Homework 19 (marked homework 21 on the pdf)

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.

NTLE balance is given by

    n₂/n₁ = nₑ q₁₂/A₂₁ (1 + nₑ q₂₁/A₂₁)⁻¹.

I model a two level O-III atom.

I assume density n[O²⁺]/n[H] = 10⁻⁴.

I assume hydrogen is completely ionized, i.e.,

    nₑ = n[H].

I will vary n[H], so I need only find the coefficients q and A, which can be pulled from AGN3,

From combining the 3P levels,
    A₂₁(1D→3P) = [6.8e-3 + 2.0e-2 + 1.7e-6] s⁻¹ 
                = 0.0268 [s⁻¹].

The collision strengths are also given in AGN3,
    γ₁₂ = 2.29

q₂₁ = 8.629e-6 γ₁₂ Tₑ^(-1/2) g₂⁻¹ [cm³/s]
    = 1.98e-5 Tₑ^(-1/2) g₂⁻¹ [cm³/s].

Assuming electron temperature
    Tₑ = 1e4 K.

g₁ = 9,
    combining all the states of the 3P levels.

g₂ = 2*J + 1, 
    with J = 2 for 1D, so
    g₂ = 5, 

q₂₁ = 1.98e-5 * 1e-2 K^(-1/2) 1/5
    = 3.96e-08 [cm³/s].

q₁₂ = q₂₁ g₂/g₁ exp(-Eₜₒₜ/kT).

Eₜₒₜ = 26169 K.

q₁₂ = 2.20e-8 exp(-26169/1e4) [cm³/s].
    = 2.20e-8 * 0.073 [cm³/s]
    = 1.61e-9 [cm³/s].

So, the population ratio is

    n₂/n₁ = nₑ q₁₂/A₂₁ (1 + nₑ q₂₁/A₂₁)⁻¹

            = nₑ (1.61e-9 [cm³/s])/(0.0268 [s⁻¹])
                × (1 + nₑ (3.96e-08 [cm³/s])/(0.0268 [s⁻¹]))⁻¹

            = nₑ (1.61e-9/0.0268 [cm³])
                × (1 + nₑ (3.96e-08/0.0268) [cm³])⁻¹

            = nₑ (6.01e-8 [cm³]) (1 + nₑ (1.48e-6) [cm³])⁻¹.

This can now be computed across the n[H] density spectrum.

Plot attached.


2. Solve for this population using the Boltzmann equation, assuming U=9.  Add this to the plot.  

From the boltzmann equation

    n₂/n₁ = g₂/g₁ exp(-Eₜₒₜ/kt)
            = 5/9 * 0.073
            = 0.041.



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?

Critical density can be predicted as
    
    n[crit] = A₂₁/q₂₁ = 6.78e5 [cm⁻³].

This line is drawn on the plot, and shows where the level populations predicted by the NLTE solution agrees with LTE predictions.