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\rhead{Earth, Moon, and Planets Lab (Tue 7-10pm)}
\chead{}
\lhead{Exercise set three}
\renewcommand{\rightmark}{}
\lfoot{Roban Hultman Kramer} \cfoot{\thepage} \rfoot{Spring 2006}

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\begin{document}
\begin{center}
\huge Exercise set three
\end{center}

% So far in lab we have focused on two things: (1)  how does Earth's
% motion through the Solar System 

% Reminder: a solar day is how long it takes (on average) for the sun to
% return to the same position in the sky. A sidereal day is how long it
% takes the stars to return to the same position in the sky, i.e. how
% long it take the Earth to rotate 360\degrees. \textbf{What causes the
%   difference between the solar and sidereal days?}

\section{On paper: Introduction to celestial coordinates}

\subsection*{Materials} celestial sphere model

\subsection*{Instructions}

Astronomers use several different coordinate systems to describe the
locations of objects in the sky. We have been using the
``altitude-azimuth'' or ``horizontal'' system, because it is useful
for describing the apparent diurnal motion of objects in the sky.
There are two big disadvantage of the alt-az system: one is that
almost everything in the sky is constantly and rapidly changing
coordinates all the time, the other is that alt-az coordinates vary
from place to place on the Earth, even at the same moment in time.
Astronomers, therefore, tend to use coordinate systems in which at
least some celestial objects are more or less stationary. The most
common of these is the ``equatorial'' system. 

Imagine projecting the Earth's system of latitude and longitude lines
out onto the infinitely-large celestial sphere. This becomes the
celestial North pole. We'll keep the lines of latitude and refer to
this coordinate as the ``declination'', $\delta$, the angle between
the celestial equator and a point in the sky (expressed in $\pm$
degrees, arcminutes, arcseconds; $+$ for N, $-$ for S). There's still
a problem with the lines of longitude, though. 

Instead of fixing lines of longitude to a point on the Earth,
astronomers fix this coordinate to a (more or less) fixed point
relative to the stars called the Vernal Equinox (we'll talk more about
this later). The zero line of ``right ascension'' passes through this
point and the celestial poles, much like the zero line of longitude on
Earth (the ``Greenwich meridian'') passes through Greenwich, England
and the Earth's poles. You may want to refer to a celestial globe at
this point to help you visualize this coordinate system.

Right ascension (RA) is not measured in degrees, but in hours,
minutes, and seconds, increasing to the East. There are 24 hours of
right ascension around the sky, with 60 minutes per hour, and 60
seconds per minute. Note that a minutes or second of right ascension
is \textit{not} the same as an arcminute or arcsecond. 

The coordinates of a celestial object might be written like this:
$5\hours52\minutes0\seconds$, $+ 07\degrees24\arcmin57\arcsec$.

Note that the precession of the Earth's axis very slowly changes where
the North celestial pole points in the sky, and thus shifts the whole
coordinate system. For this reason, very precise astronomical
coordinates are always specified along with a year, and have to be
converted to be accurate at later times.

\textbf{Answer the following questions in your lab notebook. You may
  work with other students, but show all your work. It is very
  important to keep units straight in these problems.}

\begin{enumerate}

\item What constellation contains the celestial North pole?

\item Why is right ascension is measured in units of time (i.e. why is
  this convenient)?

\item Why does right ascension increase to the East?

\item How many degrees are in 1\hours of RA (along the celestial
  equator)? 

\item How long does it take for the Earth to rotate through 24\hours
  of RA? Think carefully about what we learned in the first lab before
  you answer. \textit{I want your answer to this question and the next
    two be precise to the second.}

\item If a star at RA $0\hours 0 \minutes 0\seconds$ is on your meridian
  at midnight, what will be the RA of a star overhead at 1:30am?

\item What star (or at least what constellation) is at
  $5\hours52\minutes0\seconds$, $+ 07\degrees24\arcmin57\arcsec$?

\item At midnight, tonight, stars with a RA of $10\hours 11\minutes
  53\seconds$ will be crossing the meridian. When will the star from
  the previous question transit?

\end{enumerate}

\section{Indoor: Calibrate your hand to measure angles}

\subsection*{Materials} you, ruler, string

\subsection*{Instructions}

Sextants allow the precise measurement of angles, but not all of us
carry one in our back pocket (plus, as you all found out, they take
some practice). So you are going to calibrate various parts of your
hand to allow you to measure angles. 

Turn your head 90\degrees and extend your arm straight out from the
shoulder. Now extend your thumb like you're hitchhiking. This will be
one of your basic measuring devices. 

\textbf{Label and record all your measurements and calculations:}

Measure the width of your thumb across the nail. Use the string to
measure the distance from your eye to your thumb at arm's length.
Repeat each measurement a couple times to get it as accurate as
possible. Calculate the angle subtended by your thumb in degrees.

Repeat this process for your index finger and the width across your
knuckles when you make a fist.

You now have three ``devices'' you can use any time to measure angles.

\section{Indoor: Measure angles with your hand}

\subsection*{Materials} you, ruler, string, your calculations of the
angles subtended by your thumb, finger, and fist, hallway with markers

\subsection*{Instructions}

Do the calibration exercise before this one.

At the West end of the 14th floor hall I have put a line of tape on
the floor. I have also put two vertical lines of orange stickers on
the wall. \textbf{Use various parts of your hand to measure the
  angular separation between the two lines, and the angular height of
  each line. Make sure to record what body part you used, the
  measurement in ``body-part'' units, and the final measurement in
  degrees. Repeat each measurement at least twice (this means you
  should have at least 6 measurements, total).}

\textbf{Get your partner's help in using the string to measure the
  distance from your eye to the wall. Measure the height of each line
  and the separation between them and calculate the angles you just
  measured. How well do your results agree? What are your sources of
  error?}

\section{Outdoor: Measure angles with your hand}

\subsection*{Materials} you, star charts, your calculations of the
angles subtended by your thumb, finger, and fist, the sky

\subsection*{Instructions}

Do the calibration exercise before this one.

\textbf{Measure the angular separation between the stars Betelgeuse
  and Rigel in Orion, between Sirius and Procyon, and between Castor
  and Pollux. Repeat each measurement at least twice.}

\section{Outdoor: Measuring the positions of planets}

\subsection*{Materials} you, star charts, your calculations of the
angles subtended by your thumb, finger, and fist, the sky

\subsection*{Instructions}

Do the calibration exercise before this one.

Find Mars and Saturn. You can go in and use the planetarium software
if you have trouble identifying them. 

\textbf{Identify nearby constellations and draw a chart (or charts)
  showing the position of each planet.}

\textbf{Measure the angles between each planet and at least three of
  the stars on your chart. You don't have to know each star by name,
  but at least draw its position in its constellation so that you
  can identify it later using planetarium software.}

\section{Outdoor: Measure angles between buildings with your hand}

\subsection*{Materials} you, your calculations of the angles subtended
by your thumb, finger, and fist, the sky

\subsection*{Instructions}

Do the calibration exercise before this one.

\textbf{Measure the angle between the Empire State Building and Time
  Warner Center. The TWC is about $4.8km$ away. If you assume the
  Empire State Building is at the same distance, how far apart are the
  two buildings? Is the real separation greater or smaller than this?}

\end{document}