## Appendix A: spectral line emission from a circumstellar diskWe consider a simple model of an inclined, slightly flared disk in
Keplerian rotation. Using Cartesian coordinates, and with the where where is the Planck function. Background radiation fields are specified through , whereas the intrinsic disk field is determined by the excitation temperature, . Thus, is the line source function, . Hence, with obvious notations is obtained from the solution of the rate equations for statistical equilibrium of radiative and collisional excitations, i.e. from the (fractional) energy level populations, , where corresponds to the number of energy levels considered. The level populations are found by inverting the matrix equation for the transition probabilities, (e.g., Spitzer 1978). Hence, in the steady state, where is a matrix of dimension
and is a vector
containing elements. In our models,
= 50 for CO with Einstein To obtain the line intensity to be received by the observer, the
source brightness distribution is convolved, at each frequency, with
the diffraction pattern of the telescope (the `beam'). For
simplicitity, the beam is here represented by a circular Gaussian with
the full width at half power The optical depth at frequency is obtained as the sum of (molecular) line and (dust) continuum optical depth, i.e. . The line optical depth is given by where the column density of hydrogen is the Einstein transition probability and
is the molecular abundance with respect to
hydrogen, H The continuum optical depth is assumed constant over the line and computed as where We have computed an analogue of and compared with the model results for HL Tau obtained by Gómez & D'Alessio (1995) and our line to continuum ratios are in very good agreement with what can be estimated from the profiles in their Fig. 3. We also found very good agreement with Yamashita et al. (1993) regarding the profile shape of the CO (1-0) line for Eri [however, our result for the integrated line intensity (in the scale) for their model is a factor of about 4 below that given in their Table 2]. © European Southern Observatory (ESO) 1998 Online publication: June 2, 1998 |