# Galaxies Lecture 24

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### The Winding Problem

• Winding Problem
• Our galaxy differentially rotates. If we assume that spiral arms hang around for about a Hubble time (${\displaystyle 10^{10}yrs}$), and if spiral arms move with the rotation of the galaxy, then we’d expect arms to have a characteristic angle of ${\displaystyle 0.15^{\circ }}$. However, we measure an angle of ${\displaystyle \sim 5^{\circ }}$.
• Either spiral arms are short-lived (in which case we should see many fewer spiral galaxies with clear arms–but we see about ${\displaystyle {\frac {1}{2}}}$), 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:
${\displaystyle \kappa ^{2}=R{\frac {d}{dR}}{(\Omega ^{2})}+4\Omega ^{2}\,\!}$

or written in terms of the Oort constants:

${\displaystyle \kappa ^{2}=-4B\Omega \,\!}$

This results in a pattern which precesses with angular velocity:

${\displaystyle \Omega _{p}=\Omega -{n\kappa \over m}\,\!}$

where ${\displaystyle n,m}$ are integers representing different harmonics. In practice, the lower harmonics are excited with greatest amplitude.