Abstract
Analytical representations of the rate of apsidal precession in the planar elliptical restricted three-body problem are considered in the case when the orbit of the disturbing body is external with respect to the orbit of the test particle. The analytical expressions are compared with the numerical data obtained for the apsidal precession rate in the form of a power series with a parameter equal to the ratio of the semi-major axis of the orbit of the test particle to that of the disturbing planet. It is shown that the analytical expressions for the rate of apsidal precession of the particle are reliable only at distances not close to the instability zone near the orbit of the disturbing planet. Near the Wisdom gap, the linear secular theory is no more valid. An empirical correction formula is proposed to calculate the apsidal procession rate at relatively high (however less than 0.5) eccentricities of the particle and disturbing planet. The proposed formulas are applied to describe the precession of orbits in real exoplanetary systems.
REFERENCES
T. A. Heppenheimer, “On the formation of planets in binary star systems,” Astron. Astrophys. 65, 421–426 (1978).
D. Whitmire, J. Matese, L. Criswell, and S. Mikkola, “Habitable planet formation in binary star systems,” Icarus 132, 196–203 (1998).
P. Thebault, F. Marzari, and H. Scholl, “Relative velocities among accreting planetesimals in binary systems: The circumprimary case,” Icarus 183, 193–206 (2006). https://doi.org/10.1016/j.icarus.2006.01.022
M. Mayor, A. Duquennoy, J.-L. Halbwachs, and J.-C. Mermilliod, “CORAVEL surveys to study binaries of different masses and ages,” in Complementary Approaches to Double and Multiple Star Research: Proc. 135th IAU Colloquium, Pine Mountain, Ga., Apr. 5–10, 1992 (Astronomical Society of the Pacific, Chelsea, 1992); in Ser.: ASP Conference Series, Vol. 32, pp. 73–81.
R. B. Larson, “Implications of binary properties for theories of star formation.,” in Proc. 200th IAU Symp. on The Formation of Binary Stars, Potsdam, Germany, Apr. 10–15, 2000 (Astronomical Society of the Pacific, San Francisco, 2001), pp. 93–106. https://doi.org/10.48550/ar**v.astro-ph/0006288
I. I. Shevchenko, Dynamical Chaos in Planetary Systems (Springer-Verlag, Cham, 2020). https://doi.org/10.1007/978-3-030-52144-8
T. V. Demidova and I. I. Shevchenko, “Spiral patterns in planetesimal circumbinary disks,” Astrophys. J. 805, 38 (2015). https://doi.org/10.1088/0004-637X/805/1/38
C. D. Murray and S. F. Dermott, Solar System Dynamics (Cambridge Univ. Press, Cambridge, 1999; Fizmatlit, Moscow, 2009).
Y. Lithwick and Y. Wu, “Theory of secular chaos and Mercury’s orbit,” Astrophys. J. 739, 31 (2011). https://doi.org/10.1088/0004-637X/739/1/31
H. Rein and D. S. Spiegel, “IAS15: A fast, adaptive, high-order integrator for gravitational dynamics, accurate to machine precision over a billion orbits,” Mon. Not. R. Astron. Soc. 446, 1424–1437 (2015). https://doi.org/10.1093/mnras/stu2164
H. Rein and S.-F. Liu, “REBOUND: An open-source multi-purpose N-body code for collisional dynamics,” Astron. Astrophys. 537, A128 (2012). https://doi.org/10.1051/0004-6361/201118085
J. Wisdom, “The resonance overlap criterion and the onset of stochastic behavior in the restricted three-body problem,” Astron. J. 85, 1122–1133 (1980). https://doi.org/10.1086/112778
M. Duncan, T. Quinn, and S. Tremaine, “The long-term evolution of orbits in the solar system. A map** approach,” Icarus 82, 402–418 (1989). https://doi.org/10.1016/0019-1035(89)90047-X
M. Ya. Marov and I. I. Shevchenko, “Exoplanets: Nature and models,” Phys. Usp. 63, 837–871 (2020). https://doi.org/10.3367/UFNr.2019.10.038673
ACKNOWLEDGMENTS
The author is grateful to I.I. Shevchenko for valuable advice and recommendations concerning this manuscript. The author thanks the reviewers for their useful detailed remarks.
Funding
This work was supported by the Russian Science Foundation, project no. 22-22-00046.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
The author of this work declares that she has no conflicts of interest.
Additional information
Translated by E. Smirnova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Berezina, A.A. Orbital Precession in the Restricted Three-Body Problem: Exact Representations. Vestnik St.Petersb. Univ.Math. 57, 130–139 (2024). https://doi.org/10.1134/S1063454124010047
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S1063454124010047