# Galaxies Lecture 11

### Rotation Curves

Today we’ll talk about the measurement of rotation curves. This will cover Chapter 8 of Binney & Merrifield. Rotation curves are usually measured using the 21 cm line. As before, we note that we have approximately linear growth in ${\displaystyle V(r)}$ as a function of radius, until we hit a regime where ${\displaystyle V(r)\sim constant}$. For linearly growing ${\displaystyle V(r)=Kr}$, we derived that ${\displaystyle \rho =constant}$.\

It is important to not that when we measure rotational curves of galaxies, we are almost always measuring the velocity of gas. This gas comes in 3 forms:

• Atomic Gas
• Characterized by electronic transitions in optical/UV wavebands. To excite these transitions, gas must be hot.
• We also see 21 cm radiation. We can estimate the inherent width of the 21 cm line by saying ${\displaystyle \Delta E\Delta t\geq \hbar }$. Using ${\displaystyle \Delta E=h\Delta \nu }$, and ${\displaystyle \Delta t=3\cdot 10^{14}s}$. Plugging this in, we find that ${\displaystyle \Delta \nu \sim 5\cdot 10^{-16}Hz}$, so we can ignore the inherent line width of the 21 cm line. However, we cannot ignore the effects of temperature on linewidth. For high-end temperatures ${\displaystyle T\sim 10^{4}}$, we get ${\displaystyle v\sim 10{km \over s}}$. The lowest ${\displaystyle T}$ we expect to see is ${\displaystyle 3K}$, which gives us ${\displaystyle v\sim 0.17{km \over s}}$. This sets a lower bound on the resolution needed for 21 cm spectrometers.
• Molecular Gas
• The most common molecule in the ISM is ${\displaystyle H_{2}}$ (99.98%). ${\displaystyle H_{2}}$ has no dipole transition (${\displaystyle J=1\to 0}$), so the strongest rotational transition in ${\displaystyle H_{2}}$ is ${\displaystyle J=2\to 0}$ at ${\displaystyle 28\mu m}$, and this is a weak (forbidden) transition.
• Since ${\displaystyle H_{2}}$ is hard to detect, we look more to other abundant molecules: ${\displaystyle CO}$, ${\displaystyle CNO}$, ${\displaystyle H_{2}0}$, ${\displaystyle NH_{3}}$. These molecules have dipole transitions, and are much easier to see. Most molecules form in high-density regions. The critical density for populating the first rotational transition of ${\displaystyle CO}$ is ${\displaystyle n\sim 10^{3}cm^{-3}}$. However, we typically observe the ${\displaystyle J_{10}}$ transition of ${\displaystyle CO}$ to be thermalized around ${\displaystyle n\sim 0.1cm^{-3}}$. This is caused by collisions with ${\displaystyle H_{2}}$. We use ${\displaystyle CO}$ as a tracer for ${\displaystyle H_{2}}$, because it forms at comparable densities, and is dissociated at comparable temperatures.
• Ionized Gas
• One region of pressure equilibrium in the ISM are ${\displaystyle 10^{4}K}$ HII regions, typically around O stars. Around O7 stars, there are stable regions of ${\displaystyle 10^{6}K}$ gas. We’ll talk more about these regions later.

In the region of a galaxy, we can typically measure ${\displaystyle I}$, and ${\displaystyle {\bar {v}}}$. We’ve discussed isophotes (relating to ${\displaystyle I}$), but now we’d like to know what lines of constant velocity look like (isovelocity contours). If a galaxy is tilted with respect to our vantage point, we’ll see circles as ellipses:

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,\!}$

and we define ${\displaystyle \sin i\equiv {b \over a}}$. If ${\displaystyle R}$ is the actual radius of the galaxy, and ${\displaystyle \theta (R)}$ is the actual rotational velocity, we’d like to determine the velocity ${\displaystyle v_{r}}$ that we would measure. As discussed previously, ${\displaystyle \theta (R)\approx constant=\theta _{0}}$. Then:

${\displaystyle v_{r}=\theta _{0}\cos \phi \sin i\,\!}$

where ${\displaystyle \phi }$ is the angle around the galaxy. If ${\displaystyle v_{r}}$ is constant (isovelocity contours), then ${\displaystyle \cos \phi }$ must be constant. Thus, isovelocity contours are lines through the center of the galaxy.

For next class, we should answer the question for ${\displaystyle \theta (R)=kR}$ (solid-body rotation).