Dynamical Lifetimes and Final Fates of Small Bodies: Orbit Integrations vs Öpik Calculations

Abstract The dynamical lifetimes of small bodies against ejection from the Solar System or collision with the Sun or a planet are often estimated by Monte Carlo codes based on the equations of Opik and using a method implemented by Arnold. Such algorithms assume that orbital changes are dominated by close encounters, and that successive encounters are uncorrelated. We have compared the results of an Opik code (H. J. Melosh and W. B. Tonks, Mete-oritics 28 , 398 (1993)) and a fast integrator (H. F. Levison and M. J. Duncan, Icarus 108 , 18 (1994)) to investigate the regimes of validity of the Opik–Arnold approach. We investigate the transfer of ecliptic comets from Neptune-crossing orbits to observable Jupiter-family comets, the dynamics of Halley-type comets, and the transport of meteorites among the terrestrial planets. In all cases, the Opik code overestimates the median lifetime of the small bodies, although both codes show a rapid initial loss of objects followed by a slow decay. For martian impact ejecta, some of which find their way to Earth as the SNC meteorites, the Opik code substantially overestimates lifetimes because of its neglect of secular resonances, which rapidly pump eccentricities (B. J. Gladman et al. , Science 271 , 1387 (1996)).

[1]  M. E. Bailey,et al.  Origin of sungrazers : a frequent cometary end-state , 1992 .

[2]  F. Marzari,et al.  Statistics of close approaches between asteroids and planets: Project SPACEGUARD , 1990 .

[3]  G. Wetherill.  Orbital evolution of impact ejecta from Mars. , 1984 .

[4]  Harold F. Levison,et al.  The Dynamical Evolution of Lunar Impact Ejecta , 1995 .

[5]  Harold F. Levison,et al.  On the Character and Consequences of Large Impacts in the Late Stage of Terrestrial Planet Formation , 1999 .

[6]  H. Melosh,et al.  The Large Crater Origin of SNC Meteorites , 1987, Science.

[7]  Alessandro Morbidelli,et al.  The Secular Resonances in the Solar System , 1994 .

[8]  John E. Chambers,et al.  Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions , 1998 .

[9]  G. Wetherill,et al.  The range of validity of the two-body approximation in models of terrestrial planet accumulation: I. Gravitational perturbations , 1984 .

[10]  Statistical and Evolutionary Aspects of Cometary Orbits , 1989 .

[11]  S. Ipatov.  Migration of Small Bodies to Earth , 1995 .

[12]  G. Hahn,et al.  Dynamics of planet-crossing asteroids: Classes of orbital behavior: Project SPACEGUARD , 1989 .

[13]  Comparison of Numerical Integration and 'Gas Dynamic' Modelling of Runaway Planetesimal Growth , 1996 .

[14]  Brett Gladman,et al.  Destination: Earth. Martian Meteorite Delivery , 1997 .

[15]  J. Burns,et al.  The Exchange of Impact Ejecta Between Terrestrial Planets , 1996, Science.

[16]  G. Wetherill.  ASTEROIDAL SOURCE OF ORDINARY CHONDRITES , 1985 .

[17]  P. Warren.  Lunar and Martian Meteorite Delivery Services , 1994 .

[18]  P. Farinella,et al.  The Dynamics of Objects in Orbits Resembling That of P/Encke , 1995 .

[19]  W. Bottke,et al.  Mapping the Effects of Distant Perturbations on Particle–Planet Interactions , 1997 .

[20]  Alessandro Morbidelli,et al.  Orbital and temporal distributions of meteorites originating in the asteroid belt , 1998 .

[21]  George W. Wetherill,et al.  Where do the Apollo objects come from , 1988 .

[22]  Collisional Probability of Periodic Comets with the Terrestrial Planets: An Invalid Case of Analytic Formulation , 1998 .

[23]  Giovanni B. Valsecchi,et al.  Outcomes of planetary close encounters: A systematic comparison of methodologies , 1988 .

[24]  Statistical Properties of Encounters among Asteroids: A New, General Purpose, Formalism , 1998 .

[25]  D. L. Rabinowitz,et al.  Are Main-Belt Asteroids a Sufficient Source for the Earth-Approaching Asteroids? , 1997 .

[26]  P. Michel,et al.  THE KOZAI RESONANCE FOR NEAR-EARTH ASTEROIDS WITH SEMIMAJOR AXES SMALLER THAN 2 AU , 1996 .

[27]  R. Wolstencroft,et al.  Energy Changes in Close Planetary Encounters , 1971 .

[28]  G. Wetherill.  Relationships Between Orbits and Sources of Chondritic Meteorites , 1969 .

[29]  Dynamics of comets in the outer planetary region. II. Enhanced planetary masses and orbital evolutionary paths. , 1997 .

[30]  S. Tremaine,et al.  Symplectic integrators for solar system dynamics , 1992 .

[31]  M. Bailey,et al.  Dynamical evolution of Halley-type comets , 1996 .

[32]  George W. Wetherill,et al.  The range of validity of the two-body approximation in models of terrestrial planet accumulation: II. Gravitational cross sections and runaway accretion , 1985 .

[33]  G. Wetherill.  Steady state populations of Apollo-Amor objects , 1979 .

[34]  Harold F. Levison,et al.  The Long-Term Dynamical Behavior of Short-Period Comets , 1993 .

[35]  G. Wetherill.  Solar System Sources of Meteorites and Large Meteoroids , 1974 .

[36]  H. Yoshida.  Construction of higher order symplectic integrators , 1990 .

[37]  E. J. Oepik,et al.  The stray bodies in the solar system. i - survival of cometary nuclei and the asteroids. , 1963 .

[38]  Giovanni B. Valsecchi,et al.  Asteroids falling into the Sun , 1994, Nature.

[39]  Y. Kozai.  Asteroids with large secular orbital variations , 1980 .

[40]  Harold F. Levison,et al.  The Dynamical Structure of the Kuiper Belt , 1995 .

[41]  J. Wisdom,et al.  Symplectic maps for the N-body problem. , 1991 .

[42]  H. Melosh,et al.  Origin of the Spacewatch Small Earth-Approaching Asteroids , 1996 .

[43]  Robert A Kolvoord,et al.  Collision lifetimes and impact statistics of near-Earth asteroids , 1993 .

[44]  Brett James Gladman,et al.  The Near-Earth Object Population , 2000 .

[45]  P. Farinella,et al.  Dynamical evolution of NEAs: Close encounters, secular perturbations and resonances , 1996 .

[46]  P. Farinella,et al.  The orbital evolution of the asteroid Eros and implications for collision with the Earth , 1996, Nature.

[47]  D. Davis,et al.  Collisional history of asteroids: Evidence from Vesta and the Hirayama families , 1985 .

[48]  E. Shoemaker,et al.  Earth-crossing asteroids - Orbital classes, collision rates with earth, and origin , 1979 .

[49]  Eiichiro Kokubo,et al.  Oligarchic growth of protoplanets , 1996 .

[50]  Patrick Michel,et al.  The Location of Linear Secular Resonances for Semimajor Axes Smaller Than 2 AU , 1997 .

[51]  P. Farinella,et al.  Secular Resonances and the Dynamics of Mars-Crossing and Near-Earth Asteroids , 1995 .

[52]  Harold F. Levison,et al.  Dynamical Lifetimes of Objects Injected into Asteroid Belt Resonances , 1997 .

[53]  J. Henrard,et al.  Secular resonances in the asteroid belt: Theoretical perturbation approach and the problem of their location , 1991 .

[54]  G. Wetherill.  Cratering of the terrestrial planets by Apollo objects , 1989 .

[55]  Patrick Michel,et al.  Dynamical evolution of two near-Earth asteroids to be explored by spacecraft: (433) Eros and (4660) Nereus. , 1996 .

[56]  E. Opik.  SYMPOSIUM: Comets: The survival of comets and cometary material , 1961 .

[57]  Patrick Michel,et al.  Dynamics of Eros , 1998 .

[58]  W. Bottke,et al.  Asteroidal collision probabilities , 1993 .

[59]  J. Arnold.  The origin of meteorites as small bodies. ii - the model. , 1965 .

[60]  Richard Greenberg.  Orbital interactions - A new geometrical formalism , 1982 .

[61]  Main Belt Asteroid Collision Probabilities and Impact Velocities , 1998 .

[62]  S. Tremaine,et al.  Planetary perturbations and the origins of short-period comets , 1990 .

[63]  Harold F. Levison,et al.  From the Kuiper Belt to Jupiter-Family Comets: The Spatial Distribution of Ecliptic Comets☆ , 1997 .

[64]  M. Nolan,et al.  Velocity Distributions among Colliding Asteroids , 1994 .

[65]  D. Asher,et al.  Orbital evolution of the large outer Solar system object 5145 Pholus , 1993 .

[66]  P. Farinella,et al.  Collision rates and impact velocities in the Main Asteroid Belt , 1992 .

[67]  MODELLING CLOSE ENCOUNTERS WITH OPIK'S THEORY , 1997 .

[68]  H. Scholl,et al.  The three principal secular resonances ν5, ν6, and ν16 in the asteroidal belt , 1989 .

[69]  Donald J. Kessler,et al.  Derivation of the collision probability between orbiting objects: the lifetimes of jupiter's outer moons , 1981 .

[70]  S. Tremaine,et al.  The Origin of Short-Period Comets , 1988 .

[71]  G. Wetherill.  Occurrence of Earth-Like Bodies in Planetary Systems , 1991, Science.

[72]  (Erratum) Long-term dynamical evolution of the brightest bolides. , 1995 .

[73]  M. E. Bailey,et al.  Rapid dynamical evolution of giant comet Chiron , 1990, Nature.

[74]  Eugene M. Shoemaker,et al.  Asteroid and comet flux in the neighborhood of the earth , 1988 .

[75]  A. Morbidelli,et al.  The Kozai resonance in the outer solar system and the dynamics of long-period comets , 1996 .

[76]  P. Michel.  Effects of Linear Secular Resonances in the Region of Semimajor Axes Smaller Than 2 AU. , 1997 .

[77]  Yoshihide Kozai,et al.  Secular perturbations of asteroids with high inclination and eccentricity , 1962 .

[78]  E. Opik.  THE SURVIVAL OF COMETS AND COMETARY MATERIAL , 1961 .

[79]  G. W. Wetherill,et al.  Collisions in the asteroid belt , 1967 .

[80]  D. Olsson-Steel.  Collisions in the Solar System – IV. Cometary impacts upon the planets , 1987 .