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

3. Model

For simplicity, the distribution of the gas and dust in the shell is assumed to depend only on the distance from the star. To explain of the brightness variations, we accept the variable circumstellar extinction model.

3.1. Structure of shell

The shell can be divided into the following regions: stellar chromosphere, H II, C II, and C I regions.

The radius of the H II region () should not be much larger than several AU. Otherwise, steady emission in the Balmer lines - the indicator of the H II regions would be observed. It is assumed that the size of the transition zone between the H II and H I regions . Note that a real picture may be more complicated as follows from the theoretical consideration of Grachev (1996).

The innermost parts of the shell are free of dust. If the inner radius of the dusty shell is determined by the process of grain sublimation, it should be about 2 - 10 AU depending on the stellar luminosity, the grain composition and size (e.g., Il'in & Voshchinnikov 1993; Friedemann et al. 1994, 1995). We assume that . The outer radius of the shell appears to be of some thousand AU (Friedemann et al. 1994, 1995; Krivova & Il'in 1996).

Pressure of radiation of an A-type star is able quickly to sweep submicron dust particles out of the vicinities of the stars. Apparently, the presence of the dusty shells can be explained by the clumps' destruction and an efficient coupling of charged grains with the circumstellar magnetic field (Voshchinnikov & Grinin 1992; Il'in & Krivov 1994).

The estimates made by Voshchinnikov & Grinin (1992) show that the size of the clumps may be  0.2 AU and the orbits of the clumps should have a large eccentricity with the minimum distance from the star of about 5 - 10 AU.

3.2. Density distribution

For a given velocity law , the gas density distribution in the shell can be written as

where is the mass-loss rate, 1.35 the average molecular weight, and the mass of an H atom.

We use the standard mass-loss rate 10 /yr typical of HAeBe stars. For AB Aur, close values were obtained by CK and Böhm & Catala (1995) from spectral data and by Skinner et al. (1993) from the radio continuum observations of free-free emission.

The velocity law in the shells of HAeBe stars is not well known. Therefore, the standard Lamers law describing the velocity field in the shells of stars of practically all types is adopted

where , and are parameters, and is the stellar radius. The following values of parameters have been used: = 5 km s^-1, = 300 km s^-1 and . They allow to approximate the average velocity law obtained by CK for the inner layers of the shell surrounding AB Aur (except for a thin layer above the stellar photosphere). Equation (2) gives nearly a constant velocity at 1 AU. Note that there is also an evidence for a deceleration of the wind at larger distances from the stars (Finkenzeller & Mundt 1984).

The density distribution in the clump is presented as

where r is the distance from the clump center, the clump length along the line of sight, the gas number density at the clump center. Quite different density profiles can be modeled by varying .

The value of parameter can be calculated from the extinction produced by a clump and the gas to dust ratio if the size of the clump is much larger than the stellar radius

where = 0.936 is the relative abundance of hydrogen, the visual extinction in the clump ¹, the gas to dust ratio, , and is the Gamma function.

The possible presence of a very dense core or an asteroid-size body at the center of the clump (Friedemann et al. 1995; Kholtygin 1995) should not change its ionization structure.

3.3. Temperature distribution

The semi-empirical model of CK used by us includes a chromosphere, the thin hot layer near the star. The temperature distribution has a maximum () at and a rapid decrease down to about 3000 - 8000 K outside the chromosphere boundary ().

The H II region is supposed to be isothermal. The narrow ( 0.01 AU) layer above the chromosphere where the temperature rapidly drops down is treated following CK.

The gas temperature in the C II region decreases outward from 8000 K to 1000 K (see, e.g., Berrilli et al. 1992). In our model, the C II region is isothermal but its temperature, , is a model parameter.

It is evident that the C I region must be colder than the C II region. As the ionization structure of both regions is practically independent of the gas temperature, one can accept that .

A consideration of the gas thermal equilibrium in the clumps (see Appendix A) shows that the gas temperature should lay between about 200 K and 1000 K. At present, a more detailed analysis does not seem to be reasonable since the description of some processes (e.g. photoelectron emission) is rather approximate and the physical conditions in the clumps are unclear. Therefore, we assume the clumps to be isothermal with being in the range given above.

3.4. Ionization equilibrium

3.4.1. Atoms and lines

As the ionization equilibrium in an H I region is studied, the elements with an ionization potential eV are taken into account. The list of low-potential elements with cosmic abundances relative to hydrogen larger than according to Cameron (1982) is given in Table 1. In H I regions, only the calcium can be twice ionized ( (Ca II) = 11.871 eV).

Table 1. Cosmic abundances and depletions of the elements with the ionization potential eV

Some of the elements considered may be strongly depleted being incorporated into dust grains. The depletion of element X is

where and are the observed and Solar system abundances relative to hydrogen. The values of published by Turner (1991) for the cold interstellar medium and by de Boer et al. (1987) for the warm diffuse medium are given in Table 1.

The characteristics of the strongest optical and ultraviolet resonance lines of the ions under consideration are collected in Table 2, where the last column gives the oscillator transition strengths.

Table 2. Resonance lines of the most abundant ions with the ionization potential eV

3.4.2. Equation of ionization equilibrium

The following processes mainly affect the ionization balance of the gas in the shell: photoionization, ionization by cosmic-ray particles, photorecombination and dielectronic recombination.

The ionization by electron collisions is very weak because of the low electron temperature. The charge transfer reactions are also unimportant as their rates are rather slow at the temperatures considered (Golovatyj et al. 1991). The emission of photoelectrons from the grain surfaces becomes significant if the ionization degree (Nishi et al. 1991). It is also evident that molecules in the clumps cannot be a source of additional electrons because their ionization potentials are too high. Therefore, we have neglected all these processes.

Thus, the equations of ionization equilibrium in the H I region of the shells can be written as follows

where is the rate of ionization by cosmic rays, the photoionization rate for element X, and are the radiative and dielectronic recombination rates, and the density of neutral and ionized atoms, respectively, and is the electron density. The approximations of the rates used in our calculations are described in Appendices B and C. In the calculations we used the standard value of the ionization rate (Spitzer & Jenkins 1975).

Equations (6) were supplemented by the conditions of charge and gas density conservation and solved iteratively from the H II region boundary outward. The obtained column density and ion number density were used to calculate the equivalent width of the spectral lines in the frame of the "quasi-nebular" approximation (see Appendix D for details).

3.4.3. Ionizing radiation

The radiation ionizing atoms is the sum of the radiation of the stellar photosphere, the chromosphere, and the H II region surrounding the star.

For the stellar photosphere, the standard LTE models were used (Kurucz 1979). Note that these models show a steep decrease of the flux at  Å and, therefore, at shorter wavelengths the spectrum of ionizing radiation should be determined by the chromosphere and/or the H II region. The fluxes for the Kurucz model with = 9000 K and = 4.0 are plotted in Fig. 1. We adopt that the star has the luminosity . Then, its radius and mass are:  cm and , respectively.

Fig. 1. Spectrum of ionizing radiation in the shell of a Herbig Ae/Be star. Dashed lines show the contributions to the total flux from the stellar photosphere (1), the chromosphere (2) and the H II region (3)

Considering the stellar chromosphere, we base on the semi-empirical model developed by CK. Since the gas temperature is relatively low K), the ionizing radiation is a result of free-free and free-bound emission of hydrogen. Other ionized atoms can be neglected because of their small abundances. The total luminosity of the chromosphere at the frequency is

where is the hydrogen emission coefficient, and all hydrogen atoms in the chromosphere are assumed to be ionized (). The method used for the calculations of the emission coefficient is described in Appendix E.

The equation similar to Eq. (7) is applied to calculate the luminosity of the H II region, but in this case, the density distribution given by Eqs. (1), (2) and a constant temperature are taken. The fluxes from the chromosphere and the H II region referred to the level of the photosphere are plotted in Fig. 1. For given density and velocity distributions (see Eqs. (1) and (2)), the flux weakly depends on . Therefore, in our calculations we used = 1 AU.

The flux of ionizing radiation outside the H II region (both in the shell and in the clumps) can be approximated as

Here, is the radiation dilution factor, and are the optical thickness caused by the dust extinction and the gas continuous absorption, respectively. The latter is

where is the photoionization cross-section of element X (see Eq. (B2)). Other sources of opacity (H , H+H, etc.) are unimportant at wavelengths shorter than 2400 Å that corresponds to the ionization threshold of Na, the atom with the lowest ionization potential (see Table 1).

We consider the optically thin shells ( 0.5), but the fraction of radiation scattered by dust may still be substantial. The calculations of Voshchinnikov et al. (1995) show that the ratio of the scattered radiation to the stellar one is about 0.2 - 0.3 at wavelengths 1000 - 2000 Å. The negligence of the scattered radiation in Eq. (8) leads to overestimating the neutral atom density, but this effect is weak. The diffuse -radiation can be also neglected since -photons would be absorbed by dust particles both in the C II and C I regions.

3.5. Dust grains

The expression of the optical thickness of a dusty layer can be written using the gas to dust ratio and the gas column density

where is the normalized extinction curve and the ratio of the total to selective extinction.

The interpretation of photometric and polarimetric observations of HAeBe stars indicates that the properties of circumstellar grains may differ from those of interstellar grains (Voshchinnikov et al. 1988, 1995, 1996). In our modelling, we used the dust mixture found for the shell around WW Vul by Voshchinnikov & Grinin (1992): the minimum grain size m, the maximum grain size m, the slope of the size distribution (), the ratio of graphite to silicate grains . For this mixture, the ratio of the total to selective extinction is . Note that the parameters of the dust mixture diverse from those of the standard MRN one ( m, 3.5, 1). Following Voshchinnikov & Grinin (1992), it is also assumed that the properties of the dust grains in the shell and clumps are the same.

3.6. Model parameters

The main parameters of our model presented in Table 3 are divided among two groups: the shell and clump parameters. They were varied within plausible boundaries. The standard values of the parameters are given in the last column of Table 3.

Table 3. Model parameters and their standard values

The element depletions in the clumps are assumed to be similar to those for the warm interstellar medium (see Table 1) as the clumps look to be fast moving warm objects. In the interclump medium of the shells the element abundances are taken to be normal ( = 0).

© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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