We report on a relaxation method to determine the electron-to-gas temperature ratio in the near-cathode region of a low-current free-burning arc operated in argon under atmospheric pressure. We used an intensified CCD camera to record the relaxation process after a fast power-interruption of the discharge with a time resolution better than 1 µs. For the CCD images we applied an ABEL inversion procedure yielding 3D information about the temperature ratio β := T e /T g with a spatial resolution of 50 µm. We deduced spatially resolved values of β from the relaxation experiment using independently measured values of T e and n e , so we could determine all free parameters in the partial LTE model.