# Galaxies Lecture 17

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When we make observations through the Milky Way with a radio telescope, we are trying to sample ${\displaystyle T_{a}(\ell ,b,v)}$ and use that information to tell us how gas is moving. However, our radio telescope has a beam width, and our spectrometer has bins of a certain width, so we essentially pixel-ize this data at some ${\displaystyle d\ell ,db,dv}$. Changing to cylindrical coordinates centered on the center of the galaxy, we can translate our pixels ${\displaystyle d\ell ,db}$ into ${\displaystyle d\phi ,dR,dz}$. Now we are in physical coordinates, and we can use the equation:

${\displaystyle N(HI)=1.823\cdot 10^{18}T_{a}dv\,\!}$

Now ${\displaystyle N(HI)=\int {\rho (\phi ,R,z)dR}}$, or we can flip around and consider columns in ${\displaystyle z}$. If we want to find the thickness of the column of hydrogen along the z axis, we can just compute the moment:

${\displaystyle \left\langle z^{2}\right\rangle =\int _{-\infty }^{\infty }{z\rho dz}\,\!}$

Using data gathered with these z measurements, we can analyze the warping modes of the plane of our galaxy. We find that the warping of our galaxy is dominated by m=0 (bowl shaped), m=1 (integral shaped), and m=2 (saddle shaped). Weinberg at UMass makes an argument that the warping is the gravitational effect of satellite galaxies (like the Large Magellenic Cloud) on the dark matter halo of our galaxy, where distortions in the dark matter halo are giving rise to coherent, low-order, modal structure in our disk.

On to a different topic: we can determine the velocity dispersion of gas in the plane of our disk in our neighborhood by looking at some direction where we shouldn’t see any postive-shifted gas, and attributing any positive velocities we see to the tail of a gaussian velocity distribution. Fitting for this, we find a dispersion in our neighborhood of about 7 km/s.

Suppose we would like to solve for some of the rotation constants of the Milky Way in the neighborhood of our sun–i.e.:

${\displaystyle {v_{r} \over R_{0}\sin \ell \cos b}=\Omega -\Omega _{0}\,\!}$

and we want ${\displaystyle v_{r}}$ for small ${\displaystyle R-R_{0}}$. To do this, we’ll Taylor expand ${\displaystyle R_{0}}$:

${\displaystyle {v_{r} \over R_{0}\sin \ell \cos b}=\overbrace {\Omega -\Omega _{0}{\bigg |}_{R_{0}}} ^{0}+(R-R_{0}){d\Omega \over dR}{\bigg |}_{R_{0}}+\overbrace {{\frac {1}{2}}(R-R_{0})^{2}{d^{2}\Omega \over dR^{2}}{\bigg |}_{R_{0}}} ^{0}\,\!}$

Now ${\displaystyle R-R_{0}=-r\cos \ell }$, so we have

{\displaystyle {\begin{aligned}{v_{r} \over \cos b}&=-rR_{0}\sin \ell \cos \ell {d\Omega _{0} \over dR_{0}}{\bigg |}_{R_{0}}\\&={-rR_{0} \over 2}\sin 2\ell {d\Omega _{0} \over dR_{0}}{\bigg |}_{R_{0}}\\&=\underbrace {-{\frac {1}{2}}R_{0}{d\Omega _{0} \over dR_{0}}{\bigg |}_{R_{0}}} _{const=A}r\sin 2\ell \\&=Ar\sin 2\ell \\\end{aligned}}\,\!}

Changing variables from ${\displaystyle \Omega }$ to ${\displaystyle \Theta }$, we have:

{\displaystyle {\begin{aligned}{d\Omega _{0} \over dR_{0}}{\bigg |}_{R_{0}}&={1 \over R_{0}}\left({d\Theta _{0} \over dR_{0}}{\bigg |}_{R_{0}}-{\Theta _{0} \over R_{0}}\right)\\A&={\frac {1}{2}}\left({\Theta _{0} \over R_{0}}-{d\Theta _{0} \over dR_{0}}{\bigg |}_{R_{0}}\right)\end{aligned}}\,\!}

This is called the Oort A constant. The first measurement of this constant confirmed Shapely’s theory that we are not in the center of the galaxy.