%Small text changes by TEH on 7/9/02.
\section{Introduction}

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. 
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. 

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.8\textwidth
\epsfbox{GRAVITY/gravityfig.eps}
\end{figure}

In this lab, we will be using the {\it AIP Orbits} program using the
computers at the ``University Computing Laboratory'' to study several 
interesting properties of gravity and its effects on 
objects in orbit around each other.  This program is quite primitive compared
to modern computer games, but lets you simulate gravity experiments quite effectively.
 The program calculates the orbits of objects by using Newton's law of gravity:
\beq
F_{gravity} = \frac{GM_1M_2}{R_{1,2}^2}
\eeq

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$_{1,2}$.  The gravitational constant G is just a small number that scales the 
force.  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.

\medskip

The above equation is basically all the computer program really knows.  Our goal 
for this lab is to prove Kepler's laws about the orbits in our Solar System by 
using the simple law of gravitation above.  We will do this by creating 
various situations and watching the orbits that objects trace out in time.

\medskip

If you get confused or the computer does something funny, ask your TA to help 
you out.  The directions below will help you use the computer and the 
program to carry out a few experiments.  If you follow the instructions closely 
you should be able to answer the questions throughout the lab.

\begin{itemize}
\item{{\it Goals:}  to discuss Kepler's three laws and use them in conjunction with the {\it AIP Orbits} computer program to explain the orbits of objects in our solar system and beyond}
\item{{\it Materials:  AIP Orbits} program, calculator}
\end{itemize} 
 
\subsection{Kepler's Laws}

Before you begin the lab, it is important to recall Kepler's three laws, 
the basics of orbital mechanics which govern the orbits of planets in our
Solar System.  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 laws are:\\
\\
(1)  ``The orbits of the planets are ellipses with the Sun at one focus.''\\
\\
(2)  ``A line from the planet to the sun sweeps over equal areas in equal 
intervals of time.''\\
\\
(3)  ``A planet's orbital period squared is proportional to its average 
distance from the Sun cubed:  P$^2$ (years) = A$^3$ (A.U.)''  [Horizons, p 69.]\\

\medskip

%We will now set out to prove that these laws are consistent with Newton's law of
%gravity by running a few simple situations with our {\it Orbits} program.

\subsection{Scientific Notation on the Computer}

The numbers we will be using in this lab, such as the mass of a planet, 
are best represented in scientific notation.  The computer displays scientific
notation in a slightly different manner than what we have seen in previous 
labs.  If we want to represent the number 1000 in scientific notation, we 
would write: 1$\times$10$^3$.  However, the computer would write: 1.000E+0003, 
where the 'E' stands for exponent (or ``10 to the'').  Here are a few more examples:
\begin{center}
\begin{small}
 3.0000E+0008 = 3$\times$10$^8$ \\
 8.2000E+0004 = 8.2$\times$10$^4$ \\
 4.1000E-0001 = 4.1$\times$10$^{-1}$ \\
 1.0000E+0001 = 10\\
-5.0000E+0000 = -5\\
\end{small}
\end{center}

\section{Gravity Experiment}

\subsection{Starting Up...}
When you sit down to your computer you will have a screen prompting 
you for your username and password.  Enter your UNIX username and your social security number as the password.  Once the computer verifies that you are a current student, it will let you continue.  If you typed your username or social security number in wrong, the computer will not let you continue.  In this case, try typing both in again.

\medskip

Put the {\it Orbits} disk into the disk drive.  There should be an icon that
says``3 1/2 Floppy(A)'' - double click on this.  Find the icon in the window that says ``Orbits.exe'' and double click on it.  The program should now begin.  If everything is OK, you may proceed.  If something has gone wrong, ask your TA for help before proceeding.

If the screen saver comes on while you are working hit any key.  The screen 
saver will go away - {\bf \underline{Do not}} restart the program from the icon in the window.  Click once on the ``Orbits'' button at the bottom of the screen.

\subsection{Program Basics}
When the Introduction screen pops up, hit Enter.  You will then be
in the Scenario Selection Screen.  The scenario we will be using is
the Orbiter Scenario.  Use the arrow keys to move the highlighted
section onto the word {\bf \underline{Orbiter}}, but don't hit Enter
yet!

In this scenario, we will study an object orbiting around the Sun.  To see 
the default parameters of the scenario, hit 'e' .  A screen will pop
up showing you the physical constants in this simulation.  This is the
Data Table.  For some of the simulations you will be changing the
numbers in this table.  

Now start up the simulation by hitting the Enter key.  When the simulation 
screen pops up, take a minute to look at it.  On the left hand side are 
the ORBITER parameters which can be changed in the orbit program.  On the right is 
the graphics window.  It should show a large disk, which represents the Sun,
and a smaller dot to the right, which is the orbiter.  The orbiter could be 
Pluto, or a comet, or anything that orbits the Sun.       

\begin{itemize}
\item{You can switch the high-lighted object in the window by using 
the F2 button.  Try that now. The high-lighted object is red.}

\item{You can get back to the Data Table at any time by hitting the
Backspace key.  In order to save the parameters at any time during
the simulation, hit both the Ctrl and F2 buttons at the same time.  Then
when you back to the Data Tables the parameters will be updated.
Try that now.} 
 
\item{Once you are in the Data Table window, you can hit the Enter 
key to take you back to the simulation window.}      

\item{The F1 button will list more commands which may be very useful
for the following exercises.  Hit F1 now to see the list of commands.
If there is ever a question about which command to use, hit F1 to get
a quick list of the different commands.  From this table you may also
read the warnings about this program.  Please do so now by hitting
Enter.}

\item{To start or stop the simulation, hit the Spacebar.  Try starting the 
simulation right now.  Then stop it again.}       

\item{You may have noticed that the orbiter moves very slowly.  To
increase the speed of the orbiter, hit the F10 key several times.
This increases the time step; you should notice that the value on the
left side of the screen changes every time you hit F10.  When you
think the orbiter is moving fast enough to make 3 orbits in 1 minute
stop increasing the time step.  Do not exceed a time-step value of
3.4E+02; if you do the computer program will generate errors which may
cause the orbiter to zip off the screen.}

\item{To clear the screen of the drawn orbits, hit F7. Go ahead and
try this now.}

\item{You can turn off the orbital tracing by hitting F3. To turn it 
back on again, hit F3 again. Try this now by running the simulation so the
orbiter traces one full orbit, then stop the program, hit F3 and then hit
the Spacebar again.  The orbiter should not be tracing its orbit.  Turn
the tracing back on by hitting F3 again.}       

\item{To get back out to the Scenario Menu from the Simulation Window, hit the 
Esc key.  This takes you out of the simulation so you can start again.}  
\end{itemize}
   
\subsection{Command Summary}

\begin{tabular}{ll}
\textbf{Command}&\textbf{Function}\\
F1&Display command screen\\
F2&Change selected body\\
F3&Display orbit paths\\
F7&Clear orbits from screen\\
F9&Decrease timestep\\
F10&Increase timestep\\
CTRL-F2&Save to data tables\\
SHIFT-F1&Zoom out\\
SHIFT-F2&Zoom in\\
SHIFT-F7&Toggle data update\\
$<$space bar$>$&Stop/start simulation\\
$<$escape$>$&Returns to scenario menu\\
$<$backspace$>$&Returns to data tables\\
\end{tabular}

\subsection{Orbiter}

Using the Orbiter Scenario, we will study some of the fundamental
properties of gravitational orbits.  Enter the scenario from the
Scenario Selection Window.  In the Simulation Window we have a large
disk representing the Sun, and a smaller dot representing the orbiter.
Follow the instructions below and answer the questions as you go.

\subsubsection{Simulation I}

\begin{itemize}
\item{Hit the keys Shift and F7 at the same time.  This will make the
program update the numbers on the left hand side of the screen
continuously as the simulation progresses.}

\item{Hit the Spacebar to start the simulation.  Watch the simulation
for several orbits and then hit the Spacebar to stop the simulation.}
\end{itemize}

\bigskip

\noindent {\bf 1.} What is the shape of the orbit traced out
by the orbiter? {\it (2 points)}

\vspace*{1cm}

\noindent {\bf 2.} Is the Sun at the center of the orbit?  If not,
where is it? {\it (2 points)}

\vspace*{1cm}

\noindent {\bf 3.} Which one of Kepler's Laws does this verify? {\it (3 points)}

\vspace*{2cm}

\subsubsection{Simulation II}

The speed of the orbiter is tabulated in terms of the velocities in
the X (horizontal) direction and the Y (vertical) direction.  Thus if
an object were moving straight up or down, it would have an X (horizontal)
velocity equal to 0.  If an object moves in the horizontal direction,
it's Y (vertical) velocity is 0.  Since the velocities are based on X and Y 
axes, the orbiter will have a positive X velocity if it is moving to the right and a negative X velocity if it is moving to the left.  Similarly, if the orbiter is moving up, it will have a positive Y velocity, and if it is moving down, it will have a negative Y velocity.  
 
\begin{itemize}
\item{Start the simulation again.  Try to stop the orbiter at its
furthest point from the Sun by hitting the Spacebar.  If the orbiter is
going too fast, hit F9 several times to slow it down.}

\item{Save the orbital data to the data tables by hitting Ctrl and F2
at the same time.  Then go to the Data Tables by hitting the
Back-Space key.}

\item{Write down the XVEL and YVEL for the orbiter (body \#2) below.}

\vspace*{1cm}

\item{Exit back to the simulation by hitting the Enter key.  Then
repeat this procedure, but now stop the orbiter at its closest approach
to the Sun.}

\item{Write down the XVEL and YVEL for the orbiter (body \#2) below.}

\vspace*{1cm}

\item{Look at the difference between the velocities XVEL and YVEL in
the two cases and then answer the questions below.}
\end{itemize}  

\bigskip

\noindent {\bf 4.} Where does the orbiter move fastest?  Where does it
move slowest? {\it (2 points)}

\vspace*{1cm}

\noindent {\bf 5.} How many times faster does the orbiter move when
closer in than when further out [divide YVEL(close) by YVEL(far) to
give you the ratio of the velocities]? {\it (3 points)}

\vspace*{1cm}

\noindent {\bf 6.} Describe this motion in terms of one of Kepler's
laws. {\it (3 points)}

\vspace*{2cm}
 
\subsubsection{Simulation III}

Now we are going to adjust the major axis of the orbiter's orbit.  This
will change the distance between the Sun and the orbiter in our
simulation.

\begin{itemize}
\item{Press Esc to go back to the Main Menu and select ``Orbiter''
again.  This will reset the simulation and change all the parameters
back to their default values.}

\item{Once you're back in the Simulation Window, write down the period
value (shown on the left window under the ORBITER parameters) below.}

\vspace*{1cm}

\item{Next, press Backspace to take you to the data table.  Change the
orbiter's (body \#2) "XPOS" to twice its default value.  This makes
the major axis of the orbit twice as long.  Be careful with the
scientific notation.}

\item{Restart the simulation by hitting the Spacebar and see what
happens.  (You may have to press Shift and F1 to zoom out.)  Watch the
orbiter for a few orbits around the Sun and then answer the questions
below.}
\end{itemize}

\bigskip

\noindent {\bf 7.} What is the new period of the orbiter? {\it (2 points)}

\vspace*{1cm}

\noindent {\bf 8.} Compare this to the period you wrote down before the
simulation was run.  Which one is larger?  What has happened to the
shape of the orbiter's orbit? {\it (3 points)}

\vspace*{1cm}

\noindent {\bf 9.} Which of Kepler's Laws does this verify?  {\it (3 points)}

\vspace*{2cm}

\subsubsection{Simulation IV}

Now we are going to change some of the other physical parameters of
the orbit simulation and see what changes occur.
 
\begin{itemize}
\item{Press Esc to go back to the Main Menu and select ``Orbiter''
again.  This will reset the simulation and change all the parameters
back to their default values.}

\item{Go to the data table by hitting Backspace.  Use the Tab button
to move over to the mass entry for body \#2 (the orbiter).  Increase
the mass to be the same mass as the body \#1 (the Sun) by typing over
the old number.  Do NOT change any other parameters at this time.}

\item{Exit back to the simulation by hitting Enter.}

\item{Start up the simulation and watch the orbit.  Check that the
time step does not exceed the upper limit of 3.4E+02.  Answer the
question below once the Earth has made a couple of orbits.}
\end{itemize}     

\noindent {\bf 10.} Now that the orbiter and Sun have the same mass, do
you see any difference in the orbit?  Describe the new orbit.  What is
going on here? {\it (6 points)}

\vspace*{3cm}

\subsubsection{Simulation V}
Now we are going to see the effect that our Sun's mass has on the
Earth's orbit and the orbits of all the planets.  As a familiar case,
we are going to look at the Earth's orbit around the Sun.

\begin{figure}[h]
\centering
\leavevmode
\epsfxsize=0.55\textwidth
\epsfbox{GRAVITY/Gravity_fig2.eps}
\end{figure}

\begin{itemize}
\item{First, press Backspace to go back to the data tables.  Enter in
the appropriate values for the Earth and Sun as listed below.  The Sun is 
body\# 1 and the Earth is body \# 2.   Refer to the above figure if you are
not sure which values in the table to change.  The
values in the table reflect the true masses, sizes and distances in
the Sun-Earth system:}
\vspace*{0.5cm}

Mass$_{Sun}$ = 1.99$\times$10$^{30}$ kg \\
Radius$_{Sun}$ = 6.96$\times$10$^{6}$ km \\
Mass$_{Earth}$ = 5.98$\times$10$^{24}$ kg \\
Radius$_{Earth}$ = 6.4$\times$10$^{3}$ km \\
Distance$_{Sun-Earth}$ = 1AU = 1.49$\times$10$^{8}$ km \\
Velocity$_{Earth}$ = 30 km/s.  \\

\vspace*{0.5cm}

\item{Press Enter to return to the simulation.  You will need to zoom
out (Shift F1) to see the entire orbit.  Also, the timestep needs to
be small so the program can calculate the orbit very accurately.
Change the timestep to 5.00E$-$03 using the F9 key to decrease the
timestep or the F10 key to increase it.}

\item{Hit the Spacebar and watch the simulation for a few orbits.}
\end{itemize}

\noindent {\bf 11.} What shape is the Earth's orbit around the Sun? (Hint:  Look at the eccentricity on the side of the screen.)
{\it (5 points)}

\vspace*{2cm}

\subsubsection{Simulation VI}

Now let's try something interesting.  What would happen to the Earth if the Sun 
were replaced by a black hole?  You may have heard of black holes before - objects
which, because of their strong gravitational pull, ``suck in'' everything that comes 
near them.  You might think that if the Sun were replaced by a black hole of
the same mass, the Earth and all the planets would be pulled into the black hole.  
Let's see if this is the case.

\begin{itemize}
\item{Stop the simulation and go back to the data table.  Check that
the masses, sizes and distances in the table on the screen are the
same as those listed in the table above.}

\item{Now change the radius of the Sun (body \#1) to 10 km.  This is
the typical radius of a black hole.  Our black hole will have the
\begin{bf}same mass\end{bf} as the Sun but will be much smaller than the Sun.}

\item{Run the simulation and watch what happens to the orbit of the Earth.}
\end{itemize}

\noindent {\bf 12.} What happened to the orbit of the Earth once you
replaced the Sun with a black hole?  Did the Earth fall in? {\it (2 points)}

\vspace*{1cm}

\noindent {\bf 13.} Explain why or why not the Earth fell into the
Sun.  {\it (6 points)}

\vspace*{3cm}

\noindent {\bf 14.} As a final question about orbits, let's apply
Kepler's Third Law to the Jupiter-Sun system.  Jupiter orbits the Sun
at a distance of 5.2 astronomical units (A=5.2) Knowing this distance
you should be able to determine Jupiter's orbital period around the
Sun.  

\vspace*{0.5cm}

a)  How long does it take (in years) for Jupiter to orbit the Sun
once? {\it (8 points)}

\vspace*{2.5cm}
 
b)  Now that you know Jupiter's orbital period, let's calculate how fast it is 
travelling through space.  The period of an object is defined as the time it 
takes to complete one orbit.  Therefore, $P=\frac{2\pi A}{v}$  where, in 
this case, $2\pi$=6.28, A=5.2 AU (as above), P is the period you just found in a), and v is 
the velocity of Jupiter.  Calculate how fast Jupiter is moving around the Sun.
Express your answer in $AU/year$ and then convert it to $miles/hour$  
{\bf (1 AU = 9.32 x $10^7$ miles)}.  Is Jupiter moving faster or slower than the average speed of a car on a highway?
{\it (15 points)}

\vspace*{2.5cm}

\section{Summary}

{\it (35 points)}  Please summarize the important concepts of this lab.  your summary should include:  

\begin{itemize}
\item{Describe the Law of Gravity and what happens to the gravitational force
as {\it a)} the mass increases, and {\it b)} the distance between the 2 objects increases} 
\item{Describe Kepler's three laws {\it in your own words}}
\item{Mention some of the things which you have learned by doing this lab}
\end{itemize}

\noindent
Use complete sentences, and proofread your summary before handing in the lab.

%\end{document}
