In this lab, we'll explore the phenomenon of orbital resonances and how they effect bodies in and around the asteroid belt. Like all the worlds in the Solar System, asteroids orbit the Sun. However, they are close enough to Jupiter, the largest world in the Solar System, that its gravity has significant effects on their orbital motion. This lab will help us to discover some of these effects.
The applet below simulates the orbital motion of bodies in the Solar System. Running it is pretty straightforward. First, click on the "Reset" button. Several things should appear on the screen. The Sun is the yellow circle in the center, the gray ring represents the approximate extent of the asteroid belt (about 2.3-3.3 AU), and the orange circle is Jupiter's orbit (at 5.2 AU). Jupiter always starts at the top of the orange circle, directly above the Sun.
Next, move the mouse over the graphic of the Solar System. The box next to the "Reset" button shows the distance in AU of the mouse pointer from the Sun. To start the simulation, pick a point and click there once. This puts an asteroid where you clicked and starts it on a circular orbit around the Sun. As the simulation runs, it traces out the path of the asteroid. Note that the asteroid may not stay on its initial circular orbit; rather, its orbit may shift, leading to a white smear showing its path. We'll explore why an asteroid may behave this way in the questions.
The applet has two more features that you'll need to use. Leaving the simulation running, check "Show Motion". The orange circle and gray ring should disappear, and you should see white and orange dots moving around. The white dot is the asteroid, and the orange dot is Jupiter. In this mode, you're seeing what the motion of these bodies would look like from above the Solar System (greatly accelerated, of course). Now, select "Slow" from the menu. The asteroid and Jupiter should orbit much more slowly, allowing you to see their motion in greater detail. Finally, if you now check "Trace orbit", Jupiter should stop moving, and the program will resume tracing the asteroid's path. To restart the program at any time, click "Reset".
In this lab, you'll use the simulation to study the motion of asteroids. Answer all the numbered questions on your own paper.
NOTE: For some reason, scrolling the applet out of the window erases it. I know it's annoying, but I haven't figured out how to fix it yet. For now, just try not to scroll while running the simulation.
When the orbital periods of a pair of bodies are integral (or rational) multiples of each other, we say the bodies are in mean-motion resonance with one another. In the case of Jupiter and an asteroid, if the asteroid goes around the Sun m times for every n times Jupiter does, we say that the asteroid is in an m:n resonance with Jupiter.
Resonances can strongly affect the motion of asteroids. Look at the following graph of mean distance from the Sun versus number of actual asteroids with that mean distance. You should notice several spots where there are almost no asteroids. These gaps are known as the Kirkwood gaps, and they all correspond to resonances with Jupiter.
Now that you have an idea of what most resonances do, let's examine a few of the anomalous ones. On the graph, you should notice a couple groups of asteroids located well outside the main belt. Interestingly, their locations also correspond to resonances with Jupiter, specifically the 3:2 (3.97 AU) and 1:1 (5.2 AU). Instead of creating gaps in the asteroid distribution, these resonances appear to be keeping asteroids in an otherwise empty region.