Chaotic dynamics around astrophysical objects with nonisotropic stresses

The existence of chaotic behavior for the geodesics of the test particles orbiting compact objects is a subject of much current research. Some years ago, Guéron and Letelier [Phys. Rev. E 66, 046611 (2002)] reported the existence of chaotic behavior for the geodesics of the test particles orbiting compact objects like black holes induced by specific values of the quadrupolar deformation of the source using as models the Erez--Rosen solution and the Kerr black hole deformed by an internal multipole term. In this work, we are interested in the study of the dynamic behavior of geodesics around astrophysical objects with intrinsic quadrupolar deformation or nonisotropic stresses, which induces nonvanishing quadrupolar deformation for the nonrotating limit. For our purpose, we use the Tomimatsu-Sato spacetime [Phys. Rev. Lett. 29 1344 (1972)] and its arbitrary deformed generalization obtained as the particular vacuum case of the five parametric solution of Manko et  al. [Phys. Rev. D 62, 044048 (2000)] characterizing the geodesic dynamics throughout the Poincaré sections method. We found only regular motion for the geodesics in the Tomimatsu-Sato $\delta =2$ solution. Additionally, using the deformed generalization of Tomimatsu-Sato $\delta =2$ solution given by Manko et al. we found chaotic motion for oblate deformation instead of prolate deformation, which is in contrast to the results by Guéron and Letelier. It opens the possibility that the particles forming the accretion disk around a large variety of different astrophysical bodies (nonprolate, e.g., neutron stars) could exhibit chaotic dynamics. We also conjecture that the existence of an arbitrary deformation parameter is necessary for the existence of chaotic dynamics.

Figure 1

(a) Boundary contour $\Phi =0$ using $E=0.94$, $L=-3.12$. There is one escape zone in the left-hand side of the picture, which correspond to small values of $x$, and a closed zone of bounded motion to the right. (b) Poincaré sections in the plane $x{P}_x$ for the values defined in (a). We have regular motion.

Figure 2

(a) Boundary contour $\Phi =0$ for $M=1.840M_{\odot }$, $J=3.683$, $b=-0.3792$, $E=0.1$ and $L=-5.6$. There is two escape zones, and a closed zone of bounded motion. (b) Poincaré sections in the plane $x{P}_x$ for the values defined in (a). We see only regular motions.

Figure 3

(a) Boundary contour $\Phi =0$ for $M=1.936M_{\odot }$, $J=4.498$, $b=-0.3080$, $E=0.95$ and $L=-9.0$. There is one escape zone, and a closed zone of bounded motion. (b) Poincaré sections in the plane $x{P}_x$ for the values defined in (a). The geodesics are only regular.

Figure 4

(a) Boundary contour $\Phi =0$ for $M=1.936M_{\odot }$, $J=4.498$, $b=0.8$, $E=0.96$ and $L=-9.9$. There are two small escape zones, and two closed zones of bounded motion. (b) Poincaré sections in the plane $x{P}_x$ for the values defined in (a). We see chaotic motion to the left-hand side and regular motion to the right-hand side.

Figure 5

(a) Boundary contour $\Phi =0$ for $M=1.936M_{\odot }$, $J=4.498$, $b=0.8$, $E=0.971$ and $L=-9.3$. There are two small escape zones, and two regions linked by a narrow connection of bounded motion. (b) Poincaré sections in the plane $x{P}_x$ for the values defined in (a). We see chaotic motion in the left-hand side of the figure and in a small external region on the right-hand side.