Spinning compact object and chaos in galactic centers

1. Problem Statement

Galactic centers are highly dynamic environments dominated by Supermassive Black Holes (SMBHs) surrounded by nuclear star clusters, molecular gas, and asymmetric matter distributions (halos/disks). The gravitational interaction between the spinning SMBH and these asymmetric distributions creates a complex, nonlinear potential.

• The Challenge: Classical Newtonian monopole potentials fail to capture relativistic effects such as frame-dragging and the shift in the Innermost Stable Circular Orbit (ISCO) caused by the black hole's spin.
• The Gap: While previous studies have used Poincaré sections and chaos indicators (like SALI and Lyapunov exponents) to analyze these systems, there is a lack of comprehensive analysis regarding the stability of equilibrium points and the basins of convergence (Newton-Raphson basins of attraction) in the context of spinning compact objects with multipolar halo asymmetry.

2. Methodology

The authors employ a hybrid modeling approach combining pseudo-Newtonian potentials with multipolar expansions to simulate the galactic center environment.

A. Mathematical Formulation

• Central Potential (ABN): The central SMBH is modeled using the Artemova–Björnsson–Novikov (ABN) pseudo-Newtonian potential. This potential effectively mimics the Kerr metric's spin-dependent features (frame-dragging and ISCO shifts) within a Newtonian framework.
  • The spin parameter $a$ ranges from $0$ (non-rotating Schwarzschild) to $1$ (Newtonian limit).
  • The potential depends on the event horizon radius $r_1$, which is a function of $a$.
• Halo Potential (Multipolar Expansion): The surrounding asymmetric halo is modeled as an axisymmetric, shell-like mass distribution. The authors expand this potential up to the dipole term ($D$) to account for vertical asymmetry (e.g., displaced nuclear clusters).
• Effective Potential: The total effective potential ($U_{eff}$) acting on a test particle includes the ABN potential, the dipole term ($Dz$), and the centrifugal potential ($L^2/2\rho^2$), where $L$ is the conserved angular momentum.

B. Analytical and Numerical Techniques

1. Equilibrium Point Analysis:
   • The authors solve $\nabla U_{eff} = 0$ to locate fixed points (equilibrium points) in the $(\rho, z)$ plane.
   • Stability Analysis: The Hessian matrix of the effective potential is computed at these points. Eigenvalues determine stability: positive eigenvalues indicate stable points; negative eigenvalues indicate instability.
2. Newton-Raphson Basins of Convergence:
   • A dense grid ($1024 \times 1024$) of initial conditions in the configuration space is iterated using the Newton-Raphson method to find which equilibrium point (attractor) they converge to.
   • Fractal Analysis: The boundaries between these basins are analyzed. Fractal boundaries indicate high sensitivity to initial conditions (chaos), while smooth boundaries indicate predictable dynamics.

3. Key Contributions

• Novel Perspective: The paper shifts the focus from traditional chaos indicators (Lyapunov exponents) to fixed-point stability and basin of attraction topology, providing a complementary view of the system's global phase space structure.
• Spin-Dependent Dynamics: It systematically quantifies how the spin parameter $a$ alters the number, position, and stability of equilibrium points in a multipolar gravitational field.
• Visualizing Chaos: The study explicitly maps the transition from fractal (chaotic) basin boundaries to smooth (predictable) boundaries as the spin parameter varies.

4. Key Results

A. Variation of Equilibrium Points

• Non-Rotating Case ($a=0$): The system possesses 6 equilibrium points (4 stable, 2 unstable).
• Rotating Case ($0 < a < 1$):
  • The introduction of even a small amount of spin causes a drastic reduction in the number of fixed points.
  • The system stabilizes at 4 equilibrium points (2 stable, 2 unstable) immediately after $a > 0$.
  • The number of points remains constant at 4 regardless of the magnitude of $a$ until the Newtonian limit is approached.
• Newtonian Limit ($a=1$): The system reduces to 2 stable equilibrium points.
• Migration: As spin increases, the outer stable points ($L_1, L_6$) migrate outward, while inner unstable points migrate inward.

B. Basins of Convergence and Chaos

• Case $a=0$ (High Chaos):
  • The configuration space is dominated by fractal basin boundaries.
  • Initial conditions are extremely sensitive; a slight change leads to convergence to different attractors.
  • Most initial conditions do not converge to any attractor within the iteration limit, indicating chaotic scattering.
• Case $a=1$ (Low Chaos/Predictable):
  • The unstable attractors ($L_2, L_3, L_4, L_5$) vanish.
  • The remaining basins ($L_1, L_6$) become significantly larger with smooth, clearly distinguishable boundaries.
  • The system becomes highly predictable, with reduced sensitivity to initial conditions.

5. Significance and Conclusion

The study demonstrates that the spin of the central compact object is a critical regulator of dynamical chaos in galactic centers.

• Stabilizing Effect: Counter-intuitively, introducing spin (moving away from the non-rotating limit) reduces the number of equilibrium points and simplifies the phase space, effectively stabilizing specific orbital families and reducing the fractal nature of the basins.
• Dynamical Architecture: The findings suggest that relativistic spin effects, when combined with multipolar mass distributions, govern the accretion processes and orbital stability in galactic nuclei.
• Future Work: The authors propose extending this analysis to higher-order multipole terms (quadrupole, octupole) to create even more realistic models of galactic centers.

In summary, this paper provides a rigorous mathematical framework showing that black hole spin acts as a "chaos suppressor" in the specific context of multipolar galactic potentials, transforming the system from a highly sensitive, fractal-dominated regime to a more stable, predictable one.