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\section{Introduction}

The origin of the science of Astronomy owes much to the need of ancient 
peoples to have a practical system that allowed them to predict the seasons. 
It is critical to plant your crops at the right time of the year---too early 
and the seeds may not germinate because it is too cold, or there is insufficient
moisture. Plant too late and it may become too hot and dry for a sensitive 
seedling to survive. In ancient Egypt, they needed to wait for the Nile to 
flood. The Nile river would flood every July, once the rains began to fall in 
Central Africa. 

Thus, the need to keep track of the annual cycle arose with the development of
agriculture, and this required an understanding of the motion of objects in
the sky. The first devices used to keep track of the seasons were large 
stone structures (such as Stonehenge) that used the positions of the rising Sun
or Moon to forecast the coming seasons. The first recognizable calendars 
that we know about were developed in Egypt, and appear to date from about 4,200
BC. Of course, all a calendar does is let you know what time of year it is---it 
does not provide you with an understanding of {\it why} the seasons occur!
The ancient people had a variety of models for why seasons occurred, but thought
that everything, including the Sun and stars, orbited around the Earth. Today, 
you will learn the real reason {\it why} there are seasons.

\begin{itemize}
\item{ {\it Goals:} To learn why the Earth has seasons.}
\item{{\it Materials:} a meter stick, a mounted globe, an elevation 
angle apparatus, string, a halogen lamp, and a few other items}
\end{itemize}

\section{The Seasons}

Before we begin today's lab, let us first talk about the seasons. In New Mexico
we have rather mild Winters, and hot Summers. In the northern parts of the 
United States, however, the winters are much colder. In Hawaii, there is
very little difference between Winter and Summer. As you are also aware, 
during the Winter there are fewer
hours of daylight than in the Summer. In Table \ref{table:tempdata} we have 
listed seasonal data for various locations around the world. Included in this 
table are the average January and July maximum temperatures, the latitude of
each city, and the length of the daylight hours in January and July. We will
use this table in Exercise \#2.

\begin{table}[ht]
\centering
\caption{{\bf Season Data for Select Cities}}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
City & Latitude & January Ave. & July Ave. & January & July \\
~ ~  & (Degrees)& Max. Temp. & Max. Temp. & Daylight & Daylight  \\
~ ~  & ~ ~ & ~ ~ & ~ ~ & Hours & Hours \\
\hline
Fairbanks, AK & 64.8N  & -2 & 72 & 3.7 & 21.8\\
\hline
Minneapolis, MN & 45.0N & 22 & 83 & 9.0 & 15.7\\
\hline
Las Cruces, NM & 32.5N & 57 & 96 & 10.1 & 14.2\\
\hline
Honolulu, HI & 21.3N & 80 & 88 & 11.3 & 13.6 \\
\hline
Quito, Ecuador& 0.0 & 77 & 77 & 12.0 & 12.0 \\
\hline
Apia, Samoa& 13.8S & 80 & 78 &  11.1 & 12.7 \\
\hline
Sydney, Australia & 33.9S & 78 & 61 & 14.3 & 10.3\\
\hline
Ushuaia, Argentina & 54.6S & 57 & 39 & 17.3 & 7.4\\
\hline
\end{tabular}
\label{table:tempdata}
\end{table}

In Table \ref{table:tempdata}, the ``N'' following the latitude means the city 
is in the northern hemisphere of the Earth (as is all of the United States and 
Europe) and thus $North$ of the equator. An ``S'' following the latitude means 
that it is in the southern hemisphere, $South$ of the Earth's equator. What do 
you think the latitude of Quito, Ecuador (0.0$^{\rm o}$) means? Yes, it is 
right on the equator. Remember, latitude runs from 0.0$^{\rm o}$ at the equator
to $\pm$90$^{\rm o}$ at the poles. If north of the equator, we say the latitude 
is XX degrees north (or sometimes ``+XX degrees''), and if south of the equator
we say XX degrees south (or ``$-$XX degrees''). We will use these terms shortly.

Now, if you were to walk into the Mesilla Valley Mall and ask a random stranger 
``why do we have seasons?'', the most common answer you would get is ``because 
we are closer to the Sun during Summer, and further from the Sun in Winter''. 
This answer suggests that the general public (and most of your classmates) 
correctly
understand that the Earth orbits the Sun in such a way that at some times of 
the year it is closer to the Sun than at other times of the year. As you have 
(or will) learn in your lecture class, the orbits of all planets around the Sun 
are ellipses. As shown in Figure \ref{fig:ellipse5} an ellipse is sort of like 
a circle that has been squashed in one direction. For most of the planets,
however, the orbits are only very slightly elliptical, and closely approximate
circles. But let us explore this idea that the distance from the Sun causes
the seasons.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.16\textheight
\epsfbox{Seasons/ellipse5.eps}
\caption{An ellipse with the two ``foci'' identified. The Sun sits at one
focus, while the other focus is empty. The Earth follows an elliptical orbit
around the Sun, but not nearly as exaggerated as that shown here!}
\label{fig:ellipse5}
\end{figure}

{\bf Exercise \#1.}  In Figure \ref{fig:ellipse5}, we show the locations
of the two ``foci'' of an ellipse (foci is the plural form of focus). We will 
ignore the mathematical details of what foci are for now, and simply note that 
the Sun sits at one focus, while the other focus is empty (see the Kepler Law 
lab for more information if you are interested). A planet
orbits around the Sun in an elliptical orbit. So, there are times when the 
Earth is closest to the Sun (``perihelion''), and times when it is furthest
(``aphelion''). When closest to the Sun, at perihelion, the distance from the 
Earth to the Sun is 147,056,800 km (``147 million kilometers''). At aphelion,
the distance from the Earth to the Sun is 152,143,200 km (152 million km).

With the meter stick handy, we are going to examine these distances.
Obviously, our classroom is not big enough to use kilometers or even meters
so, like a road map, we will have to use a reduced scale: 1 cm = 1 million km. 
Now, stick a piece of tape on the table and put a mark 
on it to set the starting point (the location of the Sun!). Carefully measure 
out the two distances (along the same direction) and stick down two more pieces of tape, one at the 
perihelion distance, one at the aphelion distance (put small dots/marks on the
tape so you can easily see them). \\

1) Do you think this change in distance is big enough to cause the seasons? 
Explain your logic. ({\bf 3 points})\\

\clearpage
2) Take the ratio of the aphelion to perihelion distances: \underline{\hskip 1.0in{}}. ({\bf 1 point}) \\

\vspace{0.15in}

Given that {\it we know} objects appear bigger when we are closer to them,
let's take a look at the two pictures of the Sun you were given as part of the
materials for this lab. One image was taken on January 23$^{\rm rd}$, 1992, and 
one was taken on the 21$^{\rm st}$ of July 1992 (as the ``date stamps'' on the 
images show). Using a ruler, {\it carefully} measure the diameter of the Sun in 
each image:\\

Sun diameter in January image = \underline{\hskip 1.0in{}} mm.\\

Sun diameter in July image = \underline{\hskip 1.0in{}} mm.\\

3) Take the ratio of bigger diameter / smaller diameter, this = \underline{\hskip 1.0in{}}. ({\bf 1 point})\\

4) How does this ratio compare to the ratio you calculated in question \#2? ({\bf 2 points})\\

\vspace{1.0in}

5) So, since an object appears bigger when we get closer to it, when is the
Earth closest to the Sun? ({\bf 2 points})\\

\vspace{0.2in}

6) At that time of year, what season is it in Las Cruces? What do you conclude
about the statement ``the seasons are caused by the changing distance between
the Earth and the Sun''? ({\bf 4 points})

\vspace{2.0in}

{\bf Exercise \#2.} Characterizing the nature of the seasons at different
locations. For this exercise, we are going to be exclusively using the data
contained in Table \ref{table:tempdata}. First, let's look at Las Cruces.
Note that here in Las Cruces, our latitude is +32.5$^{\rm o}$. That is we
are about one third of the way from the equator to the pole. In January our
average high temperature is 57$^{\rm o}$F, and in July it is 96$^{\rm o}$F.
It is hotter in Summer than in Winter (duh!). Note that there are about 10 hours
of daylight in January, and about 14 hours of daylight in July. 

7) Thus, for
Las Cruces, the Sun is ``up'' longer in July than in January.
Is the same thing true for all cities with northern latitudes?
~ ~ Yes ~ or ~ No~ ? ~ ({\bf 1 point})\\

\vspace{0.1in}

Ok, let's compare {\it Las Cruces with Fairbanks, Alaska}. Answer these questions
by filling in the blanks:\\

\vspace{0.1in}

8) Fairbanks is \underline{\hskip 1.9in{}} the North Pole than Las Cruces. ({\bf 1 point})\\

\vspace{0.1in}

9) In January, there are more daylight hours in  \underline{\hskip 1.9in{}}. ({\bf 1 point})\\

\vspace{0.1in}

10) In July, there are more daylight hours in \underline{\hskip 1.9in{}}. ({\bf 1 point})\\

\vspace{0.1in}

Now let's compare {\it Las Cruces with Sydney, Australia}. Answer these 
questions by filling in the blanks:\\

\vspace{0.1in}

11) While the latitudes of Las Cruces and Sydney are similar, Las Cruces is 
\underline{\hskip 1.0in{}} of the Equator, and Sydney is \underline{\hskip 
1.0in{}} of the Equator. ({\bf 2 points})\\

\vspace{0.1in}

12) In January, there are more daylight hours in  \underline{\hskip 1.9in{}}. 
({\bf 1 point})\\

\vspace{0.1in}

13) In July, there are more daylight hours in \underline{\hskip 1.9in{}}. ({\bf 1 point})\\

\vspace{0.1in}

14) {\bf Summarizing:} During the Wintertime (January) in both Las Cruces and 
Fairbanks there are fewer daylight hours, {\it and} it is colder. During 
July, it is warmer in both Fairbanks and Las Cruces, {\it and} 
there are more daylight hours. Is this also true for Sydney?: \underline{\hskip 1.0in{}}. ({\bf 1 point})\\

\vspace{0.1in}

15) In fact, it is Wintertime in Sydney during \underline{\hskip 1.0in{}}, and
Summertime during \underline{\hskip 1.0in{}}. ({\bf 2 points})\\

\vspace{0.1in}

16) From Table \ref{table:tempdata}, I conclude that the times of the seasons 
in the Northern 
hemisphere are exactly \underline{\hskip 1.2in{}} to those in the Southern
hemisphere. ({\bf 1 point})\\

From Exercise \#2 we learned a few simple truths, but ones that maybe you have
never thought about. As you move away from the equator (either to the north or 
to the south) there are several general trends. The first is that as you go
closer to the poles it is {\it generally} cooler at all times during the year.
The second is that as you get closer to the poles, the amount of daylight
during the Winter decreases, but the reverse is true in the Summer.

The first of these is not always true because the local climate can be moderated
by the proximity to a large body of water, or depend on the local elevation. 
For example, Sydney is milder than Las Cruces, even though they have similar 
latitudes: Sydney is on the eastern coast of Australia (South Pacific ocean) 
and has a climate like that of San Diego, California (which has a similar 
latitude and is on the coast of the North Pacific). Quito, Ecuador has a mild 
climate even though it sits right on the equator due to its high elevation--it 
is more than 9,000 feet above sea level, similar to the elevation of
Cloudcroft, New Mexico.

The second conclusion (amount of daylight) is always true---as you get closer
and closer to the poles, the amount of daylight during the Winter decreases, 
while the amount of daylight during the Summer increases. In fact, for all 
latitudes north of 66.5$^{\rm o}$, the Summer Sun is up all day (24 hrs of
daylight, the so called ``land of the midnight Sun'') for at least one day
each year, while in the Winter there are times when the Sun never rises! 
66.5$^{\rm o}$ is a special latitude, and is given the name ``Arctic Circle''. 
Note that Fairbanks is very close to the Arctic Circle,
and the Sun is up for just a few hours during the Winter, but is up for nearly
22 hours during the Summer! The same is true for the southern hemisphere: all latitudes
south of $-$66.5$^{\rm o}$ experience days with 24 hours of daylight in the 
Summer, and 24 hours of darkness in the Winter. $-$66.5$^{\rm o}$ is called the
``Antarctic Circle''. But note that the seasons in the Southern Hemisphere are
exactly opposite to those in the North. During Northern Winter, the North Pole
experiences 24 hours of darkness, but the South Pole has 24 hours of daylight.

\section{The Spinning, Revolving Earth}

It is clear from the preceding section that your latitude determines both the 
annual variation in the amount of daylight, and the time of the year when you 
experience Spring, Summer, Autumn and Winter. To truly understand why this
occurs requires us to construct a model. One of the key insights to the
nature of the motion of the Earth is shown in the long exposure 
photographs of the nighttime sky on the next two pages.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.86\textheight
\epsfbox{Seasons/circumpole2.eps}
\caption{Pointing a camera at the North Star (Polaris, the bright dot near
the center) and exposing for about one hour, the stars appear to move in 
little 
arcs. The center of rotation is called the ``North Celestial Pole", and Polaris
is very close to this position. The dotted/dashed trails in this photograph
are the blinking lights of airplanes that passed through the sky during the 
exposure.}
\label{fig:circumpole2}
\end{figure}

\begin{figure}[ht!]
\centering
\leavevmode
\epsfysize=0.86\textheight
\epsfbox{Seasons/startrails_vienna.eps}
\caption{Here is a composite of many different exposures (each about 
one hour in length) of the night sky over Vienna, Austria taken throughout the
year (all four seasons). The images have been composited using a software 
package like Photoshop to demonstrate what would be possible if it stayed dark 
for 24 hrs, and you could actually obtain a 24 hour exposure (which can only 
be truly done north of the Arctic circle). Polaris is the the smallest circle 
at the very center.}
\label{fig:vienna}
\end{figure}

What is going on in these photos? The easiest explanation is that
the Earth is spinning, and as you keep your camera shutter open, the stars 
appear to move in ``orbits'' around the North Pole. You can duplicate this
motion by sitting in a chair that is spinning---the objects in the room appear
to move in circles around you. The further they are from the 
``axis of rotation'', the bigger arcs they make, and the faster they move. An 
object straight above you, exactly on the axis of rotation of the chair, does 
not move. As apparent in Figure \ref{fig:vienna}, the ``North Star'' Polaris is 
not perfectly on the axis of rotation at the North Celestial Pole, but it is 
very close (the fact that there is a bright star near the pole is just random 
chance). Polaris has been used as a navigational aid for centuries, as it 
allows you to determine the direction of North.

As the second photograph shows, the direction of the spin axis of the Earth 
does not change during the year---it stays pointed in the same direction 
{\it all} of the time!  If the Earth's spin axis moved, the stars would not 
make perfect circular arcs, but would wander around in whatever pattern was 
being executed by the Earth's axis. 

Now, as shown back in Figure \ref{fig:ellipse5}, we said the Earth orbits
(``revolves'' around) the Sun on an ellipse. We could discuss the evidence for 
this, but to keep this lab brief, we will just assume this fact. So, now we 
have two motions: the spinning and revolving of the Earth. It is the 
combination of these that actually give rise to the seasons, as you will find
out in the next exercise.

{\bf Exercise \#3:} In this part of the lab, we will be using the mounted
globes, a piece of string, a ruler, and the halogen desk lamp. 
{\bf Warning: while the globe used here is made of fairly inexpensive parts, it
is very time consuming to make. Please be careful with your globe, as the 
paint can be easily damaged.}  Make sure that the piece of string you have 
is long enough to go slightly more than halfway around the globe at the 
equator--if your string is not that long, ask your TA for a longer piece of 
string.  As you may have guessed, this globe is a model of the Earth.
The spin axis of the Earth is actually tilted with respect to the plane of its 
orbit by 23.5$^{\rm o}$. 

Set up the experiment in the following way. Place the halogen lamp at one end
of the table (shining towards the closest wall so as to not affect your
classmates), and set the globe at a distance of 1.5 meters from the lamp. After 
your TA has dimmed the classroom lights, turn on the halogen lamp to the 
highest setting ({\it if} there is a dim, and a bright setting--some lights only
have one brightness setting). Note these lamps get 
very hot, so be careful. For this lab, we will define the top of the globe as 
the Northern hemisphere, and the bottom as the Southern hemisphere.

For the first experiment, arrange the globe so the tilted axis of the ``Earth'' is 
pointed perpendicular (or at a ``right'' angle = 90$^{\circ}$) to the 
direction of the ``Sun''. Use your best judgement. Now adjust the height of the 
desk lamp so that the light bulb in the lamp is at the same approximate height as 
the equator.

There are several colored lines on the globe that form circles which are 
concentric with the axis, and these correspond to certain latitudes.
The red line is the equator, the black line is 45$^{\rm o}$ North, while the
two blue lines are the Arctic (top) and Antarctic (bottom) circles.

{\bf Experiment \#1:} Note that there is an illuminated half of the globe, and 
a dark half of the globe. The line that separates the two is called the 
``terminator''. It is the location of sunrise or sunset. Using the piece of 
string, we want to measure the length of each arc that is in ``daylight'' and 
the length that is in ``night''. This is kind of tricky, and requires a bit of 
judgement as to exactly where the terminator is located. So make sure you have 
a helper to help keep the string {\it exactly} on the line of constant latitude,
and get the advice of your lab partners of where the terminator is (it is
probably best to do this more than once). Fill in the following table ({\bf 4 points}):

\begin{table}[ht]
\centering
\caption{Position \#1: Equinox Data Table}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Length of Daylight Arc} & {\bf Length of Nightime Arc} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:SEQ}
\end{table}

As you know, the Earth rotates once every 24 hours (= 1 Day). Each of
the lines of constant latitude represents a full circle that contains 
360$^{\rm o}$. But note that these circles get smaller in radius as you move away
from the equator. The circumference of the Earth at the equator is 40,075 km (or
24,901 miles). At a latitude of 45$^{\rm o}$, the circle of constant
latitude has a circumference of 28,333 km. At the arctic circles, the circle has
a circumference of only 15,979 km. This is simply due to our use of two 
coordinates (longitude and latitude) to define a location on a sphere. 

Since the Earth is a solid body, all of the  points on Earth rotate once
every 24 hours. Therefore, the sum of the daytime and nighttime arcs you measured
equals 24 hours! So, fill in the following table ({\bf 2 points}):
\clearpage
\begin{table}[ht]
\centering
\caption{Position \#1: Length of Night and Day}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Daylight Hours} & {\bf Nighttime Hours} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:equinox2}
\end{table}

18) The caption for Table \ref{table:SEQ} was ``Equinox data''. The word
Equinox means ``equal nights'', as the length of the nighttime is the
same as the daytime. While your numbers in Table \ref{table:equinox2} may not 
be exactly perfect, what do you conclude about the length of the nights and 
days for {\it all} latitudes on Earth in this experiment? Is this result 
consistent with the term Equinox?  ({\bf 3 points})\\
\vspace{0.7in}
 
{\bf Experiment \#2:} Now we are going to re-orient the globe so that the (top) 
polar axis  points {\it exactly away} from the Sun and repeat the process 
of Experiment \#1. Fill in the following two tables ({\bf 4 points}):

\begin{table}[ht]
\centering
\caption{Position \#2: Solstice Data Table}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Length of Daylight Arc} & {\bf Length of Nightime Arc} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:Solstice1}
\end{table}

\vspace{0.2in}

\begin{table}[ht]
\centering
\caption{Position \#2: Length of Night and Day}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Daylight Hours} & {\bf Nighttime Hours} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:sol1hours}
\end{table}

19) Compare your results in Table \ref{table:sol1hours} for +45$^{\rm o}$ 
latitude with those for Minneapolis in Table \ref{table:tempdata}. Since 
Minneapolis is at a latitude of +45$^{\rm o}$, what season does this orientation
of the globe correspond to? ({\bf 2 points})\\

\vspace{0.5in}

20) What about near the poles? In this orientation what is the length of the 
nighttime at the North pole, and what is the length of the daytime at the South 
pole? Is this consistent with the trends in Table \ref{table:tempdata}, such
as what is happening at Fairbanks or in Ushuaia? ({\bf 4 points})\\

\vspace{1.5in}

{\bf Experiment \#3:} Now we are going to approximate the Earth-Sun orientation 
six months after that in Experiment \#2. To do this correctly, the globe and
the lamp should now switch locations. Go ahead and do this if this lab is
confusing you---or you can simply rotate the globe apparatus by 180$^{\rm o}$
so that the North polar axis is tilted exactly {\it towards} the Sun. Try to 
get a good alignment by looking at the shadow of the wooden axis on
the globe. Since this is six months later, it easy to guess what season this is,
but let's prove it! Complete the following two tables ({\bf 4 points}):

\begin{table}[ht]
\centering
\caption{Position \#3: Solstice Data Table}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Length of Daylight Arc} & {\bf Length of Nightime Arc} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:Solstice2}
\end{table}

\vspace{0.2in}

\begin{table}[ht]
\centering
\caption{Position \#3: Length of Night and Day}
\begin{tabular}{|c|c|c|}
\hline
{\bf Latitude} & {\bf Daylight Hours} & {\bf Nighttime Hours} \\
\hline
Equator & & \\
\hline
45$^{\rm o}$N & &\\
\hline
Arctic Circle & & \\
\hline
Antarctic Circle &  & \\
\hline
\end{tabular}
\label{table:sol2hours}
\end{table}

21) As in question \#19, compare the results found here for the length of daytime
and nighttime for the +45$^{\rm o}$ degree latitude with that for Minneapolis. 
What season does this appear to be? ({\bf 2 points})\\

\vspace{0.5in}

22) What about near the poles? In this orientation, how long is the daylight
at the North pole, and what is the length of the nighttime at the South
pole? Is this consistent with the trends in Table \ref{table:tempdata}, such
as what is happening at Fairbanks or in Ushuaia? ({\bf 2 points})\\

\vspace{0.9in}

23) Using your results for all three positions (Experiments \#1, \#2, and \#3)
can you explain what is happening at the Equator? Does the data for Quito in 
Table \ref{table:tempdata} make sense?  Why? Explain. ({\bf 3 points})\\

\vspace{2.5in}


{\bf We now have discovered the driver for the seasons:} the Earth spins on 
an axis that is inclined to the plane of its orbit (as shown in Figure 
\ref{fig:earthorbit}). {\it But the spin axis always  points to the same place
in the sky} (towards Polaris).
Thus, as the Earth orbits the Sun, the amount of sunlight seen at a particular
latitude varies: the amount of daylight and nighttime hours change with 
the seasons. In Northern Hemisphere Summer (approximately June 21$^{\rm st}$) 
there are more daylight hours; at the start of the Autumn ($\sim$ Sept. 
20$^{\rm th}$) and
Spring ($\sim$ Mar. 21$^{\rm st}$), the days are equal to the nights. In the
Winter (approximately Dec. 21$^{\rm st}$) the nights are long, and the days
are short. We have also discovered that the seasons in the
Northern and Southern hemispheres are exactly opposite. If it is Winter in
Las Cruces, it is Summer in Sydney (and vice versa). This was clearly 
demonstrated in our experiments and is shown in Figure \ref{fig:earthorbit}.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.19\textheight
\epsfbox{Seasons/earthorbit.eps}
\caption{The Earth's spin axis always  points to one spot in the sky, {\it and}
it is tilted by 23.5$^{\rm o}$ to its orbit. Thus, as the Earth orbits the
Sun, the illumination changes with latitude: sometimes the North Pole is
bathed in 24 hours of daylight, and sometimes in 24 hours of night. The 
exact opposite is occurring in the Southern Hemisphere.}
\label{fig:earthorbit}
\end{figure}

The length of the daylight hours is one reason
why it is hotter in Summer than in Winter: the longer the Sun is above the
horizon the more it can heat the air, the land and the seas. But this is not
the whole story. At the North Pole, where there is constant daylight during 
the Summer, the temperature barely rises above freezing! Why? We will discover 
the reason for this now.

\section{Elevation Angle and the Concentration of Sunlight}

We have found out part of the answer to why it is warmer in summer than
in winter: the length of the day is longer in summer. But this is only part
of the story--you would think that with days that are 22 hours long during the 
summer, it would be hot in Alaska and Canada during the summer, but it is not. 
The other effect caused by Earth's tilted spin axis is the changing height that 
the noontime Sun attains during the various seasons. Before we discuss why this happens (as it takes 
quite a lot of words to describe it correctly), we want to explore what happens 
when the Sun is higher in the sky. First, we need to define two new terms:
``altitude'', or ``elevation angle''. As shown in the diagram in
Fig. \ref{fig:altitude}.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.19\textheight
\epsfbox{Seasons/altitude.ps}
\caption{Altitude (``Alt'') is simply the angle between the horizon, and an
object in the sky. The smallest this angle can be is 0$^{\circ}$, and the 
maximum altitude angle is 90$^{\circ}$. Altitude is interchangeably known as 
elevation.}
\label{fig:altitude}
\end{figure}

The Sun is highest in the sky at noon everyday. But how high is it? This,
of course, depends on both your latitude and the time of year. For Las Cruces, 
the Sun has an altitude of 81$^{\circ}$ on June 21$^{\rm st}$. On both
March 21$^{\rm st}$ and September 20$^{\rm th}$, the altitude of the Sun
at noon is 57.5$^{\circ}$. On December 21$^{\rm st}$ its altitude is only
34$^{\circ}$. Thus, the Sun is almost straight overhead at noon during near
the Summer Solstice, but very low during the Winter Solstice. What difference
can this possibly make? We now explore this using the other apparatus,
the elevation angle device, that accompanies this lab (the one with the 
protractor and flashlight).

\noindent
{\bf Exercise \#4:} Using the elevation angle apparatus, we now want to
measure what happens when the Sun is at a higher or lower elevation angle.
We mimic this by using a flashlight mounted on an arm that allows you to move
it to just about any elevation angle. It is difficult to exactly model
the Sun using a flashlight, as the light source is not perfectly uniform. But 
here we do as well as we can. Play around with the device. Turn on the 
flashlight and move the arm to lower and higher angles. How does the
illumination pattern change? Does the illuminated pattern appear to change
in brightness as you change angles? Explain. ({\bf 2 points})\\

\vspace{2in}

Ok, now we are ready to begin. Take a blank sheet of white paper and tape it to
the base so we have a more reflective surface. Now arrange the apparatus
so the elevation angle is 90$^{\circ}$. The illuminated spot should look 
circular. Measure the diameter of this circle using a ruler.

\vspace{0.1in}

The diameter of the illuminated circle is \underline{\hskip 1.5in{}} cm.\\

\vspace{0.1in}

Do you remember how to calculate the area of a circle? Does the formula
$\pi$R$^{\rm 2}$ ring a bell?

\vspace{0.1in}

The area of the circle of light at an elevation angle of 90$^{\circ}$ is
\underline{\hskip 1.45in{}} cm$^{\rm 2}$. ({\bf 1 point})\\

\vspace{0.1in}

Now, as you should have noticed at the beginning of this exercise, as
you move the flashlight to lower and lower elevations, the circle changes to
an ellipse. Now adjust the elevation angle to be 45$^{\circ}$. Ok, time to
introduce you to two new terms: the major axis and minor axis
of an ellipse. Both are shown in Fig. \ref{fig:ellipse2}. The minor
axis is the smallest diameter, while the major axis is the longest diameter
of an ellipse.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.16\textheight
\epsfbox{Seasons/ellipse.eps}
\caption{An ellipse with the major and minor axes defined.}
\label{fig:ellipse2}
\end{figure}

Ok, now measure the lengths of the major (``$a$'') and minor (``$b$'') axes at 
45$^{\circ}$:

\vspace{0.1in}

The major axis has a length of $a$  = \underline{\hskip 1.0in{}} cm, while the minor axis has a \\

\vspace{0.03in}
length of $b$ =  \underline{\hskip 1.0in{}} cm.

\vspace{0.1in}

The area of an ellipse is simply ($\pi$ $\times$ $a$ $\times$ $b$)/4. So, the area
of

\vspace{0.1in}

the ellipse at an elevation angle of 45$^{\circ}$ is:
 \underline{\hskip 1.5in{}} cm$^{\rm 2}$ ({\bf 1 point}).

\vspace{0.1in}

So, why are we making you measure these areas? Note that the black tube
restricts the amount of light coming from the flashlight into a cylinder.
Thus, there is only a certain amount of light allowed to come out and hit
the paper. Let's say there are ``one hundred units of light'' emitted by the 
flashlight. Now let's convert this to how many units of light hit each
square centimeter at angles of 90$^{\circ}$ and 45$^{\circ}$.

\vspace{0.1in}

At 90$^{\circ}$, the amount of light per centimeter is 100 divided by the
area of circle\\

\vspace{0.03in}
= \underline{\hskip 1.5in{}} units of light per cm$^{\rm 2}$ ({\bf 1 point}).

\vspace{0.1in}

At 45$^{\circ}$, the amount of light per centimeter is 100 divided by the
area of the ellipse\\

\vspace{0.03in}
= \underline{\hskip 1.5in{}} units of light per cm$^{\rm 2}$ ({\bf 1 point}).

\vspace{0.1in}

Since light is a form of energy, at which elevation angle is there more
energy per square centimeter? Since the Sun is our source of light, what
happens when the Sun is higher in the sky? Is its energy more concentrated,
or less concentrated? How about when it is low in the sky? Can you tell this
by looking at how bright the ellipse appears versus the circle? ({\bf 4 points})

\vspace{2.6in}


As we have noted, the Sun never is very high in the arctic regions of the
Earth. In fact, at the poles, the highest elevation angle the Sun can have is 23.5$^{\circ}$.
Thus, the light from the Sun is spread out, and cannot heat the ground
as much as it can at a point closer to the equator. That's why it is always
colder at the Earth's poles than elsewhere on the planet.

You are now finished with the in-class portion of this lab. To understand
why the Sun appears at different heights at different times of the year
takes a little explanation (and the following can be read at home unless
you want to discuss it with your TA). 
Let's go back and take a look at Fig. \ref{fig:vienna}.
Note that Polaris, the North Star, barely moves over the course of a night
or over the year---it is always visible. If you had a telescope and could
point it accurately, you could see Polaris during the daytime too. Polaris never
sets for people in the Northern Hemisphere since it is located very close to the
spin axis of the Earth. Note that as we move away from Polaris the circles
traced by other stars get bigger and bigger. But all of the stars shown in 
this photo are {\it always} visible---they never set. We call these stars 
``circumpolar''. For every latitude on Earth, there is a set of circumpolar 
stars (the number decreases as you head towards the equator).

Now let us add a new term to our vocabulary: the ``Celestial Equator''. The 
Celestial Equator is the projection of the Earth's Equator onto the sky. It is 
a great circle that spans the night sky that is directly overhead for people
who live on the Equator. As you have now learned, the lengths of the days
and nights at the equator are nearly always the same: 12 hours. But we
have also learned that during the Equinoxes, the lengths of the days and
the nights {\it everywhere} on Earth are also twelve hours. Why? Because
during the equinoxes, the Sun is {\it on the Celestial Equator}. That means
it is straight overhead (at noon) for people who live in Quito, Ecuador 
(and everywhere else on the equator). Any object that is on the Celestial
Equator is visible for 12 hours per day from everywhere on Earth. To try
to understand this, take a look at Fig. \ref{fig:celestial}. In this figure
is shown the celestial geometry explicitly showing that the Celestial
Equator is simply the Earth's equator projected onto the sky (left hand 
diagram). But the Earth is large, and to us, it appears flat. Since the
objects in the sky are very far away, we get a view like that shown in
the right hand diagram: we see one hemisphere of the sky, and the stars,
planets, Sun and Moon rise in the east, and set in the west. But note that the 
Celestial Equator exactly intersects East and West. Only objects located
on the Celestial Equator rise exactly due East, and set exactly due West.
All other objects rise in the northeast or southeast and set in the northwest
or the southwest. Note that in this diagram (for a latitude of 40$^{\circ}$) 
all stars that have latitudes (astronomers call them ``Declinations'', or 
``dec'') above 50$^{\circ}$ never set--they are circumpolar.

\begin{figure}[ht]
\centering
\leavevmode
\epsfysize=0.22\textheight
\epsfbox{Seasons/celestialgeo.ps}
\caption{The Celestial Equator is the circle in the sky that is straight 
overhead (``the zenith'') of the Earth's equator. In addition, there is
a ``North Celestial'' pole that is the projection of the Earth's North Pole
into space (that almost  points to Polaris). But the Earth's spin axis is
tilted by 23.5$^{\circ}$ to its orbit, and the Sun appears to move above
and below the Celestial Equator over the course of a year. }
\label{fig:celestial}
\end{figure}

What happens is that during the year, the Sun appears to move above and below 
the Celestial Equator. On, or about, March 21$^{\rm st}$ the Sun is on the
Celestial Equator, and each day after this it gets higher in the sky (for 
locations in the Northern Hemisphere) until 
June 21$^{\rm st}$. After that date it retraces its steps until it reaches the 
Autumnal Equinox (September 20$^{\rm th}$), after which it is then South of
the Celestial Equator. It is lowest in the sky on December 21$^{\rm st}$.
This is simply due to the fact that the Earth's axis is tilted with respect
to its orbit, and this tilt does not change. You can see this geometry by
going back to the illuminated globe model used in Exercise \#3. If you
stick a pin at some location on the globe away from the equator, turn on the 
halogen lamp, and 
slowly rotate the entire apparatus around (while keeping the pin facing the 
Sun) you will notice that the shadow of the pin will increase and decrease
in size. This is due to the apparent change in the elevation angle of the
``Sun''.

\section{Summary (35  points)}

Summarize the important points covered in this lab. Questions you should 
answer include:

\begin{itemize}
\item Why does the Earth have seasons?
\item What is the origin of the term ``Equinox''?
\item What is the origin of the term ``Solstice''?
\item Most people in the United States think the seasons are caused by the
changing distance between the Earth and the Sun. Why do you think this is?
\item What type of seasons would the Earth have if its spin axis was $exactly$
perpendicular to its orbital plane? Make a diagram like Fig. \ref{fig:earthorbit}.
\item What type of seasons would the Earth have if its spin axis was $in$
the plane of its orbit? (Note that this is similar to the situation for the
planet Uranus.)
\item What do you think would happen if the Earth's spin axis wobbled randomly
around on a monthly basis? Describe how we might detect this.
\end{itemize}

\section{Possible Quiz Questions}

\begin{flushleft}
1) What does the term ``latitude'' mean?\\
2) What is meant by the term ``Equator''?\\
3) What is an ellipse?\\
4) What are meant by the terms perihelion and aphelion?\\
5) If it is summer in Australia, what season is it in New Mexico?\\
\end{flushleft}

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

We have stated that the Earth's spin axis constantly  points to a single spot
in the sky. This is actually not true. Look up the phrase ``precession of
the Earth's spin axis''. Describe what is happening and the time scale of this
motion. Describe what happens to the timing of the seasons due to this motion.
Some scientists believe that precession might help cause ice ages. Describe 
why they believe this. 

