# Galaxies Lecture 12

For solid-body rotation (${\displaystyle \theta (R)=kR}$), ${\displaystyle v_{r}}$ contours will simply be vertical lines for a galaxy viewed edge-on, with major axis along the x-axis. This approximates the actual rotation curve of a galaxy near its center. We discussed in the last lecture how galaxies at larger radii approximate ${\displaystyle \theta (R)=k}$, with radial isovelocity contours. Thus, we may draw the velocity contours as vertical near the center of a galaxy and radial farther away. In reality, this isn’t far off–if you just blend the two together you get lines which look the the diverging lines of a magnetic dipole, and this is pretty close to what we measure. In fact, if ${\displaystyle \theta (R)}$ decreases at large radii, we may even see the closure of the field (i.e. the wires of coil magnet). All these types of diagrams together are generally called spider diagrams.

Analyzing spider diagrams can tell us a lot about the motion of stars in a galaxy. A general technique is to decompose the velocities we measure into components representing the systemic velocity (the recessional velocity of the galaxy), angular velocity (purely rotational), and infall velocity:

${\displaystyle v_{tot}=v_{sys}+v_{\phi }\cos \phi +v_{in}\sin \phi \,\!}$

We then decompose these components as a Fourier series:

${\displaystyle v_{tot}=\sum _{m=0}^{\infty }{v_{m}+v_{m\phi }\cos(m\phi )+v_{mr}\sin(m\theta )}\,\!}$

These components can then reveal circular motion, radial motion, warps, elliptical orbits, and spiral arms.

Galaxies consist of:

• stars (bulge, disk)
• HI
• ${\displaystyle H_{2}}$
• dust (${\displaystyle {1 \over 100}}$ of gas, by mass)

We can use ${\displaystyle v}$ to get a figure for the mass of a galaxy (${\displaystyle M(R)={Rv^{2} \over G}}$). We’ve mentioned already that there’s a problem of “missing matter” in galaxies. To help illustrate this, we can construct a maximum model which contains the maximum amount allowed (by ${\displaystyle M \over L}$) of each component in order to try and construct a model which recreates the rotation curves we measure. Gas components peak in velocity at small radii, and fall off slowly. However, they move rather slowly. Bulge components peak at slightly higher radii and fall off quickly. Finally, disk components peak at higher radii and at higher velocities, and fall off more slowly than the bulge, but more quickly than gas. The sum of all these components can create the initial flatness we see, but not out to large radii. For this we need dark matter, which must have a density profile:

${\displaystyle \rho _{dark}(r)=\rho _{0}\left({r \over r_{0}}\right)^{-2}\,\!}$

We observe a lack of matter on the local scale (50-100 kpc), the local group scale (500 kpc), and on the scale of galaxy clusters (2 Mpc). We assume for simplicity’s sake that they are all the same phenomenon. We’ll take a stab at guessing what dark matter is:

• Faint Stars: deep exposures of local group show no large excesses. Not likely on large scales.
• Atomic Hydrogen: DM has ${\displaystyle M\sim 10^{12}M_{\odot }}$ in 100 kpc. This column density of hydrogen would be easily detectable by 21 cm emission. It has not been observed.
• Molecular Hydrogen: would be detectable by CO tracer, or if there’s no CO, its absorption would be seen in quasar spectra. It’s not there.
• Molecular Hydrogen Iceballs: condensed ${\displaystyle H_{2}}$ couldn’t be detected as long as it never passes through the disk. We can’t rule this out.
• Dust: not enough primordial metallicity.
• Hot Ionized Gas: so hot we don’t see optical emission lines. From Fluids, remember that in hydrostatic equilibrium:
${\displaystyle {dP \over dr}=-{GM(r) \over r^{2}}\rho \,\!}$

Using the ideal gas equation ${\displaystyle P={\rho kT \over \mu m_{H}}}$, we have:

{\displaystyle {\begin{aligned}{k \over m_{H}}\left(T{d\rho \over dr}+\rho {dT \over dr}\right)&=-{GM\rho \over r^{2}}\\{Trk\rho \over \mu m_{H}}\left({1 \over \rho }{d\rho \over dr}+{1 \over T}{dT \over dr}\right)&=-{GM\rho \over r^{2}}\\{Trk\rho \over \mu m_{H}G}\left({d\ln(\rho ) \over dr}+{d\ln(T) \over dr}\right)&=M(R)\\\end{aligned}}\,\!}

Using the viral temperature ${\displaystyle {\frac {1}{2}}mv^{2}={GMm \over r^{2}}={\frac {3}{2}}kT}$, so that ${\displaystyle T={2GM \over 3rk}}$, and saying ${\displaystyle M=10^{12}M_{\odot }}$ and ${\displaystyle r=50kpc}$, we have ${\displaystyle T=7\cdot 10^{6}K}$, which says our hydrogen would emit in x-rays. We can measure the temperature of Hot Ionized Gas with x-rays and use the gravitationally inferred total mass to get an estimate of ${\displaystyle \rho }$. We find that there isn’t enough around galaxies to account for dark matter, but around clusters hot gas accounts for a significant fraction of dark matter.

Across the universe, the ratio of baryonic to dark matter is 12%.