## AppendixThe 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: where and 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 : where the line shape is assumed to be
gaussian with a velocity dispersion including
thermal width and microturbulence in the envelope
( exp[-() where and are the Einstein coefficients and is the reduced intensity:
In the LVG formalism, the mean intensity is expressed directly in term of local escape probability and local source function: with 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 in term of the local escape probability and the local source function: 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 is a modification to the local escape probability where is the usual optical depth evaluated using the populations at r. For the tangential direction ( = ) the optical depth in the LVG formalism is a good approximation to the real one: This equation is used to present the results of the models. The use
of ( 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
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
at each grid point and a calculation of the populations using the
Newton-Raphson method. The convergence is achieved when the parameters
at all grid points change by less than
10 © European Southern Observatory (ESO) 1998 Online publication: February 4, 1998 |