Abstract
Investigations of two resonant planets orbiting a star or two resonant satellites orbiting a planet often rely on a few resonant and secular terms in order to obtain a representative quantitative description of the system’s dynamical evolution. We present a semianalytic model which traces the orbital evolution of any two resonant bodies in a first- through fourth-order eccentricity or inclination-based resonance dominated by the resonant and secular arguments of the user’s choosing. By considering the variation of libration width with different orbital parameters, we identify regions of phase space which give rise to different resonant “depths,” and propose methods to model libration profiles. We apply the model to the GJ 876 extrasolar planetary system, quantify the relative importance of the relevant resonant and secular contributions, and thereby assess the goodness of the common approximation of representing the system by just the presumably dominant terms. We highlight the danger in using “order” as the metric for accuracy in the orbital solution by revealing the unnatural libration centers produced by the second-order, but not first-order, solution, and by demonstrating that the true orbital solution lies somewhere “in-between” the third- and fourth-order solutions. We also present formulas used to incorporate perturbations from central-body oblateness and precession, and a protoplanetary or protosatellite thin disk with gaps, into a resonant system. We quantify the contributions of these perturbations into the GJ 876 system, and thereby highlight the conditions which must exist for multi-planet exosystems to be significantly influenced by such factors. We find that massive enough disks may convert resonant libration into circulation; such disk-induced signatures may provide constraints for future studies of exoplanet systems.
Similar content being viewed by others
References
Ballabh G.M. (1973). Potential energy of gravitationally interacting disk galaxies. Astrophys. Space Sci. 24: 535–561
Beaugé C. (1994). Asymmetric librations in exterior resonances. Celest. Mech. Dyn. Astron. 60: 225–248
Beaugé C. and Michtchenko T.A. (2003). Modelling the high-eccentricity planetary three-body problem. Application to the GJ876 planetary system. MNRAS 341: 760–770
Beaugé C., Michtchenko T.A. and Ferraz-Mello S. (2006). Planetary migration and extrasolar planets in the 2/1 mean-motion resonance. MNRAS 365: 1160–1170
Biasco L. and Chierchia L. (2002). Effective Hamiltonian for the D’Alembert planetary model near a spin/orbit resonance. Celest. Mech. Dyn. Astron. 83: 223–237
Bills B.G. (1999). Obliquity-oblateness feedback on Mars. J. Geophys. Res. 104: 30773–30797
Blitzer L. (1984). Precession dynamics in spin-orbit coupling—a unified theory. Celest. Mech. 32: 355–364
Borderies-Rappaport N. and Longaretti P.-Y. (1994). Test particle motion around an oblate planet. Icarus 107: 129–141
Bouquillon S. and Souchay J. (1999). Precise modeling of the precession-nutation of Mars. Astron. Astrophys. 345: 282–297
Brouwer D. (1946). The motion of a particle with negligible mass under the gravitational attraction of a spheroid. Astron. J. 51: 223–231
Brouwer D. (1959). Solution of the problem of artificial satellite theory without drag. Astron. J. 64: 378–397
Brouwer D. and Clemence G.M. (1961). Methods of Celestial Mechanics. Academic Press, New York
Brumberg V.A., Evdokimova L.S. and Kochina N.G. (1970). Analytical methods for the orbits of artificial satellites of the moon. Celest. Mech. 3: 197–221
Bryden G., Różyczka M., Lin D.N.C. and Bodenheimer P. (2000). On the interaction between protoplanets and protostellar disks. Astrophys. J. 540: 1091–1101
Burns J.A., Schaffer L.E., Greenberg R.J. and Showalter M.R. (1985). Lorentz resonances and the structure of the Jovian ring. Nature 316: 115–119
Burns, J.A., Simonelli, D.P., Showalter, M.R., Hamilton, D.P., Porco, C.D., Throop, H., Esposito, L.W.: Jupiter’s ring-moon system, pp. 241–262. Jupiter. The Planet, Satellites and Magnetosphere (2004)
Cameron A.G.W. and Pine M.R. (1973). Numerical models of the primitive solar nebula. Icarus 18: 377–406
Celletti A. (1993). Stability of the synchronous spin-orbit resonance by construction of librational trapping tori. Celest. Mech. Dyn. Astron. 57: 325–328
Chabrier G. and Baraffe I. (1997). Structure and evolution of low-mass stars. Astron. Astrophys. 327: 1039–1053
Chang D. and Marsden J.E. (2003). Geometric derivation of the delaunay variables and geometric phases. Celest. Mech. Dyn. Astron. 86: 185–208
Christou A.A. and Murray C.D. (1997). A second order Laplace-Lagrange theory applied to the uranian satellite system. Astron. Astrophys. 327: 416–427
Delfosse X., Forveille T., Mayor M., Perrier C., Naef D. and Queloz D. (1998). The closest extrasolar planet. A giant planet around the M4 dwarf GL 876. Astron. Astrophys. 338: L67–L70
Efroimsky M. (2005a). Gauge freedom in orbital mechanics. New York Acad. Sci. Ann. 1065: 346–374
Efroimsky M. (2005b). Long-term evolution of orbits about a precessing oblate planet: 1. The case of uniform precession. Celest. Mech. Dyn. Astron. 91: 75–108
Efroimsky M. (2006). Long-term evolution of orbits about a precessing oblate planet. 2. The case of variable precession. Celest Mech. Dyn. Astron. 96: 259–288
Efroimsky, M., Goldreich, P.: Gauge symmetry of the N-body problem in the Hamilton-Jacobi approach. J. Math. Phys. pp. 5958–5977 (2003)
Efroimsky M. and Goldreich P. (2004). Gauge freedom in the N-body problem of celestial mechanics. Astron. Astrophys. 415: 1187–1199
Elliot, J.L., Nicholson, P.D.: The rings of Uranus. In: Greenberg, R., Brahic, A. (eds.) IAU Colloq. 75: Planetary Rings, pp. 25–72 (1984)
Elliot J.L., French R.G., Frogel J.A., Elias J.H., Mink D.J. and Liller W. (1981). Orbits of nine Uranian rings. Astron. J. 86: 444–455
Ellis K.M. and Murray C.D. (2000). The disturbing function in solar system dynamics. Icarus 147: 129–144
Ferraz-Mello S. (1988). The high-eccentricity libration of the Hildas. Astron. J. 96: 400–408
Ferraz-Mello S. (1994). The convergence domain of the Laplacian expansion of the disturbing function. Celest. Mech. Dyn. Astron. 58: 37–52
Ferraz-Mello S., Beaugé C. and Michtchenko T.A. (2003). Evolution of migrating planet pairs in resonance. Celest. Mech. Dyn. Astron. 87: 99–112
Ferrer S. and Osacar C. (1994). Harrington’s Hamiltonian in the stellar problem of three bodies: reductions, relative equilibria and bifurcations. Celest. Mech. Dyn. Astron. 58: 245–275
Fischer D.A., Marcy G.W., Butler R.P., Vogt S.S., Henry G.W., Pourbaix D., Walp B., Misch A.A. and Wright J.T. (2003). A planetary companion to HD 40979 and additional planets orbiting HD 12661 and HD 38529. Astrophys. J. 586: 1394–1408
Flynn A.E. and Saha P. (2005). Second-order perturbation theory for spin-orbit resonances. Astron. J. 130: 295–307
Ford E.B., Kozinsky B. and Rasio F.A. (2000). Secular evolution of hierarchical triple star systems. Astrophys. J. 535: 385–401
Franklin F.A. and Soper P.R. (2003). Some effects of mean motion resonance passage on the relative migration of Jupiter and Saturn. Astron. J. 125: 2678–2691
Godier S. and Rozelot J.-P. (2000). The solar oblateness and its relationship with the structure of the tachocline and of the Sun’s subsurface. Astron. Astrophys. 355: 365–374
Goldreich P. (1965a). An explanation of the frequent occurrence of commensurable mean motions in the solar system. MNRAS 130: 159–181
Goldreich P. (1965b). Inclination of satellite orbits about an oblate precessing planet. Astron. J. 70: 5–9
Goldreich P. and Peale S. (1966). Spin-orbit coupling in the solar system. Astron. J. 71: 425–437
Goździewski K. and Maciejewski A.J. (1998). Semi-analytical model of librations of a rigid moon orbiting an oblate planet. Astron. Astrophys. 339: 615–622
Greenberg R. (1977). Orbit-orbit resonances in the solar system—varieties and similarities. Vistas Astron. 21: 209–239
Greenberg R. (1981). Apsidal precession of orbits about an oblate planet. Astron. J. 86: 912–914
Groten E., Molodenski S.M. and Zharkov V.N. (1996). On the theory of Mars’ forced nutation. Astron. J. 111: 1388–1399
Gurfil, P., Lainey, V., Efroimsky, M.: Long-term evolution of orbits about a precessing oblate planet. 3. A semianalytical and a purely numerical approach. ArXiv:astro-ph/0607530v3. Celest Mech. Dyn. Astron. 99, (2007) In press
Hamilton D.P. (1994). A comparison of Lorentz, planetary gravitational and satellite gravitational resonances. Icarus 109: 221–240
Hamilton D.P. and Burns J.B. (1993). Lorentz and gravitational resonances on circumplanetary particles. Adv. Space Res. 13: 241–248
Harrington R.S. (1968). Dynamical evolution of triple stars. Astron. J. 73: 190–194
Harrington R.S. (1969). The stellar three-body problem. Celest. Mech. 1: 200–209
Hayashi C. (1981). Structure of the solar nebula, growth and decay of magnetic fields and effects of magnetic and turbulent viscosities on the nebula. Prog. Theor. Phys. Supp. 70: 35–53
Hilton J.L. (1991). The motion of Mars’ pole. I – rigid body precession and nutation. Astron. J. 102: 1510–1527
Iorio L. (2005). On the possibility of measuring the solar oblateness and some relativistic effects from planetary ranging. Astron. Astrophys. 433: 385–393
Ji J., Li G. and Liu L. (2002). The dynamical simulations of the planets orbiting GJ 876. Astrophys. J. 572: 1041–1047
Ji J., Liu L., Kinoshita H., Zhou J., Nakai H. and Li G. (2003). The librating companions in HD 37124, HD 12661, HD 82943, 47 Ursa Majoris, and GJ 876: alignment or antialignment?. Astrophys. J. 591: L57–L60
Kaula W.M. (1961). Analysis of gravitational and geometric aspects geodetic utilization of satellites. Geophys. J. 5: 104–133
Kaula W.M. (1962). Development of the lunar and solar disturbing functions for a close satellite. Astron. J. 67: 300–303
Kinoshita H. (1993). Motion of the orbital plane of a satellite due to a secular change of the obliquity of its mother planet. Celest. Mech. Dyn. Astro. 57: 359–368
Kley W. (2000). On the migration of a system of protoplanets. MNRAS 313: L47–L51
Kley W., Peitz J. and Bryden G. (2004). Evolution of planetary systems in resonance. Astron. Astrophys. 414: 735–747
Kley W., Lee M.H., Murray N. and Peale S.J. (2005). Modeling the resonant planetary system GJ 876. Astron. Astrophys. 437: 727–742
Konacki M., Maciejewski A.J. and Wolszczan A. (2000). Improved timing formula for the PSR B1257+12 planetary system. Astrophys. J. 544: 921–926
Kopal Z. (1969). The precession and nutation of deformable bodies, III. Astrophys. Space Sci. 4: 427–458
Kozai Y. (1959). The motion of a close earth satellite. Astron. J. 64: 367–377
Kozai Y. (1960). Effect of precession and nutation on the orbital elements of a close earth satellite. Astron. J. 65: 621–623
Kozai Y. (1962). Secular perturbations of asteroids with high inclination and eccentricity. Astron. J. 67: 591–598
Kuchner M.J. (2004). A minimum-mass extrasolar Nebula. Astrophys. J. 612: 1147–1151
Laughlin G., Butler R.P., Fischer D.A., Marcy G.W., Vogt S.S. and Wolf A.S. (2005). The GJ 876 planetary system: a progress report. Astrophys. J. 622: 1182–1190
Lee M.H. (2004). Diversity and origin of 2:1 orbital resonances in extrasolar planetary systems. Astrophys. J. 611: 517–527
Lee M.H. and Peale S.J. (2002). Dynamics and origin of the 2:1 orbital resonances of the GJ 876 planets. Astrophys. J. 567: 596–609
Lee M.H. and Peale S.J. (2003). Secular evolution of hierarchical planetary systems. Astrophys. J. 592: 1201–1216
Lemaître A. and Henrard J. (1988). The 3/2 resonance. Celest. Mech. 43: 91–98
Ling J.F. (1991). Application of the stroboscopic method to the stellar three-body problem. Astrophys. Space Sci. 185: 51–61
Malhotra R. (1994). Nonlinear resonances in the solar system. Physica D 77: 289–304
Malhotra R. and Dermott S.F. (1990). The role of secondary resonances in the orbital history of Miranda. Icarus 85: 444–480
Marcy G.W., Butler R.P., Fischer D., Vogt S.S., Lissauer J.J. and Rivera E.J. (2001). A pair of resonant planets orbiting GJ 876. Astrophys. J. 556: 296–301
Métris G. (1991). Mean values of particular functions in the elliptic motion. Celest. Mech. Dyn. Astron. 52: 79–84
Michtchenko T.A. and Ferraz-Mello S. (2001). Modeling the 5: 2 mean-motion resonance in the Jupiter-Saturn planetary system. Icarus 149: 357–374
Morbidelli A. (2001). Chaotic Diffusion in Celestial Mechanics. Reg. Chaotic Dyn. 6(3): 277
Morbidelli A. (2002). Modern Celestial Mechanics: Aspects of Solar System Dynamics. Taylor & Francis, London, ISBN 0415279399
Murdock J.A. (1978). Some mathematical aspects of spin-orbit resonance. Celest. Mech. 18: 237–253
Murray C.D. and Dermott S.F. (1999). Solar System Dynamics. Cambridge University Press, Cambridge
Namouni F. (2005). On the origin of the eccentricities of extrasolar planets. Astron. J. 130: 280–294
Peale S.J. (1976). Orbital resonances in the solar system. Ann. Rev. Astron. Astrophys. 14: 215–246
Peale, S.J.: Orbital resonances, unusual configurations and exotic rotation states among planetary satellites, pp. 159–223. IAU Colloq.77: Some Background about Satellites (1986)
Penna G.D. (1999). Analytical and numerical results on the stability of a planetary precessional model. Celest. Mech. Dyn. Astr. 75: 103–124
Press, W.H., Teukolsky, S.A., Vetterling, W.T., Flannery, B.P.: Numerical recipes in FORTRAN. The art of scientific computing. University Press, Cambridge; c1992, 2nd edn. (1992)
Psychoyos D. and Hadjidemetriou J.D. (2005). Dynamics of 2/1 resonant extrasolar systems application to HD82943 and GLIESE876. Celest. Mech. Dyn. Astron. 92: 135–156
Rivera E.J., Lissauer J.J., Butler R.P., Marcy G.W., Vogt S.S., Fischer D.A., Brown T.M., Laughlin G. and Henry G.W. (2005). A 7.5 M ⊕ planet orbiting the nearby Star, GJ 876. Astrophys. J. 634: 625–640
Roig F., Simula A., Ferraz-Mello S. and Tsuchida M. (1998). The high-eccentricity asymmetric expansion of the disturbing function for non-planar resonant problems. Astron. Astrophys. 329: 339–349
Rozelot J.P. and Roesch J. (1997). An upper bound to the solar oblateness. Solar Phys. 172: 11–18
Rozelot J.P., Godier S. and Lefebvre S. (2001). On the theory of the oblateness of the Sun. Solar Phys. 198: 223–240
Rubincam D.P. (2000). Pluto and Charon: a case of precession-orbit resonance?. J. Geophys. Res. 105: 26745–26756
Sasselov D.D. and Lecar M. (2000). On the snow line in dusty protoplanetary disks. Astrophys. J. 528: 995–998
Schaffer L. and Burns J.A. (1992). Lorentz resonances and the vertical structure of dusty rings – analytical and numerical results. Icarus 96: 65–84
Shinkin V.N. (2001). Approximate analytic solutions of the averaged three-body problem at first-order resonance with large oblateness of the central Planet. Celest. Mech. Dyn. Astron. 79: 15–27
Sidlichovsky M. (1983). On the double averaged three-body problem. Celest. Mech. 29: 295–305
Snellgrove M.D., Papaloizou J.C.B. and Nelson R.P. (2001). On disc driven inward migration of resonantly coupled planets with application to the system around GJ876. Astron. Astrophys. 374: 1092–1099
Sundman, K.: Sur les conditions nécessaires et suffisantes pour la convergence du développement de la fonction perturbatrice dans le mouvement plan. Öfversigt Finska Vetenskaps-Soc. (1916)
Šidlichovský M. and Nesvorný D. (1994). Temporary capture of grains in exterior resonances with the Earth: planar circular restricted three-body problem with Poynting-Robertson drag. Astron. Astrophys. 289: 972–982
Thommes E.W. and Lissauer J.J. (2003). Resonant inclination excitation of migrating giant Planets. Astrophys. J. 597: 566–580
Vakhidov A.A. (2001). Asteroid orbits in mixed resonances: some numerical experiments. Planet. Space Sci. 49: 793–797
Varadi F., Ghil M. and Kaula W.M. (1999). Mass-weighted symplectic forms for the N-body problem. Celest. Mech. Dyn. Astron. 72: 187–199
Veras, D.: Dangers of Truncating the Disturbing Function In Small Body Solar System Dynamics. In: Proceeding of the New Trends in Astrodynamics and Applications III, American Institute of Physics Conference Series, Vol. 886, pp. 175–186 (2007)
Veras D. and Armitage P.J. (2007). Extrasolar planetary dynamics with a generalized planar Laplace-Lagrange secular theory. Astrophys. J. 661: 1311–1322
Ward W.R. (1981). Solar nebula dispersal and the stability of the planetary system. I – scanning secular resonance theory. Icarus 47: 234–264
Weidenschilling S.J. (1977). The distribution of mass in the planetary system and solar nebula. Astrophys. Space Sci. 51: 153–158
Wiesel W. (1982). Saturn’s rings – resonance about an oblate planet. Icarus 51: 149–154
Winter O.C. and Murray C.D. (1997). Resonance and chaos. I. First-order interior resonances. Astron. Astrophys. 319: 290–304
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Veras, D. A resonant-term-based model including a nascent disk, precession, and oblateness: application to GJ 876. Celestial Mech Dyn Astr 99, 197–243 (2007). https://doi.org/10.1007/s10569-007-9097-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10569-007-9097-2