\documentclass[11pt]{article}

\usepackage{colortbl}
\usepackage{fancyhdr}
\usepackage{graphicx}
\usepackage{lscape}
\usepackage{pstricks,multido,pst-plot,pst-node,pst-text}
\usepackage{tabularx}
\usepackage{url}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% 
% PathoftheSun.tex
%
% Peter Newbury
% Department of Physics and Astronomy, UBC 
% newbury@phas.ubc.ca
% Twitter @polarisdotca
% http://blogs.ubc.ca/polarisdotca
%
% 7 Sept 2011
%
% This work is supported by the Carl Wieman Science Education
% Initiative cwsei.ubc.ca.
%
% Unless credit is given explicitly, all documents, graphics and images
% are licensed under a Creative Commons BY-NC-SA
% Attribution-NonCommercial-ShareAlike 3.0 Unported License. 
% 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Margins
\oddsidemargin  -0.5in
\evensidemargin 0.0in
\textwidth      7.5in

\topmargin     -0.5in
\headheight     0pt
 
\textheight     9.25in
\footskip       30pt

\parindent      0pt
% \renewcommand{\baselinestretch}{1.5}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Fancyhdr:
\pagestyle{fancy}
\lhead{}
\chead{}
\rhead{}
\renewcommand{\headrulewidth}{0pt}
\lfoot{Path of the Sun}
\cfoot{}
\rfoot{Page~\thepage}
\renewcommand{\footrulewidth}{0.4pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\newcommand{\answerbox}{%
  \psframe[linewidth=0.5pt](-11pt,-3pt)(0,8pt)}

\newcommand{\dfrac}[2]{\displaystyle\frac{#1}{#2}}
\newcommand{\mybox}{\psframe(-1ex,0)(1ex,2ex)}

\newcommand{\answerblank}[1]{%
  \underline{\makebox[5cm][r]{ #1}}}

\newcommand{\checkbox}{%
  \makebox[1.6em]{%
    \psset{unit=0.8em,linewidth=0.5pt}
    \psframe(-0.5,0)(0.5,1)}}

\newcommand{\aztick}[1]{%
  \psline(0,1ex)(0,-1ex)
  \rput[t]{U}(0,-1.5ex){\small \textsf{#1}}}
\newcommand{\alttick}[2]{%
  \rput[l]{U}(1.5ex,#1){\small \textsf{#2}}}

\newcommand{\foo}{1in}

\begin{document}

\enlargethispage{1in}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Introduction
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
  {\huge The Path of the Sun}
\end{center}

The Sun, stars and planets cross our sky in complicated patterns
that depend on the Earth's daily rotation around its tilted axis and
its annual revolution of the Earth around the Sun.

\medskip
For thousands of years, astronomers have watched the sky, figured out the patterns
and built ``computers'' like these so they could predict when and
where the Sun, stars and planets rise and set each day.

\renewcommand{\foo}{4.5cm}

\begin{center}
  \begin{tabular}{ccc}
    \includegraphics[height=\foo]{stonehenge_JHawkesCorbis} &
    \includegraphics[height=\foo]{EB1711_Armillary_Sphere_WikipediaCommons} &
    \includegraphics[height=\foo]{NAAPGui} \\
    Stonehenge & Armillary sphere & Path of the Sun simulator \\
    c. 2500 BC & c. AD 1700  & Today \\
    (Jason Hawkes, Corbis Images) & (Wikipedia Commons) & \\
  \end{tabular}
\end{center}


In this tutorial, you'll use your 21st Century computer to explore the
motion of the Sun.

\begin{enumerate}
\item Open your browser to \url{astro.unl.edu}
\item On the NAAP Labs tab, select Labs - Motions of the Sun.
\item Click the ``Paths of the Sun Simulator'' link to launch the sim.
\end{enumerate}


\renewcommand{\foo}{7ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Part 1: Orientation
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\Large Part 1: The Path of the Sun Simulator}

\medskip    

Explore the sim with your partners and answer these questions:

\bigskip
An object's {\bf altitude} tells you how high an object is in the sky,
from $0^\circ$ on the horizon to $90^\circ$ at your zenith.  What is the
altitude of the Sun at 12:00 noon on Canada Day (July 1), as seen by
observers in Vancouver (latitude $49.0^\circ$N)?

\vspace{1in}

Challenge question: An object's {\bf azimuth} tells you its direction, like
on a compass, from angle $0^\circ$ (due North) to $90^\circ$ (due East) to
$180^\circ$ (due South) to $270^\circ$ (due West) and back to
$360^\circ$ or $0^\circ$ at North again. How many degrees North of East is the
most northerly sunrise for observers in Vancouver?



\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Part 2: Sun's paths across the sky, Sep, Dec
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\Large Part 2: The Path of the Sun Across the Sky}


\medskip

If you were able to answer both those questions, you've probably figured
out how to

\begin{center}
  \begin{tabular}{clcl}
    \textbullet & drag the Sun &
    \textbullet & grab and move the sphere \\
    \textbullet & set the time &
    \textbullet & find the time \\
    \textbullet & set the latitude &
    \textbullet & set the date \\
    \textbullet & measure the altitude &
    \textbullet & measure the azimuth \\
    \textbullet & animate the daily motion &
    \textbullet & what the white, gray, yellow and blue circles are
  \end{tabular}
\end{center}

If you haven't figured these out yet, don't worry. Watch your TA use the
simulation to collect data tracking the path of today's Sun. Record
the data in the Table: 

\bigskip
\renewcommand{\foo}{36mm}
\renewcommand{\arraystretch}{1}

\begin{tabularx}{\textwidth}{|X|c|c|c|}\hline
  \rowcolor[gray]{0.8} &
  \makebox[\foo]{Today} &
  \makebox[\foo]{Autumnal Equinox} & \makebox[\foo]{Winter Solstice} \\
  \rowcolor[gray]{0.8} & \underline{\hspace{30mm}} & Sep 22 & Dec 22 \\\hline
   & & & \\
  Time of sunrise 
  \newline (time when the Sun is at altitude $0^\circ$) & & &\\
   & & & \\
   & & & \\\hline
   & & & \\
  Sun's location on horizon at sunrise,
  \newline in degrees North or South from East (azimuth $90^\circ$) & & & \\
   & & & \\\hline
   & & & \\
  Sun's highest altitude of the day,
  \newline in degrees above Southern horizon & & & \\
   & & & \\
   & & & \\\hline
   & & & \\
  Time of sunset
  \newline (time when the Sun is at altitude $0^\circ$) & & &\\
  & & & \\
   & & & \\\hline
   & & & \\
  Sun's location on horizon at sunset, 
  \newline in degrees North or South from West (azimuth $270^\circ$) & & & \\
   & & & \\\hline
   & & & \\
  Draw the Sun's path on diagram.
  \newline Label path with date, time of sunrise, time of sunset. & & & \\
   & & & \\\hline
\end{tabularx}

\bigskip

%Your task is to {\bf draw 2 paths} on the diagram showing the Sun's path
%on 2 special times of year.  For each date, find the locations on the
%horizon where the Sun rises and sets and the Sun's highest point in the
%sky.  Use these three points to draw the path.  Then find the time of
%sunrise and sunset and add them to the diagram.


\clearpage 

\addtocounter{page}{-1}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Part 2: Sun's paths across the sky, Mar, Jun
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

{\Large Part 2: The Path of the Sun Across the Sky}


\medskip

If you were able to answer both those questions, you've probably figured
out how to

\begin{center}
  \begin{tabular}{clcl}
    \textbullet & drag the Sun &
    \textbullet & grab and move the sphere \\
    \textbullet & set the time &
    \textbullet & find the time \\
    \textbullet & set the latitude &
    \textbullet & set the date \\
    \textbullet & measure the altitude &
    \textbullet & measure the azimuth \\
    \textbullet & animate the daily motion &
    \textbullet & what the white, gray, yellow and blue circles are
  \end{tabular}
\end{center}

If you haven't figured these out yet, don't worry. Watch your TA use the
simulation to collect data tracking the path of today's Sun. Record the
data in the Table: 

\bigskip
\renewcommand{\foo}{36mm}
\renewcommand{\arraystretch}{1}

\begin{tabularx}{\textwidth}{|X|c|c|c|}\hline
  \rowcolor[gray]{0.8} &
  \makebox[\foo]{Today} &
  \makebox[\foo]{Vernal Equinox} & \makebox[\foo]{Summer Solstice} \\
  \rowcolor[gray]{0.8} & \underline{\hspace{30mm}} & Mar 20 & Jun 22 \\\hline
   & & & \\
  Time of sunrise 
  \newline (time when the Sun is at altitude $0^\circ$) & & &\\
   & & & \\
   & & & \\\hline
   & & & \\
  Sun's location on horizon at sunrise,
  \newline in degrees North or South from East (azimuth $90^\circ$) & & & \\
   & & & \\\hline
   & & & \\
  Sun's highest altitude of the day,
  \newline in degrees above Southern horizon & & & \\
   & & & \\
   & & & \\\hline
   & & & \\
  Time of sunset
  \newline (time when the Sun is at altitude $0^\circ$) & & &\\
  & & & \\
   & & & \\\hline
   & & & \\
  Sun's location on horizon at sunset, 
  \newline in degrees North or South from West (azimuth $270^\circ$) & & & \\
   & & & \\\hline
   & & & \\
  Draw the Sun's path on diagram.
  \newline Label path with date, time of sunrise, time of sunset. & & & \\
   & & & \\\hline
\end{tabularx}


\bigskip

%Your task is to {\bf draw 2 paths} on the diagram showing the Sun's path
%on 2 special times of year.  For each date, find the locations on the
%horizon where the Sun rises and sets and the Sun's highest point in the
%sky.  Use these three points to draw the path.  Then find the time of
%sunrise and sunset and add them to the diagram.


\clearpage 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Part 3: Questions
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\ \vspace{-8ex}

\renewcommand{\arraystretch}{1.25}
\begin{tabularx}{\textwidth}{@{}rXrp{3cm}rp{3cm}@{}}
  Name & & ID No. & & Tutorial Day/Time &  \\\cline{2-2}\cline{4-4}\cline{6-6}
\end{tabularx}

\bigskip
\bigskip

{\Large Part 3: Questions} \ \ \ Please hand in this worksheet when you are finished.

\begin{enumerate}

\item Use the sunrise and sunset times you collected and from your sky
  diagram to find the {\bf length of the day} (the number of hours the
  Sun is up) throughout the year.  Put the values on this graph and draw
  a smooth curve between them.  Be sure to include today, too.


  \begin{center}
  \psset{xunit=10mm,yunit=2.0mm,linewidth=1pt,dimen=middle}
  \newgray{sungray}{0.9}
  \pspicture(-2,-4)(14,26)
  % y-axis
  \multido{\iy=0+4}{7}{% left side major ticks
    \psline[linewidth=1pt](-0.5ex,\iy)(0.5ex,\iy)}
  \multido{\iy=0+1}{25}{% left side minor ticks
    \psline[linewidth=0.5pt](-0.25ex,\iy)(0.25ex,\iy)}
  \multido{\iy=0+4}{7}{% right side major ticks
    \rput(12,\iy){\psline[linewidth=1pt](-0.5ex,0)(0.5ex,0)}}
  \multido{\iy=0+1}{25}{% right side minor ticks
    \rput(12,\iy){\psline[linewidth=0.5pt](-0.25ex,0)(0.25ex,0)}}
  % horizontal lines
  \psline[linewidth=0.5pt](0,8)(12,8)
  \psline[linewidth=0.5pt](0,16)(12,16)
  \psline[linewidth=0.5pt,linestyle=dashed](0,12)(12,12)
  % x-axis
  \multido{\ix=0+1}{13}{% bottom minor ticks
    \psline[linewidth=0.5pt](\ix,0.25ex)(\ix,-0.25ex)}
  \multido{\ix=0+3}{5}{% bottom major ticks
    \psline[linewidth=0.5pt](\ix,-0.5ex)(\ix,24)
    \psline(\ix,0.5ex)(\ix,-0.5ex)}
  \multido{\ix=0+1}{13}{% top minor ticks
    \rput(\ix,24){\psline[linewidth=0.5pt](0,0.25ex)(0,-0.25ex)}}
  \multido{\ix=0+3}{5}{% bottom major ticks
    \rput(\ix,24){\psline(0,0.5ex)(0,-0.5ex)}}
  % outer frame
  \psframe(0,0)(12,24)
  % month labels for all 12 months
  \rput[tl]{-45}(0,-1ex){\small \sc Dec 22}
  %\rput[tl]{-45}(1,-1ex){\small \sc Jan 20}
  %\rput[tl]{-45}(2,-1ex){\small \sc Feb 23}
  \rput[tl]{-45}(3,-1ex){\small \sc Mar 20}
  %\rput[tl]{-45}(4,-1ex){\small \sc Apr 20}
  %\rput[tl]{-45}(5,-1ex){\small \sc May 21}
  \rput[tl]{-45}(6,-1ex){\small \sc Jun 20}
  %\rput[tl]{-45}(7,-1ex){\small \sc Jul 23}
  %\rput[tl]{-45}(8,-1ex){\small \sc Aug 24}
  \rput[tl]{-45}(9,-1ex){\small \sc Sep 22}
  %\rput[tl]{-45}(10,-1ex){\small \sc Oct 24}
  %\rput[tl]{-45}(11,-1ex){\small \sc Nov 23}
  \rput[tl]{-45}(12,-1ex){\small \sc Dec 22}
  % time labels
  \uput[180](0,24){\small 24}
  \uput[180](0,20){\small 20}
  \uput[180](0,16){\small 16}
  \uput[180](0,12){\small 12}
  \uput[180](0,8){\small 8}
  \uput[180](0,4){\small 4}
  \uput[180](0,0){\small 0}
  \rput{L}(-1,12){Length of Day (hours)}
  % altitude labels
  %\uput[0](12,24){\small 90}
  %\uput[0](12,16){\small 60}
  %\uput[0](12,8){\small 30}
  %\uput[0](12,0){\small 0}
  %\rput{R}(13,12){Highest Point of Day (degrees)}
  % title
  \rput[b](6,25){Length of the day in Vancouver over the year.}
  % October 24 data
  %\qdisk(10,10){2pt}
  % January 20 data (month with Jan=1, length of day in hrs)
  %\qdisk(1,9){2pt}
  % February 23 data (month with Jan=1, length of day in hrs)
  %\qdisk(2,10.5){2pt}
  \endpspicture
\end{center}

\item Shade in the region of the graph in Question 1 that represents the
  hours of darkness when the Sun is down. What fraction of the graph did
  you shade in?\ \ Answer:\ \underline{\hspace{2in}}

\item Change the latitude to $90^\circ$N so the sim shows the motion of the
  Sun for observers {\bf at the North Pole}.

  \medskip
  When does the Sun rise today? \hrulefill

  \medskip
  When does the Sun rise on Jun~20? \hrulefill

  \medskip
  When does the Sun rise at the North Pole? \hrulefill
  
  \bigskip

\item There's something wrong with the equinoxes. In class, we say the
  Sun rises at 6:00~a.m. and sets at 6:00~p.m. but that's not what you
  found. Set the simulation for Vancouver again. Write the observations
  you made earlier and add another observation: the time when the Sun
  crosses the meridian (at azimuth $180^\circ$.) This is what we usually
  call ``noon.''

  \begin{center}
    \renewcommand{\foo}{3cm}
    \begin{tabular}{|l|c|c|c|}\hline
      Date & \makebox[\foo]{sunrise} &
      \makebox[\foo]{noon} & \makebox[\foo]{sunset}\\\hline
      Vernal Equinox & & & \\\hline
      Autumnal Equinox & & & \\\hline
    \end{tabular}
  \end{center}

  On the Vernal Equinox, the day is \underline{\ \ \ \ \ \ \ \ }~hours long
  but it's shifted \hrulefill

  \bigskip
  On the Autumnal Equinox\ \hrulefill

  \bigskip
  This unexpected shift will be explained in upcoming classes about
  Kepler's Laws and the analemma.
\end{enumerate}

\vfill
\centerline{Path of the Sun simulator: 
  \url{astro.unl.edu/naap/motion3/animations/sunmotions.html}}


\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% Sky graphic
%
% Four versions: blank,  with Sep and Dec, 
% and with Mar and Jun, all 4 dates
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\thispagestyle{empty}

\begin{landscape}
  \centerline{%
    \includegraphics[width=10in]{SkyGraphicNoDates}}
\end{landscape}

\clearpage

\thispagestyle{empty}

\begin{landscape}
  \centerline{%
    \includegraphics[width=10in]{SkyGraphicSepDec}}
\end{landscape}

\clearpage

\thispagestyle{empty}

\begin{landscape}
  \centerline{%
    \includegraphics[width=10in]{SkyGraphicMarJun}}
\end{landscape}

\clearpage

\thispagestyle{empty}

\begin{landscape}
  \centerline{%
    \includegraphics[width=10in]{SkyGraphicMarJunSepDec}}
\end{landscape}

\clearpage


\end{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\item When the Sun is high in the sky, the sunlight is more direct and
  feels hotter.  Two students are talking about how this causes the seasons:

  \begin{tabularx}{0.95\textwidth}{@{}cX@{}}
    Student A & I think Summer is caused by how high the Sun climbs in
    the sky.  In Summer, the Sun climbs way higher in the sky than in
    Winter so it gets hotter.  \\
    Student B & No, it has to do with the length of day -- the days are
    so short in Winter, it never warms up.
  \end{tabularx}

  Do you agree or disagree with either or both of the students?
  Explain your reasoning.

  %\vspace{4cm}
  \vfill
