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Astron. Astrophys. 324, 211-220 (1997)

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Appendix A: model equations

The radiative transition probability for downward transitions is [FORMULA] where [FORMULA] and [FORMULA] are the Einstein coefficients for spontaneous and stimulated emission, and [FORMULA] is the mean integrated radiation field intensity at the line frequency. For the LVG model with expansion velocity [FORMULA] increasing with radius r according to [FORMULA], then


where [FORMULA] is the escape probability for [FORMULA], and for other values of [FORMULA], [FORMULA] is given as a function of [FORMULA] by equation (A1) of Castor (1970).

Here the optical depth is


where n and g are the level populations and statistical weights respectively, N is the methanol column density and [FORMULA] the velocity width of the source. The dust optical depth is [FORMULA] (the modeling is not greatly sensitive to the dust opacity law). The escape probability for the H II region is [FORMULA], where [FORMULA], and the continuum emission intensity from the H II region is approximated by [FORMULA]. The Planck function is [FORMULA] where h is Planck's constant, [FORMULA] is the line frequency, c is the speed of light and k is Boltzmann's constant. [FORMULA] is the dust temperature, [FORMULA] is the dust filling factor,

[FORMULA] is the line excitation temperature, and [FORMULA] is the filling factor for the H II region. [FORMULA] is the cosmic background radiation temperature, and [FORMULA], so for example [FORMULA] and 17294 K for the 12.178 GHz and 6.668 GHz lines, respectively.

The brightness temperature of the line above the continuum is


where [FORMULA]

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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998