Abstract
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 Kdoublet 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 leadin discussion of the assumptions that went into this equation. The following is meant to be a dropin replacement for Section 15.5 of Mangum & Shirley (2015). We would like to thank Adam for bringing this error to our attention. 15.5. H_{2}CO Formaldehyde (H_{2}CO) is a slightly asymmetric rotor molecule with k ≃0.961 (Equation?(45)), which means that H_{2}CO is nearly a prolate symmetric rotor. The slight asymmetry in H2CO results in limiting prolate (quantum number K1) and oblate (quantum number K+1) symmetric rotor energy levels that are closely spaced in energy, a feature commonly referred to as “Kdoublet splitting.” Figure 14 shows the energy level diagram for H_{2}CO including all energy levels E≤300 K. In addition to the asymmetric rotor energy level structure H_{2}CO possesses spinrotation and spinspin hyperfine energy level structure. Magnetic dipole interaction between the H nuclei (I_{H}=0/1 for para/orthoH_{2}CO) and rotational motion of the molecule result in spinrotation 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} = I_{1} + I_{2} and F = J + I_{H}. Since the interchange of the two ^{1}H atoms (I_{1}=I_{2}=1/2) obey Fermi–Dirac statistics (total wave function is antisymmetric to this interchange) and alternating K_{a} ladders in H_{2}CO have a different rotational wavefunction symmetry, only orthoH_{2}CO levels (I_{H}=1) with K_{a} = odd integers have hyperfine splitting. For the orthoH_{2}CO 1_{10}1_{11} transition, the frequency offsets of these hyperfine transitions are Δν≤18.5 kHz. The weaker spinspin interactions between the nuclei are generally not considered. Table 10 lists the frequencies and relative intensities for the spinrotation hyperfine transitions of the H_{2}CO 1_{10}1_{11}, 2_{11}2_{12}, and 3_{12}3_{13} transitions. Figure 15 shows the synthetic spectra for the H_{2}CO 1_{10}1_{11}, 2_{11}2_{12}, and 3_{12}3_{13} transitions. Note that the hyperfine intensities are exactly equal to those calculated for the spinrotation hyperfine components of NH_{3} (see Section F). For illustration we can derive the column density equation for an orthoH_{2}CO (K_{1} odd) Kdoublet (ΔK_{1}=0) transition. For orthoH_{2}CO transitions: S = K^{2}/J_{u}(J_{u} + 1) m = 2.332 Debye = 2.331 × 10^{18} esu cm R_{i} = (se Section 10 or, for 1_{10}  1_{11}, 2_{11}  2_{12}), or 3_{12}  3_{13} 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 orthoH2CO (K_{1} odd) Kdoublet transition assuming optically thin emission using Equation (80): (Formula presented.) Note that the Kdoublet 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 cm3, a collisional selection effect overpopulates the lower energy states of the 1_{10}1_{11} through 5_{14}5_{15} transitions (Evans et al. 1975; Garrison et al. 1975). This overpopulation of the lower energy states results in a “cooling” of the J≤5 Kdoublets to an excitation temperature that is lower than that of the cosmic microwave background. This causes the J≤5 Kdoublet transitions to appear in absorption. For volume densities n(H_{2})≿10^{5.5} cm^{3} and T_{K}≃40 K this collisional pump is quenched. For these higher volume densities the J≤5 Kdoublets are driven into emission over a wide range of kinetic temperatures and abundances (see Figure 1 in Mangum et al. 2008).
Original language  English (US) 

Article number  069201 
Journal  Publications of the Astronomical Society of the Pacific 
Volume  129 
Issue number  976 
DOIs 

State  Published  Jun 1 2017 
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ASJC Scopus subject areas
 Astronomy and Astrophysics
 Space and Planetary Science
Cite this
Corrigendum : “how to calculate molecular column density” (PASP, (2015), 127, 266, 10.1086/680323). / Mangum, Jeffrey G.; Shirley, Yancy L.
In: Publications of the Astronomical Society of the Pacific, Vol. 129, No. 976, 069201, 01.06.2017.Research output: Contribution to journal › Comment/debate
}
TY  JOUR
T1  Corrigendum
T2  “how to calculate molecular column density” (PASP, (2015), 127, 266, 10.1086/680323)
AU  Mangum, Jeffrey G.
AU  Shirley, Yancy L
PY  2017/6/1
Y1  2017/6/1
N2  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 gK=2 for K ≠ 0. This would imply that we were ignoring the Kdoublet splitting in the molecule, which in this derivation we are not. The correct value for all asymmetric rotor levels is gK=1. We corrected Equation (100) and the leadin discussion of the assumptions that went into this equation. The following is meant to be a dropin 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 K1) and oblate (quantum number K+1) symmetric rotor energy levels that are closely spaced in energy, a feature commonly referred to as “Kdoublet 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 spinrotation and spinspin hyperfine energy level structure. Magnetic dipole interaction between the H nuclei (IH=0/1 for para/orthoH2CO) and rotational motion of the molecule result in spinrotation 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 IH = I1 + I2 and F = J + IH. Since the interchange of the two 1H atoms (I1=I2=1/2) obey Fermi–Dirac statistics (total wave function is antisymmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only orthoH2CO levels (IH=1) with Ka = odd integers have hyperfine splitting. For the orthoH2CO 110111 transition, the frequency offsets of these hyperfine transitions are Δν≤18.5 kHz. The weaker spinspin interactions between the nuclei are generally not considered. Table 10 lists the frequencies and relative intensities for the spinrotation hyperfine transitions of the H2CO 110111, 211212, and 312313 transitions. Figure 15 shows the synthetic spectra for the H2CO 110111, 211212, and 312313 transitions. Note that the hyperfine intensities are exactly equal to those calculated for the spinrotation hyperfine components of NH3 (see Section F). For illustration we can derive the column density equation for an orthoH2CO (K1 odd) Kdoublet (ΔK1=0) transition. For orthoH2CO transitions: S = K2/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 1018 esu cm Ri = (se Section 10 or, for 110  111, 211  212), or 312  313 see Table 10) gj = 2Ju + 1 gk = 1(for an asymmetric rotor) gI = 3/4 for K1 (Table presented) (Figure presented) We can derive the following equation for the molecular column density in H2CO as derived from a measurement of an orthoH2CO (K1 odd) Kdoublet transition assuming optically thin emission using Equation (80): (Formula presented.) Note that the Kdoublet 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 cm3, a collisional selection effect overpopulates the lower energy states of the 110111 through 514515 transitions (Evans et al. 1975; Garrison et al. 1975). This overpopulation of the lower energy states results in a “cooling” of the J≤5 Kdoublets to an excitation temperature that is lower than that of the cosmic microwave background. This causes the J≤5 Kdoublet transitions to appear in absorption. For volume densities n(H2)≿105.5 cm3 and TK≃40 K this collisional pump is quenched. For these higher volume densities the J≤5 Kdoublets are driven into emission over a wide range of kinetic temperatures and abundances (see Figure 1 in Mangum et al. 2008).
AB  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 gK=2 for K ≠ 0. This would imply that we were ignoring the Kdoublet splitting in the molecule, which in this derivation we are not. The correct value for all asymmetric rotor levels is gK=1. We corrected Equation (100) and the leadin discussion of the assumptions that went into this equation. The following is meant to be a dropin 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 K1) and oblate (quantum number K+1) symmetric rotor energy levels that are closely spaced in energy, a feature commonly referred to as “Kdoublet 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 spinrotation and spinspin hyperfine energy level structure. Magnetic dipole interaction between the H nuclei (IH=0/1 for para/orthoH2CO) and rotational motion of the molecule result in spinrotation 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 IH = I1 + I2 and F = J + IH. Since the interchange of the two 1H atoms (I1=I2=1/2) obey Fermi–Dirac statistics (total wave function is antisymmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only orthoH2CO levels (IH=1) with Ka = odd integers have hyperfine splitting. For the orthoH2CO 110111 transition, the frequency offsets of these hyperfine transitions are Δν≤18.5 kHz. The weaker spinspin interactions between the nuclei are generally not considered. Table 10 lists the frequencies and relative intensities for the spinrotation hyperfine transitions of the H2CO 110111, 211212, and 312313 transitions. Figure 15 shows the synthetic spectra for the H2CO 110111, 211212, and 312313 transitions. Note that the hyperfine intensities are exactly equal to those calculated for the spinrotation hyperfine components of NH3 (see Section F). For illustration we can derive the column density equation for an orthoH2CO (K1 odd) Kdoublet (ΔK1=0) transition. For orthoH2CO transitions: S = K2/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 1018 esu cm Ri = (se Section 10 or, for 110  111, 211  212), or 312  313 see Table 10) gj = 2Ju + 1 gk = 1(for an asymmetric rotor) gI = 3/4 for K1 (Table presented) (Figure presented) We can derive the following equation for the molecular column density in H2CO as derived from a measurement of an orthoH2CO (K1 odd) Kdoublet transition assuming optically thin emission using Equation (80): (Formula presented.) Note that the Kdoublet 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 cm3, a collisional selection effect overpopulates the lower energy states of the 110111 through 514515 transitions (Evans et al. 1975; Garrison et al. 1975). This overpopulation of the lower energy states results in a “cooling” of the J≤5 Kdoublets to an excitation temperature that is lower than that of the cosmic microwave background. This causes the J≤5 Kdoublet transitions to appear in absorption. For volume densities n(H2)≿105.5 cm3 and TK≃40 K this collisional pump is quenched. For these higher volume densities the J≤5 Kdoublets are driven into emission over a wide range of kinetic temperatures and abundances (see Figure 1 in Mangum et al. 2008).
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