Solutions To Astrophysics Problems (Stellar Atmospheres)

 1) Find the effective temperature of the Sun and the boundary temperature (T_o) and account for any difference.  

(Take the Stefan -Boltzmann constant s = 5.67 x 10^-8 W m^-2 K^-4  )

Solution:

The effective temperature is obtained using:

p( F_o ) = s(T_eff)⁴

So:  T_eff  = [p( F_o ) / s] ^1/4

Where s = 5.67 x 10^-8 W m^-2 K^-4

Is the Stefan-Boltzmann constant. Then:

T_eff  = [6.3 x 10 ⁷ Jm^-2 s^-1/ 5.67 x 10^-8 W m^-2 K^-4] ^1/4

T_eff  »  5800 K

The boundary temperature is found from:

T_eff  = (2)^1/4 T_o   = 1.189 T_o

Or:  

T_o  =   T_eff  /1.189  = 5800K/ 1.189  » 4800 K

 The boundary temperature differs because of being referenced to a different optical depth. The boundary temperature (T_o) approaches the value of the effective (or surface) temperature when t = 0, but this still exhibits a difference in layers so will not be exactly the same!

2) Prove   J = ½(I₁ + I₂)

   By integrating I in the forward and backward directions. 

Solution:  We define:

J  =  1/4p  ò _4p  I (q) dw    

In forward direction:

J(+) =  1/4p  ò^2p_o  ò^p/2 _o   I₁ sin q dq df    

In backward direction:

J(-)  =   1/4p  ò^2p_o  ò^-p/2 _o   I₂ sin q dq df    

(Note change in upper limit of 2nd integral)

J =  J(+)  +  J(-)  =  

    1/4p  ò^2p_o  ò^p/2 _o   I₂ sin q dq df  +  1/4p  ò^2p_o  ò^-p/2 _o   I₂ sin q dq df     

=  1/4p  [ 2p  (I₁ + I₂) ] =   ½ (I₁ + I₂) 

J = ½(I₁ + I₂)