How Do Radial Oscillations Affect Star Stability?

ergospherical
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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##?
 
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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 <br /> 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.
 
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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!
 
ergospherical said:
I will check again tomorrow with fresher eyes!
Have you cleared that with the fresher? :eek:
 
pbuk said:
Have you cleared that with the fresher? :eek:
That would imply that they have a choice!
 
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