Physics 440: Mathematical Methods of Physics Spring 2007

Problem Sets for Part Two: Differential Equations

PS4.

A&W, Chapter 9: 9.2.3, 9.2.5, 9.2.13, 9.3.2, 9.3.5, 9.3.8, 9.5.15ab, 9.5.15, 9.6.1, 9.6.11 (2 points each)

PS5: DUE AT THE START OF LECTURE ON WENDESDAY, 9 MAY.

A&W, Chapter 9: 9.6.18,19,and 20 (3 points each)

9.7.5, 9.7.7 (2 points each)

A&W, Chapter 10: 10.1.2, 10.1.4 (2 points each)

Supplemental Problem (3 points):

a) Using the homology relations discussed in class, derive a mass-luminosity relation for spherical, nonmagnetic, isolated stars in hydrostatic equilibrium,

assuming radiation pressure (P ~ T⁴) dominates ideal gas pressure and a mass-radius relation exists such that M ~ R. Ignore opacity effects. 

b) Repeat a), now assuming that the mean opacity is governed by Kramer’s opacity (k ~ rT^-7/2).

c) Consider hydrostatic polytropic spheres. Define the polytropic index by the relation g = 1 + 1/n. Derive the general M(R) relation, and identify two special cases.