# Galaxies Lecture 24

### The Winding Problem

• Winding Problem
• Our galaxy differentially rotates. If we assume that spiral arms hang around for about a Hubble time ($10^{10}yrs$ ), and if spiral arms move with the rotation of the galaxy, then we’d expect arms to have a characteristic angle of $0.15^{\circ }$ . However, we measure an angle of $\sim 5^{\circ }$ .
• Either spiral arms are short-lived (in which case we should see many fewer spiral galaxies with clear arms–but we see about ${\frac {1}{2}}$ ), or spirals are density waves kicked up by a precessing bar.
• Precessing Bar
• Considering the form of a gravitational potential in a rotating reference frame, there is a potential for an oscillating potential with:
$\kappa ^{2}=R{\frac {d}{dR}}{(\Omega ^{2})}+4\Omega ^{2}\,\!$ or written in terms of the Oort constants:

$\kappa ^{2}=-4B\Omega \,\!$ This results in a pattern which precesses with angular velocity:

$\Omega _{p}=\Omega -{n\kappa \over m}\,\!$ where $n,m$ are integers representing different harmonics. In practice, the lower harmonics are excited with greatest amplitude.