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1.
This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the
luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within
the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points
are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the
presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures
strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It
is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L
i
(i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable. 相似文献
2.
R. K. Das A. K. Shrivastava B. Ishwar 《Celestial Mechanics and Dynamical Astronomy》1988,45(4):387-393
The differential equations of motion of the elliptic restricted problem of three bodies with decreasing mass are derived. The mass of the infinitesimal body varies with time. We have applied Jeans' law and the space-time transformation of Meshcherskii. In this problem the space-time transformation is applicable only in the special case whenn=1,k=0,q=1/2. We have applied Nechvile's transformation for the elliptic problem. We find that the equations of motion of our problem differ from that of constant mass only by a small perturbing force. 相似文献
3.
In a recent paper, published in Astrophys. Space Sci. (337:107, 2012) (hereafter paper ZZX) and entitled “On the triangular libration points in photogravitational restricted three-body problem
with variable mass”, the authors study the location and stability of the generalized Lagrange libration points L
4 and L
5. However their study is flawed in two aspects. First they fail to write correctly the equations of motion of the variable
mass problem. Second they attribute a variable mass to the third body of the restricted three-body model, a fact that is not
compatible with the assumptions used in deriving the mathematical formulation of this model. 相似文献
4.
This paper examines the effect of a constant κ of a particular integral of the Gylden-Meshcherskii problem on the stability of the triangular points in the restricted three-body
problem under the influence of small perturbations in the Coriolis and centrifugal forces, together with the effects of radiation
pressure of the bigger primary, when the masses of the primaries vary in accordance with the unified Meshcherskii law. The
triangular points of the autonomized system are found to be conditionally stable due to κ. We observed further that the stabilizing or destabilizing tendency of the Coriolis and centrifugal forces is controlled
by κ, while the destabilizing effects of the radiation pressure remain unchanged but can be made strong or weak due to κ. The condition that the region of stability is increasing, decreasing or does not exist depend on this constant. The motion
around the triangular points L
4,5 varying with time is studied using the Lyapunov Characteristic Numbers, and are found to be generally unstable. 相似文献
5.
S. M. El-Shaboury 《Astrophysics and Space Science》1990,174(2):291-296
The differential equations of motion of the elliptic restricted problem of three bodies, an infinitesimal spherical body with decreasing mass and two tri-axial bodies are derived. We have applied Jeans's law and the space-time of Meshcherskii in the special case whenn=1,k=0,q=1/2. Also Nechvíle's transformation for the elliptic problem be applied for this case. 相似文献
6.
Trojan type orbits in the system of two gravitational centers with variable separation are studied within the framework of the restricted problem of three bodies. The backward numerical integration of the equations of motion of the bodies starting in the triangular libration pointsL
4 andL
5 (reverse problem) finds the breakdown of librations as the separation decreases because of the mass gain of the smaller component and an approach of the body of negligible, mass to the latter up to the distance below its sphere of action with a relative velocity approximately equal to the escape one on this sphere. The breakdown of librations aboutL
5 occurs under the mass gain of the smaller component considerably larger than in the case ofL
4 and implications are made for the asymmetry of the number of librators aboutL
4 andL
5 in the solar system (Greeks and Trojans). Other parameters of the libration motion near 1/1 commensurability are obtained, namely, the asymmetry of the libration amplitudes about the triangular points as well as the values of periods and amplitudes within the limits of those for real Trojan asteroids. Trojans could be supposedly, formed inside the Proto-jupiter and escape during its intensive mass loss. 相似文献
7.
The existence and stability of triangular libration points in the relativistic restricted three-body problem has been studied.
It is found that L4,5 are unstable in the whole range 0 ≤ μ ≤ 1/2 in contrast to the classical restricted three-body problem where they are stable
for 0 < μ < μ0, where μ is the mass parameter and μ0 = 0.03852....
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
8.
The existence and stability of a test particle around the equilibrium points in the restricted three-body problem is generalized
to include the effect of variations in oblateness of the first primary, small perturbations ϵ and ϵ′ given in the Coriolis and centrifugal forces α and β respectively, and radiation pressure of the second primary; in the case when the primaries vary their masses with time in
accordance with the combined Meshcherskii law. For the autonomized system, we use a numerical evidence to compute the positions
of the collinear points L
2κ
, which exist for 0<κ<∞, where κ is a constant of a particular integral of the Gylden-Meshcherskii problem; oblateness of the first primary; radiation pressure
of the second primary; the mass parameter ν and small perturbation in the centrifugal force. Real out of plane equilibrium points exist only for κ>1, provided the abscissae
x < \fracn(k-1)b\xi<\frac{\nu(\kappa-1)}{\beta}. In the case of the triangular points, it is seen that these points exist for ϵ′<κ<∞ and are affected by the oblateness term, radiation pressure and the mass parameter. The linear stability of these equilibrium
points is examined. It is seen that the collinear points L
2κ
are stable for very small κ and the involved parameters, while the out of plane equilibrium points are unstable. The conditional stability of the triangular
points depends on all the system parameters. Further, it is seen in the case of the triangular points, that the stabilizing
or destabilizing behavior of the oblateness coefficient is controlled by κ, while those of the small perturbations depends on κ and whether these perturbations are positive or negative. However, the destabilizing behavior of the radiation pressure remains
unaltered but grows weak or strong with increase or decrease in κ. This study reveals that oblateness coefficient can exhibit a stabilizing tendency in a certain range of κ, as against the findings of the RTBP with constant masses. Interestingly, in the region of stable motion, these parameters
are void for
k = \frac43\kappa=\frac{4}{3}. The decrease, increase or non existence in the region of stability of the triangular points depends on κ, oblateness of the first primary, small perturbations and the radiation pressure of the second body, as it is seen that the
increasing region of stability becomes decreasing, while the decreasing region becomes increasing due to the inclusion of
oblateness of the first primary. 相似文献
9.
M. K. Ammar 《Astrophysics and Space Science》2008,313(4):393-408
In this paper the effect of solar radiation pressure on the location and stability of the five Lagrangian points is studied,
within the frame of elliptic restricted three-body problem, where the primaries are the Sun and Jupiter acting on a particle
of negligible mass. We found that the radiation pressure plays the rule of slightly reducing the effective mass of the Sun
and changes the location of the Lagrangian points. New formulas for the location of the collinear libration points were derived.
For large values of the force ratio β, we found that at β=0.12, the collinear point L3 is stable and some families of periodic orbits can be drawn around it. 相似文献
10.
We study the equilibrium points and the zero-velocity curves of Chermnykh’s problem when the angular velocity ω varies continuously and the value of the mass parameter is fixed. The planar symmetric simple-periodic orbits are determined numerically and they are presented for three values of the parameter ω. The stability of the periodic orbits of all the families is computed. Particularly, we explore the network of the families when the angular velocity has the critical value ω = 2√2 at which the triangular equilibria disappear by coalescing with the collinear equilibrium point L1. The analytic determination of the initial conditions of the family which emanate from the Lagrangian libration point L1 in this case, is given. Non-periodic orbits, as points on a surface of section, providing an outlook of the stability regions, chaotic and escape motions as well as multiple-periodic orbits, are also computed. Non-linear stability zones of the triangular Lagrangian points are computed numerically for the Earth–Moon and Sun–Jupiter mass distribution when the angular velocity varies. 相似文献
11.
The configuration space around the triangular libration points in the Earth-Moon system is partitioned according to the stability of the motion. The regions aroundL
4 andL
5 are established where particles placed with zero initial velocity will librate. The complexity of the partitioning is revealed. 相似文献
12.
In this paper we have deduced the differential equations of motion of the restricted problem of three bodies with decreasing mass, under the assumption that the mass of the satellite varies with respect to time. We have applied Jeans law and the space time transformation contrast to the transformation of Meshcherskii. The space time transformation is applicable only in the special casen=1,k=0,q=1/2. The equations of motion of our problem differ from the equations of motion of the restricted three body problem with constant mass only by small perturbing forces. 相似文献
13.
In this paper, we study the existence of libration points and their linear stability when the three participating bodies are axisymmetric and the primaries are radiating, we found that the collinear points remain unstable, it is further seen that the triangular points are stable for 0<μ<μ c , and unstable for where , it is also observed that for these points the range of stability will decrease. In addition to this we have studied periodic orbits around these points in the range 0<μ<μ c , we found that these orbits are elliptical; the frequencies of long and short orbits of the periodic motion are affected by the terms which involve parameters that characterize the oblateness and radiation repulsive forces. The implication is that the period of long periodic orbits adjusts with the change in its frequency while the period of short periodic orbit will decrease. 相似文献
14.
F. Mignard 《Celestial Mechanics and Dynamical Astronomy》1984,34(1-4):275-287
The restricted three-body problem is generalized with the inclusion of solar radiation pressure. For small particles (typically 1 m to 1 mm) the familiar equilibrium triangular points L4 and L5 no longer exist. However libration orbits are not completely destroyed, although an effect of resonance causes their amplitude to be very large, for a particle initially at rest at either of the triangular point. Finally the results of a study of the linearized equations of motion, supplemented by a numerical integration, rule out the possibility of an accumulation of dust at the Earth-Moon lagrangian triangular points. 相似文献
15.
T. A. Heppenheimer 《Celestial Mechanics and Dynamical Astronomy》1973,7(2):177-194
Out-of-plane motion about libration points is studied within the framework of the elliptic restricted three-body problem. Nonlinear motion in the circular restricted problem is given to third order in the out-of-plane amplitudeA
z by Jacobi elliptic functions. Linear motion in the elliptic problem is studied using Mathieu's and Hill's equations. Additional terms needed for a complete third-order theory are found using Lindsted's method. This theory is constructed for the case of collinear libration points; for the case of triangular points, a third-order nonlinear solution is given separately in terms of Jacobi elliptic functions. 相似文献
16.
Due to various perturbations, the collinear libration points of the real Earth–Moon system are not equilibrium points anymore.
Under the assumption that the Moon’s motion is quasi-periodic, special quasi-periodic orbits called dynamical substitutes
exist. These dynamical substitutes replace the geometrical collinear libration points as time-varying equilibrium points.
In the paper, the dynamical substitutes of the three collinear libration points in the real Earth–Moon system are computed.
For the points L
1 and L
2, linearized motions around the dynamical substitutes are described, and the variational equations of the dynamical substitutes
are reduced to a form with a near constant coefficient matrix. Then higher order analytical formulae of the central manifolds
are constructed. Using these analytical solutions as initial seeds, Lissajous orbits and halo orbits are computed with numerical
algorithms. 相似文献
17.
Bálint Érdi 《Celestial Mechanics and Dynamical Astronomy》1978,18(2):141-161
The problem is considered within the framework of the elliptic restricted three-body problem. The asymptotic solution is derived by a three-variable expansion procedure. The variables of the expansion represent three time-scales of the asteroids: the revolution around the Sun, the libration around the triangular Lagrangian pointsL
4,L
5, and the motion of the perihelion. The solution is obtained completely in the first order and partly in the second order. The results are given in explicit form for the coordinates as functions of the true anomaly of Jupiter. As an example for the perturbations of the orbital elements the main perturbations of the eccentricity, the perihelion longitude and the longitude of the ascending node are given. Conditions for the libration of the perihelion are also discussed. 相似文献
18.
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
19.
We have two mass points of equal masses m
1=m
2 > 0 moving under Newton’s law of attraction in a non-collision parabolic orbit while their center of mass is at rest. We
consider a third mass point, of mass m
3=0, moving on the straight line L perpendicular to the plane of motion of the first two mass points and passing through their
center of mass. Since m
3=0, the motion of m
1 and m
2 is not affected by the third and from the symmetry of the motion it is clear that m
3 will remain on the line L. The parabolic restricted three-body problem describes the motion of m
3. Our main result is the characterization of the global flow of this problem. 相似文献
20.
We study the effect of eccentricity and inclination on small amplitude librations around the equilibrium points L
4 and L
5 in the circular restricted three-body problem. We show that the effective libration centres can be displaced appreciably
from the equilateral configuration. The secular extrema of the eccentricity as a function of the argument of pericentre are
shifted by ∼25 °
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献