On the stability of triangular libration points of the photogravitational restricted circular three-body problem

This paper considers the restricted circular three-body problem with respect to the radiation repulsion force acting upon a particle on the part of one of the main bodies (the Sun). The characteristic of the family of stationary particular solutions of the problem (libration points) representing the relative equilibrium positions in a rotating Cartesian system is given. On the basis of the KAM theory with the help of a computer a nonlinear analysis of the triangular libration points stability for the planar case is carried out. These libration points are proved to be strictly stable by Liapunov practically in the whole area of fulfilling the necessary stability conditions. Instability is discovered at the resonant curve of the third order and at the greater part of the resonant curve of the fourth order. The plotted results of the investigation allowed us to draw a conclusion about the Liapunov stability of the triangular libration points in a problem with respect to the radiation pressure for all the planets of the Solar system.     

On the stability of the triangular libration points for the photogravitational circular restricted problem of three bodies when both of the attracting bodies are radiating as well

Here the stability of triangular libration points when both the attracting bodies are radiating as well has been investigated under the non-resonance cases. It is found that except for some cases for all values of the radiation reduction factors and for all values of <0.0285954..., the motion will be stable.     

About stability of libration points in the restricted photogravitational three body problem

Nonlinear stability of the triangular libration point in the photogravitational restricted three body problem was investigated in the whole range of the parameters. Some results obtained earlier are corrected. The method for proper determination of cases when stability cannot be determined by four order terms of the hamiltonian was proposed.     

On the stability of triangular libration points taking into account the light pressure for the circular restricted problem of three bodies

In the present paper we have studied the resonance problem arising for critical mass values following Birkhoff's method of normalisation. It has been shown that for both the critical values the motion will remain unstable whether the critical terms are left in the Hamiltonian or carried to the coordinates in the process of normalisation.     

Existence of libration points in the generalised photogravitational restricted problem of three bodies

In this paper we have proved the existence of libration points for the generalised photogravitational restricted problem of three bodies. We have assumed the infinitesimal mass of the shape of an oblate spheroid and both of the finite masses to be radiating bodies and the effect of their radiation pressure on the motion of the infinitesimal mass has also been taken into account. It is seen that there is a possibility of nine libration points for small values of oblateness, three collinear, four coplanar and two triangular.     

Nonlinear stability of the libration points in the photogravitational restricted three body problem

The stability of the triangular libration points in the case when the first and the second order resonances appear was investigated. It was proved that the first order resonances do not cause instability. The second order resonances may lead to instability. Domains of the instability in the two-dimensional parameter space were determined.     

On the triangular libration points in photogravitational restricted three-body problem with variable mass

This paper investigates the triangular libration points in the photogravitational restricted three-body problem of variable mass, in which both the attracting bodies are radiating as well and the infinitesimal body vary its mass with time according to Jeans’ law. Firstly, applying the space-time transformation of Meshcherskii in the special case when q=1/2, k=0, n=1, the differential equations of motion of the problem are given. Secondly, in analogy to corresponding problem with constant mass, the positions of analogous triangular libration points are obtained, and the fact that these triangular libration points cease to be classical ones when α≠0, but turn to classical L ₄ and L ₅ naturally when α=0 is pointed out. Lastly, introducing the space-time inverse transformation of Meshcherskii, the linear stability of triangular libration points is tested when α>0. It is seen that the motion around the triangular libration points become unstable in general when the problem with constant mass evolves into the problem with decreasing mass.     

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.     

The libration points of a spherical satellite in the photogravitational restricted problem of three bodies

In this paper the photogravitational circular restricted problem of three bodies is considered. We have assumed that one of the finite bodies be a spherical luminous and the other be a triaxial nonluminous body. The possibility of existence of the libration points be studied.     

Stability of the triangular libration points of the circular restricted problem in the presence of resonances

The stability of triangular libration points, when the bigger primary is a source of radiation and the smaller primary is an oblate spheroid. has been investigated in the resonance cases ₁ = 2₂ and ₁ = 3₂. The motion is unstable for all the values of parameters q and A when ₁ = 2₂ and the motion is unstable and stable depending upon the values of the parameters q and A when ₁ = 3₂. Here q is the radiation parameter and A is the oblateness parameter.     

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.     

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.     

Linear stability and the resonance cases for the triangular libration points for the doubly photo-gravitational elliptic restricted problem of three bodies

In the present paper we have found the range of values of μ ande for the linear stability of the triangular points for the doubly photogravitational elliptic restricted problem of three bodies. It has been shown that some resonances of the third and the fourth order exist which will need special investigation for the determination of complete stability of the libration points under our terms of reference.     

The stability of the triangular libration points for the plane circular restricted three-body problem with light pressure

We study the fourth-order stability of the triangular libration points in the absence of resonance for the three-body problem when the infinitesimal mass is affected not only by gravitation but also by light pressure from both primaries. A comprehensive summary of previous results is given, with some inaccuracies being corrected. The Lie triangle method is used to obtain the fourth-order Birkhoff normal form of the Hamiltonian, and the corresponding complex transformation to pre-normal form is given explicitly. We obtain an explicit expression for the determinant required by the Arnold-Moser theorem, and show that it is a rational function of the parameters, whose numerator is a fifth-order polynomial in the mass parameter. Particular cases where this polynomial reduces to a quartic are described. Our results reduce correctly to the purely gravitational case in the appropriate limits, and extend numerical work by previous authors.     

Out-of-plane equilibrium points and stability in the generalised photogravitational restricted three body problem

The restricted three body problem is generalised to include the effects of an inverse square distance radiation pressure force on the infinitesimal mass due to the primaries, which are both radiating. In this paper we investigate the stability of out-of-plane equilibrium points, based on equations in variations. We have found the characteristic equation for the complex normal frequencies which is a sixth order polynomial. Thus we conclude that out-of-plane equilibrium points are unstable due to positive real part in complex roots.     

Conditions for the linear stability of libration points and effects of drag on the linear stability of triangular libration points in the restricted three-body problem

A number of criteria for linear stability of libration points in the perturbed restricted three-body problem are presented. The criteria involve only the coefficients of the characteristic equation of the tangent map of the libration points and can be easily applied. With these criteria the effect of drag on the linear stability of the triangular libration points in the classical restricted three-body problem is investigated. Some of Murray et al.'s results are improved.     

Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,II

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted three-body problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure, in the case of equal radiation factors of the two primaries. The full range of values of the common radiation factor is explored, from the gravitational caseq ₁ =q ₂ =q = 1 down to the critical value ofq = 1/8 at which the triangular equilibria disappear by coalescing on the rotating axis of the primaries. It is found that radiation pressure exerts a significant influence on the stability regions. For certain intervals of radiation values these regions become qualitatively different from the gravitational case as well as the solar system case considered in Paper I. There exist values of the common radiation factor, in the range considered, for which the triangular equilibrium points are stable for the entire range of mass distribution among the primaries and for large eccentricities of their orbits.     

Linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem,I

The linear stability of the triangular equilibrium points in the photogravitational elliptic restricted problem is examined and the stability regions are determined in the space of the parameters of mass, eccentricity, and radiation pressure. It is found that radiation pressure of the larger body for solar system cases exerts only a small quantitative influence on the stability regions.     

Existence of libration points in the restricted problem of three bodies with radiation pressure

In this paper the photogravitational forces restricted of three bodies be considered. We have assumed the infinitesimal mass of the shape of an axisymmetric body and one the finite masses be spherical luminous body while the other be an axisymmetric body non-luminous body. It is seen that there is a possibility of nine libration points for small values of oblatenesses.     

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.