$K^-$ - nucleus relativistic mean field potentials consistent with kaonic atoms

$K^-$ atomic data are used to test several models of the $K^-$ nucleus interaction. The t($rho$)$rho$ optical potential, due to coupled channel models incorporating the $Lambda$(1405) dynamics, fails to reproduce these data. A standard relativistic mean field (RMF) potential, disregarding the $Lambda$(1405) dynamics at low densities, also fails. The only successful model is a hybrid of a theoretically motivated RMF approach in the nuclear interior and a completely phenomenological density dependent potential, which respects the low density theorem in the nuclear surface region. This best-fit $K^-$ optical potential is found to be strongly attractive, with a depth of 180 pm 20 MeV at the nuclear interior, in agreement with previous phenomenological analyses.