An Integrable Model for the Dynamics of Planetary Mean Motion Resonances. (arXiv:1909.05264v1 [astro-ph.EP])

An Integrable Model for the Dynamics of Planetary Mean Motion Resonances. (arXiv:1909.05264v1 [astro-ph.EP])
<a href="http://arxiv.org/find/astro-ph/1/au:+Hadden_S/0/1/0/all/0/1">Sam Hadden</a>

I consider the dynamics of mean motion resonances between pairs of co-planar
planets and derive a new integrable Hamiltonian model for planets’ resonant
motion. The new model generalizes previously-derived integrable Hamiltonians
for first-order resonances to treat higher-order resonances by exploiting a
surprising near-symmetry of the full, non-integrable Hamiltonians of
higher-order resonances. Whereas past works have frequently relied on truncated
disturbing function expansions to derive integrable approximations to resonant
motion, I show that no such expansion is necessary, thus enabling the new model
to accurately capture the dynamics of both first- and higher-order resonances
for eccentricities up to orbit-crossing. I demonstrate that predictions of the
new integrable model agree well with numerical integrations of resonant planet
pairs. Finally, I explore the secular evolution of resonant planets’
eccentricities. I show that the secular dynamics are governed by conservation
of an AMD-like quantity.

I also demonstrate that secular frequencies depend on planets’ resonant
libration amplitude and this generally gives rise to a secular resonance inside
the mean motion resonance at large libration amplitudes. Outside of the secular
resonance the long-term dynamics are characterized small adiabatic modulations
of the resonant motion while inside the secular resonance planets can
experience large variations of the resonant trajectory over secular timescales.
The integrable model derived in this work can serve as a framework for
analyzing the dynamics of planetary MMRs in a wide variety of contexts.

I consider the dynamics of mean motion resonances between pairs of co-planar
planets and derive a new integrable Hamiltonian model for planets’ resonant
motion. The new model generalizes previously-derived integrable Hamiltonians
for first-order resonances to treat higher-order resonances by exploiting a
surprising near-symmetry of the full, non-integrable Hamiltonians of
higher-order resonances. Whereas past works have frequently relied on truncated
disturbing function expansions to derive integrable approximations to resonant
motion, I show that no such expansion is necessary, thus enabling the new model
to accurately capture the dynamics of both first- and higher-order resonances
for eccentricities up to orbit-crossing. I demonstrate that predictions of the
new integrable model agree well with numerical integrations of resonant planet
pairs. Finally, I explore the secular evolution of resonant planets’
eccentricities. I show that the secular dynamics are governed by conservation
of an AMD-like quantity.

I also demonstrate that secular frequencies depend on planets’ resonant
libration amplitude and this generally gives rise to a secular resonance inside
the mean motion resonance at large libration amplitudes. Outside of the secular
resonance the long-term dynamics are characterized small adiabatic modulations
of the resonant motion while inside the secular resonance planets can
experience large variations of the resonant trajectory over secular timescales.
The integrable model derived in this work can serve as a framework for
analyzing the dynamics of planetary MMRs in a wide variety of contexts.

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