共查询到20条相似文献,搜索用时 46 毫秒
1.
The vertical stability character of the families of short and long period solutions around the triangular equilibrium points
of the restricted three-body problem is examined. For three values of the mass parameter less than equal to the critical value
of Routh (μ
R
) i.e. for μ = 0.000953875 (Sun-Jupiter), μ = 0.01215 (Earth-Moon) and μ = μ
R
= 0.038521, it is found that all such solutions are vertically stable. For μ > (μ
R
) vertical stability is studied for a number of ‘limiting’ orbits extended to μ = 0.45. The last limiting orbit computed by
Deprit for μ = 0.044 is continued to a family of periodic orbits into which the well known families of long and short period
solutions merge. The stability characteristics of this family are also studied. 相似文献
2.
Elbaz I. Abouelmagd 《Astrophysics and Space Science》2012,342(1):45-53
In this paper, we prove that the locations of the triangular points and their linear stability are affected by the oblateness of the more massive primary in the planar circular restricted three-body problem, considering the effect of oblateness for J 2 and J 4. After that, we show that the triangular points are stable for 0<μ<μ c and unstable when , where μ c is the critical mass parameter which depends on the coefficients of oblateness. On the other hand, we produce some numerical values for the positions of the triangular points, μ and μ c using planets systems in our solar system which emphasis that the range of stability will decrease; however this range sometimes is not affected by the existence of J 4 for some planets systems as in Earth–Moon, Saturn–Phoebe and Uranus–Caliban systems. 相似文献
3.
This paper deals with the existence of libration points and their linear stability when the more massive primary is radiating and the smaller is an oblate spheroid. Our study includes the effects of oblateness of $\bar{J}_{2i}$ (i=1,2) with respect to the smaller primary in the restricted three-body problem. Under combining the perturbed forces that were mentioned before, the collinear points remain unstable and the triangular points are stable for 0<μ<μ c , and unstable in the range $\mu_{c} \le\mu\le\frac{1}{2}$ , where $\mu_{c} \in(0,\frac{1}{2})$ , it is also observed that for these points the range of stability will decrease. The relations for periodic orbits around five libration points with their semimajor, semiminor axes, eccentricities, the frequencies of orbits and periods are found, furthermore for the orbits around the triangular points the orientation and the coefficients of long and short periodic terms also are found in the range 0<μ<μ c . 相似文献
4.
This paper studies the existence and stability of equilibrium points under the influence of small perturbations in the Coriolis
and the centrifugal forces, together with the non-sphericity of the primaries. The problem is generalized in the sense that
the bigger and smaller primaries are respectively triaxial and oblate spheroidal bodies. It is found that the locations of
equilibrium points are affected by the non-sphericity of the bodies and the change in the centrifugal force. It is also seen
that the triangular points are stable for 0<μ<μ
c
and unstable for
mc £ m < \frac12\mu_{c}\le\mu <\frac{1}{2}, where μ
c
is the critical mass parameter depending on the above perturbations, triaxiality and oblateness. It is further observed that
collinear points remain unstable. 相似文献
5.
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
6.
K. E. Papadakis 《Astrophysics and Space Science》2009,323(3):261-272
In this paper we study the asymptotic solutions of the (N+1)-body ring planar problem, N of which are finite and ν=N−1 are moving in circular orbits around their center of masses, while the Nth+1 body is infinitesimal. ν of the primaries have equal masses m and the Nth most-massive primary, with m
0=β
m, is located at the origin of the system. We found the invariant unstable and stable manifolds around hyperbolic Lyapunov
periodic orbits, which emanate from the collinear equilibrium points L
1 and L
2. We construct numerically, from the intersection points of the appropriate Poincaré cuts, homoclinic symmetric asymptotic
orbits around these Lyapunov periodic orbits. There are families of symmetric simple-periodic orbits which contain as terminal
points asymptotic orbits which intersect the x-axis perpendicularly and tend asymptotically to equilibrium points of the problem spiraling into (and out of) these points.
All these families, for a fixed value of the mass parameter β=2, are found and presented. The eighteen (more geometrically simple) families and the corresponding eighteen terminating
homo- and heteroclinic symmetric asymptotic orbits are illustrated. The stability of these families is computed and also presented. 相似文献
7.
This paper investigates the periodic orbits around the triangular equilibrium points for 0<μ<μ c , where μ c is the critical mass value, under the combined influence of small perturbations in the Coriolis and the centrifugal forces respectively, together with the effects of oblateness and radiation pressures of the primaries. It is found that the perturbing forces affect the period, orientation and the eccentricities of the long and short periodic orbits. 相似文献
8.
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
9.
Antonis D. Pinotsis 《Celestial Mechanics and Dynamical Astronomy》2010,108(2):187-202
We studied systematically cases of the families of non-symmetric periodic orbits in the planar restricted three-body problem.
We took interesting information about the evolution, stability and termination of bifurcating families of various multiplicities.
We found that the main families of simple non-symmetric periodic orbits present a similar dynamical structure and bifurcation
pattern. As the Jacobi constant changes each branch of the characteristic of a main family spirals around a focal point-terminating
point in x- at which the Jacobi constant is C
∞ = 3 and their periodic orbits terminate at the corotation (at the Lagrangian point L4 or L5). As the family approaches asymptotically its termination point infinite changes of stability to instability and vice versa
occur along its characteristic. Thus, infinite bifurcation points appear and each one of them produces infinite inverse Feigenbaum
sequences. That is, every bifurcating family of a Feigenbaum sequence produces the same phenomenon and so on. Therefore, infinite
spiral characteristics appear and each one of them generates infinite new inner spirals and so on. Each member of these infinite
sets of the spirals reproduces a basic bifurcation pattern. Therefore, we have in general large unstable regions that generate
large chaotic regions near the corotation points L4, L5, which are unstable. As C varies along the spiral characteristic of every bifurcating family, which approaches its focal
point, infinite loops, one inside the other, surrounding the unstable triangular points L4 or L5 are formed on their orbits. So, each terminating point corresponds to an asymptotic non-symmetric periodic orbit that spirals
into the corotation points L4, L5 with infinite period. This is a new mechanism that produces very large degree of stochasticity. These conclusions help us
to comprehend better the motions around the points L4 and L5 of Lagrange. 相似文献
10.
John D. Hadjidemetriou 《Celestial Mechanics and Dynamical Astronomy》1992,53(2):151-183
Four 3 : 1 resonant families of periodic orbits of the planar elliptic restricted three-body problem, in the Sun-Jupiter-asteroid system, have been computed. These families bifurcate from known families of the circular problem, which are also presented. Two of them, I
c
, II
c
bifurcate from the unstable region of the family of periodic orbits of the first kind (circular orbits of the asteroid) and are unstable and the other two, I
e
, II
e
, from the stable resonant 3 : 1 family of periodic orbits of the second kind (elliptic orbits of the asteroid). One of them is stable and the other is unstable. All the families of periodic orbits of the circular and the elliptic problem are compared with the corresponding fixed points of the averaged model used by several authors. The coincidence is good for the fixed points of the circular averaged model and the two families of the fixed points of the elliptic model corresponding to the families I
c
, II
c
, but is poor for the families I
e
, II
e
. A simple correction term to the averaged Hamiltonian of the elliptic model is proposed in this latter case, which makes the coincidence good. This, in fact, is equivalent to the construction of a new dynamical system, very close to the original one, which is simple and whose phase space has all the basic features of the elliptic restricted three-body problem. 相似文献
11.
Families of Periodic Orbits Emanating From Homoclinic Orbits in the Restricted Problem of Three Bodies 总被引:2,自引:1,他引:1
We describe and comment the results of a numerical exploration on the evolution of the families of periodic orbits associated
with homoclinic orbits emanating from the equilateral equilibria of the restricted three body problem for values of the mass
ratio larger than μ
1. This exploration is, in some sense, a continuation of the work reported in Henrard [Celes. Mech. Dyn. Astr. 2002, 83, 291]. Indeed it shows how, for values of μ. larger than μ
1, the Trojan web described there is transformed into families of periodic orbits associated with homoclinic orbits. Also we describe how families
of periodic orbits associated with homoclinic orbits can attach (or detach) themselves to (or from) the best known families
of symmetric periodic orbits.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
Mercè Ollé Joan R. Pacha Jordi Villanueva 《Celestial Mechanics and Dynamical Astronomy》2004,90(1-2):87-107
The paper deals with different kinds of invariant motions (periodic orbits, 2D and 3D invariant tori and invariant manifolds of periodic orbits) in order to analyze the Hamiltonian direct Hopf bifurcation that
takes place close to the Lyapunov vertical family of periodic orbits of the triangular equilibrium point L4 in the 3D restricted three-body problem (RTBP) for the mass parameter, μ greater than (and close to) μR (Routh’s mass parameter). Consequences of such bifurcation, concerning the confinement of the motion close to the hyperbolic
orbits and the 3D nearby tori are also described. 相似文献
13.
We study numerically the asymmetric periodic orbits which emanate from the triangular equilibrium points of the restricted
three-body problem under the assumption that the angular velocity ω varies and for the Sun–Jupiter mass distribution. The
symmetric periodic orbits emanating from the collinear Lagrangian point L
3, which are related to them, are also examined. The analytic determination of the initial conditions of the long- and short-period
Trojan families around the equilibrium points, is given. The corresponding families were examined, for a combination of the
mass ratio and the angular velocity (case of equal eigenfrequencies), and also for the critical value ω = 2
, at which the triangular equilibria disappear by coalescing with the inner collinear equilibrium point L
1. We also compute the horizontal and the vertical stability of these families for the angular velocity parameter ω under consideration.
Series of horizontal–critical periodic orbits of the short-Trojan families with the angular velocity ω and the mass ratio
μ as parameters, are given. 相似文献
14.
Lennard F. Bakker Tiancheng Ouyang Duokui Yan Skyler Simmons 《Celestial Mechanics and Dynamical Astronomy》2011,110(3):271-290
We analytically prove the existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise
symmetric equal mass four-body problem. This is an extension of our previous proof of the analytic existence of a symmetric
periodic simultaneous binary collision orbit in a regularized planar fully symmetric equal mass four-body problem. We then
use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar
pairwise symmetric 1, m, 1, m four-body problem for m between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability
when 0.54 ≤ m ≤ 1, and are linearly unstable when 0 < m ≤ 0.53. 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 numerically the restricted five-body problem when some or all the primary bodies are sources of radiation. The allowed
regions of motion as determined by the zero-velocity surface and corresponding equipotential curves, as well as the positions
of the equilibrium points are given. We found that the number of the collinear equilibrium points of the problem depends on
the mass parameter β and the radiation factors q
i
, i=0,…,3. The stability of the equilibrium points are also studied. Critical masses associated with the number of the equilibrium
points and their stability are given. The network of the families of simple symmetric periodic orbits, vertical critical periodic
solutions and the corresponding bifurcation three-dimensional families when the mass parameter β and the radiation factors q
i
vary are illustrated. Series, with respect to the mass (and to the radiation) parameter, of critical periodic orbits are
calculated. 相似文献
17.
When μ is smaller than Routh’s critical value μ
1 = 0.03852 . . . , two planar Lyapunov families around triangular libration points exist, with the names of long and short
period families. There are periodic families which we call bridges connecting these two Lyapunov families. With μ increasing from 0 to 1, how these bridges evolve was studied. The interval (0,1) was divided into six subintervals (0, μ
5), (μ
5, μ
4), (μ
4, μ
3), (μ
3, μ
2), (μ
2, μ
1), (μ
1, 1), and in each subinterval the families B(pL, qS) were studied, along with the families B(qS, qS′). Especially in the interval (μ
2, μ
1), the conclusion that the bridges B(qS, qS′) do not exist was obtained. Connections between the short period family and the bridges B(kS, (k + 1)S) were also studied. With these studies, the structure of the web of periodic families around triangular libration points
was enriched. 相似文献
18.
The effect of the eccentricity of a planet’s orbit on the stability of the orbits of its satellites is studied. The model
used is the elliptic Hill case of the planar restricted three-body problem. The linear stability of all the known families
of periodic orbits of the problem is computed. No stable orbits are found, the majority of them possessing one or two pairs
of real eigenvalues of the monodromy matrix, while a part of a family with complex instability is found. Two families of periodic
orbits, bifurcating from the Lagrangian points L1, L2 of the corresponding circular case are found analytically. These orbits are very unstable and the determination of their
stability coefficients is not accurate, so we compute the largest Liapunov exponent in their vicinity. In all cases these
exponents are positive, indicating the existence of chaotic motions 相似文献
19.
In this paper, following the increase of the mass ratio μ, the vertical stability curves of the long and the short period families were studied, and the vertical bifurcation families
from these two families were computed. It is found that these vertical bifurcation families connect the long and short period
families with the spatial periodic family emanating from the equilateral equilibrium points. The evolution details of these
vertical bifurcation families were carefully studied and they are found to be similar to the planar bifurcation families connecting
the long period family with the short period family in the planar case. 相似文献