1) Estimate the temperature of the warm dust emission.

    Read the wavelength at the peak of the blackbody region.

    λₚₑₐₖ = 10¹³ Hz.

    Using Wien Law,
        λₚₑₐₖ = 2.90 × 10⁻³ m K / T.

    T = 2.90 × 10⁻³ m K / 10¹³ Hz.

    T = 96.7 K.
    
2) What is the turnover frequency of the brems emission?

    This is just gleaned from the SED.
    1.5 GHz = 1.5 × 10⁹ Hz.

3) Assume that the brems comes from ionized hydrogen at 10 4 K and
a density of 10 4 cm -3 . What is the thickness of NGC 7027 in cm?
In parsecs?

    κʹₛ = 0.0178 Z² g ν⁻² T^(-3/2) nₑ nₗ.

    g = 10.6 + 1.9*log(T) - 1.26*log(Z*ν)
      = 6.63.

    κʹₛ = 0.0178 × 1 × (3.32) × (1.5 × 10⁹ Hz)⁻² × (10⁴ K)^(-3/2) × (10⁴ cm⁻³)².

    κʹₛ = 5.25 × 10⁻¹⁸ cm⁻¹.

    For turnover frequency,
        1 = τ = κʹₛ l.

    l = τ / κʹₛ = κʹₛ⁻¹.

    l = 1.90 × 10¹⁷ cm
      = 0.0617 parsecs.


    After fixing the typo from the Allen text, this seems to be a much more reasonable result. The actual radius (from wikipedia) is approximately .068 parsec for this object, so this is very close to that answer.