Charge dependence of cusp formation in a phenomenological KZ-type black-hole metric

Abstract We examine the equatorial capture shadow of a phenomenological charged Konoplya–Zhidenko-type metric with $$\Delta =r^2+a^2-2Mr+Q^2-\epsilon /r$$ Δ = r 2 + a 2 - 2 M r + Q 2 - ϵ / r . The charge term is introduced in Kerr–Newman form; no field-equation derivation is asserted. Using unstable spherical photon orbits that satisfy angular admissibility and the full radial-capture condition, we calculate contacts of projected orbit images. For the theoretical benchmark $$(a/M,\epsilon /M^3)=(2,0.5)$$ ( a / M , ϵ / M 3 ) = ( 2 , 0.5 ) , the cusp-removal contact reaches the celestial axis at $$Q_{\textrm{off}}/M\simeq 1.3520$$ Q off / M ≃ 1.3520 . A sampling of spin and deformation values shows how this contact charge varies within the ansatz. In the moderate-spin family $$(a/M,\epsilon /M^3)=(0.9,0.5)$$ ( a / M , ϵ / M 3 ) = ( 0.9 , 0.5 ) , the shadow is smooth over $$0\le Q/M\le 0.6$$ 0 ≤ Q / M ≤ 0.6 , develops a cusp near $$Q/M=0.9518$$ Q / M = 0.9518 , and loses it near $$Q/M=1.3313$$ Q / M = 1.3313 . The cusp morphology therefore depends on the chosen parameter family; the values are theoretical controls, not observational constraints.