Optical and thermodynamic properties of an analytical black hole solution in the steep Dehnen $$\left( 1,4,\frac{5}{2}\right) $$ dark-matter halo

Abstract Dark-matter halos can significantly influence the physical properties of black holes, particularly when realistic density profiles are taken into account. In this work, we consider a static, spherically symmetric black hole embedded in a Dehnen $$\left( 1,4,\frac{5}{2}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo> <mml:mfrac> <mml:mn>5</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mfenced> </mml:math> dark-matter halo. This configuration is the steepest analytically tractable Dehnen model with inner density slope $$\gamma &lt;3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>γ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , providing a natural setting to probe strong-gravity effects in cuspy dark-matter environments. Unlike previous analyses that rely on linear expansions in the halo parameters, we derive the null geodesics directly from the variational principle and obtain an exact description of photon motion in the full black hole-halo spacetime. This approach allows the light-ring structure and the associated gravitational lensing properties to be determined without perturbative approximations. We further examine the thermodynamic behavior of the system by constructing the enthalpy, entropy, temperature, heat capacity, and Gibbs free energy. We find that the dark-matter environment substantially modifies the thermodynamic structure of the black hole. In particular, increasing the halo density or size enhances thermodynamic stability and can trigger phase-transition that is absent in linearized approximation.