The theory of figures of Clairaut with focus on the gravitational modulus: inequalities and an improvement in the Darwin–Radau equation

This paper contains a review of Clairaut’s theory with focus on the determination of a gravitational modulus γ defined as C - I o I o γ = 2 3 Ω 2 , where C and I o are the polar and mean moment of inertia of the body and Ω is the body spin. The constant γ is related to the static fluid Love number k 2 = 3 I o G R 5 1 γ , where R is the body radius and G is the gravitational constant. The new results are: a variational principle for γ , upper and lower bounds on the ellipticity that improve previous bounds by Chandrasekhar and Roberts (Astrophys J 138:801, 1963), and a semi-empirical procedure for estimating γ from the knowledge of m , I o , and R , where m is the mass of the body. The main conclusion is that for 0.2 ≤ I o / ( m R 2 ) ≤ 0.4 the approximation γ ≈ G 2 7 5 5 m 5 I o 3 = def γ I is a better estimate for γ than that obtained from the Darwin–Radau equation, denoted as γ DR . Moreover, an inequality in the paper implies that the Darwin–Radau approximation may be valid only for 0.3 ≤ I o / ( m R 2 ) ≤ 0.4 and within this range | γ DR / γ I - 1 | < 0.14 % .