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Quasi-stationary Tidal Evolution with Arbitrarily Misaligned Orbital and Stellar Angular Momenta with a Preliminary Numerical Investigation in the Non-Dissipative Limit

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Abstract

We review and extend the results of our 2021 paper concerning the problem of tidal evolution of a binary system with a rotating primary component with rotation axis arbitrarily inclined with respect to the orbital plane. Only the contribution of quasi-stationary tides is discussed. Unlike previous studies in this field we present evolution equations derived “from first principles.” The governing equations contain two groups of terms. The first group of terms determines the evolution of orbital parameters and inclination angles on a long time scale determined by the rate of energy dissipation often described as a “viscous” time scale though radiative damping may also be included. It can be shown that these terms are formally equivalent to corresponding expressions obtained by other authors after a map between the variables adopted is established. The second group of terms is due to stellar rotation. These terms are present even when dissipation in the star is neglected. They may lead to conservative evolution of the angles specifying the orientation of the stellar rotation axis and the orbital eccentricity vector on a relatively short time scale. The corresponding evolution is also linked to the rate of apsidal precession of the binary orbit. Unlike in our 2021 paper we consider all potentially important sources of apsidal precession in an isolated binary, namely precession arising from the tidal distortion and rotation of the primary as well as Einstein precession. We solve these equations numerically for a small sample of input parameters, leaving a complete analysis to an accompanying paper. Periodic changes to both the inclination of the rotational axis and its precession rate are found. In particular, for a particular binary parameters periodic flips between prograde and retrograde rotation are possible. Also, when the inclination angle is allowed to vary, libration of the apsidal angle becomes possible. Furthermore, when the spin angular momentum is larger than the orbital angular momentum there is a possibility of a significant periodic eccentricity changes similar those coming about from the well-known Kozai–Lidov effect. These phenomena could, in principle, be observed in systems with relatively large inclinations and eccentricities such as, e.g., those containing a compact object. In such systems both large inclinations and eccentricities could be generated as a result of a kick applied to the compact object during a supernova explosion. The paper is based on a talk presented at the astrophysical memorial seminar “Novelties in Understanding the Evolution of Binary Stars,” dedicated to the 90th anniversary of Professor M.A. Svechnikov.

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Notes

  1. Note that IP also discuss the possibility of an extended companion of low mass, whose spin is fully synchronised to the orbit.

  2. We recall that by definition the components of \({\mathbf{L}}\) in the orbit frame are \((0,0,L)\) and the components of \({\mathbf{S}}\) in the stellar frame are \((0,0,S)\).

  3. In Table 2 of IP, \(\hat {\Omega }\) erroneously appears instead of \(\tilde {\Omega }\) which should replace it.

  4. Note a misprint in IP, where there was “–” instead of “\( + \)” in (35).

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ACKNOWLEDGMENTS

We are grateful to A.J. Barker, K.A. Postnov, N.I. Shakura and the referee for useful comments.

Funding

PBI was supported in part by the grant 075-15-2020-780 “Theoretical and experimental studies of the formation and evolution of extrasolar planetary systems and characteristics of exoplanets” of the Ministry of Science and Higher Education of the Russian Federation.

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Authors and Affiliations

  1. Astro Space Centre, Lebedev Physical Institute, Moscow, Russia

    P. Ivanov

  2. DAMTP, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WA, UK

    John Papaloizou

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  1. P. Ivanov
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Correspondence to P. Ivanov or John Papaloizou.

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Ivanov, P., Papaloizou, J. Quasi-stationary Tidal Evolution with Arbitrarily Misaligned Orbital and Stellar Angular Momenta with a Preliminary Numerical Investigation in the Non-Dissipative Limit. Astron. Rep. 67, 912–928 (2023). https://doi.org/10.1134/S1063772923090068

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