# Galaxies Lecture 18

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### Oort Constants

• Oort Constants
• Our tangential velocity with respect to the galactic center can be expressed as:
$v_{T}=A\cos 2\ell r+Br\,\!$ where $A$ is a measurement of shear (determined from radial velocities), and $B$ is a measurement of the local curl (determined from tangential velocities).

{\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 $A=15{km \over s\cdot kpc}$ and $B=-10{km \over s\cdot kpc}$ . This implies $\Omega _{0}=25{km \over s\cdot kpc}$ and $A+B=5{km \over s\cdot kpc}$ .
• 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
• $R_{0}$ 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 $H_{2}O$ masers with extreme accuracy, and then watching the masers move. We find the sun is about 8 kpc from the galactic center.
• $\Omega _{0}$ is determined from A, B measurements (as described before) and by measuring the proper motion of Sgr $A^{*}$ . We find that $b\approx 0$ , but $\ell$ 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 $M_{\odot } \over yr$ , from a reservoir of $10^{9}M_{\odot }$ , so star formation would end in $10^{9}$ yrs without this inflow.