## O

Figure 4.12 Tides (greatly exaggerated) in the Earth due to the Moon (a) at some instant and (b) about 6 hours later, with the orbital motion of the Moon exaggerated five times. P is fixed to the Earth's surface.

46% of that due to the Moon. Overall, the rate at which heat is being generated by tides in the Earth today is about 30 times less than the rate at which heat is being released from long-lived radioactive isotopes. By contrast, in Io, the innermost large satellite of Jupiter, tidal energy is by far the main internal energy source.

For any two bodies, with masses M and m, the power Wtidal of tidal heating of either of them is proportional to various properties of the two bodies and their orbit, including, among other properties, those in the following expression:

where R is the radius of the body whose tidal heating is being evaluated, e is the eccentricity of their orbits, and a is the semimajor axis of the orbit of one with respect to the other. Note how rapidly tidal heating decreases as a increases.

The ultimate source of tidal energy is the rotational energy of the bodies concerned, and the energy in their orbital motions.

□ Is it then possible for tidal heating to be associated with changes in rotation rates, and changes in orbital motion? This is not only possible, but must be the case - the internal energy gained through tides must equal a loss of kinetic energy in rotation or orbital motion.

### Solar radiation

A small planetary body might for a long time have had no significant energy source except for solar radiation. In this case its interior will be at a fairly uniform temperature determined by a steady state in which the rate at which the body absorbs solar radiation is equal to the rate at which it emits radiation to space. Departures from uniformity will arise from diurnal and seasonal variations in insolation, coupled with thermal inertia. The mean surface temperature will equal

the interior value. At a given distance from the Sun the exact value of the temperature will depend on the fraction of the incident solar radiation that the body absorbs, and on the efficiency with which it emits its own radiation. For typical planetary materials at 1AU from the Sun, the value will be about 270 K. Most planetary bodies have far higher interior temperatures, so solar radiation is not an important energy source in such cases.

High interior temperatures do not necessarily mean that the rate of energy input from sources other than solar radiation greatly exceeds the solar input. A modest source confined to a small core would give high core temperatures and a temperature gradient to the surface. If the surface layer is a good thermal insulator, the interior can be hot throughout much of its volume. Temperature gradients are themselves important, because they can drive processes such as convection (Section 4.5.2).

### Question 4.8

Draw graphs showing qualitatively how the rate of energy input to a planetary body from the various energy sources outlined above might have varied during the 4600 Ma of Solar System history.

### Question 4.9

(a) Use equation (4.6) to show that the difference in the magnitude of the gravitational field across a body of radius R due to a body with a mass M a distance r away is approximately 4GMR/r3 if r is much greater than R. (It is the difference along the direction to M that is required.)

(b) Hence show that, for the Earth, the solar tidal field is 46% of the lunar tidal field. 4.5.2 Energy Losses and Transfers

As well as gaining internal energy from various energy sources, planetary bodies also lose internal energy. In the near vacuum of interplanetary space, energy is not conducted or convected away from the body, and so it is by emitting radiation that the internal energy is lost. For an atmosphereless body this radiation is from the surface. If there is a substantial atmosphere then a proportion, perhaps all, of the radiation is from the atmosphere. Overall, the radiation is somewhat like that from an ideal thermal source (Section 1.1.1), and so the wavelengths depend on the temperature of the source, as indicated in Figure 4.13. For planetary bodies the radiation is predominantly at IR wavelengths.

In order to be radiated away, the internal energy must first reach the surface or atmosphere, and there are several ways in which this happens, as follows.

### Radiative transfer

This process was outlined in Section 1.1.3, and can be thought of as an outward diffusion of photons. The rate of energy transfer increases very rapidly as temperature increases, and as the opacity of the interior decreases. Planetary interiors are too cool and too opaque for radiative transfer to be important, except perhaps in certain zones deep in the interiors of Jupiter and Saturn.

Ultraviolet Visible

Infrared (middle)

(far) Microwaves

Ultraviolet Visible

Infrared (middle)

(far) Microwaves

10 102 Wavelength/

Figure 4.13 The radiation spectra of ideal thermal sources at various temperatures. Note that the lower temperature spectra have been scaled upwards to a peak radiant power of 1.

10 102 Wavelength/

Figure 4.13 The radiation spectra of ideal thermal sources at various temperatures. Note that the lower temperature spectra have been scaled upwards to a peak radiant power of 1.

### Thermal conduction

Thermal conduction is a process by which heat is transferred through the direct contact of two regions that are at different temperatures. The more energetic particles in the higher temperature region lose some of their random energy of motion to the less energetic particles in the lower temperature region, just as a fast-moving ball transfers some of its energy of motion to a slower moving ball with which it collides. There is thus a flow of random motion - of heat - from the higher to the lower temperature region. This flow will cool the region of higher temperature unless there is energy generation within it sufficient to offset (or exceed) the heat loss.

Clearly, conduction must play a role in transporting internal energy from the interior of a planetary body to its exterior. Whether it is significant depends on the effectiveness of the two remaining mechanisms, convection and advection.

### Convection

Convection differs from conduction in that energy is transported by bulk flows, rather than on an atomic scale. It is a process of energy transfer in a substance heated from below, and requires that the substance can flow. The temperature of the heated substance increases, and it therefore expands, becomes buoyant, and ascends, displacing the overlying cooler material downwards, where it in turn is heated. The rising material loses energy to its surroundings, loses its buoyancy, and descends. This sets up the cycle exemplified in Figure 4.14 by the everyday case of heating a saucepan of liquid, where two so-called convection cells are shown in cross-section.

To obtain a closer look at convection in general, regardless of whether it is in a saucepan, or in the atmosphere or interior of a planetary body, imagine a small parcel of material being swapped with a parcel immediately above it. The hydrostatic equation (equation (4.11)) shows that the raised parcel is now in a lower pressure environment. It quickly responds by expanding until its pressure is the same as the pressure at its new level. This expansion alone causes a decrease in the temperature of the parcel - no heat has had time to flow out of it, and the

Convection cells

Convection cells

Figure 4.14 Convection in a pan of porridge, showing two convection cells.

Figure 4.14 Convection in a pan of porridge, showing two convection cells.

decrease is solely a consequence of the expansion. The lower parcel will be compressed, and this will raise its temperature, again with no flow of heat into it. If no heat has flowed into or out of a parcel, then it has undergone what is called an adiabatic process. Note that an adiabatic process does not rule out the transfer of energy, only of heat. Thus, energy is transferred through the expansions and compressions that take place. Real processes are not strictly adiabatic - there will be some heat transfer by conduction and by radiation. Nevertheless, an adiabatic process is a very useful idealisation.

The crucial question is whether the parcel's new temperature is greater or less than that of its new surroundings. Let us focus on a parcel that moves upwards. If its new temperature is greater than that of its surroundings, then, the pressures being the same, its density must be lower than its surroundings. It is therefore buoyant, so it will continue to move upwards, and convection will start.

□ What will happen if its temperature is lower than its surroundings,?

Its density will then be higher than its surroundings, and it will sink back to its point of origin -

there is no convection. Corresponding conclusions apply to a parcel that moves downwards.

Whether the temperature of a raised parcel is less than that of its surroundings depends on how rapidly the temperature of the surroundings decreases with increasing distance from the centre - this is the temperature gradient. The critical value is the adiabatic gradient. This is shown in Figure 4.15 along with two other gradients, A and B. If the actual temperature gradient is greater than the adiabatic value, as in case B in Figure 4.15 (see figure caption), then a rising parcel, which approximately follows the adiabatic gradient, is always hotter than its surroundings and it will continue to rise. Thus, convection occurs. If the actual gradient is less than the adiabatic value, as in case A in Figure 4.15, then a rising parcel is always cooler than its surroundings and it will sink back, and convection will not occur. The value of the adiabatic gradient depends on the gravitational field: the greater the field, the greater the gradient. It also depends on various thermodynamic properties of the material of the parcel, though the details will not concern us here.

If a substance can flow there is a strong tendency for the actual temperature gradient to become equal to the adiabatic value. If the actual gradient is greater (B in Figure 4.15), energy is transported at such a high rate that the deeper levels cool, thus reducing the gradient towards the adiabatic value. If the gradient is less than the adiabatic value (A in Figure 4.15) then energy is transported by conduction, and this is at such a low rate that the deeper levels get hotter, thus increasing the gradient, until convection starts.

The existence of convective currents in planetary bodies does not make the hydrostatic equation inapplicable, even though this equation requires that the gravitational force on a shell is equal to the pressure gradient force, in which case there can be no convection. The convection currents are quite slow, and therefore the imbalance between the two forces is relatively slight.

Temperature->

Figure 4.15 The adiabatic temperature gradient and two possible actual gradients. Note that B has the greatest gradient, because in this context the gradient is the rate of decrease of temperature with increasing distance from the centre.

Temperature->

Figure 4.15 The adiabatic temperature gradient and two possible actual gradients. Note that B has the greatest gradient, because in this context the gradient is the rate of decrease of temperature with increasing distance from the centre.

Moreover, averaged over large areas, the rising and sinking motions tend to cancel out, and so the hydrostatic equation applies as a global average.

### Convection in solids

It is natural to think of convection as being confined to fluids, and indeed convection does occur in many of the fluid regions of planetary bodies. But it also occurs in some regions that are solid! Solid state convection can occur if a solid behaves like a fluid, and this will be the case if the pressure exceeds the yield strength of the solid. □ In what sort of planetary bodies are the internal pressures high?

Equation (4.12) shows that high internal pressures will be found in large planetary bodies. Fluid behaviour is thus expected in a large planetary body, particularly if the yield strength is reduced by raised temperatures. The rate of convection will depend on the composition and on the buoyancy. Though the convective motions are extremely slow, e.g. of the order of 0.1 m per year in the Earth, the associated rate of outward heat transport can exceed that of conduction. In the case of the Earth, convection is particularly well developed and is the dominant process of energy loss from the interior to the crust. Note that solid state convection is far too slow for it to give rise to magnetic field generation.

There are three so-called styles of convection in rocky or icy-rocky planetary bodies. 'Classical' convection as in Figure 4.14 occurs, for example, in the Earth's mantle and crust in the form of plate tectonics (Section 8.1.2). Stagnant lid convection is where the lithosphere (the crust and the upper mantle) is too stiff to yield, and so convection occurs only at depth. This can result in a rise in internal temperatures to the point where a catastrophic disruption of the surface occurs, releasing heat, followed by restoration of the stagnant lid. This quasi-cyclic process might be occurring in Venus (Section 8.2.7). Finally, there is lithospheric delamination. The lithosphere loses material from its base, and this sinks into the more yielding lower mantle (Section 5.1), its downward motion promoting convection. As well as occurring alone, it can supplement classical convection and stagnant lid convection, as you will see in Chapter 8.

### Advection

If the uppermost part of the interior of a planetary body is solid it will be at too low a pressure and temperature to convect. But this does not leave conduction as the only mechanism of outward energy transfer. Advection is the process of upward energy flow carried by local regions of upward-moving liquids, such as molten rock, or water. If these erupt onto the surface we have volcanic effusions of various sorts. But the liquids need not reach the surface. For example, if molten rock solidifies at some depth it has still contributed to the outward flow of energy.

Table 4.5 indicates the relative importance of advection, convection, and conduction for a few planetary bodies.

### Heat transfer coefficient

Regardless of the energy transfer process, the intrinsic property of the material that determines the rate is encapsulated in a heat transfer coefficient. This is the rate at which heat crosses unit area of a material, per unit thickness, per unit temperature difference across the thickness. For many planetary and smaller bodies the heat transfer coefficients are comparatively low at the surface and in any atmosphere, and so these regions can have a significant effect on the overall rate of energy loss.

### Question 4.10

Suppose that a rocky planetary body has a weak energy source concentrated at its centre. The interior temperatures throughout are modest, decreasing with increasing radius, almost the entire decrease being across a thin surface layer.

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