# Galaxies Lecture 11

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### 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 $V(r)$ as a function of radius, until we hit a regime where $V(r)\sim constant$ . For linearly growing $V(r)=Kr$ , we derived that $\rho =constant$ .\

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 $\Delta E\Delta t\geq \hbar$ . Using $\Delta E=h\Delta \nu$ , and $\Delta t=3\cdot 10^{14}s$ . Plugging this in, we find that $\Delta \nu \sim 5\cdot 10^{-16}Hz$ , 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 $T\sim 10^{4}$ , we get $v\sim 10{km \over s}$ . The lowest $T$ we expect to see is $3K$ , which gives us $v\sim 0.17{km \over s}$ . This sets a lower bound on the resolution needed for 21 cm spectrometers.
• Molecular Gas
• The most common molecule in the ISM is $H_{2}$ (99.98%). $H_{2}$ has no dipole transition ($J=1\to 0$ ), so the strongest rotational transition in $H_{2}$ is $J=2\to 0$ at $28\mu m$ , and this is a weak (forbidden) transition.
• Since $H_{2}$ is hard to detect, we look more to other abundant molecules: $CO$ , $CNO$ , $H_{2}0$ , $NH_{3}$ . 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 $CO$ is $n\sim 10^{3}cm^{-3}$ . However, we typically observe the $J_{10}$ transition of $CO$ to be thermalized around $n\sim 0.1cm^{-3}$ . This is caused by collisions with $H_{2}$ . We use $CO$ as a tracer for $H_{2}$ , because it forms at comparable densities, and is dissociated at comparable temperatures.
• Ionized Gas
• One region of pressure equilibrium in the ISM are $10^{4}K$ HII regions, typically around O stars. Around O7 stars, there are stable regions of $10^{6}K$ gas. We’ll talk more about these regions later.

In the region of a galaxy, we can typically measure $I$ , and ${\bar {v}}$ . We’ve discussed isophotes (relating to $I$ ), 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:

${x^{2} \over a^{2}}+{y^{2} \over b^{2}}=1\,\!$ and we define $\sin i\equiv {b \over a}$ . If $R$ is the actual radius of the galaxy, and $\theta (R)$ is the actual rotational velocity, we’d like to determine the velocity $v_{r}$ that we would measure. As discussed previously, $\theta (R)\approx constant=\theta _{0}$ . Then:

$v_{r}=\theta _{0}\cos \phi \sin i\,\!$ where $\phi$ is the angle around the galaxy. If $v_{r}$ is constant (isovelocity contours), then $\cos \phi$ must be constant. Thus, isovelocity contours are lines through the center of the galaxy.

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