\section{Introduction}

One of the fundamental tasks of astronomers is figuring out how to
measure distances to astronomical objects. Understanding how far
away objects are is critical to our understanding of the size of
the Universe. It is also important to estimate distances to understand
how brightly different objects are shining; are objects which appear
fainter really putting out less energy or are they just farther away?

In fact, measuring distances to astronomical objects is a difficult
task because astronomical objects are very far away. Almost all
distance estimates in astronomy rely on first measuring the distance
to nearby objects and then using some property of these objects 
to extend distances to further objects.

A powerful method for determining the distance to nearby stars is called
parallax.  Parallax is the angle that something appears to move when
the observer looking at the object changes position.  By observing the
size of this angle and knowing how far the observer has moved, one can
determine the distance to the object.  You can experience how parallax
works by doing the following simple experiment:

Hold your thumb out in front of you at arm's length and look at it
with your left eye closed.  Now look at it with your right eye closed.
As you look at your thumb, alternate which eye you close several times.
You should see your thumb move relative to things in the background.
Your thumb is not moving but your \underline {point of view}
is moving, so your thumb {\it appears} to move.

\begin{itemize}
\item{{\it Goals:}  to discuss the theory and practice of using parallax to 
find the distances to nearby stars, and use it to measure the distance to 
objects in the classroom}
\item{{\it Materials:}  classroom ``ruler'', worksheets, ruler, 
protractor, calculator, small object}
\end{itemize}

\subsection{Parallax in the classroom}

Install the ``classroom ruler'' along one side of the classroom. You
will be measuring motions against this ruler.

Now work in groups: have one person stand at one wall and have the others
stand somewhere in between along the line that goes straight to the other
wall, and hold up a skewer or a pencil at the right height so the observer
(person against the wall) can see it against the background ruler.
The observer should blink his/her eyes and measure the number of lines
on the background ``ruler'' which the object appears to move. \textbf{Note
that you can estimate the motion measurement to a fraction of tick
mark, e.g., your measurement might be 2 1/2 tick marks).}
Do this
for three different distances, with the closest being within a few feet
of the observer, and the furthest being at most half the classroom away.
Leave the object (or a mark) where the object was located each of
the three times, so you can go back later and measure the distances.
Switch places and do it again. Each person should estimate the motion for
each of the three distances. 

How many tick marks did the object move at the closest distance?
\textit{(2 points)}\blank

How many tick marks did the object move at the middle distance?
\textit{(2 points)}\blank

How many tick marks did the object move at the farthest distance?
\textit{(2 points)}\blank

What is your estimate of the \textit{uncertainty} in your measurement of
the apparent motion? For example, do you think your recorded measurements
could be off by half a tick mark? a quarter of a tick mark? You might
wish to make the measurement several times to get some estimate of the
uncertainty. \textit{(2 points)}
\smallspace

The apparent motion of the person against the background ruler is what we
are calling parallax. It is caused by looking at an object from two different
vantage points, in this case, the difference in the location of your two
eyes. Qualitatively, what do you see? As the object gets farther away, is
the apparent motion smaller or larger?
\smallspace

What if the vantage points are further apart? For example, imagine you had
a huge head and your eyes were a foot apart rather than several inches
apart. What would you predict for
the apparent motion? Try the experiment again, this time using the object 
at one of the distances used above, but now measuring the apparent motion 
by using just one eye, but moving your whole head
a few feet from side to side to get more widely separated vantage points.

How many tick marks did the person move from the more widely separated vantage
points? \blank

For an object at a fixed distance, how does the apparent motion change as
you observe from more widely separated vantage points?
\smallspace

\subsection{Measuring distances using parallax}

We have seen that the apparent motion depends on both the distance to an
object and also on the separation of the two vantage points. We can then
turn this around: if we can measure the apparent motion and also the
separation of the two vantage points, we should be able to infer the
distance to an object. This is very handy: it provides a way of measuring
a distance without actually having to go to an object. Since we can't
travel to stars, this provides one of the only ways to measure direct
distances to stars.

We will demonstrate the capability by trying to \textbf{determine the 
distances to the objects you looked at} just based on your measurements
of their apparent motions and a measurement of the separataion of your
two vantage points (your two eyes).

\subsubsection{Angular motion of an object}

How can we measure the apparent motion of an object? As with our
background ruler, we can measure the motion as it appears against a
background object. But what are the appropriate units to use for such a
measurement? Although we can measure how far apart the lines are on our
background ruler, the apparent motion is not really properly measured
in a unit of length; if we had put our ruler further away, the apparent motion
would have been the same, but the number of tick marks it moved would
have been larger.

The apparent motion is really an \textit{angular} motion. As such,
it can be measured in \textit{degrees}, with 360 degrees in a circle.

Figure out the angular separation of the tick marks on the ruler as seen
from the opposite side of the classroom. Do this by putting one eye at
the origin of a protractor and measuring the angle from one end of the
background ruler to the other end of the ruler. You might lay a pencil
from your eye at the origin of the protractor toward each end and use
this to measure the the total angle.  Divide this angle by the total
number of tick marks to figure out the angle for each tick mark.

Number of degrees for the entire background ruler: \blank

Number of tick marks in the whole ruler: \blank

Number of degrees in each tick mark: \blank

Convert your measurements of apparent motion in tick marks above to
angular measurements by multiplying the number of tick marks by the number
of degrees per tick mark:

How many degrees did the object appear to move at the closest distance? 
\textit{(2 points)}\blank

How many degrees did the object appear to move at the middle distance? 
\textit{(2 points)}\blank

How many degrees did the object appear to move at the farthest distance? 
\textit{(2 points)}\blank

Based on your estimate of the uncertainty in the number of tick marks 
each object moved, what is your estimate of the uncertainty in the number
of degrees that each object moved?\textit{(2 points)}
\smallspace

\subsubsection{Distance between the vantage points}

Now you need to measure the distance between the two different vantage
points, in this case, the distance between your two eyes. Have your
partner measure this with a ruler. Since you see out of the pupil part
of your eyes, you want to measure the distance between your two pupils.

What is the distance between your eyes? 
\textit{(2 points)}\blank

\subsubsection{Using parallax measurements to determine the distance to an object}

To determine the distance to an object for which you have a parallax 
measurement, you can construct an imaginary triangle between the two different
vantage points and the object, as shown in Figure \ref{fig:triangle}.

\begin{figure}[h]
\centering
\leavevmode
\epsfysize=0.3\textheight
\epsfbox{PARALLAX/fig1.eps}
\caption{Parallax triangle}
\label{fig:triangle}
\end{figure}

The angles you have measured correspond to the angle $\alpha$ on the
diagram, and the distance between the vantage points corresponds to
the distance $b$ on the diagram. The distance to the object, which is
what you want to figure out, is $d$.

The three quantities are related by a trigonometry function called the
\textit{tangent}. If you split your triangle in half (dotted line), then
the tangent of $(\alpha/2)$ is equal to the quantity $(b/2)/d$:
$$\tan {\left(\alpha\over 2\right)} = {(b/2)\over d}$$
Rearranging the equation gives:
$$ d = {(b/2)\over \tan{\left(\alpha/2\right)}}$$

You can determine the tangent of an angle using your calculator by entering
the angle and then hitting the button marked \textit{tan}. There are
several other units for measuring angles besides degrees (for example, 
radians), so you have to \textbf{make sure that your calculator is set up to use
degrees} for angles before you use the tangent function.

Combine your measurements of angular distances and the distance between
the vantage points to determine the three different distances to your
partner. The units of the distances which you determine will be the same
as the units you used to measure the distance between your eyes; if you
measured that in inches, then the derived distances will be in inches.

Distance when object was at closest distance: 
\textit{(2 points)}\blank

Distance when object was at middle distance: 
\textit{(2 points)}\blank

Distance when object was at farthest distance: 
\textit{(2 points)}\blank

Based on your estimate of the uncertainty in the angular measurements and
also on the uncertainty of your measurement of the separation of your
eyes, estimate the uncertainty in your measurements of the distances to
the objects. To do so, you might wish to redo the calculation allowing
each of your measurements to change by your estimated errors.\textit{(2 points)}
\smallspace

Now go and measure the actual distances to the locations of the objects
using a yardstick, meterstick, or tape measure. How well did the parallax 
distances work? Are the differences between the actual measurements and
your parallax measurements within your estimated errors? If not, can you
think of any reasons why your measurements might have some additional 
error in them?
\textit{(5 points)}

\bigspace

\subsection{Distances to stars using parallax}
\label{sec:solsys}

Because stars are very far away, the parallax motion will be very small.
For example, the nearest star is about $1.9\times 10^{13}$
miles or $1.2\times 10^{18}$ inches away! At such a tremendous distance,
the apparent angular motion is very small. Considering the two vantage
points of your two eyes, the angular motion of the nearest star corresponds
to the apparent diameter of a human hair seen at the distance of the Sun!
This is a truly tiny angle and totally unmeasurable by your eye.

We can improve our situation by using two more widely separated vantage
points. The two points farthest apart we can use from Earth is to use
two opposite points in the Earth's orbit about the Sun.  In other words,
we need to observe a star at two different times separated by six months.
The distance between our two vantage points will then be twice the
distance between the Earth and the Sun; this distance is about 190
million miles. Figure \ref{fig:orbit} shows the idea.

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.3\textheight
\epsfbox{PARALLAX/fig2.eps}
\caption{Parallax Method for Distance to a Star}
\label{fig:orbit}
\end{figure}

Using 190 million miles as the distance $b$, we find that the apparent
angular motion of even the nearest star is only about 0.0005 degrees.
This is also unobservable using your naked eye, which is why we cannot
directly observe parallax by looking at stars with our naked eye. However,
this angle is relatively easy to measure using modern telescopes and
instruments.

\clearpage

\section{Questions}

\begin{enumerate}

\item How does the parallax angle change as an object is moved further away?
Given that you can usually only measure an angular motion to some accuracy,
would it be easier to measure the distance to a nearby star or
a more distant star?  Why?  {\it (5 points)}
\bigspace

\item Relate the experiment you did in lab to the way parallax is used to
measure the distances to nearby stars in astronomy.  Describe the
process an astronomer has to go through in order to determine the
distance to a star using the parallax method.  What do your two eyes 
represent in that experiment?   {\it (7 points)}
\bigspace

\item Imagine that you observe a star field twice one year, separated by six
months and observe the configurations of stars shown in Figure \ref{fig:stars}:
\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.7\textwidth
\epsfbox{PARALLAX/stars.eps}
\caption{Star field seen at two times of year six months apart.}
\label{fig:stars}
\end{figure}

The star marked $P$ appears to move between your two observations because
of parallax.  So you can consider the two pictures to be like our lab
experiment where the left picture is what is seen by one eye and the right
picture what is seen by the other eye. All the stars except star $P$ do
not appear to change position; they correspond to the background ruler in
our lab experiment. 
If the angular distance between stars $A$ and $B$ is
0.5 arcminutes (remember, 60 arcminutes = 1 degree), then how
far away would you estimate that star $P$ is? Proceed by estimating the
amount that star $P$ moves between the two pictures relative to the distance
between stars $A$ and $B$. This gives you the apparent angular motion. You
also know the distance between the two vantage points (which is the Earth
at two opposite sides of its orbit) from the number given in Section
\ref{sec:solsys}. You can then use the parallax equation to estimate
the distance to star $P$.
{\it (12 points)}
\bigspace

\item Imagine that you did the classroom experiment by putting your
partner all the way against the far wall. How big would the apparent
motion be relative to the tick marks? What would you infer about the
distance to your partner? Why do you think this estimate is incorrect?
What can you infer about where the background objects in a parallax 
experiment need to be located?
{\it (10 points)}

\bigspace

\end{enumerate}

\section{Summary}

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

\begin{itemize}
\item{A brief description on the basic principles of
parallax and how astronomers can use parallax to determine the
distance to nearby stars}
\end{itemize}

\noindent
Also think about and answer the following
questions: 
\begin{itemize}
\item{Does the parallax method work for all stars we can see in
our Galaxy and why?}
\item{Why do you think it is important for astronomers
to determine the distances to the stars which they study?}
\end{itemize}

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

%\end{document}
