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%{\bf \Large The Orbit of Mercury}
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\begin{Large}
\section{The Orbit of Mercury}
\end{Large}

\subsection{Introduction}
Of the five planets known since ancient times (Mercury, Venus, Mars, Jupiter,
and Saturn), Mercury is the most difficult to see. In fact, of the 7
billion people on the planet Earth it is likely that fewer than 1,000,000
(0.0002\%) have {\it knowingly} seen the planet Mercury. The reason for this 
is that Mercury orbits very close to the Sun, about one third of the Earth's 
average distance. Therefore it is always located very near the Sun, and can
only be seen for short intervals soon after sunset, or just before sunrise.
It is a testament to how carefully the ancient peoples watched the sky that
Mercury was known at least as far back as 3,000 BC. In Roman mythology Mercury 
was a son of Jupiter, and was the god of trade and commerce. He was also the 
messenger of the gods, being ``fleet of foot'', and commonly depicted as having winged sandals.
Why this god was associated with the planet Mercury is obvious: Mercury
moves very quickly in its orbit around the Sun, and is only visible for
a very short time during each orbit. In fact, Mercury has the shortest
orbital period (``year'') of any of the planets. You will determine Mercury's 
orbital period in this lab. [Note: it is very helpful for this lab
exercise to review Lab \#1, subsection 1.3.]\\

\begin{itemize}
\item{ {\it Goals:} to learn about planetary orbits}
\item{{\it Materials:} a protractor, a straight edge, a pencil and calculator}
\end{itemize}

Mercury and Venus are called ``inferior'' planets because their orbits are
interior to that of the Earth. While the planets Mars, Jupiter, Saturn, Uranus,
Neptune, and Pluto are called ``superior'' planets, as their orbits lie
outside that of the Earth. Because the orbits of Mercury and Venus are smaller
than the Earth's, these planets can never be located very far from the
Sun as seen from the Earth. As discovered by Galileo in 1610 (see Fig. 
\ref{venus}),
the planet Venus shows phases that look just like those of the Moon. Mercury
also shows these same phases. As can be envisioned from Figure 
\ref{venus}, when 
Mercury or Venus
are on the far side of the Sun from Earth (a configuration called ``superior''
conjunction), these two planets are seen as ``full''. Note, however, that
it is almost impossible to see a ``full'' Mercury or Venus because at this
time the planet is very close to, or behind the Sun. When Mercury or Venus
are closest to the Earth, a time when they pass between the Earth and the
Sun (a configuration termed ``inferior'' conjunction), we would see a
``new'' phase. During their new phases, it is also very difficult to see
Mercury or Venus because their illuminated hemispheres are pointed away
from us, and they are again located {\it very} close to the Sun in the 
sky.\\
\begin{figure}[ht]
\centerline{{\includegraphics[width=7cm]{Mercury/venusphase.pdf}}}
\caption{A diagram of the phases of Venus as it orbits around the Sun.
The planet Mercury exhibits the same set of phases as it too is an ``inferior''
planet like Venus.}
\label{venus}
\end{figure}

The best time to see Mercury is near the time of ``greatest elongation''.
At the time of greatest elongation, the planet Mercury has its largest
{\it angular} separation from the Sun as seen from the Earth. There are six
or seven greatest elongations of Mercury each year. At the time of greatest
elongation, Mercury can be located up to 28$^{\circ}$ from the Sun, and
sets (or rises) about two hours after (or before) the Sun. In Figure 
\ref{elong}, we
show a diagram for the greatest elongation of Mercury that occurred on August
14, 2003. In this diagram, we plot the positions of Mercury and the Sun at the 
time of sunset (actually just a few minutes before sunset!). As this diagram
shows, if we started our observations on July 24$^{\rm th}$, Mercury would
be located close to the Sun at sunset. But as the weeks passed, the angle 
between Mercury and the Sun would increase until it reached its maximum 
value on
August 14$^{\rm th}$. After this date, the separation between the Sun and
Mercury quickly decreases as it heads towards inferior conjunction
on September 11$^{\rm th}$. \\

You can see from Figure \ref{elong} that Mercury is
following an orbit around the Sun: it was ``behind'' the Sun (superior
conjunction) on July 5$^{\rm th}$, and quickly races around its orbit
until the time of greatest elongation, and then passes between the Earth and
the Sun on September 11$^{\rm th}$. If we used a telescope and made careful
drawings of Mercury throughout this time, we would see the phases shown
in Figure \ref{phases}. On the first date in Figure \ref{elong} (July 24$^{\rm th}$), Mercury
was still on the far side of the Sun from the Earth, and almost had a full
phase (which it only truly has at the time of superior conjunction). The
disk of Mercury on July 24$^{\rm th}$ is very small because the planet is far 
away from the Earth. As time passes, however, the apparent size of the disk of 
Mercury grows in size, while the illuminated portion of the disk decreases. 
On August 14$^{\rm th}$, Mercury reaches greatest elongation, and the disk is 
half-illuminated. At this time it looks just like the first quarter Moon! As 
it continues to catch up with the Earth, the distance between the two planets 
shrinks, so the apparent size of Mercury continues to grow. As the angular 
separation between Mercury and the Sun shrinks, so does the amount of the 
illuminated hemisphere that we can see. Eventually Mercury becomes a crescent, 
and at inferior conjunction it becomes a ``new'' Mercury.

\begin{figure}[!ht]
\centerline{{\includegraphics[width=10cm]{Mercury/august14elong.pdf}}}
\caption{The eastern elongation of August, 2003. Mercury was at superior
conjunction on July 5$^{\rm th}$, and quickly moved around its orbit increasing
the angular separation between it and the Sun. By July 24$^{\rm th}$, Mercury
could be seen just above the Sun shortly after sunset. As time passed,
the angular separation between the Sun and Mercury increased, reaching
its maximum value on August 14$^{\rm th}$, the time of greatest
Eastern elongation. As Mercury continued in its orbit it comes closer to
the Earth, but the angular separation between it and the Sun shrinks. 
Eventually, on September 11$^{\rm th}$,  the time of inferior conjunction, 
Mercury passed directly between the Earth and the Sun.}
\label{elong}
\end{figure}
\begin{figure}[ht]
\centerline{{\includegraphics[width=9cm]{Mercury/mercuryphasesBW.pdf}}}
\caption{A diagram showing the actual appearance of Mercury during the
August 2003 apparition. As Mercury comes around its orbit from superior
conjunction (where it was ``full''), it is far away from the Earth, so it 
appears small (as on July 
24$^{\rm th}$). As it approaches greatest elongation (August 14$^{\rm th}$)
it gets closer to the Earth, so its apparent size grows, but its phase 
declines to half (like a first quarter moon). Mercury continues to close
its distance with the Earth so it continues to grow in size--but note that
the illuminated portion of its disk shrinks, becoming a thin crescent on August
30$^{\rm th}$. As Mercury passes between the Earth and Sun it is in its
``new'' phase, and is invisible.}
\label{phases}
\end{figure}

\subsubsection{Eastern and Western Elongations} \label{subsec1}
The greatest elongation that occurred on August 14, 2003 was a ``greatest
Eastern elongation''. Why? As you know, the Sun sets in the West each
evening. When Mercury is visible {\it after} sunset it is located to the
East of the Sun. It then sets in the West {\it after} the Sun has set. As you 
can imagine, however, the same type of geometry can occur in the morning
sky. As Mercury passed through inferior conjunction on September 11$^{\rm th}$,
it moved into the morning sky. Its angular separation from the Sun increased
until it reached ``greatest Western elongation'' on September 27$^{\rm th}$,
2003. During this time, the phase of Mercury changed from ``new'' to
``last quarter'' (half). After September 27$^{\rm th}$ the angular
separation between the Sun and Mercury shrinks, as does the apparent size
of the disk of Mercury, as it reverses the sequence shown in Figure 
\ref{phases}.
A diagram showing the geometry of eastern and western elongations is shown
in Figure \ref{elong2}. [Another way of thinking about what each of these means, and an
analogy that might come in useful when you begin plotting the orbit of
Mercury, is to think about where Mercury is relative to the Sun at Noon.
At Noon, the Sun is due south, and when facing the Sun, East is to the left,
and West is to the right. Thus, during an Eastern elongation Mercury is
to the left of the Sun, and during a Western elongation Mercury is to the
right of the Sun (as seen in the Northern Hemisphere).]
\begin{figure}[ht]
\centerline{{\includegraphics[width=12cm]{Mercury/elongdeffig2.pdf}}}
\caption{A diagram showing the geometry of greatest Western elongations
(left side), and greatest Eastern elongations (right side). If you see
Mercury--or any other star or planet--above the Western horizon after sunset,
that object is located to the East of the Sun. The maximum angular separation
between Mercury and the Sun at this time is called the greatest Eastern 
elongation. A greatest Western elongation occurs when Mercury is seen in 
the East {\it before} sunrise.}
\label{elong2}
\end{figure}

\subsubsection{Why Greatest Elongations are Special}
We have just spent a lot of time describing the greatest
elongations of Mercury. We did this because the time of greatest elongation
is very special: it is the only time when we know where an inferior
planet is in its orbit (except in the rare cases where the planet ``transits''
across the face of the Sun!). We show why this is true in the next figure. In
this figure, we have represented the orbits of Mercury and the Earth as
two circles (only about one fourth of the orbits are plotted). We have
also plotted the positions of the Earth, the Sun, and Mercury. At the time
of greatest elongation, the angle between the Earth, Mercury and the Sun
is a right angle. If you were to plot Mercury at some other position in
its orbit, the angle between the Earth, Mercury and the Sun would not be
a right angle. Therefore, the times of greatest elongation are special,
because at this time we know the exact angle between the Earth, Mercury, 
and the Sun. [You can also figure out from this diagram why 
Mercury has only one-half of its disk illuminated (a phase of ``first 
quarter'').]\\

Of course, plotting only one elongation is not sufficient to figure out
the orbit of Mercury--you need to plot many elongations. In today's lab
exercise, you will plot thirteen greatest elongations of Mercury, and 
trace-out its orbit. There are a lot of angles in this lab, so you need to
get comfortable with using a protractor. Your TA will help you figure
this out. But the most critical aspect is to not confuse eastern and
western elongations. Look at Figure \ref{elong3} again. What kind of elongation is this?
Well, as seen from the Earth, Mercury is to the left of the Sun. As
described earlier (in the square brackets at the end of subsection \ref{subsec1}), if 
Mercury is to the left of the Sun, it is an {\it eastern elongation}.\\
\begin{figure}[ht]
\centerline{{\includegraphics[width=9cm]{Mercury/mercuryfig3fromgif.pdf}}}
\caption{A diagram showing the orbital geometry of the Earth and Mercury
during a greatest Eastern elongation. The orbits of the Earth and Mercury are 
the two large circles. The
line of sight to Mercury at the time of greatest elongation is indicated.
Note that at this time the angle between the Earth, Mercury, and the Sun
is a {\it right} angle. The direction of motion of the two planets is
shown by the arrows on the orbits.}
\label{elong3}
\end{figure}

\subsection{The Orbits of Earth and Mercury}
As shown in the previous diagram, both the Earth and Mercury are orbiting the
Sun. That means that every single day they are at a different position in their 
orbits. Before we can begin this lab, we must talk about how we can account
for this motion! A year on Earth, the time it takes the Earth to complete
one orbit around the Sun, is 365 days. If we assume that the Earth's orbit
is a perfect circle, then the Earth moves on that circle by about 1 degree 
per day. Remember that a circle contains 360 degrees (360$^{\circ}$). If
it takes 365 days to go 360$^{\circ}$, the Earth moves 360$^{\circ}$/365 =
0.986 degrees per day ($^{\circ}$/day). For this lab, we will assume that
the Earth moves exactly one degree per day which, you can see, is very close
to the truth. How far does the Earth move in 90 days? 90 degrees! How
about 165 days? \\

\subsubsection{The Data}
In Table \ref{elongdata}, we have listed the thirteen dates of the greatest 
elongations of Mercury, as well as the angle of each greatest elongation. {\bf Note that the
elongations are either East or West!} In the third column, we have listed
something called the Julian date. Over long time intervals, our common 
calendar is very hard to use to figure out how much time has elapsed. For
example, how many days are there between March 13$^{\rm th}$, 2001 and 
December 17$^{\rm th}$ 2004? Remember that 2004 is a leap year! This
is difficult to do in your head. To avoid such calculations, astronomers
have used a calendar that simply counts the days that have passed since
some distant day \#1. The system used by astronomers sets Julian date 1 to
January 1$^{\rm st}$, 4713 BC (an arbitrary date chosen in the sixteenth
century). Using this calendar, March 13$^{\rm th}$, 2001 has a Julian date
of 2451981. While December 17$^{\rm th}$ 2004 has a Julian date of
2453356. Taking the difference of these two numbers (2453356 $-$ 2451981)
we find that there are 1,375 days between March 13$^{\rm th}$, 2001 and 
December 17$^{\rm th}$ 2004. \\

\noindent
{\bf Exercise \#1: Fill-in the Data Table}\\
The fourth and fifth columns of the table are blank, and must be filled-in
by you. The fourth column is the number of days that have elapsed between
elongations in this table (that is, simply subtract the Julian date of
the previous elongation from the {\it following} elongation). We have worked
the first one of these for you as an example. The last column lists how
far the Earth has moved in degrees. This is simply the number of days times
the number 1.0! As the Earth moves one degree per day. (If you wish, instead 
of using 1.0, you could multiply this number by 0.986 to be more accurate.
You will get better results doing it that way.) 
So, if there are 42 days between elongations, the Earth moves 42 degrees in 
its orbit (or 41.4 degrees using the correct value of 0.986 $^{\circ}$/day).
({\bf 10 points})\\

\begin{table}[ht]
\caption{Elongation Data For Mercury}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}\hline
\# &~ ~ Date ~ ~& Elongation Angle & Julian Date & ~ Days ~ & ~Degrees ~\\
\hline
\#1 &Sep. 1, 2002 & 27.2 degrees east ~& 2452518 & $-$ & $-$ \\
\hline
\#2 &Oct. 13, 2002 & 18.1 degrees west ~& 2452560 & 42 & 42 \\
\hline
\#3 &Dec. 26, 2002 & 19.9 degrees east ~& 2452634 &  ~ ~ &  \\
\hline
\#4 &Feb. 4, 2003 & 25.4 degrees west ~& 2452674 &  ~ ~ &  \\
\hline
\#5 & Apr. 16, 2003 & 19.8 degrees east ~& 2452745 & ~ ~ &  \\
\hline
\#6 & Jun. 3, 2003 & 24.4 degrees west ~& 2452793 & ~ ~ &  \\
\hline
\#7 & Aug. 14, 2003 & 27.4 degrees east ~& 2452865 &  ~ ~ &  \\
\hline
\#8 & Sep. 27, 2003 & 17.9 degrees west ~& 2452909 &  ~ ~ &  \\
\hline
\#9 &Dec. 09, 2003 & 20.9 degrees east ~& 2452982 & ~ ~ &  \\
\hline
\#10 &Jan. 17, 2004 & 23.9 degrees west ~& 2453021 & ~ ~ &  \\
\hline
\#11 &Mar. 29, 2004 & 18.8 degrees east ~& 2453093 & ~ ~ &  \\
\hline
\#12 &May 14, 2004 & 26.0 degrees west ~& 2453139 & ~ ~ &  \\
\hline
\#13 &Jul. 27, 2004 & 27.1 degrees east ~& 2453213 & ~ ~ &  \\
\hline
\end{tabular}
\end{center}
\label{elongdata}
\end{table}

\noindent
{\bf Exercise \#2: Plotting your data. }\\
Before we describe the plotting
process, go back and review Figure \ref{elong3}. Unlike that diagram, you do not
know what the orbit of Mercury looks like--this is what you are going to
figure out during this lab! But we do know two things: the first is that the
 Earth's orbit
is nearly a perfect circle, and two, that the Sun sits at the exact center of
this circle. On the next page is a figure containing a large circle
with a dot drawn at the center to represent the Sun. At one position on
the large circle we have put a little ``X'' as a reference point. The
large circle here is meant to represent the Earth's orbit, and the ``X''
is simply a good starting point. \\

To plot the first elongation of Mercury
from our data table, using a pencil, draw a line connecting the X, and the Sun
using a straight edge (ruler or protractor). The first elongation in 
the table (September 1, 2002) is 27.2 degrees East. Using your protractor,
put the ``X'' that marks the Earth's location at the center hole/dot on
your protractor. Looking back to Figure \ref{elong3}, that elongation was also an
{\it eastern} elongation. So, using that diagram as a guide, measure an angle 
of 27.2 degrees on your protractor and put a small mark on the worksheet. Now, 
draw a line from the
Earth's location through this mark just like the ``line of sight'' arrow
in Figure \ref{elong3}. Now, rotate your protractor so that the 90 degree mark
is on this line and towards the position of the Earth, while the reference 
hole/dot
is on the same line. Slide the protractor along the line until the 0$^{\circ}$
(or 180$^{\circ}$) reference line intersects the center of the Sun. Mark this
spot with a dark circle. This is the position of Mercury!\\

This is the procedure that you will use for all of the elongations, so if this 
is confusing to you, have your TA come over and clarify the technique for you
so that you don't get lost and waste time doing this incorrectly.\\

Ok, now things become slightly more difficult---the Earth moves! Looking back
to Figure \ref{elong3}, note the arrows on the orbits of Earth and Mercury. This
is the direction that both planets are moving in their orbits. For the
second elongation, the Earth has moved 42 degrees. We have to locate where
the Earth is in its orbit before we can plot the next elongation. So, now
put the center hole/dot of your protractor on the Sun. Line up the 0/180
degree mark with the first line that connected the Earth and Sun. Measure
an angle of 42 degrees (in the correct direction) and put a small mark.
Draw a line through this mark that intersects the position of the Sun, the
mark, and the orbit of the Earth. Put an X where this line intersects the
Earth's orbit. This is the spot from where you will plot the next 
elongation of Mercury.   \\

Now, repeat the process for plotting this next elongation angle. Note, however,
that this elongation is a western elongation, so that the line of sight arrow 
this time will be to the right of the Sun. It is extremely important to
remember that on eastern elongations the line of sight arrow to Mercury
goes to the left of the Sun, while during western elongations it goes to
the right of the Sun.\\

[Hints: It is helpful to label each one of the X's you place on the Earth's
orbit with the elongation number from Table \ref{elongdata}. This will allow you to
go back and fix any problems you might find. Note that you will have a
large number of lines drawn in this plot by the time you get finished. Use a
sharp
pencil so that you can erase some/all/pieces of these lines to help clean-up
the plot and reduce congestion. You might also find it helpful to simply
put a ``left'' or a ``right'' each time you encounter East and West in
Table \ref{elongdata} to insure you plot your data correctly.]\\

\begin{center}
{\bf \large Now, plot all of the data ({\bf 28 points})!}
\end{center}
\clearpage
\begin{center}
\begin{figure}[!ht]
\centerline{{\includegraphics[width=16.cm]{Mercury/worksheet.pdf}}}
\end{figure}
\end{center}
\clearpage

\noindent
{\bf Exercise \#3: Connecting the dots. }\\
Note that planets
move on smooth, almost circular paths around the Sun. So try to connect
the various positions of Mercury with a smooth arc. Do all of your points
fit on this closed curve? If not, identify the bad points and go back and
see what you did wrong. Correct any bad elongations.\\

1) Is Mercury's orbit circular? Describe its shape. ({\bf 5 points})

\vspace{2 in}
\noindent
{\bf Exercise \#4: Finding the {\it semi-major axis} of Mercury's orbit.}\\
Using a ruler, find the position on Mercury's orbit that is closest to the
Sun (``perihelion'') and mark this spot with an ``X''. Now find the point on 
the orbit of Mercury that is furthest from the Sun (``aphelion'') and mark
it with an ``X''. Draw
a line that goes through the Sun that comes closest to connecting these
two positions--note that it is likely that these two points will not lie on a 
line that intercepts the Sun. {\it Just attempt to draw the best possible line 
connecting these two points that passes through the Sun.}\\

\noindent
2) Measure the length of this line. Astronomers call this line the
major axis of the planet's orbit. Divide the length you have just measured by
two, to get the ``semi-major'' axis of Mercury's orbit:
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_  (mm).
Measure the diameter of the Earth's orbit and divide that number by two
to get the Earth's semi-major axis: \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ (mm).\\

\noindent
Divide the semi-major axis of Mercury by that of the Earth: 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_ AU.  Since the semi-major axis of 
the Earth's orbit is defined to be 
``one astronomical unit'', this ratio tells us the size of Mercury's semi-major
axis in astronomical units (AU). ({\bf 5 points})

\vspace{0.2in}
\noindent
3) As you have probably heard in class, the fact that the orbits of the 
planet's are ellipses, and not circles, was discovered by Kepler in about 1614.
Mercury and Pluto have the most unusual orbits in the solar system in that
they are the most non-circular. Going back to the line you drew that
went through the Sun and that connected the points of perihelion and aphelion, 
measure the lengths of the two line segments:\\
\vspace{0.2in}

\begin{center}
Perihelion  (p) =  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ mm.      Aphelion  (q) = 
\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ mm.
\end{center}
\vspace{0.2in}

\noindent
Astronomers use the term {\it eccentricity} (``e'') to measure how out-of-round
a planet's orbit is, and the eccentricity is defined by the equation:
\vspace{0.3in}

\noindent
e = (q $-$ p)/(p $+$ q) =  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\vspace{0.3in}
\noindent
Plug your values into this equation and calculate the eccentricity of
Mercury's orbit. ({\bf 5 points})\\
\vspace{0.3in}

\noindent
4) The eccentricity for the Earth's orbit is e = 0.017. How does your
value of the eccentricity for Mercury compare to that of the Earth? Does
the fact that we used a circle for the Earth's orbit now seem justifiable?
({\bf 5 points})\\

~
~
\vspace{2.5in}

\noindent
{\bf Exercise \#5: The orbital period of Mercury.}
 Looking at the positions
of Mercury at elongation \#1, and its position at elongation \#2, approximately
how far around the orbit did Mercury move in these 42 days? Estimate
how long you think it would take Mercury to complete one orbit around
the Sun:  \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ days. {\bf (2 points)}\\

\noindent
Using Kepler's laws, we can estimate the orbital period of a planet (for
a review of Kepler's laws, see lab \#5). Kepler's third law
says that the orbital period squared (P$^{\rm 2}$) is proportional to the 
cube of the semi-major axis ($a^{\rm 3}$): P$^{\rm 2}$ $\propto$ $a^{\rm 3}$. This is
a type of equation you might not remember how to solve (if you
have not done so already, review Lab \#1, subsection 1.3). But let's take
it in pieces:\\

\noindent
$a^{\rm 3}$ = $a \times a \times a$ = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_

\vspace{0.25in}
\noindent
Plug-in your value of $a$ for Mercury and find its cube.\\

\noindent
To find the period of Mercury's orbit, we now need to take the square
root of the number you just calculated (see your TA if you do not know
whether your calculator can perform this operation). ({\bf 3 points})\\

\noindent
\beq
P = \sqrt{a^3} = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.
\eeq

\noindent
Now, the number you just calculated probably means nothing to you. But
what you have done is calculate Mercury's orbital period as a fraction of the
Earth's orbital period (that is because we have been working in AU,
a unit that is defined by the Earth-Sun distance). Since the Earth's
orbital period is exactly 365.25 days, find Mercury's orbital period by
multiplying the number you just calculated for Mercury by 365.25:\\

P$_{\rm orb}$(Mercury) = \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ days.

\vspace{0.3in}
\noindent
5) How does the orbital period you just calculated using Kepler's laws
compare to the one you estimated from your plot at the beginning of this 
exercise? ({\bf 2 points})
\clearpage

\subsection{Summary (35 points)}

Before you leave lab, your TA will give you the real orbital period of
Mercury, as well as its true semi-major axis (in AU) and its orbital eccentricity.

\begin{itemize}
\item{Compare the precisely known values for Mercury's orbit with the ones you 
derived.  How well did you do?}
\item{What would be required to enable you to do a better job?}
\item{Describe how you might go about making the observations on your
own so that you could create a data table like the one in this lab. Do you 
think this could be done with just the naked eye and some sort of instrument 
that measured angular separation? What else might be necessary?}
\end{itemize}

\subsection{Possible Quiz Questions}

\begin{itemize}
\item What does the term ``inferior planet'' mean?
\item What is meant by elongation angle?
\item Why do Mercury and Venus show phases like the Moon?
\end{itemize}

\subsection{Extra Credit (ask your TA for permission before attempting, 5
points)}

In this lab you have measured three of the five quantities that
completely define a planet's orbit. The other two quantities are the orbital 
inclination,
and the longitude of perihelion. Determining the orbital inclination,
the tilt of the plane of Mercury's orbit with respect to the Earth's orbit, is
not possible using the data in this lab. But it is possible to determine
the longitude of perihelion. Astronomers define the zero point of solar
system longitude as the point in the Earth's orbit at the time of the
Vernal Equinox (the beginning of Spring in the northern hemisphere).
In 2004, the Vernal Equinox occurred on March 20. If you notice, one of
the elongations in the table (\#11) occurs close to this date. Thus, you
can figure out the true location of the Vernal Equinox by moving back
from position \#11 by the right number of degrees. The longitude of
Mercury's perihelion is just the angle {measured counterclockwise} from
the Earth's vernal equinox. You should find that your angle is larger than
180 degrees. Subtract off 180 degrees. How does your value compare with
the precise value of 77$^{\circ}$? 
\clearpage
~ \\
%\end{document}
