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\begin{Large}
\section{\bf Kepler's Laws}
\end{Large}

\subsection{Introduction}
Throughout human history, the motion of the planets in the sky was a mystery:
why did some planets move quickly across the sky, while other planets moved
very slowly? Even two thousand years ago it was apparent that the motion of the
planets was very complex. For example, Mercury and Venus never strayed very 
far from the Sun, while the Sun, the Moon, Mars, Jupiter and Saturn generally 
moved from the west to the east against the background stars (at this point in 
history, both the Moon and the Sun were considered ``planets''). The
Sun appeared to take one year to go around the Earth, while the Moon only took
about 30 days. The other planets moved much more slowly. In addition to this
rather slow movement against the background stars was, of course, the daily
rising and setting of these objects. How could all of these motions occur? 
Because these objects were important to the cultures of the time---even 
foretelling the future using astrology. Being able to predict their motion
was considered vital. \\

The ancient Greeks had developed a model for the 
Universe in which all of the planets and the stars were embedded in perfect
crystalline spheres that revolved around the Earth at uniform, but slightly
different speeds. This is the ``geocentric'', or Earth-centered model. But 
this model did not work very well--the speed of the planet across the sky
changed. Sometimes, a planet even moved backwards! It was left to
the Egyptian astronomer Ptolemy (85 $-$ 165 AD) to develop a model for 
the motion of the planets (you can read more about the details of the Ptolemaic 
model in your textbook). Ptolemy developed a complicated system to explain
the motion of the planets, including ``epicycles'' and ``equants'', that
in the end worked so well, that no other models for the motions of the planets
were considered for 1500 years! While Ptolemy's model worked well, the
philosophers of the time did not like this model--their Universe was perfect,
and Ptolemy's model suggested that the planets moved in peculiar, imperfect
ways.\\

In the 1540's Nicholas Copernicus (1473 $-$ 1543) published his work suggesting
that it was much easier to explain the complicated motion of the planets
if the Earth revolved around the Sun, and that the orbits of the planets
were circular. While Copernicus was not the first
person to suggest this idea, the timing of his publication coincided with
attempts to revise the calendar and to fix a large number of errors in 
Ptolemy's model that had shown up over the 1500 years since the model was 
first introduced. But the ``heliocentric'' (Sun-centered) model of 
Copernicus was slow to win acceptance, since it did not work as well as
the geocentric model of Ptolemy.\\

Johannes Kepler (1571 $-$ 1630) was the first person to truly understand how the
planets in our solar system moved. Using the highly precise observations by
Tycho Brahe (1546 $-$ 1601) of the motions of the planets against the background
stars, Kepler was able to formulate three laws that described how the planets
moved. With these laws, he was able to predict the future motion of these
planets to a higher precision than was previously possible. Many credit
Kepler with the origin of modern physics, as his discoveries were what
led Isaac Newton (1643 $-$ 1727) to formulate the law of gravity. Today we 
will investigate Kepler's laws and the law of gravity.\\

\subsection{Gravity}

Gravity is the fundamental force governing the motions of astronomical
objects.  No other force is as strong over as great a distance.  Gravity
influences your everyday life (ever drop a glass?), and keeps the planets,
moons, and satellites orbiting smoothly.  Gravity affects everything in
the Universe including the largest structures like super clusters of galaxies
down to the smallest atoms and molecules.
Experimenting with gravity is difficult to do.  You can't just go around
in space making extremely massive objects and throwing them together from
great distances.  But you can model a variety of interesting systems
very easily using a  computer. By using a computer to model the
interactions of massive objects like planets, stars and galaxies, we can
study what would happen in just about any situation.  All we have to know are
the equations which predict the gravitational interactions of the objects.\\

The orbits of the planets are governed by a single equation formulated by
Newton:

\beq
F_{gravity} = \frac{GM_1M_2}{R^2}
\eeq

A diagram detailing the quantities in this equation is shown in Fig.
\ref{fgravity}. Here F$_{gravity}$ is the gravitational attractive force 
between two objects whose masses are M$_1$ and M$_2$.  The distance between 
the two objects is ``R''.  The gravitational constant $G$ is just a small 
number that scales the size of the force. {\bf The most important thing about gravity is that 
the force depends only on the masses of the two objects and the distance 
between them.} This law is called an Inverse Square Law because the distance 
between the objects is {\it squared}, and is in the denominator of the 
fraction.  There are several laws like this in physics and astronomy.\\
 
\begin{figure}[h]
\centerline{{\includegraphics[width=8cm]{Kepler/fgravity.jpg}}}
\caption{The force of gravity depends on the masses of the two objects
(M$_{\rm 1}$, M$_{\rm 2}$), and
the distance between them (R).}
\label{fgravity}
\end{figure}

Today you will be using a computer program called ``Planets and Satellites'' by
Eugene Butikov to explore Kepler's laws, and how planets,  
double stars, and planets in double star systems move.  This program uses
the law of gravity to simulate how celestial objects move.\\

\begin{itemize}
\item{{\it Goals:}  to understand Kepler's three laws and use them in 
conjunction with the computer program ``Planets and Satellites'' to explain 
the orbits of objects in our solar system and beyond}
\item{{\it Materials:  Planets and Satellites} program, a ruler, and a calculator}
\end{itemize}

\subsection{Kepler's Laws}

Before you begin the lab, it is important to recall Kepler's three laws,
the basic description of how the planets in our
Solar System move.  Kepler formulated his three laws in the early 1600's, when he
finally solved the mystery of how planets moved in our Solar System.  These
three (empirical) laws are:\\
\\
I.  ``The orbits of the planets are ellipses with the Sun at one focus.''\\
\\
II.  ``A line from the planet to the Sun sweeps out equal areas in equal
intervals of time.''\\
\\
III.  ``A planet's orbital period squared is proportional to its average
distance from the Sun cubed:  P$^2$  $\propto$ $a^3$'' \\

Let's look at the first law, and talk about the nature of an ellipse. What
is an ellipse? An ellipse is one of the special curves called a ``conic
section''. If we slice a plane through a cone, four different types of 
curves can be made: circles, ellipses, parabolas, and hyperbolas. This
process, and how these curves are created is shown in Fig. \ref{conic}.\\

\begin{figure}[htpb]
\centerline{{\includegraphics[width=12cm]{Kepler/conic.jpg}}}
\caption{Four types of curves can be generated by slicing a cone with
a plane: a circle, an ellipse, a parabola, and a hyperbola. Strangely, these
four curves are also the allowed shapes of the orbits of planets, asteroids,
comets and satellites!}
\label{conic}
\end{figure}

\begin{figure}[htpb]
\centerline{{\includegraphics[width=12cm]{Kepler/ellipse.jpg}}}
\caption{An ellipse with the major and minor axes identified.}
\label{ellipseK}
\end{figure}

Before we describe an ellipse, let's examine a circle, as it is a simple
form of an ellipse. As you are aware, the circumference of a circle is simply
2$\pi$R. The radius, R, is the distance between the center of the circle and
any point on the circle itself. In mathematical terms, the center of the
circle is called the ``focus''. An ellipse, as shown in Fig. \ref{ellipseK},
is like a flattened circle, with one large diameter (the ``major'' axis) and
one small diameter (the ``minor'' axis). A circle is simply an ellipse
that has identical major and minor axes. Inside of an ellipse,
there are two special locations, called ``foci'' (foci is the plural of focus,
it is pronounced ``fo-sigh'').
The foci are special in that the sum of the distances between the foci
and any points on the ellipse are always equal. 
Fig. \ref{ellipse2} is an ellipse with the two foci identified,
``F$_{\rm 1}$'' and ``F$_{\rm 2}$''. \\

~ 

{\bf Exercise \#1:} On the ellipse in Fig. \ref{ellipse2} are two X's.
Confirm that that sum of the distances between the two foci
to any point on the ellipse is always the same by measuring the distances
between the foci, and the two spots identified with X's. Show your
work. ({\bf 2 points})\\

\vspace{1.0in}

{\bf Exercise \#2:} In the ellipse shown in Fig. \ref{ellipse3}, two points
(``P$_{\rm 1}$'' and ``P$_{\rm 2}$'') are identified that are not located at 
the true positions of the foci. Repeat exercise \#1, but confirm that
P$_{\rm 1}$ and P$_{\rm 2}$ are not the foci of this ellipse. ({\bf 2 points})

\vspace{1.0in}

\begin{figure}[htpb]
\centerline{{\includegraphics[width=12cm]{Kepler/ellipse2.jpg}}}
\caption{An ellipse with the two foci identified.}
\label{ellipse2}
\end{figure}

\begin{figure}[htpb]
\centerline{{\includegraphics[width=12cm]{Kepler/ellipse3.jpg}}}
\caption{An ellipse with two non-foci points identified.}
\label{ellipse3}
\end{figure}

Now we will use the {\bf Planets and Satellites} program to examine Kepler's
laws. It is possible that the program will already be running when
you get to your computer. If not, however, you will have to start it up.
If your TA gave you a CDROM, then you need to insert the CDROM into the CDROM
drive on your computer, and open that device. On that CDROM will be an icon
with the program name. It is also possible that Planets and Satellites has
been installed on the computer you are using. Look on the desktop for an
icon, or use the start menu. Start-up the program, and you should see
a title page window, with four boxes/buttons (``Getting Started'', 
``Tutorial'', ``Simulations'', and ``Exit''). Click on the ``Simulations''
button. We will be returning to this level of the program to change
simulations. Note that there are help screens and other sources of information
about each of the simulations we will be running--do not hesitate to explore
those options.\\

~

{\bf Exercise \#3:} Kepler's first law. Click on the ``Kepler's Law button''
and then the ``First Law'' button inside the Kepler's Law box. A window
with two panels opens up. The panel on the left will trace the motion of
the planet around the Sun, while the panel on the right sums the distances
of the planet from the foci. Remember, Kepler's first law states ``the
orbit of a planet is an ellipse with the Sun at one focus''. The Sun in
this simulation sits at one focus, while the other focus is empty  (but
whose location will be obvious once the simulation is run!).\\

At the top of the panel is the program control bar. For now, simply
hit the ``Go'' button. You can clear and restart the simulation by hitting
``Restart'' (do this as often as you wish). After hitting Go, note that
the planet executes an orbit along the ellipse. The program draws
the ``vectors'' from each focus to 25 different positions of the planet
in its orbit. It draws a blue vector from the Sun to the planet, and a yellow
vector from the other focus to the planet. The right hand panel sums the
blue and yellow vectors. [Note: if your computer runs the simulation too
quickly, or too slowly, simply adjust the ``Slow down/Speed Up'' slider for
a better speed.]\\

Describe the results that are displayed in the right hand panel for this first 
simulation. ({\bf 2 points}).

\vspace{1.3in}

Now we want to explore another ellipse. In the extreme left hand side of the
control bar is a slider to control the ``Initial Velocity''. At start-up
it is set to ``1.2''. Slide it up to the maximum value of 1.35 and hit
Go.\\

Describe what the ellipse looks like at 1.35 vs. that at 1.2. Does the sum
of the vectors (right hand panel) still add up to a constant? ({\bf 3 points})

\vspace{2.5in}

Now let's put the Initial Velocity down to a value of 1.0. Run the simulation.
What is happening here? The orbit is now a circle. Where are the two
foci located? In this case, what is the distance between the focus and
the orbit equivalent to? ({\bf 4 points})

\vspace{2.8in}

The point in the orbit where the planet is closest to the Sun is called
``perihelion'', and that point where the planet is furthest from the Sun
is called ``aphelion''. For a circular orbit, the aphelion is the same
as the perihelion, and can be defined to be anywhere!  Exit this simulation (click on ``File'' and ``Exit'').\\

~

{\bf Exercise \#4:} Kepler's Second Law: ``A line from a planet to the Sun 
sweeps out equal areas in equal intervals of time.'' From the simulation
window, click on the ``Second Law'' after entering the Kepler's Law window.
Move the Initial Velocity slide bar to a value of 1.2. Hit Go.\\

Describe what is happening here. Does this confirm Kepler's second law? How?
When the planet is at perihelion, is it moving slowly or quickly? Why do
you think this happens? ({\bf 4 points})

\vspace{4.0in}

Look back to the equation for the force of gravity. You know from personal
experience that the harder you hit a ball, the faster it moves. The
act of hitting a ball is the act of applying a force to the ball. The larger
the force, the faster the ball moves (and, generally, the farther it travels).
In the equation for the force of gravity, the amount of force generated
depends on the masses of the two objects, and the distance between them.
But note that it depends on one over the square of the distance: 1/R$^{\rm 2}$.
Let's explore this ``inverse square law'' with some calculations.\\
\begin{itemize}
\item If R = 1, what does 1/R$^{\rm 2}$ = \underline{\hskip 2.0in}?\\
\item If R = 2, what does 1/R$^{\rm 2}$ = \underline{\hskip 2.0in}?\\
\item If R = 4, what does 1/R$^{\rm 2}$ = \underline{\hskip 2.0in}?\\
\end{itemize}

What is happening here? As R gets bigger, what happens to 1/R$^{\rm 2}$?
Does 1/R$^{\rm 2}$ decrease/increase quickly or slowly?
 ({\bf 2 points})

\vspace{1.5in}

The equation for the force of gravity has a 1/R$^{\rm 2}$ in it, so
as R increases (that is, the two objects get further apart), does the force
of gravity {\it felt} by the body get larger, or smaller? Is the force of gravity stronger at
perihelion, or aphelion? Newton showed that the speed of a planet in its
orbit depends on the force of gravity through this equation: \\

\beq
V = \sqrt{(G(M_{\rm sun} + M_{\rm planet})(2/r - 1/a))}
\eeq
 where ``r'' is the radial distance
of the planet from the Sun, and ``a'' is the mean orbital radius (the semi-major 
axis). Do you think the planet will move faster, or slower when it is closest 
to the Sun? Test this by assuming that r = 0.5a at perihelion, and r = 1.5a at 
aphelion, and that a=1! [Hint, simply set G(M$_{\rm sun}$ + M$_{\rm planet}$) = 1
to make this comparison very easy!] Does this 
explain Kepler's second law? ({\bf 4 points})

\vspace{2.8in}

What do you think the motion of a planet in a circular orbit looks like?
Is there a definable perihelion and aphelion? Make a prediction for what
the motion is going to look like--how are the triangular areas seen for
elliptical orbits going to change as the planet orbits the Sun in a circular 
orbit? Why? ({\bf 3 points})

\vspace{2.2in}

Now let's run a simulation for a circular orbit by setting the Initial Velocity
to 1.0. What happened? Were your predictions correct? ({\bf 3 points})

\vspace{2.5in}
Exit out of the Second Law, and start-up the Third Law simulation.\\

~ 

{\bf Exercise 4:} Kepler's Third Law: ``A planet's orbital period squared is 
proportional to its average distance from the Sun cubed:  P$^2$ 
$\propto$ $a^3$''. As we have just learned, the law of gravity states that the
further away an object is, the weaker the force. We have already found that
at aphelion, when the planet is far from the Sun, it moves more slowly than at 
perihelion.  Kepler's
third law is merely a reflection of this fact--the further a planet is from
the Sun (``a''), the more slowly it will move.
The more slowly it moves,
the longer it takes to go around the Sun (``P''). The relation is
 P$^2$ $\propto$ $a^3$, where P is the orbital period in years, while $a$ is the
average distance of the planet from the Sun,
and the mathematical symbol for proportional is represented by ``$\propto$''.
To turn the proportion sign into an equal sign requires the multiplication
of the $a^3$ side of the equation by a constant: P$^2$ = $C ~ \times$ $a^3$. 
But we can get rid of this constant, ``$C$'', by making a ratio. We will do this below.\\

In the next simulation, there will be two planets: one in a smaller orbit, which
will represent the Earth (and has $a$ = 1), and a planet in a larger orbit
(where $a$ is adjustable). Start-up the Third Law simulation and hit Go.
You will see that the inner planet moves around more quickly, while the
planet in the larger ellipse moves more slowly. Let's set-up the math 
to better understand Kepler's Third Law. We begin by constructing the ratio of 
of the Third Law equation (P$^2$ = $C ~ \times$ $a^3$) for an arbitrary 
planet divided by the Third Law equation for the Earth:\\

\beq
\frac{P^2_P}{P^2_E} = \frac{C \times a^3_P}{C \times a^3_E}
\eeq

In this equation, the planet's orbital period and average distance are denoted
by $P_{P}$ and $a_{P}$, while the orbital period of the Earth and its
average distance from the Sun are $P_{E}$ and $a_{E}$. As you know from
from your high school math, any quantity that appears on both the top and 
bottom of a fraction can be canceled out. So, we can get rid of the pesky
constant ``$C$'', and Kepler's Third Law equation becomes: \\

\beq
\frac{P^2_P}{P^2_E} = \frac{a^3_P}{a^3_E}
\eeq

But we can make this equation even simpler by noting that if we use years for 
the orbital period ($P_{E}$ = 1), and Astronomical Units for the average 
distance of the Earth to the Sun ($a_{E}$ = 1), we get: \\

\beq
\frac{P^2_P}{1} = \frac{a^3_P}{1} ~ ~ ~ or ~ ~ ~ {P^2_P} = {a^3_P}
\label{eq:k3}
\eeq

(Remember that the cube of 1, and the square of 1 are both 1!) \\

Let's use equation (\ref{eq:k3}) to make some predictions. If the average 
distance of Jupiter from the Sun is about 5 AU, what is its orbital period? 
Set-up the
equation:

\beq
{P^2_J} = {a^3_J} = {5^3} = {5 \times 5 \times 5} = 125
\eeq

So, for Jupiter, $P^{2}$ = 125. How do we figure out what $P$ is? We have
to take the square root of both sides of the equation:

\beq
\sqrt{P^2} = P = \sqrt{125} = 11.2 ~ ~ years
\eeq
The orbital period of Jupiter is approximately 11.2 years. Your turn: \\

~

If an asteroid has an average distance from the Sun of 4 AU, what is its
orbital period? Show your work. ({\bf 2 points})

\vspace{2.0in}

In the Third Law simulation, there is a slide bar to set the average distance
from the Sun for any hypothetical solar system body. At start-up, it is set to 4 AU. Run
the simulation, and confirm the answer you just calculated. Note that for
each orbit of the inner planet, a small red circle is drawn on the outer
planet's orbit. Count up these red circles to figure out how many times the
Earth revolved around the Sun during a single orbit of the asteroid. Did
your calculation agree with the simulation? Describe your results. ({\bf 2 
points})
\vspace{2.3in}

If you were observant, you noticed that the program calculated the number
of orbits that the Earth executed for you (in the ``Time'' window), and you do 
not actually have to count up the little
red circles. Let's now explore the orbits of the nine planets in our
solar system. In the following table are the semi-major axes of the nine
planets. Note that the ``average distance to the Sun'' ($a$) that we have
been using above is actually a quantity astronomers call the ``semi-major axis''
of a planet. $a$ is simply one half the major axis of the orbit ellipse.
Fill in the missing orbital periods of the planets by running the Third Law 
simulator for each of them. ({\bf 3 points})

\begin{table}[ht]
\label{table:properties}
\caption{The Orbital Periods of the Planets}
\begin{center}
\begin{tabular}{|c|c|c|}\hline
Planet & $a$ (AU) & P (yr)\\
\hline
Mercury& 0.387 & 0.24 \\
\hline
Venus & 0.72 &  \\
\hline
Earth & 1.000 & 1.000\\
\hline
Mars  & 1.52 & \\
\hline
Jupiter& 5.20& \\
\hline
Saturn & 9.54  & 29.5 \\
\hline
Uranus & 19.22 & 84.3\\
\hline
Neptune & 30.06& 164.8\\
\hline
Pluto & 39.5 & 248.3 \\
\hline
\end{tabular}
\end{center}
\end{table}
\vspace{0.2in}
Notice that the further the planet is from the Sun, the slower it moves, and
the longer it takes to complete one orbit around the Sun (its ``year'').
Neptune was discovered in 1846, and Pluto was discovered in 1930 (by Clyde
Tombaugh, a former professor at NMSU). How many orbits (or what fraction 
of an orbit) have Neptune and Pluto completed since their discovery? ({\bf 3 
points})
\vspace{2.0in}

\subsection{Going Beyond the Solar System}

One of the basic tenets of physics is that all natural laws, such as gravity,
are the same everywhere in the Universe. Thus, when Newton used Kepler's laws
to figure out how gravity worked in the solar system, we suddenly had the tools
 to understand
how stars interact, and how galaxies, which are large groups of billions
of stars, behave: the law of gravity works the same way for a planet orbiting
a star that is billions of light years from Earth, as it does for the planets in our
 solar system.
Therefore, we can use the law of gravity to construct simulations for all types
of situations---even how the Universe itself evolves with time! For the
remainder of the lab we will investigate binary stars, and planets in binary 
star systems. \\

First, what is a binary star? Astronomers believe that about one half of
all stars that form, end up in binary star systems. That is, instead of 
a single star like the Sun, being orbited by planets, a pair of stars are
formed that orbit around each other. Binary stars come in a great variety
of sizes and shapes. Some stars orbit around each other very slowly, with
periods exceeding a million years, while there is one binary system
containing two white dwarfs (a white dwarf is the end product of the life of a 
star like the Sun) that has an orbital period of 5 minutes! \\

To get to the simulations for this part of the lab, exit the Third Law
simulation (if you haven't already done so), and click on button ``7'', the 
``Two-Body and Many-Body'' simulations. We will start with the ``Double Star'' 
simulation. Click Go. \\

In this simulation there are two stars in orbit around each other, a massive
one (the blue one) and a less massive one (the red one). Note how the
two stars move. Notice that the line connecting them at each point in the
orbit passes through one spot--this is the location of something called
the ``center of mass''. In Fig. \ref{cm} is a diagram explaining the
center of mass. If you think of a teeter-totter, or a simple balance, the 
center of mass is the point where the balance between both sides occurs.
If both objects have the same mass, this point is halfway between them.
If one is more massive than the other, the center of mass/balance point
is closer to the more massive object. \\

\begin{figure}[h]
\centerline{{\includegraphics[width=8cm]{Kepler/cmdiag.jpg}}}
\caption{A diagram of the definition of the center of mass. Here, object one
(M$_{\rm 1}$) is twice as massive as object two (M$_{\rm 2}$). Therefore, 
M$_{\rm 1}$ is closer to the center of mass than is M$_{\rm 2}$. In the
case shown here, X$_{\rm 2}$ = 2X$_{\rm 1}$.}
\label{cm}
\end{figure}

Most binary star systems have stars with similar masses (M$_{\rm 1}$ $\approx$
M$_{\rm 2}$), but this is not always the case. In the first (default) binary 
star simulation, M$_{\rm 1}$ = 2M$_{\rm 2}$. The ``mass ratio'' (``$q$'') in 
this case is 0.5, where mass ratio is defined to be $q$ = 
M$_{\rm 2}$/M$_{\rm 1}$. Here,
M$_{\rm 2}$ = 1, and M$_{\rm 1}$ = 2, so $q$ = M$_{\rm 2}$/M$_{\rm 1}$ = 1/2
= 0.5. This is the number that appears in the ``Mass Ratio'' window of the 
simulation.
 
\vspace{0.5in}

{\bf Exercise 5:} Binary Star systems. We now want to set-up some special
binary star orbits to help you visualize how gravity works. This requires
us to access the ``Input" window on the control bar of the simulation window
to enter in data for each simulation.
Clicking on Input brings up a menu with the following parameters:
Mass Ratio, ``Transverse Velocity'', ``Velocity (magnitude)'', and
``Direction''. Use the slide bars (or type in the numbers) to set Transverse Velocity = 1.0,
Velocity (magnitude) = 0.0, and Direction = 0.0. For now, we simply want
to play with the mass ratio.\\

Use the slide bar so that Mass Ratio = 1.0. Click
``Ok''. This now sets up your new simulation. Click Run. Describe the 
simulation. What are the shapes of the two orbits? Where is the center
of mass located relative to the orbits? What does $q$ = 1.0 mean? Describe
what is going on here. ({\bf 4 points})

\vspace{3.0in}

Ok, now we want to run a simulation where only the mass ratio is going to
be changed. Go back to Input and enter in the correct mass ratio for
a binary star system with M$_{\rm 1}$ = 4.0, and M$_{\rm 2}$ = 1.0. Run
the simulation. Describe what is happening in this simulation. How
are the stars located with respect to the center of mass? Why? [Hint: see Fig. 
\ref{cm}.] ({\bf 4 points})

\vspace{3.0in}

Finally, we want to move away from circular orbits, and make the orbit
as elliptical as possible. You may have noticed from the Kepler's law
simulations that the Transverse Velocity affected whether the orbit was round
or elliptical. When the Transverse Velocity = 1.0, the orbit is a circle.
Transverse Velocity is simply how fast the planet in an elliptical orbit
is moving $at~perihelion$ relative to a planet in a circular orbit of the
same orbital period. The maximum this number can be is about 1.3. If it goes 
much
faster, the ellipse then extends to infinity and the orbit becomes a parabola.
Go back to Input and now set the Transverse Velocity = 1.3. Run the simulation.
Describe what is happening. When do the stars move the fastest? The slowest?
Does this make sense? Why/why not? ({\bf 4 points})

\vspace{3.0in}

The final exercise explores what it would be like to live on a planet
in a binary star system--not so fun! In the ``Two-Body and Many-Body''
simulations window, click on the ``Dbl. Star and a Planet'' button. Here we 
simulate the motion of a planet going around the less massive star in a binary
system. Click Go. Describe the simulation---what happened to the planet?
Why do you think this happened? ({\bf 4 points})

\vspace{3.0in}

In this simulation, two more windows opened up to the right of the main 
one. These are what the simulation looks like if you were to sit on the
surface of the two stars in the binary. For a while the planet orbits one
star, and then goes away to orbit the other one, and then returns. So, sitting
on these stars gives you a different viewpoint than sitting high above the
orbit. Let's see if you can keep the planet from wandering away from its
parent star. Click on the ``Settings'' window. As you can tell, now that
we have three bodies in the system, there are lots of parameters to play
with. But let's confine ourselves to two of them: ``Ratio of Stars Masses''
and ``Planet--Star Distance''. The first of these is simply the $q$
we encountered above, while the second changes the size of the planet's orbit.
The default values of both at the start-up are $q$ = 0.5, and Planet--Star 
Distance = 0.24. Run simulations with $q$ = 0.4 and 0.6. Compare them
to the simulations with $q$ = 0.5. What happens as $q$ gets larger, and
larger? What is increasing? How does this increase affect the force
of gravity between the star and its planet? ({\bf 4 points})

\vspace{3.7in}

See if you can find the value of $q$ at which larger values cause the planet
to ``stay home'', while smaller values cause it to (eventually) crash into 
one of the stars (stepping up/down by 0.01 should be adequate). ({\bf 2 points})

\vspace{1.8in}


Ok, reset $q$ = 0.5, and now let's adjust the Planet--Star Distance. In
the Settings window, set the Planet--Star Distance = 0.1 and run a simulation.
Note the outcome of this simulation. Now set Planet--Star Distance = 0.3.
Run a simulation. What happened? Did the planet wander away from its
parent star? Are you surprised? ({\bf 4 points})

\vspace{3.1in}

Astronomers call orbits where the planet stays home, ``stable orbits''.
Obviously, when the Planet--Star Distance = 0.24, the orbit is unstable.
The orbital parameters are just right that the gravity of the parent star
is not able to hold on to the planet. But some orbits, even though the 
parent's hold on the planet is weaker, are stable--the force of gravity
exerted by the two stars is balanced just right, and the planet can happily
orbit around its parent and never leave. Over time, objects in unstable
orbits are swept up by one of the two stars in the binary. This can even
happen in the solar system. In the Comet lab, you can find some images
where a comet ran into Jupiter. The orbits of 
comets are very long ellipses, and when they come close to the Sun, their 
orbits can get 
changed by passing close to a major planet. The gravitational pull of the
planet changes the shape of the comet's orbit, it speeds up, or slows down
the comet. This can cause the comet to
crash into the Sun, or into a planet, or cause it to be ejected completely out 
of the solar system. (You can see an example of the latter process by changing 
the Planet--Star Distance = 0.4 in the current simulation.)\\

\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}\\

\subsection{Take Home Exercise (35 points total)}

On a clean sheet of paper, please summarize the important concepts of this 
lab. Use complete sentences, and proofread your summary before handing in the 
lab. Your response should include:  
\begin{itemize}
\item{Describe the Law of Gravity and what happens to the gravitational force
as {\it a)} as the masses increase, and {\it b)} the distance between the two
 objects increases} 
\item{Describe Kepler's three laws {\it in your own words}, and describe how
you tested each one of them.}
\item{Mention some of the things which you have learned from this lab}
\item{Astronomers think that finding life on planets in binary systems
is unlikely. Why do they think that? Use your simulation results to
strengthen your argument.}
\end{itemize}

\subsection{Possible Quiz Questions}
\begin{enumerate}
\item Describe the difference between an ellipse and a circle.
\item List Kepler's three laws.
\item How quickly does the strength (``pull'') of gravity get weaker
with distance?
\item Describe the major and minor axes of an ellipse.
\end{enumerate}

\subsection{Extra Credit (ask your TA for permission before attempting this,
5 points)}

Derive Kepler's third law (P$^{\rm 2}$ = $C ~ \times ~ a^{\rm 3}$) for a 
circular orbit. First, what is the circumference of a circle of radius $a$? If 
a planet moves at a constant speed ``$v$'' in its orbit, how long does it take 
to go once around the circumference of a circular orbit of radius $a$? [This is 
simply the orbital period ``P''.] Write down the relationship that exists 
between the orbital period ``P'', and ``$a$'' and ``$v$''. Now, if we only
knew what the velocity ($v$) for an orbiting planet was, we would have
Kepler's third law. In fact, deriving the velocity of a planet in an orbit
is quite simple with just a tiny bit of physics (go to this page to see
how it is done: http://www.go.ednet.ns.ca/$\sim$larry/orbits/kepler.html). Here
we will simply tell you that the speed of a planet in its orbit is 
$v$ = (GM/$a$)$^{\rm 1/2}$, where ``G'' is the gravitational constant mentioned 
earlier, ``M'' is the mass of the Sun, and $a$ is the radius of the orbit. 
Rewrite your orbital period equation, substituting for $v$. Now, one side 
of this equation has a square root in it--get rid of this by squaring both 
sides of the equation and then simplifying the result. Did you get 
P$^{\rm 2}$ = $C ~ \times ~ a^{\rm 3}$? What does the constant ``$C$'' have
to equal to get Kepler's third law?
~ ~ ~ \\
