# Galaxies Lecture 16

### Galactic Rotation

• Galactic Rotation
• If we consider the Local Standard of Rest (LSR) to be at -r along the x axis and the galaxy to be more or less contained in a circle of radius r centered on the origin, then we should find that for observations within the solar circle, quantrants I and II should be receding (positive v), and III and IV should be approaching (negative v).
${\displaystyle v_{r}=(\Omega (R)-\Omega _{0})R_{0}\sin \ell \,\!}$

where ${\displaystyle \Omega _{0}}$ is the angular velocity of the sun around the galaxy.

• For a given line of sight, there is a maximum ${\displaystyle v_{r}}$. For other measured velocities, there is an ambiguity.

We derived differential rotation in the Milky Way:

${\displaystyle \theta ={R \over R_{0}}\left({V_{r} \over \sin \ell \cos b}+\theta _{0}\right)\,\!}$

where ${\displaystyle \ell }$ is the galactic longitude, ${\displaystyle b}$ is the galactic latitude, and ${\displaystyle v_{r}}$ is the projection of ${\displaystyle \theta }$ along the line of sight, minus the projection of ${\displaystyle \theta _{0}}$. Recall that galactic longitude is measured relative to our reference at the edge of the galaxy, not as an angle around the galactic center. This equation assumes circular, cylindrical rotation around the galactic center.

When looking through the galactic plane, we should see gas interior to our orbit to have a positive doppler shift in lines on sight toward the direction of our orbit, and negative doppler shifts in the opposite direction of our orbit. Exterior to our orbit, the signs of both of these cases changes. According to our model of cylindrical motion, we should see only the thermal doppler broadening around the inherent wavelength of our spectral line if we look straight up in galactic longitude. As an aside, we know that the temperature of H in our galaxy cannot exceed 10,000 K, which corresponds to a line width of about 10 ${\displaystyle km \over s}$. If we see wider line widths, they have to be a result of turbulent motion.

Anyway, when we look straight straight up (or straight down), we find that gas is falling towards us. Thus, we don’t have purely cylindrical motion. Likewise, when we look at galactic longitude ${\displaystyle 180^{\circ }}$, we see a line which is close to zero doppler shift, but still slightly negative. Thus, we don’t have perfectly circular motion.

Things get worse when we look towards the galactic center. We see emission, but we also see absorption, and at many different doppler shifts. When we look slightly above and below the galactic center in latitude, much of this complexity disappears. Thus, this complexity is associated with the galactic center, and is the result of extremely non-circular orbits.

Since we know where the velocity curves peak along the line of sight (where R is a minimum), we can infer a radius and measure a maximum velocity. Thus, we can make our own velocity curves and find that the rotation curve of our galaxy is flat.

In these plots of velocity along the line-of-sight, we see features which indicate there are high-velocity clouds falling into the midplane, and that there is a warp to the midplane of the galaxy.