2017 D2 | Dark matter in a low surface brightness galaxy

A low surface brightness (LSB) galaxy is a diffuse galaxy whose surface brightness, seen from Earth, is at least one magnitude fainter than the night sky. Some of its matter is baryonic (neutral hydrogen gas and stars), but most of its mass is invisible dark matter.

The table gives data for the LSB galaxy UGC 4325, assumed to be edge-on. At each distance $r$ from the centre we have $\lambda_{\rm obs}$, the observed wavelength of the Hα line (the Hubble expansion has already been removed); $V_{\rm gas}$, the rotation speed the gas alone would produce (from HI surface densities); and $V_*$, the rotation speed the stars alone would produce (from R-band photometry).

The dark matter mass inside radius $r$ is defined from the dark matter rotation speed $V_{\rm DM}$:

The dark matter density is modelled by a cored profile, which gives

where $\rho_0$ is the central density and $r_C$ the core radius.

(D2.1) The laboratory wavelength of Hα is $\lambda_{\rm emit} = 656.281$ nm. Compute $V_{\rm obs}$ and $V_{\rm DM}$ (km/s) at each $r$, and the dynamical mass $M(r)$ and dark matter mass $M_{\rm DM}(r)$ in solar masses. (21)

(D2.2) Plot $V_{\rm obs}$, $V_{\rm DM}$, $V_{\rm gas}$ and $V_*$ against $r$ with smooth curves. Order the contributions to the observed velocity from largest to smallest. (16)

(D2.3) Use a data point at small $r$ and one at large $r$ to estimate $\rho_0$ and $r_C$. For large $x$, $\arctan x\approx\pi/2$; for small $x$, $\arctan x\approx x - x^3/3$. (7)

(D2.4) Linearise the mass equation so that $\rho_0$ can be found from a straight-line fit. Plot the graph and find $\rho_0$ in $M_\odot\,{\rm kpc^{-3}}$. If you could not find $r_C$, use $r_C = 3.2$ kpc. (19)

(D2.5) Compute $\ln\rho_{\rm DM}(r)$ and plot it against $r$. (12)