\section{Introduction}

Scientists often collect quite a large amount of data when performing
various types of experiments.  One method which scientists use to help
convert their data into something which they can easily analyze and
interpret is graphing.  Graphing is a way of visualizing data in order
to determine relationships between events in the experiment as well as
record what happened during the experiment.

\begin{itemize}
\item{{\it Goals:}  to use graphs to see the spatial distribution of the 
50 nearest and 50 brightest galaxies and determine the
most and least common galaxy type among them}
\item{{\it Materials:}  ruler, graph paper}
\end{itemize}

\subsection{Maps and Graphs}

Graphs are useful tools because they help to visualize data which sometimes
can help you see patterns which might not otherwise be evident.

For example, consider the following table of (made-up) positions of stars
in the constellation ``Home".  Positions of stars are specified by a sort
of astronomical longitude and latitude called \textit{right ascension}
and \textit{declination}. 

\begin{table}
\begin{center}
\caption{Positions of Stars in the Constellation ``Home''}
\label{tab:home}
\bigskip
\begin{tabular}{rr|rr|rr|rr}
\hline
R.A.  & Dec. &
R.A.  & Dec. &
R.A.  & Dec. &
R.A.  & Dec. \\
\hline
\hline
\input INTROII/mom.tex
\hline
\end{tabular}
\end{center}
\end{table}

Can you infer the shape of this constellation by looking at this table?
\smallspace

Now we will make a graph of the positions. The first thing you need to
do is to figure out the range of the different values, so you can make
your graph with axes which will allow all the points to fit on the graph.
What are the largest and smallest values of R.A. and Dec from all of the
stars in Table \ref{tab:home}?
\smallspace

Construct a graph with R.A. on the x-axis and Dec. on the y-axis using
graph paper. You should make each square on the graph paper equal to
some amount of R.A/Dec so that all of the points will fit on the graph.
For this to be a useful map, you want all of the squares to represent
the same distance, so make sure you draw your axes so that this is true.

Now start to plot the stars from Table \ref{tab:home}. (Once everyone
gets started and realizes that this will take a long time, it will show the
benefits of using a \htmladdnormallink{computer}{../INTROII/a110plot.htm}) .
See if you can detect something
about the shape which you couldn't infer from the tabular data. Can you
see the advantage of making a graphical representation of the data?
\smallspace

Giving the position of the stars in right ascension and declination
(i.e., the \textit{direction} in which they are seen)
doesn't fully specify their position, for example, in the Galaxy. What
additional piece of information is missing to fully specify the location
of each star?
\smallspace

\subsection{The distribution of galaxies in three dimensions}

Next we will consider the distribution of galaxies in \textit{three} 
dimensions. We could fully specify the positions of objects in space
by giving their direction (right ascension and declination) and
their distance. Alternatively, we could imagine some invisible rectangular
grid in space, centered on our Galaxy, and give the (X,Y,Z) position of
each galaxy. These are two examples of different \textit{coordinate systems}.

Table \ref{tab:near} gives the (X,Y,Z) positions of the 50 
\textit{nearest} galaxies
to the Milky Way.
Table \ref{tab:bright} gives the (X,Y,Z) positions of the 50 \textit{brightest}
galaxies (after correcting for distance, so these are really the galaxies
which produce the most starlight intrinsically)
in the nearby universe. We would like to investigate how they
are distributed in space. Are they clumped together or spread apart? Do
they uniformly fill a spherical volume around us, or are they distributed
in some structure?

\begin{table}
\begin{center}
\caption{Positions of 50 Nearest Galaxies in the Nearby Universe}
\label{tab:near}
\bigskip
\begin{tabular}{lrrrr|lrrrr}
\hline
Name  & Type &X & Y & Z &
Name  & Type &X & Y & Z \\
\hline
\hline
\input INTROII/near.tex
\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\begin{center}
\caption{Positions of 50 Brightest Galaxies in the Nearby Universe}
\label{tab:bright}
\bigskip
\begin{tabular}{lrrrr|lrrrr}
\hline
Name  & Type &X & Y & Z &
Name  & Type &X & Y & Z \\
\hline
\hline
\input INTROII/bright.tex
\hline
\end{tabular}
\end{center}
\end{table}
It's difficult to make a true 3-dimensional map, since a piece of paper
only has 2 dimensions.  To understand the distribution of nearby galaxies, 
let's make a 3-dimensional model! Take 25 to 30 of the nearest galaxies and
find out their minimum and maximum X, Y, and Z positions; don't include
any galaxies which have any coordinate larger than 1.25. Then layout your
classroom so that the different dimensions are represented. Let each
student be assigned a galaxy, then go to the appropriate position in the
classroom. What can we infer about the distribution of nearby galaxies?

Next we will look at the distribution of the \textit{brightest} galaxies. We
will resort to looking at two different \textit{projections} of the galaxy
distribution and see what we can learn from them.

Let's make a plot of the (X,Y) positions of each galaxy and a plot
of the (Y,Z) positions using the
\htmladdnormallink{computer}{../INTROII/a110plot.htm}. 
Describe the distributions of the points in each plot. (10 points)

\bigspace

\htmladdnormallink{Figure}{../html/diagrams/a110labs/INTROII/nbg-proj.htm} 
\ref{fig:nbg-proj}
shows the same projections, but with many more 
galaxies plotted. The blue-shaded regions are areas where we have a hard
time seeing galaxies even if they are there, because these are directions
in which we need to look through the disk of our own Milky Way galaxy, and
the dust in our own galaxy obscures our view. Consequently, the lack of
galaxies in the blue shaded areas does not mean there are no galaxies there.
\begin{latexonly}
\begin{figure}
\centering
\epsfysize=\textheight
\epsfbox{INTROII/nbg-proj.ps}
\caption{Locations of galaxies in the nearby universe as seen in two different 
projections.}
\label{fig:nbg-proj}
\end{figure}
\end{latexonly}

Do the brightest 50 galaxies, as shown in the plot you made, trace out
the same structure as all of the galaxies including the fainter ones?
(10 points)
\smallspace

What do you think these plots are telling you about the distribution of
galaxies in three dimensions?(10 points)

\smallspace

\subsection{The Galaxy Morphology System}

There is more to galaxies than just their location in space. As you may have
discussed in class, galaxies come in various different shapes and sizes. 
There are three basic types of galaxies: spirals, ellipticals and
irregulars.  The spiral galaxies can be further divided into a few
different categories depending on how tightly wound the spiral arms
are.  The elliptical galaxy classification is also broken down into
smaller categories depending on how elliptical the galaxy is.
Irregulars are more difficult to separate from one another, so this
class is not subdivided.  Below is a diagram showing Hubble's 
breakup of the galaxy categories.   Since it is shaped like a tuning fork, 
this diagram is commonly called the ``Hubble Tuning Fork diagram'':

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.75\textwidth
\epsfbox{INTROII/IntroII_fig5.ps}
\end{figure}

One can assign a numerical value to the different morphological
classes.  A 10 or higher corresponds to the irregular galaxy class; the spiral
galaxies fall into the range from 0 to 9.  Elliptical galaxies start
at -5 and go up to 0 (S0 galaxies are transition-type galaxies and are
grouped here as ellipticals).   Table \ref{tab:type} is a table listing
the morphological classes and corresponding T value for the Hubble system:

\begin{table}[h]
\begin{center}
\caption{Galaxy Classification System} 
\label{tab:type}
\bigskip
\begin{tabular}[h]{lrrrrrrrrrrr}
\hline
 &Ell's&&&Sp's&&&&&&Irreg's&\\
Type &E&E-S0& S0&S0/Sa&Sa&Sa-b&Sb&Sb-c & Sc & Sc-Irr & Irr \\
T  & -5 & -3  & -2 & 0  & 1 & 2 & 3 & 4 & 6 & 8  & 10  \\
\hline
\hline
\end{tabular}
\end{center}
\end{table}

Using the data in Table \ref{tab:bright},
how many of the 50 brightest nearby galaxies are 
spirals? How many are ellipticals? How many are irregulars? (15 points)
\bigspace

Take your plot of the locations of the different galaxies and attempt to
include information about the type of the galaxy as well. You might want
to plot different types with different colors, or with different types
of symbols (e.g. circles for ellipticals, crosses for spirals...) (10 points)

Do you notice anything about how the different types of galaxies are
distributed? Are spirals or ellipticals more likely to lie in clumps
of galaxies? What do you think this might mean? (10 points)
\bigspace

\section{Summary}

{\it (35 points)}  Please summarize the important concepts introduced in this 
lab.  You should discuss:

\begin{itemize}
\item The advantages of making graphs to convey information.
\item Your general impressions of how galaxies in the nearby universe are
distributed in space, paying attention to the different types of galaxies
as well.
\end{itemize}

\noindent
Use complete sentences, and proofread your summary before handing in the lab. 

\begin{latexonly}

\begin{figure}[h]
\leavevmode
\epsfbox{INTROII/graph.eps}
\end{figure}
\begin{figure}[h]
\leavevmode
\epsfbox{INTROII/graph.eps}
\end{figure}

\end{latexonly}

%\end{document}
