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By Mark D. Ardema

Unlike different books in this topic, which are inclined to pay attention to 2-D dynamics, this article specializes in the applying of Newton-Euler how you can complicated, real-life 3-D dynamics difficulties. it's hence excellent for non-obligatory classes in intermediate dynamics.

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The function ,(re ) can then be graphed, and the behavior of the graph can be used to determine the admissible values of r eo For the case of the inner orbits, it is necessary that , ~ 1, and for the outer orbits, , ~ 1. Only those values of r e yielding, in these respective ranges are acceptable. Moreover, r e are further restricted to correspond to rational or near rational values of ,. If the assumptions are realistic, then a majority of the observed Kuiper belt resonances should be recovered.

And Lohinger, E. (1993) Celest. M ech. Dynam. Astron. 56 , pp. 307-314 8. Voglis, N. and Contopoulos, G. (1994) Invariant Spectra of Orbits in Dynamical Systems, J. Phys A: Math. Gen. 27 , pp. 4899-4909 9. Contopoulos, G. and Voglis, N. (1966) Spectra of Stretching Numb ers and Helicity Angl es in Dyn amical Systems, it Celest. Mech. Dynam. Astron. 64, pp . 1-20 10. , Nesvorny, D. (1994) Temporary Capture of Grains in Exterior Resonances with th e Earth ,Astron. A strophys. 289, pp . 972-982 11.

Usually the first asteroidal frequency obtained from k + ih analyses denoted 9 is the frequency of perihelion motion, the first frequency from p + iq denoted s is the frequency of the motion of the node s. There are exceptions when the most prominent frequency is equivalent to some of the planetary frequencies . When we detect that, we list the second frequency (the new frequency which is not present in the planetary system without asteroids) in Tab. 2. Tab. 1 shows the 9 and s for planets from our calculations and for Longstop 1A [131 .

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