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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:
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.
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:
