The Homepage of Liron McLey

Snow on the way to Lake Louise, AB, Canada

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Background     The Equations     Data feed     Numerical method     Compiling and running     Validation of the code


The Equations

One can derive the equations describing the behaviour of such waves in the stellar interior. Using the hydrodynamic equations and assuming small adiabatic oscillations around a spherically symmetric equilibrium state and linearity, one arrives at a fourth order system of ordinary differential equations for the following dependent variables \xi _{r}, \; p', \; \Phi ', \; d\Phi ' / dr. This fourth order system is quite difficult to solve. However, it can be reduced to a second order system by assuming the perturbations in the gravitational potential are very small and can be neglected. This is legitimate under the following circumstances:

  • When l is large
  • When the mode order |n| is large
    This is the Cowling approximation. Under these assumptions the equations reduce to,

    \frac{d\xi_{r}}{dr}=-\left(\frac{2}{r}-\frac{1}{\Gamma_{1}}H_{p}^{-1}\right)\xi_{r}+\left(\frac{S_{l}^{2}}{\omega^{2}}-1\right)\frac{p'}{\rho c^{2}}
    \frac{dp'}{dr}=\left(\omega^{2}-N^{2}\right)\rho\xi_{r}-\frac{1}{\Gamma_{1}}H_{p}^{-1}p'
    \xi_{h}\left(r\right)=\frac{1}{r\omega^{2}\rho}p'\;\;\;(for\; l\neq0)

    where H_{p}^{-1}=-\frac{dln\, p}{dr} is the pressure scale height, \xi_{r} is the radial displacement, p' is the pressure perturbation, when l \neq 0 \xi_{h} is the horizontal displacement (the directions orthogonal to the radial direction) and the rest of the variables are the unperturbed values of the equilibrium state (\rho is the density, etc.).
    My project focuses on solving these equations by obtaining both their eigenfrequencies (for which the equations have a non vanishing solution) and their eigenfunctions.



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    Last Updated: 10th of June 2014