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MS 0735.6+7421 is a galaxy cluster which hosts a central radio galaxy with avery steep spectrum, produced by one of the most powerful known jetted activegalactic nuclei (AGN). The radio plasma, ejected at nearly light speed from thecentral AGN, have displaced the intra-cluster medium, leaving two pairs ofcavities observable in the X-ray, associated to two different outbursts, andhave distributed energy to the surrounding medium. In this work we haveperformed for the first time a detailed, high-resolution spectral study of thesource at radio frequencies and investigated its duty cycle to be compared withprevious X-ray estimates. We have used new observations at 144 MHz producedwith the LOw Frequency ARray (LOFAR) together with archival data at higherfrequencies. At LOFAR frequency, the source presents two large outer radiolobes, wider than at higher frequencies, and a smaller Intermediate lobelocated south-west of the core. A new inspection of X-ray data, allowed us toidentify an intermediate cavity, associated with that lobe, indicating thepresence of a further phase of jet activity. The radio lobes have a steepspectrum even at LOFAR frequencies, reaching $\alpha_{144}^{610}=2.9$ in theouter lobes and $\alpha_{144}^{610}=2.1$ in the Intermediate lobe. Fitting thelobe spectra using a single injection model of particle ageing, we derived atotal age of the source between 170 and 106 Myr, in agreement with the buoyancyand sound crossing time-scales derived from X-ray data. We then reconstructedthe duty cycle of the source. There were three phases of jet activity, with theAGN being active for most of the time with only brief quiescent phases,ensuring the repeated heating of the central gas. Finally, energetic estimatesrevealed that a source of additional pressure support must be present tosustain the bubbles against the pressure of the external medium.

Constraining the AGN duty cycle in the cool-core cluster MS 0735.6+7421 with LOFAR data