The precession and nutation of deformable bodies,III

In preceding papers of this series (Kopal, 1968; 1969) the Eulerian equations have been set up which govern the precession and nutation of self-gravitating fluid globes of arbitrary structures in inertial coordinates (space-axes) as well as with respect to the rotating body axes; with due account being taken of the effects arising from equilibrium as well as dynamical tides.In Section 1 of the present paper, the explicit form of these equations is recapitulated for subsequent solations. Section 2 contains then a detailed discussion of the coplanar case (in which the equation of the rotating configuration and the plane of its orbit coincide with the invariable plane of the system); and small fluctuations in the angular velocity of axial rotation arising from the tidal breathing in eccentric binary systems are investigated.In Section 3, we consider the angular velocity of rotation about theZ-axis to be constant, but allow for finite inclination of the equator to the orbital plane. The differential equations governing such a problem are set up exactly in terms of the time-dependent Eulerian angles and , and their coefficients averaged over a cycle. In Section 4, these equations are linearized by the assumption that the inclinations of the equator and the orbit to the invariable plane of the system are small enough for their squares to be negligible; and the equations of motion reduced to their canonical form.The solution of these equations — giving the periods of precession and nutation of rotating components of close binary systems, as well as the rate of nodal regression which is synchronised with precession — are expressed in terms of the physical properties of the respective system and of its constituent components; while the concluding Section 6 contains a discussion of the results, in which the differences between the precession and nutation of rigid and fluid bodies are pointed out.     

Precession of the Orbital Plane of Binary Pulsars and Significant Variabilities

There are two ways of expressing the precession of orbital plane of a binary pulsar system, given by Barker & O'Connell, Apostolatos et al. and Kidder, respectively. We point out that these two ways actually come from the same Lagrangian under different degrees of freedom. Damour & Schafer and Wex & Kopeikin applied Barker & O'Connell's orbital precession velocity in pulsar timing measurement. This paper applies Apostolatos et al.'s and Kidder's orbital precession velocity. We show that Damour & Schafer's treatment corresponds to negligible Spin-Orbit induced precession of periastron, while Wex & Kopeikin and this paper both found significant (but not equivalent) effects. The observational data of two typical binary pulsars, PSR J2051-0827 and PSR J1713+0747, apparently support a significant Spin-Orbit coupling effect. Specific binary pulsars with orbital plane nearly edge on could discriminate between Wex & Kopeikin and this paper: if the orbital period derivative of the double-pulsar system PSRs J0737-3039 A and B, with orbital inclination angle i = 87.7129 deg, is much larger than that of the gravitational radiation induced one, then the expression in this paper is supported, otherwise Wex & Kopeikin's is supported.     

Tidal fluxes and evolution of binary star systems

Explicit analytical expressions for the tidal velocities in the stars which are components of the binary system are obtained, taking into account the viscosity of the stellar matter, the star's rotation, the eccentricity of the orbit and tilt of the axis of rotation to the orbital plane.Equations which determine the tidal evolution of the star's axial and orbital motions are obtained and analyzed. Numerical solutions of these equations are analyzed.     

Rotation of close binary system components

The rotation of close binary system components is investigated. The principal physical characteristics as well as the equatorial rotational velocities and the axial and orbital inclinations for 46 close binary systems were determined. It is found that the rotation axes of the individual stars in a pair cross the orbital plane under different angles. As a rule, the rotation and orbital periods of a vast majority of the systems investigated here do not coincide.     

Analytical solution for the three-dimensional motion of a circumbinary planet around a binary star

By using the method of separating rapid and slow subsystem, we obtain an analytical solution for a stable three-dimensional motion of a circumbinary planet around a binary star. We show that the motion of the planet is more complicated than it was obtained for this situation analytically by Farago and Laskar (2010). Namely, in addition to the precession of the orbital plane of the planet around the angular momentum of the binary (found by Farago and Laskar (2010)), there is simultaneously the precession of the orbital plane of the planet within the orbital plane. We show that the frequency of this additional precession is different from the frequency of the precession of the orbital plane around the angular momentum of the binary. We demonstrate that this problem is mathematically equivalent both to the problem of the motion of a satellite around an oblate planet and to the problem of a hydrogen Rydberg atom in the field of a high-frequency linearly-polarized laser radiation, thus discovering yet another connection between astrophysics and atomic physics. We point out that all of the above physical systems have a higher than geometrical symmetry, which is a counterintuitive result. In particular, it is manifested by the fact that, while the elliptical orbit of the circumbinary planet (around a binary star) or of the satellite (around an oblate planet) or of the Rydberg electron (in the laser field) undergoes simultaneously two types of the precession, the shape of the orbit does not change. The fact that a system, consisting of a circumbinary planet around a binary star, possesses the hidden symmetry should be of a general physical interest. Our analytical results could be used for benchmarking future simulations.     

On the precession of accretion discs in X-ray binaries

In this Letter, recent results on the nodal precession of accretion discs in close binaries are applied to the discs in some X-ray binary systems. The ratio between the tidally forced precession period and the binary orbital period is given, as well as the condition required for the rigid precession of gaseous Keplerian discs. Hence the minimum precessional period that may be supported by a fluid Keplerian disc is determined. It is concluded that near-rigid body precession of tilted accretion discs can occur and generally reproduce observationally inferred precession periods, for reasonable system parameters. In particular, long periods in SS 433, Her X-1, LMC X-4 and SMC X-1 can be fitted by the tidal model. It is also found that the precession period that has been tentatively put forward for Cyg X-2 cannot be accommodated by a tidally precessing disc model for any realistic choice of system parameters.     

Precession of the Orbit of a Planet around Stars Revolving along: Low-Eccentricity Orbits in a Binary System: Analytical Solution

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.     

Observational implications of precessing protostellar discs and jets

We consider the dynamics of a protostellar disc in a binary system where the disc is misaligned with the orbital plane of the binary, with the aim of determining the observational consequences for such systems. The disc wobbles with a period approximately equal to half the orbital period of the binary and precesses on a longer time-scale. We determine the characteristic time-scale for realignment of the disc with the orbital plane as a result of dissipation. If the dissipation is determined by a simple isotropic viscosity then we find, in line with previous studies, that the alignment time-scale is of the order of the viscous evolution time-scale. However, for typical protostellar disc parameters, if the disc tilt exceeds the opening angle of the disc, then tidally induced shearing within the disc is transonic. In general, hydrodynamic instabilities associated with the internally driven shear result in extra dissipation that is expected to drastically reduce the alignment time-scale. For large disc tilts the alignment time-scale is then comparable with the precession time-scale, while for smaller tilt angles δ , the alignment time-scale varies as (sin δ )⁻¹. We discuss the consequences of the wobbling, precession and rapid realignment for observations of protostellar jets and the implications for binary star formation mechanisms.     

Apsidal Motion in Binaries: Rotation of the Components

A sample of 51 separated binary systems with measured apsidal periods and rotational velocities of the components is examined. The ranges of the angles of inclination of the equatorial planes of the components to the orbital plane are estimated for these systems. The observed apsidal velocities can be explained by assuming that the axes of rotation of the stars are nonorthogonal to the orbital plane in roughly 47% of the systems (24 of the 51) and the rotation of the components is not synchronized with the orbital motion in roughly 59% of the systems (30 of 51). Nonorthogonality and nonsynchrony are defined as deviations from 90° and a synchronized angular velocity, respectively, at levels of 1 or more.     

Role of the Rossiter-McLaughlin effect in the study of close binaries

This paper is devoted to binary stars belonging to the class of eclipsing-variable systems.Photometric and spectroscopic analysis of eclipses allows us to determine geometric parameters of the orbit and physical characteristics of stellar components as well as inclinations of stellar equators to the orbital plane. Estimations of inclinations can be obtained from measurement of the Rossiter-McLaughlin effect, which is discussed using examples of some eccentric binaries with an anomalous apsidal effect. Our task is to find the complete spectrum of solutions of the equation of apsidal motion, depending on the inclinations of the polar axes of the components to the orbital one for these systems, based on their individual spectroscopic and photometric observational data. The matrix of solutions allows us to select those pairs of polar inclinations that provide agreement with the observational apsidal period.     

Ground-based detection of terrestrial extrasolar planets by photometry: the case for CM Draconis

The detection of extrasolar planets by measuring a photometric drop in the stellar brightness due to a planetary transit can be statistically improved by observing eclipsing binary systems and photometrically improved by observing small component systems. In particular the system CM Draconis, with two dM4 components, would allow the detection of extrasolar planets in the size range of Earth-to-Neptune requiring a ground-based photometric precision of about 0.08% to 1.1% (photometric precision of about 0.3% is routinely achievable with 1-meter class telescopes at the magnitude of CM Draconis, 11.07 inR-filter). In addition, the transit of extrasolar planets in a binary star system provides a unique, quasi-periodic signal that can be cross-correlated with the observational data to detect sub-noise signals. We examine the importance of making such observations to an understanding of the formation and evolution of terrestrial-type planets in main-sequence star systems. Terrestrial planets could have formed with substancially shorter periods in this lower luminosity system, for example, and might be expected to have accreted essentially in the binary orbital plane (however, non-coplanar planets may also eventually be detectable due to precession). We also report on a network of medium-sized telescopes at varying longitudes that have been organized to provide such constraints on terrestrial-planet formation processes and discuss the extention of near-term observations to other possible binary systems, as well. Finally, we discuss a more speculative, future observation that could be performed on the CM Draconis system that would be of exobiological as well as astrophysical interest.     

存在非同步转动的双星演化模型

通过计算双星演化中的角动量转移,研究了潮汐作用下双星系统自转与公转周期的变化以及潮汐作用对双星演化的影响.结果表明,密近双星系统在主序演化时,潮汐摩擦会在较短的时间内使自转与公转达到比较接近的状态,此后经过一个较长时间的调整才能使自转与公转达到同步转动.物质交换阶段开始后,半相接双星系统更容易出现非同步转动,而相接双星系统物质交换很难破坏系统的同步状态.同时比较了非同步双星模型与同步双星模型演化曲线在赫罗图上的不同,结果表明非同步模型在物质交换阶段主星演化曲线向赫罗图光度和有效温度高的方向移动.最后通过对统计的观测数据进行分析后发现,采用该模型可以解释观测上双星超过潮汐锁定时标后仍然存在非同步转动的现象.     

The effects of viscous friction on the precession and nutation of celestial bodies

The aim of the present study has been to set the system of differential equations which govern the precession and nutation of self-gravitating globes of compressible viscous fluid, due to the attraction exerted on the rotating configuration by its companion; and to construct their approximate solution which are correct to terms of the second order in small dependent variables of the problem. Section 2 contains an explicit formulation of the effects of viscosity arising in this connection, given exactly as far as the viscosity remains a function of radial distancer only; but irrespective of its magnitude. In Section 3 the equations of motion will be linearized for the case of near-circular orbits and small inclinations andi of the equator of the rotating configuration, and of its orbital plane, to the invariable plane of the system; while in Section 4 further simplifications will be introduced which are legitimate for studies of secular (or long-periodic) motions of the nodes and inclinations. The actual solutions of so simplified a system of equations are constructed in Section 5; and these represent a generalization of the results obtained in our previous investigation (Kopal, 1969) of the inviscid case.The physical significance of the new results will be discussed in the concluding Section 6. It is demonstrated that the axes of rotation of deformable components in close binary systems are initially inclined to the orbital plane, viscous dissipation produced by dynamical tides will tend secularly to rectify their positions until perpendicularity to the orbital plane has been established, and the equators as well as orbit made to coincide with the invariable plane of the system-in a similar manner as other effects of tidal friction are bound eventually to synchronize the velocity of axial rotation with that of orbital revolution in the course of time.An application of the results of the present study to the dynamics of the Earth-Moon system discloses that the observed inclination of 1°.5 of the lunar equator to the ecliptic cannot be regarded as being secularly constant, but representing the present deviations from perpendicularity of oscillatory motion of very long period.The Lunar Science Institute is operated by the Universities Space Research Association under Contract No. NSR-09-051-001 with the National Aeronautics and Space Administration. This paper constitutes the Lunar Science Institute Contribution No. 85.     

Restriction of motions in wide pairs in the Galactic field

The motions of the components of wide binary stars in the solar neighborhood in the regular Galactic gravitational field on time scales ～10¹⁰ yr have been studied numerically. The regions of restricted motions of the components in wide pairs have been found depending on the initial conditions: the magnitude of the relative velocity of the components, their mutual distance, and the inclination of the relative velocity vector to the Galactic plane. The size of the main part of the region of restricted motions is approximately equal to the tidal radius. Profound changes in the eccentricity of the binary orbit occur at inclinations close to 90°, which can lead to close approaches of the stars with a pericenter distance less than 1 AU. In the case of retrograde motions (the binary rotates in a direction opposite to the Galactic rotation), there is a region of restricted motions extending at least to 10 pc. Examples of the trajectories of relative motion of the stars and the change in osculating orbital elements are given for systems with restricted motions.     

Precession and nutation in the η Carinae binary system: evidence from the X-ray light curve

It is believed that η Carinae is actually a massive binary system, with the wind–wind interaction responsible for the strong X-ray emission. Although the overall shape of the X-ray light curve can be explained by the high eccentricity of the binary orbit, other features like the asymmetry near periastron passage and the short quasi-periodic oscillations seen at those epochs have not yet been accounted for. In this paper we explain these features assuming that the rotation axis of η Carinae is not perpendicular to the orbital plane of the binary system. As a consequence, the companion star will face η Carinae on the orbital plane at different latitudes for different orbital phases and, since both the mass-loss rate and the wind velocity are latitude dependent, they would produce the observed asymmetries in the X-ray flux. We were able to reproduce the main features of the X-ray light curve assuming that the rotation axis of η Carinae forms an angle of  29°± 4°  with the axis of the binary orbit. We also explained the short quasi-periodic oscillations by assuming nutation of the rotation axis, with an amplitude of about  5°  and a period of about 22 days. The nutation parameters, as well as the precession of the apsis, with a period of about 274 years, are consistent with what is expected from the torques induced by the companion star.     

Tidal Angular Momentum in Close Binary Systems in presence of Wind Driven Non-conservative Mass Transfer with Uniform Mass Accretion Rate

A very well-known property of close binary stars is that they usually rotate slowly than a similar type single star. Massive stars in close binary systems are supposed to experience an exchange of mass and angular momentum via mass transfer and tidal interaction, and thus the evolution of binary stars becomes more complex than that of individual stars. In recent times, it has become clear that a large number of massive stars interact with binary companions before they die. The observation also reveals that in close pairs the rotation tends to be synchronized with the orbital motion and the companions are naturally tempted to invoke tidal friction. We here introduce the effect of tidal angular momentum in the model of wind driven non-conservative mass transfer taking mass accretion rate as uniform with respect to time. To model the angular momentum evolution of a low mass main sequence companion star can be a challenging task. So, to make the present study more interesting, we have considered initial masses of the donor and gainer stars at the proximity of bottom-line main sequence stars and they are taken with lower angular momentum. We have produced a graphical profile of the rate of change of tidal angular momentum and the variation of tidal angular momentum with respect to time under the present consideration.     

On synchronism between axial rotation and orbital motion in close binary systems

In this paper we examine the possible outcome of the tidal evolution of a close binary system using a method from which the outline has already been given by Counselman (1973). If the value of the total angular momentum of the system is sufficiently large, two equilibrium states corresponding to synchronism between stellar rotation and orbital motion are possible. In one of these states the total energy attains no extreme value. The considered evolution can be visualized geometrically by the motion of a point along a hyperbolic cylinder in three-dimensional space. A comparison with some observational data reveals that most of the synchronously rotating detached systems have attained a stable equilibrium state of minimum total energy for the given value of total angular momentum.     

Dynamical tides in close binary systems

The aim of the present paper will be to investigate the circumstances under which an irreversible dissipation of the kinetic energy into heat is generated by the dynamical tides in close binary systems if (a) their orbit is eccentric; (b) the axial rotation of the components is not synchronized with the revolution; or (c) the equatorial planes are inclined to that of the orbit.In Section 2 the explicit form of the viscous dissipation function will be set up in terms of the velocity-components of spheroidal deformation arising from the tides; in Section 3, the principal partial tides contributing to the dissipation will be detailed; Section 4 will be devoted to a determination of the extent of stellar viscosity — both gas and radiative; while in the concluding Section 5 quantitative estimates will be given of the actual rate at which the kinetic energy of dynamical tides gets dissipated into heat by viscous friction in stellar plasma.The results disclose that the amount of heat produced per unit time by tidal interaction between components of actual close binaries equals only about 10^–10th part of their nuclear energy production; and cannot, therefore, affect the internal structure of evolution of the constituent stars to any appreciable extent. Moreover, it is shown that the kinetic energy of their axial rotation can be influenced by tidal friction only on a nuclear, rather than gravitational (Kelvin) time-scale — as long as plasma or radiative viscosity constitute the sole sources of dissipation. However, the emergence of turbulent viscosity in secondary components of late spectral types, which have evolved away from the Main Sequence, can accelerate the dissipation 10⁵–10⁶ times, and thus give rise to appreciable changes in the elements of the system (particularly, in the orbital periods) over time intervals of the order of 10⁵–10⁶ years. Lastly, it is pointed out that, in close binary systems consisting of a pair of white dwarfs, a dissipation of the kinetic energy through viscous tides in degenerate fermion-gas could produce enough heat to account, by itself, for the observed luminosity of such objects.     

Perturbations in close binary systems produced by the tides lagging in latitude

The aim of this investigation is to present the periodic and secular perturbations of the orbital elements of close binary systems due to tidal lag in latitude. The variational equations of the problem of plane motion will be set up in terms of the rectengular componentsR, S, andW of the disturbing accelerations. These equations are highly nonlinear with respect to the orbital elements and we present analytic approximations to the effects produced by the perturbing acceleration due to dynamical tides lagging in latitude. The perturbed elements of the orbit have been expressed by means of Hansen coefficients in the compact form of summations.