DIRECT CALCULATION OF THE RADIATIVE EFFICIENCY OF AN ACCRETION DISK AROUND A BLACK HOLE

We begin the description of HARM3D with an explicit statement of the equations governing our model. Contrasts with Gammie et al. (2003; HARM) and Villiers & Hawley (2003; GRMHD) will be highlighted along the way. We use Greek letters for spacetime indices, and Roman letters for spacelike indices. The signature of the metric is the same as the one used in Misner et al. (1970) (i.e., −+ + +), and geometrized units are used such that G = c = M = 1.

The MHD GRMHD equations of motion include the continuity equation,

$$\begin{equation} \nabla _\mu (\rho u^\mu) = 0, \end{equation} \tag{ 1 }$$

the equations of local energy conservation

$$\begin{equation} \nabla _\mu {T^{\mu }}_\nu = 0, \end{equation} \tag{ 2 }$$

and Maxwell's equations

$$\begin{equation} \nabla _\nu {^{^*}\!\!F}^{\mu \nu } = 0, \end{equation} \tag{ 3 }$$

$$\begin{equation} \nabla _\nu F^{\mu \nu } = J^\mu. \end{equation} \tag{ 4 }$$

Here, ρ is the rest-mass density, u^μ is the 4-velocity of the fluid, F^μν is the Faraday tensor times $1/\sqrt{4\pi }$, ${^{^*}\!\!F}^{\mu \nu }$ is the dual of this tensor or the Maxwell tensor times $1/\sqrt{4\pi }$, and J^μ is the 4-current.⁴ The total stress–energy tensor is the sum of the fluid part,

$$\begin{equation} T^{\mu \nu }_\mathrm{fluid} = \rho h u^\mu u^\nu + P g^{\mu \nu }, \end{equation} \tag{ 5 }$$

and the electromagnetic part

$$\begin{eqnarray} T^{\mu \nu }_\mathrm{EM} &=& F^{\mu \lambda } {F^\nu }_\lambda - \case{1}{4} g^{\mu \nu }\!F^{\lambda \kappa } F_{\lambda \kappa } = {{\|b\|^2}}u^\mu u^\nu\nonumber\\ && + \case{1}{2} {{\|b\|^2}}g^{\mu \nu } - b^\mu b^\nu, \end{eqnarray} \tag{ 6 }$$

where g_μν is the metric, h = (1 +   +  P/ρ) is the specific enthalpy, P is the pressure, is the specific internal energy density, $b^\mu = {^{^*}\!\!F}^{\nu \mu } u_\nu$ is the magnetic field 4-vector, and ||b||² ≡ b^μb_μ is twice the magnetic pressure P_m.⁵

Equations (1)–(3) can be expressed in flux conservative form

$$\begin{equation} {\partial }_t {\bf U}({{\mathbf {P}}}) = -{\partial }_i {\bf F}^i({{\mathbf {P}}}) + \mathbf {S}({{\mathbf {P}}}) \, \end{equation} \tag{ 7 }$$

where U is a vector of "conserved" variables, Fⁱ are the fluxes, and S is a vector of source terms. Explicitly, these are

$$\begin{equation} {\bf U}({{\mathbf {P}}}) = \sqrt{-g}\big[ \rho u^t, {T^t}_t + \rho u^t, {T^t}_j, B^k \big]^T \end{equation} \tag{ 8 }$$

$$\begin{equation} {\bf F}^i({{\mathbf {P}}}) = \sqrt{-g} [ \rho u^i, {T^i}_t + \rho u^i, {T^i}_j, (b^i u^k - b^k u^i) ]^T \end{equation} \tag{ 9 }$$

$$\begin{equation} \mathbf {S}({{\mathbf {P}}}) = \sqrt{-g} \big[ 0, {T^\kappa }_\lambda {\Gamma ^\lambda }_{t \kappa }, {T^\kappa }_\lambda {\Gamma ^\lambda }_{j \kappa }, 0 \big]^T \, \end{equation} \tag{ 10 }$$

where g is the determinant of the metric, Γ^λ_μκ is the metric's affine connection, and $B^i = {^{^*}\!\!F}^{it}$ is our magnetic field.⁶ The quantity (ρu^t + T^t_t) is evolved instead of T^t_t as a means of reducing the magnitude of the internal energy's numerical error (Gammie et al. 2003). Note that the source term for the energy equation is nonzero only when the metric is time-dependent (as evidenced by its proportionality to Γ^λ_tκ). The equations of motion are closed by an equation of state, P = (Γ − 1)ρ, where Γ is the adiabatic index, set to 5/3 in this work. The primitive variables, ${{\mathbf {P}}}=\lbrace \rho, P, \tilde{u}^i\rbrace$, are recovered using an optimized version of the "2D" algorithm described in Noble et al. (2006; note that the magnetic field variables are treated as conserved variables, so no recovery is required for them). The primitive velocity is the flow's velocity as viewed by a zero angular momentum observer (ZAMO):

$$\begin{equation} \tilde{u}^i = u^i + \alpha W g^{ti}, \end{equation} \tag{ 11 }$$

where $\alpha = 1/\sqrt{-g^{tt}}$ is the lapse function and W = αu^t is the Lorentz factor.

In the initial state of the simulation, the matter is in an axisymmetric hydrostatic torus that orbits the black hole with a specific angular momentum profile slightly shallower than Keplerian and u^r = u^θ = 0. The disk is centered about the equator of the black hole's spin and is initially assumed to be isentropic. In curved spacetimes, the angular frequency, Ω = u^ϕ/u^t, is not a simple function of the specific angular momentum, l = −u_ϕ/u_t. For example, one can show that when u^r = u^θ = 0 in Boyer-Lindquist coordinates, Ω = (g^tϕ − g^ϕϕl)/(g^tt − g^tϕl). In order to solve the time-independent Euler equations, we must therefore specify l(r, θ). Following Chakrabarti (1985) and De Villiers et al. (2003), we do this by assuming that Ω ∼ λ^−q, where λ² = l/Ω. The solution is simplified by setting λ to its Schwarzschild value $\lambda = \sqrt{-g^{tt}/g^{\phi \phi }}$, which is exact when a = 0 but leads to a solution marginally out of equilibrium when a ≠ 0; the slight departure from equilibrium insignificantly affects the disk's evolution because the magnetic field quickly becomes dynamically important. Ultimately, we arrive at an equation for l(r, θ): l/l_in = (λ/λ_in)^2−q, where l_in = l(r_in, π/2) and λ_in = λ(r_in, π/2).

With the intention of closely mimicking the initial conditions of simulation KDP of De Villiers et al. (2003), we put the torus pressure maximum at r = 25M and choose an angular momentum distribution parameter q = 1.67. The torus inner boundary is r_in = 15M, with l_in = 4.576. These parameters yield a disk very similar to that of De Villiers et al., but with a slightly larger l_in.

The solution to Euler's equations provides us with h and u^μ. The rest-mass density is then calculated from the equations of state—P = (Γ − 1)ρ and P = Kρ^Γ—and h: ρ = [(h − 1)(Γ − 1)/(KΓ)]^(1/(Γ−1)). We suppose that the gas is nonrelativistic, choosing Γ = 5/3 and K = 0.01. Integrating over the volume of the initial gas distribution, we find a total rest mass of 353. This is 20% larger than that in simulation KDP, a shift due to our slightly different choice of l_in. Note that the code units of gas mass are completely arbitrary.

The initial magnetic field lies entirely within the torus and follows contours of constant density. The magnitude of the magnetic field is set so that the volume-weighted integrated magnetic pressure is 100 times less than the volume-weighted integrated gas pressure.

The atmosphere surrounding the disk is unmagnetized and static. The atmosphere's density and pressure are set to their smallest allowed values, which are chosen so that the floor state is in approximate pressure equilibrium: ρ_floor = 7 × 10⁻⁹ρ_maxr^−3/2 and P_floor = 7 × 10⁻¹¹ρ_maxr^−5/2(Γ − 1), where ρ_max is the initial maximum value of the rest-mass density in the disk.

A magnetized accretion disk is subject to the magnetorotational instability (MRI), which transfers angular momentum outward. This transfer taps into the available free energy of differential rotation, creating the magnetic fields and poloidal velocity fluctuations that make up the resulting MHD turbulence. This turbulence is dissipative; magnetic and kinetic energy is lost numerically at the gridscale. Equation (7), however, ensures that in the numerical solution all that dissipated energy is converted to heat. If that heat were retained by the fluid, the disk would become ever hotter and geometrically thicker. Ultimately the thermal energy would either be accreted by the hole or be carried out from the disk by a wind. By adding a loss term to the energy equation, we can estimate either the luminosity of those systems in which radiation is efficient or the total heat generated in those systems in which it is not. We assume that the radiation described by this loss term is optically thin. It therefore acts as a passive sink in the local energy conservation Equation (2):

$$\begin{equation} \nabla _\mu {T^{\mu }}_\nu = -\mathcal {F}_\nu, \end{equation} \tag{ 12 }$$

where $\mathcal {F}_\nu$ is the amount of radiated energy-momentum per unit 4-volume in the coordinate frame. To describe the radiation, we make the simplest assumption: that the emission is isotropic in the fluid's frame:

$$\begin{equation} \mathcal {F}_\nu = {{\mathcal {L}}}u_\nu \end{equation} \tag{ 13 }$$

where the "cooling function" ${{\mathcal {L}}}$ is the rate energy is radiated per unit proper time in the fluid frame.

The NT assumptions include complete prompt radiation of all locally dissipated heat. We cannot exactly replicate that in a simulation, for the gas must retain some thermal energy. However, we can arrange for the great majority of the heat to be radiated by constructing a cooling function that keeps the temperature of the gas at a small fraction of the virial temperature. In so doing, we can also control the disk's aspect ratio H/r, a parameter often considered significant in analytic disk models.

In different contexts, different definitions of the scale height H are sometimes used. For a thin isothermal disk in a Newtonian potential, the density profile is Gaussian, ρ ∝ exp [ − z²/(2H²_G)], with H_G = c_i/Ω, for isothermal sound speed c_i. Another common measure of the scale height is the half-width at half-maximum (HWHM), $H_{HWHM} = \sqrt{2\ln 2}H_G$. A third is the vertical density moment,

$$\begin{equation} H \equiv \int d\theta d\phi \sqrt{-g} \, \rho \sqrt{g_{\theta \theta }} |\theta - \pi /2| / \int d\theta d\phi \, \sqrt{-g} \rho. \end{equation} \tag{ 14 }$$

When the profile is Gaussian, $H = \sqrt{2/\pi }H_G = 0.798 H_G$.

We prefer the moment definition because it is a direct measure of the characteristic mass-weighted disk thickness, it is robust with respect to fluctuations, and it is closely related to the characteristic scale lengths of hydrostatic balance. Ideally, it would be computed in the fluid-frame, but in the interest of computational economy we define it in the coordinate frame. Moreover, when any of these definitions of disk thickness is taken in ratio to the radius, it should be recalled that the radial coordinate r is not a proper distance. Unfortunately, there is no obvious adequate substitute.

In any event, given this definition, the temperature that should produce a desired aspect ratio H/r in Newtonian gravity is

$$\begin{equation} T_* = \frac{\pi }{2}\left[ \frac{H}{r} r \Omega (r) \right]^2. \end{equation} \tag{ 15 }$$

In our simulation, we evaluate T_* in the disk body using the relativistic orbital frequency Ω(r > r_isco) = 1/(r^3/2 + a/M). In a more completely relativistic treatment, Ω(r) would be replaced by Ω_KR^1/2_z(r), where Ω_K is the Newtonian Keplerian rotation frequency and R_z is the relativistic correction factor for the vertical gravity (Abramowicz et al. 1997; notation as in Krolik 1999a).⁷ Inside the ISCO, we define Ω as the orbital frequency of a particle with the specific energy and angular momentum of the circular orbit at the ISCO:

$$\begin{equation} \Omega (r < r_\mathrm{isco}) = \frac{ g^{\phi \mu }(r,\theta =\pi /2) \ K_\mu }{ g^{t \mu }(r,\theta =\pi /2) K_\mu }. \end{equation} \tag{ 16 }$$

Here K_μ is the 4-velocity of the ISCO orbit.

To ensure that the disk stays near the target temperature, the cooling rate should be rapid (i.e., ∼Ω), but drop to zero when the temperature falls below T_*. All of these criteria are satisfied by a cooling function with the form

$$\begin{equation} {{\mathcal {L}}}= s \Omega \rho \epsilon [ Y - 1 + | Y - 1 | ]^q, \end{equation} \tag{ 17 }$$

where Y = (Γ − 1)/T_* and s is a constant of proportionality. Note that the term in the square brackets serves as a switch, so that ${{\mathcal {L}}}=0$ whenever Y < 1. The exponent q controls how rapidly the cooling rate grows when the temperature exceeds the target. We found that q = 1/2 cools the plasma efficiently while maintaining a stable evolution, and we set s = 1. Only those fluid elements on bound orbits, where (1 + + P/ρ)u_t > − 1, are cooled.

In addition to controlling the vertical thickness of the disk, the cooling function provides a self-consistent way of comparing emission from the simulated disk with that expected in a standard NT model. When making this comparison, we use the angle-averaged fluid-frame luminosity per unit area (of an annulus located at the equator) from our 3D simulation data:

$$\begin{equation} F_\mathrm{ff}(r) = \frac{ \int \!\!\int dx^{(\phi)} dx^{(\theta)} {{\mathcal {L}}}}{ \int dx^{(\phi)} |_{\theta =\pi /2}}, \end{equation} \tag{ 18 }$$

where each component of the vector dx^(μ) = e^(μ)_ν dx^ν represents the extent of a cell's dimension as measured in the fluid element's rest frame, and e^(μ)_ν is the orthonormal tetrad that transforms vectors in the Boyer-Lindquist coordinate frame to the local fluid frame (see Beckwith et al. 2008a for explicit expressions for the tetrad). The vector dx^ν is the Boyer-Lindquist coordinate frame version of the Kerr-Schild vector dx^ν_KS = [0, Δr, Δθ, Δϕ](r, θ, ϕ), where Δr, Δθ, Δϕ are the radial, poloidal, and azimuthal extents of our simulation's finite volume cell located at (r, θ, ϕ).

We also wish to calculate the radiated luminosity measured by a distant observer in order to include the effect of photon losses into the black hole. This is done by ray-tracing through the spacetime and integrating the radiative transfer equation along geodesics. Redshift factors from differences in inertial reference frames are automatically taken into account and include such effects as gravitational redshift and relativistic beaming. As with the cooling function, we assume that the fluid is optically thin and—consequently—ignore scattering and absorption.

To briefly summarize our method, we trace a large number of rays from observers at infinity at 8 polar angles and integrate the transfer equation along each ray. Since we aim only at estimating the bolometric luminosity of the disk, and not at computing its spectrum, we assume that all radiated energy is emitted at a single frequency equal to the Doppler shifted frequency of observation. From the transfer solution along each ray, we construct images of the disk as it would be seen by each of those observers, and then sum the radiation they receive. In order to compute the radiation reaching infinity for the NT model (whose photons are also subject to possible capture by the black hole and Doppler shifting), we place an emissivity designed to match the NT surface brightness in the two planes of cells nearest the equatorial plane. Assuming that the four-velocities of those cells are exactly those of circular orbits at those radii, we then compute the luminosity at infinity in this model by the same ray-tracing technique as employed on our simulation data. Additional details are given in the Appendix.

Equation (7) is solved using finite volume techniques on a uniform grid in the so-called "Modified Kerr-Schild" (MKS) coordinate system described in Gammie et al. (2003). It is based on the Kerr-Schild (KS) coordinate system that eliminates the coordinate singularity at the horizon. The modification allows us to adjust the radial and angular discretization through a continuous coordinate transformation. We set the MKS parameter h_MKS = 0.3 (Gammie et al. 2003), which makes the poloidal cell scale at the axis about 5.7 times larger than that at the equator and allows us to resolve greater detail in the accretion disk than would be possible with the same number of equally spaced grid cells.

The simulation reported here used 192 × 192 × 64 cells in the radial, poloidal, and azimuthal directions, respectively, with r ∈ [1.28M, 120M], θ ∈ [0.05π, 0.95π], ϕ ∈ [0, π/2]. The radial extent is as large as the one used in KDP except our coordinates penetrate the horizon by five cells. We tested whether our polar angle discretization adequately resolved the fastest-growing mode of the MRI by calculating—in the local fluid frame—the fastest-growing mode's wavelength and the local poloidal size of a cell. Averaged over azimuth and time (over t = [7000M, 15000M]), the fastest-growing mode was resolved by at least seven cells at all radii. The absolute minimum number of cells per wavelength for all time and radii is never smaller than four. For the time discretization, we have found a Courant factor of 0.8 is adequate when used with the existing step size control method in HARM.

In the original MKS coordinate system, cells are placed all the way to the axis. We have introduced a new reflecting boundary condition that allows us to excise the coordinate singularity there. With the boundary placed at an angle of 0.05π from the axis (as in KDP), the excision enlarges the time step we can take, speeding up the evolution by about a factor of four relative to simulations using grids without the cut-out. The radial boundary conditions are the same as in the released version of HARM, and we use periodic boundary conditions for the azimuthal boundaries.

The equations of motion are integrated using almost the same high-resolution shock-capturing methods as described in Gammie et al. (2003). We still use HARM's Lax-Friedrichs numerical flux formula, as it is more diffusive than the HLL formula (Harten et al. 1983) and seems to be stabler for our purposes. However, the piecewise linear reconstruction is replaced with a parabolic interpolation method (Colella & Woodward 1984) as our means of reconstructing values at cell faces. As in HARM, we use an MC (monotonized central-differenced) slope limiter. Parabolic reconstruction improves stability in low-density regions where ||b||²/ρ ≫ 1, such as are found in the funnel (McKinney 2006).

We also use parabolic interpolation in the "Flux-CT" scheme of Tóth (2000) that preserves the divergence constraint. Originally, the electromotive forces (EMFs) at the cell faces were calculated by a second-order accurate two-point averaging procedure. This method failed to dissipate a cell-scale sawtooth instability seen in the magnetic field along the intersection between the inner radial and poloidal boundaries. Parabolic interpolation of the EMFs, however, is successful at quelling the instability.

Even with the improvements described so far, stably evolving plasma whose total energy is dominated by magnetic and kinetic energies is difficult. In a conservative code like HARM3D, the critical step is deriving good primitive variables, P, from the conserved quantities. For instance, the magnetic pressure is typically a few orders of magnitude larger than the gas pressure in the funnel. The gas pressure is recovered from inverting the equation for the total energy, T^t_t, which involves subtracting T^t_EMt and the fluid's inertia term from T^t_t. In the funnel, this operation is essentially a subtraction of two large numbers whose result will likely be the size of either term's truncation error. The subtraction can result in either positive or negative pressures. This is known as the "positive-pressure problem" in hydrodynamics and MHD and has been studied extensively (Ryu et al. 1993; Balsara & Spicer 1999). We have found that even when positive pressures are recovered, numerical errors may result in pressure fluctuations that differ by orders of magnitude between adjacent cells. These fluctuations create pressure gradients that can accelerate matter and add energy to the system artificially.

In order to treat the positive pressure problem and correct for other unphysical states that may arise (e.g., W < 0), we have completely redesigned HARM's recovery procedure. The most significant change is the inclusion of the conservation of entropy equation

$$\begin{equation} \nabla _\mu (\mathcal {S} u^\mu) = 0 \end{equation} \tag{ 19 }$$

where

$$\begin{equation} \mathcal {S} \equiv \frac{P}{\rho ^{\Gamma -1}}. \end{equation} \tag{ 20 }$$

Following a method similar to that of Balsara & Spicer (1999), we integrate Equation (19) in parallel with Equation (7). Whenever the standard primitive variable method fails to converge, $\tilde{u}^i$ is unphysical, or ρ < 10⁻²||b||², we use a new inversion method which is identical to the standard one except the total energy equation is replaced by Equation (20). Even though this inversion method is guaranteed to yield a positive pressure, it can either fail to converge to a solution or yield an unphysical $\tilde{u}^i$. If either happens, we interpolate P using data from neighboring cells for which we have successfully calculated P. Finally, we impose a floor on the pressure and density and ensure W ⩽ 50 by renormalizing $\tilde{u}^i$.

We note that using Equation (19) leads to a method that no longer conserves total energy to round-off error, but the impact of these departures from strict conservation is limited. The entropy equation is substituted for the energy equation only where the fluid is very strongly magnetically dominated, and only when no energy-conserving method yields a physical solution. In the simulation reported here, the net injection or loss of mass, energy and angular momentum is only ∼0.001–0.007 times the flux of these quantities through the numerical domain. After the period of initial transients, the places where nonconservative effects can be found are almost exclusively restricted to the edge of the axial cut-out and the region roughly 45° from the polar axis within the ergosphere.

We have verified that our new code is second-order accurate for smooth solutions and satisfactorily passes the tests described in Gammie et al. (2003). A quantitative comparison of our code's performance to that of GRMHD will be left for future work.

We set the parameters of our cooling function so that the ratio of the sound speed to the local orbital speed would produce a disk with a constant aspect ratio H/r = 0.13. In Figure 1, we show how well the temperature was held to T_* by comparing the time-averaged volume-weighted temperature in the bound accretion flow to the local value of T_*. In the main disk body, this mean value was about (0.93–0.95)T_*, but it rises sharply inside the ISCO. In other words, our cooling function succeeded in holding the disk temperature very close to (in fact, slightly below) the target temperature, but inside the ISCO, where the inflow time becomes comparable to or shorter than the cooling time, the temperature rises well above T_*.

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How well our temperature regulation led to a disk aspect ratio matching the goal value of 0.13 can be seen in Figure 2. The actual H/r was slightly above the goal (≃0.14) through most of the simulation volume, but with a tendency to diminish inward inside r = 20M. At r = 10M, H/r ≃ 0.12; by the time the flow reaches the ISCO, it is only ≃0.07. Comparison with the curve showing how the scale height changes as a result of including the relativistic correction to the vertical gravity (as discussed in Section 2.3) demonstrates that this thinning at small radius can be largely attributed to neglect of that effect. Thus, use of our cooling function achieved its principal goal: to place the scale height of the disk under explicit control.

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Because our cooling function has a target temperature depending only on radius, at any particular radius the gas in the main body of the disk is nearly isothermal, and the density profile is therefore close to Gaussian (Figure 3). At higher altitudes above the midplane, the density falls slower than the Gaussian, presumably due to magnetic support. For this reason, the moment scale height is slightly greater than the HWHM (Figure 2).

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We chose a value of H/r small enough that a key approximation of the NT theory could be approximately replicated in the simulation: the prompt radiation of dissipated heat. However, if the disk is to have a finite thickness, it cannot radiate all its heat. The parameters we chose for the cooling function yielded an accretion rate-weighted mean specific enthalpy that was well described by h ≃ 1 + 0.031(r/M)^−0.8. At large radius, where the Newtonian approximation applies, the ratio of h − 1 to the net binding energy is ≃0.06(r/M)^0.2, while at the ISCO this ratio is ≃0.1. Thus, this toy-model does assure that the majority of the dissipated heat is radiated.

If the accretion flow were in a strict steady state, the local (i.e., shell-integrated) mass accretion rate $\dot{M}(r)$ would be the same at all radii at all times and the mass interior to a given radius would likewise be constant. In these turbulent disks fed by a finite mass reservoir, the most we can hope for is that the time-average local accretion rate is nearly constant as a function of r through most of the accreting region, and the mass of the inner disk, after an initial period of growth, eventually levels off and fluctuates within some range. The degree to which we approach these goals is shown in Figures 4 and 5. In the left-hand panel of Figure 4, we see that the accretion rate (measured at the event horizon) varies by roughly a factor of five in an extremely irregular way. Nonetheless, as shown in the right-hand panel, the time-averaged $\dot{M}(r)$ is very nearly constant from the horizon to r ≃ 14M for the latter 8000M of the simulation. The reason why we choose the interval 7000M–15000M for averaging is shown in Figure 5. As this figure demonstrates, it takes roughly the first 7000M of the simulation for the mass of the inner disk to reach a rough plateau. Because the mass interior to a given radius fluctuates, we chose the starting point for time-averaged quantities to be the point at which essentially all the inner disk had reached at least 90% of its final mass, which is approximately t = 7000M.

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However, for the purposes of estimating the radiative efficiency, we require a tighter definition of inflow equilibrium. This is because we wish to contrast the computed radiation rate with the NT rate at an accuracy of a few percent or better. In the NT model, 23% of the total light is emitted between 12M and 25M, where our simulation shows significant departures from inflow equilibrium; a further 27% comes from outside 25M, where our simulation is not an accretion flow and we do not compute the luminosity at all. For these reasons, when we contrast the NT luminosity with that produced in the simulation, we adjust the local accretion rates to mimic inflow equilibrium and attach a carefully chosen representation of large-radius emission where needed (see below for details).

The fluid-frame emissivity F_ff(r) found in the simulation and the NT prediction for this quantity are displayed in Figure 6. In the leftmost panel, we show how they compare when the NT emissivity uses the time-averaged accretion rate at the horizon over the same interval for which the simulation data were averaged, i.e., t = 7000M–15000M. For precise comparison of the two radiation models, we remove the effects of deviations from inflow equilibrium by altering the NT emissivity so that the value at any given radius corresponds to the time-averaged accretion rate at that radius, as determined by the simulation. Put another way, the fluid-frame surface brightness in a truly time-steady NT model may be written as $(3/4\pi)(GM\dot{M}/r^3)R_R(r)$ (notation as in Krolik 1999a); we adjust this to $(3/4\pi)(GM\dot{M}(r)/r^3)R_R(r)$, with $\dot{M}(r)$ the time-averaged accretion rate at radius r in the simulation. By doing so, we compare the two radiation models in a way that factors out any contrasts due solely to fluctuations in the accretion rate. The adjusted version is shown in the right-hand panel of Figure 6.

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As this pair of figures shows, the two models coincide closely in the main disk body, but contrast sharply near and within the ISCO, which is at r ≈ 2.3M for this spin. Because the NT model is founded on energy and angular momentum conservation in a time-steady disk, this coincidence is no surprise where the influence of the NT no-stress boundary condition is small. The principal departure between the two, a systematic offset in which the simulation curve lies ≃10% below the NT curve, is due to the small fraction of the dissipated heat that the gas must retain to provide vertical pressure support in the disk. Near the ISCO, the accretion-weighted mean specific enthalpy is ≃0.018 greater than unity. This thermal energy is 12% of the binding energy at the ISCO (0.155 per unit rest-mass).

At small radius, however, the fluid-frame surface brightness of the simulation differs substantially from the NT model. At r = 3M, the simulation surface brightness is greater by 40%; at the ISCO, although the NT model would predict no radiation, the surface brightness indicated by the simulation is roughly the same as at r = 3M; close to the horizon, the surface brightness rises to about twice the maximum predicted by the classical model.

Another way to characterize the contrast between the simulation results and the NT model is through the intermediary of another analytic model. In the model of Agol & Krolik (2000), it is supposed that a finite stress is exerted at the ISCO, but all other assumptions follow those of NT. This model (which we will abbreviate as AK) is parameterized by the additional efficiency Δ due to the non-zero stress at the ISCO; in the curves shown in the two panels of Figure 6, Δ = 0.01, a value chosen as an approximate best fit between the AK model and the simulation data. In the region immediately outside the ISCO, where the AK model is defined, it does a reasonable job of reproducing the simulation results, particularly when allowance is made for the retained heat.

Only some of this radiation reaches infinity, and any that does arrives with a significant Doppler shift, most often toward the red. Using the techniques described in Section 2.3 and the Appendix, we computed the luminosity received at infinity per unit radial coordinate dL/dr, which is shown in Figure 7. Like the emissivity in the fluid frame, dL/dr for the simulation data in the main disk body closely tracks the NT prediction. The only difference between the two is that the simulation data version lies slightly (≃10%) below the NT curve: this offset is simply another reflection of the offset already seen in the fluid-frame emissivity due to the nonzero heat content of a physical disk. At small radii, the shelf in the fluid-frame emissivity is transformed into an inward extension of significant luminosity that extends from r ≃ 4M to r ≃ 2M. Although the fluid-frame emissivity extends farther inward, its efficiency in creating luminosity at infinity is cut off by a combination of increasing redshift and probability of photon capture by the black hole.

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At larger radii, departures from inflow equilibrium become significant. To compute accurately ∫dr dL/dr, the data of our simulation must be both adjusted so as to correspond to true inflow equilibrium and supplemented by an extension to larger radius to account for the substantial radiation from radii larger than 25M. Because the time-averaged fluid-frame emissivity in the simulation tracks the NT model so closely for 5M ⩽ r ⩽ 12M, we define the simulation luminosity as its dL/dr integrated from the horizon to r = 12M plus the NT luminosity at the mean accretion rate from r = 12M outward.

Given that definition, we find that the efficiency with which this simulation generated light reaching infinity, averaged from 7000M to 15000M, was 0.151. This number is 6% greater than the NT figure, which is 0.143 after allowing for photon capture.

As discussed in the Introduction, our principal goal in this initial simulation was to achieve as close a test as possible of the effect of the ISCO stress boundary condition on the radiative efficiency. We must now evaluate the degree to which our only approximate replication of the other NT assumptions affected this test. Time- and azimuthal-averaging should provide a good approximation to a stationary state and axisymmetry; incomplete radiation of the dissipated heat is our principal concern here.

We have already seen that our radiation rate closely tracks the NT radiation rate in the main disk body, but is about 10% lower. Thus, to extrapolate to complete radiation, the emission from this portion of the flow should be increased by this amount. Near the ISCO, where the effects of the stress boundary condition become important, we cannot use this comparison method to estimate the magnitude of the retained heat. Instead, we observe first that at the ISCO the mean Thomson optical depth through the disk in our simulation was $\simeq 500\dot{m}$, where $\dot{m}$ is the accretion rate in Eddington units. The corresponding diffusion time is $\simeq 0.7\dot{m}$ orbits. At the same place, the inflow rate is ≃0.6Ω = 1.2π/P_orb. Thus, the photon diffusion time near the ISCO in a real disk should be shorter than the inflow time—and shorter than our toy-model cooling time Ω⁻¹—for all accretion rates below Eddington. A second standard of comparison may be derived from the magnitude of the retained heat. We found earlier that the accretion-weighted mean specific enthalpy is ≃1 + 0.018 at r ≃ 2M. That the retained heat is ≃10% of the binding energy there is consistent with the fact that ≃10% of the heat dissipated in the main disk body is left unradiated. Combining these two arguments, we might expect that in the limit of truly complete radiation of dissipated heat, the efficiency could have been greater by as much as 0.02, rising perhaps to ≃0.17, 20% above the classical number as adjusted for photon capture.

Additional heat is created in the plunging region (the mean accreted specific enthalpy rises from 1.02 at the ISCO to ≃1.03 at the horizon), but, as we have already seen, the fraction of photons escaping from regions so close to the horizon to infinity is relatively small, so only a small part of the additional 0.01 in rest-mass equivalent is likely to reach distant observers.

We might also ask what effect truly radiating all the heat would have on electromagnetic energy fluxes. To approach this question, we begin by considering it from the point of view of the classical (NT) theory of accretion, where much attention is paid to the r–ϕ component of the stress tensor T^μ_ν, but little is said about other components except for the assumption that the stress tensor is orthogonal to the four-velocity, u_μT^μ_ν = 0. As Beckwith et al. (2008a) pointed out, this assumption is consistent with the sort of stress NT had in mind, i.e., ordinary viscosity, but not necessarily with other physical stress mechanisms. In particular, it is inconsistent with MRI-driven MHD turbulence: the electromagnetic stress tensor contains a term ||b||²u^μu_ν, which is manifestly not orthogonal to the four-velocity; in addition, the turbulence entails another (generally rather smaller) contribution to the stress tensor (ρh + ||b||²)δu^μδu_ν, where δu^μ is the fluctuating part of the four-velocity. Described in more qualitative terms, the classical theory accounts for the energy flow due to the work done by the stress, but not the energy flow due to the advection, by the mean flow, of an energy density associated with the stress mechanism.

As numerous numerical studies of the MRI-driven turbulence have shown, the fluid-frame ratio α_mag ≡ 2〈b^rb_ϕ〉/〈||b||²〉 ≃ 0.2–0.3 in the disk body, rising by factors of a few in the plunging region (e.g., Hawley & Krolik 2002). At the order of magnitude level, the ratio of the advected magnetic energy flux to the magnetic work is ∼u^r/(α_magru^ϕ), which is very small in the disk body, but rises sharply near the ISCO and in the plunging region. In this simulation, we find that the time-averaged advected magnetic energy flux per unit rest-mass is 0.03 at the ISCO, a significant contribution to the energy budget.

To complete our extrapolation to complete radiation therefore means that we need to determine how u^r near the ISCO might change when that limit is taken at fixed accretion rate. Fixing the accretion rate means that the vertically integrated stress does not change, and we do not need to estimate how the stress would change as a function of the disk's thermal state. If u^r near the ISCO depends primarily on the shape of the potential, the advected magnetic energy flux per accreted rest-mass would remain roughly the same. On the other hand, if u^r in this region depends on the gas thermal content in the sense that it increases with increasing temperature, more complete radiation would also lead to a smaller rate of magnetic energy advection, and therefore to a larger net outward Poynting flux and a larger amount of energy available for dissipation. Which of these possibilities lies closer to the truth (and under which circumstances) remains to be determined.