共查询到20条相似文献，搜索用时 31 毫秒
1.
Shannon L. Coffey André Deprit Etienne Deprit 《Celestial Mechanics and Dynamical Astronomy》1994,59(1):37-72
We say that a planet is Earth-like if the coefficient of the second order zonal harmonic dominates all other coefficients in the gravity field. This paper concerns the zonal problem for satellites around an Earth-like planet, all other perturbations excluded. The potential contains all zonal coefficientsJ
2 throughJ
9. The model problem is averaged over the mean anomaly by a Lie transformation to the second order; we produce the resulting Hamiltonian as a Fourier series in the argument of perigee whose coefficients are algebraic functions of the eccentricity — not truncated power series. We then proceed to a global exploration of the equilibria in the averaged problem. These singularities which aerospace engineers know by the name of frozen orbits are located by solving the equilibria equations in two ways, (1) analytically in the neighborhood of either the zero eccentricity or the critical inclination, and (2) numerically by a Newton-Raphson iteration applied to an approximate position read from the color map of the phase flow. The analytical solutions we supply in full to assist space engineers in designing survey missions. We pay special attention to the manner in which additional zonal coefficients affect the evolution of bifurcations we had traced earlier in the main problem (J
2 only). In particular, we examine the manner in which the odd zonalJ
3 breaks the discrete symmetry inherent to the even zonal problem. In the even case, we find that Vinti's problem (J
4+J
2
2
=0) presents a degeneracy in the form of non-isolated equilibria; we surmise that the degeneracy is a reflection of the fact that Vinti's problem is separable. By numerical continuation we have discovered three families of frozen orbits in the full zonal problem under consideration; (1) a family of stable equilibria starting from the equatorial plane and tending to the critical inclination; (2) an unstable family arising from the bifurcation at the critical inclination; (3) a stable family also arising from that bifurcation and terminating with a polar orbit. Except in the neighborhood of the critical inclination, orbits in the stable families have very small eccentricities, and are thus well suited for survey missions. 相似文献
2.
The present paper addresses the existence of J
2 invariant relative orbits with arbitrary relative magnitude over the infinite time using the Routh reduction and Poincaré
techniques in the J
2 Hamiltonian problem. The current research also proposes a novel numerical searching approach for J
2 invariant relative orbits from the dynamical system point of view. A new type of Poincaré mapping is defined from different
central manifolds of the pseudo-circular orbits (parameterized by the Jacobi energy E, the polar component of momentum H
z
and the measure of distance Δr between the fixed point and its central manifolds) to the nodal periods T
d
and the drifts of longitude of the ascending node during one period (ΔΩ), which differs from Koon et al.’s (AIAA 2001) definition on central manifolds parameterized by the same fixed point. The Poincaré mapping is surjective because it compresses
the three-dimensional variables into two-dimensional images, and the mapping degenerates into a bijective mapping in consideration
of the fixed points. An iteration algorithm to the degenerated bijective mapping is proposed from the continuation procedure
to perform the ergodic representation of E- and H
z
-contour maps on the space of T
d
–ΔΩ. For the surjective mapping with Δr ≠ 0, different pseudo-circular or elliptical orbits may share the same images. Hence, the inverse surjective mapping may
achieve non-unique variables from a single image, which makes the generation of J
2 invariant relative orbits possible. The pseudo-circular or elliptical orbits generated from the surjective mapping will be
defined in different meridian planes. Hence, the critical contribution of the present paper is the assignment of J
2 invariant relative orbits to different invariant parameters E and H
z
depending on the E- and H
z
-contour map, which will hold J
2 invariant relative orbits for extended durations. To investigate the high-order nonlinearity neglected by previous studies,
a formation configuration with a large magnitude of 500 km is successfully generated from the theory developed in the present
work, which is beyond the scope of the linear conditions of J
2 invariant relative orbits. Therefore, the existence of J
2 invariant relative orbit with an arbitrary relative magnitude over the infinite time is achieved from the dynamical system
point of view. 相似文献
3.
I. Stellmacher 《Celestial Mechanics and Dynamical Astronomy》1982,28(4):351-365
Résumé On développe une méthode de construction d'orbites périoldiques dans un système d'axes tournants, pour un satellite gravitant autour d'un sphéroide. Les orbites sont quasi circulaires,i est l'inclinaison sur le plan équatorial de la planète. Pour les petites inclinaisons, la solution est donnée jusqu'aux termes enJ
2
2
etJ
4.Ce modèle peut être appliqué aux satellites de Saturne. Des valeurs observées des longitudes des noeuds ascendants de Mimas et Téthys, on donne une estimation des valeurs deJ
2 etJ
4 du potentiel de Saturne. La valeur deJ
2 est très sensible aux valeurs adoptées pour le rayon équatorial de la planète.
Construction of periodic orbits of satellites in a moving system of axes, I
We give an algorithm for the construction of periodic orbits in a rotating frame for the cases of satellites moving around an oblate planet.The orbits are near to the circular case; the asymptotic developments of the periodic solutions are completely calculated for the termsJ 2 andJ 4 of the potential. The solutions for small inclinations are given up toJ 2 2 .The families of solutions depend on three parameters: the semi-major axis, the inclination of the generating orbit and the initial position on this orbit.These solutions can be applied to the motion of the Saturnian satellites. From the observed longitudes of the ascending nodes of Mimas and Tethys, we estimate the valuesJ 2 andJ 4 of the Saturnian potential, the value ofJ 2 very strongly depends on the adopted value of the planet's equatorial diameter.相似文献
4.
Periodic orbits in the Stormer problem are studied using the symmetry lines of the Poincaré map introduced by De Vogelaere. Many known facts are explained by mean of these lines. The dynamics of four special symmetry lines when the Stormer parameter 1 changes is presented, and we obtain a clear global view of the structure of the simple periodic orbits and their bifurcations, including the asymmetrical ones. New asymmetrical multiple periodic orbits are obtained. 相似文献
5.
Lidia Jiménez-Lara Adolfo Escalona-Buendía 《Celestial Mechanics and Dynamical Astronomy》2001,79(2):97-117
We present some qualitative and numerical results of the Sitnikov problem, a special case of the three-body problem, which offers a great variety of motions as the non-integrable systems typically do. We study the symmetries of the problem and we use them as well as the stroboscopic Poincarée map (at the pericenter of the primaries) to calculate the symmetry lines and their dynamics when the parameter changes, obtaining information about the families of periodic orbits and their bifurcations in four revolutions of the primaries. We introduce the semimap to obtain the fundamental lines l
1. The origin produces new families of periodic orbits, and we show the bifurcation diagrams in a wide interval of the eccentricity (0 0.97). A pattern of bifurcations was found.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
6.
N. D. Caranicolas 《Celestial Mechanics and Dynamical Astronomy》1989,47(1):87-96
We derive an algebraic mapping for an autonomous, two-dimensional galactic type Hamiltonian in the 1/1 resonance case. We use the mapping to study the stability of the periodic orbits. Using the x — p
x
Poincaré surface section, we compare the results of the mapping with those found by the numerical integration of the full equations of motion. For small values of the perturbation the results of the two methods are in very good agreement while satisfactory agreement is obtained for larger perturbations. 相似文献
7.
In a previous work we studied the effects of (I) the J
2 and C
22 terms of the lunar potential and (II) the rotation of the primary on the critical inclination orbits of artificial satellites.
Here, we show that, when 3rd-degree gravity harmonics are taken into account, the long-term orbital behavior and stability
are strongly affected, especially for a non-rotating central body, where chaotic or collision orbits dominate the phase space.
In the rotating case these phenomena are strongly weakened and the motion is mostly regular. When the averaged effect of the
Earth’s perturbation is added, chaotic regions appear again for some inclination ranges. These are more important for higher
values of semi-major axes. We compute the main families of periodic orbits, which are shown to emanate from the ‘frozen eccentricity’
and ‘critical inclination’ solutions of the axisymmetric problem (‘J
2 + J
3’). Although the geometrical properties of the orbits are not preserved, we find that the variations in e, I and g can be quite small, so that they can be of practical importance to mission planning. 相似文献
8.
Osman M. Kamel 《Earth, Moon, and Planets》1989,47(1):73-89
The construction of a third order J-S theory is presented. The Hori theory of planetary perturbations is employed. No Critical J-S terms due to the 2:5 commensurabilities and its multiples exist, when we take into account the periodic terms of order 0, 1, 2 with respect to the eccentricity- inclination. In this case the Lie series transformation degenerates and is meaningless. The J-S equations of motion for secular perturbations are solved when we neglect in our treatment, the Poisson terms of degree > 2 in the Poincaré canonical variables H
u
, K
u
, P
u
Q
u
(u = 1, 2). The Jacobi-Radau referential is adopted, and the theory is expressed in terms of the canonical variables of H. Poincaré.Now at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, U.S.A. 相似文献
9.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1977,16(1):61-76
It is proved that monoparametric families of periodic orbits of theN-body problem in the plane, for fixed values of all masses, exist in a rotating frame of reference whosex axis contains always two of the bodiesP
1 andP
2. A periodic motion of theN-body problem is obtained as a continuation ofN–2 symmetric periodic orbits of the circular restricted three-body problem whose periods are in integer dependence, by increasing the masses of the originallyN–2 massless bodiesP
3, ...,P
k. The analytic continuation, for sufficiently small values of theN–2 bodiesP
3
...P
k
and finite values for the masses ofP
1 andP
2 has been proved by the continuation method and the solution itself has been found explicitly to a linear approximation in the small masses. Also, the possible application of the above periodic orbits to the study of the Solar system and of stellar systems is mentioned. 相似文献
10.
The critical inclination is of special interest in artificial satellite theory. The critical inclination can maintain minimal
deviations of eccentricity and argument of pericentre from the initial values, and orbits at this inclination have been applied
to some space missions. Most previous researches about the critical inclination were made under the assumption that the oblateness
term J
2 is dominant among the harmonic coefficients. This paper investigates the extension of the critical inclination where the
concept of the critical inclination is different from that of the traditional sense. First, the study takes the case of Venus
for instance, and provides some preliminary results. Then for general cases, given the values of argument of pericentre and
eccentricity, the relationship between the multiplicity of the solutions for the critical inclination and the values of J
2 and J
4 is analyzed. Besides, when given certain values of J
2 and J
4, the relationship between the multiplicity of the solutions for the critical inclination and the values of semimajor axis
and eccentricity is studied. The results show that for some cases, the value of the critical inclination is far away from
that of the traditional sense or even has multiple solutions. The analysis in this paper could be used as starters of correction
methods in the full gravity field of celestial bodies. 相似文献
11.
Formulae containing the elements of the variational matrix are obtained which determine the linear isoenergetic stability parameters of three-dimensional periodic orbits of the general three-boy problem. This requires the numerical integration of the variational equations but produces the stability parameters with the effective accuracy of the numerical integration. The conditions for stability, criticality, and bifurcations are briefly examined and the stability determination procedure is tested in the determination of some three-dimensional periodic orbits of low inclination bifurcating from vertical-critical coplanar orbits. 相似文献
12.
In this paper by means of a Poincaré map, we prove the existence of symmetric periodic orbits of the elliptic Sitnikov problem. Furthermore, using the presence of the Bernoulli shift as a subsystem of that Poincaré map, we prove that not all the periodic orbits of the Sitnikov problem are symmetric periodic orbits.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
13.
Poincaré's surface of section method is used to find and classify the main periodic orbits in a two-dimensional galactic potential first introduced by Hénon and Heiles. The stability of these periodic orbits is studied. Numerical integration with Bulirsch-Stoer method is used. 相似文献
14.
E. A. Perdios 《Celestial Mechanics and Dynamical Astronomy》2007,99(2):85-104
This paper deals with the Sitnikov family of straight-line motions of the circular restricted three-body problem, viewed as
generator of families of three-dimensional periodic orbits. We study the linear stability of the family, determine several
new critical orbits at which families of three dimensional periodic orbits of the same or double period bifurcate and present
an extensive numerical exploration of the bifurcating families. In the case of the same period bifurcations, 44 families are
determined. All these families are computed for equal as well as for nearly equal primaries (μ = 0.5, μ = 0.4995). Some of the bifurcating families are determined for all values of the mass parameter μ for which they exist. Examples of families of three dimensional periodic orbits bifurcating from the Sitnikov family at double
period bifurcations are also given. These are the only families of three-dimensional periodic orbits presented in the paper
which do not terminate with coplanar orbits and some of them contain stable parts. By contrast, all families bifurcating at
single-period bifurcations consist entirely of unstable orbits and terminate with coplanar orbits. 相似文献
15.
P. S. Soulis K. E. Papadakis T. Bountis 《Celestial Mechanics and Dynamical Astronomy》2008,100(4):251-266
We study the existence, linear stability and bifurcations of what we call the Sitnikov family of straight line periodic orbits
in the case of the restricted four-body problem, where the three equal mass primary bodies are rotating on a circle and the
fourth (small body) is moving in the direction vertical to the center mass of the other three. In contrast to the restricted
three-body Sitnikov problem, where the Sitnikov family has infinitely many stability intervals (hence infinitely many Sitnikov critical orbits), as the “family parameter” ż0 varies within a finite interval (while z
0 tends to infinity), in the four-body problem this family has only one stability interval and only twelve 3-dimensional (3D) families of symmetric periodic orbits exist which bifurcate from twelve
corresponding critical Sitnikov periodic orbits. We also calculate the evolution of the characteristic curves of these 3D
branch-families and determine their stability. More importantly, we study the phase space dynamics in the vicinity of these
orbits in two ways: First, we use the SALI index to investigate the extent of bounded motion of the small particle off the
z-axis along its interval of stable Sitnikov orbits, and secondly, through suitably chosen Poincaré maps, we chart the motion
near one of the 3D families of plane-symmetric periodic orbits. Our study reveals in both cases a fascinating structure of
ordered motion surrounded by “sticky” and chaotic orbits as well as orbits which rapidly escape to infinity. 相似文献
16.
We study the problem of critical inclination orbits for artificial lunar satellites, when in the lunar potential we include,
besides the Keplerian term, the J
2 and C
22 terms and lunar rotation. We show that, at the fixed points of the 1-D averaged Hamiltonian, the inclination and the argument
of pericenter do not remain both constant at the same time, as is the case when only the J
2 term is taken into account. Instead, there exist quasi-critical solutions, for which the argument of pericenter librates around a constant value. These solutions are represented by smooth
curves in phase space, which determine the dependence of the quasi-critical inclination on the initial nodal phase. The amplitude
of libration of both argument of pericenter and inclination would be quite large for a non-rotating Moon, but is reduced to
<0°.1 for both quantities, when a uniform rotation of the Moon is taken into account. The values of J
2, C
22 and the rotation rate strongly affect the quasi-critical inclination and the libration amplitude of the argument of pericenter.
Examples for other celestial bodies are given, showing the dependence of the results on J
2, C
22 and rotation rate. 相似文献
17.
We consider a system of a harmonic and an unharmonic oscillator with a weak cubic coupling. We study the non-degenerate bifurcations of periodic orbits for the resonant tori of the unperturbed system for which the twist condition holds. We demonstrate that this system also exhibits for certain values of the small parameter non-twist bifurcations, where the rotation number of the Poincaré map attains a minimum value. 相似文献
18.
We study two and three-dimensional resonant periodic orbits, usingthe model of the restricted three-body problem with the Sun andNeptune as primaries. The position and the stability character ofthe periodic orbits determine the structure of the phase space andthis will provide useful information on the stability and longterm evolution of trans-Neptunian objects. The circular planarmodel is used as the starting point. Families of periodic orbitsare computed at the exterior resonances 1/2, 2/3 and 3/4 withNeptune and these are used as a guide to select the energy levelsfor the computation of the Poincaré maps, so that all basicresonances are included in the study. Using the circular planarmodel as the basic model, we extend our study to more realisticmodels by considering an elliptic orbit of Neptune and introducingthe inclination of the orbit. Families of symmetric periodicorbits of the planar elliptic restricted three-body problem andthe three-dimensional problem are found. All these orbitsbifurcate from the families of periodic orbits of the planarcircular problem. The stability of all orbits is studied. Althoughthe resonant structure in the circular problem is similar for allresonances, the situation changes if the eccentricity of Neptuneor the inclination of the orbit is taken into account. All theseresults are combined to explain why in some resonances there aremany bodies and other resonances are empty. 相似文献
19.
Osman M. Kamel 《Earth, Moon, and Planets》1990,49(3):259-267
We construct the outline of a third order secular theory for the four major planets. We apply the Hori-Lie technique to solve the problem. We take into consideration both parts of the perturbing function. Our canonical variables are those of H. Poincaré. Our periodic terms are the only 2:5 and 1:2 critical terms of J-S and U-N respectively. Terms of degree higher than the second in the Poincaré canonical variables H, K, P, Q are neglected. 相似文献
20.
V. V. Markellos 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):215-224
The critical orbits, corresponding to bifurcations of the generating family and its branches, are considered more closely and the part off is investigated that has branches of very high order only. Three families of periodic solutions of the elliptic problem are also determined in an effort to follow the evolution of the stability region aroundf when the eccentricity of the primaries is increased to non-zero values. 相似文献