# Galaxies Lecture 23

### Spiral Structure (Density Waves)

This material is covered in Chapter 6 of the new Binney and Tremaine, and in Chapter 3 of Binney and Tremaine for a discussion of epicyclic frequencies.

We define the pitch angle to be the angle between the tangent of a circular orbit and the spiral at that point. Most grand-design spirals (spirals which are symmetric, usually have 2 arms, and stretch from the center to the edge of a galaxy) have pitch angles between ${\displaystyle 5^{\circ }}$ and ${\displaystyle 20^{\circ }}$, with ${\displaystyle \sim 10^{\circ }}$ being typical. There is a good discussion of this in 1964 ApJ (Lin & Shu).

Spiral arms last for about a Hubble time. At the distance of our sun from the center of the galaxy, this is something like 50 orbits. Using that the circular velocity is given by:

${\displaystyle v_{c}=\left({GM \over R}\right)^{\frac {1}{2}}\,\!}$

we can compute the total angular momentum of a galaxy:

{\displaystyle {\begin{aligned}L_{tot}&=2\pi \int {R^{2}v_{c}(R)\Sigma (R)dR}\\&=2\pi (GM)^{\frac {1}{2}}\int {R^{\frac {3}{2}}\Sigma (R)dR}\\\end{aligned}}\,\!}

The energy of rotation is:

{\displaystyle {\begin{aligned}E_{tot}&=2\pi \int {({\frac {1}{2}}v_{c}^{2}(R)+\Phi (R))\Sigma (R)dR}\\&=-\pi GM\int {\Sigma (R)dR}\\\end{aligned}}\,\!}

The change in these quantities as a result of a small mass ${\displaystyle {\delta m}}$ being transported outward (from ${\displaystyle R_{i}\to R_{f}}$), is:

{\displaystyle {\begin{aligned}\Delta L&=\delta m\left[(GMR_{f})^{\frac {1}{2}}-(GMR_{i})^{\frac {1}{2}}\right]\to diverges\\\Delta E&=-\delta m\left({GM \over R_{f}}-{GM \over R_{i}}\right)\to bounded\\\end{aligned}}\,\!}

Thus, with a small piece of mass, we can transport any amount of angular momentum out of the system, but not an arbitrary amount of energy.

Looking at stellar structure in a galaxy (M51) with a grand design spiral, we see that spiral structure is very faint, but present, in the stars which are responsible for the bulk of the gravitational potential in the plane of the galaxy. Looking at gas emission, we find that the inner edges of spiral arms have molecular gas, the middle of the arms consists of atomic gas, and the outer edges have ionized gas. If we guess that molecular gas collapses into stars which ionize the rest of the gas, this suggests that gas enters the insides of spiral arms and flows outward across the arm. The width of the ionized portion of the spiral structure is determined by how long molecular clouds last and are able to produce massive O9 stars to create HII regions. Once these molecular clouds stop producing massive stars (which don’t live very long), HII regions rapidly recombine to atomic gas, and as this atomic gas gets pulled and compressed into the next spiral arm, the likewise increasing density of dust acts as an effective coolant, counteracting heating from compression and allowing atomic gas to cool to a level that allows molecular clouds to form.

From what we observe, spiral structure is nearly stationary. This means that spiral structure must undergo solid-body rotation. Thus, as we go farther out, the rotational speed of the spiral structure must increase, and at the corotation resonance point, will achieve the same orbital speed as the gas and stars.

To from grand design spirals, there must be a non-axisymmetric perturbation, usually caused by a companion galaxy or a bar, and there must be a linearly increasing rotation curve.