Existence for Stable Rotating Star-Planet Systems

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 $P(ρ)=Kρ^γ$ 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 $γ> 2$, 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 $\frac{3}{2} < γ\leq 2$, 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.

💡 Research Summary

**
The paper addresses the mathematical existence and qualitative properties of uniformly rotating star‑planet configurations, a regime in which the planet’s mass is much smaller than that of the star. The physical model is the three‑dimensional Euler–Poisson system with a polytropic pressure law (P(\rho)=K\rho^{\gamma}). The authors follow the variational framework introduced by McCann for binary stars and adapt it to the star‑planet setting, where the small mass ratio (m) (planet mass divided by total mass) plays a central role.

The main object of study is the energy functional \