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*To*: Meteorite List <meteorite-list@meteoritecentral.com>*Subject*: Resonance - Part 4 of 5*From*: Bernd Pauli HD <bernd.pauli@lehrer1.rz.uni-karlsruhe.de>*Date*: Wed, 26 May 1999 00:19:21 +0200*Old-X-Envelope-To*: <meteorite-list@meteoritecentral.com>*Resent-Date*: Tue, 25 May 1999 18:23:22 -0400 (EDT)*Resent-From*: meteorite-list@meteoritecentral.com*Resent-Message-ID*: <hc5icC.A.kpE.IJyS3@mu.pair.com>*Resent-Sender*: meteorite-list-request@meteoritecentral.com

Gravitational Hypothesis The gravitational hypothesis, the most widely accepted explanation for the origin of the Kirkwood gaps, says the gaps are formed by purely gravitational interactions with Jupiter. Both Stanley Dermott and Carl Murray in 1981 and 1983, and Jack Wisdom in 1982 and 1983 have provided strong evidence to support the gravitational hypothesis. In the first study, 757 main-belt, non-family asteroids were used. To analyze the properties of asteroids and their orbits, Dermott and Murray defined a parameter d as the measure of the displacement of an asteroid from the nearest strong resonance. An exact resonance corresponds to d = 0, while /d/ = 1 is the greatest displacement from resonance that an asteroid can have. A direct plot of absolute magnitude, B(1,0) versus d is shown in Fig. 5-9. Figure 5-10 shows a smoothed number density contour plot based on the same data. Since the number density increases continuously as d increases, this implies the effects of resonance pervade the whole distribution of main-belt asteroids. Dermott and Murray showed that observational selection produces an excess of high magnitude, low inclination asteroids. Thus, they reduced their data set to 144 asteroids, constituting a bias-free set, from which they concluded there is no magnitude-frequency distribution change near the Kirkwood gaps. This result was mentioned as tending to disprove the collisional hypothesis. More importantly, they found the tendency for both eccentricity and inclination to increase with increasing /d/ is a fundamental property of the Kirkwood gaps (Fig. 5-11). Since the magnitude of the gravitational disturbing function acting on an asteroid increases with e and i, this result strongly suggests that Jupiter's gravitational effect creates the Kirkwood gaps. They conclude that at least some of the Kirkwood gaps have been formed since the time of formation of the solar system, and that they are not simply regions of small asteroid number density since the effects of resonance pervade the entire asteroid belt. They confirmed and extended these conclusions in 1983. Fig. 5-12 shows quite dramatically the lack of asteroids at resonant positions in the belt, particularly at 3.3 AU and 4.0 AU where resonances overlap. The few asteroids that remain in a libration region tend to have eccentricities much higher than the average of their neighbors. Near the 3/1 commensurability, for example, only two asteroids are known with librating orbits: Alinda (e = 0.55) and Quetzalcoatl (e = 0.58). According to the gravitational mechanism, Iarge eccentricities can be expected for librating orbits whose rate of change of perihelion longitude is near zero. As shown by Scherbaum & Kazantsev (1985), this is precisely what is found. Wisdom (1982) used a set of 300 'test asteroids' in the neighborhood of the 3/1 commensurability to determine if gravitational influences over a period of 2 million years could produce a gap. Such a gap formed, but was narrower than the actual one. Using improved computational techniques, he repeated the test. "Of the 300 test asteroids in the random distribution, 89 were found to have chaotic trajectories and only 11 were quasiperiodic librators. All but five of the chaotic trajectories became Mars crossing within 300,000 years and only one had not reached an eccentricity of 0.3 within 1 million years. The predicted gap is now in satisfactory agreement with the full distribution of real asteroids." (Wisdom, 1983). As Fig. 5-13 clearly shows, the boundaries of the chaotic zone discovered by Wisdom match the boundaries of the real 3/1 Kirkwood gap quite well. ---------- 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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