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1.
Conxita Pinyol 《Celestial Mechanics and Dynamical Astronomy》1995,61(4):315-331
The main goal of this paper is to give an approximation to initial conditions for ejection-collision orbits with the more massive primary, in the planar elliptic restricted three body problem when the mass parameter µ and the eccentricity e are small enough. The proof is based on a regularization of variables and a perturbation of the two body problem.This work was partially supported by DGICYT grant number PB90-0695. 相似文献
2.
The theory of Burdet's focal elements is outlined. The differential equations are presented, and the initial value problem is described together with the transformation to rectangular coordinates and classical elements. The focal elements are well defined for zero eccentricity and inclination. They can be adopted for the computation of elliptic, parabolic and hyperbolic motion. For the numerical integration of near-geostationary orbits a comparison of the efficiency is made between focal elements, KS theory and rectangular coordinates. For this class of orbits, a higher accuracy has been obtained by integrating elements than integrating rectangular coordinates. 相似文献
3.
Consecutive collision orbits in the limiting case µ = 0 of the elliptic restricted three-body problem are investigated. in particular those in which the infinitesimal mass collides twice with the smaller (massless) primary. A timing condition is presented that allows the extension of previous results to the case of arbitrary relative orientation of the orbits of the infinitesimal mass and the smaller primary. The timing condition is expressed in two general forms - in terms of orbit parameters and eccentric (or hyperbolic) anomalies at the times of collision - for the specific cases of elliptic. parabolic or hyperbolic orbits of the infinitesimal mass. Some families of solutions are presented. 相似文献
4.
5.
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
6.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1993,56(1-2):71-80
We present numerical results of the so-called Sitnikov-problem, a special case of the three-dimensional elliptic restricted three-body problem. Here the two primaries have equal masses and the third body moves perpendicular to the plane of the primaries' orbit through their barycenter. The circular problem is integrable through elliptic integrals; the elliptic case offers a surprisingly great variety of motions which are until now not very well known. Very interesting work was done by J. Moser in connection with the original Sitnikov-paper itself, but the results are only valid for special types of orbits. As the perturbation approach needs to have small parameters in the system we took in our experiments as initial conditions for the work moderate eccentricities for the primaries' orbit (0.33e
primaries
0.66) and also a range of initial conditions for the distance of the 3
rd
body (= the planet) from very close to the primaries orbital plane of motion up to distance 2 times the semi-major axes of their orbit. To visualize the complexity of motions we present some special orbits and show also the development of Poincaré surfaces of section with the eccentricity as a parameter. Finally a table shows the structure of phase space for these moderately chosen eccentricities. 相似文献
7.
In the present paper, an efficient iterative method of arbitrary integer order of convergent ≥2 based on the homotopy continuation techniques for the solution of the initial value problem of space dynamics using the universal Y functions is presented. The method is of dynamic nature in the sense that, ongoing from one iterative scheme to the subsequent one, only additional instruction is needed. Most importantly, the method does not need any prior knowledge of the initial guess. This is a property which avoids the critical situations between divergent to very slow convergent solutions that may exist in other numerical methods which depend on initial guesses. A computational package for digital implementation of the method is given, together with numerical applications for elliptic, hyperbolic, and parabolic orbits. The accuracy of the results for all orbits is O(10–16). (© 2016 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
8.
R. Dvorak 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):369-378
This is a numerical study of orbits in the elliptic restricted three-body problem concerning the dependence of the critical orbits on the eccentricity of the primaries. They are defined as being the separatrix between stable and unstable single periodic orbits. As our results are adapted to the existence of planetary orbits in double stars we concentrated first on the P-orbits (defined to surround both primaries). Due to the complexity of the elliptic problem there is no analytical approach possible. Using the results of some 300 integrated orbits for 103 to 3. 103 periods of the primaries we established lower and upper bounds for the critical orbits for different values of the eccentricity. 相似文献
9.
R. G. Langebartel 《Astrophysics and Space Science》1981,75(2):437-454
The restricted three-body Hamiltonian is partitioned into a two-center type principal part and its accompanying perturbational part. The mathematical analysis, involving the Jacobian elliptic functions, is adapted for the case of figure-eight orbits winding around the two given mass points. For many such orbits the elliptic function modulusk is small and can serve as a small parameter.Fourier expansions in terms of a parameter related tot are obtained for the intermediate orbit functions which provide representations in terms of elementary functions. 相似文献
10.
Sergey Bolotin 《Celestial Mechanics and Dynamical Astronomy》2005,93(1-4):343-371
We consider the plane restricted elliptic 3 body problem with small mass ratio and small eccentricity and prove the existence
of many periodic orbits shadowing chains of collision orbits of the Kepler problem. Such periodic orbits were first studied
by Poincaré for the non-restricted 3 body problem. Poincaré called them second species solutions. 相似文献
11.
R. A. Serafin 《Celestial Mechanics and Dynamical Astronomy》1985,36(3):273-280
We consider the collision probability for comets with the Sun under the suppositions of different velocity distributions and various initial conditions. We solve the problem applying Laplace's method and using Schiaparelli's hyperboloid of visibility. The probabilities obtained in this manner are given separately for elliptic and hyperbolic orbits. 相似文献
12.
Orbits around Mercury are influenced by the strong elliptic third-body perturbation, especially for high eccentricity orbits, the periapsis altitude changes dramatically. Frozen orbits whose mean eccentricity and argument of perigee remain constants are obviously a good choice for space missions, but the forming conditions are too harsh to meet practical needs. To deal with this problem, a continuous control method that combines analytical theory and parameter optimization is proposed to build an artificial frozen orbit. The artificial frozen orbits are investigated on the basis of double averaged Hamiltonian, of which the second and third zonal harmonics and the perturbation of elliptic third-body gravity are considered. In this paper, coefficients of perturbations which satisfy the conditions of frozen orbits are involved as control parameters, and the relevant artificial perturbations are compensated by the control strategy. So probes around Mercury can be kept on frozen orbit under the influence of continuous control force. Then complex method of optimization is used to search for the energy optimized artificial frozen orbits. The choosing of optimal parameters, the objective function setting and other issues are also discussed in the study. Evolution of optimal control parameters are given in large ranges of semi-major axis and eccentricity, through the variation of these curves, the fuel efficiency is discussed. The result shows that the control method proposed in this paper can effectively maintain the eccentricity and argument of perigee frozen. 相似文献
13.
Hideki Asada 《Celestial Mechanics and Dynamical Astronomy》2007,97(3):151-164
We present an exact solution of the equations for orbit determination of a two body system in a hyperbolic or parabolic motion.
In solving this problem, we extend the method employed by Asada, Akasaka and Kasai (AAK) for a binary system in an elliptic
orbit. The solutions applicable to each of elliptic, hyperbolic and parabolic orbits are obtained by the new approach, and
they are all expressed in an explicit form, remarkably, only in terms of elementary functions. We show also that the solutions
for an open orbit are recovered by making a suitable transformation of the AAK solution for an elliptic case. 相似文献
14.
Xiaodong Liu Hexi Baoyin Xingrui Ma 《Celestial Mechanics and Dynamical Astronomy》2011,109(3):303-320
Frozen orbits are always important foci of orbit design because of their valuable characteristics that their eccentricity
and argument of pericentre remain constant on average. This study investigates quasi-circular frozen orbits and examines their
basic nature analytically using two different methods. First, an analytical method based on Lagrangian formulations is applied
to obtain constraint conditions for Martian frozen orbits. Second, Lie transforms are employed to locate these orbits accurately,
and draw the contours of the Hamiltonian to show evolutions of the equilibria. Both methods are verified by numerical integrations
in an 80 × 80 Mars gravity field. The simulations demonstrate that these two analytical methods can provide accurate enough
results. By comparison, the two methods are found well consistent with each other, and both discover four families of Martian
frozen orbits: three families with small eccentricities and one family near the critical inclination. The results also show
some valuable conclusions: for the majority of Martian frozen orbits, argument of pericentre is kept at 270° because J
3 has the same sign as J
2; while for a minority of ones with low altitude and low inclination, argument of pericentre can be kept at 90° because of
the effect of the higher degree odd zonals; for the critical inclination cases, argument of pericentre can also be kept at
90°. It is worthwhile to note that there exist some special frozen orbits with extremely small eccentricity, which could provide
much convenience for reconnaissance. Finally, the stability of Martian frozen orbits is estimated based on the trace of the
monodromy matrix. The analytical investigations can provide good initial conditions for numerical correction methods in the
more complex models. 相似文献
15.
Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy. 相似文献
16.
Poincaré's continuation method is applied to the elliptic restricted problem for the computation of four families of doubly symmetric three-dimensional periodic orbits emanating from similar orbits corresponding to zero eccentricity. The results are given in four tables and the orbits are characterized in regard to their stability. 相似文献
17.
For equatorial orbits about an oblate body, we show that the Lie series for the elliptic elementse,f,l and
diverge when the oblateness exceeds a critical multiple of the transformed eccentricity constant. The use of similar truncated series expansions for such elliptic elements by Brouwer accounts for the first-order errors at low eccentricity in his derived coordinates for an artificial satellite. 相似文献
18.
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
19.
N. Caranicolas 《Journal of Astrophysics and Astronomy》1989,10(2):197-202
Using theoretical arguments we study some properties of a time independent, two-dimensional Hamiltonian system in the 1:1
resonance case. In particular we find the frequency of the oscillation near the elliptic points, the ratio of the semiaxes
of the ellipses surrounding them and the inclination angle of the asymptotes of the hyperbolic orbits in the original variablesx–p
x. The results are in satisfactory agreement with those given by numerical integration of the equations of motion. Some possible
applications for the motion of stars in elliptical galaxies are also discussed. 相似文献
20.
I. A. Robin 《Celestial Mechanics and Dynamical Astronomy》1981,23(1):97-106
We show that the procedure employed in the circular restricted problem, of tracing families of three-dimensional periodic orbits from vertical self-resonant orbits belonging to plane families, can also be applied in the elliptic problem. A method of determining series of vertical bifurcation orbits in the planar elliptic restricted problem is described, and one such series consisting of vertical-critical orbits (a
v=+1) is given for the entire range (0,1/2) of the mass parameter . The initial segments of the families of three-dimensional orbits which bifurcate from two of the orbits belonging to this series are also given. 相似文献