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1.
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   

2.
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   

3.
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   

4.
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   

5.
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   

6.
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   

7.
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8.
The aim of the present paper will be to establish the explicit form of the equations which govern the internal structure of stars rotating with constant angular velocity formulated in terms of Clairaut coordinates (cf. Kopal, 1980) in which the radial coordinate is replaced by the total potential, which for equilibrium configurations remains constant over distorted level surfaces. The introductory Section 1 contains an account of previous work on rotating stars, commencing with Milne (1923), von Zeipel (1924) and Chandrasekhar (1933), who all employed orthogonal coordinates for their analysis. In Section 2 we shall apply to this end the curvilinear Clairaut coordinates introduced already in our previous work (cf. Kopal, 1980, 1981); and although these are not orthogonal, this disadvantage is more than offset by the fact that, in their terms, the fundamental equation of our problem will assume the form of ordinary differential equations, subject to very simple boundary conditions. The explicit form of these equations — exact to terms of fourth order in surficial distortion caused by centrifugal force—will be obtained in Section 3; while in the concluding Section 4 these will be particularized (for the sake of comparison with work of previous investigators) to stars of initially polytropic structure. These will prove to be much simpler in Clairaut coordinates than they were in any previously used frame of reference. Lastly, in Appendix A we shall present the explicit forms, in Clairaut coordinates, of the differential operators which were needed to establish the results given in Sections 3–4; while Appendix B will summarize other auxiliary algebraic relations of which use was made to formulate our fourth-order theory developed in Section 3.  相似文献   

9.
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.  相似文献   

10.
The aim of the present paper will be to utilize the results obtained in the preceding papers of this series for the development of practical procedures for obtaining the elements of any eclipsing system from the observed photometric data by their analysis in the frequency-domain, for any type of eclipses, any proximity of the two components, and any degree of the law of limbdarkening of the eclipsed star.In Section 2, which follows a brief introduction to the subject, procedures will be developed which should permit us to perform such an analysis — by hand or automatic machine computation — for the case of mutual eclipses in binary systems the components of which can be regarded as spheres; and whose apparent discs are characterized by an arbitrary radially symmetrical distribution of surface brightness. In Section 3 we shall generalize these procedures to systems consisting of arbitrarily distorted stars.Paper dedicated to Professor Hannes Alfvén on the occasion of his 70th birthday, 30 May, 1978.  相似文献   

11.
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   

12.
The aim of the present paper will be to generalize the methods for computation of the elements of eclipsing binary systems in the frequency-domain, summarized in our recent Paper I (Kopal, 1981), to the case ofclose systems, in which photometric proximity effects become conspicuous and must be taken into account before the methods previously outlined in Paper I become directly applicable.Following a brief introduction to the subject given in Section 1, Section 2 summarizes (and comments upon) the difficulties previously encountered in separation of the photometric proximity and eclipse effects. In Section 3 we develop an alternative new approach to the problem by modulation of the light curves throughout the entire orbital cycle, intended to filter out proximity effects from the observed light changes and isolate those due to eclipses; while in Section 4 we shall present a numerical application of the new method to an analysis of the observed light changes of the eclipsing system W Ursae Maioris.In Section 5 we shall present a quantitative investigation of the photometric effects of distortion on the light changes of close eclipsing systems within eclipses-the most complicated part of the whole problem-with numerical application to the system of U Sagittae carried out in the concluding Section 6.Appendices 1–3 contain numerical data which should facilitate application of the methods developed and illustrated in Sections 3–4; while Appendix 4 will be reserved for a mathematical proof of certain expansions used in Section 5, which would have been too discursive for the main text.  相似文献   

13.
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   

14.
The aim of the present paper will be to develop a theory of the radial-velocity changes of the components of close binary systems, with special attention to phenomena arising from finite dimensions of such components and their mutual distortion as well as irradiation. It is particularly stressed that the deformability of fluid stars and gas motions in their atmospheres can give rise to systematic differences between the observed radial velocities of such stars and those of their mass centres.In Section 2 (which follows a brief statement of the problem outlined in Section 1) we shall introduce the coordinate systems subsequently employed to treat various aspects of our problem: Section 3 will be concerned with an extraction of information from the radial-velocity component of absolute motions of the mass-centres of such stars; and in Section 4 we shall generalize the classical work by an investigation of radial velocities at any point of the apparent disks of distorted components, and their relation to the motion of their centres of mass. Section 5 will contain an evaluation of the effects of distortion, on radial velocity, averaged over the entire visible disk of the respective star at different phases; and in Section 6 we shall extend the same treatment to stars undergoing eclipses.An investigation of the effects, on the observed radial velocities, of atmospheric streaming caused by mutual irradiation of the two stars is being postponed for a subsequent communication.  相似文献   

15.
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   

16.
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.  相似文献   

17.
The aim of the present paper will be to pioneer a new approach to the analysis of the light changes of eclipsing binary systems in the frequency domain, and to point out its merits in comparison with a conventional treatment of the same problem in the time-domain which has been developed so far. Following an introductory section in which the broad features of our problem will be set forth, Section 2 will contain an outline, and critique, of the time-domain approach. Section 3 will give an explicit treatment of the light changes arising from total and annular eclipses in the frequency domain — a problem which we succeeded in solving in close algebraic form. Section 4 will extend this treatment to the case of partial eclipses; and in the concluding Section 5 the relative merits of our new results will be discussed in broader perspective. Sections 3 and 4 contain explicit results pertaining to mutual eclipses of spherical stars exhibiting uniformly bright discs. An extension of these results to the case of arbitrary limb-darkening, and taking account of mutual distortion of both components, will be given in subsequent communications.  相似文献   

18.
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   

19.
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   

20.
The aim of the present paper will be to translate the essential parts of the theory of Fourier analysis of the light changes of eclipsing variables into more practical terms; and describe procedures (illustrated by numerical examples) which should enable their users to obtain the desired results with maximum accuracy and minimum loss of information by processes which can be fully automated.In order to unfold in steps how this can be done, the scope of the present paper-the first of two-will be restricted to an exposition of the analysis of light changes caused by eclipses of spherical stars; while between minima due to this cause the light of the system should remain sensibly constant. An extension of our analysis to incorporate photometric effects arising from mutual distortion of the components of close eclipsing systems between minima as well as within eclipses is being postponed for the second communication.In developing this subject we shall single out for the user's attention only those parts of the whole theory which are of direct relevance to practical work. Their justification can be largely found in sources already published; and new developments essential for our work, not yet made public, will be relegated to several Appendices at the end of the text, in order not to render its text too discursive and deflect the reader's attention from the main theme of its narrative.After a brief outline of the subject given in Section 1, Section 2 will introduce the reader to practical aspects of the Fourier analysis of the light curves; and Section 3 will be devoted to its use to determine the numerical values of the momentsA 2m of the light curves which constitute the cornerstones for all subsequent work. Section 4 will describe an algebraization of the process of determination of the elements for the case of total (annular) eclipses; while Section 5 will do the same for partial eclipses. The concluding Section 6 will be devoted to an error analysis of our problem, and to an outline of the way by which the errors of the individual observations will compound to the uncertainty of the final results. Lastly, Appendices 1–5 concluding the paper will contain additional details of some aspects of our work, or proofs of new processes made use of to obtain our results, whose earlier inclusion would have made the main text too discursive.  相似文献   

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