Global dynamics of planetary systems with the MEGNO criterion (original) (raw)
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Dynamical instability and statistical behaviour of N-body systems
Planetary and Space Science, 1998
In this paper, with particular emphasis on the N-body problem, we focus our attention on the possibility of a synthetic characterization of the qualitative properties of generic dynamical systems. The goal of this line of research is very ambitious, though up to now many of the attempts have been discouraging. Nevertheless we shouldn't forget the theoretical as well practical relevance held by a possible successful attempt. Indeed, if only it would be concevaible to single out a synthetic indicator of (in)stability, we will be able to avoid all the consuming computations needed to empirically discover the nature of a particular orbit, perhaps sensibly different with respect to another one very near to it. As it is known, this search is tightly related to another basic question: the foundations of Classical Statistical Mechanics. So, we reconsider some basic issues concerning the (argued) relationships existing between the dynamical behaviour of many degrees of freedom (mdf) Hamiltonian systems and the possibility of a statistical description of their macroscopic features. The analysis is carried out in the framework of the geometrical description of dynamics (shortly, Geometrodynamical approach, GDA), which has been shownto be able to shed lights on some otherwise hidden connections between the qualitative behaviour of dynamical systems and the geometric properties of the underlying manifold. We show how the geometric description of the dynamics allows to get meaningful insights on the features of the manifold where the motion take place and on the relationships existing among these and the stability properties of the dynamics, which in turn help to interpret the evolutionary processes leading to some kind of metastable quasi-equilibrium states. Nonetheless, we point out that a careful investigation is needed in order to single out the sources of instability on the dynamics and also the conditions justifying a statistical description. Some of the claims existing in the literature have here been reinterpreted, and few of them also corrected to a little extent. The conclusions of previous studies receive here a substantial confirmation, reinforced by means of both a deeper analytical examination and numerical simulations suited to the goal. These latter confirm well known outcomes of the pioneering investigations and result in full agreement with the analytical predictions for what concerns the instability of motion in generic classical Hamiltonian systems. They give further support to the belief about the existence of a hierarchy of timescales accompanying the evolution of the system towards a succession of ever more detailed local (quasi) equilibrium states. These studies confirm the absence of a direct (i.e. trivial) relationship between average curvature of the manifold (or even frequency of occurrence of negative values) and dynamical instability. Moreover they clearly single out the full responsibility of fluctuations in geometrical quantities in driving a system towards Chaos, whose onset do not show any correlation with the average values of the curvatures, being these almost always positive. These results highlight a strong dependence on the interaction potential of the connections between qualitative dynamical behaviour and geometric features. These latter lead also to intriguing hints on the statistical properties of generic many dimensional hamiltonian systems. It is also presented a simple analytical derivation of the scaling law behaviours, with respect to the global parameters of the system, as the specific energy or the number of degrees of freedom, of the relevant geometrodynamical quantities entering the determination of the stability properties. From these estimates, it emerges once more the strong peculiarity of Newtonian gravitational interaction with respect to all the topics here addressed, namely, the absence of any threshold, in energy or number of particles (insofar N≫ 1) or whatsoever parameter, distinguishing among different regimes of Chaos; the peculiar character of most geometrical quantities entering in the determination of dynamical instability, which is clearly related to the previous point and that it is also enlightening on the physical sources of the statistical features of N-body self gravitating systems, contrasted with those whose potential is stable and tempered, in particular with respect to the issue of ergodicity time. We will also address another elementary issue, related to a careful analysis of real gravitational N-body systems, which nevertheless provides the opportunity to gain relevant conceptual improvements, able to cope with the peculiarities mentioned above.
Solar System dynamics, beyond the two-body-problem approach
AIP Conference Proceedings, 2006
When one thinks of the solar system, he has usually in mind the picture based on the solution of the two-body problem approximation presented by Newton, namely the ordered clockwork motion of planets on fixed, non-intersecting orbits around the Sun. However, already by the end of the 18th century this picture was proven to be wrong. As discussed by Laplace and Lagrange (for a modern approach see or [2]), the interaction between the various planets leads to secular changes in their orbits, which nevertheless were believed to be corrections of higher order to the Keplerian elliptical motion.
The non-resonant, relativistic dynamics of circumbinary planets
Monthly Notices of the Royal Astronomical Society, 2010
We investigate a non-resonant, three-dimensional (spatial) model of a hierarchical system composed of a point-mass stellar (or substellar) binary and a low-mass companion (a circumbinary planet or a brown dwarf). We take into account the leading relativistic corrections to the Newtonian gravity. The secular model of the system relies on the expansion of the perturbing Hamiltonian in terms of the ratio of semi-major axes α, averaged over the mean anomalies. We found that a low-mass object in a distant orbit may excite a large eccentricity of the inner binary when the mutual inclination of the orbits is larger than about 60 •. This is related to the strong instability caused by a phenomenon that acts similarly to the Lidov-Kozai resonance (LKR). The secular system may be strongly chaotic and its dynamics unpredictable over long-term timescales. Our study shows that in the Jupiter-or brown-dwarf-mass regime of the low-mass companion, the restricted model does not properly describe the long-term dynamics in the vicinity of the LKR. The relativistic correction is significant for the parametric structure of a few families of stationary solutions in this problem, in particular for direct orbit configurations (with mutual inclination less than 90 •). We found that the dynamics of hierarchical systems with small α ∼ 0.01 may be qualitatively different in the realms of Newtonian (classic) and relativistic models. This holds true even for relatively large masses of the secondaries.
Dynamical conditions in planets and stars
Earth, Moon and Planets, 1992
With the available data in planets, stars and galaxies, it is studied the functions of angular momenta J(M) and amounts of action A,(M) (associated to the non rotational terms in the kinetic energy). The results indicate that independently of how are these functions J(M), A,(M) their ratio A,IJ remains a near invariant. It is independent also from the type of angular momenta: intrinsic "spins" of the bodies or the total angular (orbital) momenta of the bodies forming a system; for instance, the Solar System and the planets. The relation A,(M) for the Solar System are analogous to these in the FGK stars of the main sequence, and the relation .7(M) (also for the Solar System) is analogous to the lower possible limit for binary stars. The different types of binary stars from the short period, detached systems to contactary systems, gives a range of functions J(M), A,(M) that are the same that one can expect in stars with planetary systems. According to the detection limits given for planetary companions by Campbell, Walker and Yang (1988) (masses of less than 9 Jupiter masses and orbital periods of less than 50 years) we calculate the limits for J(M) and A,(M) This gives a lower limit A,/J 3 1 associated to stars with planetary systems as 61 Cygni and to short period detached binaries. The upper limit A,IJ% 16 correspond to planetary systems as the ours and probably to cataclysmic binaries. There are reasons to suspect that systems as the ours and in range 4 s AC/J G 16 (with a lower limit analogous to contactary binaries as Algols and W Ursa Majoris) must be the most common type of planetary systems. The analogies with the functions J(M) A,(M) for galaxies suggest cosmogonical conditions in the stellar formation. Independently of this, one can have boundary conditions for the Jacobi problem when applied to a collapsing cloud. Namely, from the initial stage (a molecular cloud) to the final stage (a formed stellar system: binary or planetary) the angular momenta and amounts of action decayed to 10m4 the initial values, but in such a form that A,(t)/J(t) remains a near invariant.
Chaos in the one-dimensional gravitational three-body problem
Chaos: An Interdisciplinary Journal of Nonlinear Science, 1993
We have investigated the appearance of chaos in the 1-dimensional Newtonian gravitational three-body system (three masses on a line with −1/r pairwise potential). In the center of mass coordinates this system has two degrees of freedom and can be conveniently studied using Poincaré sections. We have concentrated in particular on how the behavior changes when the relative masses of the three bodies change. We consider only the physically more interesting case of negative total energy. For two mass choices we have calculated 18000 full orbits (with initial states on a 100 × 180 lattice on the Poincaré section) and obtained dwell time distributions. For 105 mass choices we have calculated Poincaré maps for 10×18 starting points. Our results show that the Poincaré section (and hence the phase space) divides into three well defined regions with orbits of different characteristics: 1) There is a region of fast scattering, with a minimum of pairwise collisions. This region consists of 'scallops' bordering the E = 0 line, within a scallop the orbits vary smoothly. The number of the scallops increases as the mass of the central particle decreases. 2) In the chaotic scattering region the interaction times are longer, and both the interaction time and the final state depend sensitively on the starting point on the Poincaré section. For both 1) and 2) the initial and final states consists of a binary + single particle. 3) The third region consists of quasiperiodic orbits where the three masses are bound together forever. At the center of the quasiperiodic region there is a periodic orbit discovered (numerically) by Schubart in 1956. The stability of the Schubart orbit turns out to correlate strongly with the global behavior.
Monthly Notices of the Royal Astronomical Society, 2009
We construct a secular theory of a coplanar system of N planets not involved in strong mean motion resonances, and which are far from collision zones. Besides the point-to-point Newtonian mutual interactions, we consider the general relativistic corrections to the gravitational potential of the star and the innermost planet, and also a modification of this potential by the quadrupole moment and tidal distortion of the star. We focus on hierarchical planetary systems. After averaging the model Hamiltonian with a simple algorithm making use of very basic properties of the Keplerian motion, we obtain analytical secular theory of high order in the semimajor axes ratio. A great precision of the analytic approximation is demonstrated with the numerical integrations of the equations of motion. A survey regarding model parameters (the masses, semimajor axes, spin rate of the star) reveals a rich and non-trivial dynamics of the secular system. Our study is focused on its equilibria. Such solutions predicted by the classic secular theory, which correspond to aligned (mode I) or anti-aligned (mode II) apsides, may be strongly affected by the gravitational corrections. The so-called true secular resonance, which is a new feature of the classic two-planet problem discovered by Michtchenko & Malhotra, may appear in other, different regions of the phase space of the generalized model. We found bifurcations of mode II from which emerge new, yet unknown in the literature, secularly unstable equilibria and a complex structure of the phase space. These equilibria may imply secularly unstable orbital configurations even for initially moderate eccentricities. The point mass gravity corrections can affect the long-term stability in the secular timescale , which may directly depend on the age of the host star through its spin rate. We also analyse the secular dynamics of the υ Andromedae system in the realm of the generalized model. Also in this case of the three-planet system, new secular equilibria may appear.
On the dynamical stability of planets in double stars
The importance of stability studies of planetary motion in binaries arises from the fact that double and multiple star systems are more numerous than single stars - at least in the solar neighborhood. Another impulse to carry out such dynamical studies was the discovery of planets in binaries, where we distinguish between two types of motion: P-type and S-type orbits. A dynamical stability study of two binary systems (γ Cephei and Gliese 86) is shown in this investigation, where we examined the region between the two stars in order to find stable zones where other planets might exist. For the determination of the stable zones we used two chaos indicators (1. the Fast Lyapunow Indicator - FLI and 2. the Mean Exponential Growth factor of Nearby Orbits - MEGNO) and additionally straight-forward numerical computations by applying the Lie integration method. In the general stability study of S-type motion we show the results for a double star with mass-ratio 0.2 which can be applied to t...
Behaviour of a Weakly Perturbed Two-Planetary System on a Cosmogonic Time-Scale
Orbital evolution of planetary systems similar to our Solar one represents one of the most important problems of Celestial Mechanics. In the present work we use Jacobian coordinates, introduce two systems of osculating elements, construct the Hamiltonian expansions in the Poisson series in all elements for the planetary three-body problem (including the problem of Sun-Jupiter-Saturn). Further we construct the averaged Hamiltonian by the Hori-Deprit method with accuracy upto second order with respect to the small parameter, the generating function, change of variables formulae, and right-hand sides of averaged equations. The averaged equations for the Sun-Jupiter-Saturn system are integrated numerically at the time-scale of 10 Gyr. The motion turns out to be almost periodical. The low and upper limits for eccentricities are 0.016, 0.051 (Jupiter), 0.020, 0.079 (Saturn), and for inclinations to the ecliptic plane (degrees) are 1.3, 2.0 (Jupiter), 0.73, 2.5 (Saturn). It is remarkable that the evolution of the ascending node longitudes may be secular (Laplace's plane) as well as librational one (ecliptic plane).
Dynamical Aspects of Exoplanetary Systems
2013
The detection of more than 130 multiple planet systems makes it necessary to interpret a broader range of properties than are shown by our Solar system. This thesis covers aspects linked to the proliferation in recent years of multiple extrasolar planet systems. A narrow observational window, only partially covering the longest orbital period, can lead to solutions representing unrealistic scenarios. The best-fit solution for the three-planet extrasolar system of HD 181433 describes a highly unstable configuration. Taking into account the dynamical stability as an additional observable while interpreting the RV data, I have analysed the phase space in the neighbourhood of the statistical best-fit. The two giant companions are found to be locked in the 5:2 MMR in the stable best-fit model. I have analysed the dynamics of the system HD 181433 by assessing different scenarios that may explain the origin of these eccentric orbits, with particular focus on the innermost body. A scenario ...
2006
Periodic Orbits and Variational Methods.- On the variational approach to the periodic n-body problem.- On families of periodic solutions of the restricted three-body problem.- Hip-hop solutions of the 2N-body problem.- Double choreographical solutions for n-body type problems.- From the circular to the spatial elliptic restricted three-body problem.- Stability of axial orbits in galactic potentials.- Perturbation Theory and Regularization.- KAM tori for N-body problems: a brief history.- Analysis of the chaotic behaviour of orbits diffusing along the Arnold web.- The scattering map in the planar restricted three body problem.- On final evolutions in the restricted planar parabolic three-body problem.- Quaternions and the perturbed Kepler problem.- Dynamics of Solar and Extrasolar Systems.- The 3:2 spin-orbit resonant motion of Mercury.- Symmetric and asymmetric librations in extrasolar planetary systems: a global view.- The influence of mutual perturbations on the eccentricity excit...