Mass-Radius Relationships for Solid Exoplanets

Abstract: We use new interior models of cold planets to investigate the mass-radius relationships of solid exoplanets, considering planets made primarily of iron, silicates, water, and carbon compounds. We find that the mass-radius relationships for cold terrestrial-mass planets of all compositions we considered follow a generic functional form that is not a simple power law: $\log_{10} R_s = k_1 + 1/3 \log_{10}(M_s) - k_2 M_s^{k_3}$ for up to $M_p \approx 20 M_{\oplus}$, where $M_s$ and $R_s$ are scaled mass and radius values. This functional form arises because the common building blocks of solid planets all have equations of state that are well approximated by a modified polytrope of the form $\rho = \rho_0 + c P^n$. We find that highly detailed planet interior models, including temperature structure and phase changes, are not necessary to derive solid exoplanet bulk composition from mass and radius measurements. For solid exoplanets with no substantial atmosphere we have also found that: with 5% fractional uncertainty in planet mass and radius it is possible to distinguish among planets composed predominantly of iron or silicates or water ice but not more detailed compositions; with $\sim$~5% uncertainty water ice planets with $\gtrsim 25%$ water by mass may be identified; the minimum plausible planet size for a given mass is that of a pure iron planet; and carbon planet mass-radius relationships overlap with those of silicate and water planets due to similar zero-pressure densities and equations of state. We propose a definition of "super Earths'' based on the clear distinction in radii between planets with significant gas envelopes and those without.

• The paper establishes that mass–radius relationships deviate from a simple power law, proposing a modified functional form that applies to planets up to around 20 Earth masses.
• The study reveals that similar equations of state for iron, silicates, water, and carbon-based materials allow for composition discrimination with approximately 5% measurement precision.
• The research demonstrates that detailed interior models are not necessary for bulk composition analysis, streamlining exoplanet classification for future observational missions.

Mass-Radius Relationships for Solid Exoplanets

The exploration of exoplanetary characteristics has been accelerated by the increasing discovery of diverse extrasolar planets. The study "Mass-Radius Relationships for Solid Exoplanets," authored by Seager et al., provides an essential contribution to the understanding of these celestial bodies by investigating definitive mass-radius relationships for solid exoplanets. Utilizing theoretical models under the assumption of cold, zero-temperature or 300 K temperature conditions, the research delineates the relationships for planets predominantly composed of iron, silicates, water, and carbon-based compounds.

Key Insights and Numerical Findings

1. Functional Form of Mass-Radius Relationships: The study reveals that the mass-radius relationships for cold terrestrial-mass planets are not simply characterized by a linear power law. Instead, the relationships adhere to a modified functional form: $$\log_{10} R_s = k_1 + 1/3 \log_{10}(M_s) - k_2 M_s^{k_3}$$. This form results from the mass equilibrium between the polytropic-like pressure support in the planet interior and the gravitational forces. The authors provide scaling factors to transform these dimensionless parameters into physical values, allowing for practical application to celestial bodies up to approximately 20 Earth masses ($20 M_{\oplus}$).
2. Equation of State (EOS) Considerations: A novel insight from this research is the similarity in the equations of state (EOSs) across various solid planet components. This homogeneity is attributable to the physical nature of electron degeneracy and the dominance of Coulomb forces at the pressure range of interest. The result is a consistent mass-radius relationship characterized by a modest variation across different assumed planet compositions.
3. Compositional Distinctions: The study emphasizes the distinctions between potential planet compositions detectable through precision measurements. The minimal uncertainties in planet mass and radius—advisably around 5%—prove adequate to discern water-ice planets from their rockier or denser counterparts. Planets found above defined mass-radius curves for pure materials can be interpreted as possessing significant gaseous envelopes, indicative of different formation histories or subsequent atmospheric evolution.
4. Lack of Necessity for Highly Detailed Interior Models: The conclusions extend to inferring that detailed planetary interior models, accounting for temperature structures and phase changes, are not strictly required to derive bulk compositions of exoplanets from their mass and radius measurements. The EOS employed are well-suited for these analyses due to their robust description of material behavior under compression. This is particularly beneficial given the constraints on obtaining precise observational data for exoplanets.
5. Implications for "Super Earths": A notable classification emerges for "super Earths," defined as planets devoid of significant gaseous envelopes observed within the size constraints established for homogeneous water-ice planets.

Implications and Future Prospects

This research lays a fundamental framework for interpreting observational data from upcoming exoplanetary missions, such as astrophysical surveys involving precise photometric and radial velocity measurements. As stellar mass and radius measurement techniques evolve—anticipated advances via missions like GAIA—the ability to discern more granular details about planetary compositions will improve.

The findings not only provide a basis for understanding potential compositions but also highlight the need for atmospheric and evolutionary models to complement bulk compositional analyses. Such advancements are essential for discerning the complete narrative of planet formation and its resultant diversity, especially considering planets that may possess high amounts of carbon or exhibit atypical atmospheric features.

Future theoretical endeavors may entail developing temperature-dependent EOSs for pressures beyond current terrestrial limits, enriching the fidelity of mass-radius models. This will allow for the expansion of the models' applicability to exoplanets with dynamic atmospheres, further bridging the gap between theoretical advancements and tangible astrophysical observational techniques.