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Astron. Astrophys. 323, 189-201 (1997)

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Appendix A: thermal equilibrium

The gas temperature inside the clump can be determined from the solution to the thermal balance equation


where [FORMULA] and [FORMULA] are the heating and cooling rates. The main gain in the heating is provided by electrons from the photoionization of the neutral carbon ([FORMULA]) and the photoelectron emission from the surfaces of dust grains ([FORMULA]). The contribution of the atomic carbon photoionization to the heating is (e.g., Rudzikas et al. 1991)


Here, [FORMULA] is the threshold frequency, [FORMULA] the mean intensity. The approximation of the photoionization cross-section [FORMULA] 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 H2 molecules by the UV radiation has been also estimated. The rôle of this effect is rather small for the clumps with the considered values of parameters.

At the temperatures [FORMULA] 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, [FORMULA] is the occupancy of the level k, [FORMULA] the [FORMULA] transition probability and [FORMULA] the energy of the transition. Note that all transitions between the fine structure levels of the ion are summed. The following atoms and ions have been included in the calculations: H, C, C [FORMULA], N, O, S, S [FORMULA], Fe and Fe [FORMULA]. The values of [FORMULA] have been found as the solutions to the statistical balance equations. The rates of the radiative and collisional transitions were obtained from the atomic constants published by Tielens & Hollenbach (1985) and Golovatyj et al. (1991).

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 [FORMULA] K. The same with H2 molecules is also insignificant in the considered range of density and temperature values (Groenewegen 1993). The cooling due to the excitation of forbidden and resonance lines is effective only if [FORMULA].

The gas temperature [FORMULA] 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 [FORMULA] 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 [FORMULA] 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) [FORMULA] 5000 K if a clump is at the distance [FORMULA] 0.1 AU from the star.

Appendix B: photoionization rate

The photoionization rate of an atom or ion X of the ground state is


The photoionization cross-sections are approximated as


where [FORMULA] are constants and [FORMULA]. 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%.


Table 6. Constants for the approximation of photoionization cross-sections

For Fe ions, we used the approximation for the cross-sections from Verner et al. (1993).

Appendix C: recombination rates

The rate of radiative recombination [FORMULA] 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 [FORMULA] and [FORMULA] were taken from Golovatyj et al. (1991) or calculated by us.

For ions of C, Mg, Al and Si, the rates of dielectronic recombination [FORMULA] 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 [FORMULA] 8000 K.

Appendix E: equivalent width of spectral lines

The 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 [FORMULA] is the intensity at the inner boundary of the shell ([FORMULA]).

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 [FORMULA] is the line absorption coefficient, [FORMULA] 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 [FORMULA], [FORMULA] is the line central frequency in the rest frame, v is equal to the terminal velocity of the wind [FORMULA] and the projection of the clump orbital velocity on the line of sight [FORMULA] for the shell and the clump, respectively.

The absorption coefficient at the line center [FORMULA] is connected with the oscillator strength [FORMULA] (see Table 2)


The Doppler half-width [FORMULA] is determined by the velocities of gas motions


where [FORMULA] and [FORMULA] are the velocities of thermal and microturbulent motions, respectively.

Appendix F: emission coefficient of a hydrogen plasma

The emission coefficient of a hydrogen-like ion with a charge Z was evaluated using the expressions from Sarmiento & Canto (1985)




[FORMULA] = [FORMULA] [ [FORMULA] ], [FORMULA] 13.56 eV, [FORMULA] is the Gaunt factor for free-free transitions. The function [FORMULA] can be written as


Here, [FORMULA] is the Gaunt factor for free-bound transitions to the level n. The lower limit [FORMULA] has to be determined from the condition [FORMULA].

Detailed calculations of the Gaunt factor [FORMULA] have been performed by Carson (1988). Analytical approximations of [FORMULA] are known only for [FORMULA] (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 ([FORMULA])


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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998