Jihad Touma

The Chaotic Obliquity of Mars

Numerical integration of the rotation of Mars shows that the obliquity of Mars undergoes large chaotic variations. These variations occur as the system evolves in the chaotic zone associated with a secular spin-orbit resonance.

Evolution of the Earth-Moon system

The tidal evolution of the Earth-Moon system is reexamined. Several models of tidal friction are first compared in an averaged Hamiltonian formulation of the dynamics. With one of these models, full integrations of the tidally evolving Earth-Moon system are carried out in the complete, fully interacting, and chaotically evolving planetary system. Classic results on the history of the lunar orbit are confirmed by our more general model. A detailed history of the obliquity of the Earth which takes into account the evolving lunar orbit is presented.

RESONANCES IN THE EARLY EVOLUTION OF THE EARTH-MOON SYSTEM

Most scenarios for the formation of the Moon place the Moon near Earth in low-eccentricity orbit in the equatorial plane of Earth. We examine the dynamical evolution of the Earth-Moon system from such initial configurations. We find that during the early evolution of the system, strong orbital resonances are encountered. Passage through these resonances can excite large lunar orbital eccentricity and modify the inclination of the Moon to the equator. Scenarios that resolve the mutual inclination problem are presented. A period of large lunar eccentricity would result in substantial tidal heating in the early Moon, providing a heat source for the lunar magma ocean. The resonance may also play a role in the formation of the Moon.

Stellar dynamics around black holes in galactic nuclei

We classify orbits of stars that are bound to central black holes in galactic nuclei. The stars move under the combined gravitational influences of the black hole and the central star cluster. Within the sphere of influence of the black hole, the orbital periods of the stars are much shorter than the periods of precession. We average over the orbital motion and end up with a simpler problem and an extra integral of motion: the product of the black hole mass and the semimajor axis of the orbit. Thus the black hole enforces some degree of regularity in its neighborhood. Well within the sphere of influence, (i) planar, as well as three dimensional, axisymmetric configurations–both of which could be lopsided–are integrable, (ii) fully three dimensional clusters with no spatial symmetry whatsover must have semi–regular dynamics with two integrals of motion. Similar considerations apply to stellar orbits when the black hole grows adiabatically. We introduce a family of planar, non–axisymmetric potential perturbations, and study the orbital structure for the harmonic case in some detail. In the centered potentials there are essentially two main families of orbits: the familiar loops and lenses, which were discussed in Sridhar and Touma (1997, MNRAS, 287, L1-L4). We study the effect of lopsidedness, and identify a family of loop orbits, whose orientation reinforces the lopsidedness, an encouraging sign for the construction of self–consistent models of eccentric, discs around black holes, such as in M31 and NGC 4486B.

Gauss's method for secular dynamics, softened

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NONLINEAR CORE-MANTLE COUPLING

We explore the nonlinear dynamics of a forced core-mantle system. We show that the free axisymmetric motion of a uniform-vorticity —uid core coupled to a rigid mantle (the model) is Poincarec -Hough integrable. We derive an approximate Hamiltonian for the core tilt mode that includes the leading nonlinear contribution. We then include gravitational perturbations in the analysis. We identify the principal nonlinear prograde and retrograde resonances and the characteristic excitation associated with each. We compare the nonlinear excitation with the excitation expected in the corresponding linear model. The nonlinear model indicates that for each principal commensurability there are multiple overlapping resonances, and so varying degrees of chaotic behavior are predicted. Chaotic behavior at the principal coremantle commensurabilities is con—rmed with surfaces of section. We then present the results of numerical evolutions done with a generalization of our (1994) Lie-Poisson integrator to allow for a Poincarec core, core-mantle friction, and tidal dissipation. We use our analytical and numerical models to Hough explore the evolution of Earth through the prograde core-mantle resonances and to explore the evolution of Venus through the retrograde resonances. Heating of the core-mantle boundary during resonance passage is much greater for Venus than for Earth. We raise the question whether heating during coremantle resonance passage could be responsible for the global resurfacing of Venus.

Counter-rotating stellar discs around a massive black hole: self-consistent, time-dependent dynamics

We formulate the collisionless Boltzmann equation for dense star clusters that lie within the radius of influence of a massive black hole in galactic nuclei. Our approach to these nearly Keplerian systems follows that of Sridhar & Touma: Delaunay canonical variables are used to describe stellar orbits and we average over the fast Keplerian orbital phases. The stellar distribution function (DF) evolves on the longer time-scale of precessional motions, whose dynamics is governed by a Hamiltonian, given by the orbit-averaged self-gravitational potential of the cluster. We specialize to razor-thin, planar discs and consider two counter-rotating (‘±’) populations of stars. To describe discs of small eccentricities, we expand the ± Hamiltonian to fourth order in the eccentricities, with coefficients that depend self-consistently on the ± DFs. We construct approximate ± dynamical invariants and use Jeans’ theorem to construct time-dependent ± DFs, which are completely described by their centroid coordinates and shape matrices. When the centroid eccentricities are larger than the dispersion in eccentricities, the ± centroids obey a set of four autonomous equations ordinary differential equations. We show that these can be cast as a two-degree-of-freedom Hamiltonian system which is non-linear, yet integrable. We study the linear instability of initially circular discs and derive a criterion for the counter-rotating instability. We then explore the rich non-linear dynamics of counter-rotating discs, with focus on the variety of steadily precessing eccentric configurations that are allowed. The stability and properties of these configurations are studied as functions of parameters such as the disc mass ratios and angular momentum.

Stellar dynamics around a massive black hole – I. Secular collisionless theory

We present a theory in three parts, of the secular dynamics of a (Keplerian) stellar system of mass

The disruption of multiplanet systems through resonance with a binary orbit.

M

The Doubling of Stellar Black Hole Nuclei

orbiting a black hole of mass