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Russia Moscow
Section Mechanics
Title On the exo-planet precession under torqes due to three celestial bodies with the evolution of the satellite's orbit
Author(-s) Krasil’nikov P.S.a
Affiliations Moscow Aviation Institutea
Abstract We investigate the non-resonant evolution of the axial tilt of hypothetical exo-Earth in the gravitational field of a star, planet's satellite (exo-Moon) and outer planet (exo-Jupiter). The exo-Earth is assumed to be rigid, axially symmetric ($A=B$) and almost spherical. We assume the orbits of the both exo-planets to be Keplerian ellipses with focus in the star, the orbit of exo-Moon to be an evolving Keplerian ellipse with slowly changing of ascending node longitude and periapsis argument. Assuming the frequencies of the unperturbed orbital elliptical motion to be of the order of unity, we obtain the canonical averaged equations describing the perturbed oscillations of the exo-Moon spin axis. These equations contain parameters changing slowly over time. Using the smallness of the planets' masses relative to the mass of the star, we have obtained simplified equations of oscillations of the exo-Earth spin axis by the small parameter method. Time integration of simplified equations gives the axial tilt of exo-Moon as a function of time. It is shown that the torques from the exo-Jupiter create a secular, long-period oscillation mode in axial tilt with a frequency equals to frequency of unperturbed spin axis precession of the exo-Earth. The impact of the exo-Moon on the evolution of the exo-Earth spin axis is that short-period harmonics appear in the oscillations of the axial tilt. The frequency of such oscillations is close to the precession frequency of the ascending node longitude of the exo-Moon orbit. We have calculated the evolution of exo-Earth axial tilt for two exo-planetary systems, i.e., for a system similar to the solar system, and for a planetary exo-system 7 Canis Majoris. The effect of destabilization (stabilization) of the exo-Earth tilt oscillations due to the torques exerted by exo-Moon and exo-Jupiter is described.
Keywords axial tilt of exo-planet, planetary exo-system, averaged equations, effect of destabilization
UDC 521.92, 517.928.7
MSC 70F15, 70K65
DOI 10.35634/vm220210
Received 18 May 2022
Language Russian
Citation Krasil’nikov P.S. On the exo-planet precession under torqes due to three celestial bodies with the evolution of the satellite's orbit, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2022, vol. 32, issue 2, pp. 319-337.
  1. Armstrong J.C., Barnes R., Domagal-Goldman S., Breiner J., Quinn T.R., Meadows V.S. Effects of extreme obliquity variations on the habitability of exoplanets, Astrobiology, 2014, vol. 14, no. 4, pp. 277-291.
  2. Cowan N.B., Voigt A., Abbot D.S. Thermal phases of Earth-like planets: estimating thermal inertia from eccentricity, obliquity, and diurnal forcing, The Astrophysical Journal, 2012, vol. 757, no. 1, article 80.
  3. Heller R., Leconte J., Barnes R. Tidal obliquity evolution of potentially habitable planets, Astronomy and Astrophysics, 2011, vol. 528, article 27.
  4. Ferreira D., Marshall J., O'Gorman P.A., Seager S. Climate at high-obliquity, Icarus, 2014, vol. 243, pp. 236-248.
  5. Kilic C., Raible C.C., Stocker T.F. Multiple climate states of habitable exoplanets: The role of obliquity and irradiance, The Astrophysical Journal, 2017, vol. 844, no. 2, article 147.
  6. Milankovitch M. Kanon der Erdbestrahlung und seine Anwendung auf das Eiszeitenproblem (Canon of insolation and the ice-age problem), Belgrade: Serbian Academy, 1941.
  7. Williams D.M., Pollard D. Extraordinary climates of Earth-like planets: three-dimensional climate simulations at extreme obliquity, International Journal of Astrobiology, 2003, vol. 2, issue 1, pp. 1-19.
  8. Drysdale R.N., Hellstrom J.C., Zanchetta G., Fallick A.E., Sánchez Goñi M.F., Couchoud I., McDonald J., Maas R., Lohmann G., Isola I. Evidence for obliquity forcing of glacial termination. II, Science, 2009, vol. 325, no. 5947, pp. 1527-1531.
  9. Williams D.M., Kasting J.F. Habitable planets with high obliquities, Icarus, 1997, vol. 129, no. 1, pp. 254-267.
  10. Li G., Batygin K. On the spin-axis dynamics of a Moonless Earth, The Astrophysical Journal, 2014, vol. 790, no. 1, article 69.
  11. Shan Y., Li G. Obliquity variations of habitable zone planets Kepler-62f and Kepler-186f, The Astronomical Journal, 2018, vol. 155, no. 6, article 237.
  12. Quarles B., Barnes J.W., Lissauer J.J., Chambers J. Obliquity evolution of the potentially habitable exoplanet Kepler-62f, Astrobiology, 2020, vol. 20, issue 1, pp. 73-90.
  13. Laskar J., Joutel F., Boudin F. Orbital, precessional, and insolation quantities for the Earth from $-20$Myr to $+10$Myr, Astronomy and Astrophysics, 1993, vol. 270, nos. 1-2, pp. 522-533.
  14. Laskar J., Robutel P. The chaotic obliquity of the planets, Nature, 1993, vol. 361, no. 6413, pp. 608-612.
  15. Laskar J., Joutel F., Robutel P. Stabilization the Earth's obliquity by the Moon, Nature, 1993, vol. 361, no. 6413, pp. 615-617.
  16. Quillen A.C., Chen Y.-Y., Noyelles B., Loane S. Tilting Styx and Nix but not Uranus with a spin-precession-mean-motion resonance, Celestial Mechanics and Dynamical Astronomy, 2018, vol. 130, issue 2, article 11.
  17. Spiegel D.S., Raymond S.N., Dressing C.D., Scharf C.A., Mitchell J.L. Generalized Milankovitch cycles and long-term climatic habitability, The Astrophysical Journal, 2010, vol. 721, no. 2, pp. 1308-1318.
  18. Ward W.R., Hamilton D.P. Tilting Saturn. I. Analytic model, The Astronomical Journal, 2004, vol. 128, no. 5, pp. 2501-2509.
  19. Ward W.R., Hamilton D.P. Tilting Saturn. II. Numerical model, The Astronomical Journal, 2004, vol. 128, no. 5, pp. 2510-2517.
  20. Krasil'nikov P.S., Amelin R.N. On the precession of Saturn, Cosmic Research, 2018, vol. 56, no. 4, pp. 306-316.
  21. Krasil'nikov P.S. Vrashcheniya tverdogo tela otnositel'no tsentra mass v ogranichennoi zadache trekh tel (Rotations of a rigid body about the center of mass in the restricted three-body problem), Moscow: Moscow Aviation Institute, 2018.
  22. Krasil'nikov P.S., Podvigina O.M. On evolution of the planet's obliquity in a non-resonant planetary system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 549-564 (in Russian).
  23. Correia A.C.M. Stellar and planetary Cassini states, Astronomy and Astrophysics, 2015, vol. 582, article 69.
  24. Lhotka C. Steady state obliquity of a rigid body in the spin-orbit resonant problem: Application to Mercury, Celestial Mechanics and Dynamical Astronomy, 2017, vol. 129, issue 4, pp. 397-414.
  25. Colombo G. Cassini's second and third laws, The Astronomical Journal, 1966, vol. 71, no. 9, p. 891.
  26. Ward W.R. Tidal friction and generalized Cassini's laws in the solar system, The Astronomical Journal, 1975, vol. 80, pp. 64-70.
  27. Beletskii V.V. Resonance rotation of celestial bodies and Cassini's laws, Celestial Mechanics, 1972, vol. 6, no. 3, pp. 356-378.
  28. Lidov M.L., Neishtadt A.I. The method of canonical transformations in the problems of rotation of celestial bodies and Cassini's laws, Preprint no. 9. Institute of Applied Mathematics of RAS, 1973 (in Russian).
  29. Beletskii V.V. Dvizhenie sputnika otnositel'no tsentra mass v gravitatsionnom pole (The motion of a satellite relative to the center of mass in a gravitational field), Moscow: Moscow State University, 1975.
  30. Peale S.J. The proximity of Mercury's spin to Cassini state 1 from adiabatic invariance, Icarus, 2006, vol. 181, issue 2, pp. 338-347.
  31. Tisserand F.F. Traité de mécanique céleste. Tome II. Théorie de la figure des corps célestes et de leur mouvement de rotation, Paris: Gauthier-Villars et fils, 1891.
  32. Smart W.M. Celestial mechanics, London-New York-Toronto: Longmans, Green and Co, 1953.
  33. Krasinsky G.A. Critical inclinations in planetary problem, Celestial Mechanics, 1972, vol. 6, issue 1, pp. 60-83.
  34. Lidov M.L. The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies, Planetary and Space Science, 1962, vol. 9, issue 10, pp. 719-759.
  35. Lidov M.L. Approximate analysis of the evolution of orbits of artificial satellites, Problemy dvizheniya iskusstvennykh nebesnykh tel, Moscow: Academy of Sciences of the Soviet Union, 1963.
  36. Balk M.B. Elementy dinamiki kosmicheskogo poleta (Elements of space flight dynamics), Moscow: Nauka, 1965.
  37. Krasil'nikov P.S., Zakharova E.E. Nonresonant rotations of a satellite about its center of mass in the restricted $N$-body problem, Cosmic Research, 1994, vol. 31, no. 6, pp. 596-604.
  38. Podvigina O.M., Krasilnikov P.S. Evolution of the obliquity of an exoplanet: A non-resonant case, Icarus, 2020, vol. 335, 113371.
  41. Shen W., Chen W., Sun R. Earth's temporal principal moments of inertia and variable rotation, Geo-spatial Information Science, 2008, vol. 11, no. 2, pp. 127-132.
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