Revised research about chaotic dynamics in Manko et al. spacetime

A recent work by Dubeibe et al. [Phys. Rev. D 75, 023008 (2007)] stated that chaos phenomenon of test particles in gravitational field of rotating neutron stars which are described by Manko, Sanabria-Goméz, and Manko (Manko et al.) metric can only occur when the stars have oblate deformation. But the chaotic motions they found are limited in a very narrow zone which is very close to the center of the massive bodies. This paper argues that this is impossible because the region is actually inside of the stars, so the motions cannot exist at this place. In this paper, we scan all parameters space and find chaos and unstable fixed points outside of stars with big mass-quadrupole moments. The calculations show that chaos can only occur when the stars have prolate deformation. Because real deformation of stars should be oblate, all orbits of test particles around the rotating neutron stars described by Manko et al. solutions are regular. The case of nonzero dipolar magnetic moment has also been taken into account in this study.

Figure 1

$\mathcal{Q}$ as a function of $b$ while $M=2.904$, $a=1.549$. The mass deformation is oblate when $-1.355<b<4.453$, and for other $b$, is prolate. There are three minimum quadrupolar deformation values while $b=0.466$, $-2.85125$, and 7.032.

Figure 2

The Poincaré sections of a set of parameters with oblate deformation. It has two apart regions, one is the narrow chaotic zone on the left-hand side, the other is the wide regular motion zone. But the chaotic orbits in fact are inside of the central body.

Figure 3

The Poincaré sections of a set of parameters with prolate deformation. It clearly shows chaos at the exterior of the central star (black hole or neutron star).

Figure 4

An unstable fix point and a chain of islands at parameters: $E=0.95$, $L=3.0$, $a=0.6$, and $b=3.0$ with unit mass.

Figure 5

The FLI of the hyperbolic equilibrium point in Fig. 4. An exponential raise of FLI presents it as a chaotic motion.