Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2018 / S2 / 2

At this small redshift, $D\simeq cz/H_0\simeq 20.0$ Mpc. The outer angular radius gives $R_{\mathrm{out}}\simeq D\theta \simeq 21.3$ kpc. Approximating the dynamical mass by a spherical equivalent,The supplied Tully-Fisher relation gives $M_V\simeq -22.49$, and hence $L_V\simeq 9.9\times 10^{10}{L}_{V,\odot }$. Treating the population as solar-like gives the estimate ${{\rm Y}}_V\simeq 1M_{\odot }/L_{V,\odot }$, soThe dynamical estimate exceeds that by about a factor of five; a massive extended dark-matter component is the natural interpretation. Solar-like spectra do not exactly fix the population mass-to-light ratio, and a disk is not spherical, but order-unity corrections do not remove this large outer mass excess under the stated estimate.

The effective radius is one tenth of the optical outer radius: $R_e\simeq 2.13$ kpc. A flat rotation speed then gives $M_{\mathrm{dyn}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$. Half the luminosity, with the same mass-to-light ratio, implies $M_{\mathrm{luminous}}(<R_e)\simeq 4.9\times 10^{10}{M}_{\odot }$.The concentration of light into a small fraction of the optical radius suggests a high-surface-brightness, centrally concentrated disk galaxy, plausibly an early-type spiral or lenticular with a substantial bulge. The kinematic and photometric data do not uniquely distinguish those morphologies. Disk geometry, an unknown inclination correction, or a varying stellar mass-to-light ratio would alter the exact numerical inner comparison.

Using the same luminosity estimate, the distance modulus is $12.5-(-22.49)=34.99$, implying $D\simeq 99.4$ Mpc. Pure Hubble expansion at that distance predicts about $7450\mathrm{km}{\mathrm{s}}^{-1}$. The measured velocity therefore impliesIts line-of-sight peculiar velocity is toward us, of the order expected for orbital motions in a rich galaxy cluster. A galaxy falling toward a cluster centre from its far side could have this sign. Its actual front/back location or three-dimensional orbit is not determined without the cluster's mean velocity, distance and projected position. Tully-Fisher scatter and the assumption of identical luminosities also limit that inference.

Let $I$ denote a characteristic surface brightness and ${\rm Y}=M/L$ the relevant dynamical mass-to-light ratio. With comparable profiles, $L\propto I{R}^2$ and $M\propto V^{2}R/G$. Eliminating $R$ givesIf both $I$ and ${\rm Y}$ are independent of speed, converting luminosity to magnitude gives $M_V=\text{constant}-10{\log }_{10}V$. The empirical coefficient $-8$ instead means $L\propto V^{3.2}$, and thereforeFor constant surface brightness this requires ${\rm Y}\propto V^{0.4}$; for constant mass-to-light ratio it requires $I\propto V^{0.8}$. A mixture of these changes, including varying stellar populations or dark-matter fractions, can give the same observed slope. The simple derivation assumes comparable characteristic radii and structural coefficients.