Yi-Sui Sun

On the Sitnikov problem

A mapping which reflects the properties of the Sitnikov problem is derived. We study the mapping instead of the original differential equations and discover that there exists a hyperbolic invariant set. The theoretical prediction of the disorder region agrees remarkably with numerical results. We also discuss the LCEs and KS-entropy of the dynamical system.

The dynamics of Neptune Trojan – I. The inclined orbits

The stability of Trojan type orbits around Neptune is studied. As the first part of our investigation, we present in this paper a global view of the stability of Trojans on inclined orbits. Using the frequency analysis method based on the fast Fourier transform technique, we construct high-resolution dynamical maps on the plane of initial semimajor axis a0 versus inclination i0. These maps show three most stable regions, with i0 in the range of (0 ◦ ,1 2 ◦ ), (22 ◦ ,3 6 ◦ ) and (51 ◦ ,5 9 ◦ ), respectively, where the Trojans are most probably expected to be found. The similarity between the maps for the leading and trailing triangular Lagrange points L4 and L5 confirms the dynamical symmetry between these two points. By computing the power spectrum and the proper frequencies of the Trojan motion, we figure out the mechanisms that trigger chaos in the motion. The Kozai resonance found at high inclination varies the eccentricity and inclination of orbits, while the ν8 secular resonance around i0 ∼ 44 ◦ pumps up the eccentricity. Both mechanisms lead to eccentric orbits and encounters with Uranus that introduce strong perturbation and drive the objects away from the Trojan like orbits. This explains the clearance of Trojan at high inclination (>60 ◦ ) and an unstable gap around 44 ◦ on the dynamical map. An empirical theory is derived from the numerical results, with which the main secular resonances are located on the initial plane of (a0, i0). The fine structures in the dynamical maps can be explained by these secular resonances.

Apsidal corotation in mean motion resonance: the 55 Cancri system as an example

The inner two planets around 55 Cancri were found to be trapped in a 3 : 1 mean motion resonance (MMR). In this paper, we study the dynamics of this extrasolar planetary system. Our numerical investigation confirms the existence of the 3 : 1 resonance and implies a complex orbital motion. Different stable motion types, with and without apsidal corotation, are found. Owing to the high eccentricities in this system, we apply a semi-analytical method based on a new expansion of the Hamiltonian of the planar three-body problem in the discussion. We analyse the occurrence of apsidal corotation in this MMR and its influence on the stability of the system.

The dynamics of Neptune Trojans – II. Eccentric orbits and observed objects

In a previous paper, we presented a global view of the stability of Neptune Trojans (NTs hereafter) on inclined orbits. As the continuation of the investigation, we discuss in this paper the dependence of the stability of NT orbits on the eccentricity. For this task, high-resolution dynamical maps are constructed using the results of extensive numerical integrations of orbits initialized on fine grids of initial semimajor axis (a 0 ) versus eccentricity (e 0 ). The extensions of regions of stable orbits on the (a 0 , e 0 ) plane at different inclinations are shown. The maximum eccentricities of stable orbits in the three most stable regions at low (0°, 12°), medium (22°, 36°) and high (51°, 59°) inclination are found to be 0.10, 0.12 and 0.04, respectively. The fine structures in the dynamical maps are described. Via the frequency-analysis method, the mechanisms that portray the dynamical maps are revealed. The secondary resonances, at the frequency of the librating resonant angle λ - λ 8 and the frequency of the quasi 2:1 mean-motion resonance (MMR hereafter) between Neptune and Uranus, are found to be deeply involved in the motion of NTs. Secular resonances are detected and they also contribute significantly to the triggering of chaos in the motion. In particular, the effects of the secular resonance ν 8 , ν 18 are clarified. We also investigate the orbital stabilities of six observed NTs by checking the orbits of hundreds of clones generated within the observing error bars. We conclude that four of them are deeply inside the stable region, with 2001 QR322 and 2005 T074 being the exceptions. 2001 QR322 is in the close vicinity of the most significant secondary resonance. 2005 T074 is located close to the boundary separating stable orbits from unstable ones, and it may be influenced by a secular resonance.

Chaotic motion of comets in near-parabolic orbit: Mapping approaches

There exist many comets with near-parabolic orbits in the Solar System. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable (Oort, 1950). The theory assumes that there is a cometary cloud at a distance 103 – 105 AU from the Sun and that perturbing forces from planets or stars make orbits of some of these comets become of near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the Solar System forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighbourhood of any separatrix i.e. the trajectory with zero frequency of the unperturbed motion of an Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study on near-parabolic motion is very important. Petroskys elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion (Petrosky, 1988). In this paper we derive a similar mapping directly and discuss its dynamical characters.

The origin of the high-inclination Neptune Trojan 2005 TN53

Aims. We explore the formation and evolution of the highly inclined orbit of Neptune Trojan 2005 TN 53 . Methods. With numerical simulations, we investigated a possible mechanism for the origin of the high-inclination Neptune Trojans as captured into the Trojan-type orbits by an initially eccentric Neptune during its eccentricity damping and rapid inward migration, then migrating to the present locations locked in Neptunes 1:1 mean motion resonance. Results. Two 2005 TN 53 -type Trojans out of our 2000 test particles were produced with inclinations above 20°, moving on tadpole orbits librating around Neptunes leading Lagrange point.

Evidence for Lévy Random Walks in the Evolution of Comets from the Oort Cloud

With a simple map model, derived within the framework of the planar circular restricted three-body problem (SunuuJupiteruucomet), we study the dynamical evolution of near-parabolic comets under the perturbation of Jupiter. The commonly adopted random walk assumption about the energy evolution of the comets is examined. Numerical results show that for the comets on Jupiter-crossing orbits, due to the large energy changes with Jupiter per passage, the statistical evolution of the cometary energy follows a Lévy random walk, thus statistically the final energy parameter that a comet reaches is linked to the number of passages by a power law Kf ∼ nf. The mechanism that generates the Lévy random walk is explained in this model.

A perturbed extension of hyperbolic twist mappings

AbstractIn this paper we discuss a perturbed extension of hyperbolic twist mappings to a 3-dimensional measure-preserving mapping

The Role of Hyperbolic Invariant Sets in Stickiness Effects

\begin{array}{*{20}c} {T:\left\{ {\begin{array}{*{20}c} {x_{n + 1} = s(x_n \cos \varphi _n - y_n \sin \varphi _n ) + A\cos z_n ,} \\ {y_{n + 1} = s^{ - 1} (x_n \sin \varphi _n + y_n \cos \varphi _n ) + B\sin z_n ,} \\ {z_{n + 1} = z_n + C\cos (x_{n + 1} + y_{n + 1} ) + D,(\bmod 2\pi )} \\ \end{array} } \right.} \\ {\varphi _n = (x_n^2 + y_n^2 )^k } \\ \end{array}