The Theory of Bodily Tides . I .

The Gerstenkorn-MacDonald-Kaula theory of bodily tides assumes constancy of the geometric lag angle δ , while the Singer-Mignard theory asserts constancy of the time lag ∆t . Each of these two models is tacitly based on a particular law of the geometric lag’s dependence upon the main frequency χ of the tidal flexure. In the theory of Gerstenkorn (1955), MacDonald (1964), and Kaula (1964) δ ∼ χ0 , while in the theory of Singer (1968) and Mignard (1979, 1980) δ ∼ χ1 . The actual dependence of the lag on the frequency is more complicated and is determined by the rheology of the planet. Each particular functional form of this dependence will unambiguously fix the form of the frequency dependence of the tidal quality factor, Q(χ). Since at present we know the shape of the function Q(χ) , it enables us to reverse our line of reasoning and to single out the appropriate actual dependence δ(χ) . This dependence entails considerable alterations in the theory of tides. Our new model addresses only the land tides, not the ocean ones, and therefore is intended to be a tool for exploring the dynamics of the Martian satellites. It is applicable to the Earth-Moon system only at the aeons preceding the formation of oceans. 1 The Gerstenkorn-MacDonald-Kaula and Singer-Mignard theories of bodily tides If a satellite is located at a planetocentric position ~r, it generates a tidal bulge that either advances or retards, depending on the interrelation between the planetary spin rate ωp and the tangential part of satellite’s velocity ~v divided by r ≡ |~r|. It is convenient to imagine that the bulge emerges beneath a fictitious satellite located at ~rf = ~r + ~f , (1) where the position lag ~f is given by ~f = ∆t ( ~ ωp × ~r − ~v ) , (2)