Engineering topics
Publications and source records attributed to Wisdom, J..
Cassini has identified a geologically active province a the south pole of Saturn's moon Enceladus. The shape of Enceladus suggests a possible intense heating epoch in the past by capture into a 1:4 secondary spin/orbit resonance.
The large amplitude oscillations of the shape of Neptune's Great Dark Spot are well reproduced by simple dynamical models of an isolated vortex embedded in a background shear flow. From the time series of the aspect ratio and inclination of the vortex values are estimated for the background shear and the mean vorticity of the Great Dark Spot, and a lower bound is placed on the value of the Rossby deformation radius. These models imply the existence of a planetary-scale zone of deterministic chaotic advection in the atmosphere of Neptune.
A special-purpose computer is used to integrate the orbits of the outer five planets for more than 100 Myr into the future and more than 100 Myr into the past. The strongest features in the Fourier transforms of the orbital elements of the Jovian planets can be identified with the frequencies predicted by linear secular theory. Many of the weaker features in the Fourier spectra are identified as linear combinations of the basic frequencies. Serious differences are noted between the present measurements and the predictions of Bretagnon (1974). The amplitude of the 3.796 Myr period libration of Pluto's longitude of perihelion is modulated with a period of 34 Myr. Very long periods, on the order of 137 Myr, are also seen. The orbit of Pluto is stable for the duration of the integration; the maximum Liapunov characteristic exponent is less than 10 to the -6.8 power/yr.
The solution by Sessin and Ferraz-Mello (1984) of the Hori auxiliary system for the motion of two planets with periods nearly commensurate in the ratio 2:1 is considerably simplified by the introduction of canonical variables. An analogous canonical transformation simplifies the elliptic restricted problem.
A semianalytic perturbation theory for motion near the 3/1 commensurability in the planar elliptic restricted three-body problem is presented. The predictions of the theory are in good agreement with the features found on numerically generated surfaces of section; a global understanding of the phase space is achieved. The principal cause of the large chaotic zone near the 3/1 commensurability is identified, and a new criterion for the existence of large-scale chaotic behavior is presented.
A numerical integration was employed to show that the 3/1 Kirkwood gap region between Jupiter and the sun can serve as the source for asteroidal debris that strikes the earth as chondrites. A previous study by Wetherill (1968) considered planetary close encounters, secular resonances, collisional destruction in space and subsequent atmospheric abrasion of the chondrites. The factors would have resulted in chaotic trajectories for asteroids in the asteroid belt, i.e., eccentricities large enough to make the objects earth-crossing. A sampling of trajectories which could reach large eccentricities is integrated in terms of deterministic solutions of Hamiltonian systems of equations which behave erratically and are dependent on the initial conditions. The examinations successfully predicted a trajectory with a peak eccentricity of 0.652, a time interval of 340,000 yr, and a semimajor axis between 2.4-2.53 AU that was earth-crossing.
Under the assumption that the satellite is rotating about a principal axis that is normal to its orbit plane, a plot of spin rate-versus-orientation for Hyperion at the pericenter of its orbit has revealed a large, chaotic zone surrounding Hyperion's synchronous spin-orbit state. The chaotic zone is so large that it surrounds the 1/2 and 2 states, and libration in the 3/2 state is not possible. Rotation in the chaotic zone is also attitude-unstable. As tidal dissipation drives Hyperion's spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone, becoming attitude-unstable and tumbling. It is therefore predicted that Hyperion will be found to be tumbling chaotically.
The sudden eccentricity increases discovered by Wisdom (1982) are reproduced in numerical integrations of the planar ecliptic restricted three-body problem, verifying that this phenomenon is real. Mapping derivations are qualitatively reviewed and the maximum Liapunov characteristic exponent and its importance for determining the character of a trajectory are explained. The results of a number of calculations of this exponent using the differential equations for the unaveraged three-body problem are shown and compared to equivalent calculations using a mapping. In all cases the two approaches agree whether the orbits are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 3/1 commensurability, both in the planar-ecliptic problem and in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 3/1 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.
A mapping of the phase space onto itself with the same low-order resonance structure as the 3/1 commensurability in the planar-elliptic restricted three-body problem is obtained. This mapping is about 1,000 times faster than the usual method of numerically integrating the averaged equations of motion. It exhibits some surprising behavior that might provide a key to the origin of the Kirkwood gaps. It is noted that a test asteroid placed in the gap may evolve for a million years with low eccentricity (less than 0.05) and then suddenly jump to large eccentricity (greater than 0.3), becoming a Mars crosser. The removal of the asteroid by a close encounter with Mars would then be possible. As a first test of this hypothesis, a distribution of 300 test asteroids in the area of the 3/1 commensurability was evolved for two million years. When the Mars crossers are removed, the distribution of initial conditions reveals a gap at the location of the 3/1 Kirkwood gap.
The resonance overlap criterion for the onset of stochastic behavior is applied to the planar circular-restricted three-body problem with small mass ratio (mu). Its predictions for mu = 0.001, 0.0001, and 0.00001 are compared to the transitions observed in the numerically determined Kolmogorov-Sinai entropy and found to be in remarkably good agreement. In addition, an approximate scaling law for the onset of stochastic behavior is derived.