Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2016 / S26 / 3 / 2 / Solution

The 21.1 cm line is neutral hydrogen's hyperfine spin-flip transition, observed with radio telescopes and interferometers in the cool/warm neutral gas disk, often beyond the bright stellar disk. The mean wavelength gives $z=22.0/21.1-1\simeq 0.0427$ and, using the stated low-redshift Hubble approximation, $d\simeq 180$ Mpc. A 16-arcsecond angular radius gives $R\simeq 14$ kpc. For a thin circular disk, $\cos i=8/32$, so $i\simeq 75.5^{\circ }$ from face-on.

The original spectrum reaches roughly 21.98 and 22.02 cm: $v_{\mathrm{los},\max }\simeq c(0.02/22.0)\simeq 273$ km s${}^{-1}$. Deprojecting gives $\boxed{v_{\max }\simeq 282\mathrm{km}{\mathrm{s}}^{-1}}$. A spherical dynamical estimate $M(<R)\simeq v^{2}R/G$ gives $\boxed{M\sim 2.6\times 10^{11}{M}_{\odot }}$. A thin massive disk is not spherical: its exact rotation-to-total-mass coefficient depends on radial mass profile. The no-exterior-matter assumption does not uniquely fix that coefficient, so the quoted result is the conventional order-of-magnitude dynamical estimate. Cosmological luminosity and angular-diameter distances differ at this redshift; the source's simple Hubble-distance convention is being used consistently.