ASTR 498N            Homework #4        Hints      

 

P    O&C Problem 9.16

 

The star is spherically symmetric,so align ds with the z-axis.   The ingredients are:

 

 

 

 

 

 

This will allow you to write the transfer equation in spherical coordinates [part (a)].

 

To derive the radiation pressure gradient [part (b)], follow O&C p. 286:  multiply both sides of the transfer equation by  and integrate over all solid angles.  Remember that the source function is isotropic and refer to O&C Eqs. (9.6) and (9.7).

 

 

 

P    Explain why the bound-free opacity is proportional to  while the free-free opacity is proportional to  [O&C Eqs. (9.19) and (9.20)].

 

Don’t worry about the Z dependences, but do explain physically why both the b-f  and  f-f opacities depend on the number density of free electrons.  Then follow Lecture 6 to derive an expression for the number density of free electrons in terms of the mass fractions X, Y, and Z (analogous to the expression we derived for the number of free particles) and show that this expression reduces to (X+1).