Corrigendum

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

Jeffrey G. Mangum, Yancy L Shirley

Research output: Contribution to journalComment/debate

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 gK=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 gK=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 (IH=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 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 anti-symmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only ortho-H2CO levels (IH=1) with Ka = 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/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 10-18 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 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)≿105.5 cm-3 and TK≃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).

Original languageEnglish (US)
Article number069201
JournalPublications of the Astronomical Society of the Pacific
Volume129
Issue number976
DOIs
StatePublished - Jun 1 2017

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rotors
energy
energy levels
nuclei
quantum numbers
derivation
formaldehyde
molecules
cosmic microwave background radiation
symmetry
asymmetry
pump
replacement
temperature
diagram
magnetic dipoles
ladders
lists
excitation
integers

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 journalComment/debate

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title = "Corrigendum: “how to calculate molecular column density” (PASP, (2015), 127, 266, 10.1086/680323)",
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 gK=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 gK=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 (IH=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 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 anti-symmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only ortho-H2CO levels (IH=1) with Ka = 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/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 10-18 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 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)≿105.5 cm-3 and TK≃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).",
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T2 - “how to calculate molecular column density” (PASP, (2015), 127, 266, 10.1086/680323)

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AU - Shirley, Yancy L

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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 K-doublet 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 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 (IH=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 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 anti-symmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only ortho-H2CO levels (IH=1) with Ka = 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/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 10-18 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 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)≿105.5 cm-3 and TK≃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).

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 K-doublet 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 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 (IH=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 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 anti-symmetric to this interchange) and alternating Ka ladders in H2CO have a different rotational wavefunction symmetry, only ortho-H2CO levels (IH=1) with Ka = 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/Ju(Ju + 1) m = 2.332 Debye = 2.331 × 10-18 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 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)≿105.5 cm-3 and TK≃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).

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