3 Methodology - separating the interstellar reddening and circumstellar excess

Figure 1: Data extracted from Dougherty et al. (1994). The ratio $\beta$ was calculated by means of a least squares fit, shown above as a solid line. Note that "sd'' is the rms deviation of the residuals about the fitted line in the ordinate axis

The flux from a star is reduced by interstellar extinction by a factor ( $\exp[-\tau_{\rm ext}(\lambda)]$), where $\tau_{\rm ext}(\lambda)$ 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 & 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, $(M_{\lambda_1}-M_{\lambda_2})_{\rm obs}$, of a Be star consists of three components - the star's intrinsic colour, $(M_{\lambda_1}-M_{\lambda_2})_{0}$, the excess due to circumstellar material, $E(M_{\lambda_1}-M{\lambda_2})_{\rm cs}$ and the interstellar reddening, $E(M_{\lambda_1}-M_{\lambda_2})_{\rm is}$. Using our JHK filters we can construct two observed colours:

 

$$(J-H)_{\rm obs}=(J-H)_{0}+E(J-H)_{\rm cs}+E(J-H)_{\rm is}$$ (1)

 

$$(H-K)_{\rm obs}=(H-K)_{0}+E(H-K)_{\rm cs}+E(H-K)_{\rm is}.$$ (2)

A unique solution for these equations is possible by assuming a universal interstellar reddening law of the form:

 

$$E(J-H)_{\rm is} =\alpha E(H-K)_{\rm is},$$ (3)

and also by assuming that the colours of the disc around the Be star can be related in a similar fashion (see, for example Fig. 5 from Dougherty et al. 1994, re-plotted, with a least squares fit applied as our Fig. 1):

 

$$E(J-H)_{\rm cs} =\beta E(H-K)_{\rm cs}.$$ (4)

The value of $\alpha$ may be simply derived from the interstellar extinction law of Rieke & Lebofsky (1985), giving $\alpha=1.7\pm0.1$. To derive $\beta$, 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 $\chi_{\rm reduced}^2=1.02$, presented in their Fig. 5 gives $\beta=0.61\pm0.02$ (see Fig. 1). We note that the rms deviation in the ordinate direction of the graph, $E[J-H]_{\rm cs}$ vs. $E[H-K]_{\rm cs}$ is sd=0.04, while that for the graph $E[H-K]_{\rm cs}$ vs. $E[J-H]_{\rm cs}$ 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, r=0.8, when applied to this data, implying a high correlation between the circumstellar excess colours.

By combining Eqs. (1) to (4) we are able to analytically solve to separate the interstellar and circumstellar components. We find:

    

$$E(H-K)_{\rm is}=\frac{(H-K)_{\rm obs}-\frac{1}{\beta} (J-H)_{\rm obs}-(H-K)_{0}+\frac{1}{\beta}(J-H)_{0}}{(1-\frac{\alpha}{\beta})}$$ (5)

$$E(H-K)_{\rm cs}=\frac{(H-K)_{\rm obs}-\frac{1}{\alpha}(J-H)_{\rm obs}-(H-K)_{0}+\frac{1}{\alpha}(J-H)_{0}}{(1-\frac{\beta}{\alpha})}$$ (6)

$$E(J-H)_{\rm is}=\frac{(J-H)_{\rm obs}-\frac{1}{\beta}(H-K)_{\rm obs}-(J-H)_{0}+\frac{1}{\beta}(H-K)_{0}}{(1-\frac{\beta}{\alpha})}$$ (7)

$$E(J-H)_{\rm cs}=\frac{(J-H)_{\rm obs}-\frac{1}{\alpha}(H-K)_{\rm obs}-(J-H)_{0}+\frac{1}{\alpha}(H-K)_{0}}{(1-\frac{\alpha}{\beta})}\cdot$$ (8)

The intrinsic colours (H-K)₀ and (J-H)₀ come from Koornneef (1983). The solutions of Eqs. (5-6) are tabulated in Table 1. We present only the (H-K) solutions because the (J-H) results are not independent, the ratios $\alpha$ and $\beta$ relating the two. Colour excesses for (J-H) can be simply calculated using Eqs. (7-8).

Figure 2: Interstellar extinction versus Na D2 5890 Å line EW (left panel), where the fitted line is a least squares fit weighted to the ordinate axis errors, and versus circumstellar excess (right panel). Note that as expected no correlation between the interstellar Na EW and the circumstellar excess is present

The errors generated from our calculations are twofold, (i) random errors from our observational data and the intrinsic colours, which enables us to quantify the scatter and (ii) systematic errors from the ratios $\alpha$ and $\beta$, which shift the calculated best fit lines to their upper and lower extremities. We calculate a systematic error of $\sim$5% in $E(H-K)_{\rm cs}$ and $\sim$4% in $E(H-K)_{\rm is}$.

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 width (EW) of the interstellar sodium D₂ 5890 Å line, listed in Col. 3 of Table 1. We note that there is an error of 10% on the Na EW. This was measured from the red optical spectra of the sample (see Paper V) using the FIGARO routine ABLINE. In Fig. 2 we plot the EW of this line against our derived interstellar reddening and circumstellar excess. As expected there appears to be a correlation with $E(H-K)_{\rm is}$ although not with $E(H-K)_{\rm cs}$. To quantify this we performed non-parametric correlation tests (Spearman rank). The results for all such tests carried out in this paper are presented in Table 2. We note here that Spearman rank correlation confidences are normally compared with a critical correlation coefficient, $r_{\rm s}$, which imply a significance level for the correlation. We list this significance level for each test in Table 2. However we have also chosen to express our results as a standard deviation ($\sigma$) 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 $r_{\rm s}$ 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 2 the confidence level of each test. The positive correlation between sodium EW and the interstellar extinction is confirmed at a >$4.5\sigma$ confidence level while any correlation between sodium and $E(H-K)_{\rm cs}$is at a confidence level of less than $1\sigma$. This result gives us confidence that our method does indeed separate the interstellar and circumstellar components of the infrared excess.

Figure 3: (Left panel) a plot of $E(B-V)_{{(H-K)}_{\rm is}}$, the IR (H-K) interstellar reddening, converted to an optical (B-V) colour versus sodium EW(Å). (Right Panel) a plot of $(B-V)_{\rm is+cs}$ versus the Na D2 5890 Å line EW. The fitted lines are least squares fits minimised in the ordinate axis

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 & Lebosky (1985). The interstellar excess converted from an (H-K) colour to a (B-V) equivalent colour is denoted by $E(B-V)_{{(H-K)}_{\rm is}}$. This is plotted against $E(B-V)_{\rm cs+is}$, i.e. incorporating both interstellar reddening and circumstellar excess (see Fig. 4), where $\Delta E(B-V)_{{(H-K)}_{\rm is}}\gg\Delta E(B-V)_{\rm is+cs}$ and so it is $E(B-V)_{{(H-K)}_{\rm is}}$ that has been minimised. $E(B-V)_{\rm cs+is}$ is derived from historical observational data (see Paper I) and the intrinsic (B-V) colours of B stars (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 (r=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 Fig. 4). There is also a systematic error (as described above) of 0.2 mags associated with the plot in the ordinate direction. This implies boundary conditions of $-0.03<E(B-V)_{\rm cs}<0.37$ 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 $E(B-V)_{\rm cs}\sim0.1$ magnitudes.

Figure 4: A plot of $E(B-V)_{{(H-K)}_{\rm is}}$ (the IR (H-K) interstellar reddening converted to an optical (B-V) colour) versus the optical $(B-V)_{\rm is+cs}$ colour, which incorporates both interstellar reddening and circumstellar excess

In the 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 $E(B-V)_{\rm is+cs}$ versus the sodium EW (see Fig. 3, right panel) is r=0.56. For $E(B-V)_{{(H-K)}_{\rm is}}$ versus sodium EW (Fig. 3, left panel) the Spearman rank correlation coefficient is r=0.45. However the greatest correlation is between $E(B-V)_{\rm is+cs}$ and $E(B-V)_{{(H-K)}_{\rm is}}$ (see Fig. 4) with r=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).