Astron. Astrophys. 337, 819-831 (1998)

Appendix A: optically thin line profiles from planar disks including the effects of absorption

This appendix details some of the technical aspects associated with the calculation of the line profiles in Sect. 2.2for realistic disk velocity fields. The calculation of the emission from the disk is based on Eqs. (7) for the location of the isovelocity zones and (10) for the flux of emission from the disk, including the effects of occultation and finite star depolarization.

The major new ingredient in the computations is the consideration of absorption. The amount of stellar continuum radiation observed in a small interval of frequency within the broadened line depends on two things: the fraction of the star's surface that is covered by disk material and the optical depth of that material. In regard to the former, the observed flux of continuum emission consists of a fraction that is direct, or unattenuated, and a fraction that is attenuated. The attenuating material is located on the near side of the star and covers an area in the disk that is traced by given of Eq. (17). This attenuating regions covers only the lower facing stellar hemisphere. However, for a small interval of frequency in the line, only a relatively fraction of that covering area contributes to the absorption.

The expression for the continuum emission is

where is the stellar continuum flux at the wavelength of the line and is assumed constant over the range of line frequencies, is the projected area of the star, is the geometrical area in the disk (i.e., as viewed pole-on) presented by the absorbing material with opacity in the interval , is the projection of that area along the line-of-sight and is equal to , and is the optical depth. Note that can never fall below 0.5 since the disk can at most cover half of the projected stellar surface.

The optical depth is given by

where corresponds to a point in the disk where . An expression for the optical depth can be derived from (A2) for any given disk velocity field. The optical depth reduces to , where consists of scaling parameters such as and , and is some function of position in the disk, the specifics of which depend on the particular velocity law. The point here is that the parameters included in also appear in the scaling constant associated with the line emission from the disk. This is a natural consequence of the optically thin assumption. Consequently, the amount of emission and the degree of absorption are not independent but are coupled through the parameters common to . For the profile calculations of Sect. 2.2, it was noted that was chosen to be the same in both the linear expansion and Keplerian rotation cases. This choice effectively determines the ratio of for expansion to that for rotation, which is of order unity. Absorption does not generally modifythe observed emission profile significantly (except near line center), because even if tends toward zero, the coverage of the stellar surface presented by the disk for a given frequency interval in the line is typically a small fraction of .

© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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