Jack Wisdom

Cassini Observes the Active South Pole of Enceladus

Cassini has identified a geologically active province at the south pole of Saturns moon Enceladus. In images acquired by the Imaging Science Subsystem (ISS), this region is circumscribed by a chain of folded ridges and troughs at ∼55°S latitude. The terrain southward of this boundary is distinguished by its albedo and color contrasts, elevated temperatures, extreme geologic youth, and narrow tectonic rifts that exhibit coarse-grained ice and coincide with the hottest temperatures measured in the region. Jets of fine icy particles that supply Saturns E ring emanate from this province, carried aloft by water vapor probably venting from subsurface reservoirs of liquid water. The shape of Enceladus suggests a possible intense heating epoch in the past by capture into a 1:4 secondary spin/orbit resonance.

Symplectic maps for the n-body problem

The present study generalizes the mapping method of Wisdom (1982) to encompass all gravitational n-body problems with a dominant central mass. The rationale for the generalized mapping method is discussed as well as details for the mapping for the n-body problem. Some refinements of the method are considered, and the relationship of the mapping method to other symplectic integration methods is shown. The method is used to compute the evolution of the outer planets for a billion years. The resulting evolution is compared to the 845 million year evolution of the outer planets performed on the Digital Orerry using standard numerical integration techniques. This calculation provides independent numerical confirmation of the result of Sussman and Wisdom (1988) that the motion of the planet Pluto is chaotic.

Chaotic behavior and the origin of the 31 Kirkwood gap

The sudden eccentricity increases discovered by J. Wisdom (Astron J.87, 577–593, 1982) are reproduced in numerical integrations of the planar-elliptic restricted three-body problem, verifying that this phenomenon is real. Maximum Lyapunov characteristic exponents for trajectories near the 31 commensurability are computed both with the mappings presented in Wisdom (1982) and by numerical integration of the planar-elliptic problem. In all cases the agreement is excellent, indicating that the mappings accurately reflect whether trajectories are chaotic or quasiperiodic. The mappings are used to trace out the chaotic zone near the 31 commensurability, both in the planar-elliptic problem and to a more limited extent in the three-dimensional elliptic problem. The outer boundary of the chaotic zone coincides with the boundary of the 31 Kirkwood gap in the actual distribution of asteroids within the errors of the asteroid orbital elements.

The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem

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.

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.

The chaotic rotation of Hyperion

Abstract A plot of spin rate versus orientation when Hyperion is at the pericenter of its orbit (surface of section) reveals a large chaotic zone surrounding the synchronous spin-orbit state of Hyperion, if the satellite is assumed to be rotating about a principal axis which is normal to its orbit plane. This means that Hyperions rotation in this zone exhibits large, essentially random variations on a short time scale. The chaotic zone is so large that it surrounds the ½ and 2 states, and libration in the 3/2 state is not possible. Stability analysis shows that for libration in the synchronous and ½ states, the orientation of the spin axis normal to the orbit plane is unstable, whereas rotation in the 2 state is attitude stable. Rotation in the chaotic zone is also attitude unstable. A small deviation of the principal axis from the orbit normal leads to motion through all angles in both the chaotic zone and the attitude unstable libration regions. Measures of the exponential rate of separation of nearby trajectories in phase space (Lyapunov characteristic exponents) for these three-dimensional motions indicate the the tumbling is chaotic and not just a regular motion through large angles. As tidal dissipation drives Hyperions spin toward a nearly synchronous value, Hyperion necessarily enters the large chaotic zone. At this point Hyperion becomes attitude unstable and begins to tumble. Capture from the chaotic state into the synchronous or ½ state is impossible since they are also attitude unstable. The 3/2 state does not exist. Capture into the stable 2 state is possible, but improbable. It is expected that Hyperion will be found tumbling chaotically.

Chaotic Evolution of the Solar System

The evolution of the entire planetary system has been numerically integrated for a time span of nearly 100 million years. This calculation confirms that the evolution of the solar system as a whole is chaotic, with a time scale of exponential divergence of about 4 million years. Additional numerical experiments indicate that the Jovian planet subsystem is chaotic, although some small variations in the model can yield quasiperiodic motion. The motion of Pluto is independently and robustly chaotic.

Numerical Evidence That the Motion of Pluto Is Chaotic

The Digital Orrery has been used to perform an integration of the motion of the outer planets for 845 million years. This integration indicates that the long-term motion of the planet Pluto is chaotic. Nearby trajectories diverge exponentially with an e-folding time of only about 20 million years.

A perturbative treatment of motion near the 3/1 commensurability

Abstract 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 unusual features of the motion discovered by J. Wisdom (1982, Astron. J. 87, 577–593; 1983a, Icarus 56 , 51–74) are explained. 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.

Urey prize lecture: Chaotic dynamics in the solar system☆

Abstract Newtons equations have chaotic solutions as well as regular solutions. There are several physical situations in the solar system where chaotic solutions of Newtons equations play an important role. There are examples of both chaotic rotation and chaotic orbital evolution. Hyperion is currently tumbling chaotically. Many of the other irregularly shaped satellites in the solar system have had chaotic rotations in the past. This episode of chaotic tumbling could have had a significant effect on the orbital histories of these satellites. Chaotic orbital evolution seems to be an essential ingredient in the explanation of the Kirkwood gaps in the distribution of asteroids. The phase space boundary of the chaotic zone at the 3 1 mean-motion commensurability with Jupiter is in excellent agreement with the boundary of the observed population of asteroids. Chaotic trajectories at the 3 1 commensurability have the correct properties to provide a dynamical route for the transport of meteoritic material from the asteroid belt to Earth. There is a large chaotic zone at the 2 1 commensurability, where there is a Kirkwood gap, but the phase space near the Hilda group of asteroids at the 3 2 commensurability is dominated by quasiperiodic behavior. Chaotic trajectories in the 2 1 chaotic zone reach very high eccentricities by a path that temporarily takes them to high inclinations. The long-term evolution of Pluto is suspiciously complicated, but objective criteria have not yet indicated that the motion is chaotic.