1975 -- 2003

The research group on Solar System Dynamics of IAG-USP included among its science projects, since the 90's, the Dynamics of Planetary Systems. This decision was triggered by previous studies on the influence of the 5/2 Jupiter-Saturn near resonance on the stability of asteroids and the problems initially considered were related to the dynamics of the outer planets. The discovery of the first extra solar planetary system (u And), in 1999, reinforced that decision.

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:

After 15 years, the research interests shifted to Solar System Dynamics with emphasis in systems showing chaos or capture. In this period the following topics were successfully considered:


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

  • (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 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
    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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    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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    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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    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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    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:

    November, 2003