Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 60 / 4

Older disc stellar populations generally have larger velocity dispersions, partly through accumulated galactic disk heating, so random-motion stresses supply more of their support against gravity. The radial Jeans equation then requires a smaller mean rotation than the circular speed, producing larger stellar asymmetric drift than in dynamically colder young stars.

Low-mass main sequence stars live for much of the Milky Way's age, retaining representatives of many generations, while phase mixing disperses their birth associations around their orbits. In an approximately axisymmetric galactic disk, a sufficiently complete local census can therefore sample the mixed annular stellar population at the Sun's radius despite its small volume. This is a representative local-radius sample, not automatically an unbiased whole-disc census: different vertical scale heights, selection effects and radial migration require corrections when inferring integrated population fractions.

Metal absorption lines depend on the temperatures, ionization states and luminosity weights of the contributing stellar populations, as well as their stellar metallicity. A young stellar population dominated by hot luminous stars can have weak metal absorption lines even at substantial abundance, so weak lines alone do not establish low metallicity, an aspect of the age-metallicity degeneracy.

Typical present elliptical galaxies are dominated by old stellar populations: their brightest short-lived stars have disappeared, leaving faint long-lived stars and stellar remnants, so luminosity per unit stellar mass is low. Ongoing star formation in many spiral galaxies and irregular galaxies supplies luminous young stars, giving a smaller stellar mass-to-light ratio in a fixed band; dark matter, the initial mass function and aperture must also be specified if the quoted ratio is dynamical.

Many open clusters are weakly bound and are dispersed by two-body relaxation, stellar evaporation and tidal heating, including encounters with molecular clouds in the galactic disk, so their present age histogram preferentially retains young survivors. More massive compact globular clusters can survive much longer, while the released low-mass stars remain as old field stellar populations: disruption removes clusters from the catalogue without removing their stars from the disc.

In the radial-residual convention $\delta r=a_4\cos 4\theta$, boxy isophotes have $a_4<0$ and disky isophotes have $a_4>0$, since negative $a_4$ extends the diagonal directions and positive $a_4$ extends the fitted axes. Massive elliptical galaxies tend to have slowly rotating, anisotropic stellar structures associated with dry galaxy mergers, whereas rotating embedded galactic disks, often associated with dissipation in gas-rich galaxy mergers, favour disky structure in lower-mass elliptical galaxies. These are population trends, as in Central structure measurements, rather than a universal mass rule, and brightness-residual Fourier coefficients must first be converted to this radial convention.

The neutral-hydrogen 21-centimeter line traces cold gas far beyond the bright stellar galactic disk, where a dark matter halo contributes strongly to the galaxy rotation curve. With relatively small random velocities, this gas approximately follows circular orbits, allowing its Doppler velocities to measure the circular speed after inclination and pressure corrections, whereas stellar dispersions require an orbital-anisotropy model. Its extended radial reach and comparatively simple force tracing make neutral gas especially useful for outer-halo constraints, although warps and noncircular motions can invalidate that simplicity.

Many lower-luminosity elliptical galaxies have dense central stellar cusps, giving far more photons and kinematic tracers near a central black hole than the shallow depleted cores common in massive elliptical galaxies, as measured in Central galaxy profiles. Thus central surface brightness can make the smaller black hole easier to measure, despite its smaller mass, whereas sparse central tracers and uncertain stellar orbital anisotropy weaken measurements in a large core. This is not a universal mass ordering: the black-hole sphere of influence $r_{\inf }\simeq G{M}_{\bullet }/{\sigma }^2$ must still be sufficiently resolved in angle, so distance, velocity dispersion and angular resolution also matter.