The mass-dependent interplay of active galacitc nuclei and supernova feedback in shaping the $L_{\rm X}$--$T$ relation of early-type galaxies
The observed X-ray luminosity--temperature ($L_{\rm X}$--$T$) relation of hot gas in early-type galaxies deviates significantly from the prediction of purely gravitational heating, providing a key constraint on non-gravitational processes such as supernova (SN) and active galactic nucleus (AGN) feedback. We investigate the physical origin of this relation using high-resolution 3D hydrodynamical simulations with the multiscale AGN-regulated cosmic ecosystem resolver in 3D (MACER3D) framework, which we applied to a dwarf elliptical, a massive elliptical, and a cluster-central galaxy. For comparison, we performed controlled simulations that included AGN winds and SN feedback in isolation, excluding cosmological inflow and environmental effects. The dominant regulation mechanism depends strongly on the halo mass. In the cluster-central case, neither AGN winds nor SN feedback alone can sufficiently suppress the gas density and $L_{\rm X}$. When both are included, their nonlinear coupling suppresses the X-ray emission, producing ($L_{\rm X}$, $T$) values below the observed relation; this discrepancy can be resolved by incorporating AGN jet feedback. In massive elliptical galaxies, the inclusion of AGN feedback brings the model predictions into broad agreement with the observed $L_{\rm X}$--$T$ relation, indicating that AGN feedback dominates SN feedback. At the low-mass end, dwarf galaxy models also follow the observed trend. In this regime, models with either SN or AGN feedback alone predict low $L_{\rm X}$. When both are included, AGN wind-driven transport of SN-enriched gas to intermediate radii enhances the metallicity and radiative cooling, thereby increasing $L_{\rm X}$. This coupled process establishes a fountain-like circulation, in which gas is repeatedly lifted and recycled within the galaxy.