1. Abstract Direct high resolution imaging of the galactic center black hole will be possible in the near future, providing a test of strong field general relativity, and constraining black hole parameters and accretion models. Achieving these goals will require theoretical predictions for the appearance of the inner portion of the accretion flow. Global general relativistic MHD (GRMHD) simulations are currently the most realistic dynamical models of accreting black hole systems. To compare results from such simulations with observations, radiative transfer must be performed as a post-processor on the simulation data. We use a semi-analytic ray tracing code to compute optically thin synchrotron emission in the form of images, spectra and light curves using tilted GRMHD data from Fragile et al (2007). At sub-millimeter frequencies of interest for future very long baseline interferometry (VLBI) measurements, the synchrotron emission is dominated by the inner radii (~10 GM/c 2 ). Doppler beaming for an inclined observer in conjunction with strongly peaked synchrotron intensity in the inner radii may explain the extremely small observed size of Sgr A* at 1.3mm from Doeleman et al (2008). Methods Fragile et al (2007; 2008) presented results from a GRMHD simulation of accretion onto a spinning black hole (a=.9), where the black hole spin axis is misaligned from the initial torus symmetry axis by 15 degrees. The gravitational torque causes a solid body precession of the torus, with no noticeable Lense-Thirring alignment due to the relatively thick torus (h/r~.15). We use data from the simulation 915h, which used a grid with maximum equivalent resolution of 128 3, but under resolved the poles (see Fig. 1). The simulation was run for 10 orbital times at r=25M, in units with G=c=1 To compute images of the accretion flow, rays are traced backwards in time from a camera at infinity by converting coordinates in the image plane to photon (null geodesic) orbital parameters. Geodesic trajectories are calculated in the Kerr metric using the code described in Dexter & Agol (2009). We interpolate the simulation data to compute fluid variables along the path (see Fig 2). The code is time dependent, and we also include the light travel time delays along the ray by sampling simulation data from the appropriate time steps. Figure 2. Rays are tabulated as a series of points on null geodesics (solid lines). The intensity along each ray is computed as a path integral using interpolated fluid quantities in the region where the ray intersects the accretion flow. Diagram taken from Schnittman et al (2006). Figure 1. Grid geometry used in the GRMHD simulation, with lower resolution near the pole. This figure is in the torus frame, which is tilted with respect to the black hole symmetry axis. Taken from Fragile et al (2007). Results To produce images, spectra and light curves relevant for current and future observations of the galactic center black hole, we calculate the synchrotron emissivity along each ray from the interpolated fluid quantities. Code units are converted to cgs by scaling the mass of the torus to the estimated accretion rate onto Sgr A* (Schnittman et al 2006), 10 -8 M sun /yr. The time and length scales are determined by the black hole mass, taken to be 4x10 6 M sun. The thermal temperature is used, and the synchrotron function is computed using the analytic fits from Mahadevan et al (1996). We also calculate a frequency-integrated emissivity proportional to the fluid mass density, which demonstrates the importance of relativistic effects (Schnittman et al 2006). In all cases, emission outside of r=25M is ignored. Images from the density emissivity for the last simulation time step at two different observer inclinations, i=0 (face on) and i=60 degrees are shown in Fig. 3. All images shown here are scaled linearly to the maximum intensity of the individual panel, from blue to red to yellow to white. There are two strong lensing features. The first is from rays which pass under the black hole and intersect the back side of the accretion flow, while the second is from those which orbit the black hole multiple times. When the observer is inclined, Doppler beaming causes a strong asymmetric peak in the intensity on the side of the accretion flow where the orbital velocity has a large component towards the observer. (a) (b) (a) (d) (c) (b) Figure 3. Images of the final time step of the accretion flow from Fragile et al (2007) with frequency-integrated emissivity taken as proportional to density at inclinations of (a) 0 and (b) 60 degrees relative to the black hole spin axis. The image scale is 25M, and the intensities are scaled linearly to the maximum for each panel. In each case, the tilted image has one more strong lensing ring than in equivalent untilted images. The rings are due to geodesics which orbit the black hole multiple times. Figure 4. Optically thin synchrotron images of the final time step of the simulation from Fragile et al (2007) for Sgr A* parameters at (a) 3, (b) 30, (c) 300 and (d) 3000 GHz. The inclination is 60 degrees, and the panel size is 15M. At higher frequencies, the image appears smaller due to a sharp peak in intensity in the inner radii coupled with Doppler beaming from gas moving towards the observer. The images from optically thin synchrotron emission are shown for i=60 degrees in Fig. 4, and for i=0 in Fig. 5 for several frequencies. The panel size is now 15M. At low frequency, the flow is optically thick and the images shown here will be modified significantly. Still, qualitatively the intensity is much more sharply peaked at higher frequencies, along the tail of the synchrotron spectrum (Fig. 6). Figure 6. Synchrotron spectra for Sgr A* parameters neglecting self- absorption in cgs units. At higher inclination, Doppler beaming leads to higher observed luminosities. (a)(b) (c) (d) In addition to the emissivity, we calculate an optical depth due to synchrotron self-absorption, and find that the flow is marginally optically thin above ~10 12 Hz (Fig. 7). Adding absorption is a goal for future work. Figure 7. Maximum optical depth vs. observed frequency for the same situation as in Fig. 6. The horizontal line indicates an optical depth of unity. The optically thin approximation used here is only somewhat valid even at the highest frequencies considered. Figure 5. Optically thin synchrotron images of the final time step of the simulation from Fragile et al (2007) for Sgr A* parameters at (a) 3, (b) 30, (c) 300 and (d) 3000 GHz. The observer is looking down the black hole spin axis, and the panel size is 15M. At higher frequencies, the image appears smaller due to a sharp peak in intensity in the inner radii. Finally, Fig. 8 shows a light curve for a little over one full orbit at r=25M. At higher observed frequencies, the variability is much more significant. This is due to the small number of image pixels contributing to the overall intensity. Small fluctuations in intensity along these rays lead to large observed variability. To ensure that this is a real effect, higher geodesic resolution must be used, both in terms of number of points on each ray and more rays on the camera. In addition, synchrotron self-absorption may affect this result, although at high frequencies the flow is only borderline optically thick. Acknowledgments J.D. acknowledges a Graduate Fellowship from the Kavli Institute of Theoretical Physics at the University of California, Santa Barbara under National Science Foundation Grant No. PHY05-51164. This work was partially supported by NASA Grant NNX08AX59H. Discussion We have produced the first time-dependent synchrotron images, spectra and light curves from a 3D GRMHD code. At frequencies of interest for VLBI, the intensity is strongly peaked in the inner radii. Combined with the Doppler beaming at high inclination, this could explain the extremely small measured size of Sgr A* from recent VLBI measurements (Doeleman et al 2008). More detailed observations can be compared directly with such images, and potentially constrain black hole parameters, our inclination to the galactic center, or accretion models. In future work we plan to include synchrotron self absorption to compute a more realistic low frequency spectrum, as was done for an axisymmetric GRMHD simulation in Noble et al (2007). Then the spectrum from the simulation at many inclinations and time steps can be compared to observations. In addition, we will be able to calculate sizes for the images shown in Figs. 4-5, and compare them with the measurements from Doeleman et al (2008) and future VLBI observations. Polarization measurements are a strong independent constraint to be satisfied by models of Sgr A*. The polarization angle can be modified significantly by strong field relativity, and this will also be incorporated into our predicted images. The observed degree of linear polarization could favor certain geometries, and the Faraday rotation angle places a constraint on the densities in the inner region of the accreting volume. References Dexter, J. & E. Agol, A Fast New Code for Computing Photon Orbits in a Kerr Spacetime, Astrophysical Journal, submitted Doeleman, S.S. et al, Event-horizon-scale Structure in the Supermassive Black Hole Candidate at the Galactic Centre, Nature, 455, 78 (2008) Fragile, P.C., O.M. Blaes, P. Anninos & J.D. Salmonson, Global General Relativistic MHD Simulation of a Tilted Black-Hole Accretion Disk, Astrophysical Journal, 668, 417 (2007) Fragile, P.C. & O.M. Blaes, Epicyclic Motions and Standing Shocks in Numerically Simulated Tilted Black-Hole Accretion Disks, Astrophysical Journal, 687, 757 (2008) Mahadevan, R, R. Narayan & I. Yi, Harmony in Electrons: Cyclotron and Synchrotron Emission by Thermal Electrons in a Magnetic Field, Astrophysical Journal, 465, 327 (1996) Noble, S.C., P.K. Leung, C.F. Gammie & L.G. Book, Simulating the Emission and Outflows from Accretion Discs, Classical and Quantum Gravity, 24, 259 (2007) Schnittman, J.D., J.H. Krolik & J.F. Hawley, Light Curves from an MHD Simulation of a Black Hole Accretion Disk, Astrophysical Journal, 651, 1031 (2006) For further information Please contact jdexter@u.washington.edu. Time-dependent synchrotron images (movies) and a PDF version of this poster can be found at http://students.washington.edu/jdexter/cosmos.http://students.washington.edu/jdexter/cosmos Figure 8. Light curves for synchrotron emission at various frequencies. At high frequency, the variability is much more pronounced, likely due to the small amount of the image contributing significantly to the intensity. For reference, the orbital time at r=10M is ~200M. Jason Dexter 1,2, Chris Fragile 3, Eric Agol 4 and Omer Blaes 5 1 Department of Physics, University of Washington, Seattle, WA, 98195 2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106 3 Department of Physics & Astronomy, College of Charleston, Charleston, SC, 29424 4 Department of Astronomy, University of Washington, Seattle, WA, 98195 5 Department of Physics, University of California, Santa Barbara, CA, 93106 The appearance of Sgr A* from a tilted GRMHD simulation