Relation between Mass and Radius of Exoplanets Distinguished by their Density

Abstract

The formation of the solar system has been studied since the 18th century and received a boost in 1995 with the discovery of the first exoplanet, 51 Pegasi b. The investigations increased the number of confirmed planets to about 5,300 to date. The possible internal structure and composition of these planets can be inferred from the relationship between planet mass and radius, M-R. We have analyzed the M-R relation of a selected sample of iron-rock and ice-gas planets using a fractal approach to their densities. The application of fractal theory is particularly useful to define the physical meaning of the proportionality constant and the exponent in an empirical M-R power law in exoplanets, but this does not necessarily mean that they have an internal fractal structure. The M-R relations based on this sample are M$=(1.46 \pm 0.08) \text{R}^{2.6 \pm 0.2}$ for the rocky population ($3.6 \leq \rho \leq 14.3$ g/cm$^3$), with $1.5 \leq \text{M} \leq 39$ M$_{\Earth}$, and M $=(0.27 \pm 0.04) \text{R}^{2.7 \pm 0.2}$ for ice-gas planets ($0.3 \leq \rho \leq 2.1$ g/cm$^3$) with $5.1 \leq \text{M} \leq 639$ M$_{\Earth}$ (or $\simeq 2 \text{M}_{\text{J}}$) and orbital periods greater than 10 days. Both M-R relations have in their density range a great predictive power for the determination of the mass of exoplanets and even for the largest icy moons of the solar system. The average fractal dimension of these planets is D$=2.6 \pm 0.1$, indicating that these objects likely have a similar degree of heterogeneity in their densities and a nearly similar composition \textcolor{red}{in each sample}. The M-R diagram shows a ``gap'' between ice-gas and iron-rock planets. This gap is a direct consequence of the density range of these two samples. We empirically propose an upper mass limit of about 100 M$_{\Earth}$, so that an M-R relation for an ice-gas planets in a narrow density range is defined by M $\propto$ R$^{3}$.