Astron. Astrophys. 344, 342-354 (1999)

Appendix A: the [C II] 158 µm PDR emission in three dimensions

In spherical polar coordinates , with the stellar source of ultraviolet radiation at the origin, the regions of [C II] 158 µm emission are situated at , viz.

where is the true radial distance and the observed projection onto the plane of the sky (xy-plane). The z-axis is along the line of sight and the azimuthal angle is given by the observed position angle, p. The observable s is also given by (see: Sect. 4.1)

so that

The local line flux, , is related to the stellar source through a luminosity , viz.

Combining with and inserting Eqs. (A3) we find for the z-coordinate

and for the polar angle

The local values of the luminosity are determined from energy conservation considerations. The fraction of the stellar UV luminosity intercepted by the [C II] 158 µm-source at r is given by

where

The size of the [C II] 158 µm source is approximated by . A fraction of this intercepted stellar UV-luminosity is radiated isotropically in the [C II ] 158 µm line, so that

Hence, at distance D, the observed line flux is

where the stellar UV-flux, , has been expressed in units of (1.6 10^-3 erg cm^-2 s^-1). The source fills a fraction of the LWS beam, i.e. , so that Eq. (A10) reduces to

With the definition of in Eq. (A4) one finds that

and

For given spectral type and luminosity class of the star and for empirically determined values of , values of can be obtained from Eq. (A12). These must satisfy for the values of implied by Eq. (A6) [root must be real], yielding estimates of . Values of are then given by Eqs. (A4) and (A13). Similarly, the vector r can be constructed from Eqs. (A3) and (A5) and its direction cosine from Eq. (A6).

© European Southern Observatory (ESO) 1999

Online publication: March 10, 1999
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