Equation of motion for oscillations about a stable orbit

[Daniel Sellers]

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

U_eff = -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!

 

[Daniel Sellers]

Ok I did a little more work with 'the radial equation':

μr^⋅⋅ = d/dr(U)

= γ/r² - L²/μr³

L² = μ²r⁴ω²

which implies

r^⋅⋅ = γ/r² - ω²r

Which looks a lot like a differential equation which gives simple harmonic motion except for the γ term.

Now, when dU/dr = 0 which is the condition that I used to find the radius, γ = L²/(μr₀) where r₀ is the stable orbit radius found in the first part.

So

r^⋅⋅ = ω²r²/r₀ - ω²r

Soooooo what do I do with this? How do I derive and SHO equation which shows the small oscillations asked for and has the correct period?

Just need someone to point out what I'm missing or give me the last step I think

 

[Daniel Sellers]

Okay I got it. Start with radial equation and let r = r0 + α where α << r₀. Then do a Taylor expansion about r₀ and the first derivative of U is zero, the second is known.
You end up with the equation of an oscillator and you can read off the frequency without even solving the differential equation.
Its also true for any SHO that the frequency is equal to the second serivative of potential devided by the mass.

The frequency found for the radial oscillations is equal to the orbital frequency (as is true for all bounded Kepler orbits)

Thanks guys, good talk