scholarly journals A general solution to non-collinear equilibria in terms of largest root (κ) of confocal oblate spheroid

2016 ◽  
Vol 4 (1) ◽  
pp. 1
Author(s):  
M Javed Idrisi
Keyword(s):  
General Solution ◽  
Body Problem ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Mean Motion ◽  
Three Body ◽  
Infinitesimal Mass

<p>This paper deals with the existence of non-collinear equilibria in restricted three-body problem when less massive primary is an oblate spheroid and the potential of oblate spheroid is in terms of largest root of confocal oblate spheroid. This is found that the non-collinear equilibria are the solution of the equations <em>r</em><sub>1</sub> = <em>n</em><sup>-2/3</sup> and κ = 1 – <em>a</em><sup>2</sup>, where <em>r</em><sub>1</sub> is the distance of the infinitesimal mass from more massive primary, <em>n</em> is mean-motion of primaries, <em>a</em> is semi axis of oblate spheroid and κ is the largest root of the equation of confocal oblate spheroid passes through the infinitesimal mass.</p>

scholarly journals Location and stability of the triangular Lagrange points in photo-gravitational elliptic restricted three body problem with the more massive primary as an oblate spheroid

2019 ◽  
Vol 7 (2) ◽  
pp. 57
Author(s):  
A. Arantza Jency ◽  
Ram Krishan Sharma
Keyword(s):  
Body Problem ◽  
Critical Mass ◽  
Oblate Spheroid ◽  
Motion Equation ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lagrangian Points ◽  
Mean Motion ◽  
The Mean ◽  
Three Body

The triangular Lagrangian points of the elliptic restricted three-body problem (ERTBP) with oblate and radiating more massive primary are studied. The mean motion equation used here is different from the ones employed in many studies on the perturbed ERTBP. The effect of oblateness on the mean motion equation varies. This change influences the location and stability of the triangular Lagrangian points. The points tend to shift in the y-direction. The influence of the oblateness on the critical mass ratio is also altered. But the eccentricity limit  for stability remains the same.   

scholarly journals Stationary solutions, critical mass, Tadpole orbits in the circular restricted three-body problem with the more massive primary as an oblate spheroid

2020 ◽  
Vol 13 (39) ◽  
pp. 4168-4188
Author(s):  
A Arantza Jency
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Critical Mass ◽  
Equations Of Motion ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Mean Motion ◽  
The Mean ◽  
Three Body

Background: The location and stability of the equilibrium points are studied for the Planar Circular Restricted Three-Body Problem where the more massive primary is an oblate spheroid. Methods: The mean motion of the equations of motion is formulated from the secular perturbations as derived by(1) and used in(2–4). The singularities of the equations of motion are found for locating the equilibrium points. Their stability is analysed using the linearized variational equations of motion at the equilibrium points. Findings: As the effect of oblateness in the mean motion expression increases, the location and stability of the equilibrium points are affected by the oblateness of the more massive primary. It is interesting to note that all the three collinear points move towards the more massive primary with oblateness. It is a new result. Among the shifts in the locations of the five equilibrium points, the y–location of the triangular equilibrium points relocate the most. It is very interesting to note that the eccentricities (e) of the orbits around L1 and L3 increase, while it decreases around L2 with the addition of oblateness with the new mean motion. The decrease in e is significant in Saturn-Mimas system from 0.95036 to 0.87558. Similarly, the value of the critical mass ratio mc, which sets the limit for the linear stability of the triangular points, further reduces significantly from 0:285: : :A1 to 0:365: : :A1 with the new mean motion. The mean motion sz in the z-direction increases significantly with the new mean motion from 9A1/4 to 9A1/2.

scholarly journals Pulsating curves of zero velocity for infinitesimal mass around oblate and triaxial rigid body of triangular equilibrium points in elliptical restricted three body problem

2017 ◽  
Vol 5 (1) ◽  
pp. 29
Author(s):  
Nutan Singh ◽  
A. Narayan
Keyword(s):  
Rigid Body ◽  
Body Problem ◽  
Equilibrium Points ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Zero Velocity ◽  
Triaxial Rigid Body ◽  
Three Body ◽  
Infinitesimal Mass

This paper explore pulsating Curves of zero velocityof the infinitesimal mass around the triangular equilibrium points with oblate and triaxial rigid body in the elliptical restricted three body problem(ER3BP).

scholarly journals Stability of the Sitnikov's circular restricted three body problem when the primaries are oblate spheroid

BIBECHANA ◽  
2014 ◽  
Vol 11 ◽  
pp. 149-156
Author(s):  
RR Thapa
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Body ◽  
Independent Variable ◽  
The Stability ◽  
Three Body ◽  
Special Case

The Sitnikov's problem is a special case of restricted three body problem if the primaries are of equal masses (m1 = m2 = 1/2) moving in circular orbits under Newtonian force of attraction and the third body of mass m3 moves along the line perpendicular to plane of motion of primaries. Here oblate spheroid primaries are taken. The solution of the Sitnikov's circular restricted three body problem has been checked when the primaries are oblate spheroid. We observed that solution is depended on oblate parameter A of the primaries and independent variable τ = ηt. For this the stability of non-trivial solutions with the characteristic equation is studied. The general equation of motion of the infinitesimal mass under mutual gravitational field of two oblate primaries are seen at equilibrium points. Then the stability of infinitesimal third body m3 has been calculated. DOI: http://dx.doi.org/10.3126/bibechana.v11i0.10395 BIBECHANA 11(1) (2014) 149-156

scholarly journals Solutions of the Sitnikov's circular restricted three body problems when primaries are oblate spheroid

BIBECHANA ◽  
2013 ◽  
Vol 10 ◽  
pp. 92-99
Author(s):  
RR Thapa ◽  
MR Hassan
Keyword(s):  
Body Problem ◽  
Oblate Spheroid ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

The solutions of sitnikov's circular restricted three body problem has been tried to obtain by using Lindstedt poincare method if the primaries are oblate spheroid. DOI: http://dx.doi.org/10.3126/bibechana.v10i0.9340   BIBECHANA 10 (2014) 92-99

A third order canonical approximation of the general solution of the restricted problem of three bodies in the neighbourhood of L4

1966 ◽  
Vol 25 ◽  
pp. 170-175
Author(s):  
A. Deprit
Keyword(s):  
General Solution ◽  
Restricted Problem ◽  
Body Problem ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Third Order ◽  
The Third ◽  
Transformation Of Variables ◽  
Definition Of ◽  
Three Body

A canonical transformation of variables is introduced in the plane restricted three-body problem which gives the Hamiltonian in the form of a power series with normalized second order terms. Then a generating function is constructed, step by step, that permits the definition of new action and angle variables, such that the Hamiltonian is independent of the angle variables. This procedure has been done explicitly up to the third order terms.

scholarly journals Robe's Restricted Three-Body Problem with Variable Masses and Perturbing Forces

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Jagadish Singh ◽  
Oni Leke
Keyword(s):  
Body Problem ◽  
Equilibrium Points ◽  
Second Primary ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Small Perturbations ◽  
Triangular Points ◽  
Numerical Exploration ◽  
Three Body ◽  
Infinitesimal Mass

The linear stability of equilibrium points of a test particle of infinitesimal mass in the framework of Robe's circular restricted three-body problem, as in Hallan and Rana, together with effect of variation in masses of the primaries with time according to the combined Meshcherskii law, is investigated. It is seen that, due to a small perturbation in the centrifugal force and an arbitrary constant of a particular integral of the Gylden-Meshcherskii problem, every point on the line joining the centers of the primaries is an equilibrium point provided they lie within the shell. Further, a number of pairs of equilibrium points lying on the -plane and forming triangles with the centers of the shell and the second primary exist, for some values of . The points collinear with the center of the shell are found to be stable under some conditions and the range of stability depends on the small perturbations and , while the triangular points are unstable. Illustrative numerical exploration is given to indicate significant improvement of the problem in Hallan and Rana.

scholarly journals Beyond-Newtonian dynamics of a planar circular restricted three-body problem with Kerr-like primaries

2020 ◽  
Vol 501 (1) ◽  
pp. 713-729
Author(s):  
Shounak De ◽  
Suparna Roychowdhury ◽  
Roopkatha Banerjee
Keyword(s):  
Body Problem ◽  
Newtonian Potential ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Spin Effects ◽  
Newtonian Dynamics ◽  
Zero Values ◽  
Newtonian System ◽  
Three Body ◽  
Infinitesimal Mass

ABSTRACT The dynamics of the planar circular restricted three-body problem with Kerr-like primaries in the context of a beyond-Newtonian approximation is studied. The beyond-Newtonian potential is developed by using the Fodor–Hoenselaers–Perjés procedure. An expansion in the Kerr potential is performed and terms up to the first non-Newtonian contribution of both the mass and spin effects are included. With this potential, a model for a test particle of infinitesimal mass orbiting in the equatorial plane of the two primaries is examined. The introduction of a parameter, ϵ, allows examination of the system as it transitions from the Newtonian to the beyond-Newtonian regime. The evolution and stability of the fixed points of the system as a function of the parameter ϵ is also studied. The dynamics of the particle is studied using the Poincaré map of section and the Maximal Lyapunov Exponent as indicators of chaos. Intermediate values of ϵ seem to be the most chaotic for the two cases of primary mass ratios (=0.001, 0.5) examined. The amount of chaos in the system remains higher than the Newtonian system as well as for the planar circular restricted three-body problem with Schwarzschild-like primaries for all non-zero values of ϵ.

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