as-500/hw16/text

1) What is the partition function for O III at 10 4 K?

    The partition function for O III is approximately independent of temperature, provided the density is not very small, and can be read from table 3-2B in Stellar Atmospheres.

    U[O III] = 10^0.95 ≈ 8.9.

    I computed this by taking the sum in an excel spreadsheet.. I get ~9.009. Like this:

    Config.   Term J  "Level (eV)" gᵢ  E₁ᵢ              Weight
                            
    2s22p2   3P    0   0           1   0               1
                   1   0.0140323   3   0.0140323       2.951544
                   2   0.0379607   5   0.0379607       4.784523
    2s22p2   1D    2   2.513565    5   2.513565        0.270513
    2s22p2   1S    0   5.354349    1   5.354349        0.002002
    2s2p3    5S°   2   7.479321    5   7.479321        0.000850
    2s2p3    3D°   3   14.88123    7   14.88123        2.214550E-07
                   2   14.88472    5   14.88472        1.575428E-07
                   1   14.88532    3   14.88532        9.445992E-08
    2s2p3    3P°   2   17.65299    5   17.65299        6.342369E-09
    ...

    Temperature 1.00E+04    U   9.00943346
    

2) What is the density of the 1 S level if the total O III density is 1e10 cm -3 ?
    
    nₜₒₜ/nₛ = g₁ₛ/U.
    gᵢ = 2*Jᵢ+1, g₁ₛ = 1.
    n₁ₛ/nₜₒₜ = g₁ₛ*exp(-E₁﹐₁ₛ/kT)/8.9 = 0.000222.
    n₁ₛ = 2.22×10⁶ cm⁻³.

3) Assume a temperature of 10 4 K, about that of a typical A star like Vega or Sirius.
What is the hydrogen ionization fraction H + /H total at hydrogen densities of 10 10 ,
10 15 and 10 20 cm -3 ?

    I'm actually stuck on how to solve this. I can get the algebraic manipulation to

    nₕ/(nₕ₊)² = 1/nₕ₊ + 1/saha,
    but I can't figure out how to solve for nₕ/(nₕ₊). Once I get that, then I just plug the saha equation in the appropriate place and get the density. Most people are saying that they just are not putting the square in and getting about the right answer anyway, so I would appreciate my credit being considered within that context.

    At least for partial credit, the saha equation evaluates to
    6.948×10²². I have the densities, so once this is solved I'll have the density of nₕ₊.


    ***Late Finish:

        This is a quadratic equation that reads

    (nₕ₊)² + nₕ₊*saha - nₕ*saha = 0, which has positive solution
    nₕ₊ = 1/2 (√(saha)*√(4*nₕ+saha) - saha)

    This gives solutions

    nₕ₊ = {1.00×10¹⁰, 4.35×10¹⁴, 1.83×10¹⁷}, so
    nₕ₊/nₕ = {1.00, 43.5, 1.83×10³}.


    This is not right, however. I know these should be different by about 5 orders of magnitude. I am not sure what is wrong. I am copying in my spreadsheet values, because maybe you can help identify where my values went wrong. Thanks.


2*pi*me*k/h/h = 17998640157.
(2*pi*me*k/h/h)^3/2 = 2.41×10¹⁵.

nH      1.00E+10    1.00E+15    1.00E+20    
UH      2           2           2   
UH+     1           1           1       
ge      2           2           2       
Eion    13.605693   13.605693   13.605693009        
T       1.00E+04    1.00E+04    1.00E+04        
kT      0.86173303  0.86173303  0.86173303      
saha    3.3565×10¹⁴ 3.3565×10¹⁴ 3.3565×10¹⁴ 
nH+     1.00E+10    4.35E+14    1.83E+17        
nH+/nH  1.00E+00    4.35E-01    1.83E-03