Short Time Lyapounov Indicators in the Case of a Sun-Jupiter-Saturn-Asteroid System

Details Short Time Lyapounov Indicators in the Case of a Sun-Jupiter-Saturn-Asteroid System Short Time Lyapounov Indicators in the Case of a Sun-Jupiter-Saturn-Asteroid System

Sep 1999

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1999dps....31.1208b&link_type=abstract

American Astronomical Society, DPS meeting #31, #12.08

Astronomy and Astrophysics

Astronomy

Scientific paper

In our previous papers (Sandor et al., 1999a; Sandor et al., 1999b) we have discussed the application of the short time indicators in the planar circular Restricted Three Body Problem (RTBP) and in the Elliptic Restricted Three-Body problem (ERTBP) in order to distinguish between chaotic and regular domains of the phase space in theese problems. The method of stretching numbers was introduced by Voglis and Contopoulos (1994). This method allows a quick distinction between ordered and chaotic regions. We also applied the method of stretching numbers to the elliptic restricted three-body problem. As an extension of our investigation, in the present paper we apply the method of stretching numbers to a realistic Sun-Jupiter-Saturn-Asteroid (SJSA) problem. We represent the structure of the phase-space in the a-e plane, where a is the semimajor axis and e is the eccentricity. For an individual N curve, where N is the average value of stretching numbers. The values of the semimajor axis has been taken from the interval [3.2,5.2] (AU) for a fixed value of the eccentricity of the test particle between e=0 and e=0.4. For a good visualization of the regular and chaotic regions in the a-e plane we have processed the curves of average values calculating the absolut value of their ``derivative'' |frac {Delta N}{Delta a}|, where Delta a = a_{i+1}-ai is the difference between two consecutive initial semimajor axis and Delta N is the corresponding change of the average value of stretching numbers. If this derivative is larger than a certain value (in our case 0.002), the corresponding region between two neighbouring initial conditions is classified as chaotic. The usefulness of this method is based on the very fast and effective way how it approximates the location and size of the regular and chaotic regions. We have found that the structure of the phase-space is very similar in the RTBP and in the ERTB but there is a significant difference in the case of the SJSA. The grant OTKA F030147 of the National Research Foundation is acknowledged.