- #1
ergospherical
- 1,035
- 1,316
- 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##?
\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##?