We present a new 2 Ms Chandra observation of the cluster merger Abell 2146, which hosts two huge M~2 shock fronts each ~500 kpc across. For the first time, we resolve and measure the width of cluster merger shocks. The best-fit width for the bow shock is 17+/-1 kpc and for the upstream shock is 10.7+/-0.3 kpc. A narrow collisionless shock will appear broader in projection if its smooth shape is warped by local gas motions. We show that both shock widths are consistent with collisionless shocks blurred by local gas motions of 290+/-30 km/s. The upstream shock forms later on in the merger than the bow shock and is therefore expected to be significantly narrower. From the electron temperature profile behind the bow shock, we measure the timescale for the electrons and ions to come back into thermal equilibrium. We rule out rapid thermal equilibration of the electrons with the shock-heated ions at the 6 sigma level. The observed temperature profile instead favours collisional equilibration. For these cluster merger shocks, which have low sonic Mach numbers and propagate through a high $\beta$ plasma, we find no evidence for electron heating over that produced by adiabatic compression. Our findings are expected to be valid for collisionless shocks with similar parameters in other environments and support the existing picture from the solar wind and supernova remnants. The upstream shock is consistent with this result but has a more complex structure, including a ~2 keV increase in temperature ~50 kpc ahead of the shock. Comment: 18 pages, 17 figures, 1 table, accepted to MNRAS
