Except in very simple cases, it is not possible to solve the coupled molecular physics and radiation transfer problems for masers, or their pumping radiation, analytically. For more realistic problems, we need to resort to numerical solutions. There are many general-purpose radiative transfer codes available. However, a substantial fraction of these require modification to work in situations that may produce masers: inverted populations yield negative absorption coefficients, optical depths and source functions: situations that will cause many codes to fail.

Large velocity gradient approximation

The large velocity gradient (LVG) or Sobolev approximation (Sobolev, 1957) is a means of casting the radiative transfer problem into an entirely local form. In LVG, the integrations that appear in the formal solution of the radiative transfer equation can be carried out, so the line mean intensity can be expressed explicitly as a function of the energy-level populations from the same transition. Elimination of the mean intensities in favour of the population expressions leads to a set of master equations which are non-linear algebraic equations in the populations. The LVG approximation is therefore not really a numerical method, but a clever approximation that allows much simpler numerical methods to be used than suggested by the original problem.

Theory

We begin by selecting the radiation transfer equation for transport along a ray element ds, Eq. (3.78). We do not, at present, assign any particular geometry to the problem, and one of the greatest advantages of the LVG approximation is that it is almost geometry free.