Three-dimensional bifurcations of periodic solutions around the triangular equilibrium points of the restricted three-body problem

Vertically critical, planar periodic solutions around the triangular equilibrium points of the Restricted Three-Body Problem are found to exist for values of the mass parameter in the interval [0.03, 0.5]. Four series of such solutions are computed. The families of three-dimensional periodic solutions that branch off these critical orbits are computed for µ = 0.3 and are continued till their end. All orbits of these families are unstable.     

Periodic motion around the triangular equilibrium points of the photogravitational restricted problem of three bodies

In this article the effect of radiation pressure on the periodic motion of small particles in the vicinity of the triangular equilibrium points of the restricted three body problem is examined. Second order parametric expansions are constructed and the families of periodic orbits are determined numerically for two sets of values of the mass and radiation parameters corresponding to the non-resonant and the resonant case. The stability of each orbit is also studied.     

A two-parameter survey of periodic orbits in the restricted problem of three bodies

Within the context of the restricted problem of three bodies, we wish to show the effects, caused by varying the mass ratio of the primaries and the eccentricity of their orbits, upon periodic orbits of the infinitesimal mass that are numerical continuations of circular orbits in the ordinary problem of two bodies. A recursive-power-series technique is used to integrate numerically the equations of motion as well as the first variational equations to generate a two-parameter family of periodic orbits and to identify the linear stability characteristics thereof. Seven such families (comprised of a total of more than 2000 orbits) with equally spaced mass ratios from 0.0 to 1.0 and eccentricities of the orbits of the primaries in a range 0.0 to 0.6 are investigated. Stable orbits are associated with large distances of the infinitesimal mass from the perturbing primary, with nearly circular motion of the primaries, and, to a slightly lesser extent, with small mass ratios of the primaries.Conversely, unstable orbits for the infinitesimal mass are associated with small distances from the perturbing primary, with highly elliptic orbits of the primaries, and with large mass ratios.     

Motion near the triangular points in the elliptic restricted problem of three bodies

In this paper the first variational equations of motion about the triangular points in the elliptic restricted problem are investigated by the perturbation theories of Hori and Deprit, which are based on Lie transforms, and by taking the mean equations used by Grebenikov as our upperturbed Hamiltonian system instead of the first variational equations in the circular restricted problem. We are able to remove the explicit dependence of transformed Hamiltonian on the true anomaly by a canonical transformation. The general solution of the equations of motion which are derived from the transformed Hamiltonian including all the constant terms of any order in eccentricity and up to the periodic terms of second order in eccentricity of the primaries is given.     

Stability of triangular equilibrium points in the elliptic restricted problem of three bodies with radiating and triaxial primaries

This paper studies the stability of infinitesimal motions about the triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. The perturbation technique developed by Bennet (Icarus 4:177, 1965b) has been used for determination of characteristic exponents. This technique is based on Floquet’s Theory for determination of characteristic exponents in the system with periodic coefficients. The results of the study are analytical and numerical expressions are simulated for the transition curves bounding the region of stability in the μ–e plane, accurate to O(e ²). The unstable region is found to be divided into three parts. The effect of radiation parameter is significant. For small values of e, the results are in favor with the numerical analysis of Danby (Astron. J. 69:166, 1964), Bennet (Icarus 4:177, 1965b), Alfriend and Rand (AIAA J. 6:1024, 1969). The effect of radiation pressure is significant than the oblateness and triaxiality of the primaries.     

On the triangular equilibrium points in the photogravitational relativistic restricted three body problem

The photogravitational restricted three body within the framework of the post-Newtonian approximation is carried out. The mass of the primaries are assumed changed under the effect of continuous radiation process. The locations of the triangular points are computed. Series forms of these locations are obtained as new analytical results. In order to introduce a semi-analytical view, a Mathematica program is constructed so as to draw the locations of triangular points versus the whole range of the mass ratio μ taking into account the photogravitational effects and/or the relativistic corrections. All the obtained figures are analyzed. The size of relativistic effects of about.08 normalized distance unit is observed.     

A note on the equilibrium points of the restricted problem of three bodies

Using Sylvester's theorem on matrices, an elegant expression is obtained for the solutions of the restricted problem of three bodies in the neighborhood of the equilibrium points.     

Almost-square orbits in the restricted problem of three bodies

In the restricted problem of three bodies, it has been discovered that, in a nonrotating frame, almost perfectly square orbits can result. Numerical investigations of these orbits have been made, and a brief explanation of their behaviour is given.     

A grid search for families of periodic orbits in the restricted problem of three bodies

A method is described for the numerical determination of families of periodic orbits in the planar restricted problem of three bodies. The families are sought in their representation as curves in a two-dimensional space of parameters. A grid search is applied to the study of the evolution of satellite motion when the mass parameter is varied. Only that part of the space of parameters is investigated for which one of them, the relative energy constant, takes values larger than that corresponding to the inner Lagrangian pointL ₂. Critical values of the mass parameter are determined for which new families of simple or double periodic orbits appear inside the closed ovals of zero velocity.     

Three-dimensional regions of stability about the triangular equilibrium points

Within the context of the restricted problem of three bodies, an analytic upper bound on the three-dimensional regions of stability about the triangular equilibrium points is derived for general initial velocity limits and a wide class of bounding regions. This upper bound is illustrated and compared to numerical investigations for two bounding regions using the Earth-Moon mass ratio.     

Existence of periodic orbits of the second kind in the elliptic restricted problem of three bodies

Hale's method is used to show the existence of symmetric periodic orbits of the second kind for the particular case of the elliptic restricted problem of three bodies. In this treatment we also obtain a new proof of the existence of periodic orbits of the first and second kinds in the circular restricted problem.     

Some analytical results about periodic orbits in the restricted three body problem with dissipation

We present some analytical results about the existence of periodic orbits for the planar restricted three body problem with dissipation considered recently by Celletti et?al. (CMDA 109, 265, 2011) We show that, under fairly general conditions on the dissipation term, the circular orbits cannot be continued to the dissipative framework. Moreover, we give general conditions for the occurrence or not of a Hopf bifurcation around the libration points L ₄ and L ₅. Our results are consistent with the numerical findings of Celletti et?al.     

Resonance stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries

The main aim of this paper is to study the existence of resonance and linear stability of the triangular equilibrium points of the planar elliptical restricted three body problem considering the photo gravitational effect of both the primaries in circular and elliptical case. A practical application of this case could be the study of the dynamical system around the binary systems. For this the Hamiltonian function, convergent in nature and describing the motion of the infinitesimal body in the neighborhood of the triangular equilibrium solutions is derived. Also, the Hamiltonian for the system is expanded in powers of the generalized components of momenta. Further, canonical transformation has also been used to study the stability of the triangular equilibrium points. The study primarily focuses on establishing the relation for determining the range of stability at and near the resonance frequency ω ₂=1/2 around the binary systems using simulation technique. It is observed that the parametric resonance is only possible at the resonance frequency ω ₂=1/2 in both circular and elliptical cases.     

Nonlinear stability of the triangular libration points for the photo gravitational elliptic restricted problem of three bodies

In the present paper we have studied the stability of the triangular libration points for the doubly photogravitational elliptic restricted problem of three bodies under the presence of resonances as well as under their absence. Here we have found the conditions for stability.     

Three-dimensional periodic solutions around equilibrium points in Hill's problem

The three-dimensional periodic solutions originating at the equilibrium points of Hill's limiting case of the Restricted Three Body Problem, are studied. Fourth-order parametric expansions by the Lindstedt-Poincaré method are constructed for them. The two equilibrium points of the problem give rise to two exactly symmetrical families of three-dimensional periodic solutions. The family_HL _2v ^e originating at L₂ is continued numerically and is found to extend to infinity. The family originating at L₁ behaves in exactly the same way and is not presented. All orbits of the two families are unstable.     

Motion and stability of triangular equilibrium points in elliptical restricted three body problem under the radiating primaries

This paper studies the motion and orbital stability of the infinitesimal mass in the vicinity of the equilateral (triangular) Lagrangian points of the elliptic restricted three body problem, considering photo gravitational effects of both the primaries. The stability of the triangular points is studied under the effects of radiating primaries around the binary system (Achird, Luyten, αCen AB, Kruger-60, Xi-Bootis); using simulation technique by drawing different curves of zero velocity.     

Continuation of periodic orbits: Three-dimensional circular restricted to the general three-body problem

It is proved that a periodic orbit of the three-dimensional circular restricted three-body problem can be continued analytically, when the mass of the third body is sufficiently small, to a periodic orbit of the three dimensional general three-body problem in a rotating frame. The above method is not applicable when the period of the periodic orbit of the restricted problem is equal to 2k (k any integer), in the usual normalized units. Several numerical examples are given.     

On the period of the periodic orbits of the restricted three body problem

We will show that the period T of a closed orbit of the planar circular restricted three body problem (viewed on rotating coordinates) depends on the region it encloses. Roughly speaking, we show that, \(2 T=k\pi +\int _\Omega g\) where k is an integer, \(\Omega \) is the region enclosed by the periodic orbit and \(g:{\mathbb {R}}^2\rightarrow {\mathbb {R}}\) is a function that only depends on the constant C known as the Jacobian constant; it does not depend on \(\Omega \). This theorem has a Keplerian flavor in the sense that it relates the period with the space “swept” by the orbit. As an application we prove that there is a neighborhood around \(L_4\) such that every periodic solution contained in this neighborhood must move clockwise. The same result holds true for \(L_5\).     

Effects of radiation and triaxiality of primaries on triangular equilibrium points in elliptic restricted three body problem

This paper studies the motion of an infinitesimal mass around triangular equilibrium points in the elliptic restricted three body problem assuming bigger primary as a source of radiation and the smaller one a triaxial rigid body. A practical application of this case could be the study of motion of a satellite under the effect of Sun and Earth. We have exploited the method of averaging used by Grebnikov (Nauka, Moscow, revised 1986) throughout the analysis of stability of the system. The critical mass ratio depends on the radiation pressure, oblateness, eccentricity and semi major axis of the elliptic orbits and the range of stability decreases as the radiation parameter increases.