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Astronomical Applications Department, U.S. Naval Observatory - lyap2 (Page 4)

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Astronomical Applications Department, U.S. Naval Observatory - lyap2
quences of increasing Jupiter's mass to speed
up perturbation effects.
The dynamical system utilized for this study
is the three-dimensional elliptic restricted three-
body (ERTB) problem, with Jupiter as the sec-
ondary mass m
. We integrated the equations
of motion in the rotating-pulsating frame (cf.
Szebehely 1967, Szebehely and Giacaglia 1964)
with a Bulirsch-Stoer extrapolation method
(e.g. Stoer and Bulirsch 1980, Press et al.
1992). See Murison (1989) for details on the
numerical performance of the integrator. The
(constant) eccentricity used for Jupiter in all
cases was e
=0.04848. This model is conven-
ient because of its simplicity. The ERTB prob-
lem is the simplest dynamical model which is
still complex enough to exhibit all the important
behavior required for this study. Comparisons
with a model that includes the effects of Saturn
on the motion of Jupiter (cf. Lecar and Frank-
lin, 1973) yielded no significant differences that
would affect our results in the region of the
asteroid belt considered here. In particular, an
integration of (2311) El Leoncito yielded a T
of 460 T
, fully consistent with our value of 422
. Relying on a reasonable and simple statisti-
cal interpretation, as well as indications from
recent nonlinear dynamics literature, we conjec-
ture that the property of certain orbits to
exhibit long-term "stability" despite short
Lyapunov times is in fact to be expected, as we
argue in the next section.
Along with each orbit, a second "test" orbit
was integrated. This second orbit started a dis-
tance 10
in phase space from the reference
orbit and was used for calculating the
Lyapunov exponent. Renormalization of the
test orbit with respect to the reference orbit
occurred whenever the phase space distance
exceeded 10
, thus avoiding saturation prob-
lems. See e.g. Benettin et al. (1976) and Wolf
et al. (1985) for details on calculating Lyapu-
nov exponents.
We obtained orbital elements for the 25
We performed numerical integrations for
each of the 25 asteroids. Predicted event times
were determined by application of the T
relation, eq. (1), to the calculated Lyapunov
exponent. The length of these integrations was
Jovian years (~1.2 Myr) ­ sufficient to
determine whether or not the predicted event
time is within several standard deviations of the
age of the solar system, T
. Longer integra-
tions, though certainly possible, are in fact
unnecessary, since the shortest Lyapunov times
(those most likely to be in serious disagreement
with the age of the solar system) are accurately
In order to investigate the T
relation for
a high mass ratio, we integrated 440 fictitious
outer-belt orbits over a wide range of initial
semimajor axis with m
=10 M
. The initial
eccentricity of these orbits was 0.05, the longi-
tude of pericenter 320 deg, inclination 3 deg,

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