An integral theorem on the equilibrium of a star

Abstract

In this paper an integral theorem on the equilibrium of a star is proved which gives the lower limit to the value of P_c/ρ_c^(n+I)/n, assuming that both ρ and P/ρ^(n+I)/n do not increase outward. As a special case of the theorem (n = 3) it is shown that for a gaseous star of a given mass in radiative equilibrium, in which ρ and [κη̅]^r_R do not increase outward, the minimum value of I-β_c is the constant value of (I-β) ascribed to a standard model configuration of the same mass. For n=∞ the theorem gives the minimum central temperature for a gaseous star with negligible radiation pressure.