The tidal braking forces of a primary also apply to satellites. If h2 of a satellite on a planet is greater than 2.0 but that of the primary is less than 2.0, one would expect to find the planet's rotation halted with respect to the satellite but continuing with respect to the primary. The planet's solar day and synodic month would be of the same length. For this condition to be compatible with habitability, however, the period would have to be such as to produce a solar day less than 96 hours in duration—a figure rather arbitrarily chosen as the longest day for habitability.
The relationship between a sidereal month P, the year Y, and the synodic month S, which may be written
is such that to a rough approximation for small values of P relative to Y, S may be taken as equal to P. This simplifies the presentation of general conclusions.
If the tidal braking force due to a satellite, hM2, is greater than 2.0 and the relative rotation of the planet with respect to the satellite has stopped, the tides on the planet due to the satellite will be fixed (not moving across the surface of the planet), and thus they will not be an environmental variable. There may be tides due to the primary, however, and a new limiting condition will appear when the tides produced by the primary reach a destructive level incompatible with land life; that is, if the erosive power of tides becomes excessively high, all the dry land on the planetary surface will disappear, and the planetary surface will become a continuous deep ocean swept twice daily by tides of enormous magnitude. At what tidal magnitude would this occur? The Moon produces on the Earth mid-ocean tides approximately only a foot in height,' yet local coastal tides are much higher because of the piling up of water in shallow bays; it might then be assumed that mid-ocean tides of the order of 10 to 20 feet would probably begin to be of sufficient magnitude to erode away all of the Earth's land masses over a period of many years. For present purposes, let us assume that the destructive tide limit is represented by h equal to 20.
Using this criterion, we can represent, by areas shown in Figure 27, all combinations of the tidal braking force due to the primary, hs, and tidal braking force due to the satellite, hM. In this figure, region 1 would contain freely rotating planets; region 2, planets with rotation halted with respect to a satellite; region 3, planets with rotation halted with respect to a satellite but with destructive tides due to the primary; and region 4, planets with rotation halted with respect to the primary. All habitable planets must fall within regions 1 and 2; and those falling within region 2 would be habitable only if their periods of rotation were less than 96 hours.
For a planet having the characteristics of the Earth, the limitations on satellite mass and distance are shown in Figure 28. The line marked "Roche's limit" marks a region inside which the smaller of the two bodies would tend to break into fragments as a result of the tide-raising forces of the larger.
It is interesting to note that within a certain range of satellite masses, twin habitable planets that did not rotate with respect to each other could exist.
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