We present deep LOFAR observations between 120-181 MHz of the Toothbrush (RX J0603.3+4214), a cluster that contains one of the brightest radio relic sources known. Our LOFAR observations exploit a new and novel calibration scheme to probe 10 times deeper than any previous study in this relatively unexplored part of the spectrum. The LOFAR observations, when combined with VLA, GMRT, and Chandra X-ray data, provide new information about the nature of cluster merger shocks and their role in re-accelerating relativistic particles. We derive a spectral index of $alpha = -0.8 pm 0.1$ at the northern edge of the main radio relic, steepening towards the south to $alpha approx - 2$. The spectral index of the radio halo is remarkably uniform ($alpha = -1.16$, with an intrinsic scatter of $leq 0.04$). The observed radio relic spectral index gives a Mach number of $mathcal{M} = 2.8^{+0.5}_{-0.3}$, assuming diffusive shock acceleration (DSA). However, the gas density jump at the northern edge of the large radio relic implies a much weaker shock ($mathcal{M} approx 1.2$, with an upper limit of $mathcal{M} approx 1.5$). The discrepancy between the Mach numbers calculated from the radio and X-rays can be explained if either (i) the relic traces a complex shock surface along the line of sight, or (ii) if the radio relic emission is produced by a re-accelerated population of fossil particles from a radio galaxy. Our results highlight the need for additional theoretical work and numerical simulations of particle acceleration and re-acceleration at cluster merger shocks.
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