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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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)
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)
Burns J.A.: Elementary derivation of the perturbation equations of Celestial Mechanics. Am. J. Phys. 44, 944–949 (1976)
Burns J.A.: Orbital evolution. In: Burns, J.A. (eds) Planetary Satellites, pp. 113–156. University of Arizona Press, Tucson (1977)
Chandrasekhar S.: Ellipsoidal Figures of Equilibrium. Yale University Press, New Haven (1969)
Danby J.M.A.: Fundamentals of Celestial Mechanics. Willman-Bell, Richmond (1992)
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)
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)
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)
Descamps P., Marchis F.: Angular momentum of binary asteroids: Implications for their possible origin. Icarus 193, 74–84 (2008)
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)
Efroimsky M., Williams J.G.: Tidal torques: A critical review of some techniques. Celest. Mech. Dyn. Astron. 104, 257–289 (2009)
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)
Gerstenkorn H.: Über Gezeitenreibung beim Zweikörperproblem. Zeitschrift fur Astrophysik 36, 245–274 (1955)
Goldreich P.: On the eccentricity of satellite orbits in the solar system. Mon. Not. R. Astron. Soc. 126, 257–268 (1963)
Goldreich P.: History of the lunar orbit. Rev. Geophys. Space Phys. 4, 411–439 (1966)
Goldreich P., Sari R.: Tidal evolution of rubble piles. Astroph. J. 691, 54–60 (2009)
Goldreich P., Soter S.: Q in the solar system. Icarus 5, 375–389 (1966)
Holsapple K.A., Michel P.: Tidal disruptions: A continuum theory for solid bodies. Icarus 183, 331–348 (2006)
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)
Kaula W.M.: Tidal dissipation by solid friction and the resulting orbital evolution. Rev. Geophys. 2, 661–685 (1964)
Lambeck K.: On the orbital evolution of the Martian satellites. J. Geophys. Res. 84, 5651–5658 (1979)
Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1927)
MacDonald G.J.F.: Tidal friction. Rev. Geophys. Space Phys. 2, 467–541 (1964)
MacRobert T.M.: Spherical Harmonics. Pergamon Press, Oxford (1967)
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)
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)
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)
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)
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)
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)
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)
Mignard F.: The evolution of the lunar orbit revisited. I. Moon and Planets 20, 301–315 (1979)
Mignard F.: The evolution of the lunar orbit revisited. II. Moon and Planets 23, 185–201 (1980)
Mignard F.: The lunar orbit revisited. III. Moon Planets 24, 189–207 (1981)
Munk W.H., MacDonald G.J.F.: The Rotation of the Earth. Cambridge University Press, Cambridge (1960)
Murray C.D., Dermott S.F.: Solar System Dynamics. Cambridge University Press, Cambridge (1999)
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)
Peale S.J.: Origin and evolution of the natural satellites. Annu. Rev. Astron. Astrophys. 37, 533–602 (1999)
Pravec P., Harris A.W.: Binary asteroid population 1. Angular momentum content. Icarus 190, 250–259 (2007)
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)
Redmond J.C., Fish F.F.: The luni-tidal interval in Mars and the secular acceleration of Phobos. Icarus 3, 87–91 (1964)
Richardson D.C., Bottke W.F., Love S.G.: Tidal distortion and disruption of Earth-crossing asteroids. Icarus 134, 47–76 (1998)
Richardson D.C., Walsh K.J.: Binary minor planets. Annu. Rev. Earth Planet Sci. 34, 47–81 (2006)
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)
Rubincam, D.P.: The early history of the lunar inclination. NASA-GSFC Rep. X-592-73-328, Goddard Space Flight Center, Greenbelt, Md. (1973)
Rubincam D.P.: Radiative spin-up and spin-down of small asteroids. Icarus 148, 2–11 (2000)
Scheeres D.J.: Changes in rotational angular momentum due to gravitational interactions between two finite bodies. Celest. Mech. Dyn. Astron. 81, 39–44 (2001)
Scheeres D.J.: Stability of the planar full 2-body problem. Celest. Mech. Dyn. Astron. 104, 103–128 (2009)
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)
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)
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)
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)
Sharma I.: The equilibrium of rubble-pile satellites: The Darwin and Roche ellipsoids for gravitationally held granular aggregates. Icarus 200, 636–654 (2009)
Sharma I.: Equilibrium shapes of rubble-pile binaries: The Darwin ellipsoids for gravitationally held granular aggregates. Icarus 205, 638–657 (2010)
Sharma I., Jenkins J.T., Burns J.A.: Tidal encounters of ellipsoidal granular asteroids with planets. Icarus 183, 312–330 (2006)
Smith J.C., Born G.H.: Secular acceleration of Phobos and Q of Mars. Icarus 27, 51–53 (1976)
Sridhar S., Tremaine S.: Tidal disruption of viscous bodies. Icarus 95, 86–99 (1992)
Szeto A.M.K.: Orbital evolution and origin of the martian satellites. Icarus 55, 133–168 (1983)
Taylor P.A., Margot J.L.: Tidal evolution of solar system binaries. Bull. Am. Astron. Soc. 39, 439 (2007)
Taylor, P.A., Margot, J.L.: Binary asteroid systems: Tidal end states and estimates of material properties. (2010, submitted)
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)
Walsh K.J., Richardson D.C.: Binary near-Earth asteroid formation: Rubble pile model of tidal disruptions. Icarus 180, 201–216 (2006)
Walsh K.J., Richardson D.C., Michel P.: Rotational breakup as the origin of small binary asteroids. Nature 454, 188–191 (2008)
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)
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Taylor, P.A., Margot, JL. Tidal evolution of close binary asteroid systems. Celest Mech Dyn Astr 108, 315–338 (2010). https://doi.org/10.1007/s10569-010-9308-0
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-010-9308-0