Mean Exponential Growth Factor Of Nearby Orbits Research Articles

Chaos in multiplanetary extrasolar systems

ABSTRACT Here, we present an initial look at the dynamics and stability of 178 multiplanetary systems that are already confirmed and listed in the NASA Exoplanet Archive. To distinguish between the chaotic and regular nature of a system, the value of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) indicator for each system was determined. Almost three-quarters of them could be labelled as long-term stable. Only 45 studied systems show chaotic behaviour. We consequently investigated the effects of the number of planets and their parameters on the system stability. A comparison of results obtained using the MEGNO indicator and machine-learning algorithm SPOCK (Stability of Planetary Orbital Configuration Klassifier) suggests that the SPOCK could be used as an effective tool for reviewing the stability of multiplanetary systems. A similar study was already published by Laskar and Petit. We compared their analysis based on the AMD (Angular Momentum Deficit) criterion with our results. The possible discrepancies are discussed.

A numerical study of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in planetary systems

ABSTRACT We present a numerical study on the stability of the 1/2, 2/1, and 1/1 retrograde mean motion resonances in the three-body problem composed of a solar mass star, a Jupiter mass planet, and an additional body with zero mass (elliptic restricted three-body problem) or masses corresponding to either Neptune, Saturn, or Jupiter (planetary three-body problem). For each system, we obtain stability maps using the n-body numerical integrator REBOUND and computing the chaos indicator mean exponential growth factor of nearby orbits (MEGNO). We show that families of periodic orbits exist in all configurations and they correspond to the libration of either a single resonant argument or all resonant arguments (fixed points). We compare the results obtained in the elliptic restricted three-body problem with previous results in the literature, and we show the differences and similarities between the phase space topology for these retrograde resonances in the circular restricted, elliptic restricted, and planetary three-body problems.

Analysis of the dynamical evolution of the Quadrantid meteoroid stream

We investigate numerically the dynamical evolution of simulated meteoroid stream of the Quadrantids ejected from the parent body of the asteroid (196256) 2003 EH1. The main goal of this work is to identify mean motion and secular resonances and to study the mutual influence of resonance relations and close encounters with the major planets. Since the dynamics of this asteroid is predictable only on short time intervals, and not only close and/or multiple close encounters with major planets, but also the presence of at least one unstable resonance can lead to chaotic motion of test particles, we studied their resonant dynamics. The dynamical evolution of the test particles expects possible scenario for resonant motion. We conjecture that the reasons of chaos are the overlap of stable secular resonances and unstable mean motions resonances and close and/or multiple close encounters with the major planets. The estimate of the stability of orbits in which the particles in simulations moved was carried out by analyzing the behavior of the parameter MEGNO (Mean Exponential Growth factor of Nearby Orbits). The larger part of the identified resonances is stable. We found a peculiar behavior for this stream. Here, we show that the orbits of some ejected particles are strongly affected by the Lidov–Kozai mechanism that protects them from close encounters with Jupiter. Lack of close encounters with Jupiter leads to a rather smooth growth in the parameter MEGNO and the behavior imply the stable motion of simulation particles of the Quadrantids meteoroid stream.

Dynamical evolution and stability maps of the Proxima Centauri system

Proxima Centauri was recently discovered to host an Earth-mass planet of Proxima b, and a 215-day signal which is probably a potential planet c. In this work, we investigate the dynamical evolution of the Proxima Centauri system with the full equations of motion and semi-analytical models including relativistic and tidal effects. We adopt the modified Lagrange-Laplace secular equations to study the evolution of eccentricity of Proxima b, and find that the outcomes are consistent with those from the numerical simulations. The simulations show that relativistic effects have an influence on the evolution of eccentricities of planetary orbits, whereas tidal effects primarily affects the eccentricity of Proxima b over long timescale. Moreover, using the MEGNO (the Mean Exponential Growth factor of Nearby Orbits) technique, we place dynamical constraints on orbital parameters that result in stable or quasi-periodic motions for coplanar and non-coplanar configurations. In the coplanar case, we find that the orbit of Proxima b is stable for the semi-major axis ranging from 0.02 au to 0.1 au and the eccentricity being less than 0.4. This is where the best-fitting parameters for Proxima b exactly fall. Additional simulations show that the robust stability of this system would favor an eccentricity of Proxima b less than 0.45 and that of Proxima c below 0.65. In the non-coplanar case, we find that mutual inclinations of two planets must be lower than $50^{\circ}$ in order to provide stability. Finally, we estimate the mass of Proxima c to be $3.13~{M_ \oplus } \le {m_c} \le 70.7~{M_ \oplus }$ when $1.27~{M_ \oplus } \le {m_b} \le 1.6~{M_ \oplus }$, if ${i_{mutual}} \le {50^{\circ}}$ and $\Delta \Omega = 0^{\circ}$.

Second-Order Chaos Indicators MEGNO2 and OMEGNO2: Theory

Modifications of the Mean Exponential Growth factor of Nearby Orbits (MEGNO) linear variational method called MEGNO2 and OMEGNO2 indicators are introduced. The modifications are based on taking into account not only the linear, but also the nonlinear part of the increment of the phase flow in the divergence among nearby trajectories according to the second-order formulas. The new indicators allow one to determine more quickly the nature of the orbits under study in dynamical systems with zero or small Lyapunov exponents in comparison with the first-order variational indicators. They improve the analysis of regular regions and, in particular, periodic orbits as well as prevent the appearance of spurious structures in the resulting mappings.

The dynamical evolution and the force model for asteroid (196256) 2003 EH1

This paper is devoted to the dynamics of asteroid (196256) 2003 EH1 which belongs to the Amor group. It is known that asteroid 2003 EH1 is associated with one of the main annual meteor showers – the Quadrantids. In this work we analyze the influence of various perturbing factors on the asteroid motion. The perturbations’ estimation was done by five different methods based on the nominal orbit evolution and the size of the initial confidence region. The most significant influences on the dynamical evolution of 2003 EH1 are gravitational forces from the Sun, major planets and the Moon, and the relativistic effects (RE) of the Sun. The Earth, the Sun and Jupiter oblateness; gravitational perturbations from Pallas, Ceres, Vesta and Pluto; and the RE of planets, the Moon, and Pluto are of less importance.The researches of chaoticity and evolution of asteroid 2003 EH1 were examined by integrating its motion equations along with 500 clones. The time interval (1000–4000 years) has been determined by integration precision estimation. We estimated the mean exponential growth factor of nearby orbits (MEGNO) and found that MEGNO < 2 only in the interval from 1700 year to 2300 year. After 2300 year the MEGNO parameter increases that indicates motion instability. It shows that the orbit may be considered as regular on the time interval of ±300 years from now, and as chaotic outside this interval. We suppose that the reasons are the frequent close approaches of the asteroid with Jupiter and the overlap of apsidal-nodal resonances.

The reversibility error method (REM): a new, dynamical fast indicator for planetary dynamics

We describe the Reversibility Error Method (REM) and its applications to planetary dynamics. REM is based on the time-reversibility analysis of the phase-space trajectories of conservative Hamiltonian systems. The round-off errors break the time reversibility and the displacement from the initial condition, occurring when we integrate it forward and backward for the same time interval, is related to the dynamical character of the trajectory. If the motion is chaotic, in the sense of non-zero maximal Characteristic Lyapunov Exponent (mLCE), then REM increases exponentially with time, as exp \lambda t, while when the motion is regular (quasi-periodic) then REM increases as a power law in time, as t^{\alpha}, where {\alpha} and {\lambda} are real coefficients. We compare the REM with a variant of mLCE, the Mean Exponential Growth factor of Nearby Orbits (MEGNO). The test set includes the restricted three body problem and five resonant planetary systems: HD 37124, Kepler-60, Kepler-36, Kepler-29 and Kepler-26. We found a very good agreement between the outcomes of these algorithms. Moreover, the numerical implementation of REM is astonishing simple, and is based on solid theoretical background. The REM requires only a symplectic and time-reversible (symmetric) integrator of the equations of motion. This method is also CPU efficient. It may be particularly useful for the dynamical analysis of multiple planetary systems in the KEPLER sample, characterized by low-eccentricity orbits and relatively weak mutual interactions. As an interesting side-result, we found a possible stable chaos occurrence in the Kepler-29 planetary system.

Second-order variational equations forN-body simulations

First-order variational equations are widely used in N-body simulations to study how nearby trajectories diverge from one another. These allow for efficient and reliable determinations of chaos indicators such as the Maximal Lyapunov characteristic Exponent (MLE) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). In this paper we lay out the theoretical framework to extend the idea of variational equations to higher order. We explicitly derive the differential equations that govern the evolution of second-order variations in the N-body problem. Going to second order opens the door to new applications, including optimization algorithms that require the first and second derivatives of the solution, like the classical Newton's method. Typically, these methods have faster convergence rates than derivative-free methods. Derivatives are also required for Riemann manifold Langevin and Hamiltonian Monte Carlo methods which provide significantly shorter correlation times than standard methods. Such improved optimization methods can be applied to anything from radial-velocity/transit-timing-variation fitting to spacecraft trajectory optimization to asteroid deflection. We provide an implementation of first- and second-order variational equations for the publicly available rebound integrator package. Our implementation allows the simultaneous integration of any number of first- and second-order variational equations with the high-accuracy IAS15 integrator. We also provide routines to generate consistent and accurate initial conditions without the need for finite differencing.

Whfast: a fast and unbiased implementation of a symplectic Wisdom–Holman integrator for long-term gravitational simulations

We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.

Chaotic dynamics of the planet in HD 196885 AB

Depending on the planetary orbit around the host star(s), a planet could orbit either one or both stars in a binary system as S-type or P-type, respectively. We have analysed the dynamics of the S-type planetary system in HD 196885 AB with an emphasis on a planet with a higher orbital inclination relative to the binary plane. The mean exponential growth factor of nearby orbits (MEGNO) maps are used as an indicator to determine regions of periodicity and chaos for the various choices of the planet's semimajor axis, eccentricity and inclination with respect to the previously determined observational uncertainties. We have quantitatively mapped out the chaotic and quasi-periodic regions of the system's phase space which indicate a likely regime of the planet's inclination. In addition, we inspect the resonant angle to determine whether alternation between libration and circulation occurs as a consequence of Kozai oscillations, a probable mechanism that can drive the planetary orbit to a very large inclination. Also, we demonstrate the possible higher mass limit of the planet and improve upon the current dynamical model based on our analysis.

Application of chaos indicators in the study of dynamics of S-type extrasolar planets in stellar binaries

The orbits of two individual planets in two known binary star systems, \gamma Cephei and HD 196885 are numerically integrated using various numerical techniques to assess the chaotic or quasi-periodic nature of the dynamical system considered. The Hill stability (HS) function which measures the orbital perturbation of a planet around the primary star due to the secondary star is calculated for each system. The maximum Lyapunov exponent (MLE) time series are generated to measure the divergence/convergence rate of stable manifolds, which are used to differentiate between chaotic and non-chaotic orbits. Then, we calculate dynamical Mean Exponential Growth factor of Nearby Orbits (MEGNO) maps from solving the variational equations along with the equations of motion. These maps allow us to accurately differentiate between stable and unstable dynamical systems. The results obtained from the analysis of HS, MLE, and MEGNO maps are analysed for their dynamical variations and resemblance. The HS test for the planets shows stability and quasi-periodicity for at least ten million years. The MLE and the MEGNO maps have also indicated the local quasi-periodicity and global stability in a relatively short integration period. The orbital stability of the systems is tested using each indicator for various values of planet inclinations (i_{pl} \le 25^\circ) and binary eccentricities. The reliability of HS criterion is also discussed based on its stability results compared with the MLE and MEGNO maps.

COMPARATIVE STUDY OF VARIATIONAL CHAOS INDICATORS AND ODEs' NUMERICAL INTEGRATORS

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon &amp; Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba &amp; Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince &amp; Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

Influence of Earth’s shadowing effects on space debris stability

Solar radiation pressure affects the evolution of high area-to-mass geostationary space debris. In this work, we extend the stability study of Valk et al. (2009) by considering the influence of Earth’s shadows on the short- and long-term time evolutions of space debris. To assess the orbits stability, we use the Mean Exponential Growth factor of Nearby Orbits (MEGNO), which is an efficient numerical tool to distinguish between regular and chaotic behaviors. To reliably compute long-term space debris motion, we resort to the Global Symplectic Integrator (GSI) of Libert et al. (2011) which consists in the symplectic integration of both Hamiltonian equations of motion and variational equations. We show how to efficiently compute the MEGNO indicator in a complete symplectic framework, and we also discuss the choice of a symplectic integrator, since propagators adapted to the structure of the Hamiltonian equations of motion are not necessarily suited for the associated variational equations. The performances of our method are illustrated and validated through the study of the Arnold diffusion problem. We then analyze the effects of Earth’s shadows, using the adapted conical and cylindrical Earth’s shadowing models introduced by Hubaux et al. (2012) as the smooth shadow function deriving from these models can be easily included into the variational equations. Our stability study shows that Earth’s shadows greatly affect the global behaviour of space debris orbits by increasing the size of chaotic regions around the geostationary altitude. We also emphasize the differences in the results given by conical or cylindrical Earth’s shadowing models. Finally, such results are compared with a non-symplectic integration scheme.

A comparison of different indicators of chaos based on the deviation vectors: application to symplectic mappings

The aim of this research work is to compare the reliability of several variational indicators of chaos on mappings. The Lyapunov Indicator (LI); the Mean Exponential Growth factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI); the Fast Lyapunov Indicator (FLI); the Dynamical Spectra of stretching numbers (SSN) and the corresponding Spectral Distance (D); and the Relative Lyapunov Indicator (RLI), which is based on the evolution of the difference between two close orbits, have been included. The experiments presented herein allow us to reliably suggest a group of chaos indicators to analyze a general mapping. We show that a package composed of the FLI and the RLI (to analyze the phase portrait globally) and the MEGNO and the SALI (to analyze orbits individually) is good enough to make a description of the systems' dynamics.

Analytical relation between two chaos indicators: FLI and MEGNO

Abstract We report an intrinsic relation between the mean exponential growth factor of nearby orbits (MEGNO) and the fast Lyapunov indicator (FLI), two variational methods that have been widely applied to stability issues in astronomy. For both continuous-time and discrete-time systems, we arrive at an analytic formula that expresses the MEGNO in terms of the FLI and its time-average. This connection, unknown for more than 10 years, allows us to understand the differences and similarities in the performance of both indicators.

Testing a fast dynamical indicator: The MEGNO

To investigate non-linear dynamical systems, like for instance artificial satellites, Solar System, exoplanets or galactic models, it is necessary to have at hand several tools, such as a reliable dynamical indicator.The aim of the present work is to test a relatively new fast indicator, the Mean Exponential Growth factor of Nearby Orbits (MEGNO), since it is becoming a widespread technique for the study of Hamiltonian systems, particularly in the field of dynamical astronomy and astrodynamics, as well as molecular dynamics.In order to perform this test we make a detailed numerical and statistical study of a sample of orbits in a triaxial galactic system, whose dynamics was investigated by means of the computation of the Finite Time Lyapunov Characteristic Numbers (FT-LCNs) by other authors.

Global dynamics of high area-to-mass ratios GEO space debris by means of the MEGNO indicator

In this paper we provide an extensive analysis of the global dynamics of high-area-to-mass ratios geosynchronous (GEO) space debris, applying a recent technique developed by Cincotta and Simó [Cincotta, P.M., Simó, C.Simple tools to study global dynamics in non-axisymmetric galactic potentials–I. Astron. Astrophys. (147), 205–228, 2000.], Mean Exponential Growth factor of Nearby Orbits ( MEGNO), which provides an efficient tool to investigate both regular and chaotic components of the phase space. We compute a stability atlas, for a large set of near-geosynchronous space debris, by numerically computing the MEGNO indicator, to provide an accurate understanding of the location of stable and unstable orbits as well as the timescale of their exponential divergence in case of chaotic motion. The results improve the analysis presented in Breiter et al. [Breiter, S., Wytrzyszczak, I., Melendo, B. Long-term predictability of orbits around the geosynchronous altitude. Advances in Space Research 35, 1313–1317, 2005] notably by considering the particular case of high-area-to-mass ratios space debris. The results indicate that chaotic orbits regions can be highly relevant, especially for very high area-to-mass ratios. We then provide some numerical investigations and an analytical theory that lead to a detailed understanding of the resonance structures appearing in the phase space. These analyses bring to the fore a relevant class of secondary resonances on both sides of the well-known pendulum-like pattern of geostationary objects, leading to a complex dynamics.

The long-term stability of extrasolar system HD 37124. Numerical study of resonance effects

We describe numerical tools for the stability analysis of extrasolar planetary systems. In par- ticular, we consider the relative Poincare variables and symplectic integration of the equations of motion. We apply the tangent map to derive a numerically efficient algorithm of the fast indicator Mean Exponential Growth factor of Nearby Orbits (MEGNO), a measure of the maximal Lyapunov exponent, that helps to distinguish chaotic and regular configurations. The results concerning the three-planet extrasolar system HD 37124 are presented and discussed. The best-fitting solutions found in earlier works are studied more closely. The system involves Jovian planets with similar masses. The orbits have moderate eccentricities, nevertheless the best-fitting solutions are found in dynamically active region of the phase space. The long-term stability of the system is determined by a net of low-order two-body and three-body mean mo- tion resonances. In particular, the three-body resonances may induce strong chaos that leads to self-destruction of the system after Myr of apparently stable and bounded evolution. In such a case, numerically efficient dynamical maps are useful to resolve the fine structure of the phase space and to identify the sources of unstable behaviour.

Conditions of Dynamical Stability for the HD 160691 Planetary System

In our previous paper we showed that the currently determined orbital parameters placed four recently announced planetary systems (HD 12661, HD 38529, HD 37124, and HD 160691) in very different situations from the point of view of dynamical stability. In the present paper we deal with the last of these systems, whose orbital parameters of the outer planet are yet uncertain. We discover a stabilizing mechanism that could be the key to its existence. The paper is devoted to the study of this mechanism by a global dynamics analysis in the orbital parameter space related to the HD 160691 system. We obtained our results using a new technique called the mean exponential growth factor of nearby orbits (MEGNO), and verified them with the fast Lyapunov indicator technique (FLI). In order to be dynamically stable, the HD 160691 planetary system has to satisfy the following conditions: (1) it should have a 2 : 1 mean motion resonance, (2) combined with an apsidal secular resonance, (3) in a configuration Pc(ap)-S-Pb(ap) (which means that the planets c and b may be considered as initially located at their apoastron around the central star S), and (4) it must satisfy specific conditions for the respective sizes of the eccentricities. High eccentricity for the outer orbit (ec > 0.52) is the most probable necessary condition, while the eccentricity of the inner orbit eb becomes relatively unimportant when ec > 0.7. We also show that there is an upper limit for planetary masses (in the interval permitted by the undetermined line-of-sight inclination factor sin il) due to the dynamical stability mechanism. More generally, in this original orbital topology, where the resonance variables θ1 and θ3 librate about 180° while θ2 librates about 0°, the HD 160691 system and its mechanism have revealed aspects of the 2 : 1 orbital resonances that have not been observed nor analyzed before. The present topology with antialigned apsidal lines, combined with the 2 : 1 resonance, is indeed more wide-ranging than the particular case of the HD 160691 planetary system. It is a new theoretical possibility that is suitable for a stable regime despite relatively small semimajor axes with respect to the important masses in interactions.

Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

In this paper we deal with an alternative technique to study global dynamics in Hamiltonian systems, the mean exponential growth factor of nearby orbits (MEGNO), that proves to be efficient to investigate both regular and stochastic components of phase space. It provides a clear picture of resonance structures, location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the Lyapunov characteristic number. Here the MEGNO is applied to a rather simple model, the 3D perturbed quartic oscillator, in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure. Examples of application to multi-dimensional canonical maps are also included.