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Next: Kepler's Laws and Gravitation Up: AY110 lab manual Previous: Introduction to the Geology

Subsections

Kepler's Laws and Gravitation I.

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

In this lab, we will be using the 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:

Fgravity = $\displaystyle {\frac{{GM_1M_2}}{{R_{1,2}^2}}}$ (6.1)

Here F gravity is the gravitational attractive force between two objects whose masses are M1 and M2. The distance between the two objects is R1, 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 squared and is in the denominator of the fraction. There are several laws like this in physics and astronomy.


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.


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.

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: P2 (years) = A3 (A.U.)'' [Horizons, p 69.]


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: 1×103. However, the computer would write: 1.000E+0003, where the 'E' stands for exponent (or "10 to the''). Here are a few more examples:

3.0000E+0008 = 3×108
8.2000E+0004 = 8.2×104
4.1000E-0001 = 4.1×10-1
1.0000E+0001 = 10
-5.0000E+0000 = -5

Gravity Experiment

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.


Put the 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 - Do not restart the program from the icon in the window. Click once on the "Orbits'' button at the bottom of the screen.

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 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.

Command Summary

Command 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

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.

Simulation I


1. What is the shape of the orbit traced out by the orbiter? (2 points)




2. Is the Sun at the center of the orbit? If not, where is it? (2 points)




3. Which one of Kepler's Laws does this verify? (3 points)







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.


4. Where does the orbiter move fastest? Where does it move slowest? (2 points)




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]? (3 points)




6. Describe this motion in terms of one of Kepler's laws. (3 points)







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.


7. What is the new period of the orbiter? (2 points)




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? (3 points)




9. Which of Kepler's Laws does this verify? (3 points)







Simulation IV

Now we are going to change some of the other physical parameters of the orbit simulation and see what changes occur.

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? (6 points)









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

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







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.

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




13. Explain why or why not the Earth fell into the Sun. (6 points)









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.



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








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 (1 AU = 9.32 x 107 miles). Is Jupiter moving faster or slower than the average speed of a car on a highway? (15 points)








Summary

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

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

next up previous
Next: Kepler's Laws and Gravitation Up: AY110 lab manual Previous: Introduction to the Geology
Tom Harrison 2008-07-09