Evaluation of the photometric perturbations in close binary systems

In the present study the photometric perturbations arising from the mututal tides of the two components in a close binary system are calculated and discussed. The evaluation is made for each star separately as well as for both stars together in the case of occulation eclipses terminating in totality.     

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.     

Periodic perturbations in close binary systems due to rotational distortion

This paper deals with the perturbations which rotation can produce to the orbital elements of a close binary system. The rectangular components,R, S andW of the disturbing accelerations due to rotation have been substituted to the Gauss form of Lagrange's planetary equations to yield the first order approximation. The results obtained are exact for any value of orbital eccentricity between the values 0<e<1 and for arbitrary inclinations of the rotational axes to the orbital plane.First and second order approximations are given for the special case when equators are coplanar to the orbit.     

Periodic perturbations in close binary systems due to tidal distortion (non-lagging tides)

In a previous paper (Zafiropoulos and Kopal, 1982; hereafter referred to as Paper I) we have studies the effects of rotational distortion on the orbital elements. The aim of the present paper is to investigate the secular and periodic perturbations of the orbital elements due to tidal distortion. For tidal distortion when tides do not lag, the Gaussian form of Lagrange's planetary equations has been employed to yield the first- and second-order approximations. The results obtained include the effects produced by the second, third and fourth harmonic distortions. The first order approximation for non-lagging tides has been expressed by means of Hansen coefficients.     

Re-analysis of close binary systems

Line-forming regions around close binaries with strong winds ( /4r _* v 10^–4 g cm^–2) are large in extent compared with the stars, large enough to screen them. Their orbitally-modulated Doppler shifts can overestimate the mass function, because of a larger rotational lever arm. In particular, most of the black-hole candidates need not involve companions more massive than a neutron star.The solar-wind problem is reconsidered. An extrapolation to Wolf-Rayet stars suggests that their winds are centrifugally driven. Their mass-loss rates tend to have been overestimated.Seemingly single (massive) stars can hide a (compact) companion.     

Roundchrom emissivity in close binary systems

Roundchrom is a common chromosphere enveloping both components of close binary system. In present article, the problem of the construction of outward boundary of the roundchrom is examined, particularly, identifying this boundary with zero-velocity equipotential Lagrangian surface corresponding to a definite value of Jacoby constantC. The roundchroms for seven RS CVn type close binary systems as well for an interacting binary HD 207739 are constructed. The main parameters of these roundchroms are obtained, particularly, the electron concentrations are of the order of 10¹⁰ cm^–3.     

Dynamical tides in close binary systems

Differential equations governing the dynamical tides in close binary systems consisting of centrally condensed components of viscous gas are split up (Section 2) in their real and imaginary parts, the ratio of which defines the tidal lag. In Sections 3 and 4 these equations will be particularized to a case in which the central mass-point of each star is surrounded by an evanescent envelope the density of which decreases as the inverse square of the central distance. It is shown that self-gravitating configurations built up in accordance with this model are incapable of performing free nonradial oscillations with a frequency comprised between 0 ² ; but explicit expressions for forced oscillations representing dynamical tides are given for an arbitrary form of the external field of force. Equations for the imaginary components of the displacement, constructed for the same model in Section 4, disclose that if the viscosity of stellar material is identified with that of hydrogen plasma, the tidal lag due to a viscous dissipation of kinetic energy may produce dynamical effects, the cumulative outcome of which becomes appreciable on the Kelvin time-scale, but over short intervals of time their stationary photometric effects should be negligible. The latter can become observationally significant only for stars in which turbulent viscosity under near-adiabatic conditions becomes and important factor.     

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.     

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.     

Dynamical tides in close binary systems

The aim of the present paper will be to deduce the explicit form of differential equations which govern dynamical tides in close binary systems, with simplifications which are permissible for the mass-point model (Section 2), as well as for one exhibiting finite but high internal density concentration (Section 3). It is pointed out that, whereas the exact formulation of the problem leads to a simultaneous system of equations of sixth order (fourth in the inviscid case), this order reduces to four (or two for inviscid fluids) for the mass-point model; and to five (three for inviscid case) if the density concentration is high but finite.In the last section of this paper the coefficientsC _i,j which specify the amplitudes of the individual partial tides are explicitly formulated as functions of the time.     

Figures of equilibrium in close binary systems

The equilibrium configurations of close binary systems are analyzed. The autogravitational, centrifugal and tidal potentials are expanded in Clairaut's coordinates. From the set of the total potential angular terms an integral equations system is derived. The reduction of them to ordinary differential equations and the determination of the boundary conditions allow a formulation of the problem in terms of a single variable.     

On the intercomponent emission in close binary systems

An empirical relationship is discovered for RS CVn type close binary systems between their absolute luminosity, L(MgII), of the ultraviolet magnesium doublet 2800 MgII, and the intercomponent distancea of the system. It has the following form: L(MgII) a ⁿ(Figure 1). It is shown that for the overwhelming majority of binary systemsn = 1 (Figure 4). This correlation presents itself as a direct confirmation of the intercomponent origin of the observed emission, particularly, in the magnesium doublet in close binary systems. The basic relationship of intercomponent emission is derived in the form: L(MgII) = 1.0 × 10³² a ergs s^–1. At the same time, accidental statistical divergences from this correlation are possible on both sides: asn > 1 as welln < 1 (Figure 4). The correlationn = 1 determines also the character, - i.e. cylindric for a stream - of the transfer of gaseous matter from one component of the system to the other, and in the general gas dynamics of the intercomponent medium.The existence of a new category of stellar atmosphere, - which we callroundchrom, is predicted, representing the common chromosphere of a superclose binary system, surrounding or blending both components of the system (Figure 3). The boundaries between the three most important divisions of magnesium doublet emission - chromosphere of single stars, roundchrom of superclose binary systems and intercomponent space - are established for RS CVn type systems. Finally, a number of new problems, both observational and theoretical, are brought forward.     

On the mass-radius relation in close binary systems

In a close binary system with circular orbits in which mass transfer occurs when material from one star fills up and possibly spills over its Roche lobe, the usual mass-radius relation cannot be used as it would lead to unreasonable results in some situations.     

Critical equipotential surfaces in close binary systems

The well-known problem of reckoning the critical surfaces (equipotential zero-velocity surfaces) in the close binary systems is approached by an independent method. The formulation of the problem is based on the assumptions of the binary's matter consisting of ionized hydrogen, the system possessing black-body radiation, a potential magnetic field, being in adiabatic equilibrium. Total pressure and total internal energy are examined. The model, implying synchronous rotation of the components, is described by hydromagnetic equations. For a statical case, however, it is representable by the equation of motion alone. Next, the temperature field is reproduced whereby the ratioP _r/P_g= is playing part of a free parameter. The resulting potential functions, applied to particular binaries, furnish the Lagrangian collinear points, critical surfaces and potentials over them in terms of . The families of surfaces thus obtained, compared with those springing from the Roche model, differ qualitatively in their geometry, position of the collinear equilibrium points, number of possible equilibrium states and the values of critical potentials. At identifying the allowed and forbidden regions of the gas motion new areas have been disclosed across which the gas outflow can take place and more possibilities of shell forming both around the individual components and the system as a whole. As the gas enthalpy and radiation are increased, the surfaces' geometry is undergoing changes. The method enables the intensity of gas velocity to be ascertained at any point in the system.The results of the method outlined here complement the picture of possible equilibrium states in the close binary systems in the presence of radiation and magnetic field.     

Irradiation pressure effects in close binary systems

We test an analytic model for the two-point correlations of galaxy clusters in redshift space using the Hubble volume N -body simulations. The correlation function of clusters shows no enhancement along the line of sight, owing to the lack of any virialized structures in the cluster distribution. However, the distortion of the clustering pattern arising from coherent bulk motions is clearly visible. The distribution of cluster peculiar motions is well described by a Gaussian, except in the extreme high-velocity tails. The simulations produce a small but significant number of clusters with large peculiar motions. The form of the redshift-space power spectrum is strongly influenced by errors in measured cluster redshifts in extant surveys. When these errors are taken into account, the model reproduces the power spectrum recovered from the simulation to an accuracy of 15 per cent or better over a decade in wavenumber. We compare our analytic predictions with the power spectrum measured from the APM cluster redshift survey. The cluster power spectrum constrains the amplitude of density fluctuations, as measured by the linear rms variance in spheres of radius 8  h ⁻¹ Mpc, denoted by σ ₈. When combined with the constraints on σ ₈ and the density parameter Ω derived from the local abundance of clusters, we find a best-fitting cold dark matter model with     and     , for a power spectrum shape that matches that measured for galaxies. However, for the best-fitting value of Ω and given the value of Hubble's constant from recent measurements, the assumed shape of the power spectrum is incompatible with the most readily motivated predictions from the cold dark matter paradigm.     

Revised photometric elements of three semi-detached binary systems

TheV andB light curves of three Algol-type eclipsing binaries (UZ Cyg, VW Cyg, AQ Peg), observed by Ammanet al. (1979), have been analysed using the Wilson and Devinney (1971) model. We find that all of them are semi-detached systems, with A-type primary components and K-type evolved secondary which fill the corresponding lobe, although VW Cyg should perhaps be considered as an sd-d system.     

Calculations of proximity effects in close binary systems

The proximity effects in the light curves of close binary systems are investigated with the aid of facilities provided by a high-speed, digital computer. The treatment follows, to a large extent, that summarized by Kopal (1959) and makes extensive use of the alpha-functions and related integrals provided by that author. The ellipticity and reflection effects are studied individually and different expressions are checked and compared. Test data are drawn continually from the well known system SZ Cam for which a rectified (in a more realistic sense) light curve is ultimately produced.