The main application of the second fundamental model of resonance is the study of resonant encounters that may occur in the course of the orbital evolution of two celestial bodies. As seen above, orbital evolution takes place in the case of migrating planets in the protoplanetary disk. The outcome of a resonant encounter can either be resonant capture or passing through a resonance without capture.

The Hamiltonian (5.9) depends only on the parameter 5. The action J is defined for a given value of 5 as

The variable J is the area enclosed by the phase trajectory on the (0 — O) or (x — y) planes. If, during one period of the phase trajectory, the variation of 5 is negligible, J can be regarded as constant, and it is called the adiabatic invariant of the system. Then the area enclosed by the phase trajectory (J) is preserved, until it encounters the separatrix of the resonance. If the trajectory encounters the separatrix, the period of the trajectory tends to infinity, therefore the variation of 5 is no longer negligible, meaning that J will no longer be an adiabatic invariant of the system. Thus when an orbit passes through a separatrix, the value of J changes suddenly. During its further evolution the new value of J is preserved.

In our case, the process of the resonant encounter is achieved through the migration of an outer planet towards the inner one when their mutual orbits are close to a mean-motion commensurability. Initially, the corresponding phase trajectory (of the inner planet) encloses a certain area. During the inward motion of the outer planet the shape of the trajectory is changing slowly (adiabatically) caused by the variation of 5, but the area enclosed by it is preserved. When the separatrix of the resonance gets close to the actual trajectory, a resonant encounter occurs, the outcome of which depends on the momentary conditions.

Since \/20 « e, there is a critical eccentricity ec corresponding to the critical separatrix (at 5 = 0). Figure 5.6 shows the case when e < ec. The shaded areas enclosed by the trajectories, corresponding to different 5, are the same. In Fig. 5.6c the critical separatrix (5 = 0) appears. Since e < ec, the area enclosed by the trajectory is smaller than the area enclosed by the critical separatrix, thus the orbit lies inside the separatrix, which means libration. Thus the orbit is captured into the 2:1 resonance. Then, by increasing 5, the orbit evolves according to Fig. 5.6 (d,e); its eccentricity increases, while the libration amplitude decreases, exactly as seen in the hydrodynamical simulations shown above.

The critical eccentricity can be approximated as [13]

which in the case of two Jupiter-mass planets and a solar-mass star is ec & 0.112. Due to the eccentricity damping mechanism acting in protoplanetary disks, the eccentricities of the embedded giant planets are very small, therefore the condition einner < ec is typically satisfied when an inner giant is captured by an outer one migrating inwards.

If the initial eccentricity of the orbit is larger than the critical eccentricity, e > ec, the result of the resonant encounter cannot be so easily forecasted. The area enclosed by the initial trajectory is larger than the area enclosed by the critical separatrix. By increasing S, there exists a separatrix, which encloses an area equal to the area enclosed by the trajectory. On the other hand, if the trajectory encounters a separatrix, the action J changes suddenly. Either J changes to the area enclosed between the two branches of the separatrix, or changes to the area enclosed by the inner part of the separatrix. In the first case the orbit is captured into the 2:1 resonance and its eccentricity does not change, in the second case the trajectory passes through the resonance without being captured, while its eccentricity decreases suddenly. The more detailed treatment of the resonant capture, also including the cases of higher order resonances and capture probability estimates for e > ec, can be found in [13].

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