[LSN/ME-12002] Motion in Spherically Symmetric Potentials
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I Lesson Overview
In this lesson you will learn the following.
- Conservation of angular momentum (direction) implies orbits lie in a plane So the problem gets reduced to solving for two coordinates in a plane.
- The conservation of angular momentum (absolute value) implies Kepler's second law holds for any spherically symmetric potential.
- Conservation of energy and angular momentum (magnitude) helps in reducing the problem to quadratures.
- Basic knowledge of differential and integral calculus and vectors.
- Solving the equations of motion of a planet around the sun can be is equivalent to the solution of a single body of reduced mass \(\mu\) where \[\mu = \frac{mM}{M+m}.\] and \(M, m\) are masses of the sun and the planet respectively.
- For a particle in a spherically symmetric potential, the angular momentum is conserved.
II Main Lessons
\(\S\) 1 Kepler Orbits
\(\S\) 2 Question for You to try it now, in two different ways
Show that the latus rectum of the elliptic Kepler orbit is just the radius of circular orbit for energy \(E\) and angular momentum \(L\).
\(\S\) 3 Equation of Planetary Orbits
\(\S\) 4 Orbit Parameters
\(\S\) 5 Hyperbolic Orbits
\(\S\) 6 Question for You to try it now
Are there orbits for zero angular momentum? Is the orbit, if it exists, is bounded or unbounded? Answer different ranges of energies.
III EndNotes