Formation of Emission Lines

  • Tomokazu Kogure
  • Kam-Ching Leung
Part of the Astrophysics and Space Science Library book series (ASSL, volume 342)


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
$$ W = \frac{\varpi } {{4\pi }}. $$
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
$$ \omega = 2\pi \int_0^\theta {\sin \theta d\theta = 2\pi \left( {1 - \cos \theta } \right)} . $$
Then the dilution factor is given by
$$ W = \frac{1} {2}\left( {1 - \cos \theta } \right). $$
Converting cosθ to the ratio of stellar radius R and distance r, we have
$$ W = \frac{1} {2}\left\{ {1 - \sqrt {1 - \left( {\frac{R} {r}} \right)^2 } } \right\}. $$
In case of rR, we have
$$ W \sim \frac{1} {4}\frac{{R^2 }} {{r^2 }}. $$
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.
Figure 4.1

Dilution factor at point P is defined as the ratio of solid angle subtended to the photospheric disk, relative to the whole solid angel 4π.

Table 4.1

The relation between the dilution factor W and x = r/R


Emission Line Optical Depth Line Profile Planetary Nebula Stellar Disk 
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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Tomokazu Kogure
    • 1
  • Kam-Ching Leung
    • 2
    • 3
    • 4
  1. 1.Kyoto UniversityYawata, KyotoJapan
  2. 2.Institute of Astronomy and AstrophysicsAcademia SinicaTaiwan, China
  3. 3.Department of Physics & AstronomyUniversity of Nebraska-LincolnLincolnUSA
  4. 4.Brace LaboratoryUSA

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