Parabolic Orbits Research Articles

Winding strings, Kaluza-Klein monopoles and runge-lenz vectors

We study the interaction of a class of closed solitonic string states (“winders”) which wind around the Kaluza-Klein circle with a Kaluza-Klein monopole. We find that winders are attracted towards monopoles with an inverse-square law force and even when moving relativistically they pursue exactly elliptical parabolic or hyperbolic orbits. The simplicity of the motion is due to the existence of an extra conserved Runge-Lenz vector in addition to the conserved angular momentum. The Poisson algebra of these conserved vectors closes on so (3) ⊕ s R 3 or so (3), so (3) ⊕ s R or so (3) ⊕ so (2, 1). We also find a class of instanton solutions of the euclidean equations in the field of two or more monopoles. The possibility that monopoles catalyse the decay or annihilation of winders is discussed.

Transient chaos and perturbed long range interactions

The motion of a particle in a 1-dimensional perturbed long range potential is studied. A McGehee-type transformation resolves the degeneracy of the stationary point at infinity and the Melnikov method becomes applicable. For a special situation of the planar 3 body problem the Melnikov functions are calculated. There is a chaotic region of the phase space near the parabolic orbit of the uperturbed system.

The cometary cloud in the solar system and the R�sibois-Prigogine singular invariants of motion

A relation between nonintegrability of nonlinear dynamical systems with a continuous Fourier spectrum and irreversibility is investigated in terms of the Liealgebraic formalism. Resibois and Prigogine's singular invariants of motion play an essential role. As an application of the formalism, we solve the restricted three-body problem for the case of nearly parabolic motion of the third body. This gives a model of the motion of a comet in the solar system. The results indicate that there is (deterministic) chaos in the motion of a comet in a nearly parabolic orbit. A possible physical implication of the chaotic motion is the existence of a cometary cloud surrounding the solar system. The theoretical results are compared with numerical results, and show good agreement.

Area-preserving mappings and deterministic chaos for nearly parabolic motions

The present work investigates a mechanism of capturing processes in the restricted three-body problem. The work has been done in a set of variables which is close to Delaunay's elements but which allows for the transition from elliptic to hyperbolic orbits. The small denominator difficulty in the perturbation theory is overcome by embedding the small denominator in an analytic function through a suitable analytic continuation. The results indicate that motions in nearly parabolic orbits can become chaotic even though the model is deterministic. The theoretical results are compared with numerical results, showing an agreement of about one percent. Some possible applications to cometary orbits are also given.

The near-parabolic flux and the origin of short-period comets

The origin of short-period comets, usually believed to arise by planetary perturbations acting on nearly parabolic orbits, has been unclear for many years1. Given the observed near-parabolic flux, the predicted number of such comets is too small by a factor of ∼100 (refs 2–15). It has been suggested, therefore, that the adopted near-parabolic flux is in error16, that a physical effect such as fragmentation3,17,18 could operate, or that short-period comets do not come from the outer Oort cloud at all9,10,13,19. It has been proposed13–15,20,21 that the problem could be resolved by postulating a comet belt or massive inner core to the Oort Cloud. Here I recalculate the flux of nearly parabolic orbits from the Oort Cloud, including the recently discovered dominant injection of comets by the galactic tide22–25. The observed number of short-period comets is consistent with the Oort Cloud having a massive inner core26.

Chaos and cometary clouds in the solar system

Chaotic motion of comets in nearly parabolic orbits is predicted. A physical implication of chaotic motion is interpreted as the existence of a cometary cloud surrounding the solar system. The formation of the cometary cloud is due to a strong resonance effect between the period of Jupiter and the motion of the comets. The dynamical structure of the cloud is described.

Comet Halley and its historic passages during the past millennium

Comet Halley, which is again visiting our solar neighborhood during fall 1985 and spring 1986, was seen with a telescope for the first time on August 15, 1682, by John Flamsteed, assistant observer at the Greenwich Observatory near London. Jean Picard and Philippe de La Hire saw it at Paris on August 26, 1682, shining as if it were a star of the second magnitude. Johannes Hevelius, at Danzig (now Gdansk, Poland), estimated the length of its tail at that time at 12°‐16°. The astronomers of that epoch considered the comet to be a new one that should be added to the list of all those previously observed. The English astronomer Edmond Halley, who was haunted by the idea that all comets need not necessarily describe parabolic orbits, calculated the orbital elements of the new comet and compared them to those of comets that had appeared in 1607 and in 1531 (the heavenly paths of which had been noted by Johannes Kepler and by Christian Severin Logomontanus and Petrus Apianus, respectively). Upon examining the data in Table 1, one can not but be surprised by the near identical value of the elements.

Tidal disruption of dissipative planetesimals

Tidal disruption is a potentially important process for the accumulation of the planets from planetesimals. The fact that stable equilibria do not exist for circular orbits inside the Roche limit has often been hypothesized to mean that any object that passes within the Roche limit is totally disrupted. We have disproven this hypothesis by solving the dynamic problem of the tidal disruption of a dissipative planetestimal during a close encounter with a protoplanet. The solution consists of a numerical integration of the three-dimensional, nonlinear equations of motion, including an approximate treatment of viscous dissipation in the solid regions of the planetesimal. The numerical methods have been extensively tested on a series of one-, two- (Jeans), and three-(Roche) dimensional test problems involving the equilibrium of a body subjected to tidal forces. The results may be scaled to planetesimals of arbitrary size, providing that the scaled equation of state applied. The calculations show that a strongly dissipative planetesimal which passes by the Earth on a parabolic orbit with a perigee within the Roche limit (≈3 R Earth) is not tidally disrupted (even for grazing incidence), and loses no more than a few percent of its mass. This result applies to bodies of radius R which have a kinematic viscosity ν ⪢ 10 12( R/1000km) 2 cm 2sec −1. Less dissipative planetesimals ( ν ≈ 10 13( R/1000 km) 2 cm 2sec −1) may lose up to about 20% of their mass. There are two coupled reasons why this result differs from previous hypotheses: (1) in a dynamic encounter, there is insufficient time to disrupt the planetesimal, and (2) even in circular orbit, the small velocities in the solid region imply that many orbital periods are necessary to completely disrupt the planetesimal. Hence solid and partially molten planetesimals will not experience substantial tidal disruption; completely molten bodies may be sufficiently inviscid to undergo tidal disruption.

Formation of the prelunar accretion disk

According to the single-impact hypothesis for forming the Moon, the angular momentum needed for the present Earth-Moon system can be imparted to the proto-Earth by a collision with a body having one-tenth of the mass or more. The collision must vaporize a large amount of rock which must stay in the form of vapor after expanding in density by a factor of several, so that pressure gradients can accelerate significant amounts of the matter into orbital motion about the proto-Earth. A successful theory must put considerably more than a lunar mass into orbit, having considerably more angular momentum than is needed to assemble a lunar mass in orbit at 3 Earth radii. Such a collision has been simulated by a particular form of a particle-in-cell representation of hydrodynamics and 78 cases have been run representing variations in a variety of parameters. A significant fraction of the cases were successful in creating a satisfactory prelunar accretion disk. A fairly common characteristic of these cases was the presence of an excess velocity in the collision (above that of a parabolic orbit), implying that the projectile involved in the collision existed in an Earth-crossing orbit of significant ellipticity. A majority of the mass of the prelunar accretion disk is contributed by the projectile.

The stability of binary star systems during encounters with a third star

The interaction of a binary star system and a third star not moving on a closed orbit is discussed in terms of three-body zero velocity surfaces. The ranges of orbital parameters for which the binary system is stable against disruption or collision by a third star moving on a parabolic or hyperbolic orbit are determined. It is found that the regions of stable motion decreased as the mass and eccentricity of the passing star increased.

Homoclinic orbits and oscillation for the planar three-body problem

Sitnikov proved the existence of oscillation and capture for a special motion of the restricted three-body problem, m, = 0 [ 141. Alekseev made a systematic study of this situation [ 11. He also proved the same result for all nonzero masses. (See Section 5 below for a discussion of this case.) A good exposition of this example following lectures of Conley is contained in [9] using the stable manifold result for a degenerately hyperbolic closed orbit of McGehee [ 81. Easton and McGehee proposed a planar three-body example with negative energy which could (possibly) exhibit similar oscillation and capture [3]. While Sitnikov’s example has two degrees of freedom, this planar example, after all the integrals and symmetries are removed, has three degrees of freedom. They studied a model example which completely decouples when the third body is near infinity and proved oscillation and capture exist for this model example. Their work left the following three steps undone: (1) show the parabolic orbits form a submanifold for the real equations, (2) show the a-parabolic orbits and w-parabolic orbits are transverse in a strong sense (see the definition of a hyperbolic homoclinic orbit defined below), and (3) show that symbolic dynamics can be used to show oscillation and capture exist using only the fact that the binary asymptotically decouples and not that it completely decouples when the third body is near infinity. For negative energy h, as the third particle goes to infinity parabolically, the asymptotic motion of q = r2 rl is that of a two-body problem with energy h. After regularization the set of all two-body motions with energy h < 0 is the Hopf flow on the three sphere, S3. Let IVs(S3) (resp. wU(S”)) be the set of w-parabolic orbits (resp. w-parabolic orbits). Easton has now shown that Ws(S3) and wl((S3) are Lipschitz manifolds [2]. This paper proves they are real analytic manifolds and “Cm at infinity” (at S”)

Parabolic orbits in the planar three body problem

It is not known whether there exist oscillatory and capture orbits in the planar 3-body problem. Sitnikov [6] proved such orbits exist for the restricted 3-body problem. Alekseev [I] extended this work and related it to the existence of homoclinic orbits. McGehee and Easton ]3] studied a model problem closely related to the planar 3-body problem and proved the existence of oscillatory and capture orbits near certain orbits biasymptotic to an invariant 3-sphere. The goal of this paper is to construct an invariant 3-sphere “at infinity” in the planar 3-body problem and to prove that the stable and unstable manifolds of this sphere are Lipschitz manifolds. To complete the study of oscillatory and capture orbits one must investigate the way in which the stable and unstable manifolds of the 3-sphere intersect each other. This is a difficult question which is not treated here. However, using methods of Melnikov and using one of the masses as a perturbation parameter such an investigation might be possible. The stable manifold theorem which is proven here is not standard since the invariant 3-sphere is not hyperbolic. Geometric methods of McGehee [ 4 ] are extended and applied to obtain the theorem. A special case of the system of equations that we study has the form J, =x: dY)lX, + E,(x, Y)L f, = 4 dY>l-x* + E,(x, Y)L (0.1) 4’ =

The stability of masses during three-body encounters

The dynamical interactio of a binary system and a third body not moving on a closed orbit arises in a large number of physical situations. The C2H condition for determining Hill stability of coplanar bound three-body systems is extended to cover situations where the outer body moves on a parabolic or hyperbolic orbit. Regions where such a body is stable against exchange or collision with other components of the system are determined for a number of important cases where closed solutions are possible.

Parabolic orbits in the planar three body problem : Robert W. Easton, Department of Mathematics, University of Colorado, USA

Parabolic orbits in the planar three body problem : Robert W. Easton, Department of Mathematics, University of Colorado, USA

Capture/Escape Boundary in the Collinear Restricted Three-Body Problem

AbstractWe study the cantorian structure of the successive intersections of the invariant manifolds of infinity (parabolic orbits) with a certain surface of section. The first of these intersections is computed numerically. The structure of the set of orbits of capture or escape after n binary collisions is given.

The passage of a star by a massive black hole

view Abstract Citations (52) References (29) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The passage of a star by a massive black hole Nolthenius, R. A. ; Katz, J. I. Abstract The gridless, smoothed particle hydrodynamic method of Lucy (1977) and Gingold and Monaghan (1977, 1980, a, b, 1981) is used to solve the three-dimensional problem posed by the effects on a 1.0 solar mass star of a 10,000 solar mass black hole's passage, in an initially parabolic orbit with a variety of pericenter distances. It is found that the tidal forces induce rotation and radial and nonradial pulsations in the star, and that the loss of orbital energy to these internal motions leads to the star's capture by the black hole. While the outer layers of the star are disrupted at small pericenter distances, the entire star is destroyed at even smaller distances. These results are applicable to X-ray sources, active galactic nuclei, and quasars Publication: The Astrophysical Journal Pub Date: December 1982 DOI: 10.1086/160511 Bibcode: 1982ApJ...263..377N Keywords: Black Holes (Astronomy); Equations Of Motion; Stellar Motions; Stellar Oscillations; Stellar Rotation; Active Galactic Nuclei; Angular Momentum; Astrodynamics; Equations Of State; Hydrodynamics; Particle Motion; Quasars; Stellar Mass; X Ray Sources; Astrophysics full text sources ADS |

The kinematics of the broad-line emission gas in quasars and Seyfert nuclei

We propose a kinematic model in which the broad-line emission clouds orbit the central continuum source in nearly parabolic orbits in order to produce a symmetric Ly..cap alpha.. profile in spite of the angular asymmetry in the Ly..cap alpha.. emission from an individual optically-thick cloud. Theoretical profiles are generated and compared with logarithmic line shapes, and the pros and cons of this model and others proposed are discussed.

The stellar perturbations of orbital elements of long-period comets

Perturbations by nearby stars on orbital elements of comets with large heliocentric distances are calculated by the impulse approximation and by accurate numerical integration using Cowell's method. It is shown that the agreement is sufficiently close for elliptic orbits but rather poor for parabolic orbits. The perturbation of inclination is a few tens of degrees while that of perihelion distance, q, is from a few to a few tens of AU depending on mutual configurations. For very close encounters, δq can be a few hundred AU. Some cases have been found where an initially direct orbit is converted into a retrograde one and vice versa. For random encounters, statistical parameters which specify the distributions of orbital elements are calculated by the Monte Carlo method. For direct orbits, the perturbations are such as to increase i, while if the initial orbit is retrograde, the tendency is for i to decrease. An almost complete randomization of inclinations will take place within a few million yr. Again, q tends to increase by a few AU but the dispersion is much greater. Although the inclination is significantly perturbed, longitude and latitude of perihelion point are hardly perturbed. This may have an implication for cometary origin.

On the restrited three-body problem when the mass parameter is small

We study some aspects of the restricted three-body problem when the mass parameter μ is sufficiently small. First, we describe the global flow of the two-body rotating problem, μ=0, and we use it for the analysis of the collision and parabolic orbits when μ≳0. Also we show that for any fixed value of the Jacobian constant and for any e>0, there exists a μ0>0 such that if the mass parameter μ∈[0,μ0], then the set of bounded orbits which are not contained in the closure of the set of symmetric periodic orbits has Lebesgue measure less than e.

On the manifolds of total collapse orbits and of completely parabolic orbits for the n-body problem

In this paper we show for most choices of masses in the coplanar n-body problem and for certain masses in the three-dimensional n-body problem that the set of initial conditions leading to complete collapse forms a smooth submanifold in phase space where the dimension depends upon properties of the limiting configuration. A similar statement holds for completely parabolic motion. By a proper scaling, these two motions become dual to each other in the sense that one forms the unstable set while the other forms the stable set of a particular set of points in the new phase space. Most of the paper is devoted to solving the Painleve-Wintner problem which asserts that these types of orbits cannot enter in an infinite spin.