Skip to main content
Sandbox Physics
How black holes move energy through cluster gas

L1 · Bubbles, waves and cluster heating

How much enthalpy does a rising bubble transfer?

Sandbox Physics ·

A light pressure-balanced bubble expands as it rises into lower pressure. It can still carry most of its original energy after travelling far. Following enthalpy tells us how much energy it has transferred along the way.

Define one bubble first

Radius rr increases outward from the cluster centre. The bubble has volume VV and internal energy UU. Pressure pp is in pascals, volume in cubic metres and energy in joules. The gravitational acceleration magnitude gg is positive and points inward.

Assume a light bubble that stays approximately at pressure balance as it expands and rises. Its internal equation of state has a constant heat-capacity ratio. We use four-thirds inside the relativistic bubble; the surrounding monatomic thermal gas has five-thirds.

U=pVγb−1,H≡U+pV=γbγb−1pVU=\frac{pV}{\gamma_b-1},\qquad H\equiv U+pV=\frac{\gamma_b}{\gamma_b-1}pV
Enthalpy adds the internal energy to the work needed to make room for the cavity.

Reading: Rafferty et al. (2006) · Bîrzan et al. (2004)

Where the factor four comes from

For a heat-capacity ratio of four-thirds, internal energy is three times pressure times volume. Making room adds one more such contribution. Calculate each cavity separately before adding a pair.

The internal equation of state is an assumption about what supports the cavity. An ordinary monatomic gas would instead give a factor of two and a half. Uncertainty in the measured volume and surrounding pressure also enters the enthalpy estimate.

γb=43⟹U=3pV,H=4pV\gamma_b=\frac43\quad\Longrightarrow\quad U=3pV,\qquad H=4pV
The coefficient follows from the contents. The image helps estimate volume.

Reading: Rafferty et al. (2006)

Follow an adiabatic pressure change

After detachment, first neglect exchange of heat and matter. Expansion work comes from internal energy. Applying the first law to a small volume change gives the adiabatic relation.

dU=−p dV⟹pVγb=constantdU=-p\,dV\quad\Longrightarrow\quad pV^{\gamma_b}=\mathrm{constant}
The heat-capacity ratio is held fixed and pressure balance is maintained.
VVd=(pdp)1/γb,HHd=(ppd)(γb−1)/γb\frac{V}{V_d}=\left(\frac{p_d}{p}\right)^{1/\gamma_b},\qquad\frac{H}{H_d}=\left(\frac{p}{p_d}\right)^{(\gamma_b-1)/\gamma_b}
A subscript marks detachment. Increasing volume partly offsets the falling pressure.
γb=43:HHd=(ppd)1/4\gamma_b=\frac43:\qquad\frac{H}{H_d}=\left(\frac{p}{p_d}\right)^{1/4}
Halving external pressure leaves about 84.1 percent of the enthalpy; about 15.9 percent has been transferred.

A worked example

Take a hypothetical detachment pressure 5.00×10−11 Pa5.00\times10^{-11}\,\mathrm{Pa} and volume 1.00×1060 m31.00\times10^{60}\,\mathrm{m^3}. Initial enthalpy is 2.00×1050 J2.00\times10^{50}\,\mathrm J. These are specified exercise parameters, not a new measurement of Perseus.

At half the initial pressure, volume becomes about 1.682 times larger. A spherical radius grows by about 1.189. Remaining enthalpy is about 1.682 times ten to the fiftieth joules, leaving about 3.18 times ten to the forty-ninth joules available to drive surrounding gas.

QuantityMultiple of initial value
External pressure0.500
Bubble volume1.68
Spherical radius1.19
Bubble enthalpy0.841

Transferred energy can remain in motion

In an approximately hydrostatic atmosphere, the pressure gradient supports the gas weight. Differentiating the bubble enthalpy connects its decrease with buoyancy work.

dpdr=−ρg,dH=V dp⟹−dHdr=ρgV\frac{dp}{dr}=-\rho g,\qquad dH=V\,dp\quad\Longrightarrow\quad-\frac{dH}{dr}=\rho gV
The right side is the weight of displaced gas. This measures released mechanical energy before specifying its dissipation time.
K˙=−Kτ,Q(t)=E0(1−e−t/τ)\dot K=-\frac K\tau,\qquad Q(t)=E_0\left(1-e^{-t/\tau}\right)
For a specified exponential dissipation model, a packet of wake energy gradually becomes heat. After one timescale about 63.2 percent has thermalized.

Reading: Rafferty et al. (2006)

Try the observation

Separate the bubble path from delayed heating

  1. Predict whether a longer dissipation time changes the bubble path or mainly the arrival of heat.
  2. Compare Rising pair and Heat arrives later at the same physical time.
  3. Use the unchanged geometry and changed radial heat/wake energy to explain where the energy is stored.

Exercises and explained answers

Replace the bubble contents with an ordinary monatomic gas. What coefficient multiplies pressure times volume, and what fraction remains after pressure halves?

For γb=5/3\gamma_b=5/3, the coefficient is 5/25/2 and the retained fraction is 2−2/5≃0.7582^{-2/5}\simeq0.758. The internal equation of state changes the energy represented by the same pressure and volume.

What fraction of one deposited wake-energy packet has thermalized after two dissipation times? Must the bubble wait there?

The fraction is 1−e−2≃0.8651-e^{-2}\simeq0.865. The bubble can continue rising while energy deposited earlier dissipates at the earlier location.

Primary sources and calculation notes

  1. Bîrzan et al. (2004), A systematic study of radio-induced X-ray cavities in clusters, groups, and galaxies

    Frozen 30 April 2004 version: Section 4.1 on three cavity ages; Section 5.2, equations 7–9 on buoyancy and enthalpy transfer; Section 5.3 compares mechanical power with radiative losses. The Perseus input values come from the separate Rafferty 2006 tables.

  2. Rafferty et al. (2006), The feedback-regulated growth of black holes and bulges

    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.

FeedbackSomething broken, confusing, or missing on this page?Tell us
What kind of feedback?

Sent with this page’s address, its Lab ID, the site language and your device type. Please leave out personal details. Feedback is kept for up to 12 months.Privacy policy