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) = I₀ + m I₁    (Eddington approximation)

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