NUMERICAL SIMULATION OF HOT ACCRETION FLOWS. II. NATURE, ORIGIN, AND PROPERTIES OF OUTFLOWS AND THEIR POSSIBLE OBSERVATIONAL APPLICATIONS

We perform both HD and MHD calculations of two-dimensional axisymmetric accretion flows around black holes. The equations we use are exactly the same as those of SPB99 and Stone & Pringle (2001). For the convenience of readers, we copy them here. The HD equations are

$$\begin{equation} \frac{d\rho }{dt}+\rho \nabla \cdot \mathbf {v}=0, \end{equation} \tag{ 1 }$$

$$\begin{equation} \rho \frac{d\mathbf {v}}{dt}=-\nabla p-\rho \nabla \psi +\nabla \cdot \mathbf {T}, \end{equation} \tag{ 2 }$$

$$\begin{equation} \rho \frac{d(e/\rho)}{dt}=-p\nabla \cdot \mathbf {v}+\mathbf {T}^2/\mu. \end{equation} \tag{ 3 }$$

The MHD equations are

$$\begin{equation} \frac{d\rho }{dt}+\rho \nabla \cdot \mathbf {v}=0, \end{equation} \tag{ 4 }$$

$$\begin{equation} \rho \frac{d\mathbf {v}}{dt}=-\nabla p-\rho \nabla \psi +\frac{1}{4\pi }(\nabla \times \mathbf {B})\times \mathbf {B}, \end{equation} \tag{ 5 }$$

$$\begin{equation} \rho \frac{d(e/\rho)}{dt}=-P\nabla \cdot \mathbf {v}+\eta \mathbf {J}^2, \end{equation} \tag{ 6 }$$

$$\begin{equation} \frac{\partial \mathbf {B}}{\partial t}=\nabla \times (v\times \mathbf {B}-\eta \mathbf {J}). \end{equation} \tag{ 7 }$$

In the above equations, ρ, P, v, e, $\mathbf {B}$, and $\mathbf {J}=(c/4\pi)\nabla \times \mathbf {B}$ are the mass density, pressure, velocity, internal energy, magnetic field, and the current density, respectively. We adopt an adiabatic equation of state P = (γ − 1)e with γ = 5/3. ψ is the gravitational potential. We adopt the Paczyńsky & Wiita potential ψ = −GM/(r − r_s), where M is the central black hole mass, G is the gravitational constant, and r_s ≡ 2GM/c². The self-gravity of the accretion flow is neglected. We use the spherical coordinates (r, θ, ϕ) to solve these equations. We choose units such that c = M = G = 1.

In the HD equations, $\mathbf {T}$ is the anomalous stress tensor. The final two terms in Equations (2) and (3) represent anomalous stress and its corresponding heating, respectively. Since we do not have a magnetic field, we add two terms to transfer the angular momentum and convert energy, so that we may mimic the real MHD case (see below). The viscosity coefficient μ = νρ and ν is the kinematic viscosity coefficient. We adopt the form ν = αr^1/2 because it corresponds to the usual "α" description (refer to "Run K" in SPB99). We set α = 0.01. Following SPB99, to better mimic the MHD case we assume that the only non-zero components of $\mathbf {T}$ are the azimuthal components (see SPB99 for details). In the MHD equations, the final terms in Equations (6) and (7) are the magnetic heating and dissipation rate mediated by a finite resistivity η. Since the energy equation here is actually an internal energy equation, numerical reconnection inevitably results in loss of energy from the system. By adding the anomalous resistivity η, the energy loss can be captured in the form of heating in the current sheet (Stone & Pringle 2001).

The MHD simulation is, of course, more realistic than the HD one. Then what are the motivations for performing the HD simulation? The reasons are threefold. The first reason is mainly of academic interest. We hope to compare with and understand previous HD works, both analytical and numerical. The second is to study the effects of the initial conditions. Although we can use MHD simulations to accomplish this, it will obviously be much more expensive. More importantly, we believe that in terms of the effect of the initial condition, the results for HD and MHD simulations should be intrinsically the same. Lastly, by comparing the HD and MHD simulations we hope to be able to deepen our understanding of the production and properties of outflows.

We investigate four models. Three models (models A, B, and C) are HD and one (model D) is MHD. The first three models have different initial conditions. The initial condition of model D is identical to that of model A, except that a magnetic field is included. Our intention is that the models should be as simple and as clean as possible. Based on this consideration, except for the two models that are devoted to studying the effect of initial conditions (i.e., models B and C), the other two (models A and D) are the most "popular" in the literature. Their details are as follows.

Model A. This model is almost identical to "Run K" in SPB99. The initial condition is a rotating torus with constant angular momentum, which is identical to that adopted in Stone et al. (1999). Readers are referred to SPB99 for the detailed description of the torus. In contrast to SPB99, where the Newtonian potential was adopted, we adopt the Paczyńsky & Wiita (1980) potential. This introduces the length scale, so we put the center of the density maximum of the torus at ∼100r_s. We set the maximum density of the torus ρ_max = 1.0, the density of the surrounding medium ρ₀ = 10⁻⁴, and pressure p₀ = ρ₀/r. The torus is a bound system; therefore the Bernoulli parameter Be < 0 (see the dotted line in Figure 5 for its value at the equatorial plane).

Model B. A rotating torus has no radial velocity while a realistic accretion flow has. In this sense, we think that perhaps a better initial condition is that the gas has both radial and rotational velocities. This is one of our motivations for the initial conditions in models B and C. In model B, to obtain the initial condition, we first determine the one-dimensional global solution of two-temperature ADAFs. This corresponds to solving the one-dimensional version of Equations (1)–(3), although the equations are for one-temperature accretion flow. We use the calculation results (e.g., density and temperature) for ions as the initial conditions in our simulation. We choose to solve the two-temperature rather than one-temperature case simply because we think the former is more realistic; however a one-temperature solution should not make any notable difference. The solution extends from the inner boundary to the outer boundary, and it has both rotation and radial velocities. The details of the calculations are described in Yuan (1999). Specifically, we choose the outer boundary condition so that Be < 0 in the solution, but larger than that in model A. We do not choose Be > 0 in model B in order to discriminate this model from model C, where Be > 0. We then expand this one-dimensional solution to two dimensions by assuming that the vertical density profile follows an exponential distribution but all other quantities such as temperature and velocity retain their original one-dimensional values. The thickness of the accretion flow is determined by the scale height H = c_s/Ω_k.

Model C. We adopt the "injection-type" initial condition in this model. The gas is injected at the outer boundary. The properties of the injected gas, such as temperature, rotational and radial velocities, are completely determined by the self-similar solution of Narayan & Yi (1994). In this case, Be > 0 (Narayan & Yi 1994).

Model D. This is an MHD model. The model is fully identical to "Run F" in Stone & Pringle (2001).³ Namely, the initial condition is a rotating torus with the density maximum located at r = 100r_s, exactly the same as model A. So the initial Be is identical in the two models. The difference is that an initial magnetic field is added. The field is poloidal, confined to the interior of the torus. The loops are parallel to the density loops. Readers are referred to Stone & Pringle (2001) for details. One reason we adopt this model is because, as Hawley & Krolik (2002) argued, compared to the initial toroidal field configuration, the poloidal configuration is more realistic.

We will see that while the main results remain similar for the different initial conditions, some results do differ. In particular, as we will show in Section 3.3, the sign of Be is largely dependent on the value of Be in the initial condition. What initial condition is more realistic? This is a difficult question: the answer may depend on different environments. These environments include, among other things, the accretion flow formed in the Roche Lobe overflow or stellar wind in the case of X-ray binaries, the centers of galaxies in the case of various types of AGNs, the molecular cloud in the case of star formation, or even a rotation disk in the case of planetary formation. In terms of the Bernoulli parameter of the initial gas, if the radiative energy loss can be neglected during accretion, we expect Be ≳ 0 since it comes from infinity. This may be the case for most elliptical galaxies, such as M87, in which hot interstellar medium (ISM) fuels the black hole, and some spiral galaxies such as Sgr A*. In these cases, models B and C are likely more realistic than models A and D. But if radiation is important, as perhaps it is in the case of most spiral galaxies and black hole X-ray binaries where the gas accreted is cool and dense, we usually consider that at large radius the accretion flow is described by a standard thin disk. This disk may be truncated and replaced by a hot accretion flow within a transition radius if the accretion rate is not too large (see Section 7.1 for more details). In this case the sign of Be of the inner hot accretion flow is unclear, although it is certain that Be < 0 for the outer truncated thin disk. This is because the physics of the transition from the outer thin disk to the inner hot accretion flow is still an unsolved problem. The two models proposed include evaporation and radial turbulent transport. There must be energy transfer either vertically (in the former model) or radially (in the latter model). Therefore the value of Be must be increased after the transition. Similar uncertainty exists for the hot corona sandwiching the thin disk.

We use the ZEUS code (Stone & Norman 1992a, 1992b) in spherical geometry to solve the equations. The two modifications to the code are to include the shear stress terms (in the HD case) and the implementation of the Paczyńsky & Wiita (1980) potential. The computational grids extend from an inner boundary at r = 1.2r_s to r = 500r_s. The inner boundary is small enough to make sure that our solution is transonic. The standard outflow boundary condition is adopted at both the inner and outer radial boundaries. In the angular direction, the boundary conditions are set by symmetry at the poles. To adequately resolve the flow, we adopt the same non-uniform grid in both the radial and angular directions as in SPB99 and Stone & Pringle (2001).

For the aim of the present paper we are interested only in the steady state of the solution. We judge the radial range within which the steady solution is achieved by examining whether the net accretion rate (Equation (14)) is a constant of radius. The general descriptions of the simulation results of models A and D can be found in SPB99 and Stone & Pringle (2001). For model B, at the beginning of the simulation, we find a shock is formed at the inner boundary and it propagates outward. The flow achieves a steady state after 0.7 orbital times at the outer boundary. In model C, initially the whole computational domain is filled with a very low density ISM. Gas is steadily injected into the calculation domain from the equatorial plane at the outer boundary. The injected gas quickly moves inward and forms an accretion flow. After about nine orbital times at the outer boundary, the accretion flow achieves a quasi-steady state.

When we evaluate the radial distribution of physical quantities, such as Be, velocity, and temperature, we need to calculate the angle-integrated average. We adopt the mass flux-weighted value, which is defined as (for quantity q)

$$\begin{equation} \langle q(r)\rangle =\frac{2\pi r^2\int ^\pi _0\rho q \max (v_r,0)\sin {\theta }d\theta }{2\pi r^2\int ^\pi _0\rho \max (v_r,0)\sin {\theta }d\theta } \end{equation} \tag{ 8 }$$

for outflow and

$$\begin{equation} \langle q(r)\rangle =\frac{2\pi r^2\int ^\pi _0\rho q \min (v_r,0)\sin {\theta }d\theta }{2\pi r^2\int ^\pi _0\rho \min (v_r,0)\sin {\theta }d\theta } \end{equation} \tag{ 9 }$$

for inflow. We have also tried the density-weighted values, which are defined as

$$\begin{equation} \langle q_{d}(r)\rangle =\frac{2\pi r^2\int ^\pi _0\rho q \max (v_r,0)/v_r\sin {\theta }d\theta }{2\pi r^2\int ^\pi _0\rho \max (v_r,0)/v_r\sin {\theta }d\theta } \end{equation} \tag{ 10 }$$

and

$$\begin{equation} \langle q_{d}(r)\rangle =\frac{2\pi r^2\int ^\pi _0\rho q \min (v_r,0)/v_r\sin {\theta }d\theta }{2\pi r^2\int ^\pi _0\rho \min (v_r,0)/v_r\sin {\theta }d\theta } \end{equation} \tag{ 11 }$$

for outflow and inflow, respectively. We find the results are very similar. In the following, we compare the various properties of inflow and outflow.

Following SPB99, the mass inflow and outflow rates, $\dot{M}_{\rm in}$ and $\dot{M}_{\rm out}$, are defined as the following time-averaged and angle-integrated quantities:

$$\begin{equation} \dot{M}_{\rm in}(r) = 2\pi r^{2} \int _{0}^{\pi } \rho \min (v_{r},0) \sin \theta d\theta , \end{equation} \tag{ 12 }$$

$$\begin{equation} \dot{M}_{\rm out}(r) = 2\pi r^{2} \int _{0}^{\pi } \rho \max (v_{r},0) \sin \theta d\theta. \end{equation} \tag{ 13 }$$

The net mass accretion rate is

$$\begin{equation} \dot{M}_{\rm net}(r)=\dot{M}_{\rm in}(r)-\dot{M}_{\rm out}(r). \end{equation} \tag{ 14 }$$

The net rate is the accretion rate that finally falls onto the black hole. Note that the above rates are obtained by time-averaging the integrals rather than integrating the time averages.

Figure 1 shows the radial distributions of the inflow and outflow rates and net rate for the four models. The radial profile of the inflow rate in all four models can be described by $\dot{M}_{\rm in}(r)\propto r^{s}$. Regarding the value of s, Paper I presents a more precise measurement since the dynamical range of simulations presented there is much larger than in the present work. The radial profile of density is also measured, and we find ρ(r)∝r^−p. Moreover, the relationship s = 1.5 − p is well satisfied, which indicates that the flattening of density is caused by the inward decrease of the accretion rate. Another notable result is that the accretion rate profiles for all four models are roughly similar. In Paper I, we discussed this consistency in a much more comprehensive and detailed way by combining all published HD and MHD simulation works. Also shown in the figure by the thick solid lines are the outflow rates with a positive Be. We will discuss this issue later.

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We study the angular distribution of the mass rate by time-averaging the following outflow and inflow rates as a function of θ:

$$\begin{equation} \dot{M}_{\rm out}(\theta)=2\pi r^2 \rho \max (v_r,0)\sin \theta \Delta \theta , \end{equation} \tag{ 15 }$$

$$\begin{equation} \dot{M}_{\rm in}(\theta)=2\pi r^2 \rho \min (v_r,0)\sin \theta \Delta \theta. \end{equation} \tag{ 16 }$$

Figures 2 and 3 show the angular distributions of the inflow rate and the outflow rate, respectively. We can see that their angular distributions are in general quite broad; there is no angle where there is only inflow or outflow in the time-average sense. The distributions of models A and D are similar; they are nearly symmetric to the equatorial plane. The distributions of models B and C are similar, peaked at an angle θ away from the equatorial plane. The discrepancy between these two groups is obviously because of the initial conditions. Models A and D have the same initial condition, while models B and C are similar since in both models the gas has an initial radial velocity. Comparing these two figures, we can see that, for all four models, the angular distributions of inflow and outflow roughly match, in the sense that in the region where the inflow is strong the outflow is weak. For example, for model A, most of the inflow occurs within θ = 70°–90°, while the outflow is very weak in this region. Most of the outflow occurs at the surface of the accretion flow, peaking at θ = 50° and 140°. Model D is similar to model A.

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For models B and C, the distributions of both inflow and outflow rates are not symmetric to the equatorial plane and there is one peak, located at θ ∼ 50° and 130° depending on the radius and model. But again outflow is strong in the region where the inflow is weak. For example, for model B, at r = 50r_s, most of the inflow occurs below the equatorial plane, around θ ∼ (110–140)° whereas most of the outflow occurs above the equatorial plane, around θ ∼ (40–70)°. This suggests that the inflow and outflow rates are dominated by large-scale bulk motion rather than fluctuations, and the bulk of the inflowing gas is turned around and becomes outflow. As we will discuss later, the outflowing motion is caused by the buoyant force. We speculate that the reason for the difference in the angular distributions between models B and C and models A and D may be due to the difference values of Be in the initial condition. For models B and C, the Bernoulli parameter of the initial gas is significantly higher, which means that the gas is more "energetic." This results in apparently more "violent" motion.

The Bernoulli parameter is the sum of kinetic energy, enthalpy, and potential energy:

$$\begin{equation} Be=\frac{v^2}{2}+\frac{\gamma P}{(\gamma -1)\rho }-\frac{GM}{r-r_s}. \end{equation} \tag{ 17 }$$

Here the magnetic field is not included. In the case of inviscid flow, the Bernoulli parameter of a fluid element along a streamline is conserved. A positive Be means that the accretion flow has sufficient energy to overcome the gravitational energy and does work to its surroundings to escape to infinity.

Figure 4 presents the two-dimensional contours of Be for the accretion flow (both inflow and outflow) for the four models. The red regions denote Be > 0. We can see that the results of the four models are quite different. For model A Be < 0 in most of the area, as stated in SPB99 and Yuan & Bu (2010). The latter found that at the outer boundary only about 1% of the outflow has a positive Be. For model B, the region with a positive Be becomes larger. For model C, Be is positive in almost all of the area. The main reason for such differences is that the initial conditions are different, i.e., different models have different initial Be. The value of Be along the equatorial plane of the initial condition in the four models is shown by the dotted lines in Figure 5. For models A and D, Be is the smallest and the gas is strongly bound, i.e., Be < 0. In fact, we find that in the initial torus the temperature T of gas is about 10 times lower than the virial value. For model B, the value of Be of the initial condition is larger, but still less than zero. For model C, Be > 0 for all the injected gas. Since radiation is neglected in our simulation, the total energy of the gas in the computational domain should be roughly conserved (if we neglect energy loss through the outer boundary). Therefore it is natural to expect that the final value of the Be is a function of Be of the initial condition.

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Although their initial condition is identical, the contours of models D and A have one significant difference. The value of Be close to the rotation axis in model D is positive; while it is negative in model A. Such a difference should be because of the magnetic effect. In model D, we find that a large-scale ordered poloidal magnetic field is present in the small-θ region (and only in that region). Therefore, compared to model A, additional energy flux, coming from the rotational energy of the accretion flow, is transferred from small radius outward along the field line, which increases the value of Be there. This is the Blandford & Payne (1982) mechanism.

The thick solid lines in each plot of Figure 1 represent the radial profile of the outflow rate with a positive Be. We can see that for models A and B a small fraction of the outflow has a positive Be. For model C, Be > 0 for almost all of the outflow. For model D, a very tiny fraction of the outflow has Be > 0. The result of model D is consistent with Narayan et al. (2012). This is because the initial conditions in the two works are similar.

While the value of Be for different models is quite different as shown by Figure 4, we see from Figure 1 that the strength of the outflow is similar for the four models, independent of the value of Be. This indicates that the outflow is not produced because Be > 0. Of course, for these outflows to reach infinity, Be > 0 is still required.

Figure 5 shows the radial distribution of the flux-weighted Be of the outflow (dashed line) and inflow (solid line). In all models Be increases inward, including the case of model C. The apparent violation of Be conservation (i.e., Be is not a constant of radius) is because of the energy flux from small to large radii associated with the viscous stress. The dotted lines in the figure show the value of Be in the initial condition. We can immediately find that the value of Be for the outflow is always larger than that of the inflow for models A, B, and C. Even the difference between inflow and outflow seems to be similar for these three models. For model D, however, the value of Be is almost equal for inflow and outflow. This again suggests that the mechanism of the production of outflow is different for HD and MHD flows.

We are especially interested in the value of Be for the outflow since it determines whether the outflow can escape to infinity and the terminal velocity of the outflows if it can (Section 3.5). Unfortunately from Figure 5 we can see that the results are quite diverse for the four models, depending on the initial conditions. For models A and D, the Be of the initial torus is the smallest; the value of Be for the outflow in these two models is also roughly the smallest.⁴ For model C, Be > 0 for the initially injected gas and subsequently Be > 0 for both inflow and outflow. Unfortunately, as we have discussed in Section 2.2, it is difficult to determine what kind of initial condition is more realistic. But as we have argued there, in the case that the accretion flow starts from very large radius, it is likely that the initial condition with Be close to or larger than zero will be picked up. In this case we expect that Be > 0 for the outflow. To illustrate this point, we have calculated the value of Be for the outflow of model A in Paper I. In that model, the initial condition is the same rotating torus as in models A and D of the present paper, except that the torus is located at a much larger radius, 10⁴r_s. Figure 6 shows the results. We see that Be > 0 for all the outflow. In another example, Li et al. (2012) started their hot accretion flow at 10 times the Bondi radius, so Be > 0 for the initial condition. They found Be > 0 for all their outflow (J. Ostriker 2012, private communication). Finally, as we shall see in Section 7, observations seem to indicate the existence of outflow from hot accretion flows. In the following discussions, we therefore assume that the outflow from a hot accretion flow can always escape to infinity, i.e., Be > 0.

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We next present some speculations which further support the above assumption of Be > 0. Note that the importance of the initial condition does not mean that the final solution can be arbitrary. It must be restricted in a certain range. From Figure 5, we see that the diversity of the solution at the inner region of the accretion flow away from the outer boundary seems to become smaller compared to that of the initial condition. There seems to exist a "common" (but not unique) solution at the inner region, which can be very roughly described by

$$\begin{equation} Be\approx (-0.1\sim 0)v_k^2 \end{equation} \tag{ 18 }$$

for inflow and

$$\begin{equation} Be\approx (0.1\sim 0.2)v_k^2 \end{equation} \tag{ 19 }$$

for outflow. At large radii, the solutions strongly deviate from the above "common" solution because they are still "tied" to the initial condition there. Based on this result, we speculate that if the outer boundary is set at a very large radius and the location of the torus is at a large radius, the solution at the inner region may suffer less from the initial condition and becomes more "converged." In other words, the degree of the dependence of the solution on the initial condition depends on how far the gas is away from the black hole. In almost all current simulations, including the present work, the outer boundary is rather small; thus it is hard for the accretion flow to fully evolve and deviate from the initial condition. Only at small radii is the accretion flow better evolved; thus solutions there can better approach the "genuine" solution. For model B, at both small and large radii, the simulated solution is close to the initial condition. This is likely because the initial condition of model B is "good," i.e., close to the "genuine" solution. It is interesting to note that in this model the value of Be/v²_k is roughly constant, consistent with the self-similar solution (Narayan & Yi 1994). So this seems to again indicate that model B is more realistic than the other three models. Based on these considerations, when we calculate the terminal velocity in Section 3.5, we will assume Equation (19). We emphasize again that the above consideration is rather speculative. It is crucial to explore it further by simulations with a much larger outer boundary. This will be our future work. On the other hand, we believe that the value of Be determined by Equation (19) should be correct in order of magnitude; then the calculated terminal velocity (Equation (21)) will be correct within a factor of three.

Comparing models A and D, we can see that the value of Be in model D is significantly smaller than that in model A, although they have the same initial condition. This is because in model D some energy is converted into magnetic energy which is not included in our value of Be.

In the case that the outer boundary is small, or additional physics such as radiation is included, the value of Be for the outflow should be determined mainly by the initial condition. While there is some deviation, this should still be limited because of energy conservation. If the value of Be of the initial condition is far below zero, we expect that there will be no outflow with Be > 0, i.e., no jet. This result supplies an important clue to the following long-standing observational puzzle, namely why a relativistic jet can only be formed in a hot accretion flow and not in the standard thin disk. Black hole X-ray binaries usually come in various states (e.g., McClintock & Remillard 2006). The two most prominent states are hard and soft states, which are described by hot and cold accretion flows, respectively (e.g., see reviews by Zdziarski & Gierliński 2004; Done et al. 2007). Radio observations clearly show that relativistic jets exist only in the hard state, not in the soft state. That is, a jet can only be formed from a hot accretion flow, not from the thin disk. We speculate that this may be because of the difference in the values of Be for the two types of accretion flows. Because of strong radiative cooling, Be in the case of the thin disk is obviously much lower than that in the hot accretion flow. Following this argument, we predict that for hot accretion flows the ratio of the mass-loss rate in the jet and the accretion rate should decrease with increasing accretion rate. When the accretion rate is high, radiative cooling is strong and Be should be low. This is consistent with the modeling result of Yuan & Cui (2005).

Figure 7 shows the angular distribution of Be at r = 10r_s and 50r_s for the four models. At a given radius, part of the outflow has positive Be, but part has negative Be. For both models A and D, Be is largest away from the equatorial plane and smallest around the equatorial plane. The difference between these two models is that in model D the contrast of Be is much larger and the region of large Be is much closer to the axis. This is because of the existence of the large-scale poloidal field close to the axis, as we have explained in the previous section. The distributions of models B and C are different. It is interesting to note that Be is largest at the angles θ where the mass outflow rate is largest (Figure 3).

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Figure 8 shows the flux-weighted radial distributions of the temperature of the inflows and outflows for the four models. One distinct result is that for the three HD models, the outflow temperature is higher then the inflow temperature by nearly T_vir. Here the virial temperature is defined as

$$\begin{equation} T_{\rm vir}\equiv \frac{GMm_p}{3kr}. \end{equation} \tag{ 20 }$$

Since the temperatures of both the inflow and the outflow are nearly virial, the internal energy plays an important role in Be. This is why the value of Be for the outflow is higher than that for the inflow (Figure 5). This is strong evidence for the convection origin of outflow in HD flows, i.e., the fluid elements with temperature higher than that of the surrounding medium will move outward due to buoyancy. For model D, however, the temperatures of the inflow and outflow are almost identical. This suggests that an MHD flow is not convectively unstable, as we will confirm by stability analysis. Correspondingly, the mechanism of producing outflow in model D is not convection.

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Figure 9 shows the flux-weighted radial velocity of inflow and outflow for the four models. For the profiles of inflow, we see that models B and C are similar, as are models A and D are similar. The former can be described roughly by v_r/v_k ∼ constant, or v_r∝r^−0.5. This scaling is consistent with the self-similar solution of ADAF (Narayan & Yi 1994). For models A and D, the radial velocity scaling with radius is much steeper, i.e., the radial velocity increases faster inward compared to the Keplerian velocity. The discrepancy between the two groups of models is caused by the difference in the initial conditions. In models A and D, the radial velocity in the torus is initially zero; it is thus difficult for the gas at large radii (≳ 100r_s) to achieve a large velocity because of the boundary effect. On the other hand, because of the very strong gravity, the radial velocity has to increase rapidly inward at r ≲ 10r_s. These two effects make the profile steeper. If the initial torus in models A and D is located at a much larger radius, i.e., the dynamical range is much larger, the results for these two models will be similar to those for models B and C. This has actually been confirmed by our simulation in Paper I, where the torus is located at 10⁴r_s.

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However, the profile of the radial velocity of the outflow is completely different. It is not so sensitive to the initial condition. The ratio of radial velocity to Keplerian velocity remains roughly constant with radius, and this ratio is similar for all four models, i.e., v_r/v_k ≈ (0.2–0.4), regardless of the differences in the initial conditions and whether the magnetic field is included or not. Given that the mechanism of producing outflows in HD and MHD flows is different, as we will discuss later, the reason for the same outflow radial velocities remains to be explored. We would like to emphasize that the systematic difference in the profile of radial velocity between inflow and outflow again strongly indicates that the outflow is not simply due to turbulent fluctuation or convective motion. There is a systematic outward flux of mass.

Figure 10 shows the angular distribution of the radial velocity, averaged for all accretion flows including both inflows and outflows. It is clear that, in the time-average sense, the angular distributions of the radial velocities for models A and D are quite symmetric to the equatorial plane. Taking r = 50r_s (i.e., the solid lines in each plot) as an example, the inflow concentrates around the equatorial plane, 60° ≲ θ ≲ 120° while the outflow occurs at 15° ≲ θ ≲ 60° and 120° ≲ θ ≲ 165°. For models B and C, the distributions are not symmetric to the equatorial plane. When the inflow concentrates above (below) the plane, i.e., θ ≲ 90° (θ ≳ 90°), the outflow then concentrates below (above) the plane. We note that the angular distributions of radial velocity of the inflow and the outflow shown in this figure are consistent with the angular distributions of inflow and outflow rates, shown in Figures 2 and 3. Another notable feature is that, for all the four models, close to the axis average the radial velocity is negative, i.e., it is inflow.

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One important question is the terminal radial velocity of the outflow if it reaches infinity. We are interested in this question since outflow is potentially very important for AGN feedback. We suppose that, once launched, the viscous stress in the outflow and the mixing of mass, energy, and angular momentum between streamlines can be ignored. For such an adiabatic expanding inviscid flow, Be is conserved. When r is large enough, the last two terms in Equation (17) vanish. If the angular momentum is conserved, v_ϕ ≈ 0 so the velocity is mainly in the radial direction. Therefore the terminal velocity is

$$\begin{equation} v_{\rm term}\sim v_r \sim \sqrt{2Be(r_{\rm out})}. \end{equation} \tag{ 21 }$$

Combining with Equation (19), we have

$$\begin{equation} v_{\rm term}\sim 0.5 v_k (r_{\rm out}). \end{equation} \tag{ 22 }$$

That is to say, the radial terminal velocity of the outflow is roughly half of the Keplerian velocity at the outer radius of the hot accretion flow where most of the outflow originates.

A positive sign of Be is only a condition for the inviscid HD outflow to escape to infinity. When viscosity and a magnetic field are present, energy may be transferred from small to large radii and the magnetic force may also do work to the outflow. In this case, Be defined by Equation (17) is not conserved, and it may be not necessary for the outflow to have Be > 0 to escape to infinity. The terminal velocity given by Equation (21) will be a lower limit. We do not consider these complications in the present work. The exact solution of the terminal velocity as a function of its initial properties in the presence of a magnetic field will be an interesting question to study.

It has long been speculated that the outflow can carry away energy and angular momentum (e.g., Shakura & Sunyaev 1973; Blandford & Begelman 1999). This requires that the specific energy and specific angular momentum of the outflow should be larger than those of the inflow. We have discussed the Bernoulli parameter in the previous section and found that the value of Be for the outflow is systematically larger than that for the inflow for models A, B, and C, but for model D the values of Be for the inflow and the outflow are the same. In this section we examine angular momentum transfer by the outflow.

Figure 11 shows the distributions of specific angular momentum for both the inflow and the outflow. The results for the three HD models are similar, but different from model D. For the three HD models, the angular momentum of the outflow is larger than that of the inflow at small radii, consistent with Stone et al. (1999). This result indicates that the outflow can transfer angular momentum at small radii. At large radii, however, the angular momentum of the outflow is smaller than that of the inflow. This means that the radial profile of the angular momentum of the outflow is flatter than that of the inflow, and the angular momentum of the outflow tends to be conserved during its outward propagation. For both inflow and outflow, the angular momentum is significantly lower than the Keplerian value at the equatorial plane.

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Model D is completely different. We see from the figure that throughout the accretion flow the specific angular momentum of the outflow is very close to and even in some regions slightly larger than the Keplerian value at the equatorial plane. This is significantly larger than that of the inflow, whose value is substantially sub-Keplerian.⁵ This indicates that in an MHD flow, the outflow can transfer angular momentum much more efficiently compared to the HD flows. This is related to the mechanism of producing outflow. We will return to this question in Section 6.

Based on the above results, we can estimate the radial momentum flux and kinetic power of the outflow. The terminal radial momentum flux is

$$\begin{equation} \dot{p}_w=\dot{M}_{\rm out}(r_{\rm out}) v_{\rm term}\sim 0.5\dot{M}_{\rm out}(r_{\rm out})v_k(r_{\rm out}). \end{equation} \tag{ 23 }$$

Here $\dot{M}_{\rm out}(r_{\rm out})$ is the outflow rate at the outer boundary of the hot accretion flow r_out.

It is interesting to compare this momentum flux with the radiation flux L/c. Adopting the result from Paper I, the inflow rate at r = 10r_s is $\dot{M}_{\rm in}(10r_s)=\dot{M}_{\rm in}(r_{\rm out})(10r_s/r_{\rm out})^{0.5}=\dot{M}_{\rm out}(r_{\rm out}) (10r_s/r_{\rm out})^{0.5}$. The radiative flux is then

$$\begin{equation} L/c\sim \eta \dot{M}_{\rm out}(r_{\rm out})(10r_s/r_{\rm out})^{0.5}c. \end{equation} \tag{ 24 }$$

Here η is the radiative efficiency of the hot accretion flow defined as $\eta \equiv L_{\rm bol}/\dot{M}_{\rm in}(10r_s)c^2$. So the ratio of momentum flux of the outflow to the radiation flux is

$$\begin{equation} \frac{\dot{p}_w}{L/c}\approx \frac{0.1}{\eta }. \end{equation} \tag{ 25 }$$

The efficiency η is mainly a function of accretion rate and the parameters δ and α (see Xie & Yuan 2012 for details). For δ = 0.5, as indicated from the detailed modeling of Sgr A* (Yuan et al. 2003), and α = 0.1, we have (Xie & Yuan 2012)

$$\begin{equation} \eta =0.05 (\dot{M}_{\rm in}(10r_s)/4\times 10^{-3}\dot{M}_{\rm Edd})^{0.2} \end{equation} \tag{ 26 }$$

for $\dot{M}_{\rm in}(10r_s)=(4 \times 10^{-3}\hbox{--}10^{-5})\dot{M}_{\rm Edd}$ and

$$\begin{equation} \eta =0.05(\dot{M}_{\rm in}(10r_s)/4\times 10^{-3}\dot{M}_{\rm Edd})^{0.6} \end{equation} \tag{ 27 }$$

for $\dot{M}_{\rm in}(10r_s)=(4\times 10^{-3}\hbox{--}10^{-2})\dot{M}_{\rm Edd}$. Above $10^{-2}\dot{M}_{\rm Edd}$, up to ${\sim} 0.1 \dot{M}_{\rm Edd}$, the accretion flow is two-phase. The detailed study of the dynamics and radiation of this kind of accretion flow remains to be explored. The approximate calculation presented in Xie & Yuan (2012) gives η ≈ 0.08. So in general we expect

$$\begin{equation} \dot{p}_w\gtrsim L/c. \end{equation} \tag{ 28 }$$

The involvement of a jet is still controversial. Assuming it is dominated by normal plasma, we estimate the momentum flux of the jet:

$$\begin{equation} \dot{p}_{\rm jet}=\Gamma _{\rm jet}^2\eta _{\rm jet} \left(\frac{10r_s}{r_{\rm out}}\right)^{0.5}\dot{M}_{\rm in}(r_{\rm out})c=9\Gamma _{\rm jet}^2\eta _{\rm jet}\dot{p}_w, \end{equation} \tag{ 29 }$$

with η_jet the ratio of the mass-loss rate in the jet to the mass accretion rate at 10r_s, and Γ_jet the Lorentz factor of the jet.

The value of η_jet is highly uncertain. From the detailed modeling of the black hole X-ray binary XTE J1118+480 (Yuan et al. 2005), we get η_jet ∼ 1%. For comparison, Fender et al. (2001) estimated the jet power in the same source based on the observed radio luminosity. The range of jet power obtained is quite large but the "geometric average" of its upper and lower limits is consistent with the Yuan et al. (2005) value within a factor of three. The Lorentz factors of jets are different for AGNs and black hole X-ray binaries, and reasonable values are Γ_jet ∼ a few and Γ_jet ∼ 1, respectively (Gallo et al. 2003). For the jet in M 87, the estimated jet power (Equation (33)) based on η_jet ≈ 1%, combined with r_out = 5 × 10⁵ and $\dot{M}_{\rm in }(r_{\rm out})=0.1\,{M}_{\odot }\,{{\rm yr}^{-1}}$ from Di Matteo et al. (2003) and Γ_jet ≈ 6 (Biretta et al. 1999), is L_jet ∼ 7 × 10⁴² erg s⁻¹. This value is completely consistent with the independent estimation of Reynolds et al. (1996). So the momentum flux of the outflow is comparable to that of a jet for AGNs, while it is a factor of 10 larger than that of a jet for black hole X-ray binaries.

The kinetic energy power of the outflow is estimated to be

$$\begin{equation} \dot{E}_w=\frac{1}{2}\dot{M}_{\rm out}(r_{\rm out})v_{\rm term}^2\approx 0.1\dot{M}_{\rm out}(r_{\rm out})v_k^2(r_{\rm out}). \end{equation} \tag{ 30 }$$

Here we use the approximate flux-weighted radial velocity which is not precise but correct in order of magnitude. Since v²_k(r_out) = GM/r_out = c²/(2r_out/r_s), we have

$$\begin{equation} \dot{E}_w=\frac{\dot{M}_{\rm out}(r_{\rm out})c^2}{20r_{\rm out}/r_s}. \end{equation} \tag{ 31 }$$

We now compare $\dot{E}_w$ with several other powers. The accretion power at 10r_s is

$$\begin{equation} P_{\rm acc}=\left(\frac{10r_s}{r_{\rm out}}\right)^{0.5}\dot{M}_{\rm in}(r_{\rm out})c^2\approx 60\dot{E}_w\left(\frac{r_{\rm out}}{r_s}\right)^{0.5}\gg \dot{E}_w. \end{equation} \tag{ 32 }$$

The jet power is

$$\begin{equation} \dot{E}_{\rm jet}=\Gamma _{\rm jet}^2\eta _{\rm jet} \left(\frac{10r_s}{r_{\rm out}}\right)^{0.5}\dot{M}_{\rm in}(r_{\rm out})c^2=60\Gamma _{\rm jet}^2\eta _{\rm jet}\dot{E}_w\left(\frac{r_{\rm out}}{r_s}\right)^{0.5}. \end{equation} \tag{ 33 }$$

The radiation power of the accretion flow is

$$\begin{equation} \dot{E}_{\rm rad}=\eta \left(\frac{10r_s}{r_{\rm out}}\right)^{0.5}\dot{M}_{\rm in}(r_{\rm out})c^2=60\eta \dot{E}_w\left(\frac{r_{\rm out}}{r_s}\right)^{0.5}. \end{equation} \tag{ 34 }$$

The outer boundary of the hot accretion flow r_out in the cases of both AGNs and black hole X-ray binaries is uncertain. In the case of AGNs, one possibility is that the outer boundary is the Bondi radius. This may be the case if the accretion flow starts out hot, as is likely in elliptical galaxies. Another possibility is that the accretion flow starts out cool, so it is described at large radii by a standard thin disk. This may be the case for spiral galaxies. If the accretion rate is not too high, this thin disk will be truncated and replaced by a hot accretion flow at a transition radius (Yuan 2007). This transition radius is then the outer boundary of the hot accretion flow, r_out. Detailed modeling shows that it is a function of the mass accretion rate or, equivalently, the luminosity of the system (Yuan & Narayan 2004). When the luminosity is very low, say L ≲ 10⁻⁶L_Edd, as in the case of Sgr A*, r_out ∼ r_Bondi ∼ 10⁵r_s. If the luminosity is higher, L ≳ 10⁻⁴L_Edd, r_out may be in the range from several tens to several thousands of r_s.

From the above assumptions on parameters, we conclude that the power of the jet and radiation is usually much stronger than that of the outflow, $\dot{E}_{\rm jet}\gg \dot{E}_{\rm rad} \gg \dot{E}_w$. But note that this does not mean the mechanical feedback of the outflow is not important compared to the jet. Since the jet is well collimated, it just drills through the surrounding gas with little deposition of energy and momentum within the galaxy. Rather, the jet may act as the dominant feedback mechanism for galaxy clusters. For the evolution of a single galaxy, the solid angle of the outflow is very large so that it can fully interact with the ISM. It has been shown that the momentum feedback of the outflow dominates the growth of the central black hole (Ostriker et al. 2010). In Section 7, we will discuss two additional examples of such interactions.

Outflows from AGNs are generally detected via absorption lines. If we hope to explain observations using the model proposed in this paper, we may ask if the hot accretion flow model can be applied to luminous AGNs and how the absorption line can be produced if the temperature of the flow is as hot as virial. Before we answer these questions, we first briefly introduce the accretion model for black hole binaries. An individual black hole X-ray binary comes in several states, mainly hard, soft, and steep power-law (or very high) states. These three states are associated with continuous jets, no jets, and episodic jets, respectively (Fender 2006). The luminosity of the hard state spans a large range, from very low values up to ∼10%L_Edd or even 30%L_Edd (e.g., Zdziarski & Gierliński 2004). The soft state, on the other hand, only exists above ∼2%L_Edd. Sources with 2%L_Edd ≲ L ≲ 10%L_Edd are not certain because of the "hysteresis" (Zdziarski & Gierliński 2004). The soft state is described by a standard thin disk extending to the innermost stable circular orbit, while the hard state is described by an inner hot accretion flow, which contributes most of the bolometric luminosity of the hard state, and an outer truncated thin disk (e.g., Narayan 1996; Esin et al. 1997; Yuan et al. 2005; Yuan & Zdziarski 2004; Oda et al. 2012; see Narayan 2005; McClintock & Remillard 2006; Done et al. 2007 for reviews). The transition radius between the hot and cool accretion flows (r_tr) is a function of accretion rate or luminosity (Yuan & Narayan 2004). The model of the very high state is still unclear.

Compared to black hole X-ray binaries, the accretion model for AGNs is far less certain. Since the physics of accretion is irrelevant with black hole mass, by analogy to the black hole binaries, it is reasonable to assume that low-luminosity AGNs with L up to (2–10)%L_Edd should correspond to the "hard state" and are powered by a hot accretion flow. This picture has received observational support (see Yuan 2007; Ho 2008 for reviews). The model of luminous AGNs is more complicated and may not be easily associated with the standard thin disk. This is because, on the one hand, some luminous AGNs may correspond to the very high state whose model is unknown; on the other hand, the observational evidence for the thin disk is not as good as in the soft state of black hole X-ray binaries. Moreover, it seems that the hard X-ray emission of luminous AGNs, which cannot be explained by a standard thin disk, is much stronger than that in the soft state of black hole X-ray binaries. This indicates that hot gas must exist. One suggestion is to invoke a corona sandwiching the thin disk (e.g., Galeev et al. 1979; Haardt & Maraschi 1991). If the dynamics of the corona can be described by a hot accretion flow, the study we present in this paper will be applicable.

Almost all works, including this one, assume that the hot accretion flow is one-phase. Hot accretion flow is likely thermally unstable under short-wavelength perturbations (Kato et al. 1997; Wu 1997; Yuan 2003). Yuan (2003) found that when the accretion rate of the accretion flow $\dot{M} \gtrsim \alpha ^2 \dot{M}_{\rm Edd}$, the timescale of growth of perturbation will be shorter than the accretion timescale; consequently, cold dense clumps may be formed within the hot phase. So we expect that when $\dot{M} \gtrsim \alpha ^2 \dot{M}_{\rm Edd}$, the accretion flow and outflow should be two-phase. In this case, the presence of an absorption line is expected. If α = 0.1, the corresponding $\dot{M}\sim 0.01 \dot{M}_{\rm Edd}$ and L ∼ 10⁻³L_Edd.

7.2.1. The Fermi Bubble in the Galactic Center

The first example is the two "Fermi bubbles" revealed by Fermi-LAT, extending 50° above and below the Galactic center (Su et al. 2010). The bubbles have approximately uniform surface brightness with sharp edges and are symmetric about the Galactic plane. The bubbles are spatially correlated with the hard-spectrum microwave excess known as the Wilkinson Microwave Anisotropy Probe (WMAP) haze. The initial analysis in Su et al. (2010) indicates that the bubbles are most likely created by some large episode of energy injection in the Galactic center. The age and energy of the bubble have been roughly estimated by Su et al. (2010) to be 10⁷ yr and E ∼ (10⁵⁴–10⁵⁵) erg, respectively.

Several models have been proposed to explain the origin of the Fermi bubble. They usually associate the bubble with recent activity of the supermassive black hole Sgr A* (e.g., Guo & Mathews 2012; Zubovas & Nayakshin 2012). Guo & Mathews (2012) proposed that the bubble was inflated by a past jet in Sgr A*. They performed some numerical simulations and roughly reproduced the morphology of the bubble. The problems with this scenario are that the jet is more likely an outflow since it is not collimated, and the required mass flux in the jet is super-Eddington, which may be unreasonable. In another model, Zubovas & Nayakshin (2012) required that the bolometric luminosity of Sgr A* in the past was very large, approaching L_Edd, i.e., Sgr A* was a very luminous quasar. In this case, a powerful outflow could have been produced by the radiation pressure, which then inflated the Fermi bubble. We note, however, that the required high luminosity of Sgr A* is in conflict with the study of Totani (2006), who argued based on other observations that Sgr A* in the past should still have been within the regime of ADAF, although the mass accretion rate was three to four orders of magnitude higher than the present value.

Our scenario is that the bubbles are inflated by the outflow from an ADAF. Shocks are formed when the outflow interacts with the ISM, and electrons are accelerated to relativistic energies in the shock front. These electrons emit synchrotron and inverse Compton radiation by scattering with cosmic microwave background photons. The former explains the WMAP haze while the latter is responsible for the Fermi bubbles. We now estimate whether the outflow can reasonably power the bubble. Taking E ∼ 10⁵⁴ erg and 10⁷ yr as an example, the required bubble power is 3 × 10³⁹ erg s⁻¹. From Equation (31), the required accretion rate is

$$\begin{equation} \dot{M}_{\rm out}\sim 10^{-5}r_{\rm out}\dot{M}_{\rm Edd}. \end{equation} \tag{ 36 }$$

Here $\dot{M}_{\rm Edd}\equiv 10L_{\rm Edd}/c^2\sim 9\times 10^{-2}\,{M}_{\odot }\,{\rm yr}^{-1}$ is the Eddington accretion rate of Sgr A*. Note that according to Yuan & Narayan (2004) r_out is a function of $\dot{M}_{\rm out}$. A reasonable set of parameters would be r_out ∼ 10³r_s and $\dot{M}_{\rm out}\sim 10^{-2}\dot{M}_{\rm Edd}$. The current accretion rate of Sgr A* at the Bondi radius (r_out ∼ 10⁵) is $\dot{M}\sim 10^{-6}\,{M}_{\odot }\,{\rm yr}^{-1}\sim 10^{-5}\dot{M}_{\rm Edd}$ (Yuan et al. 2003). So we require that the past activity of Sgr A* was about 1000 times higher in terms of the accretion rate. This is in good agreement with the estimate of Totani (2006). Numerical simulation work is in progress (G. Mou et al. 2012, in preparation).

7.2.2. M 87

One popular model for producing AGN winds is radiation driving. However, detailed studies of individual sources often exhibit some problems. In the following we present several examples.

NGC 3783. This is a Seyfert 1 galaxy, with mass of the black hole ∼(3 ± 1) × 10⁷ M_☉, bolometric luminosity L_bol ∼ 3 × 10⁴⁴ erg s⁻¹ ∼ 0.1L_Edd. Chelouche & Netzer (2005) carefully calculated the physical properties of the highly ionized outflow and compared their theoretical calculation with the Chandra data. The main results they obtained are: (1) the highly ionized outflow in this source is not driven by the radiation pressure (they suggested a thermal pressure gradient); (2) the rate of outflow is ${\sim} 0.01\hbox{--}0.1 \,{M}_{\odot }\,{\rm yr}^{-1}\approx 0.02\hbox{--}0.2 \dot{M}_{\rm Edd}$ ($\dot{M}_{\rm Edd}\equiv 10L_{\rm Edd}/c^2\approx 0.5 \,{M}_{\odot }\,{\rm yr}^{-1}$); (3) the velocity of the outflow is ∼1000 km s⁻¹; (4) the global covering factor of the flow is ∼20%.

NGC 4151. Kraemer et al. (2005) presented a detailed analysis of the intrinsic X-ray absorption in this source using Chandra data and found the presence of a very highly ionized outflow in addition to lower ionization gas. The authors pointed out that the outflows are so highly ionized and the luminosity of these galaxies is only ∼4%L_Edd so they are unlikely to be accelerated by radiation pressure. They pointed out that this kind of high ionization outflow is very common among Seyfert galaxies and its existence is the key to understanding the origin of mass outflow.

3C 111. Tombesi et al. (2011b) detected blueshifted absorption lines in 3C 111 based on their Suzaku observations. The mass of the black hole is (2–30) × 10⁸ M_☉ and the bolometric luminosity is L_bol ≈ 8 × 10⁴⁵ erg s⁻¹ ∼ (2–30)%L_Edd. Detailed modeling indicates that the location of absorption material is constrained at <0.006 pc ∼ (20–300)r_s and the velocity and mass outflow rate of the outflow are ∼0.1c and $\dot{M}_{\rm out}\sim 1\,M_{\odot } \,{\rm yr^{-1}}$, respectively. The latter is similar to the mass accretion rate $\dot{M}_{\rm acc}\sim L_{\rm bol}/\eta c^2$ (for η = 0.1). The momentum flux of the outflow is comparable to the momentum flux of radiation, $\dot{M}_{\rm out}\sim L_{\rm bol}/c$. However, the Thomson scattering optical depth of the outflow is found to be very small, τ ∼ 0.05. Therefore, for the radiation mechanism to work, we have to look for other opacities. This seems to be quite difficult given the high ionization of the outflow, similar to NGC 4151. In addition, while the rather small upper limit on the location of the outflow indicates its accretion flow origin and rules out other models, this location is more than one order of magnitude smaller than that predicted in the radiation driving model of Murray et al. (1995).

Low-luminosity AGNs. Outflow can be more easily detected in luminous AGNs partly because these sources are suitable for high-resolution spectroscopy. Recently, outflows have also been detected in low-luminosity AGNs. Crenshaw & Kraemer (2012) investigated the outflow from a sample of nearby AGNs, focusing on determining their mass outflow rate and the kinetic luminosity. Although they have to restrict their sample to apparently luminous AGNs and they do not include ultrafast outflow, of the ten nearby Seyfert 1 galaxies in their sample, six sources still have their bolometric luminosities below 5%L_Edd. Their detailed study of the UV and X-ray absorbers clearly shows that a strong outflow exists in these sources. The bolometric luminosity of one source, NGC 4395, is even as low as 10⁻³L_Edd. These low-luminosity AGNs are described by an ADAF rather than a standard thin disk. Thus it is questionable whether the radiation driving model works for them. Crenshaw & Kraemer found that the mass outflow rate exceeds the mass accretion rate by a factor of 10–1000. The kinetic luminosities of the outflows are approximately 0.5%–5% of their bolometric luminosities for half of the moderate-luminosity AGNs in their sample.

Now we estimate whether the above examples can possibly be explained by our outflow model. We want to emphasize that such estimations are necessarily very crude.

NGC 3783. During the propagation of outflow, the mass flux may increase and the velocity may decrease because of the contamination by the ISM, but the momentum flux should be conserved. Assuming an X-ray luminosity L_x ∼ f_xL_bol ∼ 0.1f_xL_Edd and assuming that the hard X-ray emission is produced by a hot corona which can be modeled by a hot accretion flow, the required $\dot{M}(10r_s)\sim 0.1 \eta ^{-1}f_x\dot{M}_{\rm Edd}$ (here η is the radiative efficiency of the hot accretion flow). So $\dot{M}_{\rm out}(r_{\rm out})=0.1\eta ^{-1}f_x\dot{M}_{\rm Edd}(r_{\rm out}/10r_s)^{0.5}$, $v_{\rm term}\sim 0.5 v_k(r_{\rm out})\sim c/2\sqrt{2r_{\rm out}/r_s}$. The momentum flux is then $\dot{p}_w\sim 0.02f_x\eta ^{-1}\dot{M}_{\rm Edd}c \sim 2\times 10^{-3}\dot{M}_{\rm Edd}c$ for f_x = 10⁻² and η ∼ 0.1. This is consistent with the observed $\dot{p}_w\sim (0.7\hbox{--}7)\times 10^{-3}\dot{M}_{\rm Edd}c$. In addition, assuming r_out ∼ 100r_s, the outflow rate is $\dot{M}_{\rm out}(r_{\rm out})\sim 0.03 \dot{M}_{\rm Edd}$, which also satisfies the requirement that the outflow rate produced from the hot accretion flow should be smaller than that observed because of contamination by the ISM.

3C 111. The location of the absorption material is very close to the black hole, so we expect little contamination of the outflow. First, there is obviously no problem with our model producing outflows from a radius as small as ∼100r_s. Second, assuming r_out ∼ 100r_s (since luminosity is relatively high), since (100r_s/10r_s)^0.5 ∼ 3, this explains why $\dot{M}_{\rm out}(100r_s)$ is not so different from the accretion rate at 10r_s. Third, the velocity of outflow is ∼0.5v_k(100r_s) ∼ 0.1c, in good agreement with observations. Finally, the approximate equality of the momentum flux between radiation and outflow is exactly what we expect (Equation (28)).

Low-luminosity AGNs. First, for low-luminosity AGNs, a reasonable assumption is that r_out ∼ 10³–10⁵r_s, depending on the exact value of the accretion rate and the temperature of the fueling material. In this case, the ratio of the mass outflow rate and accretion rate (at 10r_s) is $\dot{M}_{\rm out}(r_{\rm out})/\dot{M}_{\rm in}(10r_s)\sim (r_{\rm out}/10r_s)^{0.5}=10\hbox{--}100$. Given further the contamination by the ISM, this explains the observed ratio of the outflow rate and accretion rate. Second, from Equation (34), for η ∼ 0.05, we have $\dot{E}_w\sim 1\%\dot{E}_{\rm rad}$, again in good agreement with observations.

A number of high ionized winds have been detected in black hole and neutron star X-ray binaries in recent years (see references in Neilsen & Homan 2012). GRO J1655-40 is one of the best examples. This source was observed by Chandra twice during its 2005 outburst. The second observation was made when the source was in the soft state. A series of absorption lines from a dense and highly ionized wind have been revealed. Detailed spectral analyses and theoretical studies have been performed by different groups and confirmed repeatedly that this outflow was driven not by radiation or thermal pressure, but likely by a magnetic mechanism (Miller et al. 2006, 2008; Kallman et al. 2009; Luketic et al. 2010). Another observation was made 20 days earlier than this one, when the source was just at the beginning of the transition from hard to soft state. This observation only detected one absorption line (Fe xxvi). Neilsen & Homan (2012) analyzed the data and concluded that the outflow in the hard state also cannot be driven by radiation pressure because the ionization parameter is too high. Their photoionization modeling indicates that the difference between the Chandra spectra in the two observations cannot possibly be explained by a change in the ionizing spectrum. Instead, the properties of the wind in the two states must be changed. This is reasonable since the accretion flow in the two states is different.

Most recently, King et al. (2012) have investigated the kinetic power of outflows and jets across the black hole mass scale. They found that both powers scale with the radiative power. More importantly, they found that the scaling relations are consistent with each other within error bars, although the normalization of the jet relation is higher than that of the outflow. They argued that such a formal consistency suggests a common origin mechanism for the outflow and jets. Since it is generally believed that jets are of magnetic origin, this would imply that outflow is also of magnetic origin.