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

TLDR: In this paper, Liouville's theorem for radiation, of which the generalized etendue is a consequence, implies that 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) is constant along the ray path, where μ is the refractive index, where d2Ω and d2A are the ranges, respectively, of solid angle and area that define a ray (actually a bundle of rays).

Abstract: Liouville's theorem for radiation, of which the generalized etendue is a consequence, implies μ2 d2Ω d2A = constant along the ray path, where μ is the refractive index and d2Ω and d2A 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 TBof 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 TB 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 TBf and that the factor f must be significantly less than unity.