The orbital evolution of the whole sample sample of short-period comets was computed by numerical integrations for a time interval of 2000 yr centered on the present epoch. This data base is intended to serve in various studies involving the statistics of orbital evolution and correlation with physical parameters or discovery circumstances. We present some results concerning the following aspects: the evolution of the orbital elements and their past-future asymmetry, statistics on the discovery of comets and on the encounters of comets with Jupiter.

Recent numerical calculations [1,2,3] have shown that Jupiter-family comets, which are on low inclination orbits, cannot originate from the gravitational scatter of long-period comets. Work by Quinn, Tremaine & Duncan [1] shows that objects originally on low-inclination, Neptune crossing orbits will evolve into a population of objects with orbital parameters consistent with those of Jupiter-family comets. However, they point out that the timescale to deplete this initial population of planet-crossing objects is short. Therefore, they conclude there must be a system of objects that are evolving into planet-crossers on the timescale of the age of the solar system. The most likely source of these objects is a region just beyond the orbit of Neptune, the Kuiper belt.

Among the comets that were observed to break in two or more fragments, only a few of them are periodic. So far the dynamic study of the relative motion of a secondary nucleus with respect to the primary has supposed that a cometary fragment is subject to a small and continuous radial nongravitational force after separation at rest. This force acts against the solar attraction and varies according to an inverse square law. A small impulse at break up may also be invoked in some case. A different approach is followed in this paper when dealing with a fragment of a periodic comet: after separation the motion of a secondary nucleus is characterized by nongravitational forces which vary according to the same g(r) law currently used for the primary.

Results of the study of comets P/Biela and P/du Toit-Hartley show that the motion of their fragments after separation is characterized by nongravitational parameters which are larger than those of the parent bodies. Both fragments lasted for about 2 full revolutions and three returns.

We analyze the incoming direction of 180 long period comets, taken into account the gravitational influence of the planets and the Galaxy as a whole. We have made an improvement of some previous works on the matter, by inclusion of comets observed up to 1989. An asymmetry of aphelia is detected that cannot be explained as due to observational selection. We conclude that the so called Dynamical Friction induced by the solar motion through the interstellar medium could be the proper answer to this observed fact.

The present paper deals with a general dynamical qualitative study of the rotational motion for cometary-type bodies submitted to gravitational torques. Numerical experiments of the evolution of comet nucleus attitude have been then performed, including the Sun and Jupiter's disturbing torques in the model. Results show small effects of the solar gravitational perturbation for Halley-type orbits. Only a very close-approach with Jupiter induces notable effects. The latter configuration presents some interesting sensitivity to initial conditions.

The recent observations, of energetic heavy ions in the vicinity of Comet Halley and Comet Giacobini-Zinner, are rather intriguing. Moreover the solar coronal heating is still not well understood and is somewhat puzzling. Since both these observations can not be properly explained by means of classical processes, we have to invoke anomalous heating and acceleration effects due to chaotic and turbulent plasma processes.

Meteor showers are seen at regular and frequent intervals on Earth. They are caused by meteoroids (that is small dust grains) in a coherent stream, all moving on similar heliocentric orbits, burning up on encountering the atmosphere of the Earth. Such streams contain 1012 or more meteoroids, with the mass of the visible meteoroids ranging up to about 1 g. The main evolutionary effect on such streams is gravitational perturbations by the planets. Though grain-grain collision may be catastrophic for the two grains involved, it has no effect on the remainder of the stream, other than the fact that there are now two less grains in it. Solar radiation has some effect, but this can be included in the equations of motion. Because of the large numbers of particles involved, meteoroid streams represent a laboratory where many of our dynamical concepts can be tested.

At a basic level, meteoroid streams represent a collective dynamical phenomenon in which all members display roughly the same behavior. One of the fundamental questions which can be investigated is whether the behavior of the mean orbit of the whole stream represents the mean behavior of the stream members. Within the boundaries of some meteor streams lie regions where the orbits are in high order resonance with Jupiter. This also represents a phenomenon of interest. Finally, the possibility exists that some streams are in chaotic regions and it is interesting to investigate whether or not meteoroids in such regions do display chaotic behavior.

Unusually long activity of the Taurid meteor complex, extending over 3-5 months according to new estimations based on various orbital similarity criteria, has been evoking controversies about the possible origin and dynamical evolution of this unique complex of meteor streams. It even casts doubts on the reality of the extension of the complex. In the present paper orbital elements and the extension of the Taurid meteor complex are re-examined on the bases of the most precise photographic meteor orbits available from the IAU Meteor Data Center in Lund. The results are evaluated and discussed from the viewpoint of various proposals on the origin and dynamical evolution of the complex.

The Quadrantid meteor shower has been recognized for more than 150 years. The dynamics of the corresponding stream is peculiar due to the high orbital inclination and, for some particles, the closeness of the 2/1 mean motion resonance with Jupiter. It has been the subject of many investigations relating to the structure of the stream and its nodal retrogression as well as its long-term history and its likely cometary origin. Thus Hamid and Youssef (1963) found that the jovian secular perturbations lead to very large changes in the inclination and perihelion distance of typical stream particles with a period around 4000 yrs. From a more extensive study by Williams et al. (1979) it was obvious that this period is not unique but may vary considerably between different particles. Related to this behaviour is also the investigation by Froeschlé and Scholl (1982) who performed an extensive study of three-dimensional orbits at the 2/1 resonance. The orbits remain confined in the resonance zone and are stable in Hill's sense. Close encounters with Jupiter are avoided through the action of three main protection mechanisms: σ libration around 0, ω libration aroud 90°, and e–ω coupling, although most orbits exhibit large variations in inclination and eccentricity. Investigating in more detail a Quadrantid-like meteor stream, Froeschlé and Scholl (1986) also found a nonuniform nodal retrogression and an unusual progression. This behaviour causes a formation of arcs which was not found for other meteor streams in resonance with Jupiter – however, almost never having large inclinations.

The Quadrantid stream covers a region of space which contains many strong resonances and commensurabilities with the Jovian orbit. We have numerically integrated the orbital evolution of over one hundred actual meteoroids backwards to BC 5000. The evolution is quit complex, but most of the meteoroids are quite well behaved with rapid but smooth changes in the orbital elements. One meteoroid however shows sharp sudden changes in its orbital parameters and these changes are generally indicative of the presence of chaos.

Advances in infrared astronomy and in computing power have recently opened up an interesting area of the solar system for dynamical exploration. The survey of the sky made by The Infrared Astronomical Satellite (IRAS) in 1983 revealed the complex structure of the zodiacal dust cloud. We now know the inclination and nodes of the plane of symmetry of the cloud with respect to the ecliptic and we have evidence that the cloud is not rotationally symmetric with respect to the Sun. Of even more interest is the discovery by IRAS of prominent dust bands that circle the Sun in planes near-parallel to the ecliptic. In 1984, we suggested (Dermott et al., Nature, 312, 505-509) that the solar system dust bands discovered by IRAS are produced by the gradual comminution of the asteroids in the major Hirayama asteroid families. The confirmation of this hypothesis has involved: (1) The development of a new secular perturbation theory that includes the effects of Poynting-Robertson light drag on the evolution of the dust particle orbits; (2) The production of a new high resolution Zodiacal History File by IPAC (the Infrared Processing and Analysis Center at Caltech); (3) The development of the SIMUL code: a three-dimensional numerical model that allows the calculation of the thermal flux produced by any particular distribution of dust particle orbits. SIMUL includes the effects of planetary perturbations and PR drag on the dust particle orbits and reproduces the exact viewing geometry of the IRAS telescope. We report that these tools allow us to account in detail for the observed structure of the dust bands. They also allow us to show that there is evidence in the IRAS data for the transport of asteroidal dust from the main belt to the Earth by Poynting-Robertson light drag.

Some features of the dynamics of particles affected by drag, in the field of the Sun and planets, are presented here. In particular mean motion and secular resonances are investigated. When dust particles are considered as a whole in the zodiacal cloud, a simple secular theory can explain much of its geometry. Dynamics of particles near an inner planet is mostly dispersive, but an average behavior can be deduced from some analytical and numerical considerations.

We have searched for stable stationary solutions of the resonant elliptic restricted problem of three bodies with Stokes drag. Results for the 1/2 external Jovian resonance are shown as example, including a comparison with numerical results of an N-body simulation.

Dynamical evolution of a plane system of non-interacting particles, which are in a certain smoothed-out regular field, is of a certain interest in a number astronomical applications. Among them are the plane subsystems of the spiral galaxies, the rings of the giant planets and protoplanetary nebulae. The problem of determining the stationary state of these systems has been studied enough [1]. But one seldom comes across studies concerning with the reaction of the non-stationary field.

Two important theories in Dynamical Systems were constructed in the sixties: the hyperbolic theory for general systems and the KAM (after Kolmogorov, Arnold and Moser) theory for conservative systems as the ones that appear in Celestial Mechanics. Most of our discussions here concern dissipative (or locally dissipative) systems, although most questions are now being posed for area preserving maps (symplectic maps in higher dimensions). Moreover, one can argue that understanding dynamically small dissipative perturbations of conservative systems is of much importance: indeed it has been recently shown that a KAM curve (tori in higher dimension) can be destroyed and in fact engulfed in the basin of attraction of a Hénon-like strange attractor as defined below.

We present a method for the study of the Krein signature in perturbed Hamiltonian integrable systems. The method is developed up to first order in the small parameter. We apply this method to a particular instance of the two-body problem in which the semi-major axis is not affected by the perturbation.

We review mappings mainly devised for the study of the dynamics of comets and asteroids. An attempt of a typology according to the method used to devise the mapping and to its deterministic or stochastic character is made.

We construct a two step algebraic mapping from Sessin's simplified model for the first order resonance. The orbits obtained with this mapping are compared to the ones calculated with the exact solution. We also derive a reduced Hamiltonian. A plane Poincaré mapping, using delta periodic function, is constructed and compared to the reduced Hamiltonian contour curves showing the splitting of the separatrix due to delta perturbation technique.

When planetary orbits are numerically integrated for a long time by conventional integrators, the most serious problem is secular errors in the energy and the angular momentum of the planetary system due to discretization (truncation) errors. The secular errors in the energy and the angular momentum mean that the semi-major axes, the eccentricities, and the inclinations of planetary orbits have a secular error which grows linearly with time. Recently symplectic integrators and linear symmetric multistep integrators are found not to produce the secular errors in the energy and the angular momentum due to the discretization errors. Here we describe briefly both methods and discuss favorable properties of these integrators for a long-term integration of planetary orbits.

Symplectic integrators are numerical integration methods for Hamiltonian systems, which conserves the symplectic 2-form exactly. With use of symplectic integrators there is no secular increase in the error of the energy because of the existence of a conserved quantity closed to the original Hamiltonian. Higher order symplectic integrators are obtained by a composition of 2nd order ones.