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1.
This paper is the third in a series of articles devoted to one of the basic problems of celestial mechanics: the study of the evolution of solar-type planetary systems. In the previous papers a brief review of the history and current state of the problem was given; the plan of the study was outlined; the Jacobi coordinates and the related osculating elements were introduced; the form of the Poisson expansion of the Hamiltonian in all elements was given; and the expansion coefficients for the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem were obtained (though with impure accuracy) by a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. In the present paper the expansion of the Hamiltonian of the two-planetary Sun–Jupiter–Saturn problem into the Poisson series in all elements is constructed with the help of the PSP Poisson series processor, which is capable of required accuracy. 相似文献
2.
The Institute of Theoretical Astronomy in St. Petersburg and the Astronomical Institute in Bratislava are preparing a new edition of the Catalogue of short-period comets. This edition will be supplemented by short-period comets discovered after the year 1983 and comprises some new features, e. g. the evolution of orbital elements between the years 1750 and 2050, and the perihelion passages of comets within the 1994–2050 years. A new method has been employed for the determination of nongravitational parameters from the osculating elements of a comet based on all its observed returns.The method has been tested on the comets P/Comas Solá and P/Forbes with all returns, except the last one. The results have been compared with the osculating elements of the last return and those used in the old edition of the Catalogue of short-period comets. The new method enables a good prediction of osculating elements for the future, at least for the next return. 相似文献
3.
This is the second paper in a series of articles devoted to one of the basic problems of celestial mechanics: the evolution of solar-type planetary systems. In the first paper (Kholshevnikov et al., 2001), we reviewed the history and the current state of the issue, outlined the scheme of the study, introduced Jacobi coordinates and related osculating elements, and indicated the form of the Hamiltonian expansion into a Poisson series in all elements. In this paper, the expansion coefficients are found according to a simple algorithm that is reduced to the calculation of multiple integrals of elementary functions. At the first stage, we restricted our analysis to the two-planetary problem (Sun–Jupiter–Saturn). The general case will be investigated in a forthcoming paper. 相似文献
4.
In the method of variation of parameters we express the Cartesian coordinates or the Euler angles as functions of the time
and six constants. If, under disturbance, we endow the “constants” with time dependence, the perturbed orbital or angular
velocity will consist of a partial time derivative and a convective term that includes time derivatives of the “constants”.
The Lagrange constraint, often imposed for convenience, nullifies the convective term and thereby guarantees that the functional
dependence of the velocity on the time and “constants” stays unaltered under disturbance. “Constants” satisfying this constraint
are called osculating elements. Otherwise, they are simply termed orbital or rotational elements. When the equations for the
elements are required to be canonical, it is normally the Delaunay variables that are chosen to be the orbital elements, and
it is the Andoyer variables that are typically chosen to play the role of rotational elements. (Since some of the Andoyer
elements are time-dependent even in the unperturbed setting, the role of “constants” is actually played by their initial values.)
The Delaunay and Andoyer sets of variables share a subtle peculiarity: under certain circumstances the standard equations
render the elements nonosculating. In the theory of orbits, the planetary equations yield nonosculating elements when perturbations
depend on velocities. To keep the elements osculating, the equations must be amended with extra terms that are not parts of the disturbing function [Efroimsky, M., Goldreich, P.: J. Math. Phys. 44, 5958–5977 (2003); Astron. Astrophys. 415, 1187–1199 (2004); Efroimsky, M.: Celest. Mech. Dyn. Astron. 91, 75–108 (2005); Ann. New York Acad. Sci. 1065, 346–374 (2006)]. It complicates both the Lagrange- and Delaunay-type planetary equations and makes the Delaunay equations
noncanonical. In attitude dynamics, whenever a perturbation depends upon the angular velocity (like a switch to a noninertial
frame), a mere amendment of the Hamiltonian makes the equations yield nonosculating Andoyer elements. To make them osculating,
extra terms should be added to the equations (but then the equations will no longer be canonical). Calculations in nonosculating
variables are mathematically valid, but their physical interpretation is not easy. Nonosculating orbital elements parameterise
instantaneous conics not tangent to the orbit. (A nonosculating i may differ much from the real inclination of the orbit, given by the osculating i.) Nonosculating Andoyer elements correctly describe perturbed attitude, but their interconnection with the angular velocity
is a nontrivial issue. The Kinoshita–Souchay theory tacitly employs nonosculating Andoyer elements. For this reason, even
though the elements are introduced in a precessing frame, they nevertheless return the inertial velocity, not the velocity
relative to the precessing frame. To amend the Kinoshita–Souchay theory, we derive the precessing-frame-related directional
angles of the angular velocity relative to the precessing frame. The loss of osculation should not necessarily be considered
a flaw of the Kinoshita–Souchay theory, because in some situations it is the inertial, not the relative, angular velocity
that is measurable [Schreiber, K. U. et al.: J. Geophys. Res. 109, B06405 (2004); Petrov, L.: Astron. Astrophys. 467, 359–369 (2007)]. Under these circumstances, the Kinoshita–Souchay formulae for the angular velocity should be employed (as
long as they are rightly identified as the formulae for the inertial angular velocity). 相似文献
5.
Time evolution of the interplanetary dust particle under the action of the solar electromagnetic radiation (Poynting-Robertson effect) is investigated. Evolution of the initially circular orbit in terms of the orbital elements present in the standard equations for their secular changes is considered. It is pointed out that the osculating eccentricity is practically constant during the motion in spite of generally accepted opinion that the standard equations for the secular changes of orbital elements represent time evolution of the osculating elements. 相似文献
6.
M. A. Vashkov’yak 《Astronomy Letters》2005,31(1):64-72
We describe an approximate numerical-analytical method for calculating the perturbations of the elements of distant satellite orbits. The model for the motion of a distant satellite includes the solar attraction and the eccentricity and ecliptic inclination of the orbit of the central planet. In addition, we take into account the variations in planetary orbital elements with time due to secular perturbations. Our work is based on Zeipel’s method for constructing the canonical transformations that relate osculating satellite orbital elements to the mean ones. The corresponding transformation of the Hamiltonian is used to construct an evolution system of equations for mean elements. The numerical solution of this system free from rapidly oscillating functions and the inverse transformation from the mean to osculating elements allows the evolution of distant satellite orbits to be studied on long time scales on the order of several hundred or thousand satellite orbital periods. 相似文献
7.
Mean orbital elements are obtained from osculating ones by removing the short periodic perturbations. Large catalogues of
asteroid mean elements need to be computed, as a first step in the computation of proper elements, used to study asteroid
families. The algorithms for this purpose available so far are only accurate to first order in the masses of the perturbing
planet; the mean elements have satisfactory accuracy for most of the asteroid belt, but degraded accuracy in the neighbourhoods
of the main mean motion resonances, especially the 2:1. We investigate a number of algorithms capable of improving this approximation;
they belong to the two classes of Breiter-type methods and iterative methods. The former are obtained by applying some higher
order numerical integration scheme, such as Runge–Kutta, to the differential equation whose solution is a transformation removing
the fast angular variables from the equations; they can be used to compute a full second order theory, however, only if the
full second order determining function is explicitly computed, and this is computationally too cumbersome for a complicated
problem such as the N-body. The latter are fixed point iterative schemes, with the first order theory as an iteration step,
used to compute the inverse map from mean to osculating elements; formally the method is first order, but because they implement
a fixed frequency perturbation theory, they are more accurate than conventional single iteration methods; a similar method
is already in use in our computation of proper from mean elements. Many of these methods are tested on a sample of asteroid
orbits taken from the Themis family, up to the edge of the 2:1 resonance, and the dispersion of the values of the computed
mean semimajor axis over 100 000 years is used as quality control. The results of these tests indicate that the iterative
methods are superior, in this specific application, to the Breiter methods, in accuracy and reliability. This is understood
as the result of the cancellations occurring between second order perturbation terms: the incomplete second order theory,
resulting from the use of a Breiter method with the first order determining function only, can be less accurate than complete,
fixed frequency theories of the first order. We have therefore computed new catalogues of asteroid mean and proper elements,
incorporating an iterative algorithm in both steps (osculating to mean and mean to proper elements). This new data set, significantly
more reliable even in the previously degraded regions of Themis and Cybele, is in the public domain.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
Sławomir Breiter 《Celestial Mechanics and Dynamical Astronomy》1997,67(3):237-249
The analytical solution for the perturbations of an artificial satellite due to the zonal part of the geopotential is presented.
The Hamiltonian is fully normalized up to the second order by a single averaging transformation and the generating function
is given explicitly. The formulas allow an arbitrarily high degree of geopotential harmonics to be included. The transformation
from mean to osculating variables or vice versa is performed by means of a numerical method proposed by the author in a previous
paper (Breiter,1997): periodic perturbations are computed by means of a Runge-Kutta method of order 2 instead of being explicitly
derived from a generator.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
9.
R. Vilhena de Moraes 《Celestial Mechanics and Dynamical Astronomy》1988,45(1-3):281-284
A semi-analytical method is presented to study the system of differential equations governing the rotational motion of an artificial satellite. Gravity gradient and non gravitational torques are considered. Operations with trigonometric series were performed using an algebraic manipulator. Andoyer's variables are used to describe the rotational motion. The osculating elements are transformed analytically into a mean set of elements. As the differential equations in the mean elements are free of fast frequency terms, their numerical integration can be performed using a large step size. 相似文献
10.
We consider the definitions and resulting equations of motion for the Lagrangian orbital elements associated with conventional osculating orbit theory for central forces. The analysis indicates that the definitions themselves lead to difficulties which are most apparent in the circular limit. An alternate set of defining relations is presented which eliminates the problems associated with osculating elements. The remaining equation of motion based on these new definitions is reduced to quadratures. This solution completely expresses the orbits for central force problems with no restriction on the eccentricity. Both bounded and open orbits are considered. A generalized Laplace-Runge-Lenz vector is developed and a number of example solutions are presented. 相似文献
11.
When the problem of the rotation of a non-rigid body is studied, the usual procedure consists of adding perturbations to the Hamiltonian of the rigid solid. In some cases, as occurs with the centrifugal deformation, the new perturbations contains potentials which depend on the velocity, but usually one alter neither the definition of the canonical variables nor the method for obtaining the Hamiltonian. Although this procedure gives good estimates and its formulation is simpler, it is incorrect from a theoretical point of view.In this paper we rigorously develop a Hamiltonian formulation of the problem, considering potentials that depend on the velocity. Thus the differences between the two procedures are clearly shown, giving special emphasis to the case of the elastic Earth, for which we show that the differences obtained cannot be ignored within the accuracy limits at present required. 相似文献
12.
A new theory for the calculation of proper elements, taking into account terms of degree four in the eccentricities and inclinations, and also terms of order two in the mass of Jupiter, has been derived and programmed in a self contained code. It has many advantages with respect to the previous ones. Being fully analytical, it defines an explicit algorithm applicable to any chosen set of orbits. Unlike first order theories, it takes into account the effect of shallow resonances upon the secular frequencies; this effect is quite substantial, e.g. for Themis. Short periodic effects are corrected for by a rigorous procedure. Unlike linear theories, it accounts for the effects of higher degree terms and can thus be applied to asteroids with low to moderate eccentricity and inclination; secular resonances resulting from the combination of up to four secular frequencies can be accounted for. The new theory is self checking : the proper elements being computed with an iterative algorithm, the behaviour of the iteration can be used to define a quality code. The amount of computation required for a single set of osculating elements, although not negligible, is such that the method can be systematically applied on long lists of osculating orbital elements, taken either from catalogues of observed objects or from the output of orbit computations. As a result, this theory has been used to derive proper elements for 4100 numbered asteroids, and to test the accuracy by means of numerical integrations. These results are discussed both from a quantitative point of view, to derive an a posteriori accuracy of the proper elements sets, and from a qualitative one, by comparison with the higher degree secular resonance theory. 相似文献
13.
Joseph J. F. Liu Philip M. Fitzpatrick 《Celestial Mechanics and Dynamical Astronomy》1975,12(4):463-487
Differential equations are derived for studying the effects of either conservative or nonconservative torques on the attitude motion of a tumbling triaxial rigid satellite. These equations, which are analogous to the Lagrange planetary equations for osculating elements, are then used to study the attitude motions of a rapidly spinning, triaxial, rigid satellite about its center of mass, which, in turn, is constrained to move in an elliptic orbit about an attracting point mass. The only torques considered are the gravity-gradient torques associated with an inverse-square field. The effects of oblateness of the central body on the orbit are included, in that, the apsidal line of the orbit is permitted to rotate at a constant rate while the orbital plane is permitted to precess (either posigrade or retrograde) at a constant rate with constant inclination.A method of averaging is used to obtain an intermediate set of averaged differential equations for the nonresonant, secular behavior of the osculating elements which describe the complete rotational motions of the body about its center of mass. The averaged differential equations are then integrated to obtain long-term secular solutions for the osculating elements. These solutions may be used to predict both the orientation of the body with respect to a nonrotating coordinate system and the motion of the rotational angular momentum about the center of mass. The complete development is valid to first order in (n/w
0)2, wheren is the satellite's orbital mean motion andw
0 its initial rotational angular speed. 相似文献
14.
15.
André Deprit 《Celestial Mechanics and Dynamical Astronomy》1991,51(3):201-225
A new canonical transformation is proposed to handle elliptic oscillators, that is, Hamiltonian systems made of two harmonic oscillators in a 1-1 resonance. Lissajous elements pertain to the ellipse drawn with a light pen whose coordinates oscillate at the same frequency, hence their name. They consist of two pairs of angle-action variables of which the actions and one angle refer to basic integrals admitted by an elliptic oscillator, namely, its energy, its angular momentum and its Runge-Lenz vector. The Lissajous transformation is defined in two ways: explicitly in terms of Cartesian variables, and implicitly by resolution of a partial differential equation separable in polar variables. Relations between the Lissajous variables, the common harmonic variables, and other sets of variables are discussed in detail. 相似文献
16.
A set of differential equations is derived that has a number of advantages in special perturbation work. In particular, the equations remain valid for all values of the orbital eccentricity and inclination including zero. They are therefore applicable to parabolic- and hyperbolic-type orbits as well as elliptic-type; a scheme for use when the orbit is rectilinear or nearly so is provided. The equations are also much simpler in form than the Lagrange planetary equations and the transformations of the osculating elements to and from the rectangular coordinates are straightforward. 相似文献
17.
Martin Lara Juan F. San-Juan Luis M. López Paul J. Cefola 《Celestial Mechanics and Dynamical Astronomy》2012,113(4):435-452
The long-term effects of a distant third-body on a massless satellite that is orbiting an oblate body are studied for a high order expansion of the third-body disturbing function. This high order may be required, for instance, for Earth artificial satellites in the so-called MEO region. After filtering analytically the short-period angles via averaging, the evolution of the orbital elements is efficiently integrated numerically with very long step-sizes. The necessity of retaining higher orders in the expansion of the third-body disturbing function becomes apparent when recovering the short-periodic effects required in the computation of reliable osculating elements. 相似文献
18.
V. A. Brumberg L. S. Evdokimova N. G. Kochina 《Celestial Mechanics and Dynamical Astronomy》1971,3(2):197-221
The motion of a close artificial satellite of the Moon is considered. The principal perturbations taken into account are caused by the nonsphericity of the Moon and the attraction of the Earth and the Sun. To begin with, the expansions of the disturbing functions due to the nonsphericity of the primary body and the action of the disturbing mass-point body have been derived. The second expansion is produced in terms of the Keplerian elements of a satellite and the spherical coordinates of the disturbing body. Both expansions are valid for an arbitrary reference plane. The motion of a satellite of the Moon is studied in the selenocentric coordinate system referred to the Lunar equator and rotating with respect to the fixed ecliptic system. However, the coordinate exes in the equatorial plane are chosen so that the angular speed of rotation of the system is small. The motion of the satellite is described by means of the contact elements which enable one to utilize the conventional Lagrange's planetary equations and may be regarded as the generalization of the notion of the osculating elements to the case of the disturbing function depending not only o the coordinates and the time but on the velocities as well. Two methods are proposed to represent the motion of Lunar satellites over long intervals of time: the von Zeipel method and the Euler method of analytical integration with application of the variation-of-elements technique at every step of integration. The second method is exposed in great detail.Presented at the Meeting of Commission 7 of the IAU on Analytical Methods for the Orbits of Artificial Celestial Objects 14-th General Assembly of the IAU, Brighton, 1970. 相似文献
19.
The formulas for the Poisson bracket of a perturbed two-body problem and a perturbed planetary problem are found in different systems of Keplerian elements. As with canonical parametrization, the Poisson bracket is equal to a linear combination of partial brackets, but it contains coefficients depending on semimajor axis, eccentricity, and inclination. A simple relation between the Poisson brackets and matrices of coefficients of Lagrange-type equations determining the variations of osculating elements is derived. The Poisson bracket of D'Alembertian functions is proved to be a D'Alembertian one by itself. 相似文献
20.
For precise control, to minimize the fuel consumption, and to maximize the lifetime of satellite formations a precise analytic
solution is needed for the relative motion of satellites. Based on the relationship between the relative states and the differential
orbital elements, the state transition matrix for the linearized relative motion that includes the effects due to the reference
orbit eccentricity and the gravitational perturbations is derived. This method is called the Geometric Method. To avoid any
singularities at zero eccentricity and zero inclination, equinoctial variables are used to derive the relative motion state
transition matrices for both mean and osculating elements. This approach can be extended easily to include other perturbing
forces. 相似文献