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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 = Wx x + Wy y + Wz z

 

where

Wx = sinq cosj

Wy = sinq sinj

Wz = 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) = I0 + m I1 (Eddington approximation)

(b) I(m) = I0 d(m - m0) where m0 = 1/Ö3