V. Minimum Radius And Mass For A Planetary Body

It has been seen that a planetary body has a maximum radius for a given composition and so also a maximum mass. A lower limit for the radius, and so for the mass, can also be recognised for a given composition as we now show. The body, of mass M and radius R, is described as a planetary body because the same arguments apply whether it is orbiting the Sun (and so is a planet) or orbiting a planet (and so is a satellite).

The controlling internal influence is the gravity, that is the self-gravity of the mass. The upper mass limit has been seen to arise when the internal gravitational energy first becomes comparable to the energy of the constituent atoms so that they become ionised. At this point the body ceases to obey the rule MR-3 = constant which applies to planetary material. The lower limit is met when the mass is so small that the energy of internal gravity per atom has the same magnitude as the energy of the atomic interaction,e (int). Between these mass limits the material comprising the body behaves as a viscous fluid.

The self-gravity has the energy E ~ GM2 / R: there are N ~ M / A mP atoms in the body giving the gravitational energy per atom eg = (E / N) ~ G M A mP / R. This must be at least comparable to the inter-atomic energy, eI, giving a minimum mass and radius for a given chemical composition. Explicitly

ei ~ eg > eg(min) ~ G M(min) A mP / R(min) .

Then, introducing the mean material density r = 3 M / (12.57dR3)

R(min) ~ [3eI / 12.57GdAmP]1/2

In terms of numbers, eI has a value in the range 10-22- 10-21 J: this range is found to be quite insensitive to the precise atoms involved.

For silicates, A ~ 30, d ~ 3x103 kg m-3, G = 6.67x10-11 and eI ~ 10 -21 J, so that R(min) ~ 3x105m. The corresponding mass is M(Rmin) ~ 3.4x1020 kg. For a largely water-ice composition the radius may be as low as R(min) ~ 1.6x105 m and the corresponding mass as low as M(Rmin) ~ 1019 kg. Down to this size the body will maintain a spherical shape. Such mass-radius combinations are fully consistent with those observed for the smallest spherical planetary bodies, namely the small satellites Mimas (of Saturn: R = 195 km, M = 3.8x1019 kg) and Miranda (of Uranus: R = 235 km, M = 6.89x1019 kg). The larger asteroids also fit these values.

References

G.H.A.Cole, Physics of Planetary Interiors, Adam Hilger, Bristol, 1984

For an application to the study of asteroids see D.H. Hughes and G.H.A.Cole, Mon. Not. Roy. Ast. Soc., vol 277, pp 99-105, 1995

Copyright G.H.A. Cole, 2000, Last Updated 24/02/00

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