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*To*: Meteorite List <meteorite-list@meteoritecentral.com>*Subject*: Resonance - Part 2 of 5*From*: Bernd Pauli HD <bernd.pauli@lehrer1.rz.uni-karlsruhe.de>*Date*: Wed, 26 May 1999 00:19:03 +0200*Old-X-Envelope-To*: <meteorite-list@meteoritecentral.com>*Resent-Date*: Tue, 25 May 1999 18:21:44 -0400 (EDT)*Resent-From*: meteorite-list@meteoritecentral.com*Resent-Message-ID*: <P_vZ5.A.LmE.XIyS3@mu.pair.com>*Resent-Sender*: meteorite-list-request@meteoritecentral.com

Asteroids can even be in resonance with one another. Opik (1970) examined the effect of Ceres on Pallas, which have nearly identical mean distances from the Sun, mean angular motions and periods of revolution. This 1/1 commensurability results in conjunctions approximately every 2,640 years (the last one being about 313 years ago). Opik concludes that the two asteroids are little affected by resonance, having retained their orbital characteristics since their formation in the same "ring of diffuse matter." The stability of a resonance is determined by its phase angle. If it librates about some angle, usually 0 or 180°, the resulting resonances will be very stable. Such is the case for 4/3, 3/2, and 2/1. At 1/1, the phase angle librates about 60°, a case that will be considered in Chapter 15. As an example, consider the asteroid 1362 Griqua at the 2/1 resonance. In 1970, Brian Marsden did a study of Griqua's orbit. Fig. 5-8 shows the oscillations of the phase angle for a period of 3,000 years. Griqua librates about 0° with an amplitude of 100° to 120° in a period of close to 400 years. The numbers at the maxima and minima of the curve show the least distances from Jupiter in AU. These take place when the libration is near its extremes. Marsden concluded that Griqua's nodal period is 34,000 years. Liu and Innanen (1985) have found that the resonant phenomenon becomes more pronounced as the orbital eccentricity of the asteroid increases. In addition, the variations of the elements a, e and i are larger in the regions of the resonant zone which lie just inside or outside the boundary of the libration region. While the mathematical theory of resonances is beyond the scope of this book, it is important to understand the basic principles and their shortcomings. Most models have been based on the three-body problem (Sun-Jupiter-asteroid), which is a reasonable approximation since the disturbing influence of Jupiter far exceeds that of the other planets (Giffen, 1973). In the study of planetary motions, the theory of secular perturbations can be used over intervals of 10^5 years. This theory fails in the case of commensurable motion, however, due to the presence of the infamous small divisors in the series expansions of the perturbations. The alternative of numerically integrating the equations of motion directly also fails for long time intervals due to cumulative round-off errors. Thus, almost every theoretical and numerical study of the asteroid resonances rely on the so-called averaging principle. It was first introduced by Lagrange and Laplace, and was explicitly stated by Gauss, who replaced each planet by a ring of mass whose density at each longitude was inversely proportional to the planet's velocity at that longitude. Following an extension of this idea by Poincaré in 1902, J. Schubart developed a model in the 1960's for the investigation of commensurable motion at a resonance. This averaging principle remains an intuitive assertion which is, strictly speaking, untrue. According to V.I. Arnold, "This principle is neither a theorem, an axiom, nor a definition, but a physical proposition. Such assertions are often fruitful sources of mathematical theorems." The averaging principle is used simply because it is too expensive to study unaveraged equations of motion. The averaging process eliminates short-period variations, making the equations tractable. With this caveat in mind, we can now look at the four theories put forward to explain the Kirkwood gaps. ---------- Archives located at: http://www.meteoritecentral.com/list_best.html For help, FAQ's and sub. info. visit: http://www.meteoritecentral.com/mailing_list.html ----------

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