Existence for Stable Rotating Star-Planet Systems

TL;DR

This paper proves the existence of stable, rotating star-planet systems modeled by Euler-Poisson equations, analyzing their properties and bounds for different equations of state and mass ratios.

Contribution

It establishes the existence of local energy minimizers for star-planet systems with small mass ratios using a variational approach and Wasserstein metrics.

Findings

Existence of minimizers for small mass ratios under specific conditions.

Radii of minimizers tend to zero for b3 > 2.

Upper bounds for expansion rates of radii when b3  1.5 to 2.

Abstract

This paper investigates the existence and properties of stable, uniformly rotating star-planet systems, i.e. mass ratio is sufficiently small. It is modeled by the Euler-Poisson equations. Following the framework established by McCann for binary stars \cite{McC06}, we adopt a variational approach, and prove the existence of local energy minimizers with respect to the Wasserstein $L^\infty$ metric, under the assumed equation of state $P(\rho)=K\rho^\gamma$ and under the condition that the mass ratio $m$ is sufficiently small, corresponding to a star-planet system. Such minimizers correspond to solutions of the Euler-Poisson system. We consider two cases. For $\gamma > 2$, we not only prove existence but also show, via scaling arguments, that the radii (to be precise, the bounds of the supports of the minimizers) tend to zero. For $\frac{3}{2} < \gamma \leq 2$, we estimate an upper bound for the (potential) expansion rates of the radii, and it turns out that the existence result remains valid in this case as well. Finally, we provide estimates for the distances between different connected components of supports of minimizers and propose a conjecture regarding the number of connected components.