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75 lines
3.3 KiB
Plaintext
75 lines
3.3 KiB
Plaintext
1) What is the partition function for O III at 10 4 K?
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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.
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U[O III] = 10^0.95 ≈ 8.9.
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I computed this by taking the sum in an excel spreadsheet.. I get ~9.009. Like this:
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Config. Term J "Level (eV)" gᵢ E₁ᵢ Weight
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2s22p2 3P 0 0 1 0 1
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1 0.0140323 3 0.0140323 2.951544
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2 0.0379607 5 0.0379607 4.784523
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2s22p2 1D 2 2.513565 5 2.513565 0.270513
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2s22p2 1S 0 5.354349 1 5.354349 0.002002
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2s2p3 5S° 2 7.479321 5 7.479321 0.000850
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2s2p3 3D° 3 14.88123 7 14.88123 2.214550E-07
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2 14.88472 5 14.88472 1.575428E-07
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1 14.88532 3 14.88532 9.445992E-08
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2s2p3 3P° 2 17.65299 5 17.65299 6.342369E-09
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...
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Temperature 1.00E+04 U 9.00943346
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2) What is the density of the 1 S level if the total O III density is 1e10 cm -3 ?
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nₜₒₜ/nₛ = g₁ₛ/U.
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gᵢ = 2*Jᵢ+1, g₁ₛ = 1.
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n₁ₛ/nₜₒₜ = g₁ₛ*exp(-E₁﹐₁ₛ/kT)/8.9 = 0.000222.
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n₁ₛ = 2.22×10⁶ cm⁻³.
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3) Assume a temperature of 10 4 K, about that of a typical A star like Vega or Sirius.
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What is the hydrogen ionization fraction H + /H total at hydrogen densities of 10 10 ,
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10 15 and 10 20 cm -3 ?
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I'm actually stuck on how to solve this. I can get the algebraic manipulation to
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nₕ/(nₕ₊)² = 1/nₕ₊ + 1/saha,
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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.
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At least for partial credit, the saha equation evaluates to
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6.948×10²². I have the densities, so once this is solved I'll have the density of nₕ₊.
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***Late Finish:
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This is a quadratic equation that reads
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(nₕ₊)² + nₕ₊*saha - nₕ*saha = 0, which has positive solution
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nₕ₊ = 1/2 (√(saha)*√(4*nₕ+saha) - saha)
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This gives solutions
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nₕ₊ = {1.00×10¹⁰, 4.35×10¹⁴, 1.83×10¹⁷}, so
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nₕ₊/nₕ = {1.00, 43.5, 1.83×10³}.
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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.
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2*pi*me*k/h/h = 17998640157.
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(2*pi*me*k/h/h)^3/2 = 2.41×10¹⁵.
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nH 1.00E+10 1.00E+15 1.00E+20
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UH 2 2 2
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UH+ 1 1 1
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ge 2 2 2
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Eion 13.605693 13.605693 13.605693009
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T 1.00E+04 1.00E+04 1.00E+04
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kT 0.86173303 0.86173303 0.86173303
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saha 3.3565×10¹⁴ 3.3565×10¹⁴ 3.3565×10¹⁴
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nH+ 1.00E+10 4.35E+14 1.83E+17
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nH+/nH 1.00E+00 4.35E-01 1.83E-03
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