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

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

\section{Analysis: Plotting the positions of planets}

\subsection*{Materials} data from Set 3 ``Outdoor: Measuring the
positions of planets'', ruler, pencil, star chart of region with RA and Dec

\subsection*{Instructions}

You should have the angular distances between two or three stars and
each planet you observed. In this exercise you will use that
information to plot the position of each planet on a star chart.
\textbf{Each of you should make a separate chart.}

First, 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. \textbf{Record your scale. Why isn't
  the scale exactly the same for the whole chart? Why do you use the
  lines of declination and not right ascension (think about a globe)?}

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 degree you
would use $d= x s$.

First check your measurements of angular separations of pairs of
stars. \textbf{Make a little table:} for each pair of stars list: the
stars, the separation you measured in the sky, the separation you
measure from the chart.

Convert your measurements of star-planet angular separations into
chart units ($\mathrm{cm}$). Use some paper to make a makeshift
compass and \textit{in pencil} trace an arc of the appropriate radius
around each star. \textbf{\textit{In pen} draw a point to indicate the
  location of each planet. Label it with the date of the observation.} 

\section{Long-term outdoor: Tracking the positions of planets}

\subsection*{Instructions}

This exercise will be performed outside of lab. You will use the
skills you have learned in lab so far to track the positions the
visible planets over the course of a semester. 

At least once a week (\textit{this means you need at least eight
  observations by the end}) you should measure the angular separations
between each visible planet and at least three nearby stars. Make sure
the stars are ones you can identify on the chart you will use for
plotting. It also helps if the stars you choose ``surround'' the
planet and are not too close to each other. Be aware of the weather
and take advantage of clear nights when they occur.

To summarize:

\textbf{Once a week go outside, find Mars and Saturn, record date and time,
record the angular separations between each planet and several nearby,
identifiable stars. Later you should plot the positions of the planets
on your chart to make sure you are getting useful data.}

At the end of the semester you will use these observations to
construct a final chart showing the movement of the planets and to
measure the speed of their motion. 

\section{Indoor: Parallax through a window}

\subsection*{Instructions}

Imagine riding in a train or car and watching the scenery go
by. Nearby objects whizz by the window, while objects farther away
seem to move more slowly, and distant clouds or stars on the horizon
seem to hang motionless in the sky. Astronomers take advantage of this
effect to measure the distances to solar-system objects and to nearby
stars. These measurements are the basis of everything we know about
distances in Astronomy.

I have set up a baseline (two points marked on the floor) and a target
(taped to the window) for you to use to make some parallax
measurements. \textbf{First measure the length of your baseline. Let's
  call this $b$.} Now pick a distant object that can be seen through
the window from both ends of the baseline. Call this your ``background
object''. \textbf{Now measure the angles between the target and the
  background object at each end of the baseline. Call these $\theta_1$
  and $\theta_2$.} 

The total angle you will use is $\theta = \theta_1 + \theta_2$.
\textbf{Calculate the distance $r$ to your target using $\theta$ and $b$
  (see below).}

For small angles a useful approximation is $\theta = b/r$ where $b$
and $r$ are the baseline and distance (measured in the same units) and
$\theta$ is the angle in radians. You can convert to degrees by
remembering that $\mathrm{degrees} = 180 \mathrm{radians} / \pi$. So
for the above problem you would use the formula $r = (b/\theta)
(180/\pi)$.


\section{On paper: Periods, velocities, and radii of circular orbits}

\subsection*{Instructions}

The centripetal acceleration of an object moving in a circle of radius
$r$ and speed $v$ is $a = \frac{v^2}{r}$. The force due to gravity
between objects of mass $m_1$ and $m_2$ is $F_g = \frac{G m_1
  m_2}{r^2}$. Write an equation for the speed $v$ of an object in a
circular orbit of radius $R$ around a body of mass $M$ (hint: $F = m
a$). Now write an equation for the period $P$ of that circular orbit
(hint: $c = 2\pi r$; $t=d/v$).

\textit{Let's say the Sun were twice as massive as it is.}

\textbf{If you wanted to recreate the Solar System with circular
  orbits of the original radii what parameters would you have to
  change? By how much?}

\textbf{If you wanted to recreate the Solar System with circular
  orbits of the original \textit{period} what parameters would you
  have to change?  By how much?}

\textbf{If you wanted to recreate the Solar System with circular
  orbits of the original \textit{orbital velocities} what parameters
  would you have to change?  By how much?}

\section{Computer: Orbit Explorer}

\subsection*{Materials} computer with ``Orbit Explorer'' or other
orbit simulation software

\subsection*{Instructions}

Do this exercise after the previous one.

Play around with the orbit simulation software. Make sure you
understand what all of the settings mean. Recreate the Earth-Moon
system and watch it in motion. Zoom in on the Earth. \textbf{Does it
  move?  Why or why not? What happens if you make the Moon twice as
  massive?  What happens if you make the Earth twice as massive,
  instead?}

Recreate the Solar System. Zoom in on the sun. \textbf{Which planet is
  most responsible for the motion of the Sun? What is the next most
  influential planet?}

Now set up a Solar System with just the Earth and a Sun that is twice
as massive as normal. \textbf{Verify your answers to the previous
  exercise.}


\end{document}