\section{21 centimeter Observations}

\par

To measure the Doppler shift in the 21 centimeter line, we first had
to calibrate our equipment.  We used a precision signal generator set
at 710.2025 MHz, to generate a signal in the second harmonic at the
frequency of interest: namely, 1420.405 MHz.  This produced a sharp
peak in the center of our 30-channel display.  Next, we experimented
with shifting the frequency by a known amount and measuring the
correponding shift in our output.  We found that each channel
coreponds to roughly .1 MHz, with a 3 Mhz range, as expected, across
all 30 channels.
\par
Once our equipment had been calibrated, we turned the telescope on the
galaxy.  We took spectra at $5\deg$ intervals along the galactic
equator, to the range of our telescope.  In each case, we collected
8000 samples of data and compared them to a reference spectrum taken
out of the galactic plane. Subtracted the data spectrum from the
reference spectrum at each point, and plotted the intensity of the
signal as a function of frequency.  Our data can be seen in Apendix
(number).
\par
In order to obtain meaningful results from the data, it was necessary
to determine the channel correspondng to the 21 centimeter line with
the greatest Doppler shift.  In essence, this is the channel at which
the intensity first begins to slope up sharply to its peak.
Initially, we tried to set this channel by looking for any intensity
above a certain baseline level, but this proved to be difficult.
The overall intensity varied from graph to graph. In addition, the
background level exhibited peaks and valleys and slope of its own.
We tried another approach.  Essentially, we differentiated each graph
by subtracting each channel from its successor.  Then we simply looked
for any increase in slope above a certain level.
\par
Once we determined the average channel of the greatest Doppler-shifted
21 centimeter line for each of the points along the galactic equator,
we can begin to apply the math outlined in section (sectionref).  We
turned the shifted freqencies into velocities, factored in our
corrections, and computed the angular velocity of the galaxy at the
radius given by $R = R_\sun\sin\lambda, $ where $\lambda$ is the
longitude in galactic coordinates of our observation point.
\par
We then plotted the angular velocity of the galaxy verses the radius
for our data up to galactic longitude $ 90 \deg $ (see \ref{avel}). We
note that our graph compares favorably to the data \cite{hulst}.
Similarly, we plot the tangential velocity curve of the galaxy
(\ref{tvel}), and find that it also parallels to existing data
\cite{hulst}.
\par
\begin{figure}[hb]
\begin{center}
\leavevmode
\epsfxsize=0.8\hsize
\epsfbox{/mit/bert/Class/JuniorLab/paper/avel.epsi}
\end{center}
\caption{Angular Velocity Curve of the Galaxy}
\label{avel}
\end{figure}
\begin{figure}[t]
\begin{center}
\leavevmode
\epsfxsize=0.8\hsize
\epsfbox{/mit/bert/Class/JuniorLab/paper/tvel.epsi}
\end{center}
\caption{Tangential Velocity Curve of the Galaxy}
\label{tvel}
\end{figure}
We note that our data deviates from accepted models of the rotational
curve of the galaxy at the low end of our graph.  Most likely, this is
because we are at the low end (galactic longitude ~$10-20\deg$), we
are looking at the central bulge of the galaxy. Made up of older stars
and less interstellar hydrogen, the central bulge also has a different
rotation rate than the spiral arms.  In addition, it is at these
longitudes that the background levels of emission are most
pronounced.  A large, sloping baseline contributes to error in the
estimation of the lowest-frequency peak.
\par
Despite these errors, our data still shows the compelling evidence
that our galaxy is cannot be modelled by a simple point-mass model.
The flatness of the galaxy's rotation curve above 4-6 kiloparsecs
first suggested to astronomers that there might be more mass to the
galaxy than we could see. Today, the study of dark matter is at the
forefront of astrophysics. The study of 21 centimeter hydrogen
emissions helped put it there, and no doubt the study of 21 centimeter
hydrogen emissions will contiue to be a valuable tool in the study of
our galaxy.



