= Solution
<accretion> onto a massive <black hole> releases <luminosity> $L\simeq\eta\dot M c^2$. Even a small coupling of this <energy> to surrounding gas can matter because a galactic binding <energy> is only of order $M_{\rm gas}\sigma^2$, with stellar <velocity> dispersion $\sigma\ll c$. <active-galactic-nucleus feedback> means that the <energy> or <momentum> changes the gas supply, cooling or star formation, potentially regulating further <accretion>. Large <energy> availability alone does not establish efficient coupling.
In a radiatively efficient bright nucleus, <radiation pressure> and winds can push on dusty or ionized gas. Outflows can remove fuel or shock-heat it; the outcome depends on cooling, covering factor, clumping and whether the flow escapes. In a mechanically dominated state, <relativistic jets> inflate radio cavities in a hot halo or <galaxy cluster>. Their expansion, shocks, <sound waves> and subsequent motions can distribute <energy> and suppress a <cooling flow>. The two descriptions identify useful channels rather than a universal sharp division between all nuclei.
An approximately pressure-balanced cavity containing a relativistic fluid has <enthalpy>
$$
E_{\rm cav}=\frac{\Gamma}{\Gamma-1}pV\simeq4pV\quad(\Gamma=4/3),\qquad
P_{\rm cav}\sim\frac{4pV}{t_{\rm age}}.
$$
<X-ray> <pressure> estimates, cavity volume and buoyant or sound-crossing ages permit comparison with radiative cooling losses. Radio images check the jet association; <X-ray> fronts and ripples help locate transferred <energy>. Deep Chandra observations of Perseus reveal such structures and motivate mechanical heating of the cooling atmosphere.<A deep Chandra observation of the Perseus cluster: shocks and ripples by A. C. Fabian and others>[Abstract and discussion of radio-bubble-associated fronts, ripples and heating]{ref}
Projection, nonrelativistic cavity contents, <pressure> gradients, uncertain ages and intermittent duty cycles affect the <power> estimate. Injected cavity <energy> need not all dissipate where cooling occurs. A successful negative feedback loop requires that falling gas increases <accretion>, heating then reduces cooling or inflow, and the delay does not destroy regulation. This can explain reduced cooling and the shortage of excessively massive star-forming galaxies; correlations between black-hole <mass> and bulge properties are supporting context rather than standalone proof of causation. \b[The key comparison is coupled heating versus local cooling and binding, including where and when <energy> is deposited.]
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