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1.
We have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when one of the primaries
is an oblate body. We have determined the periodic orbits for different values of μ, h and A (h is energy constant, μ is mass ratio of the two primaries and A is an oblateness factor). These orbits have been determined by giving displacements along the tangent and normal to the mobile
coordinates as defined by Karimov and Sokolsky (Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of oblateness by
taking some fixed values of μ, A and h. As starters for our method, we use some known periodic orbits in the classical restricted three body problem. 相似文献
2.
We consider periodic halo orbits about artificial equilibrium points (AEP) near to the Lagrange points L
1 and L
2 in the circular restricted three body problem, where the third body is a low-thrust propulsion spacecraft in the Sun–Earth
system. Although such halo orbits about artificial equilibrium points can be generated using a solar sail, there are points
inside L
1 and beyond L
2 where a solar sail cannot be placed, so low-thrust, such as solar electric propulsion, is the only option to generate artificial
halo orbits around points inaccessible to a solar sail. Analytical and numerical halo orbits for such low-thrust propulsion
systems are obtained by using the Lindstedt Poincaré and differential corrector method respectively. Both the period and minimum
amplitude of halo orbits about artificial equilibrium points inside L
1 decreases with an increase in low-thrust acceleration. The halo orbits about artificial equilibrium points beyond L
2 in contrast show an increase in period with an increase in low-thrust acceleration. However, the minimum amplitude first
increases and then decreases after the thrust acceleration exceeds 0.415 mm/s2. Using a continuation method, we also find stable artificial halo orbits which can be sustained for long integration times
and require a reasonably small low-thrust acceleration 0.0593 mm/s2. 相似文献
3.
A systematic numerical exploration of the families of asymmetric periodic orbits of the restricted three-body problem when
a) the primary bodies are equal and b) for the Earth-Moon mass ratio, is presented. Decades families of asymmetric periodic
solutions were found and three of the simplest ones, in the first case, and ten of the second one are illustrated. All of
these families consist of periodic orbits which are asymmetric with respect to x-axis while are simple symmetric periodic orbits with respect to y-axis (i.e. the orbit has only one perpendicular intersection at half period with y-axis). Many asymmetric periodic orbits, members of these families, are calculated and plotted. We studied the stability of
all the asymmetric periodic orbits we found. These families consist, mainly, of unstable periodic solutions but there exist
very small, with respect to x, intervals where these families have stable periodic orbits. We also found, using appropriate Poincaré surface of sections,
that a relatively large region of phase space extended around all these stable asymmetric periodic orbits shows chaotic motion. 相似文献
4.
Periodic orbits in the photogravitational restricted problem with the smaller primary an oblate body
In this paper, we have studied periodic orbits generated by Lagrangian solutions of the restricted three body problem when
more massive body is a source of radiation and the smaller primary is an oblate body. We have determined periodic orbits for
fixed values of μ, σ and different values of p and h (μ mass ratio of the two primaries, σ oblate parameter, p radiation parameter and h energy constant). These orbits have been determined by giving displacements along the tangent and normal to the mobile co-ordinates
as defined by Karimov and Sokolsky (in Celest. Mech. 46:335, 1989). These orbits have been drawn by using the predictor-corrector method. We have also studied the effect of radiation pressure
on the periodic orbits by taking some fixed values of μ and σ. 相似文献
5.
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
6.
Richard Schwarz Markus Gyergyovits Rudolf Dvorak 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):139-148
The orbits of real asteroids around the Lagrangian points L4 and L 5of Jupiter with large inclinations (i > 20°) were integrated for 50 Myrs. We investigated the stability with the aid of the
Lyapunov characteristic exponents (LCE) but tested also two other methods: on one hand we integrated four neighbouring orbits
for each asteroid and computed the maximum distance in every group, on the other hand we checked the variation of the Delaunay
element H of the asteroid. In a second simulation – for a grid of initial eccentricity versus initial inclination – we examined the
stability of the orbits around both Lagrangian points for 20° < i < 55° and 0.0 < e < 0.20. For the initial semimajor axes
we have chosen the one ofJupiter(a = 5.202 AU). We determined the stability with the aid of the LCEs and also the maximum
eccentricity of the orbits during the whole integration time. The region around L4 turned out to be unstable for large inclinations and eccentricities (i > 55° and e > 0.12). The stable region shrinks for
orbits around L5: we found that they become unstable already for i > 45° and e > 0.10. We interpret it as a first hint why we observe more
Trojans around the leading Lagrangian point. The results confirm the stability behaviour of the real Trojans which we computed
in the first part of the paper. 相似文献
7.
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
8.
Aimed at the initial value problem of the particular second-order ordinary differential equations,y
=f(x, y), the symmetric methods (Quinlan and Tremaine, 1990) and our methods (Xu and Zhang, 1994) have been compared in detail by integrating the artificial earth satellite orbits in this paper. In the end, we point out clearly that the integral accuracy of numerical integration of the satellite orbits by applying our methods is obviously higher than that by applying the same order formula of the symmetric methods when the integration time-interval is not greater than 12000 periods. 相似文献
9.
Cédric Langbort 《Celestial Mechanics and Dynamical Astronomy》2002,84(4):369-385
In this paper, we study circular orbits of the J
2 problem that are confined to constant-z planes. They correspond to fixed points of the dynamics in a meridian plane. It turns out that, in the case of a prolate body, such orbits can exist that are not equatorial and branch from the equatorial one through a saddle-center bifurcation. A closed-form parametrization of these branching solutions is given and the bifurcation is studied in detail. We show both theoretically and numerically that, close to the bifurcation point, quasi-periodic orbits are created, along with two families of reversible orbits that are homoclinic to each one of them. 相似文献
10.
M. A. Vashkov’yak 《Solar System Research》2010,44(6):527-540
A new analytical solution of the system of differential equations describing secular perturbations and long-period solar perturbations
of mean orbits of outer satellites of giant planets was obtained. As distinct from other solutions, the solution constructed
using von Zeipel’s method approximately takes into account, in the secular part of the perturbing function, the totality of
fourth order with respect to the small parameter m of the ratio of the mean motions of the primary planet and the satellite. This enables us to describe more accurately the
evolution of satellite orbits with large apocentric distances, which in the course of evolution may exceed the halved radius
of the Hill sphere of the planet with respect to the Sun. Among these are the orbits of the two outermost Neptunian satellites
N10 (Psamathe) and N13 (Neso). For these satellites, the parameter m amounts to 0.152 and 0.165, respectively. Different from a purely analytical solution, the proposed solution requires preliminary
calculations for each satellite. More precisely, in doing so, we need to construct some simple functions to approximate more
complex ones. This is why we use the phrase “constructive analytical.” To illustrate the solution, we compare it with the
results of the numerical integration of the strict motion equations of the satellites N10 and N13 over time intervals 5–15
thousand years. 相似文献
11.
A. G. Mavraganis 《Astrophysics and Space Science》1988,146(1):163-168
We study the existence of three-dimensional symmetric orbits in a magnetic-binary system. We point out that only two kinds of such orbits exist, depending on the orientation of both magnetic momentsM
i,i=1, 2; one with respect to the plane,y=0 and one with respect to thex-axis of the rotating-coordinate system. 相似文献
12.
One of the possible early states of the Earth-Moon system was a system of several large satellites around the Earth. The dynamical
evolution of coplanar three-body systems is studied; a planet (Earth) and two massive satellites (proto-moons) with geocentric
orbits of slightly different radii. Such configurations may arise in multiple satellite systems receding from a planet due
to tidal friction. The numerical integration of the equations of motion shows that initially circular Keplerian orbits are
soon transformed into disturbed elliptic orbits which are intersecting. The life-time of such a coplanar system between two
probable physical collisions of satellites is roughly from one day to one year for satellite systems with radii less than
20R⊕, and may reach 100 yr for three-dimensional systems. This time-scale is short in comparison with the duration of the removal
of satellites due to tides raised on the planet, which is estimated as 106–108 yr for the same orbital dimensions. Therefore, the life-time of a system of several proto-moons is mainly determined by their
tidal interactions with the Earth. For conditions which we have considered, the most probable result of the evolution was
coalescence of satellites as the consequence of the collisions. 相似文献
13.
G. A. Krasinsky 《Celestial Mechanics and Dynamical Astronomy》2002,84(1):27-55
Differential equations describing the tidal evolution of the earth's rotation and of the lunar orbital motion are presented in a simple close form. The equations differ in form for orbits fixed to the terrestrial equator and for orbits with the nodes precessing along the ecliptic due to solar perturbations. Analytical considerations show that if the contemporary lunar orbit were equatorial the evolution would develop from an unstable geosynchronous orbit of the period about 4.42 h (in the past) to a stable geosynchronous orbit of the period about 44.8 days (in the future). It is also demonstrated that at the contemporary epoch the orbital plane of the fictitious equatorial moon would be unstable in the Liapunov's sense, being asymptotically stable at early stages of the evolution. Evolution of the currently near-ecliptical lunar orbit and of the terrestrial rotation is traced backward in time by numerical integration of the evolutional equations. It is confirmed that about 1.8 billion years ago a critical phase of the evolution took place when the equatorial inclination of the moon reached small values and the moon was in a near vicinity of the earth. Before the critical epoch t
cr
two types of the evolution are possible, which at present cannot be unambiguously distinguished with the help of the purely dynamical considerations. In the scenario that seems to be the most realistic from the physical point of view, the evolution also has started from a geosynchronous equatorial lunar orbit of the period 4.19 h. At t < t
cr
the lunar orbit has been fixed to the precessing terrestrial equator by strong perturbations from the earth's flattening and by tidal effects; at the critical epoch the solar perturbations begin to dominate and transfer the moon to its contemporary near-ecliptical orbit which evolves now to the stable geosynchronous state. Probably this scenario is in favour of the Darwin's hypothesis about originating the moon by its separation from the earth. Too much short time scale of the evolution in this model might be enlarged if the dissipative Q factor had somewhat larger values in the past than in the present epoch. Values of the length of day and the length of month, estimated from paleontological data, are confronted with the results of the developed model. 相似文献
14.
Bruce C. Douglas Chreston F. Martin Ronald G. Williamson Carl A. Wagner 《Celestial Mechanics and Dynamical Astronomy》1969,1(2):252-270
An error analysis of resonant orbits for geodesy indicates that attempts to use resonance to recover high order geopotential coefficients may be seriously hampered by errors in the geopotential. This effect, plus the very high correlations (up to .99) of the resonant coefficients with each other and the orbital period in single satellite solutions, makesindividual resonant orbits of limited value for geodesy. Multiple-satellite, single-plane solutions are only a slight improvement over the single satellite case. Accurate determination of high order coefficients from low altitude resonant satellites requires multiple orbit planes and small drift-periods to reduce correlations and effects of errors of non-resonant geopotential terms. Also, the effects of gravity model errors on low-altitude resonant satellites make the use of tracking arcs exceeding two to three weeks of doubtful validity. Because high-altitude resonant orbits are less affected by non-resonant terms in the geopotential, much longer tracking arcs can be used for them. 相似文献
15.
The existing explanations for the asteroid distribution in the main belt (between the orbits of Mars and Jupiter) are based
on numerical integration of resonance orbits in models with more than two degrees of freedom. We suggest an approach based
on the investigation of the families of periodic solutions of the planar circular restricted three-body problem, i.e., a model
with two degrees of freedom. This work shows that (a) the distribution of asteroids near the (p + 1)/p resonances and position of the outer boundary of the main asteroid belt can be explained within the planar circular restricted
three-body problem and (b) this problem does not explain the asteroid distribution near other resonances. 相似文献
16.
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
17.
Spatial Structure of Regular and Chaotic Orbits in Self-Consistent Models of Galactic Satellites 总被引:2,自引:2,他引:0
In several previous papers we had investigated the orbits of the stars that make up galactic satellites, finding that many
of them were chaotic. Most of the models studied in those works were not self-consistent, the single exception being the Heggie
and Ramamani (1995) models; nevertheless, these ones are built from a distribution function that depends on the energy (actually,
the Jacobi integral) only, what makes them rather special. Here we built up two self-consistent models of galactic satellites,
freezed theirs potential in order to have smooth and stationary fields, and investigated the spatial structure of orbits whose
initial positions and velocities were those of the bodies in the self-consistent models. We distinguished between partially chaotic (only one non-zero Lyapunov exponent) and fully chaotic (two non-zero Lyapunov exponents) orbits and showed that, as could be expected from the fact that the former obey an additional
local isolating integral, besides the global Jacobi integral, they have different spatial distributions. Moreover, since Lyapunov
exponents are computed over finite time intervals, their values reflect the properties of the part of the chaotic sea they
are navigating during those intervals and, as a result, when the chaotic orbits are separated in groups of low- and high-valued
exponents, significant differences can also be recognized between their spatial distributions. The structure of the satellites
can, therefore, be understood as a superposition of several separate subsystems, with different degrees of concentration and
trixiality, that can be recognized from the analysis of the Lyapunov exponents of their orbits. 相似文献
18.
Yu. D. Medvedev 《Solar System Research》2011,45(5):386-391
Observations at the first opposition are used to determine the orbits of 16 near-Earth asteroids with two or more observed
oppositions. The orbits are improved by the differential method. This paper considers two modifications of the improvement
technique, which are compared to the classical method based on the principle of the least square method (LSM). The first modification
uses the principle of least absolute deviations (LAD). In the second modification, the differences O - C (between the observed and calculated positions) are transformed to fit into a new coordinate system whereby the axes go parallel
and perpendicular to the asteroid’s apparent path (the modified differential method (MDM)). The orbits determined from one
opposition by the classical LSM, LAD, and MDM are compared to a more accurate orbit calculated by the LSM from all the available
oppositions. The calculations show that in 13 cases out of 16, the asteroid orbits calculated by LAD are more accurate than
those calculated by the classical LSM. The improvement by the modified differential method, which includes the O - C transformation, does not produce any perceptible increase in accuracy when compared to the orbits calculated by the classical
method. 相似文献
19.
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
20.
B. Barbanis 《Celestial Mechanics and Dynamical Astronomy》1990,48(1):57-77
We investigate the escape regions of a quartic potential and the main types of irregular periodic orbits. Because of the symmetry of the model the zero velocity curve consists of four summetric arcs forming four open channels around the lines y = ± x through which an orbit can escape. Four unstable Lyapunov periodic orbits bridge these openings.We have found an infinite sequence of families of periodic orbits which is the outer boundary of one of the escape regions and several infinite sequences of periodic orbits inside this region that tend to homoclinic and heteroclinic orbits. Some of these sequences of periodic orbits tend to homoclinic orbits starting perpendicularly and ending asymptotically at the x-axis. The other sequences tend to heteroclinic orbits which intersect the x-axis perpendicularly for x > 0 and make infinite oscillations almost parallel to each of the two Lyapunov orbits which correspond to x > 0 or x < 0. 相似文献