Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 349 (The Life and Death of Galaxies)

With the convention in the question, the constant velocity-anisotropy parameter is $\beta =1-{\sigma }_t^2/{\sigma }_r^2$. Since $V_c^2=rd\Phi /dr$, the Spherical Jeans equation becomesFor $\nu ={\nu }_0{r}^{-\alpha }$ and $V_c^2=V_0^2{r}^{2\gamma }$, its general integrating factor solution isThe homogeneous term represents a boundary pressure. For an extended scale-free system the physical boundary condition normally removes it, leavingPositivity requires $\alpha -2\beta -2\gamma >0$. A complete physical model must also have a nonnegative galactic distribution function and sensible inner and outer boundary behaviour; for example, strong radial anisotropy is restricted by density-slope--anisotropy inequalities.

Observationally, the tracer density can be estimated from star counts only after correcting distances, extinction, survey selection, and incompleteness. Spectroscopy supplies mainly line-of-sight velocities; proper motions add transverse information but become less precise for distant halo stars. The equation shows the mass--anisotropy--density degeneracy directly: the same measured ${\sigma }_r$ can result from a larger $V_c$, a steeper tracer slope $\alpha$, or a different $\beta$. Even globally constant power laws therefore do not determine the galactic mass profile unless some of these quantities are independently constrained.

If $\alpha$, $\beta$, or $\gamma$ changes near a break radius, the solution at one radius also depends on the outer boundary integral. A break in observed dispersion may be attributed to a mass-profile feature, a tracer-density break, or a change in orbital anisotropy. Separate tracer populations, full three-dimensional velocities, higher velocity moments, and measurements over a wide radial range help break this degeneracy.

More flexible alternatives model a nonnegative solution of the Collisionless Boltzmann equation itself. An action-based galactic distribution function gives an analytic or parametrized $f(\mathbf{J})$; a Schwarzschild orbit-superposition model assigns nonnegative weights to an orbit library; and a made-to-measure stellar-dynamical model adjusts particle weights to reproduce observations. These methods retain more phase-space information than Jeans moments, although their flexibility introduces model choices and regularization.

Direct evidence for stellar feedback includes expanding ionized shells and superbubbles around young associations, hot X-ray-emitting gas in supernova remnants, broad or split emission lines, and blueshifted absorption showing cool and warm outflows. P-Cygni profiles reveal massive-star winds, while extraplanar filaments and metal-enriched gas demonstrate transport away from star-forming disks. Indirect evidence includes the galaxy mass--metallicity relation, low baryon fractions and suppressed star formation in dwarf galaxies, chemically enriched circumgalactic gas, and correlations of outflow speed and mass loading with the star formation rate.

Rapid gas removal changes the gravitational potential before stellar and dark-matter orbits can respond adiabatically. Positions and velocities are initially unchanged, but orbital binding energies rise; orbits expand and become more eccentric, and some particles escape. Repeated burst--outflow--reaccretion cycles can irreversibly transfer energy to collisionless matter and turn a central dark-matter cusp into a core. This matters because dwarf-galaxy rotation curves are used to test dark-matter microphysics: a feedback-made core can mimic a non-cold or self-interacting dark-matter signature.

Write the initial potential energy as $W=-aG{M}^2/R$. The virial theorem gives $T=-W/2$. If a well-mixed fraction $ϵ$ remains after instantaneous mass loss, the immediate kinetic and potential energies are $T_a=ϵT$ and $W_a={ϵ}^{2}W$, soAfter revirialization at $R^{\prime }$, $E_f=W_f/2=-aG{ϵ}^2{M}^2/(2R^{\prime })$. Equating energies gives the impulsive mass-loss expansion lawThe remnant is bound only for $ϵ>1/2$; loss of half or more of the gravitating mass disrupts this idealized system.

For many infinitesimal, individually revirialized losses, put $ϵ=1+dM/M$ in the impulsive result. To first order, $dR/R=-dM/M$. Integration yields the adiabatic mass-loss expansion lawSlow loss causes finite expansion for every positive remaining mass fraction and has no sharp disruption threshold.

Finally consider an initially circular orbit of radius $R$ around a point mass $M$. Its speed and specific angular momentum obey $v^2=GM/R$ and $h^2=GMR$. Immediately after $M\to ϵM$, these remain unchanged, while the new specific energy isUsing the orbital eccentricity relation $e^2=1+2E^{\prime }{h}^2/(G^2{ϵ}^2{M}^2)$ givesIt is an ellipse for $ϵ>1/2$, parabolic at $ϵ=1/2$, and unbound for smaller $ϵ$, in agreement with the virial argument.

The Kennicutt–Schmidt law is the empirical relationfor disk-averaged total gas, with a nearly linear molecular-gas relation in many resolved observations. Atomic gas is mapped through the H I 21-cm line, molecular gas mainly through carbon-monoxide line emission and a CO-to-${\mathrm{H}}_2$ conversion factor, and star formation through combinations of ultraviolet continuum, H-alpha recombination emission, and infrared dust emission. Inclination, dust attenuation, the initial mass function, tracer lifetimes, and conversion factors must be treated consistently.

The gas-depletion time $M_g/{\dot{M}}_{\ast }$ is typically of order a gigayear, whereas a giant molecular cloud has a dynamical or free-fall time of order a few megayears. Star formation is therefore inefficient per collapse time, commonly at the percent level, rather than converting an entire cloud in one free fall.

For the first closed-box model of galactic chemical evolution, neglect returned mass or absorb it into the definitions. Thenand henceThus a region reaching $Z_{\odot }=0.014$ after an enrichment time $t_{\odot }$ with $y_Z=0.006$ hasabout $4.3\mathrm{Gyr}$ for a fiducial $t_{\odot }\simeq 10\mathrm{Gyr}$ enrichment age of the Galactic disk.

The second prescription implies ${\dot{M}}_{\ast }=M_g^2/(M_{g0}{\tilde{\tau }}_{\ast })$. Solving the gas-consumption equation givesTherefore

For long-lived stars, $dN$ is proportional to $d{M}_{\ast }=-d{M}_g$. In the exponential model,In the second model, $1+t/{\tilde{\tau }}_{\ast }=e^{Z/y_Z}$; multiplying its star-formation rate by $dt/dZ$ gives exactly the same result:The metallicity distribution is fixed by the closed-box relation $M_g/M_{g0}=e^{-Z/y_Z}$ and is independent of the star-formation history. Merely changing the time law therefore does not cure the G-dwarf problem; gas inflow, outflow, variable yields, or selection effects must alter the closed-box assumptions.

A star-forming galaxy contains short-lived, massive O and B stars whose hot photospheres dominate the ultraviolet and blue continuum and ionize surrounding gas. Once star formation ceases, these stars disappear quickly and an older, cooler stellar population produces a redder spectrum with stronger stellar absorption features and a prominent 4000-angstrom break.

In star-forming regions, direct stellar continuum is accompanied by nebular free-bound and free-free continuum, hydrogen and helium recombination lines, collisionally excited metal lines, and infrared emission from dust that absorbed shorter-wavelength photons. Supernova remnants and cosmic rays add synchrotron radio emission, while hot shocked gas can emit X-rays.

The Strömgren sphere model assumes a steady ionizing source in uniform, static, pure hydrogen of number density $n_H$, with a sharp ionization front enclosing fully ionized gas. If $N_i=(4\pi /3)R^3{n}_H$ is the number of ions, photon conservation givesThe equilibrium Strömgren radius and recombination time areConsequently the radius obeysWriting $x=(R/R_S{)}^3$ turns this into $dx/d(t/t_{\mathrm{rec}})=1-x$. For an initially neutral medium, $x(0)=0$, and the Ionization-front growth of a Strömgren sphere isThe front initially expands rapidly because few ions are recombining and asymptotically approaches $R_S$ as recombinations balance ionizations. This photon-counting solution precedes any pressure-driven hydrodynamic expansion of the H II region.