4. The chaotic obliquity of Venus

In figure 5, the frequency of precession of the axis of rotation of Venus is traced according to its obliquity. The trajectory corresponds to an initial obliquity of 1 degree and initial period of 3 days. This corresponds to a precession frequency of approximately 16 seconds of degrees per annum (81 000 years period). The precession is then in resonance with the oscillations of the plane of the orbit of the planet resulting from the gravitational perturbations from the other planets. These perturbations produce a very broad zone (in grey in the figure) in which the movement of the axis of planet will be chaotic, and can undergo very strong oscillations that may bring the obliquity beyond 70 degrees (the Earth would currently be in this situation in the absence of the Moon) (Laskar and Robutel, 1993).

Once that the obliquity reaches a large value, the dissipative effects of tidal and core-mantle friction can make the planet turn over, and bring it in the retrograde state Fp-. Because of the existence of this chaotic zone, which is crossed by the axis of planet during its history, all the initial conditions can lead to one of the four final states (F0+, Fp+, F0-, Fp -).

The unusual Venus rotation thus does not require the assumption of a strong impact at the end of the formation of the Solar system which would have made the planet turn over, as it was sometimes proposed. In fact, by taking into account only dissipative effects of tidal and core-mantle friction, a large part of the acceptable initial conditions lead Venus to the current state, whatever its initial obliquity, but they are two possible scenarios leading at the same final apparent state.
- In the first case, the planet slows down, stops, and starts again in the other direction, whereas its obliquity tends towards 0°.
- In the second scenario, the planet is turned over and finishes with an obliquity of 180°.

Figure 5: Possible example of evolution of the Venus obliquity during its history. The frequency of precession (in seconds of arc per annum) is traced according to the obliquity of the axis (in degrees). The initial obliquity is one degree, and initial period of three. The frequency of initial precession is 16 seconds of degrees per annum, but because of dissipative effects of tidal and core mantle friction, the planet slows down and the frequency of precession decreases. The obliquity enters a very important chaotic zone then (in grey), resulting from the planetary perturbations. The obliquity can then strongly increase, until the dissipative effects lead it away from the chaotic zone, for a strong value of the obliquity. The various dissipative effects can then bring the obliquity towards 180 degrees (Correia and Laskar, 2003).

We thus end to a form of paradox where the retrograde rotation of Venus is the most probable result of a natural evolution, and not the result (always possible) of a strong accidental collision; but in same time, for a large part of the prograde initial conditions, it is possible to arrive in this state by two entirely different ways, that are not possible to distinguish with the observation of the present final state of the planet (Correia and Laskar, 2001).