Abstract
We study systems of close orbiting planets evolving under the influence of tidal circularization. It is supposed that a commensurability forms through the action of disk induced migration and orbital circularization. After the system enters an inner cavity or the disk disperses the evolution continues under the influence of tides due to the central star which induce orbital circularization. We derive approximate analytic models that describe the evolution away from a general first order resonance that results from tidal circularization in a two planet system and which can be shown to be a direct consequence of the conservation of energy and angular momentum. We consider the situation when the system is initially very close to resonance and also when the system is between resonances. We also perform numerical simulations which confirm these models and then apply them to two and four planet systems chosen to have parameters related to the GJ 581 and HD 10180 systems. We also estimate the tidal dissipation rates through effective quality factors that could result in evolution to observed period ratios within the lifetimes of the systems. Thus the survival of, or degree of departure from, close commensurabilities in observed systems may be indicative of the effectiveness of tidal disipation, a feature which in turn may be related to the internal structure of the planets involved.
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Barnes R., Jackson B., Raymond S.N., West A.A., Greenberg R.: The HD 40307 planetary system: super-earths or mini-Neptunes?. Astrophys. J. 695, 1006–1011 (2009)
Bonfils X., Forveille T., Delfosse X. et al.: The HARPS search for southern extra-solar planets. VI. A Neptune-mass planet around the nearby M dwarf Gl 581. Astron. Astrophys. 443, L15–L18 (2005)
Brouwer D., Clemence G.M.: Methods of Celestial Mechanics, pp. 416–421. Academic Press, New York (1961)
Brunini A., Cionco R.G.: The origin and nature of Neptune-like planets orbiting close to solar type stars. Icarus 177, 264–268 (2005)
Goldreich P., Soter S.: Q in the Solar System. Icarus 5, 375–389 (1966)
Lovis, C., Sgransan, D., Mayor, M., et al.: The HARPS search for southern extra-solar planets. XXVII. Up to seven planets orbiting HD 10180: probing the architecture of low-mass planetary systems. Astron. Astrophys. (2010) (in press)
Lynden-Bell D., Pringle J.E.: The evolution of viscous discs and the origin of the nebular variables. Mon. Not. R. Astron. Soc. 168, 603–637 (1974)
Mayor M., Bonfils X., Forveille T. et al.: The HARPS search for southern extra-solar planets. XVIII. An Earth-mass planet in the GJ 581 planetary system. Astron. Astrophys. 507, 487–494 (2009a)
Mayor M., Udry S., Lovis C. et al.: The HARPS search for southern extra-solar planets. XIII. A planetary system with 3 super-Earths (4.2, 6.9, and 9.2 \({{M}_{\oplus}}\)). Astron. Astrophys. 493, 639–644 (2009b)
Murray C.D., Dermott S.F.: Solar System Dynamics, pp. 254–255. Cambridge University Press, Cambridge (1999)
Papaloizou J.C.B.: Disc-planet interactions: migration and resonances in extrasolar planetary systems. Celest. Mech. Dyn. Astron. 87, 53–83 (2003)
Papaloizou J.C.B., Larwood J.D.: On the orbital evolution and growth of protoplanets embedded in a gaseous disc. Mon. Not. R. Astron. Soc. 315, 823–833 (2000)
Papaloizou J.C.B., Szuszkiewicz E.: On the migration-induced resonances in a system of two planets with masses in the Earth mass range. Mon. Not. R. Astron. Soc. 363, 153–176 (2005)
Papaloizou J.C.B., Szuszkiewicz E.: Conditions for the occurrence of mean-motion resonances in a low mass planetary system. EAS Publ. Ser. 42, 333–343 (2010)
Papaloizou J.C.B., Terquem C.: Dynamical relaxation and massive extrasolar planets. Mon. Not. R. Astron. Soc. 325, 221–230 (2001)
Papaloizou J.C.B., Terquem C.: On the dynamics of multiple systems of hot super-Earths and Neptunes: tidal circularization, resonance and the HD 40307 system. Mon. Not. R. Astron. Soc. 405, 573–592 (2010)
Paardekooper S.-J., Mellema G.: Halting type I planet migration in non-isothermal disks. Astron. Astrophys. 459, L17–L20 (2006)
Paardekooper S.-J., Papaloizou J.C.B.: On disc protoplanet interactions in a non-barotropic disc with thermal diffusion. Astron. Astrophys. 485, 877–895 (2008)
Paardekooper S.-J., Papaloizou J.C.B.: On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration. Mon. Not. R. Astron. Soc. 394, 2283–2296 (2009)
Raymond S.N., Barnes R., Mandell A.M.: Observable consequences of planet formation models in systems with close-in terrestrial planets. Mon. Not. R. Astron. Soc. 384, 663–674 (2008)
Schlaufman K.C., Lin D.N.C., Ida S.: The signature of the ice line and modest Type I migration in the observed exoplanet mass-semimajor axis distribution. Astrophys. J. 691, 1322–1327 (2009)
Sinclair A.T.: The orbital resonance amongst the Galilean satellites of Jupiter. Mon. Not. R. Astron. Soc. 171, 59–72 (1975)
Tanaka H., Takeuchi T., Ward W.R.: Three-dimensional interaction between a planet and an isothermal gaseous disk. I. Corotation and lindblad torques and planet migration. Astrophys. J. 565, 1257–1274 (2002)
Terquem C., Papaloizou J.C.B.: Migration and the formation of systems of hot super-Earths and Neptunes. Astrophys. J. 654, 1110–1120 (2007)
Udry S., Bonfils X., Delfosse X. et al.: The HARPS search for southern extra-solar planets. XI. Super-Earths (5 and 8 \({{M}_{\oplus}}\)) in a 3-planet system. Astron. Astrophys. 469, L43–L47 (2007)
Vogt S.S., Butler R.P., Rivera E.J. et al.: The Lick-Carnegie exoplanet survey: a 3.1 \({{M} _{\oplus}}\) Planet in the Habitable Zone of the Nearby M3V Star Gliese 581. Astrophys. J. 723, 954–965 (2010)
Ward W.R.: Protoplanet migration by Nebula tides. Icarus 126, 261–281 (1997)
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Papaloizou, J.C.B. Tidal interactions in multi-planet systems. Celest Mech Dyn Astr 111, 83–103 (2011). https://doi.org/10.1007/s10569-011-9344-4
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DOI: https://doi.org/10.1007/s10569-011-9344-4