Background

1. Methane Planets and the Mass-Radius Diagram Morris Podolak, RavitHelled, Amit Levi, EranVos Dept. of Geophysical, Atmospheric, & Planetary Sciences Tel Aviv University Ramat Aviv, Israel • hydrogen atmosphere causes a net increase in the radius of the planet despite the compact carbon core. The planet with the carbon core is the stable configuration. • The amount of methane that is actually present in the planet will vary with its mass. So long as the planetary mass is low enough so that dissociation does not occur, the mass fraction of methane will be unity. But once the critical mass for dissociation is reached, the mass fraction of methane decreases rapidly as is shown in Fig. 3. This is accompanied by the appearance of a hydrogen atmosphere. Such an atmosphere might be a diagnostic for planets with CH4 dissociation. • Effect of Silicate Core • 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 participate in the dissociation, and the smaller the difference in radius at the transition point. Background One of the major efforts of planetary science today is to put all of the newly discovered exo-planets into some useful context. This is often done via a mass-radius relationship that can be computed for different compositions (see, e.g. Lissauer et al. 2011; Fressin et al. 2012). Curves of planetary radius as a function of mass for different compositions have been computed by a number of authors (see, e.g. Seager et al. 2008). For the most part, these curves are smooth and monotonic. However, if phase changes are allowed discontinuities can appear. A mild example of this can be found in the case of hydrogen planets, where the transition from molecular to metallic H at around 100 Gpa causes a kink in the mass-radius curve for planets of the order of 100 Mearth (see, e.g. Swift et al. 2012). Pressure can also cause a dissociation of CH4. There may be some intermediate products, but at high enough pressures, of the order of 100 – 300 GPa (Chau et al. 2011; Gao et al. 2010), the CH4 molecule dissociates to C + 4H. If we assume that the hydrogen diffuses through the shell of undissociated CH4 to the atmosphere, then pure CH4 planets above a certain mass will have a pure carbon core, a methane shell and a hydrogen atmosphere. We explore the consequences of this below using the SESAME equations of state for carbon, methane, and hydrogen. In the immediate vicinity of the transition mass, multiple solutions are possible, something that was first pointed out by Ramsey (1948). An example of this is shown in Fig. 2 for a 5.85 Mearth planet at T = 50K and an assumed dissociation pressure of 100 GPa. For this case several solutions exist of which the two extremes are shown. For the red curve the central pressure is just below 100 GPa and the CH4 does not dissociate. For the blue curve the central pressure is ~350 GPa and substantial dissociation occurs. The presence of a Pure Methane Planet For a pure methane planet there will be a sharp discontinuity in the mass radius relation as can be seen in Fig. 1. The solid curves are for T = 50 K isotherms, while the dashed curves are for T = 500 K isotherms. Curves are shown for dissociation pressures of 100 GPa (blue) and 300 GPa (red). As long as the planet is undifferentiated the radius is insensitive to the temperature. When the planet is massive enough to produce an H2 atmosphere, the radius increases abruptly and is much more sensitive to temperature. Figure 2. Density as a function of radius for 5.85 Mearth planet for T = 50K and Pdiss = 100 GPa. Figure 4. Mass – radius relation for a methane planet with no core (blue) and with a silicate core equal to half the planetary mass (red). The discontinuity due to methane dissociation is much less prominent. • Conclusions • A mass – radius diagram is invaluable for understanding and classifying exoplanets. However this important tool must be treated with care. Even a simple composition consisting of one species, such as methane, can lead to complications due to phase changes. Caution must be used in deriving conclusions from such diagrams. CH4 References Chau, R., Hamel, S., and Nellis, W. J. (2011). Nat. Commun. 2, 203-205. Gao, G., Oganov, A. R., Ma, Y. et al. (2010). J. Chem. Phys.133, 144508-1 – 144508-5 Lissauer, J. J., Fabrycky, D. C., Ford, E. B., et al. (2011). Nature470, 53-58. Fressin, F., Torres, G., Rowe, J. F., et al. (2012). Nature 482, 195-198. Ramsey, W. H. (1948). MNRS, 108, 406-413. Seager, S., Kuchner, M., Hier-Majumder, C. A., and Militzer, B. (2008). Astrophys. J., 669, 1279-1297. Swift, D. C., Eggert, J. H., Hicks, D. G., et al. (2012). Astrophys. J., 744, 59-68. Pdiss = 100 GPa Pdiss = 300 GPa H2 Figure 3. Mass fraction as a function of planet mass for a dissociation pressure of 100 GPa (blue) and 300 GPa (red). The curves are nearly the same for isotherms of 50K (solid) and 500K (dashed). Solid curves show CH4 mass fraction, dashed curves show H2 mass fraction. Figure 1. Radius as a function of mass for a pure CH4 planet with an isotherm of 50 K (solid) and 500 K (dashed)., and a CH4 dissociation pressure of 100 GPa (blue) and 300 GPa (red) .