## Project for ASTR7741

*Fall Semester, 2007*

**
Radial Oscillations of n = 0 and n = 1 Polytropes
**

**Part I:** Derive the Mathematical Form of the Relevant Eigenvalue Problem
By far the best reference on this topic is chapter 38 of the following textbook:

- "Stellar Structure and Evolution" by R. Kippenhahn and A. Weigert (hereafter, KW)

Starting from equations (2.22) and (2.23) of Padmanabhan (Vol. II), that is,

- (dr/dM
_{r}) rdr/dt = - dP/dM_{r} - GM_{r}/(4pr^{4})
- dr/dM
_{r} = (4prr^{2})^{-1}

along with the polytropic equation of state, that is,
derive BOTH forms of the "oscillation"
equation that are presented as eqs. (38.8) and (38.34) of KW.
Show that the eigenfrequencies (w_{0}
and w_{1}) -- along with the corresponding
eigenfunctions x_{0} and x_{1} --
given by eqs. (38.27) and (38.28) of KW are solutions to the stated eigenvalue
problem in the case of a uniform-density sphere.

**Part II:** Pulsation Modes for n=0 and n=1 Polytropic Spheres

Solve the relevant eigenvalue problem for two other cases:

- Find the eigenfrequency and eigenfunction for the second overtone mode
of the uniform-density sphere.
- Find the fundamental mode eigenfrequency and eigenfunction for an
n = 1 polytrope.

In both cases, plot your resulting eigenfunction in a manner that can be
easily compared to Figs. 38.1 and 38.2 of KW.