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1.
Zdeněk Kopal 《Astrophysics and Space Science》1987,134(1):55-71
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. 相似文献
2.
Zdeněk Kopal 《Astrophysics and Space Science》1981,76(1):187-212
The aim of the present paper will be to detail the explicit form of the equations which govern first-order oscillations of fast-rotating globes of self-gravitating fluids; with due account taken of the effects arising from the centrifugal as well as Coriolis force. As such configurations oscillate in general about distorted figures of equilibrium, the equations governing them can be conveniently expressed in terms of the Clairaut coordinates, associated with distorted spheroidal figures, and introduced in our previous paper (Kopal, 1980) for this purpose.In Section 2 which follows a brief outline of our problem, the equilibrium properties of fast-rotating configurations or arbitrary structure will be formulated. In Section 3 we shall carry out a separation of the variables in the equations of motion, and reduce the partial differential equations of the problem to an equivalent system of ordinary differential equations, by an expansion of expressions for the velocity componentsU, V, W in terms of tesseral harmonicsY
n
m
(, ). The explicit form of such a system, including the effects of all tesseral harmonics of orders up tom=n=4, will be specified in Section 3 for configurations whose equilibrium form is a sphere; while in Section 4 this latter condition will be relaxed to allow for the equilibrium configuration to become a rotational spheroid.In the concluding Section 5 we shall convert the complex form of our equations of motion into real terms, amenable to a solution-analytical or numerical-in terms of real variables; and shall establish the boundary conditions necessary for a specification of the characteristic frequencies of oscillation. 相似文献
3.
Zdeněk Kopal 《Astrophysics and Space Science》1991,186(2):253-276
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. 相似文献
4.
Zdeněk Kopal 《Astrophysics and Space Science》1987,133(1):157-175
The aim of the first part of this investigation will be to establish the explicit form of the linearized systems of differential equations governing arbitrary oscillations (of amplitudes small enough for their squares and higher powers to be negligible) of the rotating Roche model in Clairaut's coordinates (in which their radial component is identified with the total potential). By solving these equations in a closed form we shall prove that this model is incapable of performing such oscillations (for any type of symmetry) about equipotential surfaces representing the figures of equilibrium, as soon as the centrifugal force will cause their equilibrium form to depart from a sphere.In the second part of this paper we shall set up the closed forms of the Laplace equation in Clairaut (non-orthogonal) as well as Roche (orthogonal) coordinates associated with the rotating Roche model; and by a construction of their solution establish successively the explicit forms of the respective harmonic functions associated with such figures (as a generalization of Legendre functions which are similarly associated with a sphere. 相似文献
5.
Zdeněk Kopal 《Astrophysics and Space Science》1969,4(3):330-364
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. 相似文献
6.
Zdeněk Kopal 《Astrophysics and Space Science》1980,70(2):407-424
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. 相似文献
7.
The aim of the present paper will be to establish the explicit form of the equations of radiative transfer, in plane-parallel atmospheres surrounding the stars which are distorted by axial rotation or tides, in curvilinear coordinates which parallel the distorted surface; with particular attention to the circumstances under which the effects arising from limb- and gravity-darkening are multiplicative and admit of algebraic separation. In Section 2 (which follows a general outline of our problem) the fundamental equations of the radiativetransfer problem will be formulated for the ‘grey’ case; and rewritten in Section 3 in terms of non-orthogonal coordinates in which the potential over a level surface in hydrostatic equilibrium replaces the radial coordinate of spherical polars. In Section 4 we shall proceed to construct an explicit solution of the corresponding transfer problem in a plane-parallel approximation; and to prove that the effects of limb- and gravity-darkening remain factorizable only to terms which are linear in the cosines μ of the angle of foreshortening. Lastly, in Section 5 we shall list additional problems, arising in this connection, which still await appropriate treatment. 相似文献
8.
The Roche Coordinates in three dimensions and their application to problems of double-star astronomy
Zdeněk Kopal 《Astrophysics and Space Science》1970,8(2):149-171
In a preceding paper (Kopal, 1969; in what follows referred to as Paper I) we introduced a new system of curvilinear coordinates-hereafter referred to as Roche Coordinates — in which spheres of constant radius in spherical polars have been replaced by surfaces of constant potential of a rotating gravitational dipole; while the angular coordinates are orthogonal to the equipotentials. In Paper I we established an explicit form of such a transformation, and related the Roche coordinates with polar coordinates (with which they coalesce in the immediate neighbourhood of each one of the two finite mass-points) in the plane case. The aim of the present investigation will be to generalize the definition of the Roche coordinates to three dimensions.The opening Section 1 of this paper will contain a general outline of the proposed three-dimensional transformation; and in Section 2 details of this transformation will be explicitly worked out correctly to quantities of first order in superficial distortion — an approximation which should prove adequate in regions surrounding the two finite masses; while in Section 3 we shall evaluate (to this degree of accuracy) the metric coefficients of the respective transformation, and its direction cosines, in both polar and curvilinear coordinates. Section 4 will then contain a formulation of the fundamental equations of hydrodynamics in terms of the three-dimensional Roche coordinates; and their advantages for a treatment of certain classes of dynamical problems encountered in doublestar astronomy will be illustrated in the concluding Section 5 by an investigation of the vibrational stability of the Roche model. We shall show that this model is capable of performing free radial oscillations which remain barotropic only if its equilibrium form is spherical (i.e., in the absence of any external mass in the neighbourhood); but not if it is distorted to any extent by rotation or tides. 相似文献
9.
The aim of the present paper will be to extend our previous investigation of the vibrational stability of rotating configurations (Kopal, 1981) to a similar investigation of the stability of the components of close binary systems which not only rotate, but also distort each other by tidal action. To this end, differential equations which govern first-order oscillations of arbitrary spherical-harmonic symmetry will be set up in Clairaut coordinates in which the radial coordinate is replaced by the potential which remains constant over level surfaces of equilibrium configurations; introduced by us in an earlier paper (Kopal, 1980), and their form detailed for surface distorted by second-, third-, and fourth-harmonic tides raised by the external mass; and their boundary conditions established. A solution of such differential boundary-value problems arising in connection with the stars of arbitrary structure remains, of course, a task for automatic computers. It may only be added that the tide-generating potential Ψ T established in this paper should enable us to study, by the same method, not only free, but also forced oscillations of the components of close binary systems, arising from orbital eccentricity of the respective couples, dynamical tides, or other causes likely to be operative in such systems. 相似文献
10.
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. 相似文献
11.
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. 相似文献
12.
Zdeněk Kopal 《Astrophysics and Space Science》1974,27(2):389-418
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). 相似文献
13.
14.
Zdeněk Kopal 《Astrophysics and Space Science》1975,38(1):191-241
The methods of analysis of the light changes of eclipsing variables in the frequency-domain, developed in our previous papers (Kopal 1975a, b, c, d) for an interpretation of mutual eclipses in systems consisting of spherical stars, have now been extended to analyse the light variations — between minima as well as within eclipses — ofclose binaries whose components are distorted by axial rotation and mutual tidal action. Following a brief introduction (Section 1) in which the need of this new approach will be expounded, in Sections 2 and 3 we shall deduce the theoretical changes of close eclipsing systems between minima (Section 2) as well as within eclipses (Section 3), which in Sections 4 and 5 will be analysed in the frequency-domain; and explicit formulae obtained which should enable us to separate the photometric proximity and eclipse effects directly from the observed data as they stand-without the need of any preliminary ‘rectification’. Section 6 will contain the explicit forms of the expressions for photometric perturbations in the frequency-domain, due to rotational and tidal distortion of both stars; and the concluding Section 7 will then be concerned with practical aspects of the application of these new methods to an analysis of the observed light changes of close eclipsing systems — in which the proximity and eclipse effects cannot be distinguished from each other by mere inspection. 相似文献
15.
Zdeněk Kopal 《Astrophysics and Space Science》1969,5(3):360-384
The aim of the present paper will be to introduce a new system of curvilinear coordinateshereafter referred to as Roche coordinates-in which spheres of constant radius are replaced by equipotential surfaces of a rotating gravitational dipole (which consists of two discrete points of finite mass, revolving around their common center of gravity); while the remaining coordinates are orthogonal to the equipotentials. It will be shown that the use of such coordinates offers a new method of approach to the solution of certain problems of particle dynamics (such as, for instance, the construction of certain types of trajectories in the restricted problem of three bodies); as well as of the hydrodynamics of gas streams in close binary systems, in which the equipotential surfaces of their components distorted by axial rotation and mutual tidal interaction constitute essential boundary conditions.Following a general outline of the problem in Section 1, the Roche coordinates associated with the equipotentials of a rotating gravitational dipole will be constructed in the plane case (Section 2), and their geometrical properties discussed. In Section 3, we shall transform the fundamental equations of hydrodynamics to their forms appropriate in the curvilinear Roche coordinates. The metric coefficients of this transformation will be formulated in a closed form in Section 4 in terms of the respective partial derivatives of the potential; while in Section 5 analytic expressions for the Roche coordinates will be given in the orbital plane of the dipole, which are exact as far as the distortion of the equipotential curves from circular form can be described by the second, third and, fourth harmonics.The concluding Section 6 will be devoted to a formulation of the equations of a mass-point in the restricted problem of three bodies in the Roche coordinates. Three special cases will be considered: (a) motion in the neighborhood of the equipotential curves; (b) motion in the direction normal to such curves; and (c) motion in the neighbourhood of the Lagrangian points. It will be shown that motion in one coordinate is possible only in limiting cases which will be enumerated; but twodimensional motions in which one velocity component is very much smaller than the other invite further study.A generalization of the plane Roche coordinates to three dimensions, with application to additional classes of problems, is being postponed for a subsequent paper. 相似文献
16.
Having formulated the Clairaut second-order differential equations up to the fourth order in superficial distortion due to Hensen's coefficients in the previous article (El-Sharawyet al., 1989 III, hereafter denotes by SM3), we are now in a position to solve them. In this paper we shall discuss the methods of solving the Clairaut theory, to give an explicit form about the distortion of the surfaces of Jupiter and Saturn, numerically up to the fourth-order. 相似文献
17.
William E. Haymes 《Astrophysics and Space Science》1973,22(1):165-192
In this paper we shall investigate the energy of close binary systems of constant momentum takng into consideration the first-order effects of rotation and tidal attraction of the components of finite size. The equations for the momentum and the energy of the system will be set up in Section 2, making use of terms including the effects of finite size of the components of finite degree of central condensation. In Section 3 perturbation theory is applied to these equations using the results of Kopal (1972b) as our initial values. In Section 4 we shall compare our results with the initial values and then discuss variations in our constants and the application to various real systems. 相似文献
18.
M. Bahgat El-Shaarawy 《Earth, Moon, and Planets》1989,45(2):115-126
The aim of this paper is to investigate numerical solutions of third-order Clairaut theory, under the boundary conditions given in our previous work (El-Shaarawy, 1974). This solution gives an explicit form of the shape and rotational distortion, due to third-order sectorial harmonic terms, of the equipotential surfaces of the two rapidly rotating planets, Jupiter and Saturn at the different levels inside these planets owing to a certain internal density distribution model (Zharkov, 1975). We considered each of them as a heterogeneous self-gravitating fluid mass in hydrostatic equilibrium. 相似文献
19.
Zdeněk Kopal 《Astrophysics and Space Science》1975,35(1):171-183
The aim of the present paper will be to extend our new methods of analysis of the light curves, of eclipsing binary systems, consisting of spherical components, by Fourier approach to eclipses oftransit type — which arise when the eclipsing component happens to be smaller of the two. Our present principal concern will be transit eclipses, terminating in annular phase, of stars characterized by arbitrary radially-symmetrical distribution of brightness over their apparent discs — a phenomenon which will cause the light of the system to vary continuously during annular phase. In the first section which follows this abstract, an outline of the problem at issue will be given. Section 2 has been devoted to an analysis of light changes arising in the course of partial phases of transit eclipses; and the concluding Section 3 will contain an analysis of the corresponding light changes, during annular phase. Unlike for occultation eclipses considered in our previous paper (cf. Kopal, 1975b), the momentsA 2m of the light curves due to eclipses of transit type can again be expressed in terms of the geometrical elements of such eclipses in a closed form for limb darkening characterized by any value ofn; but the use of such functions will require auxiliary tables (now in preparation) for applications to practical cases. A parallel treatment of partial eclipses of the occultation or transit type — eclipses which stop short of totality or annular phase — is being postponed for a subsequent communication. 相似文献
20.
Zdenêk Kopal 《Astrophysics and Space Science》1969,4(4):427-458
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. 相似文献