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Existence for Stable Rotating Star-Planet Systems

Hangsheng Chen

TL;DR

This work rigorously constructs rotating star–planet equilibria in the Newtonian Euler–Poisson framework with a polytropic EOS by extending McCann’s variational approach for binary stars to small mass ratios. It develops a constrained variational problem in the Wasserstein $L^{\infty}$ metric, showing the existence of local energy minimizers $\rho(m)=\rho_m+\rho_{1-m}$ that correspond to rotating solutions with $v(x)=\omega\hat{e}_z\times x$ and $\omega=J/I(\rho(m))$, solving the reduced Euler–Poisson equations. The analysis establishes scaling and convergence properties: scaling densities converge to the unit-mass non-rotating minimizer, and for $\gamma>2$ the planet’s support collapses as $m\to 0$ (while for $\tfrac{3}{2}<\gamma\le 2$ the expansion is bounded); it proves uniform density and potential bounds and derives quantitative bounds on the sizes of supports and separations of components. It then proves the existence of star–planet minimizers for small mass ratio, with the two components residing in prescribed interior domains and yielding rotating equilibria, and provides explicit upper bounds on distances between potential connected components, along with a conjecture that only two components (one star and one planet) occur in minimal configurations.

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 $P(ρ)=Kρ^γ$ 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.

Existence for Stable Rotating Star-Planet Systems

TL;DR

This work rigorously constructs rotating star–planet equilibria in the Newtonian Euler–Poisson framework with a polytropic EOS by extending McCann’s variational approach for binary stars to small mass ratios. It develops a constrained variational problem in the Wasserstein metric, showing the existence of local energy minimizers that correspond to rotating solutions with and , solving the reduced Euler–Poisson equations. The analysis establishes scaling and convergence properties: scaling densities converge to the unit-mass non-rotating minimizer, and for the planet’s support collapses as (while for the expansion is bounded); it proves uniform density and potential bounds and derives quantitative bounds on the sizes of supports and separations of components. It then proves the existence of star–planet minimizers for small mass ratio, with the two components residing in prescribed interior domains and yielding rotating equilibria, and provides explicit upper bounds on distances between potential connected components, along with a conjecture that only two components (one star and one planet) occur in minimal configurations.

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 metric, under the assumed equation of state and under the condition that the mass ratio is sufficiently small, corresponding to a star-planet system. Such minimizers correspond to solutions of the Euler-Poisson system. We consider two cases. For , 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 , 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.
Paper Structure (19 sections, 42 theorems, 96 equations)

This paper contains 19 sections, 42 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Variational formulation and Statement of Main Results
  3. Variational formulation
  4. Statement of Main Results
  5. Preliminary Results
  6. Results for 2-body systems
  7. Results for Non-rotating Bodies
  8. Existence of Constrained Minimizers of $E_{J}(\rho)$ on ${W}_{m}$
  9. Bound of the Supports of Density Functions
  10. Convergence of scaling densities in $L^{\frac{4}{3}}\left( \mathbb{R}^{3} \right)$ and $L^{1}\left( \mathbb{R}^{3} \right)$
  11. Uniform bound of scaling densities in $L^{\infty}\left(\mathbb{R}^{3}\right)$
  12. Estimate for the supports of densities
  13. Existence for Star-Planet Systems
  14. Estimate for the Center of Mass Separation
  15. Existence theorem
  16. ...and 4 more sections

Key Result

Lemma 2.4

Let $\rho\geq 0$, $\sigma \geq 0$ be the density functions in $\mathbb{R}^3$ with mass $\int \rho=m_1<\infty, \int \sigma=m_2<\infty$. $\bar{x}(\rho)$ and $\bar{x}(\sigma)$ denote the centers of mass, $I(\rho)$ and $I(\sigma)$ denote the moments of inertia, function $r$ is given in (notations).

Theorems & Definitions (115)

  • Definition 2.1: Notations
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Expansion of Moment of Inertia Che26G1
  • Remark 2.5
  • Lemma 2.6: Differentiability of Energy $E_J(\rho)$ Che26G1
  • Remark 2.7
  • Definition 2.8: Wasserstein $L^{\infty}$ distance
  • Lemma 2.9: Simple Properties of the Wasserstein $L^{\infty}$ Metric McC06
  • Remark 2.10
  • ...and 105 more