Mapping the three-body system - decay time and reversibility

In this paper we carry out a quantitative analysis of the three-body systems and map them as a function of decaying time and intial conguration, look at this problem as an example of a simple deterministic system, and ask to what extent the orbits are really predictable. We have investigated the behavior of about 200 000 general Newtonian three body systems using the simplest initial conditions. Within our resolution these cover all the possible states where the objects are initially at rest and have no angular momentum. We have determined the decay time-scales of the triple systems and show that the distribution of this parameter is fractal in appearance. Some areas that appear stable on large scales exhibit very narrow strips of instability and the overall pattern, dominated by resonances, reminds us of a traditional Maasai warrior shield. Also an attempt is made to recover the original starting conguration of the three bodies by backward integration. We find there are instances where the evolution to the future and to the past lead to different orbits, in spite of time symmetric initial conditions. This implies that even in simple deterministic systems there exists an Arrow of Time.Comment: 8 pages, 9 figures. Accepted for publication in MNRAS. Includes low-resolution figures. High-resolution figures are available as PNG

A Hybrid N-Body Code Incorporating Algorithmic Regularization and Post-Newtonian Forces

We describe a novel N-body code designed for simulations of the central regions of galaxies containing massive black holes. The code incorporates Mikkola's 'algorithmic' chain regularization scheme including post-Newtonian terms up to PN2.5 order. Stars moving beyond the chain are advanced using a fourth-order integrator with forces computed on a GRAPE board. Performance tests confirm that the hybrid code achieves better energy conservation, in less elapsed time, than the standard scheme and that it reproduces the orbits of stars tightly bound to the black hole with high precision. The hybrid code is applied to two sample problems: the effect of finite-N gravitational fluctuations on the orbits of the S-stars; and inspiral of an intermediate-mass black hole into the galactic center.Comment: 12 pages, 15 figures, accepted for publication in MNRA

Asteroids in the Inner Solar System I - Existence

Ensembles of in-plane and inclined orbits in the vicinity of the Lagrange points of the terrestrial planets are integrated for up to 100 million years. The integrations incorporate the gravitational effects of Sun and the eight planets (Pluto is neglected). Mercury is the least likely planet, as it is unable to retain tadpole orbits over 100 million year timescales. Both Venus and the Earth are much more promising, as they possess rich families of stable tadpole and horseshoe orbits. Our survey of Trojans in the orbital plane of Venus is undertaken for 25 million years. Some 40% of the survivors are on tadpole orbits. For the Earth, the integrations are pursued for 50 million years. The stable zones in the orbital plane are larger for the Earth than for Venus, but fewer of the survivors are tadpoles. Both Venus and the Earth also have regions in which inclined test particles can endure near the Lagrange points. For Venus, only test particles close to the orbital plane are stable. For the Earth, there are two bands of stability, one at low inclinations (i < 16 degrees) and one at moderate inclinations (between 24 degrees and 34 degrees). The inclined test particles that evade close encounters are primarily moving on tadpole orbits. Our survey of in-plane test particles near the Martian Lagrange points shows no survivors after 60 million years. Low inclination test particles do not persist, as their inclinations are quickly increased until the effects of a secular resonance with Jupiter cause de-stabilisation. Numerical integrations of inclined test particles for timespans of 25 million years show stable zones for inclinations between 14 and 40 degrees.Comment: 20 pages, 21 figures, Monthly Notices (in press

Dynamics of two planets in co-orbital motion

We study the stability regions and families of periodic orbits of two planets locked in a co-orbital configuration. We consider different ratios of planetary masses and orbital eccentricities, also we assume that both planets share the same orbital plane. Initially we perform numerical simulations over a grid of osculating initial conditions to map the regions of stable/chaotic motion and identify equilibrium solutions. These results are later analyzed in more detail using a semi-analytical model. Apart from the well known quasi-satellite (QS) orbits and the classical equilibrium Lagrangian points L4 and L5, we also find a new regime of asymmetric periodic solutions. For low eccentricities these are located at $(\sigma,\Delta\omega) = (\pm 60\deg, \mp 120\deg)$, where \sigma is the difference in mean longitudes and \Delta\omega is the difference in longitudes of pericenter. The position of these Anti-Lagrangian solutions changes with the mass ratio and the orbital eccentricities, and are found for eccentricities as high as ~ 0.7. Finally, we also applied a slow mass variation to one of the planets, and analyzed its effect on an initially asymmetric periodic orbit. We found that the resonant solution is preserved as long as the mass variation is adiabatic, with practically no change in the equilibrium values of the angles.Comment: 9 pages, 11 figure

Disruption of the three-body gravitational systems: Lifetime statistics

We investigate statistics of the decay process in the equal-mass three-body problem with randomized initial conditions. Contrary to earlier expectations of similarity with "radioactive decay", the lifetime distributions obtained in our numerical experiments turn out to be heavy-tailed, i.e. the tails are not exponential, but algebraic. The computed power-law index for the differential distribution is within the narrow range, approximately from -1.7 to -1.4, depending on the virial coefficient. Possible applications of our results to studies of the dynamics of triple stars known to be at the edge of disruption are considered.Comment: 13 pages, 2 tables, 3 figure

On the dissolution of star clusters in the Galactic centre. I. Circular orbits

We present N-body simulations of dissolving star clusters close to galactic centres. For this purpose, we developed a new N-body program called nbody6gc based on Aarseth's series of N-body codes. We describe the algorithm in detail. We report about the density wave phenomenon in the tidal arms which has been recently explained by Kuepper et al. (2008). Standing waves develop in the tidal arms. The wave knots or clumps develop at the position, where the emerging tidal arm hits the potential wall of the effective potential and is reflected. The escaping stars move through the wave knots further into the tidal arms. We show the consistency of the positions of the wave knots with the theory in Just et al. (2009). We also demonstrate a simple method to study the properties of tidal arms. By solving many eigenvalue problems along the tidal arms, we construct numerically a 1D coordinate system whose direction is always along a principal axis of the local tensor of inertia. Along this coordinate system, physical quantities can be evaluated. The half-mass or dissolution times of our models are almost independent of the particle number which indicates that two-body relaxation is not the dominant mechanism leading to the dissolution. This may be a typical situation for many young star clusters. We propose a classification scheme which sheds light on the dissolution mechanism.Comment: 18 pages, 20 figures; accepted by MNRA

The 1990 MB: The first Mars Trojan

Asteroid 1990 MB was discovered by D. H. Levy and H. E. Holt during the course of the Mars and Earth Crossing Asteroid and Comet Survey. An orbit based on a 9 day arc and the asteroid's location near Mars' L5 (trailing Lagrangean) longitude led E. Boswell to speculate that it might be in 1:1 resonance with Mars, analogous to the Trojan asteroids of Jupiter. Subsequent observations strengthened the possibility, and later calculations confirmed it. Thus 1990 MB is the first known asteroid in 1:1 resonance with a planet other than Jupiter. The existence of 1990 MB (a small body most likely between 2 and 4 km in diameter) provides remarkable confirmation of computer simulations. These self consistent n-body simulations demonstrated this sort of stability for Trojans of all the terrestrial planets over at least a 2 million year time base. The discovery of 1990 MB suggests that others of similar or smaller diameter may be found. Using hypothetical populations of Mars Trojans, their possible sky plane distributions were modeled as a first step in undertaking a systematic observational search of Mars' L4 and L5 libration regions

Symplectic integration of space debris motion considering several Earth's shadowing models

In this work, we present a symplectic integration scheme to numerically compute space debris motion. Such an integrator is particularly suitable to obtain reliable trajectories of objects lying on high orbits, especially geostationary ones. Indeed, it has already been demonstrated that such objects could stay there for hundreds of years. Our model takes into account the Earth's gravitational potential, luni-solar and planetary gravitational perturbations and direct solar radiation pressure. Based on the analysis of the energy conservation and on a comparison with a high order non-symplectic integrator, we show that our algorithm allows us to use large time steps and keep accurate results. We also propose an innovative method to model Earth's shadow crossings by means of a smooth shadow function. In the particular framework of symplectic integration, such a function needs to be included analytically in the equations of motion in order to prevent numerical drifts of the energy. For the sake of completeness, both cylindrical shadows and penumbra transitions models are considered. We show that both models are not equivalent and that big discrepancies actually appear between associated orbits, especially for high area-to-mass ratios