Relativistic R3BP Research Articles

Locations of L 4,5 of a dust grain type II comet tail in Solar-Jupiter system in the photogravitational relativistic R3BP

A calculating formula when q∈(−∞, 1] for the locations of the triangular points L 4,5 in the relativistic restricted three-body problem is proposed when the bigger primary is a source of radiation with a mass reduction factor q. As an application, we consider a dust grain in type II comet tail in Solar-Jupiter system with a mass reduction factor q∈(−1.2, 0.5). An investigation in this neighborhood reveals that the shifts due to the relativistic terms on the locations of a dust grain particle are very small with the shifts in ξ— coordinate smaller than those of η— coordinate. It is further observed that increasing negative values of q results in moving L 4,5 further away from the line joining the primaries and towards the smaller primary. Increasing positive values of q, however results in a shift of L 4,5 towards both the line joining both primaries and towards the smaller primary.

On Motion around the Collinear Equilibrium Points in the Relativistic R3BP with a Smaller Triaxial Primary

This paper studies the motion of an infinitesimal mass near the collinear equilibrium points in the framework of relativistic restricted three-body problem (R3BP) when the smaller primary is a triaxial body. It is observed that the positions of the collinear points are affected by the relativistic and triaxiality factors. The collinear points are found to remain unstable. Numerical studies in this connection, with the Sun-Earth, Sun-Pluto and Earth-Moon systems have been carried out to show the relativistic and triaxiality effects.

On the stability of triangular points in the relativistic R3BP when the bigger primary is oblate and the smaller one radiating with application on Cen X-4 binary system

Abstract In the framework of the relativistic R3BP, we examine the effects of oblateness of the primary body and radiation pressure of the secondary on the positions and stability of the triangular points L4,5. It is found that the parameters involved all affect the positions and increase in any of the parameters leads to a reduction in the size of the region of stability. Thus establishing their destabilizing tendencies. The presence of positive real roots or positive real part in complex roots affirms the instability of L4,5 of the problem when applied to Cen X-4.

Collinear Equilibrium Points in the Relativistic R3BP when the Bigger Primary is a Triaxial Rigid Body

This study examines the effect of the relativistic factor as well as the triaxiality effect of the bigger primary on the positions and stability of the collinear points in the frame work of the post-Newtonian approximation. Using semi-analytical and numerical approach the collinear points are found to be unstable. A numerical exploration in this connection, with the Earth-Moon system, reveals that the relativistic factor has an effect on these positions. It is also found that under the combined effect of relativistic factor and triaxiality, the collinear point L1 moves towards the primaries with the increase in triaxiality, while L2 and L3 move away from the bigger primary. It is also seen that in most of the cases in the presence of triaxiality, the effect of relativistic factor on the positions of L1 and L3 is not observable; however it has an observable effect on the position of L2 in the presence of triaxiality except for the case 2.

On the stability of L4,5 in the perturbed relativistic R3BP with a triaxial bigger primary

In the present paper, we endeavor to study the stability of triangular points under the influence of small perturbations in the Coriolis and centrifugal forces, together with the triaxiality of the bigger primary in the framework of the relativistic R3BP. It is observed that the locations of these points are affected by the relativistic factor, triaxiality and a small perturbation in the centrifugal force, but are unaffected by that of the Coriolis force. It is also seen that for these points the range of stability region increases or decreases according as equation (14) without is greater or less than zero.

On the stability of the triangular points in the relativistic R3BP with a bigger triaxial primary

This paper studies the motion of a third body (test particle) in the vicinity of the triangular points L4,5 by considering the more massive as a triaxial body in the frame work of the relativistic restricted three-body problem (R3BP). It is seen that the positions and stability of the triangular points are affected by both relativistic and triaxiality factors. It turns out both the coordinates of the infinitesimal mass axe affected. It is seen that for these points, the range of stability region increases or decreases according as p>0 or p<0 where p depends upon the triaxiality and relativistic factors. Furthermore we have studied the periodic orbits around the triangular points in the range 0 < µ < µc. It is found that these orbits axe elliptical; the frequencies of long and short orbits of the periodic motion,the eccentricities,semi-major and semi-minor axes, orientation and coefficients of long and short periodic terms are all affected by triaxiality and relativistic factors.

Existence and stability of triangular points in the relativistic R3BP when the bigger primary is a triaxial rigid body and a source of radiation

We study the effect of triaxiality and radiation of the bigger primary on the positions and stability of the triangular points in the relativistic R3BP. It is found that the locations of the triangular points are affected by the relativistic terms apart the radiation force and the triaxiality of the bigger primary. It is also seen that for these points, the range of stability region increases or decreases according as without which depends upon the relativistic terms, the radiation and triaxiality coefficient is greater than or less than zero. A practical application of this model could be the study of the motion of a dust grain particle near the Sun-Earth system. A practical application of this model could be the study of the motion of a dust grain particle near the Sun-Earth system.

Existence and linear stability of triangular points in the perturbed relativistic R3BP when the bigger primary is an oblate spheroid

We study the effects of oblateness and small perturbations in the Coriolis and centrifugal forces on the locations and stability of the triangular points in the relativistic R3BP. It is observed that the positions are affected by the oblateness, relativistic, and a small perturbation in the centrifugal force, but are unaffected by that of Coriolis force. It is also seen that the relativistic terms, oblateness, small perturbations in the centrifugal and Coriolis forces influence the critical mass ratio. It is also noticed that all the former three and the latter one possess destabilizing and stabilizing behavior respectively. However, the range of stability increases or decreases according to as p &gt;0 or p&lt;0 where p depends upon the relativistic, oblateness and small perturbations in the Coriolis and centrifugal forces.

Stability of triangular points in the relativistic R3BP when the bigger primary is an oblate spheroid

In this paper, we study the effect of oblateness of the more massive primary in the relativistic R3BP. We observe that the locations of the triangular points and their stability are affected by the relativistic and oblateness factors. It is also noticed that the oblateness factor possesses destabilizing behavior. Therefore, the size of the region of stability decreases with increase in the value of the oblateness factor.Further, a numerical study on the locations of the triangular points and the critical mass for the Earth-Moon , Jupiter and its Moons, Saturn and its Moons systems is given.

Existence and Stability of Triangular Points in the Relativistic R3bp When the Primaries are Triaxial Rigid Bodies and Sources of Radiation

This paper deals with the triangular points and their linear stability in the relativistic R3BP when the primaries are triaxial rigid bodies and sources of radiation. It is observed that the locations of the triangular points are affected by the relativistic terms, radiation pressure forces and the triaxiality of the primaries. It is also seen that for these points the range of stability region increases or decreases according as p >0 or p<0 where p depends upon the relativistic terms, the radiation and triaxiality coefficients.

On the stability of triangular points in the relativistic R3BP with oblate primaries and bigger radiating

We consider a version of the relativistic R3BP which includes the effects of oblateness of the primaries and radiation of the bigger primary as well on the stability of triangular points. We observe that the positions of the triangular points and their stability are affected by the relativistic effect apart from the radiation and oblateness of the primaries. It is further seen for these points that the range of stability region increases or decreases according as the part of the critical mass value, depending upon relativistic terms, radiation and oblateness coefficients, is positive or negative. A numerical exploration shows that in the Sun–Saturn, Sun–Uranus, Sun–Neptune systems, the oblateness has no influence on their positions and range of stability region; whereas it has a little influence on the Sun–Mars, Sun–Jupiter systems. On the other hand, we found that radiation pressure has an observable effect on the solar system.

On the collinear point L3 in the generalized relativistic R3BP

The photogravitational oblate restricted three-body problem is carried out within the post-Newtonian framework. The primaries are assumed to be oblate as well as radiant sources. The perturbed location of the collinear point \( L_{3}\) is computed. A series form of \( L_{3}\) location is obtained as a new analytical results. In order to introduce a semi-analytical view, a Mathematica program is constructed so as to draw the locations of collinear point \( L_{3}\)versus the mass ratio \( \mu\in (0, 0.5)\) taking into account one or more of the considered perturbations. All the obtained figures are analyzed.

Motion around [formula omitted] in the relativistic R3BP with smaller triaxial primary

This paper deals with the triangular points L4,5 of the relativistic restricted three-body problem (R3BP) when the smaller primary is assumed triaxial. It is noticed that the locations and stability of the triangular points are affected by both relativistic and triaxiality pertubations. It can be easily seen that the range of stability region of these points is reduced by the effects of relativistic and triaxiality factors and more especially decreases with the increase of triaxiality factor.

On the equilibrium point L1 in the photogravitational oblate relativistic R3BP with application on Sun-planet systems

In the present article the effects of the relativistic, photogravitational and oblateness perturbations in restricted three-body problem, in brief R3BP are investigated. The relativistic effects and the dynamical shape of both radiant primaries are taken into account. The perturbed location of the collinear point L 1 is computed. Series forms of this location are obtained as new analytical results. In order to introduce a semi-analytical view, a mathematica code, using the Mathematica9 package software, is constructed so as to draw the location of collinear point L 1 versus the mass ratio μ ∈ (0, 0.5), taking into account one or more of the considered perturbations. All the obtained figures are analyzed. Some specific systems of R3BP formed of the Sun as the massive primary and a planet as the less massive primary are studied.

Effect of Perturbations in the Coriolis and Centrifugal Forces on the Stability of L 4 in the Relativistic R3BP

The centrifugal and Coriolis forces do not appear as a result of physically imposed forces, but are due to a special property of a rotation. Thus, these forces are called pseudo-forces or ‘fictitious forces’. They are merely an artifact of the rotation of the reference frame adopted. This paper studies the motion of a test particle in the neighbourhood of the triangular point L4 in the framework of the perturbed relativistic restricted three-body problem (R3BP) when small perturbations are conferred to the centrifugal and Coriolis forces. It is found that the position and stability of the triangular point are affected by both the relativistic factor and small perturbations in the Coriolis and centrifugal forces. As an application, the Sun–Earth system is considered.

On the Stability of L 4,5 in the Relativistic R3BP with Radiating Secondary

This paper discusses the motion of a test particle in the neighbourhood of the triangular points L4,5 by considering the less massive primary (secondary) as a source of radiation in the framework of the relativistic restricted three-body problem (R3BP). It is found that the positions and stability of the triangular point are affected by both relativistic and electromagnetic radiation factors. It turns out that both the coordinates of the infinitesimal mass are affected, contrary to the classical where this happens only for one coordinate. A practical application of this model could be the study of dynamical evolution of dust particles in orbits around a binary system with a dark degenerate first primary and a secondary stellar companion.

Effect of radiation pressure on the stability of L 4,5 in the relativistic R3BP

This paper examines the motion of a test particle in the vicinity of the triangular points L4,5 by considering the more massive primary as a source of radiation in the framework of the relativistic restricted three-body problem (R3BP). It is found that the position and stability of the triangular point are affected by both the relativistic factor and radiation pressure.

Motion around L 4 in the perturbed relativistic R3BP

This paper studies the motion of a test particle (infinitesimal mass) in the neighborhood of the triangular point L4 in the framework of the perturbed relativistic restricted three-body problem (R3BP). The problem is perturbed in the sense that a small perturbation is given to the centrifugal force. It is found that the position and stability of the triangular point are affected by both the relativistic factor and a small perturbation in the centrifugal force.