Optical depth using Bremsstahlung emission coefficient

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Kayla Martin
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Homework Statement
I am a bit confused how to go about this question.
Relevant Equations
See below.
Printed homework problem on an HII region: part (a) asks to show that the optical depth through the centre is tau_v = (v/v_T)^-2.1, given the bremsstrahlung emission coefficient j_v.

Equations I think may be relevant:

Handwritten-style derivation of free-free emissivity j_v, Kirchhoff's law step to alpha_v, and the optical depth integral tau_v with numerical constants and bullet-point notes.
 
My god, something I can actually help with! (I'm an infrequent visitor here who just sponges you guys with my own questions!)
So hopefully in your "class" it was explained how at the turnover frequency that tau=1 (this can be seen better by plotting the log I vs log nu
Putting that (tau=1) into the optical depth equation (last on your pic), and re-arrange the equation for GHz ^ (-2.1), now put that back into the same optical depth equation (in other words you are substituting your result for the GHz ^ (2.1) when the optical depth was 1. You'll see everything cancel leaving just the result the question asks for`. Hope that helps!
 
weak_phys said:
My god, something I can actually help with! (I'm an infrequent visitor here who just sponges you guys with my own questions!)
So hopefully in your "class" it was explained how at the turnover frequency that tau=1 (this can be seen better by plotting the log I vs log nu
Putting that (tau=1) into the optical depth equation (last on your pic), and re-arrange the equation for GHz ^ (-2.1), now put that back into the same optical depth equation (in other words you are substituting your result for the GHz ^ (2.1) when the optical depth was 1. You'll see everything cancel leaving just the result the question asks for`. Hope that helps!
Am I able to direct message you to ask you another question that relates to this part seeing as you seem to know what you're doing?
 
Kayla Martin said:
Am I able to direct message you to ask you another question that relates to this part seeing as you seem to know what you're doing?
Of course, i don't know what time zone you're in but if i go to sleep before i hear from you i'll be sure to check in tomorrow.
 
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