Sharma et al. 2007, for a discussion of temperature ratios in a collisionless accretion flow model; their work suggests that may be a natural value). The equation of state is gas pressure ( is the proper internal energy density), with , appropriate to a plasma with and (we will discuss our procedure for extracting an electron temperature later). The parameters of the accreting plasma model, then, are and .

The numerical realization of the physical model uses the GRMHD code harm, a conservative shock-capturing scheme with constrained transport to preserve (Gammie et al. 2003). All models in this paper are axisymmetric; we will explore 3D models in a subsequent publication. Our grid is uniform in modified spherical Kerr-Schild coordinates (Gammie et al. 2003), which permit the flow to be followed through the event horizon. The coordinates are logarithmic in the Kerr-Schild radius and nonuniform in Kerr-Schild colatitude (Boyer-Lindquist and Kerr-Schild and are identical).333The modified Kerr-Schild coordinates ,,, are related to spherical Kerr-Schild coordinates by , , , and . We set . The resolution is .

The inner boundary of the computational domain is at , i.e. just inside the event horizon. The outer boundary is at , or an angular radius of . Since low frequency emission is believed to arise at larger radius, this means that we are unable to model the low frequency (radio and mm) portion of the SED.

We must also supply initial conditions and boundary conditions. For numerical convenience we adopt the same initial equilibrium torus used by Gammie et al. 2003, McKinney & Gammie 2004, and others. This torus has an inner boundary at and a rest-mass density maximum at .444For we set so that scale height has similar for all the models. It is seeded with a weak, purely poloidal field that follows the isodensity contours and has minimum (see Gammie et al. (2003) for details). Small perturbations are added to the internal energy. The torus quickly becomes turbulent due to the magnetorotational instability (Balbus & Hawley 1991). At the accretion flow soon reaches a nearly (statistically) stationary state that is independent of the initial conditions (except for the magnetic field geometry; see Hawley & Krolik 2002, Beckwith et al. 2008). If our numerical model accurately represents the physical model, this inner accretion flow should be similar to the inner portion of a much more extended accretion flow. We use outflow boundary conditions at both the inner and outer boundaries, and the usual polar boundary conditions at and . We integrate for , or 8 orbital periods at .

harm (and similar codes) fail if or are small in comparison to the kinetic and magnetic energy densities, or the density in nearby zones. To prevent this we impose a hard “floor,” so that and .

To “observe” the numerical model we must specify the observer’s distance and the inclination of the black hole spin to the line of sight. Because the dynamical simulation is scale-free but the radiative transfer calculation is not, we need to specify the simulation length unit , time unit , and mass unit (equivalently: the mass accretion rate). Since we set the peak density in the GRMHD model to in simulation units, the peak density is in cgs units. The mass unit is not set by (which appears in the dynamical model only in the combination ) because the flow mass is , and has negligible effect on the gravitational field. is therefore a free parameter.

To calculate the SED we use the relativistic Monte Carlo scheme grmonty. A detailed description of the algorithm and tests are given in Dolence et al. (2009). The code fully accounts for synchrotron emission and absorption, and Compton scattering. It uses a “stationary flow” approximation, computing the spectrum through each time slice of simulation data as if it were time-independent. This is an approximation because the light crossing time is comparable to the dynamical time. It is done because tracking photons through the time-dependent simulation data is still too computationally expensive. We will evaluate the quality of this approximation once we are able to calculate fully self-consistent spectra. An average spectrum is formed by averaging over single slice spectra from each of different realizations of the simulation (the realizations differ in the random number seed used to generate initial conditions).

To image the model we use the relativistic ray-tracing code ibothros, which accounts only for synchrotron emission and absorption, again using a stationary flow approximation (see Noble et al. 2007). An average image is created using the same averaging procedure as for the spectra.

To sum up, the model parameters (aside from those that describe the initial conditions) are , , , , , and . and are set by the observations of stellar orbits; we briefly explore the consequence of varying them below. we will set for each model by requiring that time-averaged flux . The remaining three free parameters are , , and .

In what follows we explore models with , with and , i.e. and , , and , and and .

The observational constraints on the model are (aside from the flux density): the submillimeter spectral slope , the upper limit on the quiescent NIR flux density (Genzel et al. 2003, Melia & Falcke 2001, Hornstein et al. 2002, Schödel et al. 2007, Dodds-Eden et al. 2009) and the upper limit on the quiescent luminosity between and keV (Baganoff et al. 2003). Since the measured is produced inside (), and our simulation domain covers only the inner , we exclude models for which is close to the “quiescent emission” . 555We use the source brightness profile (Baganoff et al. 2003) and estimate that of the X-ray luminosity comes from the central pixel of size . This is still far larger than our computational domain, so we require that the X-ray luminosity in our models not exceed .

4 Results

We have studied the combinations of model parameters listed in Table 1 (=1), Table 2 (=3), and Table 3 (10). The model that best satisfies the observational constraints has (model D), , and . We will present this “best-bet” model in some detail before going on to the full parameter survey (see §4.2) to give the reader a physical sense for the models.

4.1 Best-bet model

In what follows we will consider time and realization averaged SEDs and images. But to get a sense for physical conditions in the accretion flow, consider a single time-slice at . Figure 1 shows the run of , and in the time slice. Evidently is a typical equatorial plane density, G is a typical field strength, and is a typical electron temperature. Notice that the field changes from a tangled, turbulent structure near the midplane to a more organized structure in the “funnel” over the poles of the black hole. Temperature generally increases away from the midplane.

In Figure 2 we show the SED computed from the same timeslice used in Figure 1 (thick line) along with the average SED (thin line). The figure also shows a selection of radio, NIR and X-ray observational data points (references given in the figure caption). The SED has a peak at due to thermal synchrotron emission. Below it fails to fit the data because that emission is produced outside the simulation volume. A second peak in the far UV is due to the first Compton scattering order, and at () the photons are produced by two or more scatterings.

Where do the photons originate in each band? Figure 3 maps the points of origin for photons in the synchrotron peak ( to ), in the NIR () and in the X-ray (2-8 keV). Most of the submillimeter emission originates near the midplane at . NIR photons are produced in hot regions with high magnetic field intensity and high gas temperature, and these are concentrated close to the innermost stable circular orbit (ISCO; for ) i.e. they come from between . All photons responsible for the formation of the first Compton bump are up-scattered between but the 2-8 keV emission is produced mainly by scatterings in the hottest parts of the disk at . Emission can be seen around the borders of the funnel in each panel, but this is at a low level and is associated with unreliable temperatures assigned by harm to the funnel region.

The photons that form the submillimeter peak () originate at , where Gauss, and . 666 Angle brackets indicate an average over grid zones weighted with the photon number ‘detected’ in a given frequency band. The emissivity calculated from these averaged values peaks at Hz, which is quite close to the actual peak in (where ). The averaged values also yield ; the submillimeter peak plasma is dominated by gas pressure rather than magnetic pressure.

The optically thin synchrotron photons in the energy range between the synchrotron peak and first-order Compton scattering bump ( Hz) tend to arise in current sheets at , where ,  Gauss, and or higher. The corresponding peaks in the mid-IR, at . These current sheets have higher entropy than the surrounding plasma, consistent with the idea that they are heated by (numerical) reconnection. The prominence of the current sheets is likely an artifact of axisymmetry. Similar current sheets are seen in axisymmetric shearing box models of disks that are not present in three dimensional shearing box models.777Our best guess is that well resolved 3D models will exhibit heating that is more evenly distributed. Direct, physical heating of the electrons can take place in this strongly shearing region via plasma instabilities acting on anisotropic pressure, as discussed by Sharma et al. (2007).

The synchrotron photons Compton scatter at where . The X-ray emission at 2-8 keV is formed by scatterings from plasma with at . For this temperature the average energy amplification per scattering is , consistent with the seed photons with energies (). This means that many of the seed photons are produced in current sheets, and so some uncertainty attaches to the Compton scattered flux. We know observationally that Sgr A* produces frequent flares with fluxes larger than those produced by our quiescent-source model, so there is a source of seed photons in this energy band, albeit a fluctuating one.

A small fraction of photons are emitted from the funnel wall at large radii () where the gas temperature is . This is also likely an artifact of the inability of harm and similar codes to track the internal energy of a fluid when the internal energy is much smaller than the other energy density scales. Nevertheless, this raises the interesting question of what the electron distribution function should be in the funnel. High energy electrons might be naturally generated within this tenuous plasma by steepening of MHD waves excited by turbulence near the equatorial plane.

4.2 Parameter survey

In Figure 4, we present averaged spectra for models with different spins (referred to as A, B, C, D, E and F; see Tables 12, and 3), inclination angles and in the upper, middle and bottom panels, respectively and temperature ratio and from left to right. All SEDs are averaged over time and runs as described in § 3.

The tables indicate whether the model is consistent with observations. The model can fail in one of four ways: it can produce the wrong submillimeter spectral slope ; it can overproduce the quiescent NIR flux; it can overproduce the quiescent X-ray flux; and it can be too large at to be consistent with the VLBI data. The last constraint we will discuss separately in the next section. It may be useful to recall that is adjusted in each model so that the flux is .

The model can also fail by cooling too rapidly to be consistent with our neglect of cooling in the dynamical model. The Tables list a radiative efficiency , where is the bolometric luminosity (integrated over solid angle), and for comparison a thin disk efficiency at the same . ranges between for to for (the thin disk efficiency for the latter is ). Only in the , model is the radiative efficiency sufficiently high that cooling is likely to have a significant effect on the GRMHD model. We will consider models with cooling in a future publication.

Very few of the time averaged SEDs based on a single-temperature () models produce the correct . The exception is edge-on tori () around fast spinning black holes (model E and F). These models are ruled out, however, because they overproduce NIR and X-ray flux.

For only the model with seen at agrees with the data. This is the best-bet model discussed in § 4.1. For , models with spins below (A, B and C) are ruled out by the inconsistent spectral slope, and models with higher spins (E and F), although consistent with the observed , overproduce the quiescent NIR and X-ray emission. All models with observed at and are ruled out by the inconsistent .

For , we find that all models with are ruled out by both and violation of NIR and X-ray limits. For lower inclination angles () a few models (E and F with , and A, B, C, and D at ) reproduce the observed . These models are consistent with X-rays and NIR limitations. Models E and F for are ruled out by NIR and X-ray limitations whereas models A, B, C and D for produce which is too small.

What is the physical origin of these constraints?

The dependence on arises largely because as increases the inner edge of the disk — the ISCO – reaches deeper into the gravitational potential of the black hole, where the temperature and magnetic field strength are higher. In the disk mid-plane, the temperature is a fraction of the virial temperature and scales with radius . , while the density , below the pressure maximum. Holding all else constant (which we do not: we hold the flux constant) this implies a higher peak frequency for synchrotron emission, a constant Thomson depth (in our models the Thomson depth at the ISCO is roughly constant, since the path length but the density ), and a larger energy boost per scattering , as can be seen in comparing models with different spin in Figure 4. The X-ray flux therefore increases with because at the ISCO increases.

The dependence on is mainly due to synchrotron self-absorption, which is strongest at high inclination. For example, because the , model is optically thick at the emission is produced in a synchrotron photosphere well outside . The typical radius of the synchrotron photosphere ranges between 15 for low spin models () and 8 for high spin models (). The flux can then be produced only with large ; as increases the optically thin flux in the NIR increases due to increasing density and field strength. The scattered spectrum also depends on since the energy boost per scattering is .

The inclination dependence is, interestingly, a relativistic effect. is nearly independent of (it varies by , except for , which due to optical depth effects has much larger variation), so models with different inclination are nearly identical. Nevertheless the X-ray flux varies dramatically with , increasing by almost 2 orders of magnitude from to . This occurs because Compton scattered photons are beamed forward parallel to the orbital motion of the disk gas. The variation of mm flux with is due to self-absorption. The mm flux reflects the temperature and size of the synchrotron photosphere. At lower the visible synchrotron photosphere is hotter than at high .

There is an additional constraint due to Faraday rotation measurements, but this constraint is qualitatively different because we do not directly calculate Faraday rotation in our model. Instead we adopt the constraints on which are inferred, via a separate model, from the Faraday rotation data (Bower et al. 2005, Marrone et al. 2006). increases, in a nonlinear way, with increasing . For . For . All these values are consistent with the Faraday rotation constraints, although the highest , , models are only marginally consistent.

There are a few other general trends worth mentioning. In all models the average optical depth drops below at to . For and high BH spins the emission in NIR band (2 ) is formed by the direct synchrotron emission while the emission results from a first-order scattering. For low BH spins the emission in NIR is due to first-order Compton scatterings and the X-ray is second-order scattering. For and independently of the BH spin the NIR emission is formed by a first-order Compton scatterings and X-rays- by second-order scatterings.

4.3 Images and the size of the emitting region

We compute the 230 GHz intensity maps of our models using a ray tracing code (Noble et al. 2007) and we average them in the same manner as the spectra. To estimate the size of the emitting region we calculate the eigenvalues of the matrix formed by taking the second angular moments of the image on the sky (i.e. the length of the “principal axes”). The eigenvalues along the major () and minor () axis are given in Table 4. In Figure 5 we show averaged images for models with SEDs that are consistent with the data.

The source size depends on the model parameters. For , we find a critical mass accretion rate (the exact value depends on and ) below which the size of the emitting region decreases monotonically with increasing . Above , the size of the emitting region increases with increasing . The increasing trend can be explained by the appearance of the synchrotron photosphere at for larger mass accretion rates at high inclination. For and , at the black hole horizon is cloaked by the photosphere and cannot be observed by VLBI (notice that this model is ruled out for other reasons). For a constant and the size always decreases with increasing , because the emissivity of the central regions increases with .

The size of the emitting region for our best-bet model is consistent with the observed FWHM (inferred from VLBI data using a two-parameter Gaussian model). For the sizes of the images are inconsistent with the VLBI measurement, except model D, which is only marginally consistent. Notice that this moment-based analysis is crude. It would be better to “observe” the model with the same baselines used in gathering the VLBI data (this would add a new parameter, the position angle). Our analysis is particularly ill suited to low models that are ring-like and therefore poorly fit by a Gaussian model.

4.4 Varying distance and mass

In our discussion we have fixed the mass and distance of Sgr A*, but these are uncertain to . How would changing these parameters change our results?

First, consider how depends on and . Near the submillimeter peak, (since and ) if the model is optically thin (although there are usually optically thick lines of sight through the model even if the mean optical depth at the submillimeter peak is ). We therefore expect that .

Consider varying and in our best-bet model (model C; and ). We find that when changing the distance from kpc to kpc if we fix . In particular, for and kpc, respectively. For , and for and respectively, which gives when changing black hole mass from . This is crudely consistent with our expectations based on an optically thin source.

Finally we change the mass and distance simultaneously according to the observational relation =constant (Ghez et al. 2008). For and , whereas for kpc and , . We find that at , , ; at , ; at , , . The spectral slope and X-ray luminosity therefore vary .

Our best-bet model remains consistent with the data, then, if we vary with and within the range permitted by observation. Models with and with BH spin and become acceptable if and are lowered, but only the model with would be (marginally) consistent with VLBI measurements of the Sgr A*size. In sum, and are tightly constrained; varying them within the narrow range of values permitted by observations does not change the main conclusions of this work.

5 Summary

Under the assumption that the accretion flow at the galactic center is optically thin, geometrically thick, and lightly magnetized, we have presented constraints on , , and for Sgr A*. We find that models with and describe the sub-mm spectral observations () better than models with . We find that the model with , black hole spin and the close to edge-on inclination angles is consistent with the broadband SED observational data and the size of Sgr A* measured by VLBI. In this case the silhouette of the black hole is difficult to observe because Doppler boosting of the disk emission places almost all the emission on one side of the black hole.

If, on the other hand, the electrons are heated relatively inefficiently () then models with observed at , or observed at are consistent with the observed SED. The sizes of the emitting regions in these models, however, seem to be inconsistent with the VLBI measurements, except again at .

Our best-bet estimate of the black hole spin () disagrees with Broderick et al. (2008) (following Yuan et al. 2009) who found and ( errors) based on a careful analysis of images of RIAF models. The discrepancy may be a consequence of different emissivity (ours is based on Leung et al. 2009, in preparation), and different underlying models for the run of temperature, density profile, magnetic field strength, and geometry of the flow. We also do not include non-thermal emission as in Broderick et al. (2008). The results for and at low spin values agree with the previous study, but according to our moment analysis these models are inconsistent with the VLBI data. An analysis of images at that folds the models through the VLBI observation process is needed to definitely exclude models based on the VLBI data.

Our and constraints are different from those presented in Noble et al. (2007), because here we allow . We do not find a good fit to the observational data for single-temperature models which were studied in the earlier work. We confirm the trend that the bolometric luminosity increases with the increasing black hole spin.

The models studied here differ in many respects from those considered earlier by Mościbrodzka et al. (2007). The earlier models were based on low angular momentum, nonrelativistic hydrodynamic models for the accretion flow that extended over a wide range in radii. The models described here are fully relativistic MHD models that extend over a limited range in radius and use fully relativistic radiative transfer.

There are still significant uncertainties in our models. These uncertainties fall into four categories: the unimportant; those which may be important and be easily eliminated with a small additional effort; those which may be important and require a major effort, but are in principle straightforward to eliminate; and those which are serious and require new physical understanding.

In the interests of full disclosure, our unimportant approximations are: (1) we use a law equation of state rather than a Synge-type equation of state that would more accurately represent our two-temperature relativistic gas. Shiokawa et al. 2009 (in preparation) show that the associated changes in spectra are small; (2) for our cyclo-synchrotron emissivity and absorptivity are imperfect because most of the emission comes from electrons with , where our approximate expression begins to break down (Leung et al. 2009, in preparation) 888For models with and the error in SED associated with our approximate emissivity function is less than at all frequencies. For the errors are less than at because the emissivity-weighted mean temperature is lower and self-absorption is important. The errors are taken from comparison of Leung et al. 2009 emissivity formula with directly integrated cyclo-synchrotron harmonics for lowest values of found in our simulations.; (3) we neglect bremsstrahlung, which is expected to be important only far from the horizon; (4) we neglect double Compton scattering. The cross section for the double Compton process is (fine-structure constant) smaller than single Compton scattering and can be neglected here (also ); (5) we neglect induced Compton scattering. Induced Compton scattering is important for , where is the brightness temperature. In Sgr A*  K and so it is indeed negligible.

The significant approximations that could be fixed with some additional effort include: (1) our neglect of cooling. It is straightforward to run our GRMHD models with cooling, but then they are no longer scale-free. With cooling turned on we would need to fix by evolving the GRMHD model at a trial , calculating the spectrum, and repeating until . (2) axisymmetry. Three dimensional (3D) models are available but far more expensive to evolve. Use of a 3D models would permit us to evolve models with (3) a wider range of radii, so that millimeter emission could be included and even the sub-mm emission could be more accurately modeled. We have not run axisymmetric models with radially extended accretion flows because they tend to develop pathologies (strong, radially extended magnetic filaments). (4) our neglect of nonthermal electrons. These could be readily included using a phenomenological prescription for the shape and amplitude of the nonthermal portion of the electron distribution function (Özel et al. 2000, Yuan et al. 2003, Chan et al. 2009), (5) our use of the steady-state approximation in calculating SEDs and images. This would require time-dependent radiative transfer, which is straightforward in principle but computationally expensive.

Other approximations can be fixed only with significant additional effort: (1) our neglect of pair production. This would require a model for the radiation field near the pair production threshold. Our preliminary estimates suggest that in many of our models pair production is substantial. One advantage of incorporating pair production is that it might permit us to eliminate our numerical floor and therefore more accurately evolve the low density funnel region; (2) our treatment of thermal energy in the funnel. harm tends to produce high temperatures in the tenuous funnel plasma, some of which are clearly numerical artifacts caused by application of the density floor and other, more subtle, numerical issues associated with the small ratio of thermal energy density to other energy densities; (3) our simplistic, two-temperature thermal model for the plasma. This includes our neglect of conduction and anisotropy of the plasma.

New physical understanding would be required to predictively model (1) nonthermal parts of the distribution function and (2) the initial magnetic field configuration. Nonthermal particles can of course be included in a phenomenological prescription, but the particle injection and acceleration processes are still not fully understand. As we have already mentioned, prior work shows that the GRMHD models depend nontrivially on the initial field configuration. We have adopted a simple, numerically appealing initial configuration, but the long-term evolution of the large-scale field is ill understood.

Finally, notice that there are observational constraints from polarization data and from light curves (statistically, the one and two-point statistics of the light curves at each frequency, and the cross correlations between different frequencies). Treating the polarization data requires accurate emissivities and absorptivities, as well as models that extend well past the radius where , which is where most of the intrinsic Faraday rotation occurs. The light curves require full, time-dependent radiative transfer, since the dynamical time is comparable to the light crossing time.

This work was supported by the National Science Foundation under grants AST 00-93091, PHY 02-05155, and AST 07-09246, through TeraGrid resources provided by NCSA and TACC, and by a Richard and Margaret Romano Professorial scholarship, a Sony faculty fellowship, and a University Scholar appointment to CFG. The authors are grateful to Stu Shapiro, Fred Lamb, Dan Marrone, Shep Doeleman, and Vincent Fish for discussion and comments.

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run consistent
[deg] w/ obs.?
5 3.97 -2.04 30.9 NO
A 0.5 45 3.55 -1.78 30.7 0.0821 NO
85 3.81 -1.21 31.3 NO
5 1.6 -1.94 31.1 NO
B 0.75 45 1.39 -1.69 31.0 0.112 NO
85 1.5 -1.12 31.6 NO
5 0.94 -1.92 31.3 NO
C 0.875 45 0.87 -1.65 31.3 0.145 NO
85 0.91 -1.07 32.3 NO
5 0.86 -1.67 31.9 NO
D 0.94 45 0.78 -1.41 32.1 0.179 NO
85 0.80 -0.87 33.1 NO
5 0.85 -1.51 32.3 0.11 NO
E 0.97 45 0.78 -1.25 32.7 0.1 0.213 NO
85 0.80 -0.68 33.8 0.1 NO
5 0.64 -1.51 32.9 0.19 NO
F 0.98 5 45 0.58 -1.25 33.1 0.17 0.245 NO
85 0.59 -0.69 34.2 0.18 NO

Note. – The columns from left to right are: run ID, dimensionless spin of the black hole, inclination angle of the observer with respect to the black hole spin axis, averaged rest mass accretion rate, spectral slope between 230-690  GHz(), and luminosity in the X-rays (at Hz), the radiative efficiency , the thin disk efficiency for the same , and whether the model is consistent with the data.

Table 1: Summary of MHD and SED for models with
run consistent
[deg] w/ obs.?
5 10.0 -2.09 30.5 NO
A 0.5 45 8.8 -1.72 32.4 0.0821 NO
85 10.7 -0.75 31.3 NO
5 3.9 -1.99 31.0 NO
B 0.75 45 3.5 -1.59 31.0 0.112 NO
85 4.1 -0.67 31.7 NO
5 2.28 -1.97 31.1 NO
C 0.875 45 2.07 -1.54 31.2 0.145 NO
85 2.43 -0.56 32.3 NO
5 1.9 -1.68 31.5 NO
D 0.94 45 1.7 -1.27 31.8 0.179 NO
85 1.86 -0.44 32.9 YES
5 1.86 -1.37 32.2 NO
E 0.97 45 1.68 -1.01 32.5 0.213 NO
85 1.83 -0.21 33.8 NO
5 1.4 -1.48 32.6 0.11 NO
F 0.98 45 1.23 -1.13 32.9 0.10 0.245 NO
85 1.29 -0.26 34.3 0.10 NO

Note. – Columns same as in Table 1.

Table 2: Summary of MHD and SED for models with
run consistent
[deg] w/ obs.?
5 90.8 -1.37 30.1 NO
A 0.5 45 117.1 -0.2 31.4 0.0821 YES
85 369.0 1.38 33.5 NO
5 38.3 -1.05 30.0 NO
B 0.75 45 50.2 0.04 31.8 0.112 YES
85 190.6 1.49 34.6 NO
5 19.5 -1.15 31.4 NO
C 0.875 45 23.6 -0.07 32.0 0.145 YES
85 41.4 1.19 34.2 NO
5 13.7 -0.93 31.8 NO
D 0.94 45 15.2 -0.05 32.3 0.179 YES
85 31.2 1.17 34.4 NO
5 13.6 -0.40 32.5 YES
E 0.97 45 14.3 0.2 33.1 0.213 NO
85 26.5 1.19 35.2 NO
5 9.07 -0.6 32.7 YES
F 0.98 45 9.16 -0.08 33.2 0.245 NO
85 15.5 1.12 35.4 NO

Note. – Columns same as in Table 1.

Table 3: Summary of MHD and SED for models with =10.
run
5 41.7 41.6 38.1 38.0 35.9 35.9
A 0.5 45 31.2 28.4 28.9 25.9 32.7 31.6
85 23.1 20.7 23.3 21.9 37.6 33.9
5 38.8 38.7 35.4 35.3 34.6 34.6
B 0.75 45 28.6 25.9 26.7 23.7 31.2 30.2
85 20.7 19.4 20.8 20.4 36.9 32.3
5 39.3 39.2 35.2 35.0 31.9 31.8
C 0.875 45 29.3 26.7 26.4 23.4 29.3 28.0
85 20.6 19.4 20.4 19.7 30.6 30.1
5 37.2 37.1 32.2 32.1 28.0 28.0
D 0.94 45 27.4 25.3 24.0 21.5 24.7 23.9
85 19.2 18.4 18.5 17.2 27.1 26.4
5 37.0 36.9 31.2 31.1 26.5 26.4
E 0.97 45 27.4 25.7 23.4 21.6 23.6 23.4
84 19.6 18.2 18.1 17.3 25.8 24.6
5 36.5 36.4 31.3 31.2 26.1 26.1
F 0.98 45 27.2 25.2 23.4 21.2 22.8 21.7
85 18.8 18.4 16.5 17.6 23.8 22.6

Note. – The columns from left to right are: run ID, dimensionless spin of the black hole, inclination angle , size of the emitting region in terms of standard deviation in the major () and minor () axis for (col. 4 and 5), (col. 6 and 7), and (col. 6 and 9) in units of . For a Gaussian model, the VLBI data require FWHM (Doeleman et al. 2008), or .

Table 4: The size of the emitting region at .
Figure 1: Disk structure in our best-bet model with (model D) and with . The number density, the magnetic field strength, and the electron temperature are shown in the left, middle, and right panel respectively. The axis scale units are .The figure presents a single time slice.
Figure 2: SEDs computed based on a single time slice (thick line) (see Figure 1 for the distributions of the physical variables corresponding to the same time) along with the time averaged spectrum (thin line) of our best-bet model. Observational points are taken from: Falcke et al. 1998, An et al. 2005, Marrone et al. 2006 at radio, Genzel et al. 2003 at NIR (1.65, 2.16, and 3.76 ) and Baganoff et al. 2003 at X-rays (2-8 keV). The upper limits in the NIR band are taken from Melia & Falcke 2001 (30, 24.5 and 8.6 ), Schödel et al. 2007 (8.6 ) and Hornstein et al. 2007 (2 ). The points in the NIR at flaring state are from Genzel et al. 2003 (1.65, 2.16, and 3.76 ) , and Dodds-Eden et al. 2009 (3.8 ). An example of X-ray flare ( ) is taken from Baganoff et al. 2001.
Figure 3: Maps showing the point of origin for photons in our best-bet model. We show the logarithm of the sum of the photon weights in each zone, which is proportional to the number of photons seen at (left), (middle) and 2-8 keV (right) band. The axis scale units are . The figure presents a single time slice at . Note that color bands differ in scales.
Figure 4: SEDs computed for and 10 in left, middle and right panels, respectively and and in bottom, middle and top panel, respectively. Models A, B, C, D, E and F, have different (see Tables 12, and 3). Each SED is a result of averaging over 200 individual SEDs taken from last of the GRMHD runs (4 runs and 50 dumps for each spin). Observational points and upper limits as in Figure 2. Models consistent with the observations are marked with thick lines.
Figure 5: Images of the accretion flow at for models with SEDs that are consistent with observations of Sgr A*. The images has been averaged over time and over four separate realizations of each model. Intensities are given in units of , where the pixel size is . The images show inner . Left top panel shows model D with at . Left middle and bottom panels show high spin models E and F, respectively, for and . The right panels show models A, B, and D for and in the upper, middle, and lower panel respectively. The white circle marks FWHM=37 of a symmetric Gaussian brightness profile centered at the image centroid.
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