\section{Introduction}

Measurements, calculations, physical principles and estimations (or
educated guesses) lie at the heart of all scientific endeavors.
Measurements allow the scientist to quantify natural happenings,
conditions, and characteristics.  However, measurements can be hard
to make for practical reasons. We will investigate some of the
issues with taking measurements in this lab.

In addition, an important part about the measurement of something
is an understanding about the \textit{uncertainty} in that measurement.
No one, including scientists, ever make measurments with perfect accuracy, 
and estimating
the degree to which a result is uncertain is a fundamental part of
science. Using a result to prove or disprove some theory can only be
done after a careful consideration of the uncertainty of the result.

\begin{itemize}
\item{{\it Goals}: to discuss the concepts of estimation, measurement and
measurement error, and to use these, along with some data from the Hubble 
Space Telescope, to estimate the number of observable galaxies in the Universe}
\item{{\it Materials}: Hubble Deep Field image}
\end{itemize}

\section{Exercise Section}

\subsection{Direct Measurement, Measurement Error}

We will start out by counting objects much closer to home than galaxies!

\indent
How many chairs do you think there are in your classroom? You have one minute!
\blank

How did you determine this? 
\smallspace

How does your number compare with that of other groups? What does this say
about the uncertainty in the results?
\smallspace

Now do an exact count of the number of chairs - you have three minutes. 
Note the advantage of working with a group! By comparing results from
different groups, what is the uncertainty in the result?
\smallspace

\subsection{Estimation}

Now we extend our measurement to a larger system where practical considerations
limit us from doing a direct count.

How many chairs do you think there are in the entire University? You might
wish to consider the 
\htmladdnormallink{campus map}{../html/diagrams/a110labs/INTROI/campus-map.htm}
\begin{latexonly}
in Figure \ref{fig:campus-map}.
\begin{figure}
\centering
%\leavevmode
\epsfxsize=\textwidth
\epsfbox{INTROI/campus-map.ps}
\caption{A map of the NMSU campus from the NMSU WWW site}
\label{fig:campus-map}
\end{figure}
\end{latexonly}
\blank

How did you determine your number?

\smallspace

How accurate do you think your number is?

\smallspace

How might you estimate the uncertainty in your number?

\smallspace

\subsection{How many galaxies are there in the Universe?}

\indent
Considering how you estimated the number of chairs in the classroom and
on campus, consider and write down several alternative ways of estimating the 
number of galaxies in the Universe.
\bigspace

Let's consider the issue by looking at a picture of the sky taken with
the Hubble Space Telescope. This telescope is the most capable of existing
telescopes for viewing very faint objects. In an effort to observe the
faintest galaxies, astronomers decided to spend 10 entire days training
this telescope on 
\htmladdnormallink{one small region of the sky}{../html/images/galaxies/high-z/hdf.htm} to observe the faintest galaxies and learn about them.

\begin{latexonly}
The image that was obtained is shown in Figure \ref{fig:hdf}.
\begin{figure}
\centering
\leavevmode
\epsfxsize=\textwidth
\epsfbox{INTROI/IntroI_fig1.eps}
\caption{A reproduction of the Hubble Deep Field image.}
\label{fig:hdf}
\end{figure}
\end{latexonly}
\begin{htmlonly}
\htmladdnormallink{HDF image}{../html/images/galaxies/high-z/hstdeep.htm}
\end{htmlonly}

\medskip

First, let's figure out how long it would take for the Space Telescope
to take pictures like this over the entire sky. 

To do this, we need to
talk about how we measure distances and areas on the sky, concepts that we will
need in some of other labs as well. When one measures, for example,
the distance between two stars as seen from Earth, one measures what
is known as an angular distance. A standard unit of this angular
distance is the familiar unit of the degree; there are 360 degrees in
a full circle. As an example, the distance between an object which is
straight overhead and one which is on the horizon is 90
degrees. However, when one makes astronomical observations with big
telescopes, one usually sees an area which is only a small fraction of
a degree on a side. To make things easier to write, astronomers
sometimes use units known as arcminutes and arcseconds.  There are 60
arcminutes in a degree, and 60 arcseconds in an arcminute.

\medskip

\begin{enumerate}
\item  We can now use this information to calculate how many
pictures the Space Telescope would have to make to cover the entire
sky. The picture from the Space Telescope 
covers a region that is about 1 arcminute on a side. Our first conversion 
is from arcminutes to degrees (this has been partially done for you): ({\it 3 points})

\bigskip
\beq
1 \; arcminute \times \frac{1 \; degree}{60 \; arcminutes} =
{\underline{\hskip 2in{}} } \; degrees
\eeq
\bigskip

\item  The {\it area} of the entire picture is measured in square
degrees, so we take the number of degrees found in question 1 and square 
it to get: ({\it 3 points})

\smallspace

\item  Now there are 4.13 $\times$ 10$^4$ square degrees in the
sky.  From this you can figure out how many pictures you would need to
take to cover the whole sky: ({\it 5 points})

\smallspace

\item  Finally if it takes 10 days for each picture, we can figure
out how long it would take to cover the whole sky with similar
pictures: ({\it 4 points})

\smallspace

\item  With a unit conversion of 365 days/year, we can determine
the number of years it would take: ({\it 4 points})

\smallspace

Clearly, this is a very long time! This is an interesting point to
note: astronomers can only take deep pictures of a small fraction of
the sky. So it is not practical to count galaxies by taking pictures
of the entire sky.

So how can we proceed to figure out how many total galaxies there are?
We can make an {\it estimate} by guessing that the number of galaxies
in any particular picture will be the same regardless of where we
point. We can then estimate the total number of galaxies in the sky by
counting the number of galaxies in this one picture, and multiplying it
by the number of pictures that it would take to cover the whole sky.


\item  Take a look at the image of the Hubble Deep Field given to you 
by your TA. Almost every one of the objects you see in this picture is a 
distant galaxy. Count up all the galaxies in each section then add them 
up to get an estimate of the number of galaxies in this one field. Again,
you can proceed quicker by taking advantage of the multiple members of
your group; however, you might wish to have everyone in the group count
one region independently to get some idea of the measurement uncertainty. ({\it 
10 points})

\bigskip

\noindent
\begin{center}
Region A1:\blank \\
\smallskip
Region A2:\blank \\
\smallskip
Region A3:\blank \\
\smallskip
Region B1:\blank \\
\smallskip
Region B2:\blank \\
\smallskip
Region B3:\blank \\
\smallskip
Region C1:\blank \\
\smallskip
Region C2:\blank \\
\smallskip
Region C3:\blank \\

\bigskip

\noindent
There are a total of \blank galaxies in Hubble Deep Field.
\end{center}

\item  Now estimate the total number of galaxies in the whole sky,
using our calculation of the number of pictures it takes to cover the
sky which we did above. ({\it 7 points})

\smallspace

This is a pretty amazingly large number. Consider that each galaxy has
billions of stars, and think just for minute about how many total
stars there are in the Universe! It makes you feel pretty
small.... but, on the other hand, think how cool it is that humans
have evolved to the point where they can even make such an estimation!

\bigskip

\item  As we've discussed, an estimate of the \textit{uncertainty}
in a result is often as important as the result itself. Discuss several
reasons why your result may not be especially accurate. You may wish
to compare the number of galaxies in any given region which you counted
with the number counted by other groups, or consider the variation in
the number of galaxies from one region to another. Also, remember a
fundamental assumption that we made for getting our estimate, namely, 
that the number of galaxies we would
see in some other portion of the sky is the same as that which we see
in this Hubble deep field. ({\it 16 points})

\bigspace

\item  Finally, there's one more caveat to our calculated total
number of galaxies. To make our estimate, we $assumed$ that the 10 day
exposure sees every single galaxy in this portion of the sky.  With this
in mind, how would the calculation you just conducted compare to the
real number of galaxies in the Universe?  Back up your answer with a
short explanation. ({\it 13 points})

\end{enumerate}

\bigspace

\subsection{Summary}

({\it 35 points}) Please summarize the important concepts discussed in
this lab.  Your summary should include a brief discussion of

\begin{itemize}
\item{direct measurement vs. estimation}
\item{error estimates in both direct measurement and estimation}
\item{Consider the importance of the galaxy counting results discussed in
lab.  Since the Hubble Deep Field was taken in a presumably empty part
of the sky, what is the significance of finding so many galaxies in
this picture?} 
\item{Use the concepts discussed in this lab to estimate the total number of
stars that you can see in the night sky by going out at night and doing
some counting and estimating. Describe your method as well as the number
you get and provide some estimate of your uncertainty in the number.}
\end{itemize}

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

%\end{document}



