Table Of ContentPASJ:Publ.Astron.Soc.Japan,1–??,
(cid:13)c 2008.AstronomicalSocietyofJapan.
Radiative Flow in a Luminous Disk
Jun Fukue
Astronomical Institute, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582-8582
[email protected]
(Received 2005 May 28; accepted 2005 August 3)
Abstract
Radiatively-driven flow in a luminous disk is examined in the subrelativistic regime of (v/c)1, taking
account of radiation transfer. The flow is assumed to be vertical, and the gravity and gas pressure are
6 ignored. When internal heating is dropped, for a given optical depth and radiation pressure at the flow
0 base (disk “inside”), where the flow speed is zero, the flow is analytically solved under the appropriate
0 boundary condition at the flow top (disk “surface”), where the optical depth is zero. The loaded mass
2 and terminal speed of the flow are both determined by the initial conditions; the mass-loss rate increases
n as the initial radiation pressure increases, while the flow terminal speed increases as the initial radiation
a pressure and the loaded mass decrease. In particular, when heating is ignored, the radiative flux F is
J constant, and the radiation pressure P at the flow base with optical depth τ is bound in the range of
0 0
7 2/3<cP /F <2/3+τ . In this case, in the limit of cP /F =2/3+τ , the loaded mass diverges and the
0 0 0 0
flow terminal speed becomes zero, while, in the limit of cP0/F =2/3, the loaded mass becomes zero and
1
the terminal speed approaches (3/8)c, which is the terminal speed above the luminous flat disk under an
v
approximation of the order of (v/c)1. We also examine the case where heating exists, and find that the
3
flow properties are qualitatively similar to the case without heating.
4
1 Key words: accretion, accretion disks — astrophysical jets — galaxies: active — radiative transfer
1 — relativity — X-rays: stars
0
6
0
1. Introduction fromthedisksurface(e.g.,Meyer,Meyer-Hofmeister1982;
/
h Cannizzo, Wheeler 1984; Shaviv, Wehrse 1986; Adam et
p
Accretion disks are tremendous energy sources in the al. 1988; Hubeny 1990; Ross et al. 1992; Artemova et
-
o active universe: in young stellar objects, in cataclysmic al. 1996; Hubeny, Hubeny 1997, 1998; Hubeny et al.
r variables,ingalaxticX-raysourcesandmicroquasars,and 2000, 2001; Davis et al. 2005; Hui et al. 2005; see
t
s in active galaxies and quasars (see Kato et al. 1998 for also Mineshige, Wood 1990). In these studies, however,
a a review). In particular, in a supercritical accretion disk, the verticalmovementandmasslosswerenotconsidered.
:
v the mass-accretion rate highly exceeds the critical rate, Moreover,the relativistic effect for radiation transfer was
i the disk local luminosity exceeds the Eddington one, and not considered in the current studies.
X
the mass loss from the disk surface driven by radiation Inthispaperwefirstexaminetheradiativelydrivenver-
r
a pressure takes place. tical flow – moving photosphere – in a luminous flat disk
Such a radiatively driven outflow from a luminous ac- within the framework of radiation transfer in the subrel-
cretion disk has been extensively studied in the con- ativistic regime of(v/c)1, where we incorporatethe effect
text of models for astrophysical jets by many researchers of radiationdragup to the firstorderof the flowvelocity.
(Bisnovatyi-Kogan,Blinnikov1977;Katz1980;Icke1980; At the firststage, we ignore the gravityof the centralob-
Melia, K¨onigl 1989; Misra, Melia 1993; Tajima, Fukue ject as well as the gas pressure, althoughinternal heating
1996, 1998; Watarai, Fukue 1999; Hirai, Fukue 2001; is considered in a limited way. We then consider analyt-
Fukue et al. 2001; Orihara, Fukue 2003), as on-axis jets ical solutions for such a radiative flow, and examine the
(Icke 1989; Sikora et al. 1996; Renaud, Henri 1998; Luo, flow properties.
Protheroe 1999; Fukue 2005a), as outflows confined by a In the next section we describe basic equations in the
gaseoustorus(Lynden-Bell1978;Davidson,McCray1980; vertical direction. In section 3 we examine the radia-
Sikora, Wilson 1981; Fukue 1982), or as jets confined by tiveflowwithoutinternalheating,whiletheradiativeflow
theouterfloworcorona(Soletal. 1989;Marcowithetal. with internal heating is discussed in section 4. The final
1995;Fukue 1999),and as numerical simulations (Eggum section is devoted to concluding remarks.
etal. 1985,1988). Inalmostallofthesestudies,however,
the disk radiation field was treated as an external field, 2. Basic Equations
and the radiation transfer was not solved.
The radiation transfer in the disk, on the other hand, Let us suppose a luminous flat disk, deep inside which
has been investigated in relation to the structure of a gravitationalornuclearenergyisreleasedviaviscousheat-
static disk atmosphere and the spectral energy density ing or other processes. The radiation energy is trans-
2 J. Fukue [Vol. ,
ported in the vertical direction, and the disk gas, itself, Finally, integrating the sum of equations (8) and (10)
alsomovesintheverticaldirectionduetotheactionofra- gives the momentum conservation in the present approx-
diation pressure (i.e., plane-parallel approximation). For imation,
simplicity, in the presentpaper,the radiationfield is con-
Jv+P =K (=const.), (12)
sidered to be sufficiently intense that both the gravita-
tionalfieldof,e.g.,thecentralobjectandthegaspressure and similarly from equations (8) and (9) we have the en-
canbeignored: tenuouscoldnormalplasmasinthesuper- ergy conservation,
Eddington disk, cold pair plasmas in the sub-Eddington
1 q+
disk, or dusty plasmas in the sub-Eddington disk. As Jv2+F = dτ, (13)
2 − (κ +κ )ρ
for the order of the flow velocity v, we consider the sub- Z abs sca
relativistic regime, where the terms of the first order of where the first term on the left-hand side is eventually
(v/c) are retained, in order to take account of radiation dropped,althoughweretainitheretoclarifythephysical
drag. Under these assumptions, the radiation hydrody- meanings.
namicequationsforsteadyvertical(z)flowsaredescribed We solvedequations (8), (12), and (13) for appropriate
as follows (Kato et al. 1998). boundary conditions, and we obtained analytic solutions
The continuity equation is for several special cases.
As for the boundary conditions, we imposed the fol-
ρv=J (=const.), (1)
lowing cases. At the flow base (disk “inside”) with an
where ρ is the gas density, v the vertical velocity, and J arbitrary optical depth τ , the flow velocity is zero and
0
the mass-loss rate per unit area. The equation of motion the radiation pressure is P , where subscript 0 denotes
0
is the values at the flow base. At the flow top (disk “sur-
dv κ +κ face”),wheretheopticaldepthiszero,theradiationfields
abs sca
v = [F (E+P)v], (2) shouldsatisfythe valuesabovethe luminous infinite disk.
dz c −
Namely, just above the disk with surface intensity I , the
where κ and κ are the absorption and scattering 0
abs sca radiation energy density E , the radiative flux F , and
opacities (gray), E the radiation energy density, F the s s
the radiation pressure P are E = (2/c)πI , F = πI ,
radiative flux, and P the radiation pressure. In a gas- s s 0 s 0
and P = (2/3c)πI , respectively, where subscript s de-
pressureless approximation, the energy equation is re- s 0
notesthevaluesattheflowtop. Rigorouslyspeaking,this
duced to
latter boundary condition at the flow top is for a static
0=q+ ρ j cκ E+κ 2Fv , (3) photosphere, and should not be applied to a moving pho-
abs abs
− − c tosphere, although we approximately used this condition
(cid:18) (cid:19)
inthe presentsubrelativisticregime. Theexactboundary
where q+ is the heating and j is the emissivity. For radi-
conditions are derived and discussed in a separate paper
ation fields, we have
for a fully special relativistic case.
dF Fv As shown explicitly later, in order for the flow to exist,
=ρ j cκ E+(κ κ ) , (4)
dz − abs abs− sca c the radiation pressure P0 at the flow base should be re-
(cid:20) (cid:21)
stricted in some ranges. The lower limit is set due to the
dP ρv κ +κ
= (j cκ E) ρ abs sca[F (E+P)v],(5) negativegradientoftheradiationpressuretowardtheflow
dz c2 − abs − c −
top. Theupperlimitissetwhentheloadedmassdiverges.
and the closure relation, Beyond the upper limit, the hydrostatic balance without
1 4Fv mass loss may be established in the vertical direction.
P = E+ . (6)
3 3 c2
3. Radiative Flow Without Heating
Eliminatingj andE withthe helpofequations(3)and
(6), and introducing the optical depth by
We first examine the case, where there is no viscous
dτ = (κ +κ )ρdz, (7) or nuclear heating: q+ = 0. In the luminous accretion
abs sca
− disk,itisusuallysupposedthattheheatingprocessestake
we can rearrange the basic equations up to the order of
placedeepinsidethedisk. Hence,theassumptionwithout
(v/c)1 as
heating approximately describes the radiative flow in or
dv near the surface envelope of the disk.
cJ = (F 4Pv), (8)
dτ − − In this case and under the present subrelativistic
dF q+ v regime, equation (13) means that the radiative flux F is
= +F , (9) conserved along the flow,
dτ −(κ +κ )ρ c
abs sca
dP F =Fs. (14)
c =F 4Pv, (10)
dτ − Using the boundary conditions at the flow base (v =0),
dz equation (12) is expressed as
(κ +κ )J = v. (11)
abs sca
dτ −
Jv+P =P . (15)
0
No. ] Radiative Flow in a Luminous Disk 3
Hence, equation (8) becomes 10 1.0
dv (a)
cJ = (Fs 4P0v). (16) 8 0.8
dτ − −
This equation can be analytically solved to yield the so-
6 0.6
lution for v, J v
s
4P 4 0.4
v= Fs 1 e cJ0(τ−τ0) . (17)
4P0" − # 2 0.2
Thus, the radiative flow from the luminous disk without
heating is expressed in terms of the boundary values and 0 0
0 1 2
the mass-lossrate. In addition,the flow velocityv atthe
s P
flow top (τ =0) is 0
4P
v = Fs 1 e− cJ0τ0 . (18)
s
4P0 − ! 1000 1.0
Using the boundary condition at the flow top, we fur- (b)
800 0.8
ther impose a condition on the values at boundaries. As
already mentioned, at the flow top cP =cP =(2/3)F .
s s 600 0.6
Hence, inserting boundary values into momentum equa- J v
s
tion (15), we have the following relation:
400 0.4
4P
J Fs 1 e− cJ0τ0 + 2 F =P . (19) 200 0.2
s 0
4P0 − ! 3c
0 0
Thatis,forgivenτ andP attheflowbase,themass-loss
0 0 0 2 4 6 8 10
rate J is determined in units of Fs/c2, as an eigen value. P
In figure 1 we show the mass-loss rate J (solid curves) 0
and the terminal velocity v (dashed curves) at the flow
s
top as a function of P for severalvalues of τ at the flow
0 0
base. The quantities are normalized in units of c and Fs. Fig. 1. Mass-lossrateJ (solidcurves)andterminalvelocity
As can be seen in figure 1, the mass-loss rate increases vs (dashed curves) at the flow top as a function of P0 for
as the initial radiation pressure increases, while the flow severalvaluesofτ0attheflowbase: (a)τ0=1and(b)τ0=10.
terminal speed increases as the initial radiation pressure Thequantities arenormalizedinunitsofcandFs.
and the loaded mass decrease.
Moreover, as can be seen in figure 1, and easily shown the flow top is also large. As a result, the loaded mass
from equation (19), in order for the flow to exist, the ra- J also becomes large, but the flow final speed vs is small
diation pressure P at the flow base is restricted in some due to momentum conservation (12). When the initial
0
range, radiation pressure P0 [>(2/3c)Fs] is small, on the other
hand,thepressuregradientbecomessmall,andtheloaded
2 cP 2
< 0 < +τ . (20) massis alsosmall,butthe flowfinalspeedbecomeslarge.
0
3 F 3
s In the limiting case of P = (2/3c)F , the loaded mass
0 s
At the upper limit of cP /F =2/3+τ , the loaded mass J becomes zero, but the terminal speed v is (3/8)c, as
0 s 0 s
diverges and the flow terminal speed becomes zero. On already mentioned.
the other hand, at the lower limit of cP /F =2/3, the Finally, inserting equation(17) into equation (11), and
0 s
pressure gradientvanishes, the loadedmass becomes zero integratingthe resultantequation, we obtainthe heightz
andtheterminalspeedapproaches(3/8)c. Thisisjustthe as a function of the optical depth τ:
terminalspeedofthegas,drivenbytheradiationpressure
(κ +κ )J
abs sca
under the radiation drag,above the luminous flat infinite z
c
disk in the present approximation of the order of (v/c)1
4P
(cf. Icke 1989). = Fs (τ τ) cJ 1 e cJ0(τ−τ0) , (21)
Infigure2weshowtheflowvelocityv(solidcurves)and 4cP0( 0− −4P0" − #)
the radiation pressure P (dashed curves) as a function of
where we use the boundary condition at the flow base:
the optical depth τ for several values of P at the flow
0
z=0 at τ =τ . In addition, the height z of the flow top
base in a few cases of τ . The quantities are normalized 0 s
0
(disk “surface”) is also expressed as
in units of c and F .
s
When the initial radiation pressure P0 at the flow base (κabs+κsca)J Fs 2 cP0
is large, the pressure gradient between the flow base and zs= τ0+ , (22)
c 4cP 3− F
0(cid:18) s (cid:19)
4 J. Fukue [Vol. ,
1.0 for the integration to be performed analytically, several
functional forms need to be assumed. One simple case,
(a)
0.8 whichweassumeinthepresentpaper,isafollowingform:
q+
v 0.6 P (κ +κ )ρ =q0 (=const.). (24)
abs sca
v
0.4 This is quite reasonablesince the heating q+ may be pro-
portional to the density ρ.
0.2 In this case, under the present subrelativistic regime
andusingtheboundaryconditionsattheflowbase,equa-
0 tions (8), (13), and (12) become, respectively,
0 0.2 0.4τ0.6 0.8 1.0
dv
cJ = (F 4P v), (25)
0
dτ − −
F
s
F =q (τ τ)= (τ τ), (26)
0 0 0
− τ −
1.0 0
P =P Jv, (27)
0
(b) −
0.8 whereF (=q τ )isagaintheradiativefluxattheflowtop
s 0 0
(disk“surface”). Here,weimposetheboundarycondition
0.6 P on F at the flow base as F =0 at τ =τ .
v 0
The above equation (25) can be analytically solved to
v
0.4 yield the solution for v:
4P
0.2 v= Fs (τ τ) cJFs 1 e cJ0(τ−τ0) .(28)
4P0τ0 0− −(4P0)2τ0" − #
0
0 2.0 4.0τ6.0 8.0 10.0 Thus,theradiativeflowfromtheluminousdiskwithsim-
pleheatingisalsoexpressedintermsoftheboundaryval-
ues and the mass-loss rate. In addition, the flow velocity
v at the flow top (τ =0) is
s
4P
sFPeivg(ed.raa2sl.hveaFdlulocewusrovvfeelsPo)0ciatasytavth(fesuonfllcoidtwiocbnuarvoseefsit)nhaeandofeprwtaidccaiaalsteidosenpotfphrτe0τs:su(faore)r vs= 4FPs0 −(4cPJ0F)2sτ0 1−e− cJ0τ0!. (29)
τ0=1and(b)τ0=10. Thequantitiesarenormalizedinunits Using the boundary condition at the flow top, we fur-
of c and Fs. From top to bottom of v and from bottom to
topofP,thevaluesofP0 are0.67,0.8,1,1.2,1.4in(a),and ther impose a condition on the values at the boundaries.
0.67,1,2,5in(b). In this case with simple heating, we have the following
relation:
wcPh0e/rFesw=e u2/se3+eqτu0a,tiwonhe(r1e8)t.heHelonacde,edatmthaessudpipveerrgleims,itthoef J4FPs0 −(4cPJ02)F2sτ0 1−e−4cPJ0τ0!+32cFs=P0. (30)
height of the flow/disk becomes zero. On the other hand,
atthe lowerlimit ofcP0/Fs=2/3,wherethe loadedmass Thatis,forgivenτ0 andP0 attheflowbase,themass-loss
becomeszero,theheightoftheflow/diskbecomesinfinite. rate J is determined in units of F /c2.
s
It should be noted that, at the upper limit of cP0/Fs= In figure 3 we show the mass-loss rate J (solid curves)
2/3+τ0 with a small terminal speed, and the terminal velocity vs (dashed curves) at the flow
top as a function of P for severalvalues of τ at the flow
3 2 0 0
T4= T4 +τ , (23) base in the case with simple heating. The quantities are
c 4 eff 3 0
(cid:18) (cid:19) normalized in units of c and Fs.
where T is the temperature at the flow base (i.e., P = The properties are qualitatively same as those without
c 0
aT4/3),andT the effective temperatureatthe flow top heating. The mass-loss rate increases as the initial ra-
c eff
(i.e., F =σT4 ). This is the Milne approximation. diation pressure increases, while the flow terminal speed
s eff
increases as the initial radiation pressure and the loaded
4. Radiative Flow With Heating mass decreases. Moreover, in order for flow to exist, the
radiationpressureP attheflowbaseisrestrictedinsome
0
We next examine the effect of heating: q+ =0. That range:
6
is, we consider the radiative flow from deep inside of the
2 cP 2 1
0
disk. < < + τ0. (31)
3 F 3 2
Ifthequantityq+/ρisafunctionofτ,theheatingterm s
can be integrated [see equations (9) or (13)]. In order
No. ] Radiative Flow in a Luminous Disk 5
10 1.0 1.0
(a) (a)
8 0.8 0.8
6 0.6 0.6 P
J v v
s
4 0.4 0.4
v
2 0.2 0.2
0 0 0
0 1 2 0 0.2 0.4τ0.6 0.8 1.0
P
0
1000 1.0 1.0
(b) (b)
800 0.8 0.8
600 0.6 0.6 P
J v v
s
400 0.4 0.4
v
200 0.2 0.2
0 0 0
0 2 4 6 8 10 0 2.0 4.0τ6.0 8.0 10.0
P
0
Fig. 3. Mass-lossrateJ (solidcurves)andterminalvelocity Fig. 4. Flowvelocityv(solidcurves)andradiationpressure
vs(dashedcurves)inthecasewithsimpleheating: (a)τ0=1 P (dashedcurves)inthecasewithsimpleheating: (a)τ0=1
and (b) τ0=10. The quantities are normalizedin units of c and (b) τ0=10. The quantities are normalized inunits of c
andFs. andFs. Fromtoptobottom ofv andfrombottom totopof
P,the values ofP0 are0.7,0.8,0.9, 1,1.1in(a), and0.7,1,
2,5in(b).
Atthe upper limit, the loadedmassdivergesandthe flow
terminal speed becomes zero. On the other hand, at the
F (τ τ )2 cJ
lowerlimitofcP0/Fs=2/3,theloadedmassbecomeszero = s − 0 + (τ0 τ)
4cP 2τ 4P τ −
and the terminal speed approaches (3/8)c. 0(cid:26) 0 0 0
theInrafidgiuarteio4nwpereshssouwrethPefl(doawshveedloccuitryvves()soalsidacfuurnvcetsi)onanodf + cJ 2 1 1 e4cPJ0(τ−τ0) , (32)
the optical depth τ for several values of P at the flow (cid:18)4P0(cid:19) τ0" − #)
0
base in a few cases of τ in the case with simple heating. where we use the boundary condition at the flow base:
0
The quantities are normalized in units of c and F . z=0 at τ =τ . In addition, the height z of the flow top
s 0 s
In this case with simple heating, the velocity distribu- (disk “surface”) is also expressed as
tion is somewhat different from that in the case without
(κ +κ )J F τ 2 cP
heating. In the case without heating, the flow is rather abs sca z = s 0 + 0 , (33)
s
c 4cP 2 3− F
quickly accelerated to be a terminal speed, since the ra- 0(cid:18) s (cid:19)
diative flux is constant. In the case with heating, on the where we use equation (29). Hence, at the upper limit of
other hand, the flow is gradually accelerated, since the cP /F =2/3+τ /2, where the loaded mass diverges,the
0 s 0
radiative flux is increasing from the flow base toward the height of the flow/disk becomes zero. On the other hand,
flow top due to heating. atthe lowerlimit ofcP /F =2/3,wherethe loadedmass
0 s
Finally, inserting equation(17) into equation (11), and becomeszero,theheightoftheflow/diskbecomesinfinite.
integratingthe resultantequation,we obtainthe heightz It should be noted that, at the upper limit of cP /F =
0 s
as a function of the optical depth τ: 2/3+τ /2 with small terminal speed,
0
(κabs+κsca)J 3 2 1
z T4= T4 + τ , (34)
c c 4 eff 3 2 0
(cid:18) (cid:19)
6 J. Fukue [Vol. ,
where T is the temperature at the flow base (i.e., P = broad absorption lines in the UV resonance are always
c 0
aT4/3),andT the effective temperatureatthe flow top blueshiftd relative to the emission-line rest frame, which
c eff
(i.e., F =σT4 ). This is just the modified Milne approx- indicates the presence of outflows from the nucleus, with
s eff
imation. velocitiesaslargeas0.2c. Thesemoderatelyhigh-velocity
outflows are supposedto be acceleratedby line or contin-
5. Concluding Remarks uumradiation(e.g.,Murrayetal. 1995;Progaetal. 2000;
Proga,Kallman2004). Intheseobjects,theoutflowsmust
Inthispaperwehaveexaminedtheradiativeflowfroma consist of dense clouds, or be shielded from the central
luminousdisk,whiletakingaccountoftheradiativetrans- strong source, in order to operate a line-driven mecha-
fer, in the subrelativistic regime of (v/c)1. The flow ve- nism. Otherwise, the material is highly ionized by the
locity,theradiationpressuredistribution,andotherquan- central source, and, instead of line-acceleration, contin-
tities are analytically solved as a function of the optical uum acceleration may work (see, e.g., Chelouche, Netzer
depthforthe caseswithoutandwithheating. Attheflow 2003).
base (disk “inside”),where the flow speed is zero,the ini- Finally, narrow-line Seyfert 1 galaxies and luminous
tial optical depth is τ and the initial radiation pressure quasars are also believed to harbor supercritically accret-
0
is P ; in the usual accretion disk these quantities are de- ingblackholes,andhigh-velocityoutflowsarereportedas
0
termined interms of the centralmass,the mass-accretion warmabsorbers(cf. Blustinetal. 2005). Intheseobjects,
rate, and the viscous process as a function of the radius. radiativeaccelerationbycontinuummayplayadominant
That is, the optical depth is determined by the disk sur- role.
facedensity,theradiativefluxisgivenbythecentralmass The radiative flow investigated in the present paper
and the accretion rate, and the radiation pressure relates must be quite fundamental problems for accretion disk
to the disk internal structure. At the flow top (disk “sur- physics and astrophysical jet formation, although the
face”),wheretheopticaldepthiszero,theradiationfields present paper is only the first step, and there are many
are assumedto be those above a static flat disk with uni- simplifications at the present stage.
formintensity. Inordertomatchthisboundarycondition, For example, we have ignored the gravitational field
one relation is imposed on the boundary quantities, and produced by the centralobject. This means that the flow
themass-lossrateJ andterminalspeedv arebothdeter- considered in the present paper would correspondto nor-
s
mined by the initial conditions τ and P , as eigenvalues. malplasmasinthe super-Eddingtondisk,pairplasmasin
0 0
Inparticular,inorderfortheflowtoexist,theinitialradi- thesub-Eddingtondisk,ordustyplasmasintheluminous
ationpressureisboundinsomerange: attheupperlimit, disk. In other cases, or even for normal plasmas in the
the loaded mass diverges and the flow terminal speed be- super-Eddington disk, the gravitationalfield would affect
comes zero, while, at the lower limit, the loaded mass the flow properties. In particular, the gravitational field
becomes zero and the terminal speed approaches (3/8)c, in the vertical direction is somewhat complicated (e.g.,
whichistheterminalspeedabovetheluminousdiskunder Fukue 2002, 2004), and the influence of the gravitational
the approximation of the order of (v/c)1. fieldisimportant. Theeffectofgravitywillbe considered
Amongvarioustypesofastrophysicaljetsandoutflows, in a seperate paper (Fukue 2005b).
some are believed to be accelerated by continuum radi- We also ignored the gas pressure. This means that the
ation, and to which the present mechanism can be ap- radiation field is sufficiently intense. In general cases,
plied. For example, in supersoft X-ray sources, including where the gas pressure is considered, there usually ap-
accreting white dwarfs, high-velocity mass outflows were pear sonic points (e.g., Fukue 2002, 2004), and the flow
reported (e.g., Cowley et al. 1998). The wind velocity is acceleratedfromsubsonic to supersonic. The crosssec-
measured in these objects is about 3000–5000 km s−1, tion of the flow also has a similar influence. In this paper
whichisoftheorderoftheescapevelocityofwhitedwarfs. we consider a purely vertical flow, and the cross section
Inaddition,thelineprofiledependsontheinclinationan- of the flow is constant. If the cross section of the flow
gle, and the P Cyg profiles are seen. Hence, in supersoft increases along the flow, the flow properties, such as the
X-raysources,theoutflowsshouldblowofffromtheinner- transonic nature, would be influenced.
mostaccretiondisk. However,the gastemperature ofthe In the present paper, we have considered the internal
inner disk is so high that the line-driven mechanism can- heating processes in a simple form. Since the heating
notoperate,andtherefore,massoutflowsinthese objects prcesses should couple with the viscous process or other
would be driven by continuum radiation. heating processes, such as a nuclear process, we should
In SS 433, a prototype of astrophysical jets, the mass- treat internal heating more carefully.
accretionratehighlyexceedstheEddingtonone,andtwin In order to take account of the effect of radiation drag,
jets with 0.26 c would be accelerated by radiation pres- we consider the subrelativistic regime, where the terms
sure (see, e.g., Cherepashchuket al. 2005 for a recentob- of the first order of (v/c) are retained. As a result, the
sevation). The luminosity of microquasarGRS 1915+105 terminal speed (3/8)c for the subrelativistic regime ap-
exceedsthe Eddingtonone(L )during its outburst. In pears in the extreme case. This is quite consistent with
Edd
quiscentphase,itsluminosityneverdropbelow 0.3L the previous knowledge. As is well known, the terminal
Edd
∼
(Done, Gierlin´ski 2004). speed above the luminous flat disk in the full relativity
In broad absorption-line quasars (BAL QSOs), strong is [(4 √7)/3]c 0.45c (Icke 1989). To consider highly
− ∼
No. ] Radiative Flow in a Luminous Disk 7
relativistic cases, all of the terms up to (v/c)2 should be Marcowith, A., Henri, G., & Pelletier, G. 1995, MNRAS,277,
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