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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Modelling of lightcurves of minor planetary satellites
A. V. Melnikov
Pulkovo Observatory, St. Petersburg, Russia
This report presents results of modelling of lightcurves of minor planetary
satellites. Lightcurves of two satellites of Saturn, namely Hyperion (S7) and
Phoebe (S9), were modeled using algorithms and programs developed for calculat­
ing the rotational dynamics and constructing theoretical lightcurves of planetary
satellites. The modelling was based on Pulkovo sets of observations carried out by
Devyatkin et al. [1]. Hyperion's observations performed earlier by Klavetter [2]
were also used for modelling.
The following assumptions were made: the satellite is a nonspherical tri--axial
rigid body; the planet is considered to be a gravitating point. In case of Hyperion
the motion in the perturbed elliptic orbit was considered, because its orbit is
subject to strong short­period perturbations from Titan. The orbit of Phoebe
was taken to be a fixed ellipse, and mean values of the orbital elements [3] were
used. In case of Phoebe the perturbations are essentially smaller; besides, the
time interval for the modelling is less than the orbital period.
In calculations of the observed stellar magnitude it was assumed that a satel­
lite is a tri--axial ellipsoid, and its surface is orthotropic: the light flux from the
satellite is proportional to the area of projection of the visible illuminated part
of the satellite's surface on the celestial sphere. Deviations from orthotropicity
for the reflecting surface were taken into account by means of correction of the
model lightcurves for the ``Sun -- satellite -- observer'' phase angle using a definite
phase function.
The problem of fitting an observed lightcurve with the model one is solved by
varying the initial data and values of the parameters of the problem. As an initial
step a rough approximation to the observed lightcurve was found by minimizing
the sum of squares of deviations of theoretical values of the satellite's stellar
magnitude from the observed ones. Finally, the initial data and values of the
parameters were refined by the steepest descent (gradient) method.
The rotational states of satellites and values of the parameters of phase func­
tion were deduced in this way for four sets of observational data on Hyperion
(three sets of Pulkovo observations carried out by Devyatkin et al. [1] and a set
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of observations carried out by Klavetter [2]) and for a set of Pulkovo observational
data [1] on Phoebe.
In case of Hyperion, the initial data, which specify the model curve best
fitting the observed one, are situated in a chaotic component of phase space of
the rotational motion. The computation of the maximum Lyapunov characteristic
exponent (MLCE) was performed for all sets of the initial data and values of the
parameters of the problem. The computed values of the MLCE are close to its
analytical estimates calculated by means of the separatrix map theory [4]. For
all the sets, the value of the MLCE is greater then zero. The distinction of the
value of the MLCE from zero is another indicator of the chaotic character of
the rotation. One can make conclusion that Hyperion in the period covered by
observations was in the chaotic regime of the rotational motion.
The initial data for trajectories of the rotational motion of Phoebe are situated
in the regular domain of phase space of the rotational motion. The value of the
MLCE calculated for the deduced initial data and values of the parameters is
close to zero. The zero value of the MLCE again points on the regular character
of the rotation. The obtained period of rotation of Phoebe is equal to 9:3 h . This
is close to an earlier estimation by Kruse et al. [5].
This work was partially supported by the Russian Foundation of Basic Re­
search under grant 01­02­17170.
References
1. Devyatkin A. V., Gorshanov D. L., Gritsuk A. N. et al. Observations and the­
oretical analysis of lightcurves of natural satellites of planets. Astron. Vestn.,
2002, 36, No. 3 (in press, in Russian).
2. Klavetter J. J. Rotation of Hyperion. I. Observations. Astron. J., 1989, 97,
570--579.
3. Jacobson R. A. The orbit of Phoebe from earthbased and Voyager observa­
tions. Astron. & Astrophys. Suppl. Ser., 1998, 128, 7--17.
4. Shevchenko I. I. On the dynamical entropy of rotation of Hyperion. Izv. GAO
RAN, 2000, 214, 153--160 (in Russian).
5. Kruse S., Klavetter J. J., Dunham E. W. Photometry of Phoebe. Icarus.,
1986, 68, 167--175.
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