- #1
Dazed&Confused
- 191
- 3
Homework Statement
The potential energy of a particle of mass [itex] m[/itex] is [itex] V(r) = k/r + c/3r^3[/itex] where [itex]k<0[/itex] and [itex]c[/itex] is a small constant. Find the angular velocity [itex] \omega[/itex] in a circular orbit of radius [itex]a[/itex] and the angular frequency [itex]\omega'[/itex] of small radial oscillations about this circular orbit. Hence show that a nearly circular orbit is approximately an ellipse whose axes precess at an angular frequency [itex] \Omega \approx (c/|k|a^2)\omega^2[/itex]
Homework Equations
[tex]m\ddot{r} -mr\dot{\theta}^2 = -\frac{dV}{dr} [/tex]
The Attempt at a Solution
So the first part is just setting [itex]\ddot{r}[/itex] to zero so that
[tex]-\omega^2 = \left. -\frac{1}{ma}\frac{dV}{dr} \right|_a = \frac{k}{ma^3} + \frac{c}{ma^5}[/tex]
which is the answer given. For the second part I Taylor expanded [tex]
\omega^2r -\frac{dV}{dr}[/tex]
about [itex]a[/itex] took the second term to be minus the angular frequency squared. Thus I get
[tex]\omega'^2 = -\omega^2 + \left. \frac{d^2V}{d^2r}\right|_a [/tex]
however with this I get a result different from the answer. I'm not sure where the mistake lies.