Astronomy: Local Thermodynamic Equilibrium, Gaunt Factor

[Rokon117]

1. The problem statement, all variables and given/known

Homework Equations

show that a hydrogen gas cloud at 10000K is about twice as optically thick at the Balmer limit compared with the Paschen limit. Assume that the H atoms are in Local Thermodynamic Equilibrium, the Gaunt factor is unity at all energy levels, and include absorption by H to levels up to n=10. Note that hf1/k=1.578*10^5K for lyman limit photon of frequency f1.

The Attempt at a Solution

an(f)=[64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)
gbf(n)=1
sorry just don't know how to type those symbols

 

[Asim Riaz]

.a0=0.0529nmfn=1.5809*10^15 Hz at Balmer limitfp=7.9579*10^14 Hz at Paschen limitn=1 to 102. Relevant equationsan(f)=[64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)3. The attempt at a solutionan(f) at Balmer limit = [64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)= [64/(3*3^0.5) A] (pi 0.0529nm^2)n(1.5809*10^15 Hz/1.578*10^5K)^3 1= 8.106*10^-19 m^2an(f) at Paschen limit = [64/(3*3^0.5) A] (pi a0^2)n(fn/f)^3 gbf(n)= [64/(3*3^0.5) A] (pi 0.0529nm^2)n(7.9579*10^14 Hz/1.578*10^5K)^3 1= 4.049*10^-19 m^2Therefore, the hydrogen gas cloud at 10000K is about twice as optically thick at the Balmer limit compared with the Paschen limit.