Ohmic heating and field transmission in the upper atmospheres of Trappist-1 b (left column) and π Men c (right column). The upper panels show the heating rate Q at the sonic point r_c divided by the heating from photochemistry Q_ph at the same location, as function of the planetary field B_P and the stellar wind field B_sw. Blue corresponds to (Q/Q_ph)(r_c) = 0.01 and red to (Q/Q_ph)(r_c) = 100. Values below 0.01 are shown in gray. The dashed and dotted white lines in the top left panel correspond to estimates of the ambient magnetic field based on observations of the magnetic field at the surface of Trappist-1 (Reiners & Basri 2010). We note that we have no such constraints for π Men at the moment. The thin dashed lines correspond to contours of the ratio of 0.1, 1 and 10. The middle panels show the transmission of B_sw from the top of the atmosphere to the level of 1 µbar, as a function B_P (1 corresponds to a full transmission, 0 to a full screening). The bottom panels show the altitude (in pressure value) at which the maximal Ohmic heating rate is realized as a function of B_P. The pressure at the sonic point is shown by the black dashed line.