Global structure and dynamics of slowly rotating accretion flows

We consider transonic, steady, non-self-gravitating, viscous accretion flows with a finite and small angular momentum. Such flows are nearly spherical. Thus, the height-integrated differential equations can describe them well. We assume all of variables are functions only of radius and the viscous accretion flow is in steady-state and axisymmetric ($\partial/\partial t=\partial/\partial\phi=0$). The assumptions presented here are identical to those described by Narayan & Fabian (2011), except we use the assumption of vertical hydrostatic equilibrium to compute the vertical thickness of the accretion disk. We adopt the cylindrical coordinate system, i.e., $(r,\phi,z)$. Under the assumptions mentioned above, the one-dimensional vertically integrated, steady-state continuity equation read as follows

where $\rho$ is the density of the gas, $v$ is the radial inflow velocity, and $H$ denotes the vertical scale height of the gas, respectively. Following Narayan et al. (1997), the hydrostatic assumption is adopted to describe the entire flow, including the subsonic and supersonic regions¹¹1In almost all one-dimensional flow models, the vertical half-thickness of the flow is approximated using the vertical hydrostatic equilibrium, although this assumption is not very accurate close to the horizon where the accretion flow becomes supersonic.. Hence, the flow half-thickness will be written as

In the above equation, $\Omega_{\rm K}$ denotes the Keplerian angular velocity and $c_{\rm s}\equiv(p/\rho)^{1/2}$ is the isothermal sound speed and $p$ represents the gas pressure, respectively.

In analytical works, the mass accretion rate is generally assumed to be constant. So here, with the integration of equation 2, the mass accretion rate $\dot{M}$ is determined as follows

In this study, the effects of general relativity are modeled with the pseudo-Newtonian form of potential Paczynsky & Wiita (1980) as,

where $r_{\rm s}=2GM/c^{2}$ is the Schwarzschild radius of the black hole of mass $M$ ($c$ represents the speed of light and $G$ is the gravitational constant)²²2Note that in section 4.2, the galaxy potential is also included to the total potential, $\Phi$.. The Keplerian angular velocity $\Omega_{\rm K}$, as well as the specific angular momentum $l_{\rm K}$ at the radius of $r$, can be defined respectively by,

and

In a steady-state, the radial momentum equation can be expressed as

where $\Omega$ is the angular velocity of the gas on the disk mid-plane. We mainly focus on a condition of $\Omega^{2}\ll\Omega_{\rm K}^{2}$, meaning the centrifugal acceleration is extremely weak compared to the gravitational one. In a real accretion flow, the angular momentum is transferred by Maxwell stress associated with magnetohydrodynamic (MHD) turbulence driven by magneto-rotational instability (MRI; see Balbus & Hawley (1991)). In this study, we mimic the effect of MRI by adding viscous terms in the momentum and energy equations (see, e.g., Yuan et al. (2012b)). Furthermore, the MHD numerical simulations of accretion flow show that the $r\phi$ component of the magnetic stress is the most dominant component of the stress tensor (see e.g., Brandenburg et al. (1995); Hawley et al. (1995)). The kinematic viscosity coefficient is evaluated with the well-known $\alpha$-prescription Shakura & Sunyaev (1973) as

Where $\alpha$ is a constant called Shakura–Sunyaev viscosity parameter. The steady-state angular momentum equation in the presence of $r\phi$ component of the viscous stress tensor takes the form

Substituting equation (9) into equation (10) and integrating of it yields

where the integration constant $j$ represents the specific angular momentum per unit mass, which is swallowed by the black hole³³3The quantity $j$ is one of the two eigenvalues in this study and will be determined self-consistently through our physical boundary conditions. Eventually, the energy conservation equation can be expressed by the balance between the viscous heating, $Q^{+}$, the radiative cooling, $Q^{-}$, and the radial advection heat transport, $Q^{\rm adv}$ which is expressed as

As mentioned, we are interested in the flow in hot accretion mode, where the radiative cooling term is negligibly small compared to the viscous heating ones. Hence, we can write the energy equation of the flow in one dimension as

Where the specific internal energy of the gas is given by

And $\gamma$ is the ratio of specific heat of the gas, set to be $\gamma=5/3$. In the above equation, following Narayan & Yi (1994), the advection parameter is introduced as $f=Q^{\rm adv}/Q^{+}$. Consequently, the cooling term takes the $(1-f)$ factor of the heating rate, which is an excellent approximation for inefficient radiative accretion flows (RIAFs). We set $f=1$ throughout this paper (full advection case).

The flow must pass a critical point (sonic radius) in transonic accretion. The critical radius needs a regularity condition that the flow must satisfy there. Substituting equations (4) and (11) into equation (13) the derivative of sound speed will be written as

Using the above equation for $\mathrm{d}c_{\rm s}/\mathrm{d}r$ as well as the equation (4), the radial component of the momentum equation can be achieved in the form

Here, the numerator and denominator will be given respectively as,

with

The critical radius is where the flow speed equals the characteristic speed of acoustic perturbations. This is not always the same as where the flow speed is equal to the speed of sound Kato et al. (2008). In equation (17), $C_{\rm s}$ is the adiabatic sound speed which equals the characteristic speed of acoustic perturbations and is defined as

Where $c_{s}$ is the isothermal sound speed. The Mach number is given by

In the next section, we will explain the boundary conditions as well as the numerical technique for solving the system of equations.

In our accretion flow model, viscosity and angular momentum play a crucial role in the outer subsonic region. This region starts from sonic radius $r_{\rm c}$ to the outer radius, which is at the Bondi radius in this study. To find the solution in this region, we must solve the set of differential equations with the boundary value problem method. Our differential equations, equation (11), (14) and (15) consist of three first-order ordinary equations for three radial dependent variables $v(r)$, $\Omega(r)$ and $c_{\rm s}(r)$ with two unknown parameter $j$ and $r_{\rm c}$. Due to the stiffness of our differential equations, we use a relaxation method to solve them (see, Press et al. (1992)). Here, we need five boundary conditions on both ends. The problem is that only one boundary abscissa, i.,e $r_{\rm out}$ can be specified, while the other boundary $r_{\rm c}$ is the eigenvalue and needs to be determined. Mathematically, this case is called a free boundary problem. We reduce this problem to the standard point by introducing a new dependent variable as

where

We also define a new independent variable $t$ by setting,

Due to the presence of the new independent variable $t$, we have two boundaries specified so that we can solve the set of differential equations with the boundary value problem method.

As we mentioned before, the motion of the flow near the BH is physically significant. When the matter passes the sonic point, $r=r_{\rm c}$, the accretion flow tends to be supersonic. The transition from a subsonic flow to a supersonic occurs in a narrow region in the radial direction, and viscosity decreases in this region. Following NF11, we assume the viscosity vanishes in this region (see Narayan (1992), Kato & Inagaki (1994); Kley & Papaloizou (1997) for more details). Thus, we drop all terms corresponding to the viscosity from our differential equations and set $\alpha=0$. By ignoring $\alpha$ from the differential equations, the equations simplify to become a set of algebraic equations. Consequently, the specific angular momentum, $\ell_{\rm in}$, entropy, $\rm{S}_{\rm in}$ and Bernoulli parameter, $\mathcal{B}$ are constant in this zone and evaluated as:

and

We first compute the value of $\ell_{\rm in}$, $\mathrm{S}_{\rm in}$, $\mathcal{B}$, and $\dot{M}$ at the sonic point, $r=r_{\rm c}$ and then directly evaluate the solution from the above algebraic equations for the inner supersonic region. It is worth mentioning that in terms of MHD case, studying the inner region of the accretion flow close to the central black hole is important for understanding the jet ejection and also in the magnetically arrested disk, MAD (see e.g., Kogan & Ruzmaikin (1976)).

In this section, we investigate the effects of the outer boundary condition (OBC) on the dynamics and the structure of slowly rotating accretion flows. We find that OBC plays an important role. In many accretion flows, external boundary conditions like angular momentum, temperature, and density significantly influence flow properties, such as velocity and the mass accretion rate. In addition, the complex astrophysical environments such as interstellar mediums in the nuclei of galaxies, create different modes for the accreting gas at the outer boundary, like the temperature and angular momentum (Yuan (1999), Yuan et al. (2001)). So investigating the role of the OBC in accretion flows around the black holes is particularly important due to the complexity of astrophysical environments.

In the following, we focus on the mass accretion rate of the accretion flow, mainly its dependence on the angular momentum of gas at the outer boundary. The dependence of the mass accretion rate on boundary conditions can only be addressed by constructing global solutions.

In accretion systems, the most important parameter is the mass accretion rate, which the gas supply environment can determine (e.g., gas temperature, density, and angular momentum), directly influencing the boundary conditions. One of the essential scales which is being utilized the accretion rate is the Eddington accretion rate $\dot{M}_{E}=10L_{E}/c^{2}=1.39\times 10^{18}(M/M_{\odot})gs^{-1}$. If the mass accretion rate is less than $2\%$ of $\dot{M}_{E}$, we have hot accretion flow (Narayan & Yi (1994)). For $2\%\dot{M}_{E}<\dot{M}<\dot{M}_{E}$ the mass accretion rate consistent with a thin accretion disk. When the mass accretion rate is in the range $\dot{M}\gg\dot{M}_{E}$, it consistent with thick accretion disk and for slim accretion disk models, accretion rate is $\dot{M}\approx\dot{M}_{E}$ (Abramowicz et al. (1988)).The other important scale is the mass accretion rate in spherical accretion, known as the Bondi accretion rate. Bondi (1952) found that the mass accretion rate for the Bondi model is directly determined by the boundary conditions far from the accreting object, including temperature and density of surrounding gas. The Bondi rate is

The constant $\Lambda(\gamma)$ is $0.25$ for $\gamma=5/3$. The Bondi radius is as $r_{B}=GM/c_{\rm out}^{2}$. In the Bondi radius, gravity overcomes the gas pressure and begins to fall onto the black hole. $\dot{M}_{B}$ in our unit ($r_{g}=c=\rho_{\rm out}=1$) is equal to $3.65\times 10^{8}$, while $T_{\rm out}=6.5\times 10^{6}$K. Both units of mass accretion rate ($\dot{M}_{E}$ and $\dot{M}_{B}$) have applications in the present study.

Our results show that the mass accretion rate in slowly rotating accretion flow is lower than the Bondi rate, which is compatible with NF11 results. In the present study, we consider the mass accretion rate as a constant parameter at every radius; in other words, our model has no mass outflow. The mass accretion rate depends on angular momentum at the outer boundary and for $\ell_{B}>\ell_{ms}$ it value is smaller than the Bondi accretion rate and for $\ell_{B}<\ell_{ms}$ approximately equal with $\dot{M}_{B}$.

In Figure 2, the mass accretion rate is shown as a function of angular momentum at the outer boundary (Panels (a), (c), (d)). It is obvious that when the angular momentum flow is low, the mass accretion rate is very close to the Bondi rate, but when the angular momentum flow is high, the rate is lower than the corresponding Bondi rate. Narayan & Fabian (2011) have used a simple approximation for the thickness of flow. They have assumed that flow geometrically is thick and $H=r$. In this work, we have set $H\neq r$, and our results are compared to NF11 in panel (a) of Figure 2. Surprisingly this correction leads to a change in the value of the mass accretion rate. Figure 2, panel (a) shows the variation of mass accretion rate as a function of parameter $L$, angular momentum at the outer boundary. As we can see from panel (a), the mass accretion rate decreases significantly when we consider a more realistic mode ($H\neq r$) instead of a geometrically thick disk ($H=r$). Indeed keeping the assumption of the vertical hydrostatic equilibrium causes the mass accretion rate to be reduced. In Panel (b) of Figure 2, the mass accretion rate is shown as a function of viscosity $\alpha$. Our solutions contain 19 different $\alpha$ in the range $0.01\leq\alpha\leq 0.45$ and $L>1$ (Disk-like types). As it is obvious, there is a linear relationship between the mass accretion rate and parameter $\alpha$. This linear relation can also be seen in the case of ADAFs (Narayan & Yi (1994), NF11). Increasing $\alpha$ helps to accretion process and increases the mass accretion rate onto the black hole. This case study well confirms the importance of viscosity in accretion flows. In the following, we plan to investigate the effects of temperature at the outer boundary on the structure of slowly rotating accretion flows.

A halo of hot gas surrounds most large elliptical galaxies, and its temperature range is 1-10 million Kelvin. (It is in the range of the virial temperature) (Knapp (1998), Roshan & Abbassi (2014)). In this study, we assume that gas accretes from the edge of a hot gas located in the Bondi radius. Beyond the Bondi radius, there is an interstellar medium, which is in the range of $10^{6}-10^{7}$ K in real position. Our focus is on how the outer boundary gas temperature affects the accretion flow structure. Figure 2, panel (c) illustrates the variation of mass accretion rate $\dot{M}$ with the angular momentum of the outer boundary for different temperatures at the outer boundary. We plot the mass accretion rate in a unit of $\dot{M}_{E}$ versus parameter $L$. Since the Bondi accretion rate is highly dependent on the temperature of the embedded cloud, $T_{\rm out}$, it seems more reasonable to use the Eddington accretion rate, which under any circumstance is constant. Panel (c) in Figure 2 shows that by increasing the temperature of the outer boundary, the mass accretion rate slightly decreases. We consider three different values for temperature at the outer boundary as: $T_{\rm out}=6.5\times 10^{6}$ K, $T_{\rm out}=8.5\times 10^{6}$ K and $T_{\rm out}=1.2\times 10^{7}$ K. We find that regardless of $T_{\rm out}$, the mass accretion rate $\dot{M}$ perceptibly decreases as the angular momentum $L$ increases. It is clear that for a given $L$, the mass accretion rate can be very different with the changing temperature at the outer boundary. This result has also been reported in simulations (BY19). Panel (d) of Figure 2 shows the effect of the potential of the galaxy on the mass accretion rate. A very slight change in the mass accretion rate is observed when galaxy potential is included in our solutions (see section 4.2 for more detail).

The angular momentum at the outer boundary affects the structure of the flow. Here we tend to focus on the influence of angular momentum $L$ on the accretion of gas onto a black hole. The main result is to find two separate regimes, quasi-spherical accretion for small $L$ and Disk-like accretion for large $L$, respectively. There is no smooth transition between the two types of flows. For the first time, (Abramowicz & Zurek (1980)) found a discontinuity in their study of the adiabatic accretion flow with a constant specific angular momentum between Disk-like and Bondi-like types. In this study, concerning the previous works, we corroborated that this transition exists even though the viscosity of flow is considered. The following brief description is of the two types.

Disk-like type : In this flow type, $\ell_{B}>\ell_{ms}$ the sonic point occurs roughly in small radii. When the specific angular momentum is high, the centrifugal force plays a dominant role rather than the gravitational force. The gas becomes supersonic only after passing through a sonic point near the horizon. Bondi-like type : When $\ell_{B}<\ell_{ms}$, accretion flow is quasi-spherical accretion. In such a case, centrifugal force is too weak to gravity force, so the flow can not be in hydrodynamical equilibrium. This accretion flow is similar to spherical accretion when the flow has zero angular momentum. Because of this, Abramowicz & Zurek (1980) named it Bondi-like accretion. We prefer to substitute $H\simeq r$ in conservation equations of this type. The low specific angular momentum of the accretion gas (Bondi-like type) is a new state, distinguished by its large sonic radius.

Figure 3 displays the radial variations of the Mach number with different specific angular momentum at the outer boundary $r_{B}$. At the sonic point, the Mach number equals 1. The right panel shows the Mach number for the two types of accretion flow, which corresponds to the relatively low and high angular momentum at the outer boundary, respectively. Solutions are presented for $L=85$ (Disk-like type) and $L=0.11$ (Bondi-like type) for $\alpha=0.1$ that their sonic radii are (purple line) $\sim 3r_{s}$ and (pink line)$\sim 415r_{s}$, respectively. Solutions for the subsonic region are included in this panel. The left panel shows variation of Mach number for three values of $L$ when $\alpha=0.05$. For $L=15$ , location of sonic point is larger than $L=85$. Indeed, the sonic point moves to a larger radius as $L$ decreases. In this panel, solutions include both subsonic and supersonic regions. Figure 4 illustrates the global solutions of slowly rotating accretion flows for the different parameters of $L$. Solutions are presented for two types of the accretion flow, the left panel according to $L\geq 1$ ($L=100,24,11,5,1.6$), which is equivalent to a Disk-type accretion flow, and the right panel according to $L<1$ ($L=0.6,0.3,0.11,0.05,0.012$), which is equivalent to a Bondi-type accretion flow. Solutions clearly illustrate how angular momentum at the outer boundary affects the angular velocity. This figure shows that the angular velocity decreases as the angular momentum changes at the outer boundary for every two types of accretion. To a better understanding of the difference between the angular velocity of our solutions and the Keplerian angular velocity, we plot the dot-dash line that corresponds to the Keplerian angular velocity, which shows our solutions are sub-keplerian. This result emphasizes the importance of considering the outer boundary conditions of the accretion flow.

As a final point, we must note that the effect of OBCs is subject to the location of the outer boundary. When $r_{\rm out}$ is large, $T_{\rm out}$ has a limited range, therefore its effect is minimized. When applying one-temperature plasmas, the effect of variations in $T_{\rm out}$ on the structure of flows is reduced. However, the effect of $\Omega_{\rm out}$ is always evident. Typically, when $r_{\rm out}$ is small, the effect of OBCs is most apparent (Yuan et al. (2001)).

Recent studies have examined numerous approaches to model a realistic approximation for the dynamics of black hole accretion flows. One of the works involves considering the gravitational potential of host galaxies. For example, the study carried out by Mancino et al. (2022) incorporated the effects of the gravitational field of the host galaxy on the flow of Bondi accretion. In this section, we consider the contribution of the gravitational potential of both the central black hole and the stars of the host galaxy. At a larger scale (near $\sim$ 10 parsecs), in addition to the potential of the BH, the gravitational potential of stars of the host galaxy will become essential and should be included (Yang & Bu (2018b)). This is while previous studies neglected the gravitational force of stars at parsec scale NF11, NKH97. So the total gravitational potential is given by

The potential of the stars of the host galaxy is

Where $\sigma$ is the velocity dispersion of stars and $C$ is a constant (Bu et al. (2016)). The term $\phi_{Galaxy}$ will add to radial momentum equation 8 and can effect on dynamics of flow. In elliptical galaxies with central black hole to mass $M_{BH}=10^{8}M_{\odot}$ is common the velocity dispersion be about 150 - 250 $km$ $s^{-1}$. Here, we assume the velocity dispersion of stars is a constant of radius, which is $\sigma=200$ $km$ $s^{-1}$ (Kormendy & Ho (2013), Greene & Luis (2006)). The stellar velocity dispersion has a crucial contribution to the gravitational force distribution of the galaxy ( Yang & Bu (2018b)). It has been found in previous studies (Bu et al. (2016), Yang & Bu (2018b), Sun & Yang (2021)) that the gravitational effects of stars of the galaxy can become significant when the radius is beyond $1$ pc. Because of this, in our study, the effect of gravity of stars is ignored close to the black hole (the supersonic region $r\leq 1$pc). Therefore, the algebraic equations in Section 3.2 do not change. Figure 2, Panel (d) illustrates variations of the mass accretion rate ($\dot{M}/\dot{M}_{B}$) as a function of the angular momentum at the outer boundary ($L$). There is a clear difference when we turn on the effect of galaxy potential. The star’s gravity can slightly enhance the mass accretion rate onto the black hole. This finding is in agreement with simulations of YB19 and Yang & Bu (2018b). As shown in Figure 2, Panel (d), the gravitational potential of the galaxy has a greater effect on the mass accretion rate for $L\leq 20$ (roughly Bondi-like type). This could be one of the reasons most recent Bondi solutions are taking into account the galaxy’s potential alongside the black hole’s potential. (Mancino et al. (2022), Samadi et al. (2019), Ciotti & Pellegrini (2017)). Our numerical solutions with or without nuclear stars’ gravity are summarized in Tables 1. In column 1 of Tables 1, we show the values of angular momentum at the outer boundary (dimensionless). It should be noted that in this paper, we study low angular momentum accretion. In Table 1, we have two types of accretion, Disk-like, and Bondi-like. For both states, the temperature at the outer boundary is $T_{\rm out}=6.5\times 10^{6}$ K. In Table1, the temperature and gas properties in our model are the same as those in the NF11 model. The values of mass accretion rate of numerical solutions are listed in column 6 of Tables 1. It appears that the galaxy potential does significantly influence the mass accretion rate. Considering the star gravity slightly increases the mass accretion rate in our model. Our results in columns 4 and 5 indicate that the values of $r_{c}$ and $j$ do not rely much on galaxy potential and only slightly increase the radius of the sonic point. We also observe that by maintaining hydrostatic equilibrium ($H\neq r$) within our model, the eigenvalues such as the sonic radius and $j$ do not change significantly. When we compare our results with those of Narayan & Fabian (2011), we see that there is only a significant change in the value of the mass accretion rate.

Bu & Yang (2019) performed the two-dimensional simulations to investigate slowly rotating accretion flows irradiated by an LLAGN at parsec scales. They found the fitting formula based on temperature and density at the parsec scale, which could predict the luminosity of observed low-luminosity AGNs. As Bu & Yang (2019) investigated accretion flow with low angular momentum and considered the potential of the galaxy in their study, we can compare our results with theirs. To compare this work with our model, we assume the black hole mass equal to $10^{8}M_{\odot}$ ($M_{\odot}$ is solar mass) and the density at the outer boundary $\rho_{0}=10^{-24}$ $g$ $cm^{-3}$.

Figure 5 compares the results of our numerical solutions and previous studies (numerical solution NF11, simulation YB19), which shows the variations of black hole accretion rate as a function of temperature at the outer boundary. In our study, the trend in the mass accretion rates versus temperature is similar to YB19. It is clear that for a given constant gas density at the outer boundary, the mass accretion rate will become smaller by increasing of temperature at the outer boundary. There is a significant difference between our results with simulation because we do not consider the effect of outflow and radiation in our study. In this Figure, the green circles correspond to the values of the mass accretion rate of solutions derived from the model of Narayan & Fabian (2011) and the orange triangle represents our numerical solutions by considering the effect of galaxy potential. The green squares are according to the time-averaged values of the mass accretion rate of simulations in BY19, and the error bars on squares correspond to the change range of simulations due to fluctuations. In comparison to YB19 simulations, our results deviate slightly since the effect of radiation and outflow has been ignored. Hence all the gas captured at the outer boundary can fall into the black hole. The dashed line shows the Bondi accretion rate by assuming $\gamma=5/3$. The solid line shows the black hole accretion rate according to the fitting formula (Equation of (5) in BY19). It should be pointed out that the simulations of YB19 are calculated by considering the stars’ gravity. As a result of the presence of outflow in natural systems, the black hole accretion rate is significantly reduced. Since the Bondi solution does not involve any rotation or wind and is quite simple, all of our numerical solutions and simulations of BY19 are lower than the Bondi accretion rate.