A Simple Analytical Model for Rocky Planet Interiors

Abstract

This work aims at exploring the scaling relations among rocky exoplanets. With the assumption of internal gravity increasing linearly in the core, and staying constant in the mantle, and tested against numerical simulations, a simple model is constructed, applicable to rocky exoplanets of core mass fraction (CMF) $\in 0.1\sim0.4$ and mass $\in 0.1 \sim 10 M_{\oplus}$. Various scaling relations are derived: (1) core radius fraction $\rm CRF \approx \sqrt{CMF}$, (2) Typical interior pressure $P_{\text{typical}} \sim g_{\rm{s}}^2$ (surface gravity squared), (3) core formation energy $E_{\rm diff} \sim \frac{1}{10} E_{\rm grav}$ (the total gravitational energy), (4) effective heat capacity of the mantle $C_{\rm p}\approx \left( \frac{M_{\rm{p}}}{M_{\oplus}} \right) \cdot 7 \cdot 10^{27}$ J K$^{-1}$, and (5) the moment of inertia $I\approx \frac{1}{3} \cdot M_{\rm{p}} \cdot R_{\rm{p}}^2$. These scaling relations, though approximate, are handy for quick use owing to their simplicity and lucidity, and provide insights into the interior structures of rocky exoplanets. As examples, this model is applied to several planets including Earth, GJ 1132b, Kepler-93b, and Kepler-20b, and made comparison with the numerical method.