The numerical models described above are successful in reproducing many of the observed characteristics of the Raman lines in symbiotic systems, and in helping to understand how the spectra respond to changes in the model parameters; there are also several points on which they fail, suggesting ways in which they may be extended and improved.
Table 4: Quadrature polarization efficiencies and mean velocities for
model 
6825 Å lines
Table 4 (click here)
summarizes two
basic quantities derived from selected models: the
`polarization efficiency' of the line,
and its first moment (or `mean wavelength')
expressed as a velocity. By comparing the data in Table 4 (click here)
with the corresponding
observational data
presented in Paper I, we note that (i) the overall
redshifts found in the observational data are reproduced by the models
(which give reasonable quantitative agreement with the mean observed
value of 265kms
), and (ii) that the observed values of
(0.017 to 0.27; mean 0.074) are
encompassed by the range found in the models. In addition, the
reference model
returns values for photon number ratios,
and
, which are
in reasonable agreement
with observations
(average 
-5,
Allen 1980, Paper I;
,
Espey et al. 1995).
These results
are encouraging from the point of view of developing `tailored' models
for individual systems.
The intensity profiles of the observed Raman lines are broadly similar
(see Paper I); all the lines are asymmetric, with a stronger blue wing
than red. The intensity profiles are often resolved into a multiple
subpeak structure, with the most frequently observed profile consisting
of a strong central peak with less intense blue- and red-shifted
subpeaks. The reference model (Fig. 5 (click here)) has an asymmetric
profile with a stronger blue wing, but does not have resolved subpeaks.
As the mass-loss rate is reduced (Fig. 13 (click here)), the subpeak
phenomenon is observed, such as in the reference model with 
.
The widths of the observed Raman lines are difficult to measure
objectively due to their asymmetric, multipeaked nature and frequently
low contrast. Nonetheless, the observed lines are generally much
broader than the model profiles obtained by using
=10 kms
, and are comparable with the models
calculated using 
-100 kms
. This breadth could
result from intrinsically broad OVI lines, but such breadth
would `wash out' the structure in the Raman lines, and would not
explain the separation of the observed peaks. In principle,
it would be possible to obtain the observed profiles by scattering of
suitably-structured OVI emission lines in a static regime.
However, this would be entirely ad hoc, would not be consistent with
high-dispersion observations of optical emission lines, and would ignore
the more natural explanation afforded by scattering in the red-giant
wind.
We conclude that the separation of the peaks in the Raman lines is most
probably a direct function of the velocity structure of the red-giant
wind. In that case the expansion velocities in the scattering regions
must typically be closer to 
50 kms
than the canonical
value of 
10 kms
frequently adopted for
single stars. These two numbers may not be in conflict if the red
components of symbiotic systems differ from their isolated counterparts
(cf. Kenyon 1988, Whitelock & Munari 1992), or if most of the
acceleration occurs rather close to the star. In the latter case,
gravity could very plausibly slow the outflow to the canonical value
observed at large distances from single stars, although this would be in
conflict with both standard wind theory and the density structure
inferred in EG And by Vogel (1991). A direct determination of the
far-field velocity of a symbiotic system's red-giant wind would clearly
be valuable.
Figure 22: Polarization profiles calculated using an extended source. a) The
reference model (solid line) and a model with 
,
b) the reference model with 
(solid line) and with
and 
(dotted line) and c) as for
model b) but with 
The most straightforward use of the Raman lines is as binary-orbit
diagnostics. All the model polarization spectra show systematic
dependences on wavelength which are largely independent of the viewing
angle 
. Thus the rotation of observed PA with binary phase (for
orbits with 
) is virtually model independent; it results
solely from rotation of the binary geometry. Hence a time series of
polarization spectra should yield a binary period, simply by observing
the period changes in the PA of the polarization.
The magnitude of the Raman-line polarization is a more ambiguous
diagnostic, but has some simple properties as a function of binary phase
(for systems with 
). The polarization should reach a maximum
twice per binary orbit when the source-scatterer-observer angle is 
.
Similarly, the two polarization minima per period will occur at
conjunctions.
The models described above show that the Raman-line intensity is much less
sensitive to the binary phase than the polarization, particularly in systems of
extensive mass-loss. This is a result of the relatively large volume of Raman
scattering, which reduces the effect of occultation by the giant star. In
systems of lower mass loss the bulk of the scattering occurs in the illuminated
hemisphere of the giant and the line intensity shows a strong binary dependence,
with the intensity reaching a maximum when the largest fraction of the
scattering hemisphere is visible to the observer.
Observations of a small number of objects at two epochs, reported in Paper I, support these predictions of the models. The intensity profiles of the Raman lines are remarkably stable for most objects (with the exception of RR Tel, for which the variation in rectified intensity is probably due to a change in the continuum level rather than a change in the absolute line flux), while the polarization spectra show significant changes. With only two published observations, conclusions cannot be drawn about the binary orbits of these objects, but, in future, time-series spectropolarimetry will certainly yield valuable data on the symbiotic orbits (cf. Harries & Howarth 1996a).
The most troubling features of the model calculations reported here result from scrutiny of the predicted PA structure. The model structure has a quite straightforward interpretation, but the majority of our calculations do not reflect the behaviour reported in Paper I for the majority of systems. In particular, many observations show three (polarized) intensity peaks, with a PA `flip' in the redmost peak (e.g., Fig. 6 (click here)); in contrast, most models show a flip in the central peak (e.g., Fig. 5 (click here)). This may simply be a consequence of the fact that we have concentrated on a region of parameter space most appropriate to relatively close, short-period systems, with negligible absorptive opacities; the sensitivity tests do show a rather broad range of behaviour in the models, although none show the `red flip'.
Qualitatively, we expect that it may be possible to reproduce this PA behaviour in a bipolar-type outflow, with the wind velocity greatest in the direction out of the orbital plane; there is direct evidence that in at least some systems the outflow may be more nearly cylindrically, rather than spherically, symmetric (e.g., Corradi & Schwartz 1993). The inclusion of a non-spherical wind structure is straightforward in principle, but is once again expensive computationally, since the mass-column calculation requires a numerical approach except for a few special geometries. We have performed a small number of exploratory calculations, but we have not yet succeeded in obtaining a PA flip in only the third (redmost) peak.
A more fundamental difficulty is that any model with `top-bottom' symmetry can
only show emission at position angles of 90
and 0/180
(or other
orthogonal pairs, as the reference system is rotated), because of cancellation
in Stokes' U. As demonstrated in Paper I, several systems, such as H2-38,
show rotation in PA through the line profiles (that is, loops in the
Q-U plane). This rotation can be synthesized within the framework of the
models by adding some additional polarized intensity at a different PA. The
inference of significant intrinsic continuum polarization in a large fraction
of well-observed systems (Paper I) suggests that this is a plausible (if
qualitative) explanation. Moreover, rotation is much commoner in the D-type
systems discussed in Paper I than in the S-type systems (6/7, compared to
8/17). This is circumstantial evidence that dust may implicated, in which
case an alignment mechanism would be required.
The basic model contains several assumptions that must be relaxed in later work. The most obvious simplification is the omission of a proper, consistent treatment of the ionized region of the wind. Since Raman scattering arises only in neutral-hydrogen regions, none occur in the HII wind volume, but the latter will have absorptive and scattering opacities (different to those in the neutral region), including an electron-scattering component, potentially leading to polarization of both Raman and OVI photons; dust scattering and absorption may also be important. The inclusion of the ionized region is conceptually straightforward but requires significantly more computational effort; the intercept between an arbitrary photon path and the boundary region must be calculated repeatedly, and hence an analytical solution must be available if the calculation is to be feasible in reasonable times. In addition, the calculation of the optical depth along a photon path becomes more complex, since absorption and scattering opacities will vary along the photon path. None of these difficulties represent fundamental obstacles.
We have presented the results of Monte-Carlo simulations of Rayleigh+Raman scattering in spherically symmetric, steady-state red-giant winds, with a view to understanding the characteristics of the Raman lines observed in symbiotic systems. These models represent a first reconnaissance of parameter space, and so close matches with specific observations are not to be expected. Nonetheless, encouraging agreement with many observed characteristics has been achieved.
The next phase of this work should be to attempt to relax some of the simplifying assumptions made in this work and to produce `tailored' models for specific systems. This will be most easily undertaken where repeat observations constrain the orbital parameters and define the phase dependence of the line characteristics. The model described here is sufficiently flexible that it can easily be modified to exploit the additional diagnostic potential offered by the orbital modulation of line strengths resulting from intrasystem line and (quasi-)continuum absorption (cf. Vogel 1991; Shore & Aufdenberg 1993; Schmid 1995).
Acknowledgements
We thank France Allard and Peter Hauschildt for communicating opacities from their cool-star model atmospheres. The Particle Physics and Astronomy Research Council supported this work through the provision of Starlink computing facilities at UCL, where the numerical work was performed, and of a postgraduate studentship to TJH.