Equations of Motion for the Perturbed Restricted Three-Body Problem

AIAA Journal ◽  
1974 ◽  
Vol 12 (10) ◽  
pp. 1442-1442
Author(s):  
T. A. HEPPENHEIMER
Keyword(s):  
Body Problem ◽  
Equations Of Motion ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

Orbits of Trojan Asteroids

1974 ◽  
Vol 62 ◽  
pp. 63-69 ◽  
Author(s):  
G. A. Chebotarev ◽  
N. A. Belyaev ◽  
R. P. Eremenko
Keyword(s):  
Solar System ◽  
Body Problem ◽  
Equations Of Motion ◽  
Orbital Evolution ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Trojan Asteroids ◽  
Actual Motion ◽  
Three Body

In this paper the orbital evolution of Trojan asteroids are studied by integrating numerically the equations of motion over the interval 1660–2060, perturbations from Venus to Pluto being taken into account. The comparison of the actual motion of Trojans in the solar system with the theory based on the restricted three-body problem are given.

scholarly journals Orbital stability of high inclination asteroids

1996 ◽  
Vol 172 ◽  
pp. 187-192
Author(s):  
N. A. Solovaya ◽  
E. M. Pittich
Keyword(s):  
Solar System ◽  
Body Problem ◽  
Equations Of Motion ◽  
Orbital Stability ◽  
Dynamical Model ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

The orbital evolutions of fictitious asteroids with high inclinations have been investigated. The selected initial orbits represent asteroids with movement, which corresponds to the conditions of the Tisserand invariant for C = C (L1) in the restricted three body problem. Initial eccentricities of the orbits cover the interval 0.0–0.4, inclinations the interval 40–80°, and arguments of perihelion the interval 0–360°. The equations of motion of the asteroids were numerically integrated from the epoch March 25, 1991 forward within the interval of 20,000 years, using a dynamical model of the solar system consisting of all planets. The orbits of the model asteroids are stable at least during the investigated period.

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 A new perturbative solution to the motion around triangular Lagrangian points in the elliptic restricted three-body problem

2021 ◽  
Vol 133 (6) ◽  
Author(s):  
Bálint Boldizsár ◽  
Tamás Kovács ◽  
József Vanyó
Keyword(s):  
Body Problem ◽  
Mass Parameter ◽  
Equations Of Motion ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Lagrangian Points ◽  
Perturbative Solution ◽  
The Stability ◽  
Hill’S Equations ◽  
Three Body

AbstractThe equations of motion of the planar elliptic restricted three-body problem are transformed to four decoupled Hill’s equations. By using the Floquet theorem, a perturbative solution to the oscillator equations with time-dependent periodic coefficients are presented. We clarify the transformation details that provide the applicability of the method. The form of newly derived equations inherently comprises the stability boundaries around the triangular Lagrangian points. The analytic approach is valid for system parameters $$0 < e \le 0.05$$ 0 < e ≤ 0.05 and $$0 < \mu \le 0.01$$ 0 < μ ≤ 0.01 where e denotes the eccentricity of the primaries, while $$\mu $$ μ is the mass parameter. Possible application to known extrasolar planetary systems is also demonstrated.

scholarly journals Investigation of the effect of albedo and oblateness on the circular restricted four variable bodies problem

2017 ◽  
Vol 2 (2) ◽  
pp. 529-542 ◽  
Author(s):  
Abdullah A. Ansari
Keyword(s):  
Radiation Pressure ◽  
Body Problem ◽  
Equilibrium Points ◽  
Equations Of Motion ◽  
Variable Mass ◽  
Solar Radiation Pressure ◽  
Basins Of Attraction ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

AbstractThe present paper investigates the motion of the variable infinitesimal body in circular restricted four variable bodies problem. We have constructed the equations of motion of the infinitesimal variable mass under the effect of source of radiation pressure due to which albedo effects are produced by another two primaries and one primary is considered as an oblate body which is placed at the triangular equilibrium point of the classical restricted three-body problem and also the variation of Jacobi Integral constant has been determined. We have studied numerically the equilibrium points, Poincaré surface of sections and basins of attraction in five cases (i. Third primary is placed at one of the triangular equilibrium points of the classical restricted three-body problem, ii. Variation of masses, iii. Solar radiation pressure, iv. Albedo effect, v. Oblateness effect.) by using Mathematica software. Finally, we have examined the stability of the equilibrium points and found that all the equilibrium points are unstable.

scholarly journals Global regularization method for planar restricted three-body problem

2015 ◽  
pp. 39-49
Author(s):  
M.A. Sharaf ◽  
H.R. Dwidar
Keyword(s):  
Power Series ◽  
Analytical Solutions ◽  
Regularization Method ◽  
Body Problem ◽  
Equations Of Motion ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Initial Values ◽  
Three Body ◽  
Global Regularization

In this paper, global regularization method for planar restricted three-body problem is purposed by using the transformation z = x+iy = ? cos n(u+iv), where i = ??1, 0 < ? ? 1 and n is a positive integer. The method is developed analytically and computationally. For the analytical developments, analytical solutions in power series of the pseudotime ? are obtained for positions and velocities (u, v, u', v') and (x, y, x?, y?) in both regularized and physical planes respectively, the physical time t is also obtained as power series in ?. Moreover, relations between the coefficients of the power series are obtained for two consequent values of n. Also, we developed analytical solutions in power series form for the inverse problem of finding ? in terms of t. As typical examples, three symbolic expressions for the coefficients of the power series were developed in terms of initial values. As to the computational developments, the global regularized equations of motion are developed together with their initial values in forms suitable for digital computations using any differential equations solver. On the other hand, for numerical evolutions of power series, an efficient method depending on the continued fraction theory is provided.

scholarly journals ON THE RESTRICTED THREE-BODY PROBLEM

2021 ◽  
Vol 2 (336) ◽  
pp. 138-144
Author(s):  
M. Zh. Minglibayev ◽  
T. M. Zhumabek
Keyword(s):  
Differential Equations ◽  
Coordinate System ◽  
Analytical Solutions ◽  
Body Problem ◽  
Equations Of Motion ◽  
Stationary Solutions ◽  
Libration Points ◽  
Restricted Three Body Problem ◽  
Three Body Problem ◽  
Three Body

The paper analytically investigates the classical restricted three-body problem in a special non-inertial central coordinate system, with the origin at center of forces. In this coordinate system, an analytical expression of the invariant of the centre of forces is given. The existence of the invariant of the centre of forces admits the correct division of the problem into two problems. The first is a triangular restricted three-body problem. The second is a collinear restricted three-body problem. In this paper the collinear restricted three-body problem is investigated. Using the properties of the invariant of centre of forces of the restricted three-body problem in the special non- inertial central coordinate system, the basic differential equations of motion for the collinear restricted three-body problem are obtained when three bodies lie on the same line during all motion. Differential equations of the collinear restricted three-body problem in the rotating non-inertial central coordinate system in pulsating variables are derived. New differential equations of motion for the collinear restricted three-body problem in three regions of possible location of the massless body with stationary solutions corresponding to the three Euler libration points have been derived. The circular collinear restricted three-body problem is investigated in detail. The corresponding Jacobi integrals are obtained. New exact non-stationary partial analytical solutions of the obtained new differential equations of motion of the collinear restricted three-body problem have been found for the considered case.

scholarly journals Studying Close Approaches for a Cloud of Particles Considering Atmospheric Drag

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Jorge Formiga ◽  
Rodolpho Vilhena de Moraes
Keyword(s):  
Mathematical Model ◽  
Numerical Simulations ◽  
Body Problem ◽  
Equations Of Motion ◽  
Restricted Three Body Problem ◽  
The Sun ◽  
Atmospheric Drag ◽  
Three Body Problem ◽  
Three Body ◽  
The Mathematical Model

The present paper has the goal of studying close approaches between a planet and a group of particles. The mathematical model includes the presence of the atmosphere of the planet. This cloud is assumed to be created by the passage of the spacecraft in the atmosphere of the planet, which can cause the explosion of the spacecraft. The system is assumed to be formed by the Sun, the planet, and the spacecraft that explodes and becomes a cloud of particles. The Sun and the planet are assumed to be in circular orbits and the motion is planar. The equations of motion are the ones given by the circular planar restricted three-body problem combined with the forces given by the atmospheric drag. In the numerical simulations, the planet Jupiter is the celestial body used for the close approaches. The initial positions and velocities of the spacecraft and the particles are specified at the periapsis, because it is assumed that this is the point where the explosion occurs.

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