Shadow radius and classical scattering analysis of two secondary hair Horndeski black holes

We perform the shadow radius analysis of a charged Horndeski black hole (CHB) and the naked singularity (NS) with secondary scalar hair obtained from the Einstein-Horndeski-Maxwell theory. For this analysis, we include the beyond Horndeski black hole (bH) with secondary scalar hair and the magnetically charged black hole (MC) found from the Einstein-Euler-Heisenberg theory. It is worth noting that the NS versions of CHB and bH arise from the charge extension of their photon spheres, while there is no NS version for MC. One branch (i) from the CHB is a point in the horizon realization but it shows up on the photon sphere and shadow radius. The shadow radius for the CHB is the nearly same as that for the MC with a single horizon and the charge of the NS is constrained by the EHT observation. From classical scattering analysis, it turns out that i-NS and NS play different roles from CHB, bH, and MC.