3D GRMHD SIMULATIONS OF BLACK HOLE ACCRETION DISKSRamesh NarayanBH Accretion Astrophysical black holes nearly always have observable accretion disks around them These accretion disks provide information on accretion physics, e.g., different spectral states, enabling us to check our models Conversely, observations of disk emission allow us to study the BH: M, a*, event horizon Our group has estimated spin parameters of a number of stellar mass BHs in X-ray binaries by fitting the disk spectrumOur TeamJeff McClintock Ramesh NarayanShane Davis, Lijun Gou, Li-Xin Li, Jifeng Liu, Jon McKinney, Jerry Orosz, Bob Penna, Mark Reid, Ron Remillard, Rebecca Shafee, Jack Steiner, Sasha TchekhovskoyBH Masses and SpinsShafee et al. (2006); McClintock et al. (2006); Davis et al. (2006); Liu et al. (2007); Gou et al. (2009) ; Steiner et al.Source Name BH Mass (M) BH Spin (a*)LMC X-3 5.9—9.2 ~0.25XTE J1550-564 8.4—10.8 (~0.5)GRO J1655-40 6.0—6.6 0.65—0.75M33 X-714.2—17.1 0.77 ± 0.054U1543-47 7.4—11.4 0.75—0.85LMC X-19.0—11.60.85—0.97GRS 1915+105 10—18 0.98—1Theoretical Model Any method of measuring a*is only as good as the theoretical model behind it Our method assumes that the accretion disk is well described by the GR disk model of Novikov & Thorne (1973) In particular, we assume that the disk luminosity profile L(r) takes the form predicted by the NT modelNovikov & Thorne L(r)L(r) peaks at a different radius for each value of the dimensionless BH spin parameter a*Therefore, the observed spectrum depends on a*This is what enables us to estimate a*from observationsDifferent representations of the luminosity profileNovikov-Thorne ModelBut How Good is the Novikov-Thorne Model? The NT model assumes a geometrically thin disk It assumes that the “viscous” torque vanishes at the ISCO (Shakura & Sunyaev 1973; Novikov & Thorne 1973) But magnetic fields could produce significant torque at and inside the ISCO (Krolik 1999; Gammie 1999) Afshordi & Paczynski (2003) suggested that the effect is probably not important for a THIN disk (Shafee et al. 08) Can we verify this?Testing the Novikov-Thorne Model using 3D GRMHD Simulations 3D MHD simulations in the Kerr metric Magnetic fields self-consistently generate “viscous” torques via the MRI (Balbus & Hawley 1991) We must simulate geometrically thin disks – numerically very challenging Reynolds & Fabian (2008); Shafee et al. (2008); Noble, Krolik & Hawley (2009)Numerical Method We use the GRMHD code HARM (Gammie, McKinney & Toth 2003) Conservative code, runs in 3D in the stationary Kerr metric We add an ad hoc cooling where we specify the target entropy of the gas as a parameter: This parameter lets us tune the disk thickness target2KuududOur Fiducial Run A very thin disk (<|h|>/r ~ 0.05) around a non-spinning BH (a*=0) 256 x 64 x 32 grid (-wedge angle: /2) Gas is initially in a torus beyond r=20M Simulation is run for a time of 17000M Steady state after t ~ 12000MPenna et al. (2009)a*=0256x64x32256 x 64 x 32Penna et al. (2009)Mass Conservation;0M ass Flux( ) constant (steady state)integral 0 : all the fluidintegral / 2 2 / : limited to diskruu g d dMrhr  -Fiducial Run: Mass Accretion RatePenna et al. (2009)a*=0256x64x32a*=0256x64x32Angular Momentum Conservation   ;2in out2inang mmtm loss via radiationFlux( ) nearly constant( ) ( ) ( ) for comparing with NT()rrrTu b u u b b g d dJrJ r J r J rJ r u b u u g d d    -- - --         Our New Fiducial Run (a*=0): Penna et al. (2009)Jdot inJdot tota*=0256x64x32The results from the two runs appear to be similar. We view the deviations as a measure of the errorbara*=0 a*=0256x64x32512x128x32Thin Disks: Other Values of a*a*=00.70.90.98Pretty good agreement with Novikov-Thorne, except at the largest value of a*Thicker Disks with a*=0The accretion flow becomes quite sub-Keplerian as the disk thickness increasesa*=0256x64x32Angular Momentum: Summary Thin disks with h/r<0.1 behave quite a lot like the Novikov-Thorne model Deviations are larger for larger values of a*, but the dependence is modest However, deviations increase rapidly as the disk thickness increases Therefore, the NT model is not trustworthy for thick disksEnergy Conservation ;2energy loss via radiationFlux( ) : increases with radius (radiation)1 Binding energy released per unit mass1ln lntrrttTu b u u b b g d dErEMdL d Ed r d rM  -----       Fiducial Run: Energy Flux!!Very Preliminary!!a*=0256x64x32!!Preliminary Result!!a*=0256x64x32Cyan: 256 x 64 x 32 (Penna et al. 2009): ~5000MMagenta: 512 x 128 x 32 (Shafee et al. 2008): ~2000Ma*=0Thin Disks: different a*a*=00.70.90.98Thicker Disks: a*=0Distinction between the disk and the plunging region becomes washed out as the disk becomes geometrically thickera*=0Energy and Luminosity: Summary Thin disks with h/r<0.1 seem to behave like the Novikov-Thorne model Deviations are larger for larger BH spins, and may be serious as a* 1 Deviations increase rapidly as the disk thickness increases Accretion luminosity/efficiency is not very different from NT valueBottom LineSource Name BH Mass (M) BH Spin (a*)LMC X-3 5.9—9.2 ~0.25XTE J1550-564 8.4—10.8 (~0.5)GRO J1655-40 6.0—6.6 0.65—0.75M33 X-714.2—17.1 0.77 ± 0.054U1543-47 7.4—11.4 0.75—0.85LMC X-19.0—11.60.85—0.97GRS 1915+105 10—18 0.98—1Current (very preliminary!) indication: geometrically thin accretion disks behave quite a lot like the Novikov-Thorne modelSuggests that our spin estimates are probably