Magma-atmosphere interactions impact mass-radius relation of sub-Neptunes
Authors: Fan Bowen , Lee Eve , Kite Edwin 
Published: July 2021

The Kepler mission revealed that sub-Neptunes are about as common as stars. Sub-Neptunes have sizes between rocky terrestrial and gassy giant planets; their discovery defied our pre-existing notion of planet demographics. The prevailing view for sub-Neptunes was that they are mostly core by mass and atmosphere by volume [1]. (Although the alternative explanation that sub-Neptunes are waterworlds has not been ruled out [2], it does not explain the radius distribution with orbital period nor the densities of highly irradiated planets, among other problems, and so it is disfavored [3]). However, the current formation models lack material properties data (e.g. solubilities) for their core-atmosphere interface.However, the temperature and pressure at the magma-atmosphere interface can rise to >3000 K and ~5-30 GPa [4][5], high enough for dissolution of hydrogen gas into the magma [6][7]. Thus, sub-Neptune magma oceans at the core-atmosphere interface are abundant, long-lived [8], and massive [9]. The dissolution of atmosphere into the magma may explain the drop-off in exoplanet abundance at 3 times Earth radius, but the puff-up of the magma due to gas dissolution was not included.We propose a simple model to calculate sub-Neptune mass-radius relation including the puff-up effect. Key assumptions include: (1) the Fe/core mass fraction is Earth-like, and He/gas mass fraction is Solar-like; (2) photospheric radius is 6 scale heights above the radiative-convective boundary (RCB); (3) ideal mixing between the dissolved gas and magma; (4) the dissolved gas is well mixed over the silicate-layer, within which the temperature gradient is controlled by the properties of magma; (5) nonlinear solubility is constrained by limited laboratory data [9]. The equations-of-state (EoS) used are a Mg2SiO4 for the magma [10]; the H/He EoS [11]; and a simple model for Fe [12]. The model is integrated from the RCB and evolves until magma-atmospheric solubility equilibrium.We have varied the core mass, atmospheric mass and equilibrium temperature in the atmosphere. Our preliminary results are shown in Figures. The critical point for the puff-up of the core due to the dissolved gas corresponds to ~1% solubility at the magma-atmosphere boundary (Fig. 1). The puff-up effect turns out to be important up to 0.3 Earth radius (Fig. 2), .e.g, much larger than the radius error bars for a single planet in the CKS survey with Gaia DR2 data [13].Figure 1: The solubility of gas at the magma-atmosphere interface as a function of temperature and pressure, assuming ideal gas mixture. Lines correspond to atmospheric profiles of sub-Neptunes with core masses of 10 Earth masses, situated at 0.1 AU from their host stars.&#160;Figure 2: Mass-radius relation of the core. Subscripts represent conditions at the magma-atmosphere interface. The blue curve is more representative of sub-Neptunes, while the red curve coincides with an air-less rocky planet (black).In future, we will add additional constrain on gas/core mass fraction [14], forward-model the relationship between mass and photospheric radius (the observables), and generate early predictions for exoplanet masses and radii that can be used to help interpret data such as ESA&#8217;s PLAnetary Transits and Oscillations of stars (PLATO) and NASA&#8217;s Transiting Exoplanet Survey Satellite (TESS).&#160;[1] Lopez, E. D., &amp; Fortney, J. J. (2014). Understanding the mass-radius relation for sub-Neptunes: Radius as a proxy for composition.&#160;The Astrophysical Journal,&#160;792(1), 1.[2] Mousis, O., Deleuil, M., Aguichine, A., Marcq, E., Naar, J., Aguirre, L. A., ... &amp; Gon&#231;alves, T. (2020). Irradiated ocean planets bridge super-earth and sub-Neptune populations.&#160;The Astrophysical journal letters,&#160;896(2), L22.[3] Bean, J. L., Raymond, S. N., &amp; Owen, J. E. (2021). The nature and origins of sub&#8208;Neptune size planets.&#160;Journal of Geophysical Research: Planets,&#160;126(1), e2020JE006639.[4] Lee, E. J., Chiang, E., &amp; Ormel, C. W. (2014). Make super-Earths, not Jupiters: Accreting nebular gas onto solid cores at 0.1 AU and beyond.&#160;The Astrophysical Journal,&#160;797(2), 95.[5] Piso, A. M. A., Youdin, A. N., &amp; Murray-Clay, R. A. (2015). Minimum core masses for giant planet formation with realistic equations of state and opacities.&#160;The Astrophysical Journal,&#160;800(2), 82.[6] Chachan, Y., &amp; Stevenson, D. J. (2018). On the Role of Dissolved Gases in the Atmosphere Retention of Low-mass Low-density Planets.&#160;The Astrophysical Journal,&#160;854(1), 21.[7] Kite, E. S., Fegley Jr, B., Schaefer, L., &amp; Ford, E. B. (2019). Superabundance of Exoplanet Sub-Neptunes Explained by Fugacity Crisis.&#160;The Astrophysical Journal Letters,&#160;887(2), L33.[8] Vazan, A., Ormel, C. W., &amp; Dominik, C. (2018). Effect of core cooling on the radius of sub-Neptune planets.&#160;Astronomy &amp; Astrophysics,&#160;610, L1.[9] Kite, E. S., Fegley Jr, B., Schaefer, L., &amp; Ford, E. B. (2020). Atmosphere origins for exoplanet sub-Neptunes.&#160;The Astrophysical Journal,&#160;891(2), 111.[10] Hirschmann, M. M., Withers, A. C., Ardia, P., &amp; Foley, N. T. (2012). Solubility of molecular hydrogen in silicate melts and consequences for volatile evolution of terrestrial planets.&#160;Earth and Planetary Science Letters,&#160;345, 38-48.[11] Stewart, S., Davies, E., Duncan, M., Lock, S., Root, S., Townsend, J., ... &amp; Jacobsen, S. (2020, November). The shock physics of giant impacts: Key requirements for the equations of state. In&#160;AIP Conference Proceedings&#160;(Vol. 2272, No. 1, p. 080003). AIP Publishing LLC.[12] Chabrier, G., Mazevet, S., &amp; Soubiran, F. (2019). A new equation of state for dense hydrogen&#8211;helium mixtures.&#160;The Astrophysical Journal,&#160;872(1), 51.[13] Seager, S., Kuchner, M., Hier-Majumder, C. A., &amp; Militzer, B. (2007). Mass-radius relationships for solid exoplanets.&#160;The Astrophysical Journal,&#160;669(2), 1279.[14] Fulton, B. J., &amp; Petigura, E. A. (2018). The California-Kepler survey. VII. Precise planet radii leveraging Gaia DR2 reveal the stellar mass dependence of the planet radius gap.&#160;The Astronomical Journal,&#160;156(6), 264.[15] Lee, E. J. (2019). The boundary between gas-rich and gas-poor planets.&#160;The Astrophysical Journal,&#160;878(1), 36.