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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

Existence of particular solutions in the generalised restricted problem of three bodies with variable masses

The restricted problem of three bodies with variable masses is considered. It is assumed that the infinitesimal body is axisymmetric with constant mass and the finite bodies are spherical with variable masses such that the ratio of their masses remains constant. The motion of the finite bodies are determined by the Gyldén-Meshcherskii problem. It is seen that the collinear, triangular, and coplanar solutions not exist, but these solutions exist when the infinitesimal body be a spherical.     

Nonlinear stability in the generalised photogravitational restricted three body problem with Poynting-Robertson drag

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.     

Eulerian libration points of restricted problem of three oblate spheroids

In this paper we consider the circular restricted problem of three oblate spheroids. The collinear equilibrium solutions are obtained. Finally a numerical study of the influence of the non-sphericity in the location of the libration points is made.     

Libration points in the generalised elliptic restricted three body problem

In this paper the existence of collinear as well as equilateral libration points for the generalised elliptic restricted three body problem has been studied distinct from Kondurar and Shinkarik (1972) where a study has been made for the generalised circular restricted three body problem. Here the coordinates of the libration points have been found to be functions of timet.     

On the stability of the triangular libration points for the photogravitational circular restricted problem of three bodies under the resonances of the third and the fourth order

We investigate the stability of the triangular libration points when both the attracting bodies are radiating under the resonance conditions ₁ = 2₂ and 3₂.     

Equilibrium points in the photogravitational restricted four-body problem

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.     

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.     

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.     

The effect of oblateness of the bigger primary on collinear libration points in the restricted problem of three bodies

In the restricted problem of three bodies, the effect of oblateness of the bigger primary appears as an additional term in the potential. As a result, the location of libration points and the roots of the characteristic equation at these points depend not only upon the mass parameter but also on the oblateness termI of the bigger primary. Series solutions are developed in terms of andI which are used for locating the collinear libration points and for determining the mean motions and characteristic exponents at these points.The work is supported by a fellowship awarded to the second author by University Grant Commission, India.     

Existence and stability of the non-collinear libration points in the restricted three body problem when both the primaries are ellipsoid

This paper deals with the existence and stability of the non-collinear libration points in the restricted three-body problem when both the primaries are ellipsoid with equal mass and identical in shape. We have determined the equations of motion of the infinitesimal mass which involves elliptic integrals and then we have investigated the existence and stability of the non-collinear libration points. This is observed that the non-collinear libration points exist only in the interval 52^°<φ<90^° and form an isosceles triangle with the primaries. Further we observed that the non collinear libration points are unstable in 52^°<φ<90^°.     

Periodic motion around stable collinear equilibrium point in the photogravitational restricted problem of three bodies

In a binary system with both bodies being luminous, the inner collinear equilibrium pointL ₁ becomes stable for values of the mass ratio and radiation pressure parameters in a certain region. The kind of periodic motions aroundL ₁ is examined in this case. Second-order parametric expansions are given and the families of periodic orbits generated fromL ₁ are numerically determined for several sets of values of the parameters. Short- and long-period solutions are identified showing a similarity in the character of periodicity with that aroundL ₄. It is also found that the finite periodic solutions in the vicinity ofL ₁ are stable.