Reaching thermal noise at ultra-low radio frequencies Toothbrush radio relic downstream of the shock front

Context. 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 for studying the emission from low-energy populations of cosmic rays.Aims. We aim to obtain the first thermal-noise limited (similar to 1.5 mJy beam(-1)) deep continuum radio map using the Low Frequency Array's Low Band Antenna (LOFAR LBA) system. Our demonstration observation targeted the galaxy cluster RX J0603.3+4214 (known as the Toothbrush cluster). We used the resulting ultra-low frequency (39-78 MHz) image to study cosmic-ray acceleration and evolution in the post shock region considering the presence of a radio halo.Methods. We describe the data reduction we used to calibrate LOFAR LBA observations. The resulting image was combined with observations at higher frequencies (LOFAR 150 MHz and VLA 1500 MHz) to extract spectral information.Results. 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 eight hours of data, we reached an rms noise of 1.3 mJy beam(-1) at a resolution of 18 '' x11 ''.Conclusions. The procedure we developed is an important step 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 is 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 may be explained by electrons being re-accelerated downstream by background turbulence, possibly combined with projection effects with respect to the radio halo.