Paper 1 (Paper 210, Formation of Structure in the Universe, Minor topics, 26 April 2012)

gas heats when compression, shocks or absorbed radiation increase its internal energy. Important sources include accretion shocks, stellar winds, supernovae, photoionization, photoelectric emission from interstellar dust, cosmic rays and energy deposited by active galactic nuclei. photoionization heating uses the photoelectron energy above the ionization threshold; compression contributes mechanical work rather than photon absorption.

For specific internal energy

u

, an idealized thermal balance is

ρ \frac{D u}{D t} = - P \nabla \cdot v + H - C, C = n_{e} n_{H} Λ (T, Z) .

Here

H, C

are powers per volume and

Z

describes abundances. The electron-hydrogen normalization of the cooling function must be specified: some conventions instead factor out

n_{H}^{2}

. For a monatomic gas with total particle density

n

t_{cool} = \frac{3 n k _{B} T}{2 n _{e} n _{H} Λ}

when heating is negligible. Compare this with the dynamical time: rapid cooling permits loss of pressure support, whereas long cooling times allow a hot atmosphere to persist.

collisional excitation followed by spontaneous emission cools partially ionized gas; radiative recombination also produces continuum cooling. metal lines enhance cooling over substantial intermediate-temperature ranges, while thermal bremsstrahlung gives a roughly

Λ \propto T^{1/2}

high-temperature tail. Atomic hydrogen cooling becomes inefficient below roughly

1 0^{4} K

, where molecular line cooling and interstellar dust can be important. inverse Compton scattering cools hot electrons against a colder radiation field. A tabulated cooling curve assumes a composition and ionization model; photoionization or nonequilibrium ionization can change it.^[ref]

X-ray line ratios and continuum shape constrain temperature and ionization, while the emission measure constrains

\int n_{e} n_{H} d V

. Turning that into a density and radiative gas cooling time also requires a geometry; unresolved clumping, foreground absorption and multiphase gas are important controls.

Solution

inside-out collapse of a singular isothermal sphere starts from an idealized static envelope,

ρ (r) = \frac{a _{s}^{2}}{2 π G r ^{2}}, M (r) = \frac{2 a _{s}^{2} r}{G},

with constant isothermal sound speed

a_{s}

. Collapse begins centrally after formation of a central mass. An expansion wave moves outward through the initially static envelope at the sound speed:

r_{ew} = a_{s} t .

“Expansion” describes the outward-moving information front; the gas behind it moves inward. The exterior remains close to its original

r^{- 2}

equilibrium until the wave reaches it.

In the inner region, the central mass dominates and pressure becomes less important:

v_{r} ≃ - \frac{2 G M _{*}}{r}, ρ ≃ \frac{M ˙ _{*}}{4 π r ^{2} ∣ v _{r} ∣} \propto r^{- 3/2} .

(3)

The self-similar accretion scale is $\dot{M}_{*} = m_{0} a_{s}^{3} / G$ , where the dimensionless coefficient

m_{0}

follows from the dynamical similarity solution, not from counting material swept by the front. Indeed, the initial mass enclosed by the front grows at

2 a_{s}^{3} / G

, and some of that material remains in the infalling envelope. For fixed mean particle mass,

\dot{M}_{*} \propto T^{3/2}

. The original similarity calculation establishes the static outer and free-fall inner regimes.^[ref]

The model assumes spherical symmetry, isothermality and negligible rotation or magnetic stresses. An asymmetric optically thick spectral line alone does not establish infall: an optically thin systemic-velocity tracer and spatial maps help distinguish inward motion from rotation, foreground absorption and outflow.

Solution

galactic feedback couples star formation and black-hole accretion back to the gas supply. Stellar radiation, winds, supernovae and cosmic rays inject energy and momentum; active-galactic-nucleus feedback adds radiation, jets and winds powered by accretion. Heating, turbulence or gas ejection can suppress later star formation, while compression can sometimes promote it. Negative feedback permits self-regulation rather than unrestricted gas consumption.

Define the mass-loading factor

η = \dot{M}_{out} / \dot{M}_{*}

. If the launch speed scales with the escape speed and the injected energy per stellar mass is fixed, conservation of energy suggests

η \propto v_{esc}^{- 2}

; fixed injected momentum instead suggests

η \propto v_{esc}^{- 1}

. Coupling efficiencies, radiative losses and recycling change these ideal scalings. Thermal energy injected into dense gas can radiate away before driving an outflow, so an energy budget alone does not demonstrate effective feedback.

Radio-filled X-ray cavities provide a mechanical diagnostic. For a cavity containing a relativistic fluid with

γ = 4/3

, its enthalpy is

γ P V / (γ - 1) ≃ 4 P V

; dividing by an estimated age gives a mechanical-power estimate. Chandra X-ray Observatory observations of the Perseus cluster revealed fronts and ripples associated with radio bubbles, interpreted as weak shocks and pressure waves that can transport jet energy into the cooling atmosphere.^[ref]

The test requires ambient density and temperature estimates, radio/X-ray spatial comparison, projected cavity volumes and an age model. Brightness edges must be distinguished from absorption and cool rims; wave energy is not automatically thermalized locally. These controls determine whether inferred mechanical heating can actually offset radiative cooling.

d

Solution

cold dark matter is nonrelativistic and effectively collisionless on the scales of interest. Its small velocity dispersion limits free streaming, allowing small-scale gravitational wells to survive. baryons coupled to photons undergo acoustic motion and damping before recombination; dark matter does not share that photon pressure, and provides potential wells into which baryons can later fall.

For a subhorizon pressureless perturbation, once a Newtonian treatment is applicable,

\ddot{δ} + 2 H \dot{δ} - 4 π G \overset{ρ}{ˉ}_{m} δ = 0.

During radiation domination the matter contribution to the background gravity is small; neglecting it gives

δ ≃ C_{1} + C_{2} lo g t

. This slow subhorizon growth is the Mészáros effect. During matter domination,

δ = C_{+} a + C_{-} a^{- 3/2}

in an Einstein-de Sitter universe. Once

δ

approaches unity, linear theory fails and collapse, virialization and merging become important.

Modes entering the horizon during radiation domination therefore grow less before equality than longer modes entering later. This scale dependence is encoded in a transfer function, with a schematic linear matter power spectrum

P (k, a) = A k^{n_{s}} T^{2} (k) D^{2} (a)

. cold dark matter supports hierarchical assembly; a sufficiently large warm- or hot-dark-matter free-streaming length suppresses small-scale structure. Superhorizon density contrasts change under a gauge transformation in cosmological perturbation theory, so the subhorizon growth laws should not be applied indiscriminately outside the horizon. The original cold-dark-matter formation model explicitly connects weak free streaming to the spectrum that seeds galaxies and clusters.^[ref]

2

Solution

For the radial epicyclic frequency, introduce the effective potential at fixed specific angular momentum

l

Φ_{eff} = Φ + l^{2} / (2 r^{2})

. A circular orbit at

r_{c}

satisfies

Φ^{'} (r_{c}) = l^{2} / r_{c}^{3} = r_{c} Ω^{2} (r_{c})

. A small radial displacement

ξ

obeys

\ddot{ξ} = - Φ_{eff}^{''} (r_{c}) ξ

. Differentiate at fixed

l

before inserting the circular-orbit relation:

κ^{2} = \frac{3 l ^{2}}{r _{c}^{4}} + Φ^{''} (r_{c}) = 3 Ω^{2} + \frac{d}{d r} (r Ω^{2}) = 4 Ω^{2} + 2 r Ω Ω^{'} = 4 Ω^{2} (1 + \frac{r}{2Ω} Ω^{'}) .

For a flat rotation curve,

Ω \propto r^{- 1}

, so $κ = 2 Ω$ .

In a perturbation proportional to

exp [i (\int k d r + n ϕ - ω t)]

n

is the integer azimuthal mode number, giving

∣ n ∣

arms for a nonaxisymmetric pattern. Its angular pattern speed is

Ω_{p} = ω / n

for a real frequency and

n \neq = 0

;

a_{0}

is the unperturbed gas sound speed, or the specified effective fluid pressure-response speed. For growing patterns the angular phase speed uses

Re ω / n

. Axisymmetric modes have

n = 0

: their pattern speed is undefined, and one must use

(ω - n Ω)^{2} \to ω^{2}

, not set the pattern-speed expression to zero.

Define the Toomre parameter and Toomre wavenumber by

k_{T} = \frac{κ ^{2}}{2 π G σ}, Q = \frac{κ a _{0}}{π G σ} .

Then

q = ∣ k ∣/ k_{T}

gives

\frac{ω ^{2}}{κ ^{2}} = 1 - q + \frac{Q ^{2} q ^{2}}{4} = 1 - \frac{1}{Q ^{2}} + \frac{Q ^{2}}{4} (q - \frac{2}{Q ^{2}})^{2} .

(3)

For

a_{0} > 0

, the minimum is negative precisely when $Q < 1$ . Thus this condition is necessary and sufficient for an exponentially growing axisymmetric mode within the given local razor-thin fluid model;

Q = 1

is marginal. It is not a guarantee against global or nonaxisymmetric instabilities, and the collisionless stellar-disk criterion has a different pressure-response normalization.

The Toomre unstable band and fastest growth follow by minimizing

ω^{2}

∣ k ∣_{*} = \frac{π G σ}{a _{0}^{2}} = \frac{2 k _{T}}{Q ^{2}}, λ_{*} = \frac{2 a _{0}^{2}}{G σ}, γ_{*} = κ Q^{- 2} - 1 .

(4)

Here the growing branch has

ω = i γ_{*}

under the stated Fourier convention. Both signs of the radial wave number have the same growth rate. The unstable band is

q_{-} < q < q_{+}

, where

q_{\pm} = 2 (1 \pm 1 - Q^{2}) / Q^{2}

For the minimum Toomre parameter in an exponential flat-rotation disk with constant

a_{0}

, put

x = r / r_{0}

Q (r) = Q_{0} \frac{e ^{x - 1}}{x}, Q_{0} = \frac{2 Ω _{0} a _{0}}{π G σ _{0}} .

(5)

The logarithmic derivative is

d lo g Q / d x = 1 - 1/ x

, so the unique minimum occurs at

r = r_{0}

. Both

Q \to \infty

r ↓ 0

and

Q \to \infty

r \to \infty

. Hence inner and outer regions are stable within this idealized rotation law, even when an intermediate annulus is unstable. The whole disk is strictly stable in this local axisymmetric test if

Ω_{0} > \frac{π G σ _{0}}{2 a _{0}} .

(6)

Equality makes the ring at

r_{0}

marginal. The central conclusion uses the assumed flat rotation curve all the way inward; real central rotation laws need not retain that limit.

Figure 1.
Local disk stability
. **Details**.

3

Solution

The normalized cycloid has

\frac{R _{s}}{R _{m}} = \frac{1 - c o s η}{2}, \frac{t}{t _{m}} = \frac{η - s i n η}{π} .

The initially expanding perturbation reaches maximum radius at

η = π

t = t_{m}

, then formally collapses to zero radius at

η = 2 π

t = 2 t_{m}

. A finite virialized object is assigned the conventional formation epoch $t_{f} = 2 t_{m}$ ; random motions replace the pressureless point-collapse singularity. The cycloid alone does not describe that relaxation.

Figure 1.
Spherical-collapse history
. **Details**.

For the standard top-hat virialization estimate, write

U = - C_{G} G M^{2} / R

and retain the same gravitational structure factor

C_{G}

at turnaround and final equilibrium. The coherent kinetic energy vanishes at turnaround, whereas the virial theorem gives

2 K_{v} + U_{v} = 0

in the final isolated equilibrium. conservation of energy gives

U_{m} = K_{v} + U_{v} = \frac{U _{v}}{2}, R_{v} = R_{m} /2 .