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1.
The twice-averaged Hill problem with the oblateness of the central planet is considered in the case where its equatorial plane coincides with the plane of its orbital motion relative to the perturbing body. A qualitative study of this so-called coplanar integrable case was begun by Y. Kozai in 1963 and continued by M.L. Lidov and M.V. Yarskaya in 1974. However, no rigorous analytical solution of the problem can be obtained due to the complexity of the integrals. In this paper we obtain some quantitative evolution characteristics and propose an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of satellite orbit elements. The methodical accuracy has been estimated for several orbits of artificial lunar satellites by comparison with the numerical solution of the evolution system. 相似文献
2.
G. N. Duboshin 《Celestial Mechanics and Dynamical Astronomy》1972,6(1):27-39
The existence of ten first integrals for the classical problem of the motion of a system of material points, mutually attracting according to Newtonian law, is well known.The existence of the analogous ten first integrals for the more complicated problem of the motion of a system of absolutely rigid bodies, whose elementary particles mutually attract according to the Newtonian law, was established by the author (Duboshin, 1958, 1963, 1968).In his later papers (Duboshin, 1969, 1970), the problem of the motion of a system of material points, attracting each other according to a more general law, was considered and, in particular, it was shown under what conditions the ten first integrals, analogous to the classical integrals, may exist for this problem.In the present paper, the generalized problem of translatory-rotatory motion of rigid bodies, whose elementary particles acting upon each other according to arbitrary laws of forces along the straight line joining them, is discussed.The author has shown that the first integrals for this general problem, analogous to the integrals of the problem of the translatory-rotatory motion of rigid bodies, whose elementary particles acting according to the Newtonian law, exist under certain well known conditions.That is, it has been established that if the third axiom of dynamics (action = reaction) is satisfied, then the integrals of the motion of centre of inertia and the integrals of the moment of momentum exist for this generalized problem.If the third axiom is not satisfied, then the above mentioned integrals do not exist.The third axiom is a necessary but not a sufficient condition for the existence of the tenth integral-the energy integral. The tenth integral always exists if the elementary particles of the bodies acting with a force, depend only on the mutual distances between them. In this case the force function exists for the problem and the energy integral can be expressed in a well known form.The tenth integral may exist for some more general case, without expressing the principle of conservation of energy, but permitting calculation of the kinetic energy, if the configuration of a system is given.The problem, in which the elementary particles acting according to the generalized Veber's law (Tisserand, 1896) has been cited as an example of this more general case. 相似文献
3.
M. A. Vashkov’yak 《Solar System Research》2017,51(4):315-326
The well-known twice-averaged Hill problem is considered by taking into account the oblateness of the central body. This problem has several integrable cases that have been studied qualitatively by many scientists, beginning with M.L. Lidov and Y. Kozai. However, no rigorous analytical solution can be obtained in these cases due to the complexity of the integrals. This paper is devoted to studying the case where the equatorial plane of the central body coincides with the plane of its orbital motion relative to the perturbing body, while the satellite itself moves in a polar orbit. A more detailed qualitative study is performed, and an approximate constructive-analytical solution of the evolution system in the form of explicit time dependences of the eccentricity and pericenter argument of the satellite orbit is proposed. The methodical accuracy for the polar orbits of lunar satellites has been estimated by comparison with the numerical solution of the system. 相似文献
4.
J. M. Ferrándiz 《Celestial Mechanics and Dynamical Astronomy》1987,41(1-4):343-357
In this paper, we present a canonical transformation that extends the change of coordinates of Cartesian type into the associate homogeneous coordinates, and provides a redundant set of eight canonical variables to describe the orbital motion of a particle. The transformed problem has two additional integrals, since the transformation increases the number of variables. Using these variables and a time proportional to the true anomaly, the Kepler problem can be reduced to a 4-dimensional oscillator, whose frequency can be selected to be either the magnitude of the angular momentum or unity, depending on a suitable scaling.Perturbed problems are represented by perturbed harmonic oscillators, whatever the type of the orbit is, and in the special case of central force fields, the resulting equations can be linearized exactly. 相似文献
5.
This paper deals with the generalized problem of motion of a system of a finite number of bodies (material points).We suppose here that every point of the system acts on another one with a force (attractive or repulsive) directed along the straight line connecting these two points, and proportional to the product of their masses and a certain function of time, mutual distance and its derivatives of the first and second order (Duboshin, 1970).The laws of forces are different for different pairs of points and, generally speaking, the validity of the third axiom of dynamics (law of action and reaction) is not assumed in advance.With these general assumptions we find the conditions for the laws of the forces under which the problem admits the first integrals, analogous to the classic integrals of the many-body problem with the Newton's law of attraction.It is shown furthermore, that in this generalized problem it is possible to obtain an equation, analogous to the classic equation of Lagrange-Jacobi and deduce the conditions of stability or instability of the system in Lagrange's sense.The results obtained may be applied for the investigation of motion in some isolated stellar systems, where the laws of mechanics may be different from the laws in our solar system. 相似文献
6.
In the present paper, the motion of three rigid bodies is considered. With a set of new variables, and the 10 first integrals of the motion, the problem is reduced to a system of order 25 and one quadrature. The plane motions are characterized, and finally, an equation for the existence of central configurations (in particular, Lagrangian and Eulerian solutions) has been found. Besides, the case of three axisymmetric ellipsoids is studied. 相似文献
7.
Johannes Hagel 《Celestial Mechanics and Dynamical Astronomy》1995,63(2):205-225
The problem of stability of the Lagrangian pointL
4 in the circular restricted problem of three bodies is investigated close to the 1 : 2 commensurability of the long and short period libration. By stability we define boundedness of the solution for a given initial finite displacement from the equilibrium point as function of the mass parameter close to the commensurability. A rigorous treatment close to the resonance condition is possible using a transformation that diagonalizes the matrix related to the linear part of the equations of motion. The so obtained equations are further transformed to action angle type variables. Then using an isolated resonance approach, only the slowly varying terms are kept in the equations and two independent isolating first integrals can be found. These integrals finally enable us to solve the stability problem in an exact way. The so obtained results are compared to numeric integration of the equations of motion and are found to be in perfect agreement. 相似文献
8.
Perturbed two-body problems play a special role in Celestial Mechanics as they capture the dominant dynamics for a broad range
of natural and artificial satellites. In this paper, we investigate the classic Stark problem, corresponding to motion in
a Newtonian gravitational field subjected to an additional uniform force of constant magnitude and direction. For both the
two-dimensional and three-dimensional cases, the integrals of motion are determined, and the resulting quadratures are analytically
integrated. A complete list of exact, closed-form solutions is deduced in terms of elliptic functions. It is found that all
expressions rely on only seven fundamental solution forms. Particular attention is given to ensure that the expressions are
well-behaved for very small perturbations. A comprehensive study of the phase space is also made using a boundary diagram
to describe the domains of the general types of possible motion. Numerical examples are presented to validate the solutions. 相似文献
9.
E. I. Timoshkova 《Celestial Mechanics and Dynamical Astronomy》1985,36(2):105-121
The first integrals of motion of the restricted planar circular problem of three bodies are constructed as the formal power series in r1/2, r being the distance of a moving particle from the primary. It is shown that the coefficients of these series are trigonometric polynomials of an angular variable. Some particular solutions have been found in a closed form. The proposed method for constructing the formal integrals can be generalized to a spatial problem of three bodies. 相似文献
10.
András Pál 《Celestial Mechanics and Dynamical Astronomy》2014,119(1):45-54
Lie-integration is one of the most efficient algorithms for numerical integration of ordinary differential equations if high precision is needed for longer terms. The method is based on the computation of the Taylor coefficients of the solution as a set of recurrence relations. In this paper, we present these recurrence formulae for orbital elements and other integrals of motion for the planar $N$ -body problem. We show that if the reference frame is fixed to one of the bodies—for instance to the Sun in the case of the Solar System—the higher order coefficients for all orbital elements and integrals of motion depend only on the mutual terms corresponding to the orbiting bodies. 相似文献
11.
The effect of the perturbative force, solar pressure, on the motion and stability of two cableconnected satellites in the Earth's central gravitational field of force has been studied. The osculating plane of the orbit of the centre of mass has been supposed to be inclined at a constant angle with the plane of the ecliptic. The particular solutions of the nonlinear, non-homogeneous and non-autonomous equations of relative motion in three-dimensional cases have been obtained.The particular solution in which the system lies wholly along the radius vector joining the attracting centre and the centre of mass of the system under the central attracting force alone was found to be stable (cf. Singh, 1973). It has been established in our case too that the same particular solution remains stable if the parameter of the solar pressure lies within a certain limit. 相似文献
12.
The two-body problem is a twelfth-order time-invariant dynamic system, and therefore has eleven mutually-independent time-independent
integrals, here referred to as motion constants. Some of these motion constants are related to the ten mutually-independent
algebraic integrals of the n-body problem, whereas some are particular to the two-body problem. The problem can be decomposed into mass-center and relative-motion
subsystems, each being sixth-order and each having five mutually-independent motion constants. This paper presents solutions
for the eleventh motion constant, which relates the behavior of the two subsystems. The complete set of mutually-independent
motion constants describes the shape of the state-space trajectories. The use of the eleventh motion constant is demonstrated
in computing a solution to a two-point boundary-value problem. 相似文献
13.
The motion of one point mass of the classical mechanics is treated by means of the relativistic spinor regularization (KUSTAANHEIMO 1975). Most general spinor equations of motion (2.9)‒(2.10)and the differential equations(3.16)‒(3.25)for the “possible integrals” (3.2)‒(3.11)of these quations of motion are deduced. If the force is a superposition of a conservative central force and of another force perpendicular to radius vector and velocity (Chapter 4, Case D), then the theory yields scalar and spinor integrals (4.7), (4.10)‒(4.12), (4.14)‒(4.15), (4.17), (4.28) that enable a parametric representation of the orbit by quadratures, as soon as one solution of a RICCATI differential equation (4.33) has been found. 相似文献
14.
We investigate the evolution of high Earth satellite orbits under gravitational perturbations from the Sun and light pressure forces, without the Earth shadow effect. We present the disturbing function of the problem provided that the satellite is a sphere. The mean value of the disturbing function in the absence of resonances between the mean unperturbed motion of the satellite and the mean motion of the Sun has also been obtained. The semimajor axis of the satellite orbit and the mean value of the disturbing function are shown to be integrals of the averaged osculating equations. TheHill version of the problem, whereby the distance to the satellite is much smaller than the Earth–Sun distance, has been studied in detail: we have constructed the phase portraits of the oscillations at various parameters and described three types of quasiperiodic satellite trajectories—librational and rotational trajectories as well as Earth collision trajectories. Numerical simulations of trajectories have allowed the additional effects caused by light pressure to be described: the displacement of the bounded trajectory of the satellite as a whole relative to the trajectory of the classical three-body problem into a region more distant from the Sun. 相似文献
15.
R. Broucke 《Astrophysics and Space Science》1980,72(1):33-53
In this article we collect several results related to the classical problem of two-dimensional motion of a particle in the field of a central force proportional to a real power of the distancer. At first we generalize Whittaker's result of the fourteen powers ofr which lead to intergrability with elliptic functions. We enumerate six more general potentials, including Whittaker's fourteen potentials as particular cases (Sections 2 and 3).Next, we study the stability of the circular solutions, which are the singular solutions of the problem, in Whittaker's terminology. The stability index is computed as a function of the exponentn and its properties are explained, especially in terms of bifurcations with other families of ordinary periodic solutions (Sections 4, 5 and 7). In Section 6, the detailed solution of the inverse cube force problem is given in terms of an auxiliary variable which is similar to the eccentric anomaly of the Kepler problem.Finally, it is shown that the stable singular circular solutions of the central force problem generalize to stable singular elliptic solutions of the two-fixed-center problem. The stability and the bifurcations with other families of periodic solutions of the two-fixed-center problem are also described. 相似文献
16.
G. F. Chörny 《Celestial Mechanics and Dynamical Astronomy》2007,97(4):229-248
A restricted three-body problem for a dust particle, in presence of a spherical cometary nucleus in an eccentric (elliptic,
parabolic or hyperbolic) orbit about the Sun, is considered. The force of radiation pressure and the Poynting– Robertson effect
are taken into account. The differential equations of the particle’s non-inertial spatial motion are investigated both analytically
and numerically. With the help of a complex representation, a new single equation of the motion is obtained. Conversion of
the equations of motion system into a single equation allows the derivation of simple expressions similar to the integral
of energy and integrals of areas. The derived expressions are named quasiintegrals. Relative values of terms of the energy
quasiintegral for a smallest, largest, and a mean comet are calculated. We have found that in a number of cases the quasiintegrals
are related to the regular integrals of motion, and discuss how the quasiintegrals may be applied to find some significant
constraints on the motion of a body of infinitesimal mass. 相似文献
17.
The paper deals with the effect of solar pressure on the motion and stability of two satellites connected by an inextensible string in a central gravitational field of force. A system of nonlinear, nonhomogeneous, and non-autonomous equations under the rotating frame of reference in Nechvíle's coordinate system have been obtained. The general solution of the above system of equations is beyond our reach. The particular solutions have been obtained.The particular solution in which the system lies, wholly along the radius vector joining the attracting centre and the centre of mass of the system under the central attracting force along was found to be stable (Singh, 1973). Naturally we got interested in examining the effect of solar radiation pressure on the stability of this particular solution. 相似文献
18.
Aaron J. Rosengren Daniel J. Scheeres 《Celestial Mechanics and Dynamical Astronomy》2014,118(3):197-220
We consider sets of natural vectorial orbital elements of the Milankovitch type for perturbed Keplerian motion. These elements are closely related to the two vectorial first integrals of the unperturbed two-body problem; namely, the angular momentum vector and the Laplace–Runge–Lenz vector. After a detailed historical discussion of the origin and development of such elements, nonsingular equations for the time variations of these sets of elements under perturbations are established, both in Lagrangian and Gaussian form. After averaging, a compact, elegant, and symmetrical form of secular Milankovitch-like equations is obtained, which reminds of the structure of canonical systems of equations in Hamiltonian mechanics. As an application of this vectorial formulation, we analyze the motion of an object orbiting about a planet (idealized as a point mass moving in a heliocentric elliptical orbit) and subject to solar radiation pressure acceleration (obeying an inverse-square law). We show that the corresponding secular problem is integrable and we give an explicit closed-form solution. 相似文献
19.
T. G. Vozmischeva 《Celestial Mechanics and Dynamical Astronomy》2000,77(1):37-48
The generalization of a test particle motion in a central field of two immovable point-like centers to the case of a constant
curvature space, on a three-dimensional sphere, is investigated in the paper. The bifurcation set in the plane of integrals
of motion is constructed and the classification of the domains of possible motion is carried out on a two-dimensional sphere.
The regularization of the Kepler’s problem on a two-dimensional sphere is carried out.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
20.
In the following paper we argue that each wind-driving star in relative motion with respect to the ambient interstellar medium experiences a force exerted on its central wind-generating body. The exact magnitude of this force depends on the actual geometry of the counterflow configuration of stellar and interstellar winds for a particular kinematic situation which is especially sensitive to whether the interstellar flow is subsonic or supersonic. It will, however, be demonstrated here that this force is of an accelerating nature, i.e., it operates like a rocket-motor, as long as the peculiar motion of the wind-driving star with respect to the ambient interstellar medium remains subsonic.Here we use a specific analytical model to describe theoretically the specific counterflow configuration for the case of the solar system in a subsonic peculiar motion with respect to the local interstellar medium assuming irrotational and incompressible flows. We can work out a quantitative number for the accelerating force governing the Sun's motion at present. The net reaction force exerted on the solar body is then mediated by the asymmetric boundary conditions to which the distant solar wind field has to adapt.Next we study the indirect action of such a force on orbiting Keplerian objects like planets, planetesimals and comets. Since this force only influences the central solar body, but not the planets themselves, the problem is different from the treatment of a constant perturbation force perturbing the Keplerian orbits. We present a perturbation analysis treating the action of a corresponding position-dependent perturbation force resulting in secular changes of the orbital elements of Keplerian objects. It is found that changes are accumulating more rapidly in time the closer to the sun the orbiting bodies are. Main axis and perihelion distances are systematically increasing. Especially pronounced are changes in the perihelion position angle of the objects. For solar wind mass losses larger than the Sun's present value by a factor of 1000 (T-Tauri phase of the Sun,) the migration periods calculated for the planet Mercury are of the same order of magnitude as that for corresponding general relativistic migration. 相似文献