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This absence of a surface makes the Jovian radius a matter of definition. It is defined as the radius at which the atmospheric pressure is 105Pa (standard atmospheric pressure at the Earth's surface is 1.01 x 105Pa). The same definition is adopted for all the giant planets. In the case of Jupiter, 105 Pa is not far below the top of the upper cloud deck.

We do, however, encounter a surface of sorts deeper down, albeit with only about a 10% increase in density across it. This is the transition between liquid molecular hydrogen (plus atomic helium) and liquid metallic hydrogen (plus atomic helium). At pressures around 2 x 1011 Pa, the density of molecular hydrogen is about 800 kg m-3, sufficiently high that each of the hydrogen atoms in the H2 molecule is attracted to atoms in neighbouring molecules as much as to its molecular partner. Therefore, the H2 molecules break up. Moreover, the single electron that orbits the nucleus of each hydrogen atom will also become equally attracted to neighbouring atoms, and so the atoms break up too. The hydrogen will then be a 'gas' of electrons moving in a 'sea' of hydrogen nuclei. Many of the characteristic properties of metals arise from the existence of 'electron gases' within them, so the term 'metallic hydrogen' is appropriate. Metallic hydrogen was once just a theoretical prediction, but then in 1996 sufficiently high pressures were produced in the laboratory for metallic hydrogen to appear as expected. In Jupiter the transition to the metallic form takes place over a range of pressures, and divides the interior into a metallic hydrogen mantle and an overlying molecular hydrogen envelope. The range of depths over which the transition takes place is unknown. Figure 5.9 shows the top of the transition. Beneath, there might be a narrow layer of inhomogeneous composition.

One of the properties that an electron gas gives to a metal is high electrical conductivity. Thus, electric currents in the metallic hydrogen mantle are an obvious source of the planet's large magnetic dipole moment. The volume of metallic hydrogen, the hot convective interior, and the rapid spin of Jupiter are all consistent with this conclusion. A small but significant contribution to the field could arise from liquid iron and other conductors much deeper down. The detailed configuration of the field close to Jupiter is consistent with currents in the metallic hydrogen mantle.

Gravitational data indicate an increase in density towards the centre, but it is possible that this is due to self-compression in the metallic hydrogen mantle. Models consistent with the data have cores of rocky and icy materials ranging from 0 to 10mE. To satisfy the constraints, there is a complementary range from 42mE down to little more than 10mE in the amount of rocky and icy materials in the whole planet. This broad range of values results from various factors, including uncertainties in the equations of state of hydrogen and helium, the insensitivity of the gravitational coefficients to deep structures, the uncertainty in the width of the transition zone between molecular and metallic hydrogen, and so on. If the icy-rocky core really has a low mass, this could be due to erosion of the core by the high temperatures at the base of the mantle, which is not much less than the central temperature given in Table 5.4. □ If icy and rocky materials account for 20mE in the whole of Jupiter, what is the heavy element mass fraction?

Almost all the mass of icy and rocky materials consists of elements other than hydrogen and helium, i.e. the heavy elements. Jupiter's mass is 317.8mE, and so the fraction is 6.3%. This is nearly four times the solar mass fraction of about 1.6%. The modelled enrichment of Jupiter by a factor of a few is consistent with (but not the same as) the measured enrichment of the atmosphere (Section 11.1.2). The enrichment is thought to be primordial, from a kernel composed of rocky-icy materials, with a contribution from the subsequent capture of planetesimals. In the gravitational instability model of formation, the planetesimal capture rate after the formation of Jupiter would probably have been too low to give such a high heavy element fraction.

Thermal models show that the present-day high temperatures of the Jovian interior can be accounted for by two dominant energy sources - energy from accretion, plus energy from differentiation when the icy-rocky kernel acquired more mass to become the icy-rocky core. An active but small energy source is the settling of helium through the metallic hydrogen mantle. The two major energy sources became inactive about 4500 Ma ago, yet the central temperature of Jupiter is still about 2 x 104K. The reason is the large size of Jupiter. This has two consequences. First, there would have been a huge amount of accretional energy per unit mass as Jupiter formed, and so Jupiter would have become extremely hot (Section 4.5.1). Second, Jupiter has a comparatively low ratio of surface area to mass, giving a low rate of internal energy loss (Section 4.5.4) in spite of the efficient outward transfer of energy by convection.

Though convection is expected to occur throughout most of the Jovian interior, calculations also indicate that over the depth range in which the temperatures are 1200-3000 K, energy might be transported outwards by radiation rather than by convection. This would lead to cooler interiors than in Table 5.4, and to a deeper transition to the metallic hydrogen phase than in Figure 5.9. This is also the case for Saturn.

Saturn

In many ways, the interior of Saturn is similar to Jupiter, as Figure 5.9 shows. Saturn is a bit smaller, and less dense, leading to lower internal pressures. These lower pressures mean that the transition to metallic hydrogen occurs much nearer the centre, so whereas most of the hydrogen in Jupiter is in metallic form, most of that in Saturn is in molecular form. As with Jupiter, it is the top of the transition that is shown, should this occur over a (thin) shell. As in Jupiter, it is the metallic hydrogen in models of Saturn that can account for the planet's magnetic field. With the two planets rotating at about the same rate, and with comparable internal activity, the lower mass of metallic hydrogen in Saturn is predicted to lead to a smaller magnetic dipole moment. □ Is this the case?

Table 4.2 shows that the magnetic dipole moment of Saturn is about 30 times less than that of Jupiter.

The smaller size of Saturn also means that its internal temperatures after formation would have been lower than in Jupiter, and it should have cooled more rapidly. Therefore, the present-day high internal temperatures, indicated by the IR excess, cannot be accounted for solely by energy of accretion and by past differentiation as the icy-rocky kernel acquired more mass to become the icy-rocky core. An additional source of energy is needed, and this is thought to be the ongoing separation of helium from metallic hydrogen (which plays at most a small role in Jupiter). Initially, the helium in the metallic hydrogen mantle was thoroughly mixed at the atomic level, and random thermal motion prevented any settling of the helium atoms downwards - a tendency arising from the greater mass of the helium atom. As the interior cooled, the miscibility of helium in metallic hydrogen fell, and Saturn is estimated to have reached the point about 2000 Ma ago where helium began to form small liquid droplets. These could not be held by random thermal motions in uniform concentration throughout the metallic hydrogen, so downward separation of the helium droplets began. This is essentially the same process as the separation of oil from vinegar in salad dressing. Convection is believed to have slowed the settling rate, but an outer core of helium is forming, and as it does so energy of differentiation is released. An additional source of energy might be continuing growth of an icy-rocky core.

The removal of helium from the metallic hydrogen would result in helium diffusing down from the molecular hydrogen envelope, so we would expect the observable atmosphere to be depleted in helium compared with Jupiter. And indeed it is! In Jupiter the outer atmosphere is observed to consist of about 23% helium by mass, whereas for Saturn the value is about 20% helium. The greater extent of downward separation of helium in Saturn is reflected in models by larger increases in the helium mass fractions with depth. Below the molecular hydrogen envelope, which models indicate has a helium mass fraction not very different from the atmospheric value, the same models give the metallic mantle about 30%. Thus, overall, Saturn, like Jupiter, has about the same helium mass fraction as the Sun at its birth.

The lower internal pressures make the equations of state for hydrogen and helium less uncertain than in the case of Jupiter. This exposes strong evidence for a significant rocky-icy core in Saturn, and thus further evidence against the gravitational instability model of formation. It is likely that the core mass is not greater than 10-20mE, depending on the extent to which heavy elements reside outside the core. The core mass could be reduced by several mE if, as is quite possible, helium separation has produced a nearly pure helium shell around the core. The mass of heavy elements in Saturn is roughly the same as that in Jupiter.

Question 5.7

As Jupiter's interior cools, a certain energy source will become increasingly powerful. State what this source is, and why it is triggered by cooling. What effect could this source have on Jupiter's subsequent internal temperatures?

5.3.2 Uranus and Neptune

We have noted that the mean densities of Uranus and Neptune show that they are much less dominated by hydrogen and helium than are Jupiter and Saturn. The equations of state of the icy-rocky materials that dominate the interiors are collectively less well known than those of hydrogen and helium, and the range of possible models is thus larger. All models predict that these planets, like Jupiter and Saturn, are fluid throughout.

Uranus and Neptune are smaller and less massive than Jupiter and Saturn, and the overall composition of the models can be obtained, very roughly, by stripping away a good deal of the hydrogen and helium from Jupiter or Saturn. □ How is this feature explained in core-accretion theories?

The kernels of Uranus and Neptune formed considerably more slowly than those of Jupiter and Saturn, and so there was less time to capture hydrogen and helium before the proto-Sun's T Tauri phase drove away the nebular gas (Section 2.2.5). In typical models of Uranus and Neptune, hydrogen and helium account for 5-15% of the mass of each planet.

The mass ratio of icy to rocky materials, derived from solar elemental abundances, is about 3, which means that for these planets 60-75% by mass is icy and 20-25% rocky. The greater density of Neptune, 1640 kg m-3 versus 1270 kg m-3 for Uranus, in bodies of similar size, indicates that either Neptune has a greater proportion of icy-rocky materials, and/or the rocky proportion of the icy-rocky materials is higher in Neptune. Both possibilities are consistent with models in which Neptune formed closer to the Sun than Uranus did, with subsequent outward migration of both placing Neptune further out (Section 2.2.5). In the icy component, H2O, CH4, and NH3 must be the major ingredients. This mix will be rich in ions at depths where the pressures and temperatures are high enough.

The atmospheres where the composition can be observed consist by mass of about 65% hydrogen and about 23% helium - not very much less than the fractions estimated for the young Sun. As in Jupiter and Saturn, the accessible hydrogen is in molecular form and helium is in atomic form. The rest of the atmosphere is icy gases, enriched above what would be derived from solar abundances by the capture of at least 0.1 mE of planetesimals after the atmosphere was in place. In typical models, such as those in Figure 5.9, the hydrogen-helium-icy composition continues in this envelope down to an icy-rocky mantle, possibly with a rocky (but fluid) inner core. The internal pressures are too low for metallic hydrogen, and so there is no metallic hydrogen mantle.

The IR excess of Neptune indicates a hot, convective interior - convective throughout most of its volume. An interior hot enough to be liquid is also implied by Neptune's large magnetic dipole moment. The detailed configuration of the field indicates that the electric currents are located in a thin outer shell of the icy-rocky mantle. As noted above, mixtures of H2O, CH4, and NH3 can become ionised, and thus highly electrically conducting. The predicted convective interior and the observed rapid rotation of the planet complete the requirements of the dynamo theory. For Neptune to have high internal temperatures today there needs to be an active energy source. This is thought to be differentiation, though uncertainties about the internal distribution of the various icy and rocky materials make the details obscure.

Uranus rotates only slightly slower than Neptune and has nearly double Neptune's magnetic dipole moment. As with Neptune, the detailed configuration of the field indicates that the electric currents are located in a thin outer shell of the icy-rocky mantle.

The IR excess of Uranus is barely detectable, corresponding to a power outflow per unit mass about nine times less than that of Neptune. However, the two planets are so similar in so many ways that it is thought that the internal temperatures are roughly the same, and that some process is greatly reducing the rate at which the energy in Uranus is transported to the upper atmosphere, where it would be radiated away to space. It was mentioned near the beginning of Section 5.3 that the suppression of convection over some range of depths, due to a composition gradient, might be the reason for the low rate of transfer. This could be caused by a significant change in the abundance of heavier molecules over some modest range of depths. In the case of Neptune, any such region is likely to be at a greater depth, and consequently less effective. Such zones are not ruled out by the gravitational data.

Question 5.8

It is believed that radioactive isotopes are only a minor energy source (per unit mass) in the giant planets. Why is this believed to be the case?

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