Variational Principle for Planetary Interiors

TL;DR

This paper introduces a variational principle-based analytical framework for planetary interiors, simplifying complex differential equations into a single equation, and deriving universal relations and error estimates relevant to diverse planetary structures.

Contribution

It applies the variational principle to planetary interior modeling, providing a unified analytical approach and new insights like a universal mass-radius relation and a planetary virial theorem.

Findings

Derivation of a universal mass-radius relation.

Estimation of error propagation from equation of state.

Formulation of a planetary virial theorem.

Abstract

In the past few years, the number of confirmed planets has grown above 2000. It is clear that they represent a diversity of structures not seen in our own solar system. In addition to very detailed interior modeling, it is valuable to have a simple analytical framework for describing planetary structures. Variational principle is a fundamental principle in physics, entailing that a physical system follows the trajectory which minimizes its action. It is alternative to the differential equation formulation of a physical system. Applying this principle to planetary interior can beautifully summarize the set of differential equations into one, which provides us some insight into the problem. From it, a universal mass-radius relation, an estimate of error propagation from equation of state to mass-radius relation, and a form of virial theorem applicable to planetary interiors are derived.