[Back to Structure and internal dynamics of planets]

Staff - Olivier Grasset, Antoine Mocquet, Sabrina Carpy

Abstract - Un modèle théorique qui décrit la structure interne des planètes depuis 1 jusqu’à 100 masses terrestres a été mis au point. Avec ce modèle, le rayon et le moment d’inertie d’une planète se déduisent de la masse du corps concerné, en supposant que sa composition globale en éléments lourds est similaire à celle de son étoile. Ce modèle calcule la position, les profils de pression et de densité des différentes couches qui composent ce corps. Il permet de décrire les relations Masse-Rayon des différentes familles de planètes, outil indispensable au traitement des données de la sonde CoroT

Observing other planetary systems opens the perspective of discovering new types of planets, unknown in our Solar System. The recent discoveries of exoplanets with masses below 20 M_E (Earth masses) confirms the existence of Super-Earths [Léger et al., 2009, Selsis et al., 2007]. This family ncludes terrestrial planets such as Mars, Venus and Earth, water-rich planets like our giant icy moons, iron-rich planets similar to Mercury, but also mini-Neptunes (Fig.1). The abundance of low mass planets is not yet known. So far, over the 350 exoplanets which have been discovered, but only 34 are below 30 M_E . This observed distribution is biased by observational limitations and is evolving very rapidly.

So far, over the 350 exoplanets which have been discovered, more than 90 % are giant planets, and only 34 are below 30 M_E . But this observed distribution, which is biased by observational limitations, is evolving very rapidly. By analogy with the solar system, a metallic iron core is expected if the amount of iron is sufficient. One important aspect of these planets is that they are potentially habitable, in the sense of bearing a sufficient amount of liquid water at the surface (if permitted by the planetary surface temperature). It is anticipated that terrestrial concepts such as oceans and continents, volcanic and tectonic activity and even vegetal cover ("forests") are applicable to these planets.

Fig. 1 - Internal structure of the planetary families. From left to right : Super-Mercuries, silicate-rich planets (like our Earth), Ocean-planets, Neptune-like planets, gaseous giants.

It is assumed that the structure of a super-Earth is similar to that of Earth-like planets of the solar system: a metallic core, overlaid by a silicate mantle, and a water-rich outer layer. The bulk composition is then fixed by the amount of water X_w , and the atomic ratios Fe/Si and Mg/Si [Léger et al., 2009; Selsis et al., 2007]. Stars which host exoplanets, have Fe/Si and Mg/Si ratios which can differ significantly from solar ones. It is shown that our model is compatible with all these compositions, meaning that the presence of silicate-rich planets with or without water is very likely around these stars. Planetary radii have been calculated for a range of M and X_w . The first result is that R is only multiplied by 3 (Fig. 2) for M increasing from 1 to 100 M_E , whereas it should be 1001/3 = 4.6 if the sphere was homogeneous and incompressible. The exponent decreases as mass increases resulting in a significant flattening of the curves above 20 M_E . For a given M , the transition from a dry silicate-rich planet to an icy world induces an increase of the radius between 30 % (100 M_E ) and 40 % (1 M_E ). All this information can be gathered into one empirical equation [Grasset et al., 2009] describing M − R relationships in the range 1-100 M_E using α, β, γ and ε parameters depending on X_w :

log (R/R_E) = log(α) + [β + γ(M/M_E) + ε(M/M_E)²] log(M/M_E)

with R_E =6430 km is the radius of a 1M_E planet in the reference case (not the Earth’s radius).

Fig. 2: Radius of planets as a function of their mass, from the internal structure modelling [Selsis et al., 2007]. The curves [Fe], [silicates], [ices], [H₂-He] correspond to planets made of pure Fe, silicates and metallic core, pure water ice, and pure H₂-He gas,  respectively. The grey area corresponds to planets with both silicates and water.

The uncertainties on radius estimates due to model approximation have been investigated. Compositional variations in silicates and in iron alloys change the relative size of the metallic core but do not influence strongly the planetary radius (less than 2 %). Similarly, very large thermal variations within the planetary layers do not affect significant ly the size of the planet. A combination of these two factors indicates that numerical modelling provides radii estimate of dry silicate-rich planets with a precision better than 5%. Since it will be very hard to provide any constraints on the silicate composition and on the thermal profiles, one cannot expect to have a more precise constraint from numerical modelling. The M −R relation proposed here will allow the characterization of the main composition of the solid cores of exoplanets as long as both M and R can be measured, and if the contribution of the atmosphere can be removed. The different approaches that will allow such drastic conditions to be met in the future have been discussed in [Grasset et al., 2009; Sotin et al., 2007].

This internal structure model is applied to CoroT measurements, and in particular to the special case of recently detected CoroT-7b Super Earth planet.

Associated Publications

Léger, A., et al. dont O. Grasset. Transiting exoplanets from the CoRoT space mission *VIII. CoRoT- 7b: the first Super-Earth with measured radius. Astronomy & Astrophysics Published 2009.

Grasset, O., J. Schneider, and C. Sotin. A study of the accuracy of Mass-Radius relationships for silicate-rich and ice-rich planets up to 100 Earth masses. Astrophysicak Journal Vol: 693 Pages 722-733 Published 2009.

Selsis, F., et al. dont C. Sotin, O. Grasset. Could we identify hot ocean-planets with CoRoT, Kepler and Doppler velocimetry?. Icarus Vol: 191 Pages: 453- 468, Published 2007.

Sotin, C., O. Grasset, and A. Mocquet. Curve mass/radius for extrasolar Earth-like planets and ocean planets. Icarus, Vol: 191 Pages: 337-351 Published 2007.