In order to examine the distribution of the Raman line formation, the position, frequency, weight, and polarization of each Raman-scattered photon packet were stored for the reference model, viewed at quadrature. The three spatial dimensions were then collapsed to produce two-dimensional images of the model outputs, which are shown in Figs. 2 (click here), 3 (click here), 4 (click here). The corresponding spectropolarimetric results are shown in Fig. 5 (click here).
The strongest Raman emission in this model originates close to the
parent photon source, with the intensity dropping with increasing
radial distance simply because of dilution. (This result must be
treated with some caution, since the reference model neglects the
HII region which will usually be associated
with the hot source; within such a region the number of
neutral-hydrogen scatterers will be negligible.) The intensity also
falls off rapidly away from the red giant, because of the decreasing
density of the wind. The bulk of the Raman scattering therefore
occurs between the photon source and the red-giant photosphere, where
the density of both O
photons and H
scatterers is high.
Multiple Rayleigh scatterings degrade the directionality of the
OVI photon field, and hence the normalized polarized
intensity 
. Since there is no UV opacity in the reference
model, the Raman photons emerge after three Rayleigh scatterings on
average; we therefore expect the normalized polarized intensity to be
greatest at moderate to large radial distances from the photon source,
because the scatterers will see a more nearly radial radiation field
(most scatterings occuring close to the source).
Figure 2 (click here)d confirms this expectation, with normalized
intensities approaching unity at 
from the parent source. Figures 2 (click here)a and b
further confirm that the code generates the expected behaviour in Q
and U.
A second effect of Rayleigh scatterings is broaden the velocity dispersion of the OVI photons. Local scatterings have little effect in this regard, as there is only a small velocity dispersion between the source atoms and scatterers, but more distant scatterings can introduce considerable shifts in the profile; scatterings across opposite hemispheres will introduce shifts corresponding to twice the wind velocity. These shifts are always redshifts, as every point in the wind sees every other point receding from it (the `expanding universe' paradigm).
Figure 3: Results for the reference model (
6825 Å line),
viewed at quadrature (
5 (click here)).
Shown are a) the polarized intensity, 
(scaled logarithmically over
3 decades), and b) the line-of-sight component of the velocity
(scaled linearly over the range -50 to
+150 km s
)
Figure 4: The polarization of 
6825 Å
Raman photons
for the reference model viewed at quadrature, overlaid on a logarithmic
intensity scale (scaled over 3 decades). 100% polarization
corresponds to a vector of 1 
in length
Figure 5: The polarization spectrum of the reference model
(Sect. 5 (click here)), viewed at quadrature.
The error bars are one sigma. Note that the two Raman lines are
plotted on different scales
Figure 6: Observations of the 
6825 Å line in a) V455 Sco,
b) Hen 1092, c) Hen 1242, and d) RR Tel, adapted from Paper I
(Harries & Howarth 1996b).
The data shown here have been corrected for the interstellar
polarizations derived in Paper I, to facilitate comparison of the
percentage polarizations with the models. The polarized-flux and
position-angle data in the line are insensitive to such corrections,
aside from zero-point shifts
6825Å line
The spatially-integrated polarization spectrum (Fig. 5 (click here))
is basically composed of three separate peaks (at 
6821, 6831,
and 6836 Å in the 
6825 Å line), most obvious in the
polarized flux, but also identifiable in the intensity spectrum.
The polarization in the central peak is orthogonal to the other two.
The origin of this characteristic is most easily understood by noting
that the polarization-vector directions are centrosymmetric about the
parent source (Fig. 4 (click here)). Because of the cancellation of
the U parameter in the spatially integrated flux, the position angle
in the models must have a value of 90
or 0/180
(within the
errors of the numerical `noise'). The 90
features originate
`above' and `below' the parent-photon source (i.e., along directions
perpendicular to both the sight-line and the line-of-centres); thus the
spatial origin of a given peak can be deduced from its position-angle
dependence.
The bluemost peak is clearly produced in the region between the two stars (the only region of the red-giant wind seen as blue-shifted by the OVI source in the geometry of the reference model). The polarization spectrum is characterized by very high polarization in the blue line wing (as high as 80%), although there is very little line flux there. The highly polarized blue line-wing is observed in many symbiotics after the removal of continuum flux (cf. Schmid & Schild 1994; Paper I; Fig. 6 (click here)).
The central (redshifted) peak has polarization position-angle
orthogonally polarized to the bluemost peak, and thus must originate in
the two quadrants of the red-giant wind `above' and `below' the
parent-photon source. The highest-velocity peak is displaced from rest
by 
100 km s
in `parent-velocity' space. The velocity image
(Fig. 3 (click here))
confirms that this emission arises from
scattering across hemispheres, from close to the parent-photon source to
the `back' of the red giant. This third, broad peak is more obvious in
the intensity spectrum than the polarized flux (Fig. 5 (click here)),
showing that the `back' of the wind first sees most photons after they
have already undergone multiple scatterings, which
produce an effectively spatially extended source, resulting in
reduced polarization (Sect. 11 (click here)).
Although in our models all photon
packets are forced to Raman-scatter, the weighting scheme
(Sect. 2.4 (click here)) ensures that the number ratio of input OVI photons to output Raman photons is correct. In the reference model
this ratio is of order 10:1 for 
1032:6085 - that is,
90% of OVI photons escape, in agreement with the
observations reported by Espey et al. (1995).
7082 Å line
The 
7082 Å simulation (Fig. 5 (click here)) shows broadly
similar characteristics to the shorter-wavelength Raman line, in
agreement with observations (Paper I). The most obvious differences
between the lines are the factor 
6.7 difference in Stokes' I
(for an assumed 2:1 photon ratio for
:
),
and
the change in relative strength of the central
(
7090 Å)
and redshifted (
7095 Å) polarized-intensity peaks.
The observed 
intensity ratios are in
the range 
2-10, averaging 
4-5 (Allen 1980; Paper I). The
reference-model result is slightly larger than this average value,
but this mild discrepancy
could easily be resolved if
, or if
:
.
In our models, the only fixed parameters distinguishing the
6825 and 7082 Å lines are the Rayleigh- and
Raman-scattering cross-sections for the parent OVI photons,
which are smaller for the longer-wavelength line. The difference in
line intensity in the reference model is therefore directly attributable
to the escape of a larger fraction of 
1038 Å than
1032 Å photons (by a factor of 
3).
This also accounts for the differences in line structure: the
red-giant wind is quite optically thin perpendicular to the line of
centres, so there is little scattered flux from those quadrants, or
from the quadrants furthest from the red giant. The blue-shifted
polarized flux again arises from the region between the two stars,
while the redmost peak arises from multiple scattering, and is
slightly more intense (relative to the other two peaks) than the
model because the ratio of Raman:Rayleigh scatterings is
slightly higher, meaning that the average Raman photon has undergone
fewer scatterings (which reduce the `coherence' of the polarization
signal) than in the corresponding 
model.
Clearly, observed differences between the 
6825 and
7082 Å lines encode differences in the governing system parameters
(e.g., the separate values of 
and 
appropriate to the formation of the two lines). Such differences have
important diagnostic potential in principle, but the broad similarity
of the two lines in the reference model (as well as in observations;
Paper I) suggests that the model sensitivity of response of the Raman
lines to those parameters can, to first order, be adequately
demonstrated by investigating either line separately. Moreover, the
ratio depends directly on the
input 
ratio, which is not well
known; a 2:1 ratio is expected under optically-thin conditions, but
the resonance doublets of CIV, NV, and
SiIV commonly show smaller ratios in practice (e.g., Shore
& Aufdenberg 1993 and references therein). Henceforth we therefore
concentrate on the stronger, and more reliably observed,
6825 Å line.
Figure 6 (click here) shows polarization data for the S-type systems V455 Sco, Hen 1092, and Hen 1242, and for the D-type system RR Tel. These illustrate part of the diversity of polarimetric behaviour reported in Paper I.
The blueward rise in polarization seen in the model spectra is not apparent in the observations, as dilution by unpolarized continuum flux (which is not included in the model calculation) means that the polarization tends to zero in the line wings. It is possible to subtract off the diluting flux in the data (cf. Figure 3 in Paper I), but small uncertainties in continuum placement lead to large uncertainties in the polarization, particularly in the line wings. As commented in Paper I, we therefore believe that the polarized flux the more straightforward observable against which to compare the models.
The data for Hen 1092 are broadly consistent
with the results from the reference model in that they show a central peak in
PA, `flipped' at 
to the adjacent regions, although the relative
strengths of the polarized-intensity peaks differ from those in the reference
model. The data for Hen 1242, which are of better quality, also clearly show
the 
flip between the bluemost and central peaks, although the
polarized intensity of the bluemost peak is again much less than the reference
model predicts. However, in the observations shown in Fig. 6 (click here),
the central and redmost peaks show the same PA. In both V455 Sco and
RR Tel the relative strengths of the intensity and polarized-intensity
peaks are in satisfactory
agreement with the model, but the central and bluemost peaks show the same PA,
with a 
degree flip in the redmost peak. Thus these stars
show the bluemost peak flipped (Hen 1242), the central peak flipped
(Hen 1092), and the redmost peak flipped (V455 Sco, RR Tel) - the last
configuration being the commonest, present in at least half the spectra
reported in Paper I as showing a `flip'.
Given the range in observed spectropolarimetric characteristics noted above and reported on further in Paper I, as well as the number of free parameters in the model, it is not unexpected that there should be some disagreements between the data and the reference calculation. In order better to understand these disagreements, we have performed calculations which relax some important basic assumptions of the reference model.