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On Tidal Distances between Components of Wide Binary Stars in the Pleiades

On Tidal Distances between Components of Wide Binary Stars in the Pleiades The paper estimates the tidal distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} between the components of wide binary stars (BS) in the Pleiades. We used data on the parameters of the Pleiades cluster and wide BSs in the Pleiades, obtained earlier from the Gaia DR2 data. Two models of a cluster are considered in the form of a gravitating sphere, homogeneous and inhomogeneous in density. Using the integrals of angular momentum and energy of the motion of the centers of mass of the BSs, two elliptical orbits and two open ‘‘rosette-like’’ trajectories of the centers of mass of the BSs relative to the center of mass of the three-body system (the BS components and the cluster) are constructed. For three bodies in a coordinate system nonuniformly rotating relative to their center of mass with the origin at the center of mass of the BS the equations of motion are written. For the model of a homogeneous cluster, a formula is obtained for the value \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for a BS moving along an elliptical orbit. The integrals of angular momentum and energy of motion of the components of the BS and three bodies of the system are obtained. The sizes of the area under the surface of zero velocities (SZV) of the BS components along the axes of the coordinate system with the origin in the center of mass of the BS are estimated. It is shown that in the model of a homogeneous cluster, the sizes of the area under the SZV of the BS components in the pericenters of the considered orbits are smaller than in the apocenters, and on the inner and outer parts of the BS orbits there are restrictions on the size of the area under the SZV of the BS components that differ in number and magnitude. For the model of an inhomogeneous cluster, an equation is obtained for the quantity \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for the BS moving along the ‘‘rosette’’ trajectory relative to the center of mass of three bodies. The quantities \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} are numerically determined for the BS at different points of two such trajectories. With the help of the energy integral \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{1,2,3}$$\end{document} of the motion of three bodies, the sizes of the area under the SZV of the BS components are determined. With an increase in the distance of the center of mass of the BS from the center of mass of three bodies in the model of an inhomogeneous cluster, an increase in the size of the area under the SZV of the BS components was noted, as well as the existence of more complex restrictions on the size of this area than in a homogeneous cluster. Mutual distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i,j}$$\end{document} between the components of wide BSs in the Pleiades are between the maximum and minimum values of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for BSs on ‘‘rosette’’ trajectories. The increase in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i,j}$$\end{document} with the distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i}$$\end{document} of the BS components from the center of the cluster is mainly due to the motion of the BS along its trajectories in the Pleiades. Other applications of the integral \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{1,2,3}$$\end{document} for estimating the BS parameters in the Pleiades are also considered. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Astrophysical Bulletin Springer Journals

On Tidal Distances between Components of Wide Binary Stars in the Pleiades

Danilov, V. M.
Astrophysical Bulletin , Volume 78 (1) – Mar 1, 2023
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References (28)

Publisher
Springer Journals
Copyright
Copyright © Pleiades Publishing, Ltd. 2023
ISSN
1990-3413
eISSN
1990-3421
DOI
10.1134/s1990341323010042
Publisher site
See Article on Publisher Site

Abstract

The paper estimates the tidal distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} between the components of wide binary stars (BS) in the Pleiades. We used data on the parameters of the Pleiades cluster and wide BSs in the Pleiades, obtained earlier from the Gaia DR2 data. Two models of a cluster are considered in the form of a gravitating sphere, homogeneous and inhomogeneous in density. Using the integrals of angular momentum and energy of the motion of the centers of mass of the BSs, two elliptical orbits and two open ‘‘rosette-like’’ trajectories of the centers of mass of the BSs relative to the center of mass of the three-body system (the BS components and the cluster) are constructed. For three bodies in a coordinate system nonuniformly rotating relative to their center of mass with the origin at the center of mass of the BS the equations of motion are written. For the model of a homogeneous cluster, a formula is obtained for the value \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for a BS moving along an elliptical orbit. The integrals of angular momentum and energy of motion of the components of the BS and three bodies of the system are obtained. The sizes of the area under the surface of zero velocities (SZV) of the BS components along the axes of the coordinate system with the origin in the center of mass of the BS are estimated. It is shown that in the model of a homogeneous cluster, the sizes of the area under the SZV of the BS components in the pericenters of the considered orbits are smaller than in the apocenters, and on the inner and outer parts of the BS orbits there are restrictions on the size of the area under the SZV of the BS components that differ in number and magnitude. For the model of an inhomogeneous cluster, an equation is obtained for the quantity \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for the BS moving along the ‘‘rosette’’ trajectory relative to the center of mass of three bodies. The quantities \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} are numerically determined for the BS at different points of two such trajectories. With the help of the energy integral \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{1,2,3}$$\end{document} of the motion of three bodies, the sizes of the area under the SZV of the BS components are determined. With an increase in the distance of the center of mass of the BS from the center of mass of three bodies in the model of an inhomogeneous cluster, an increase in the size of the area under the SZV of the BS components was noted, as well as the existence of more complex restrictions on the size of this area than in a homogeneous cluster. Mutual distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i,j}$$\end{document} between the components of wide BSs in the Pleiades are between the maximum and minimum values of \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$|\xi_{c,1,2}|$$\end{document} for BSs on ‘‘rosette’’ trajectories. The increase in \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i,j}$$\end{document} with the distances \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$r_{i}$$\end{document} of the BS components from the center of the cluster is mainly due to the motion of the BS along its trajectories in the Pleiades. Other applications of the integral \documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$E_{1,2,3}$$\end{document} for estimating the BS parameters in the Pleiades are also considered.

Journal

Astrophysical Bulletin – Springer Journals

Published: Mar 1, 2023

Keywords: stars: kinematics and dynamics; open clusters and associations: individual: Pleiades

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