|
The flux from a star is reduced by interstellar extinction by a factor (exp[-
ext(
)]), where
ext() is the extinction optical depth. In general, if we have knowledge of the spectral
type of the object (and hence its intrinsic colour) and the observed colours, then
we can remove extinction effects from data using an interstellar extinction law
(e.g., Rieke and Lebofsky 1985). However Be stars are well known to exhibit an infrared
continuum excess, caused by free-free and free-bound emission within the disc, as well as
the usual interstellar reddening (e.g., Gehrz et al. 1974). At first sight it appears
not to be possible to separate the interstellar and circumstellar components using
infrared photometry alone. However by using the fact that the spectral indices
of the two components are different, we find that a deconvolution is possible as
follows:
The observed colour, (M
1 -M2)obs, of a Be star consists of three components - the star’s
intrinsic colour, (M1 - M2)0, the excess due to circumstellar material, E(M1 - M2)cs
and the interstellar reddening, E(M1 -M2)is. Using our JHK filters we can construct two
observed colours:
| (J - H)obs | = (J - H)0 + E(J - H)cs + E(J - H)is | (7.1) |
| (H - K)obs | = (H - K)0 + E(H - K)cs + E(H - K)is. | (7.2) |
The value of 
may be simply derived from the interstellar extinction law of Rieke and
Lebofsky (1985), giving = 1.7 ± 0.1. To derive 
, we use the circumstellar excesses of Be
stars measured by Dougherty et al. (1994) who de-redden their photometry based on a
combination of the reddening free Geneva system parameters X and Y, and the strength of
the interstellar 2200 Å feature in IUE spectra. A least squares fit to the data, giving a
reduced2 = 1.02, presented in their Figure 5 gives = 0.61 ± 0.02 (see Figure 7.1). We note
that the r.m.s. deviation in the ordinate direction of the graph, E[J -H]cs vs E[H -K]cs is
sd = 0.04, while that for the graph E[H - K]cs vs E[J - H]cs is sd = 0.05. We
are therefore confident that a 1D minimisation is sufficient. We note also that a
Spearman rank correlation test gives a Spearman rank coefficient, rs = 0.8, when
applied to these data, implying a high correlation between the circumstellar excess
colours.
By combining equations 7.2 to 7.4 we are able to analytically solve to separate the interstellar and circumstellar components. We find:
| E(H - K)is = |  | (7.5) |
| E(H - K)cs = |  | (7.6) |
| E(J - H)is = |  | (7.7) |
| E(J - H)cs = |  | (7.8) |
and relating the two. Colour excesses
for (J - H) can be simply calculated using equations 7.7-7.8.
and , which shift the
calculated best fit lines to their upper and lower extremities. We calculate a systematic error
of ~ 5% in E(H - K)cs and ~ 4% in E(H - K)is.
In order to test our de-reddening procedure, we compare the measured interstellar reddening to an independent measure of the same quantity. For this we use equivalent width2 (EW) of the interstellar sodium D2 5890Å line, listed in column 3 of Table 7.1. We note that there is an error of 10% on the Na EW. This was measured from the red optical spectra of the representative sample (see Steele and Negueruela 2002) using the FIGARO routine ABLINE. In Figures 7.2 and 7.3 we plot the EW of this line against our derived interstellar reddening and circumstellar excess. As there is such a large range of error bars evident in Figure 7.2 the plot shows two fitted lines, the solid line is a non-weighted fit whilst the dashed line is weighted to the errors in the ordinate direction. The difference between the two is minimal with gradients differing by only 0.009 and the intercepts by 0.02. However the error bars on the plot intercept more frequently with the unweighted line, therefore this fit may be more reliable in this case.
As expected there appears to be a correlation with E(H - K)is although not with
E(H - K)cs. To quantify this we performed non-parametric correlation tests (Spearman
rank), it should be noted that this test does not rely on the fitted lines. The results for all
such tests carried out in this chapter are presented in Table 7.2. We note here that Spearman
rank correlation confidences are normally compared with a critical correlation coefficient, rs,
which imply a significance level for the correlation. We list this significance level for each test
in Table 7.2. However we have also chosen to express our results as a standard deviation (
)
measure (confidence level) to allow easy comparison with parametric tests. Implicit in this is
the assumption that repeated tests of similar samples would find a normal distribution of
the derived correlation coefficients. To derive this confidence level we used the
one-tailed rs lookup tables of Wall (1996) to find the significance level and then the
one-tailed normal distribution lookup tables of Wall (1979) to find the confidence levels.
Therefore we also list in Table 7.2 the confidence level of each test. The positive
correlation between sodium EW and the interstellar extinction is confirmed at a
> 4.0 confidence level while any correlation between sodium and E(H - K)cs is
at a confidence level of <<1. This result gives us confidence that our method
does indeed separate the interstellar and circumstellar components of the infrared
excess.
To quantify the strength of any optical circumstellar excess in our sample we convert our IR
interstellar excesses to equivalent optical data using our adopted interstellar extinction law of
Rieke and Lebofsky (1985). The interstellar excess converted from an (H - K) colour to a
(B - V ) equivalent colour is denoted by E(B - V )(H-K)is. This is plotted against
E(B - V )cs+is, i.e., incorporating both interstellar reddening and circumstellar
excess (see Figure 7.6), where 
E(B - V )(H-K)is » E(B - V )is+cs and so it is
E(B - V )(H-K)is that has been minimised. E(B - V )cs+is is derived from historical
observational data (see Steele et al. 1999) and the intrinsic (B - V ) colours of B
stars from Cramer (1984). An independent test of our de-reddening procedure may
now be carried out if we assume a negligible circumstellar excess for the optical
(B - V ) colour: The colour-colour plot should produce a one-to-one correlation if the
assumption of zero optical excess is true. A correlation is again obvious (rs = 0.74),
and we note that no significant offset between the two measures of reddening is
apparent.
This implies that the assumption of negligible optical circumstellar excess appears to be reasonable at the level of <0.17 magnitudes, (the intercept of Figure 7.6). There is also a systematic error (as described above) of ±0.2mags associated with the plot in the ordinate direction. This implies boundary conditions of -0.03 < E(B - V )cs < 0.37 magnitudes. A similar result was found by Dachs et al. 1988 who find that the maximum contribution of circumstellar envelopes to observed (B -V ) colours in Be stars amounts to E(B -V )cs ~ 0.1 magnitudes.
In light of this result (negligible optical circumstellar excess) it would be interesting to determine which method (optical colours, infrared colours or sodium equivalent width) gives a better estimate of the interstellar reddening to Be stars. The Spearman rank correlation coefficient of E(B - V )is+cs versus the sodium EW (see Figure 7.5) is rs = 0.56. For E(B -V )(H-K)is versus sodium EW (Figure 7.4) the Spearman rank correlation coefficient is rs = 0.45. However the greatest correlation is between E(B -V )is+cs and E(B -V )(H-K)is (see Figure 7.6) with rs = 0.74. In other words it appears that both the traditional optical and our new infrared method are more reliable than the sodium equivalent width for determining the interstellar reddening to Be stars. In the sections that follow we prefer to use our new method, as it is based on data taken closer in time (within a few years) to the spectroscopic data than the optical data (over 30 years in many cases).
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