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1.
New methods are proposed for solving equations of motion of celestial bodies. The methods are based on the use of superosculating orbits with second- and third-order tangency to the trajectory of the real motion of a body. The construction of these orbits is related to the concept of a fictitious attracting center, whose mass varies in accordance with the first Meshchersky law. In the original reference methods, the perturbed trajectory is represented by a sequence of small arcs of superosculating orbits. The order of accuracy of the reference methods coincides with the order of tangency of the superosculating orbit used in calculations. Using Runge's rule and Richardson's extrapolation scheme leads to the methods of higher order. The efficiency of the new methods in comparison with the numerical integration of equations of motion based on the well-known fourth- and seventh-order Runge–Kutta–Fehlberg methods is illustrated by examples of the calculation of perturbed orbits of some asteroids. 相似文献
2.
V. A. Shefer 《Astronomy Letters》2007,33(4):270-282
We suggest a new approach to solving the problem of finding the orbit of a celestial body from its three spatial position vectors and the corresponding times. It allows most of the perturbations in the motion of a celestial body to be taken into account. The approach is based on the theory of intermediate orbits that we developed previously. We construct the orbit the motion along which is a combination of two motions: the motion of a fictitious attracting center whose mass varies according to Mestschersky’s first law and the motion relative to the fictitious center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden-Mestschersky problem. The constructed orbit has such parameters that their limiting values at any reference epoch define a superosculating intermediate orbit with a fourth-order tangency. We have performed a numerical analysis to estimate the accuracy of approximating the perturbed motion of two minor planets, 145 Adeona and 4179 Toutatis, by the orbits computed using two-position procedures (the classical Gauss method and the method that we suggested previously), a three-position procedure based on the Herrick-Gibbs equation, and the new method. Comparison of the results obtained suggests that the latter method has an advantage. 相似文献
3.
Based on the theory of intermediate orbits developed earlier by the author of this paper, a new approach is proposed to the solution of the problem of finding the orbit of a celestial body with the use of two position vectors of this body and the corresponding time interval. This approach makes it possible to take into account the main part of perturbations. The orbit is constructed, the motion along which is a combination of two motions: the uniform motion along a straight line of a fictitious attracting center, whose mass varies according to the first Meshchersky law, and the motion around this center. The latter is described by the equations of the Gylden–Meshchersky problem. The parameters of the constructed orbit are chosen so that their limiting values at any reference epoch determine a superosculating intermediate orbit with third-order tangency. The accuracy of approximation of the perturbed motion by the orbits calculated by the classical Gauss method and the new method is illustrated by an example of the motion of the unusual minor planet 1566 Icarus. Comparison of the results obtained shows that the new method has obvious advantages over the Gauss method. These advantages are especially prominent in cases where the angular distances between the reference positions are small. 相似文献
4.
V. A. Shefer 《Astronomy Letters》2006,32(12):845-858
The theory of superosculating intermediate orbits previously suggested by the author is developed. A new class of orbits with a fourth-order tangency to the actual trajectory of a celestial body at the initial time is constructed. Orbits with a fifth-order tangency have been constructed for the first time. The motion in the constructed orbits is represented as a combination of two motions: the motion of a fictitious attracting center with a variable mass and the motion relative to this center. The first motion is generally parabolic, while the second motion is described by the equations of the Gylden—Mestschersky problem. The variation in the mass of the fictitious center obeys Mestschersky’s first and combined laws. The new orbits represent more accurately the actual motion in the initial segment of the trajectory than an osculating Keplerian orbit and other existing analogues. Encke’s generalized methods of special perturbations in which the constructed intermediate orbits are used as reference orbits are presented. Numerical simulations using the approximations of the motions of Asteroid Toutatis and Comet P/Honda—Mrkos—Pajdu?áková as examples confirm that the constructed orbits are highly efficient. Their application is particularly beneficial in investigating strongly perturbed motion. 相似文献
5.
An analytical approach to small amplitude solutions of the extended nearly circular Sitnikov problem
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》2009,103(3):251-266
The model of extended Sitnikov Problem contains two equally heavy bodies of mass m moving on two symmetrical orbits w.r.t the centre of gravity. A third body of equal mass m moves along a line z perpendicular to the primaries plane, intersecting it at the centre of gravity. For sufficiently small distance from the
primaries plane the third body describes an oscillatory motion around it. The motion of the three bodies is described by a
coupled system of second order differential equations for the radial distance of the primaries r and the third mass oscillation z. This problem which is dealt with for zero initial eccentricity of the primaries motion, is generally non integrable and
therefore represents an interesting dynamical system for advanced perturbative methods. In the present paper we use an original
method of rewriting the coupled system of equations as a function iteration in such a way as to decouple the two equations
at any iteration step. The decoupled equations are then solved by classical perturbation methods. A prove of local convergence
of the function iteration method is given and the iterations are carried out to order 1 in r and to order 2 in z. For small values of the initial oscillation amplitude of the third mass we obtain results in very good agreement to numerically
obtained solutions. 相似文献
6.
Victor Szebehely Arthur L. Whipple 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):125-133
The restricted problem of three bodies is generalized to the restricted problem of 2+n bodies. Instead of one body of small mass and two primaries, the system is modified so that there are several gravitationally interacting bodies with small masses. Their motions are influenced by the primaries but they do not influence the motions of the primaries. Several variations of the classical problem are discussed. The separate Jacobian integrals of the minor bodies are lost but a conservative (time-independent) Hamiltonian of the system is obtained. For the case of two minor bodies, the five Lagrangian points of the classical problem are generalized and fourteen equilibrium solutions are established. The four linearly stable equilibrium solutions which are the generalizations of the triangular Lagrangian points are once again stable but only for considerably smaller values of the mass parameter of the primaries than in the classical problem. 相似文献
7.
Peter J. Shelus 《Celestial Mechanics and Dynamical Astronomy》1972,5(4):483-489
Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios. 相似文献
8.
O. M. Kochetova 《Solar System Research》2004,38(1):66-75
Masses of 19 asteroids have been determined from the analysis of their gravitational effect on the motion of perturbed bodies. The following asteroids were selected as perturbed bodies: (1) those which had single close encounters with the perturbing asteroid; (2) those whose mean motion was in a 1 : 1 commensurability with that of the perturber and which had close or moderate recurrent encounters with the perturber. The perturber mass was determined from observations of several tens of perturbed asteroids that were selected from these two groups. The selection criterion was the error of the mass determined from observations of only one asteroid. Positional observations of the asteroids on the interval 1900–2002 were used. The masses were determined with errors by an order-half an order of magnitude smaller than the masses found. The results are compared with those of other authors. 相似文献
9.
In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented.
Three bodies of masses m
1, m
2 and m
3 (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the
system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction
of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families
and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal
and finally when we have three unequal masses. Series, with respect to the mass m
3, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of
the mass parameters are also calculated. 相似文献
10.
The restricted three-body problem is reconsidered by replacing the point-like primaries of the classical problem by a pair of axisymmetric rigid bodies which have a plane of symmetry perpendicular to their axes, and the infinitesimal mass by a gyrostat. The conditions for the circular motion of the primaries around their center of mass are stated and they yield the classification of all possible orientations of these bodies into four groups according to the value of their angular velocity. Then the equations of motion of the gyrostat are derived and solved for the equilibrium configurations of the system. 相似文献
11.
In this article we treat the 'Extended Sitnikov Problem' where three bodies of equal masses stay always in the Sitnikov configuration.
One of the bodies is confined to a motion perpendicular to the instantaneous plane of motion of the two other bodies (called
the primaries), which are always equally far away from the barycenter of the system (and from the third body). In contrary
to the Sitnikov Problem with one mass less body the primaries are not moving on Keplerian orbits. After a qualitative analysis
of possible motions in the 'Extended Sitnikov Problem' we explore the structure of phase space with the aid of properly chosen
surfaces of section. It turns out that for very small energies H the motion is possible only in small region of phase space and only thin layers of chaos appear in this region of mostly
regular motion. We have chosen the plane (
) as surface of section, where r is the distance between the primaries; we plot the respective points when the three bodies are 'aligned'. The fixed point
which corresponds to the 1 : 2 resonant orbit between the primaries' period and the period of motion of the third mass is
in the middle of the region of motion. For low energies this fixed point is stable, then for an increased value of the energy
splits into an unstable and two stable fixed points. The unstable fixed point splits again for larger energies into a stable
and two unstable ones. For energies close toH = 0 the stable center splits one more time into an unstable and two stable ones. With increasing energy more and more of
the phase space is filled with chaotic orbits with very long intermediate time intervals in between two crossings of the surface
of section. We also checked the rotation numbers for some specific orbits.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
J. Hagel 《Celestial Mechanics and Dynamical Astronomy》1990,50(3):209-230
The circular restricted problem of three bodies is investigated analytically with respect to the problem of deriving a second integral of motion besides the well known Jacobian Integral. The second integral is searched for as a correction the angular momentum integral valid in the two body case. A partial differential equation equivalent to the problem is derived and solved approximately by an asymptotic Fourier method assuming either sufficiently small values for the dimensionless mass parameter or sufficiently large distances from the barycentre. The solution of the partial equation then leads to a function of the coordinates, velocities and time being nearly constant, which means that its variation with time is about 40–300 times less than that of the pure angular momentum. By averaging over the remaining fluctuating part of the quasi-integral we are able to integrate the first order equations using a renormalization transformation. This leads to an explicit expression for the approximate solution of the circular problem which describes the motion of the third body orbiting both primaries with nonvanishing initial eccentricity (eccentric planetary type orbits). One of the main results is an explicit formula for the frequency of the perihelion motion of the third body which depends on the mass parameter, the initial distance of the third body from the barycentre and the initial eccentricity. Finally we study orbits of the P-Type, being defined as solutions of the restricted problem with circular initial conditions (vanishing initial eccentricity). 相似文献
13.
T. J. Kalvouridis 《Astrophysics and Space Science》1996,246(2):219-227
The paper deals with the restricted photogravitational 2+2 body problem when the primaries are oblate spheroids. A study of the effect of the oblateness on the equilibrium positions and on the areas of the permissible motion of the minor bodies, is also made. 相似文献
14.
K. E. Papadakis 《Earth, Moon, and Planets》2008,103(1-2):25-42
This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L 1, L 2 and L 3 correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations. 相似文献
15.
Zdenek Sekanina 《Icarus》1978,33(2):415-427
A model is proposed for single close encounters between two small masses, m1and m2, which orbit a much larger mass, M. The main new feature of the model is the assumption of conic motion of the center of mass of m1and m2 in the gravitational field of M. Comparisons of the model with the three-body equations of motion indicate that the model is a useful approximation for m1, m2 ? 10?5M. The model is therefore applicable for encounters between bodies of the order of an earth mass or smaller in the presence of the sun. Comparisons are also made of outcomes obtained by the model with outcomes of numerical integration for a large variety of close encounters. The above comparisons reveal that for many purposes the model is an adequate approximation for those encounters with ? ≥ 4, where ? is the eccentricity of the hyperbolic orbit of m1about m2. 相似文献
16.
A simple method for numerical integration of the equations of motion of small bodies of the Solar System is proposed, which is especially efficient in studying the orbits with small perihelion distances. The evolution of orbits of 121 numbered asteroids with perihelion distances q < 1.2 AU is investigated over the time interval of years 2000–2100 with allowance made for the gravitational influence of nine planets and three largest asteroids. The circumstances of close encounters of asteroids with the Earth and other terrestrial planets are presented. 相似文献
17.
We deal with the problem of a zero mass body oscillating perpendicular to a plane in which two heavy bodies of equal mass orbit each other on Keplerian ellipses. The zero mass body intersects the primaries plane at the systems barycenter. This problem is commonly known as theSitnikov Problem. In this work we are looking for a first integral related to the oscillatory motion of the zero mass body. This is done by first expressing the equation of motion by a second order polynomial differential equation using a Chebyshev approximation techniques. Next we search for an autonomous mapping of the canonical variables over one period of the primaries. For that we discretize the time dependent coefficient functions in a certain number of Dirac Delta Functions and we concatenate the elementary mappings related to the single Delta Function Pulses. Finally for the so obtained polynomial mapping we look for an integral also in polynomial form. The invariant curves in the two dimensional phase space of the canonical variables are investigated as function of the primaries eccentricity and their initial phase. In addition we present a detailed analysis of the linearized Sitnikov Problem which is valid for infinitesimally small oscillation amplitudes of the zero mass body. All computations are performed automatically by the FORTRAN program SALOME which has been designed for stability considerations in high energy particle accelerators. 相似文献
18.
T.J. Kalvouridis 《Astrophysics and Space Science》1998,259(1):77-90
The restricted gravitational 2 + 2 body problem, is a particular case of the N body problem and it may be used to approximate
the dynamical behaviour of binary asteroids or dual sattelites moving in the gravitational field of two primaries Pi, i =
1,2. By considering oblate primaries, five parameters are needed to describe the model, namely the reduced mass μ of the primary
P2, the reduced masses μ1 and μ2 of the minor bodies and the oblatenesses Ii, i = 1,2 of the primaries. This work deals with
the effect of those parameters on the location of the stationary solutions.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
19.
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
20.
The equations of motion of the 2+2 body problem (two interacting particles in the gravitational field of two much more massive primaries m1 and m2 in circular keplerian orbit) have an integral analogous to the Jacobi integral of the circular 2+1 body problem. We show here that with 2+2 bodies this integral does not give rise to Hill stability, i.e. to confinement for all time in a portion of the configuration space not allowing for some close approaches to occur. This is because all the level manifolds are connected and all exchanges of bodies between the regions surroundingm
1,m
2 and infinity do not contradict the conservation of the integral. However, it is worth stressing that some of these exchanges are physically meaningless, because they involve either unlimited extraction of potential energy from the binary formed by the small bodies (without taking into account their physical size) or significant mutual perturbations between the small masses without close approach, a process requiring, for the Sun-Jupiter-two asteroids system, timescales longer than the age of the Solar System. 相似文献