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.     

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.     

The Roche Coordinates in three dimensions and their application to problems of double-star astronomy

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.     

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.     

Distribution of brightness over apparent discs of distorted stars

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.     

Vibrational stability of the components of close binary systems

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.     

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.     

Vibrational stability of rotating stars

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.     

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.     

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

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

The precession and nutation of deformable bodies

The aim of the present paper will be to derive from the fundamental equations of hydrodynamics the explicit form of the Eulerian equations which govern the motion about the centre of gravity of self-gravitating bodies, consisting of compressible fluid of arbitrary viscosity, in an arbitrary external field of force. If the problem is particularized so that the external field of force represents the attaction of the sun and the moon, this motion would represent the luni-solar precession and nutation of a fluid viscous earth; if, on the other hand, the external field of force were governed by the earth (and the sun), the motion would define the physical librations of the moon regarded as a deformable body. The same equations are, moreover, equally applicable to the phenomena of precession and nutation of rotating fluid components in close binary systems, distorted by mutual tidal action; and the present paper contains the first formulation of the effects of viscosity on such phenomena.Investigation supported in part by the U.S. National Aeronautics and Space Administration under Contract No. NASW-1470.     

Effects of rotation on internal structure of the stars

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.     

地球动力学扁率及其与岁差章动的关系

由岁差常数求得的日月岁差是天文学的重要参数之一,它和地球动力学扁率相联系。地球动力学扁率在章动理论的计算中也是一个重要的物理量。介绍了由不同的观测方法和模型给出的地球动力扁率值,并讨论了它也岁差的关系和对章动计算的影响。在刚体地球章动振幅的计算中,地球动力学扁率值起着尺度因子的作用,要改善刚体地球章动振幅的计算,需要修改目前的黄经总岁差值。非刚体地球章动的转换函数中所采用的简正模和常数都直接或间接地依赖地球动力学扁率值。在IAU1980章动理论中,计算刚体地球章动振幅所使用的地球动力学扁率值计算转换函数中简正模频率和常数所使用的地球动力学扁率值并不一致。随着观测和计算精度的提高,地球动力学扁率值的不一致将影响章动振幅的计算。在建立刚体地球章地动理论中,如何解释地球动力学扁率值的差异,如何选取地球动力学扁率值,还有待进一步的研究。     

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.     

The roche coordinates and their use in hydrodynamics or celestial mechanics

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.     

Rotational distortion of stars having arbitrary structure described by fourth-order sectorial harmonics

In a series of papers, the equilibrium configurations of highly rotating fluid bodies have been derived. The deformation of these inhomogeneous self-gravitating fluid, of arbitrary internal structure are due to centrifugation potential. These level surfaces are expressed in terms of fourth-order sectorial harmonics.In this paper, the main equations of the problem — such as the surface of the distorted body, the gravitational potential at an arbitrary point and the disturbing potential — have been expanded to the fourth-order in terms of the even-order sectorial harmonics.This work will hereafter be referred to as Paper I.     

An analysis of the radial velocity curves of close binary systems,I

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.     

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.     

关于章动序列计算中的地球动力学扁率的取值

刚体地球章动序列和非刚体地球章动的转换函数都和地球动力学扁率有关。ＩＡＵ１９８０章动理论中采用了一个不一致的地球动力学扁率值,从而影响了章动振幅的计算。本文介绍了章动序列计算中地球动力学扁率的取值。由地球模型１０６６Ａ或ＰＲＥＭ得到的地球动力学扁率值比由岁差观测得到的约小１％,并且不可靠。当考虑体静力学平衡被破坏时新的地球物理模型,可得到与岁差常数相一致的地球动力学扁率值。地球动力学扁率值Ｈ＝０．