6.5 Methodology

Fabregat and Torrejon (1998) propose a procedure to separate the intrinsic radiation, the circumstellar excess and the interstellar reddening that make up a star’s observed Strömgren colour. In this section the procedure will be tested using the representative sample. Using this method Fabregat and Torrejon (1998) were able to derive circumstellar colours for a range of Be stars earlier than B5.

The circumstellar excess values derived by Fabregat and Torrejon (1998) are between ~ (-0.4 0.16)mags however the errors are not explicitly stated and so it is not possible to tell if they lie explicitly in range found by Dachs et al. (1988) ( 0.1mags). However errors will be generated on the representative data sample presented here. The results presented by Fabregat and Torrejon (1998) could be construed as agreeing with the hypothesis presented in Section 6.2 (that the equivalent width increases with disc density; and therefore with increased continuum excess) as a > 0 (where a is the gradient of a circumstellar excess versus equivalent width plot), however the gradients are very shallow and no errors are explicitly presented on the fits and so a could be seen as being a 0.

Several authors (see Dachs et al., 1986, 1988; Kaiser, 1989, and references therein and also Chapter 7) have shown that the equivalent width of H is related to the continuum emission of a circumstellar envelope, with Torrejon et al. (1997) showing that the correlation is even better for H. The Strömgren  colour is able to measure this emission photometrically, as it is by construction linearly related to the H equivalent width (Golay, 1974), thus a method of separating the circumstellar excess from the intrinsic colour should be possible.

Fabregat and Torrejon (1998) lay down a method for doing this whence this description is distilled; it has been noted by several authors (see e.g.,  Alfaro and Delgado, 1991; Crawford, 1978) that it is necessary to split Be stars into different spectral types to accurately derive the required quantities from the observed data, as such this technique is only valid for the spectral range 09-B5. In this spectral range is calculated to have an average value of ( 2.6), this is verified with data from the representative sample. The excess in is denoted and is defined as the difference between the observed index and the intrinsic index of the star. Due to the circumstellar disc the intrinsic value is not known initially and must be estimated using an iterative method. The first approximation of is therefore = - _*, where _* is initially set to 2.6, the mean index.

This value of is then used to compute the value of the circumstellar excess in (b-y) and c₁, denoted E(b - y)_cs and E(c₁)_cs respectively, from an empirical relationship previously determined in a series of papers (see Fabregat and Torrejon, 1998; Fabregat et al., 1996; Torrejon et al., 1997). Those fits are,

E(b - y)cs	= -0.339	(6.17)
E(c1)cs	= 0.661.	(6.18)

Using these circumstellar excess values as a starting point the method proposed by Crawford (1978), who states that the c₁ is quite insensitive to interstellar reddening, is now used to obtain the interstellar excess and intrinsic colours. By plotting the c₁ colour against the (b - y) colour an upper boundary edge, termed the blue-envelope line, is evident (see Figure 1 of Crawford, 1978). The movement away from this blue-envelope in the direction of increasing (b - y) colour is seen to be the interstellar reddening. The equation of the blue-envelope line is derived directly from Figure 1 of Crawford (1978), the interstellar reddening and intrinsic colours for the representative sample may then be derived. Balona (1984) presents an equation for determining the index from the de-reddened c₁ index,

(6.19)

where c₀ is the de-reddened Strömgren colour c₁. The equation is derived from fitting the observational Hertzsprung-Russell diagram showing vs c₀. This is iteratively repeated until convergence, with being recalculated using the result of Equation (6.19). The fundamental plots of Fabregat and Torrejon (1998) are now re-plotted for these data, see Figure 6.5. These plots investigate the relationship between the circumstellar excess in the (b - y) colour and the H & H equivalent width measured from spectra.

Throughout this chapter least squares fits will be applied to the data that appear to have a linear correlation. The plotted fits will be (i) a solid line indicating that the line does not take into account errors and (ii) a dashed line indicating a fit weighted to the errors in the ordinate axis.

Also throughout this chapter Spearman rank correlation (Press et al., 1992) tests, will be performed on each plot in order to study the association between parameters. The test is non-parametric and so no assumption about the form of dependence is imposed. The Spearman rank coefficient is denoted “r_s” and its range is 0 < r_s < 1 with high values indicating significant correlation. To find out how significant, r_s is referred to a table of critical values (see Wall, 1996, for a good example) which supply a significance level. If r_s exceeds a critical value, the hypothesis that the variables are unrelated is rejected at that level of significance (Wall, 1996).

In this analysis one-tailed tests may be performed, since it is expected from the physics previously determined that a rise in circumstellar excess will occur for rise in n and result in a rise in the equivalent width, i.e., the approximate result is known. This does not nullify using non-parametric tests as no parameterisation of the form of the increase has been imposed. Wall (1996) explains that in the case where the form of relationship between parameters is not clear there is little lost by using a non-parametric test over a parametric one, as the efficiency is 91%. This means that if r_s is applied to set of normally distributed data it would take 100 data points for r_s to achieve the same level of significance as “r” (a parametric test known as the Pearson correlation coefficient) operating on only 91 data points. As an example if the number of data points N = 55, and r_s = 0.28, using two-tailed tables, the hypothesis that the variables are unrelated is rejected at the 5% level of significance (Wall, 1996), i.e., they are related at a 95% or the 2 level.

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