**31st Annual Meeting of the DPS, October 1999**

*Session 12. Near Earth Asteroids Posters*

Poster Group I, Monday-Wednesday, October 11, 1999, , Kursaal Center
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## [12.08] Short Time Lyapounov Indicators in the Case of a Sun-Jupiter-Saturn-Asteroid System

*R.F. Balla (ELTE), Zs. S\'andor (KOHAS)*

In our previous papers (Sándor et al., 1999a; Sándor 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 <s>_{N} curve, where
<s>_{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''
|\Delta <s>_{N}/\Delta a|, where \Delta a =
a_{i+1}-a_{i} is the difference between two consecutive
initial semimajor axis and \Delta <s>_{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.

The author(s) of this abstract have provided an email address
for comments about the abstract:
brobi@innin.elte.hu

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