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Mass relative to Sun's mass

Figure 25. Some characteristics of the smaller main-sequence stars.

The tide-generating potential of an astronomical body may be shown to be proportional to

MARji2

and the maximum height h of equilibrium tides in deep water on the surface of an Earth-like planet may be shown to be proportional to maRb1

where MA is the mass of the disturbing body, MB is the mass of the disturbed body, RB is the radius of the disturbed body, and r is the distance between the two bodies (Webster, 1925). If we let the maximum height of tides in deep water on the Earth due to the Moon equal unity when dimensions are expressed in feet, then h = 0.85 MjR,?! M lsr'. Actually, on the Earth, tides in the middle of the ocean are only a foot or so in height above mean sea level, but the fact that there is a finite quantity of water confined in more or less rigid connecting basins with highly irregular rim and bottom shapes results in a piling up of water to heights of many feet on shores and in bays. Also, strong tidal currents are produced along some coastlines and through certain channels and straits, and very complicated flows of water are generated on various parts of the Earth's surface, depending on local topographical relationships.

The height of tides alone would be important in our discussion of habitable planets in general, since all such planets must have more or less extensive bodies of water on their surface. But probably more important are the rotation-retarding effects of tides, a subject still only imperfectly understood, although it has been studied intensively for the past two hundred years. It is now widely accepted that the dissipation of energy by tidal friction in all of the shallow seas of the Earth is quantitatively adequate to account for the observed slowing of the Earth's rotation. Other factors may also be important, such as bodily tides in the Earth and changes in the Earth's moment of inertia due to secular changes within the Earth, changes in the oceans or the sea level, tides in the atmosphere, or interactions between the magnetic fields of Earth and Sun.

Jeffreys developed a theoretical relationship between heights of tides and tidal torques indicating that tidal rotation-retarding torques should be proportional to A2 (Urey, 1952).

From Table 9 which gives calculated values of h and h2 for various pairs of bodies of the solar system where the disturbed body is terrestrial in type, it may be seen that high values of h2 always result in a stopping of the rotation of the disturbed body with respect to the disturbing body. The break point seems to come somewhere between A2 = 1.2 and 2.0, since we know that the Earth's rotation has been retarded somewhat, principally by the action of the Moon, while the rotation of Venus is apparently quite slow, although its period of rotation is not known with any accuracy. It is quite probable that the retardations in the rotations of Mercury and Venus (assuming that Venus has no oceans) have been produced by bodily tides. Although some of the large satellites of the giant planets also produce high values of h2 on their primaries, and may have produced some retardations in their rotations, the tides would be largely atmospheric in these cases and probably not nearly as effective in retarding

Table 9. Tidal Retardation Effects in the Solar System

Effect of body A on Rotation of body B B with re spect to A

Body A Body B h h2

Sun

Mercury

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