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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
The Lyapunov spectra in spin--orbit dynamics
V. V. Kouprianov, I. I. Shevchenko
Pulkovo Observatory, St. Petersburg, Russia
We study the problem of the chaotic rotation of a minor planetary satellite
around its mass center, in the planar and spatial settings. In the planar setting
a satellite rotates around the axis of its maximum moment of inertia which is
orthogonal to the orbit plane, while in the spatial setting it may rotate in any
direction. Whether the motion is chaotic or regular, one can infer by means of
calculation of the Lyapunov characteristic exponents (LCEs) [1, 2]. The LCEs
describe rates of exponential divergence of close trajectories in phase space.
For a set of eleven satellites of Mars, Jupiter, Saturn, and Neptune (the
satellites with known values of inertial and orbital parameters) the full Lya­
punov spectra of chaotic rotation are computed numerically. Among these
satellites, currently only Hyperion (S7) is confirmed to rotate chaotically. The
present rotation modes of five other satellites, namely Helene (S12), Atlas (S15),
Prometheus (S16), Pandora (S17), and Proteus (N8), are unknown. The rest of
the satellites in our set rotate regularly in synchronous resonance; but, in any
case, each of them had passed through the stage of chaotic rotation. Calculation
of the LCE spectra for these satellites extends our knowledge of their dynamical
history.
A satellite is modelled as a tri--axial rigid body moving in a fixed elliptic
orbit. The dynamic system is described by the Euler equations [3]. The param­
eters of the problem are the orbital eccentricity of a satellite e and the ratios
of its principal central moments of inertia A=C and B=C (A ! B ! C). These
ratios characterize the dynamical asymmetry of a satellite. The LCE spectrum
is computed by the HQR method in the version of von Bremen et al. [4]. This
method is based on the QR decomposition of the tangent map matrix using the
Householder transformations.
The maximum LCEs for the satellites in our set are estimated analytically
using the separatrix map theory in the model of nonlinear resonance (here syn­
chronous spin--orbit resonance) as a perturbed nonlinear pendulum [5, 6]. This
approach is based on the hypothesis by Chirikov [1] that the dynamical entropy
of the separatrix map is constant in the high--frequency limit of perturbation.
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A key role in the method belongs to the average dependence of the dynamical
entropy of the separatrix map upon – (the ratio of the perturbation frequency
to the frequency of small oscillations on the resonance) in the whole range of the
values of – (0 ! – ! +1).
Comparison of the results of numerical and analytical estimations of the max­
imum LCEs shows a good agreement in the case of planar rotation. It is shown
also that the theory developed for the planar case is most probably still applicable
in the case of spatial rotation, if the dynamical asymmetry of the satellite is suffi­
ciently small, A=C ?
ё 0:8, or/and the orbital eccentricity is relatively large, e ?
ё 0:02
(but not too large, in order for the dynamical model to be valid). Otherwise the
theory should be different.
It is plausible that chaotic dynamics of strongly asymmetric satellites in nearly
circular orbits, in the case of spatial rotation, is determined mainly by interaction
of internal coupling resonances. In order to check this, we have recomputed the
LCE spectra for our set of satellites, formally assigning the orbital eccentricity
to zero. Comparison with the LCEs in case of the actual eccentricities provides
a hint for the development of a future theory.
This work was supported by the Russian Foundation of Basic Research
(project number 01­02­17170).
References
1. Chirikov B. V. A universal instability of many­dimensional oscillator sys­
tems. Phys. Rep., 1979, 52, 263--379.
2. Lichtenberg A. J., Lieberman M. A. Regular and chaotic dynamics. New
York: Springer, 1992.
3. Beletsky V. V. The motion of an artificial satellite about its mass center.
M.: Nauka, 1965 (in Russian).
4. Von Bremen H. F., Udwadia F. E., Proskurowski W. An efficient QR based
method for the computation of Lyapunov exponents. Physica D, 1997, 101,
1--16.
5. Shevchenko I. I. On the dynamical entropy of the rotation of Hyperion.
Izvestia GAO, 2000, 214, 153--160.
6. Shevchenko I. I. On the maximum Lyapunov exponents of the chaotic ro­
tation of natural planetary satellites. Kosmich. Issled., 2002, 40, No. 3 (in
press, in Russian).
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