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Orbit Stability Two particles whose reduced mass is u interact via a po- 5. tential energy U = ;kr2 , where r is the distance between them. (a) Make a sketch showing U(r), the centrifugal potential energy Uct(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. Hint: This requires that dUeff/dr be zero. (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r- ro) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro- 3

Orbit Stability Two particles whose reduced mass is u interact via a po- 5. tential energy U = ;kr2 , where r is the distance between them. (a) Make a sketch showing U(r), the centrifugal potential energy Uct(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. Hint: This requires that dUeff/dr be zero. (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r- ro) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro- 3

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Orbit Stability Two particles whose reduced mass is u interact via a po- 5. tential energy U = ;kr2 , where r is the distance between them. (a) Make a sketch showing U(r), the centrifugal potential energy Uct(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. Hint: This requires that dUeff/dr be zero. (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r- ro) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro- 3
Transcribed Image Text:Orbit Stability Two particles whose reduced mass is u interact via a po- 5. tential energy U = ;kr2 , where r is the distance between them. (a) Make a sketch showing U(r), the centrifugal potential energy Uct(r), and the effective potential energy Ueff (r). (Treat the angular momentum l as a known, fixed constant.) (b) Find the "equilibrium" separation ro, the distance at which the two particles can circle each other with constant r. Hint: This requires that dUeff/dr be zero. (c) By making a Taylor expansion of Ueff (r) about the equilibrium point ro and neglecting all terms in (r- ro) and higher, find the frequency of small oscillations about the circular orbit if the particles are disturbed a little from the separation ro- 3
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