ROTATING ACCRETION FLOWS: FROM INFINITY TO THE BLACK HOLE

We use ZEUS-2D v2.0 (Stone & Norman 1992) to solve the equations of hydrodynamics,

$$\begin{eqnarray} &&\left(\frac{\partial }{\partial t}+ \boldsymbol{v\cdot \nabla } \right)\rho + \rho \boldsymbol{\nabla \cdot v} =0, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} &&\rho \left(\frac{\partial }{\partial t}+ \boldsymbol{v\cdot \nabla } \right)\boldsymbol{v}=-\boldsymbol{\nabla }p - \rho \boldsymbol{\nabla }\Phi , \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} &&\rho \left(\frac{\partial }{\partial t}+ \boldsymbol{v\cdot \nabla } \right)\frac{e}{\rho }=-p\boldsymbol{\nabla \cdot v}, \end{eqnarray} \tag{ 4 }$$

where ρ is mass density, $\boldsymbol{v}$ is flow velocity, e is energy density, gas pressure p ≡ (γ − 1)e, where γ is the adiabatic index, and the gravitational potential Φ ≡ −GM_BH/r. We run global hydrodynamical simulations of axisymmetric rotating accretion flows in spherical polar coordinates. The simulations include thermal Bremsstrahlung cooling, shear viscosity to capture drag due to turbulent stresses, and modified boundary conditions to allow for disk and wind outflows. The source terms for momentum and energy in the ZEUS hydrocode are modified by the addition of the terms within square brackets below:

$$\begin{equation} \rho \left(\frac{\partial \boldsymbol{v}}{\partial t}\right)_{\rm sources} = -\boldsymbol{\nabla } p -\rho \boldsymbol{\nabla } \Phi -\nabla \cdot \boldsymbol{Q}+ [\boldsymbol{\nabla \cdot \sigma ^{\prime }}]\,, \end{equation} \tag{ 5 }$$

and

$$\begin{eqnarray} \left(\frac{\partial e}{\partial t}\right)_{\rm sources} &= &{-} p \boldsymbol{\nabla \cdot v} -\boldsymbol{Q\cdot \nabla v} \nonumber\\ && +[\boldsymbol{\sigma ^{\prime }\cdot \nabla v} - \dot{e}_{\rm Brem} ]\,, \end{eqnarray} \tag{ 6 }$$

in which $\boldsymbol{\sigma ^{\prime }}$ is the viscous shear tensor and $\boldsymbol{q}$ is the heat flux due to thermal conduction of electrons. ZEUS' standard tensor artificial viscosity $\boldsymbol{Q}$ is applied with shocks spread over ≈2 zones, and we use a Forward Time Centered Space differencing scheme for our modified source terms. We neglect the non-azimuthal components of the viscous shear tensor $\boldsymbol{\sigma ^{\prime }}$, as it is believed that the poloidal shear is subdominant to the azimuthal shear in the nonlinear regime of the MRI (see Stone et al. 1999). The azimuthal components of the viscous shear tensor are

$$\begin{equation} \boldsymbol{\sigma ^{\prime }}_{r\phi }=\nu \rho r \frac{\partial }{\partial r}\left(\frac{v_{\phi }}{r}\right), \end{equation} \tag{ 7 }$$

$$\begin{equation} \boldsymbol{\sigma ^{\prime }}_{\theta \phi }=\frac{\nu \rho \sin \theta }{r} \frac{\partial }{\partial \theta }\left(\frac{v_{\phi }}{\sin \theta }\right), \end{equation} \tag{ 8 }$$

where ν is the kinematic viscosity. The Bremsstrahlung cooling term (Svensson 1982; Ball et al. 2001)

$$\begin{equation} \dot{e}_{\rm Brem}\equiv \alpha _f r_e^2m_ec^3n^2(32/3)(2/\pi)^{1/2}\left(\frac{k_BT}{m_ec^2} \right)^{1/2}, \end{equation} \tag{ 9 }$$

where α_f is the fine structure constant and r_e is the classical electron radius. We use this abbreviated form for the Bremsstrahlung cooling to capture the qualitative effects of thermal emission; in fact in high temperature regions the electron component approaches relativistic speeds and Bremsstrahlung radiation increases substantially above Equation (9). The cooling is limited by a floor in the temperature, equal to the gas temperature at the outer boundary, in order to maintain stability. The advective transport terms in ZEUS are unmodified.

In our simulations gas flows in through the outer boundary, located at R_out = 10 R_B, at the Bondi rate, $\dot{M}_{\rm B}$. We modify the outer boundary condition locally to an outflow boundary condition in whichever grid zones at the outer boundary that the flow is moving radially outward, however. We note that several other authors have studied accretion flows with an outer boundary near the Bondi radius (e.g., Pen et al. 2003; Pang et al. 2011). Our inner boundary, located at R_in = 10⁻³ R_B, is a standard no torque outflow boundary condition. R_in is located at 91 R_s, and we have checked the convergence of our results with respect to the location of the inner boundary (see the Appendix). The flow is initialized throughout with the Bondi profile for density, energy density, and radial velocity. Our standard simulations are run with T_∞ = 2 × 10⁷ K. This is close to the temperature of gas heated radiatively by a typical quasar output spectrum (Sazonov et al. 2005). The simulations are run with adiabatic index γ = 1.5, except where otherwise noted, and we verify explicitly that this assumption does not affect our main results (see Figure 7). The real gas will have γ = 5/3 above R ∼ 300–500 R_s. At lower radii, in the non-collisional case, electrons become relativistic while protons are nonrelativistic, leading to lower effective γ. Our choice for γ is made in order to allow us to well separate the sonic and centrifugal radii for supersonic infall. The density at infinity, ρ_∞, is a parameter that determines the dimensionless quantity $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$, where $\dot{M}_{\rm Edd}\equiv 4\pi G M_{\rm BH}m_p / \epsilon \sigma _T c$ is the Eddington luminosity and we assume an efficiency = 0.1. This dimensionless mass accretion rate parameter scales as $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}\propto M_{\rm BH}\rho _\infty /c_{s,\infty }^3$. The dimensionless cooling fraction $\dot{e}_{\rm Brem} \Delta t_{\rm char}/nkT \propto M_{\rm BH}\rho _\infty /c_{s,\infty }^4$, where we have taken characteristic time $\Delta t_{\rm char}=M_{\rm BH}/c_{s,\infty }^4$. The black hole mass and gas density at infinity appear together in the above two quantities, and the free parameter in the accretion flow is actually the product M_BHρ_∞ (see Chang & Ostriker 1985). This parameter effectively determines the strength of the Bremsstrahlung cooling, with $\dot{M}_{\rm B}/\dot{M}_{\rm Edd} \ll 1$ corresponding to no cooling and $\dot{M}_{\rm B}/\dot{M}_{\rm Edd} \gg 1$ corresponding to strong cooling. For convenience we pick M_BH = 2 × 10⁸ M_☉ and allow ρ_∞ to vary freely. At ρ_∞ = 2.3 × 10⁻²¹ g cm⁻³, the Eddington ratio is $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=0.1$.

Gas is initialized with constant specific angular momentum throughout, except near the rotation axis, where it tapers to zero. The inflowing gas at the outer boundary has the same angular momentum, without tapering, ensuring that after initial transients, the gas has uniform angular momentum throughout. For adiabatic inviscid flows, the Bernoulli constant

$$\begin{equation} Be\equiv \frac{1}{2}\left(v_r^2+v_\theta ^2\right)+\frac{1}{2}\frac{j^2}{\varpi ^2}-\frac{GM}{r}+\frac{c_s^2}{\gamma -1}, \end{equation} \tag{ 10 }$$

where ϖ is cylindrical radius, is a conserved quantity (see, e.g., Narayan & Yi 1994). It immediately follows that there is a centrifugal radius interior to which the gas cannot penetrate if gas starts from rest at infinity with $Be=\gamma c_{s,\infty }^2/(\gamma -1)$. The specific angular momentum is another free parameter in our simulations, but we pick $j_\infty =\sqrt{0.02}\,R_{\rm B}c_{s,\infty }$ so that the centrifugal radius R_c = 0.02 R_B is well resolved and lies at a radius 20 times that of the inner boundary. Hence we are computing within the domain given by Equation (1). The limit of weak angular momentum with R_c ≪ R_in approaches the Bondi solution. For our choice of specific angular momentum, the introduction of shear viscosity and angular momentum transport allows for gas to migrate from the centrifugal radius down to the inner boundary and to accrete. The accreting gas carries only a small fraction (<20%) of the initial angular momentum j_∞ through the inner boundary. In our simulations we set the kinematic viscosity ν to a constant nonzero value ν = 10⁻³c_{s, ∞}R_B. The effective α coefficient, α ≡ ν/c_sH_p, where c_s is the local sound speed and H_p the local pressure scale height (see Section 3.3), is then spatially variable, but is typically of order ∼0.01 near the centrifugal barrier where the density peaks. Stone et al. (1999) found that the properties of their non-radiative hydrodynamic flow do not depend strongly on the assumed form for kinematic viscosity, and Stone & Pringle (2001) found weakly magnetized flows to be qualitatively similar to the viscous hydrodynamic ones. Incidentally, the limit of large specific angular momentum with R_c ≳ R_B should approach that of a giant viscous disk with large radial extent, but we have not explored this regime in any detail. The Bremsstrahlung cooling also modifies the flow profile, and for cold disks the inflow can be highly supersonic. In these cold solutions the sonic radius is typically intermediate between but still well separated from the centrifugal and Bondi radii.

Our standard simulations are run with 21 equally spaced grid zones in polar angle θ varying from θ = 0 to θ = π/2, enforcing symmetry across the equatorial plane. In the radial direction we use a non-uniform logarithmic grid with 32 zones per decade ranging from R_in = 10⁻³ R_B to R_out = 10 R_B, with 128 total grid zones. As a code check, we ran the spherically symmetric test problem with no cooling. Our code is able to maintain the Bondi profile for mass density and radial velocity to within 5% for various values of the adiabatic index γ. The internal energy density is less well conserved in the very inner portions of the flow (≈10 innermost cells) due to artificial viscous heating, but is still within a factor of ≈2 of the Bondi solution. The simulations are run until all time-averaged gross quantifiers of the flow structure approach quasi-steady values. All diagnostics of the flow are averaged over many tens of Bondi times t_B ≡ R_B/c_{s, ∞}, typically between times t = 40 t_B and t = 90 t_B (see the Appendix). Flow patterns are illustrated at time t = 90 t_B except where otherwise noted. The code is mass conservative, so mass flowing inwards from large radii must accumulate on the grid or outflow either through the inner boundary or back out to large radii via the combination of polar and equatorial outflow.

We also run a number of tests to check the proper implementation of the additional physics we have added to ZEUS. We verified the proper thermal cooling of a stationary constant density gas, as well as the diffusion of thermal energy when a temperature gradient exists. Finally, we have reproduced qualitatively the viscous evolution of an initial ring of matter (Pringle 1981). The ring spreads in radius as matter moves inwards, but the bulk of the angular momentum is transferred outwards. Kinetic energy lost to viscous dissipation is added to the internal energy of the gas, with the viscous substep of the code conserving energy to within 10%.

For adiabatic flows with sufficiently high angular momentum that the centrifugal barrier is located far from the black hole, Equation (10) immediately tells us that there can be no accretion onto the black hole. Any inflowing high angular momentum material must either accumulate beyond the centrifugal radius or be deflected back outwards and ejected in outflows (Proga & Begelman see 2003a, and references therein; see also Hawley et al. 1984a, 1984b; Hawley & Smarr 1986).

In the adiabatic regime with constant angular momentum j throughout, there is a stationary (v_r = v_θ = 0) settling solution satisfying the momentum equation (see Papaloizou & Pringle 1984; Fishbone & Moncrief 1976). Starting from the momentum equation,

$$\begin{equation} {-}\frac{1}{\rho }\boldsymbol{\nabla }p - \boldsymbol{\nabla }\Phi +\frac{j^2}{\varpi ^3} \hat{\boldsymbol{\varpi }}=0, \end{equation} \tag{ 11 }$$

the adiabatic relation p∝ρ^γ gives

$$\begin{equation} \boldsymbol{\nabla } \left(\frac{c_s^2}{\gamma -1} + \Phi +\frac{j^2}{2\varpi ^2} \right) =0. \end{equation} \tag{ 12 }$$

This yields an equation for the sound speed,

$$\begin{equation} \frac{c_s^2}{c_{s,\infty }^2}=1+(\gamma -1)\frac{GM}{c_{s,\infty }^2r}-\frac{\gamma -1}{2}\frac{j^2}{c_{s,\infty }^2\varpi ^2}, \end{equation} \tag{ 13 }$$

valid where the right-hand side is positive, as well as the density profile,

$$\begin{equation} \frac{\rho }{\rho _{\infty }}=\left[1+(\gamma -1)\frac{GM}{c_{s,\infty }^2r}-\frac{\gamma -1}{2}\frac{j^2}{c_{s,\infty }^2\varpi ^2} \right]^{1/(\gamma -1)}. \end{equation} \tag{ 14 }$$

The density is zero near the polar axis in the region where the right-hand side of Equation (13) is negative. This solution is actually dynamically unstable to non-axisymmetric perturbations (Papaloizou & Pringle 1984), which we do not allow for in our axisymmetric code. Neglecting these instabilities is likely a reasonable assumption, however, as we have angular momentum gradients that can stabilize the flow (Papaloizou & Pringle 1987).

Our multidimensional adiabatic simulations closely resemble this stationary settling solution, with sound speed profile matching Equation (13) to within a few percent exterior to the centrifugal barrier. Figure 1(a) shows the structure of the flow within the Bondi radius at Eddington ratio $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=10^{-4}$. The red contours are logarithmic contours of density, given in cgs units, and we overlay black contours of logarithmic density for the adiabatic settling solution. The solid black line shows the zero density surface in the stationary solution. Blue arrows indicate vectors of mass flux in the R–z plane, with length proportional to the magnitude of the mass flux vector. Subsequent diagrams of flow patterns showing logarithmic density contours and mass flux vectors are illustrated in a similar fashion. The pressure associated with the accumulation of matter in an extended torus impedes the inflow of matter, as in the stationary solution. Indeed, the mass inflow rate at the Bondi radius is reduced by more than one order of magnitude to ${\sim } 0.06\,\dot{M}_{\rm B}$ at t = 45 t_B. There is evidence of time-dependent circulation, however, partially due to our choice of initial conditions, and the flow is not purely stationary with zero inflow rate at the Bondi radius. Matter flows inwards at intermediate latitudes and turns around at radii exterior to the centrifugal barrier, flowing back outwards in the polar and equatorial regions. Figure 1(b) shows the flow structure interior to 0.1 R_B, with centrifugal radius located at R_c = 0.02 R_B. There is again evidence of time-dependent circulation, but the largest differences in the flow relative to the stationary solution occur near the polar axis where the stationary solution has zero mass density. There are very small inflows in the polar region due to angular momentum transport associated with numerical viscosity, but the accretion rate is extremely low, $\dot{M}_{\rm accretion}< 10^{-3}\,\dot{M}_{\rm B}$, i.e., any accretion is due to numerical error. In the absence of physical angular momentum transport the gas in the centrifugal barrier is stationary in the r and θ directions, cannot accrete, and its pressure prevents new gas from falling down to small radii. Our results indicate that in the limit of very long and accurate integrations the solution would approach the stationary solution given in Equations (13) and (14), since there is no engine to drive circulation for adiabatic flows.

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We find two general states for rotating axisymmetric accretion flows with radiative cooling and viscous angular momentum transport: hot thick disks with low accretion having polar and disk outflows, and cold thin disks with high accretion and weak polar and disk outflows.

Figure 2(a) shows the flow structure interior to the Bondi radius for the hot disk with $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=10^{-4}$. Red contours are logarithmic contours of density, blue vectors represent mass flux in the R–z plane, and the solid black line indicates the zero density surface in the adiabatic settling solution. Gas flows inwards over a wide range of intermediate polar angles, and there are equatorial and polar outflows. The mass ejected in the polar outflow, ${\sim } 0.02\,\dot{M}_{\rm B}$, is similar in magnitude to the adiabatic case, and is again caused by inflowing matter turning around exterior to the centrifugal torus. The detailed properties of this circulation may depend on the length and accuracy of our numerical integrations. The equatorial outflow is driven primarily by azimuthal stresses associated with the shear viscosity, and it has magnitude ${\sim } 0.25\,\dot{M}_{\rm B}$ at the Bondi radius. The mass inflow rate through the Bondi radius increases over that in the inviscid case and essentially balances this equatorial outflow. Figure 2(b) shows red contours of specific angular momentum in units of $j_\infty =\sqrt{0.02}\,R_{\rm B}c_{s,\infty }$, with blue vectors indicating the flux of angular momentum. The purple contour shows where j = j_∞; equatorward the gas has higher specific angular momentum, and poleward the gas has lower specific angular momentum. The equatorial outflow carries off the angular momentum of the gas that was accreted (Kolykhalov & Syunyaev 1979; Inogamov & Sunyaev 2010) and some of the angular momentum lost by the polar outflow.

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Figure 3(a) shows logarithmic density contours and velocity vectors interior to 0.1 R_B. The black line indicates the zero density surface in the stationary solution. Inflowing matter dumps angular momentum into the equatorial regions, and an outflow is driven beginning just beyond the centrifugal radius, located at R_c = 0.02 R_B. The inflowing gas is able to penetrate below the centrifugal radius only due to the angular momentum transport, i.e., the assumed weak MRI induced viscosity. In Figure 3(b), which shows logarithmic density contours and velocity vectors interior to half the centrifugal radius, we see that the gas accretes over a range of latitudes from a time-dependent, vertically extended, sub-Keplerian disk. The accretion rate is very low, however, with magnitude ${\sim } 4 \times 10^{-3}\,\dot{M}_{\rm B}$ of order α times the mass inflow rate. There is some accretion due to the angular momentum transport, but the bulk of the gas is too energetic, with $Be>c_{s,\infty }^2/(\gamma -1)$, and has too much angular momentum for much of it to accrete. This conclusion is in close agreement with that of Stone et al. (1999).

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In the cold disk solutions, the non-adiabatic gas radiates energy and most of the gas cannot travel back out to infinity on energetic grounds since $Be<c_{s,\infty }^2/(\gamma -1)$. In the absence of angular momentum transport the gas continuously accumulates near the centrifugal barrier. Gas pressure is insufficient to drive matter inward onto the central black hole and there is no steady solution. The introduction of angular momentum transport allows for significant accretion from the centrifugal torus, however. Figure 4(a) shows logarithmic density contours and mass flux vectors for the cold disk solution at Eddington ratio $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=0.1$, interior to the Bondi radius. The solid black line shows the zero density surface in the adiabatic settling solution given in Equation (14). Matter accumulates in an extended torus whose pressure slows mass inflow, similar in spirit to the adiabatic settling solution. Inflow proceeds at low polar angles θ ≲ π/4, however, with total inflow rate ${\sim } 0.45\,\dot{M}_{\rm B}$ at the Bondi radius. Equatorial gas exterior to the centrifugal barrier is still driven outwards by the viscous engine, but the outward propagation of the equatorial outflow is slowed by the strong radiative cooling. In the absence of external pressure, the equatorial outflow would continue to propagate outwards indefinitely (Kolykhalov & Syunyaev 1980). Our constant pressure outer boundary condition slows the outflow, however, and we find that the equatorial outflow is unable to propagate to the Bondi radius on time scales of order ∼100 t_B. This solution is not technically a steady-state solution, as angular momentum is continuously dumped into the radially extended torus, which acts as a large sink of angular momentum. If we integrated for much longer times, however, the equatorial outflow must continue to propagate outwards to conserve momentum in a steady-state. Figure 4(b) shows logarithmic density contours and mass fluxes interior to 0.1 R_B. The flow forms a geometrically thin disk, and angular momentum transport allows matter to continue to flow inwards from this disk. We illustrate the cold thin disk near the centrifugal barrier in greater detail in Figure 5. Red contours are again logarithmic density contours and blue vectors represent mass flux. Accretion proceeds through the inner boundary from the cold, thin, sub-Keplerian disk, at a rate nearly equal to the mass inflow rate through the Bondi radius. This case resembles the standard thin disk (Shakura & Sunyaev 1973).

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Figure 6 shows the quasi-steady mass accretion rate through the inner boundary, the equatorial and polar wind mass outflow rates through the Bondi radius, and the mass inflow rate through the Bondi radius, all as functions of the parameter $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$. Crosses on the accretion rate curve indicate the Eddington ratios for which our simulations have converged with respect to temporal duration. For $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}\lesssim 0.01$, the Bremsstrahlung cooling is weak and plays little dynamic role in the flow. The flow converges to a low accretion rate solution for $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}\lesssim 10^{-3}$. The non-zero viscosity allows only a small fraction, ∼0.4%, of the inflowing matter at the Bondi radius to accrete onto the central black hole. The mass inflow rate at the Bondi radius $\dot{M}_{\rm inflow}$ is itself only ${\sim } 0.29 \,\dot{M}_{\rm B}$ because the centrifugal torus and outflows significantly impede the inflow (see Figures 2 and 3). The gas density on the equator is much larger than in the polar region, and the equatorial outflow dominates the outward mass and momentum flux at the Bondi radius. The equatorial outflow nearly matches the mass inflow rate at the Bondi radius, whereas the polar outflow is much less significant and is an order of magnitude weaker than either. It is important to note, however, that viscous heating near the centrifugal radius leads to a buildup of thermal energy and can lead to intermittent episodes of convective overturning. The overturning episodes can last for ∼10 t_B, with outward mass fluxes over a broad range of polar angles and a significant fraction of the characteristic mass inflow rates. The filled boxes in Figure 6 represent estimates of the time-averaged polar mass outflows including these overturning episodes, which by definition are not included in our quasi-steady average mass fluxes (see the Appendix for more detail).

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The strength of the radiative cooling increases with the parameter $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$, and in the range $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}\sim 0.01\hbox{--}0.1$ there is an abrupt transition in the properties of the flow to the strong cooling regime. For $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}\sim 0.1$ the inflowing gas is impeded in the equatorial regions by gas pressure in an extended centrifugal torus (see Figure 4), and matter flows in through the Bondi radius at roughly half the Bondi rate. Nearly the entirety of this inflowing gas is accreted onto the central black hole and the solutions approach the standard Shakura & Sunyaev (1973) picture. Most of the inflowing gas does not have enough energy to travel back out to infinity, and only a very small portion bounces off the centrifugal barrier and is ejected in a polar wind. The persistence of the polar wind is uncertain, and much higher resolution studies with additional physics such as magnetic fields and radiation are necessary to study the detailed properties of polar outflows. We emphasize again that the high Eddington ratio solutions are only quasi-steady (see Section 3.2), and the equatorial outflows would be stronger if we integrated for times ≳ 1000 R_B/c_{s, ∞}.

We have further verified that the low accretion rate solutions depend weakly on our choice of adiabatic index. Figure 7 shows the mass inflow rate from large radii, equatorial and polar outflow rates at R_B, and the accretion rate as a function of γ. Mass fluxes depend weakly on γ in the range 1.4–1.65, and our ansatz γ = 1.5 should not significantly affect our results.

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The high accretion rate flows from Figure 6 with $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}>0.1$ are cold and form a geometrically thin disk, and the low accretion rate flows with $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}<0.01$ are hot and form vertically extended disks. Figure 8(a) shows the emission weighted temperature,

$$\begin{equation} \langle T\rangle \equiv \frac{\int \dot{e}_{\rm Brem} T dV}{\int \dot{e}_{\rm Brem} dV}, \end{equation} \tag{ 15 }$$

as a function of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$. We show three characteristic temperature curves, from top to bottom averaged over the region interior to the centrifugal barrier, the total region within 0.1 R_B, and in the centrifugal torus. The centrifugal torus is extremely dense and cools to a lower temperature (the floor that we set) than other parts of the flow at high Eddington ratios. The gas interior to the centrifugal barrier is hottest because the gas has penetrated deeper into the gravitational potential well. For low $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}<0.01$ the gas in the inner parts of the flow is hot and a sizable fraction of the centrifugal temperature

$$\begin{equation} T_{\rm c}\equiv \frac{\mu m_p}{k_b \gamma }\frac{2 GM_{\rm BH}}{ R_{\rm c}}=100\,T_\infty , \end{equation} \tag{ 16 }$$

where μ = 0.5 is the mean molecular weight. Electrons become weakly relativistic at such temperatures, but we neglect in our computations the increase of the Bremsstrahlung energy losses for relativistic electrons. We measure the pressure scale height of the disk as the vertical distance at the centrifugal radius over which the pressure drops by a factor of e. The pressure scale height of the disk for the hot solutions with vertically extended disks is H_p/R_c ∼ 0.75 (see Figure 8(b)), and they have α ∼ 0.01 at the centrifugal barrier. The high accretion rate solutions with cold thin disks have pressure scale at the centrifugal radius H_p/R_c ∼ 0.15, covering ∼5 cells in the highest resolution simulation (see the Appendix). For a cold thin disk in hydrostatic equilibrium and with angular momentum $j=\sqrt{0.02}\,R_{\rm B}c_{s,\infty }$ we expect H_p/R_c = c_s/v_ϕ = 0.07. Since we fix the kinematic viscosity ν in our code, our cold disks have α ∼ 0.4 rather than α ∼ 0.01 at the centrifugal barrier. The code has difficulty running the cold solutions at lower viscosity, as there is significantly greater mass deposited near the centrifugal radius before accretion can proceed. The gas has cooled strongly and energetically cannot reach infinity, however, and we expect it to accrete independently of the strength of the viscosity, as long as it is non-zero. In any case, radiative transfer becomes important at Eddington ratios of ∼few × 0.01, physics that we do not include here. The precise location of the transition between thin and thick disks, as well as the density, luminosity, and optical depth of the thin disks, will depend on these choices.

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The conical polar outflows, driven by inflowing matter bouncing off the centrifugal barrier, are in general weak, but they exist over a wide range of Eddington ratios. Figure 8(c) gives the half-opening angle of the polar outflow at the Bondi radius as a function of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$. The half-opening angle is typically θ_half-open ∼ 5°. We show also the half-angles at which half of the mass and half of the momentum in the polar outflows are ejected, again computed at the Bondi radius. The zero density surface in the adiabatic settling solution lies at an angle of ∼3° at the Bondi radius. Figure 8(d) shows the mass-flux weighted average wind velocity in the polar outflow, measured at the Bondi radius, as a function of $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$. The wind velocity can be a significant fraction of the centrifugal velocity

$$\begin{equation} v_{\rm c}\equiv (2GM_{\rm BH}/R_{\rm c})^{1/2}, \end{equation} \tag{ 17 }$$

with typical wind velocity 〈v_wind〉 ∼ 0.15 v_c. This value is similar to that found in Yuan et al. (2012a). The outflow opening angle and velocity depend on the radius at which we measure these quantities. The outflow propagates into a constant density and pressure medium in our setup, and at sufficiently large distances the outflow will transfer its momentum and energy to the external medium. Beyond this region the outflow would become subsonic and probably circulate back to join the inflowing gas at intermediate latitudes. We quote values at the Bondi radius to give a quantitative estimate for the outflow properties at a definite radius.

We now discuss in greater detail the radiative properties of our accretion flows. If a spherically symmetric distribution of gas is in free-fall with velocity

$$\begin{equation} v_{\rm ff}=(2GM_{\rm BH}/r)^{1/2} \end{equation} \tag{ 18 }$$

and number density

$$\begin{equation} n(r)=\frac{\dot{M}_{\rm B}}{4\pi c R_s^2 \eta }(r/R_s)^{-3/2}, \end{equation} \tag{ 19 }$$

where η is the mean mass per electron, then the optical depth

$$\begin{equation} \tau \equiv \sigma _T \int n(r) dr \propto \dot{M}_{\rm B}/\dot{M}_{\rm Edd}. \end{equation} \tag{ 20 }$$

We compute the optical depth along three different sight lines with fixed polar angle θ = 0°, 45°, 90°, starting from the inner boundary at R_in = 91 R_s. We find that τ scales linearly with $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}$ (see Figure 8(e)). The optical depths approach 10⁻² just below $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=10^{-2}$, where the flow switches between the hot, vertically extended disk and the cold, thin disk, and there is a jump by a factor of ∼100 in τ₉₀ at this transition. Further, making use of the temperature profile of an adiabatic gas in free-fall,

$$\begin{equation} T\simeq T(R_s)(r/R_s)^{-3(\gamma -1)/2}, \end{equation} \tag{ 21 }$$

the total integrated Bremsstrahlung luminosity

$$\begin{eqnarray} L_{\rm Brem}/L_{\rm Edd} &\propto \frac{1}{L_{\rm Edd}}\int n(r)^2 T^{1/2} r^2 dr \end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray} &\propto (\dot{M}_{\rm B}/\dot{M}_{\rm Edd})^2. \end{eqnarray} \tag{ 23 }$$

Figure 8(f) shows the Bremsstrahlung luminosity in our simulations integrated over the regions between R_in = 91 R_s and 0.1 R_B, and between R_in = 91 R_s and the centrifugal torus. We obtain the scaling $L_{\rm Brem}/L_{\rm Edd} \propto (\dot{M}_{\rm B}/\dot{M}_{\rm Edd})^2$ for both integrated luminosities, and L_Brem → 10⁻⁷ L_Edd at $\dot{M}_{\rm B}/\dot{M}_{\rm Edd}=10^{-2}$. As we move the inner boundary at R_in closer to the black hole, we expect the luminosity, temperature, and optical depth interior to the centrifugal radius to increase. Other work including general relativistic and magnetohydrodynamic physics and extending down to the horizon for non-spinning black holes suggests that essentially all matter flowing inwards at our inner boundary R_in = 91 R_s will ultimately accrete, however (Narayan et al. 2012).