Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 65 / 4

A typical globular cluster has undergone appreciable two-body relaxation, since its stellar relaxation time is $t_{\mathrm{rel}}\sim 0.1(N/\ln N)t_{\mathrm{cross}}$: for $N\sim 10^5$ and stellar crossing time $t_{\mathrm{cross}}\sim 10^6\mathrm{yr}$ this is about $10^9\mathrm{yr}$, shorter than its age, allowing a relatively relaxed, tidally limited structure. A galaxy with $N\sim 10^{11}$ and $t_{\mathrm{cross}}\sim 10^8\mathrm{yr}$ instead has $t_{\mathrm{rel}}\sim 4\times 10^{16}\mathrm{yr}$ and behaves as a collisionless stellar system, retaining differences from formation, galaxy mergers, and star formation. Relaxation erases more memory in clusters than in galaxies, although real globular clusters also differ in concentration and evolutionary state, so structural similarity is approximate.

At small separations the extended dark-matter haloes overlap, and Chandrasekhar dynamical friction transfers orbital energy to gravitational wakes while tidal interactions also redistribute orbital energy. A characteristic decay time scales as $t_{\mathrm{df}}\sim v{r}^2/(Gm\ln \Lambda )$, so sufficiently massive close pairs evolve rapidly towards a galaxy merger and spend relatively little time at small separation. Short residence time depletes the close-pair population; the quoted separation scale is characteristic of the population, not a universal exclusion radius.

The stellar population has appreciable random velocity dispersion, and its declining radial density creates an outward stress gradient that supplies some support against gravity; cold gas has much less random-motion support and follows nearly the local circular speed. Neglecting velocity-ellipsoid tilt, the radial Jeans equation gives $v_c^2-{\overline{v}}_{\varphi }^2={\sigma }_R^2[-d\ln (n_{\ast }{\sigma }_R^2)/d\ln R-1+{\sigma }_{\varphi }^2/{\sigma }_R^2]$, generally positive for the stellar disk. The slower mean stellar rotation is asymmetric drift.

A nonrotating elliptical galaxy need not have an isotropic velocity ellipsoid: unequal random-motion stresses in different directions can support an oblate or triaxial shape without substantial mean rotation. The tensor virial theorem, $2T_{ij}+W_{ij}=0$ for a stationary isolated system with appropriate boundary terms, balances this directional kinetic stress against the directional gravitational stress. Anisotropic random motions can supply the flattening, whereas an isotropic nonrotating equilibrium has much stronger restrictions on its shape.

A flux-limited galaxy selection samples a much larger volume for bright galaxies: in the nearby Euclidean limit $d_{\max }\propto \sqrt{L}$, hence $V_{\max }\propto L^{3/2}$, so very numerous faint dwarf galaxies are usually detectable only nearby. With a Schechter function $\varphi (L)\propto L^{{\alpha }_S}{e}^{-L/L_{\ast }}$, detected counts per logarithmic luminosity interval scale as $L^{{\alpha }_S+5/2}{e}^{-L/L_{\ast }}$ and, for ${\alpha }_S\sim -1$ to $-1.3$, peak near $L\sim (1.2\text{–}1.5)L_{\ast }$ rather than at the numerous faint end. The competition between increasing survey volume and the bright-end cutoff concentrates detected luminosities near $L_{\ast }$, with the precise width depending on the selection and survey depth.

The differential gravitational field of the Milky Way removes loosely bound outer material by tidal stripping, giving globular clusters and satellite dwarf galaxies a characteristic tidal radius. For a low-mass satellite of mass $m$ on a circular orbit of radius $R$ in a spherical host, the Jacobi tidal radius is $r_t\simeq R\{m/[(3-d\ln M_h/d\ln R)M_h(<R)]{\}}^{1/3}$; eccentric orbits often make the strongest restriction near pericentre. Tidal limitation explains the sharp outer falloff, although escaping stars can form extended tidal tails, so a fitted luminosity cutoff is not an exact boundary of all associated material.

Massive stars rapidly return oxygen, magnesium, neon and other alpha elements through core-collapse supernovae, whereas Type Ia supernovae supply a substantial delayed iron contribution. If the galactic bulge and stellar halo formed much of their stellar mass rapidly, their stars locked in enhanced alpha-element to iron ratios before much delayed iron enrichment, while prolonged star formation in the local galactic disk included more iron-rich gas. The abundance contrast chiefly records different enrichment timescales, with the initial mass function and gas exchange also affecting the ratios.

In a compact starburst galaxy, stellar feedback can approach the radiation-pressure limit of a dusty starburst, $\kappa F/c\sim g$, so stronger luminosity drives a galactic outflow rather than allowing indefinitely increasing star formation rate. Optically thick starburst-disk calculations give a characteristic infrared flux near $10^{13}{L}_{\odot }{\mathrm{kpc}}^{-2}$; with $L_{\mathrm{IR}}\sim 10^{10}{L}_{\odot }[{\dot{M}}_{\ast }/(M_{\odot }{\mathrm{yr}}^{-1})]$ and an emitting area of order $1{\mathrm{kpc}}^2$, this corresponds to ${\dot{M}}_{\ast }\sim 10^3{M}_{\odot }{\mathrm{yr}}^{-1}$ and would consume $10^{10}{M}_{\odot }$ of gas in only $10^7\mathrm{yr}$. Feedback and a finite fuel supply produce a characteristic upper scale for compact bursts, rather than a universal ceiling on the total rate independent of star-forming area and gas supply.