Fig. 7.4 Examples of models of the internal structure of Uranus (dotted line) and Neptune (3 models, continuous, chain-dotted and dashed lines). In the case of Neptune, the size of the central core is poorly constrained (After Marley, 1999)
Figure 7.5 shows the typical characteristics of models of the interiors of the giant planets, and their uncertainties. [After Guillot, 2006]
We have seen that the giant planets, except Uranus, have a source of internal energy (see Sect. 7.1.1). The most plausible explanation is the gravitational energy accumulated during the formation of the planet, converted into heat in the interior and released outwards in the cooling phase. During a initial cooling phase, the planets also contracted, liberating an even more significant amount of gravitational energy. In the current phase, when the internal pressure hardly depends on temperature, the cooling takes place with an essentially constant radius. This mechanism is known as Kelvin-Helmholtz cooling (Marley, 1999).
Knowing the energy excess L for each planet, we can obtain an estimate of the planet's duration of cooling, assuming homogeneous cooling over the course of time, by the use of the following equation:
where Te is the actual effective temperature, T0 the effective temperature in the absence of an internal source (and thus linked to the amount of solar energy that is received), and Ti is an average internal temperature. Cv is the mean specific heat per gramme. The radius of the planet, R, is assumed to be constant throughout the contraction, and Ti is assumed to be similar to Te.
The time required to cool from the initial temperature Tei to the current temperature Te is designated T. Because Tei ^ Te, the choice of Tei is not critical in calculating the integral. We obtain the following result:
For the calculation to make sense, it is essential that the value obtained for t should be similar to the age of the Solar System, i.e., 5000 million years.
For Jupiter the result is conclusive: To obtain t = 4.5 x 109 years, we must start with an initial temperature Tei above 340 K, which is a reasonable assumption. For Saturn, in contrast, the value obtained for t is only half, which means that the excess internal energy in Saturn is too great to be explained by Kelvin-Helmholtz cooling. Here, the mechanism by which helium separates out from the metallic hydrogen becomes involved, and this could be responsible for some of Saturn's internal energy. Because Saturn, less massive than Jupiter, is always colder, the condensation of helium occurs earlier in the cooling phase, and the effect is therefore more obvious (see Fig. 7.2).
More elaborate models have been able to simulate the evolution of Jupiter and Saturn, assuming a completely convective transfer and also the existence of a radiative zone (Guillot, 2006). Figure 7.6 illustrates the evolution of the temperature and radius as a function of time for Jupiter.
In the cases of Uranus and Neptune, the problem is the opposite of that with Saturn: The calculated times are greater than the age of the Solar System. One would therefore expect to observe an excess internal energy greater than what actually measured for these two planets. The problem is particularly severe for Uranus, because no internal-energy excess whatsoever has been detected. One possible cause could
1 10 100 1000 104 Time/106 years
1 10 100 1000 104 Time/106 years
Fig. 7.6 Contraction and cooling in Jupiter. The temperature at 1 bar and the effective temperature are shown as a function of time. Tb, T* and R* are the current values of these parameters. The vertical dotted line indicates the age of the Solar System (After Guillot, 2006)
be the inhibition of convection, mentioned earlier, which could prevent energy from being released towards the exterior.
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