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1.
This paper analyzes Robe??s circular restricted three-body problem when the hydrostatic equilibrium figure of the first primary is assumed to be an oblate spheroid, the shape of the second primary is considered as a triaxial rigid body, and the full buoyancy force of the fluid is taken into account. It is found that there is an equilibrium point near the center of the first primary, another equilibrium point exists on the line joining the centers of the primaries and there exist infinite number of equilibrium points on an ellipse in the orbital plane of the second primary. It is also observed that under certain conditions, all these equilibrium points can be stable. The most interesting and distinguishable results of this study are the existence of elliptical points and their stability. 相似文献
2.
This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid, and the shape of the second primary is also an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable. 相似文献
3.
The existence of all the equilibrium points, their location and stability in the Robe's (1977) restricted three—body problem have been studied. It is seen that the center of the first primary is always an equilibrium point, whatever be the values of the density parameter K, eccentricity parameter eand mass parameter . The other equilibrium points exist only when K 0 and e = 0 that is when the second primary, a mass point, moves around the first, a spherical shell filled with fluid, in a circular orbit. When K > 1, there is one additional equilibrium point lying on the line joining the center of the first primary and the second primary. When K + = 1, there are infinite number of equilibrium points in the x—yplane lying on a circle of radius one and center as the second primary, provided the points are inside the spherical shell. When K < 0 and K + > 0, there are two more equilibrium points lying in the x—z plane forming triangles with the center of the shell and the second primary. Results of the stability of the equilibrium point (–,0,0), center of the first primary are the same as those given by Robe (1977). Circular points and triangular points are always unstable. The equilibrium point collinear with the center of the shell and the second primary is stable provided and Ksatisfy the inequality
.Thus, Robe's elliptic restricted three-body problem has only one equilibrium point for all values of the parameters Kand and Robes circular restricted three-body problem can have two, three or infinite number of equilibrium points depending upon the values of Kand . This is contrary to the classical-restricted problem where there are five equilibrium points, which are finite in number. Further, only the points collinear with the center of the shell and the second primary are stable in the Robe's problem where as in the classcial problem collinear points are unstable and triangular points are stable.This revised version was published online in October 2005 with corrections to the Cover Date. 相似文献
4.
This paper examines the existence and stability of the out-of-plane equilibrium points of a third body of infinitesimal mass when the equations of motion are written in the three dimensional form under the set up of the Robe’s circular restricted three-body problem, in which the hydrostatic equilibrium figure of the first primary is an oblate spheroid and the second one is a triaxial rigid body under the full buoyancy force of the fluid. The existence of the out of orbital plane equilibrium points lying on the xz-plane is noticed. These points are however unstable in the linear sense. 相似文献
5.
《New Astronomy》2015
This paper examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations are given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. The linear stability of this configuration is examined; it is observed that the first axial point is unstable while the second one is conditionally stable and the circular points are unstable. 相似文献
6.
Stability of generalized elliptic restricted four body problem with radiation and oblateness effects
《New Astronomy》2020
This paper studies the existence and stability of non-collinear equilibrium points in the elliptic restricted four body problem with bigger primary as a source of radiation and other two primaries having equal masses as oblate spheroid. In the elliptic restricted four body problem, three of the bodies are moving in elliptical orbit around their common centre of mass fixed at the origin of the coordinate system, while the fourth one is infinitesimal. Three pairs of non-collinear points are obtained symmetric with respect to x-axis. We found the equilibrium points are stable in linear sense. We also investigate the pulsating zero velocity surfaces and basin of attraction for varying value of oblateness coefficient and radiation pressure parameter. 相似文献
7.
Peter J. Brown Alice A. Breeveld Stephen Holland Paul Kuin Tyler Pritchard 《Astrophysics and Space Science》2014,353(1):89-96
This paper studies the motion of an infinitesimal mass in the framework of Robe’s circular restricted three-body problem in two cases; the first case is when the hydrostatic equilibrium figure of the first primary is an oblate spheroid, the shape of the second primary is considered as an oblate spheroid with oblateness coefficients up to the second zonal harmonic, while the first primary is a Roche ellipsoid in the second case and the full buoyancy of the fluid is taken into account. In case one; it is observed that there are two axial libration points on the line joining the centres of the primaries, points on the circle within the first primary are also libration points under certain conditions. It is further found that the first axial point is stable, while the second one is conditionally stable, and the circular points are unstable. It is found in case two that there is exist only one libration point (0,0,0) this point is stable. 相似文献
8.
The nonlinear stability of triangular equilibrium points has been discussed in the generalised photogravitational restricted
three body problem with Poynting-Robertson drag. The problem is generalised in the sense that smaller primary is supposed
to be an oblate spheroid. The bigger primary is considered as radiating. We have performed first and second order normalization
of the Hamiltonian of the problem. We have applied KAM theorem to examine the condition of non-linear stability. We have found
three critical mass ratios. Finally we conclude that triangular points are stable in the nonlinear sense except three critical
mass ratios at which KAM theorem fails. 相似文献
9.
L. M. Perko 《Celestial Mechanics and Dynamical Astronomy》1983,30(2):115-132
This paper establishes the existence and first order perturbation approximation of an infinite number of one-parameter families of symmetric periodic solutions of the restricted three body problem that are analytic continuations of symmetric periodic solutions of Hill's problem for small values of the mass ratio μ>0. 相似文献
10.
In this paper, the restricted problem of three bodies is generalized to include a case when the passively gravitating test particle is an oblate spheroid under effect of small perturbations in the Coriolis and centrifugal forces when the first primary is a source of radiation and the second one an oblate spheroid, coupled with the influence of the gravitational potential from the belt. The equilibrium points are found and it is seen that, in addition to the usual three collinear equilibrium points, there appear two new ones due to the potential from the belt and the mass ratio. Two triangular equilibrium points exist. These equilibria are affected by radiation of the first primary, small perturbation in the centrifugal force, oblateness of both the test particle and second primary and the effect arising from the mass of the belt. The linear stability of the equilibrium points is explored and the stability outcome of the collinear equilibrium points remains unstable. In the case of the triangular points, motion is stable with respect to some conditions which depend on the critical mass parameter; influenced by the small perturbations, radiating effect of the first primary, oblateness of the test body and second primary and the gravitational potential from the belt. The effects of each of the imposed free parameters are analyzed. The potential from the belt and small perturbation in the Coriolis force are stabilizing parameters while radiation, small perturbation in the centrifugal force and oblateness reduce the stable regions. The overall effect is that the region of stable motion increases under the combine action of these parameters. We have also found the frequencies of the long and short periodic motion around stable triangular points. Illustrative numerical exploration is rendered in the Sun–Jupiter and Sun–Earth systems where we show that in reality, for some values of the system parameters, the additional equilibrium points do not in general exist even when there is a belt to interact with. 相似文献
11.
This paper presents a generalized problem of the restricted three body studied in Abdul Raheem and Singh with the inclusion that the third body is an oblate spheroidal test particle of infinitesimally mass. The positions and stability of the equilibrium point of this problem is studied for a model in which the primaries is the binary system Struve 2398 (Gliese 725) in the constellation Draco; which consist of a pair of radiating oblate stars. It is seen that additional equilibrium points exist on the line collinear with the primaries, for some combined parameters of the problem. Hence, there can be up to five collinear equilibrium points. Two triangular points exist and depends on the oblateness of the participating bodies, radiation pressure of the primaries and a small perturbation in the centrifugal force. The stability analysis ensures that, the collinear equilibrium points are unstable in the linear sense while the triangular points are stable under certain conditions. Illustrative numerical exploration is given to indicate significant improvement of the problem in Abdul Raheem and Singh. 相似文献
12.
Generoso Aliasi Giovanni Mengali Alessandro A. Quarta 《Celestial Mechanics and Dynamical Astronomy》2012,114(1-2):181-200
Different types of propulsion systems with continuous and purely radial thrust, whose modulus depends on the distance from a massive body, may be conveniently described within a single mathematical model by means of the concept of generalized sail. This paper discusses the existence and stability of artificial equilibrium points maintained by a generalized sail within an elliptic restricted three-body problem. Similar to the classical case in the absence of thrust, a generalized sail guarantees the existence of equilibrium points belonging only to the orbital plane of the two primaries. The geometrical loci of existing artificial equilibrium points are shown to coincide with those obtained for the circular three body problem when a non-uniformly rotating and pulsating coordinate system is chosen to describe the spacecraft motion. However, the generalized sail has to provide a periodically variable acceleration to maintain a given artificial equilibrium point. A linear stability analysis of the artificial equilibrium points is provided by means of the Floquet theory. 相似文献
13.
W. J. Robinson 《Celestial Mechanics and Dynamical Astronomy》1974,10(1):17-33
This paper is based on the restricted problem of three bodies with the unusual feature that the lightest particle is replaced by a rigid body. The attitude stability of the body is considered when its centre of mass is located at one of the equilibrium points. The stable attitude is determined when the satellite is stationary relative to the primaries. It is found that for some bodies there are two such attitudes, and these are determined. 相似文献
14.
《New Astronomy》2024
We numerically study a version of the synchronous circular restricted three-body problem, where an infinitesimal mass body is moving under the Newtonian gravitational forces of two massive bodies. The primary body is an oblate spheroid while the secondary is an elongated asteroid of a combination of two equal masses forming a rotating dipole which is synchronous to the rotation of the primaries of the classic circular restricted three-body problem. In this paper, we systematically examine the existence, positions, and linear stability of the equilibrium points for various combinations of the model's parameters. We observe that the perturbing forces have significant effects on the positions and stability of the equilibrium points as well as the regions where the motion of the particle is allowed. The allowed regions of motion as determined by the zero-velocity surface and the corresponding isoenergetic curves as well as the positions of the equilibrium points are given. Finally, we numerically study the binary system Luhman-16 by computing the positions of the equilibria and their stability as well as the allowed regions of motion of the particle. The corresponding families of periodic orbits emanating from the collinear equilibrium points are computed along with their stability properties. 相似文献
15.
Galaxies with supermassive binary black holes: (I) a possible model for the centers of core galaxies
The dynamics of galactic systems with central binary black holes is studied. The model is a modification from the restricted three body problem, in which a galactic potential is added as an external potential. Considering the case with an equal mass binary black holes, the conditions of existence of equilibrium points, including Lagrange Points and additional new equilibrium points, i.e. Jiang-Yeh Points, are investigated. A critical mass is discovered to be fundamentally important. That is, Jiang-Yeh Points exist if and only if the galactic mass is larger than the critical mass. The stability analysis is performed for all equilibrium points. The results that Jiang-Yeh Points are unstable could lead to the core formation in the centers of galaxies. 相似文献
16.
17.
In this paper we have found secular solutions at the triangular equilibrium point in the generalized photogravitational restricted
three body problem. The problem is generalised in the sense that smaller primary is an oblate spheroid and more massive primary
as source of radiation. The triangular point has long or short-period retrograde elliptical orbits. The critical mass parameter
decreases with the increase in oblateness and radiation pressure.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
18.
《New Astronomy》2016
This article examines the effects of the zonal harmonics on the out-of-plane equilibrium points of Robe's circular restricted three-body problem when the hydrostatic equilibrium shape of the first primary is an oblate spheroid, the shape of the second primary is an oblate spheroid with oblateness coefficients up to the second zonal harmonic, and the full buoyancy of the fluid is considered. It is observed that the size of the oblateness and the zonal harmonics affect the positions of the out-of-plane equilibrium points L6 and L7. It is also observed that these points within the possible region of motion are unstable. 相似文献
19.
We present a numerical study of the set of orbits of the planar circular restricted three body problem which undergo consecutive
close encounters with the small primary, or orbits of second species. The value of the Jacobi constant is fixed, and we restrict
the study to consecutive close encounters which occur within a maximal time interval. With these restrictions, the full set
of orbits of second species is found numerically from the intersections of the stable and unstable manifolds of the collision
singularity on the surface of section that corresponds to passage through the pericentre. A ‘skeleton’ of this set of curves
can be computed from the solutions of the two-body problem. The set of intersection points found in this limit corresponds
to the S-arcs and T-arcs of Hénon’s classification which verify the energy and time constraints, and can be used to construct
an alphabet to describe the orbits of second species. We give numerical evidence for the existence of a shift on this alphabet
that describes all the orbits with infinitely many close encounters with the small primary, and sketch a proof of the symbolic
dynamics. In particular, we find periodic orbits that combine S-type and T-type quasi-homoclinic arcs. 相似文献
20.
We study numerically the photogravitational version of the problem of four bodies, where an infinitesimal particle is moving under the Newtonian gravitational attraction of three bodies which are finite, moving in circles around their center of mass fixed at the origin of the coordinate system, according to the solution of Lagrange where they are always at the vertices of an equilateral triangle. The fourth body does not affect the motion of the three bodies (primaries). We consider that the primary body m 1 is dominant and is a source of radiation while the other two small primaries m 2 and m 3 are equal. In this case (photogravitational) we examine the linear stability of the Lagrange triangle solution. 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 on the orbital plane are given. The existence and the number of the collinear and the non-collinear equilibrium points of the problem depends on the mass parameters of the primaries and the radiation factor q 1. Critical masses m 3 and radiation q 1 associated with the existence and the number of the equilibrium points are given. The stability of the relative equilibrium solutions in all cases are also studied. In the last section we investigate the existence and location of the out of orbital plane equilibrium points of the problem. We found that such critical points exist. These points lie in the (x,z) plane in symmetrical positions with respect to (x,y) plane. The stability of these points are also examined. 相似文献