Ultra-low frequency observations (<100 MHz) are particularly challenging because they are usually performed in a low signal-to-noise ratio regime due to the high sky temperature and because of ionospheric disturbances whose effects are inversely proportional to the observing frequency. Nonetheless, these observations are crucial to study the emission from low-energy populations of cosmic rays. We aim to obtain the first thermal-noise limited (~ 1.5 mJy/beam) deep continuum radio map using the LOFAR Low Band Antenna (LBA) system. Our demonstration observation targeted the galaxy cluster RX J0603.3+4214 (the Toothbrush cluster). We used the resulting ultra-low frequency (58 MHz) image to study cosmic-ray acceleration and evolution in the post shock region, as well as their relation with the presence of a radio halo. We describe the data reduction we have used to calibrate LOFAR LBA observations. The resulting image is combined with observations at higher frequencies (LOFAR 150 MHz and VLA 1500 MHz) to extract spectral information. We obtained the first thermal-noise limited image from an observation carried out with the LOFAR LBA system using all Dutch stations at a central frequency of 58 MHz. With 8 hours of data, we reached an rms noise of 1.3 mJy/beam at a resolution of 18 x 11. The procedure we have developed is an important step forward towards routine high-fidelity imaging with the LOFAR LBA. The analysis of the radio spectra shows that the radio relic extends to distances of 800 kpc downstream from the shock front, larger than what allowed by electron cooling time. Furthermore, the shock wave started accelerating electrons already at a projected distance of <300 kpc from the crossing point of the two clusters. These results can be explained if electrons are reaccelerated downstream by background turbulence possibly combined with projection effects.
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