Periodic orbits in the restricted four-body problem with two equal masses     

Jaime Burgos-García  Joaquín Delgado 《Astrophysics and Space Science》2013,345(2):247-263

The restricted (equilateral) four-body problem consists of three bodies of masses m ₁, m ₂ and m ₃ (called primaries) lying in a Lagrangian configuration of the three-body problem i.e., they remain fixed at the apices of an equilateral triangle in a rotating coordinate system. A massless fourth body moves under the Newtonian gravitation law due to the three primaries; as in the restricted three-body problem (R3BP), the fourth mass does not affect the motion of the three primaries. In this paper we explore symmetric periodic orbits of the restricted four-body problem (R4BP) for the case of two equal masses where they satisfy approximately the Routh’s critical value. We will classify them in nine families of periodic orbits. We offer an exhaustive study of each family and the stability of each of them.  相似文献   

2D calculations of the collapse of magnetized gas cloud     

N. V. Ardeljan  G. S. Bisnovatyi-Kogan  S. G. Moiseenko 《Astrophysics and Space Science》1996,239(1):1-13

All the families of planar symmetric simple-periodic orbits of the photogravitational restricted plane circular three-body problem, are determined numerically in the case when the primaries are of equal mass and radiate with equal radiation factors (q ₁=q₂=q). We obtain a global view of the possible patterns of periodic three-body motion while the full range of values of the common radiation factor is explored, from the gravitational case (q=1) down to near the critical value at which the triangular equilibria disappear by coalescing with the inner equilibrium pointL ₁ on the rotating axis of the primaries. It is found that for large deviations of its value from the gravitational case the radiation factorq can have a strong effect on the structure of the families.  相似文献   

Families of Asymmetric Periodic Orbits in the Restricted Three-body Problem     

K. E. Papadakis 《Earth, Moon, and Planets》2008,103(1-2):25-42

This paper studies the asymmetric solutions of the restricted planar problem of three bodies, two of which are finite, moving in circular orbits around their center of masses, while the third is infinitesimal. We explore, numerically, the families of asymmetric simple-periodic orbits which bifurcate from the basic families of symmetric periodic solutions f, g, h, i, l and m, as well as the asymmetric ones associated with the families c, a and b which emanate from the collinear equilibrium points L ₁, L ₂ and L ₃ correspondingly. The evolution of these asymmetric families covering the entire range of the mass parameter of the problem is presented. We found that some symmetric families have only one bifurcating asymmetric family, others have infinity number of asymmetric families associated with them and others have not branching asymmetric families at all, as the mass parameter varies. The network of the symmetric families and the branching asymmetric families from them when the primaries are equal, when the left primary body is three times bigger than the right one and for the Earth–Moon case, is presented. Minimum and maximum values of the mass parameter of the series of critical symmetric periodic orbits are given. In order to avoid the singularity due to binary collisions between the third body and one of the primaries, we regularize the equations of motion of the problem using the Levi-Civita transformations.  相似文献   

Families of periodic orbits in the planar three-body problem     

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   

Families of periodic orbits in the restricted four-body problem     

A. N. Baltagiannis  K. E. Papadakis 《Astrophysics and Space Science》2011,336(2):357-367

In this paper, families of simple symmetric and non-symmetric periodic orbits in the restricted four-body problem are presented. Three bodies of masses m ₁, m ₂ and m ₃ (primaries) lie always at the apices of an equilateral triangle, while each moves in circle about the center of mass of the system fixed at the origin of the coordinate system. A massless fourth body is moving under the Newtonian gravitational attraction of the primaries. The fourth body does not affect the motion of the three bodies. We investigate the evolution of these families and we study their linear stability in three cases, i.e. when the three primary bodies are equal, when two primaries are equal and finally when we have three unequal masses. Series, with respect to the mass m ₃, of critical periodic orbits as well as horizontal and vertical-critical periodic orbits of each family and in any case of the mass parameters are also calculated.  相似文献   

Surface fitting of three-dimensional closed solutions of the restricted three-body problem     

P. G. Kazantzis 《Astrophysics and Space Science》1978,57(2):499-509

The orbits of a family of three-dimensional periodic orbits in the restricted problem of three bodies form a surface. In this paper we determine the equation of this surface in the case of the orbits of double symmetry of the family which emanates from the equilibrium pointL ₁. This equation is obtained numerically by a least squares approximation method.  相似文献   

A variable mass of the primaries and librations around the triangular points     

G P. Horedt 《Celestial Mechanics and Dynamical Astronomy》1974,10(3):319-326

The delimitations of the librational motion around the Lagrangian triangular pointsL ₄,L ₅ are investigated within the framework of the restricted circular three body problem according to Brown's and Thüring's theory. The isotropic mass variation of the primaries does not exceed the order of the small primary and the derivatives of the masses with respect to the time are negligible second order quantities. The amplitude of the maximum elongations with respect to the small primary remains unchanged. The expression for the maximum variation of the distance of the particle from the large mass has the same form as in the classical problem with constant masses.  相似文献   

Effect of Stellar Wind and Poynting–Robertson Drag on Photogravitational Elliptic Restricted Three Body Problem     

A. Chakraborty  A. Narayan 《Solar System Research》2018,52(2):168-179

The existence and linear stability of the planar equilibrium points for photogravitational elliptical restricted three body problem is investigated in this paper. Assuming that the primaries, one of which is radiating are rotating in an elliptical orbit around their common center of mass. The effect of the radiation pressure, forces due to stellar wind and Poynting–Robertson drag on the dust particles are considered. The location of the five equilibrium points are found using analytical methods. It is observed that the collinear equilibrium points L₁, L₂ and L₃ do not lie on the line joining the primaries but are shifted along the y-coordinate. The instability of the libration points due to the presence of the drag forces is demonstrated by Lyapunov’s first method of stability.  相似文献   

Asymptotic orbits in the (<Emphasis Type="Italic">N</Emphasis>+1)-body ring problem     

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   

Stability of the photogravitational restricted three-body problem with variable masses     

Jagadish Singh  Oni Leke 《Astrophysics and Space Science》2010,326(2):305-314

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   

On Sitnikov-like motions generating new kinds of 3D periodic orbits in the R3BP with prolate primaries     

C. Douskos  V. Kalantonis  P. Markellos  E. Perdios 《Astrophysics and Space Science》2012,337(1):99-106

The existence of new equilibrium points is established in the restricted three-body problem with equal prolate primaries. These are located on the Z-axis above and below the inner Eulerian equilibrium point L ₁ and give rise to a new type of straight-line periodic oscillations, different from the well known Sitnikov motions. Using the stability properties of these oscillations, bifurcation points are found at which new types of families of 3D periodic orbits branch out of the Z-axis consisting of orbits located entirely above or below the orbital plane of the primaries. Several of the bifurcating families are continued numerically and typical member orbits are illustrated.  相似文献   

Non-symmetric doubly-asymptotic orbits at the outer collinear equilibrium pointL 3     

E. Perdios 《Astrophysics and Space Science》1993,199(2):185-188

In the framework of the solar system case (with only the larger primary radiating) of the photogravitational restricted three-body problem we compute and present some non-symmetric asymptotic orbits connecting the outer collinear equilibrium pointL ₃ with the neighbourhood of one of the triangular equilibrium pointsL _{4, 5}. Such orbits have not been found previously in the restricted problem.  相似文献   

Dynamics in the Center Manifold Around L2 in the Quasi-Bicircular Problem     

M. A. Andreu 《Celestial Mechanics and Dynamical Astronomy》2002,84(2):105-133

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The equilibrium solutions of the restricted problem of three rigid bodies with variable mass     

S. M. Elshaboury 《Astrophysics and Space Science》1989,155(2):209-214

In this paper we consider the circular planar restricted problem of three rigid bodiesS _i(i=1, 2, 3), two of them are axisymmetric ellipsoids and a third bodyS ₃ is a spherical satellite with decreasing mass, under the gravitational forces. The effect of small perturbations in the Coriolis force and the centrifugal forces on the location of equilibrium points has been studied. It is found only in the case when the primaries have equal differences between their respective principal moments of inertial the pointsL ₄ andL ₅ form nearly equilateral tringles with the primaries. The equilibrium pointsL ₁,L ₂,L ₃ remain collinear an ies on the line joining the primaries.  相似文献   

A highly symmetric restricted nine-body problem and the linear stability of its relative equilibria     

Marcelo Marchesin  Deborah da Paixão Vasconcellos 《Astrophysics and Space Science》2014,352(2):443-460

We study a highly symmetric nine-body problem in which eight positive masses, called the primaries, move four by four, in two concentric circular motions such that their configuration is always a square for each group of four masses. The ninth body being of negligible mass and not influencing the motion of the eight primaries. We assume all the nine masses are in the same plane and that the masses of the primaries are \(m_{1}=m_{2}=m_{3}=m_{4}=\tilde{m}\) and m ₅=m ₆=m ₇=m ₈=m and the radii associated to the circular motion of the bodies with mass \(\tilde{m}\) is λ∈[λ ₀,1] and for the bodies with mass m is 1. We prove the existence of central configurations which characterize such arrangement of the primaries and we study the influence of the parameter λ, the ratio of the radii of the two circles, on the masses m and \(\tilde{m}\) . We use a synodical system of coordinates to eliminate the time dependence on the equations of motion. We show the existence of equilibria solutions symmetrically distributed on the four quadrants and their dependence on the parameter λ. Finally, we show that there can be 13, 17 or 25 equilibria solutions depending on the size of λ and we investigate their linear stability.  相似文献   

Equilibrium points in the photogravitational restricted four-body problem     

J. P. Papadouris  K. E. Papadakis 《Astrophysics and Space Science》2013,344(1):21-38

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   

On Brown's conjecture     

J. Henrard 《Celestial Mechanics and Dynamical Astronomy》1983,31(2):115-122

We examine the conjecture made by Brown (1911) that in the restricted three body problem, the long period family of periodic orbits aroundL ₄, ends on a homoclinic orbit toL ₃. By numerical integration we establish that for the mass ratio Sun-Jupiter such a homoclinic orbit toL ₃ does not exist but that there exists a family of homoclinic orbits to periodic orbits aroundL ₃.  相似文献   

Robe’s restricted problem of 2+2 bodies when the bigger primary is a Roche ellipsoid and the smaller primary is an oblate body     

Bhavneet Kaur  Rajiv Aggarwal 《Astrophysics and Space Science》2014,349(1):57-69

In this problem, one of the primaries of mass m ₁ is a Roche ellipsoid filled with a homogeneous incompressible fluid of density ρ ₁. The smaller primary of mass m ₂ is an oblate body outside the Ellipsoid. The third and the fourth bodies (of mass m ₃ and m ₄ respectively) are small solid spheres of density ρ ₃ and ρ ₄ respectively inside the Ellipsoid, with the assumption that the mass and the radius of the third and the fourth body are infinitesimal. We assume that m ₂ is describing a circle around m ₁. The masses m ₃ and m ₄ mutually attract each other, do not influence the motions of m ₁ and m ₂ but are influenced by them. We have extended the Robe’s restricted three-body problem to 2+2 body problem under the assumption that the fluid body assumes the shape of the Roche ellipsoid (Chandrashekhar in Ellipsoidal figures of equilibrium, Chap. 8, Dover, New York, 1987). We have taken into consideration all the three components of the pressure field in deriving the expression for the buoyancy force viz (i) due to the own gravitational field of the fluid (ii) that originating in the attraction of m ₂ (iii) that arising from the centrifugal force. In this paper, equilibrium solutions of m ₃ and m ₄ and their linear stability are analyzed. We have proved that there exist only six equilibrium solutions of the system, provided they lie within the Roche ellipsoid. In a system where the primaries are considered as Earth-Moon and m ₃,m ₄ as submarines, the equilibrium solutions of m ₃ and m ₄ respectively when the displacement is given in the direction of x ₁-axis or x ₂-axis are unstable.  相似文献   

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem     

Jagadish Singh  Oni Leke 《Astrophysics and Space Science》2013,344(1):51-61

This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q _i , and changing oblateness of the primaries A _i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α _i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.  相似文献   

Three-dimensional periodic orbits about the triangular equilibrium points of the restricted problem of three bodies     

C. G. Zagouras 《Celestial Mechanics and Dynamical Astronomy》1985,37(1):27-46

The third-order parametric expansions given by Buck in 1920 for the three-dimensional periodic solutions about the triangular equilibrium points of the restricted Problem are improved by fourthorder terms. The corresponding family of periodic orbits, which are symmetrical w.r.t. the (x, y) plane, is computed numerically for =0.00095. It is found that the family emanating from L₄ terminates at the other triangular point L₅ while it bifurcates with the family of three-dimensional periodic orbits originating at the collinear equilibrium point L₃. This family consists of stable and unstable members. A second family of nonsymmetric three-dimensional periodic orbits is found to bifurcate from the previous one. It is also determined numerically until a collision orbit is encountered with the computations.  相似文献