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

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Appendix

The main point of our model is the removal of the Sobolev approximation in solving the radiative transfer equation of maser lines. The maser intensity is the solution of the usual radiative transfer equation:

[EQUATION]

where [FORMULA] and [FORMULA] are local emission and absorption coefficients, ds is the differential ray path. Complete redistribution is assumed which implies that the absorption and emission line profiles are identical. In that case, the quantity used in the statistical equilibrium equation is the average intensity [FORMULA]:

[EQUATION]

where the line shape [FORMULA] is assumed to be gaussian with a velocity dispersion [FORMULA] including thermal width and microturbulence in the envelope ([FORMULA] exp[-([FORMULA])2 ], FWHM = 1.67 [FORMULA]). In order to calculate the average intensity [FORMULA] we use the same approach described in the paper of Dickel & Auer (1994). The OH volume is divided into shells with a large number of radial grid points. A set of impact parameters p are chosen tangential to the shells as shown in the Fig. 16. Because the maser emission is essentially beamed into the radial direction, a large number of impact parameters p distributed over the range p = 0 to p = [FORMULA] are chosen to ensure the correct calculation of the averaged intensity of the maser lines. The radiative transfer equation is solved along the line of sight at each p. The averaged intensity [FORMULA] is evaluated by replacing the integral over angle by a weighted sum of [FORMULA] for all rays that intersect a shell and a simple integration over frequency. For any maser transition j [FORMULA] k the net rate of molecules going to the lower maser level due to spontaneous and induced emission is :

[EQUATION]

or

[EQUATION]

where [FORMULA] and [FORMULA] are the Einstein coefficients and [FORMULA] is the reduced intensity:

[EQUATION]


[FIGURE] Fig. 16. Radial grid points used to calculate the maser average intensity.

In the LVG formalism, the mean intensity is expressed directly in term of local escape probability [FORMULA] and local source function:

[EQUATION]

with

[EQUATION]

By inserting the expression (  A6) into the expression (  A4) we find the usual form of the radiative term in the rate equation (Eq.   10) . In the case of maser lines, the average intensities calculated from Eq. (  A2) are coupled to the emission and absorption coefficients in the entire envelope. Because we make use of the Newton-Raphson method to solve the rate equations, explicit derivatives of each term in the rate equation with respect to the local populations are needed. One easy way to do this is to express the average intensity [FORMULA] in term of the local escape probability and the local source function:

[EQUATION]

[FORMULA] is a proportional factor introduced to make the correspondence between the exact average intensity and that calculated using the LVG formalism. This substitution ensures that the term in the rate equation which corresponds to the average intensity in each maser line is equal to that calculated exactly by Eq. (  A2). The parameter [FORMULA] is a modification to the local escape probability

[EQUATION]

where [FORMULA] is the usual optical depth evaluated using the populations at r. For the tangential direction ([FORMULA] = [FORMULA]) the optical depth in the LVG formalism is a good approximation to the real one:

[EQUATION]

This equation is used to present the results of the models. The use of [FORMULA] ([FORMULA] 1) in expression (  A9) is necessary to prevent numerical overflows in the unsaturated core of OH masers where the population difference is much larger than in the saturated region. The values of [FORMULA] for all maser transitions are fixed in all calculations. A solution for the overall populations in the envelope is obtained by iterating between a calculation of the average intensity in the maser lines and the parameter [FORMULA] at each grid point and a calculation of the populations using the Newton-Raphson method. The convergence is achieved when the parameters [FORMULA] at all grid points change by less than 10-3 between two successive iterations .

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© European Southern Observatory (ESO) 1998

Online publication: February 4, 1998
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