Apsidal corotation

They are such that R and ai are constant (except for the short period terms eliminated by the averaging and for contributions of higher orders). Constant R s mean semi-major axes and eccentricities constant in these solutions o and 02 constant mean that Aw = ai — <72 is constant - that is, the periapses are moving with same velocities so that their mutual separation do not vary. This frozen relative state in resonant systems is known as Apsidal corotation. 

The above equations were used to find Apsidal corotation solutions in the case of planets in 2/1 and 3/1 mean-motion resonances. The relationship between eccentricities and mass ratios in some of these solutions is shown in Figure 8. The top panels correspond to symmetric solutions. In the left-hand side panel, the periapses are anti-aligned. This is the... 

Figures 10 show the variation of the eccentricities, critical angles ai = 2A2 — i wi and Aw in the same time interval as the previous figures. They show that, after capture, the two critical angles become trapped in the neighborhood of 0 and 7r, respectively and, consequently, the angle Aw is trapped in the neighborhood of 7r. The capture into a symmetric Apsidal corotation with anti-aligned periapses is thus simultaneous with the capture into the resonance.

Figure 10 also shows that, after some time, 02 jumps from 7r to 0 and the Apsidal corotation becomes one with aligned periapses. This change is not the result of a discontinuous process. The left-hand side plot shows that the change happens when the eccentricity e2 is zero. Thus, we may describe the process by a momentary circularization of the orbit such that, when it becomes an ellipse again, the periapses is not... 

The eccentricity ei increases monotonically. When it reaches 0.46, the asymmetric Apsidal corotation changes back to a symmetric configuration with aligned periapses. 

The contents of this paper include, with variable emphasis, the topics of a series of lectures whose main title was Routes to Order Capture into resonance . This was indeed the subject of the last section above. The study of this subject has, however, shown that - unlike the restricted three-body problem - capture into resonance drives the system immediately to stationary solutions known as Apsidal corotations . The whole theory of these solutions was also included in the paper from the beginning - that is, from the formulation of the Hamiltonian equations of the planetary motions and the expansion of the disturbing function in the high-eccentricity planetary three-body problem. The secular theory of non-resonant systems was also given. Motions with aligned or anti-aligned periapses, resonant or not, resulting from non-conservative processes (tidal interactions with the disc) in the early phases of the life of the system, seem to be frequent in extra-solar planetary systems. 

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