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## Orbits and Effective Potential: Instructor's Guide

### Main Ideas

• Orbits
• Effective Potential
• Differential Equations

Estimated Time: 30 minutes

Use a pre-made Java program and Maple worksheet to visualize orbital motion in central forces problems. In particular, students vary parameters such as the total energy and angular momentum and explore the orbital shapes that result from different central force functions.

### Prerequisite Knowledge

Students should be familiar with the ideas of effective potential and angular momentum in the context of central forces problems. Generally we expect students to have worked through solutions to the central force equations of motion before beginning this activity.

### Activity: Introduction

Students are instructed how to download and open the Java program orbits.jar. To avoid slowdowns from the file server, it is best to download this program to the local computers before running it. We ask students to do this at the beginning of this activity.

Each student is given the handout handout and asked explore the behavior or different central force problems by following the instructions in the handout.

Two different worksheets are provided. Both solve the differential equations for the motion of and object in a central potential and plot the resulting orbits in various ways. The first, cforbits.mw allows the user to set initial conditions for both the radial and angular initial position and velocities. This worksheet gives students control over the variables which are likely to be most familiar to them. The second orbits_LandE.mw allows the student to control the values of the initial radial velocity and angular position as well as the total energy and angular momentum. This worksheet encourages students to think in terms of constants of motion. While either or both of the worksheets can be used, the purpose of this activity is to give students the opportunity to thoughtfully explore the parameter space for a particle in central force motion.

For those who don't have Maple or prefer to use a more user friendly Java applet, you can download orbits.jar to explore the orbits of a particle in a gravitational potential. This simulation will run faster if you download the .jar file to the local machine before running it.

### Activity: Student Conversations

All of these files help students to see the relationship between the radial motion of the particle, displayed on the effective potential plot, and the angular ($\phi$) motion, displayed on the orbit plot. Students often have difficulty connecting these two motions in their head. As a result, they often misunderstand the effective potential plot. This worksheet ties in well with the Interpreting Effective Potential Plots kinesthetic activity.

There are three suggested alternate potentials listed at the bottom of the Maple worksheet. Below are some tips for working with each of these.

• The Harmonic Oscillator is generally well behaved
• The first order general relativistic correction, $U(r)=-\frac{k}{r} + \frac{\delta}{r^3}$ works best for small values of $\delta$, e.g., $0.1>\delta>0.01$
• Solutions for the spiral orbits $U(r)= -\frac{k}{r^2}$ can be very sensitive. In particular, if the angular momentum is too small, the orbit will rapidly spiral in toward the center of the well where the potential diverges to $-\infty$. For values of the angular momentum that are too large, the orbits spiral away from the potential very rapidly. In some cases, the inward spiraling orbits cause the differential equation solver to return an error to avoid this carefully choose values for $k$, $l$, and the number of orbits. We recommend that if you use this potential, you first play with the potential and suggest values of $k$, $l$, and numorbits for your students to try.

### Extensions

The orbits_changing.mws worksheet allows students to explore how objects might change from one orbit to another. This worksheet is currently a work in progress but you may find it useful in your class. Changes in orbits are achieved by applying a time varying force in the phi direction.

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