Mass-radius relationships and contraction of condensed planets by cooling or despinning

Abstract

Condensed planets contract or expand as their temperature changes. With the exception of the effect of phase changes, this phenomenon is generally interpreted as being solely related to the thermal expansivity of the planet's components. However, changes in density affect pressure and gravity and, consequently, the planet's compressibility. A planet's radius is also linked to its rate of rotation. Here again, changes in pressure, gravity and compressibility are coupled. In this article we clarify how the radius of a condensed planet changes with temperature and rotation, using a simple and rigorous thermodynamic model. We consider condensed materials to obey a simple equation of state which generalizes a polytopic EoS as temperature varies. Using this equation, we build simple models of condensed planet's interiors including exoplanets, derive their mass-radius relationships, and study the dependence of their radius with temperature and rotation rate. We show that it depends crucially on the value of $\rho_s g R/K_s$ ($\rho_s$ being surface density, $g$ gravity, $R$ radius, $K_s$ surface incompressibility). This non-dimensional number is also the ratio of the dissipation number which appears in compressible convection and the Grune\"isen mineralogic parameter. While the radius of small planets depends on temperature, this is not the case for large planets with large dissipation numbers; Earth and a super-Earth like CoRoT-7b are in something of an intermediate state, with a moderately temperature-dependent radius. Similarly, while the radius of these two planets are functions of their rotation rates, this is not the case for smaller or larger planets.