Theory of the light curves of eclipsing systems with oscillating components,II

In continuation of our preceding investigation (Kopal, 1982c) of photometric consequences of free oscillations — radial or nonradial — of the eclipsing component on the light changes of close binary systems, the aim of the present paper will be to extend the scope of our inquiry to include the effects of similar oscillations of the component which undergoes eclipse, with arbitrary period and phase.In Sections 2 and 3 which follow a brief introduction to the particular aspect of the problem treated in this paper, a theory will be given of photometric effects caused by arbitrary harmonic oscillations of the components of eclipsing binary systems distorted by equilibrium tides, or by axial rotation with constant angular velocity. A translation of these effects into the frequency domain constitutes a problem which is solvable in a closed form only if the eclipse in question ends in totality. In any other case, the desired effects can be mathematically described only in terms of infinite series (of satisfactory asymptotic properties); and a construction of their explicit forms will be given in the Appendix.     

On secular stability of tidally distorted stars of arbitrary structure

The aim of the present paper will be to develop a theory which should make it possible to investigate secular stability of close binary systems, consisting of tidally-distorted components of arbitrary internal structure, by a minimization of the potential energy of the system as a whole. In the second section which follows brief introductory remarks, appropriate expressions for the total potential energy of a close binary will be formulated. Section 3 will be concerned mainly with the nature of the tide-generating potential, and its effects on the shape of each star. In Section 4, the amplitudes of partial tides raised by this potential will be specified, for stars of arbitrary structure, correctly to terms of second order in superficial distortion; and in Section 5 we shall investigate the effects of interaction between rotation and tides to the same degree of approximation. The concluding Section 6 will then contain an explicit formulation of different constituents adding up to the total potential energy of the system, which can be used as a basis for its secular stability by the methods outlined already in our previous investigation (Kopal, 1973).     

The precession and nutation of deformable bodies,II

In a previous paper of this series (Kopal, 1968a) the Eulerian equations have been set up which govern the precession and nutation of selfgravitating bodies of viscous fluid in inertial coordinates which are at rest in space. In order to facilitate their solution, in the present investigation we shall transform these equations to the rotating body-axes; and shall explicitly evaluate all their coefficients arising as a result of second-harmonic dynamical tides.Following the introductory Section 1 which contains a mathematical statement of the problem, the requisite transformation of coordinates will be outlined in Section 2, and applied to the equations of motion in Section 5. The corresponding moments and products of inertia appropriate for selfgravitating configurations of arbitrary internal structure will be formulated in Section 4; while the deformation terms arising from second-harmonic dynamical tides raised on centrally-condensed configurations will be evaluated in Sections 3 and 6. The concluding Section 7 will then contain a specification of the components of the disturbing force.The next stage of our investigation — namely, a construction of the actual solutions of the equations governing precession and nutation of fluid bodies in different cases of astrophysical interest — has been postponed for a separate paper.     

Self-similar solutions in the theory of flare-ups in novae,I

The aim of this paper will be to present the geometrical elements of a group of totally-eclipsing close binary systems of W UMa-type, evaluated in the frequency-domain by a method developed recently by one of us (Kopal, 1982a, b) which takes account of photometric effects caused by mutual distortion of the components within eclipses. The outcome discloses that the geometry of such systems conforms more closely to that of a pair of contact configurations of the respective mass-ratio than that previously deduced from the same photometric evidence by more elementary methods; and strengthens our earlier surmise (Kopal, 1955) that the components of such systems are not actually separated from each other by any significant gap.     

Theory of the light curves of eclipsing systems with oscillating components: I

The aim of the present investigation will be to develop a theory of the light changes of the eclipsing systems in which one (or both) components oscillate(s) freely-radially or non-radially-in period(s) which may (though need not) be synchronized with that of their orbit. The light curves exhibited by systems with oscillating components will, in general, beasymmetric; and these oscillations may, in fact, be the cause of asymmetries observed in many eclipsing systems with evolved components (for which such oscillations-far from being anomalous-should be the rule rather than exception). The present paper will be concerned with an investigation of photometric phenomena arising from oscillations of theeclipsing components of close binary systems-cutting off (partly or wholly) the star (spherical or distorted) which undergoes eclipse.In Section 2, which follows a brief introductory survey aiming to place the entire subject in its historical perspective, a theory will be given of the light curves of close binary systems, in the frequency-domain, which are affected byradial oscillations of the eclipsed or eclipsing star; while Sections 3 and 4 will be devoted to a similar treatment ofnon-radial oscillation of the secondary (eclipsing) component distorted by equilibrium tides, or axial rotation with constant angular velocity. As will be shown, a frequency-domain approach will enable us to describe all these phenomena in algebraic terms; with auxiliary results required to this end relegated to three Appendices to the main text (in order not to render the latter too discursive and impede the main line of the argument).A similar treatment of photometric phenomena which may arise in such systems from oscillations of their components which undergo eclipse is being postponed for a subsequent communication.     

Dynamical tides in close binary systems,I

The aim of the present paper will be to develop from the fundamental equations of hydrodynamics a theory of dynamical tides in close binary systems, the components of which are regarded to consist of heterogeneous viscous fluid, and to revolve around their common centre of gravity in eccentric orbits; moreover, the equatorial planes of their axial rotation and the orbital plane need not be co-planar, but all may be inclined to the invariable plane of the system of arbitrary amounts. The changes in the pressure or density invoked by time-dependent deformation will be regarded as adiabatic; but, in the equilibrium state, both the density and viscosity of the material of our components may be arbitrary functions of the radial distance.Following a brief exposition in Section 2 of the fundamental equations linearized to small oscillations — be these free or forced — in Section 3 we shall particularize them to describe spheroidal deformations; with due regard to all terms arising from viscosity. Section 4 will contain a specification of the boundary conditions to be imposed upon such oscillations; and in Section 5 we shall solve the problem of non-radial oscillations of self-gravitating inviscid configurations in terms of hypergeometric series. The remaining Sections 6–8 will be devoted to a discussion of the phenomena arising from viscosity: in particular, we shall solve in a closed form the problem of non-radial oscillations of incompressible viscous globes in the terms of Bessel functions. It will be shown that the effect of viscosity — like those of compressibility — tend to de-stabilize all non-radial oscillations of homogeneous configurations.At the other extreme, a similar treatment of a mass-point model — as well as of one exhibiting high but finite degree of central condensation — is being postponed for a subsequent communication.     

Rotational distortion of the stars of arbitrary structure: Fourth approximation

Clairaut's theory of the rotational distortion of self-gravitating configurations of arbitrary structure, arising from axial rotation with constant angular velocity, previously developed (cf. Kopal, 1973) to quantities of third order in superficial distortion, has now been extended to terms of fourth order. The differential equations governing the form and exterior potential of stars so rotating have been set up by the method followed in our previous paper (Kopal, 1973) together with their boundary conditions; but their applications to practical cases are being postponed for a subsequent investigation.Work carried out partly at the Naval Research Laboratory, Washington, D.C., U.S.A.     

Use of Roche coordinates in the problems of small oscillations of rotationally distorted stellar models

The system of Roche coordinates developed by Kopal to study the problems of stars in close binary systems has been used to study the problems of small oscillations of rotationally distorted stars.     

Use of Roche coordinates in the problems of small oscillations of tidally distorted stellar models

The system of Roche coordinates developed by Kopal to study the problems of stars in close binary systems has been used to study the problems of small oscillations of tidally distorted stars.     

Roche limit for homogeneous incompressible masses

The aim of the present investigation has been to establish the minimum distance (commonly referred to as the ‘Roche limit’), to which a small satellite can approach its central star without the loss of its stability. In order to do so, we shall depart from hydrodynamical equations governing small oscillations of stellar structures, and set out to establish the limit at which their distorted form of equilibrium can no longer vibrate periodically in response to arbitrary perturbations. To this end, such equations will be rewritten in terms of curvilinear Clairaut coordinates (Kopal, 1980) in which the gravitational potential defining equilibrium surfaces plays the role of the radial coordinate; and their solution constructed for the classical Roche problem in which the oscillating satellite of infinitesimal mass consists of material which is homogeneous and incompressible, while its primary component acts gravitationally as a mass-point. The outcome of such a solution agrees satisfactorily with that previously established by Chandrasekhar (1963) on the basis of the virial theorem; but the method employed by us lends itself more readily to a generalization of the Roche limit to systems of finite mass ratios and consisting of the components of finite size.     

Tidal evolution in close binary systems

The aim of the present paper will be to give a mathematical outline of the theory of tidal evolution in close binary systems of secularly constant total momentum — an evolution activated by viscous friction of dynamical tides raised by the two components on each other. The first section contains a general outline of the problem; and in Section 2 we shall establish the basic expressions for the energy and momenta of close binaries consisting of components of arbitrary internal structure. In Section 3 we shall investigate the maximum and minimum values of the energy (kinetic and potential) which such systems can attain for given amount of total momentum; while in Section 4 we shall compare these results with the actual facts encountered in binaries with components whose internal structure (and, therefore, rotational momenta) are known to us from evidence furnished by the observed rates of apsidal advance.The results show that all such systems — be these of detached or semi-detached type — disclose that more than 99% of their total momenta are stored in the orbital momentum. The sum of the rotational momenta of the constituent components amounts to less than a percent of the total — a situation characteristic of a state close to the minimum energy for given total momentum. This appears, moreover, to be true not only of the systems with both components on the Main Sequence, but also of those possessing evolved components in contact with their Roche limits.Under such conditions, a synchronism between rotation and revolution (characteristic of both extreme states of maximum and minimum energy) is not only possible, but appears to have been actually approached — if not attained — in the majority of cases. In other words, it would appear that — in at least a large majority of known cases — the existing close binaries have already attained orbits of maximum distension consistent with their momenta; and tidal evolution alone can no longer increase the present separations of the components to any appreciable extent.The virtual absence, in the sky, of binary systems intermediate between the stages of maximum and minimum energy for given momentum leads us to conjecture that the process of dynamical evolution activated by viscous tides may enroll on a time-scale which is relatively short in comparison with their total age — even for systems like Y Cygni or AG Persei, whose total age can scarcely exceed 10⁷ yr. A secular increase of the semi-major axes of relative orbits is dynamically coupled with a corresponding variation in the velocity of axial rotation of both components through the tidal lag arising from the viscosity of stellar material. The differential equations of so coupled a system are given in Section 5; but their solution still constitutes a task for the future.The Lunar Science Institute Contribution No. 90. 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.     

Tidal evolution in close binary systems,II

The aim of the present investigation will be to determine the explicit forms of differential equations which govern secular perturbations of the orbital elements of close binary systems in the plane of the orbit (i.e., of the semi-major axisA, eccentricitye, and longitude of the periastron ), arising from the lag of dynamical tides due to viscosity of stellar material. The results obtained are exact for any value of orbital eccentricity comprised between 0e<1; and include the effects produced by the second, third and fourth-harmonic dynamical tides, as well as by axial rotation with arbitrary inclination of the equator to the orbital plane.In Section 2 following brief introductory remarks the variational equations of the problem of plane motion will be set up in terms of the rectangular componentsR, S, W of disturbing accelerations with respect to a revolving system of coordinates. The explicit form of these coefficients will be established in Section 3 to the degree of accuracy to which squares and higher powers of quantities of the order of superficial distortion can be ignored. Section 4 will be devoted to a derivation of the explicit form of the variational equations for the case of a perturbing function arising from axial rotation; and in Section 5 we shall derive variational equations which govern the perturbation of orbital elements caused by lagging dynamical tides.Numerical integrations of these equations, which govern the tidal evolution of close binary systems prompted by viscous friction at constant mass, are being postponed for subsequent investigations.Prepared at the Lunar Science Institute, Houston, Texas, under the joint support of the Universities Space Research Association, Charlottesville, Virginia, and the National Aeronautics and Space Administration Manned Spacecraft Center, Houston, Texas, under Contract No. NSR 09-051-001. This paper constitutes Lunar Science Institute Contribution no. 100.Normally at the Department of Astronomy, University of Manchester, England.     

Effect of Coriolis force on the equilibrium structures of rotating stars and stars in binary systems

Kopal (Adv. Astron. Astrophys. 9:1–65, 1972) introduced the concept of Roche equipotentials to incorporate the effects of rotation and tidal distortions on the equilibrium structure and periods of small oscillations of rotating stars and stars in binary systems. However his expression for the Roche equipotential accounts for only the effects of centrifugal and gravitational forces and does not take into account the effect of Coriolis force. In this paper we have suitably modified Kopal’s expression for Roche equipotentials to incorporate into it the effect of Coriolis force as well. The modified expression for the Roche equipotential has then been used to compute the equilibrium structures and shapes of polytropic models of rotating stars and stars in binary systems.     

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.     

Periodic perturbations in close binary systems due to tidal lag

In two previous papers (Zafiropoulos and Kopal, 1983a, b; hereafter referred to as Papers I and II) we have investigated the effects of rotational and tidal distortion (for non-lagging tides) on the orbital elements of a close binary system. The present paper deals with secular and periodic perturbations caused by dynamical tides. The componentsR, S, andW of disturbing accelerations for tidal lag have been substituted in the Gaussian form of Lagrange's planetary equations to give the first-order approximation. The results obtained have been expressed by means of Hansen coefficients and include the effects produced by the second, third and fourth harmonic dynamical tides.     

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.     

Clairaut coordinates and the vibrational stability of distorted stars

The aim of the present paper will be to generalize the concept of the Roche coordinates, introduced previously by the author (see Kopal, 1969, 1970, 1971) for a treatment of dynamical phenomena in close binary systems, to Clairaut's coordinates in which the Roche potential of a rotating dipole is replaced by the actual potential of configurations of finite density concentration and arbitrary structure.By virtue of an identification of the potential with the radial coordinate of our three-dimensional system, the Roche and Clairaut coordinates are both bound to be curvilinear if the star in question departs from spherical form. However, unlike Roche coordinates, the Clairaut coordinates introduced in this paper will not be required to constitute an orthogonal system; and, as a result of the freedom so preserved, their angular variables will be identified with the angles and of spherical polars.Such an adoption entails advantages and disadvantages. In the orthodox Roche system, the radial coordinate (i.e., the potential ) is given to us in a closed form; but their angular variables and must, in general, be obtained by an integration of partial differential equations constituting the orthogonality conditions. On the other hand for the Clairaut (non-orthogonal) system of coordinates no such integration is necessary — and, in fact, the angular variables can be adopted at will. However, their radial coordinate (i.e., the potential of a star of arbitrary structure and distortion) is no longer available in a closed form and must be constructed by a sequence of successive approximations — a process initiated in the 18th century by Clairaut (1743), which can be developed to any desired accuracy.As is well known, investigations of the stability of self-gravitating configurations of arbitrary internal structure must be conducted on the basis of fundamental equations of stellar hydrodynamics, which for small oscillations can be reduced to linear forms. In Section 2 the explicit form of these fundamental equations will be set up in Clairaut's coordinates and linearized in Section 3 to the case of small oscillations, while in Section 4 a critical comparison of the Clairaut and Roche coordinates will be made. However their application to rotating stars will be the subject of subsequent papers.     

Vibrational stability of double-star Roche model

The aim of the present paper will be to extend the methods of our previous investigations (Kopal, 1980, 1987) by employing the Clairaut coordinates (in which the radial component is identified with the total potential) to analyze the nature of small oscillations about the equilibrium form of Roche double-star model (identical, in fact, with zero-velocity surfaces of the restricted problem of three bodies).Linearized equations of this problem have been set up in Clairaut coordinates, and solved in a closed form. This solution turns out to be closely analogous to that obtained already for the rotating single-star Roche model, and discloses that (like in the preceding case) the terms secular in time appear already in the linear approximation. However, whether or not a retention of nonlinear terms in the equations of motion can regain secular stability of the respective configurations remains yet to be clarified by future investigations.     

Radiative energy transfer in extended photospheres of the components of close binary systems

The aim of the present paper will be to set up, and solve, the equations governing transfer of radiation in semi-transparent envelopes of the stars; and, in order to do so, to employ a system of curvilinear (non-orthogonal) three-dimensional coordinates in which the radial coordinate has been identified with equipotential surfaces. Such coordinates are particularly suitable to a treatment of the problems arising in close binary systems, which render the outcome more than any other amenable to observable tests, but which has so far received but very scant attention.The introductory section of this paper will contain a statement of the problem; and its mathematical formulation in terms of Clairaut coordinates (cf. Kopal, 1980, 1989, Chapter V) will be outlined in Section 2; their methods in Section 3. Section 4 will then contain an application to the problem of distribution of surface brightness (limb-darkening) over the apparent discs of distorted components of close binary systems; while in Section 5 we shall do the same for radiative flux of distorted stars as a function of the phase (gravity darkening).The concluding Section 6 will then contain an outline of additional problems arising in this connection, to which we shall turn in successive parts of this series.