Energetics of AGN Feedback
Integrating realistic active galactic nucleus (AGN) feedback into cosmological hydrodynamical simulations remains a major challenge as resolving the spatial coupling of jet energy over cosmic timescales is computationally prohibitive. We present an analytic framework that predicts the radial and polar-angle dependence of feedback energy from lobed AGNs, enabling a physically motivated, computationally efficient prescription for jet feedback. Built upon the Radio AGN in Semi-analytic Environments (RAiSE) dynamical model, our approach tracks the post-AGN jet phase evolution of radio sources through two distinct mechanisms: buoyantly rising bubbles subject to ablation, and the gravitational collapse of swept-up gas in the shocked shell. We find that these two feedback mechanisms produce strongly contrasting spatial distributions across ten representative cluster environments. Buoyant bubbles preferentially deposit energy in steep density gradients near the core radius, while the collapsing shocked gas shell drives heating within flatter cluster cores. Weak, short-lived AGN outbursts ($Q<10^{37}$ W and $t_{\rm on}<10$ Myr) confine their energy deposition to the inner 10 kpc of the cluster; longer-lived events ($t_{\rm on} \geqslant 100$ Myr) deposit less than 1% of their injected energy within 30 kpc. We find that time-averaged heating rates across multiple outbursts are sufficient to offset radiative cooling in all but the highest density cluster cores (within 100 kpc) for duty cycles of $0.08< δ\leqslant 1$. This framework provides a scalable basis for modelling anisotropic, physically motivated jet feedback that can be incorporated into next-generation cosmological simulations.