\documentstyle[12pt,psbox,osa,manuscript]{revtex}

\title{Examination of Galactic Structure by Observation of the 21cm Hydrogen Line}
\author{Matthew K. Gray}

\begin{document}
\global\firstfigfalse
\maketitle

\begin{abstract}
Observation of the radio profile of the galaxy provides substantial
information about its large scale structure.  Specifically, examining
the 21cm hydrogen line and the variation of the Doppler shift along
the galactic equator provides a straightforward way of measuring
galactic distances and velocities.  Data at regular intervals along
the galactic equator were collected.  These data give strong evidence
of the galaxy's spiral structure.

\end{abstract}

\section{INTRODUCTION}

The Milky Way Galaxy is composed of approximately $10^{11}$ stars.
Through careful observation one can attempt to determine the structure
of the galaxy through optical and radio data.  Observations of other
galaxies give evidence of a variety of different galactic structures,
including the more common eliptical and spiral structures.  Simple
optical observation of the night sky makes it clear that the Milky Way
galaxy has some sort of disklike structure, based on simple counts of
stars at different regions of the sky.
\par
More detailed information can be gathered, however, by examination of
the radio spectrum.  Specifically, one can examine the 21cm line of
hydrogen and the variation of the Doppler shift along the galactic
equator.  This variation will very quickly give strong evidence for
the spiral structure of the galaxy.

\section{APPARATUS}
\begin{figure}
\PSbox{/mit/mkgray/afs/jrlab/bert/equipment.eps hscale=.8 vscale=.8}{7in}{3in}
\caption{Apparatus Diagram}
\end{figure}
\subsection{Telescope}
For this experiment, a 2.5 meter parabolic dish radio telescope
located on the roof of the MIT Compton Laboratory (building 26) was
used.  The telescope has an angular resolution of approximately
$5^\circ$ in the vicinity of the 21cm line, permitting detailed
measurements along the entire galactic equator. The telescopes control
system is arranged to be manipulated via a computer in the telescope
control room.  The control program permits adjustments in both local
coordinates or Galactic coordinates.  Additionally, for ease of data
collection the system is configured to allow batch jobs.  This avoids
much of the tedium of long data collection runs.

\subsection{Receiver}
Due to the fact that a very narrow region of the radio spectrum in the
vicinity of the 21cm line is of interest, the receiver mixes the
signal down to a 3 MHz range about the 21cm line.  This is
accomplished by heterodyning the incoming signal with a 1543 MHz
crystal oscillator.  This yields a signal at 123 MHz (the difference
between the 1420 MHz signal and the 1543 MHz mixer).  This procedure
is repeated by mixing down to 22 MHZ with a 165.01 MHz mixer and again
with a 40 MHz mixer.  This procedure prevents unnecassarily high
sampling rates by limiting the total interesting range to 3 MHz. By
Nyquist's theorem, a sampling rate of twice the maximum frequency is
needed to have useful resolution, and by mixing down, that maximum
frequency is greatly reduced. The disadvantage of this method is it
eliminates any signal in the low frequency ranges, but for the
purposes of this experiment, that is quite acceptable, as there is
nothing of interest in that range.  The final profile has a width just
over 3 MHz and the 21cm line lies approximately at the center channel.

\subsection{Spectrometer}
This mixed down signal is fed to a digital autocorrelator and into a PC for data collection.  The apparatus has collects 32 channels of data with each corresponding to a width of 100 KHz $\pm 20 \frac{km}{s}$.  The anticipated velocities of galactic hydrogen is on the order of $.001c$ which corresponds to a Doppler shift of approximately 1.4 MHz.  This is the justification for mixing down to a total range of approximately 3 MHz end to end.
\par

\par
\section{THEORY}
Radio astronomy for a time failed to be useful due to the fact that there we no useful spectral lines as one had with optical astronomy.  Without spectral lines, distances and velocities become difficult to measure.  Shortly after World War II, however, the 21cm line of hydrogen was discovered as a useful spectral
line for doing radio astronomy and radio astronomy has grown tremendously in scope since.
\par
Monatomic hydrogen emits at very low intensity a 1420.4050 MHz radio
signal due to spin-flip transitions.  This energy difference is caused
by a magnetic coupling between the nuclear and electron spin.  This
slightly excited anti-parallel state has a lifetime in the millions of
years.  Collisional de-excitation greatly reduces this lifetime to
approximately 50 years, though still, the transition is very rare.
Though extermely low intensity, due to the extreme abundance and sheer
mass of monatomic hydrogen in the Milky Way, the signal is strong and
measurable.  This makes this line ideal for observations of galactic
structure.
\par
The hydrogen gas will rotate around the center of the galaxy with a radius of
\begin{equation}
R = R_\odot\sin\lambda
\end{equation}
where $\lambda$ is the angle of the line of sight in galactic
coordinates and $R_\odot$ is the solar distance from the galactic
center.  Motion of this gas relative to the Sun yields a Doppler shift
in this frequency, as 
\begin{math}
\nu' = \nu(1+\frac{V}{c})
\end{math}
to first order.  The observed velocity of the hydrogen will simply be a product of the difference in angular velocity and radius.  That is,
\begin{math}
{\Delta}V = (\omega - \omega_\odot)R_\odot\sin\lambda
\end{math}

\section{EFFECTIVE TERMPERATURE OF THE SUN}
Based on a 150 K calibration with a noise diode, the effective temperature of the sun is $4.09$ times that of the noise diode, or 613 K.  This is not, however, taking into account the fact that the sun subtends a substantially smaller angle than the telescope can resolve.  For this reason, the sun was allowed to pass throught he beam of the telescope, getting a complete temperature profile.  This yields a full width at half power of 1440 seconds, or $\frac{1}{60}$th of an hour.  This indicates a beam width of $6^\circ$, and the linear diamater of the sun is $.5^\circ$.  Therefore, there is a correction of
\begin{math}
c = (\frac{\sin\theta_b}{\sin\theta_\odot})^2
\end{math}
for $\theta_b$, the telescope beam angle.  This yields a temperature of the sun of about 87,952 K.

\section{21CM OBSERVATIONS}
\subsection{Calibration}
In order to calibrate the spectrometer, a correlation between readout
channel and frequency must be found.  This is accomplished by taking
spectra at three known frequencies and finding the resolution per
channel by a simple linear fit.
\begin{figure*}
\PSbox{/mit/mkgray/matlab/radio/raw2/caliblow.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\PSbox{/mit/mkgray/matlab/radio/raw2/calibmid.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\PSbox{/mit/mkgray/matlab/radio/raw2/calibhigh.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\caption{Calibration spectra}
\end{figure*}
This yields a calibration of $\nu = .1045 x n + 1418.9 MHz$, where $n$ is the channel number.  This makes channel 14 the unshifted hydrogen line.
\subsection{Spectra}
Data was collected at various galactic longitudes.  Given the
calibration, these spectra are converted to frequency spectra.  To
eliminate background noise, this involved taking a spectrum along the
galactic equator and off of the galactic equator and subtracting to
give a useful data spectrum.  Typical spectra, as gathered at
$30^\circ, 60^\circ$, and $80^\circ$ are shown below.  Each peak in these spectra corresponds to a mass of hydrogen gas moving relative to the Sun with a velocity corresponding to the shift from the stationary hydrogen line at 1420.4 MHz
\begin{figure*}
\PSbox{/mit/mkgray/matlab/radio/raw2/dotplot30.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\PSbox{/mit/mkgray/matlab/radio/raw2/dotplot60.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\PSbox{/mit/mkgray/matlab/radio/raw2/dotplot80.ps voffset=-60 hscale=.3 vscale=.3}{2in}{2in}
\caption{Sample Spectra with frequencies}
\end{figure*}
The primary galactic arm is visible as the large peak in each of these
spectra.  Further structure is visible in the nearby peaks.  As is
visible by looking at these three profiles, the variation of velocity
across values of galactic longitude is not extremely large, but is
measurable.

\section{RESULTS}
\subsection{Line Profiles}
Data was collected at intervals more frequent than the resolution of
the telescope ($6^\circ$). Selected line profiles at galactic
longitudes ranging from $16^\circ$ to $88^\circ$ with velocity (in
$\frac{km}{s}$) on the abcissa and intensity on the oordinate are
shown on the following page.  Typical velocities are less than 100
$\frac{km}{s}$. In range from approximately $30^\circ$ to $75^\circ$ a
second spiral branch is visible.
\begin{figure}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp16.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp20.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp25.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp30.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp36.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp40.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp45.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp50.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp56.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp60.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp72.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp77.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp80.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\PSbox{/mit/mkgray/matlab/radio/raw2/lp88.ps voffset=-60 hscale=.21 vscale=.21}{1.5in}{1.5in}
\vspace{.25in}
\caption{Line Profiles}
Unfortunately, the resolution on these peaks is hindered by a number
of environmental factors, including sky noise (believed to be coming
from Logan Airport) and buildings (16 and 10) occasionally obsructing
the portion of the sky that was trying to be observed.  Despite this,
numerous spectra were collected.  
\subsection{Structure}
Examining the distant structure, one can measure a separation between arms corresponding to approximately $100 \frac{km}{s}$.  This corresponds to a separation in space of $.7R_\odot$.
\end{figure}

\section{CONCLUSIONS}
The figure below shows the radio visible hydrogen as a function of angle and velocity.  This indirectly gives us a look at the galactic structure.  Specifically, if we assume constant differential rotation, we can directly map velocity into distance from the galactic center as
\begin{math}
v = (\omega - \omega_0)R_0\sin\gamma\\
\omega_0 = 250,000 \frac{m}{s}\\
\omega =\frac{R_0\omega_0}{R}
\end{math}
\par
In this figure the dual ridges show the close and distant galactic arms.  Sample times were constant, so variations in the ridge height correspond to variations in seeing over the observation period.
\vspace{.6in}
\par
\vspace{.6in}
\begin{figure}
\PSbox{/mit/mkgray/matlab/radio/raw2/cool3dthing.ps hscale=.4 vscale=.4}{2in}{2in}
\caption{Surface representation of galactic structure}
\end{figure}
It would seem that the data presented here does in fact support the concept of a spirally structured galaxy.  To claim that it independantly indicates spiral structure is likely too bold, however, in combination with other radio data, optical data and observations of other galaxies, it would seem to be a strong indicator for the spiral structure of the Milky Way Galaxy.
\begin{references}
\bibitem{1} J. Kraus, ``Radio Astronomy,'' McGraw-Hill Book Company, New York, 1966.
\bibitem{2} F. Graham Smith, ``Radio Astronomy,'' Penguin Books, Baltimore, MD, 1960.
\bibitem{3} H.C. van de Hust, ``Radio Astronomy,'' IAUS, Cambridge University Press, Cambridge, England, 1957.
\end{references}

\end{document}
