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Fig. 5


Each point on the (e,ϵ) plane represents an elliptical orbit with eccentricity e and energy − ϵ. The ABC triangle with vertices A , B , C is a locus of all confocal ellipses lying entirely in-between two bounding spheres of fixed radii ua and ub and centered on the focal point. The two families of mutually crossing lines through point (− 1,0) and through point (1,0) give rise to a new coordinate system (α,β) on the (e,ϵ) plane. When and the coordinates cover a u-dependent quadrilateral integration domain abcB with vertices a , b , c , and B, which is the locus of all orbits (e,ϵ) crossing at least once a sphere of a given radius ua < u < ub.

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