## Info

Gerasimenko

a For orbiters and landers a range of dates is given if the encounter spanned more than a month. Payloads are from NASA, except: b from the USSR, c from the European Space Agency, d from Japan.

If the volume of a body is V and its mass is M, then the mean density is given by

For a sphere, V = (4/3)^R3 where R is the radius. Planets and satellites are not perfectly spherical; Saturn, for example, is clearly flattened by its rotation (Plate 16(a)). We can, however, measure the shapes of planetary bodies sufficiently accurately for the actual volumes to be obtained.

There are various ways of measuring the mass M. You have seen in Section 1.4.4 how the mass of the Sun can be obtained from the semimajor axis and period of the orbit of a planet. The same procedure can be applied to get the mass of a planet itself, from the orbit of a satellite or spacecraft around it. If the mass of the satellite or spacecraft is m, then from Newton's laws of motion and gravity we get an equation like equation (1.6) (Section 1.4.5), which can be rearranged as

where a is the semimajor axis of the orbit of m with respect to M, P is the orbital period, and G is the gravitational constant. If m is much less than M, it can be omitted from equation (4.2), and we get the value of M without knowing the value of m.

If m is not negligible in comparison with M, e.g. the Moon in comparison with the Earth, then we can still get M by finding the centre of mass of the system. It is the centre of mass that moves in an elliptical orbit around the Sun, and M and m are each in orbit around the centre of mass, as you saw in Section 1.4.5. The position of the centre of mass can be determined from these orbital motions, and the ratio of M and m is then given by an equation like equation (1.7) (Section 1.4.5)

where rM and rm are the distances of M and m from the centre of mass at any instant. From equations (4.2) and (4.3) we can obtain both masses, and we can then use equation (4.1) to get the mean density of each body in turn.

The two bodies do not have to be in orbit around each other. Though the equations are different, the masses can be obtained from the change in their trajectories as they pass each other. If one body has a much smaller mass than the other, then only the trajectory of the smaller body will change appreciably. This is the case, for example, when a spacecraft passes near a planet.

Table 4.2 lists the mean densities, and other data, for the planets and larger satellites. For a wider range of bodies, Figure 4.1 shows the mean density along with the radius. Radius and density are more indicative of composition than density alone, as you can see if we consider two bodies with the same mean density but with greatly different radii. You might think that, with equal mean densities, the two bodies could have the same composition. This is not so. The internal pressures in the larger and thus more massive body will be much greater than in the less massive body.

□ What effect does this have on the mean densities?

This results in greater compression in the more massive body resulting in a greater density. The hypothetical uncompressed mean density of the more massive body must be lower than that of the other body, and therefore the more massive body must contain a higher proportion of intrinsically lower density substances.

In Figure 4.1 some distinct groups can be recognised. On the basis of size alone the planets can be divided into four giants, four terrestrial planets, and Pluto. With the mean densities added, this broad division is reinforced by a strong indication of compositional differences between the groups. Consider the giants. It is clear that these bodies are so large that the compression is considerable. If the compression were slight their mean densities would be lower than those in Figure 4.1 by a large factor. The mean densities of the other bodies would be much less reduced, and so the already clear distinction between the giants and the rest would sharpen. The giant planets thus form a quite distinct group.

For the remaining bodies in Figure 4.1, the uncompressed mean densities of Venus and the Earth would be about 20% less than the values shown, with smaller reductions for the rest. Therefore, the groupings among the non-giants are not sharply defined. The bodies in the centre right region, namely the four terrestrial planets plus the Moon, Io, and Europa, constitute the terrestrial bodies. The remaining large satellites, namely Ganymede, Callisto, Titan, and Triton, group with Pluto to constitute the icy-rocky bodies, a term that reflects their composition. The other bodies in Figure 4.1 are the two largest asteroids, namely Ceres and Pallas, and nearly all of the intermediate-sized satellites. These satellites have mean densities comparable with those of the icy-rocky bodies, whilst those of the two asteroids are more comparable with the terrestrial bodies, indicating broad compositional affinities in each case.

In Section 3.1.4 you saw that there is a threshold size below which a body could have an irregular shape. Bodies large enough to be approximately spherical are often called planetary bodies.

Table 4.2 Some physical properties of the planets and larger satellites"

Object

Equatorial

Flattening

Mass/ 1020kg

Mean

Sidereal

,d J2

C/MRl

Mag.

Energy

0 0