On the Chaotic Orbits of Disk-Star-Planet Systems

Abstract

Following Tancredi et al.'s criteria of chaos, two ways of setting initial velocities are used in numerical surveys to explore possible chaotic and regular orbits for disk-star-planet systems. We find that the chaotic boundary does not depend much on the disk mass for type I initial conditions, but can change a lot for different disk masses for type II initial conditions. A few sample orbits are further studied. Both the Poincaré surface of section and the Lyapunov exponent indicator are calculated, and they are consistent with each other. We also find that the influence from the disk can change the locations of equilibrium points and the orbital behaviors for both types of initial conditions. Because chaotic orbits are less likely to become stable resonant orbits, we conclude that the protostellar disk plays important roles for the capture and depletion histories of resonant orbits of both the asteroid and Kuiper belts during the formation of the solar system.

The Linear Stability of Collinear Equilibrium Points and Resonances

“…1b) which shows that the function K(x) is an increasing function of x having distinct sign in the interval 1 − µ < x < ∞ this implies that there exists a point x L2 (say), in (1 − µ, ∞) such that K(x L2 ) = 0. Similarly, in case (2), when x ∈ (0, 1 − µ), then x + µ − 1 < 0 and x + µ > 0. Thus, from equation (9) we get, we again have x + µ − 1 > 0 and x + µ > 0 but x is negative and less than µ.…”

“…Consider a small change in this coordinate such as x = x e + X, y = y e + Y , where X = P e λt , Y = Qe λt are very small quantities and P, Q are constants and λ is parameter to be determined. Substituting these coordinates into equations (1) and (2). Keeping in mind that displacements are sufficiently small, we have expanded U x and U y by Taylor series neglecting the second and higher order terms, we getẌ…”

“…Existence of a disc or belt like structure in the problems of celestial mechanics is a common phenomenon. For example, in the solar system there are asteroid belt and Kuiper belt [2], [3], in the extra-solar planetary systems the disc of dust [4], [5], [6] and circumbinary rings in case of binary system [7]. The disc or belt-like structure affects the dynamical evolution of these systems.…”

“…It can change the location of equilibrium points and also the orbital behaviors. The influence of the belt for planetary system is observed by [2], [3] and found that probability of equilibrium points is larger, near the inner part of the belt than outer one. Different aspects of the problem with and without disc are studied by many authors as [8], [9], [10], [7], [11], [12], [13], [14], [15], [16] etc., under some assumptions on mass parameter and angular velocity of the system.…”

“…Authors are interested in the problem due to a number of recent applications in addition to past one, in different areas such as celestial mechanics, chemistry and extra solar planetary system [29], [4], [5], [2]. Since, the collinear equilibrium points play a crucial role in space dynamics.…”