A&A 426, 323-328 (2004)
J. C. Brown1 - J. P. Cassinelli2 - Q. Li1,3 - A. F. Kholtygin4,6 - R. Ignace5
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
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/ffor 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
On the other hand, the available rate of delivery of momentum is
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 ,
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
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
The mass loss rate
and the momentum delivery rate
The wind optical depth for starlight due to lines treated in the
gray approximation is (for individually thick clumps)
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
(the actual value adopted for
being some frequency
average over lines).
Thus the optical depth integral expands to
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
which can be expressed
single clump volume
at r) as
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
with subscript "o'' denoting the smooth case, we get
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),
from Eqs. (6) and (23), and finally that f=1. These conditions are
The corresponding emission measure expression is now as before but based on the new clumping dependent value
in Eq. (34) of
which leads to
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
be the space density of
scatterers, then the covering factor at r is the total solid angle of all the scatterers divided by ,