6.2 The Theory

The solid line of Figure 6.2 shows the continuum emission produced by a stellar photosphere (Kurucz, 1979). The points indicate the circumstellar excess of a Be star. It can been seen that the excess at 2.5m (IR) is extremely small whilst at ~ 0.4m (visible) the effect is almost non-existent, Dachs et al. (1988) find that the maximum contribution from the circumstellar disc in the (B-V) colour (E(B - V )_cs) ~ < 0.1mags. The Strömgren filters lie in the region ~ 0.35-0.49m so if there is any excess it is should be below 0.1mags.

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To understand the form of the relationship between circumstellar excess and the equivalent width of emission lines the physics is now considered. The relative intensity of flux (I) received from the disc of a Be star is described by

(6.5)

where n is the number density of ions and dv is the volume of radiating material.

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In terms of flux the equivalent width of a line is described by

(6.6)

where f_tot.() is the total observed flux across the emission line at and f_cont.() is the flux in the underlying continuum at a wavelength (). The wavelengths, ₁ and ₂, bound the line of interest. Figure 6.3 shows the effect of increasing the density of particles in an arbitrary Be star disc without the contribution of the stellar photosphere;

• (a) The baseline, where both the continuum emission and line emission are optically thin, is the starting point.
• (b) If n is increased only slightly to a point where neither the line or continuum become optically thick, both the continuum and line flux increase, as a result the line equivalent width remains constant.
• (c) Increasing n to the point where an emission line becomes optically thick whilst the continuum does not, results in the line flux remaining constant, the continuum flux however will continue to increase resulting in the line equivalent width decreasing.
• (d) At the point where n becomes so large as to make both the line and continuum emission optically thick both the line and continuum fluxes remain constant, the equivalent width therefore remains constant.

This indicates that the equivalent width of emission lines can not increase by considering the Be stars’ discs alone. In term of the plots to be presented later in the chapter, colour excess against equivalent width, this means that the gradient (a) must be less than or equal to zero; (a 0).

If the continuum emission is also considered then the flux from the disc can be given can be given in terms of this emission,

(6.7)

where f_* is the flux from the star, n is the number density of particles in the disc and is some fraction. The total continuum excess, inclusive of that from the star is then,

(6.8)

whereas the total line flux is,

(6.9)

where both and are fractional and << . From this simple model it is possible to see that the equivalent width can increase; if increases, whilst the total continuum flux increases, the line flux increases faster. For example if is set to 0.1 and is set to 0.01, then if n = sqrt1,

fcont.	= 1.1	fline	= 0.01,	(6.10) increasing n such that, n = ,
fcont.	= 1.4	fline	= 0.04,	(6.11)

The percentage increase of f_cont. is 27%, whilst the percentage increase of f_line is 400%. The model is therefore re-done to include this effect.

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Figure 6.4 shows the effect of increasing the density of particles in an arbitrary Be star disc inclusive of the stellar photosphere;

• (a) The baseline, where both the continuum emission and an arbitrary line emission are optically thin, is the starting point.
• (b) n is now increased to a point where neither the line or continuum become optically thick, both the continuum and line flux increase, however the line flux will increase quicker than the continuum flux (see Equations 6.8 - 6.11) as a result the line equivalent width increases.
• (c) Increasing n further to the point where an emission line becomes optically thick whilst the continuum does not, results in the equivalent width decreasing, as while continuum flux can continue to increase the line flux will remain constant.
• (d) At the point where n becomes so large as to make both the line and continuum emission optically thick both the line and continuum fluxes remain constant, the equivalent width therefore remains constant.

Thus it is possible, within the regime where continuum and line emission are both optically thin, for the equivalent width to increase as the disc density increases, and in the context of the graphs to be presented it means that the gradient may take any value.

The important difference between Figure 6.3 and Figure 6.4, is that Figure 6.4 corresponds to an emission line and optical continuum excess, (i.e., not dominated by the disc), whereas Figure 6.3 corresponds to a disc dominated source.

Whilst strictly this premise is only true of a single star observed over an extended time-scale, it can also be applied to a sample of stars, as there appears to be no correlation between disc density and spectral type (see e.g.,  Cote and Waters, 1987).