-0.5cm -1cm 0.8cm 21.5cm 0.5cm 1.2cm

\
Jun [Fukue]{} and Masayuki [Ioroi]{}\
[*Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582*]{}\
[*E-mail(JF): fukue@cc.osaka-kyoiku.ac.jp*]{}\

[**Abstract**]{}

We investigate Hoyle–Lyttleton accretion for the case where the central source is a luminous accretion disk. In classical Hoyle-Lyttleton accretion onto a “spherical” source, accretion takes place in an axially symmetric manner around a so-called accretion axis. In the spherical case the accretion rate $\dot{M}$ is given as $\dot{M}_{\rm HL} (1-\Gamma)^2$, where $\dot{M}_{\rm HL}$ is the accretion rate of the classical Hoyle–Lyttleton accretion onto a non-luminous object and $\Gamma$ the luminosity of the central object normalized by the Eddington luminosity. If the central object is a compact star with a luminous accretion disk, the radiation field becomes “non-spherical”. In such a case the axial symmetry around the accretion axis breaks down; the accretion radius $R_{\rm acc}$ generally depends on an inclination angle $i$ between the accretion axis and the symmetry axis of the disk and the azimuthal angle $\varphi$ around the accretion axis. Hence, the accretion rate $\dot{M}$, which is obtained by integrating $R_{\rm acc}$ around $\varphi$, depends on $i$. In the case of a pole-on accretion ($i=0$), although the axisymmetry around $\varphi$ is retained, the accretion radius becomes smaller than that of the spherical case. The accretion rate is therefore smaller than that of the spherical case. We found that the accretion rate is approximately expressed as $\dot{M} \sim \dot{M}_{\rm HL} (1-\Gamma)(1-2\Gamma)$. In the case of an edge-on accretion ($i=90^{\circ}$), the accretion radius depends strongly on $\varphi$ and is somewhat larger than that of the spherical case. Depending on the central luminosity $\Gamma$, the shape of the accretion cross-section varies from a circle ($\Gamma=0$), an ellipse, a hollow ellipse ($\Gamma \sim 0.5$), and a twin lobe ($\Gamma {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.65$). The accretion rate is larger than that of the spherical case and approximately expressed as $\dot{M} \sim \dot{M}_{\rm HL} (1-\Gamma)$ for $\Gamma \leq 0.65$ and $\dot{M} \sim \dot{M}_{\rm HL} (2-\Gamma)^2/5$ for $\Gamma \geq 0.65$. Once the accretion disk forms and the anisotropic radiation fields are produced around the central object, the accretion plane will be maintained automatically (the direction of jets associated with the disk is also maintained). Thus, the anisotropic radiation field of accretion disks drastically changes the accretion nature, that gives a clue to the formation of accretion disks around an isolated black hole. Hoyle–Lyttleton type accretion onto the accretion disk may take place in various astrophysical situations, including the Galactic Center X-ray source 1E 1740.7$-$2942.

[**Key words:**]{} Accretion — Accretion disks — Radiation mechanisms — X-rays: individual (1E 1740.7$-$2942)

[**1. Introduction**]{}

Accretion onto a gravitating body is one of the most important phenomena in recent astrophysics, since it produces and radiates a remarkable energy via the release of the gravitational energy (e.g., Kato et al. 1998). If the central object is a compact star, a significant fraction of the rest mass energy of the accreting gas is liberated. Furthermore, an energetic mass outflow from the accretion system often takes place. Thus, the central engine of the accretion system exerts a strong influence on the environments through intense radiation and mass outflow.

If there is an isolated object in interstellar space, the gravitational field of the object attracts gas particles. If there exists a relative velocity between the object and the gas, the gas particles move in hyperbolic orbits, which intersect downstream of the point mass on the axis of symmetry (the accretion axis). The intersecting particles are allowed to collide and coalesce on the accretion axis. Particles colliding sufficiently close to the point mass lose enough of their kinetic energy to be trapped by the central object. This is the classical picture of Hoyle–Lyttleton accretion (Hoyle, Lyttleton 1939; Bondi, Hoyle 1944).

Let us suppose a graviating body with mass $M$ and luminosity $L$, placed in a gas flow with velocity $v_{\infty}$ and density $\rho_{\infty}$ at infinity. If the velocity $v_{\infty}$ is much larger than the sound speed of the gas at infinity, the effect of pressure can be neglected, except at the very center. The Hoyle–Lyttleton accretion radius $R_{\rm HL}$, where the effect of radiation pressure is ignored, is $$\begin{aligned}
   R_{\rm HL} &=& \frac{2GM}{v_{\infty}^2}
\nonumber \\
   &=& 2.65 \times 10^{15} 
            \frac{M}{10M_{\odot}}
            \left( \frac{v_\infty}{10{\rm ~km~s}^{-1}} \right)^{-2}
            {\rm ~cm}.\end{aligned}$$ Here, we bear in mind a stellar mass black hole moving slowly in the interstellar space. In addition, the relevant timescale of the classical Hoyle–Lyttleton accretion is $$\begin{aligned}
   t_{\rm HL} &=& \frac{R_{\rm HL}}{v_\infty}
\nonumber \\
   &=& 84.0 
            \frac{M}{10M_{\odot}}
            \left( \frac{v_\infty}{10{\rm ~km~s}^{-1}} \right)^{-3}
            {\rm ~yr}.
\label{tHL}\end{aligned}$$

Inside the accretion radius the potential energy exceeds the kinetic energy and the gas is trapped by the gravitating object. Thus, the mass accretion rate ${\dot M}_{\rm HL}$ onto the gravitating object is $$\begin{aligned}
   {\dot M}_{\rm HL} &=& \pi R_{\rm HL}^2 \rho_{\infty} v_{\infty}
           = \frac{4\pi\rho_{\infty}G^2 M^2}{v_{\infty}^3}
\nonumber \\
   &=& 3.70 \times 10^{18}
            \left( \frac{M}{10M_{\odot}} \right)^2
            \frac{n_\infty}{10^5 {\rm ~cm}^{-3}}
            \left( \frac{v_\infty}{10{\rm ~km~s}^{-1}} \right)^{-3}
            {\rm ~g~s}^{-1},\end{aligned}$$ where $n_\infty$ is the number density at infinity. In addition, the growth time of the central object is $$\begin{aligned}
   t_{\rm growth} &=& \frac{M}{\dot{M}_{\rm HL}}
\nonumber \\
   &=& 1.70 \times 10^8 
            \left( \frac{M}{10M_{\odot}} \right)^{-1}
            \left( \frac{n_\infty}{10^5 {\rm ~cm}^{-3}} \right)^{-1}
            \left( \frac{v_\infty}{10{\rm ~km~s}^{-1}} \right)^{3}
            {\rm ~yr}.
\label{tgrowth}\end{aligned}$$ If we know the mass accretion rate, we can estimate the (accretion) luminosity of the radiation emitted from the compact object. Therefore, knowledge about the accretion radius is important to compare theories with observations. The effect of gas pressure on Hoyle–Lyttleton accretion has been investigated by many researchers (Bondi 1952; Hunt 1971; Shima et al. 1985; Fryxell et al. 1987; Fryxell, Taam 1988; Taam, Fryxell 1989; Ho et al. 1989; Matsuda et al. 1991; Ruffert, Arnett 1944; Ruffert 1994, 1996; Font, Ibáñez 1998a, b). The effect of radiation pressure, however, has not been taken into account well (e.g., Taam et al. 1991). Due to radiation pressure, the accretion radius and the accretion rate are significantly reduced from the Hoyle–Lyttleton estimate. That is, the accretion radius modified by the radiation pressure becomes $$R_{\rm HL}^{\rm rad}=\frac{2GM(1-\Gamma)}{v_{\infty}^2}
   =R_{\rm HL}(1-\Gamma),
\label{eqn:bondi}$$ when the gas is fully ionized and transparent. Here, $\Gamma$ is the normalized luminosity, which is defined by the ratio of the luminosity $L$ of the gravitating object to the Eddington luminosity $L_{\rm E}$ ($=4 \pi cGMm_{\rm p}/\sigma_{\rm T}$): $$\Gamma = \frac{L}{L_{\rm E}},$$ where $m_{\rm p}$ is the proton mass and $\sigma_{\rm T}$ the Thomson scattering cross-section. In this case with radiation pressure, $\dot{M}=\dot{M}_{\rm HL}(1-\Gamma)^2$.

Recently, the effect of radiation drag (Compton drag) in Hoyle–Lyttleton accretion around a luminous source was also considered by Nio et al. (1998). In the ionized gas the radiation-drag force becomes important if the velocity of the gas is comparable to the light speed and the luminosity of the central object is comparable to the Eddington luminosity. Radiation drag is also important if the specific cross-section — “opacity” — of particles is sufficiently large like, e.g., dust. Nio et al. (1998) found that the accretion radius decreases due to radiation pressure, while it increases due to radiation drag. In addition, the accretion radius including radiation drag becomes larger as the incident velocity becomes fast. In the limit of small $v_\infty$, the above equation (\[eqn:bondi\]) is modified to $R_{\rm HL}^{\rm rad}/R_{\rm HL} = 1 - \Gamma (1-2v_\infty/c)$.

In all of these studies the central sources have been supposed to be “spherical” in the sense that both of the gravitational and radiation fields have a spherical symmetry. In actual accretion systems like the Galactic Center X-ray source 1E 1740.7$-$2942, however, as a consequence of mass accretion, an accretion disk often forms around the central gravitating body. Hence, in such a case the radiation field is not spherical but “anisotropic”. Namely, the radiative flux $F$ at a distance $R$ from the center depends on a polar angle $\theta$ as $$F= \frac{L}{2\pi R^2} \cos \theta,
\label{diskF}$$ where $L$ is the total luminosity of the disk. The accretion radius and therefore the mass accretion rate will be changed due to the violation of the spherical symmetry. Indeed, the anisotropic radiation field drastically changes the nature of accretion. The purpose of this paper is to examine the Hoyle–Lyttleton accretion onto an accretion disk.

In the next section we describe situations and basic equations. Numerical results are presented in section 3. The canonical luminosity is derived in section 4. Astrophysical implications and applications are discussed in section 5. The final section is devoted to concluding remarks. The physical quantities of the X-ray source 1E 1740.7$-$2942 are deduced in the appendix.

[**2. Situations and Basic Equations**]{}

[*2.1. Radiation Fields*]{}

Let us consider a point mass with mass $M$ and a surrounding accretion disk with luminosity $L$ (figure 1). The system is assumed to be immersed in uniform gas flow with velocity $v_\infty$ and density $\rho_\infty$. We also assume that the gas is fully ionized and transparent to radiation from the object. The central accretion disk is generally inclined to the gas flow (the inclination angle between the accretion axis and the symmetry axis of the disk is $i$). We adopt cylindrical coordinates $(r, \varphi, z)$, where the $z$-axis ([*accretion axis*]{}) is in the direction of the upstream gas particles (see figure 1). Moreover, the coordinates of a particle coming from infinity are $(x, y, z)$ in Cartesian coordinates and $(R, \theta, \varphi)$ in spherical coordinates ($R=\sqrt{r^2+z^2})$. In this system we first derive the radiative flux vector at a point P, where a particle is located.

————\
figure 1\
————\

In Cartesian coordinates, the unit vector ${\mbox{{\boldmath}$i$}}$ in the direction of the symmetry axis of the disk, and the unit vector ${\mbox{{\boldmath}$s$}}$ in the direction of a particle are respectively $$\begin{aligned}
   {\mbox{{\boldmath}$i$}}&=& \left( \sin i, 0, \cos i \right),
\\
   {\mbox{{\boldmath}$s$}}&=& \left( \frac{r}{R}\cos\varphi,
                   \frac{r}{R}\sin\varphi, \frac{z}{R} \right).\end{aligned}$$ Hence, the angle $\psi$ between the symmetry axis ${\mbox{{\boldmath}$i$}}$ and the particle direction ${\mbox{{\boldmath}$s$}}$ is given by $$\cos \psi = {{\mbox{{\boldmath}$i$}}} \cdot {{\mbox{{\boldmath}$s$}}}
             = \frac{1}{R} 
               \left( r\cos \varphi \sin i + z \cos i \right).$$

The radiative flux $F_R$ in the $R$-direction at a point (distance $R$) is expressed as $$F_R = \frac{L \cos \psi}{2\pi R^2}.
\label{FR}$$ Consequently, the radial component $F_r$ and the vertical component $F_z$ in cylindrical coordinates are, respectively,

$$\begin{aligned}
   F_r &=& F_R \frac{r}{R},
\\
   F_z &=& F_R \frac{z}{R}.\end{aligned}$$

It should be noted that if the distance from the center is of the order of the disk size, the radiation fields of the disk becomes much more complicated (cf. Tajima, Fukue 1998) and the above formula becomes inadequate.

[*2.2. Basic Equations*]{}

The motion of the gas particle in radiation fields is described by $$\frac{d{\mbox{{\boldmath}$v$}}}{dt}
     =-{\mbox{{\boldmath}$\nabla$}}\phi + \frac{\sigma_{\rm T}}{mc}
       ({\mbox{{\boldmath}$F$}}-E {\mbox{{\boldmath}$v$}}-{\bf P} \otimes {\mbox{{\boldmath}$v$}})
\label{vector}$$ upto the first order of $v/c$, where ${\mbox{{\boldmath}$v$}}$ is the particle velocity, $\phi$ the gravitational potential, $m$ the particle mass (proton mass for the normal plasma), $E$ the radiation energy density, ${\mbox{{\boldmath}$F$}}$ the radiative flux vector, and [**P**]{} the radiation stress tensor (Hsieh, Spiegel 1976; Fukue et al. 1985; see also Kato et al. 1998).

To examine the effect of radiation drag, Nio et al. (1998) considered the radiation energy and the radiation stress tensor as well as the radiative flux. In the present study we only consider the radiative flux to avoid the complication and to focus our attention on the anisotropic feature of the radiation field. Thus, the equation of motion in cylindlical coordinates $(r, \varphi, z)$ is expressed as

$$\begin{aligned}
    \frac{\displaystyle dv_r}{\displaystyle dt} 
    &=& -\frac{\displaystyle GMr}{\displaystyle R^3}
        +\frac{\displaystyle \sigma_{\rm T}}{\displaystyle mc}
         F_R \frac{\displaystyle r}{\displaystyle R}, 
\\
    \frac{\displaystyle dv_z}{\displaystyle dt} 
    &=& -\frac{\displaystyle GMz}{\displaystyle R^3}
        +\frac{\displaystyle \sigma_{\rm T}}{\displaystyle mc}
         F_R \frac{\displaystyle z}{\displaystyle R}.  
\label{eqn:basiceq0}\end{aligned}$$

Using equation (\[FR\]), the above equation of motion becomes

$$\begin{aligned}
    \frac{\displaystyle dv_r}{\displaystyle dt} 
    &=&-\frac{\displaystyle GMr}
             {\displaystyle R^3}(1-\Gamma_{\rm eff}),
\\
    \frac{\displaystyle dv_z}{\displaystyle dt} 
    &=&-\frac{\displaystyle GMz}
             {\displaystyle R^3}(1-\Gamma_{\rm eff}),\end{aligned}$$

where $\Gamma_{\rm eff}$ is the [*effective*]{} normalized luminosity defined by $$\begin{aligned}
   \Gamma_{\rm eff} &\equiv& 2 \Gamma \cos\psi
\nonumber \\
             &=& 2\frac{L}{L_{\rm E}}
                 \frac{ r\cos \varphi \sin i + z \cos i }
                      {R}.\end{aligned}$$ Measuring the length and velocity in units of $R_{\rm HL}$ and $v_\infty$, respectively, and using the relevant timescale $t_{\rm HL}$, the equation of motion is finally rewritten in the dimensionless form:

$$\begin{aligned}
    \frac{\displaystyle d\hat{v}_r}{\displaystyle d\hat{t}} 
    &=&-\frac{\displaystyle \hat{r}}
             {\displaystyle 2\hat{R}^3}(1-\Gamma_{\rm eff}),
\\
    \frac{\displaystyle d\hat{v}_z}{\displaystyle d\hat{t}} 
    &=&-\frac{\displaystyle \hat{z}}
             {\displaystyle 2\hat{R}^3}(1-\Gamma_{\rm eff}).
\label{eqn:basiceq}\end{aligned}$$

For various combinations of initial positions $(r, \varphi)$ of particles at sufficiently large $z$, we numerically integrate equation (17) using the forth-order Runge–Kutta–Gill method and calculate trajectories of particles. We estimate the accretion radius by the same procedure as that of Hoyle and Lyttleton (1939). That is, according to parameters, particles intersect the accretion axis at the downstream of the axis. If the $z$-velocity is less than the escape velocity at that point ($z=z_0$), defined by $\hat{v}_{\rm esc}=
\sqrt{(1-\Gamma_{\rm eff})/\hat{z}}
\left|_{r=0, z=z0}\right.$, these particles are judged to be captured and accreted on the gravitating body.

The parameters of the present problem are the normalized luminosity $\Gamma$, the incident velocity $v_\infty$ (although $v_\infty$ is just the unit and therefore renormalized), and the disk inclination angle $i$. In the present study we only examine two extreme cases; a pole-on accretion ($i=0$) and an edge-on one ($i=90^{\circ}$). Other general cases with arbitrary $i$ lie between these two cases.

[**3. Accretion Radius and Accretion Rate**]{}

Figure 2 shows the accretion radius $R_{\rm acc}$ normalized by the Hoyle–Lyttleton accretion radius $R_{\rm HL}$ ($=2GM/v_{\infty}^2$) as a function of the normalized luminosity $\Gamma$ ($=L/L_{\rm E}$).

————\
figure 2\
————\

The dotted straight line is for the case of a spherical source, that is proportional to $(1-\Gamma)$, as shown in equation (\[eqn:bondi\]). The dashed curve is the pole-on accretion case ($i=0$), where the accretion radius is independent of $\varphi$. As already stated in equation (\[diskF\]), the disk radiation field is not spherical but orients towards the polar direction. Hence, in the pole-on accretion of $i=0$, the central source (disk) is effectively luminous, compared with the spherical case. As a result, the accretion radius is remarkably smaller than that of the spherical case, as seen in figure 2.

The solid curves are the edge-on accretion case ($i=90^{\circ}$), where the accretion radius strongly depends on $\varphi$. In this case the situation is much more complicated, because the radiation strength varies along the particle path and depends on $\varphi$. Roughly speaking, however, in the edge-on accretion of $i=90^{\circ}$, the disk is effectively less luminous, and the accretion radius is larger than that of the spherical case.

In figure 3 the shapes of cross section in the case of $i=90^{\circ}$ are shown for various $\Gamma$. Since incoming particles in the disk plane ($\varphi=90^{\circ}$) receive no radiative flux, the accretion radius in the disk plane is just the classical Hoyle–Lyttleton one $R_{\rm HL}$. On the other hand, particles travelling over the disk are so influenced by the disk radiation field that the accretion radius becomes smaller than $R_{\rm HL}$. Moreover, particles passing through just above the pole ($\varphi=0$) cannot accrete for $\Gamma {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.65$. As a result, depending on the normalized luminosity $\Gamma$, the shape of the accretion cross-section varies from a circle ($\Gamma=0$), an ellipse compressed perpendicular to the disk plane, a hollow ellipse ($\Gamma \sim 0.5$), and a twin lobe in the disk plane ($\Gamma {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.65$).

————\
figure 3\
————\

Since in the present situation the accretion radius $R_{\rm acc}$ generally depends on $\varphi$, the cross sectional area of accretion is more convenient to discuss the accretion nature. In addition, the cross sectional area $A_{\rm acc}$ of accretion is directly related to the accretion rate by $\dot{M}= A_{\rm acc} \rho_\infty v_\infty$.

In figure 4 we show the cross sectional area $A_{\rm acc}$, normalized by that of the Hoyle–Lyttleton accretion, $\pi R_{\rm HL}^2$, as a function of the normalized luminosity $\Gamma$ (this also expresses $\dot{M}/\dot{M}_{\rm HL}$). As expected from the accretion radius calculated above, the accretion rate in the pole-on case is smaller than that of the spherical case. The accretion rate in the edge-on case, on the other hand, is larger than that of the spherical case. It should be noted that in the edge-on case accretion is possible even if $\Gamma$ is unity!

————\
figure 4\
————\

Finally, from these numerical results, the cross sectional area $A_{\rm acc}$ is approximately expressed as $$A_{\rm acc} = \pi R_{\rm HL}^2 f(\Gamma, i),$$ where $$f\left(\Gamma, i\right) = \left\{
     \begin{array}{ll}
       (1-\Gamma)^2 & ~~~{\rm for~a~spherical~case} \\
       (1-\Gamma)(1-2\Gamma) & ~~~{\rm for~}i=0 \\
       (1-\Gamma) & ~~~{\rm for~}i=90^{\circ}, \Gamma \leq 0.65 \\
       (2-\Gamma)^2/5 & ~~~{\rm for~}i=90^{\circ}, \Gamma \geq 0.65.
     \end{array}
   \right.
\label{f}$$

[**4. Canonical Luminosity**]{}

In the previous section, we treat the central luminosity $L$ and the accretion rate $\dot{M}$, seperately. In the accretion phenomena, however, these quantities are closely connected via the accretion process. That is, if the Hoyle–Lyttleton accretion takes place on an isolated compact star, the accretion rate $\dot{M}$ determines the central (accretion) luminosity $L$. We here estimate the [*canonical*]{} luminosity $L_{\rm can}$ (or equivalently $\Gamma_{\rm can}$) under such a situation. For simplicity, we assume a steady state of accretion, although there may exist a time lag for the gas to accrete along the accretion axis toward the center (e.g., Taam et al. 1991).

As shown in the previous section, the cross sectional area $A_{\rm acc}$ is given by $A_{\rm acc}=\pi R_{\rm HL}^2 f(\Gamma, i)$, and therefore, the accretion rate $\dot{M}$ ($=A_{\rm acc}\rho_\infty v_\infty$) is given by $$\frac{\dot{M}}{\dot{M}_{\rm HL}} = f(\Gamma, i)
\label{dM1}$$ for the present situation (see figure 4 for a representation of the function $f$). On the other hand, the accretion luminosity $L$ follows the mass accretion rate $\dot{M}$ via $L=\eta \dot{M}c^2$ (and therefore, $\Gamma=\eta \dot{M}c^2/L_{\rm E}$), where $\eta$ is the efficiency of the release of the accretion energy. Introducing the critical accretion rate $\dot{M}_{\rm E}$ by $\dot{M}_{\rm E} \equiv L_{\rm E}/(\eta c^2)$, we finally have $$\frac{\dot{M}}{\dot{M}_{\rm HL}} 
   = \frac{\dot{M}_{\rm E}}{\dot{M}_{\rm HL}}\Gamma
   = \frac{1}{\dot{m}_{\rm HL}}\Gamma,
\label{dM2}$$ where $\dot{m}_{\rm HL}$ is the dimensionless parameter of the system defined by $$\begin{aligned}
   \dot{m}_{\rm HL} &\equiv& \frac{\dot{M}_{\rm HL}}{\dot{M}_{\rm E}}
   = \frac{\eta cGM\sigma_{\rm T}\rho_\infty}{m_{\rm p}v_\infty^3}
\nonumber \\
   &=& 0.2648
       \frac{\eta}{0.1}
            \frac{M}{10M_{\odot}}
            \frac{n_\infty}{10^5 {\rm ~cm}^{-3}}
            \left( \frac{v_\infty}{10{\rm ~km~s}^{-1}} \right)^{-3}.
\label{dmHL}\end{aligned}$$ This parameter $\dot{m}_{\rm HL}$ is just $\alpha$ in Taam et al. (1991), where the spherical case was considered.

For a given configuration of the system (source type or inclination) and a given parameter $\dot{m}_{\rm HL}$, these two relations (\[dM1\]) and (\[dM2\]) have an intersection point on the $(\Gamma, \dot{M})$-plane (cf. figure 4). Moreover, this solution is stable in the sense that the increase of the accretion rate leads to the increase of the luminosity, then the accretion rate decreases. This intersection (solution) thus gives the [*canonical luminosity*]{} $L_{\rm can}$ (or $\Gamma_{\rm can}$) and the [*canonical accretion rate*]{} $\dot{M}_{\rm can}$. In figure 5 the canonical luminosity are shown as a function of $\dot{m}_{\rm HL}$. The canonical accretion rate is obtained by $\dot{M}_{\rm can}/\dot{M}_{\rm E} 
= L_{\rm can}/L_{\rm E} = \Gamma_{\rm can}$ or by $\dot{M}_{\rm can}/\dot{M}_{\rm HL}
=\Gamma_{\rm can}/\dot{m}_{\rm HL}$.

————\
figure 5\
————\

As seen in figure 5, the canonical luminosity $\Gamma_{\rm can}$ increases with $\dot{m}_{\rm HL}$. Compared with the spherical case (dotted curve), in the pole-on case (dashed one) the canonical luminosity is small, whereas it is large in the edge-on case (solid one). It is easy to obtain $\Gamma_{\rm can}$ algebraically as a function of $\dot{m}_{\rm HL}$: $$\Gamma_{\rm can} = \left\{
     \begin{array}{ll}
       \frac
            {\displaystyle 2\dot{m}_{\rm HL}+1
              -\sqrt{4\dot{m}_{\rm HL}+1} } 
            {\displaystyle 2\dot{m}_{\rm HL}}
       & ~~~{\rm for~a~spherical~case} \\
       \frac
            {\displaystyle 3\dot{m}_{\rm HL}+1
              -\sqrt{\dot{m}_{\rm HL}^2+6\dot{m}_{\rm HL}+1} }
            {\displaystyle 4\dot{m}_{\rm HL}}
       & ~~~{\rm for~}i=0 \\
       \frac{\displaystyle \dot{m}_{\rm HL}}
            {\displaystyle \dot{m}_{\rm HL}+1}
       & ~~~{\rm for~}i=90^{\circ}, \Gamma \leq 0.65 \\
       \frac
            {\displaystyle 4\dot{m}_{\rm HL}+5
              -\sqrt{40\dot{m}_{\rm HL}+25} }
            {\displaystyle 2\dot{m}_{\rm HL}}
       & ~~~{\rm for~}i=90^{\circ}, \Gamma \geq 0.65.
     \end{array}
   \right.$$

As accretion proceeds, the mass of the central object gradually increases, while producing the enormous luminosity. This growth timescale $t_{\rm growth}$ is roughly given by equation (\[tgrowth\]). Since the parameter $\dot{m}_{\rm HL}$ increases with mass $M$ and the canonical luminosity $\Gamma_{\rm can}$ also increases with $\dot{m}_{\rm HL}$, the canonical luminosity $\Gamma_{\rm can}$ increases as the mass increases. In other words, after sufficiently large time, $\Gamma_{\rm can}$ approaches unity.

[**5. Discussion**]{}

In this section we shall briefly discuss the astrophysical implications and applications of the present Hoyle–Lyttleton accretion onto accretion disks.

[*5.1. Maintenance of the Disk Plane*]{}

As shown in the previous sections, the anisotropic radiation fields of accretion disks may drastically change the accretion nature of the classical Hoyle–Lyttleton accretion.

First, let us suppose the edge-on accretion ($i=90^{\circ}$). Because of the anisotropy of radiation fields, the accretion radius over the disk poleward becomes small, compared with that in the disk plane (see figure 3). In other words, accretion takes place mainly in the disk plane. This property, that [*the prefered accretion plane coincides with the disk plane*]{}, thus continues to maintain the accretion disk. Naively speaking, of course, the net angular momentum of the accreting gas in the disk plane is zero. Hence, the spin direction of the disk may change on a timescale of $t_{\rm HL}$. However, the disk always forms with a fixed axis of symmetry. These properties are essentially equal in general cases with arbitrary inclination angle, except for the pole-on accretion case.

Now, let us investigate the longterm behavior of the accretion. As is seen from figure 4, the accretion rate of the edge-on case is larger than other cases; the edge-on accretion is the most effective. Hence, if the disk was initially inclined to the gas flow with arbitrary inclination angle, the inclination angle would gradually increase, as accretion proceeds. [*Ultimately, the disk configuration will settle down into a state, where the disk axis becomes perpendicular to the accretion axis*]{}. The final outcome would be edge-on accretion.

Finally, let us look at the origin and maintenance of accretion disks, which form around the gravitating object during the Hoyle–Lyttleton accretion process. At the first stage, where only the gravitating body exists in the interstellar space, there is no prefered direction of accretion. So, the classical Hoyle–Lyttleton accretion takes place symmetrically around the accretion axis. After the Hoyle–Lyttleton accretion proceeds, some possible fluctuations in the gas density or velocity break the symmetry of accretion around the accretion axis. [*Due to this spontaneous breaking of symmetry, an accretion disk forms and a prefered direction is chosen*]{}. Once the disk forms, the prefered direction and the disk plane will be fixed and maintained, as discussed above. If astrophysical jets emanate from the disk, the direction of jets is also constant.

[*5.2. Application to 1E 1740.7$-$2942*]{}

We apply the present model to the Galactic Center X-ray source 1E 1740.7$-$2942.

Electron-positron pair annihilation lines from the Galactic Center region have been detected since the 1970’s (e.g., Johnson et al. 1972; see Morris 1989 for a review). The intensity of annihilation lines is about $10^{-3}$photons s${}^{-1}$ cm${}^{-2}$, and this yields a positron annihilation rate of $10^{43}$ s${}^{-1}$ with the distance to the Galactic Center (Johnson, Haymes 1973; Leventhal et al. 1978). The lack of significant redshift and the narrowness of the line width suggest that the pair annihilation takes place in a relatively cold gas. Recent observations by the GRANAT satellite identified the pair annihilation source to be the hard X-ray source 1E 1740.7$-$2942 near the Galactic Center (Sunyaev et al. 1991; Bouchet et al. 1991). Subsequently, this X-ray source 1E 1740.7$-$2942 was found to be embedded in the molecular cloud which belongs to the inner molecular layer of the Galactic Center region (Mirabel et al. 1991; see also Ramaty et al. 1992). This cloud has a size of $\sim$ 3pc and a mean density of $\sim 5\times 10^4 {\rm cm}^{-3}$, therefore, it has a mass of $\sim 5 \times 10^4 M_{\odot}$. Finally, observation with the Very Large Array (Mirabel et al. 1992) showed a double-sided radio jet from this X-ray source 1E 1740.7$-$2942. The projected length of this radio jet is $\sim$ 1pc.

Several observational facts suggest the following picture for the X-ray source 1E 1740.7$-$2942 and the radio jet (Mirabel et al. 1991, 1992). The X-ray source 1E 1740.7$-$2942 may be a stellar mass black hole without companion — an [*isolated black hole*]{} (the X-ray spectrum of 1E 1740.7$-$2942 resembles that of Cyg X-1). From a molecular cloud, the interstellar gas accretes onto this black hole via the Hoyle–Lyttleton process, an accretion disk forms around the black hole, and electron-positron pairs are created in the vicinity of the black hole. The electron-positron pair plasma is ejected perpendicularly to the disk plane (cf. Tajima, Fukue 1998), and then injected into the molecular cloud in the form of a double-sided jet (the radio emission from jets is supposed to be synchrotron radiation from electrons and positrons). Electron-positron pairs travel at high velocities for a few years and then they are slow down and annihilated in the high-density cold molecular cloud (the annihilation lines are narrow).

Now, using the observational facts described above and physical quantities deduced from observations (in the appendix), we shall estimate the plausible accretion rate and the intrinsic luminosity of 1E 1740.7$-$2942.

We first evaluate the relative velocity $v_\infty$ at infinity by geometrical reasoning and by looking at activity duration. As already discussed, in Hoyle–Lyttleton accretion onto the accretion disk, the symmetry axis of the disk generally tends to lie in the plane perpendicular to the flow direction. If this is true for the case of 1E 1740.7$-$2942, the black hole moves perpendicular to the line-of-sight, and the transverse velocity is of the order of $v_\infty$. VLA observations show that the jets in 1E 1740.7$-$2942 are rather straight and no prominent bending is seen. This geometrical properties mean that the ratio of the transverse velocity to the jet velocity is less than $\sim 10^4$. Since the jet velocity is estimated as $0.3c$ in the appendix, the transverse velocity and therefore $v_\infty$ is about $v_\infty {\raisebox{0.3mm}{\em $\, <$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}10 {\rm ~km~s}^{-1}$. On the other hand, from the projection length of jets ($\sim$ 1 pc) and the jet velocity ($0.3c$), the central activity producing jets lasts at least 10 yrs. Due to the possible irregularities, the direction of the disk rotation changes intermittently. Indeed, the electron-positron annihilation features disappear in a timescale of a few years. These dynamical variations occur in the timescale of $t_{\rm HL}$ given in equation (\[tHL\]). Hence, the activity timescale again means that $v_\infty \sim 10 {\rm ~km~s}^{-1}$ as long as $M \sim 10M_{\odot}$.

Thus, for 1E 1740.7$-$2942, since the mass $M$ is reasonably assumed to be $10M_{\odot}$, a mean density of the ambient cloud is observed to be $\sim 5\times 10^4 {\rm cm}^{-3}$, and the velocity $v_\infty$ is derived to be ${\raisebox{0.3mm}{\em $\, <$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}10 {\rm ~km~s}^{-1}$, we can estimate the parameter $\dot{m}_{\rm HL}$ as $\dot{m}_{\rm HL} {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.1$.

Furthermore, if the canonical luminosity is established in 1E 1740.7$-$2942, then the intrinsic luminosity is $\Gamma {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.1$ (or $L {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}1.25 \times 10^{38} {\rm ~erg~s}^{-1}$ for $M=10M_{\odot}$). The observational luminosity, on the other hand, is about $3 \times 10^{37} {\rm ~erg~s}^{-1}$ (or normalized luminosity is 0.02). Since the observational luminosity is the projected one and equal to $2\Gamma \cos i$, the inclination angle of the disk is $i {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}80^{\circ}$, that is consistent with the geometrical configuration.

Similary, the canonical mass accretion rate is estimated to be $\dot{M}/\dot{M}_{\rm HL} {\raisebox{0.3mm}{\em $\, <$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.8$ (or $\dot{M}/\dot{M}_{\rm E}=\Gamma {\raisebox{0.3mm}{\em $\, >$} \hspace{-3.3mm}
   \raisebox{-1.8mm}{\em $\sim \,$}}0.1$, $\dot{M} \sim 10^{19}{\rm ~g~s}^{-1}$). Since the mass ejection rate of jets is to be $M_{\rm jet} = 10^{16}{\rm ~g~s}^{-1}$ (appendix), a small fraction $(\sim 10^{-3})$ of accretion mass is ejected by jets.

[**6. Concluding Remarks**]{}

We have examined Hoyle–Lyttleton accretion onto the central accretion disk. Such a situation is very likely to be realized in the universe. Because the radiation field of accretion disks is [*anisotropic*]{}, the accretion nature is quite different from that of the spherical case, qualitatively as well as quantitatively. In contrast to the traditional spherical model, the accretion radius $R_{\rm acc}$ generally depends on an inclination angle $i$ between the accretion axis and the symmetry axis of the disk and the azimuthal angle $\varphi$ around the accretion axis, as well as the mass $M$, the normalized luminosity $\Gamma$, and the relative velocity $v_\infty$. Hence, the effective cross section of accretion changes, and therefore, the accretion rate $\dot{M}$ depends on $i$ as well as $M$, $\Gamma$, and $v_\infty$. In the case of a pole-on accretion ($i=0$), the accretion radius becomes smaller than that of the spherical case, although the axisymmetry around $\varphi$ is retained. The accretion rate is therefore smaller than that of the spherical case. In the case of an edge-on accretion ($i=90^{\circ}$), the accretion radius depends strongly on $\varphi$ and is somewhat larger than that of the spherical case. The accretion rate is larger than that of the spherical case. Since accretion mainly takes place in the disk plane, the anisotropic radiation fields of the central accretion disks automatically act to maintain the disk plane (the direction of jets associated with the disk is also maintained).

We make some comments on recent numerical simulations. In 2D non-axisymmetric simulations of black-hole accretion, the so-called flip-flop instability occurs and the accretion disk forms transiently (Fryxell, Taam 1988; Taam, Fryxell 1989; Matsuda et al. 1991). In 3D simulations, however, the flip-flop instability disappears and the accretion cone remains quite stable (Ruffert, Arnett 1994; Ruffert 1994, 1996). In 3D relativistic computations, accretion also takes place in a stationary way without instability (Font, Ibáñez 1998a, b). According to these recent results of 3D numerical simulations, an accretion disk hardly forms via black-hole accretion, even if transiently. In some astronomical systems such as 1E 1740.7$-$2942, on the other hand, it is believed that an accretion disk and jets exist around an isolated black hole. The present mechanism interprets the discrepancy between the numerical simulations and observational facts. The accretion disk, once formed, must break the axial symmetry of black-hole accretion!

We have neglected the effect of radiation drag. If the radiation drag force is included, the accretion radius will generally increase (Nio et al. 1998). When the relative velocity is large, such radiation drag becomes important. In the present study we have assumed that the size of the accretion disk is negligible compared to the accretion radius. This assumption is adequate for a slow accretor \[cf. equation (1)\]. However, when the relative velocity is large, the accretion radius could become as small as the disk. In such a case, the size effect will be significant, the radiation fields of the disk is complicated (Tajima, Fukue 1998), and the present results would be modified quantitatively. Magneto-hydrodynamical and relativistic effects are also ignored. Finally, the present mechanism can be applied to the dust accretion onto isolated stars, including protostars and late type stars. These problems are left as future work.

[**Reference**]{}

=0.5cm=1

Bondi H. 1952, MNRAS 112, 195

=0.5cm=1

Bondi H., Hoyle F. 1944, MNRAS 104, 273

=0.5cm=1

Bouchet, L. et al. 1991, ApJL 383, L45

=0.5cm=1

Font J.A., Ibáñez J.M. 1998a, ApJ 494, 297

=0.5cm=1

Font J.A., Ibáñez J.M. 1998b, MNRAS 298, 835

=0.5cm=1

Fryxell B.A., Taam R.E., McMillan S.L.W. 1987, ApJ 315, 536

=0.5cm=1

Fryxell B.A., Taam R.E. 1988, ApJ 335, 862

=0.5cm=1

Fukue J. 1986, PASJ 38, 167

=0.5cm=1

Fukue J., Kato S., Matsumoto R. 1985, PASJ 37, 383

=0.5cm=1

Ho C., Taam R.E., Fryxell B.A., Matsuda T., Koide H., Shima E. 1989, MNRAS 238, 1447

=0.5cm=1

Hoyle F., Lyttleton R.A. 1939, Proc.Camb.Phil.Soc. 35, 405

=0.5cm=1

Hsieh S.-H., Spiegel E.A. 1976, ApJ 207, 244

=0.5cm=1

Hunt R. 1971, MNRAS 154, 141

=0.5cm=1

Johnson III W.N., Harnden F.R., Haymes R.C. 1972, ApJL 172, L1

=0.5cm=1

Johnson III W.N., Haymes R.C. 1973, ApJ 184, 103

=0.5cm=1

Kato S., Fukue J., Mineshige S. 1998, Black-Hole Accretion Disks (Kyoto University Press, Kyoto) chap16

=0.5cm=1

Leventhal M., MacCallum C.J., Stang P.D. 1978, ApJL 225, L11

=0.5cm=1

Matsuda T. Sekino N., Sawada K., Shima E., Livio M., Anzer U., Börner G. 1991, A&A 248, 301

=0.5cm=1

Mirabel I.F., Morris M., Wink J., Paul J., Cordier B. 1991, A&A 251, L43

=0.5cm=1

Mirabel I.F., Rodriguez L.F., Cordier B., Paul J., Lebrun F. 1992, Nature 358, 215

=0.5cm=1

Morris M. (ed) 1989, The Center of the Galaxy (Kluwer, Dordrecht)

=0.5cm=1

Nio T., Matsuda T., Fukue J. 1998, PASJ 50, 495

=0.5cm=1

Ramaty R., Leventhal M., Chan K.W., Lingenfelter R.G. 1992, ApJL 392, L63

=0.5cm=1

Ruffert M. 1994, ApJ 427, 342

=0.5cm=1

Ruffert M., Arnett D. 1994, ApJ 427, 351

=0.5cm=1

Ruffert M. 1996, A&A 311, 817

=0.5cm=1

Shima E., Matsuda T., Takeda H., Sawada K. 1985, MNRAS 217, 367

=0.5cm=1

Sunyaev R. et al. 1991, A&A 247, L29

=0.5cm=1

Taam R.E., Fryxell B.A. 1989, ApJ 339, 297

=0.5cm=1

Taam R.E., Fu A., Fryxell B.A. 1991, ApJ 371, 696

=0.5cm=1

Tajima Y., Fukue J. 1998, PASJ 50, 483

=0.5cm=1

[**Appendix. Physical Quantities of 1E 1740.7$-$2942**]{}

In this appendix we shall evaluate the physical quantities of the electron-positron pair jet, i.e., the velocity and number density, using the observational data and appropriate assumptions.

Let us first suppose that the observed radio jets from the X-ray source 1E 1740.7$-$2942 are composed of an electron-positron pair plasma traveling with high velocity (Fukue 1986; Tajima, Fukue 1998). Entrainment from the ambient gas must be negligible. Or, the redshifted annihilation lines which are produced in the entrainment region would be detected. The electron-positron pairs are carried from the X-ray source 1E 1740.7$-$2942 to the annihilation point in the molecular cloud without significant loss due to pair annihilation, since the annihilation efficiency in the pair jet is low (Fukue 1986). Thus, the mass is conserved along the jet: $$2\pi a^2 n_{\rm e}m_{\rm e}cu = \dot{M}_{\rm jet},
\label{A1}$$ where $a$ is the radius of the pair jet, $n_{\rm e}$ the number density of electrons (which is equal to that of positrons), $m_{\rm e}$ the mass of electrons, $c$ the light speed, $u (=\gamma v/c)$ the four velocity of the pair jet, and $\dot{M}_{\rm jet}$ the mass ejection rate. Of these quantities, the mass ejection rate $\dot{M}_{\rm jet}$ can be derived from the observed annihilation rate: the annihilation rate of $10^{-3}$photons s${}^{-1}$ cm${}^{-2}$ corresponds to the mass ejection rate of $\sim 10^{16} {\rm g \, s}^{-1}$. From the VLA map (Mirabel et al. 1992) the radius $a$ of the pair jet is about 0.1–0.2pc. Hence, equation (\[A1\]) gives one relation between $n_{\rm e}$ and $u$. At the distance of $\sim 1$ pc from the central X-ray source 1E 1740.7$-$2942, the pair jet terminates and electrons and positrons are annihilated there (Mirabel et al. 1991). The momentum carried by the pair jet should be balanced with that of the interstellar gas. The former is $2(n_{\rm e}m_{\rm e}c^2 + n_{\rm e}kT_{\rm e})u^2 + 2n_{\rm e}
kT_{\rm e}$, where $k$ is the Boltmann constant and $T_{\rm e}$ the electron temperature of the pair jet. The factor 2 includes the contribution from both electrons and positrons. Since it is reasonable that the pair jet is sufficiently cold (Fukue 1986), this reduces to $2n_{\rm e}m_{\rm e}c^2u^2$, which just expresses the ram pressure of the pair jet. On the other hand, the interstellar gas pressure is $nkT$, where $n$ is the number density of the interstellar gas in the molecular cloud and $T$ is its temperature. Then, the condition that the ram pressure of the pair jet is balanced with the gas pressure of the ambient molecular gas becomes $$2n_{\rm e}m_{\rm e}c^2u^2 = nkT.
\label{A2}$$ In this equation (\[A2\]), the number density $n$ of the interstellar gas in the molecular cloud is of the order of $5\times 10^4 {\rm ~cm}^{-3}$ (Mirabel et al. 1991). The temperature $T$ of the molecular gas is typically 10–20 K. Hence, equation (\[A2\]) gives another relation between $n_{\rm e}$ and $u$. Solving equations (\[A1\]) and (\[A2\]), we have typically $$\begin{aligned}
     u & = & \pi a^2 nkT/\dot{M}_{\rm jet} c \nonumber \\
       & = & 0.275 \left(\frac{a}{\rm 0.2pc}\right)^2 
             \frac{n}{5 \times 10^4 {\rm cm}^{^3}} 
             \frac{T}{10{\rm K}} 
  \left(\frac{\dot{M}_{\rm jet}}{10^{16}{\rm g \, s}^{-1}}\right)^{-1},
\label{A3}\end{aligned}$$ $$\begin{aligned}
     n_{\rm e} & = & \dot{M}_{\rm jet}^2/2\pi^2 a^4 m_{\rm e} nkT 
\nonumber \\
      & = & 5.55 \times 10^{-4} 
      \left(\frac{a}{\rm 0.2pc}\right)^{-4} 
      \left(\frac{n}{5 \times 10^4 {\rm cm}^{^3}}\right)^{-1} 
\nonumber \\
      &&\times
      \left(\frac{T}{10{\rm K}}\right)^{-1} 
      \left(\frac{\dot{M}_{\rm jet}}{10^{16}{\rm g \, s}^{-1}}\right)^2 
     {\rm ~cm}^{-3}.\end{aligned}$$

The velocity of the electron-positron pair jet from the X-ray source 1E 1740.7$-$2942 is evaluated to be mildly relativistic. It is emphasized that the typical velocity given by equation (\[A3\]) is close to that of SS433 (0.26$c$).

Figure Captions

Fig. 1. Configuration of the system. A point object with mass $M$, which is surrounded by an accretion disk with luminosity $L$, is moving against an ambient gas of density $\rho_\infty$ at relative speed $v_\infty$. The accretion disk is generally inclined to the gas flow with inclination angle $i$. Left: the side view from the $y$-direction. Right: the top view from the $z$-direction.

Fig. 2. Accretion radius normalized by the Hoyle–Lyttleton one, $R_{\rm acc}/R_{\rm HL}$, as a function of the normalized luminosity $\Gamma$. The dotted line, which linearly decreases with $\Gamma$, is for the spherical case. The dashed curve is for the case of pole-on accretion ($i=0$). The solid curves are for the case of edge-on accretion ($i=90^{\circ}$), where $\varphi=90^{\circ}$ to 0 from top to bottom.

Fig.3. Shapes of accretion cross section in the case of edge-on accretion ($i=90^{\circ}$) for various $\Gamma$. The outer circles represent the classical Hoyle–Lyttleton radii, while the inner circles denote those of the spherical case.

Fig.4. Cross sectional area normalized by the Hoyle–Lyttleton value, $A_{\rm acc}/\pi R_{\rm HL}^2$, as a function of the normalized luminosity $\Gamma$. The dotted curve is for the spherical case. The dashed curve is for the case of pole-on accretion ($i=0$). The solid curve is for the case of edge-on accretion ($i=90^{\circ}$). This plot also represents the mass accretion rate normalized by the Hoyle–Lyttleton one, $\dot{M}/\dot{M}_{\rm HL}$.

Fig.5. Canonical luminosity normalized by the Eddington one, $\Gamma_{\rm can}$ ($=L_{\rm can}/L_{\rm E}$), as a function of $\dot{m}_{\rm HL}$ ($=\dot{M}_{\rm HL}/\dot{M}_{\rm E}$). The dotted curve is for the spherical case. The dashed curve is for the case of pole-on accretion ($i=0$). The solid curve is for the case of edge-on accretion ($i=90^{\circ}$).
