The uncertainty of the close approach distance of a Potentially Hazardous Object (PHO), either an asteroid or a comet, can be represented on the Modified Target Plane (MTP), a modification of the one used by Öpik. The MTP is orthogonal to the geocentric velocity at the closest approach along the nominal orbit, solution of the least square fit to the observations. The confidence regions of this solution in the 6-D space of orbital elements (for an epoch close to the observations) are well approximated by a family of concentric ellipsoids, if the observed arc is not too short. In the linear approximation these ellipsoids are mapped on the MTP into concentric ellipses, which can be computed by solving for the state transition matrix.

For a PHO observed at only one opposition, with a close approach expected after many revolutions, the ellipses on the MTP become extremely elongated and the linear approximation may fail. In this case the confidence boundaries on the MTP, i.e. the nonlinear images of the confidence ellipsoids, may not be well approximated by the ellipses. The Monte Carlo method (Muinonen and Bowell, 1993) can be used to find nonlinear confidence regions, but the computational load is very heavy: to estimate a low probability event the number of test cases must be larger than the inverse of the probability. We propose a new method to compute semilinear confidence boundaries on the MTP (Milani and Valsecchi, 1998), based on the theory developed to compute confidence boundaries for predicted observations (Milani, 1999). This method is a good compromise between reliability and computational load, and can be used for real time risk assessment.

Results from a numerical integration survey of all 179 currently-known inner solar system asteroids with a ≤ aMars, q ≥ aMercury are presented. A surprising number of asteroids are currently in, or very near, mean-motion resonances with Mercury, Venus, Earth, or Mars. Some of the resonance associations are of high order. Most of the resonance associations are relatively short-lived, with the asteroids wandering in and out of resonance on timescales of hundreds to several thousand years.

The fragmentation of cometary nuclei is a frequent phenomenon, but the dynamics of the fragments is not yet well understood. During the close approach of comet C/1996 B2 Hyakutake to the Earth (0.1 AU) on late March 1996, images were taken with the 1 meter telescope of Pic du Midi observatory. Bright condensations were observed near the nucleus on images taken between March, 22,1996 and March, 31, 1996. It was suggested that these features were mini-comæ surrounding fragments receding from the nucleus (Lecacheux et al., 1996). A model was developped for the motion of cometary fragments in the orbital plane of the comet, and the simulations were compared with the observations (Desvoivres et al, 1998).

In the model, we consider that the nucleus of the comet and a fragment are under the influence of the gravity of the Sun, of their mutual gravity, and of non-gravitational forces (NGF) due the loss of mass induced by solar heating. From an estimation of those NGF, we compute numerically the trajectories of the fragment and of the nucleus with respect to their common center of mass (CoM). Then, the motion of the center of mass is studied in an heliocentric reference frame using the theory of perturbed keplerian motion.

Up to now ~ 400 asteroids are known which move close to the Lagrangian equilibrium points L4 (246) and L5 (167) of Jupiter. In this investigation the orbits of all known Trojans were integrated numerically for 10 million years using the Lie Series integrator with adaptive stepsize (Hanslmeier and Dvorak, 1984) in the dynamical model of the outer planetary system. The goal of the study was to extend the computation of the proper elements for all known Trojans for a longer time interval; the respective results are compared to already existing ones (e.g. Bien & Schubart, 1987; Milani, 1993).

The determination of the characteristic quantities proper eccentricity ep, proper inclination ip and libration D was done by numerical filtering techniques using the method of Labrouste (Burger, 1998) and by a very precise frequency analysis (Chapront, 1997). Our results differ from the ones calculated for a shorter time interval by Milani (1993) between 5% and 10%. In Fig.1 we plotted the nomogram for the three proper elements mentioned above for L4 and the ep for L5. As a new interesting result we found Trojans with 0.15 < ep < 0.22 for L4 (upper left) and L5 (upper right) which answers an open question of an existing gap in this range of the ep. For L4 Trojans there is a well defined maximum in the nomogram for ep ∼ 0.06; for the L5 Trojans the maximum is flat and shifted versus ep ∼ 0.07. The proper elements sin(ip) for L4 (bottom left graph) show a maximum for Trojans moving in the plane of Jupiter’s orbit and a decrease versus larger inclinations, although a plateau for 14° < i < 27° is visible; this confirms that the Trojan orbits are more inclined that the orbits of the main belt asteroids.

We study the dynamics of mean motion resonance with Jupiter in the 4:1 gap using only gravitational methods. This mechanism is capable of explaining this Kirkwood gap in an uniform way (see Ferraz-Mello, 1994; Ferraz-Mello et al., 1994; Moons, 1997; Yoshikawa, 1989). We considered the asteroidal motion in two and three dimensions and we carried out our investigations integrating numerically the full equations of motion and taking into account Mars, Jupiter and Saturn as disturbing planets. The orbital evolution of asteroids was obtained considering the elements variation. The numerical investigations were carried out using symplectic integrators. These integrations were stopped when the asteroid had close encouters with Mars or Jupiter, this occurs when the distance between the planet and the asteroid is of the order of 0.01 AU or less, or when the eccentricity increases up to 0.9. We studied real and fictitious asteroids on a time scale of 5 × 107 yr. The initial osculating elements of perturbing planets and their inverse masses were taken from the Ephemerides of Minor Planets (EMP) at the epoch of JD 2450000.5. The initial data corresponding to the real asteroids were also taken from the EMP. The starting elements of fictitious asteroids were, in all analyzed cases, a = acrit = 2.064; AU, i = 2°.5 and e = 0.01 (in the majority of cases). The other initial elements are shown in Table II. We have also studied fictitious asteroids with i = 0°, a = acrit and e = 0.01 (Table I). The present analysis leads to the following results: (1) The motions are unstable. The eccentricity, in the majority of cases, has very large increase. It may grow up to 0.9 in 106 yr. The semi major axis has large variations then, owing to both effects some fictitious asteroids reach the 3:1 resonance while others reach 7:2 resonance in a few million years, they are very chaotic regions. (2) The eccentricities of fictitious asteroids become large by the effect of the secular resonance v6, i.e. when (ϖ − ϖSat) ≅ 0, the rate of this resonance is 26.217”/year with period ~ 4.9 × 104 years (Bretagnon, 1974). (3) The fictitious asteroids studied with a = acrit, e < 0.05 and i < 3° are removed of this gap mainly by the effects of the secular resonances v6 and v16 (see Moons and Morbidelli, 1995; Williams, 1969). (4) There are close encounters with Mars or eventually with the Earth (not considered here) in a time scale of 106 - 107 yr. (5) For certain initial conditions some fictitious asteroids are temporally captured by Mars and in some cases for a long time. (6) If a = acrit,e = 0.3 and the inclination is less than 3°, Mars and asteroid’s perihelion are very close ( ~ 0.06AU ). This situation helps the capture. (7) The (a,e)-plane was used to determine the dynamical behaviour of all asteroids and we found that the 4:1 resonance is very strong. The Lyapunov times are very short.

We consider here the spin—orbit coupling influence on the relative orbital motion of two bodies interacting gravitationally. We assume that one of the bodies is spherically symmetric and the other possesses a plane of dynamical symmetry. In the full non-linear settings, this problem permits coplanar motion when the mass center of the spherically symmetric body moves in the plane. We used this simple model for a qualitative estimation of the changes of the relative orbit in two cases: A) the Sun-asteroid case (the fast rotating rigid body), B) a small satellite of a big planet in resonant rotation.

The motion is described in the rigid body fixed frame. An appropriate change of physical units (Goźdiewski,1998a) leads to nondimensional dynamical variables and parameters. After that the Hamiltonian of the problem, written in polar variables, is the following

where (I1, I2, I3) are the principal moments of inertia, (r, φ) are the relative polar coordinates of the point mass in the body frame, (Pr, pφ) are the canonical momenta, (G3 represents the constant of total angular momentum, ε = (ro/r)2, and ro is the mean radius of the body.

The Kirkwood gaps in the main asteroidal belt (2 – 3.5 AU) coincide with the mean motion resonances with Jupiter (4/1, 3/1, 5/2, 7/3, 2/1). Similarly, several narrower gaps are observed in the outer asteroid belt (3.5 – 4 AU) at places of 11/6, 9/5, 7/4 and 5/3 Jovian resonances (Holman and Murray 1996). As it is now generally accepted, the formation and preservation of these gaps is due to the chaos of the resonant space and efficient ejection of the primordial and collisionaly injected bodies towards high eccentricities and planet-crossing orbits.

The Jovian mean motion resonances are not the most important in what concerns the chaos of the observed (i.e. remaining) asteroid population. It was estimated by Šidlichovský and Nesvorný (1998) that about 40% of known objects have the Lyapunov time less than 105 years. It was later found (Nesvorný and Morbidelli 1998, 1999; Morbidelli and Nesvorný 1999) that the resonances responsible for this chaos are, in decreasing order of importance: 1) three-body resonances with Jupiter and Saturn, 2) exterior resonances with Mars, 3) moderate order Jovian resonances, and 4) three-body resonances with Mars and Jupiter.

Two statistical effects in the long-term chaotic asteroidal dynamics are considered, namely the power-law character of the dependence of recurrence times on local Lyapunov times and the power-law decay in the tails of the recurrence distributions. The dependences in both cases are shaped by effects of anomalous transport, due to the presence of the chaos border in phase space, and by statistical selection effects.

We study the dynamics of real asteroids at the 2/1 resonance with Jupiter, and find that Griqua-like objects have a short life-time inside the resonance, while Zhongguo group can survive for many 108yr.

High-eccentricity asteroidal librations are modelled using the high-eccentricity non-planar asymmetric expansion (Roig et al 1997). This second-degree expansion gives us the potential of the perturbing forces acting on a resonant asteroid in a first order resonance in explicit form, as a quadratic polynomial in the canonical non-singular variables. Secular and short periodic perturbations are introduced in the model, giving a more realistic description of the dynamics.

The reducing Sessin’s transformation (Sessin, 1981; Sessin & Ferraz-Mello, 1984) is used to include the main effect of Jupiter’s ecc entricity in the main part of the Hamiltonian. It leads to an integrable first-order approximation known as the second fundamental model for resonance (Henrard & Lemaitre 1983) or Andoyer Hamiltonian (Andoyer 1903).

The two main arenas of astrophysical dynamics are celestial mechanics and stellar dynamics. The former deals with the motion of few bodies in a near-Kepler potential; the latter with the motion of many bodies in a non-Kepler potential. I would like to discuss the hybrid problem of many bodies in a near-Kepler potential, which is relevant to a number of astrophysical systems, including protoplanetary disks and the centers of galaxies containing massive black holes.

Consider a spherical system of radius r, containing N bodies of mass m orbiting in the gravitational field of a body of mass M ≫ N m. Assume that the orbits have moderate eccentricities and random orientations and imagine taking a time exposure of the system over several orbits. Each body is then smeared into an approximate Kepler ellipse, which precesses slowly on a timescale tprec. Each ellipse exerts a force on other bodies at comparable radius, f ~ Gm/r2. The mean force from all the ellipses is Fm ~ N f. The mean force determines the precession time through the relation L ~ rFmtprec, where L ~ (GMr)1/2 is the specific angular momentum of an orbit. Thus tprec ~ (M/Nm)tcr, where tcr ~ (r3/GM)1/2 is the crossing time.

Although interarm features in the Population I disk of spiral galaxies frequently give an impression of a three-fold symmetry (Patsis et al., 1997), true three-armed spiral structures in the old stellar disk are seldomly seen. Such systems are of special interest since they display unique conditions which favor the growth of m=3 modes. The face-on spiral NGC 7137 shows a clear three-armed pattern on K-band images and was used as a prototype for the potential of these systems.

A K-band map of NGC 7137 was decomposed in axisymmetric components and a synthetic rotation curve was generated [see (Grosbøl and Patsis, 1998)]. The maximum rotational velocity was taken to be ≈150 km sec−1 corresponding to the mean value given by (Rubin et al., 1982) for this type of galaxy.

Because of their analytical simplicity and regularity, Stäckel potentials are attractive tools for modelling galaxies. The third integral I3 is explicitly known in a Stäckel potential, and can be used as an approximation to the effective third integral, in order to construct three-integral models (cf. Dejonghe, et al., 1996, A&A 306, 363).

Moreover, Stäckel potentials turn out to yield good global descriptions for either axisymmetric or triaxial systems without central mass concentration (de Zeeuw 1985, MNRAS 216, 273, de Zeeuw & Lynden-Bell 1985, MNRAS 215, 713), and even for some systems with a black hole included (Sridhar & Touma 1997, MNRAS 292, 657).

One long-standing concern though, is that Stäckel potentials form only a very small subspace in the family of all potentials. The main orbit families found by numerical integration in general triaxial potentials are present in a Stäckel potential (Schwarzschild 1981, ApJ 232, 236, de Zeeuw 1985, MNRAS 216, 273), but there is obviously no place in an integrable potential for smaller orbital families or stochastic orbits. However, since regular orbits are the rule rather than the exception, a potential which yields a good representation of those orbits is certainly a good basis for building models.

Number count profiles of many Galactic and some extra-galactic satellite systems show evidence for associated stars beyond the cut-off in density that is identified as the point of tidal limitation (e.g. Irwin & Hatzidimitriou 1995, Grillmair et al. 1995). These “extra-tidal” stars are assumed to be debris lost from the satellite due to heating or stripping by the Galactic tidal field or (in the case of globular clusters) evaporation of stars over the tidal boundary. In this contribution we present a method for using these features to measure the mass loss rate from the satellite, and test it on the results of numerical simulations of satellite disruption. A more detailed discussion of all aspects of this work can be found in Johnston, Sigurdsson & Hemquist (1998).

In the numerical simulations, the satellite’s evolution along an orbit in a three component rigid model of the Galaxy is followed using a self-consistent field (SCF) code (developed by Hernquist & Ostriker 1992) to calculate the mutual interactions of stars in the satellite. Figure 1 shows the annularly averaged number surface density from one example simulation “observed” from the viewpoint of the center of the Galaxy after several Gigayears. The closed (open) symbols show the profile recovered if only bound (all) stars are considered. Clearly there is a break in the open symbols at the radius rbreak where the analysis becomes dominated by unbound stars.

When a body moves through a medium of smaller particles, it suffers a deceleration due to dynamical friction (Chandrasekhar 1943). Dynamical friction is inversely proportional to the relaxation time, which can be defined as the time needed for the orbits to experiment an energy exchange of the order of their initial energies, as a result of the perturbations produced by stellar encounters. Chaotic orbits, present in non-integrable systems, have exponential sensitivity to perturbations, a feature that makes them to relax in a time much shorter than regular ones, which suggests that dynamical friction would increase in the presence of chaotic orbits (Pfenniger 1986). We present preliminary results of numerical experiments used to check this idea, investigating the orbital decay, caused by dynamical friction, of a rigid satellite which moves within a larger stellar system (a galaxy) whose potential is non-integrable. Triaxial models with similar density distributions but different percentages of chaotic orbits are considered. This last quantity depends on the central concentration of the models. If the potential corresponds to triaxial mass models with smooth cores, the regular orbits have shapes that can be identified with one of the four families of regular orbits in Stäckel potentials (box and three types of tubes). Chaotic orbits behave very much like regular orbits for hundreds of oscillations at least. In this case, the galaxy is represented by the triaxial generalization of the γ-models with γ = 0 (Merritt & Fridman 1996). However, the situation is very different in triaxial models with divergent central densities (cusps) or black holes, a feature that is in agreement with the observations. While the tube orbits are not strongly affected by central divergencies, the boxlike orbits are often rendered chaotic (Gerhard & Binney 1985). The rimescale in which the chaos manifests itself in the orbital motion is short compared to a Hubble time. In this models, the compact object is taken as a Plummer sphere.

Observations only partially constrain dynamical models of globular clusters and similar systems, in large part because distances and velocities are seen only in projection, and only at a single (unknown) orbital phase. As a result, some dynamical assumptions are necessary if the cluster dynamics and mass distribution are to be inferred from kinematic data — typically that mass follows light, or that the stars observed are the sole source of the gravitational field in which they move, or that orbital energies are completely thermalized, or that orbital angular momenta are exponentially distributed — most often some combination of these is invoked. Such assumptions, reasonable as they may seem, are almost never justified by the data per se for the cluster in question, and thus diminish the credibility of resulting estimates of cluster dynamics and, especially, of the presence and extent of dark matter.

Orbital characteristics of 80 HIPPARCOS mildly metal-poor stars are derived assuming a Galactic model slightly modified from that of Ostriker and Caldwell (1979). A comparison with other works is performed and a relationship between the dynamical properties of the stars and their well-determined spectroscopie abundances is investigated.

The M dwarf L722-22 (= LHS 1047) was discovered to be a binary system by Ianna 20 years ago. The analysis of the ground- based data indicated a mass 0.06M⊙ for the secondary. This is below the nominal stellar mass limit of 0.08M⊙. The importance of potential “brown-dwarf” candidates, and the fact that the masses of both components place them near the end of the main sequence, made this system a prime object for further, intensive, study.

This close (separation 0."3), faint (V = 11.m5, 14.m4) binary was near the limit for ground-based work. The residuals of an individual night’s photographic data were typically at the 50% level. Also, the photographic images are completely blended. The few one-dimensional speckle data points yielded a merged, asymmetric image profile. Finally, this system is too faint for HIPPARCOS. Our proposal for Hubble Space Telescope Fine Guidance Sensor (FGS) observing was approved in 1992.

In binary stars, tidal friction dissipates a fraction of the orbital energy at constant angular momentum and will circularise binary orbits on a rather short rimescale compared with the nuclear rimescale, provided that at least one star of the binary has a radius comparable to the separation between binary components. This dissipation effectively ceases once the orbit is circularised. In a hierarchical triple system such dissipation cannot cease entirely, as neither inner nor outer orbit can become exactly circular because of the perturbation of the third distant body. Thus in such systems tidal friction can lead to a steady secular decrease of the inner semimajor axis, accompanied by transfer of angular momentum from the inner to the outer pair, persisting over the whole nuclear lifetime of the system. The situation can be even more dramatic if two orbits have high relative inclination i > 40° It can be shown analytically and numerically (see e.g., Kozai 1962, Marchal 1990, Kiseleva 1996 and references therein) that for triple systems with high relative inclination there is a quasi-periodic change of the inner eccentricity (on a timescale ) during which it reaches a maximum value . This value only depends on the inclination i between the two orbital planes; other parameters affect only the timescale. For example, if we approximate a triple stellar system like β Per (Algol) (m1 = 0.8M⊙, m2 = 3.7M⊙, Pin = 2.87 days; m3 = 1.7M⊙, Pout = 1.86 yr, eout = 0.23; i = 100°) as three point masses, then the inner eccentricity ein cycles rather smoothly between 0 and 0.985, while i fluctuates between 100° and 140°. We call these fluctuations ‘Kozai cycles’. Such ‘Kozai cycles’ do not actually occur in this semi-detached system: they can be damped to a small value by tidal friction, but in fact they are also strongly reduced by the non-dissipative effect of the quadrupole moments of the two stars in the inner pair. This effect produces apsidal motion which is much more rapid than the apsidal motion due to the third star, and so prevents the Kozai cycles from operating.

We consider elliptical galactic models, whose dynamical system consists of a three-dimensional isotropic harmonic oscillator plus a potential given by a homogeneous polynomial of degree four with an additional discrete symmetry. We identify families of simple periodic orbits by studying the reduced phase space.