Structure-preserving numerical simulations of test particle dynamics around slowly rotating neutron stars within the Hartle-Thorne approach

In this paper, we explore the chaotic signatures of the geodesic dynamics for particles moving in the slowly rotating Hartle-Thorne spacetime; an approximate solution of vacuum Einstein field equations describing the exterior of a massive, deformed, and slowly rotating compact object. We employ a numerical study to examine the geodesics of prolate and oblate deformations for generic orbits and find the plateaus of the rotation curve, which are associated with the existence of Birkhoff islands in the Poincar\'e surface of the section, where the ratio of the radial and polar frequency of geodesics remains constant throughout the island. We investigate various phase-space structures, including hyperbolic points and chaotic regions in the neighborhood of resonant islands. Moreover, chaotic behavior is observed to be governed by the stickiness phenomenon, where chaotic orbits remain attached to stable ones for an extended duration before eventually diverging and are attracted toward the surface of the neutron star. The precision of the numerical integration used to simulate the particle's trajectories plays a crucial role in the structure of the Poincar\'e surface of the section. We present a comparison of several efficient structure-preserving numerical schemes of order four applied to the considered nonintegrable dynamical system and we investigate which schemes possess the canonical property of the Hamiltonian flow. In particular, we compare the performance of the symplectic Runge-Kutta integrator with the G-symplectic general linear method. Among the class of nonsymplectic integrators, we employ the explicit Runge-Kutta method and an explicit general linear method with a standard projection technique to project the numerical solution onto the desired manifold. The projection scheme admits the integration without any drift from the desired manifold and is computationally cost effective. We are concerned with two crucial aspects; long-term behavior and CPU time consumption.

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