
When studying the orbits of an object of mass m around a planet of mass M it is useful to create an “effective radial potential” that is only a function of the raidus and has the form
where L is the object’s orbital angular momentum. With a little algebra and dimensional analysis (don’t do it now) we can write this in the more convenient form
(a) Sketch what the curve of U_{eff}(r) looks like as a function of r identifying the salient points (maxima or minima, crossings of 0) by giving an expression for them in terms of the constants σ and ε.
(b) Expand U_{eff}(r) in a power series about the location of its minimum value, r_{0}, to second order. Sketch the same figure you drew in (a) but this time add onto it a sketch of the second order power series approximation you have generated.
(c) Identify the “effective spring constant”, k, that would give an approximation for U_{eff}(r) in the neighborhood of its minimum,
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Last revision 27. December, 2010