The zero velocity surfaces in the photogravitational restricted three-body problem
TL;DR: In this paper, the surfaces of the restricted three-body problem when the more massive body is luminous are studied and the properties of the function Ω which determines these surfaces are given.
Abstract: The surfaces of zero velocity of the restricted three-body problem when the more massive body is luminous, are studied. The properties of the function Ω which determines these surfaces are given. It is found that the topological properties of the zero velocity surfaces while not affected by the variation of the mass parameter, are essentially varied when the radiation pressure parameter changes values. Closed regions where the motion can be trapped are described while periodic motions about the “out of plane” equilibrium points seem to be probable.
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TL;DR: In this paper, a modern state of the three-body problem is discussed, where an infinitesimal mass is effected not only by gravitation but also by light pressure from one (or both) of the primaries.
Abstract: We discuss a modern state of the three-body problem when an infinitesimal mass is effected not only by gravitation but also by light pressure from one (or both) of the primaries. This problem, called the photogravitational one, attracted much attention during the last ten years. Many aspects of the libration point locations and their stability for all values of radiation pressure and mass ratios are shown and discussed. A retrospective chronological review of the results is given.
37 citations
Cites background or methods or result from "The zero velocity surfaces in the p..."
...A more detailed investigation of these points including their stability in the linear approximation was performed by Kunitsyn and Tureshbaev (1985d, 198G) and by Ragos and Zagouras (1988)....
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...…of the results obtained by Simmons et al. (1985) on the stability of the coplanar points for p # 0.5 was initiated by Tureshbaev (1986) and Ragos and Zagouras (1988) as aimed a t the following two questions: 1) Is it true that the points LG and L7 are stable only if both the primaries…...
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...A three-dimensional picture of the topological features of the zero-velocity surface for a photogravitational problem with one radiating center was given by Ragos and Zagouras (1988) by means of computer calculations....
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...A more detailed investigation of these orbits around the stable points Ls and L7 was carried out by Ragos and Zagouras (1988)....
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...But in fact this agrees with the conclusions of Tureshbaev (1986) and Ragos and Zagouras (1988) because of the nature of evolution of the stability region of the points Ls and L7 when p decreases....
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TL;DR: In this paper, the photogravitational restricted four-body problem is employed to describe the motion of an infinitesimal particle in the vicinity of three finite radiating bodies.
Abstract: The photogravitational restricted four-body problem is employed to describe the motion of an infinitesimal particle in the vicinity of three finite radiating bodies. The fourth body $P_{4}$
of infinitesimal mass does not affect the motion of the three bodies (
$P_{1}, P_{2}, P_{3}$
) that are always at the vertices of an equilateral triangle. We consider that two of the bodies (
$P_{2}$
and $P_{3}$
) have the same radiation and mass value $\mu$
while the dominant primary body $P_{1}$
is of mass $1 - 2\mu$
. The equilibrium points (
$L_{1}^{z},L_{2}^{z}$
) lying out of the orbital plane of the primaries as well as the allowed regions of motion as determined by the zero velocity curves are studied numerically. Finally the stability of these points is studied and they are found to be unstable.
33 citations
TL;DR: In this article, the Stromgren families with the radiation pressure parameter of the more massive body was studied in the Sun-Jupiter system and the stability of each periodic solution was also studied.
Abstract: The evolution of the periodic orbits around the collinear equilibrium positions, belonging to the Stromgren families a, b and c, with the radiation pressure parameter of the more massive body is studied in the Sun-Jupiter system. These families are determined for a single value of the radiation pressure parameter and particularly when the radiation force of the more massive body is equal to one half of the gravitational attraction. Then the critical stability orbits of each family are transferred with the radiation parameter. The stability of each periodic solution is also studied.
28 citations
TL;DR: The zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies and their parametric evolution is the subject of as discussed by the authors, which provides valuable information concerning the regions of the permissible particle motion and the existence of equilibrium positions.
Abstract: The zero-velocity surfaces in the three-dimensional ring problem of N + 1 bodies and their parametric evolution is the subject of this paper. These surfaces, which are also known as Hill's or Jacobian surfaces, provide us with valuable information concerning the regions of the permissible particle motion and the existence of equilibrium positions.
28 citations
01 May 1980
TL;DR: In the case of the radiating body (such as the sun) is much smaller than the mass of the other main gravitating body, the triangular libration points in the plane photogravitational restricted circular three-body problem will be Lyapunov-stable almost everywhere within the region in which the necessary stability conditions are satisfied.
Abstract: In the case where the mass of the radiating body (such as the sun) is much smaller than the mass of the other main gravitating body (such as the galactic nucleus), the triangular libration points in the plane photogravitational restricted circular three-body problem will be Lyapunov-stable almost everywhere within the region in which the necessary stability conditions are satisfied Instability will, however, occur on the curves corresponding to the inner third- and fourth-order resonances
25 citations
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31 Jan 1971
TL;DR: The three-body problem was studied in this paper, where Covarinace of Lagarangian Derivatives and Canonical Transformation were applied to the problem of estimating the perimeter and the velocity of the system.
Abstract: The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The Regularizing Transformation.- Application to the Three-Bdy Problem.- An Estimate of the Perimeter.- An Estimate of the Velocity.- Sundman's Theorem.- Triple Collision.- Triple-Collision Orbits.- Periodic Solutions: The Solutions of Lagrange.- Eigenvalues.- An Existence Theorem.- The Convergence Proof.- An Application to the Solution of Lagrange.- Hill's Problem.- A Generalization of Hill's Problem.- The Continuation Method.- The Fixed-Point Theorem.- Area-Preserving Analytic Transformations.- The Birkhoff Fixed-Point Theorem.- Stability: The Function-Theoretic Center Problem.- The Convergence Proof.- The Poincare Center Problem.- The Theorem of Liapunov.- The Theorem of Dirichlet.- The Normal Form of Hamiltonian Systems.- Area-Preserving Transformations.- Existence of Invariant Curves.- Proof of Lemma.- Application to the Stability Problem.- Stability of Equilibrium Solutions.- Quasi-Periodic Motion and Systems of Several Degrees of Freedom.- The Recurrence Theorem.
1,075 citations
TL;DR: New ideas are controversial when they challenge orthodoxy, but orthodoxy changes with time, often surprisingly fast as mentioned in this paper. But I believe we have now reached the point where we can, if we so choose, build new habitats far more comfortable, productive and attractive than is most of Earth.
Abstract: New ideas are controversial when they challenge orthodoxy, but orthodoxy changes with time, often surprisingly fast. It is orthodox, for example, to believe that Earth is the only practical habitat for Man, and that the human race is close to its ultimate size limits. But I believe we have now reached the point where we can, if we so choose, build new habitats far more comfortable, productive and attractive than is most of Earth.
193 citations
TL;DR: In this paper, the effects of an inverse square distance radiation pressure force on the infinitesimal mass due to the large masses, which are both arbitrarily luminous, were considered.
Abstract: The restricted 3-body problem is generalised to include the effects of an inverse square distance radiation pressure force on the infinitesimal mass due to the large masses, which are both arbitrarily luminous. A complete solution of the problems of existence and linear stability of the equilibrium points is given for all values of radiation pressures of both liminous bodies, and all values of mass ratios. It is shown that the inner Lagrange point, L1, can be stable, but only when both large masses are luminous. Four equilibrium points, L6, L7, L8, and L9 can exist out of the orbital plane when the radiation pressure of the smaller mass is very high. Although L8 and L9 are always linearly unstable, L6 and L7 are stable for a small range of radiation pressures provided that both large masses are luminous.
192 citations