Implications of liouville's theorem on the apparent brightness temperatures of solar radio bursts

Abstract

Liouville's theorem for radiation, of which the generalized étendue is a consequence, implies μ² d² Ω d² A = constant along the ray path, where μ is the refractive index and d² Ω and d² A are the ranges, respectively, of solid angle and of area that define a ray (actually a bundle of rays). Implications of this concept on the propagation of radio waves from the actual to the apparent source in the solar corona (i.e., the scatter image of the true source) are discussed. The implications for sources of fundamental plasma radiation include: (1)The observed solid angle ΔΩ (defining the directivity) and apparent area ΔA of the source are compatible with Liouville's theorem only if the apparent source (the scatter image of the true source) corresponds to the envelope of subsources with a small ‘filling factor’ f. (2) The brightness temperature T _Bof the actual source is greater than that of the apparent source by f ^-1. (3) For sources of fundamental plasma radiation the factor f is very small (≲ 10^-2). (4) A long-standing discrepancy between the observed low value of T _B at meter/decameter wavelengths for the quiet Sun and the known coronal temperature may be resolved by noting that the implied coronal temperature is given by T _B f and that the factor f must be significantly less than unity.

A brief discussion is included of the relation between Liouville's theorem, the generalized étendue, Milne's laws, occupation numbers, extension in phase, and suppression of emission by a medium with refractive index unequal to unity.