Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 346 (Formation of Galaxies)

At $10^8$--$10^9\mathrm{Hz}$, the diffuse radio continuum is dominated by Galactic synchrotron emission from relativistic cosmic-ray electrons spiralling in the Galactic magnetic field. It therefore traces an energetic-particle population and the magnetic field rather than gas density alone. Thermal free--free radiation from ionized gas becomes more important at higher radio frequencies and traces the emission measure $\int n_e^{2}dl$.

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The neutral-hydrogen 21-centimeter line is the ground-state hyperfine transition of atomic hydrogen. In the optically thin regime its velocity-integrated brightness is proportional to the H I column density, while the Doppler-resolved line maps line-of-sight velocity. Optical-depth measurements against a continuum source additionally constrain the spin temperature, so the line primarily traces atomic-gas density and kinematics.

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Low rotational transitions of carbon monoxide are excited by collisions in cold molecular clouds. CO survives in shielded gas and is readily observable, so the carbon-monoxide tracer of molecular hydrogen uses its integrated intensity with $X_{\mathrm{CO}}$ to infer the otherwise difficult-to-observe ${\mathrm{H}}_2$ column. Line ratios also constrain excitation temperature and density, but the dominant large-scale use is to map molecular-gas mass and velocity.

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Interstellar dust absorbs ultraviolet and optical starlight and thermally reradiates it. Far-infrared emission comes mainly from large cool grains near thermal equilibrium and probes dust column density times a temperature-dependent emissivity. Mid-infrared emission emphasizes warmer grains, stochastically heated small grains, and aromatic features near star-forming regions. These bands therefore trace both dust mass and the intensity of the radiation field that heats it.

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Near-infrared light is dominated over much of the Milky Way by old, cool, low-mass stars, with extra emission from young stars and hot dust in active regions. Because interstellar dust attenuates it much less strongly than optical light, near-infrared surface brightness and star counts are useful tracers of stellar mass and the obscured Galactic bulge and bar.

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Optical continuum light maps photospheric emission from stars and therefore depends on stellar density, luminosity, temperature, age, and composition. Recombination and forbidden lines additionally trace warm ionized gas. Strong wavelength-dependent absorption and scattering by interstellar dust obscure the inner disk, so an optical map is also a map of the foreground extinction.

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Galactic X rays include thermal bremsstrahlung and line emission from gas at roughly $10^6$--$10^8\mathrm{K}$, together with nonthermal emission from compact binaries, pulsars, and supernova remnants. They consequently trace hot plasma, shocks, accretion, and other high-energy populations; soft X rays are strongly absorbed by intervening gas.

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Diffuse Galactic gamma-ray emission is produced by neutral-pion decay after collisions of cosmic-ray nuclei with gas, electron bremsstrahlung in gas, and inverse-Compton scattering of relativistic electrons from starlight and microwave photons. Compact objects and supernova remnants add resolved sources. Gamma rays therefore probe cosmic-ray populations, their target gas or radiation fields, and particle acceleration.

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The differing maps follow from radiative transfer and from the different source functions of the bands. Low-frequency radio synchrotron emission passes through dust and follows cosmic-ray electrons and magnetic fields over a thick halo. Infrared emission follows dust mixed with dense gas and reprocessed starlight, whereas optical light follows unobscured stars and is strongly reshaped by extinction. The same matter distribution can therefore appear smooth at radio wavelengths, thin and clumpy in the infrared, and broken by dark dust lanes in the optical.

The Galactic scale height of a material tracer is set by vertical gravity balanced by its random, thermal, turbulent, magnetic, or cosmic-ray support. It also depends on source lifetime and transport. Cold molecular gas and dust have small velocity dispersions and settle into a thin layer. Relativistic electrons diffuse or advect far from their sources and radiate in a vertically extended magnetic field, so Galactic synchrotron emission is broad. Cosmic rays similarly fill a halo, and inverse-Compton gamma rays can be produced wherever they meet an extended radiation field; gamma-ray absorption within the Galaxy is also weak. Molecular and infrared emission remain concentrated around the thin cold-gas and young-star disk.

The spatial correlation between infrared and CO emission follows because molecular clouds contain dust and are the sites of massive-star formation. Young stars heat nearby dust, while CO maps their molecular fuel. Converting integrated CO intensity to ${\mathrm{H}}_2$ column density gives a molecular-gas surface-density profile; converting extinction-corrected infrared luminosity to a star formation rate gives the corresponding star-formation profile. Kinematic distances from velocity-resolved CO, with care near the Galactic center and for the near--far distance ambiguity, then estimate the radial relation between gas and star formation.

The three-dimensional Galactic bar is reconstructed by combining complementary distance and velocity information. Extinction-corrected near-infrared star counts reveal its projected stellar density, and standard-candle populations such as red-clump stars show that its near end is brighter and closer than its far end. CO and H I longitude--velocity diagrams trace noncircular gas streams and shocks in a barred potential. Maser parallaxes, proper motions, radial velocities, and dynamical forward models then constrain the bar angle, length, pattern speed, and vertical box or peanut shape.

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Let $\bar{\rho }(t)$ be the homogeneous density. A Lagrangian coordinate volume $d^{3}q$ contains the same mass as its image $d^{3}x$ under the Zel'dovich approximation, so mass conservation givesSince $1+\delta =\rho /\bar{\rho }$, the density contrast is

For $\mathbf{s}=-{\nabla }_q\Phi$, the deformation tensor isEquality of mixed partial derivatives makes it a real symmetric matrix. The real spectral theorem therefore supplies an orthonormal eigenbasis and real eigenvalues, which we denote by $-{\lambda }_1,-{\lambda }_2,-{\lambda }_3$. In that basis,and henceBefore the first crossing all factors are positive, allowing the absolute values to be omitted.

When $1-D{\lambda }_i=0$, the map loses rank in the corresponding principal direction. Its Jacobian determinant vanishes, trajectories meet, and shell crossing creates a cosmological caustic. The single-stream pressureless density formally diverges and the approximation no longer describes the subsequent multistream dynamics. If only one ${\lambda }_i$ is positive, one axis first collapses while the other two remain extended, producing a sheet or pancake. Collapse along a second and then a third principal axis produces filaments and nodes. Spatial variation of the eigenvalues joins these objects into the cosmic web around underdense voids.

The approximation succeeds because it reproduces linear growing-mode evolution exactly, preserves the initial tidal displacement and anisotropic collapse, and follows matter along nearly inertial comoving trajectories instead of expanding only the density at a fixed point. Large scales remain weakly nonlinear and are insensitive to the detailed dynamics after crossing. Its limitations begin at shell crossing: it permits streams to pass through one another, cannot produce virialized halos, and omits velocity dispersion, vorticity, gas pressure, shocks, feedback, and strongly nonlinear self-gravity.

For the one-dimensional displacement,so mass conservation givesThe earliest crossing occurs where the cosine is maximal:for integers $n$. The denominator vanishes at these isolated points.

Write $u=q-q_n$ at $t=t_c$. The Taylor series givesThus $|u|\sim (6|x-x_n|/k^2{)}^{1/3}$ andThe first caustic is therefore a cubic cusp with

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Put $x=r/r_s$. The enclosed mass of the Navarro--Frenk--White profile isSince $y/(1+y{)}^2=(1+y{)}^{-1}-(1+y{)}^{-2}$,The circular-orbit condition $v_c^2/r=GM(r)/r^2$ then gives the galaxy rotation curve

For $x\ll 1$,and thereforeFor $x\gg 1$,soThe NFW curve thus rises from the center, peaks, and eventually declines slowly. Typical spiral-galaxy rotation curves remain approximately flat over the observed outer disk, although an NFW halo combined with baryons can approximate such a plateau over a finite interval.

At the maximum, $x_m=2.16$, so the scale radius of a Navarro--Frenk--White profile isWriting $f(x)=\log (1+x)-x/(1+x)$, the characteristic density of a Navarro--Frenk--White profile isWith $G=4.3009\times 10^{-6}\mathrm{kpc}(\mathrm{km}{\mathrm{s}}^{-1}{)}^2{M}_{\odot }^{-1}$ and $f(2.16)=0.4670$, this is

An NFW cusp has $\rho \propto r^{-1}$ and hence $v_c\propto r^{1/2}$. Many dwarf and low-surface-brightness galaxies instead favor a nearly constant-density core, for which $M(r)\propto r^3$ and $v_c\propto r$; after matching the outer speed, this gives the more slowly rising observed inner curve. This is the cusp--core problem. Repeated burst-driven gas outflows can fluctuate the central potential and transfer orbital energy to dark matter, while self-interacting dark matter provides another possible core-forming mechanism. Beam smearing and noncircular motions are observational systematics but do not explain every case.

For an exponential galactic disk,soSince $7\mathrm{kpc}\simeq 2.3R_d$ is near the disk maximum, assigning the full $220\mathrm{km}{\mathrm{s}}^{-1}$ there givesIf the disk mass is increased while the observed total curve is fixed, the halo contribution must decrease in the inner galaxy. An NFW fit therefore needs a lower characteristic density, a larger scale radius and lower concentration, or a correlated combination of these changes.

A maximum disk model raises the stellar mass-to-light ratio as far as the rotation curve permits, conventionally making the disk contribute about $85\%$ of the speed near $2.2R_d$. High stellar surface densities, population-synthesis mass-to-light ratios, microlensing optical depths, and fast bars that have suffered little dynamical-friction braking can support a large disk contribution. Directly measured vertical stellar dispersions often favor submaximal disks, especially in low-surface-brightness galaxies, while stability and cosmological halo constraints may also disfavor an excessively massive disk.

The disk--halo degeneracy follows fromOver a finite radial range, raising the stellar mass-to-light ratio and lowering or broadening the halo can leave the same sum. The degeneracy can be reduced by measuring the disk surface density dynamically from vertical velocity dispersion and scale height, or by constraining the stellar mass-to-light ratio with resolved populations, colors, and a specified initial mass function. Strong or weak gravitational lensing, gas-layer flaring, stellar streams, and rotation data far beyond the optical disk provide further independent halo constraints.

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For either component $j\in \{h,s\}$ and $0<\gamma <3$, direct integration of the power law density givesLet $B_j=4\pi G{A}_j/(3-\gamma )$. Since $d{\Phi }_j/dr=G{M}_j/r^2=B_j{r}^{1-\gamma }$,The additive constants depend on the chosen reference and cannot generally be set by requiring $\Phi \to 0$ at infinity for an untruncated power law.

For a satellite on a circular orbit, the linearized effective radial force gives the tidal radiusHere ${\Omega }^2=G{M}_h(<r_o)/r_o^3=B_h{r}_o^{-\gamma }$ and ${\Phi }_h^{\prime \prime }=(1-\gamma ){\Omega }^2$, soSubstitution of the two mass profiles yieldsand equivalently

Now set $\gamma =1$. ThenThe host circular speed is $v=B\sqrt{r_o}$ with $B=\sqrt{2\pi G{A}_h}$. The magnitude of Chandrasekhar dynamical friction becomesa constant. The specific angular momentum is $j=r_{o}v=B{r}_o^{3/2}$. Since the drag torque gives $dj/dt=-r_o{a}_{\mathrm{df}}$,and thereforeIntegration giveswhich is finite.

If stripping is switched off, hold the satellite mass at its initial value $M_0=\mu r_{o0}^2$. The frictional acceleration is then $a_{\mathrm{df}}=a_0{r}_{o0}^2/r_o^2$, where $a_0$ is the constant acceleration in the stripped calculation. The same torque equation givesThusWithout tidal stripping, the satellite retains its mass while the background density increases inward, so dynamical friction strengthens rapidly. Stripping instead gives $M_s\propto r_o^2$ and removes the very mass that creates the gravitational wake.

For a singular isothermal host, ${\rho }_h=A_h{r}^{-2}$, $M_h\propto r$, and $v=B$ is constant. The tidal formula with a satellite ${\rho }_s=A_s{r}^{-1}$ givesConsequently $a_{\mathrm{df}}\propto M_s{\rho }_h/v^2\propto r_o^2$. Since now $j=B{r}_o$, the torque equation yieldsIts solution satisfies $r_o^{-2}=r_{o0}^{-2}+2C_{2}t$, so $r_o$ approaches zero only as $t^{-1/2}$ and never arrives in finite time. This is dynamical-friction stalling by tidal stripping: the inward tidal field strips the satellite so aggressively that its wake and drag disappear. Observationally, disrupted satellites should deposit stars in streams and the stellar halo, surviving low-mass remnants can remain at finite radii for very long times, and merger times inferred from a constant satellite mass can be severe underestimates.

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