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We present the detailed analysis of the thermal, diffuse emission of theproto-intracluster medium (ICM) detected in the halo of the Spiderweb Galaxy atz=2.16, within a radius of $\sim$ 150 kpc. We combined deep X-ray data fromChandra and millimeter observations of the Sunyaev-Zeldovich (SZ) effectobtained by ALMA. Thanks to independent measurements of the pressure profilefrom ALMA SZ observation and the electron density profile from the availableX-ray data, we derived, for the first time, the temperature profile in the ICMof a z>2 protocluster. It reveals the presence of a strong cool core(comparable to the local ones) that may host a significant mass depositionflow, consistent with measured local star formation values. We also find mildevidence of an asymmetry in the X-ray surface brightness distribution, whichmay be tentatively associated with a cavity carved into the proto-ICM by theradio jets or, alternatively, may be due to the young dynamical status of thehalo. The cooling time of baryons in the core of the Spiderweb Protocluster isestimated to be $\sim$ 0.1 Gyr, implying that the baryon cycle in the firststages of the protocluster formation is characterised by a high-duty cycle anda very active environment. In the case of the Spiderweb protocluster, we arewitnessing the presence of a strongly peaked core that is possibily hosting acooling flow with a mass deposition rate up to 250-1000 $M_{\odot}$/yr,responsible for feeding both the central supermassive black hole and the highstar formation rate observed in the Spiderweb Galaxy. This phase is expected tobe rapidly followed by active galactic nucleus feedback events, whose onset mayhave already left an imprint in the radio and X-ray appearance of the Spiderwebprotocluster, eventually driving the ICM into a self-regulated, long-termevolution in less than one Gyr.

Feeding and feedback processes in the Spiderweb proto-intracluster medium