On Asymmetric Periodic Solution of the Plane Restricted Problem of Three Bodies, and Bifurcations of Families

1978 ◽  
pp. 319-323 ◽  
Author(s):  
P. J. Message ◽  
D. B. Taylor
Keyword(s):  
Periodic Solution ◽  
Restricted Problem

On nearly-commensurable periods in the restricted problem of three bodies, with calculations of the long-period variations in the interior 2:1 case

1966 ◽  
Vol 25 ◽  
pp. 197-222 ◽  
Author(s):  
P. J. Message
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Large Amplitude ◽  
Restricted Problem ◽  
Small Amplitude ◽  
Minor Planets ◽  
Long Period ◽  
Numerical Integrations ◽  
Period Variations ◽  
The Mean

An analytical discussion of that case of motion in the restricted problem, in which the mean motions of the infinitesimal, and smaller-massed, bodies about the larger one are nearly in the ratio of two small integers displays the existence of a series of periodic solutions which, for commensurabilities of the typep+ 1:p, includes solutions of Poincaré'sdeuxième sortewhen the commensurability is very close, and of thepremière sortewhen it is less close. A linear treatment of the long-period variations of the elements, valid for motions in which the elements remain close to a particular periodic solution of this type, shows the continuity of near-commensurable motion with other motion, and some of the properties of long-period librations of small amplitude.To extend the investigation to other types of motion near commensurability, numerical integrations of the equations for the long-period variations of the elements were carried out for the 2:1 interior case (of which the planet 108 “Hecuba” is an example) to survey those motions in which the eccentricity takes values less than 0·1. An investigation of the effect of the large amplitude perturbations near commensurability on a distribution of minor planets, which is originally uniform over mean motion, shows a “draining off” effect from the vicinity of exact commensurability of a magnitude large enough to account for the observed gap in the distribution at the 2:1 commensurability.

scholarly journals A Theory of the Trojan Asteroids

1978 ◽  
Vol 41 ◽  
pp. 145-145
Author(s):  
B. Garfinkel
Keyword(s):  
Periodic Solution ◽  
Restricted Problem ◽  
Elliptic Integral ◽  
Mass Parameter ◽  
Small Mass ◽  
Polar Coordinates ◽  
Lagrangian Point ◽  
Elementary Functions ◽  
The Family ◽  
The Mean

AbstractThe paper constructs a long-periodic solution for the case of 1:1 resonance in the restricted problem of three bodies. The polar coordinates r and 0 appear in the formHere λ is the mean synodic longitude, m is the small mass-parameter, k is the integer nearest to the ratio ω2/ω1 of the fundamental angular frequencies of the motion, and ck is a Fourier coefficient of a certain periodic function. Only elementary functions enter r(λ) and θ(λ), while the calculation of λ(t) requires the inversion of a hyper-elliptic integral t(λ).The internal resonant terms, carrying the critical divisor D, impart to the orbit an epicyclic character, in qualitative accord with the results of the numerical integration by Deprit and Henrard (1970). Our solution is valid except in the vicinity of the singularities at D = 0 and λ = 0.The presence of the resonant terms invalidates the Brown conjecture (1911) regarding the termination of the family of the tadpoleshaped orbits at the Lagrangian point L3. However, this conjecture holds for the mean orbits defined by r = r(λ), θ = θ(λ), and it also holds in the limit as m → 0.

scholarly journals Recent Progress in the Theory of Trojan Asteroids

1979 ◽  
Vol 81 ◽  
pp. 251-256
Author(s):  
Boris Garfinkel
Keyword(s):  
Periodic Solution ◽  
Restricted Problem ◽  
Recent Progress ◽  
Mass Parameter ◽  
Trojan Asteroids

In order to provide the necessary background for this report, this section summarizes the previously published “Theory of the Trojan Asteroids, Part I”. Treating the system as the case of 1:1 resonance in the restricted problem of three bodies, the author constructs a formal long-periodic solution of 0(m), where m is the mass-parameter of the system, assumed to be sufficiently small.

Periodic Trojan orbits in the elliptic restricted problem

1966 ◽  
Vol 25 ◽  
pp. 176-186
Author(s):  
E. Rabe
Keyword(s):  
Periodic Solution ◽  
Periodic Solutions ◽  
Restricted Problem ◽  
Angular Motion ◽  
Basic Period ◽  
Short Period ◽  
Elliptic Restricted Problem ◽  
The Mean

As previously shown the generally non-periodic librations about the equilateral points of the plane elliptic restricted problem depend on three frequencies. After elimination of the short-period oscillations, the solutions depend on the libration frequencynand the mean angular motionNof Jupiter. Whenn/Nis commensurable the librations become periodic. One periodic solution corresponds to the actual Trojans, but additional planar solutions exist also with periods equal to multiples of the basic period. For some specific values of the eccentricityethese solutions are accompanied by an associated sequence of non-planar periodic solutions. The values ofedepend slightly on the inclinationi.

The orbit of Pluto over a long interval of time

1966 ◽  
Vol 25 ◽  
pp. 227-229 ◽  
Author(s):  
D. Brouwer
Keyword(s):  
Periodic Orbit ◽  
Numerical Integration ◽  
Restricted Problem ◽  
Three Dimensional ◽  
The Third ◽  
Circular Restricted Problem ◽  
Resonance Relation

The paper presents a summary of the results obtained by C. J. Cohen and E. C. Hubbard, who established by numerical integration that a resonance relation exists between the orbits of Neptune and Pluto. The problem may be explored further by approximating the motion of Pluto by that of a particle with negligible mass in the three-dimensional (circular) restricted problem. The mass of Pluto and the eccentricity of Neptune's orbit are ignored in this approximation. Significant features of the problem appear to be the presence of two critical arguments and the possibility that the orbit may be related to a periodic orbit of the third kind.

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