Appendix A: model equations
The radiative transition probability for downward transitions is where and are the Einstein coefficients for spontaneous and stimulated emission, and is the mean integrated radiation field intensity at the line frequency. For the LVG model with expansion velocity increasing with radius r according to , then
where is the escape probability for , and for other values of , is given as a function of 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 the velocity width of the source. The dust optical depth is (the modeling is not greatly sensitive to the dust opacity law). The escape probability for the H II region is , where , and the continuum emission intensity from the H II region is approximated by . The Planck function is where h is Planck's constant, is the line frequency, c is the speed of light and k is Boltzmann's constant. is the dust temperature, is the dust filling factor,
is the line excitation temperature, and is the filling factor for the H II region. is the cosmic background radiation temperature, and , so for example and 17294 K for the 12.178 GHz and 6.668 GHz lines, respectively.
The brightness temperature of the line above the continuum is
© European Southern Observatory (ESO) 1997
Online publication: May 26, 1998