Saturday, June 22, 2013

Advanced Astrophysics (4) Solutions

1) The specific intensity is defined from:

p( Fo ) = 2 p (I(cos (q))

And for q = p/4, then cos (p/4) =  Ö2/2 and:

I = p( Fo ) / 2 p (Ö2/2) =  p( Fo ) / p (Ö2)

Therefore:

I = (6.3 x 10 7 Jm-2 s-1) / p(Ö2)


I » 1.4 x  10 7 Jm-2 s-1


 
2)  The effective temperature is obtained using:

p( Fo ) = s(Teff)4

So:  Teff  = [p( Fo ) / s] 1/4

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

Then:
 
Teff  = [6.3 x 10 7 Jm-2 s-1/ 5.67 x 10-8 W m-2 K-4] 1/4

Teff  »  5800 K

The boundary temperature is found from:

Teff  = (2)1/4 To   = 1.189 To



Or:   To  =   Teff  /1.189  = 5800K/ 1.189  » 4800 K


The boundary temperature differs because of being referenced to a different optical depth. The boundary temperature (To) 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!

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