Conceptual Understanding in Introductory Physics XX: The Laplace-Runge-Lenz Vector

This post is inspired by the October 2015 AstroNotes in The Physics Teacher. I have sometimes introduced vectors into my introductory astronomy course and students were able to do most of the things described below. We never discussed angular momentum or the Laplace-Runge-Lenz vector, but the other quantities were familiar. I was not permitted in my article to comment on whether or not, and to what extent, students found this work helpful because there is no published study on this topic (i.e. vectors in introductory astronomy, something that most would say is mathematical taboo) and because I could produce no hard evidence. These restrictions were quite disturbing given that I made no extraordinary claims to begin with. Such is the world of publishing. Nor was I permitted to go into details of how the geometric properties of vectors encode a lot of physics without the need for coordinate systems and the problems with traditional approaches to this topic. Such is the world of publishing.

Consider a standard two-body gravitational interaction with one object orbiting the other. Treat the mass of the orbiting object as less than the mass of the other object. For both a circular orbit and an elliptical orbit, draw an appropriate diagram showing both objects, the orbital path (assume counterclockwise orbital motion), the orbiting object at four randomly chosen points on the orbit, and each of the following vector quantities appropriately placed and approximately scaled relative to each other. Use a different style of arrow for each quantity.

  • the orbiting object’s position relative to the other object
  • the gravitational force on the orbiting object
  • the orbiting object’s momentum
  • the orbiting object’s angular momentum relative to the other object
  • the Laplace-Runge-Lenz vector (sometimes called the eccentricity vector)
  • the torque on the orbiting object

At each of those four points, using only the geometric properties of vectors and no numerical calculations, describe what is happening to the orbiting object’s kinetic energy and what is happening to the two-body system’s gravitational potential energy. Compare and contrast the differences in these descriptions for the two orbits.


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