Mean motion

Abstract: 1. Elementary Celestial and Hamiltonian Mechanics 2. Quasi-Integrable Hamiltonian System 3. Kam Tori 4. Single Resonance Dyanmics 5. Numerical Tools for the Detection of Chaos 6. Interactions Among Resonances 7. Secular Dynamics of the Planets 8. Secular Dynamics of Small Bodies 9. Mean Motion Resonances 10. Three Body Resonances 11. Secular Dynamics Inside Mean Motion Resonances 12. Global Dynamical Structure of the Belts of Small Bodies.

Abstract: The gravitational waves, emitted by a compact object orbiting a much more massive central body, depend on the central body’s spacetime geometry. This paper is a first attempt to explore that dependence. For simplicity, the central body is assumed to be stationary, axially symmetric (but rotating), and reflection symmetric through an equatorial plane, so its (vacuum) spacetime geometry is fully characterized by two families of scalar multipole moments Ml and Sl with l=0,1,2,3,..., and it is assumed not to absorb any orbital energy (e.g., via waves going down a horizon or via tidal heating). Also for simplicity, the orbit is assumed to lie in the body’s equatorial plane and to be circular, except for a gradual shrinkage due to radiative energy loss. For this idealized situation, it is shown that several features of the emitted waves carry, encoded within themselves, the values of all the body’s multipole moments Ml, Sl (and thus, also the details of its full spacetime geometry). In particular, the body’s moments are encoded in the time evolution of the waves’ phase Φ(t) (the quantity that can be measurd with extremely high accuracy by interferometric gravitational-wave detectors), and they are also encoded in the gravitational-wave spectrum ΔE(f) (energy emitted per unit logarithmic frequency interval). If the orbit is slightly elliptical, the moments are also encoded in the evolution of its periastron precession frequency as a function of wave frequency, Ωρ(f); if the orbit is slightly inclined to the body’s equatorial plane, then they are encoded in its inclinational precession frequency as a function of wave frequency, Ωz(f). Explicit algorithms are derived for deducing the moments from ΔE(f), Ωρ(f), and Ωz(f). However, to deduce the moments explicitly from the (more accurately measurable) phase evolution Φ(t) will require a very difficult, explicit analysis of the wave generation process—a task far beyond the scope of this paper.

Abstract: The dynamical parameters conventionally used to specify the orbit of a test particle in Kerr spacetime are the energy E, the axial component of the angular momentum, Lz, and Carter's constant Q. These parameters are obtained by solving the Hamilton–Jacobi equation for the dynamical problem of geodesic motion. Employing the action-angle variable formalism, on the other hand, yields a different set of constants of motion, namely, the fundamental frequencies ωr, ωθ and ω associated with the radial, polar and azimuthal components of orbital motion, respectively. These frequencies, naturally, determine the time scales of orbital motion and, furthermore, the instantaneous gravitational wave spectrum in the adiabatic approximation. In this paper, it is shown that the fundamental frequencies are geometric invariants and explicit formulae in terms of quadratures are derived. The numerical evaluation of these formulae in the case of a rapidly rotating black hole illustrates the behaviour of the fundamental frequencies as orbital parameters, such as the semi-latus rectum p, the eccentricity e or the inclination parameter θ− are varied. The limiting cases of circular, equatorial and Keplerian motion are investigated as well and it is shown that known results are recovered from the general formulae.