Spectra and velocities

For $v\ll c$,

Positive means receding (redshift). For larger speeds, $1+z = \sqrt{(1+\beta)/(1-\beta)}$. Use the air wavelength when the data are in air (Hα at 6562.82 Å in air, 656.281 nm in the 2017 problem; follow the value the problem gives).

The 2017 galaxy spectra show shifts of only 0.1–0.2 nm on 656 nm, so keep all the decimals: a 0.001 nm reading error is 0.5 km/s.

For circular orbits in a spherical mass distribution,

Independent components add in quadrature: $v_{\rm obs}^2 = v_{\rm gas}^2 + v_*^2 + v_{\rm DM}^2$. A flat rotation curve ($v$ constant) means $M\propto r$, so $\rho\propto r^{-2}$ at large radius. Visible matter alone would give $v\propto r^{-1/2}$ (Keplerian) outside the light. The difference is the evidence for dark matter.

For a star in a circular orbit with period $P$ and RV semi-amplitude $K$:

The mass function is a lower limit on $M_2$ (it equals $M_2$ only if $\sin i = 1$ and $M_1 = 0$). To get $M_2$ you need $M_1$ and $i$ from elsewhere: $\log g$ and a radius give $M_1 = gR^2/G$; a rotation period and $v\sin i$ give $\sin i = Pv\sin i/(2\pi R)$ if the rotation is synchronised.

If $M_2$ exceeds about 2.5 $M_\odot$ and the companion emits nothing, it is a black hole candidate (2018 D2).

An orbiting pulsar's apparent spin period is Doppler shifted, $P = P_0(1 + v_r/c)$, and the line-of-sight acceleration $a = \dot v_r$ is measured from the drift of $P$. For a circular orbit the pair $(P, a)$ traces an ellipse with semi-axes $P_t = P_0v/c$ and $a_t = v^2/r$, which is the basis of 2016 D1.

An expanding shell produces emission at the line's rest wavelength and blueshifted absorption from gas between us and the star: a P Cygni profile. The absorption minimum gives the expansion velocity. Integrating the velocity over time gives the shell's radius:

and with a distance, an angular size $\theta = 2R/d$. The 2019 nova problem used this. A pulsating star works the same way, with $\Delta R = -p\int(v_r - \gamma)\,dt$ (the projection factor $p\approx1.3$; the sign because a positive radial velocity means the surface is moving away from us, i.e. contracting).