Planetary close encounters: Probability distributions of resultant orbital elements and applications to Hidalgo and Chiron

A method is described for computing the probability distributions of the new orbital elements (a, e, i, q, q′) of a minor body which is subject ot close encounters with a planet. By including the frequency of such encounters the rate at which one class of orbit is transposed into a new class (e.g., Mars-crossing asteroids changed into Apollos) can be estimated. By applying this technique to the cases of Hidalgo and Chiron its uses are illustrated, and its limitations due to the two, two-body approximation utilized are pointed out.     

Planetary close encounters: Importance of nearly tangent orbits

It is shown that close encounters between Jupiter and minor bodies are generally more efficient if the initial orbit of the small body is nearly tangent to that of the planet. Starting from the analysis of the results of previous numerical simulations, some indications on the mobility of the small bodies in the semiaxis-eccentricity diagram are given.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratory di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Planetary close encounters between Jupiter and about 3000 fictitious minor bodies

Single close encounters between Jupiter and about 3000 hypothetical minor bodies, initially on elliptical orbits, have been studied computing the evolution of the three-body system Sun-Jupiter-object, by means of a new numerical method of integration. The fictitious population processed contains almost all the orbits which allow a close approach to the planet. The efficiency of a single encounter in varying the orbital parameters of the objects resulted to be generally poor, as it is shown by the distributions of the orbital parameter variations. Collisions and ejections from the solar system on hyperbolic orbits are little numerous; some temporary satellite capture have been recognised. The results of this work show that any attempt to study the close encounter event by means of two distinct two-body problems is physically meaningless because the mid-range perturbations, disregarded in such cases, are very far from being negligible.     

Excitation of oscillation modes by tides in close binaries: constraints on stellar and orbital parameters

The parameter space favourable for the resonant excitation of free oscillation modes by dynamic tides in close binary components is explored using qualitative considerations to estimate the order of magnitude of the tidal force and the frequency range covered by the tidally induced oscillations. The investigation is valid for slowly rotating stars with masses in the interval between 2 and  20 M_⊙  , and an evolutionary stage ranging from the beginning to the end of the main sequence. Oscillation modes with eigenfrequencies of the order of five times the inverse of the dynamical time-scale  τ_dyn  of the star, i.e. the lowest-order p -modes, the f -mode and the lowest-order g ⁺-modes, are found to be outside the favourable parameter space since their resonant excitation requires orbital eccentricities that are too high for the binary to stay detached when the components pass through the periastron of their relative orbit. Resonances between dynamic tides and g ⁺-modes with frequencies of the order of half of the inverse of the dynamical time-scale of the star on the other hand are found to be favourable for orbital periods up to  ∼200τ_dyn  , provided that the binary mass ratio q is larger than 1/3, and the orbital eccentricity e is larger than ∼0.25. This favourable range comes down to orbital periods of up to 5–12 d in the case of  2–20 M_⊙  zero-age main-sequence binary components, and orbital periods of up to 21–70 d in the case of terminal main-sequence binary components.     

Collisions and close encounters involving massive main-sequence stars

We study close encounters involving massive main-sequence stars and the evolution of the exotic products of these encounters as common-envelope systems or possible hypernova progenitors. We show that parabolic encounters between low- and high-mass stars and between two high-mass stars with small periastrons result in mergers on time-scales of a few tens of stellar free-fall times (a few tens of hours). We show that such mergers of unevolved low-mass stars with evolved high-mass stars result in little mass-loss  (∼0.01 M_⊙)  and can deliver sufficient fresh hydrogen to the core of the collision product to allow the collision product to burn for several million years. We find that grazing encounters enter a common-envelope phase which may expel the envelope of the merger product. The deposition of energy in the envelopes of our merger products causes them to swell by factors of ∼100. If these remnants exist in very densely populated environments  ( n ≳ 10⁷ pc⁻³)  , they will suffer further collisions which may drive off their envelopes, leaving behind hard binaries. We show that the products of collisions have cores rotating sufficiently rapidly to make them candidate hypernova/gamma-ray burst progenitors and that ∼0.1 per cent of massive stars may suffer collisions, sufficient for such events to contribute significantly to the observed rates of hypernovae and gamma-ray bursts.     

QYMSYM: A GPU-accelerated hybrid symplectic integrator that permits close encounters

We describe a parallel hybrid symplectic integrator for planetary system integration that runs on a graphics processing unit (GPU). The integrator identifies close approaches between particles and switches from symplectic to Hermite algorithms for particles that require higher resolution integrations. The integrator is approximately as accurate as other hybrid symplectic integrators but is GPU accelerated.     

Planetary orbital equations in externally-perturbed systems: position and velocity-dependent forces

The increasing number and variety of extrasolar planets illustrates the importance of characterizing planetary perturbations. Planetary orbits are typically described by physically intuitive orbital elements. Here, we explicitly express the equations of motion of the unaveraged perturbed two-body problem in terms of planetary orbital elements by using a generalized form of Gauss’ equations. We consider a varied set of position and velocity-dependent perturbations, and also derive relevant specific cases of the equations: when they are averaged over fast variables (the “adiabatic” approximation), and in the prograde and retrograde planar cases. In each instance, we delineate the properties of the equations. As brief demonstrations of potential applications, we consider the effect of Galactic tides. We measure the effect on the widest-known exoplanet orbit, Sedna-like objects, and distant scattered disk objects, particularly with regard to where the adiabatic approximation breaks down. The Mathematica code which can help derive the equations of motion for a user-defined perturbation is freely available upon request.     

Asteroid close encounters characterization using differential algebra: the case of Apophis

A method for the nonlinear propagation of uncertainties in Celestial Mechanics based on differential algebra is presented. The arbitrary order Taylor expansion of the flow of ordinary differential equations with respect to the initial condition delivered by differential algebra is exploited to implement an accurate and computationally efficient Monte Carlo algorithm, in which thousands of pointwise integrations are substituted by polynomial evaluations. The algorithm is applied to study the close encounter of asteroid Apophis with our planet in 2029. To this aim, we first compute the high order Taylor expansion of Apophis’ close encounter distance from the Earth by means of map inversion and composition; then we run the proposed Monte Carlo algorithm to perform the statistical analysis.     

Rotationally-perturbed orbital elements of close binary systems

The aim of this investigation is to present the secular and periodic perturbations of the six orbital elements of a close binary system due to rotational distortion. In our study we consider very small inclinationst of the orbital plane of the system, whereas the eccentricity of the orbit may assume any value between 0<e<1. The final formulae for the various elements have been expressed by means of the unperturbed true anomaly measured from the ascending node.     

Determination of asteroid masses from their close encounters with Mars

An obstacle to the asteroid mass determination lies in the difficulty in isolating the gravitational perturbation exerted by a single asteroid on the planets, being strongly correlated and mixed up with those of many other asteroids. This hindrance may be avoided by the method of analysis presented here: an asteroid mass is estimated in correspondence with its close encounters with Mars where the acceleration it induces on the planet can be sufficiently disentangled from those generated by the remaining asteroid masses to calculate. We test this technique in the analysis of range observations to Mars Global Surveyor and Mars Express performed from 1999 to 2007. For this purpose, we adopt the dynamical model of the planetary ephemeris INPOP06 (Fienga et al., 2008), which includes the gravitational influences of the 300 most perturbing asteroids of the Martian orbit. We obtain the solutions of 10 asteroid masses that have the largest effects on this orbit over the period examined: they are generally in good agreement with determinations recently published.     

Planetary core size: A seismological approach

A key parameter for understanding the geodynamics of a terrestrial planet is the size of its core. Numerical evaluation of 28 different interior structure models of Mercury, Venus, Earth, the Moon, and Mars suggests that there is an almost linear relationship between the core radius and the extent of the seismic P-wave core shadow. A scaling law is derived from a simple mantle density and velocity model that permits the interpretation of respective seismic measurements on terrestrial planetary bodies.     

Evolution of NEO rotation rates due to close encounters with Earth and Venus

In this paper we study the statistical effect of planetary flybys on the rotation rates and states of Near Earth Objects (NEOs). Our approach combines numerical and analytical methods within a Monte Carlo model that simulates the evolution of the NEO spin rates. We take as input for the simulation a source distribution of spin states and evolve it to find their steady state distribution. In performing this evolution we track the changes in the spin rate and state distribution for the different components of the NEO population. We show that the cumulative effect of planetary encounters is to spin up the overall population of NEOs. This spin up effect holds on average only, and particular members of the population may experience an overall decrease in rotation rate. This effect is clearly seen across all components of the NEO population and is significant both statistically and physically. For initially slow rotators the spin up effect is strong, lowering the mean rotation period by 32%. For faster rotating populations the effect is less, lowering the spin period by 15% for the intermediate case, 6% for fast rotating rubble piles, and 8% for fast rotating monoliths. Physically, the spin up effect pushes 1% of the fast rotating rubble-pile NEOs over the disruption limit, while 6% of these bodies experience a sub-disruption event that could modify their physical structure. For monolithic NEOs, the spin up effect is self-limiting, reaching a minimum spin period of 1.1 hr, with a strong cut-off between 2-3 hr. This has two implications. First, it may not be necessary to invoke the rubble-pile hypothesis to recover a cut-off in spin period. Second, it shows that planetary flybys cannot account for the extremely rapid rotation rates of some small NEOs. We also tested a different balance between the effects of Earth and Venus by treating the Aten sub-class of asteroids separately. Due to increased interactions with the planets, the spin up effect is more pronounced (10%) and disruptions increase by a factor of three. The slow rotation tails of the spin distributions are increased to longer periods, in general, with rotation periods of over 100 hr occurring for a few tenths of a percent for some component populations. Thus, this mechanism may account for some of the noted excess in slow rotators among the NEOs. Planetary flybys also cause NEOs to enter a tumbling state, with approximately 0.5% of the population being placed into a long-axis rotation mode. Finally, based on the evolution of spin states of different components of the NEO population, we compared the evolved states with the measured distribution of NEOs to estimate the relative populations of these components that comprise the NEOs.     

A model for binary-binary close encounters and collisions from a dynamical point of view

The goal of this paper is to provide a model for binary-binary interactions in star clusters, which is based on simultaneous binary collision of a special case of the one-dimensional 4-body problem where four masses move symmetrically about the center of mass. From the theoretical point of view, the singularity due to binary collisions between point masses can be handled by means of regularization theory. Our main tool is a change of coordinates due to McGehee by which we blow-up the singular set associated to total collision and replace it with an invariant manifold which includes binary and simultaneous binary collisions, and then gain a complete picture of the local behavior of the solutions near to total collision via the homothetic orbit.     

Conservation of the Tisserand parameter at close encounters of interplanetary objects with Jupiter

This article has a long story. It was first written between 1979 and 1981, at the beginning of our cooperation with ubor Kresák. The goal of the paper was to check the conservation of various formulations of the Tisserand parameter and to investigate the relationships between theT-values and the dynamical behaviours at close encounters with Jupiter. A first draft of the paper was subsequently enlarged and revised in 1983, as new findings led us to the addition of another substantial section. Then, the paper has remained in a draft version up to now, being modified from time to time because of our committments with the computations of long-term evolutions of short-period comets. We want to honour the memory of ubor letting this article finally come out.Andrea Carusi and Giovanni B. Valsecchi, Rome, 1994     

WZ Sge stars/TOADs and (soft) X-ray transients: close encounters of the same kind

The outbursts of WZ Sge stars (or TOADs), are compared to those seen in the (soft) X-ray transients. Both types of outbursts exhibit strong similarities: large amplitudes, long recurrence times, occurrence of superhumps, and of rebrightenings or reflares at the end or just after the main outburst. This suggests that the same kind of mechanism is at work to produce these outbursts. I also briefly discuss their differences and whether superoutbursts may exist among the very long (≳month) outbursts in U Gem stars. This report is basically similar to that by Kuulkers (1999), with a few modifications.     

Orbital evolution of the Gefion and Adeona asteroid families: close encounters with massive asteroids and the Yarkovsky effect

Asteroid families are groupings of minor planets identified by clustering in their proper orbital elements; these objects have spectral signatures consistent with an origin in the break-up of a common parent body. From the current values of proper semimajor axes a of family members one might hope to estimate the ejection velocities with which the fragments left the putative break-up event (assuming that the pieces were ejected isotropically). However, the ejection velocities so inferred are consistently higher than N-body and hydro-code simulations, as well as laboratory experiments, suggest. To explain this discrepancy between today’s orbital distribution of asteroid family members and their supposed launch velocities, we study whether asteroid family members might have been ejected from the collision at low speeds and then slowly drifted to their current positions, via one or more dynamical processes. Studies show that the proper a of asteroid family members can be altered by two mechanisms: (i) close encounters with massive asteroids, and (ii) the Yarkovsky non-gravitational effect. Because the Yarkovsky effect for kilometer-sized bodies decreases with asteroid diameter D, it is unlikely to have appreciably moved large asteroids (say those with D > 15 km) over the typical family age (1-2 Gyr).For this reason, we numerically studied the mobility of family members produced by close encounters with main-belt, non-family asteroids that were thought massive enough to significantly change their orbits over long timescales. Our goal was to learn the degree to which perturbations might modify the proper a values of all family members, including those too large to be influenced by the Yarkovsky effect. Our initial simulations demonstrated immediately that very few asteroids were massive enough to significantly alter relative orbits among family members. Thus, to maximize gravitational perturbations in our 500-Myr integrations, we investigated the effect of close encounters on two families, Gefion and Adeona, that have high encounter probabilities with 1 Ceres, by far the largest asteroid in the main belt. Our results show that members of these families spreads in a of less than 5% since their formation. Thus gravitational interactions cannot account for the large inferred escape velocities.The effect of close encounters with massive asteroids is, however, not entirely negligible. For about 10% of the simulated bodies, close encounters increased the “inferred” ejection velocities from sub-100 m/s to values greater than 100 m/s, beyond what hydro-code and N-body simulations suggest are the maximum possible initial ejection velocity for members of Adeona and Gefion with D > 15 km. Thus this mechanism of mobility may be responsible for the unusually high inferred ejection speeds of a few of the largest members of these two families.To understand the orbital evolution of the entire family, including smaller members, we also performed simulations to account for the drift of smaller asteroids caused by the Yarkovsky effect. Our two sets of simulations suggest that the two families we investigated are relatively young compared to larger families like Koronis and Themis, which have estimated ages of about 2 Byr. The Adeona and Gefion families seems to be no more than 600 and 850 Myr old, respectively.     

Internal stellar rotation and orbital period modulation in close binary systems

The orbital period modulation, observed in close binary systems with late-type secondary stars, is considered in the framework of a general model that allows us to test the hypothesis proposed by Applegate. It relates the orbital period variation to the modulation of the gravitational quadrupole moment of their magnetically active secondary stars produced by angular momentum exchanges within their convective envelopes. By considering the case of RS CVn binary systems, it is found that the surface angular velocity variation of the secondary component required by Applegate's hypothesis is between 4 and 12 per cent, i.e. too large to be compatible with the observations and that the kinetic energy dissipated in its convection zone ranges from 4 to 43 times that supplied by the stellar luminosity along one cycle of the orbital period modulation. Similar results are obtained for other classes of close binary systems by applying a scaling relationship based on a simplified internal structure model. The effect of rapid rotation is briefly discussed finding that it is unlikely that the rotational quenching of the turbulent viscosity may solve the discrepancy. Therefore, the hypothesis proposed by Applegate is not adequate to explain the orbital period modulation of close binary systems with a late-type secondary.