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NASA-CR-204982 



The Astrophysical Journal, 455 :448-455, 1995 December 20 

'.;0 1995. The American Astronomical Society. All rights reserved. Printed in U.S.A. 






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RADIATION PRESSURE-DRIVEN MAGNETIC DISK WINDS IN BROAD ABSORPTION LINE 

QUASI-STELLAR OBJECTS 

Martin de Kool 

Max Planck Institut fur Astrophysik, Karl Schwarzschild Strasse 1, 85740 Garching bei Miinchen, Germany; dekool@mpa-garching.mpg.de 

AND 

Mitchell C. Begelman 1 

JILA, University of Colorado, Campus Box 440, Boulder, CO 80309-0440; mitch@jila.colorado,edu 
Received 1995 March 10; accepted 1995 June 22 

ABSTRACT 

We explore a model in which QSO broad absorption lines (BALs) are formed in a radiation pressure- 
driven wind emerging from a magnetized accretion disk. The magnetic field threading the disk material is 
dragged by the flow and is compressed by the radiation pressure until it is dynamically important and strong 
enough to contribute to the confinement of the BAL clouds. We construct a simple self-similar model for such 
radiatively driven magnetized disk winds, in order to explore their properties. It is found that solutions exist 
for which the entire magnetized flow is confined to a thin wedge over the surface of the disk. For reasonable 
values of the mass-loss rate, a typical magnetic field strength such that the magnetic pressure is comparable to 
the inferred gas pressure in BAL clouds, and a moderate amount of internal soft X-ray absorption, we find 
that the opening angle of the flow is approximately 0.1 rad, in good agreement with the observed covering 
factor of the broad absorption line region. 

Subject headings: accretion, accretion disks — MHD — quasars: absorption lines 



1. INTRODUCTION 

About 10% of QSOs exhibit strong absorption in the UV 
resonance lines of highly ionized species like N v, C iv, and 
Si iv, which are always blueshifted relative to the emission-line 
rest frame. These lines indicate the presence of outflows from 
the active nucleus, with velocities ranging up to 0.1c. The 
properties of these broad absorption lines (BALs) and their 
interpretation have been reviewed and discussed extensively in, 
e.g., Weymann, Turnshek, & Christiansen (1985, hereafter 
WTC); Turnshek (1988); Begelman, de Kool, & Sikora (1991, 
hereafter BdKS); Weymann et al. (1991); and Hamann, 
Korista, & Morris (1993). Simple arguments lead to the follow- 
ing model-independent constraints on the physical conditions 
and geometry of the region in which the BALs are formed 
(hereafter referred to as the BALR). 

1. The ionization parameter of the BAL gas lies in the range 
0.01 < U < 1 ([/ being the ratio of the number density of 
photons above the Lyman limit to the hydrogen density) and 
does not change rapidly with the velocity of the absorbing 
material. This leads to limits on the density of the BAL gas. 

2. Depending on assumptions about line saturation and 
degree of covering of the continuum source, estimates of the 
total column density of the BALR range from N H ~ 10 20 5 to 
A/ H ~ 10 22 cm~ 2 . 

3. BAL gas occupies only a very small fraction of the 
volume of the BALR, the filling factors typically being < 10" 5 . 
Individual BAL clouds are very thin in the direction of our line 
of sight. Typical dimensions ~ 10' ' cm can be derived. 

4. The BALR has at least some part which is located outside 
the broad emission-line region (BELR), because the flux in the 
Lya BEL is usually significantly reduced as a result of absorp- 



1 Also at Department of Astrophysical, Planetary, and Atmospheric Sci- 
ences, University of Colorado, Boulder. 



tion by the blueshifted N v line, and sometimes the blue wing 
of the C iv BEL is also absorbed. 

5. The absence of observable emission from the high- 
velocity material seen in absorption indicates that the global 
covering factor of the BALR is <0.1. Combined with the fact 
that 10% of QSOs exhibit BALs, this leads to the conclusion 
that most QSOs have a BALR. 

Apart from these general constraints, relatively little 
progress has been made in identifying the physical processes 
underlying the formation of the BALs. The following basic 
questions still need to be answered: 

1. What is the origin of the BAL clouds? 

2. What force accelerates the clouds to such high velocities? 

3. How do the clouds maintain such a high internal pres- 
sure, or in other words, what confines the clouds? 

These subjects were already discussed in the first major 
review of BAL QSOs (WTC). There it was assumed that the 
confinement had to be caused by a hot gas between the clouds, 
with a temperature of the order of the Compton temperature of 
the active galactic nucleus (AGN) radiation field. Since it can 
easily be shown that such an intercloud medium would exert 
drag forces on the clouds far in excess of anything achievable 
by radiation pressure, it was conjectured that these drag forces 
had to be responsible for the acceleration, the hot medium 
being in the form of a wind dragging the clouds along. BdKS 
attempted to develop a detailed model along these lines. 
Although they were able to obtain reasonable line profiles for 
certain sets of assumptions, the difficulties associated with con- 
finement and survival of clouds dragged by a hot medium 
could not be satisfactorily addressed. This led them to suggest 
that confinement by magnetic fields instead of hot gas may be 
the only physically reasonable solution. 

BAL models involving acceleration by UV resonance line 
radiation pressure were in fact among the first to be developed, 



448 



BAL QSO MAGNETIC DISK WINDS 



449 



on the basis of the analogy between BAL profiles and P Cygni 
profiles that are observed from the winds of early-type stars 
(Drew & Boksenberg 1984). However, these models ran into 
problems, since they assumed that both the global covering 
factor and the filling factor of the absorbing material are 1 and 
that the gas provides its own confinement. In this case, 
unphysically large mass-loss rates are required, and many fea- 
tures of the observed line profiles cannot be reproduced. Con- 
sequently, this model was not generally accepted. 

The case for acceleration by UV line radiation pressure was 
revived recently (Arav & Li 1994; Arav, Li, & Begelman 1994) 
by considering outflows with a very low filling factor of cool 
gas and a massless confining medium. Once these assumptions 
are made, this model is very successful in explaining observed 
BAL properties. 

1. The momentum flux in the BAL clouds is very similar to 
the momentum absorbed from the radiation field in the BALs 
(see also Korista et al. 1992), as expected for winds that are not 
too optically thick. 

2. Modeling the dynamics of such flows with techniques 
similar to those used for O-star winds shows (Arav & Li 1994; 
Arav et al. 1994) that a wind with the observed column den- 
sities reaches terminal velocities in the observed range if the 
flow starts at or just outside the BELR ( ~ 10' 8 cm). 

3. Some objects show clear evidence of extra acceleration at 
velocities where the N v A 1240 doublet starts to scatter Lyoc 
emission-line photons, so that the emission line can contribute 
to the acceleration (Arav & Begelman 1994). Possible evidence 
for line-locking effects was also discussed by Weymann et al. 
(1991) and Korista et al. (1993). 

The most outstanding observable difference between BAL 
QSOs and non-BAL QSOs that is not directly related to the 
lines themselves is the absence of radio-loud BAL QSOs 
(Stocke et al. 1992; Francis, Hooper, & Impey 1993). Some 
authors (Stocke et al. 1992) have argued that this must reflect 
an intrinsic difference between BAL and non-BAL QSOs, in 
which the same energy source that powers the jet in radio-loud 
objects gives rise to a fast wind in radio-quiet objects. This 
wind is then thought to be responsible for the BALs by strip- 
ping material from clouds. Some observational support for this 
picture is derived from the fact that BAL QSOs, although not 
radio-loud, still show radio emission that is consistent with 
what is expected from a wind. A competing hypothesis is that 
the absence of radio-loud BAL QSOs can be explained by 
selection effects resulting from beaming of the radio emission 
(de Kool 1993), under the assumption that the BALR is associ- 
ated with the accretion disk and that BALs are only visible if 
our line of sight passes very close to the surface of the disk, i.e., 
perpendicular to a possible radio jet. This straightforwardly 
explains the absence of core-dominated radio-loud BAL 
QSOs. However, to explain the absence of extended, lobe- 
dominated radio-loud BAL QSOs, one has to make the addi- 
tional assumption that the extended emission is also weakly 
beamed, and although this seems consistent with the observed 
distribution of the flux ratios between the lobes, this assump- 
tion is controversial. The beaming hypothesis avoids having to 
invoke ad hoc differences between BAL and non-BAL QSOs, 
which is attractive because, apart from the radio emission, 
there seems to be very little difference in QSO properties of 
BAL and non-BAL objects (e.g., Weymann et al. 1991). The 
accretion disk also seems a natural choice for the source of 
BAL material. Additional evidence that the BALR is only 
visible in QSOs where the line of sight is close to the accretion 



disk comes from the polarization measurements of Glenn, 
Schmidt, & Foltz (1994). 

Combining all the above considerations, a model for the 
BALR that ties it to the accretion disk and relies on magnetic 
confinement and acceleration by line radiation pressure seems 
attractive. In this paper we will explore such a model of a line 
radiation pressure-driven wind from a magnetized accretion 
disk. Note that this model has some similarities to the BELR 
model of Emmering, Blandford, & Shlosman (1992), which 
explains the BELs as resulting from a hydromagnetically driven 
disk wind. However, we shall show that the dynamical role of 
the magnetic field in our model is significantly different. 

In § 2 of this paper we will give a general description of the 
model and make some order-of-magnitude estimates to show 
that the model is feasible. In § 3 we will present self-similar 
solutions of the simplified problem of optically thin winds and 
use these to illustrate the properties of such flows. Finally, in 
§ 4 we will discuss our results. 

2. DESCRIPTION OF THE MODEL 

In Figure 1 we show a schematic representation of our BAL 
QSO model. At radii where the BAL outflow must originate 
(comparable to the radius of the BELR or beyond), the accre- 
tion disk is vertically self-gravitating and consequently must 
consist of a collection of relatively cool, mostly molecular 
clumps (e.g., Schlosman & Begelman 1989). Arguing along 
similar lines to Emmering et al. (1992), we assume that these 
clouds are threaded by a magnetic field, not unlike clouds in 
the interstellar medium in the disk of our Galaxy. Due to 
encounters or magnetic buoyancy, some clouds may acquire a 
small velocity perpendicular to the disk plane and become 
exposed to the UV radiation from the central source. As they 
emerge from the disk, they are immediately heated to 
a "warm" thermal equilibrium state, with temperatures 
~ 10 4 K. 

As shown below, we expect the magnetic field above the disk 
to be ordered, with mainly poloidal field lines. The heated 
cloud will be prevented from expanding sideways by the mag- 
netic field and will expand mainly in the radial direction, 
forming a thin filament. In order to keep the cloud material 
confined, we have to assume that it is not completely optically 
thin in the radial direction so that the acceleration by radiation 
pressure can act as an effective gravity, compressing the cloud 
in the radial direction. In fact, the observed thickness of clouds 
in the radial direction (~ 10 11 cm ; see above) is a strong indica- 
tion that the acceleration is ultimately responsible for the con- 
finement: estimating the acceleration as g ct{ ~ v 2 /R with i; and 
R the typical velocity and size of the BALR, it is easily shown 
that 10" cm corresponds to about one scale height in a gas 
with the temperature of the BAL clouds (~3 x 10 4 K). Thus, 
we emphasize here that the small thickness of the BAL clouds 
is the natural size for clouds with a temperature of a few times 
10 4 K being accelerated to velocities of 0.1c- over a length scale 
of 10 18 cm. This argument applies not only to radiative accel- 
eration of optically thick clouds, but also to any mechanism 
that accelerates the clouds by a surface force, e.g., as in the 
model in which the clouds are dragged by a hot wind. 

We will now estimate the relative magnitudes of the three 
main forces acting on a cloud : gravity, radiation pressure, and 
magnetic forces. The radiation force exerted on a gram of opti- 
cally thin gas can be expressed as 



F = F 

rad gra* 



— Y- 



(2.1) 



450 



DE KOOL & BEGELMAN 



Vol. 455 



BAL cloud 




Fig. 1. — This figure illustrates the basic idea behind our model. Cool clouds that are threaded by magnetic field rise from the clumpy, self-gravitating accretion 
disk at a distance of ~ 10 ' 8 cm " 2 from the central black hole (e.g., as a result of gravitational scattering of clouds or from the Parker instability). As they are exposed 
to the continuum source, they are heated to a warm phase (T ~ 10* K) and rapidly accelerated outward by UV-line radiation pressure. The magnetic field carried by 
the flow is compressed against the disk up to the point where it becomes dynamically important, and the magnetic pressure gradient perpendicular to the disk forces 
the flow to spread to some opening angle # BAL . This occurs when the magnetic pressure is some significant fraction of the radiation pressure, so that the field is 
automatically strong enough to confine the BAL clouds at the required gas pressure. 



where L is the QSO luminosity, L Edd is the Eddington lumi- 
nosity, k t is the Thomson opacity, and k is the mean line 
opacity (averaged over the UV continuum). Because the 
opacity in UV resonance lines is so large relative to the 
Thomson opacity (k/k t can be as large as 10 3 or 10 4 ; see, e.g., 
Arav & Li 1994; Arav et al. 1994), it is easy to show that the 
radiation force is much larger than gravity, provided that the 
luminosity of the QSO is >10~ 3 times the Eddington lumi- 
nosity and that the cloud is not extremely optically thick 
(N H ~ 10 20 cm" 2 ). Thus, all clouds with a column density 
below this limit will be accelerated outward, dragging the mag- 
netic field along. Since the radiation force is much larger than 
gravity, the clouds will very quickly reach speeds far in excess 
of the local escape speed and the local Keplerian speed in the 
disk. This implies that the outflow will be almost purely poloi- 
dal and that the winding up of field lines with the development 
of a strong azimuthal field, which typifies hydromagnetically 
driven winds that are not subject to a strong radiation force 
(e.g., Blandford & Payne 1982; Emmering et al. 1992), will not 
occur. 

Although the magnetic field lines will be combed out radially 
by the effect of radiation pressure, magnetic pressure forces will 
prevent the flow from being squashed flat against the disk. The 



balance between magnetic and radiation forces will determine 
the thickness of the flow, yielding a flow geometry like that 
illustrated schematically in Figure 2. The magnetic force per 
unit mass, acting on the flow, is given by F mag ~ p l P mag /d, 
where d is the length scale over which the magnetic field 
changes (see Fig. 2) and P mag is the magnetic pressure. We 
estimate the mean flow density by p ~ p H N H /R, where N H is 
the hydrogen column density of the BALR. The corresponding 
force resulting from radiation pressure can be written in the 
form F rad ~ icP rad = 400(ic/10 3 K: T )P rad , where P rad is the radi- 
ation pressure. By demanding that in the equilibrium solution 
the magnetic and radiative forces must be comparable, we 
arrive at the estimate 



n; 



(2.2) 



21 is the column density of the BALR normalized to 

2 



where N 
10 21 cm 

If we assume, in addition, that the magnetic field is 
responsible for the confinement of the BAL clouds, we must 
have P mag ~ P gas = E" 'P rad , where 3 is the ionization param- 
eter in a slightly different definition than that used for U above. 




Fig. 2. — The geometry of the estimate leading to eq. (2.3) 



No. 2, 1995 



BAL QSO MAGNETIC DISK WINDS 



The two are related by an expression of the form E =f(T)U. 
Since we know P rad from the QSO luminosity and E from the 
ionization equilibrium (S BAL ~ 10, e.g., Krolik, McKee, & 
Tarter 1981), we can estimate the magnetic pressure. Substitut- 
ing P mJP, a a ~ 0.1 into equation (2.2) and solving for d/R, we 
obtain 



J-0.W 



10 K T . 



(2.3) 



Thus, we can expect that clouds emerging from the disk are 
accelerated almost radially outward, compressing the magnetic 
field lines on which the clouds are moving up to a point where 
magnetic forces become comparable to the radiation force. 
Equation (2.3) indicates that for column densities in the range 
inferred from observations this will occur when the flow is 
compressed to a thickness d ~ 0.01-0.3R. Equation (2.3) is 
probably an underestimate, since we did not take account of 
the fact that the magnetic forces only need to balance the 
component of the radiation force perpendicular to the field 
lines. To give a more accurate estimate requires knowledge of 
the detailed shape of the field lines. In § 3, we shall derive the 
field line shape and flow opening angle from a rigorous 
analysis of a self-similar model. 

Clearly, there are many uncertainties associated with this 
model that prevent us from constructing detailed models. 
Apart from the difficulties associated with two-dimensional 
line radiation transport, there are at present no reliable theo- 
retical estimates available for the distribution of magnetic 
fields and mass loss over the surface of the accretion disk, the 
main quantities that will determine the structure of the flow. 
Because of this, we will only consider a very restricted class of 
models here, the self-similar ones (see Blandford & Payne 
1982). The solutions we obtain in this case can be used to 
illustrate the properties of the kind of magnetic disk winds we 
are considering here. 

3. SELF-SIMILAR SOLUTIONS FOR OPTICALLY THIN WINDS 

3.1. The Basic Equations 
We will consider azimuthally symmetric outflows in spher- 
ical geometry, where the toroidal components of the magnetic 
field and velocity are zero. In this way we neglect all effects of 
rotation of the underlying accretion disk, which is a good 
approximation since we have shown that the radial velocity of 
the wind will be much higher than the Keplerian speed. The 
equations of motion (in spherical polar coordinates) we want 
to solve are 



Sjr Vj,dv,_vl 

1 r dr r dO r 



= Pa,-. 



±(BedB, 
4n\r dB 



r 



SB, 

dr 



(3.1a) 



/ dve v^dv, v i v L \_±( B r dB B t B 8B t 



(3.1b) 

To obtain self-similar solutions, we require that all quantities 
X can be written in the form 

X = X r "«£(©), 

with X a constant. For our optically thin wind approx- 



451 

imation, we take a rad = a r 2 . Since the magnetic field is pol- 
oidal and divergence free, we can derive it from a potential, ¥ : 



V47T 8V 

r ~ r 2 sin 6 36 



B a = 



4n d*¥ 



r sin 6 dr 
If we define 4* = 4» r~'f{e), then 



(B„ B,) = VP ° 



4n 



r 2 sin 



r-°(f, «f) , 



(3.2a) 



(3.2b) 



(3.3) 



where a prime denotes differentiation with respect to 0. If we 
define «P(0, n/2) = 0, then V(R, n/2) is proportional to the total 
magnetic flux through the surface of the disk for radii smaller 
than R, and we see that a has to be negative. From the condi- 
tion that all terms in equation (3.1a) scale with the same power 
of r, we obtain that v oc r " ' /2 and p oc r ~ (3 + 2a) . Mass conserva- 
tion states that pv is divergence free and can also be derived 
from a potential, x- 



(3.4a) 



. . ^, 4n By 

r sin 6 30 



(pv)e = 



'4n dx 
r sin 6 dr 



(3.4b) 



where /(«, n/2) is proportional to the integrated mass flux from 
the surface of the disk for r < R. Since flux-freezing implies that 
the velocity is always parallel to the magnetic field, lines of 
constant x must coincide with lines of constant *¥, implying 



x = xm = w 



(3.5) 



Substituting the scalings of v and p obtained above into equa- 
tion (3.4a) and using equation (3.5), we find that self-similar 
solutions exist only if = 2 + (3/2a). Thus, for a prescribed 
distribution of magnetic flux over the surface of the disk, self- 
similar solutions exist only for one distribution of mass flux 
from the disk. 

In addition to/(0), there is one more free angular function 
which needs to be calculated, which we define in terms of the 
density. Let 



p = r 



-(3 + 2a) 



g(6) sin 



(3.6) 



This strange-looking definition will lead to more compact 
equations. Substituting this into equations (3.4a)-(3.4b), we 
obtain 



v t ) = V' br- l > 1 Pg(f',ar). 



(3.7) 



An energy equation can be constructed by multiplying equa- 
tion (3.1a) by \jp and equation (3.1b) by B 9 /(pB r ) and adding 
the two, leading to 

1 Pi J3 

fl - = 2£ (B ' 2 + -> + ^tf + «'. 2 >- (3-8) 

The magnetic terms cancel because they do no work on the 
flow. Canceling factors of r~ 2 and rearranging terms after 
substituting equation (3.7) into equation (3.8), we can write 



452 

equation (3.8) in the form 



DE KOOL & BEGELMAN 



Vol. 455 



2(10 + 0W) 2 + oe 2 / 2 ] = « S - - 



4»S'b 2 /? 2 



[ 2a 
IW 2 /? 



^ + 9 2 [(/') 2 + « 2 / 2 ]^ (3-9) 



where we have used the fact that 2a /CF 2 ,"£) 2 /? 2 ) is a constant. 
Equation (3.9) is easily integrated. To fix the constant of inte- 
gration, let us consider flows that start from rest at the disk 
(0 = nil), with / and /' finite. Therefore, we have g(n/l) = 0, 
and we may arbitrarily take/(n/2) = 1. In this case, integrating 
equation (3.9) yields 



1 + 



la 



g 2 [if) 2 + a 2 / 2 ] =/ 



rl/u 



(3.10) 



Equation (3.10) is the self-similar form of the energy equation. 
Next, substituting equations (3.2a)-(3.2b), (3.4a)-(3.4b), and 
(3.6) into equation (3.1b), we obtain 

^-WtffJ^afl + y 



= -/' 
Defining the new variables 



(1 +a)af d ( f 
sin 38 Vsin 



= /"(I/") 



y=f 






equation (3.10) becomes 

(1 - y) - H'V + y' 2 ) = , 
and equation (3.1 1 becomes) 



(3.11) 



(3.12) 



(3.13) 



sin 



= («+!) 



i + * 



+ 



cos 9 y' 



sin 



(3.14) 



Using equation (3.13), we eliminate w from equation (3.14) 
(taking the negative root, since eq. [3.12] implies that w is 
negative) to obtain a second-order ordinary differential equa- 
tion for y. 



2C sin 9(1 - >')" 2 >' 

(y 2 + y' 2 ?' 2 
V + y 2 ) 1 - y 



C o sin0(l-y) i;2 

cv 2 + y 2 ) 1 ' 2 



cos 9 , . 

+ — - y + (1 + a)y 1 + 
sin 9 



(3.15) 

We see that the solution depends only on the constants a and 
C , the latter being given by 



_ « + (3/4) 3/2 fi- 



(3.16) 



The equation requires two boundary conditions. The first one 
has already been used to derive equation (3.10): /(jt/2) = 1, 
implying y(n/l)= 1. For the second condition, we have a 
choice We can prescribe >•' at the surface of the disk, which 



turns the solution into an initial value problem. This is equiva- 
lent to prescribing the angle the field lines have with respect to 
the disk plane when they emerge from the disk. Alternatively, 
we can demand that the flow be confined to a region between 
the disk and a minimum angle 9 b , i.e., we demand y(6 b ) = 0, in 
which case we have to solve a two-point boundary value 
problem. The solutions of equation (3.15) can contain critical 
points when the velocity component in the 9 direction becomes 
equal to the Alfven speed: 



1C sin 0(1 - y) ll2 y = 4npv 2 e s M 2 = , 



(y 2 + / 



'2x3/2 



B 2 + B 2 



(3.17) 



The same type of critical point also occurs in the self-similar 
hydromagnetic wind models of Blandford & Payne (1982). 

3.2. Solutions: General Constraints 
We are interested in solutions of equation (3.15) in which the 
magnetically confined outflow takes place between the disk 
and a minimum angle 9 b . The range of parameters that will 
lead to physically acceptable solutions can be limited by con- 
sidering the behavior at the boundary 9„. In order to match to 
the vacuum, the magnetic pressure at the boundary must 
vanish, implying/^) = f'(B h ) = 0. Moreover, the field must be 
parallel to the boundary, implying///' -» as 9 -> 9„. In terms 
of the variable y, we have (for a < 0) 



y(0„) = 



► as 0-0,, 



(3.18) 



Making these substitutions in equation (3.15) and retaining 
only the leading terms, we have 



y 



2C sin 8y\ C sin Oy (ly_ 



:+ 1 



+ 



— - v + l+ot— • (3.19) 
sin ' y 

If y" is finite at the boundary, we can make a Taylor expansion 
around the boundary to study the behavior of the solution 
there. However, when doing so it is easily seen that this 
approximate solution implies that y" - <x at the boundary, 
showing that the assumption of finite y" is inconsistent. Thus, 
we expand the solution in A = 8 - 8 b by assuming that 

y"-/lA" as 0-0„. (3.20) 

Since we want / and /' to go to zero at the boundary, the 
singularity in y" must be integrable, leading to the constraint 
that < n < 1 . Then, substituting into equation (3. 19) that 

y'-^A 1 ■", y- ^ ;A 2 -", (3-2D 



i-n 



(1 -Ml-n) 



we obtain 



2C o sin0 6 (l -n) 2 A2u .,' 



cos h 



. - + (1 +«)- 

sin 8 b 1 - \i 1 - H 



(3.22) 



It is impossible to satisfy the equation (3.22) when \i < \. If 
H > \, we have A 2 " " ' -» 0, implying 



1 +2a 



ft = 



(3.23) 



No. 2, 1995 



BAL QSO MAGNETIC DISK WINDS 



453 



and 

to 



1 < a < — f. If n = \ identically, equation (3.22) reduces 



C sin 6 b 

3A 2 



-(2 + 3a), 



(3.24) 



and because the left-hand side of equation (3.23) is positive, we 
must have a < — §. 

We constrain the solutions further by demanding that the 
density go to zero at the boundary, or p(0 h ) = 0. We have 

poc/^ 1 g- , ocA ,1 -"»- 2,I+1 > (2 -'". (3.25) 

00, so 



If equation (3.23) applies, we have pocA~ ,1 ~" ) 
we reject this solution, and the solution with \i = \ is the 
only remaining one. For this solution, we find from equation 
(3.25) that the density at the boundary goes to zero as long as 
a< -f. 

3.3. Scaling of the Solutions for Highly Compressed Flows 
Consider the basic equation (3.15). When the radiation pres- 
sure is very strong, or equivalently the constant C is large, the 
flow will be confined to a narrow wedge over the suface of the 
disk. In this case, we can expect that y' ^> y. Using this approx- 
imation, changing the independent variable from 6 to x = 
C /3 [>/2 - ff] and taking sin 6 v 1 and cos 6 » xC ' 3 , equa- 
tion (3.15) can be written as 



+ Co 2/3 xy + (l +a) : 



y 



(3.26) 



where an overdot represents differentiation with respect to x. 
Now suppose that x and all the dotted derivatives of y are of 
order 1, and that C is large. Then the terms containing Co 2/3 
can be neglected in equation (3.26), leading to 



1 - 



2y(l - y) 



1/2" 



J iii-y) 



1/2 



+ (l+«) : 



(3.27) 



Once this equation is solved, the solution for a given C can be 
determined by scaling the solution with the transformation 
6 = (jt/2) - Cq ly3 x. In this limit, the opening angle of the 
outflow 4> BAL = (k/2) - 6 b will scale with Cq " 3 . The numerical 
solutions below show that this scaling is quite accurate for 
C >1. 

3.4. A Numerical Solution 

We will now present a numerical solution of equation (3.15) 
to show the character of the solutions. Consider the case 
a = — 1. This choice for a is inspired by the observation that 
the ionization parameter of BAL outflows does not vary 
strongly with outflow velocity, so that the magnetic pressure 
responsible for confinement should scale as r 2 , which in our 
model implies a = — 1 . 

To solve equation (3.15) (note that for a = - 1 the last term 
on the right-hand side vanishes) we need two boundary condi- 
tions. The first one is that y = at the disk surface. For the 
second one we use the result of our discussion in § 3.2 that at 
the boundary b , y behaves as in equation (3.21) with \i = j 
and A given by equation (3.24). The location of b is not known 
a priori. For a = - 1 and n = j, the right-hand side of equa- 
tion (3.22) vanishes for A -»• 0, so that the left-hand side must 
also vanish, implying that the boundary point b is also a 
critical point. 



The numerical solution for a given C is determined by 
shooting from the two boundaries to an interior fitting point. 
Taking first guesses for y' at the disk surface and for the loca- 
tion of 6 b , we integrate equation (3. 1 5) outward from the disk 
and inward from 6 b , in the latter case using the expansion in 
equation (3.21) for the first integration step because y" diverges 
there. The correct solution is now found by iterating this pro- 
cedure until the values of / at the disk surface and 6 b are such 
that the two solutions match at an interior fitting point. 

Figure 3 shows the shape of the field lines obtained for C = 
1. The straight line from the origin is the limiting angle b , 
which for C = 1 has the value 0.552. In Figure 4 we illustrate 
the runs of the Mach number M e defined in equation (3.17), the 
density, and the magnetic pressure with 6 for a given radius. In 
Figure 5 the opening angle of the outflow r/> BAL is plotted as a 
function of C , clearly illustrating that the scaling law $ BAL oc 
C 1/3 is a very good approximation for C > 1. 

3.5. Physical Scales 

In this section we will investigate whether the model 
described above can yield the observed quantitative properties 
of the BAL region. We want to express the constant C in more 
easily interpretable physical quantities. For simplicity, we will 
again constrain ourselves to models with a. = — 1. First we 
determine a physical value for the constant a giving the 
strength of the radiation pressure 



a = GM BH [ — II 



10 36 M 8 ^ 3 ^_ 1 (cgs), (3.28) 



where M 8 is the mass of the central black hole in units of 10 8 
Mq, .#3 is k/k t in units of 10 3 , and if '_ , is the luminosity in 
units of 0.1L Edd . Using the fact that 4ji 3/2x V(R) is the magnetic 
flux through the surface of the disk integrated from r = to 
r = R, and that the largest contribution to this integral comes 





constant flux surfaces for C, = 


= i 


1U 


■ . . | . . . | . . i y i p i 


' / ' / 


8 


/ / 


/ / - 


6 


/ / / / 


/ / 


4 


/ / / / / // 




2 


/ ////////// 


//// 





Z,XxX^<x^^ 


<^ 



10 



Fig. 3. — The shape of the compressed field lines for a model with a - - 1 
and C = 1. The straight line coming from the origin shows the boundary 
anele. which is 0. — O.S52 for these narameter*; 



angle, which is b = 0.552 for these parameters 



454 



DE KOOL & BEGELMAN 



Vol. 455 



1.0 


-T T . , I 1 ■ , 1 1 1 | 1 . 


,,..., 


1 — ri 1 




w 


P 


t . 


0.8 


\ 




/ 

/ 
/ 
/ 
/ 








/ 


0.6 




/ / 


P 

1 mag • 












,-'/ 


- 






.'/ 








.'/ 


- 






.■> 




0.4 


^v 




- 






.■'/ 


. 






•'/ 






.-■'/ 




' 




.--' /■ 








s 














0.2 


_ ~ s 
s 

.■■' • 




- 




.■■' ^ ' 








' " 




N. 


0.0 


, , L A ". . . 1 . . ■ 1 ■ ■ 


. 1 ... 1 





0.5 



0.4 



0.6 0.8 



1.0 

e 



1.2 



1.4 



1.6 



Fie;. 4. — The run of density p, magnetic pressure P m „, and Mach number 
M„ as a function of I). The density and magnetic pressure are arbitrarily 
normalized. The density goes to infinity in the disk plane, i.e., for 8 = n/2. 



from r % R, we have T = [(4k) 1 ' 2 ] l R 2 B e , and we can esti- 
mate 



4»„ 



RB e 



(3.29) 



Similarly, 4n i!2 x is the integrated mass loss from the disk for 
r < R, so that from equation (3.5) we have 



4" 



*0.1 



M 



RB l » 



(3.30) 



Substituting these expressions for a , *F , and b into the defini- 
tion of C , we obtain 

(3.31) 



C *100M 26 Rr8 5 ' 2 B e 



,Ml 



.#\ 12 2"1\ 



with M lb the mass-loss rate in units of 10 2f> g s 



\ R 18 the 
radius of the BALR normalized to 10' 8 cm, and B„ 2 2 the 
vertical component of the magnetic field at the disk surface in 
units of 10 2 G. Estimating (/'//) % tf> BA Y % C£' 3 at the disk 
surface, the total magnetic pressure becomes 



P = 

mag 



Bl + B 2 



87t 



B[ 

87t 



1 + 



871 



(1 + C 2 ' 3 ) . (3.32) 



If we require the magnetic pressure to be of the order of the gas 



pressure in BAL clouds, P„ 



P BAL ~10- 4 M 8 i? x R- i( 



dyn cm \ and take C > 1, we obtain 

<t>B\L 



^OAM^Rl'iMli'Jti 



1/2 u>m 



if'_' 2 



(3.33) 



From equation (3.33), we see that these simple assumptions 
lead to the conclusion that a rather high mass-loss rate of the 
order of 1 M yr" ' is needed to explain the observed covering 
factor of the BALR. Although a mass-loss rate of this order is 
not unreasonable in comparison with the accretion rate 
inferred for the central black hole, it is at the upper end of what 
the UV line profiles indicate. It could also lead to more opti- 
cally thick winds, in which the effective opacity would be 



0.0 



en 
O 




-0.5 



1 

log C 

Fig. 5. — The scaling of the opening angle of the outflow BAL = (n/2) — 6 b 
with the parameter C . The straight line is a line with slope — ^; the crosses are 
calculated by solving the full eq. (3.15) for different values of C . It is seen that 
the scaling derived for C p 1 is quite good even outside its formal range of 
validity. 

reduced below the value used in equation (3.33), thus increas- 
ing 0bal again- 

A straightforward way out of this problem is to assume that 
the ionizing flux that determines the pressure in the BALR is 
smaller than would be expected from a simple extrapolation of 
the UV flux that does the line driving. It is easy to show that a 
reduction of the number of ionizing photons enters linearly 
into the expression <t> KAL , so that a reduction of this flux by a 
factor of 10 caused by some highly ionized absorber would 
suffice to obtain the correct opening angle for a mass-loss rate 
of 0.1 JW G yr~ \ in much better agreement with observed BAL 
profiles. Evidence for the existence of such highly ionized 
absorbers with the necessary column density (N H ~ 10 2O -10 22 
cm 2 ) is seen in the X-ray spectra of several Seyfert galaxies 
(Netzer 1 993 ; Netzer, Turner, & George 1995). 

4. DISCUSSION 

In this paper we have studied a very simple model of a 
radiatively driven magnetic disk wind that could be a general 
feature of all high-luminosity QSOs. Clearly the simplifications 
are so severe that we cannot make detailed predictions about 
line profiles, ionization states, etc. However, some general 
properties of our models, such as the fact that the observed 
opening angle of the BAL outflow can be reproduced for a 
reasonable set of physical parameters, can be expected to apply 
in more detailed models as well. 

Several other observed features of BAL QSOs seem to fit 
naturally into our model. The model predicts that as the line of 
sight to the continuum source makes a smaller angle with 
respect to the disk surface, it passes through regions with a 
smaller ratio of radiation pressure to magnetic (i.e., confining) 
pressure, implying a lower ionization parameter. This suggests 
that the appearance of a QSO may change from a high- 
ionization BAL QSO to a low-ionization BAL QSO as the 
angle between the line of sight to the center and the disk gets 



No. 2, 1995 



BAL QSO MAGNETIC DISK WINDS 



455 



smaller. Some observational characteristics of low-ionization 
BAL QSOs, such as their large infrared-to-optical flux ratios 
(Low et al. 1989) and the evidence for dust absorption in their 
spectra (Spray berry & Foltz 1992), lend support to this picture. 

The model also provides an explanation for the finding of 
Voit, Weymann, & Korista (1993) that, based on ionization 
equilibrium calculations, the high-velocity material is shielding 
the lower velocity, lower ionization gas in the line of sight to 
the central continuum source. Such a situation, with the high- 
velocity gas being closer to the center than the low-velocity 
gas, is exactly what is predicted by the radiatively accelerated 
disk wind model. 

Recently a new model for BAL QSOs was proposed by 
Murray et al. (1995), which attempts to explain the BALs using 
a radiatively driven disk wind with a filling factor of one, i.e., 
without any clouds. The great advantage of such a model is 
that one does not have to deal with the problems associated 
with the confinement and survival of the very small BAL 
clouds, and it would obviate the need for any magnetic fields in 
the outflow. However, to construct a model with a continuous 
wind it was necessary to make two assumptions that deviate 
strongly from the " standard " model. The first is that the size of 
the BALR is about a factor of 100 smaller than the scales 
considered here, and the second is that the ionization param- 
eter in the BALR is about 100 times larger. To reconcile the 
latter with the observed ionization state of the BALR, a very 
highly ionized and high column density (JV H ~ 10 24 cm 2 ) 
absorbing screen is assumed to exist between the central con- 
tinuum source and the BALR. 

In our opinion, this has at least two consequences that are 
very difficult to reconcile with observations. First, the small 
scale of the BALR required by Murray et al. (~ 10 16 cm) makes 
it very hard to explain the observed lack of variability in the 
BAL profiles on a timescale of several years (Barlow, Junk- 
karinen, & Burbidge 1989; Barlow et al. 1992), since the cross- 
ing time of the BALR would be of the order of months. Since 
most observed BAL profiles are highly structured, such 
changes on the flow timescale should be easily observable. 
Even if the structure we see is caused by some structure in the 
disk influencing the mass loss, so that the flow crossing time- 
scale is not relevant, the rotation period of the structures on 
the disk would still be shorter than the timescales over which 
no variability is observed. Second, although the class of BAL 
QSOs as a whole may be underluminous in X-rays (Green et 
al. 1995), the strong X-ray shielding of the BAL outflow 
required in this case to avoid overionizing the wind does not 



seem consistent with the fact that several BAL QSO have been 
observed with normal optical-to-X-ray flux ratios (Bregman 
1984; Gioia et al. 1986; Singh, Westergaard, & Schnopper 
1987; de Kool & Meurs 1994; Green et al. 1995). 

From the theoretical side, some aspects of the model also 
need closer study. It is not clear that the hard ionizing flux can 
be suppressed as strongly as needed in the model, since a large 
fraction of the absorbed energy will be reemitted in the soft 
X-ray band (Netzer 1993). Since the evidence that the BALR 
coincides with or lies outside the BELR is incontrovertible, the 
model also implies that the BELR lies at much smaller radii 
than previously thought and will have to be ionized by a 
heavily absorbed spectrum. Attempts at modeling the observed 
ionization state of the BELR with such ionizing spectra have 
not been very successful so far (H. Netzer, private 
communication). 

While a model invoking clouds is necessarily more compli- 
cated, we are unconvinced that it is inherently less plausible 
than a continuous wind model, especially in light of the diffi- 
culties mentioned above. In arguing the disadvantages of the 
cloud model, Murray et al. claim that there are no physical 
effects which would set a size scale for clouds. However, we 
have shown (§ 2) that the inferred sizes of BAL clouds coincide 
with the pressure scale height of the absorbing gas in the accel- 
erating reference frame of the wind. Thus, very small cloud 
sizes may be a generic feature of any BAL wind model in which 
the accelerating force is mainly applied at the surface of the 
absorbing material. This would include radiative acceleration 
of clouds with moderate to high optical depths, as well as 
dragging of clouds by magnetic stresses or a hot intercloud 
medium. 

The magnetized wind model proposed by us may be difficult 
to extend to the point at which a detailed comparison with 
observations is possible, but it does not lead to clear contradic- 
tion with observation, and it does not require us to revise the 
currently accepted picture of QSO structure. Further theoreti- 
cal investigations are clearly necessary, especially regarding the 
details of the magnetic confinement mechanism, and we are 
currently investigating this issue. 

We thank Hagai Netzer for his comments on the effects of 
ionized absorbers. This work was partially supported by 
National Science Foundation grant AST 91-20599 and 
National Aeronautics and Space Administration grants 
NAGW-3554 and NAGW-3838. M. dK. thanks the Fellows of 
JILA for hospitality and resources during a visit. 



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