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Astron. Astrophys. 331, 317-327 (1998)

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3. The radiative transfer model

In order to investigate both the radiative and collisional processes we need a more elaborate model. The circumstellar envelope is assumed to be spherical. We use the LVG approximation to treat the radiative transfer in the FIR rotational lines. Since the maser lines are very sensitive to the physical conditions in the circumstellar envelope a non-local treatment is used for maser lines in the ground state. Collisional rates between OH and [FORMULA] are taken from Offer et al. (1994). The ratio of ortho- [FORMULA] to para- [FORMULA] is assumed to be equal to 3. Line transition probabilities are taken from Burdyuzha & Varshalovich (1973) and Destombes et al.(1977).

The radiation field consists of the cosmic background radiation with a temperature [FORMULA] = 2.7K and of the intensity of FIR emission due to dust grains which are assumed to be optically thin. The dust absorption coefficient may be written as follows:

[EQUATION]

where [FORMULA], [FORMULA] and [FORMULA] are the density, cross section ([FORMULA] = [FORMULA]) and the absorption efficiency of the grain, respectively. In this paper we take [FORMULA], [FORMULA] and [FORMULA] are the absorption efficiency and frequency at [FORMULA] and p may vary between 1 - 2. The ratio [FORMULA] is adjusted to fit the observed FIR continuum emission of the envelope. The expansion velocity as a function of radius in the envelope is very uncertain. We take the velocity expansion law suggested by Deguchi & Nguyen-Q-Rieu (1990):

[EQUATION]

where [FORMULA] is the terminal velocity and [FORMULA] is the radius of the central star. The velocity gradient decreases with radius. The value of the velocity gradient [FORMULA] = dlnV/dln r is set to 0.01 when it becomes smaller than 0.01. The exact value of the velocity gradient should have only a minor effect on the final results at least for FIR rotational lines.

The populations of OH molecules are calculated by solving the equations of statistical equilibrium at each grid point. Our formulation of the overlap effect is similar to that presented by Bujarrabal et al. (1980) and by Collison & Nedoluha (1993). The rate equations in the LVG approximation can be written in the form:

[EQUATION]

where [FORMULA] is the population of magnetic sublevels, [FORMULA] is the statistical weight of the hyperfine level, [FORMULA] is the deexcitation rate between levels j and k, [FORMULA] is the gas temperature and [FORMULA] is the escape probability for the transition j [FORMULA] k

[EQUATION]

where [FORMULA] is the optical depth in the direction which makes an angle [FORMULA] with the radius vector r, [FORMULA] and [FORMULA] (see Fig. 16) is given by:

[EQUATION]

The term [FORMULA] includes the contribution of the thermal emission from dust grains and of the overlapping lines if line overlap occurs. The integral involving [FORMULA] (r') is evaluated along a ray of angle [FORMULA].

[EQUATION]

where S is the source function:

[EQUATION]

[FORMULA] is the dust temperature and

[EQUATION]

[FORMULA] is the optical depth of the l [FORMULA] m line which overlaps the j [FORMULA] k line in the direction [FORMULA]. The optical depth [FORMULA] and the source function [FORMULA] of the overlapping line l [FORMULA] m are evaluated exactly at the position [FORMULA] where this line is redshifted by the velocity shift [FORMULA] between the two lines. The term [FORMULA] can be easily generalized in the case of multiple overlaps. As pointed out by Collison & Nedoluha (1995), using the LVG approximation to calculate the FIR average intensity usually gives good results in comparison with direct calculations. The solution for the level populations is obtained by iterating between a calculation of the term [FORMULA] at all grid points and a calculation of the populations throughout the OH shell. By doing this we have explicitly taken into account the variation of physical conditions in the envelope. The emergent profiles of OH-FIR rotational lines are then calculated from the populations in the envelope. The intensity along a line of sight with an impact parameter p directed from the observer toward the envelope is given by:

[EQUATION]

[FORMULA], [FORMULA] are the positions in the envelope where photons of the j [FORMULA] k and l [FORMULA] m lines are emitted. This formula can be easily generalized in the case of multiple overlaps. The profile observed by an observer at a distance D to the source is then calculated:

[EQUATION]

The LVG approach simplifies enormously the radiative transfer problem and allows us to calculate quickly the OH populations over the whole envelope and to derive various observable quantities. But this formalism may lead to significant errors since the expansion velocity in the envelope is nearly constant implying the interaction of different parts of the shell. This situation may have great effect on the population inversion of the ground state OH masers. Following Collison & Nedoluha (1995), we use a model in which the radiative transfer equations of the 4 ground state maser transitions are solved non-locally while all other transitions (FIR hyperfine lines and microwave lines in the upper states) are treated under the LVG formalism. The populations from the LVG model described above are used as an input to the calculations of the populations of the maser levels. More details can be found in Appendix A and in the paper of Collison & Nedoluha (1995). Once the overall populations at all grid points are obtained, the maser profiles and luminosities can be calculated.

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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