METHANE PLANETS AND THEIR MASS–RADIUS RELATION

The internal structure is given by the basic equations of planetary structure: the equation of hydrostatic equilibrium and the equation of mass conservation. There should also be an equation describing energy transport that provides the temperature profile, but the transport properties are not well known, and so for simplicity we assume an isotherm. The sensitivity to this assumption can be assessed by comparing different isotherms, as shown below.

The structure equations are integrated using a fourth-order Runge–Kutta procedure. We integrate from a central pressure P_c to a low surface pressure P_s (typically set to 1 bar) and compute the planetary mass and radius. For pure CH₄ planets, if P_c is greater than the methane dissociation pressure, hereafter, P_diss (see below), we use the equation of state (EOS) for pure carbon. We continue the integration until the pressure falls below P_diss. The mass of the body at this point is the mass of the carbon core, M_C. The dissociation of CH₄ produces an additional $1/3{{M}_{C}}$ of hydrogen which we assume has diffused to the outer envelope of the planet to form a hydrogen atmosphere. Any remaining mass is in the form of an intermediate shell of CH₄.

In principle, each planetary mass should correspond to a unique P_c value, and we can find a single-valued function $M({{P}_{c}})$ for the planetary mass versus central pressure. However this is not always the case. In the immediate vicinity of ${{P}_{c}}={{P}_{diss}}$, central pressures just below and just above P_diss give the same mass (e.g., Ramsey 1948). In this case, the planet with a carbon core is the stable configuration.

For planets with a silicate/iron core (see Section 2.4) we follow the same procedure, except that the innermost region is assumed to be SiO₂/Fe. We integrate out from the center until the mass of the body equals the mass of the prescribed silicate/iron core. After that, if the pressure is greater than P_diss we use the EOS for carbon, otherwise we use the EOS for CH₄. Also, in this case, we guarantee that the mass of the hydrogen atmosphere is one-third the mass of the carbon shell.

We use the SESAME EOS for carbon, methane, and hydrogen (e.g., Kerley 1972). When cores are included, we use the SESAME EOS for SiO₂ and iron (Fe). In order to account for the effect of methane dissociation, we must assume the pressure at which dissociation occurs. Several studies suggest that methane dissociates at a pressure of ∼300 GPa (Ancilotto et al. 1997; Gao et al. 2010; Chau et al. 2011); thus, the exact dissociation pressure of CH₄ is still unknown and needs to be confirmed through experiments and theoretical calculations. In order to investigate the sensitivity of the derived internal structure (and M–R relation) to the dissociation pressure, we also consider a lower dissociation pressure of 170 GPa. The lower the dissociation pressure is, the lower the planetary mass for which a jump occurs in the M–R relation. In fact, before CH₄ breaks down to carbon and hydrogen, some intermediate products of hydrocarbons can be formed: methane will first form ethane (C₂H₆) at 95 GPa, butane (C₄H₁₀) at 158 GPa, and finally carbon (diamond) and hydrogen (Gao et al. 2010).

If hydrogen diffuses through the shell of undissociated CH₄ to the planetary atmosphere, then pure CH₄ planets above a certain mass would have a pure carbon core, a methane inner shell, and a (molecular) hydrogen atmosphere. In this work, we assume that once dissociation occurs differentiation takes place. Modeling the actual differentiation is beyond the scope of this paper. Some pioneering work in that direction was recently presented by Levi et al. (2014).

For a pure methane planet, there is a sharp discontinuity in the M–R relation above the mass where dissociation occurs. This is presented in the two top panels of Figure 1. Since, unlike for refractory materials, the temperature is not negligible, we consider four different temperatures: 50, 500, 1000, and 1500 K. The isotherm of 50 K should be taken as the extreme-low-temperature. The other isotherms are comparable to the atmospheric temperatures of closed-in exoplanets. In reality, planets are not isothermal but are expected to have a temperature profile with the temperature increasing toward the center. As can be seen from the figure, the temperature mostly affects the planetary radius when dissociation occurs, with higher temperatures leading to a more extended atmosphere. Therefore, the effect of temperature on the M–R relation is strongest in the outer region and the effect of a hotter interior is expected to be relatively small.

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The derived M–R relations are shown for P_diss of 170 GPa (left) and 300 GPa (right). The solid, dashed, dotted, and dashed–dotted curves correspond to T = 50, 500, 1000, and 1500 K, respectively. As long as the planet is undifferentiated, the radius is insensitive to temperature. When the planetary mass is high enough to produce a hydrogen (H₂) atmosphere, the radius increases abruptly and becomes much more sensitive to temperature. The amount of methane in the planet varies with its mass. As long as the planetary mass is low enough so that dissociation does not occur, the mass fraction of methane is unity and the radius is insensitive to the temperature; thus, once the critical mass (pressure) for dissociation is reached, the mass fraction of methane decreases rapidly.

Hot methane planets that dissociate and develop a molecular hydrogen envelope, could, in principle, lose some of their atmospheres. For a sun-like star, the four temperatures of 50, 500, 1000, and 1500 K correspond to radial distances of about 30, 0.33, 0.083, and 0.037 AU, respectively. The most vulnerable planets are those with the high temperatures (i.e., T = 1000 K and T = 1500 K). Planets with small masses have ${{P}_{c}}\lt {{P}_{{\rm diss}}}$, and therefore are not expected to have hydrogen atmospheres. For the cases we investigate, this means masses less than ∼8 ${{M}_{\oplus }}$ (see Figure 1). More massive planets can dissociate CH₄, but for these masses hydrogen escape is less efficient due to the relatively large gravity. We can estimate the mass loss rate for the different cases by using the energy-limited mass loss estimates (e.g., Lecavelier Des Etangs 2007; Murray-Clay et al. 2009; Lopez et al. 2012; Owen & Jackson 2012; Kurosaki et al. 2014; Mordasini et al. 2015). For planets with temperatures of 50 and 500 K, it is found that evaporation is negligible. For temperatures of 1000 and 1500 K and planets with masses larger than about 7 ${{M}_{\oplus }}$, the mass that is lost after 10 Gyr is about 2% and 10% of the total planetary mass, respectively.¹ Smaller planets, however, can lose a significant amount of mass, especially if they are close to their host star. This means that low-mass planets with hydrogen atmospheres are unlikely to exist in old systems. Thus, the mass loss estimate assumes a pure-hydrogen atmosphere, and since small planets cannot dissociate (due to low internal pressures), they are expected to have methane atmospheres for which the mass loss rate can be much lower. Further investigations of the mass loss rates of various atmospheric compositions is required to understand the survivability of close-in small methane planets (e.g., Lammer et al. 2012).

Since planets are not expected to consists of pure methane, we next consider the existence of a silicate (SiO₂) core and investigate how it affects the derived M–R relation. The presence of a silicate core reduces the importance of this effect. As the silicate mass fraction of the planet increases, the magnitude of the radius change decreases. This is because the silicate core occupies part of the high pressure region where the methane dissociation should take place. The larger the silicate core, the smaller the volume of methane that participates in the dissociation, and the smaller the difference in radius at the transition point. The effect of the silicate core is demonstrated in the bottom panels of Figure 1 where we compare the M–R relation for methane planets with no core (solid black) and with silicate cores that are equal to 30% (dotted gray), 50% (dashed blue), and 80% (dashed–dotted red) of the total planetary mass for T = 500 K for P$_{{\rm diss}}$ of 170 GPa (left) and 300 GPa (right). While we use SiO₂ to represent the dense core material, the core could be composed of other elements such as Fe or MgO. The results are not expected to change much qualitatively since the effect is similar as long as the core material is significantly denser than methane. A comparison of the inferred internal structure for an SiO₂ core versus an Fe core is shown in Figure 3 (right panels). As can be seen from the bottom panels of Figure 1, the discontinuity due to CH₄ dissociation is much less prominent when the planet consists of a massive core. When the silicate core consists of 30% and 50% of the total planetary mass, dissociation occurs at smaller masses. This is because the core results in a higher internal pressure which allows dissociation at lower masses than in the no-core case. Thus, at some point, when the core mass is significantly larger than the methane mass, methane will exist only in the outer region (envelope) where there pressure is low. In that case, dissociation occurs at higher masses compared to the no-core case. Indeed, for the case of 80%, core dissociation is found to occur at higher masses. Here too the kink in the M–R relation occurs for larger masses for the higher P$_{{\rm diss}}$.