Carl D. Murray

The dynamics of tadpole and horseshoe orbits: I. Theory

Abstract The properties of the tadpole and the horseshoe orbit solutions of both the circular and elliptic restricted three-body problem with small mass ratio are examined using analytical and numerical methods. We show how the trajectory of a particle in a near-circular orbits is critically dependent on the initial radial separation between the particle and satellite orbits. This separation can also be used to derive an approximation to the minimum particle-satellite distance. A useful relation between the shape of a particles path in a reference frame in which the perturbing satellite is stationary and the shape of the particles associated zero-velocity curve is presented. We also determine the circumstances in which horseshoe paths rather than the more common tadpole paths are to be expected. By numerical integration of a number of horseshoe orbits we have investigated the effects of changes in eccentricity and longitude of pericenter over repeated satellite encounters. Such changes will determine the long-term stability of particles in horseshoe orbits.

The dynamics of tadpole and horseshoe orbits: II. The coorbital satellites of saturn

Abstract The recently discovered coorbital satellites of Saturn, 1980S1 and 1980S3, are shown to be librating in horseshoe orbits. By considering the effects of tangential forces on the semimajor axes of the satellite orbits, we derive an accurate relation between the sum of the satellite masses and (a) their minimum angular separation, (b) the variation of their angular separation with time and (c) the libration period. Observations of (b) and (c) are the most practical methods of determining the satellite masses. The orbits of the coorbital satellites of Dione and Tethys are discussed. We demonstrate the possibility of calculating a new value for the mass of Dione and we show that one of the coorbital satellites of Tethys could be moving in a horseshoe orbit even though another satellite is librating in a tadpole orbit about the leading Lagrangian equilibrium point L 4 . The origin of coorbital satellites and the stability of their orbits are discussed.

Origin of the eccentricity gradient and the apse alignment of the ϵ ring of Uranus

Abstract The suggestion of P. Goldreich and S. Tremaine (1979, Nature 277, 97–99; 1979 Astron. J. 84, 1638–1641) that the apse alignment of the eccentric Uranian ϵ ring is maintained by self-gravitation alone is criticized. If self-gravitation were the only factor involved, then the extreme variation in width of the ring would be a mere chance product: this we do not accept. We consider that the ring particles move in a near monolayer which is close packed at pericenter. We suggest that it is the close packing of the particles which prevents differential precession. We describe how differential precession, particle collisions, and self-gravitation, acting together, can transform a narrow, eccentric ring of uniform width into a ring with a large, positive eccentricity gradient. The model we present is supported by the observed occultation profiles of the ϵ ring. The F ring of Saturn which, as suggested by S. F. Dermott, C. D. Murray, and A. T. Sinclair (1980, Nature 284, 309–313), probably consists of a number of narrow rings of variable width (J. A. Simpson, T. S. Bastian, D. L. Chenette, G. A. Lentz, R. B. McKibben, K. R. Pyle, and A. J. Tuzzolino (1980b, J. Geophys. Res.)) should provide further tests of our model.

Nodal regression of the quadrantid meteor stream: An analytic approach

Abstract The mean orbit of the Quadrantid meteor stream has a high eccentricity and inclination with an aphelion close to the orbit of Jupiter. The nodal regression rate, a quantity which has been well determined from observations, cannot be calculated with sufficient accuracy using standard low-order expansions of the disturbing function. By using a high-order expansion of the disturbing function we show how the behavior of the longitude of ascending node of the Quadrantid stream is a result of both secular and resonant effects. Our analysis illustrates how the proximity of the streams orbit to the 2: 1 commensurability with Jupiter dominates the short-term variations in orbital elements.

Solar System Dynamics: Structure of the Solar System

Theres not the smallest orb which thou beholdst But in his motion like an angel sings, Still quiring to the young-eyed cherubins; Such harmony is in immortal souls; William Shakespeare, Merchant of Venice, V, i Introduction It is a laudable human pursuit to try to perceive order out of the apparent randomness of nature; science is, after all, an attempt to make sense of the world around us. Moving against the background of the “fixed” stars, the regularity of the Moon and planets demanded a dynamical explanation. The history of astronomy is the history of a growing awareness of our position (or lack of it) in the universe. Observing, exploring, and ultimately understanding our solar system is the first step towards understanding the rest of the universe. The key discovery in this process was Newtons formulation of the universal law of gravitation; this made sense of the orbits of planets, satellites, and comets, and their future motion could be predicted: The Newtonian universe was a deterministic system. The Voyager missions increased our knowledge of the outer solar system by several orders of magnitude, and yet they would not have been possible without knowledge of Newtons laws and their consequences. However, advances in mathematics and computer technology have now revealed that, even though our system is deterministic, it is not necessarily predictable. The study of nonlinear dynamics has revealed a solar system even more intricately structured than Newton could have imagined.

Solar System Dynamics: The Restricted Three-Body Problem

Twos company, threes a crowd. Proverb Introduction In Chapter 2 we showed how the problem of the motion of two masses moving under their mutual gravitational attraction can be solved analytically and that the resulting motion is always confined to fixed geometrical paths that are closed in inertial space. We will now extend our analysis to consider the gravitational interaction of three bodies, paying particular attention to the problem in which the third body has negligible mass compared with the other two. The simplicity and elusiveness of the three-body problem in its various forms have attracted the attention of mathematicians for centuries. Among the giants of mathematics who have tackled the problem and made important contributions are Euler, Lagrange, Laplace, Jacobi, Le Verrier, Hamilton, Poincare, and Birkhoff. The books by Szebehely (1967) and Marchal (1990) provide authoritative coverage of the literature on the subject as well as derivations of the important results. Today the three-body problem is as enigmatic as ever and although much has been discovered already, the recent developments in nonlinear dynamics and the spur of new observations in the solar system have meant a resurgence of interest in the problem and the derivation of new results. If two of the bodies in the problem move in circular, coplanar orbits about their common centre of mass and the mass of the third body is too small to affect the motion of the other two bodies, the problem of the motion of the third body is called the circular, restricted, three-body problem .