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86 lines
2.6 KiB
Plaintext
86 lines
2.6 KiB
Plaintext
Homework 19 (marked homework 21 on the pdf)
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1. Solve the NLTE balance equation (equation 3.24 in AGN3, also given on page 6, side 2, of the lecture notes) for the population of the ¹D level of O III, n(¹D), cm⁻³. That equation gives the dimensionless population ratio n(¹D) / n(3P) we want the population. Assume that the total density of O²⁺ is 10⁻⁴ of the hydrogen density, and that the electron density is equal to the hydrogen density. Vary the hydrogen density n(H) between 10⁻² to 10¹⁰ cm⁻³ in 1 dex steps. Plot log n(¹D) as the x-axis and log n(H) as the y-axis.
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NTLE balance is given by
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n₂/n₁ = nₑ q₁₂/A₂₁ (1 + nₑ q₂₁/A₂₁)⁻¹.
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I model a two level O-III atom.
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I assume density n[O²⁺]/n[H] = 10⁻⁴.
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I assume hydrogen is completely ionized, i.e.,
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nₑ = n[H].
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I will vary n[H], so I need only find the coefficients q and A, which can be pulled from AGN3,
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From combining the 3P levels,
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A₂₁(1D→3P) = [6.8e-3 + 2.0e-2 + 1.7e-6] s⁻¹
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= 0.0268 [s⁻¹].
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The collision strengths are also given in AGN3,
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γ₁₂ = 2.29
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q₂₁ = 8.629e-6 γ₁₂ Tₑ^(-1/2) g₂⁻¹ [cm³/s]
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= 1.98e-5 Tₑ^(-1/2) g₂⁻¹ [cm³/s].
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Assuming electron temperature
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Tₑ = 1e4 K.
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g₁ = 9,
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combining all the states of the 3P levels.
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g₂ = 2*J + 1,
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with J = 2 for 1D, so
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g₂ = 5,
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q₂₁ = 1.98e-5 * 1e-2 K^(-1/2) 1/5
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= 3.96e-08 [cm³/s].
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q₁₂ = q₂₁ g₂/g₁ exp(-Eₜₒₜ/kT).
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Eₜₒₜ = 26169 K.
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q₁₂ = 2.20e-8 exp(-26169/1e4) [cm³/s].
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= 2.20e-8 * 0.073 [cm³/s]
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= 1.61e-9 [cm³/s].
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So, the population ratio is
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n₂/n₁ = nₑ q₁₂/A₂₁ (1 + nₑ q₂₁/A₂₁)⁻¹
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= nₑ (1.61e-9 [cm³/s])/(0.0268 [s⁻¹])
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× (1 + nₑ (3.96e-08 [cm³/s])/(0.0268 [s⁻¹]))⁻¹
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= nₑ (1.61e-9/0.0268 [cm³])
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× (1 + nₑ (3.96e-08/0.0268) [cm³])⁻¹
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= nₑ (6.01e-8 [cm³]) (1 + nₑ (1.48e-6) [cm³])⁻¹.
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This can now be computed across the n[H] density spectrum.
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Plot attached.
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2. Solve for this population using the Boltzmann equation, assuming U=9. Add this to the plot.
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From the boltzmann equation
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n₂/n₁ = g₂/g₁ exp(-Eₜₒₜ/kt)
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= 5/9 * 0.073
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= 0.041.
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3. Find the critical density and indicate that on the plot. How does this compare with the density where the ¹D level has come into LTE?
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Critical density can be predicted as
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n[crit] = A₂₁/q₂₁ = 6.78e5 [cm⁻³].
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This line is drawn on the plot, and shows where the level populations predicted by the NLTE solution agrees with LTE predictions.
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