Download Bondi Accretion onto a Luminous Object

PASJ: Publ. Astron. Soc. Japan 53, 687–692, 2001 August 25
c 2001. Astronomical Society of Japan.
Bondi Accretion onto a Luminous Object
Jun F UKUE
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582
[email protected]
(Received 2001 March 16; accepted 2001 June 27)
Abstract
Spherical accretion of ionized gas onto a gravitating object is examined under the influence of central radiation. In classical Bondi accretion onto a non-luminous source with mass M, the accretion rate ṀB is expressed
−3
, where ρ∞ is the density at infinity and cs∞ the sound speed at infinity. We first
as ṀB = λ(γ ) × 4π G2 M 2 ρ∞ cs∞
found that the normalized accretion rate λ(γ ) is approximated by λ(γ ) = −(5/4)γ + (19/8), instead of a rigorous
expression. When the central object is a “spherical” source, the accretion rate Ṁ reduces to ṀB (1 − Γ)2 , where Γ
is the central luminosity normalized by the Eddington one. If the central luminosity is produced by the accretion
energy, the steady canonical luminosity is determined and the normalized luminosity does not exceed unity, as expected. On the other hand, when the central object is a “disk” source, such as an accretion disk, the accretion rate
becomes Ṁ/ṀB = 1 − 2Γd + (4/3)Γ2d for Γd ≤ 1/2, and Ṁ/ṀB = 1/(6Γd ) for Γd > 1/2, where Γd is the normalized
disk luminosity. We also found steady canonical solutions, where the normalized luminosity can exceed unity for
sufficiently large accretion rates. The anisotropic radiation field of accretion disks greatly modifies the accretion
nature.
Key words: accretion, accretion disk — black hole physics — radiation mechanism: general — X-rays: stars
1. Introduction
Mass accretion onto a gravitating object is one of the fundamental processes of modern astrophysics, since the accretion
process releases an enormous amount of gravitational energy.
A mass accretor is just a “gravitating power house” in the universe.
There are several modes of accretion: spherical or disklike. There are several situations concerning the mass accretor: static or moving. There are also several physical processes relating to accretion: adiabatic, isothermal, rotation, radiation, conduction, convection, magnetic, relativity, and so on.
Various combinations of these processes have been extensively
studied by many researchers, though not all. Even in the simple
spherical case, several important situations often go unrecognized.
Since the classical paper of Bondi (1952), various aspects
on spherical accretion have been investigated by many authors
(see, e.g., Holzer, Axford 1970; Kato et al. 1998 for a review).
Among them, the influence of radiation of a central object has
been examined: for radiating energy loss (e.g., Shapiro 1973),
for an optically thick case (Tamazawa et al. 1975; Burger, Katz
1980; Flammang 1982), with radiation drag (Umemura, Fukue
1994), for dust–gas fluid (Fukue 2001), and for the Hoyle–
Lyttleton case (Taam et al. 1991; Nio et al. 1998; Fukue, Ioroi
1999; Hanamoto et al. 2001). Up to now, however, nobody has
examined the dynamical effect of the radiative force produced
by the central luminosity on an optically thin Bondi flow.
As for the case of Hoyle–Lyttleton accretion, recent investigations revealed that the radiative force drastically modifies
the flow pattern, particularly for the disk source (Fukue, Ioroi
1999; Hanamoto et al. 2001). In the case of Bondi accretion,
we also expect that the flow would be greatly modified by the
central luminosity. In the present paper, we thus examine a
spherical Bondi-type flow of an optically thin gas under the
influence of the central luminosity.
In the next section the basic equations are presented. In section 3 the classical Bondi flow is briefly summarized with a
few expressions. We examine a Bondi-type flow onto a spherical source in section 4 and onto a disk source in section 5. The
final section is devoted to concluding remarks.
2. Basic Equations
Let us suppose stationary, spherically-symmetric accretion
of ionized gas onto a central gravitating body of mass M. The
flow is assumed to be optically thin, and affected by the radiation field of the central object, if it shines. The magnetic field
and self-gravity of the gas are neglected.
The basic equations for such an accretion flow are described
as follows (Kato et al. 1998 for a review; see also Fukue 2001
and references therein):
(a) Continuity equation
The equation for mass conservation is
4π r 2 ρv = −Ṁ,
(1)
where ρ is the gas density, v the gas infall velocity, and Ṁ the
gas-accretion rate.
(b) Equation of motion
When the central object or the accreting gas radiates at luminosity L, the ionized gas suffers from radiation pressure. The
equation of motion in the radial direction for the present case
is generally written as
v
dv
1 dp GM(1 − Γeff )
=−
−
,
dr
ρ dr
r2
(2)
688
J. Fukue
where p is the gas pressure and Γeff is the normalized luminosity, defined for each case.
(c) Energy equation
The energy equation is expressed as
1 d
1 d 2 2 p
(3)
r
v +p 2
r v = q,
r 2 dr
γ −1
r dr
where γ is the ratio of the specific heats and q is the heating
rate. In this paper we focus our attention on the dynamical
effect of radiation and neglect the irradiation heating/cooling
for simplicity; i.e., q = 0.
(d) Equation of state
Finally, the equation of state to close the equation set is
p=
R
ρT ,
µ
(4)
where R is the gas constant, µ the mean molecular weight,
and T the gas temperature. Throughout this paper, the mean
molecular weight is set to be 0.5.
(e) Wind equation
According to the usual procedure for transonic flow, we derive the wind equation, an ordinary differential equation of the
first order on the variable. From the equation of motion (2) and
the energy equation (3), we can derive √
a set of wind equations
on velocity v and sound speed cs (≡ ∂p/∂ρ). After some
manipulations, we obtain wind equations for the present case:
2
GM(1 − Γeff )
v cs2 −
dv
r
r2
=
,
(5)
dr
(v 2 − cs2 )
2 2 GM(1 − Γeff )
2
(γ − 1)cs − v +
dcs2
r
r2
=
.
(6)
2
2
dr
(v − cs )
Here and hereafter, we use the relation on the sound speed:
cs2 = γp/ρ = γ (R/µ)T .
The critical point rc , where the flow becomes transonic, is
located at
GM(1 − Γeff ) 5 − 3γ GM(1 − Γeff )
rc =
=
,
(7)
2
2
2csc
4
cs∞
and the sound speed there is
2
=
csc
2
c2 ,
5 − 3γ s∞
(8)
where cs∞ is the sound speed at infinity.
(f) Method of solution
Consequently, the basic equations of the present accretion
onto a luminous central object are equations (1), (5), and (6)
on variables ρ, v, and cs . The parameters are apparently
γ , µ(= 0.5), Ṁ, and Γeff , but the accretion rate is determined
as an eigenvalue of the differential equation.
In order to solve these basic equations, we should impose boundary conditions. For accreting flow from a infinite
distance to the center, the temperature and sound speed become asymptotically constant at large distance from the center:
cs → cs∞ and T → T∞ at r → ∞. Similarly, ρ → ρ∞ at r → ∞
(i.e., the number density n → n∞ at r → ∞). The gas velocity
v is then given by continuity equation (1):
[Vol. 53,
v → v∞ = −
Ṁ
4π r 2 ρ∞
(9)
at r → ∞. Thus, we must arbitrarily give cs∞ and ρ∞ as additional parameters.
Starting from these boundary conditions, we can solve the
basic equations inwards by the shooting method. Namely, with
a trial value of the accretion rate Ṁ, the basic equations are
integrated inwards. For an incorrect value of Ṁ, the integrated solution misses the critical point. Then, the trial value
is changed. When the value of Ṁ is sufficiently close to the
correct value, the solution approaches the critical point. In the
present case of a system of differential equations, a transonic
point is generally a saddle type with a first-order singularity
and a nodal type with a higher order singularity does not appear (cf. Kato et al. 1998). We thus get over the critical point
by using a linear approximation. Beyond the critical points, the
integration of wind equations is resumed. During the abovementioned procedures, the gas-accretion rate Ṁ is determined
as an eigenvalue.
3. Classical Bondi Accretion
Before examining the case where the central object is shining, we briefly review Bondi flow (Bondi 1952) with a few additional expressions. For Bondi flow, where the central object
is non-luminous, the normalized luminosity is set to be Γeff = 0.
For Bondi flow, we can normalize the basic equations as follows. Since we consider the accretion flow (v < 0), let us take
the absolute value of velocity: v → |v|. The length, veloc2
,
ity, and density are respectively measured in units of GM/cs∞
cs∞ , and ρ∞ .
Then, the basic equations [(1), (5), and (6)] are rewritten as
r̂ 2 ρ̂ v̂ = λ(γ ),
2 2 1
v̂ ĉs − 2
d v̂
r̂
r̂
=
,
d r̂
(v̂ 2 − ĉs2 )
2 2 1
2
(γ − 1)ĉs − v̂ + 2
d ĉs2
r̂
r̂
=
.
d r̂
(v̂ 2 − ĉs2 )
(10)
(11)
(12)
Here, λ(γ ) is the normalized accretion rate, which is determined as an eigenvalue of the problem. The boundary conditions are ĉs∞ → 1, ρ̂∞ → 1, and v̂ → λ/r̂ 2 .
Under these normalizations, the relevant scale length becomes
GM
M
T∞ −1
14
−1
= 7.98 × 10 cm × γ
. (13)
2
cs∞
10 M
104 K
Furthermore, the accretion rate (Bondi rate) becomes
ṀB ≡ λ(γ )
4π (GM)2 ρ∞
3
cs∞
= 2.74 × 10−13 M yr−1 λ(γ )γ −3/2
2 T∞ −3/2 n∞ M
.
×
10 M
104 K
1 cm−3
(14)
No. 4]
Bondi Accretion onto a Luminous Object
4. Spherical Source
Table 1. Normalized accretion rate.
γ
λ
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.66666
1.12
0.995
0.872
0.7495
0.625
0.500
0.367
0.250
689
We now consider a luminous accretor, where the radiation
force as well as the gravitational force exert a strong influence
on the accreting gas.
We first examine a spherical source, where the central object
is a spherically symmetric radiator with luminosity L. In this
case the normalized luminosity Γeff in the basic equations is
read as
L
,
(16)
Γeff = Γ ≡
LE
where Γ is the central luminosity normalized by the Eddington
luminosity, LE (= 4π cGM/κ), of the central object.
In such a spherical flow with radiation force, the gravity is
reduced by a factor of (1 − Γ). In other words, mass M should
be replaced by M(1 − Γ) in the force balance. As shown in
equation (14), the accretion rate ṀB of the Bondi flow depends
on M with its square. Hence, the accretion rate Ṁ of the spherical flow onto a spherical source with normalized luminosity Γ
is expressed as
Ṁ = (1 − Γ)2 ṀB .
(17)
This nature is very similar to the Hoyle–Lyttleton accretion
onto a luminous source (Fukue, Ioroi 1999; Fukue 1999). As
expected, the mass accretion rate becomes zero at Γ = 1, where
the effective gravity vanishes (see the upper panel of figure 2).
Next, we obtain the canonical luminosity under the steady
state solutions. In some cases the central luminosity L is independent of the mass accretion rate Ṁ. In usual accretion
phenomena, however, the central (accretion) luminosity is determined by the accretion rate via L = ηṀc2 (and therefore,
Γ = ηṀc2 /LE ), where η is the efficiency of the release of the
accretion energy. Introducing the critical accretion rate ṀE by
ṀE ≡ LE /(ηc2 ), we have
Fig. 1. Normalized accretion rate λ as a function of γ for spherical
Bondi flow. The crosses are the numerical values, while the straight
line is a fitting curve: λ = −(5/4)γ + (19/8).
In table 1 and figure 1 we show the values of the normalized accretion rate λ as a function of γ , according to
Bondi (1952) and additional numerical calculations. Bondi
(1952) obtained these values, making free use of algebraic
and graphic techniques, for several special cases (1, 1.2, 7/5,
1.5, 5/3). We, on the other hand, can obtain these values
by using a numerical technique as eigen values of the system for general cases. Both are, of course, in perfect agreement. Although Bondi (1952) derived a rigorous expression:
λ(γ ) = (1/4)[2/(5−3γ )](5−3γ )/[2(γ −1)] , we found a useful fitting
formula for λ. In figure 1 the crosses are the numerical values,
while the straight line is a fitting curve which, we found, is
expressed by
19
5
.
λ(γ ) = − γ +
4
8
(15)
Ṁ
ṀE
1
=
Γ=
Γ,
ṁ
ṀB ṀB
B
(18)
where ṁB is a dimensionless parameter of the system, defined
by
ṁB ≡
ṀB ηṀB c2
=
.
LE
ṀE
Numerically,
ṀE = 2.21 × 10−7 M yr−1
(19)
η −1 M 2
,
0.1
10 M
η −1 M 0.1
10 M
−3/2 n∞ T∞
.
×
104 K
105 cm−3
(20)
ṁB = 0.124λγ −3/2
(21)
For a given parameter ṁB , two relations (17) and (18) have
an intersection point on the (Γ, Ṁ)-plane (see the upper panel
of figure 2). This is a steady canonical solution (cf. Fukue,
Ioroi 1999 for a Hoyle–Lyttleton case).
In the lower panel of figure 2, the canonical luminosity normalized by the Eddington one, Γcan (= Lcan /LE ), and the canonical accretion rate normalized by the Bondi one, ṁ (= Ṁ/ṀB ),
690
J. Fukue
[Vol. 53,
Let us estimate the optical depth of the flow. The flow
is roughly divided into the outer quasi-hydrostatic region of
rc r rout , where the density is almost uniform, and the inner freefall region of rin r rc , where the infall velocity is
approximately a freefall one. The optical depth τ is roughly
evaluated as follows:
rout
rout
rc
κ Ṁ
dr
+
τ =
κρdr ∼
κρ∞ dr
2
rin
rin 4π r v
rc
2
κ Ṁ
2
+ κρ∞ (rout − rc )
∼
√
√ −√
rin
rc
4π GM(1 − Γ)
2
κ Ṁ
√
√ + κρ∞ rout
4π GM(1 − Γ) rin
−1/2
= 2λ (1 − Γ)3/2 τ∞ r̂in
+ τ∞ r̂out .
(23)
Here, we used equations (14) and (17) for the accretion rates,
and τ∞ is the relevant optical depth:
2
)ρ∞
τ∞ ≡ κ(GM/cs∞
×
T∞
104 K
M
10 M
−1 n
= 5.33 × 10−5 γ −1
∞
105 cm−3
.
(24)
Since τ∞ is sufficiently small for appropriate parameter ranges,
the flow is optically thin in the almost relevant region.
5. Disk Source
We next examine a disk source, where the central object is a
disk-like radiator with luminosity Ld . In this case the radiation
field produced by the disk is not spherical, but “anisotropic”,
in the sense that the radiative flux F at a distance R from the
center depends on the polar angle θ as
Ld
2 cos θ.
(25)
4π R 2
Hence, the normalized luminosity Γeff in the basic equations
should be read as
Ld
Γeff = 2Γd cos θ =
2 cos θ,
(26)
LE
F=
Fig. 2. Upper panel: Accretion rate Ṁ vs. normalized luminosity Γ
for a spherical accretor. The thick solid curve is the accretion rate of
spherical flow onto a spherical source. The two solid lines are accretion luminosities produced by the accretion processes for two different
values of ṁB . The intersection point gives a steady canonical solution
under a given parameter ṁB . Lower panel: Normalized canonical luminosity Γcan (= Lcan /LE ) and normalized accretion rate ṁ (= Ṁ/ṀB )
as a function of a parameter ṁB (= ṀB /ṀE ). The solid curve denotes
the former, whereas the dashed one means the latter.
are shown as a function of a parameter ṁB (= ṀB /ṀE ), by a
solid curve and a dashed one, respectively. For a spherical accretor, the normalized canonical luminosity Γcan increases with
ṁB , but does not exceed unity, as expected. As ṁB and Γcan
increase, the normalized accretion rate ṁ decreases, since the
radiation pressure suppresses the mass accretion.
In this spherical case, we can obtain algebraically the canonical luminosity. Namely, from equations (17) and (18), we have
1 2
1
1+
Γcan = 1 +
−
− 1.
(22)
2ṁB
2ṁB
where Γd (≡ Ld /LE ) is the disk luminosity normalized by the
Eddington luminosity.
As can be easily seen from above expression (26), the factor
(1 − Γeff ) has a strong polar-angle dependence. That is, for 0 <
Γd < 1/2, (1 − Γeff ) is always positive, although it is somewhat
reduced in the polar direction. For Γd > 1/2, there appears a
cone of avoidance, where (1 − Γeff ) becomes negative or the
radiation pressure overcomes gravity, in the region of
θ < θF
where
cos θF = 1/(2Γd ).
(27)
The dependence of (1−Γeff ) on the polar angle θ is shown in
figure 3 for several values of Γd (0, 0.1, 0.3, 0.5, 0.7, 1). As the
normalized disk luminosity Γd increases, the shape expressing
a factor (1−Γeff ) varies from a circle (Γd = 0), a flattened ellipse
(0 < Γd < 1/2), and a twin lobe (Γd ≥ 1/2) with a cone of
avoidance.
No. 4]
Bondi Accretion onto a Luminous Object
691
Fig. 3. Factor (1 − Γeff ) in the meridional cross-section for several
Γd . As the normalized disk luminosity Γd increases, the shape expressing this factor varies from a circle (Γd = 0), a flattened ellipse
(0 < Γd < 1/2), and a twin lobe (Γd ≥ 1/2) with a cone of avoidance.
As noticed in the case for Hoyle–Lyttleton accretion onto
an accretion disk (Fukue, Ioroi 1999), an anisotropic radiation
field of accretion disks drastically changes the accretion nature. This is also true for the present Bondi accretion. One
typical example is the existence of a cone of avoidance; for
Γd > 1/2, mass accretion becomes impossible in the polar direction, while it takes place in the equatorial direction.
Under such an anisotropic radiation field, rigorously speaking, the assumption of spherical symmetry for Bondi-type flow
will be violated. As a result, the flow pattern must deviate
from the spherically symmetric case, particularly in the central region. However, the deviation from spherical symmetry
is generally small outside the critical point, where the flow is
quasi-hydrostatic, and the mass-accretion rate is determined at
the critical point. Hence, we assume spherical symmetry, at
least outside of the critical point, for Bondi accretion flow onto
a disk source.
If we assume the spherical symmetry, the streamlines are
also assumed to be radial at least outside the critical point.
In this case the accretion rate d Ṁ in some conical sector of
a “spherical” flow onto a disk source is expressed as
dΩ
,
(28)
4π
where dΩ is the unit solid angle.
Integration of equation (28) over the whole solid angle yields
the mass-accretion rate. In the case of Γd < 1/2, integration in
the polar direction is done for 0 ≤ θ ≤ π/2, whereas it should
be done for θF ≤ θ ≤ π/2 in the case of Γd > 1/2 with the
cone of avoidance. After performing the integration, we finally
have the accretion rate as a function of the normalized disk
luminosity:
d Ṁ = (1 − Γeff )2 ṀB
Fig. 4. Upper panel: Accretion rate Ṁ vs. normalized luminosity Γd
for a disk accretor. The thick solid curve is the accretion rate of “spherical” flow onto a disk source. The two solid lines are the accretion
luminosities produced by the accretion processes for two different values of ṁB . The intersection point gives a steady canonical solution
under a given parameter ṁB . Lower panel: Normalized canonical luminosity Γcan (= Lcan /LE ) and normalized accretion rate ṁ (= Ṁ/ṀB )
as a function of a parameter ṁB (= ṀB /ṀE ). The solid curve denotes
the former, whereas the dashed one means the latter.

4 2

Ṁ  1 − 2Γd + 3 Γd
=
1
ṀB 

6Γd
for Γd ≤ 1/2
for Γd ≥ 1/2.
(29)
It should be noted that the mass accretion rate does not become
zero at Γd = 1, but has a finite value at large Γd (see the upper
panel of figure 4). This is because for a disk source the mass
accretion is always possible, at least in the equatorial region.
Next, we again obtain the canonical luminosity for a disk
source under steady-state solutions. In the sub-Eddington
regime the central (accretion) luminosity is related to the accretion rate via Ld = ηṀc2 , similar to the spherical case. In
the supercritical regime, where the accretion rate exceeds the
692
J. Fukue
critical one and the disk is described by the supercritical disk
(Watarai, Fukue 1999; Fukue 1999; Hanamoto et al. 2001), the
accretion luminosity is modified. In general, we should use an
approximate expression (cf. Hanamoto et al. 2001):
1
Ṁ
1 ηṀc2
Γd = 2 ln 1 +
= 2 ln 1 + ṁB
,
(30)
2 LE
2
ṀB
where ṁB is a dimensionless parameter of the system defined
by equation (19). Although this approximate expression depends weakly on the viscous parameter α of accretion disks,
the qualitative behavior does not change very much.
For a given parameter ṁB , two relations (29) and (30) have
an intersection point on the (Γd , Ṁ)-plane (see the upper panel
of figure 4). This is a steady canonical solution (cf. Fukue,
Ioroi 1999 for a Hoyle–Lyttleton case).
In the lower panel of figure 4, the canonical luminosity normalized by the Eddington one, Γcan (= Lcan /LE ), and the canonical accretion rate normalized by the Bondi one, ṁ(= Ṁ/ṀB ),
are shown as a function of parameter ṁB (= ṀB /ṀE ), by a solid
curve and a dashed one, respectively. On the contrary to the
spherical case, for a disk accretor, the normalized canonical
luminosity increases with ṁB , and can exceed unity for sufficiently large ṁB . Similar to the spherical case, as ṁB and Γcan
increase, the normalized accretion rate ṁ decreases. Compared
with the spherical case, however, the mass-accretion rate at
the same values of the normalized luminosity is somewhat enhanced.
account the influence of a central luminosity. We summarize
the main results:
1. When the central object is a “spherical” source, the accretion rate Ṁ reduces to ṀB (1 − Γ)2 , where Γ is the
normalized central luminosity. If the central luminosity
is produced by the accretion energy, the steady canonical
luminosity is determined and the normalized luminosity
does not exceed unity, as expected.
2. When the central object is a “disk” source, the accretion
rate becomes Ṁ/ṀB = 1 − 2Γd + (4/3)Γ2d for Γd ≤ 1/2,
and Ṁ/ṀB = 1/(6Γd ) for Γd > 1/2, where Γd is the
normalized disk luminosity. We also found the steady
canonical solutions, where the normalized luminosity
can exceed unity for sufficiently large accretion rates.
In this paper we emphasized that the anisotropic radiation
field of accretion disks greatly modifies the accretion nature.
We have assumed spherical symmetry for the case of a disk
accretor. Although this assumption does not change the present
result in a qualitative sense, a more sophisticated treatment is
necessary in order to obtain detailed quantitative results. The
flow behavior in the region of the cone of avoidance which
appeared for a disk accretor is also left as a future work.
6. Concluding Remarks
In this paper we have examined the spherical accretion of
ionized gas onto a central gravitating object, while taking into
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