Mass–radius Relation for Rocky Planets Based on Prem

Abstract

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: R R ⊕ = (1.07 − 0.21 · CMF) · ( M M ⊕ ) 1/3.7 , where CMF stands for Core Mass Fraction. It is applicable to 1∼8 M ⊕ and CMF of 0.0∼0.4. Applying this formula to Earth and Venus and several known small exoplanets with radii and masses measured to better than ∼30% precision gives a CMF fit of 0.26 ± 0.07.

The Transiting Multi-planet System HD 3167: A 5.7 M⊕ Super-Earth and an 8.3 M⊕ Mini-Neptune

“…Figure 11 displays the position of HD 3167b on the mass-radius diagram compared to the sub-sample of small transiting planets (R R 4  Å ) whose masses and radii have been derived with a precision better than 20%. Theoretical models from Zeng et al (2016) are overplotted using different lines and colors. The precision of our mass determination (∼8%) allows us to conclude that HD 3167b is a rocky terrestrial planet with a composition consisting of ∼50% silicate and ∼50% iron.…”

Three’s Company: An Additional Non-transiting Super-Earth in the Bright HD 3167 System, and Masses for All Three Planets

“…N, V, and E mark the solar system planets. The curves show the mass-radius correlation for compositions ranging from 100% iron to 100% water from Zeng et al (2016). Planet b is likely predominately rocky, and planet c is volatile-rich.…”

“…Figure 11 shows HD3167b in comparison with other small exoplanets with masses measured to better than 50% precision; the lines show the composition models of Zeng et al (2016). We randomly draw 100,000 planet masses and radii from our posterior distributions, and compare them to the mass-radius relation of Fortney et al (2007) for pure rock, finding results that are consistent with the models of Zeng et al (2016). Assuming that the planet is a mixture of rock and iron, we compute the iron mass fraction from each random draw using Equation (8) of Fortney et al (2007).…”

PREDICTIONS OF THE ATMOSPHERIC COMPOSITION OF GJ 1132b

“…We note that there is additionally a well-known degeneracy in determining the planet's composition from the density. Using the online tool of Zeng et al (2016), we find that the nominal mass of the planet allows for up to about 20 wt% of the planet to be water. Note that this extreme value results in a nearly zero silicate mass fraction, which is highly unphysical, as giant impact simulations show that mantle stripping can produce planets with at most a70% core mass fraction (Marcus et al 2010).…”

“…The core mass and radius assuming a two-component model (silicate + metal, no water) are determined with the online tool of Zeng et al (2016). Values for the planet properties are given in Table 1.…”