Periodic orbits and quasinormal modes of a black hole surrounded by King dark matter halo

Abstract We examine the Hawking radiation sparsity and quasinormal mode (QNM) spectra of a Schwarzschild black hole surrounded by the King dark matter distribution. The modified metric, derived from the King density profile, yields altered expressions for the lapse function, mass, and Hawking temperature. The presence of dark matter reduces the Hawking temperature, indicating suppressed thermal emission. Analysis of radiation sparsity reveals its dependence on the horizon radius, with larger scale radius R and central density $$\rho _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> enhancing sparsity compared to the constant value in the Schwarzschild case. The photon sphere and shadow radius increase with R and $$\rho _0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> </mml:math> , suggesting enlarged black hole shadows under dark matter influence. Hawking emission rates exhibit a downward shift in both peak intensity and frequency. The QNM spectra, obtained via the Mashhoon and 3rd-order Wentzel–Kramers–Brillouin (WKB) methods, show that both the oscillation frequencies and damping or decay rate decrease with the dark matter parameter $$\alpha = 8\pi \rho _0 R^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>8</mml:mn> <mml:mi>π</mml:mi> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . This study also examines how King dark matter halo affects periodic particle motion around a black hole, showing that it alters orbital stability and enhances precession, leading to transitions from bound to unbound motion. These results collectively demonstrate that King dark matter substantially modifies the thermodynamic and dynamical properties of black holes, offering potential observational imprints in black hole shadow and gravitational wave studies.