Limitations in determining v_turb and T_kin from the two direct methods. Panels a) and b) show the relative error in v_turb for a given (v_turb/c_s) as a function of the temperature a) and line width b) corresponding to the direct method with a known excitation temperature and the co-spatial method, respectively. Panel c) shows the relative error in T_kin from the co-spatial method as a function of relative error in line width. The left panel does not take into account errors in the line width. The dashed contour lines show 10, 5, and 3σ limits.