\documentclass[psfig,preprint]{aastex}
\usepackage{graphicx}
\usepackage[verbose]{wrapfig}
\begin{document}
\tolerance=10000
\def\simlt{\lower.5ex\hbox{$\; \buildrel < \over \sim \;$}}
\def\simgt{\lower.5ex\hbox{$\; \buildrel > \over \sim \;$}}

\title{THE HI AND OH LINES, THE GALAXY, and SUPERNOVA SHOCKS}

\tableofcontents

\section{GOALS} 

The goals of this lab consist of some collection of the following,
depending on the project chosen: \begin{itemize}

  \item Perform a large-scale observing survey project balancing
    considerations of telescope time, integration time, sky coverage,
    data reduction, report writing, and personal time commitment versus
    research goals and quality of results and their presentation.

\item Use the collected data to infer astrophysical phenomenae or
  quantities such as the properties of supernova-driven interstellar
  bubbles, the Galaxy, and molecular masers.

\item Measure maser polarization using Stokes parameters and Mueller
  matrix techniques.

\item Present data and/or using greyscale and/or color images.

\item Learn about using colors in computer graphics including plots and
  images. 

\item Learn about problems in using color for presentations,
  specifically contrast issues and colorblindness.

\end{itemize}


\section{INTRODUCTION}

	Diffuse atomic gas permeates all corners of interstellar space.
On the largest scales, its distribution reveals a lot about Galactic
structure and dynamics. On smaller scales, its morphology is dominated
by shocks from energetic stellar winds and supernovae. On still smaller
scales, magnetic fields, pressure differences, and thermal instability
shape the gas into thin filaments and sheets. On still smaller scales,
the densities get large from self-gravitation; the gas turns molecular
and stars form. 

Star formation occurs in regions that are dense enough for gravity to
pull together matter to form the star. In such clouds, the interstellar
dust shields the inside of he cloud, eliminating the ultraviolet light
that otherwise dissociates molecules. The most abundant molecule is
H$_2$, which (owing mainly to its being a homonuclear diatomic molecule)
has no spectral lines that are easily observed. However, there exist
many other molecules, including simple diatomics (of
which OH and CO are the most easily observed examples); triatomics
(e.g.\ water, H$_2$O); polyatomics (e.g.\ ammonia (NH$_3$) and methanol
(CH$_4$OH)), and many others. Some of these molecules radiate as
astronomical masers. These include OH, H$_2$O, and CH$_3$OH. OH masers
lie within the frequency range of our telescope.

For this lab, we outline several projects in HI and also one OH project
that are suitable for our 4.5-m diameter Leuschner dish. The HI projects
fall into two categories. One explores large-scale Galactic structure
and dynamics by obtaining HI spectra along great circles, presenting the
data as position-velocity images, and interpreting those maps as
discussed below. The other category maps specific interstellar
structures (e.g., expanding shells produced by multi-supernovae) by
sampling a grid of spectra over a large area, presenting the data as
grey-scale images of velocity-integrated line intensity (i.e., antenna
temperature $T_A$) or as images in which image brightness indicates
velocity-integrated $T_A$ and color indicates velocity, and interpreting
the data.

The OH project measures the spectra of a few OH masers and, also, the
very broad OH in the Galactic center. We measure polarization by using
crosscorrelation spectroscopy to measure all four Stokes parameters
simultaneously.

We'd like each group to pick a single project and carry it through to
completion by the time the lab reports are due, which is the last day of
instruction (May 5). All of the HI projects involve making maps and
images. One of the features of this lab is to learn about image
processing and using colors in images.

For the great-circle projects (\S \ref{galacticplane} and \S
\ref{galacticpoles}), we gain large angular scale information
in the sky by observing on great circles. Getting nice results
requires data over the largest portions of the great circles you can
get. For a full 360-degree circle, that's 180 pointings at $2^\circ$
spacings.  Mostly, though, we can't cover a full great circle because
some of the circle lies too far south, so we do as much as we can. 

\section{SCHEDULE}

The end of the semester is approaching like a freight train. And our new
telescope at Leuschner isn't even functional yet! Imagine yourself tied
onto the tracks by a madman: you've gotta finish before it hits. And the
madman won't let you even begin serious measurements before the dish is
ready! Here's a schedule. If we don't keep up with this by making the
telescope work during the first week, and if you don't keep up with
this, we're in trouble! \begin{enumerate}

\item Week 1, 08apr-14apr: Obtain and reduce test spectra from our new
  spectrometer. Develop software so that a properly calibrated spectrum
  (intensity calibration and Doppler calibrated/corrected) can be
  created from two frequency-switched spectra in a single procedure
  call. When the telescope is finally pointable, take the data and use
  this software to make a plot of the completely calibrated HI line
  spectrum at the position in Galactic coordinates $(\ell, b) =
  (120^\circ, 0^\circ)$.  We will compare the spectra of all groups at
  the class on 14 April, and if your group doesn't have a spectrum to
  show this will be considered very bad form.

Pick a group project. Figure out the RA and Dec boundaries of
  your object, when the object is up, and how you will get the data
  for your project. Coordinate with other groups to figure out how you
  can share telescope time. 
With luck, the telescope will be working by the end of the week. Be
prepared for this circumstance, and if it comes to pass, get started on
your project!

\item Week 2, 15apr-21apr: Get as much data as you can. Reduce the data.
If you have enough data, begin the map-making process.

\item Week 3, 22apr-28apr: Get the rest of your data. Figure out
  how you want to present the data as images. Make some first-try
  images. 

\item Week 4, 16apr-03may: Finalize data. Make final images. Write
report. Report is due the last day of class, 05may.
\end{enumerate}

\section{ON MAPPING AREAS: HOW TO SPATIALLY SAMPLE} \label{sampling} 

All of the HI projects examine the angular dependence of the 21-cm line
profile, so you need closely-spaced observations to make a map or an
image. How closely should the observations be spaced?  The usual
criterion is two samples per HPBW (Half-Power BeamWidth), because
observing at more closely spaced angles just gives redundant information
and takes extra time, while more widely spaced angles loses small-scale
structural information. (This statement can be justified more exactly by
Fourier transforming the structure of the sky in terms of angular
frequency and then imposing the Nyquist criterion\dots).  The HPBW is
$\sim 3^\circ$; using a somewhat larger spacing of $2^\circ$ makes
things go faster and it's being a round number is nice.

For the mapping projects (not the great-circle ones), we need to cover
areas of 1000 to 2000 square degrees. Consider the Orion-Eridanus
Superbubble (\S \ref{eridanus} below), which wants to map the the region
covering roughly $(\ell=160^\circ \rightarrow 220^\circ, b=-70^\circ
\rightarrow -10^\circ)$. You might think you'd set up a regular grid in
$(\ell,b)$ with $(\Delta \ell, \Delta b)$ both equal to
$2^\circ$. That's find for $\Delta b$, but you'd be sampling too closely
in $\ell$. That's because the true (great-circle) angular distance
between two points is $\Delta \ell \cdot \cos(b)$; there's a
foreshortening for $b \ne 0^\circ$ (e.g.: when you're at the Earth's
North Pole, you can go completely around a small circle of constant $b$
with just a few steps.). So you can save lots of observing time by
observing at $\Delta \ell = {2^\circ \over \cos(b)}$ instead of $\Delta
\ell = 2^\circ$.

The true angular area is not the total extent in $\ell$ times the total
extent in $b$ because of the foreshortening discussed in the above
paragraph. So the actual angular area of the Orion-Eridanus area area
isn't the range in $\ell$ times the range in $b$, i.e.\ $60^\circ
\times 60^\circ = 3600$ square degrees (which would require 900
pointings), but rather only about 2800 square degrees.  At $2^\circ$
sampling, that's about 700 pointings.

These projects require what seems like a lot of pointings,
but---hey!---the telescope is automated. (With luck, that is\dots) With
the magic of a computer-controlled telescope you can do these
observations in just one or two days. But it will require planning and
coordination with other groups---all of you are in the same boat! (Or
should we say, ``in the same dish''?) 

%I suggest some trial observations
%to get started, both making sure you are observing correctly and also
%getting your datataking and reduction software in shape.

\section{THE GALACTIC PLANE: THE GREAT CIRCLE AT LATITUDE ZERO 
\boldmath{$(b=0^\circ$)}} 
\label{galacticplane}

	Point the telescope at a series of Galactic longitudes ($\ell$)
along the Galactic plane (Galactic latitude $b = 0^\circ$), over the
full longitude range that is observable for our telescope.  Getting nice
results requires data over the largest section of the Galactic plane you
can get, which is roughly (or somewhat less than) $\sim -10^\circ \simlt
\ell \simlt 250^\circ$. This observation has two science goals:
\begin{enumerate}

	\item {\it The Galactic rotation curve}. Determine the rotation
curve of the Galaxy for the portion of the Galaxy inside the Sun and use
the result to estimate the {\it gravitational} mass $M_{grav}$ of the
Galaxy that lies inside the Solar circle; for this, you just assume the
usual $v^2 = {G M_{grav} \over R}$.  In fact, you can even get the
radial dependence $M_{grav}(R)$! Also, use your observations to estimate
the {\it gaseous} mass $M_{gas}(R)$ of the Galaxy.  What fraction of the
total Galactic mass comes from interstellar gas? For this you'll need to
know the radius of the Solar circle: it's 8.5 kpc.  See {\it Commentary}
below for a discussion. 

\item {\it The Galactic center.} Our Galactic center, like others, has a
  big black hole, with infalling matter and jets and lots of anamolous
  velocities. You can see a lot of this activity in  your map. Using the
  spatial extents and the velocities, you can estimate a black hole
  mass. To get an accurate estimate, you need to look at motions of
  matter lying really close to the black hole, which you can't do
  because you don't have the angular resolution. But you can at least
  make an estimate!

	\item {\it Spiral Structure of the Galaxy}. All your life,
you've been told that we live in a spiral galaxy.  Are they lying? In
principle, to detect spiral structure you make a position-velocity plot
along the Galactic plane [that's the image $T_A(\ell,V_{LSR})$ for
$b=0^\circ$].  Bright regions are regions of excess density---or regions in
which lots of HI is packed into a narrow velocity range (see \S
\ref{coldensity} below). 

	Do you see spirals in your position-velocity plot? How do you
tell? There's only one easy way.  Make a model of a spiral arm and see
how it projects onto position-velocity space.  To do this, you need to
know the rotation curve.  You've measured it for the inner Galaxy; for
the outer Galaxy (outside the Solar circle) assume that the rotation
curve is constant beyond the solar circle [$V(R)$=220 km/s for $R > 8.5$
kpc].  In polar coordinates, the equation for a spiral arm is

\begin{equation}
R_{arm}  =  R_0 e^ {\kappa (\phi - \phi_0)} 
\end{equation}

\noindent where $R_0$, $\kappa$, and $\phi_0$ are free parameters;
$\kappa$ is the tangent of the {\it pitch angle}.  You project this into
position-velocity space using equation \ref{vdopp} and your knowledge of
the Galactic geometry.  

{\it Comment:} Finding spiral arms inside the Solar circle is very
difficult, and has thwarted efforts by astronomers over the past several
decades. Finding them in the outer Galaxy is not so bad.

\end{enumerate}

\section{IS THE GALACTIC PLANE FLAT?}

	External galaxies are often distorted (like the brim of a Fedora). 
These distortions are probably produced primarily by nearby neighbors. 
How about our own Galaxy? To determine this we want to look at the
vertical structure of the Galactic plane and see where the peaks occur;
a flat Galaxy would everywhere have the peaks at $b=0^\circ$. 

The most comprehensive way to observe and display these distortions is
with a map that covers the biggest possible swath in Galactic longitude
$\ell$ for the range of Galactic latitude ($b$) from $-20^\circ$ to
$+20^\circ$. This is a huge area---almost $10^4$ square degrees! At
$2^\circ$ sampling, you'd need $\sim 2500$ profiles.  To make the
project more manageable, you can increase the increment in $\ell$, say
to 4 degrees, while keeping the same, fully-sampled increment in
$b$. Resolution is $l$ is less important because---like a
fedora---the warp doesn't change rapidly with $l$.

This project affords a great opportunity to make a spectacular color
image in $(\ell,b)$ coordinates: brightness showing the amount of gas,
color the Galactic rotation.  Officially, the science goals here are to
estimate the thicknesses of the Galactic disk and, also, to obtain an
approximate Fourier representation of the warp---which is not so
straightforward given the absence of data in the ``Southern'' Galaxy.

\section{THE GALACTIC POLES: GREAT CIRCLES AT CONSTANT LONGITUDE $\ell$}
\label{galacticpoles}

How is the interstellar gas distributed within the Galaxy---for example,
is it ``disklike'' or more spherically distributed? Are there any
systematic motions other than just ``Galactic rotation''? (You'd be
surprised!).  To answer these questions, determine the column density
and mean velocity of interstellar gas as a function of Galactic latitude
$b$. 

	To answer this completely you'd need to do a sky survey! That's
a big job.  So let's do something easier. We have one project (\S
\ref{galacticplane}) that looks around a great circle, the Galactic
equator (constant $b$). Other projects could check out one or more
orthogonal great circles (at constant longitudes $\ell$) going from one
Galactic pole to the other and back again (or, if not the whole 360
degree circle, as much as possible).  Particularly appropriate sets of
constant-longitude great circles are the pairs $\ell = (220^\circ,
40^\circ)$ (partly because you can get the whole great circle) and $\ell
= (130^\circ, 310^\circ)$ (because at positive latitudes there's weak
high-velocity gas).

	The science goals here are to characterize the thickness of the
gas layer and to determine the vertical kinematics of the gas, and if
possible an estimate of the energy involved in the vertical motions.

\section{MAPPING THE NORTH CELESTIAL POLE} \label{celestial}

The region near the North Celestial pole contains a large shell,
probably produced by one or more supernovae, and also to has angular
scales that are well-matched to our telescope and the available time for
your project. For this, you'd map the region covering roughly
$(\ell=105^\circ \ {\rm to} \ 160^\circ), (b=15^\circ \ {\rm to}
\ 50^\circ)$. Interesting features should produce antenna temperatures
$\sim 10$ K. This is about 1600 square degrees and, with 2 degree
spacing, requires about 400 profiles. With our telescope, this is
probably the most spectacular and contrasty object we can map in rhe
sky.

\section{MAPPING THE ORION-ERIDANUS SUPERBUBBLE} \label{eridanus}

The ``Orion-Eridanus Superbubble'' was, and continues to be, produced by
energetic stellar winds and supernovae that were located in immediate
vicinity of the Orion nebula. For this, you'd map the region covering
roughly $(\ell=160^\circ \rightarrow 220^\circ, b=-70^\circ \rightarrow
-10^\circ)$. Interesting features should produce antenna temperatures
$\sim 20$ K. This is almost 2800 square degrees and, with 2 degree
spacing, requires almost 700 profiles. See \S \ref{sampling}!

The goal here is to map the HI in the 3-d space of $(l, b, v)$ (that's
Galactic longitude, latitude, and velocity) and then---the challenging
part---present the results in one or more color images to show the
hollowed-out shell with the swept-up gas piled up at the edges.

\section{MAPPING THE NORTH POLAR SPUR and its associated EXPANDING HI SHELL}

This is a huge shell produced (and continuing) by several energetic
stellar winds and supernovae that were located in the Sco/Oph
association of stars. The shell contains not only HI but also
relativistic electrons, so the shell is visible in both the 21-cm line
and synchrotron emission. We can't easily see the latter, but we can see
the HI---and the pattern of its velocity follows that of an expanding
shell.

For this, you'd map the region covering roughly $(\ell=210^\circ
\rightarrow 20^\circ, b=0^\circ \rightarrow 90^\circ)$. (The longitude
range is $170^\circ$: $210^\circ \rightarrow 360^\circ$ plus $0^\circ
\rightarrow 20^\circ$). This is roughly 1/4 of the whole sky, so this is
a huge area---almost 10000 square degrees! (This shell has an angular
diameter of $120^\circ$). But (unfortunately) you can't measure the
equivalent 2000 HI profiles because most of this area is too far
south. At least, this makes the project do-able! In doing your
observations, you need to cover as far south as you possibly can. See
\S \ref{sampling}!

The goal here is to map the HI in the 3-d space of ${l, b, v}$ (that's
Galactic longitude, latitude, and velocity) and then---the challenging
part---present the results in one or more color images to show the
hollowed-out shell with the expanding swept-up gas piled up at the
edges.

\section{MAPPING A BIG HIGH-VELOCITY CLOUD}

When we look up, away from the Galactic plane, we see infalling
gas---some a very high velocities. It's called ``High-Velocity Gas''.
The line is weak (about 1 to 1.5 K), so you need high sensitivity---you
need to use much longer integration times than for the above projects,
at least a few minutes per point.  This project needs to map the region
bounded roughly by $(\ell=60^\circ \rightarrow 180^\circ), b=20^\circ
\rightarrow 60^\circ)$. This is about 3700 square degrees and needs
about 900 profiles at $2^\circ$ spacing. See
\S \ref{sampling}!

The goal here is to map the HI in the 3-d space of ${l, b, v}$ (that's
Galactic longitude, latitude, and velocity) and then---the challenging
part---present the results in one or more color images to show the
hollowed-out shell with the expanding swept-up gas piled up at the
edges.

\section{MAPPING THE MAGELLANIC STREAM}

The Magellanic Stream is an intergalactic tidal stream produced by
gravitational interaction between the Magellanic Clouds and the Milky
Way. The signal is weak ($\sim$ a few tenths K), so you need to use much
longer integration times than for the above projects---a minimum of 10
minutes per point. You need to map the pie-shaped area bounded roughly
by $(\ell=60^\circ \rightarrow 110^\circ), b=-90^\circ \rightarrow
-30^\circ)$. This is about 1250 square degrees and needs about 310
profiles at $2^\circ$ spacing. See \S \ref{sampling}!  Also, the LSR
velocity of this gas ranges from roughly $-400 \ {\rm to} \ -100$ km/s
along its length, so you need frequency switch by a large enough
interval!

The goal here is to map the HI in the 3-d space of ${l, b, v}$ (that's
Galactic longitude, latitude, and velocity) and then---the challenging
part---present the results in one or more color images to show the
hollowed-out shell with the expanding swept-up gas piled up at the
edges.

\section{THE OH MOLECULE}

This project is more oriented towards computing and signal processing
than ``astronomy'', and uses cross correlation to derive the complete
set of polarization Stokes parameters. It doesn't use images, except
possibly for interference detection and excision, and as such would not
appeal to everyone.

The OH molecule is an important one for star formation and molecular
clouds because it is one of the first to form when a diffuse atomic
cloud starts to become denser; in other words, OH marks the transition
between atomic and molecular gas. It persists at higher densities. The
ground rotational/vibrational state exhibits ``$\Lambda$ doubling'',
which produces two spectral lines; each of these is hyperfine-split
(just like the 21-cm line), and the net result is four lines centered at
1612.2310, 1665.4018, 1667.3590, 1720.5300 MHz. At sufficiently high
densities, OH can be ``pumped'' by collisions, IR, or UV radiation to
make these lines shine as masers. The strongest maser line tends to be
the 1665 line. 1667 is the strongest nonmaser line.

The maser lines are strong, but our telescope is small, and the
strongest source, W49, will produce $< 1$ K antenna temperature. We have
interference, which makes the line difficult---maybe impossible---to
detect. We have never been able to detect it, but with our new dish and
spectrometer it should be do-able. 

The goal here is to detect the 1665 MHz maser line and to measure its
velocities and complete state of polarization in a number of HII
regions, where it appears in maser emission, and towards the Galactic
center, where it appears in very broad absorption. The HII regions are
W3, W33, W43, and W49. We will use W49 as a polarization calibrator,
because one of its velocity components is 100\% circularly polarized,
and having that calibration measure the polarization state of the other
velocity components in W49 and, also, in the other HII regions. The
coordinates of these HII regions are in the {\it Simbad Astronomical
  Database}\footnote{To reach {\it Simbad}, Google it, and once you
  reach the site click on `by identifier'.}.
While you're at it, also measure the OH line towards the
Galactic center. It should show up as broad absorption lines against the
continuum of the Galactic center.


\section{BASICS OF THE 21-CM LINE.} \label{basics}

\subsection{Column Density and Mass} \label{coldensity}

	The 21-cm line, with frequency 1420.405751786 MHz, comes from
atomic hydrogen (HI).  In the terrestrial environment, H atoms quickly
become H$_2$ molecules; in interstellar space, where densities are far
lower than in the best vacuum systems on Earth and there are ``lots'' of
UV photons that dissociate H$_2$, H remains atomic unless it resides
inside dark clouds where it is shielded from starlight.

	The intensity of the 21-cm line is directly proportional to the
column density of H atoms as long as the opacity of the line is small;
this is a reasonably good approximation (but not perfect, particularly
in the Galactic plane where the line is strong). With this
approximation, 

\begin{equation} \label{nh1}
N_{HI} = 1.8 \times 10^{18} \int T_{B}(v) dv \ {\rm cm^{-2}}. 
\end{equation}

\noindent Here $N_{HI}$ is the column density of H atoms---the number of
atoms in a 1 cm$^2$ column along the line of sight; $T_{B}(v)$ is the
brightness temperature of the 21-cm line, which is a function of 
velocity $v$; and $v$ is the velocity in km s$^{-1}$. The velocity $v$
is produced by the Doppler effect, so a frequency shift $\Delta \nu$
from line center corresponds to velocity $v = - c {\Delta \nu \over
\nu}$ or, for the 21-cm line, $v = -{\Delta \nu \over 4.73 {\rm kHz}}$.
Note the minus sign! It means that positive velocities are receding---a
very convenient convention for astronomy, because of the expansion of
the Universe. 

          Note an important corollary to equation \ref{nh1}: If we
consider a small velocity interval $\Delta v$, then the number of H
atoms in the column in that velocity range is just 

\begin{equation} \label{nh2}
N_{HI}(v \rightarrow v + \Delta v) = 1.8 \times 10^{18} T_{B}(v) \Delta
v \ {\rm cm^{-2}} \ .  
\end{equation}

\noindent This is very important, because it gives us the opportunity to
make maps of the interstellar gas at different velocities and to
determine how it moves.

	To get the column density we need to get the brightness
temperature $T_B$, which is not the same as our antenna temperature
$T_A$. The relationship between these depends on the relative size of
the source and the beam. This is equation (9) in the ``Fount of All
Knowledge!'' handout, namely 

\begin{equation} \label{taeqn}
T_A \approx T_B {\Omega_s \over \Omega_s + \Omega_b} \ ,
\end{equation}

\noindent where $\Omega_s$ and $\Omega_b$ are the solid angles of the
source and beam, respectively. From this, we see that if the source
fills the beam ($\Omega_s \gtrsim \Omega_b$) then $T_A \approx \langle
T_B\rangle$ (the brackets denote the average over solid angle);
while if it is much smaller ($\Omega_s \ll \Omega_b$) then $T_A \approx
{\langle T_B \Omega_s\rangle \over \Omega_b}$, i.e.\ $T_A$ is smaller by the ratio of
the solid angles. 

	To get the mass of HI from the column density, you need to know
the area of the region in {\it linear} measure, which is just $\Omega \
d^2$, where $d$ is the distance.  Each HI atom has mass $m_H$, so the
total mass of HI in a region of size $\Omega$ is just

\begin{mathletters} \label{masseqn}
\begin{equation} 
M_{HI} = m_H \ d^2 \ \langle N_{HI} \Omega\rangle \ .
\end{equation}

\noindent Here $\langle N_{HI} \Omega \rangle$ is shorthand for the
general relation, which uses an integral:

\begin{equation}
\langle N_{HI} \Omega \rangle = \int_{\rm region} N_{HI} d \Omega
\end{equation}
\end{mathletters}

\noindent Substituting for $N_{HI}$ from equation \ref{nh2}, we have

\begin{equation}  \label{telemass}
M_{HI}(v) = 1.8 \times 10^{18} \Delta v \ d^2 \  m_H \ \langle T_B  \Omega\rangle \ {\rm
gm} \ . 
\end{equation}

\noindent Units are cgs. Sometimes there is a well-defined source with
boundaries; in this case, to get the mass of the whole source, we use
the solid angle occupied by the source, $\Omega_s$.

	Sometimes (in fact, often) we want to know the mass seen {\it by
the telescope} for some particular observation.  For an extended region
with $\Omega \gg \Omega_b$, $T_A = T_B$ and the telescope sees the solid
angle $\Omega_b$, so the product $\langle T_B \Omega\rangle = T_A
\Omega_b$ because angular area seen by the telescope is limited by its
beam size.  In contrast, if the HI emission is limited to a small
region---a small source with $\Omega_s \ll \Omega_b$---then $T_A \approx
{\langle T_B \Omega_b\rangle \over \Omega_s}$ so the product $\langle
T_B \Omega_s\rangle = T_A \Omega_b$; the telescope sees the whole
source.  

For both extremes, the product $T_B \Omega = T_A \Omega_b$. 
It's not only these two extremes, but for {\it all} sources the
product $\langle T_B \Omega \rangle= T_A \Omega_b$.  Thus, it is always true that the
mass seen by the telescope is

\begin{equation}  \label{telemass1}
M_{HI}(v) = 1.8 \times 10^{18} \Delta v \ d^2 \ m_H \ T_A  \Omega_b  \ {\rm
gm} \ . 
\end{equation}

\noindent The distance $d$ is usually a function of velocity $v$,
especially in the Galactic plane. 

	$T_A(v)$ is directly measured and $\Omega_b$ is the telescope
beam area, which is known, so it's easy to calculate $M_{HI}(v)$---but
only if you know the distance! Often you don't know the distance, but
you might have a hunch for a reasonable value for the distance. Suppose
this is 100 parsecs. In this
case, people usually evaluate the mass for this distance and, when
giving the mass, say ``The mass is $xxx \times d_{100}$, where $d_{100}$
is the distance in units of 100 pc''. 

\subsection{Converting Galactic Rotation to Doppler Velocity}

          Measurement of the Doppler shift caused by differential
Galactic rotation for $0^\circ < \ell < 90^\circ$ allows a direct
determination of the Galactic rotation curve inside the Solar circle,
using the ``tangent point'' method. See Burton's article ``Structure
of our Galaxy from Observations of HI'' in {\it Galactic and
Extragalactic Radio Astronomy, second edition} (editors: G.L. Verschuur
and K.I. Kellermann). In essence, you use the following equation.

	We won't go through the derivation here; we cover it in class,
and it is also done in Burton's article,
page 303-304.  Let $V_{Dopp}$ be the Doppler velocity (this is also the
observed LSR velocity), $V(R)$ be the Galactic rotation velocity at
Galactocentric distance $R$, and let the subscript $\odot$ denote values
at the Solar circle.  Then we have
\begin{equation} \label{vdopp}
V_{Dopp} = \left[ {V(R) \over R} - {V(R_\odot) \over R_\odot}\right]
R_\odot \sin(\ell)
\end{equation}
\noindent The values at the solar circle are $V(R_\odot) \approx 220$
km/s, $R_\odot \approx 8.5$ kpc.

\section{ON MEASURING POLARIZATION}

Polarization is completely specified by the Stokes parameters. Stokes
parameters are linear combinations of power measured in {\it orthogonal
  polarizations}. There are four:

\begin{mathletters}
\begin{equation}
I = E_X^2 + E_Y^2 = E_{0^\circ}^2 + E_{90^\circ}^2 
\end{equation}
\begin{equation}
Q = E_X^2 - E_Y^2 = E_{0^\circ}^2 - E_{90^\circ}^2 
\end{equation}
\begin{equation}
U =  E_{45^\circ}^2 - E_{-45^\circ}^2 
\end{equation}
\begin{equation}
V = E_{LCP}^2 - E_{RCP}^2 
\end{equation}
\end{mathletters}

\noindent We like to write the {\it Stokes vector}
\begin{eqnarray*}
\label{IQUV}
{\bf S} = \left[
\begin{array}{c}
    I \\ Q \\ U \\ V \\
\end{array}
\; \right] \; .
\end{eqnarray*}

\noindent The first, Stokes $I$, is total intensity. It is the sum of
          {\it any two orthogonal} polarizations.  The second two,
          Stokes $Q$ and $U$, completely specify linear polarization.
          The last, Stokes $V$, completely specifies circular
          polarization. One often specifies fractional polarized power
          instead of polarized power itself, by dividing all Stokes
          parameters by $I$; in this case, of course, $I=1$.

As shown in Figure \ref{pol}, left panel, linear polarization is
conventionally specified by two different parameters, the polarized
intensity $P_{QU}$ and the position angle $\chi$. These are directly
related to the $Q$ ande $U$ Stokes parameters:

%\begin{wrapfigure}{r}[2.5in]{5.8in}
\begin{mathletters}
\begin{equation}
P_{QU} = ( Q^2 + U^2 )^{1/2} 
\end{equation}
\begin{equation}
\chi = 0.5 \tan^{-1} {U \over Q}
\end{equation}
\end{mathletters}

\noindent with the inverse relationships

\begin{mathletters}
\begin{equation}
Q = P_{QU}  \cos (2 \chi)
\end{equation}
\begin{equation}
U = P_{QU}  \sin (2 \chi)
\end{equation}
\end{mathletters}
%\vspace{-35ex}
\begin{figure} [!h]
\begin{center}
\includegraphics[width=2.5in]{fig3.ps}
\includegraphics[width=2.5in]{fig4.ps}
\end{center}
\caption{Left: The position angle of polarization. Right: Adding $E_X$
 (same as $E_0$) and $E_Y$ ($E_{90}$) gives $E_{45}$. \label{pol}}
\end{figure}
%\end{wrapfigure}

{\it Real} radioastronomers measure all four Stokes parameters
simultaneously! After all, extracting two orthogonal polarizations
provides {\it all} the information; you can synthesize {\it all} other E
fields from the two measured ones! Example (Figure \ref{pol}, right
panel): Sample $(E_X,E_Y)$ and synthesize $E_{45}$ from $(E_X,E_Y)$. To
generate $E_{45}$, add $(E_X,E_Y)$ with no phase difference. To generate
$E_{LCP}$, add $(E_X,E_Y)$ with a $90^\circ$ phase difference.

Carrying through the algebra for the two linears:
It's clear that
\begin{mathletters}
\begin{equation}
E_{45^\circ} = { E_{0^\circ} + E_{90^\circ} \over \sqrt 2}
\end{equation}
\begin{equation}
E_{-45^\circ} = { E_{0^\circ} - E_{90^\circ} \over \sqrt 2}
\end{equation}
\end{mathletters}


Write the two linear Stokes parameters:

$$Q = E_X^2 - E_Y^2 = E_{0^\circ}^2 - E_{90^\circ}^2$$

$$U =  E_{45^\circ}^2 - E_{-45^\circ}^2  = 2E_XE_Y$$

%\begin{mathletters}
%\begin{equation}

\noindent {\it Stokes $U$ is given by the crosscorrelation $E_XE_Y$.}
To get $V$, throw a $90^\circ$ phase factor into the correlation. Our
new spectromer provides this cross-product, so this means we can be {\it
  real} radio astronomers! We have a handout explaining all the gory
details for any group that pursues this project.

\end{document}

