Lie Group Theory of Multipole Moments and Shape of Stationary Rotating Fluid Bodies

TL;DR

This work develops a non-perturbative, Lie-group based framework for the hydrostatic equilibrium of uniformly rotating self-gravitating fluids. By employing exponential maps of fluid diffeomorphisms and a shift-operator Neumann series, it derives a closed nonlinear master equation for the shape and height functions, circumventing divergences inherent in Legendre expansions and enabling arbitrary-order deformations. The formalism yields consistent boundary conditions, Love numbers, and multipole moments, with exact solutions for Maclaurin/Jacobi and polytropic unit-index cases, and provides a systematic spectral decomposition via Wigner algebra for nonlinear couplings. The approach offers precise, scalable tools for planetary and stellar modeling, improving internal-structure inference and gravitational-field predictions under strong rotation and across diverse equations of state.

Abstract

We present a rigorous framework for determining equilibrium configurations of uniformly rotating self-gravitating fluid bodies. This work addresses the longstanding challenge of modeling rotational deformation in celestial objects such as stars and planets. By integrating classical Newtonian potential theory with modern mathematical tools, we develop a unified formalism that improves both the precision and generality of shape modeling in astrophysical contexts. Our method employs Lie group theory and exponential mapping to characterize vector flows associated with rotational deformations. We derive functional equations for perturbations in density and gravitational potential, resolved analytically using the shift operator and Neumann series. This extends Clairaut's classical linear theory into the nonlinear regime. The resulting formulation yields an exact nonlinear differential equation for the shape function, describing hydrostatic equilibrium under rotation without assuming slow rotation. This generalized Clairaut equation incorporates nonlinear effects and accommodates large rotational speeds. We validate the theory by deriving exact solutions, including the Maclaurin spheroid, Jacobi ellipsoid, and the unit-index polytrope. We also introduce spectral decomposition techniques to analyze radial harmonics of the shape function and gravitational perturbations. Using Wigner's formalism for angular momentum addition, we compute higher-order spectral corrections and derive boundary conditions for radial harmonics. This enables accurate computation of Love numbers and gravitational multipole moments, offering a comprehensive, non-perturbative approach to modeling rotational deformations in astrophysical systems.