## Appendix A: thermal equilibriumThe gas temperature inside the clump can be determined from the solution to the thermal balance equation where and are the heating and cooling rates. The main gain in the heating is provided by electrons from the photoionization of the neutral carbon () and the photoelectron emission from the surfaces of dust grains (). The contribution of the atomic carbon photoionization to the heating is (e.g., Rudzikas et al. 1991) Here, is the threshold frequency, the mean intensity. The approximation of the photoionization cross-section is given by Eq. (B2). The photoelectron emission from dust grains has been treated according to Tielens & Hollenbach (1985). Using the expressions from Draine (1978) and Tielens &
Hollenbach (1985), the heating caused by the photodissociation and
photoexcitation of the H At the temperatures 1000 K, the main source of the gas cooling appears to be the line emission of atoms and ions which are excited by electron impacts. Then, the gas cooling rate is equal to Here, is the occupancy of the level
Other sources of the gas cooling are not expected to be essential.
The excitation of the fine structure levels by the collisions with H
atoms is important if K. The same with
H The gas temperature has been calculated from Eq. (A1). For the sake of simplicity, each clump was described by a single value of the temperature calculated at its center. It is found that the values of mainly depend on the gas to dust ratio and the distance between the clump and the star. Other clump parameters do not influence the gas temperature. For the gas to dust ratio from 0.1 to 10 of its standard value (see Table 3) and the value of from 3 AU to 50 AU, the temperature was found to be in the range from 200 K to 1000 K. Note that in the model of Sorelli et al. (1996) 5000 K if a clump is at the distance 0.1 AU from the star. ## Appendix B: photoionization rateThe photoionization rate of an atom or ion X of the ground state is The photoionization cross-sections are approximated as where are constants and . The values of the constants were obtained by least square fitting of the experimental data or the results of theoretical calculations. They are presented in Table 6 (see also Golovatyj et al. 1991). The accuracy of the approximation given by Eq. (B2) is of about 10 - 20%.
For Fe ions, we used the approximation for the cross-sections from Verner et al. (1993). ## Appendix C: recombination ratesThe rate of radiative recombination is the sum of the recombination rates for all levels of an atom or ion. It is usually represented in the following form The values of constants and were taken from Golovatyj et al. (1991) or calculated by us. For ions of C, Mg, Al and Si, the rates of dielectronic recombination have been published by Nussbaumer & Storey (1984, 1986, 1987). The rates for Na and Ca ions are not yet available (Nussbaumer 1992). However, a consideration of the structure of their autoionization levels shows that the rôle of dielectronic recombination should be unimportant provided 8000 K. ## Appendix E: equivalent width of spectral linesThe equivalent width of an absorption line can be calculated if the local density of the atom (or ion) is known at any point in the line of sight. As the source function vanishes in the shell, the equivalent width is (see, e.g., Spitzer 1978) where is the intensity at the inner boundary of the shell (). In general case, the observed absorption line is the superposition of the lines that originate in the shell and clump. Its optical thickness is where is the line absorption coefficient, number density of the atoms (or ions) at the ground level. We assumed that both in the shell and the clump the line absorption coefficient has the Doppler profile Here , is the line
central frequency in the rest frame, The absorption coefficient at the line center is connected with the oscillator strength (see Table 2) The Doppler half-width is determined by the velocities of gas motions where and are the velocities of thermal and microturbulent motions, respectively. ## Appendix F: emission coefficient of a hydrogen plasmaThe emission coefficient of a hydrogen-like ion with a charge
= [ ], 13.56 eV, is the Gaunt factor for free-free transitions. The function can be written as Here, is the Gaunt factor for free-bound
transitions to the level Detailed calculations of the Gaunt factor have been performed by Carson (1988). Analytical approximations of are known only for (see, e.g., Kaplan & Pikel'ner 1979). We approximated the results of Carson (1988) in a wide range of the frequency and temperature values as follows () © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |