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: ${\frac{R}{R_\oplus}} = (1.07-0.21\cdot \text{CMF})\cdot (\frac{M}{M_\oplus})^{1/3.7}$, 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 $0.26\pm0.07$.

• The paper derives a semi-empirical mass-radius relation for two-layer rocky exoplanets based on the Preliminary Reference Earth Model (PREM) to bridge observational data and terrestrial planet models.
• Using PREM and the Birch-Murnaghan EOS, the study provides a specific formula relating a rocky planet's mass and radius to its core mass fraction (CMF) for masses up to 8 Earth masses.
• Results indicate many small exoplanets likely possess Earth-like compositions and core-to-mantle ratios, consistent with theoretical planet formation scenarios and exoplanet observations.

Mass-Radius Relation for Rocky Planets based on PREM

The paper "Mass-Radius Relation for Rocky Planets based on PREM," authored by Li Zeng, Dimitar Sasselov, and Stein Jacobsen, addresses a closer scrutiny of the compositional and structural characteristics of small rocky exoplanets, drawing into focus the comparison with terrestrial planets like Earth and Venus. It primarily derives a semi-empirical mass-radius relation for two-layer rocky exoplanets that effectively bridges observational insights and terrestrial planet models.

Methodology and Equation of State (EOS)

This study introduces a mass-radius relation for rocky planets utilizing the Preliminary Reference Earth Model (PREM), which offers a detailed seismic model of Earth's interior. Recognizing the importance of precise modeling in exoplanetary science, the authors derive this relation based on PREM, acknowledging discrepancies often stemming from direct extrapolation of simplified models that apply equations of state (EOS) of pure solids such as $\epsilon$-Fe and Mg-perovskite/post-perovskite.

The investigation considers the non-uniform density profiles derived seismically for Earth and outlines distinct EOS for the core and mantle by extrapolating from the PREM. Specifically, the Birch-Murnaghan 2nd order EOS plays an instrumental role in modeling the density responses of the Earth's low- and high-pressure regions, reflecting the complex phase transitions and diverse mineral compositions within. A central approximating feature of this EOS framework is its asymptotic convergence with the Thomas-Fermi-Dirac (TFD) model above $\sim$1 TPa, denoting high-pressure conditions where electron degeneracy pressure dominates.

Mass-Radius Relation and Core Mass Fraction (CMF)

A key output of this study is the resulting mass-radius formula for rocky planets, where core mass fraction (CMF) is an important determinant:

$$\frac{R}{R_\oplus} = (1.07-0.21\cdot \text{CMF})\cdot \left(\frac{M}{M_\oplus}\right)^{1/3.7}$$

This relation applies to planets within the mass range of 1 to 8 Earth masses and CMF values between 0 and 0.4. The analysis highlights a CMF fit of $0.26 \pm 0.07$ for Earth, Venus, and various exoplanets, consistent with terrestrial-like compositions.

Comparative Insights and Implications

The results of this study imply that many known small exoplanets exhibit a compositional structure similar to Earth, characterized by an analogous ratio of mantle to core. This mass-radius correlation suggests that these rocky exoplanets maintain Earth-like CMFs despite potential variations in surface conditions due to intense stellar irradiation in close-in orbits.

The findings align with observations of terrestrial debris in white dwarf spectra, confirming Earth-like elemental abundances in extra-solar planetary materials. The analysis also echoes theoretical models of planet formation, pointing to similar condensation sequences in nebular environments primarily comprising Fe, Mg, Si, and O.

Conclusion and Future Directions

This semi-empirical formulation conveys crucial implications for the study of rocky exoplanets and their potential diversity—an important consideration for exoplanet classification and habitability studies. Moving forward, enhanced precision in mass and radius measurements for these exoplanets will refine the model's accuracy further and inform the understanding of core-to-mantle differentiation across exoplanetary systems. Continuing advancements in observational technology, coupled with robust analytical methods such as those presented in this study, will further deepen our comprehension of planetary structure and formation.