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\centerline{\bf Astronomy 101U Homework \#2 -- The Sun and the Seasons}
\bigskip
(Original conception by Bruce Balick.  Modified by Doug Ingram with
help from Woody Sullivan.  Modified again and updated by Doug Ingram
It is originally modelled on a lab using the Voyager software for
the Mac.  In that version, the students took the data themselves
using the program to guide them.)
\bigskip

\noindent {\bf I. Introduction}

Familiarity with pp. 18-26 of the text is essential before starting
this lab.  You might want to go back and review it.  We all notice
the seasons--the long, bright, balmy days are a sharp contrast to
the winter days.  Somehow the Sun is higher and brighter in the summer
than in the winter, and shadows are longer in the winter.  We also
notice that seasons are much more pronounced further from the equator:
Hawaii's summer and winter days are much less of a contrast than the
summer and winter in Alaska.  Why?  We shall explore this and other
questions about the relationship of the Sun to the seasons in this
homework.

\medskip
As we did last time, we'll start with a series of observations and
then try to incorporate them into a model that accomodates them
and explains other observations as well.

\medskip
\noindent {\bf II. Observations}

Here is a summary of some basic observations to start with:

\begin{description}
\item
1 -- The northern hemisphere gets warmer during the months of June,
     July and August (``northern summer'') while the southern
     hemisphere gets cooler at the same time (``southern winter'').

\item
2 -- Shadows tend to be longer and days tend to be shorter during the
     winter months.

\item
3 -- The Sun seems to rise and set further north during the summer.

\item
4 -- The Sun seems to appear higher in the sky at local noon during
     the summer.
\end{description}

\bigskip
Now let's make a detailed table of observations of the Sun, following
its motion in the sky from three different locations on the Earth in
two month intervals.  We'll look at the following quantities:

\begin{description}
\item
- Rise and set time in Pacific Standard Time (and day length).
\item
- Maximum altitude of the Sun during the day.
\item
- Rise and set azimuth of the Sun.
\end{description}

\vfill\eject
\noindent Location:  Seattle, Washington (latitude 48\deg North)
\begin{center}
\begin{tabular}{c c c c c c c}
\underline{Date} & \underline{Rise Time} & \underline{Set Time} &
\underline{Day Length} & \underline{Max altitude} &
\underline{Rise Azimuth} & \underline{Set Azimuth} \\
2/22 & 7:03 AM & 5:42 PM & 10h 42m & 32\deg & 14\deg S of E & 15\deg S of W \\
4/22 & 5:07 AM & 7:09 PM & 14h 02m & 55\deg & 20\deg N of E & 19\deg N of W \\
6/22 & 4:12 AM & 8:10 PM & 15h 58m & 66\deg & 37\deg N of E & 37\deg N of W \\
8/22 & 5:16 AM & 7:08 PM & 13h 52m & 54\deg & 18\deg N of E & 18\deg N of W \\
10/22 & 6:40 AM & 5:07 PM & 10h 27m & 31\deg & 16\deg S of E & 16\deg S of W \\
12/22 & 7:55 AM & 4:21 PM & 8h 26m & 19\deg & 35\deg S of E & 35\deg S of W \\

\end{tabular}
\end{center}

\bigskip

\noindent Location:  Fairbanks, Alaska (latitude 65\deg North)
\begin{center}
\begin{tabular}{c c c c c c c}
\underline{Date} & \underline{Rise Time} & \underline{Set Time} &
\underline{Day Length} & \underline{Max altitude} &
\underline{Rise Azimuth} & \underline{Set Azimuth} \\
2/22 & 7:27 AM & 4:44 PM & 9h 17m & 16\deg & 25\deg S of E & 24\deg S of W \\
4/22 & 3:49 AM & 7:52 PM & 16h 03m & 38\deg & 33\deg N of E & 33\deg N of W \\
6/22 & 12:59 AM & 10:47 PM & 21h 48m & 49\deg & 75\deg N of E & 75\deg N of W \\
8/22 & 4:02 AM & 7:44 PM & 15h 42m & 37\deg & 30\deg N of E & 30\deg N of W \\
10/22 & 7:07 AM & 4:03 PM & 8h 56m & 14\deg & 25\deg S of E & 25\deg S of W \\
12/22 & 9:59 AM & 1:41 PM & 3h 42m & 2\deg & 65\deg S of E & 65\deg S of W \\

\end{tabular}
\end{center}

\noindent Location:  Singapore (latitude 1\deg North)
\begin{center}
\begin{tabular}{c c c c c c c}
\underline{Date} & \underline{Rise Time} & \underline{Set Time} &
\underline{Day Length} & \underline{Max altitude} &
\underline{Rise Azimuth} & \underline{Set Azimuth} \\
2/22 & 6:45 AM & 6:50 PM & 12h 05m & 78\deg & 11\deg S of E & 10\deg S of W \\
4/22 & 6:28 AM & 6:37 PM & 12h 09m & 79\deg & 12\deg N of E & 12\deg N of W \\
6/22 & 6:30 AM & 6:42 PM & 12h 12m & 68\deg & 23\deg N of E & 23\deg N of W \\
8/22 & 6:33 AM & 6:41 PM & 12h 08m & 80\deg & 12\deg N of E & 12\deg N of W \\
10/22 & 6:16 AM & 6:21 PM & 12h 05m & 78\deg & 11\deg S of E & 11\deg S of W \\
12/22 & 6:31 AM & 6:34 PM & 12h 03m & 65\deg & 23\deg S of E & 23\deg S of W \\

\end{tabular}
\end{center}

These tables contain a lot of information, so study them carefully and think
about some of the patterns you see.

\medskip
\noindent {\bf III. Analysis}

Answer all of the questions below on your own paper.  Please use graph
paper for any plots you are asked to make.

\medskip
\begin{description}
\item
(1) (15 points) One popular theory behind the origin of the seasons is
    that the summer is warmer because the Earth is closer to the Sun
    during the summer months.  At least one of the basic observations we
    gave in the previous section, however, contradicts this model.  Pick
    one of the basic observations that you think most clearly contradicts
    this model and explain why it does.

\item
(2) (10 points) Make a graph of the day length (in hours) versus month
    (Feb, Apr, June, etc).  Plot the data from all three cities on the 
    same graph (in different colors for each city...make it clear which
    points/colors go with which city).  \underline{Annotate} your graph,
    pointing out any interesting features (such as points of intersection,
    maxima and minima) and note what each feature corresponds to (for 
    example, the June 22 point corresponds to the summer solstice).

\item
(3) (15 points) On two separate graphs, plot for each of the three cities
    the azimuths of sunrises vs month and the maximum solar altitude vs
    month.  Again, note the important features on each graph.

\item
(4) (20 points) The following diagram is extremely important but
    difficult to understand.  It is a picture of the Sun's path relative
    to our horizon from Seattle on different dates during the year.
    It is meant to show not only how the maximum altitude of the Sun
    changes over the course of the year but also why the day length
    changes during the year (the portion of the Sun's path above the
    horizon varies in length) and why the azimuths of rise and set change
    during the year.  Using the data you've got, draw similar diagrams
    for Fairbanks and Singapore.

[Diagram omitted from LaTeX version of the lab]
\end{description}

\noindent
Now explain, using the diagrams you've drawn and your own words, why
you see difference in the shapes of your graphs from (2) and (3).  
Specifically, answer the following:

\begin{description}
\item
(5) (15 points) Why is the day length roughly constant in Singapore as
    opposed to in Fairbanks and Seattle?  Explain which city has the
    longest summer days and why.

\item
(6) (15 points) Why does the graph of maximum solar altitude have the
    opposite shape in Singapore as it does in the other two cities?

\item
(7) (10 points) In which of the three cities would you expect your
    shadow to at noon in winter to be the longest?  Explain why.

\end{description}
\vfil\eject
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