Astron. Astrophys. 337, 832-846 (1998)

2. The physical model

In Papers I and II of this series the time dependent photoevaporation of a 1.6  circumstellar disk around a 8.4  star was calculated under a variety of physical conditions. The ionizing flux of the central source and its "hardness" as well as the stellar wind parameters (mass loss rate and terminal velocity) were varied. States of these models at selected evolutionary times are the basis for our diagnostic radiation transfer calculations.

2.1. Continuum transport

To determine the continuum spectral energy distribution (SED) over a frequency range from the radio region up to the optical, we take into account three major radiation processes: thermal free-free radiation (i.e. bremsstrahlung of electrons moving in the potential of protons in the H II -region), thermal dust radiation and the radiation emitted from the photosphere of an embedded source.

2.1.1. Free-free radiation

For this process we adopt the approximation for the emission coefficient (Spitzer 1969):

Here, and are the particle densities of electrons and protons. All other symbols have their usual meanings. We approximate the Gaunt factor for a non-relativistic plasma by:

where () is Euler's constant. Assuming the validity of Kirchhoff 's Law , the absorption coefficient for thermal free-free radiation can be written ():

2.1.2. Dust emission

We adopt the `dirty ice' dust model developed by Preibisch et al. (1993), which includes two refractory components: amorphous carbon grains (aC) and silicate grains as well as volatile ice coatings on the surface of the silicate grains at temperatures below 125 K (Core Mantle Particles, CMP's). The icy coatings contain 7% of the available amorphous carbon and consist of water and ammonium with a volume ratio of 3:1. At temperatures above 125 K the silicate core and approximately 11 amorphous carbon particles are released into the dusty gas for each CMP. In Table 1 the sublimation temperature , the mean radius and the number of grains per gram gas are listed for the different species.

Table 1. Parameters for the grain species used in the dust model of Preibisch et al. (1993).

The absorption coefficient [] for the individual dust components is given by:

where the mean absorption efficiency for grain type "d" has been determined using Mie theory for spherical grains of a given size distribution. Fig. 1 displays the absorption efficiencies for the different dust components as a function of frequency. Each dust component's contribution to the source function due to thermal emission is also calculated under the assumption that .

Fig. 1. Mean absorption efficiencies for the different dust components. Solid line: amorphous carbon, dotted line: silicate, dashed line: CMP's.

2.1.3. Net continuum absorption and emission

Both emission processes mentioned above occur simultaneously within the same volume. Thus the net absorption coefficient and source function are:

2.2. Forbidden lines

In order to calculate profiles of the forbidden lines for the elements oxygen and nitrogen ([O II ] 3726, [O III ] 5007 and [N II ] 6584), we adopt the following procedure. First, the equilibrium ionization structure of these elements is calculated over the volume of consideration. Next, the occupation densities of metastable levels due to collisional excitation by electrons is determined. We take into account Doppler shifts due to bulk gas motions and thermal Doppler broadening to calculate the profile function :

with the thermal Doppler width:

Here R is the gas constant, µ the atomic weight of the relevant ion, and is the transition frequency Doppler-shifted by the radial velocity of the gas relative to the observer.

The emission coefficient of the transition , which enters into the equation of radiative transfer, is then given by:

where is the Einstein coefficient for spontaneous emission. Note that we have neglected radiative excitation and stimulated emission in this approximation.

2.2.1. Ionization equilibrium

The equations for ionization equilibrium for two neighboring ionization stages, and , are:

We solve these equations simultaneously for the up to the ionization stage for both oxygen and nitrogen.

Radiative ionization. The rate of radiative ionization is calculated from the flux of incident photons and the absorption cross section integrated over all ionizing frequencies. We use the radiation field of the central source and neglect scattering to determine :

An analytical expression for the absorption cross section is given in Henry (1970).

Collisional ionization. This ionization process is important in hot plasmas, where the mean kinetic energy of the electrons is comparable to the ionization potentials of the ions. N I , for example, has an ionization potential 14.5 eV; the corresponding Boltzmann temperature is  170 000 K. The coefficient for collisional ionization is approximated by the analytical expression in Shull & van Steenberg (1982).

Radiative recombination. This is the inverse process to radiative ionization. For the recombination coefficient we use the formula given in Aldrovandi & Pequinot (1973, 1976).

Dielectronic recombination. The probability for recombination is enhanced when the electron being captured has a kinetic energy equal to the energy necessary to excite a second electron in the shell of the capturing ion. The density of excited levels in the term scheme of the ions grows with energy. Thus, this process becomes more and more important with increasing temperature. We use two analytical expressions for : one for temperatures between 2000 K and 60 000 K (Nussbaumer & Storey 1983) and one for higher temperatures (Shull & van Steenberg 1982).

Charge exchange. The exchange of electrons during encounters of atoms and ions, e.g. is also important. Arnaud & Rothenflug (1985) give an expression for the coefficients . Special care is necessary in the case of the reaction . Due to the similarity of the ionization energies of hydrogen and oxygen ( eV) the backward reaction is also very effective. At sufficiently high electron temperatures this leads to the establishment of an ionization ratio , even in the absence of ionizing radiation. We explicitly include both reactions in Eq. (10) via the term . An expression for this coefficient can also be found in Arnaud & Rothenflug (1985).

2.2.2. Collisional excitation of metastable states

Neglecting the effects of radiative excitation and stimulated emission, we solve the equations of excitation equilibrium for the population densities (sums over all values "j" for which the conditions under the summation signs are fulfilled):

together with the condition . We use the formulae for the activation and deactivation coefficients given in e.g. Osterbrock (1989):

and

where denotes the collision strength for the transition , and the statistical weights of both states involved and the energy difference between them. For the we use the tables given in Osterbrock (1989).

2.3. Balmer lines

Our neglect of line absorption of Balmer photons by hydrogen is justified as long as the density of photons is sufficiently low to insure that the hydrogen 2p state is not significantly populated. This is equivalent to the assumption that photons generated in the nebula by recombination either are quickly destroyed, e.g. by dust absorption or by hydrogen absorption followed by 2-photon emission, or are able to escape sufficiently rapidly, e.g. by a random walk in frequency (Osterbrock 1961). The emission coefficient of the Balmer lines is given by:

The effective recombination coefficients used in this work were adopted from Hummer & Storey (1971).

2.4. Radiation from the central star

As argued in Paper I the resulting UV spectrum of a star accreting material via an accretion disk is very uncertain. For simplicity we have assumed that the photospheric emission of the central source (star + transition zone) can be approximated by a black body of given temperature in the frequency range of interest ( nm). determines the "hardness" of the ionizing photons, thus affecting both the nebula temperature and the ionization fraction of oxygen and nitrogen. We use the same values for as in Papers I and II for the hydrodynamic models.

Nevertheless, the successful spectral classification of the ionizing star in the UCHII region G29.96-0.02 by Watson & Hanson (1997) gives rise to the hope that more information on the spectral properties of young, still accreting massive stars will be available in the future.

© European Southern Observatory (ESO) 1998

Online publication: August 27, 1998
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