Planetary Orbits and Kepler's
Laws
This Lab requires the use of the computer program called Orbit
Maker - it can be found on the computers in any CAC PC lab on
campus.
I. Kepler's 1st Law: Planets orbit the sun in ellipses
(with the sun located at one of the foci)
Here's an Ellipse:
- Semi-Major Axis (a): Half the length of the
longest dimension (side)
- Semi-Minor Axis (b): Half the length of the
shortest dimension (side)
- Eccentricity (e): A number between 0 and 1 that describes
how "squashed" the ellipse is (an ellipse with a small
eccentricity, near 0, is very round, whereas an ellipse with a large
eccentricity, near 1, is very elongated)
Here are some ellipses for you:
a.
b.
c.
Questions (note: you do not need to measure anything, this is
qualitative)
- List these ellipses in order of INCREASING eccentricity (i.e.
write: "a, b, c", or "c, b, a", or ...).
- List these ellipses in order of INCREASING semi-major axis (i.e.
write: "a, b, c", or "c, b, a", or ...).
II. Kepler's 2nd Law: The line between a planet and the Sun
sweeps out equal areas in equal times.
This law is a bit strange as stated, however, its meaning should become
clear soon. In a weird way, it qualitatively describes
the speed of a planet at different parts of its orbit.
Using Orbit Maker, set up the following orbit. The Sun (or
some other star) is represented by "star1",
"star2" is a planet. Adjust the Scale setting so that you
can see the whole orbit, and set the Step value so that one orbit is
completed in a reasonable amount of time.
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
5 |
0 |
- As you watch the planet orbiting around the Sun, describe the shape of
its orbit. According to Kepler's 1st Law, the Sun is located
at one of the foci of the planet's elliptical orbit. What is located at the
other one?
- Does the planet travel at the same speed during the entire orbit? If
not, describe how its speed changes at different points in its orbit (i.e.
describe where it is when it's going fastest, and where it is when it's going
slowest). Where does the planet spend most of its time - close to the Sun,
far from the Sun?
Adjust the Step value so that it takes roughly 30-60 seconds
for the planet to complete its orbit. Rest the orbit.
- Draw an ellipse that roughly corresponds to the ellipse on the
screen. Include the location of the Sun and the initial location of
the planet (please label these points). Start the motion; let the
planet orbit for about 5 seconds then stop it. Mark its new position
on your ellipse. Keep incrementing its orbit by 5 sec intervals and
continue to record its position each time, until the orbit is
complete. Draw lines between each location of the planet and the Sun
(just like in the figure at the beginning of this section).
- Look at the area contained in each sector (between two lines).
Are they comparable (the same)? How does this fit in with Kepler's
2nd Law? Based on your observations, do you think Kepler's
2nd Law is correct?
III. Kepler's 3rd Law: For any planet in the Solar
System, P2 = a3
Where:
P = orbital period (how long it takes to finish one orbit) in
YEARS (Earth-years, that is)
a = semi-major axis (the average distance between the Sun and
the planet) in ASTRONOMICAL UNITS (AU)
This law describes quantitatively, how a planet orbits the Sun. It says
that as the average distance between the planet and the Sun INCREASES (or as
a gets larger), then the time it takes for the planet to complete
its orbit also INCREASES (P gets larger).
All right, let's make the Solar System. Well, part of it
anyway...
Enter these settings ("star1"=Sun,
"star2"=Venus, "star3"=Earth,
"star4"=Mars, "star5"=Jupiter):
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Venus) |
0.0001 |
0.723 |
0 |
0 |
0 |
7.39 |
0 |
| star3 (Earth) |
0.0001 |
1 |
0 |
0 |
0 |
6.28 |
0 |
| star4 (Mars) |
0.0001 |
1.52 |
0 |
0 |
0 |
5.1 |
0 |
| star5 (Jupiter) |
0.0001 |
5.21 |
0 |
0 |
0 |
2.75 |
0 |
- Reset the screen and notice where the Earth ("star3") is
on the right side along with all the other planets. Start the planets
orbiting, let the Earth make a complete orbit, then stop the motion;
notice where the other planets are in their orbits. Which planets
have completed at least one orbit? Where is Jupiter
("star5") along in its orbit? (Has it gone very far?)
Which planets complete their orbits the fastest: those closer to the
Sun (inner planets), or those farther away (outer planets)?
- How does the motion you see relate to Kepler's 3rd Law
(the equation P2=a3)? Does what
you see match what the equation predicts? (What does the equation
predict? - HINT: the x parameter in Orbit Maker is related to
the distance from the Sun measured in AU's.)
IV. Orbital Mechanics
These next parts of the lab show how different physical parameters
affect the shapes of planetary orbits. Rerun the planet from section
II again; here are the settings from Orbit Maker:
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
5 |
0 |
Make sure that you can see the whole orbit, and that the planet takes
a reasonable amount of time to complete its orbit
Watch the planet orbit for a moment - you will be comparing the period
and shape of this orbit to other ones.
First let's increase the mass of the Sun and see what effect that
has on the orbit of the planet. Change Orbit Maker so that it
has the following settings (BE SURE TO USE THE SAME SCALE AND STEP VALUES THAT YOU HAD FOR THE PREVIOUS ORBIT!):
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
2 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
5 |
0 |
- How does the orbit of this planet compare with the other?
(i.e. Is it larger/smaller? More/less eccentric? Does the planet move
faster/slower?) Describe any changes you observe.
- Now examine how different initial velocities effect the orbits.
Change Orbit Maker to these settings.
(NOTE: the initial velocities, vy,
are different), but the distances, x, are the same, and the Sun's mass
is back to normal)
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
5 |
0 |
| star3 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
4 |
0 |
| star4 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
3 |
0 |
This time play around with the Scale and Step and adjust accordingly.
Make sure that you can entirely see all three orbits.
Compare qualitatively the motions of the three planets. (Make sure you
include size of orbit, eccentricity, orbital period, etc.)
- Now, to determine how different semi-major axes effect the
orbits, create these settings. (NOTE: this time, the distances, x,
are different and the velocities, vy, are the same)
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Planet) |
0.0001 |
3 |
0 |
0 |
0 |
3 |
0 |
| star3 (Planet) |
0.0001 |
5 |
0 |
0 |
0 |
3 |
0 |
| star4 (Planet) |
0.0001 |
7 |
0 |
0 |
0 |
3 |
0 |
As with the previous problem, compare the three orbits qualitatively.
Include the same information as the previous problem. Are the changes in
the three orbits the same as the previous problem, or are they different?
- Summarize by explaining how the mass of the Sun, the distance from
the Sun, and the initial velocity of the planet (i.e. its initial
vy) affect the shape of the orbit and the time it takes to
complete the orbit.
V. Optional Section: Binary Star Systems, and Fiddling with the
Solar System (not a good idea!!!)
- This question is a qualitative question dealing with some really
weird star systems. Not all star systems are "nice" like
our own Solar System, some have very strange and unusual orbits (but
they can still be described by modifications of Kepler's Laws!). The
following are just a few examples.
- Here is a binary star system (two stars orbiting each other), with
one star a lot more massive than the other. Create these settings in
Orbit Maker:
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 |
30.7 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 |
1.25 |
0.8 |
0 |
0 |
0 |
39.72 |
0 |
Describe what the orbits in this star system look like.
- Now enter these settings for another binary star system, which is
made up of two stars that are the same mass:
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 |
1.25 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 |
1.25 |
1 |
0 |
0 |
0 |
5 |
0 |
Describe what the shapes of the orbits in this system are like (are they
still ellipses? How do their semi-major axes compare?). Compare it to the
system in part a.
- Recently there has been a lot of talk about planets discovered
around other Sun-like stars (extra-solar planets). These new
"solar systems" are very different from our own. Almost all
of them have planets the mass of Jupiter (or larger) in orbits much
closer to their star than Jupiter is to our Sun. Additionally many of
the planets have very elliptical orbits. We are going to
"magically" transport one of those extra-solar planets to
the Solar System and let it orbit around the Sun with the other
planets. Your job will be to see what effect it has on the orbits of
the inner planets (Venus, Earth, and Mars). Here are the settings for
the inner Solar System planets again: "star1" is the Sun,
"star2" is Venus, "star3" is the Earth, and
"star4" is Mars. This time "star5" will be the
extra-solar planet.
Enter these settings into Orbit Maker:
| Name |
Mass |
x |
y |
z |
vx |
vy |
vz |
| star1 (Sun) |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
| star2 (Venus) |
0.0001 |
0.723 |
0 |
0 |
0 |
7.39 |
0 |
| star3 (Earth) |
0.0001 |
1 |
0 |
0 |
0 |
6.28 |
0 |
| star4 (Mars) |
0.0001 |
1.52 |
0 |
0 |
0 |
5.1 |
0 |
star5 (massive planet
with
small, eccentric orbit) |
0.005 |
0.3 |
0 |
0 |
0 |
15 |
0 |
Let Orbit Maker run, watch how the system changes. Write down
your observations.
(Note: you may want to increase the Step value so the planets orbit
rather fast, also try turning the trails on and off periodically to
get a better sense of how the orbits change.)
Here are some things to consider in your observations: What effect
does this new planet have on the inner Solar System? What is the
ultimate fate of Venus? What about Mars? How about the Earth? Does
this new planet make the Solar System a "nice place to
live"? What do you think happened to any possible
"Earths" that may have been formed in that planet's own
"solar system"? Any other comments?
After the system has "settled down" zoom in on the sun (go
to something like Scale = 1 AU), what effect does this new planet
("planet5") have on the sun? (Note: astronomers can observe
this effect by carefully watching other stars, and are using it to
find more extra-solar planets - they actually can see it as varying
Doppler shifts in the spectra of the star.)
More info on extra-solar planets can be found by visiting:
- Summarize (on a separate sheet) the facts and ideas presented in this lab, including nay additional questions you may have.
Also visit:
Doomsday
Asteroid - this is a JAVA program that works a lot like Orbit Maker,
except the planets are all ready entered (you can choose which ones to
display), you get to play around with an asteroid (enter the parameters
right and you can get the asteroid to crash into the Earth, Sun, or other
planets) you can also change your viewpoint and see what an "Earth(or any
other planet)-centered" solar system would look like (my what a mess!).
*This lab is based off of the original lab written by
Karen Lewis, Oct. 1999
