A SOLAR-PUMPED FLUORESCENCE MODEL FOR LINE-BY-LINE EMISSION INTENSITIES IN THE B–X, A–X, AND X–X BAND SYSTEMS OF ¹²C¹⁴N

In a multilevel system, the line transfer between molecular ro-vibrational levels, u (upper) and l (lower), is determined by transition rates appropriate to their specific populations, n_u and n_l. This transfer involves collisional and radiative processes and is commonly described by the statistical equilibrium (SE) equation:

$$\begin{eqnarray}&&\displaystyle \frac{{{dn}}_{u}}{{dt}}=\sum _{l\ne u}^{N}\,{n}_{l}{R}_{{lu}}-{n}_{u}\sum _{l\ne u}^{N}\,{n}_{l}{R}_{{ul}}=0,\end{eqnarray} \tag{ 1 }$$

where R_ul and R_lu are the rate coefficients for loss from and entry into level u, respectively, resulting from collisions (both exciting and de-exciting), absorption of photons (excitation), and stimulated and spontaneous emission (de-excitation). External effects include solar and cosmic microwave background (CMB) radiation and local excitation caused by electron and neutral collisions. We defer consideration of thermal emission from dust and the comet nucleus to a future publication.

Schleicher (2010) discusses the omission of key vibrational and electronic levels to explain the mismatch of models and astronomical measurements for the violet system and cites the minor role of collisions in fluorescent equilibrium models (e.g., based on studies by Arpigny 1964 and Ishii & Tamura 1979). However, care must be taken in some specific cases. Collisions shape rotational populations in the ground vibrational state in the inner coma, where neutral gas density exceeds the critical density (whose value depends on water production rates, thermal and outflow velocities, and the molecular transition under consideration; Weaver & Mumma 1984; Xie & Mumma 1992; Bensch & Bergin 2004; Zakharov et al. 2007). At greater nucleocentric distances, collisions become less important and rotational populations increasingly approach fluorescence equilibrium. Observations taken with a large FOV tend to reflect the latter condition, but observations of the mid- and near-nucleus coma taken with a small FOV (e.g., most NIR observations, or near-nucleus observations by spacecraft) are dominated by collisions with neutrals (mainly H₂O, CO, and CO₂) and with electrons (details depend on production rates and other parameters; e.g., Feldman et al. 2015)—each situation must be considered on a case-by-case basis. These different processes and environments can be treated with the radiative transfer (RT) and SE equations.

The RT equation describes the emission and scattering of photons along a line in the direction of propagation (see van der Tak et al. 2007, for further details) and is defined as

$$\begin{eqnarray}&&\displaystyle \frac{{{dI}}_{\nu }}{d{\tau }_{\nu }}={S}_{\nu }-{I}_{\nu }\end{eqnarray} \tag{ 2 }$$

where ${S}_{\nu }$ is the source function, ${I}_{\nu }$ represents the specific intensity of electromagnetic radiation, and $d{\tau }_{\nu }={\alpha }_{\nu }$ds is the optical depth, with the source function S_ul(ν) depending on the emission ε_ul(ν) and absorption α_ij(ν) coefficients as S_ul(ν) = ε_ul(ν)/α_ul(ν). These coefficients are defined by

$$\begin{eqnarray}&&{\varepsilon }_{{ul}}(v)=\displaystyle \frac{{{hv}}_{0}}{4\pi }{n}_{l}{A}_{{ul}}\phi (v),\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&{\alpha }_{{ul}}(v)=\displaystyle \frac{{{hv}}_{0}}{4\pi }({n}_{l}{B}_{{lu}}\phi (v)-{n}_{u}{B}_{{ul}}\phi (v)).\end{eqnarray} \tag{ 4 }$$

These equations involve Einstein coefficients for spontaneous emission and absorption, A_ul and B_lu, and for stimulated emission, B_ul. Brooke et al. (2014) published Einstein coefficients and line strengths for many bands of the A²Π–X²Σ⁺ and B²Σ⁺–X²Σ⁺ electronic systems and for ro-vibrational transitions within the X²Σ⁺ state of CN (they used the pgopher program of Western 2010), and they considered the J dependence of the transition dipole moment matrix elements—the Herman–Wallis effect (see their paper for further details).

Collisional processes are defined by the rate C_ul = n_cpk_ul, with k_ul being the collisional rate coefficient and n_cp the density of the collision partner(s). For our model, we considered collisions of CN with electrons (e^–, hereafter e) and water (we defer the consideration of diverse molecular collision partners to a future paper). We obtained detailed e-CN cross sections from the Leiden Atomic and Molecular Database³ and scaled the electron density and temperature profiles from measurements in the coma of 1P/Halley, following the approach advocated by Biver (1997) (see also Xie & Mumma 1992).

Since detailed CN–H₂O cross sections are not yet available, we followed the standard procedure of assuming a constant cross section that is independent of relative velocity and ro-vibrational transition, equal to 40 Ų (or 4 × 10⁻¹⁵ cm²; Kleine et al. 1994), corresponding to a collision rate of 1.93 × 10⁻¹⁰ cm³ s⁻¹ at an assumed kinetic temperature of 100 K. As noted by Kleine et al., this value could be larger due to the long-range interaction between polar molecules (Green 1985).

Assuming spherically symmetric release from the nucleus and uniform outflow velocity, the density distribution of primary water can be estimated using a quantitative description presented by Haser (1957):

$$\begin{eqnarray}&&{n}_{p}(\rho )=\displaystyle \frac{Q}{4\pi {\rho }^{2}{\upsilon }_{\exp }}({e}^{-\rho /{\lambda }_{p}}),\end{eqnarray} \tag{ 5 }$$

where ρ is the nucleocentric distance (radius), v_exp is the expansion velocity, and λ_p is the photodissociation scale length equal to v_exp/β, where the photodissociation rate is β₀ at 1 AU and scales as $\beta ={\beta }_{0}/{{R}_{{\rm{h}}}}^{2}$. We took the lifetime-defining photodissociation rate coefficients (β₀) from Huebner et al. (1992). Equation (5) is based on several simplifying assumptions; for example, it omits the increase in expansion velocity with nucleocentric distance, assumes kinetic velocities to be constant, and omits H₂O release from icy-mantled grains in the near-nucleus coma region.

The origin of cometary CN radicals is a subject of extensive debate (e.g., Fray et al. 2005), and its roots are reviewed and discussed in Paganini et al. (2010). HCN is a known source of CN in cometary exospheres. Photodissociation of HCN leads to direct formation of H and CN, with a branching ratio of 97% (Huebner et al. 1992), and the production rates of HCN and CN are consistent in some comets. However, the production rate of CN often greatly exceeds the production rate of HCN, demonstrating that other production mechanisms are present; dust grains were identified as the principal precursor of CN in the study of 85 comets by A'Hearn et al. (1995), but see the earlier discussion in Section 1.

With the assumption of a single gaseous precursor (HCN), the density of CN is defined by

$$\begin{eqnarray}&&{n}_{d}(\rho )=\displaystyle \frac{Q}{4\pi {\rho }^{2}{\upsilon }_{\exp }}\displaystyle \frac{{\lambda }_{d}}{{\lambda }_{p}-{\lambda }_{d}}({e}^{-\rho /{\lambda }_{p}}-{e}^{-\rho /{\lambda }_{d}}).\end{eqnarray} \tag{ 6 }$$

If written in the form of modified Bessel functions, Equation (6) becomes

$$\begin{eqnarray}{n}_{d}(\rho ) & = & \displaystyle \frac{Q}{4\pi \rho {\upsilon }_{\exp }}\displaystyle \frac{{\lambda }_{d}}{{\lambda }_{p}-{\lambda }_{d}}\\ & & \times \left({\displaystyle \int }_{0}^{\rho /{\lambda }_{p}}{K}_{0}(y){dy}-{\displaystyle \int }_{0}^{\rho /{\lambda }_{p}}{K}_{0}(y){dy}\right),\end{eqnarray} \tag{ 7 }$$

where K₀(y) is the modified Bessel function of the first kind (Combi et al. 2004 and references cited therein).

This system of linear equations is handled by the public version of an RT code Radex (van der Tak et al. 2007), which uses a treatment of optical depth effects based on an escape probability method. The key inputs to Radex are (1) frequency range for the calculation (GHz), (2) kinetic temperature of the coma (K), (3) number of collision partners (we chose H₂O and e), (4) density of collision partners, (5) temperature of the CMB (2.73 K), (6) the column density of CN, and (7) the FWHM line widths of CN spectral lines. The solution ultimately yields the level populations of CN in the ground vibrational level, at the corresponding nucleocentric distances.

If we assume thermodynamic equilibrium, the rotational level population in the ground vibrational state is defined by a Boltzmann distribution, simplifying the computational leverage. This assumption is valid for most cases in the NIR, if the FOV encompasses distances where collisions with water are the dominant excitation mechanism; however, in the electron collision zone (between the contact and recombination surfaces, R_cs and R_rec; cf. Gombosi 2015) level populations change slightly, ultimately producing variations in estimated g-factors. Table 1 shows a comparison of the relative populations obtained, for two cases: (1) assuming a Boltzmann distribution and (2) assuming different FOV radii using Radex (see also Figure 1). The ends of our 178 long NIRSPEC slit extract correspond to a cometocentric distance of ∼645 km at the tangent point along the line of sight (typical for our extraction with Keck-2/NIRSPEC); the 2000 km distance corresponds to the region where electrons dominate excitations (between R_cs and R_rec) in our test case (i.e., a comet at heliocentric (R_h) and geocentric (Δ) distances of 1 AU; see Figure 1). We observe odd–even variations, which depend on the spin statistics (degeneracy) and energy levels of each particular rotational level.

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The values given in Table 1 differ from those of Schleicher (2010), who assumed fluorescence equilibrium. This difference is expected when different excitation mechanisms drive the excitation (collisions versus fluorescence equilibrium). The distribution of rotational levels (characterized by T_rot; see Equations (8) and (9) in Section 2.2) directly impacts the estimated fluorescence rates (g-factors). In Section 2.2 we provide the resulting g-factors, and in Section 2.3 we present comparisons with previous computations.

The determination of g-factors has been broadly explained in the literature, so we give only a short overview here (see Crovisier & Encrenaz 1983; Weaver & Mumma 1984; Zucconi & Festou 1985; Bockelée-Morvan 1987, for further details). Using level populations obtained for the lower energy levels (P_j described in Section 2.1), we estimate the direct pumping rates (g-factor, photons molecule⁻¹ s⁻¹) resulting from solar infrared irradiation, using

$$\begin{eqnarray}&&{g}_{{lu}}=\displaystyle \frac{1}{8\pi {{hv}}_{{ul}}^{3}}{P}_{l}\left(\displaystyle \frac{{w}_{l}}{{w}_{u}}\right){A}_{{ul}}J({v}_{{ul}}),\end{eqnarray} \tag{ 8 }$$

where w_l and w_u are the lower- and upper-level statistical weights, respectively, ν_ul is the frequency, and J(ν_ul) is the solar flux density.

We remind the reader that the Swings effect is considered in our models, but not until Section 2.4 do we analyze the effect of heliocentric velocity. Prior to Section 2.4, g-factors consider a heliocentric velocity (${\dot{{\rm{R}}}}_{{\rm{h}}}$) of 0 km s⁻¹.

Fluorescence excitation depends strongly on variations of solar radiation with wavelength, which are strong in the CN band systems owing to the high abundance of CN in the solar atmosphere. Using an accurate estimate of the solar flux is key for accurate determinations of the excitation process. We generated disk-averaged solar spectra by combining data from the Atmospheric Chemistry Experiment (ACE) and a purely theoretical model (Kurucz 1997). We followed the synthetic methodology described in Villanueva et al. (2011), where our hybrid model convolves the generated disk-averaged solar spectra with a modeled limb-darkening profile. The ACE spectrum features very high resolving power (v/δv ∼ 1 × 10⁵ from 750 to 4400 cm⁻¹; Hase et al. 2010), spanning most of the X–X band system, while the theoretical model covers the A–X and B–X band systems. Examples of the convolved spectra are shown in Figure 2; Fraunhofer lines of solar CN (X²Σ⁺—X²Σ⁺) are shown as an inset.

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Using Equation (8), we estimate the effective (excitation) pumping rate of rotational levels in v'' = 0 by direct infrared pumping. Later on, we estimate the radiative decay from excited ro-vibrational levels in higher vibrational states (v' = 1, 2, 3, ...15), including excitation and cascade from B and A electronic states into X-state ro-vibrational levels (to account for nonresonant fluorescence), following selection rules and Equation (9), described next.

For CN, the selection rules governing electric dipole transitions require that ΔJ = J' − J'' = 0, ±1, and the parity of the states involved in the transitions must change. For transitions involving ²Σ–²Σ levels, there is an additional selection rule: ΔN = ±1, and ΔJ = 0 is strictly forbidden. For them (e.g., B–X and X–X), there is no Q-branch, and the band system is characterized by R- and P-branches corresponding to transitions ΔJ = +1 and ΔJ = −1, respectively. The effect of these selection rules requires three components for each spectral line: R₁, R₂, and satellite transition ^RQ₂₁ $({\rm{\Delta }}J\ne {\rm{\Delta }}N)$, or P₁, P₂, and ^PQ₁₂ shown in Figure 3.

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Once fluorescence is traced, we estimate the detailed cascading fluorescence rate for each ro-vibrational level using

$$\begin{eqnarray}&&{g}_{{ul}}={A}_{{ul}}\displaystyle \frac{{\sum }_{j,{\rm{v}}=0}\,{n}_{j}{g}_{{ju}}}{{\sum }_{{\rm{v}}}{\sum }_{j}\,{A}_{{uj}}}.\end{eqnarray} \tag{ 9 }$$

In Table 2, we provide a list of g-factors at R_h = 1 AU, supposing the following: pure excitation of ro-vibrational transitions in X²Σ⁺ by solar flux alone (see note "a" in Table 2), and excitation of ro-vibrational transitions in X²Σ⁺ including the pumping contribution from the B–X and A–X systems (see note "b" in Table 2). The net result demonstrates that optical pumping through the red and violet bands is responsible for a significant fraction of the predicted fluorescent intensity in the CN (1–0) band.

We emphasize direct excitation and decay among ro-vibrational levels within the X-state alone and radiative pumping and decay from the A and B states. In Tables 3–4 and Figure 4, we show an example of the contribution (i.e., branching ratios, percentage) from the lower three vibrational states in the A- and B-states to the X-state. The g-factors from the main bands of CN resulting from our full-band model are displayed in Figure 5. In Figure 6 we show an expanded view of the infrared ro-vibrational emission bands: (1–0), (2–1), and (3–2) near 5 μm. A complete listing of line-by-line g-factors, including for the B–X and A–X band systems, can be provided by the authors upon request.

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As mentioned in Section 1, previous studies have presented calculations of g-factors for the observed violet (e.g., Zucconi & Festou 1985; Schleicher 2010) and red (Zucconi & Festou 1985; Fink 1994) band systems. Moreover, Figure 5 in Zucconi & Festou (1985) and Figure 15 in Kleine et al. (1994) presented detailed calculations of ro-vibrational lines within the violet system of ¹²C¹⁴N and their variation with heliocentric radial velocity.

In Tables 5–9 we compare results from our model with those of previous studies. For instance, our A–X band rates agree with calculations by Fink (1978, see Table 3 in his paper), as do branching ratios given in our Table 4 and in Fink's paper (see his Table 5). On the other hand, the estimate of the A–X (0–0) band efficiency by Zucconi & Festou (1985) is about a factor of 3 smaller than those obtained by Fink and this work.

The integrated fluorescent intensities for the violet bands (Table 7) show good agreement with earlier work. However, an in-depth comparison between the two most complete studies presenting several band g-factors (ours and Schleicher 1983) show significant differences (see Tables 6 and 8). Furthermore, the line-by-line g-factors differ significantly. Compared with Zucconi & Festou (1985) and Kleine et al. (1994), our individual g-factors for ro-vibrational lines in the B–X (0–0) band (Table 7) differ by less than a factor of 2 for the R2–R5 lines, by a factor of 3 in R0, and by up to a factor of 16 for R1.

Based on these comparisons, the aforementioned differences could be attributed to the strong dependency on our assumed collisional environment and to the distribution of rotational levels in the ground vibrational state (characterized by T_rot; see Equations (8) and (9)) that can influence these comparisons directly. Moreover, we expect differences in input parameters such as the solar spectrum, in methodologies used to estimate the rotational population in v'' = 0, and in Einstein coefficients. Given the lack of further details in the models presented by these authors, these are mere assumptions, which cannot be backed up by empirical evidence at the present time.

The rotational temperature defines the distribution of detailed fluorescence rates by shaping the relative level population (see Equation (8)). This distribution is reflected in (1–0) P- and R-branches in Figure 6, where the stronger g-factors of low-J ro-vibrational transitions decrease with increasing temperature, while transitions involving higher rotational levels experience a slight increase and the overall breadth of the distribution increases too. This behavior is also observed in g-factors that account for cascade pumping of (1–0) emission by B–X and A–X band fluorescence (right panels). The fluorescence cascade also excites slightly weaker lines of higher vibrational bands (in this case, 2–1 and 3–2) that should be diagnostic of this cascade in actual cometary spectra.

Our CN model accounts for the changes in solar flux available for fluorescence due to Doppler displacement of Fraunhofer lines relative to the CN line positions in the comet's rest frequency (n_ul, cm⁻¹) (the Swings effect; Swings 1941). The effective pumping frequency depends on the comet's radial velocity as

$$\begin{eqnarray}&&{v}_{\mathrm{eff}}={v}_{{ul}}\left(1-\displaystyle \frac{{\dot{{\rm{R}}}}_{{\rm{h}}}}{c}\right).\end{eqnarray} \tag{ 10 }$$

Tatum & Gillespie (1977) and Tatum (1984) extended Swings foundational work for the violet system by using a whole-disk solar spectrum at optical wavelengths to show variations in the integrated band intensity. Kleine et al. (1994) demonstrated the significant variation of fluorescence efficiencies with heliocentric radial velocities for several R-lines of the B–X (0–0) band, confirming the importance of an accurate solar spectrum. This is reflected in Figure 7 and Equation (10). Further studies by Zucconi & Festou (1985) and Schleicher (2010) showed similar outcomes. Ultimately, while we observed a strong dependence of g-factors on rotational temperature, the few percentage variation of these rates with heliocentric velocity can create subtle changes in the retrieval of rotational temperatures and production rates.

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