Model orbits of asteroid 3040 Kozai

The orbital evolutions of the asteroid 3040 Kozai and model asteroids with similar orbits have been investigated. Their osculating orbits for an epoch 1991 December 10 were numerically integrated forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all inner planets, Jupiter, and Saturn.The orbit of the asteroid Kozai is stable. Its motion is affected only by long-period perturbations of planets. With change of the argument of perihelion of the asteroid Kozai, the evolution of the model asteroid orbits changes essentially, too. The model orbits with the argument of perihelion changed by the order of 10% show that asteroids with such orbital parameters may approach the Earth orbit, while asteroids with larger changes may even cross it, at least after 10,000 years. Long-term orbital evolution of asteroids with these orbital parameters is very sensitive on their angular elements.     

The orbital evolution of NEA 30825 1900 TG1

The orbital evolution of the near-Earth asteroid (NEA) 30825 1990 TG1 has been studied by numerical integration of the equations of its motion over the 100 000-year time interval with allowance for perturbations from eight major planets and Pluto, and the variations in its osculating orbit over this time interval were determined. The numerical integrations were performed using two methods: the Bulirsch-Stoer method and the Everhart method. The comparative analysis of the two resulting orbital evolutions of motion is presented for the time interval examined. The evolution of the asteroid motion is qualitatively the same for both variants, but the rate of evolution of the orbital elements is different. Our research confirms the known fact that the application of different integrators to the study of the long-term evolution of the NEA orbit may lead to different evolution tracks.     

A survey of orbits of co-orbitals of Mars

Many asteroids with a semimajor axis close to that of Mars have been discovered in the last several years. Potentially some of these could be in 1:1 resonance with Mars, much as are the classic Trojan asteroids with Jupiter, and its lesser-known horseshoe companions with Earth. In the 1990s, two Trojan companions of Mars, 5261 Eureka and 1998 VF₃₁, were discovered, librating about the L₅ Lagrange point, 60° behind Mars in its orbit. Although several other potential Mars Trojans have been identified, our orbital calculations show only one other known asteroid, 1999 UJ₇, to be a Trojan, associated with the L₄ Lagrange point, 60° ahead of Mars in its orbit. We further find that asteroid 36017 (1999 ND₄₃) is a horseshoe librator, alternating with periods of Trojan motion. This asteroid makes repeated close approaches to Earth and has a chaotic orbit whose behavior can be confidently predicted for less than 3000 years. We identify two objects, 2001 HW₁₅ and 2000 TG₂, within the resonant region capable of undergoing what we designate “circulation transition”, in which objects can pass between circulation outside the orbit of Mars and circulation inside it, or vice versa. The eccentricity of the orbit of Mars appears to play an important role in circulation transition and in horseshoe motion. Based on the orbits and on spectroscopic data, the Trojan asteroids of Mars may be primordial bodies, while some co-orbital bodies may be in a temporary state of motion.     

Analytical method for the effects of the asteroid belt on planetary orbits

Analytic expressions are derived for the perturbation of planetary orbits due to a thick constant density asteroid belt. The derivations include extensions and adaptations of Plakhov's analytic expressions for the perturbations in five of the orbital elements for closed orbits around Saturn's rings. The equations of Plakhov are modified to include the effect of ring thickness and additional equations are derived for the perturbations in the sixth orbital element, the mean anomaly. The gravitational potential and orbital perturbations are derived for the asteroid belt with and without thickness, and for a hoop approximation to the belt. The procedures are also applicable to Saturn's rings and the newly discovered rings of Uranus.The effects of the asteroid belt thickness on the gravitational potential coefficients and the orbital motions are demonstrated. Comparisons between the Mars orbital perturbations obtained using the analytic expressions and those obtained using numerical integration are discussed. The effects of the asteroid belt on the Earth based ranging to Mars are also demonstrated.     

Determining asteroid masses from perturbations on Mars

The orbit of Mars is perturbed more than 5 m, a value compatible with the accuracy of the Viking lander ranging data, by about three dozen asteroids. In addition to larger asteroids throughout the belt, significant perturbations of long period are generated by smaller objects near commensurabilities with Mars. The largest periodic terms induced by 1 Ceres and 2 Pallas have amplitudes of 0.8 and 0.2 km, respectively, both with 10-year periods. Due to a near commensurability, 4 Vesta causes a 5-km, 52-year term. While the Viking ranges will yield significant mass determinations for the largest three asteroids, and some of the smaller bodies should be detectable, it will be difficult to seperate the smaller bodies with useful accuracies. Accurate discrimination must await range data from future missions to Mars or other bodies in the neighborhood of the asteroid belt. The Viking ranges can also yield improved masses for the outer planets (except Pluto), an application which is being exploited by groups analyzing these data. Uncertainties in the asteroid masses limit the ultimate accuracy of the Viking determinations of both the long time scale motion of the system the inner four planets with respect to an inertial frame and the rate of change of the gravitational constant.     

Numerical Simulation of the Orbital Evolution of Near-Earth Asteroids Close to Mean Motion Resonances

The dynamics of near-Earth asteroids near mean motion resonances with the Earth or other planets is considered. The probability domains of the motion of some near-Earth asteroids close to low-order resonances are presented. The investigations have been carried out by means of a numerical integration of differential equations, taking into account the perturbations from the major planets and the Moon. For each investigated object an ensemble of 100 test particles with orbital elements nearby those of the nominal orbit has been constructed and its evolution has been retraced over the time interval (–3000, +3000 years). The initial set of orbits has been generated on the basis of probable variations of the initial orbital elements obtained from the least square analysis of observations.This revised version was published online in October 2005 with corrections to the Cover Date.     

1984 AB—A unique Mars-crossing asteroid

Asteroid 1984 AB, discovered in January 1984, proved to be a unique object with a close dynamical relationship to Mars. A brief history of the discovery and subsequent “evolution” of the orbit as it was refined is presented. The preliminary orbit of 1984 AB indicated that it might be a Mars Trojan, and an extended discussion of this interesting possibility is presented, but this hypothesis had to be dismissed after further observations had refined the orbit. The semimajor axis and orbital eccentricity are very similar to that of Mars. No other known Mars-crossing asteroid exists with an orbit as closely associated to Mars.     

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.     

On the stability of a planet between Mars and the asteroid belt: Implications for the Planet V hypothesis

The stability of an additional planet between the orbit of Mars and the asteroid belt is examined in the context of the Planet V hypothesis. In this model, the Solar System initially contained a fifth terrestrial planet, “Planet V,” which was removed after ∼700 Myr, a possible trigger for the late heavy bombardment on the inner planets. The model is investigated using 96 N-body integrations of the 8 major planets with an additional body between Mars and the asteroid belt. In more than 1/4 of simulations, Planet V survives for 1000 Myr. In most other cases, Planet V collides with the Sun or hits another planet after several hundred Myr, leaving 4 surviving terrestrial planets. In 24/96 simulations, Planet V is lost by ejection or collision with the Sun while the other four terrestrial planets survive without undergoing a collision. In 18 cases, Planet V is removed at least 200 Myr after the beginning of the simulation. The endstate depends sensitively on the mass of Planet V. Collision with the Sun is likely when Planet V's mass is 0.25 Mars masses or less. When Planet V is more massive than this, collisions involving it and/or other terrestrial planets become commonplace. In unstable systems, the times of first encounter and first collision/ejection depend on the initial aphelion distance of Mars. Reducing Mars's aphelion distance increases these times and also increases the fraction of systems surviving for 1000 Myr. When Mars's current orbit is used, the stability of Planet V increases when these two planets are widely separated initially. Planet V's aphelion distance Q typically begins to cross the asteroid belt within a few tens to a few hundred Myr, and its orbit last leaves the belt several hundred Myr later in most cases. The total time spent with Q>2.1 AU is typically less than 200 Myr.     

Stability of the solar system: Evidence from the asteroids

An analysis of the distribution of the orbital periods of the asteroids has shown that there is a preference for these periods to be near-commensurate with that of Mars. We suggest that this preference is associated with a formation process and implies that the orbital period of Mars has not changed greatly since the time of asteroid formation. We deduce from this that the solar system is highly stable and long-period gravitational perturbations have probably had little influence on the gross evolution of the solar system.Paper presented to the NATO Advanced Study Institute on Lunar Studies, Patras, Greece, September 1971.     

Multi-SunSynchronous Orbits in the Solar System

Performances of a planetary observation system are strongly related to the choice of the orbit used. Trajectories with characteristics of periodicity are very useful for the assessment of time-varying phenomena and thus Periodic SunSynchronous and Periodic Multi-SunSynchronous Orbits are particularly suitable to this end. In this paper, the research into these kinds of orbits, previously proposed for the Earth and Mars, has been extended to planets of the Solar System and to their principal moons. In general, these trajectories are typically obtained under the hypothesis that the J₂ harmonic is predominant with respect to the other orbital perturbations, since this allows an analytical solution. However, the hypothesis of J₂ predominant is not always verified in the Solar System and so analytical techniques must be replaced by numerical simulations. Interesting results have been obtained for the planets Mars and Jupiter and for the moons Europa, Callisto and Titan, where periodic trajectories with reduced revisit times and low altitudes have been found. These solutions allow the observation of time-varying phenomena with high spatial and temporal resolution.     

Palomar-Leiden minor planets: Proper elements,frequency distributions,belt boundaries,and family memberships

Proper elements have been calculated for 1227 higher accuracy orbits from the Palomar-Leiden Survey of faint minor planets. Tabulations are given for the special orbits: Earth and deep Mars crossers, Trojans, Hildas, and one 2:1 librator. The frequency distributions of the proper semimajor axis, eccentricity, sine inclination, longitudes of perihelion and node plus their rates, and the closest distances of approach to Mars and Jupiter are displayed as histograms and discussed. The distribution of the closest approach distance to Mars drops off sharply near zero while that for Jupiter vanishes near 1.1 AU. Mars and Jupiter have apparently caused these boundaries and the asteroid belt must have been larger early in the history of the solar system. 3.5% of the sample can impact Mars. Most of these potential impactors are shallow crossers which require occasional fortuitous alignments of the secular terms to intersect Mars' orbit so that long lifetimes result and moderate populations remain. As these fortuitous alignments occur with near simultaneity for many, but not all, asteroids the shallow crossers in the observed size range will episodically bombard Mars. Proper elements have been used to recognize families and 49% of the sample of minor planets fall into these families. The proper elements and family assignments are tabulated.     

Orbital history of the Martian satellites with inferences on their origin

Recent Viking results indicate the Martian satellites are composed of carbonaceous chondritic material, suggesting that Phobos and Deimos were once asteroids captured by Mars. On the other hand, the low eccentricities and inclinations of their orbits on the equator of Mars argue against that hypothesis. This paper presents detailed calculations of the tidal evolution of Phobos and Deimos, considering dissipation in both Mars and its satellites simultaneously and using a new method applicable for any value of the eccentricity. In particular, including precession of the satellites' orbits indicates that they have always remained close to their Laplacian plane, so that the orbital planes of Phobos and Deimos switched from near the Martian orbital plane to the Martian equator once the perturbations due to the planetary oblateness dominated the solar perturbations, as they do presently. The results show that Deimos has been little affected by tides, but several billion (10⁹) years ago, Phobos was in a highly eccentric orbit lying near the common plane of the solar system. This outcome is obtained for very reasonable values of dissipation inside Mars and inside Phobos. Implications for the origin of the Martian satellites are discussed.     

Explicit Semianalytical Theory of Asteroid Motion

A new semianalytical theory of asteroid motion is presented. The theory is developed on the basis of Kaula's expansion of the disturbing function including terms up to the second order with respect to the masses of disturbing bodies. The theory is constructed in explicit form that gives the possibility to study separately the influence of different perturbations in the dynamics of minor planets. The mean-motion resonances with major planets as well as mixed three-body resonances can also be taken into account. For the non-resonant case the formulas obtained can be used for deriving the second transformation to calculate the proper elements of an asteroid orbit in closed form with respect to inclinations and eccentricities. This revised version was published online in July 2006 with corrections to the Cover Date.     

A dynamical investigation of the conjecture that Mercury is an escaped satellite of Venus

The possibility that Mercury might once have been satellite of a Venus, suggested by a number of anomalies, is investigated by a series of numerical computer experiments. Tidal interaction between Mercury and Venus would result in the escape of Mercury into a solar orbit. Only two escape orbits are possible, one exterior and one interior to the Venus orbit. For the interior orbit, subsequent encounters are sufficiently distant to avoid recapture or large perturbations. The perihelion distance of Mercury tends to decrease, while the orientation of perihelion librates for the first few thousand revolutions. If dynamical evolution or nonconservative forces were large enough in the early solar system, the present semimajor axes could have resulted. The theoretical minimum quadrupole moment of the inclined rotating Sun would rotate the orbital planes out of coplanarity. Secular perturbations by the other planets would evolve the eccentricity and inclination of Mercury's orbit through a range of possible configurations, including the present orbit. Thus the conjecture that Mercury is an escaped satellite of Venus remains viable, and is rendered more attractive by our failure to disprove it dynamically.     

Orbital Evolution of Extrasolar Planets

The recent discovery of extrasolar planets and planetary systems has raised many new research problems for astronomers. It has become apparent that the newly discovered systems differ significantly from the Solar System. In particular, many massive planets of other stars, in contrast to Jupiter, have large orbital eccentricities. In the present paper, we investigate several dynamic implications of this finding. Numerical integration results show that the orbits of low-mass planets in such systems usually have large evolving eccentricities. If the motion remains regular and no close encounters occur, the orbital evolution can be described analytically by using secular perturbations of Laplace–Lagrange equations. In terms of the Lagrange variables, the trajectories are circles, and the semimajor axis remains constant. The loss of the regularity of motion is normally followed by a nonmonotone synchronous increase in the semimajor axis and eccentricity, and the orbit becomes similar to that of a large-period comet. Narrow resonance-related regions include more complex motions.     

Influence of the largest asteroids on the orbital motions of terrestrial planets: application to the Earth and Mars

The level of precision of modern numerical ephemeris of the Solar System necessitates taking into account the gravitational influence of the largest asteroids on the terrestrial planets. This can be done in a straightforward manner when assuming that the mass of the asteroid is well known. Nevertheless, this is rarely the case, even for the largest asteroids. In this paper, we use recent determinations of the masses of Ceres, Pallas, and Vesta to both qualitatively and quantitatively determine the action of these asteroids on the orbital parameters of the Earth and Mars. This is done by the numerical integration by comparing the orbital motions of the perturbed planet when adding or not the perturbing asteroid to the classical 9 bodies problem (the Sun + the eight planets). Some preliminary results are discussed. Published in Russian in Astronomicheskii Vestnik, 2009, Vol. 43, No. 1, pp. 83–86. The text was submitted by the autors in English.     

The determination of the satellite orbit of Mariner 9

This paper presents a comprehensive analysis of the Mars orbital phase of the Mariner 9 trajectory as determined from Earth based radio data. Both the method and accuracy of the orbit determination process are reviewed. Analysis is presented to show the effects of Mars gravity model and node in the plane of the sky errors on the accuracy of orbit determination. In addition the long term evolution of the orbit from insertion through the first 500 revolutions is presented, and decomposed into effects from the Mars garvity field,n-body perturbations, and solar radiation pressure. Since the orbit period is nearly commensurable with the Mars rotational period, the orbit experiences significant resonance perturbations. The primary perturbation is in-track with a maximum amplitude of 1000 km and a wavelength of 39 spacecraft revolutions.This paper was presented at the AIAA/AAS Astrodynamics Conference, Palo Alto, California, September 11 and 12, 1972. At this time Mariner 9 operations were still underway. The operational life of Mariner 9 ended October 27, 1972, when the supply of nitrogen gas, used for attitude stabilization, was depleted. This paper represents one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, under NASA Contract No. NAS 7-100.     

The influence of the correlations between an asteroid’s orbital parameters on the estimation of the probability of planetary collision by the Monte Carlo method

The probability of an asteroid colliding with a planet can be estimated by the Monte Carlo method, in particular, through the statistical simulation of the possible initial conditions for the motion of an asteroid based on the probability density distribution set by the respective covariance matrix to be further projected with the orbital model onto the supposed time point of the collision. Hence, the collision probability is calculated as the ratio between the number of projected (virtual) asteroids striking the planet and their total number. The main problem is that different elements of the initial conditions (orbit or state vector) are correlated and, therefore, cannot be simulated independently. These correlations are reflected in the nondiagonal covariance matrix of the solution. The matrix is diagonalized by an orthogonal transformation. In the uncertainty domain constructed from the diagonal matrix elements, the initial values for each of the six orbital elements are simulated independently from the other elements, but with the accounting for their normal distribution. The program for calculating the normal distribution is based on the central limit theorem. Each sample of the initial values for the six orbital elements is transferred to the initial reference frame using an inverse transformation. Then, numerical integration is used to track the asteroid’s motion along the respective orbit to predict a possible impact event. Asteroids 99942 Apophis and 2007 WD5 are used as examples to show that disregarding the correlations when diagonalizing the covariance matrix to set the initial conditions may seriously distort the collision probability estimates. The paper gives the probabilities of the collisions of Apophis with the Earth and asteroid 2007 WD5 with Mars calculated by the author from observation sets showing nonzero collision probabilities. The author’s estimates are compared to those calculated by NASA.     

Quasi-satellite orbits in the general context of dynamics in the 1:1 mean motion resonance: perturbative treatment

Our investigation is motivated by the recent discovery of asteroids orbiting the Sun and simultaneously staying near one of the Solar System planets for a long time. This regime of motion is usually called the quasi-satellite regime, since even at the times of the closest approaches the distance between the asteroid and the planet is significantly larger than the region of space (the Hill’s sphere) in which the planet can hold its satellites. We explore the properties of the quasi-satellite regimes in the context of the spatial restricted circular three-body problem “Sun–planet–asteroid”. Via double numerical averaging, we construct evolutionary equations which describe the long-term behaviour of the orbital elements of an asteroid. Special attention is paid to possible transitions between the motion in a quasi-satellite orbit and the one in another type of orbits available in the 1:1 resonance. A rough classification of the corresponding evolutionary paths is given for an asteroid’s motion with a sufficiently small eccentricity and inclination.