## Dynamics of Close Binary SystemsThe aim of the present book will be to provide a comprehensive account of our present knowledge of the theory of dynamical phenomena exhibited by elose binary systems; and on the basis of such phenomena as have been attested by available observations to outline probable evolutionary trends of such systems in the course of time. The evolution of the stars - motivated by nuelear as weIl as gravitation al energy sources - constitutes nowadays a well-established branch of stellar astronomy. No theo ries of such an evolution are as yet sufficently specific - let alone infallible - not to require continual tests by a confrontation of their consequences with the observed prop erties of actual stars at different stages of their evolution. The discriminating power of such tests depends, of course, on the range of information offered by the test objects. Single stars which move alone in space are now known to represent only a minority of objects constituting our Galaxy (cf. Chapter 1-2); and are, moreover, not very revealing of their basic physical characteristics - such as their masses or absolute dimensions. If there were no binary systems in the sky, the only star whose vital statistics would be fully known to us would be our Sun. |

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### Contents

BINARY STARS IN THE SKY | 1 |

11 BinaryStar Population in our Galaxy | 10 |

BIBLIOGRAPHICAL NOTES | 16 |

FIGURES OF EQUILIBRIUM | 18 |

II2 Rotational Distortion | 26 |

II3 Tidal Distortion | 36 |

II4 Interaction between Rotation and Tides | 46 |

II5 Effects of Internal Structure | 58 |

A DEFINITION | 253 |

C RELATION BETWEEN ORBITAL PERIOD AND TIMES OF THE MINIMA | 261 |

V5 Effects of Variable Mass | 264 |

A GENERALIZED EQUATIONS OF MOTION | 265 |

B ISOTROPIC MASS LOSS | 268 |

C NONISOTROPIC MASS LOSS | 272 |

V6 Perturbations by a Third Body | 276 |

A EQUATIONS OF THE PROBLEM | 277 |

II6 Gravity Density Distribution and Moments of Inertia | 68 |

BIBLIOGRAPHICAL NOTES | 77 |

DYNAMICAL TIDES | 80 |

III1 Equations of the Problem | 81 |

B RADIATING SYSTEMS | 87 |

III2 Linearized Equations | 93 |

A VISCOUS SPHEROIDAL DEFORMATIONS | 95 |

B BOUNDARY CONDITIONS | 101 |

C HOMOGENEOUS MODEL | 103 |

Dynamical Tides | 110 |

A MASSPOINT MODEL | 111 |

B CENTRALLYCONDENSED MODEL | 116 |

C DISTURBING POTENTIAL | 117 |

III4 Dissipation of Energy by Dynamical Tides | 126 |

A VISCOUS FRICTION | 130 |

B APPLICATION TO BINARY SYSTEMS | 132 |

BIBLIOGRAPHICAL NOTES | 136 |

GENERALIZED ROTATION | 138 |

IV1 Equations of Motion for Deformable Bodies | 140 |

A EULERIAN EQUATIONS | 147 |

IV2 Rotation of Deformable Bodies | 154 |

A EFFECTS OF DEFORMATION | 156 |

B MOMENTS AND PRODUCTS OF INERTIA | 161 |

C COEFFICIENTS OF DEFORMATION | 165 |

IV3 Effects of Viscosity | 170 |

SPHERICAL CONFIGURATIONS | 178 |

SPHEROIDAL CONFIGURATIONS | 183 |

D DISSIPATION OF ENERGY BY TIDAL FRICTION | 187 |

IV4 Nonuniform Rotation | 191 |

B NONSTEADY ROTATION | 197 |

BIBLIOGRAPHICAL NOTES | 200 |

DYNAMICS OF CLOSE BINARIES | 201 |

V1 Equations of the Problem | 202 |

A PERTURBATION EQUATIONS | 203 |

TIDAL LAG | 209 |

V2 Perturbations of the Orbital Plane Precession and Nutation | 211 |

A EFFECTS OF VISCOSITY | 212 |

B LINEARIZED CASE | 215 |

C SECULAR AND LONGPERIODIC MOTION | 222 |

D SOLUTION OF EQUATIONS | 225 |

V3 Perturbations in the Orbital Plane | 232 |

TIDAL DISTORTION | 236 |

COMPARISON WITH OBSERVATIONS | 243 |

V4 Period Changes in Eclipsing Binary Systems | 249 |

A GENERALIZED LAW OF AREAS | 250 |

B SHORTRANGE PERTURBATIONS | 279 |

C LONGRANGE PERTURBATIONS | 293 |

D EFFECTS OF THE LIGHT EQUATION | 304 |

BIBLIOGRAPHICAL NOTES | 309 |

THE ROCHE MODEL | 312 |

VI1 Roche Equipotentials | 313 |

A SURFACES OF ZERO VELOCITY | 318 |

VI2 Geometry of Roche Surfaces | 322 |

B ROCHE LIMIT | 325 |

C GEOMETRY OF THE ECLIPSES | 331 |

D EXTERNAL ENVELOPES | 336 |

VI3 The Roche Coordinates | 339 |

A ROTATIONAL PROBLEM | 340 |

B TIDAL PROBLEM | 343 |

C DOUBLESTAR PROBLEM | 346 |

BIBLIOGRAPHICAL NOTES | 360 |

STABILITY OF THE COMPONENTS OF CLOSE BINARY SYSTEMS | 361 |

VIII Criteria of Stability | 362 |

A ROTATIONAL PROBLEM | 369 |

B DOUBLESTAR PROBLEM | 374 |

VII3 Dynamical StabUity | 380 |

A VIBRATIONS OF THE ROCHE MODEL | 383 |

V1I4 Concluding Remarks | 387 |

BIBLIOGRAPHICAL NOTES | 389 |

ORIGIN AND EVOLUTION OF BINARY SYSTEMS | 390 |

VIII1 Evolution of the Star | 391 |

VIII2 Classification of Close Binary Systems | 400 |

A CHARACTERISTIC PARAMETERS | 402 |

VIII3 Nuclear Evolution of Close Binary Systems | 405 |

A DETACHED SYSTEMS | 406 |

B SEMIDETACHED SYSTEMS | 413 |

CCONTACT SYSTEMS | 430 |

VIII4 Tidal Evolution of Close Binary Systems | 437 |

A ENERGY AND MOMENTUM | 441 |

B EVOLUTION WITH CONSTANT MOMENTUM | 446 |

C COMPARISON WITH OBSERVATIONS | 451 |

D TIDAL EVOLUTION | 458 |

VIII5 Beginnings and Ends | 464 |

SUBDWARF BINARIES | 467 |

C XRAY BINARIES AND BLACK HOLES | 475 |

BIBLIOGRAPHICAL NOTES | 487 |

REFERENCES | 491 |

502 | |

507 | |

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### Common terms and phrases

Algol amplitudes angular velocity approximation apsidal arising Astron Astrophys axial rotation axis becomes centre of mass Chapter close binary systems coefficients components of close constant constitute coordinates corresponding curves denotes density differential equations discloses dissipation dynamical tides eclipsing systems eclipsing variables effects elements Equa equilibrium evolution evolutionary explicit form expressed fact follows foregoing Equations function given by Equations gravitational harmonics inertia insertion integral interior investigations kinetic energy known Kopal Lagrangian points luminosity Main Sequence mass-point mass-ratio massive momentum moreover neutron star observed orbital eccentricity orbital period orbital plane orbital revolution oscillations partial perturbations potential energy problem quantities radiation radiative radius radius-vector relative orbit represent right-hand side Roche limit secular side of Equation solution spectral type spectroscopic spherical stage stars stellar stellar evolution subdwarf subgiant surface theory tidal distortion tion true anomaly UMa-type values velocity components viscosity white dwarfs X-ray zero