Download PDF

Celestial Mechanics and Dynamical Astronomy

, Volume 108, Issue 4, pp 315–338 | Cite as

Tidal evolution of close binary asteroid systems

Open Access
Original Article
First Online:
Received:
Revised:
Accepted:

Abstract

We provide a generalized discussion of tidal evolution to arbitrary order in the expansion of the gravitational potential between two spherical bodies of any mass ratio. To accurately reproduce the tidal evolution of a system at separations less than 5 times the radius of the larger primary component, the tidal potential due to the presence of a smaller secondary component is expanded in terms of Legendre polynomials to arbitrary order rather than truncated at leading order as is typically done in studies of well-separated system like the Earth and Moon. The equations of tidal evolution including tidal torques, the changes in spin rates of the components, and the change in semimajor axis (orbital separation) are then derived for binary asteroid systems with circular and equatorial mutual orbits. Accounting for higher-order terms in the tidal potential serves to speed up the tidal evolution of the system leading to underestimates in the time rates of change of the spin rates, semimajor axis, and mean motion in the mutual orbit if such corrections are ignored. Special attention is given to the effect of close orbits on the calculation of material properties of the components, in terms of the rigidity and tidal dissipation function, based on the tidal evolution of the system. It is found that accurate determinations of the physical parameters of the system, e.g., densities, sizes, and current separation, are typically more important than accounting for higher-order terms in the potential when calculating material properties. In the scope of the long-term tidal evolution of the semimajor axis and the component spin rates, correcting for close orbits is a small effect, but for an instantaneous rate of change in spin rate, semimajor axis, or mean motion, the close-orbit correction can be on the order of tens of percent. This work has possible implications for the determination of the Roche limit and for spin-state alteration during close flybys.

Keywords

Gravity Extended body dynamics Tides Asteroids Binary asteroids 
Download to read the full article text

Notes

Acknowledgments

The authors are indebted to the two referees whose detailed reviews and insightful suggestions improved the clarity and quality of the manuscript. The authors are especially grateful to Michael Efroimsky for many discussions on the finer points of tidal theory and Celestial Mechanics. This work was supported by NASA Planetary Astronomy grants NNG04GN31G and NNX07AK68G to Jean-Luc Margot.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

References

  1. A’Hearn M.F., Belton M.J.S., Delamere W.A., Kissel J., Klaasen K.P., McFadden L.A., Meech K.J., Melosh H.J., Schultz P.H., Sunshine J.M., Thomas P.C., Veverka J., Yeomans D.K., Baca M.W., Busko I., Crockett C.J., Collins S.M., Desnoyer M., Eberhardy C.A., Ernst C.M., Farnham T.L., Feaga L., Groussin O., Hampton D., Ipatov S.I., Li J., Lindler D., Lisse C.M., Mastrodemos N., Owen W.M., Richardson J.E., Wellnitz D.D., White R.L.: Deep impact: Excavating comet tempel 1. Science 310, 258–264 (2005)CrossRefADSGoogle Scholar
  2. Behrend R., Bernasconi L., Roy R., Klotz A., Colas F., Antonini P., Aoun R., Augustesen K., Barbotin E., Berger N., Berrouachdi H., Brochard E., Cazenave A., Cavadore C., Coloma J., Cotrez V., Deconihout S., Demeautis C., Dorseuil J., Dubos G., Durkee R., Frappa E., Hormuth F., Itkonen T., Jacques C., Kurtze L., Laffont A., Lavayssière M., Lecacheux J., Leroy A., Manzini F., Masi, G., Matter, D., Michelsen, R., Nomen, J., Oksanen, A., Pääkkönen, P., Peyrot, A., Pimentel, E., Pray, D., Rinner, C., Sanchez, S., Sonnenberg, K., Sposetti, S., Starkey, D., Stoss, R., Teng, J.P., Vignand, M., Waelchli N.: Four new binary minor planets: (854) Frostia, (1089) Tama, (1313) Berna, (4492) Debussy. Astron. Astroph. 446, 1177–1184 (2006)CrossRefADSGoogle Scholar
  3. Bills B.G., Neumann G.A., Smith D.E., Zuber M.T.: Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos. J. Geophys. Res. 110, 2376–2406 (2005)Google Scholar
  4. Burns J.A.: Elementary derivation of the perturbation equations of Celestial Mechanics. Am. J. Phys. 44, 944–949 (1976)CrossRefMathSciNetADSGoogle Scholar
  5. Burns J.A.: Orbital evolution. In: Burns, J.A. (eds) Planetary Satellites, pp. 113–156. University of Arizona Press, Tucson (1977)Google Scholar
  6. Chandrasekhar S.: Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven (1969)MATHGoogle Scholar
  7. Danby J.M.A.: Fundamentals of Celestial Mechanics. Willman-Bell, Richmond (1992)Google Scholar
  8. Darwin G.H.: On the bodily tides of viscous and semi-elastic spheroids, and on the ocean tides upon a yielding nucleus. Philos. Trans. R. Soc. London 170, 1–35 (1879a)CrossRefGoogle Scholar
  9. Darwin G.H.: On the precession of a viscous spheroid, and on the remote history of the Earth. Philos. Trans. R. Soc. London 170, 447–538 (1879b)CrossRefGoogle Scholar
  10. Darwin G.H.: On the secular changes in the elements of the orbit of a satellite revolving about a tidally distorted planet. Philos. Trans. R. Soc. London 171, 713–891 (1880)CrossRefGoogle Scholar
  11. Descamps P., Marchis F.: Angular momentum of binary asteroids: Implications for their possible origin. Icarus 193, 74–84 (2008)CrossRefADSGoogle Scholar
  12. Descamps P., Marchis F., Michalowski T., Vachier F., Colas F., Berthier J., Assafin M., Dunckel P.B., Polinska M., Pych W., Hestroffer D., Miller K.P.M., Vieira-Martins R., Birlan M., Teng-Chuen-Yu J.P., Peyrot A., Payet B., Dorseuil J., Léonie Y., Dijoux T.: Figure of the double asteroid 90 Antiope from adaptive optics and lightcurve observations. Icarus 187, 482–499 (2007)CrossRefADSGoogle Scholar
  13. Efroimsky M., Williams J.G.: Tidal torques: A critical review of some techniques. Celest. Mech. Dyn. Astron. 104, 257–289 (2009)CrossRefMathSciNetADSGoogle Scholar
  14. Ferraz-Mello S., Rodríguez A., Hussmann H.: Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited. Celest. Mech. Dyn. Astron. 101, 171–201 (2008)MATHCrossRefADSGoogle Scholar
  15. Gerstenkorn H.: Über Gezeitenreibung beim Zweikörperproblem. Zeitschrift fur Astrophysik 36, 245–274 (1955)MATHMathSciNetADSGoogle Scholar
  16. Goldreich P.: On the eccentricity of satellite orbits in the solar system. Mon. Not. R. Astron. Soc. 126, 257–268 (1963)MATHADSGoogle Scholar
  17. Goldreich P.: History of the lunar orbit. Rev. Geophys. Space Phys. 4, 411–439 (1966)CrossRefADSGoogle Scholar
  18. Goldreich P., Sari R.: Tidal evolution of rubble piles. Astroph. J. 691, 54–60 (2009)CrossRefADSGoogle Scholar
  19. Goldreich P., Soter S.: Q in the solar system. Icarus 5, 375–389 (1966)CrossRefADSGoogle Scholar
  20. Holsapple K.A., Michel P.: Tidal disruptions: A continuum theory for solid bodies. Icarus 183, 331–348 (2006)CrossRefADSGoogle Scholar
  21. Holsapple K.A., Michel P.: Tidal disruptions. II. A continuum theory for solid bodies with strength, with applications to the Solar System. Icarus 193, 283–301 (2008)CrossRefADSGoogle Scholar
  22. Kaula W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1964)CrossRefADSGoogle Scholar
  23. Lambeck K.: On the orbital evolution of the Martian satellites. J. Geophys. Res. 84, 5651–5658 (1979)CrossRefADSGoogle Scholar
  24. Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1927)MATHGoogle Scholar
  25. MacDonald G.J.F.: Tidal friction. Rev. Geophys. Space Phys. 2, 467–541 (1964)CrossRefADSGoogle Scholar
  26. MacRobert T.M.: Spherical Harmonics. Pergamon Press, Oxford (1967)MATHGoogle Scholar
  27. Marchis F., Descamps P., Berthier J., Hestroffer D., Vachier F., Baek M., Harris A.W., Nesvorný D.: Main belt binary asteroidal systems with eccentric mutual orbits. Icarus 195, 295–316 (2008a)CrossRefADSGoogle Scholar
  28. Marchis F., Descamps P., Baek M., Harris A.W., Kaasalainen M., Berthier J., Hestroffer D., Vachier F.: Main belt binary asteroidal systems with circular mutual orbits. Icarus 196, 97–118 (2008b)CrossRefADSGoogle Scholar
  29. Margot J.L., Nolan M.C., Benner L.A.M., Ostro S.J., Jurgens R.F., Giorgini J.D., Slade M.A., Campbell D.B.: Binary asteroids in the near-earth object population. Science 296, 1445–1448 (2002)CrossRefADSGoogle Scholar
  30. Margot J.L., Nolan M.C., Negron V., Hine A.A., Campbell D.B., Howell E.S., Benner L.A.M., Ostro S.J., Giorgini J.D., Marsden B.G.: 1937 UB (Hermes). IAU Circ. 8227, 2 (2003)ADSGoogle Scholar
  31. Margot, J.L., Pravec, P., Nolan, M.C., Howell, E.S., Benner, L.A.M., Giorgini, J.D., F., J.R., Ostro, S.J., Slade, M.A., Magri, C., Taylor, P.A., Nicholson, P.D., Campbell, D.B.: Hermes as an exceptional case among binary near-earth asteroids. In: IAU General Assembly (2006)Google Scholar
  32. Merline W.J., Close L.M., Shelton J.C., Dumas C., Menard F., Chapman C.R., Slater D.C., Keck Telescope W.M. II: Satellites of minor planets. IAU Circ. 7503, 3 (2000)ADSGoogle Scholar
  33. Michałowski T., Bartczak P., Velichko F.P., Kryszczyńska A., Kwiatkowski T., Breiter S., Colas F., Fauvaud S., Marciniak A., Michałowski J., Hirsch R., Behrend R., Bernasconi L., Rinner C., Charbonnel S.: Eclipsing binary asteroid 90 Antiope. Astron. Astroph. 423, 1159–1168 (2004)CrossRefADSGoogle Scholar
  34. Mignard F.: The evolution of the lunar orbit revisited. I. Moon and Planets 20, 301–315 (1979)MATHCrossRefADSGoogle Scholar
  35. Mignard F.: The evolution of the lunar orbit revisited. II. Moon and Planets 23, 185–201 (1980)CrossRefADSGoogle Scholar
  36. Mignard F.: The lunar orbit revisited. III. Moon Planets 24, 189–207 (1981)MATHCrossRefADSGoogle Scholar
  37. Munk W.H., MacDonald G.J.F.: The Rotation of the Earth. Cambridge University Press, Cambridge (1960)Google Scholar
  38. Murray C.D., Dermott S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)MATHGoogle Scholar
  39. Ostro S.J., Margot J.L., Benner L.A.M., Giorgini J.D., Scheeres D.J., Fahnestock E.G., Broschart S.B., Bellerose J., Nolan M.C., Magri C., Pravec P., Scheirich P., Rose R., Jurgens R.F., De Jong E.M., Suzuki S.: Radar imaging of binary near-Earth asteroid (66391) 1999 KW4. Science 314, 1276–1280 (2006)CrossRefADSGoogle Scholar
  40. Peale S.J.: Origin and evolution of the natural satellites. Annu. Rev. Astron. Astrophys. 37, 533–602 (1999)CrossRefADSGoogle Scholar
  41. Pravec P., Harris A.W.: Binary asteroid population 1. Angular momentum content. Icarus 190, 250–259 (2007)CrossRefADSGoogle Scholar
  42. Pravec P., Kusnirak P., Warner B., Behrend R., Harris A.W., Oksanen A., Higgins D., Roy R., Rinner C., Demeautis C., van den Abbeel F., Klotz A., Waelchli N., Alderweireldt T., Cotrez V., Brunetto L.: 1937 UB (Hermes). IAU Circ. 8233, 3 (2003)ADSGoogle Scholar
  43. Redmond J.C., Fish F.F.: The luni-tidal interval in Mars and the secular acceleration of Phobos. Icarus 3, 87–91 (1964)CrossRefADSGoogle Scholar
  44. Richardson D.C., Bottke W.F., Love S.G.: Tidal distortion and disruption of Earth-crossing asteroids. Icarus 134, 47–76 (1998)CrossRefADSGoogle Scholar
  45. Richardson D.C., Walsh K.J.: Binary minor planets. Annu. Rev. Earth Planet Sci. 34, 47–81 (2006)CrossRefADSGoogle Scholar
  46. Richardson J.E., Melosh H.J., Lisse C.M., Carcich B.: A ballistics analysis of the Deep Impact ejecta plume: Determining comet Tempel 1’s gravity, mass, and density. Icarus 190, 357–390 (2007)CrossRefADSGoogle Scholar
  47. Rubincam, D.P.: The early history of the lunar inclination. NASA-GSFC Rep. X-592-73-328, Goddard Space Flight Center, Greenbelt, Md. (1973)Google Scholar
  48. Rubincam D.P.: Radiative spin-up and spin-down of small asteroids. Icarus 148, 2–11 (2000)CrossRefADSGoogle Scholar
  49. Scheeres D.J.: Changes in rotational angular momentum due to gravitational interactions between two finite bodies. Celest. Mech. Dyn. Astron. 81, 39–44 (2001)MATHCrossRefADSGoogle Scholar
  50. Scheeres D.J.: Stability of the planar full 2-body problem. Celest. Mech. Dyn. Astron. 104, 103–128 (2009)MATHCrossRefMathSciNetADSGoogle Scholar
  51. Scheeres D.J., Ostro S.J., Werner R.A., Asphaug E., Hudson R.S.: Effects of gravitational interactions on asteroid spin states. Icarus 147, 106–118 (2000)CrossRefADSGoogle Scholar
  52. Scheeres D.J., Marzari F., Rossi A.: Evolution of NEO rotation rates due to close encounters with Earth and Venus. Icarus 170, 312–323 (2004)CrossRefADSGoogle Scholar
  53. Scheeres D.J., Fahnestock E.G., Ostro S.J., Margot J.L., Benner L.A.M., Broschart S.B., Bellerose J., Giorgini J.D., Nolan M.C., Magri C., Pravec P., Scheirich P., Rose R., Jurgens R.F., De Jong E.M., Suzuki S.: Dynamical configuration of binary near-Earth asteroid (66391) 1999 KW4. Science 314, 1280–1283 (2006)CrossRefADSGoogle Scholar
  54. Schellart W.P.: Shear test results for cohesion and friction coefficients for different granular materials: Scaling implications for their usage in analogue modelling. Tectonophys 324, 1–16 (2000)CrossRefGoogle Scholar
  55. Sharma I.: The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates. Icarus 200, 636–654 (2009)CrossRefADSGoogle Scholar
  56. Sharma I.: Equilibrium shapes of rubble-pile binaries: The Darwin ellipsoids for gravitationally held granular aggregates. Icarus 205, 638–657 (2010)CrossRefADSGoogle Scholar
  57. Sharma I., Jenkins J.T., Burns J.A.: Tidal encounters of ellipsoidal granular asteroids with planets. Icarus 183, 312–330 (2006)CrossRefADSGoogle Scholar
  58. Smith J.C., Born G.H.: Secular acceleration of Phobos and Q of Mars. Icarus 27, 51–53 (1976)CrossRefADSGoogle Scholar
  59. Sridhar S., Tremaine S.: Tidal disruption of viscous bodies. Icarus 95, 86–99 (1992)CrossRefADSGoogle Scholar
  60. Szeto A.M.K.: Orbital evolution and origin of the martian satellites. Icarus 55, 133–168 (1983)CrossRefADSGoogle Scholar
  61. Taylor P.A., Margot J.L.: Tidal evolution of solar system binaries. Bull. Am. Astron. Soc. 39, 439 (2007)Google Scholar
  62. Taylor, P.A., Margot, J.L.: Binary asteroid systems: Tidal end states and estimates of material properties. (2010, submitted)Google Scholar
  63. Vokrouhlický D., Čapek D.: YORP-induced long-term evolution of the spin state of small asteroids and meteoroids: Rubincam’s approximation. Icarus 159, 449–467 (2002)CrossRefADSGoogle Scholar
  64. Walsh K.J., Richardson D.C.: Binary near-Earth asteroid formation: Rubble pile model of tidal disruptions. Icarus 180, 201–216 (2006)CrossRefADSGoogle Scholar
  65. Walsh K.J., Richardson D.C., Michel P.: Rotational breakup as the origin of small binary asteroids. Nature 454, 188–191 (2008)CrossRefADSGoogle Scholar
  66. Weidenschilling S.J., Paolicchi P., Zappalà V.: Do asteroids have satellites?. In: Binzel, R.P., Gehrels, T., Matthews, M.S. (eds) Asteroids II, pp. 643–660. University of Arizona Press, Tucson (1989)Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  1. 1.Arecibo Observatory, HC 3AreciboUSA
  2. 2.Departments of Earth & Space Sciences and Physics & AstronomyUniversity of CaliforniaLos AngelesUSA

Personalised recommendations