On Clairaut's theory and its extension for planetary hydrostatic equilibrium derived using gravitational multipole formalism.

Clairaut's theory that relates the Earth's oblate figure and internal ellipticity to its gravity under rotational-hydrostatic equilibrium has reigned classical geodesy over the centuries. In this paper, we (i) derive from first principles the classical Clairaut's theory for the polar oblateness of a rotating planet under axi-symmetric rotational-hydrostatic equilibrium and (ii) extend the development to the triaxial case for the equatorial ellipticity of a tidally locked synchronous-rotating moon under rotational-tidal-hydrostatic equilibrium. Typical derivations of the classical Clairaut's theory presented in the literature being rather laborious even to first order, we instead exploit two concise forms of methodology: the gravitational multipole formalism on the physics side, and the Jacobian determinant for the Clairaut coordinate transformation on the mathematics side. The outcome is a logical and straightforward derivation of Clairaut's theory to first order in its entirety, encompassing all the equations and related formulas in geodesy bearing Clairaut's name. That further allows a natural extension to a tidally locked moon. In particular it is demonstrated that the same Clairaut's differential equation applies to both cases governing both the polar oblateness and the equatorial ellipticity. [ABSTRACT FROM AUTHOR]