Equations of State

Construction of a realistic model for internal structure requires a knowledge of the equations of state, derived from experimental and theoretical studies, linking the pressure of a group of components to its temperature, density and composition. At the very high pressures and temperatures encountered in planetary interiors (and a fortiori in brown dwarfs), the perfect gas law, valid at the surface, no longer applies.

For pressures less than or equal to approximately 2 Mbar, it is possible to carry out experimental simulations by research with shock waves. A projectile impacts the sample being studied at high velocity, and by compression creates, for a very short time (< 100 ns), temperature and pressure conditions comparable with those being studied.

7.1.4.1 Hydrogen

For pressures below 1 Mbar, the behaviour of molecular hydrogen is well understood, both theoretically and experimentally. At higher pressures, hydrogen passes into a phase known as 'metallic hydrogen' in which the atoms are ionized by pressure. According to experiment, the transition occurs around 1.4 Mbar, at a temperature of about 3000 K. At still higher pressures, liquid metallic hydrogen consists of an ionized mixture of protons and electrons at temperatures above 10 000 K. The phase diagram of hydrogen is shown in Fig. 7.2.

At the boundary between molecular hydrogen and metallic hydrogen, some authors have suggested the presence of a discontinuous phase transition called the 'plasma phase transition', based on the different nature of their action potentials: the potential is weakly repulsive in metals and strongly repulsive in insulators. It has not been possible to detect this discontinuous transition in the laboratory, but it should be noted that if it exists, it would have important consequences for the internal structure of planets, because it would create an impenetrable barrier to convection. In for Jupiter and Saturn are shown by the solid lines. The filled triangles and circles correspond to experimental results. The hatched region indicates the limit at which helium separates from liquid hydrogen (see Sect. 7.1.4.1) (After Guillot, 2006)

temperatures and pressures. The actual T (P) profiles

Fig. 7.2 The phase diagram for hydrogen at high

Fig. 7.2 The phase diagram for hydrogen at high

-1,0 -0.5 0.0 0.5 1.0 1.5 2.0 log P (Mbar)

such a case, the atmospheric abundances of the elements, measured in the outer layers would not be representative of the deeper layers.

7.1.4.2 Helium and Other Elements

Laboratory experiments provide data up to a pressure of several hundred kilobars. The equation of state for a mixture of hydrogen and helium is calculated from those of H and of He, using the following equations:

in which Y is the helium fraction by mass and Smx the mixing entropy. Under certain temperature conditions and mixing ratios, helium is no longer soluble in hydrogen (see Fig. 7.2). The helium then separates out and falls as droplets down into the interior, as far as regions that are still hotter at which mixing becomes possible again. The effect of this mechanism is to deplete the outer mixture of helium and enrich the interior. It is accompanied by an exchange of energy towards the outside. This phenomenon may currently be occurring in the atmosphere of Saturn, and possibly also, to a lesser extent, in Jupiter. It may be partially responsible (together with the gravitational energy associated with the contraction) for the internal energy observed with those two planets.

The other elements that must be taken into account are ices (H2O, CH4, NH3) and rocky materials (silicates, magnesium oxides, and metals). In the case of the ices, laboratory experiments have been conducted at pressures up to about 2 Mbar and at a temperature of 4000 K. The elements are in liquid form up to a pressure of approximately 300 kbar, but are then ionized, forming an electrically conduction fluid. Empirical P(p) relationships have been established for the ices and rocky material.

7.1.5 Construction of Models of Internal Structure

Static models of internal structure may be built from the following equations: • the equation of hydrostatic equilibrium in the presence of rotation:

dP 2

where g(r) is the gravitational acceleration at distance r, and rn is the angular velocity;

• the abundance profiles of the different components xi = xi(P).

It should be mentioned that these models sacrifice reliance on an energy equation in favour of matching the effects of rotation through the gravitational moments and adopting a T (P) relation.

The model, based on initial assumptions about the density and temperature profiles, should satisfy the observational constraints (see Sect. 7.1.1): i.e., the equatorial radius, the gravitational moments, and the abundances of the elements. The best fit is obtained by successive iterations. The final model does not, however, give a unique solution, mainly because of the uncertainties in the gravitational moments.

7.1.5.1 Jupiter and Saturn

Most of the models of Jupiter and Saturn assume that the interior consists of three regions: molecular hydrogen (a zone depleted in helium, see 7.1.4.2), metallic hydrogen (a zone enriched in helium), and a central core. Figure 7.3 shows an example of the density profile as a function of the normalized radius for Jupiter and Saturn (Marley, 1999). In this particular case, Jupiter's central core is approximately 5-10 Earth masses, while that of Saturn is significantly smaller. Other models (Guillot, 2006) imply that Saturn has a core with a mass comparable with that of Jupiter. With Marley's model (Fig. 7.3), the transition from the core to metallic hydrogen occurs at 39Mbar; that for Saturn is at 13Mbar and 11 900 K.

Fig. 7.3 Examples of models of the internal structure of Jupiter and Saturn. The inner core consists of two parts, a rocky inner component surrounded by an outer component, rich in ices. In both cases, the radius of the overall central core is about one tenth of the planetary radius. The transition between metallic hydrogen and molecular hydrogen occurs at 85 per cent of the radius for Jupiter and 60 per cent of the radius for Saturn (After Marley, 1999)

Fig. 7.3 Examples of models of the internal structure of Jupiter and Saturn. The inner core consists of two parts, a rocky inner component surrounded by an outer component, rich in ices. In both cases, the radius of the overall central core is about one tenth of the planetary radius. The transition between metallic hydrogen and molecular hydrogen occurs at 85 per cent of the radius for Jupiter and 60 per cent of the radius for Saturn (After Marley, 1999)

7.1.5.2 Uranus and Neptune

The models assume a three-layer structure: rocky central core, a layer of ices, and an envelope of molecular hydrogen and helium.

Uranus and Neptune have comparable masses and sizes, so one might expect that their internal structure would be similar. However, the measurements of the gravitational moments obtained by Voyager 2 have shown that the internal structure of the two planets is different, with heavy elements less concentrated towards the centre in Neptune (Fig. 7.4). In both cases, the mass of hydrogen and helium is just a few Earth masses, compared with about 300 and 80 Earth masses for Jupiter and Saturn, respectively.

In the case of Uranus, it is not possible to interpret the measurements of the gravitational moments by a model with three homogeneous layers, which suggests that a significant fraction of the planet's interior is not mixed in a homogeneous manner. This peculiarity may explain why the internal energy of Uranus is very weak, because the process whereby heat would escape by convection would be inhibited (Guillot, 2006).

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