Precession of the Orbit of a Planet around Stars Revolving along: Low-Eccentricity Orbits in a Binary System: Analytical Solution |
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| Institution: | 1. Department of Mathematical Sciences, University of Zululand, Private Bag X1001, Kwa-Dlangezwa, KwaZulu-Natal 3886, South Africa;2. Faculty of Natural Sciences, Mangosuthu University of Technology, Umlazi, South Africa;1. Chemical Engineering Division, School of Engineering, University of Bradford, Bradford, West Yorkshire BD7 1DP, UK;2. Middle Technical University, Baghdad, Iraq;3. Chemical Engineering Department, College of Engineering, Tikrit University, Iraq;1. National University of Computer and Emerging Sciences, Islamabad, Lahore Campus, Pakistan;2. Department of Mathematics, Shanghai University, Shanghai 200444, PR China;1. Department of Mathematics, Sri Aurobindo College, University of Delhi, New Delhi, Delhi 110017, India;2. Department of Mathematics, Deshbandhu college, University of Delhi, New Delhi, Delhi 110019, India;3. Department of Commerce, Sri Aurobindo College, University of Delhi, New Delhi, Delhi 110017, India;4. Department of Physics, Deshbandhu college, University of Delhi, New Delhi, Delhi 110019, India |
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| Abstract: | In our previous paper (hereafter, paper I) we presented analytical results on the non-planar motion of a planet around a binary star for the cases of the circular orbits of the components of the binary. We found that the orbital plane of the planet (the plane containing the “unperturbed” elliptical orbit of the planet), in addition to precessing about the angular momentum of the binary, undergoes simultaneously the precession within the orbital plane. We demonstrated that the analytically calculated frequency of this additional precession is not the same as the frequency of the precession of the orbital plane about the angular momentum of the binary, though the frequencies of both precessions are of the same order of magnitude. In the present paper we extend the analytical results from paper I by relaxing the assumption that the binary is circular – by allowing for a relatively small eccentricity ε of the stars orbits in the binary. We obtain an additional, ε-dependent term in the effective potential for the motion of the planet. By analytical calculations we demonstrate that in the particular case of the planar geometry (where the planetary orbit is in the plane of the stars orbits), it leads to an additional contribution to the frequency of the precession of the planetary orbit. We show that this additional, ε-dependent contribution to the precession frequency of the planetary orbit can reach the same order of magnitude as the primary, ε-independent contribution to the precession frequency. Besides, we also obtain analytical results for another type of the non-planar configuration corresponding to the linear oscillatory motion of the planet along the axis of the symmetry of the circular orbits of the stars. We show that as the absolute value of the energy increases, the period of the oscillations decreases. |
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