**DEPARTMENT
OF ASTRONOMY**

**RESEARCH
GROUP IN DYNAMICS OF PLANETARY SYSTEMS**

SOLAR SYSTEM DYNAMICS

MATHEMATICAL AND DYNAMICAL ASTRONOMY

This research group exists at IAG-USP since 1975. It began as a research group on Mathematical and Dynamical Astronomy whose activities ranged from astronomical problems related to the motion of planetary satellites to the mathematical problems originated by this and other astronomical problems. In the first 10 years, results were obtained on several topics:

- (1) The analysis of series of observations with irregular time spacing;
- (2) the construction of Hamiltonian averaging theories to study resonant problems;
- (3) the analysis of observations of the Galilean satellites of Jupiter;
- (4) the study of the motion of the Uranian satellites.

- (1) Cosmogony of Kirkwood gaps and groups;
- (2) capture in resonance due to dissipative forces.

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DETERMINATION OF PERIODS FROM UNEVENLY SPACED TIME SERIES

This is the subject whose results have had the largest impact at that time. The analysis of unevenly spaced time series originated from astronomical observations is one big problem in Astronomy. Astronomers do not observe following their will. Planets, satellites, asteroids, stars, etc. are not visible the whole year but just on some times. Besides, climatic conditions may frustrate an observational program. For these reasons, the study of a series of astronomical observations cannot be done using ordinary Fourier transforms, which are tools to study time series evenly spaced and infinite. It is easy to show that trigonometric functions do not form an orthogonal and basis to best fit points sampled irregularly. Besides, one of the components of the basis shall necessarily be a constant. Classical formulas were anyway used notwithstanding being beyong their validity hypotheses. To solve this problem, the DCDFT (Data Compensated Discrete Fourier Transform), with a basis orthogonalized and including the constant component, was proposed. Other authors had also proposed routines with an orthogonal basis, but not including the contant component. DCDFT is being used for the analysis of time series (observations) in several disciplines and has been shown to be essential in some main cases

The DCDFT has been also successfully used in the analysis of data generated by numerical simulations when looking for the precise determination of low-frequency oscillations. In this case, notwithstanding the uniform distribution of the data, the series have finite number of terms and the analysis with the FFT (for instance) lead to errors in the parameters of oscillations whose period is of the same order of magnitude(1) Determination of period and amplitudes of low frequency signals. (2) Analysis of time series with low signal-to-noise ratio; (3) Successive determination of the main frequencies in a signal through harmoni filtering.

the total time span of the series.

DCDFT
has been recently reformulated by G.Foster (AAVSO-Harvard, USA) that, taking
advantage of the much better performance of nowadays computers, substitute
the previous analytical construction of a data compensated basis, by more
general numerical procedures. His method, the CLEANest, is mathematically
equivalent to the DCDFT when just one frequency is considered, but allows
the simultaneous determination of several frequencies. Other extensions
of the DCDFT are the completed-basis wavelets, developed by G. Foster,
and the time-frequency analysis developed by T. Gallardo (Univ. Montevideo)
(codes available).

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HAMILTONIAN AVERAGING THEORIES AND RESONANCE.

The investigations
in this topic were strongly influenced by the visit of Prof. Gen-Ichiro
Hori, (Univ. Tokyo) to IAG-USP in 1976. The research group was already
using Lie series to average some resonant Hamiltonian systems, but following
a very classical direction, similar to that found in the developments with
Jacobian generating functions. Some contributions come from this period.
The first was purely theoretical. The original paper by Hori introduces
a pseudo-time presented in a not satisfactory way which, with an obvious
exaggeration, was considered by some people as an error. S.Ferraz-Mello,
by mean of Cauchy's theory of characteristics showed that Hori's theory
was absolutely correct. He also proved that the Auxiliary System introduced
by Hori was just part of Cauchy's equations of the characteristics and
that the so-called pseudo-time was not but the independent variable introduced
by Cauchy's theory. Another contribution, due to W. Sessin, was the discovery
of the transformation later called "reducing transformation", which allows
an integrable Auxiliary System including the terms of first-order in the
eccentricities to be obtained in the problem of the resonant motion of
two planets (or, one asteroid disturbed by one planet). Other contributions
in this period come from the analyses of the problems with 2 simultaneous
resonances (T.Yokoyama, UNESP). More recently Hori's theory was extended
to resonant systems with several degrees of freedom allowing the solution
of Bohlin's problem in more general cases.

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CHAOS AND COSMOGONY OF THE KIRKWOOD GAPS AND GROUPS

The IAG-USP
research group in Solar System Dynamics, for a long time concentrated all
efforts in the study of Asteroid Dynamics. The main results were related
to the absence of permanent asteroids in 3/1 and 2/1 resonance with Jupiter
in contrast with the existence of many asteroids in the 3/2 resonance.
The first theory allowing an explanation of the 3/1 Kirkwood gap, was due
to J.Wisdom (MIT): asteroids in the 3/1 resonance may chaotically diffuse
and may have their eccentricities increased up to 0.3, thus intercepting
the orbit of Mars and becoming able to have a close approach to the planet
and escaping from the resonance thanks to the energy exchanged during the
approach. One difficulty of the classical models used is the non-convergence
of the series used to represent the disturbing forces. In the case of resonant
asteroids, the first model valid at high eccentricities was developed by
S.Ferraz-Mello and J.C.Klafke. This model has shown that other modes of
motion exist in the 3/1 resonance (around the stable corotation point)
and that chaotic diffusion can increase the eccentricities to more than
0.9. With such an eccentricity, the orbit may cross also the orbits of
Venus and the Earth, planets 10 times larger than Mars being thus more
efficient in the scattering of asteroids out of the resonance. Asteroids
in the 3/1 resonance diffusing chaotically to earth-crossing orbits may
be responsible by great collisions with these planets in the first Gyr
of existence of the Solar System (they were soon destroyed). The large
craters of the Moon may be the scar of the collisions in that period. In
the 2/1 resonance, the reality has been shown to be very different and
the trials to extend Wisdom model to this resonance were not successful.
The study of that resonance at IAG-USP started with numerical simulations
with on-line low-pass filtering. The results of T. A. Michtchenko have
shown the regularity of the asteroid motions in large regions inside the
resonance when only Jupiter is in a fixed orbit. Later, S.Ferraz-Mello,
D.Nesvorný, F.Roig and T.A.Michtchenko used more complete models
including the perturbations of the orbit of Jupiter due to Saturn. Several
different techniques were used: Frequency variation maps, Hadjidemetriou's
symplectic maps, symplectic integrators, etc. The results have clearly
shown that the chaotic diffusion of the orbits exist but is slow; in some
regions an asteroid can remain for 1 Gyr or even more. This diffusion is
enhanced by the fact that Jupiter and Saturn move lose to the 5/2 commensurability
of periods. The results were extended to the 3/2 resonance and have shown
that the dynamics of the asteroids in that resonance (the Hildas) is very
similar to that of the asteroids in the 2/1 resonance but the chaotic diffusion
is much slower, allowing the Hildas to remain in the resonance for times
much larger than the age of the Solar System, thus explaining the important
population observed there.

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CAPTURE IN RESONANCE DUE TO DISSIPATIVE FORCES

Problems
in which particles are spiraling towards the central star due to dissipative
forces and have the fall stopped by capture in an orbit in resonance with
a large planet were studied by C. Beaugé and S. Ferraz-Mello.
This was considered as an explanation for the existence of the dust cloud
around the star Beta Pictoris. This cloud should no longer exist because
of the continuous energy dissipation of the cloud particles due to Poynting-Robertson
effect. A similar phenomenon could have occurred during the formation of
the planets, when a large amount of gas still existed around the Sun and
the planetesimals were spiraling to the Sun because of gas drag. The existence
of one planet being formed can stop the fall. In this case, not only the
drag force the planetesimals to move in an orbit in resonance with the
embryo but also drive them to have coherent motions (stable corotation)
favoring their accretion to form a new planet. Some numerical simulations
have shown the formation of a planet in 5/2 resonance with the planet resulting
from the former embryo.

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DYNAMICS OF PLANETARY SYSTEMS.

The planetary
systems considered are those in the outer part of our Solar System: Jupiter,
Saturn, Uranus and Neptune, and also those of the several planetary systems
recently discovered around several main sequence stars and one pulsar.
The investigations on the giant planets of our system were leaded by T.A.Michtchenko.
The aim of these investigations was to understand the dynamics of systems
formed by two planets with periods nearly commensurable. The 5/2, 2/1 and
3/2 resonances were studied with many details. Besides, a chart of the
chaotic zones in the neighborhood of the large planets of our Solar System
was calculated. The same was done in the neighborhood of the planets of
the pulsar PSR 1257+13. S.Ferraz-Mello, T.A.Michtchenko e C.Beaugé
(Obs. Córdoba, Argentina) have extended results of Celestial Mechanics
to the new extrasolar planetary systems. The results obtained so far explain
the pairs of planets whose semi-major axes have the same direction. This
is observed in two cases: the so-called secular resonances, where the perihelia
of the two planets oscillate around privileged directions (they can be
aligned or anti aligned), and in the resonances due to commensurability
of periods. In this case, besides systems oscillating around stationary
solutions with aligned perihelia (Dv=0)
or anti aligned (Dv=180
degrees), stationary solutions were discovered in which the angle Dv
is
frozen at other (any) fixed values.

It was
also shown that under the action of dissipative forces the orbits may be
captured into resonance and slowly evolve to the a stationary situation
with corotating perihelia. Some new techniques developed during these investigations
proved to be essential:

- A Frequency Analysis tool using the Fast Fourier Transform of the simulations outputs allowing the charting of the zones where the motions are chaotic and their intensity (T.A.Michtchenko);
- A Modeling of the Potential Energy of the gravitational interaction between the planets valid for planets in high-eccentricity orbits (C.Beaugé, T.A.Michtchenko)

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November, 2003