rotation-period of the primary becomes exactly the same as the orbital-period of the satellite. When this condition is attained, the tides can no longer retard the rotation-period of the planet. So far, therefore, as the inner moon of Mars is concerned, it must long ago have ceased to retard the rotation of the primary. For, the orbital-period of this satellite being far shorter than the present rotation-period of Mars, its tidal action would tend to accelerate instead of retarding the time of rotation of the planet. So far as the outer moon is concerned, it is evident that its tidal action must tend to retard the rotation-period of Mars; but, in consequence of its greater remoteness, the magnitude of its influence must be small compared with that of the inner satellite. It is, therefore, difficult to conceive how the tidal influences of the moons of this planet can explain the anomalous fact that its rotation-period is longer than the orbital-period of one of its satellites.
In connection with the idea of the rotation-period of Mars having, at some former time, been much shorter than it is at present, it may be noticed that the great compression or ellipticity of this planet is totally inconsistent with its observed rotation-period.[1]
In 1784 Sir William Herschel estimated the ellipticity of Mars at 1⁄16. Schröter refused to admit this result; he contended that, if the ellipticity existed, it would not exceed 1⁄80. Bessel failed to discover any appreciable ellipticity of Mars, even with the celebrated heliometer of Königsberg, On the other hand, Arago's measurements, executed at the Observatory of Paris, from 1811 down to 1847, all confirm the existence of an ellipticity in this planet of about 1⁄30. ("Astronomic Populaire," tome iv., p. 130. Paris, 1867.) More recent observations give somewhat contradictory results. Professor Kaiser, of Leyden, makes the ellipticity 1⁄114 Main, of the Radcliffe Observatory, deduced 1⁄39; and Dawes's measurements give negative results.
To show the discordance of these results with what may be deduced from the theory of gravitation, it must be recollected that the ellipticity of a rotating planet depends upon the ratio of the centrifugal force at its equator to the force of gravity at the same place. Thus, to compare the earth and Mars—
| Let | r | and | r' | = | equatorial radii | of | earth | and | Mars | respectively. |
| " | t | " | t’ | = | time of rotation | " | " | " | " | " |
| " | Q | " | Q’ | = | mass | " | " | " | " | " |
| " | f | " | f’ | = | centrifugal force at equator | " | " | |||
| " | g | " | g’ | = | force of gravity | " | " | " | " | " |
- ↑ The oblateness or compression or ellipticity of an oblate spheroid is the difference of its equatorial and polar radii, divided by its equatorial radius. Thus, if a and b are the equatorial and polar radii respectively, then ellipticity =
a b a
