Possible histories of the obliquity of Mercury

The possible evolutionary paths of the spin axis of Mercury are determined as a function of the permanent deformation and initial condition& Transition into and subsequent evolution within the (3/2) spin reso nance are also considered. From this analysis, the evolution of the Moon and satellites of the major planets is also inferred. The situation is complicated by the orbital precession, which leads to three stable positions of the spin axis that remain fixed in the plane defined by the orbit normal and the orbital precessional an- gular velocity (Cassini states). The positions of these states in this plane are functions of several param- eters and are numbered as follows: State 1 is always located on the opposite side of the orbital angular momentum K from the vector - , state 2 is on the opposite side of - from K and lies between - and -K and state 3 is always near -K. The tides always select one or two of these Cassini states, where the selection is determined by the ratio of the precessional angular velocity of the spin in a fixed orbit to the precessional angular velocity of the orbit (fl0 / ). If 1, state 1 near the orbit normal is virtually the only possible endpoint of evolution from any initial obliquity. This situation applies to Mercury, which apparently occupies state 1 very close to the orbit normal. There is a chance of probability that for this case Mercury could have been trapped in state 2 near an obliquity of 90 . If / < 1, state 2, which is now near - , is the only possible endpoint from an arbitrary initial obliquity. The Moon, which we know to be in state 2, is the important example of this extreme. If / 1, the choice between states 1 and 2 is determined by initial conditions or the conditions at the time of capture into a spin resonance. The major satellites of Jupiter may be an application. The tides always remove a body from state 3 near an obliquity of 1 s0 , although an atmospheric thermal tide or a core-mantle interaction could select and pre- serve state 3. If the orbital parameters vary, the body will remain very close to the instantaneous position of a Cassini state if the precession of the spin about this state is rapid compared with the orbit variations. Mercury, in its present position at state 1, satisfies this condition of adiabatic invariance and we expect to fmd Mercury very close to the instantaneous position of this state. The opposite extreme applies to the Mdon orbit variations are fast compared to the spin precession. The Moon remains very close to the average position of the Cassini state. Substantial fluctuations about both the average and instantaneoiis positions of a Cassini state follow when / 1.