J. Frouard

Northwestern University, Evanston, Illinois, United States

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Publications (11)15.65 Total impact

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    ABSTRACT: Mercury's spin state is peculiar, in that it is locked into the 3:2 spin-orbit resonance. Its rotation period, 58 days, is exactly two thirds of its orbital period. It is accepted that the eccentricity of Mercury (0.206) favours the trapping into this resonance. More controversial is how the capture took place. A recent study by Makarov has shown that entrapment into this resonance is certain if the eccentricity is larger than 0.2, provided that we use a realistic tidal model, based on the Darwin-Kaula expansion of the tidal torque, including both the elastic rebound and anelastic creep of solids. We here revisit the scenario of Mercury's capture into the supersynchronous spin-orbit resonances. The study is based on a realistic model of tidal friction in solids, that takes into account the rheology and the self-gravitation of the planet. Developed in Efroimsky, it was employed by Makarov et al. to determine the likely spin state of the planet GJ581d, with its eccentricity evolution taken into account. It was also used in the afore-cited work to study the tidal spin-down and to find the likely end-state of a Mercury-like planet with its eccentricity fixed. We now go ahead by considering the evolution of Mercury's eccentricity. We find that the realistic tidal model, as opposed to the constant time lag and constant phase lag models, changes dramatically the statistics of the probable final spin-orbit states. First, after only one crossing of the 3:2 resonance this resonance becomes the most probable end-state. Second, if a capture into any resonance takes place, the capture is final, several crossings of the same state being forbidden. Third, within our model the trapping of Mercury happens much faster than previously believed. The swift capture justifies our treatment of Mercury as a homogeneous, unstratified body whose liquid core had not yet formed by the time of trapping.
    06/2013;
  • A. Compère, J. Frouard, B. Carry
    05/2012;
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    Julien Frouard, Audrey Compère
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    ABSTRACT: We study the stability of the (87) Sylvia system and of the neighborhood of its two satellites. We use numerical integrations considering the non-sphericity of Sylvia, as well as the mutual perturbation of the satellites and the solar perturbation. Two numerical models have been used, which describe respectively the short and long-term evolution of the system. We show that the actual system is in a deeply stable zone, but surrounded by both fast and secular chaotic regions due to resonances. We then investigate how tidal and BYORP effects modify the location of the system over time with respect to the instability zones. Finally, we briefly generalize this study to other known triple systems and to satellites of asteroids in general, and discuss about their distance from mean-motion and evection resonances.
    Icarus 12/2011; 220(1). · 3.16 Impact Factor
  • J. Frouard, A. Compère
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    ABSTRACT: The triple system 87 Sylvia consisting of two small satellites (Romulus and Remus) orbiting around an asteroid in nearly circular orbits is studied. We model it using a four-body system Sylvia-Romulus-Remus- Sun with a spherical harmonics expansion up to the 4th degree and order for the gravitational potential of Sylvia. We integrate the equations of motion in two ways in order to study short and long periods; a complete one with an imposed fixed rotation rate for Sylvia on its principal moment on inertia, and an averaged one over the mean longitudes. We find that the semi-major axis of the satellites are bounded by meanmotion resonances (between the mean longitudes of the satellites) and by evection resonances (between the longitude of pericenter of Romulus and the longitude of the Sun).
    10/2011;
  • J. Frouard, A. Vienne, M. Fouchard
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    ABSTRACT: Context. The dynamical region of the Jovian irregular satellites presents an interesting web of resonances that are not yet fully understood. Of particular interest is the influence of the resonances on the stochasticity of the individual orbits of the satellites, as well as on the long-term chaotic diffusion of the different families of satellites. Aims: We make a systematic numerical study of the satellite region to determine the important resonances for the dynamics, to search for the chaotic zones, and to determine their influences on the dynamics of the satellites. We also compare these numerical results to previous analytical works. Methods: Using extensive numerical integrations of the satellites along with an indicator of chaos (MEGNO), we show global and detailed views of the retrograde and prograde regions for various dynamical models of increasing complexity and indicate the appearance of the different types of resonances and the implied chaos. Results: Along with secular and mean motion resonances that shape the dynamical regions of the satellites, we report a number of resonances involving the Great Inequality, and which are present in the system thanks to the wide range of the values of frequencies of the pericenter available for the satellites. The chaotic diffusion of the satellites is also studied and shows the long-term stability of the Ananke and Carme families, in contrast to the Pasiphae family. Tables 1 and 2 are available in electronic form at http://www.aanda.org
    Astronomy and Astrophysics 01/2011; 532. · 5.08 Impact Factor
  • J. Frouard, M. Fouchard, A. Vienne
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    ABSTRACT: Among resonances commonly influential to the dynamics of satellites, the evection resonance introduces an important correction to the precession frequency of the satellite, as it is well known for the Moon's problem. However, the dynamic of the resonance itself, which is important for satellites stability and capture topics, including its libration and circulation regions, and its elliptic and hyperbolic points, has not been extensively studied. Here we investigate its dynamic with an improved analytic model, making comparisons with previous works, and resort to numerical methods and integrations to study and localize the different features of the resonance. This resonance is found in the outer orbital region near the orbital stability limit. However we also study and localize an other libration region that can be found much more closer to the parent planet when its oblateness is taken into account in the model.
    11/2010; -1:127.
  • Julien Frouard, M. Fouchard, A. Vienne
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    ABSTRACT: Interested in the global dynamical structure of the Jovian irregular satellites and their chaotic evolutions, we have performed numerical integrations of the satellites along with an indicator of chaos (MEGNO), to give a clear representation of the dynamical structure of the regions surrounding the satellites, and the consequences on the individual evolutions of the bodies. In particular, as the long-term integrations give indications about the chaotic diffusion of the satellites in the phase space, dynamical maps show the reasons of the detected chaos and the underlying resonant dynamics acting on the satellite system.
    05/2010;
  • J. Frouard, M. Fouchard, A. Vienne
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    ABSTRACT: Context. The evection resonance appears to be the outermost region of stability for prograde satellite orbiting a planet, the critical argument of the resonance indeed being found librating in regions surrounded only by chaotic orbits. The dynamics of the resonance itself is thus of great interest for the stability of satellites, but its analysis by means of an analytical model is not straightforward because of the high perturbations acting on the dynamical region of interest. Aims: It is thus important to show the results and the limits inherent in analytical models. We use numerical methods to test the validity of the models and analyze the dynamics of the resonance. Methods: We use an analytical method based on a classical averaged expansion of the disturbing function valid for all eccentricities as well as numerical integrations of the motion and surfaces of section. Results: By comparing analytical and numerical methods, we show that aspects of the true resonant dynamic can be represented by our analytical model in a more accurate way than previous approximations, and with the help of the surfaces of section we present the exact location and dynamics of the resonance. We also show the additional region of libration of the resonance that can be found much closer to the planet due to its oblateness.
    Astronomy and Astrophysics 01/2010; · 5.08 Impact Factor
  • J. Frouard, M. Fouchard, A. Vienne
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    ABSTRACT: For a long time, the estimation of the Lyapunov Characteristic Exponents (LCEs) had been used in Celestial Mechanics to caracterize the chaoticity of orbits. With the aim of gaining speed and accuracy in detecting this chaoticity, several indicators based on the theory of Lyapunov exponents have been developped. Here we present a comparison in terms of precision, CPU speed, and practicability of several of these indicators ; the FLI (Froeschlé et al, 1997) , MEGNO (Cincotta & Simó, 2000), and the GALI (Skokos et al, 2007). The GALI3 (using three tangent vectors) is the version of the GALI used here. While the FLI and MEGNO have been commonly used, the GALI has not yet been applied to Celestial Mechanics. However, this indicator has its own qualities and specificities. The final aim of the comparison of these indicators is the production of stability maps in the case of irregular satellites of giant planets, the examples and applications are shown in this sense.
    10/2008; -1:121.
  • Julien Frouard, M. Fouchard, A. Vienne
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    ABSTRACT: Until now, the study of the chaoticity of the Jovian irregular satellites has been restricted to several ones and investigated on a limited integration time. We have extended these studies to the whole number of satellites and in time integration. We present the results of long-term numerical integrations of the satellites to search for chaotic behavior, giving an indication of the origin of the detected chaos.
  • Julien Frouard, Tadashi Yokoyama
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    ABSTRACT: An analytical expansion of the disturbing function arising from direct planetary perturbations on the motion of satellites is derived. As a Fourier series, it allows the investigation of the secular effects of these direct perturbations, as well as of every argument present in the perturbation. In particular, we construct an analytical model describing the evection resonance between the longitude of pericenter of the satellite orbit and the longitude of a planet, and study briefly its dynamic. The expansion developed in this paper is valid in the case of planar and circular planetary orbits, but not limited in eccentricity or inclination of the satellite orbit.
    Celestial Mechanics and Dynamical Astronomy 115(1). · 2.32 Impact Factor