**Fall 2003** **Special Topics:
Astrophysics (Phys 493)**

Dr. I.
Fernini

**Homework
# 3**

** **

**Due Monday,
October 27 - Late HW won't be accepted anymore.**

**Problem
1:**

By
expressing the unit direction vector W in Cartesian
coordinates, i.e.,

W = W_{x} **x** + W_{y} **y** + W_{z} **z**

where

W_{x} = sinq cosj

W_{y}
= sinq
sinj

W_{z}
= cosq

prove the following identities:

(a)
ò
dW
= 4p

(b)
ò
W
dW
= 0

(c)
ò
W
(W
×
A) dW
= (4p/3)
A

(d)
ò (W
×
A) (W
×
B) dW
= (4p/3)
(A ×
B)

where A and B are any vectors which are independent of W.

**Problem 2:**

Show that the anisotropic factor f = K/J = 1/3 for each of the following
radiation fields:

(a) I(m) = I_{0} + m I_{1} (Eddington approximation)

(b) I(m) = I_{0} d(m - m_{0}) where m_{0}
= 1/Ö3