---
title: "Compare radial transport with radiative cooling"
description: "Define shells, powers and energies before comparing buoyant transport, delayed dissipation and a spherical thermal pulse."
canonical_url: "https://sandboxphysics.com/stories/agn-feedback/cooling-and-transport/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/agn-feedback/cooling-and-transport/"
last_modified: "2026-11-12"
---

[How black holes move energy through cluster gas](https://sandboxphysics.com/stories/agn-feedback/)

L2 · Bubbles, waves and cluster heating

# Compare radial transport with radiative cooling

Sandbox Physics · 2026-11-12

A large cavity energy is only a starting point. Its release radius and repetition time matter too. Comparing heat received and radiation lost in the same radial shells reveals which regions still lose energy.

## Before you begin

Read the cavity-enthalpy chapter first. You need the difference between power and energy and the volume of a spherical shell. Fully ionized hydrogen and free-free emission provide a tractable example.

## Fix the aperture and time interval

Radius $r$ increases outward. Each shell has an inner and an outer radius. Radiation power is measured in watts, accumulated losses in joules. Both curves must use the same outer boundary $R$ and time interval.

Power is energy per second. Energy is accumulated over an interval. Compare cavity enthalpy with a radiation energy over a specified time, or divide enthalpy by a stated time to obtain a power.

$$
L_{{\rm rad},k}(t)=\int_{r_k}^{r_{k+1}}\epsilon(r,t)\,4\pi r^2dr,\qquad E_{{\rm rad},k}=\int_{t_0}^{t_1}L_{{\rm rad},k}(t)\,dt
$$

Volume integration gives shell power; time integration gives the radiated energy.

Reading: Rafferty et al. (2006)

## Doubling density speeds up cooling

Fully ionized hydrogen has one electron per proton. Both species contribute thermal pressure, while free-free emission depends on opportunities for electron-ion encounters. At fixed temperature, doubling density doubles internal energy per volume but quadruples radiation power.

$$
u=3n_e k_BT,\qquad\epsilon_{\rm ff}=n_e^2\Lambda_{\rm ff}(T),\qquad t_{\rm cool,U}=\frac{u}{\epsilon_{\rm ff}}
$$

This local cooling time uses internal energy. An enthalpy-based definition or another composition changes its coefficient; check the definition when reading a paper.

| Change at fixed temperature | Multiple of initial value |
| --- | --- |
| Electron density | 2 |
| Internal energy per volume | 2 |
| Emission power per volume | 4 |
| Cooling time defined here | 0.5 |

## Heat can arrive after the bubble

Each shell receives enthalpy released while a bubble passes through it. First store this in wake motion, then allow it to dissipate over a specified time. A longer dissipation time leaves more energy in motion and less already thermalized along the same bubble path.

$$
\dot K_k=\dot W_k-\frac{K_k}{\tau_d},\qquad\dot Q_k=\frac{K_k}{\tau_d}
$$

Released cavity power enters the wake, and dissipated wake power enters heat. Summing shells preserves the transferred energy.

$$
H_{\rm pair}(t)+\sum_k\bigl[K_k(t)+Q_k(t)\bigr]=0.95E_{\rm in}(t)
$$

This is the bubble branch of the specified source example. The bubble accounting uses a fixed-temperature atmosphere.

## Spend each part of the input once

The teaching example assigns ninety-five percent of the total input to cavities. Five percent is stored and released during eighteen to nineteen million years inside a central three-kiloparsec sphere. The spherical gas then evolves under the Euler equations, gravity and radiative cooling. The partition is a declared model choice.

An eighteen-million-year mean total power $10^{43}\,\mathrm{erg\,s^{-1}}$ supplies $5.680368\times10^{50}\,\mathrm J$. Cavities receive $5.3963496\times10^{50}\,\mathrm J$ and the thermal channel $2.840184\times10^{49}\,\mathrm J$. Their sum returns the total input.

Giving the bubbles the full input and then adding five percent again would count extra energy. Nor is the bubble wake deposited a second time into the Euler gas. The two calculations share their initial atmosphere and input, with their mutual backreaction omitted.

Reading: Sandbox Physics

## Subtract matching cooling to identify the wave

The weak pulse propagates through gas that is cooling and slowly readjusting. Its total velocity includes background motion. A matching calculation starts from the same atmosphere with cooling alone, and the two are compared at identical times and finite-volume cells.

The difference identifies pressure and velocity changes produced by the pulse. Both states are nonlinear solutions; a new source power requires a new calculation. At the final sixty-million-year frame, the largest spatial relative error is about 3.32 percent, and the same-grid half-timestep error is about 4.86 parts per million.

These checks admit this gentle numerical wave. Sustained heating at the needed radii also depends on inflow, dissipation and the source response over repeated events. A real cavity catalogue provides a useful next check of the energy and time scales.

Reading: Sandbox Physics · Rafferty et al. (2006)

## Try the observation

### [Hold time fixed when comparing transport and heat](https://sandboxphysics.com/labs/agn-cavity-transport/)

1. At one bubble time, compare heat already received with normal radiation losses.
2. Switch to the wave view and hold a probe at one radius through actual frames.
3. Explain why passage of a wave and sufficient local heat are different physical tests.

## Exercises and explained answers

Why does gas with twice the density cool faster even though it holds more internal energy at the same temperature?

Internal energy doubles, but electron-ion encounters make emission quadruple. Internal energy divided by radiation power is therefore halved.

One outburst carries enough cavity enthalpy for five million years of radiation. Similar events repeat every twenty million years. Is that enough for sustained balance?

Under these specified values, the average input covers one quarter of the radiation loss. More frequent events or other energy sources are needed; radius and dissipation time also affect local gas even when average powers match.

[Previous: How much enthalpy does a rising bubble transfer?](https://sandboxphysics.com/stories/agn-feedback/enthalpy-of-a-rising-bubble/) [Next: Replay a Perseus cavity pair against cooling losses](https://sandboxphysics.com/stories/agn-feedback/perseus-cavity-replay/)

## Primary sources and calculation notes

1. [Rafferty et al. (2006), The feedback-regulated growth of black holes and bulges](https://arxiv.org/pdf/astro-ph/0605323v2)

   Frozen 20 July 2006 version: Table A1 Perseus rows, Table 2 Perseus row/footnotes, Sections 3.1.2–3.1.3. The tutorial replays the inner pair with a separately stated age estimator.

2. [Sandbox Physics, frozen gentle-source numerical evidence (2026)](https://sandboxphysics.com/stories/agn-feedback/materials/wave-refinement-manifest.json)

   Actual 512/1024/2048/4096 refinement history, original shock criteria and matching controls. The source power and scientific gates were kept fixed; the final scope remains no resolved AGN shock.