The galaxy population shows a characteristic bimodal distribution based on the star formation activity and is sorted into star-forming or quiescent. These two subpopulations have a tendency to be located in different mass halos. The circumgalactic medium (CGM), as the gas repository for star formation, might contain the answer to the mystery of the formation of such bimodality. Here we consider the bimodality of the galaxy population and study the difference between the properties of the hot CGM around star-forming and quiescent galaxies. We used the X-ray data from the first four SRG/eROSITA all-sky surveys (eRASS:4). We selected central star-forming and quiescent galaxies from the Sloan Digital Sky Survey DR7 with stellar mass $10.0< or halo mass $11.5< 200m /M_ within spectroscopic redshift spec <0.2$, and we built approximately volume-limited galaxy samples. We stacked the X-ray emission around star-forming and quiescent galaxies, respectively. We masked detected point sources and carefully modeled the X-ray emission from unresolved active galaxy nuclei (AGN) and X-ray binaries (XRB) to detect the X-ray emission from the hot CGM. We measured the X-ray surface brightness ($S_ X, CGM $) profiles and integrated the X-ray emission from hot CGM within $R_ 500c $ ($L_ X, CGM $) to provide the scaling relations between $L_ X, CGM $ and galaxies' stellar or halo mass. We detect extended X-ray emission from the hot CGM around star-forming galaxies with $ and quiescent galaxies with $ extending out to 500c $. The $S_ X, CGM $ profile of quiescent galaxies follows a beta model with $ 0.4$, where beta quantifies the slope of the profile. Star-forming galaxies with median stellar masses $ *,med /M_ = 10.7, 11.1, 11.3$ have X, CGM erg/s$, while for quiescent galaxies with $ *,med /M_ X, CGM erg/s$. Notably, quiescent galaxies with $ *,med /M_ > 11.0$ exhibit brighter hot CGM than their star-forming counterparts. In halo mass bins, we detect similar X-ray emission around star-forming and quiescent galaxies with $ 200m /M_ > 12.5$, suggesting that galaxies in the same mass dark matter halos host equally bright hot CGM. We emphasize that the observed X, CGM - M_ 500c $ relations of star-forming and quiescent galaxies are sensitive to the stellar-to-halo mass relation (SHMR). A comparison with cosmological hydrodynamical simulations (EAGLE, TNG100, and SIMBA) reveals varying degrees of agreement, contingent on the simulation and the specific stellar or halo mass ranges considered. Either selected in stellar mass or halo mass, the star-forming galaxies do not host brighter stacked X-ray emission from the hot CGM than their quiescent counterparts at the same mass range. The result provides useful constraints on the extent of feedback's impacts as a mechanism for quenching star formation as implemented in current cosmological simulations.
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