## 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/dMr) rdr/dt = - dP/dMr - GMr/(4pr4)
• dr/dMr = (4prr2)-1
along with the polytropic equation of state, that is,
• P = Kr(1+1/n)
derive BOTH forms of the "oscillation" equation that are presented as eqs. (38.8) and (38.34) of KW. Show that the eigenfrequencies (w0 and w1) -- along with the corresponding eigenfunctions x0 and x1 -- 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.