Volume 145, number 5 PHYSICSLETTERSA 16 April 1990 CHAOS IN THE KEPLER PROBLEM AND LONG-PERIOD COMET DYNAMICS M.Ya. NATENZON, A.I. NEISHTADT, R.Z. SAGDEEV, G.K. SERYAKOV and G.M. ZASLAVSKY Space Research Institute, Academy of Sciences of the USSR, Moscow 11781 O, USSR Received i November 1989;accepted for publication 5 December 1989 Communicatedby V.M. Agranovich Numerical investigationsof chaotizing perturbation due to Jupiter of the long-periodcomet motion are performed.The phase portrait of this problem is built. The comet diffusion is described. 1. Introduction Comet dynamics in the field of the sun and a large planet (Jupiter, for example) poses a certain case of the well known classical three-body problem (see ref. [ 1 ] ). Long period comets represent a special case of this problem. Currently there exists the point of view that a source of comets is the hypothetical Oort cloud. This cloud lies in the 104-2 × 105 AU region [2]. There is a large number of papers dealing with the Oort cloud comet problem. The comet dynamics as a plane three-body problem was considered in refs. [ 3,4 ], the map to which this problem can be reduced was built, and it was shown that the comet motion may be chaotic. This principal result may influence essentially the evolution problem of the Oort cloud [ 5 ]. For this case the detailed analysis of the comet dynamics is of essential interest and completes the well known numerical modeling of the Oort cloud [ 6-8 ]. Nevertheless investigations of comet dynam- ics from the dynamic system theory viewpoint is in- teresting enough and is continued in refs. [9-11 ]. The present paper is directed to the same goal - sub- sequent analysis of the so-called plane restricted cir- cular three-body problem. Numerical integration data are presented in this work, both as more precise es- timates of the comet energy change due to Jupiter interaction and the more precise Kepler map. The essential features of diffusion dynamics due to the strong intermittency property are also discussed. A simple understanding of the chaotic comet dynamics may be gained by considering the falling of a ball onto a platform in a gravity field (fig. 1 ) and regarding its collision as elastic [ 12,13 ]. As the platform os- cillates with oscillation amplitude a and velocity am- plitude U, under the condition 2U>ag (g is the gravitational acceleration) the phase of the platform oscillations at the moment of collision is a random one. The ball jumps onto the platform without order and on average goes up and up. Its average kinetic energy in the return point grows as (v 2) ~t 2/3 and the average altitude accordingly grows as (h) ~ t 2/3. The return time obviously grows but the acceleration does not stop. This example comments to some extent on the dynamics of long period com- ets having perihelia near the Jupiter region. Roughly I I Fig. 1. The stochastic acceleration of a ball jumping upon a platform. 0375-9601 / 90/$ 03.50 © Elsevier Science Publishers B.V. ( North-Holland ) 2 5 5