Download Radiato-Magneto-Thermal Winds from an Accretion Disk

PASJ: Publ. Astron. Soc. Japan 56, 181–192, 2004 February 25
c 2004. Astronomical Society of Japan.
Radiato-Magneto-Thermal Winds from an Accretion Disk
Jun F UKUE
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582
[email protected]
(Received 2003 September 30; accepted 2003 December 15)
Abstract
We examine a hydrodynamical wind, which emanates from an accretion disk and is driven by thermal, magnetic,
and radiation pressures, under a one-dimensional approximation along supposed streamlines. The disk gas is
assumed to be isothermal, the magnetic field has only a toroidal component, and the radiation field is evaluated along
the streamline. Such a disk wind is characterized by an isothermal sound speed, the Alfvén speed at the footpoint,
and the strength of radiation fields. Isothermal winds can always blow even in the cold less-luminous case, and
transonic winds are established, except for the perfectly cold case without thermal pressure. Beyond some critical
luminosity, disk winds are always supersonic, irrespective of the thermal and magnetic pressures. We found that the
2
2
= (1/2)v02 −(1/2)(GM/r0 ) + 10.5aT2 + 0.7aA0
+ 16Γeff (GM/r0 ),
terminal speed v∞ is roughly expressed as (1/2)v∞
where v0 is the initial velocity, M the mass of the central object, r0 the radius of the wind base on the disk, aT the
isothermal sound speed, aA0 the initial Alfvén speed, and Γeff the effective normalized luminosity, although the
coefficients depend on the configuration of the streamlines.
Key words: accretion, accretion disks — astrophysical jets — galaxies: active — stars: winds, outflows —
X-rays: stars
1. Introduction
Astrophysical jets and winds are observed in various classes
of astronomical objects. For example, extragalactic radio jets
have been known in many active galactic nuclei (AGNs), such
as M 87 or 3C 273, for a long time. In these extragalactic radio
jets, the magnetic field seems to be important, at least, for collimation. Relativistic jets are ejected from several galactic X-ray
sources, such as the peculiar star SS 433 (0.26 c) and superluminal sources GRS 1915 + 105 and GRO J1655−40 (0.92 c).
In these microquasars, jets are originated from the very vicinity
of black holes. In the case of the SS 433 jet, the radiation force
may be significant, since SS 433 is a very luminous source.
Mass outflows are detected in supersoft X-ray sources (SSXSs)
and cataclysmic variables (CVs). In these objects, the radiation force is believed to be essential. Bipolar outflows are also
found in young stellar objects (YSOs). In these objects, the
magnetic field is detected and believed to be a main force. In
recent years, gamma-ray bursts (GRBs) are recognized to be
some type of astrophysical jets that occurs in supernova explosions. In this case the jets are highly relativistic with a Lorentz
factor of ∼ 100. It is believed, in any case, that the central
engine of jets is a gravitating object surrounded by a gaseous
disk.
Astrophysical jets and winds emanating from accretion
disks have been extensively investigated by many researchers.
Various driving forces are proposed, including thermal,
magnetic, and radiative ones. Traditional models of astrophysical jets and winds were usually concentrated into one of these
driving forces.
For example, hydrodynamical (thermal) winds emanating
from the disk surface have been discussed by many people
(e.g., Meier 1979, 1982; Fukue 1989; Takahara et al. 1989;
Fukue, Okada 1990; Nakamura et al. 1995; Idan, Shaviv 1996).
Meier (1979, 1982) first examined supercritical winds under
a spherical approximation, while a thermal disk wind was
examined by Fukue (1989) for a given configuration of streamlines, by Takahara et al. (1989) under a self-similar treatment,
and by Idan and Shaviv (1996) as a vertical flow.
On the other hand, radiative winds have also been developed
by many people, as on-axis one-dimensional jets (e.g., Icke
1989; Sikora et al. 1996), or as disk winds (Bisnovatyi-Kogan,
Blinnikov 1977; Icke 1980; Katz 1980; Piran 1982; Melia,
Königl 1989; Misra, Melia 1993; Tajima, Fukue 1996, 1998;
Fukue 1996, 1999; Watarai, Fukue 1999; Hirai, Fukue 2001;
Fukue et al. 2001), and under numerical simulations (Eggum
et al. 1985, 1988). In radiative disk winds, one of the key
concepts is radiation drag (e.g., Icke 1989; Piran 1982; Melia,
Königl 1989), while another key point is the complicated radiation field produced by the luminous disk (e.g., Tajima, Fukue
1996, 1998). These may strongly influence the terminal speed
and collimation of jets.
Finally, magnetohydrodynamical winds, accelerated by
magnetic pressure or centrifugal force via magnetic tension,
have been examined by many people (e.g., Blandford, Payne
1982; Pudritz, Norman 1983, 1986; Sakurai 1985, 1987;
Shibata, Uchida 1985, 1986; Kaburaki, Itoh 1987; Shu et al.
1994; see also Weber, Davis 1967 for solar winds, Michel
1969, Okamoto 1975 for pulsar winds). Blandford and Payne
(1982) firstly examined the centrifugal-wind-type model via
poloidal fields under the self-similar treatment. Sakurai (1987)
found the consistent stationary solution for two-dimensional
MHD flows from the disk. Numerical simulations were
actively performed by Shibata and collaborators (e.g., Uchida,
Shibata 1985; Shibata, Uchida 1985, 1986; Kudoh, Shibata
1997a,b; Koide et al. 1998, 1999). MHD winds driven by only
182
J. Fukue
toroidal magnetic fields were also examined (Lovelace et al.
1987; Fukue 1990; Contopoulos 1995). MHD winds from
a relativistic source were also examined (Camenzind 1987;
Fendt, Camenzind 1996; Li et al. 1992).
In the previous paper (Fukue 2002), thus, both the radiative and thermal effects were investigated simultaneously. The
author examined disk winds, which are driven by both thermal
and radiation pressures, emanating from some point on the
disk surface under a one-dimensional approximation along
supposed streamlines and under Newtonian gravity. Such a
disk wind is characterized by the disk gravitational and radiation fields, whose behavior is rather different from the spherical case. Along the streamline of winds, the gravitational field
produced by the central object generally has a peak at some
height, unlike the spherical case where it decreases monotonically. The radiation field produced by the disk, on the other
hand, is almost constant near to the disk surface and decreases
far from the disk, again unlike the spherical case. Due to these
characteristic properties of force fields, disk winds are classified into three patterns: in the cold less-luminous case no wind
can blow, in the warm luminous case transonic winds are established, and beyond some critical luminosity disk winds are
always supersonic. Fukue (2002) found that transonic winds
can blow for the parameter range of a0 + 2Γeff 0.8, where a0
is the initial sound speed in units of the Keplerian speed at the
wind base and Γeff the normalized disk luminosity at the wind
base. Furthermore, supersonic winds blow for Γeff 0.4.
In this paper we add the effect of magnetic pressure,
and examine disk winds emanating from some point on
the disk surface, driven by thermal, magnetic, and radiation pressures, under a one-dimensional approximation along
supposed streamlines and under Newtonian gravity.
In
addition, in the present paper the isothermal case is examined,
while the adiabatic case was examined in Fukue (2002).
In the next section we describe the model setup and basic
equations. In section 3 the properties of a single flow, where
one of driving forces is dominant, are examined. In section 4
the escape condition is examined. In section 5 the terminal
speed of disk winds is approximately derived. In section 6 the
present model is briefly applied to several objects. The final
section is devoted to concluding remarks.
2.
Basic Equations
Let us suppose a geometrically thin accretion disk
surrounding a central object of mass M, and a hydrodynamical wind emanating from a disk surface (figure 1). The disk
wind is by nature a two-dimensional flow, and each streamline should be self-consistently determined under the power
balance perpendicular to the streamline. In this paper we
assume the configuration of streamlines, take out some streamline with base r0 in the wind, and treat the gas stream as onedimensional flow along this streamline. Unless we adopt an
unusual configuration, such a streamline method may give a
sufficient result, at least for the qualitative properties of the
disk wind. In this section we write down the basic equations
for the present model (cf. Fukue 1996, 1999, 2002) under such
approximations.
[Vol. 56,
Fig. 1. Configuration of the streamlines of the present model (hyperbolae).
The winds emanate from the geometrically thin disk
surrounding a central object with mass M. The streamlines are vertical
near to the disk and expands far from the disk. The base radius of each
steamline on the disk is r0 . Each line corresponds to various zb . The
coordinate along the streamline is s.
2.1. Basic Equations along the Streamline
In this paper we ignore the general-relativistic effects and
use Newtonian gravity. We also ignore the special-relativistic
effects, including the radiation drag force, and remaining terms
up to the zeroth order of (v/c). We take cylindrical coordinates
(r, ϕ, z ) with the z -axis along the rotation axis of the disk.
Furthermore, the flow is supposed to be steady and axisymmetric: ∂t = 0 = ∂ϕ . As for magnetic fields, we only consider
the toroidal component. The governing equations under these
situations and for the non-magnetic case have been written by
Fukue (2002).
We further introduce streamline coordinates (s, ϕ, t), where
the s-axis lays parallel to the supposed streamline (Fukue,
Okada 1990; Fukue 1996, 1999); s is the length along the
streamline, ϕ the azimuthal angle, and t the length perpendicular the streamline. In such streamline coordinates the velocity
perpendicular the streamline automatically becomes zero.
Since we suppose steady and axisymmetric flow, the basic
equations along the streamline are expressed as follows. The
continuity equation is written as
d
(Aρv) = 0,
(1)
ds
where A is the cross-sectional area specified later, ρ the
density, and v the velocity along the streamline. The equation
of motion is written as
1 dp GM rdr + z d z 2 dr
dv
=−
− 3
+ 3
ds
ρ ds
R
ds
r ds
κabs + κsca
Bϕ d rBϕ +
F,
(2)
−
4πρr ds
c
√
where p is the gas pressure, R = r 2 + z 2 , the specific
angular momentum, Bϕ the toroidal magnetic field, κabs and
v
No. 1]
Radiato-Magneto-Thermal Winds from an Accretion Disk
κsca the absorption and scattering opacities, respectively, and F
the radiative flux from the luminous disk along the streamline.
The flux F is almost vertical near to the disk, and diverges far
from the disk.
The gas pressure and density are related as
p = ρaT2 ,
where aT is the isothermal sound speed.
The induction equation is reduced to
d AvBϕ = 0.
ds
The equations for radiation fields become
(3)
(4)
1 d
(AF ) = ρ (j − cκabs E) ,
(5)
A ds
1
dP
= − ρ(κabs + κsca )F,
(6)
ds
c
1
(7)
P = E,
3
where E is the radiation energy density, P the radiation
pressure along the streamline, and j the emissivity. As Fukue
(2002), the right-hand side of equation (5) is dropped.
Finally, the optical depth τ along the streamline is
dτ
= (κabs + κsca )ρ.
(8)
ds
Instead of the streamline coordinate s, we use the variable
z for convenience. We replace the variable z instead of s
with
the help of a relation between the line elements: ds =
d z 1 + (dr/d z )2 .
2.2. Model Geometry
We here specify the configuration of streamlines (Fukue
2002).
If the gas originally corotates with the disk and starts to blow
off by the thermal, magnetic, and radiation pressures, the gas
stream may be vertical in the very vicinity of the disk surface.
Far from the disk, on the other hand, there would be several
cases. For example, the streamlines incline and extend, and
the gas stream approaches more or less radial flows, when the
centrifugal force dominates, or the radiation force dominates.
They would be collimated, when the pressure of the outer
corona operates, or the magnetic pinch sufficiently works. In
this paper we consider the former case of diverging streamlines (see, e.g., Fukue 1990 for several cases of streamlines
with toroidal magnetic fields). Among various configurations
satisfying these properties, we adopt a hyperbola as a simple
case; we assume that the streamline whose base r0 is located
on the disk is expressed as
z
r2
=
− 1,
(9)
zb
r02
where zb is some constant and dr/d z = (r02 /zb2 )z /r.
For this assumed streamline, the cross-sectional area A is
expressed as
2π r∂r
1
A
=
2
A0
2π
r0 ∂r0
1 + (dr/d z )
2
z
1
=
+
1
,
1 + (dr/d z )2 zb2
183
(10)
where A0 is a constant.
In this paper we ignore the angular momentum removal
in the wind, such as viscosity or radiation drag. Hence, the
specific angular momentum of the wind is constant along the
streamline, and is expressed by
= GMr0 .
(11)
2.3. Wind Equation
The basic equations for the present model are ultimately
written as follows:
Aρv = Ṁ (constant),
v
(12)
dr
2 dr
z+r
+ 3
dz
r dz
2
κabs + κsca
dr
Bϕ d rBϕ +
F 1+
−
,
4πρr d z
c
dz
(13)
1 dp GM
dv
=−
− 3
dz
ρ dz
R
p = ρaT2 ,
(14)
ABϕ v = Φ (constant),
(15)
AF = A0 F0 (constant),
(16)
1
dP
1
= − ρ(κabs + κsca )F,
c
1 + (dr/d z )2 d z
(17)
1
P = f E = E,
3
1
dτ
= (κabs + κsca )ρ.
2
1 + (dr/d z ) d z
(18)
(19)
According to the usual procedure for transonic flows, these
equations are rearranged into a linear set of the first-order
differential equations, and we derive the wind equation, an
ordinary differential equation of the first order on the variable.
After some manipulations, we have
1 dA
1 dr
dv
(v 2 − aT2 − aA2 )
= v aT2 + aA2
− aA2
dz
A dz
r dz
2
GM
dr
dr
− 3 z+r
+ 3
R
dz
r dz

2
dr
κabs + κsca
 , (20)
F 1+
+
c
dz
where aA is the Alfvén speed defined by
Bϕ2
Φ2
Φ2 1
=
.
(21)
ρ
=
4πρ 4π Ṁ 2
4π Ṁ Av
Finally, the equations are normalized.
Measuring the radius
√
and velocity in units of r0 and GM/r0 , respectively, and with
aA2 ≡
184
J. Fukue
the help of equation (16), the wind equation is rewritten in the
dimensionless forms:
1 dA
1 dr
dv
(v 2 − aT2 − aA2 )
= v aT2 + aA2
− aA2
dz
A dz
r dz
1
dr
1 dr
− 3 z+r
+ 3
R
dz
r dz

2
dr
A0
,
(22)
+ Γeff
1+
A
dz
where the symbols for the dimensionless variables, say â, are
dropped for simplicity. In these equations Γeff is a dimensionless parameter, which roughly expresses the normalized
luminosity evaluated at the wind base on the disk, and defined
by
Γeff ≡
κabs + κsca 4π r02 F0
,
κT
LE
(23)
where LE (= 4π cGM/κT ) is the Eddington luminosity of the
central object. It should be noted that this Γeff is different from
the normalized luminosity of the total disk Ld /LE , where Ld is
the total disk luminosity.
Similarly, the other quantities, such as the optical depth and
the radiation energy density, can be calculated:
2
dr
1 A0
dτ
= Γadv
1+
,
(24)
dz
v A
dz
2
2
dr
E
1 A0
d
1
1+
,
(25)
= − Γadv
d z F0 /c
f
v A
dz
where f = 1/3. Here, Γadv is also a dimensionless parameter,
which roughly expresses the normalized advection luminosity
for the relevant streamline of the wind, and is defined by
√
κabs + κsca 4π r02 Ṁc2 GM/r0
.
(26)
Γadv ≡
κT
A0 LE
c
The Alfvén speed is also normalized as
A0 1
,
(27)
A v
where Γmag is a dimensionless parameter, which expresses the
magnitude of the toroidal magnetic field, defined by
aA2 = Γmag
Γmag ≡
Φ2
.
4π ṀA0 (GM/r0 )3/2
(28)
In the followings, however, we often use the Alfvén speed aA0
on the disk surface, instead of Γmag , as a parameter to express
2
the magnitude of the magnetic field: aA0
= Γmag /v0 .
As was discussed and demonstrated in the previous papers
(Fukue 2002; cf. Fukue 1989; Fukue, Okada 1990; Fukue
1996), the behavior of the force field along the streamline is
very different from that of the spherical case. We do not repeat
them but briefly summarize the fundamental properties (see
Fukue 2002 for details).
The force fields along the streamline are not monotonic
functions in the disk wind, unlike the simple spherical case.
As a result, the transonic nature of the disk wind becomes
[Vol. 56,
somewhat complicated, and multiple critical points appear in
some cases.
When the radiative force is included, there often appear two
equilibrium points: lower points are dynamically stable, while
higher ones are unstable (Fukue 1996). That is, the matter feels
the attractive force toward the disk when it is located near to
the disk under the higher equilibrium points. Once the matter
exceeds the higher equilibrium points, it is lifted up further,
since the radiative force overcomes the gravitational one. If the
disk is sufficiently luminous, the equilibrium points disappear
and the matter flows freely from the disk.
If the pressure gradient force is included, these properties are qualitatively similar. For some ranges of magnetic
and thermal pressures, there appear (generally two) multiple
critical points; a lower one is a center type, corresponding
to the stable equilibrium point, and a higher one is a saddle
type, corresponding to the unstable equilibrium point. As
a result, for some ranges of driving forces and with some
specified boundary conditions on the disk surface, there exists
a transonic wind solution, which passes through the higher
saddle-type critical point (cf. Fukue 1989).
3.
Solutions of the Wind Equation
We now obtain wind solutions in the present model and
discuss several of their properties in the subsequent sections.
3.1. Critical Conditions
As is well known, the wind equation (22) becomes critical at
the height where the denominator and numerator vanish simultaneously.
The critical conditions (regularity condition) are
A0 1 2
2
2
2
,
(29)
v = aT + aA = aT + Γmag
A v
c
2
1
dr
2 1 dA
2 1 dr
aT + aA
− aA
−
z+r
A dz
r d z R3
dz
2 dr
A0
1 dr
+ Γeff
1+
(30)
+ 3
= 0,
r dz
A
dz
c
√
where quantities are in units of r0 and GM/r0 .
This relation (30) is shown in figure 2 for several combinations of parameters. In figure 2 the abscissa is the height zc of
the critical points, while the ordinate is the effective luminosity
Γeff . In figure 2a the isothermal sound speed aT is 0 to 9 in
steps of 1 from top to bottom, whereas Γmag = 0. In figure 2b
the parameter Γmag is 0, 10, 100, 1000, 10000 from out to in,
whereas aT = 0. The parameter zb is 10 in both cases.
For example, in the outermost large curve in figure 2a
(aT = 0), there are two intersections for a fixed-Γeff horizontal
line in the range of 0 < Γeff ≤ 0.4; the left intersection is center
and the right is saddle. However, in order for a transonic
solution to exist from the disk surface to infinity, the range
becomes narrower 0 < Γeff ≤ 0.25, as will be stated later. On the
other hand, in the innermost small curve in figure 2a (aT = 9),
there are two intersections in the range of 0 < Γeff ≤ 0.03. For
aT ≥ 10, there is no intersection.
No. 1]
Radiato-Magneto-Thermal Winds from an Accretion Disk
v ∼ v0 −
Fig. 2. Relations between the effective luminosity Γeff and the height
zc of the critical points for several values of the parameters. (a) The
isothermal sound speed aT is 0 to 9 in steps of 1 from top to bottom,
whereas Γmag = 0. (b) The parameter Γmag is 0, 10, 100, 1000, 10000
from out to in, whereas aT = 0. The parameter zb is 10 in both cases.
As can be seen in figure 2 and already mentioned in the
previous section, in order for critical points to exist, there is an
upper limit on Γeff for some fixed aT or Γmag . In this case of zb
there appear two critical points, if they exist for some combination of parameters. The left-side part of the peak (lower critical
points) is center, while the right-side (higher points) is saddle.
3.2. Wind Solutions
For appropriate boundary conditions at the disk plane (z = 0),
we solve the wind equation (22) along the streamline by using a
shooting method.
The boundary conditions adopted are: v = v0 ,
aA = aA0 = Γmag /v0 , τ = 0, and E = E0 = 2F0 /c at the disk
plane. In the very vicinity of the disk plane, the solutions are
approximately expressed as
v0
Γ z,
2 − v 2 eff
aT2 + aA0
0
185
(31)
τ∼
Γadv
z,
v0
(32)
E∼
1 Γadv
E0
−
z.
F0 /c f v0
(33)
In order for the solution to pass through the critical point
(regularity condition), the boundary values cannot be given
arbitrarily, but some condition is imposed; i.e., the initial
velocity v0 is uniquely determined for given parameters.
Several typical transonic solutions are shown in figure 3,
where the abscissa is the height z , while the ordinates are the
flow velocity v (solid curve), the isothermal sound speed aT
(dotted one), and the Alfvén speed aA (dashed one). In this
example the parameter zb is 10.
In figure 3a (thermally-driven wind) the parameters are
Γeff = 0, Γmag = 0.0001, aT = 0.5, and therefore v0 = 0.0209.
In this case the flow is mainly driven by the thermal pressure,
and accelerated at around z ∼ zb , where the flow cross-section
increases. Although the isothermal sound speed aT is constant
in the present isothermal flow, the Alfvén speed aA decreases,
as the flow cross-section and speed increase.
In figure 3b (magnetically-driven wind) the parameters are
Γeff = 0, Γmag = 1, aT = 0.0001, and therefore v0 = 0.472. In
this case the flow is mainly driven by the magnetic pressure.
The Alfvén speed aA decreases, as the flow cross-section and
speed increase. In contrast to the thermally-driven wind, the
terminal speed v∞ of the magnetically-driven wind is close to
the initial Alfvén speed aA0 at the disk. This is usually the
case, as long as the magnetic pressure is dominant. Indeed, as
was already noticed in the previous papers (e.g., Fukue 1990;
Kudoh, Shibata 1997a,b; Kato et al. 2003), when the flow is
mainly driven by the magnetic pressure force of the toroidal
magnetic fields, the terminal speed is about of the order of
the Alfvén speed at the flow base, which is of the order of the
Keplerian speed near the disk. This point is discussed later.
In figure 3c (radiatively-driven wind) the parameters are
Γeff = 0.25, Γmag = 0.001, aT = 0, and therefore v0 = 0.012555.
In this case the flow is mainly driven by the radiation pressure.
The velocity, which is constant near to the disk plane, quickly
increases around the critical point. As the flow cross-section
diverges and the radiative flux decreases, the flow speed
approaches the terminal one. Although for this parameter set
the radiatively-driven wind passes through the critical point,
the flow is accelerated in the supersonic regime for larger
luminosity Γeff .
In these examples we did not show the optical depth τ or
the radiation energy density E of the flow, since the equations
for τ and E are decoupled with the dynamical equations, and
the control parameter Γadv does not influence the flow pattern
(Fukue 2002). As the value of Γadv increases, the optical
depth also becomes large, while the radiation energy density
decreases at large distance. For such a large value of Γadv ,
where the optical depth highly exceeds unity, the radiation
energy density quickly drops and we cannot obtain a solution.
186
J. Fukue
4.
[Vol. 56,
Escape Condition for the Wind
As already stated, for transonic solutions the initial flow
velocity v0 is related to the parameters aT , Γmag (or aA0 ),
Γeff , and zb . Such relations are escape conditions for the
flow to escape from the disk to infinity. In contrast to the
previous study for the radiative-thermal wind (Fukue 2002), the
escape conditions for the present case are somewhat complicated, since the effect of the magnetic pressure is included and
there are many parameters. Hence, we examine several special
combinations of the parameters.
The escape conditions for several cases are shown in figure 4
(we fix zb = 10). In figure 4a (Γmag = 0), relations between the
initial velocity v0 and the isothermal sound speed aT are plotted
for several normalized luminosity Γeff . The values of Γeff is 0,
0.1, 0.2, and 0.25 from bottom to top. In the present isothermal
case, for some fixed Γeff , it seems that there exist transonic
solutions for appropriate combinations of v0 and aT . In the
adiabatic case examined by Fukue (2002), on the other hand,
no winds take place for the cold less-luminous cases; transonic
solutions exist for the parameter range of a0 + 2Γeff ≥ 0.8,
where a0 is the initial adiabatic sound speed. This is understood
as follows. In the adiabatic case there is no energy injection
except for radiative energy. Hence, for the cold less-luminous
cases the energy deposit is too small to blow off the disk gas
toward infinity. In the present isothermal case there implicitly
exists energy injection. Furthermore, when the flow velocity
is small, the energy deposit increases. As a result, even in the
cold case the gas can escape toward infinity if the initial speed
is small and there is enough time for the gas to receive sufficient heat energy.
In addition, in the case of Γmag = 0, transonic solutions exist
for a range of
Γeff ≤ 0.25,
Fig. 3. Typical transonic solutions as a function of z . The curves are
the flow velocity v (solid curve), the isothermal sound speed aT (dotted
one), and the Alfvén speed aA (dashed one). (a) Thermally-driven
wind (Γeff = 0, Γmag = 0.0001, aT = 0.5), where v0 = 0.0209. (b)
Magnetically-driven wind (Γeff = 0, Γmag = 1, aT = 0.0001), where
v0 = 0.472. (c) Radiatively-driven wind (Γeff = 0.25, Γmag = 0.001,
aT = 0), where v0 = 0.012555. The parameter zb is 10.
(34)
while the flow supersonically blows off for a luminous range
of Γeff ≥ 0.25. This condition is narrower than that expected
from the critical condition (figure 2). That is, in figure 2 critical
points exist for a range of Γeff ≤ 0.4. Solutions passing through
critical points in a range of 0.25 ≤ Γeff ≤ 0.4 do not reach the
disk surface at z = 0. In other words, the condition Γeff ≤ 0.25
means that there exist such transonic solutions that start from
the disk to connect infinity.
Moreover, for a fixed Γeff , the initial velocity v0 increases as
the isothermal sound speed aT increases. When aT is large, the
flow velocity at a critical point is also large. Hence, in order
to path the critical point, a higher initial velocity is required
(regularity condition).
In figure 4b (Γeff = 0), relations between the initial velocity v0
and the isothermal sound speed aT are plotted for several Γmag .
The values of Γmag is 0, 0.01, 0.1, and 1 from bottom to top.
Similar to the case in figure 4a, there always exist transonic
solutions even in the cold case with small aT .
Moreover, for a fixed Γmag , the initial velocity v0 increases as
the isothermal sound speed aT increases. When aT is large, the
flow velocity at a critical point is also large. Hence, in order
to path the critical point, a higher initial velocity is required
(regularity condition).
In figure 4c (aT = 0), the relations between the initial velocity
No. 1]
Radiato-Magneto-Thermal Winds from an Accretion Disk
187
v0 and the initial Alfvén speed aA0 are plotted for several Γeff .
The values of Γeff are 0, 0.1, 0.2, and 0.25 from bottom to top.
Although we can plot relations between v0 and Γmag , we show
relations between v0 and aA0 , since the relations are approximately straight lines in this case. In contrast to other cases in
figures 4a and 4b, there exist minimum values of aA0 for some
fixed Γeff in this case, similar to the adiabatic case. In other
words transonic winds can blow off for the parameter range of
0.86aA0 + 2Γeff ≥ 0.74,
(35)
where aA0 is the initial Alfvén speed. This approximate condition has a similar form with the adiabatic case of a0 + 2Γeff ≥
0.8, where a0 is the initial adiabatic sound speed.
Furthermore, the existence of the magnetic pressure moves
up the initial velocity v0 . That is, for a fixed Γeff , the initial
velocity v0 increases as the Alfvén speed aA0 increases. When
aA0 is large, the flow velocity at a critical point is also large.
Hence, in order to path the critical point, a higher initial
velocity is required (regularity condition).
5.
Terminal Speed of the Wind
Now, we derive an approximate expression for a terminal
speed. For transonic solutions the terminal speed v∞ of the
flow depends on the initial condition and various parameters:
v0 , aT , Γmag (or aA0 ), Γeff , and zb . It is very complicated
to examine all of the parameters simultaneously. We thus
examine (i) one of parameters aT , aA0 , Γeff for a fixed zb ,
(ii) two of parameters aT , aA0 , Γeff for a fixed zb , and (iii) the
dependence on the configuration zb .
In figure 5 relations between the terminal speed and other
quantities for transonic solutions with a single driving force
are shown.
In figure 5a (thermally-driven wind) the parameters are
Γeff = 0 and Γmag = 0.0001, and the terminal speed v∞ , the
velocity vc at the critical point, the initial velocity v0 , the
Alfvén speed aAc at the critical point, and the initial Alfvén
speed aA0 are shown as a function of the isothermal sound
speed aT . In this case, except for a cold regime of small aT ,
the terminal speed is roughly expressed as
1 2
(36)
v ∼ 10.5aT2 .
2 ∞
This approximate relation is shown in figure 5a by the thick
dotted line.
In figure 5b (magnetically-driven wind) the parameters are
Γeff = 0 and aT = 0.0001, and the terminal speed v∞ , the velocity
vc at the critical point, the initial velocity v0 , the Alfvén speed
aAc at the critical point, and the initial Alfvén speed aA0 are
shown as a function of Γmag . In this case, as already mentioned,
we have
Fig. 4. Relations among the parameters for transonic solutions. (a)
Relations between v0 and aT for several Γeff in the case of Γmag = 0. The
values of Γeff is 0, 0.1, 0.2, and 0.25 from bottom to top. (b) Relations
between v0 and aT for several Γmag in the case of Γeff = 0. The values
of Γmag is 0, 0.01, 0.1, and 1 from bottom to top. (c) Relations between
v0 and aA0 for several Γeff in the case of aT = 0. The values of Γeff is 0,
0.1, 0.2, and 0.25 from bottom to top. The other parameter is zb = 10.
v∞ ∼ aA0 ∼ (Γmag /v0 )1/2 ,
(37)
where we use equation (27). This nature is consistent with the
previous study by Fukue (1990), where the steady disk flow
is accelerated by toroidal magnetic fields, or by Kudoh and
Shibata (1997a,b), where there exist poloidal as well as toroidal
magnetic fields. In a recent simulation with toroidal magnetic
fields, furthermore, Kato et al. (2003) confirmed this nature.
188
J. Fukue
[Vol. 56,
In figure 5c (radiatively-driven wind) the parameters are
Γmag = 0.001 and aT = 0.5, and the terminal speed v∞ , the
velocity vc at the critical point, the initial velocity v0 , the
Alfvén speed aAc at the critical point, and the initial Alfvén
speed aA0 are shown as a function of Γeff . In this case the
terminal speed is approximately expressed as
v∞ ∼ 5aT + 4Γeff .
(38)
This approximate relation is shown in figure 5c (aT = 0.5) by
the thick dotted curve.
Next, we examine the two-parameter case. In figure 6
relations between the terminal speed and other quantities for
transonic solutions with two driving forces are shown.
In figure 6a (radiatively-thermally-driven wind; Γmag = 0),
the terminal speed v∞ is shown as a function of the isothermal
sound speed aT for several values of the normalized luminosity
Γeff . In this case any relation becomes a straight line, and the
terminal speed is expressed as
1 2
v ∼ 10.5aT2 + 16Γeff .
(39)
2 ∞
This is the generalized form of relations (36) and (38).
In figure 6b (magnetically-thermally-driven wind; Γeff = 0),
2
the terminal speed v∞ is shown as a function of aA0
for several
values of aT . In this case the relations are roughly straight lines,
and the terminal speed is roughly expressed as
1 2
2
v ∼ 10.5aT2 + 0.7aA0
.
(40)
2 ∞
This is the generalized form of relations (36) and (37).
In figure 6c (magnetically-radiatively-driven wind; aT = 0),
2
the terminal speed v∞ is shown as a function of aA0
for several
values of Γeff . In this case the terminal speed is approximately
expressed as
1 2
2
v ∼ (0.7 + 0.5Γeff ) aA0
+ 16Γeff .
(41)
2 ∞
This is the generalized form of relations (37) and (38).
Combining all of the above expressions, in the case of zb =
10, the terminal speed v∞ is approximately expressed as
1 2
1
1
v∞ ∼ v02 − + 10.5aT2
2
2
2
2
+ (0.7 + 0.5Γeff ) aA0
+ 16Γeff
Fig. 5. Relations between the terminal speed v∞ and other quantities for transonic solutions with a single driving force.
The
solid curves represent the flow velocities, whereas the dashed ones
denote the Alfvén speeds. (a) Thermally-driven wind (Γeff = 0
and Γmag = 0.0001). The thick dotted line shows an approximate
relation. (b) Magnetically-driven wind (Γeff = 0 and aT = 0.0001). (c)
Radiatively-driven wind (Γmag = 0.001 and aT = 0.5). Other parameter
is zb = 10. The thick dotted curve shows an approximate relation.
(42)
in the dimensionless form, where the speed is measured in units
of the Keplerian speed at the wind base on the disk. It should be
mentioned that in the adiabatic case without magnetic pressure
(Fukue 2002) the terminal speed is approximately expressed as
2
(1/2)v∞
∼ (1/2)v02 + [1/(γ − 1)]a02 − (1/2) + 15.8Γeff , where
γ is the ratio of specific heats, and a0 is the initial adiabatic
sound speed at the wind base. Hence, for the adiabatic case the
terminal speed becomes
1
1 2
1
1
v ∼ v2 − +
a2
2 ∞ 2 0 2 γ −1 0
2
+ (0.7 + 0.5Γeff ) aA0
+ 16Γeff
(43)
in the dimensionless form. In these expressions, (1/2)v02
means the initial kinetic energy at the wind base, and −(1/2)
(= −GM/r0 + 2 /2r02 in the usual form) represents the initial
gravitational and centrifugal energies at the wind base.
No. 1]
Radiato-Magneto-Thermal Winds from an Accretion Disk
189
It should be noted that, for the magnetic part in
equation (42),
1/2
Φ2
ρ0
(44)
v∞ ∼ aA0 ∼
4π Ṁ 2
using equation (21), where Φ is the magnetic flux for toroidal
fields, Ṁ the mass-loss rate, and ρ0 the density at the wind
base. On the other hand, in the case of magnetic flows with a
poloidal field, Michel’s scaling law (Michel 1969) is known:
2 2 1/3
Ψ Ω0
,
(45)
v∞ ∼
Ṁ
where Ψ is the magnetic flux for poloidal fields, Ṁ the massloss rate, and Ω0 the angular speed at the wind base. In
the present case without poloidal fields, Michel’s scaling law
cannot be applied. Instead, we have the scaling law in the form
of equation (44).
Finally, we examine the dependence on the structural
parameter zb . Up to now, we fixed this parameter as zb = 10 r0 .
For a large value of zb , the streamline is almost vertical, while
the streamline is remarkably curved for a small value of zb .
Hence, the value of zb determines the structural properties of
streamlines qualitatively.
In addition, in equation (42), the dependence of the terminal
speed on each element of the driving forces is mainly a sum of
each term expressing thermal, magnetic, and radiative effects.
Hence, we examine in turn the dependence on zb of the coefficient of each term.
Figure 7 shows relations between the terminal speed and
the parameters for several values of the structural parameter
zb under a single driving force.
In figure 7a we show the relations for a thermally-driven
wind, where Γmag = 0 and Γeff = 0, for several values of zb .
These results are approximately fitted by
1 2
v ∼ (15 − 4.5 log zb ) aT2 .
(46)
2 ∞
In figure 7b we show the relations for a magnetically-driven
wind, where Γeff = 0 and aT = 0, for several values of zb . These
results are approximately fitted by
1
1 2
2
v ∼ 0.7 + log 2 + 1 aA0
.
(47)
2 ∞
zb
In figure 7c, finally, we show the relations for a radiativelydriven wind, where Γmag = 0 and aT = 0, for several values of
zb . These results are approximately fitted by
Fig. 6. Relations between the terminal speed v∞ and other quantities for transonic solutions with two driving forces. (a) Radiatively-thermally-driven wind (Γmag = 0). The values of Γeff is 0, 0.1,
0.2, and 0.25 from bottom to top. (b) Magnetically-thermally-driven
wind (Γeff = 0). The values of aT is 0.2, 0.4, 0.6, 0.8, and 1 from bottom
to top. (c) Magnetically-radiatively-driven wind (aT = 0). The values
of Γeff is 0, 0.1, 0.2, and 0.25 from bottom to top. Another parameter
is zb = 10.
1 2
v ∼ 1.56zb Γeff .
(48)
2 ∞
We ultimately obtain an approximate expression for the
terminal speed v∞ as
1 2
1
1
v ∼ v2 −
2 ∞ 2 0 2
+ (15 − 4.5 log zb ) aT2
1
2
+ 0.7 + log 2 + 1 + 0.5Γeff aA0
zb
+ 1.56zb Γeff ,
(49)
190
J. Fukue
[Vol. 56,
where the length and velocity are measured
in units of the
√
initial radius r0 and the Keplerian speed GM/r0 at the wind
base, respectively.
On the right-hand side of this approximate expression, the
first two are the initial kinetic and binding energies at the wind
base on the disk surface. The third term is the thermal part, and
becomes large for a smaller zb . That is, as zb becomes small,
the streamline expands so quickly that the thermal energy is
effectively converted to kinetic energy. The fourth term is the
magnetic part. Although the coefficient weakly depends on
the other parameters, the initial magnetic energy is converted
to kinetic energy. The final term is the radiative part, and
this means the work done by the radiation pressure, similar
to Fukue (2002). This radiative term becomes large with zb ,
since the work done by radiation pressure becomes large for
vertical-like streamlines.
6.
Application to Astronomical Sources
In this section, we briefly apply the present model of disk
winds, including thermal, magnetic, and radiation pressures, to
several astrophysical objects.
The present unit and parameters are expressed in terms of
the physical unit as follows. When we consider the accretion disk around a Schwarzschild black hole of mass M (the
Schwarzschild radius is rg = 2GM/c2 ), let the radius r0 of the
wind base be
M
r0 = xrg = 3.0 × 1013 cm 8
x.
(50)
10 M
Then, the unit of velocity becomes
GM
c
= √ = 2.1 × 1010 cm s−1 x −1/2 .
(51)
r0
2x
The isothermal sound speed aT in a dimensionless form is
expressed as
aT2
p0 /ρ0
T
x,
=
=
GM/r0 GM/r0 2.7 × 1012 K
(52)
where we use the equation of state and suppose an ionized
hydrogen. Furthermore, the Alfvén speed aA0 at the wind base
is
2
2
2
Bϕ0
Bϕ0
/4π aT2
/4π
aA0
T
=
=
x, (53)
12
GM/r0
p0 GM/r0 2.7 × 10 K p0
where the subscript 0 denotes the value at the wind base. As
for the effective luminosity Γeff , the disk flux F0 at the wind
base depends on the disk model. In the case of self-similar
supercritical accretion disks with a conical cross section of an
opening angle δ (Fukue 2000), the flux F0 is expressed as F0 =
(3/4) tan δ(LE /4π r02 ). Then,
Fig. 7. Relations between the terminal speed and the parameters for
several values of the structural parameter zb . (a) Thermally-driven wind
(Γmag = 0 and Γeff = 0). The values of zb is 1, 2, 5, and 10 from top to
bottom. (b) Magnetically-driven wind (Γeff = 0 and aT = 0). The values
of zb is 1, 2, 5, and 10 from top to bottom. (c) Radiatively-driven wind
(Γmag = 0 and aT = 0). The values of zb is 1, 2, 5, 10, and 100 from
bottom to top.
Γeff =
κabs + κsca 4π r02 F0 3 κabs + κsca
=
tan δ.
κT
LE
4
κT
(54)
It should be noted that the present treatment is Newtonian.
Hence, when the velocity is ∼ c, there is an error of the order
of factor two.
No. 1]
Radiato-Magneto-Thermal Winds from an Accretion Disk
Fig. 8. Example of disk winds from an optically-thin ADAF source.
The adopted parameters aT = 1, aA0 = 0.989 (Γmag = 0.6), and
Γeff = 0. The initial velocity is v0 = 0.6135 and the terminal velocity is
v∞ = 4.73. The parameter zb is 10.
6.1. Optically-Thin ADAF Sources
In several astrophysical objects, including the Galactic
Center, low (hard) states of galactic black hole binaries,
some stages of quasars and microquasars, an accretion rate
is supposed to be very low. As a result, the central disk
would be an optically-thin advection-dominated accretion flow
(optically-thin ADAF), which was proposed by, e.g., Ichimaru
(1977), Narayan and Yi (1994). In such an optically-thin
ADAF, the gas temperature is very high to be of the order of the
virial temperature, and the thermal pressure may be dominant,
while the radiation pressure is less important. For example,
in the case of the Galactic Center Sgr A∗ , the accretion rate
is of the order of 10−5 of the Eddington rate (e.g., Narayan
et al. 1995). There may exist a magnetic field, but the strength
of the magnetic pressure is unknown, and the equipartition is
usually assumed (e.g., Narayan, Yi 1995; Kato et al. 1998).
The virial temperature at r0 is T = 2.7 × 1012 K/x. Hence, with
the help of equations (52), (53), and (54), for radiato-magnetothermal winds emanating from optically-thin ADAF sources,
we assume that aT ∼ 1 (virial temperature), aA0 ∼ 1 (equipartition), and Γeff ∼ 0.
An example is shown in figure 8, where the solid curve
represents the flow velocity v, the dashed one denotes the
Alfvén speed aA , and the dotted one is the isothermal sound
speed aT . The adopted parameters are aT = 1, Γmag = 0.6
(aA0 = 0.989), and v0 = 0.6135. The terminal speed is then
v∞ = 4.73.
The wind is mainly driven by thermal and magnetic
pressures, and in a wide range of the parameter there exist
transonic solutions, where the critical height is around zb (= 10),
Since the thermal pressure is sufficiently high and the magnetic
field is assumed to be equipartition, the terminal speed is much
higher than the Keplerian speed at the wind base on the disk.
191
Fig. 9. Example of disk winds from a supercritical accretion source.
The adopted parameters aT = 0.01, aA0 = 0.01 (Γmag = 0.00001, and
Γeff = 1. The initial velocity is v0 = 0.1 and the terminal velocity is
v∞ = 5.51. The parameter zb is 10.
For example, in the case of figure 8, for r0 = 100 rg (x = 100),
the velocity unit is 0.07 c and v∞ = 0.33 c, while the velocity
unit is 0.22 c and v∞ ∼ c for r0 = 10 rg .
6.2. Supercritical Accretion Sources
In several astrophysical objects, including SS 433, narrowline Seyfert 1 galaxies, some stages of quasars and microquasars, and ultraluminous X-ray sources, an accretion rate is
supposed to be very high and of the order of the Eddington
rate. As a result, the central disk would be an optically-thick
advection-dominated accretion flow or a supercritical accretion disk (Abramowicz et al. 1988). In such a supercritical
accretion disk, the gas temperature is not so high, and radiation pressure is quite important. Since we do not know the
strength of magnetic field, we assume the equipartition as in
usual case. In the case of the supercritical disk around the
galactic stellar-mass black holes, the temperature of the inner
disk is about 107–8 K. Hence, for radiato-magneto-thermal
winds emanating from the inner region (x ∼ 10) of supercritical accretion sources, we assume that aT ∼ 0.01, aA0 ∼ 0.01
(equipartition), and Γeff = 1.
An example is shown in figure 9, where the solid curve
represents the flow velocity v, the dashed one denotes the
Alfvén speed aA , and the dotted one is the isothermal sound
speed aT . The adopted parameters are aT = 0.01, Γmag =
0.00001 (aA0 = 0.01), and v0 = 0.1. The terminal speed is then
v∞ = 5.51.
The wind is mainly accelerated by radiation pressure, and
usually the flow is initially supersonic without critical points.
Irrespective of the strength of the magnetic pressure, the
terminal speeds have similar values, which are much higher
than the Keplerian speed at the wind base, since the work done
by the radiation pressure is the same for a fixed configuration
(zb = 10). For example, in the case of figure 9, for r0 = 10 rg
192
J. Fukue
(x = 10), the velocity unit is 0.22 c and v∞ ∼ c.
7.
Concluding Remarks
In this paper we have examined disk wind driven by thermal,
magnetic, and radiation pressures under the one-dimensional
approximation along each streamline of winds.
In the present paper we a priori specify and give the configuration of the streamlines. In real winds the streamline must
be determined and adjusted by the force balance perpendicular
to the streamline, self-consistently. However, as long as the
streamline is vertical near to the disk plane and expands far
from the disk, the conclusions of the present model are qualitatively valid.
The main results are summarized as follows:
1. There appear multiple critical points for the disk winds,
as in the previous case (Fukue 2002), and the present
disk wind is characterized by an isothermal sound speed
aT , the Aflvén speed aA0 at the wind base, and the
normalized luminosity Γeff .
2. Isothermal winds can always blow even in the cold
less-luminous case, and transonic winds are established, except for the perfectly cold case without thermal
pressure. Beyond some critical luminosity (Γeff 0.25),
i.e., in the case of a sufficiently luminous disk, winds
always become supersonic, irrespective of the thermal
and magnetic pressures.
3. Several conditions and relations, including an approximate expression for the terminal speed, are obtained
for the present disk wind. For the configuration
parameter of zb = 10, the terminal speed is approx2
imately expressed as (1/2)v∞
∼ (1/2)v02 − (1/2) +
2
2
10.5aT + (0.7 + 0.5Γeff ) aA0 + 16Γeff , where the length
and velocity are measured
by the initial radius r0 and the
√
Keplerian speed GM/r0 there, respectively.
4. The present model is applied to optically-thin ADAF
sources and supercritical accretion sources. In the
former case the wind is mainly driven by thermal and
magnetic pressures, while it is mainly driven by radiation pressure in the latter case.
We neglected the relativistic effects in both the kinematical
and gravitational ones. If the disk rotational velocity and the
flow speed are of the order of the light speed, radiation drag
becomes important. If the central object is relativistic, the
gravitational and radiation fields are changed. These problems
remain for the future work.
The author would like to thank an anonymous referee for
valuable comments and useful suggestions. This work has been
supported in part by a Grant-in-Aid for the Scientific Research
Funds (15540235 JF) of the Ministry of Education, Culture,
Sports, Science and Technology.
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