- #1
Daniel Sellers
- 117
- 17
Homework Statement
A) By examining the effective potential energy find the radius at which a planet with angular momentum L can orbit the sun in a circular orbit with fixed r (I have done this already)
B) Show that the orbit is stable in the sense that a small radial nudge will cause only small radial oscillations. [look at d^2U/dr^2] and show that the period of oscillations is equal to the planets orbital period.
Homework Equations
Ueff = -Gm1m2/r + L2/(2μr2)
ω = L/μr2
ϒ = Gm1m2
The Attempt at a Solution
[/B]
It is easy to find the radius of stable orbit:
dU/dt = γ/r2 - L2/(μr3) = 0
r = L2/(γr3)
I can also show that the orbit is stable
d2U/dt2 = γ/r3 > 0 so the radius above is a minimum.
The first derivative being zero also allows me to write
γ = L2/(μr)
which means that
T2 = 4π2μr3/γ
Which matches the equation derived in the text for the period of any bound orbit with constant angular momentum.
The problem! :
I am supposed to find the equation of simple harmonic oscillation in the radial direction, show that the oscillations are small, and show that the period of the radial oscillations is equal to the orbital period. I am at a loss as to how to do this. The last equation which might be relevant is
r(Φ) = 1/(1+εcosΦ) where ε is the eccentricity, which is zero for a circular orbit but obviously not quite zero for a circular orbit that has been given a 'small radial nudge'
I think I have done enough work correctly to get credit for this problem, but I want the correct answer very badly. Can anyone suggest an approach?
Thanks!