Formation of Emission Lines

Abstract

When we consider the radiation fields of stellar envelopes exposed to the photospheric UV radiation, the basic feature is their anisotropic nature. The degree of anisotropy is expressed by the geometrical dilution factor, W, defined as the ratio of the solid angle, ω, of the photosphere seen from a point of interest, P, relative to the total solid angle 4π. Thus we have

$$ W = \frac{\varpi } {{4\pi }}. $$

((4.1.1))

If point P is located at a distance r from the star’s center, and θ be the angle subtended by the stellar radius at point P, as seen in Figure 4.1, the solid angle ω is expressed as

$$ \omega = 2\pi \int_0^\theta {\sin \theta d\theta = 2\pi \left( {1 - \cos \theta } \right)} . $$

((4.1.2))

Then the dilution factor is given by

$$ W = \frac{1} {2}\left( {1 - \cos \theta } \right). $$

((4.1.3))

Converting cosθ to the ratio of stellar radius R and distance r, we have

$$ W = \frac{1} {2}\left\{ {1 - \sqrt {1 - \left( {\frac{R} {r}} \right)^2 } } \right\}. $$

((4.1.4))

In case of r ≫ R, we have

$$ W \sim \frac{1} {4}\frac{{R^2 }} {{r^2 }}. $$

((4.1.5))

The value of W ranges from 0.5 at the stellar surface (r = R) to 10⁻¹⁵ in extended planetary nebulae. In stellar envelopes the typical value of W is in between 10⁻¹ and 10⁻⁵. The relation between W and x = r/R is partly given in Table 4.1.