Radial oscillations of a star

[ergospherical]

Homework Statement

The spherically symmetric oscillations of a star satisfy the Sturm-Liouville equation\begin{align*}
\frac{d}{dr} \left[ \frac{\gamma p}{r^2} \frac{d}{dr}(r^2 \xi_r) \right] - \frac{4}{r} \frac{dp}{dr} \xi_r + \rho \omega^2 \xi_r = 0
\end{align*}

Relevant Equations

N/A

So far I have not made much meaningful progress beyond two equations; \begin{align*}
\rho \frac{D\mathbf{u}}{Dt} = - \nabla p \implies \rho \left( \frac{\partial}{\partial t} + u \frac{\partial}{\partial r} \right)u = - \frac{\partial p}{\partial r}
\end{align*}and thermal energy:\begin{align*}
\frac{Dp}{Dt} = -\gamma p \nabla \cdot \mathbf{u} \implies \left( \frac{\partial}{\partial t} + u \frac{\partial}{\partial r} \right)p = -\gamma p \frac{\partial}{\partial r}(r^2 u)
\end{align*}I'm somewhat confused why there's no time dependence in the form given. What exactly is $\xi_r$?

 

[pasmith]

You are dealing with oscillations, and there is an $\omega^2$ in the ODE. That suggests variables are assumed to be functions of $r$ alone times $e^{i\omega t}$. I suspect $\xi_r$ denotes (small) radial displacement from an equilibrium position, in which case $$u_r = \frac{\partial}{\partial t}(\xi_r(r)e^{i\omega t}) = i\omega \xi_r(r) e^{i\omega t}.$$ The ODE you are trying to derive is linear and the governing PDEs are not, which suggests that the system has been linearised about the static equiblibrium state.

 

[ergospherical]

Thank you. I applied $\partial/\partial t$ to my first equation and $\partial/\partial r$ to my second equation, and subsequently eliminated $\partial^2 p / \partial r \partial t$. Then I substituted for $u = i\omega \xi_r e^{i \omega t}$ and kept only terms to linear order in $\xi_r$. After some manipulation,
\begin{align*}
(\xi_r p')' + (\gamma p(r^2 \xi_r)')' + \rho \omega^2 \xi_r = 0
\end{align*}Seems likely that I made at least one slip. I will check again tomorrow with fresher eyes!

 

[pbuk]

I will check again tomorrow with fresher eyes!

Have you cleared that with the fresher?

 

[ergospherical]

Have you cleared that with the fresher?

That would imply that they have a choice!