Advanced Astrophysics (5) Solutions

1) If I is taken over all space (i.e. 4 p steradians), the mean intensity is:

J = 1/4p  ò _4p  I  dw   = 1/4p  (1.4 x  10 ⁷ Jm^-2 s^-1) 4p

So: J » 1.4 x  10 ⁷ Jm^-2 s^-1

2) The net flux (H) passing through the Sun’s surface is estimated using:  sT_o ⁴/ p = 2H.

Therefore: H = sT_o ⁴/2 p

 

Where: sT_o ⁴ =  (5.67 x 10^-8 W m^-2 K^-4)(4800K)⁴

sT_o ⁴ =   3.0 x 10 ⁷ W m^-2

H = (3.0 x 10 ⁷ W m^-2) /2 p = 4.8 x 10⁶ W m^-2

3) Consider the solar half-sphere and the energy going into it each second.  We know on average photons are absorbed after traveling a distance with optical depth t =1 in the propagation direction. Averaged over all directions this corresponds to a vertical optical depth of t = 2/3. Based on this find:

a) The energy going into the half sphere each second.

Solution: 

    This is just:   s(T_eff)⁴   =  p F = D 2p S =D 2p I

b) The change in (a) over each absorption and re-emission over vertical optical depth.

Solution:

Over a vertical optical depth one has t = 2/3, then:

D 2p S/D t   =   p F/ (2/3)  = 3p F/ 2   

or:

DS/D t   =   3p F/ 2 (2p) = 3F/4

c) The total absorption and total emission and the relationship between them, defined over all space.

Solution:

Over all space, the total emission = 4p S and the total absorption = 4p J, and by radiative equilibrium:

4p S = 4p J  so that S = J.