# Galaxies Lecture 18

### Oort Constants

• Oort Constants
• Our tangential velocity with respect to the galactic center can be expressed as:
${\displaystyle v_{T}=A\cos 2\ell r+Br\,\!}$

where ${\displaystyle A}$ is a measurement of shear (determined from radial velocities), and ${\displaystyle B}$ is a measurement of the local curl (determined from tangential velocities).

{\displaystyle {\begin{aligned}A-B&=\Omega _{0}\\A+B=-\left(R_{0}{\frac {d}{dR}}{\Omega }{\bigg |}_{R_{0}}+\Omega _{0}\right)=-{\frac {d}{dR}}{\theta }{\bigg |}_{R_{0}}\\\end{aligned}}\,\!}
• Measurements show ${\displaystyle A=15{km \over s\cdot kpc}}$ and ${\displaystyle B=-10{km \over s\cdot kpc}}$. This implies ${\displaystyle \Omega _{0}=25{km \over s\cdot kpc}}$ and ${\displaystyle A+B=5{km \over s\cdot kpc}}$.
• We are on the inside edge of a spiral arm. Gas falling into the arm provides a shear beyond normal, so our A is abnormally high.
• There are additionally Oort Constants C and K which measure shear along the tangential axis, and the divergence of the local velocities.
• Measurements of Constants
• ${\displaystyle R_{0}}$ is determined from RR Lyrae stars, globular clusters, proper motion of masers in the galactic center, and from statistical parallax. Statistical parallax is done using VLBI to get the position of ${\displaystyle H_{2}O}$ masers with extreme accuracy, and then watching the masers move. We find the sun is about 8 kpc from the galactic center.
• ${\displaystyle \Omega _{0}}$ is determined from A, B measurements (as described before) and by measuring the proper motion of Sgr ${\displaystyle A^{*}}$. We find that ${\displaystyle b\approx 0}$, but ${\displaystyle \ell }$ changes, so we deduce that it’s motion is our motion.
• Gas Flow
• CO measurements at the galactic anticenter show inflow at 2-3 km/s.
• This gas flow replentishes gas which is depleted from star formation.
• Star formation proceedes at 1-5 ${\displaystyle M_{\odot } \over yr}$, from a reservoir of ${\displaystyle 10^{9}M_{\odot }}$, so star formation would end in ${\displaystyle 10^{9}}$ yrs without this inflow.