An Etymological Dictionary of Astronomy and Astrophysics
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A method allowing for a simplified solution to the → radiative transfer equation at frequencies of spectral lines in media moving with a high velocity gradient. This method assumes that the macroscopic velocity gradients are more important than local random variations of thermal line width: dv/dr > v_th/l, where dv/dr is the velocity gradient, v_th is the thermal broadening of the line, and l the length scale. The Sobolev approximation is only valid if the conditions of the gas do not change over the → Sobolev length. Under the Sobolev approximation, each point in the medium is isolated from other points, and the → radiative transfer problem becomes a local one and therefore much easier to solve.

Named after the Russian astronomer Viktor Viktorovich Sobolev, Moving Envelopes of Stars [in Russian], Leningr. Gos. Univ., Leningrad (1947) [translated by S. Gaposchkin, Harvard Univ. Press, Cambridge, Mass. (1960)]; → approximation.

In the → Sobolev approximation, the length over which the conditions of the gas do not change and the approximation is valid. It is expressed by: l_s = v_th/(dv/dr), where v_th is the thermal line width and (dv/dr) the velocity gradient. In other words, the length over which the profile function of a line is shifted through a distance equal to its own width by the macroscopic velocity gradients that exist in the moving medium.

→ Sobolev approximation; → length.