\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 events,
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 measurements 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!

~ ~ \\

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 campus map shown in Figure \ref{fig:campus-map}.

\begin{figure}
\centering
\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}

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?

\clearpage

\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 one small region of the sky to observe the faintest galaxies 
and learn about them. 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}

\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 our 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): ({\bf 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: ({\bf 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: ({\bf 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: ({\bf 4 points})

\smallspace

\item  With a unit conversion of 365 days/year, we can determine
the number of years it would take: ({\bf 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. ({\bf 
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. ({\bf 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. ({\bf 8 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. ({\bf 8 points})

\end{enumerate}
\bigspace
\section{The Mass and Density of the Universe (Contained in Galaxies)}

In the preceding we have estimated the number of galaxies in the Universe.
In the final section of this lab, we now want to explore the implications
of this calculation by making an estimate of the matter density of the Universe.
In your lecture sections, and some of the earlier labs (like the Terrestrial
Planet lab, esp. Table 4.2) you have probably encountered the concept of 
density: density = Mass/Volume. 
Astronomers usually use the unit of gm/cm$^{\rm 3}$ for density.
We can now make an estimate for the density of matter contained in all of the
galaxies in our Universe. We will
start with very large numbers, and end up with an extremely tiny number.
It is quite likely that your calculator cannot handle such numbers.
To make this calculation easier, we will use some round numbers so that
you can do the calculation by hand using the techniques outlined in Lab 1 (if 
you get stuck with how to multiply numbers with exponents, refer back to section
1.4). This
is a challenging exercise, but one that gives you an answer that you might
not expect!

\noindent
\begin{enumerate}
\setcounter{enumi}{9}
\item In question 7 above, you estimated the total number of galaxies in the
sky. If we assume that these galaxies are similar to (though probably somewhat
smaller than) our Milky Way galaxy, we can calculate the total mass of all of
the galaxies in the Universe. Over the course of this semester you will
learn
that the Milky Way has about 100 billion stars, and most of these stars are
about the mass of the Sun, or somewhat smaller. The mass of the Sun is
2 $\times$ 10$^{\rm 33}$ gm. Let's assume that the average galaxy in the 
Universe has 1/2 the number of stars that the Milky Way has: 50 billion.
Fifty billion in scientific notation is 5 x 10$^{\rm 10}$. To calculate
the mass (in gm) of all of the galaxies in the Universe, we need to
solve this equation:

\begin{flushleft}
Mass of Galaxies in Universe = (\# of Galaxies)$\times$(Average Mass of a Galaxy)
\end{flushleft}

You calculated the \# of Galaxies in question 7. We need to multiply that
number by the Average Mass of a Galaxy. The Average Mass of a Galaxy (in gm)
is simply:

\begin{flushleft}
Average Mass of a Galaxy = (\# of stars in a galaxy)$\times$(average mass of a star)
\end{flushleft}

If the number of stars in a galaxy is 5 x 10$^{\rm 10}$, and the average
mass of a star is 2 X 10$^{\rm 33}$ gm, what is the average mass of a galaxy? 
{\bf (2 points)}

\begin{flushleft}
Average Mass of a Galaxy = (\underline{\hskip 1.5in{}})$\times$(\underline{\hskip 1.5in{}}) \\
\vspace{0.2in}
= \hspace{2in} \underline{\hskip 2in{}} gm
\end{flushleft}

With this number, you can now calculate the total mass of all of the galaxies
in the Universe {\bf (2 points)}:

\begin{flushleft}
Mass of Universe = (\underline{\hskip 1.5in{}})$\times$(\underline{\hskip 1.5in{}}) =\\
\vspace{0.2in} 
\hspace{2in} \underline{\hskip 2in{}} gm
\end{flushleft}

\item  We have just calculated the total mass of galaxies in the Universe, and are halfway 
to our goal of figuring out the density of galactic matter in the Universe. Since density = M/V,
and we now have M, we have to figure out V, the Volume of the Universe. This
is a little more difficult than getting M, so make sure you are confident of
your answer to each of the following steps before proceeding to the next. We 
are going to make some
assumptions that will simplify the calculation of V. First off, we will assume
that the Universe is a sphere. The volume of a sphere is simply four thirds
``pi'' R cubed: V$_{\rm sphere}$ = 4$\pi$R$^{\rm 3}$/3. To figure out the
volume of the Universe we need to calculate ``R'', the radius of the Universe.

So, how can we estimate R? In your lecture class you will find out that the
most distant parts of the Universe are moving away from us at nearly the 
speed of light (the observed expansion of the Universe is covered in the 
Hubble's Law lab). Let's assume that the largest distance an object can have
in our Universe 
is given by the speed of light $\times$ the age of the Universe. Remember, if 
a car travels at 50 mph for one hour it will cover 50 miles: Distance = 
velocity $\times$ time. We can use this equation to estimate the radius of the 
Universe: R$_{\rm Universe}$ = velocity $\times$ time = (speed of light) $\times$ (age of Universe).

The speed of light is a very large number: 3 X 10$^{\rm 10}$ cm/s, and the
age of the Universe is also large: 13 billion years. To calculate the
radius of the Universe in cm, we must convert the age of the Universe in
years to an age in seconds (s). First, how many seconds are there in a year? 
Let's do the calculation:

\begin{flushleft}
Seconds in year = (seconds in day) $\times$ (days in year) = (60 $\times$ 60 
$\times$ 24) $\times$ 365 =\\
\vspace{0.2in}
\hspace{2in} \underline{\hskip 1.5in{}} s/yr 
\end{flushleft}

\noindent
Since this is only estimate, feel free to round off any decimals to whole
numbers. Now that we have the number of seconds in a year, we can convert
the age of the Universe from years to seconds:

\begin{center}
Age of Universe in seconds = (Age of Universe in Years) $\times$ (seconds in a 
year) \\
\vspace{0.2in}
= (13 $\times$ 10$^{\rm 9}$ yr) $\times$ \underline{\hskip 1.5in{}} s/yr = 
\underline{\hskip 1.5in{}} s.
\end{center} 

\noindent
Ok, we now have the ``time'' part of the equation distance = velocity $\times$
time. And we have already set the velocity to the speed of light: 3 $\times$
10$^{\rm 10}$ cm/s. Now we can figure out the Radius of the Universe 
{\bf (3 points)}:

\begin{center}
Radius of Universe (in cm) = (speed of light) $\times$ (Age of Universe in
seconds) = \\
\vspace{0.2in}
(3 $\times$ 10$^{\rm 10}$ cm/s) $\times$ \underline{\hskip 2in{}} s = \\
\vspace{0.2in}
 \underline{\hskip 2in{}} cm.
\end{center}

In these calculations, notice how the $units$ cancel. The units on a distance
or radius is length, and astronomers generally use centimeters (cm) to measure
lengths. Velocities
have units of length per time, like cm/s. So when calculating a
radius in cm, we multiply a velocity with units of cm/s $\times$ a time measured
in seconds, and the units of seconds cancels, leaving a length unit (cm).

We are now ready to calculate the Volume of the Universe, V = 4$\pi$R$^{\rm 3}$/3. It may be easier for you to break this into two parts, multiplying out
4/3 $\times$~$\pi$, and then taking R$^{\rm 3}$, and then multiplying those
two numbers. [Remember, $\pi$ = 3.14.]

\begin{center}
Volume of Universe = 4/3 $\times  \pi$ $\times$ R$^{\rm 3}$ =\\
\vspace{0.2in}
 \underline{\hskip 2in{}} $\times$ \underline{\hskip 2in{}} =\\
\vspace{0.2in}
\underline{\hskip 2in{}} cm$^{\rm 3}$
\end{center}

\noindent
\item Tying it all together: figuring out the average density of the Universe (at
least that contained within galaxies--astronomers believe there is more
``dark'' matter in the Universe than the regular matter that we can see 
contained in galaxies!).
We have just calculated the Volume of the Universe, and we have already
calculated the Mass of all of the galaxies in the Universe. Now we take the final step, and 
calculate the Average Galactic Matter Density of the Universe {\bf (3 points)}: 

\begin{center}
Average Density of the Universe = M$_{\rm Universe}$/V$_{\rm Universe}$ =\\
\vspace{0.2in}
(\underline{\hskip 2in{}} gm)/(\underline{\hskip 2in{}} cm$^{\rm 3}$) =\\
\vspace{0.3in}
  \underline{\hskip 2in{}} gm/cm$^{\rm 3}$
\end{center}

\item The mass of a single hydrogen atom is 1.7 x 10$^{\rm -24}$ gm. Compare
your answer for the average density of the Universe to the mass of a single
hydrogen atom. [Hint: the average amount of mass ($in~gm$) of 1 cm$^{\rm 3}$ 
of the Universe is simply the density you just calculated, but you drop the
cm$^{\rm 3}$ of the units on density to get $gm$.] Are they similar? What
does this imply about the Universe, is it full of stuff, or mostly empty?
{\bf (3 points)}
\vspace{4cm}
\end{enumerate}
\section{Summary (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.}
\item{Think back to a time before you did this lab, would you have expected
the answer to question \#13? Our Universe has many surprises!}
\end{itemize}

\noindent
Use complete sentences, and proofread your summary before handing in the lab.
\section{Possible Quiz Questions}

\begin{enumerate}
\item What is meant by the term ``estimation''?
\item Why do scientists use estimation\u201d
\item How many degrees are in a circle?
\item What is an ``arcminute''?
\item What is the ``Hubble Deep Field''?
\end{enumerate}
\section{Extra Credit (ask your TA for permission before attempting, 5 points)}

In question \#7, you estimated the number of galaxies in the Universe. In
question \#10 you found that a typical galaxy contains 50 billion stars.
Thus, you can now estimate how many stars there are in the Universe.
Recently, some mathematicians have estimated that there are between 7 $\times$
10$^{\rm 19}$ and 7 $\times$ 10$^{\rm 22}$ grains of sand on all of the
Earth's beaches--that is every single beach on every single island and
continent on the Earth. Obviously, this is a difficult estimate to make,
and thus their estimate is quite uncertain. How would you begin
to estimate the number of sand grains on the Earth's beaches? What factors need to be taken into account?\\

 Compare the number of
stars in the Universe, with the number of grains of sand on the planet
Earth. How do they compare? We still do not know the average number of
planets that are found around an average star. It is probably safe to assume 
that 10\% of all stars have at least one planet orbiting them. If so, how many 
planets are there in the Universe? 
%\end{document}
