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1.
High-order solutions of invariant manifolds associated with libration point orbits in the elliptic restricted three-body system 总被引：1，自引：0，他引：1
Hanlun Lei Bo Xu Xiyun Hou Yisui Sun 《Celestial Mechanics and Dynamical Astronomy》2013,117(4):349-384
High-order analytical solutions of invariant manifolds, associated with Lissajous and halo orbits in the elliptic restricted three-body problem (ERTBP), are constructed in this paper. The equations of motion of ERTBP in the pulsating synodic coordinate system have five equilibrium points, and the three collinear libration points as well as the associated center manifolds are unstable. In our calculation, the general solutions of the invariant manifolds associated with Lissajous and halo orbits around collinear libration points are expressed as power series of five parameters: the orbital eccentricity, two amplitudes corresponding to the hyperbolic manifolds, and two amplitudes corresponding to the center manifolds. The analytical solutions up to arbitrary order are constructed by means of Lindstedt–Poincaré method, and then the center and invariant manifolds, transit and non-transit trajectories in ERTBP are all parameterized. Since the circular restricted three-body problem (CRTBP) is a particular case of ERTBP when the eccentricity is zero, the general solutions constructed in this paper can be reduced to describe the dynamics around the collinear libration points in CRTBP naturally. In order to check the validity of the series expansions constructed, the practical convergence of the series expansions up to different orders is studied. 相似文献
2.
It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates. 相似文献
3.
H. Alinejad 《Astrophysics and Space Science》2011,332(2):325-329
We look for particular solutions to the restricted three-body problem where the bodies are allowed to either lose or gain
mass to or from a static atmosphere. In the case that all the masses are proportional to the same function of time, we find
analogous solution to the five stationary solutions of the usual restricted problem of constant masses: the three collinear
and the two triangular solutions, but now the relative distance of the bodies changes with time at the same rate. Under some
restrictions, there are also coplanar, infinitely remote and ring solutions. 相似文献
4.
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
5.
V. S. Kalantonis C. N. Douskos E. A. Perdios 《Celestial Mechanics and Dynamical Astronomy》2006,94(2):135-153
Asymptotic motion to collinear equilibrium points of the restricted three-body problem with oblateness is considered. In particular,
homoclinic and heteroclinic solutions to these points are computed. These solutions depart asymptotically from an equilibrium
point and arrive asymptotically at the same or another equilibrium point and are important reference solutions. To compute
an asymptotic orbit, we use a fourth order local analysis, numerical integration and standard differential corrections. 相似文献
6.
In this paper, we investigate a generalization of the Hill's problem to the case where no restriction is made about the nature of the field of force perturbing two small bodies in gravitational interaction. We apply the general equations obtained to the dynamics of two bodies located in the vicinity of the triangular lagrangian points of the restricted three-body problem. 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.
Krzysztof Go&#;dziewski S&#;awomir Breiter Wojciech Borczyk 《Monthly notices of the Royal Astronomical Society》2008,383(3):989-999
We describe numerical tools for the stability analysis of extrasolar planetary systems. In particular, we consider the relative Poincaré variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best-fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best-fitting solutions are found in dynamically active region of the phase space. The long-term stability of the system is determined by a net of low-order two-body and three-body mean motion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour. 相似文献
9.
The three-body problem is the most celebrated problem of classical celestial mechanics that is not soluble in finite terms by means of any of the functions at present known to mathematical analysis.In the modern celestial mechanics is known as the main problem of the theory of the satellites and it too is not soluble in finite terms.The low-altitude satellites, which move along close orbits, are encountered. They may be done case in which the centers of masses of the bodies form an isosceles or nearly equilaterial triangle with the center of the oblate planet, and another one in which they are always located in the straight line.We study the planar problem, in which the satellites move along close orbits in a plane which forms an angle with the equatorial plane of the planet; the oblateness of which exercises a great effect. The practical importance of this problem arises from its applications.Differential equations of motion are given and particular solutions are shown to exist when the centers of masses are at the vertices of a nearly equilateral triangle or are collinear. Of course, if we take the first two terms of the Legendre series with =0, we shall obtain the same results as Aksenov (1988). 相似文献
10.
The restricted three-body problem in Schwarzschild's gravitational field is analyzed. The existen- ce of the equilibrium points in the orbital plane is discussed and the corresponding positions are established. There are three collinear libration points, and, if they exist, two triangular libration points (situated in the orbital plane of the primaries). If triangular points exist, they may not form equilateral triangles; the triangles are isosceles for equal masses of the primaries, and scalene else. 相似文献
11.
A method based on the invariace under a continuous Lie group of transformations is worked out to reduce the problem of finding solutions to the cosmological equations of Jordan and Brans-Dicke theory of gravitation for the Robertson-Walker metrics and the cases of the dust universe and the vacuum universe. The reduction consists in a first-order differential equation and a quadrature for each case. Previously known cosmological solutions are re-obtained. In particular, it becomes apparent during the development of this scheme that the flat-space solutions are indeed the general solution. 相似文献
12.
Andrzej J. Maciejewski Maria Przybylska Leon Simpson Wojciech Szumiński 《Celestial Mechanics and Dynamical Astronomy》2013,117(3):315-330
This paper discusses a constrained gravitational three-body problem with two of the point masses separated by a massless inflexible rod to form a dumbbell. This problem is a simplification of a problem of a symmetric rigid body and a point mass, and has numerous applications in Celestial Mechanics and Astrodynamics. The non-integrability of this system is proven. This was achieved thanks to an analysis of variational equations along a certain particular solution and an investigation of their differential Galois group. Nowadays this approach is the most effective tool for study integrability of Hamiltonian and non-Hamiltonian systems. 相似文献
13.
Tilemahos J. Kalvouridis 《Astrophysics and Space Science》2008,317(1-2):107-117
We deal with some new aspects of the photo-gravitational Copenhagen case of the restricted three-body problem; more particularly, the distribution and the attracting domains of the stationary solutions of small particles that move in the neighborhood of two major bodies with equal masses when one or both primaries are radiation sources with constant luminosity. Under these conditions, each particle is subjected not only to gravitational forces but to the radiation emitted from the primaries as well. 相似文献
14.
A. L. Kunitsyn 《Celestial Mechanics and Dynamical Astronomy》1974,9(4):471-481
The instability criterion of a nonlinear mechanical system neutral to the first approximation is formulated for the internal resonance case which is characterized by the existence of commensurabilities between the frequencies of the system.The criterion derived is used for determining the regions of instability of Laplace's constant triangular solutions of the unrestricted three-body problem. It is shown that in the region where necessary Routh-Joukovsky's stability conditions are satisfied there may exist eight resonanceunstable sets of the masses of the three bodies. These sets may be mechanically interpreted as follows: in the case of resonance instability the barycentre of the equilateral triangle formed by the three bodies is located on one of the eight circles constructed in the geometrical centre of this triangle. 相似文献
15.
Andrzej J. Maciejewski Maria Przybylska 《Celestial Mechanics and Dynamical Astronomy》2016,126(4):297-311
This paper discusses the dynamics of systems of point masses joined by massless rigid rods in the field of a potential force. The general form of equations of motion for such systems is obtained. The dynamics of a linear chain of mass points moving around a central body in an orbit is analysed. The non-integrability of the chain of three masses moving in a circular Kepler orbit around a central body is proven. This was achieved thanks to an analysis of variational equations along two particular solutions and an investigation of their differential Galois groups. 相似文献
16.
Douglas C. Heggie 《Celestial Mechanics and Dynamical Astronomy》1974,10(2):217-241
The work of Aarseth and Zare (1974) is extended to provide aglobal regularisation of the classical gravitational three-body problem: by transformation of the variables in a way that does not depend on the particular configuration, we obtain equations of motion which are regular with respect to collisions between any pair of particles. The only cases excepted are those in which collisions between more than one pair occur simultaneously and those in which at least one of the masses vanishes. However, by means of the same principles the restricted problem is regularised globally if collisions between the two primaries are excluded. Results of numerical tests are summarised, and the theory is generalised to provide global regularisations, first, for perturbed three-body motion and, second, for theN-body problem. A way of increasing the number of degrees of freedom of a dynamical system is central to the method, and is the subject of an Appendix. 相似文献
17.
C. N. Douskos 《Astrophysics and Space Science》2011,333(1):79-87
Paper presents a complete discussion of the existence, location and stability of the equilibrium points of the coplanar restricted
three-body problem with equal prolate and radiating primaries. Depending on the values of the radiation and negative oblateness
parameters, two or four additional collinear equilibrium points exist, in addition to the three Eulerian points of the classical
case, making up a total of up to seven collinear points. Four of these additional points, as well as the classical central
equilibrium point located at the origin, are stable for certain ranges of the parameters. Also, depending on the values of
the parameters, up to six additional non-collinear equilibrium points exist, in addition to the triangular Lagrangian points
of the classical case. Two of these additional points are located symmetrically above and below the origin and are stable,
while the other four are located symmetrically in the four quadrants and are unstable. 相似文献
18.
John D. Hadjidemetriou Th. Christides 《Celestial Mechanics and Dynamical Astronomy》1975,12(2):175-187
A periodic orbit of the restricted circular three-body problem, selected arbitrarily, is used to generate a family of periodic motions in the general three-body problem in a rotating frame of reference, by varying the massm 3 of the third body. This family is continued numerically up to a maximum value of the mass of the originally small body, which corresponds to a mass ratiom 1:m 2:m 3?5:5:3. From that point on the family continues for decreasing massesm 3 until this mass becomes again equal to zero. It turns out that this final orbit of the family is a periodic orbit of the elliptic restricted three body problem. These results indicate clearly that families of periodic motions of the three-body problem exist for fixed values of the three masses, since this continuation can be applied to all members of a family of periodic orbits of the restricted three-body problem. It is also indicated that the periodic orbits of the circular restricted problem can be linked with the periodic orbits of the elliptic three-body problem through periodic orbits of the general three-body problem. 相似文献
19.
Abimael Bengochea Manuel Falconi Ernesto Pérez-Chavela 《Astrophysics and Space Science》2011,333(2):399-408
We present some families of horseshoe periodic orbits in the general planar three-body problem for the case of two equal masses.
The considered system is a symmetric version of the one formed by Saturn, Janus and Epimetheus. We use a mass ratio equal
to 35×10−5, corresponding to 105 times the Saturn-Janus mass parameter of the restricted case; for this mass ratio the satellites have a significantly bigger
influence on the planet than in the classical Saturn, Janus and Epimetheus system. To obtain periodic orbits, we search those
horseshoe orbits passing through two reversible configurations. A particular kind of periodic orbits where the minor bodies
follow the same path is discussed. 相似文献
20.
Jacques Féjoz 《Celestial Mechanics and Dynamical Astronomy》2002,84(2):159-195
We use the global construction which was made in [6, 7] of the secular systems of the planar three-body problem, with regularized double inner collisions. These normal forms describe the slow deformations of the Keplerian ellipses which each of the bodies would describe if it underwent the universal attraction of only one fictitious other body. They are parametrized by the masses and the semi-major axes of the bodies and are completely integrable on a fixed transversally Cantor set of the parameter space. We study this global integrable dynamics reduced by the symmetry of rotation and determine its bifurcation diagram when the semi-major axes ratio is small enough. In particular it is shown that there are some new secular hyperbolic or elliptic singularities, some of which do not belong to the subset of aligned ellipses. The bifurcation diagram may be used to prove the existence of some new families of 2-, 3- or 4-frequency quasiperiodic motions in the planar three-body problem [7], as well as some drift orbits in the planar n-body problem [8]. 相似文献