A&A 426, 323-328 (2004)

DOI: 10.1051/0004-6361:200400500

**J. C. Brown ^{1} - J. P. Cassinelli^{2} - Q. Li^{1,3} - A. F. Kholtygin^{4,6} - R. Ignace^{5}**

1 - Department of Physics and Astronomy, University of Glasgow,
Glasgow, G12 8QQ, UK

2 -
Department of Astronomy, University of Wisconsin-Madison, USA

3 -
Department of Astronomy, Beijing Normal University, PR China

4 -
Astronomical Institute, St. Petersburg University, Saint Petersburg State University, VV Sobolev Astronomical Institute, 198504 Russia

5 -
Department of Physics, Astronomy, & Geology, East Tennessee State University, USA

6 -
Isaac Newton Institute of Chile, St. Petersburg Branch, Russia

Received 23 March 2004 / Accepted 24 June 2004

**Abstract**

The hot star wind momentum problem
is revisited, and it is shown that the conventional belief,
that it can be solved by a combination of clumping of the wind and
multiple scattering of photons, is not self-consistent for optically thick clumps. Clumping does
reduce the mass loss rate
,
and hence the momentum supply,
required to generate a specified radio emission measure
,
while multiple scattering increases the delivery of momentum from
a specified stellar luminosity *L*. However, in the case of thick clumps, when combined the two
effects act in opposition rather than in unison since clumping reduces
multiple scattering. From basic geometric considerations, it is shown
that this reduction in momentum delivery by clumping more than offsets
the reduction in momentum required, for a specified
.
Thus
the ratio of momentum deliverable to momentum required is maximal for
a smooth wind and the momentum problem remains for the thick clump case.
In the case of thin clumps, all of the benefit of clumping in reducing lies in reducing
for a given
so that extremely small filling factors
are needed.

It is also shown that clumping affects the inference of from radio not only by changing the emission measure per unit mass but also by changing the radio optical depth unity radius , and hence the observed wind volume, at radio wavelengths. In fact, for free-free opacity , contrary to intuition, increases with increasing clumpiness.

**Key words: **stars: circumstellar matter -
stars: mass-loss -
stars: winds, outflows -
stars: Wolf-Rayet

If one infers the mass loss rate
for hot massive
(especially Wolf-Rayet) stars from the radio emission measure
,
using a smooth spherical wind model, one
finds that the wind "momentum'' rate
involves ,
where *L*/*c* is the radiative momentum
outflow rate (Cassinelli & Castor 1973). Insofar as such
winds are believed to be radiatively driven, this poses a "momentum
problem'', the solution of which has long been a hot topic in the field
(Barlow et al. 1981; Abbott et al. 1986; Cassinelli
& van der Hucht 1987; Willis 1991; Lucy &
Abbbot 1993; Springmann 1994; Springmann & Puls
1995; Gayley et al. 1995; Owocki &
Gayley 1999). Estimates of
vary according to assumptions
(e.g., arguing for a high value of *L*) but values of
ranging
up to nearly 100 are mentioned (Hamann & Koesterke 1998).
There are two main strands of argument quite widely believed to combine
to solve the momentum problem, one being mainly observationally driven
and the other mainly theoretical.

The values of
associated with these large
are those
inferred from a smooth spherical wind density model, the radio emitting
material filling the volume. The contribution to
from
any volume element
is
.
If, however, the material is clumpy,
filling only a fraction
of the volume, then
is enhanced by a factor 1/*f*for a given .
The mass loss rate
required to generate
an observed
thus scales as
in
clumpy winds. For strong clumping (), this ameliorates the
momentum problem, though the *f*=10^{-4} required to reduce
by a factor of 100 seems very unlikely, so this
clumping effect alone cannot be the complete answer (e.g., Nugis &
Lamers 2000, cite clumping corrected mass-loss rates yielding
). For example, making clumps very small increases their
radio optical thickness and may make optically thin emission measures
irrelevant. There is extensive observational evidence for large scale
clumping in WR winds: the presence of narrow emission features moving
out on broad wind emission lines (e.g., Robert et al. 1989, 1991; Moffat & Robert 1991; Kholtygin
1995); broad band photometric and polarimetric fluctuations
(e.g., Brown et al. 1995; Li et al. 2000); and the absence
of strong electron scattering wings (which scales as
rather than
,
Hillier 1991).

On the theory side, it has long been recognised that the limit is only true if (all) photons are scattered once only. If the wind scattering optical depth is high, the photons can, loosely speaking, be scattered "back and forth'' across the wind delivering momentum of up to at each scattering (for thick clumps) until is progressively dissipated by Doppler reddening at each momentum-delivering scattering on the moving matter. The nature of this multiple scattering has been described with progressively greater insight over the years. In particular, Gayley et al. (1995) showed that scattering back and forth across the entire wind is not required. Instead, the momentum is delivered in a series of random semi-local scatterings of photons as they diffuse outward, provided successive scatterings involve long enough paths to sample different matter velocities. The essential feature is that of the large scattering optical depth , which enhances the momentum delivery rate to (e.g., Friend & Castor 1983; Kato & Iben 1992; Netzer & Elitzur 1993; Gayley et al. 1995), because the diffusive delivery scales with the number of scatterings as , while . Since the predominant driver is via the large opacity/cross-section associated with lines, Gayley et al. (1995) and Owocki & Gayley (1999) have suggested that the issue is not so much a momentum problem as an opacity problem.

The massive WR winds are still believed to be driven by line opacity
(e.g., Lucy & Abbott 1993). Unlike the less massive winds of
OB stars, the WR winds have significant ionization gradients, that can
substantially alter the line opacity distribution with radius in the flow
(Herald et al. 2000; Vink et al. 2000).
Consequently, as the photons move away from the star, interact with
a certain line opacity that exists at some radius *r* in the flow, and
then escape, the photons encounter a new line opacity distribution
at a different radius. Consequently, if there are gaps in the line
distribution at one radius, those gaps can be filled by a different line
distribution that exists in another part of the wind flow. The opacity
problem then represents how effectively all of these gaps are "filled''.

Photon escape at gaps in the line frequency forest reduces the flux mean opacity (or flux mean cross section per particle in our formulation) used in the gray approximation. The maximum that can be achieved by multiple scattering is reached when the number of random scatterings is so great as to Doppler shift the photons down to near zero frequency, the maximum Doppler shift per scattering being of order for wind speed . This requires ( ), implying or which is the energy conservation limit. Available calculations of multiple scattering with real opacities can yield gains of order 10, that may explain some WR winds, but not the more extreme cases in which is required.

Since reduction of
(for a given
)
by clumping
and increase of momentum delivery by multiple scattering can each
offer a factor of order 10 reduction in the momentum problem, there
seems to be growing widespread belief that the momentum problem can
be laid to rest (e.g., Conti 1995). However, this involves the
tacit assumption that these two factors can operate *independently*
and *constructively*, the impact of clumping on the effectiveness
of multiple scattering never having been addressed (Hillier & Miller
1999; although Shaviv (1998) has discussed the
related topic of how optically thick clumps increase the Eddington luminosity for novae).
Here we show, using simple geometric
arguments, that this assumption is incorrect in the case of optically
thick clumps, and that clumping, while reducing ,
also reduces
,
so making multiple scattering less effective. Essentially
this is because clumping reduces the number of scattering centres
compared to scattering off of atoms and also reduces
,
for a given
.
(Note that when discussing the effects of clumping it is
essential to keep in mind that the observed
is held fixed.
This fact is sometimes overlooked.)

We find quantitatively that, for thick clumps, the reduction in multiple scattering momentum delivery more than offsets the reduction in momentum required, the nett effect being that clumping worsens the momentum problem rather than solving it.

To illustrate the point, we first consider one thick
scattering clump of mass *M* composed of atoms/ions of mass *m*. This
is taken to have very high internal optical depth in the line-driving
wavelength range so the clump as a whole is the scattering centre.
Since we are not concerned with the wind speed profile but only with the final wind speed and momentum, we here approximate
clumps as moving radially with speed
and to have the shape of a conical slice of radial thickness
and solid angle ,
the volume of the cone being
at distance *r*. We assume the clump to be
optically thin at radio wavelengths, so its radio flux depends
on the emission measure, but optically thick to lines for the stellar
radiation at short wavelengths that are
responsible for driving the flow.

The emission measure
(which measures the radio emission rate) of a
single clump is

where the clump emission measure at , for the radius of the radio photosphere which may be hundreds of times larger than the optical photosphere radius. Note that, for a prescribed , for any chosen .

On the other hand, the available rate of delivery of momentum is

where we ignore scale factors of order unity due to the effects of gravity and of the backward scattering angular distribution function. The rate of momentum delivery required is

(3) |

It follows that, for a given , the effectiveness of momentum delivery to a single clump is

This decreases as we make smaller - i.e., as we make the clump clumpier - because the momentum required , but the momentum available , and the decline of the latter with is dominant for a single clump. That is, making the of a single clump smaller does reduce for a given but reduces even more. So shrinking one clump of a given makes it harder to drive it to terminal speed of known value .

Compressing the clump radially does help (in this single thick clump case) since reducing reduces for prescribed .

We now have to consider the effect of multiple scattering in the case of a multiple clump wind, since multiple scattering cannot occur in the case of an individual discrete clump. In doing so we take all clumps to be optically thick in the UV but thin in the radio, identical in size and mass, and use the gray opacity approximation, the clumps being driven by a spectral mean "continuum'' radiation flux. We are of course well aware that in reality there will be a distribution of clump sizes and masses. However, if one can prove that for any specific clump parameters, clumping reduces the benefit of multiple scattering, then the same must be true of the sum over any distribution of clump parameters so long as they remain thick. Put another way, the arguments that clumping a wind increases its emission measure, that multiple scattering increase momentum delivery, and that clumping reduces multiple scattering all derive essentially from geometric arguments and have nothing to do with the details of opacity or of clump size distribution (other than being thick).

Retention of the conical slice shape described above, taking and
independent of *r*, means that the clumps expand in 2-D
(transversely) rather than in 3-D, which is reasonable for a highly
supersonic wind. The constant ,
assumption also
means that, for constant ,
clumps occupy the same fraction
(constant filling factor *f*) of the volume at all *r*. For spherical
(3-D) clump expansion, linear radial expansion (
)
would result, for constant ,
in radial merging of clumps, which
corresponds to an *r*-dependent filling factor *f* with
as
clumps merge. Situations with non-constant filling factor *f*=*f*(*r*)have been discussed by Nugis, Crowther, & Willis (1998);
Hillier & Miller (1999); and Ignace et al. (2003).
We assume clumps are, on average, emitted uniformly over the stellar
surface at a rate
in clumps per second. Then the space density
of clumps at *r* is

where we again approximate constant and the radio emission measure of one clump is again given by Eq. (1). Using Eqs. (1) and (5), the total emission measure can be written

The last form is interesting, showing that the total emission measure is just the initial emission measure of one clump at times an effective number of clumps , namely that located in the range .

The mass loss rate
and the momentum delivery rate
required are

and

where we neglect scale factors of order unity as we did in Eq. (2). By Eqs. (6) and (8), we get the momentum delivery rate required for a given total wind emission measure as a function of

We want to compare this with the momentum delivery rate available from multiple scattering of stellar photons and we take this to be given by (cf. Sect. 1)

where is the mean (gray approximation) line scattering optical depth of the wind and is the number of scatterings of an escaping photon. is also the "covering factor'' or the total solid angle of all the clumps as seen from the star divided by - see Appendix.

The wind optical depth for starlight due to lines treated in the
gray approximation is (for individually thick clumps)

(11) |

where

We choose to split the range into two
sectors, *r*<*d* and *r*>*d*, where *d* is the distance at which
an individual clump becomes optically thin radially. At
*r*<*d* the individual *scatterer* is a clump of area
and thickness ,
while at *r*>*d*, it is an ion of
area
(the actual value adopted for
being some frequency
average over lines).
Thus the optical depth integral expands to

where

and thus

(14) |

Then Eq. (12) becomes

Consider the second term in expression (15). The ratio is the total area of all the atoms in a clump, and is the total area of a clump at . Since our analysis deals with individually thick clumps, we require , so we get and can neglect the second term in Eq. (15) to write

If we express in terms of by using Eq. (6), then Eq. (16) becomes

Now we get the available momentum using Eqs. (10) and (17) in terms of

Comparing Eqs. (9) and (18), we find a dimensionless measure of the effectiveness of momentum delivery, as the ratio of momentum available to momentum required, as a function of

For a given star (

The essential result is that
increases with increasing ,
i.e., with increasing clump cross section per unit mass (which is
different from the single clump case of Eq. (4)). To
minimise the momentum problem (maximise )
for a given mass *M*(and thickness ),
should be as large as possible while
for a given
the mass *M* should be as low as possible with,
in both cases,
varying according to Eq. (6) to
ensure the correct
.
If we change (e.g., increase) ,
does not change but
changes (falls) to maintain fixed
.
Consequently, to maximise
we must make the clump
mass small, the clump angle large, and the clump thickness large with
correspondingly small ,
all of these corresponding to minimising
clumping.

It is also of interest to express
in terms of the volume filling
factor
which can be expressed
(with
single clump volume
at *r*) as

(20) |

where and are solid angle and radial filling factors respectively. Alternatively, the volume filling factor can be expressed as

Comparing Eqs. (19) and (21), we see that, under our thick clump assumption, so that , so that for any values of ( ), decreasing

All of the above shows that, contrary to conventional "wisdom'', in the case of thick clumps, clumping does not help solve the momentum problem but actually makes it worse.

The case of a smooth wind can be considered a limit
of the clumpy case as the clumps blend.
However, there are infinitely many clumpy cases that approach the smooth
case as the clumps blend and it is easier to evaluate
for the smooth case directly using
.
Then
with subscript "o'' denoting the smooth case, we get

and

from which we deduce that

Explicitly comparing the momentum delivery effectiveness for the clumpy and smooth cases we have, by Eqs. (19) and (26)

which is clearly for clumps which are initially optically thick. Note also that involves the ratio of and , respectively, the cross-sections per unit mass of clumps and of atoms, while involves only . Clearly, the continuous limit corresponds to at (cf. Eq. (13)), the "clumps'' become individual atoms, becoming thick at that point. Note that is essentially the limit where driving approaches the smooth wind limit, equally applicable to optically thin clumps.

Carrying this line of inquiry further, it is helpful to see how
Eq. (19) for
approaches the smooth limit
.
We require that
from Eqs. (16) and (24),
that
from Eqs. (6) and (23), and finally that *f*=1. These conditions are
met for

(28) |

and

which, on substitution in Eq. (19) gives (Eq. (26)) as required. To interpret Eq. (29) physically, note that is the total cross sectional area of all the atoms in one clump while is the cross sectional area of one clump at . These can only be equal if clumps have the scale of individual atoms. Secondly, Eq. (30) can be expressed as . Here is the total area of all clump atoms per unit radial distance, so is the total area of all clump atoms in a scale length . Thus, since is the spherical area at , the scale defines the range of

where is the relevant cross section per proton. However, we have to note that the main radio absorption mechanism is free-free opacity which is density dependent ( ) and we have to write

where and are defined in any reference level . Then Eq. (31) becomes

so that now is given by , namely

Consequently, increasing

The corresponding emission measure expression is now as before but based on the new clumping dependent value
in Eq. (34) of
which leads to

which does increase as we increase (i.e., clumpiness) for a given but now with , instead of for the constant case. Thus although clumpiness still reduces for a given , it does so less than with constant and likewise, thick clumps are now even less helpful to the momentum problem.

The authors acknowledge support for this work from: a NATO Collaboration Grant (A.F.K., J.C.B., J.P.C.); a UK PPARC Research Grant (J.C.B.); NASA Grant Number TM4-5001X (J.P.C., J.C.B.); Royal Society Sino-British Fellowship Trust Award (Q.L.); a NSFC grant 10273002 (Q.L.); and a RFBR grant 01-02-16858 (A.F.K.). We thank the referee (Ken Gayley) whose comments led to a significant improvement of the paper.

*Y* is the fraction of the solid angle around a star that is covered
by scatterers. Let *A* and
be the cross section and solid angle
for one scatterer at *r*, so that
and let
be the space density of
scatterers, then the covering factor at *r* is the total solid angle of all the scatterers divided by ,
namely

(A.1) |

A photon travelling in a medium with typical size

(A.2) |

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