An asteroid orbits at the 2:1 mean-motion resonance with Jupiter

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Astrophysics & Cosmology · Celestial mechanics and planetary systems

Action and Symmetry, Ch Adiabatic invariants, canonical perturbation theory and integrable systems

Problem

An asteroid orbits at the 2:1 mean-motion resonance with Jupiter, so its mean motion is exactly twice Jupiter's: with Jupiter's mean motion νJ=2π/(11.86 yr)=0.5298 rad yr−1\nu_{\rm J} = 2\pi/(11.86\,\mathrm{yr}) = 0.5298\,\mathrm{rad\,yr^{-1}}, the asteroid's is νast=2νJ=1.0596 rad yr−1\nu_{\rm ast} = 2\nu_{\rm J} = 1.0596\,\mathrm{rad\,yr^{-1}}. The relevant harmonic of Jupiter's perturbation on the asteroid is the one that goes as cos⁡(2ϑJ−ϑast)\cos(2\vartheta_{\rm J} - \vartheta_{\rm ast}), i.e. m=(mast,mJ)=(−1,2)\mathbf m = (m_{\rm ast}, m_{\rm J}) = (-1, 2) in m⋅ϑ=mastϑast+mJϑJ\mathbf m\cdot\boldsymbol\vartheta = m_{\rm ast}\vartheta_{\rm ast} + m_{\rm J}\vartheta_{\rm J}. Working through the first-order canonical perturbation theory, someone writes: "The correction for this harmonic is w1,m=iH1,m/(m⋅ν)w_{1,\mathbf m} = \mathrm i H_{1,\mathbf m}/(\mathbf m\cdot\boldsymbol\nu), and m⋅ν=νast+2νJ=1.0596+2(0.5298)=2.1192 rad yr−1,\mathbf m\cdot\boldsymbol\nu = \nu_{\rm ast} + 2\nu_{\rm J} = 1.0596 + 2(0.5298) = 2.1192\,\mathrm{rad\,yr^{-1}}, which is finite and not small, so this harmonic of the perturbation series causes no difficulty even though the asteroid sits exactly on the 2:1 resonance." Find the error in this working, give the correct value of m⋅ν\mathbf m\cdot\boldsymbol\nu for the stated harmonic, and say what it implies for the perturbation series at this orbit.

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