Satellite Science: How Does Speed Affect Orbiting Altitude?

In this astronomy science project, you will use a model to study the trajectory of a satellite when varying its speed or the mass of the planet it orbits. An example of the satellite trajectory model you will build is shown in Figure 7.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 7. A finished satellite launcher used to study the trajectory of satellites as they revolve around planets.

You will roll a marble down a paper tube onto a cookie sheet. Tilting the cookie sheet will cause the marble to skew to one side, simulating the gravitational pull of a planet. A larger slope simulates a bigger gravitational pull. Changing the slope of the tube will change the initial speed of the marble, simulating different satellite speeds. First, you will set up your model and test if it works the way you want it to work. Only then can you use your model to find answers to your questions.r

Setting Up a Satellite Trajectory Model

1. Create a trajectory panel:
   1. Set the cookie sheet down, with the long, rimmed edge farthest from you. Place one sheet of finger-paint paper on your cookie sheet, aligning it with the long, rimmed side and the right side, as shown in Figure 8.
   2. Attach the piece of paper with tape to the cookie sheet.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 8. Place the paper on the cookie sheet so it lines up with the rimmed edge and the right edge of the cookie sheet.

2. Add the measuring indicators to your trajectory panel:
   1. Draw a pencil line on the top of your paper, parallel to and 2 cm below the rimmed edge of your cookie sheet. This line will be referred to as the horizontal track line.
   2. Draw a line parallel to the right edge of your paper that is 3 cm from the edge. You will read the deviation from a straight trajectory on this line.
   3. Create marks on your vertical line to help read the deviation from a straight trajectory:
      1. Write "0" where the line you drew in step 2.a. and the line you drew in step 2.b. intersect.
      2. From the 0 point down, draw small horizontal marks every centimeter and bigger marks every 5 cm.
      3. Add numbers to the bigger marks.
   4. Mark the spot on the horizontal track line that is 23 cm from the right side of your cookie sheet. The launch tube will end at that location.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 9. A vertical line with measuring marks will help you read the deviation from a straight trajectory.

3. Create a tube to launch your satellite (marble):
   1. Take a second piece of finger-paint paper.
   2. Cut off one end so the paper becomes a rectangle that is 40 cm long.
   3. Roll this paper lengthwise and slide it inside the cardboard tube to create a tube of length 40 cm, as shown in Figure 10.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 10. A rolled-up paper held in place by a cardboard tube is used as a launch tube.

4. Put the launch tube in place: The elevation difference between the two ends of your launch tube will give your satellite varying speeds. You will use clay to create an elevation difference of 5 cm from one end of the 40 cm launch tube to the other. Use Figure 11 to guide you as you place your tube.
   1. Form a ball of clay that is approximately 5 cm in diameter.
   2. Place the ball near the left edge of your cookie sheet on the horizontal track line.
   3. Push the middle of the clay ball down a little to create a U shape that is the width of the launch tube. The "U" should be facing the short left edge of your cookie sheet.
   4. Let the launch tube rest in the U shape of the clay ball, with one end of the tube on the mark indicated in step 2.d.
   5. Measure the elevation of the left end of the tube, as shown in Figure 11.
      1. It is easiest to measure the elevation difference over the length of the tube by comparing the elevation of the lowest points of each end of the tube. Note that the lowest point of the right end is at elevation 0, resting on the cookie sheet.
      2. Hold your metric ruler perpendicular to the elevated tube end, then place a pencil horizontally just below that tube end to measure the elevation.
   6. Adjust the clay ball until the elevation gain over the length of the tube is 5 cm.
   7. The U shape of the clay should keep the tube solidly in place, but you should add tape to secure the tube to the cookie sheet so it keeps its position.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 11. The launch tube is supported by a U-shaped lump of clay placed on the left side of the cookie sheet. A pencil helps accurately measure the elevation gain over the length of the launch tube.

5. Create your elevation blocks to elevate the launch side of the cookie sheet. Elevating this side of the cookie sheet will create a downward pull on the launched satellite (marble), modeling the gravitational pull from a nearby planet. Skip past the next few images to see an example of the elevated blocks in Figure 15.
   1. Use clay to form two identical pillars that are each 4 cm high. These will model a medium gravitational pull, like the pull created by Saturn.
   2. Form two identical pillars that are 12 cm high. These will model a strong gravitational pull like the pull created by Jupiter. Note: If you find it hard to create stable 12 cm high pillars with the clay, you can use a cup or glass and add clay below or on top (to hold the cookie sheet in place) to create a 12 cm high pillar.

The table summarizes the parts of the satellite trajectory model and what they represent in space. Keep it nearby as you launch your satellites (marbles) and try to understand how the model works.

Testing Your Satellite Trajectory Model

To test if the model does model real-life trajectories of satellites well, you will:

• Fine-tune your model to represent that the satellite's trajectory follows a straight line when no gravitational pull is present.
• Observe if the satellite's trajectory shows an increasing deviation from the straight-line trajectory with increasing gravitational pull.

1. First, fine-tune using the "no gravity present" (zero gravity) situation:
   1. If it isn't already, put the cookie sheet flat on your workspace to model the case where no gravity is present.
   2. Take a marble and launch this satellite by letting it roll through the tube. For an ideal launch, you should:
      1. Release the marble on the "floor" of your circular tube so it takes a straight path down inside the tube; in other words, do not drop the marble inside the tube, but place it on the inside edge and let it roll.
      2. Release the marble without giving it an extra boost so you do not increase the speed of the satellite as it exits the tube.
   3. Observe if the marble rolled along a straight line parallel to the edge of the cookie sheet edge. If not, evaluate if you need to:
      1. Adjust your tube so it points straight out, parallel to the cookie sheet edge, or
      2. Practice releasing the marble from the position described in step 1.b.

Repeat steps 1.b and 1.c. until your satellite consistently follows a straight line when released. Note that trajectories are made visible in Figure 12 to illustrate how trajectories should look when no gravity is present.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 12. Pencil indicating from where to release the marble so it follows a straight path down the launch tube (left) and the satellite trajectory model ready to be used (right).

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 13. Illustration of trajectories created for zero gravity by launching marbles in a misaligned launch tube (top yellow lines) and in a well-placed launch tube (bottom yellow lines).

2. As shown in Figure 13, above, you will use colored tracks to observe the various paths your satellites take. To create the colored tracks:
   1. Take three plastic lids, pour some yellow liquid food coloring in the first, blue in the second and a mixture of yellow and blue in the third (to create a green color). An example is shown in Figure 14.
   2. Have your marble and some paper towels handy.
   3. Place some protective paper (such as newspaper) at the end of the cookie sheet to collect the colored marble as it rolls off the cookie sheet, as can be seen in Figure 12, above. Try to ensure the marble continues to roll away from the cookie sheet so it does not roll or bounce back and create a "roll back" track on your finger-paint paper.
   4. Put on the gloves.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 14. Marble trajectories are created by first rolling the marbles in food coloring, as shown in this picture.

3. Create the "zero gravity tracks" in yellow:
   1. Roll a marble in the yellow food coloring. Make sure it has food coloring all over it, but does not drip.
   2. Launch this marble.
   3. Pick up the marble and put it back on the yellow food coloring tray.
   4. Wait for about 30 seconds so the food coloring in the cardboard tube and on the paper can dry.
   5. Repeat steps 3.a.–3.d. three times for a total of four launched satellites in the zero-gravity circumstances.
      1. If one launch exactly follows the track of a previous launch, use a pencil to keep track of the number of launches that followed that track by placing a tally mark next to the track.
   6. Clean your marble with paper towels.
   7. Wait at least 1 minute before launching marbles of a different color (see step 4) so all the color inside the cardboard tube and on the paper can dry.
4. Create the "Trajectories deflected by Saturn" in green:
   1. Place the 4 cm pillars under the long, rimmed side of the cookie sheet to which the launch tube is attached. Make sure your cookie sheet is stable. Figure 15 will help you visualize how to place the pillars. This elevation will create a downward pull on the marble, deflecting it from a straight-line trajectory. This models the deflection due to a nearby planet.
   2. Repeat steps 3.a.–3.g. using green food coloring this time.
   3. Observe the tracks being deflected from a straight trajectory. Is that in alignment with what you expected?

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 15. Pillars elevate one side of the cookie sheet to model the gravitational pull on satellites near a planet.

5. Create the "Trajectories deflected by Jupiter" in blue:
   1. Switch the 4 cm high pillars for the 12 cm high pillars. This elevation models a stronger gravitational pull of a planet like Jupiter nearby.
   2. Repeat steps 3.a.–3.g., now using blue food coloring.
   3. Are these tracks more deflected than the green tracks? Is this in line with what you expected?
6. Check if your model represents reality. Figure 16 can help you with this task.
   1. Check if the yellow tracks (representing the "zero gravity present" situation) creates straight trajectories.
   2. Verify if the blue and green trajectories (created when gravity is present) show a deviation from the straight line.
   3. Verify if the blue trajectories (created with the largest gravitational pull used in this model) show the largest deviation from the straight line.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 16. Results showing that the model acts the same as a satellite in space in important ways.

Studying the Effect of Changing Speed

Once you are sure your model works well, you can use it to study the effect of changing speed on the deflection of trajectories.

1. Create a larger elevation difference over the length of the launch tube. The speed of your marble (satellite) as it comes out of the launch tube depends on the difference in height between the two ends of the launch tube. You will increase this difference from 5 cm to 10 cm.
   1. Remove the tape holding your launch tube so you can lift the left end of the tube.
   2. Add a chunk of clay under the U-shaped clay ball currently supporting your launch tube.
   3. Measure the elevation difference of the tube, as shown in Figure 11, above.
   4. Adjust until the elevation difference over the length of the tube equals 10 cm.
   5. Secure the launch tube with tape.
2. Take three more clean plastic lids, pour some red liquid food coloring in the first, create a mixture of red and blue in the second (to create a purple color) and yellow with a drop of red in a third (to create a light orange).
3. Create the "Trajectories for a high-speed satellite deflected by the planet Jupiter" in red:
   1. Make sure you still have the 12 cm high pillars in place, modeling a strong gravitational pull, like the one from planet Jupiter.
   2. Repeat steps 3.a.–3.g. from the previous section Testing Your Satellite Trajectory Model, now using red food coloring.
   3. What do you observe?
4. Create the "Trajectories for a high-speed satellite deflected by the planet Saturn" in purple:
   1. Switch to 4 cm high pillars, modeling a medium gravitational pull, like the one from planet Saturn.
   2. Repeat steps 3.a.–3.g. from the previous section Testing Your Satellite Trajectory Model, now using purple food coloring.
5. Create the "Trajectories for high-speed satellites when no gravity is present" in orange:
   1. Remove the pillars so the cookie sheet lays flat, modeling the absence of gravitational pull (zero gravity).
   2. Repeat steps 3.a.–3.g. from the previous section Testing Your Satellite Trajectory Model, now using orange food coloring.

Analyzing Your Data

1. Once you are done creating trajectories, remove your paper from the cookie sheet. Do not forget to add a legend explaining the color code used.
2. Copy the data table in your lab notebook.

3. Measure the deviations of the "no gravity present" trajectories:
   1. Measure the deviations of each of the four yellow trajectories (tracks).
   2. Record your measurements in the second column of your data table.
   3. Calculate the mean value of your four measurements and record it in the last line of your table. Note: If this number (your mean deviation for the case without gravity) is not 0 (or almost zero), you will need to correct all further deviation readings for this "offset". You do this by subtracting the value you just calculated (the mean deviation for the case of zero gravity) from each reading.
4. Repeat step 3 for the green, blue, orange, purple, and red trajectories (columns 3–7).
5. Create a graphical representation of your measurements:
   1. Create graphs showing how the deviation changes with changing gravitational pull, keeping the speed constant.
      1. Use the mean data from "low speed" columns of your data table.
      2. Graph the deviation on the vertical (y) axis and the level of gravitational pull on the horizontal (x) axis.
      3. Does this graph support that your model works in important ways like satellites in space do?
      4. Repeat this graph using the "high speed" columns of your table. Do you see the same trend of increasing deviation with increasing gravitational pull?
      5. Optional: Add the trails data with a different, smaller mark to your graph.
   2. Create graphs showing how the deviation changes with changing speed, keeping the gravitational pull constant.
      1. Graph the mean deviation on the vertical (y) axis and the level of speed setting on the horizontal (x) axis. Start with the trials where gravity is absent (yellow and orange tracks).
      2. Create the same graph for the trials where a medium gravitational pull (like Saturn) is present (green and purple tracks) and high gravitational pull (like Jupiter) is present (blue and red tracks).
      3. Do these graphs support what you expected?
      4. Optional: Add the trails data with a different, smaller mark to your graph.

Additional Information

Although this model works well, it is important to note that it is not an exact miniature replica of the real thing. The Introduction mentioned leaving out any perturbations. In addition, the gravitational pull created by elevating one side of the cookie sheet always creates a downward pull (toward the center of Earth). Because your marble track is very short, this direction is essentially constant. However, a satellite travels a very long trajectory through space. The gravitational pull it feels is also directed to the center of the planet, but changes as it travels through space. Because of this, your marble will always follow a parabolic trajectory, while a real satellite in orbit can follow a circular, an elliptical, a parabolic, or a hyperbolic path. As shown in Figure 17, this model works when you study only the part of the trajectory close to the launch point. In this science project, the border of your paper limits you to studying only the part of the trajectory that is relatively close to the launch point.

Image Credit: Sabine De Brabandere, Science Buddies / Science Buddies
Figure 17. Gravitation of Earth pulls the marble in your model straight down, as shown on the left. This represents the gravitational pull of a planet on a satellite shown on the right. The representation only holds when looking at a limited area close to the launch point of the marble, as indicated by the rectangle.

Note also that the gravitational pull felt by the marble is essentially constant in value since there is a negligible variation in the distance of the marble to the center of Earth. This is also the case for a satellite in circular orbit around a planet. It is not the case, however, for an elliptical orbit or a parabolic or hyperbolic path. In those cases, the varying distance from the center of the planet will cause the force of gravity acting on the satellite to change not only in direction but also in value. The relationships between mass of the planet, altitude, and velocity found in this science project will still hold, but the altitude and velocity will constantly change along the path of the satellite.

As a last note, when you lift one side of the cookie sheet much more than 12  cm in order to model a larger gravitational pull, your model (as described in this science project) might not work as well. As you lift the cookie sheet, your launch tube moves out of the vertical plane and the effective elevation gain over the length of the tube (which creates the speed) changes. You will need to adapt your model so the vertical elevation gain over the length of your tube stays constant as you lift the cookie sheet.