# HI Rotation Curve

## Contents

## Observed curve

In order to plot the rotation curve, we need the radius of the orbit
of each cloud to go along with the orbital velocities you just calculated.
We start with the Galactic Longitude ** glon** of each cloud
(from the filename of each observation):

glon = [17 21 ... 65];

Make sure the order of longitudes in your array matches the order in which you measured your frequencies in ** freq**. Now calculate the orbital radius of the cloud, knowing that sine of the
angle

**is the orbital radius of the cloud**

`glon`**divided by the orbital radius of the Sun**

`R`**. Note that the**

`R_sun`**function in MATLAB assumes that angles are given in**

`sin`**radians**instead of

**degrees**:

```
R_sun = ... ; % in kpc
R = R_sun*sin( ... );
```

If you followed the directions in Orbital velocity of the cloud, you
should now have an array ** freq** of the smallest frequencies from
all 13 files. To include the Sun's orbital radius and velocity, add a 14th
element to the arrays:

vorb(14) = v_sun; % in km/s R(14) = R_sun; % in kpc

If you haven't saved the orbital velocity and orbital radius of the
Sun as ** v_sun** and

**, then just use the values 220 and 8.5.**

`R_sun`Now let's see what the rotation curve looks like:

figure(2); clf; plot( ... ); ...

## Theoretical curve

The distribution of luminous matter in the Galaxy suggests that most
of the Galaxy's mass is in the center, so we expect Keplerian motion (i.e. decreasing velocity with increasing distance, like in our Solar System) of objects beyond the center; This gives a rotation curve with
in the small radii (the inner 800 pc or so), and if **R** is large. The file ** TheoCurve.dat** on the lab website is an example of the disk rotation curve:

```
curve = load('TheoCurve.dat');
```

We can compare the observational results with the theoretical curve:

hold on plot( ... ) ylim([0 300]) legend( ... ,'Location','Best')

Now you should see why people believe there must be dark matter somewhere in the Galaxy!