Periodic solutions of a spring-pendulum system     

R. Broucke  P. A. Baxa 《Celestial Mechanics and Dynamical Astronomy》1973,8(2):261-267

A study has been made of a dynamical system composed of a pendulum and a harmonic oscillator, in order to show the remarkable resemblance with many classical celestial mechanics problems, in particular the restricted three-body problem. It is shown that the well-known investigations of periodic orbits can be applied to the present dynamics problem.Presented at the Conference on Celestial Mechanics, Oberwolfach, Germany, August 27–September 2, 1972.  相似文献   

A note on relative motion in the general three-body problem     

R. Broucke  H. Lass 《Celestial Mechanics and Dynamical Astronomy》1973,8(1):5-10

It is shown that the equations of the general three-body problem take on a very symmetric form when one considers only their relative positions, rather than position vectors relative to some given coordinate system. From these equations one quickly surmises some well known classical properties of the three-body problem such as the first integrals and the equilateral triangle solutions. Some new Lagrangians with relative coordinates are also obtained. Numerical integration of the new equations of motion is about 10 percent faster than with barycentric or heliocentric coordinates.  相似文献   

The Lagrangian theory of Stäckel systems     

R. Broucke 《Celestial Mechanics and Dynamical Astronomy》1981,24(2):185-193

The variational equations along an orbit in a conservative dynamic system with three degrees of freedom may be separated into (i) a linear system of order four involving only the normal and binormal displacements and (ii) a quadrature to produce the tangential displacement.  相似文献   

Book reviews     

R. Broucke 《Celestial Mechanics and Dynamical Astronomy》1982,28(3):345-348

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Numerical explorations of the rectilinear problem of three bodies     

R. Broucke  D. E. Walker 《Celestial Mechanics and Dynamical Astronomy》1980,21(1):73-81

We present some results of a numerical exploration of the rectilinear problem of three bodies, with the two outer masses equal. The equations of motion are first given in relative coordinates and in regularized variables, removing both binary collision singularities in a single coordinate transformation. Among our most important results are seven periodic solutions and three symmetric triple collision solutions. Two of these periodic solutions have been continued into families, the outer massm ₃ being the family parameter. One of these families exists for all masses while the second family is a branch of the first at a second-kind critical orbit. This last family ends in a triple collision orbit.Proceedings of the Sixth Conference on Mathematical Methods in Celestial Mechanics held at Oberwolfach (West Germany) from 14 to 19 August, 1978.  相似文献   

Determining Wedge Angle for a Wedge Aquifer     

A. Vanden Berg  D. H. Lennox 《Ground water》1973,11(1):25-31

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A statistical study of gamma-ray burst afterglows measured by the Swift Ultraviolet Optical Telescope     

S. R. Oates  M. J. Page  P. Schady  M. de Pasquale  T. S. Koch  A. A. Breeveld  P. J. Brown  M. M. Chester  S. T. Holland  E. A. Hoversten  N. P. M. Kuin  F. E. Marshall  P. W. A. Roming  M. Still  D. E. Vanden Berk  S. Zane  J. A. Nousek 《Monthly notices of the Royal Astronomical Society》2009,395(1):490-503

We present the first statistical analysis of 27 Ultraviolet Optical Telescope (UVOT) optical/ultraviolet light curves of gamma-ray burst (GRB) afterglows. We have found, through analysis of the light curves in the observer's frame, that a significant fraction rise in the first 500 s after the GRB trigger, all light curves decay after 500 s, typically as a power law with a relatively narrow distribution of decay indices, and the brightest optical afterglows tend to decay the quickest. We find that the rise could be either produced physically by the start of the forward shock, when the jet begins to plough into the external medium, or geometrically where an off-axis observer sees a rising light curve as an increasing amount of emission enters the observers line of sight, which occurs as the jet slows. We find that at 99.8 per cent confidence, there is a correlation, in the observed frame, between the apparent magnitude of the light curves at 400 s and the rate of decay after 500 s. However, in the rest frame, a Spearman rank test shows only a weak correlation of low statistical significance between luminosity and decay rate. A correlation should be expected if the afterglows were produced by off-axis jets, suggesting that the jet is viewed from within the half-opening angle θ or within a core of a uniform energy density  θ_c  . We also produced logarithmic luminosity distributions for three rest-frame epochs. We find no evidence for bimodality in any of the distributions. Finally, we compare our sample of UVOT light curves with the X-ray Telescope (XRT) light-curve canonical model. The range in decay indices seen in UVOT light curves at any epoch is most similar to the range in decay of the shallow decay segment of the XRT canonical model. However, in the XRT canonical model, there is no indication of the rising behaviour observed in the UVOT light curves.  相似文献   

An examination of the hypersaline phases of Eocene Lake Uinta,upper Green River Formation,Uinta Basin,Utah     

Michael D. Vanden Berg  Lauren P. Birgenheier 《Journal of Paleolimnology》2017,58(3):353-371

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Motion Near the Unit Circle in the Three-Body Problem     

Roger A. Broucke 《Celestial Mechanics and Dynamical Astronomy》1999,73(1-4):281-290

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Numerical integration of periodic orbits in the main problem of artificial satellite theory     

R. A. Broucke 《Celestial Mechanics and Dynamical Astronomy》1994,58(2):99-123

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ ₂ problem). The periodic orbits have been classified according to their stability and the Poincaré surfaces of section computed for different values ofJ ₂ andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (–1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincaré first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ ₂, as large as |J ₂|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.  相似文献