Mass-Radius Relation for Rocky Planets based on PREM

Several small dense exoplanets are now known, inviting comparisons to Earth and Venus. Such comparisons require translating their masses and sizes to composition models of evolved multi-layer-interior planets. Such theoretical models rely on our understanding of the Earth's interior, as well as independently derived equations of state (EOS), but have so far not involved direct extrapolations from Earth's seismic model -PREM. In order to facilitate more detailed compositional comparisons between small exoplanets and the Earth, we derive here a semi-empirical mass-radius relation for two-layer rocky planets based on PREM: ${\frac{R}{R_\oplus}} = (1.07-0.21\cdot \text{CMF})\cdot (\frac{M}{M_\oplus})^{1/3.7}$, where CMF stands for Core Mass Fraction. It is applicable to 1$\sim$8 M$_{\oplus}$ and CMF of 0.0$\sim$0.4. Applying this formula to Earth and Venus and several known small exoplanets with radii and masses measured to better than $\sim$30\% precision gives a CMF fit of $0.26\pm0.07$.