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

Author(s)

Ragazzo, Clodoaldo

• Document type: Journal article
• Source: SAO PAULO JOURNAL OF MATHEMATICAL SCIENCES; v. 14, n. 1, p. 48-pg., 2020-01-28.

Abstract

This paper contains a review of Clairaut's theory with focus on the determination of a gravitational modulus gamma defined as (C-I-o/I-o)gamma = 2/3 Omega(2), where C and I-o are the polar and mean moment of inertia of the body and Omega is the body spin. The constant gamma is related to the static fluid Love number k(2) = 3I(o) G/R-5 1/gamma, where R is the body radius and G is the gravitational constant. The new results are: a variational principle for gamma, 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 gamma 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/(mR(2)) <= 0.4 the approximation gamma approximate to G root 2(7)/5(5) m(5)/I-o(3) (def) = gamma(I) is a better estimate for gamma than that obtained from the Darwin-Radau equation, denoted as gamma(DR). Moreover, an inequality in the paper implies that the Darwin-Radau approximation may be valid only for 0.3 <= I-o/(mR(2)) <= 0.4 and within this range vertical bar gamma(DR)/gamma(I) - 1 vertical bar < 0.14%. (AU)

• FAPESP's process: 16/25053-8 - Dynamics and geometry in low dimensions
• Grantee: André Salles de Carvalho
• Support Opportunities: Research Projects - Thematic Grants