Problems for Intermediate Methods in Theoretical Physics Edward F. Redish

Orbital Oscillations: 1

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 Ueff(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 Ueff(r) in a power series about the location of its minimum value, r0, 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 Ueff(r) in the neighborhood of its minimum,

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