Corrigendum: “how to calculate molecular column density” (PASP, (2015), 127, 266, 10.1086/680323)

In the manuscript “How to Calculate Molecular Column Density” by Mangum & Shirley (2015, PASP, 127, 266), Adam Ginsburg has pointed to a logical disconnect in our derivation of the column density equation for H2CO (Equation (100)). In deriving this equation we mistakenly used the slightly asymmetric rotor approximation and set g_K=2 for K ≠ 0. This would imply that we were ignoring the K-doublet splitting in the molecule, which in this derivation we are not. The correct value for all asymmetric rotor levels is g_K=1. We corrected Equation (100) and the lead-in discussion of the assumptions that went into this equation. The following is meant to be a drop-in replacement for Section 15.5 of Mangum & Shirley (2015). We would like to thank Adam for bringing this error to our attention. 15.5. H₂CO Formaldehyde (H₂CO) is a slightly asymmetric rotor molecule with k ≃-0.961 (Equation?(45)), which means that H₂CO is nearly a prolate symmetric rotor. The slight asymmetry in H2CO results in limiting prolate (quantum number K-1) and oblate (quantum number K+1) symmetric rotor energy levels that are closely spaced in energy, a feature commonly referred to as “K-doublet splitting.” Figure 14 shows the energy level diagram for H₂CO including all energy levels E≤300 K. In addition to the asymmetric rotor energy level structure H₂CO possesses spin-rotation and spin-spin hyperfine energy level structure. Magnetic dipole interaction between the H nuclei (I_H=0/1 for para-/ortho-H₂CO) and rotational motion of the molecule result in spin-rotation hyperfine energy level splitting. Since the two H nuclei have equal coupling, the two H nuclei spins couple with each other first and the hyperfine coupling scheme is I_H = I₁ + I₂ and F = J + I_H. Since the interchange of the two ¹H atoms (I₁=I₂=1/2) obey Fermi–Dirac statistics (total wave function is anti-symmetric to this interchange) and alternating K_a ladders in H₂CO have a different rotational wavefunction symmetry, only ortho-H₂CO levels (I_H=1) with K_a = odd integers have hyperfine splitting. For the ortho-H₂CO 1₁₀-1₁₁ transition, the frequency offsets of these hyperfine transitions are Δν≤18.5 kHz. The weaker spin-spin interactions between the nuclei are generally not considered. Table 10 lists the frequencies and relative intensities for the spin-rotation hyperfine transitions of the H₂CO 1₁₀-1₁₁, 2₁₁-2₁₂, and 3₁₂-3₁₃ transitions. Figure 15 shows the synthetic spectra for the H₂CO 1₁₀-1₁₁, 2₁₁-2₁₂, and 3₁₂-3₁₃ transitions. Note that the hyperfine intensities are exactly equal to those calculated for the spin-rotation hyperfine components of NH₃ (see Section F). For illustration we can derive the column density equation for an ortho-H₂CO (K_-1 odd) K-doublet (ΔK_-1=0) transition. For ortho-H₂CO transitions: S = K²/J_u(J_u + 1) m = 2.332 Debye = 2.331 × 10^-18 esu cm R_i = (se Section 10 or, for 1₁₀ - 1₁₁, 2₁₁ - 2₁₂), or 3₁₂ - 3₁₃ see Table 10) g_j = 2J_u + 1 g_k = 1(for an asymmetric rotor) g_I = 3/4 for K_-1 (Table presented) (Figure presented) We can derive the following equation for the molecular column density in H2CO as derived from a measurement of an ortho-H2CO (K_-1 odd) K-doublet transition assuming optically thin emission using Equation (80): (Formula presented.) Note that the K-doublet transitions of H2CO are rather unusual in that, due to an unusual collisional excitation effect, these transitions are often measured in absorption against the cosmic microwave background radiation. For n(H2)???105.5 cm-3, a collisional selection effect overpopulates the lower energy states of the 1₁₀-1₁₁ through 5₁₄-5₁₅ transitions (Evans et al. 1975; Garrison et al. 1975). This overpopulation of the lower energy states results in a “cooling” of the J≤5 K-doublets to an excitation temperature that is lower than that of the cosmic microwave background. This causes the J≤5 K-doublet transitions to appear in absorption. For volume densities n(H₂)≿10^5.5 cm^-3 and T_K≃40 K this collisional pump is quenched. For these higher volume densities the J≤5 K-doublets are driven into emission over a wide range of kinetic temperatures and abundances (see Figure 1 in Mangum et al. 2008).