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\rhead{Earth, Moon, and Planets Lab (Tue 7-10pm)}
\chead{}
\lhead{Exercise set 9}
\renewcommand{\rightmark}{}
\lfoot{Roban Hultman Kramer} \cfoot{\thepage} \rfoot{Spring 2006}

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

\section{Analysis: Motion of the planets}

\subsection*{Materials} data from your planet observations, ruler,
star chart of region with RA and Dec

\subsection*{Goals}

You should have enough data to plot the position of Mars and Saturn on
several different nights over the course of the semester. The goal of
this analysis is to measure the motion of each planet against the
background stars.

If you are missing some of the data you need, just do as much of the
following procedure as you can, and explain what additional
observations you should have taken.

\subsection*{Instructions}

\subsubsection*{Individual night plots}

\begin{enumerate}
\item Make a separate chart for each night showing the positions of each
planet (see ``General plotting procedures'' below). 
\item Label each chart with the date of the observations. 
\item Measure the separation between the planets as you have plotted
  them on each chart, and convert that into degrees. Make sure to
  estimate your uncertainty. I'm looking for something like this:
  \textit{Separation between Mars and Saturn $= 31 \degrees \pm 8 \degrees$.}
\item Compare this to the separation you measured in the sky. Do they
  agree to within your uncertainty?
\end{enumerate}

\subsubsection*{Time series plot}

\begin{enumerate}
\item Make a plot with all of the best-estimate positions indicated
  and labeled by date.
\item Can you say with any certainty if any of the planets moved over
  the course of your observations?
\item Calculate the apparent speed of each planet (\textit{in hours of right
  ascension per day}) between each consecutive pair of
  observations. Movement to the East (increasing RA) should be
  positive, movement to the West should be negative.
\item Calculate the average speed over the course of all of the
  observations. If can't tell whether the planet moved, then put an upper
  limit on the speed. In other words, what is the maximum speed with
  which the planet could have moved and still have appeared to be
  stationary with your observations.
\item Ask me for a plot of the actual positions, and compare it with
  your plot. List any important differences, and what factors might
  have contributed to your error.
\end{enumerate}

\subsubsection*{General plotting procedures}

This is the procedure for plotting the position of a planet using
measured angular distances between the planet and three or more stars
of known position:

If you haven't already, figure out the angular scale of the chart:
measure the separation $x$ between lines of declination in
$\mathrm{cm}$. If the lines of declination are spaced $d$ degrees
apart, then your scale is $s=d/x$ degrees per centimeter. Now, if you
want to convert a measurement of $d$ degrees into $\mathrm{cm}$ on the
chart, you would use $x = d/s$. If you want to convert a measurement
of $x$ $\mathrm{cm}$ on the chart into degrees you would use $d= x s$.

Convert your measurements of star-planet angular separations into
chart units ($\mathrm{cm}$). Use some paper to make a makeshift
compass and trace an arc of the appropriate radius around each
star. Ideally all of your circles will intersect almost at a single
point. Draw a point to indicate your best estimate of the location of the
planet. \textit{Also draw a shape around your point to indicate your
uncertainty in the position, i.e. the extended region where you think
the planet might actually be, given the error in your measurements.}

\end{document}