Spectral Deconvolution of the 6196 and 6614 Å Diffuse Interstellar Bands Supports a Common-carrier Origin

The spectral deconvolutions and fits for all of the data used Gaussian broadening with widths derived from fits to the observed Na or K atomic line velocity profiles for the DIB sightlines (Galazutdinov et al. 2002). HD 145502 displays a bimodal Na velocity distribution. We based the width estimate on the smaller profile in order to minimize the impact of line saturation effects (Galazutdinov et al. 2002). The EW of the smaller profile is approximately 16% that of the primary profile; however, saturation of the primary feature results in an overestimate of the relative Na abundance for the smaller feature. This is borne out later, where spectral fits show that the smaller feature corresponds to a relative contribution of 6% to the HD 145502 DIB spectra. For HD 179406, we also considered the measured CH⁺ velocity profile (Kazmierczak et al. 2009), whose width is approximately 15% larger than that for the K line profile. The average of the two measured widths was used in this analysis. The values of the widths are presented in Table 1. The widths include a small, ∼0.01 cm⁻¹, contribution for the sensor resolution in which the Doppler velocity and sensor widths are combined in quadrature. Additional broadening sources could be included in the analysis, such as lifetime and vibrational anharmonicity broadening; however, as discussed below, there is no indication of a significant broadening residual in the deconvolved data.

Table 1.  Molecular Spectral Fit Parameters to the Observational Data Determined Using PGOPHER

Observation	B(cm−1)a	ΔB/B(%)	C(cm−1)	ΔC/C(%)	${T}_{\mathrm{rot}}$(K)	${J}_{\max }={K}_{\max }$	ζ	${\gamma }_{G}({\mathrm{cm}}^{-1})$ b
HD 145502 λ6614c	0.0232	−0.70	0.0096	0.0	50	26	−0.60	0.30
HD 179406 λ6614c	0.0232	−0.70	0.0096	0.0	18	26	−0.60	0.28
HD 145502 λ6196	0.0232	1.7	0.0096	0.0	7.0	26	0.62	0.32
HD 179406 λ6196	0.0232	1.7	0.0096	0.0	3.5	26	0.62	0.30
HD 145502 λ6614d	0.0232	−0.70	0.0213	0.0	150	27	0.30	0.30
HD 145502 λ6196d	0.0232	2.6	0.0213	0.0	7.0	27	0.82	0.32

Notes.

^aB is the rotational constant for the lower transition level. ^bFWHM. ^cThe splitting, Δν, and weights of the doublet components for λ6614 are defined in Figures 7 and 8. ^dNearly spherical, slightly oblate symmetric top fullerane carrier.

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The deconvolved spectrum for λ6614 from HD 145502 is presented in Figure 4. We see two slightly separated sets of similar PQR branch features, which is suggestive of doublet structure. There also appear to be comparably intense P and R branches with similarly steep outer edges. The overall shape separates into two, slightly overlapping, main regions, an approximately symmetric central PQR region spanning approximately −1.3 to 1.3 cm⁻¹, and an asymmetric red tail below approximately −1.3 cm⁻¹. In contrast to the symmetric region, the red tail does not show evidence of a doublet structure. We interpret this to mean that two blended DIBs, one for the doublet and the other for the red tail, contribute to λ6614. This interpretation is supported below through a detailed spectral analysis.

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Rather than trying to simultaneously model the three overlapping components of λ6614, the doublet and red-tail features, we found it useful to first mathematically disentangle the components using simple analytic functions. The approximate spectroscopic constants derived from subsequent spectral fitting of the analytically separated components provided a good parameter initialization for use in spectral fitting of the observed spectrum. For the doublet feature, we assumed that it was composed of two slightly offset spectra with different weights but identical shapes. The assumption of identical shapes is a convenient simplification whose validity is explored in Section 5.2. For the analytic fit, we represented the P, Q, and R branch features of a doublet component as a combination of a Gaussian and a Lorentzian function. The resulting analytic fit to the deconvolved spectrum is shown in Figure 5. The quality of the analytic fit supports the supposition that a doublet feature is a major constituent of the λ6614 spectrum.

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Figure 5 also shows that the difference between the deconvolved spectrum and the analytic model defines a residual red-tail feature with an R branch bandhead and a gradually declining red tail. This is an often-encountered DIB shape and is evidence of a separate carrier. While the primary focus of this work is on λ6614 and λ6196, we discuss the implications of the red tail, how it influences the λ6614–λ6196 correlation, and its interpretation as a separate DIB in Sections 5.4 and 5.5.

The analytic model fit to the deconvolved data confirms our previous analysis of the broadened λ6614 spectrum (Bernstein et al. 2015b), which was performed without the benefit of spectral deconvolution. In the prior analysis, we determined a doublet splitting of Δν = 0.25 cm⁻¹ and relative weights for the doublet components of w₁ = 0.59 and w₂ = 0.41 (i.e., w₁ + w₂ = 1).

We used a physics-based spectral model, PGOPHER (Western 2017), to fit the larger analytic component for λ6614. This provided good estimates of the spectroscopic parameters for subsequent use in modeling the observed spectrum. The spectroscopic parameters consist of the lower electronic state rotational constants C (for rotations about the molecular symmetry axis, which is aligned with the dipole axis in Figure 2) and B (for rotations about an axis perpendicular to the symmetry axis); the difference between the upper and lower state rotational constants, ΔB and ΔC; two effective rotational temperatures, T_rot; for the tumbling and spinning top motions, the maximum rotational quantum numbers, J_max and K_max; the Coriolis coupling constant, ζ; the Gaussian line width, γ_G; the relative weights of the two components, w₂ and w₁ (where w₁ > w₂ and w₁ + w₂ = 1); and the doublet splitting, Δν. The rotational constants, B and C, are the key to identifying potential carriers because they tightly constrain the shape and size of the carrier (Bernstein et al. 2017).

With the exception of J_max and K_max, the above set of spectroscopic parameters represents the minimum number of free parameters required to model the deconvolved doublet spectrum. Assuming a symmetric top carrier, the minimal set of parameters required to model one of the doublet spectral profiles includes B, C, ΔB, ΔC, two T_rot, and at least one broadening parameter, γ_G, to account for Doppler broadening. If sightline atomic line absorption velocity profiles have been measured, as is the case here, then γ_G is well determined from the profile width and, thus, is not a free parameter. Lifetime broadening, which is described by the Lorentz function, may also be required if the upper electronic state of the transition is very short lived. A doublet spectrum requires two additional parameters, Δν for the separation of the doublet components and the relative weight of one of the components (i.e., just one parameter needed because of the constraint w₁ + w₂ = 1). This set of parameters applies to a transition with A → A symmetry. If the transition has A → E symmetry, then an additional parameter is needed, the Coriolis coupling constant, ζ. We will see later that the Coriolis parameter is crucial to demonstrating a common-carrier origin for λ6614 and λ6196. A sharp cutoff for the rotational quantum numbers is not typically encountered. However, the narrow and sharply truncated P and R branches for λ6614 (see one of the analytic profiles in Figure 5) indicate the need for this parameter.

We screened a variety of candidate carrier classes comprising different molecular geometries (i.e., linear, spherical, oblate, and prolate symmetric tops) and transition symmetries (i.e., parallel and perpendicular, degenerate and nondegenerate vibrational states). For each geometry and transition symmetry selection, we performed a coarse spectroscopic parameter search, using the above parameter set, to ascertain if a good common-carrier spectral fit to both DIBs was feasible. We found only one combination with the potential to be quantitatively consistent with the common-carrier hypothesis. The screening suggested that the carrier is a dipolar oblate top, with a perpendicular electronic transition symmetry from a nondegenerate into a doubly degenerate vibrational state (this applies to both λ6614 and λ6196). Two classes of carbonaceous molecules are consistent with these specifications, a puckered disk-like compact PAH (e.g., corannulene, C₂₀H₁₀) and a nearly spherical fullerane (e.g., C₂₀H). Not all fulleranes satisfy the symmetric top requirement. The candidate fulleranes that satisfy the derived symmetry and size requirements are C₂₀H and C₂₈H.

Although a compact PAH and a fullerane have very different shapes, from the perspective of spectral profile fitting, they are indistinguishable. The equivalence arises from the Coriolis effect, which manifests in the transition frequencies as (1 – ζ)C (see Appendix B for the detailed spectroscopic origin of this term). Since the transition frequencies are determined by the product of two factors, any combination of factors that yield the same product will exhibit the same transition frequencies. Later, we discuss that a compact PAH and a fullerane with the same B rotational constant but different C rotational constants can produce essentially equivalent spectral profiles for λ6614 and λ6196. Hence, because of the spectroscopic equivalence, we frame our analysis and discussion primarily in terms of a compact PAH. All of the basic concepts and general results found for a compact PAH apply to a fullerane.

The PGOPHER spectral model fit to the HD 145502 λ6614 analytic profile is shown in Figure 6. The fit is good, but not exact. This is not surprising, since the analytic model is not a physics-based model. The sensitivity of the spectral fit to variations in the spectroscopic parameters is depicted in Figures 15–21 in Appendix A. There are three noteworthy aspects to the spectral fit. The first is the sharp cutoff at relatively small rotational quantum numbers, J_max = K_max = 26 (i.e., E_rot(J_max) ≪ kT_rot). When thermodynamic equilibrium is maintained through collisions, a P or R branch exhibits a much broader spectrum, by about an order of magnitude, than a Q branch. Without a J, K cutoff, the calculated P and R branches would peak at J, K ∼ 50 and would extend out to J, K ∼ 150 (see Figure 16).

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The second aspect is the sensitivity of the simulated shape and width to the Coriolis constant, as demonstrated by Figure 17. Most of the band profile differences between λ6614 and λ6196 can be explained simply by a difference in the values of the Coriolis constant. The remaining difference, primarily in the widths of the bands, is attributed to the difference in effective rotational temperatures for the bands. The third aspect is the small value of the "residual" Gaussian width, γ_G = 0.04 cm⁻¹, needed for the spectral model fit to the deconvolved spectrum. This means that the width of the observed data can be fully attributed to the combination of velocity broadening, the slightly separated doublet structure, and sensor resolution. As discussed in Section 5.2, this finding supports Webster's (2004) idea that isotopologs of a DIB carrier may produce several, slightly separated DIB bands.

We performed the PGOPHER spectral fit to the HD 145502 observation shown in Figure 7 by perturbing the spectroscopic parameters from the PGOPHER spectral fit to the analytic component. We found that the relative weights of the doublet components change slightly, from w₁ = 0.60 and w₂ = 0.40 for the analytic model to w₁ = 0.68 and w₂ = 0.32 for the spectral fit to the data, due to the inclusion of a third, small component arising from the bimodal Na velocity distribution (Galazutdinov et al. 2000). Inclusion of the third component introduced an asymmetry into the relative heights of the P and R branches, which resulted in a change of the vibration–rotation constant from ΔB/B = 0 to ΔB/B = −0.70%.

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There is a noticeable fit residual corresponding to the very narrow dip in the data at approximately −0.5 cm⁻¹. We do not believe this feature is associated with the λ6614 DIB. It is not evident in the other λ6614 spectra (Galazutdinov et al. 2002), and the width of the feature is equal to that of the sensor spectral resolution, suggesting it is an atomic emission line.

We turn our attention to the spectral fitting of λ6614 for the second sightline, HD 179406. We performed the λ6614 PGOPHER spectral fit for HD 179406, shown in Figure 8, by adjusting the λ6614 spectroscopic fit parameters determined for HD 145502. The relative weights of the doublet components are essentially identical to those for HD 145502, which, as discussed later, indicate nearly identical ¹³C abundances for the two sightlines.

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There are several notable differences in the two observed spectra that are reflected by the differences in their spectroscopic parameters. The HD 179406 spectrum is slightly narrower than that for HD 145502, which is reflected in its lower T_rot, 18 versus 50 K. Even though this is a substantial difference in rotational temperatures, its effects on the band shape and width are small because of the low rotational energy cutoff (see Figure 18). The other difference is for the doublet splitting, Δν = 0.29 and 0.35 cm⁻¹ for HD 179406 and HD 145502, respectively. The 0.06 cm⁻¹ difference in the splitting reflects the uncertainty of measuring the separation of two peaks for a sensor spectral resolution of 0.07 cm⁻¹.

We summarize here the key physical ideas with regard to the origin and effects of rotational non-equilibrium for symmetric top molecules (i.e., the DIB carriers) and then present a more detailed explanation, in Appendix B, based on the fundamental spectroscopy for symmetric tops. There are three key ideas. The first is that there are two effective rotational temperatures, as originally suggested by Glinski & Eller (2016), that characterize the populations of the rotational states for a polar symmetric top in which the dipole moment is aligned with the highest-order symmetry axis. They correspond to rotations perpendicular to and about the highest-order symmetry axis. The perpendicular rotations can emit and absorb radiation because the dipole moment is changing its orientation and thereby produces an oscillating electric field. Because de-excitation collisions are rare in the low-density ISM, the primary de-excitation pathway for a rotating molecule is through emission (see Appendix C for a quantitative comparison of radiative emission and collision rates). An excited molecule will spontaneously emit its perpendicular rotational energy until it reaches equilibrium with the ambient radiation field, which is frequently at or near the CMB at a temperature of 2.73 K but will never be colder (Bernstein et al. 2015a).

In contrast, rotations about the highest-order symmetry axis do not change the orientation of the dipole moment and therefore cannot readily, via allowed transitions, absorb or emit photons. As a result, the rotational temperature for this motion will fall somewhere between two limits. If the collision rate is zero, then its rotational temperature will reflect its nascent (i.e., initially formed) distribution of rotational states. If the collision rate is nonzero, then the rotational temperature will eventually reach thermal equilibrium with the translational temperature. However, this may take a very long time, and other timescales, such as photodissociation, may interrupt the full thermalization of the rotational states.

Although the two-temperature model has not previously been used to model DIB spectra, it has been applied to the closely related problem of modeling interstellar microwave emission spectra. Loren & Mundy (1984) used the two-temperature approximation, which they referred to as the rotational temperature equilibrium (RTE) method, to analyze the rotational emission spectrum of the symmetric top molecule CH₃CN in interstellar molecular clouds. In the two-temperature/RTE model, the relative population between K stacks and the population of J levels within a K stack are described by different effective T_rot. The J levels for a given K correspond to different end-over-end tumbling rates for a fixed spin rate about the symmetry axis. The T_rot characterizing the J-level population within a K stack is very cold because these J levels can efficiently relax via rotational radiative decay. We expect the T_rot for the different K stacks to be nearly the same, as is borne out by Figure 12 in Loren & Mundy's (1984) analysis of CH₃CN. The validity of the two-temperature model and the J_max = K_max approximation for the prototypical common carrier for the λ6614 and λ6196 DIBs are discussed in Appendix C.

The second key idea is that the value of the Coriolis constant determines which one of the two effective rotational temperatures characterizes a spectrum. For instance, in Appendix B, we show that in the limit of ζ = 1, the spectrum depends only on the very cold T_rot associated with the tumbling motion. As a result of the low rotational temperature, the band is narrow. Conversely, for ζ = −1, the spectrum is sensitive to the hotter rotational temperature associated with the spinning motion, and the band is much wider. Examples of synthetic DIB spectra corresponding to the two limits for ζ are presented in Appendix B (see Figure 22). While these are idealized limiting cases, they convey the idea that the effective rotational temperature for a DIB band will depend on its value for ζ, which can be different for each degenerate mode. Thus, two DIBs arising from the same ground state but terminating on different excited degenerate vibrational modes can exhibit very different shapes and widths.

The third key idea is that there is a steep cutoff in the maximum rotational quantum states, which we approximate as a hard cutoff, J_max = K_max. The existence of the sharp cutoff is reinforced by the spectral deconvolution of λ6614; however, the physical origin of the cutoff is a matter of conjecture. λ6196 is insensitive to the existence of the sharp cutoff because its low effective T_rot renders it insensitive to the higher J, K states associated with the cutoff (see Figure 22). Sharp rotational level cutoffs have been observed in absorption and emission spectra for many diatomic hydrides, with OH as a particularly noteworthy example (Carrington 1964). The cutoff arises from a predissociation barrier in an excited electronic state. J states at or above the predissociation barrier rapidly dissociate, leading to very broad lines with small absorption and emission spectral intensities. Conceptually, predissociation or an analogous transformation (see Appendix D) could occur for a much larger DIB molecular carrier, such as a PAH or fullerane.

In summary, the two-temperature model, the Coriolis effect, and the steep cutoff for the maximum rotational quantum states are necessary and physically credible components of a common-carrier spectral model for the λ6614 and λ6196 DIBs.

We attribute the doublet to a combination of two isotopic shifts originating from the substitution of ¹³C for ¹²C in the carrier. One shift is due to the difference in the zero-point vibrational energies between the upper and lower electronic states (Webster 2004) of a DIB transition. The other is the isotope shift in the fundamental frequency of the assumed singly excited, degenerate vibrational mode in the upper electronic state of the λ6614 DIB transition. All the other vibrations in the upper electronic state are assumed to be in their ground state, as well as all the vibrations in the lower electronic state. We estimate below the shifts and widths of the doublet component arising from one or more ¹³C substitutions.

Our estimates are based on the prototype carrier corannulene for which N_C = 20. A single ¹³C substitution is considered first, as it is the dominant substitution for typical interstellar ¹³C abundances. According to Webster (2004), substitution of a single ¹³C into a carrier with N_C = 20 will result in a zero-point energy shift of approximately +0.9 cm⁻¹. This is likely as an upper limit estimate because it rests on the assumption that the promotion of a bonding pi electron into an excited electronic state is equivalent to the complete elimination of a C–C bond. In general, we expect the decrease in electronic stability to be delocalized, resulting in a small reduction in the average bond stiffness in the upper electronic state relative to the lower, ground electronic state. We are not aware of any experimental or accurate first principles determination of the zero-point shift for a large PAH. Regardless of the precise value of the zero-point shift, all DIBs arising from ground vibrational state transitions in the upper and lower electronic states should exhibit blueshifts in their peak locations that will increase with increasing ¹³C substitution.

If not fully resolved, the ¹³C spectral component should produce a slight broadening and blue shaded asymmetry to the observed DIB profile. The variable (i.e., depends on the ¹³C abundance) broadening and asymmetry attributed to the zero-point isotope blueshift may be evident in several blueshifted DIB profiles observed by Galazutdinov et al. (2015).

Unlike the zero-point isotope shift, which involves summing over all of the vibrational modes, the isotope shift in a fundamental frequency depends specifically on which vibrational mode is excited. Since neither the actual carrier nor specific mode is known, we can only establish a plausible range for the shift of the fundamental frequency. This was accomplished using quantum chemistry computations for the fundamental frequencies of corannulene in which a single ¹³C substitution was considered for the two distinct edge positions and one distinct interior position (see Figure 2). The resulting distribution of shifts for the degenerate modes is shown in Figure 12. Not displayed is the small fraction of shifts, ∼6%, that fall within the range of −5 to −22 cm⁻¹. DIBs arising from such highly shifted modes, if observable, would appear well separated (i.e., not blended) from the λ6196 and λ6614 DIBs. The preponderance of shifts, about 80%, fall within the range of −1.9 to 0.1 cm⁻¹. Combining this range with the zero-point shift implies that the most probable combined shifts fall within the approximate range of −1.0 to +0.8 cm⁻¹. The observed shifts for λ6614 of ∼0.3 cm⁻¹, as well as the lack of an observable shift for λ6196, are consistent with the estimated range of shifts.

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Figure 12 shows that ∼2/3 of the redshifts fall within a small range of −0.7 to +0.1 cm⁻¹. If we assume that this fairly probable range encompasses the shifts of the λ6614 and λ6196 DIBs, then this implies that the zero-point shift is considerably smaller than the presumed upper limit estimate of +0.9 cm⁻¹, most likely in the range of ∼0–0.4 cm⁻¹. This reduced range for the zero-point shift makes physical sense. The widths of most DIBs fall within the range of ∼1–2 cm⁻¹ (Hobbs et al. 2008, 2009). If the typical zero-point shift were +0.9 cm⁻¹, then, contrary to observations, many DIBs would exhibit obvious doublet features.

λ6196 does not exhibit distinct doublet components, even after the application of spectral deconvolution (see Figure 9). However, we found that an extra broadening contribution, γ_G = 0.16 cm⁻¹, was required to fit the deconvolved spectrum. Also based on spectral modeling, Webster (2004) estimated a comparable zero-point broadening contribution of 0.25 cm⁻¹ for λ6196.

We obtained good spectral fits to the λ6614 DIBs based on the simplifying assumption that the doublet components have identical shapes. This assumption results in two additional spectral fitting parameters, the doublet separation, Δν, and the relative weights, w₁ + w₂ = 1, of the two components. However, one expects that the smaller doublet component, due to the ¹³C substitutions, should be wider than the larger, pure ¹²C component. This follows because single and multiple ¹³C substitutions over the different types of locations within the carrier will have different shifts, thus imparting "extra" width to the effective profile. A third adjustable parameter would be required to account for the extra width of the smaller doublet component. We conjecture that allowing the width to vary would incrementally improve the fits, and consequently, we decided not to include it in our analysis.

The relative weights of the doublet components are consistent with the isotope model. We associate w₁ with the pure ¹²C carrier and w₂ with the sum of the weights for all of the ¹³C substitutions. Table 2 presents the top four fractional isotope abundances for corannulene as a function of the ¹²C/¹³C abundance ratio, R. The abundances are based on a Poisson distribution as discussed in Webster (2004; see Equation (8)). For galactic molecular clouds, R ranges from ∼20–70, with an average of ∼45 (Savage et al. 2002). We retrieved values of w₁ = 0.67 and 0.68 for HD 145502 and HD 179406, respectively. Table 2 shows that these values for w₁ imply R ∼ 50, which is in line with the molecular cloud average of R ∼ 45. The ratio R ∼ 50 means that ∼82% of the ¹³C carriers correspond to a single ¹³C substitution, with the remaining ∼18% predominantly due to double ¹³C substitutions.

At the high end of the molecular cloud ¹³C abundance range, R ∼ 20, the appearance of the λ6614 spectrum would be dramatically different from those considered here. The majority of the carriers, about 63%, would contain one or more ¹³C. This would substantially shift the peak locations and broaden the λ6614 spectrum to the red. Additionally, the distinctive PQR branch structure would be difficult to discern.

We would expect that the isotope shifts for the sightlines would be nearly identical since they have essentially the same ¹³C abundances. However, the shifts differ by 0.06 cm⁻¹. We attribute the difference to the effects of sensor resolution. The accuracy to which the separation of two narrow Q branch peaks can be measured is limited by the measurement spectral resolution of 0.07 cm⁻¹. Variability arises because variations in the sightline radial velocities can result in a variation of up to one detector element in the apparent separation of narrow features.

In summary, isotope shifts are expected to be evident in most DIB spectra, either as separate features or as extra broadening and asymmetry. Redshifts may occur for DIB transitions from a ground vibrational and electronic state to a vibrationally excited upper electronic state. Blueshifts will occur for DIB transitions between ground vibrational levels of the lower and upper electronic states.

The DIB correlations study of Galazutdinov et al. (2002) supports the two-temperature model. They noted substantial variations of the widths of λ6196 for the sightlines investigated. In contrast, λ6614 exhibited minimal variability in its width. As discussed above, the two-temperature spectral model displays these same characteristics. We showed that the predicted shape and width for λ6614 had minimal sensitivity to T_rot due to the sharp cutoff for J_max (see Figure 18). The predicted spectrum for λ6614 is closely tied (see Appendix B) to the non-emitting rotations about the carrier's highest symmetry axis, implying that the local translational temperature should govern T_rot. Thus, even though T_rot will vary substantially among different sightlines, this will have little impact on the predicted width of λ6614. In contrast, we attribute the λ6196 spectrum to rotations about the emitting rotational axes whose T_rot will be much colder than that for λ6614 and tied closely to the cold T_rot measured for smaller polar molecules. The predicted widths for λ6196 are sensitive to small changes in temperature (see Figure 21) because even a variation of 1 K corresponds to a large fractional temperature change for a small T_rot.

Kazmierczak et al. (2009) present a more challenging observational result for the two-temperature model. They discovered a significant positive correlation between the width of the λ6196 band and the rotational temperature of a nonpolar C₂ molecule. They hypothesized that this correlation implied that the λ6196 carrier is also a nonpolar molecule because the widths for a polar molecule should be nearly invariant for all sightlines due to rotational emission. However, their hypothesis was based on the implied assumption of a single effective rotational temperature for the carrier. Whether or not the two-temperature model for a polar molecule can also exhibit this correlation is an open question. Its resolution requires the development of a first principles theoretical model for the collisional and radiative excitation and de-excitation of a polar symmetric top molecule. This is a challenging endeavor and beyond the scope of this work. However, we anticipate that such a correlation is likely to exist for the two-temperature model. This is because the excitation and de-excitation of rotation about the highest-order symmetry axis are analogous to those for a nonpolar molecule. We expect that the shape and width of any band for a polar symmetric top molecule will depend on both their hot and cold rotational temperatures, and therefore they should exhibit a positive correlation with the rotational temperature for a nonpolar molecule.

For a variety of reasons, λ6614 is one of the most studied DIBs. Its prominent PQR-like structure is indicative of a molecular carrier. It has the strongest known correlation with another DIB, λ6196. It is one of just a few DIBs that may also appear in emission in the Red Rectangle (Sarre et al. 1995). It has an extended red tail that extends much farther to the red—i.e., the so-called extended tail to the red (ETR; Dahlstrom et al. 2013; Oka et al. 2013)—than is typically observed in Herschel 36 (see Figures 7 and 8). Understanding the physical origin of the enigmatic red tail is important because it is central to resolving several outstanding issues, such as whether or not λ6614 is a blended DIB and whether or not the λ6614 and λ6196 DIBs arise from the same carrier.

A number of recent studies have explored different origins for the red tail. Several ascribed to the red tail unusually large values of the vibration–rotation constants (Oka et al. 2013; Glinski & Eller 2016). This approach did not produce quantitative fits to the DIB and ETR data for λ6614. Another study attributed the red tail to shifted hot bands of the relatively symmetric central PQR structure (Marshall et al. 2015). The approach was applied to spectra from two sightlines toward HD 147889 and HD 179406, featuring large and small red-tail contributions, respectively. Including hot bands substantially improved the fit for the large tail spectrum. Only a marginal improvement was obtained for the small tail spectrum. This approach has yet to be applied to the ETR and Red Rectangle emission DIBs. A third approach suggested that λ6614 is a blended DIB in which the central symmetric PQR structure and the red tail are due to different carriers (Bernstein et al. 2015b). This approach produced quantitative spectral fits to the DIB, ETR, and Red Rectangle emission spectra.

The enhanced spectral resolution provided by the deconvolution (see Figures 4 and 5) reveals features that are most consistent with the blended DIB hypothesis. The deconvolved spectrum suggests that the central PQR region is comprised of two slightly shifted components. This is confirmed by the analytic spectral fit that shows (1) there are similarly shaped doublet components separated by 0.35 cm⁻¹, and (2) the doublet components have comparable abundances (i.e., w₁ = 0.60 and w₂ = 0.40). The deconvolution also displays a sharp feature at a frequency of ν ∼ −1.5 cm⁻¹. If this feature was associated with the central components, then one would expect it would also exhibit a second slightly shifted sharp feature. There is no indication of a second component in the deconvolved spectrum. Furthermore, it is apparent that the smoothly declining red-tail residual cannot be quantitatively fit by a linear combination of shifted highly structured central spectral components (i.e., the hot band hypothesis). The red-tail residual has a classic molecular spectral structure, a blue bandhead with an extended red tail, which can be fit with spectroscopic constants very different from those for the PQR feature (Bernstein et al. 2015b). The spectroscopic constants for the red-tail residual also provided quantitative fits to the ETR and Red Rectangle spectra. We conclude that there are different molecular carriers for the red tail and central PQR spectral components; therefore, λ6614 is a blended DIB.

The origin of the red tail also bears strongly on whether or not λ6614 and λ6196 arise from a common carrier. If we assume that the two DIBs share a common carrier, then this argues against the hot band origin of the red tail. The hot bands originate from a vibrationally excited ground electronic state of a DIB transition and therefore they would likely produce comparably strong red tails for all DIBs sharing the same ground state. There is no evidence for a significant red tail in the λ6196 spectra (see Figures 9–11; Galazutdinov et al. 2002, 2008). There is a minor feature that appears slightly to the red of λ6196 in the Herschel 36 data. However, the shape of this feature, a distinct peak and only slight overlap with λ6196 (see Figure 6 in Krelowski 2014), suggests it is a unique DIB, not the red tail of λ6196. Thus, we conclude that the common carrier and blended DIB hypotheses are self-consistent.

Krełowski et al. (2016) demonstrated a significant variability of the ratio of EWs for λ6614 and λ6196 that was not due to measurement noise. They interpreted this to mean that there are separate but closely related carriers for the DIBs. We suggest that this "extra" variability arises from uncorrelated spectral features contaminating the DIBs. The uncorrelated features include the red tail for λ6614 and a number of smaller features for λ6196 (see Figure 10). We explore the idea that removing these uncorrelated features should measurably improve the already high correlation between the DIBs.

Our initial exploration of this idea is based on the high-S/N and high-resolution spectral profiles observed by Galazutdinov et al. (2002). The quality and resolution of these data facilitated our effort to separate the blended contributions to these DIBs. We removed the red-tail contribution from the λ6614 spectra by subtracting a normalized template red-tail model determined from our earlier spectral analysis (i.e., the red-tail residual in Figure 8). For each observed spectrum, we normalized the red-tail template to the data at ν = −1.6 cm⁻¹, which is free of contamination from the central PQR feature. For λ6196, we removed the extraneous features using a model fit to each of the deconvolved spectra (see Figure 10). We did not apply this procedure to λ6614 because its much larger EW, about four times that of λ6196, proportionally reduced the contribution of contaminating features. In Figure 13, we plot the EWs of λ6614 against those of λ6196, comparing values based on our modified profiles (using the model spectra for λ6196 and the red-tail subtracted spectra for λ6614) to values based on the observed data. The parameter values used in the plot are presented in Table 3.

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Table 3.  Modified Equivalent Widths for the λ6614 and λ6196 DIBs

Source	EW6196(mÅ)a	${\mathrm{EW}}_{6196,\mathrm{mod}}$(mÅ)b	EW6614(mÅ)a	${\mathrm{EW}}_{\mathrm{Red}\mathrm{Tail}}$ (mÅ)c	${f}_{\mathrm{Red}\mathrm{Tail}}$ d
HD 144757	10.6	10.0	45.3	8.3	0.22
HD 144217	12.1	11.6	50.8	9.7	0.24
HD 144470	12.0	10.3	57.8	13.3	0.30
HD 147165	15.1	12.2	60.2	13.5	0.29
HD 145502	15.8	13.9	57.8	13.3	0.30
HD 145502e	14.6	12.8	59.3	13.6	0.30
HD 184915	16.4	14.8	76.4	13.9	0.22
HD 179406	19.8	19.0	96.8	17.6	0.22

Notes.

^aObserved equivalent widths. ^bModified equivalent width based on λ6196 deconvolution (see text for details). ^cDerived red-tail equivalent width (see text for details). ^dFractional contribution of red tail, $f={\mathrm{EW}}_{\mathrm{Red}\mathrm{Tal}}/({\mathrm{EW}}_{6614}\mbox{--}{\mathrm{EW}}_{\mathrm{Red}\mathrm{Tail}}$). ^eNew observation at Las Campanas.

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The HD 145502 data and modified data points exhibit the largest variance from their respective linear fits. Recently, G. A. Galazutdinov & Y. Beletsky (2017, private communication) reobserved the λ6196 and λ6614 DIBs at the Las Campanas Observatory in Chile using the MIKE spectrograph (R ∼ 80,000) mounted on the Magellan/Clay telescope. The new measurements bring HD 145502 into better agreement with the correlations based on the other data points and significantly improve the Pearson coefficients for both the observed and the modified data, with the former increasing from 0.917 to 0.957 and the latter from 0.935 to 0.970. The new data are at the edges of the one sigma uncertainty bounds of the earlier measurements. Thus, the differences between the new and old measurements are most likely due to measurement statistical fluctuations.

Further support for this conclusion is provided by previous measurements of HD 145502 at the McDonald Observatory in Texas using an echelle spectrograph (R ∼ 60,000) mounted on the Canada–France–Hawaii Telescope (Krelowski & Sneden 1993). Analysis of these data by Wszołek & Godłowski (2003) yielded EWs of 14 ± 1 mÅ for λ6196 and 62 ± 3 mÅ for λ6614. These values agree well with the new measurements discussed above (see Table 3).

From Table 3, we see that there is a significant variability of the red-tail contribution to the EWs for λ6614. The ratio of the red tail and central PQR EWs, f_Red Tail, vary by a factor of ∼1.4 from lowest to highest. Other observations, for example, HD 166937 (μ Sgr; Kerr et al. 1996), display a much smaller red-tail contribution, f_{Red Tail} ∼ 0.1, than the observations presented in Table 3. We surmise that the actual variation of the red tail with respect to the central PQR feature is closer to a factor of two, supporting the idea that the red tail and the central PQR features arise from different carriers.

We conclude that the modified data exhibit a significantly improved Pearson correlation over that for the observations (0.970 versus 0.957). However, final confirmation of this initial finding requires testing with a more extensive data set than the seven sightlines considered in this study.

The number of "perfect" correlations and whether or not they arise from a common carrier has important implications with respect to the total number of DIB carriers. Only a handful (∼11) of strongly correlated DIB pairs, defined by a correlation coefficient of R_P ≥ 0.95, have been identified (Xiang et al. 2012). This is surprising since the ∼500 observed DIBs define 1.25 × 10⁵ DIB pairs. The small number of strong correlations has been interpreted to mean that almost every DIB has a unique carrier. This idea seems at odds with the widely held premise that the DIBs arise from polyatomic molecules with a size range of N_C ∼ 10–100 carbon atoms. One would expect many of these molecules to exhibit multiple strongly correlated bands. The correlations could arise from Franck–Condon sequences, multiple excited electronic states, and excited-state splitting from a variety of mechanisms, such as a spin–orbit interaction and isotopologs. We argue that there may be many strongly correlated DIBs, but the number of observable strong correlations is severely limited by S/N and DIB blending considerations.

We have formulated a simple statistical model for the number of observable strong correlations, N_Obs, based on observational properties for the DIBs. The resulting expression is

$$\begin{eqnarray}&&{N}_{\mathrm{Obs}}=(1-{f}_{B}){f}_{\mathrm{EW}}^{2}\left[\displaystyle \frac{{N}_{\mathrm{DIB}}}{{N}_{B}}\right]\left[\displaystyle \frac{{N}_{B}({N}_{B}-1)}{2}\right],\end{eqnarray} \tag{ 1 }$$

where f_B is the fraction of blended DIB bands, f_EW is the fraction of DIBs whose maximum observed EW has a sufficiently large S/N to enable the determination of a strong correlation, N_DIB is the number of known DIBs, and N_B is the average number of strongly correlated bands for a molecular carrier. The first three factors are uncorrelated probabilities, and the last term is the total number of DIB pairs. We can estimate reasonable values for all of the quantities of Equation (1). The number of known DIBs is N_DIB ∼ 500. The fraction of blended DIBs has previously been estimated to be f_B ∼ 0.5 (Bernstein et al. 2015b). Prior studies on DIB correlations have focused on DIB pairs in which each band had a maximum EW in the range of EW ∼ 50–100 mÅ (Xiang et al. 2012). Observational studies of the distribution of DIB EWs suggest that this range of EWs corresponds to f_EW ∼ 0.05–0.15 (Hobbs et al. 2008, 2009). We guesstimate that the number of correlated bands per molecule is in the range of N_B ∼ 1–10. Predictions for N_Obs based on these parameter values are presented in Figure 14.

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We have not included the potential effect of isotopologs on DIB correlation. It is conceivable that an isotopolog may be shifted outside of the narrow spectral range used to compute a DIB's EW. Since the ISM ¹³C abundance varies significantly, by about a factor of 2, this can add noise to the DIB EW distribution, thereby weakening the correlation.

We use the predictions to estimate the number of unique DIB carriers. If we assume from the observational result that N_Obs ∼ 11, then Figure 14 implies that for f_EW ∼ 0.10–0.15, N_B ∼ 5–10. The number of observable correlations is strongly tied to f_EW since it enters Equation (1) as ${{f}_{\mathrm{EW}}}^{2}$. For f_EW ∼ 0.1, the number of observable correlations is two orders of magnitude fewer than the total number of DIB pairs. This model suggests that the number of unique carriers, N_DIB/N_B, is approximately ∼50–100. The model also implies that if the limiting effects of DIB blending and S/N could be overcome, then there would be ∼1000 observable strongly correlated DIB pairs.

The spectral analysis supports two types of oblate, polar, symmetric top carriers: a compact, puckered PAH and a fullerane. Because of the ambiguity between the size-dependent rotational constants and rotational temperatures, the carrier size is not well determined from spectral analysis alone. As discussed in previous work (Bernstein et al. 2015b), the shape of a band depends on the product of the rotational constants and temperature, BT_rot, not on their individual values.

We summarize results from analyses presented in Appendices B and C that constrain T_rot and thereby enable a determination of B and the associated carrier size. The small end of the size range, which corresponds to the largest B, is limited by the lowest possible rotational temperature, T_rot = 2.73 K for the CMB. This temperature is close to the T_rot = 3.5 K that we determined for λ6196 for HD 179406 based on B ∼ 0.02 cm⁻¹. This implies a size of N_C ∼ 20 carbon atoms. In Bernstein et al. (2015a), we argued that the DIBs and AME (anomalous microwave emission) likely arise from the same collection of carriers. This led to constraints on their temperatures and rotational constants, T_rot ∼ 3–10 K and B ∼ 0.01–0.06 cm⁻¹, that are consistent with the findings of this work.

The candidate PAH carrier most consistent with the shape, size, and polarity constraints is corannulene, C₂₀H₁₀. However, we cannot rule out other potential carriers related to circumtridene, C₃₆H₁₂ whose rotational constants are about half of those for corannulene. Both molecules are compact PAHs with a large dipole moment. At present, we consider all of the ionic and symmetric top dehydrogenated variants of these molecules (e.g., symmetrically dehydrogenated C₂₀H₅ and fully dehydrogenated C₂₀) as candidate carriers. All of the candidate carriers have multiple degenerate vibrational modes. A key consideration is whether or not they have vibronic transitions that can give rise to λ6614 and λ6196.

With the exception of corannulene (Hardy et al. 2017), quantitative information on the excited electronic states of these species required for this evaluation is not yet available. Rouille et al. (2008) measured the ultraviolet-visible laboratory spectrum of corannulene and did not observe any absorption bands in the visible region. Corannulene has been proposed as a potentially abundant interstellar molecule (Thaddeus 2006) readily observable in the microwave spectral region because of its large dipole moment. Several attempts to detect interstellar corannulene in the microwave region were unsuccessful (Thaddeus 2006; Pilleri et al. 2009). We cannot comment on the results of Thaddeus (2006) because the details of the analysis were not published. However, Pilleri et al. (2009) did not account for the fast radiative decay at the high J = 135 state used for their observation. The radiative decay rate is about four orders of magnitude faster than a typical ISM collision rate (see Appendix C). This disparity in the radiative decay and collision rates would result in a negligible population of and negligible emission from the J = 135 state. Hence, it is premature to rule out the possibility of a significant abundance of corannulene based on the Pilleri et al. (2009) measurements.

Two candidate fullerane carriers, including their neutral and ionic forms, are also consistent with the size, shape, and polarity constraints, C₂₀H and C₂₈H. We expect these molecules to be slightly oblate symmetric tops with average rotational constants of approximately 0.023 and 0.013 cm⁻¹, respectively. While the modeled fullerane and compact PAH carriers have different shapes, they are equivalent from a spectral fitting perspective. They have comparable B values (see Table 1) and nearly equivalent values of (1 − ζ)C (see Appendix B). However, we presume that all variants of the fullerane and PAH candidate carriers have unique excited electronic state energies, and that only one (or no) carrier will have transition energies consistent with the λ6614 and λ6196 DIBS.

Recommendations for future investigations to clarify the existence and number of DIBs originating from a common molecular carrier include the following:

• 1.  
  Apply the λ6614–λ6196 correlation improvements to a more statistically meaningful data set (e.g., Krełowski et al. 2016). As noted in the introduction, the Pearson correlation for a number of larger data sets typically falls around r_P ∼ 0.97. Assuming the ∼0.02 correlation improvement found for the small data set considered here, this implies an improved correlation of r_P ∼ 0.99. Such a strong correlation would substantially strengthen the case for a common carrier.
• 2.  
  Increase the number of observable strongly correlated DIBs. There are two avenues to explore. The first is to use spectral deconvolution to recognize and disentangle blended DIBs as well as other "spurious" spectral features. This method would be most effective when combined with spectral modeling in order to facilitate the separation of the various spectral features. An example of this is provided in Figure 9 for λ6196 where there are multiple small and narrow spectral features that do not appear to be associated with the primary carrier spectrum. The second avenue is to improve the S/N for the observations. This would directly impact the ${{f}_{\mathrm{EW}}}^{2}$ term in Equation (1) for which even a modest improvement in f_EW, such as a factor of 1.5, would increase the number of observable strong correlations by a factor of ∼2.
• 3.  
  Develop a semi-empirical or first principles model for the distribution of rotational states for a polar symmetric top molecule in the ISM. In this paper, we demonstrated the need for a two-temperature model to approximate the J, K state populations in order to support the common-carrier hypothesis for λ6614 and λ6196. The two-temperature approach was approximately implemented within the PGOPHER spectral model. A more first principles model would offer a more rigorous test of the common-carrier hypothesis. The first principles model should include (i) velocity-dependent state-to-state collisional excitation and de-excitation cross-sections between a candidate carrier and the most abundant ISM species, H, H⁺, H₂, and He, (ii) predicted J, K populations based on the state-to-state collisional and radiative excitation and de-excitation rates for the local ISM physical conditions and interstellar radiation field, and (iii) simulated DIB absorption spectra based explicitly on the predicted ground state J, K populations. We note that the first principles model for the ground electronic state J, K populations would not obviate the need for the sharp rotational state cutoffs which we associate with the upper electronic state of the λ6614 DIB.