The Behavior of Selected Diffuse Interstellar Bands with Molecular Fraction in Diffuse Atomic and Molecular Clouds

For this exploration of the environmental dependence of the DIBs, 186 sight lines, with E_B–V ranging from 0.00 to 3.31 mag, f_H2 ranging from less than ${10}^{-4}$ to 0.83, and sampling a variety of regions in the Galactic ISM within several kpc of the Sun, were selected from our larger set of observations. As will be discussed below, the very different environmental conditions found along these 186 sight lines affect the behavior of both the known atomic and molecular species and the DIB carriers.

Most of the spectra were obtained with the ARC echelle spectrograph (ARCES) on the 3.5 m telescope at Apache Point Observatory (APO) and many of these spectra were used in the series of papers "Studies of Diffuse Interstellar Bands" (i.e., Thorburn et al. 2003; Hobbs et al. 2008, 2009; McCall et al. 2010; Friedman et al. 2011, and related papers: McCall et al. 2001; Dahlstrom et al. 2013; Welty et al. 2014). As several of those papers include detailed characterizations of the ARCES spectra and descriptions of the standard data reduction procedures, only a brief summary will be given here.

The ARCES spectra cover a wavelength region between 3500 Å and 11000 Å, and the resolving power is 33,000 with a 16 slit, corresponding to a velocity resolution of 9 km s⁻¹ (Wang et al. 2003). For each sight line, multiple exposures were combined to achieve a nominal SNR of ∼1000 (per resolution element) around 6400 Å. Lightly reddened standard stars, covering ranges in spectral type and luminosity class similar to our target sample and (ideally) with relatively low projected rotational velocity, were observed to similar SNR in order to enable recognition of any stellar lines blended with the DIBs. Cosmic-ray removal, bias subtraction, and flat fielding were done with standard techniques, and telluric lines were corrected based on humidity and air mass. The wavelength scale is set by defining the strongest component of the interstellar K i line at laboratory wavelength 7698.9645 Å as zero velocity (Morton 2003). If interstellar K i is not detected, an analogous procedure is followed using the Na i line. The reduction of the raw spectra was done by J.D.

Eight of our stars¹⁴ were observed with the Magellan Inamori Kyocera Echelle (MIKE), a double echelle spectrograph on the Clay Telescope at Las Campanas Observatory, Chile, with wavelength coverage between 3200 Å and 10000 Å (Bernstein et al. 2003). The effective resolving power is about 45,600 in the blue and 36,000 in the red for these spectra. Another five spectra¹⁵ are from the archive of the Fiberfed Extended Range Optical Spectrograph (FEROS) mounted on the ESO 1.52 m telescope at La Silla observatory, Chile (Kaufer et al. 2000), which has a resolving power of 48,000 and wavelength coverage between 3560 Å and 9200 Å. Thus, both spectrographs have similar resolving power and wavelength coverage to ARCES. The spectral extractions and stacking of multiple spectra from the MIKE and FEROS were performed using an IRAF-based pipeline constructed by Z. J.

Of the more than 500 DIBs recognized between 3500 Å and 8500 Å (Hobbs et al. 2008, 2009), eight were selected for this study. These eight DIBs are generally relatively strong (so they can be measured even in lightly reddened sight lines), fairly well separable from any other nearby DIBs (with caveats noted below), and have minimal contamination from telluric and stellar lines. Five of the eight DIBs (λλ5780.5, 5797.1, 6196.0, 6283.8, and 6613.6) have been examined in a number of previous investigations. While the prior studies have provided evidence of differences in behavior among those five well-known DIBs, we will nonetheless refer to them as "normal" DIBs. That designation is in contrast to the other three DIBs (λλ4726.8, 4963.9, 4984.8) studied in this paper that belong to the class of "C₂ DIBs" (Thorburn et al. 2003), which as far as we can tell includes less than 10% of the currently known DIBs (see D. G. York et al. 2017, in preparation). As will be seen below, the behavior of the three C₂ DIBs is in sharp contrast to that of the normal DIBs, particularly when denser regions with higher molecular fractions are present along the sight line.

2.3.1. Choice of Integrated Equivalent Widths

2.3.2. Continuum Issues

Reliable, uniform measurement of the DIB EWs depends on accurate determination of both the local continuum (e.g., taking into account the effects of any stellar absorption features, which can vary with spectral type) and the interval over which the DIB absorption is to be integrated. To illustrate some of these issues, Figure 1 shows 24 Å wide spectral segments around the eight DIBs chosen for this study, for the two DIB atlas stars HD183143 (Hobbs et al. 2009) and HD204827 (Hobbs et al. 2008). In the figure, tick marks give the positions of DIBs identified in the two sight lines, arrows show the locations of stellar absorption lines seen in the respective unreddened comparison stars, and the solid dots indicate typical integration limits (appropriate for most of the sight lines in our sample) for the DIBs. Four of the eight DIBs (λλ4726.8, 4984.8, 6196.0, and 6613.6) are relatively narrow and well-isolated from other DIBs or significant telluric absorption. They are generally not blended with stellar lines either, which makes their measurement straightforward. The narrow λ4963.9 C₂ DIB can be slightly blended with a much weaker DIB at 4965.2 Å, but the two generally are easily separable.¹⁶

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Measurement of the strong DIBs λλ5780.5, 5797.1, and 6283.8, however, requires more consideration:

• (1)  
  The λ5780.5DIB is the dominant feature in a complex of DIB absorption that contains a broader (∼20 Å wide), shallow component and a number of weaker, relatively narrow components (Hobbs et al. 2008, 2009; Dahlstrom et al. 2013 and references therein). As in most previous studies (e.g., Galazutdinov et al. 2004; Friedman et al. 2011), we focus on the dominant narrow feature at 5780.5 Å (see the integration limits in Figure 1). Because of possible significant stellar contamination of the λ5780.5 DIB for stellar types B4 and later, those measurements are excluded from our correlation analyses of this DIB.
• (2)  
  The strong, broad absorption typically assigned to the λ6283.8 DIB can include contributions from several weak, relatively narrow DIBs with different relative strengths in different sight lines. Consistent with most previous work, we measure the EW of the entire complex. This DIB sits on a telluric band, and signs of the residuals from under/over correction can be seen in some of our spectra. The uncertainties arising from these telluric residuals are not included in the errors of measurements of the λ6283.8 DIB, but flags are assigned to measurements made in spectra where the telluric correction is not satisfactory (see Table 1).
• (3)  
  Different measurement conventions for the λ5797.1 DIB have been adopted in previous studies, reflecting assumptions concerning two much weaker absorption features at 5795.2 and 5793.2 Å. Thorburn et al. (2003) and Friedman et al. (2011), for example, included the λ5795.2 DIB as part of the λ5797.1 DIB and measured them together, while Galazutdinov et al. (2004) excluded the λ5795.2 DIB by using a lower continuum (essentially assuming the λ5795.2 DIB to be part of a broader absorption feature). After detailed examination of various options (see Appendix A), we have chosen a third method for measuring the λ5797.1 DIB. In this method, the continuum is defined as in Friedman et al. (2011), but the integration limit at the short wavelength side of the DIB is placed at the local maximum between the λ5795.2 and the λ5797.1 DIBs. This choice essentially assumes that the λ5795.2 DIB is relatively narrow. Comparisons among the three methods indicate that the specific choice does not significantly affect the conclusions drawn from our correlation analyses, as long as the same method is used consistently for all sight lines, essentially because the EW is dominated in all three cases by the strong λ5797.1 feature (Appendix A). An additional complication, for some O stars with high vsini, is that the λ5797.1 DIB can be blended with a stellar C IV line on the redward wing. In many of those cases, an adequate correction for the stellar line could be performed by dividing the observed spectrum by a fitted symmetric Gaussian profile to the stellar line. Sight lines where such a correction was not adequate¹⁷ (e.g., for spectroscopic binaries observed at multiple epochs, where the stellar line profiles in the summed spectra are asymmetric), were excluded from the correlation analyses involving the λ5797.1 DIB.

Table 1.  Spectral Types and Values of E_B–V, N(H), N(H₂), f_H2, and EWs of the Eight DIBs for 186 O and B Stars

HD	Spectral Type	EB–V(mag)	Log10N(H)	Log10N(H2)	fH2	Wλ(4726) (mÅ)	Wλ(4963) (mÅ)	Wλ(4984) (mÅ)	Wλ(5780) (mÅ)	Wλ(5797) (mÅ)	Wλ(6196) (mÅ)	Wλ(6283) (mÅ)	Wλ(6613) (mÅ)
886	B2IV	0.01 ± 0.01						<0.4	<4.3	4.5 ± 0.7	0.9 ± 0.2		<1.8b
1544	B0.5III	0.43 ± 0.03				65.6 ± 3.0	4.8 ± 0.4	3.2 ± 0.6	388.3 ± 2.5	91.1 ± 1.1	39.4 ± 0.4	807.7 ± 11.2	162.1 ± 1.6
2905	B1Iae	0.33 ± 0.05	21.28 ± 0.09	20.27 ± 0.14	0.2 ± 0.1	49.6 ± 1.8b	6.3 ± 0.2	4.6 ± 0.5	309.9 ± 1.7	89.7 ± 0.8	27.7 ± 0.2	596.3 ± 5.5	125.0 ± 1.2
10516	B2Vep	0.20 ± 0.05	<20.54	19.08 ± 0.14	>0.065	17.1 ± 1.5b	<1.1b	1.8 ± 0.3	58.6 ± 1.1	9.8 ± 0.5	5.7 ± 0.3	123.2 ± 2.6	10.5 ± 0.7b
11415	B3III	0.05 ± 0.02	20.53 ± 0.10a	<19.32a	<0.11	26.1 ± 2.2b	1.0 ± 0.4b	<1.0	29.9 ± 1.7	4.8 ± 0.5		35.2 ± 2.8	4.2 ± 0.5
13267	B5Iae	0.42 ± 0.03				63.7 ± 2.9	9.3 ± 0.5	3.2 ± 0.4	279.2 ± 1.1b	78.0 ± 1.2	26.4 ± 0.5	818.8 ± 14.4	96.3 ± 1.1
BD+56 508	B2III	0.53 ± 0.03				144.7 ± 4.5b	21.0 ± 0.4b	4.2 ± 0.2	363.2 ± 5.3	108.1 ± 0.7	27.7 ± 0.4	935.1 ± 12.5	144.2 ± 0.8b
14134	B3Ia	0.58 ± 0.03				109.9 ± 2.6	16.6 ± 0.3	5.5 ± 0.7	307.3 ± 1.6	95.8 ± 0.6	30.5 ± 0.3	819.2 ± 13.5	126.9 ± 0.8
14143	B2Ia	0.67 ± 0.03				134.7 ± 2.6	25.8 ± 1.1b	4.1 ± 0.4	323.9 ± 3.2	116.7 ± 0.8	37.4 ± 0.3	894.8 ± 15.6	129.2 ± 1.2
14633	O8.5V	0.11 ± 0.03	20.60 ± 0.06	<19.11 ± 0.00	<0.061	18.7 ± 1.8	<1.6	<1.6	39.4 ± 1.8	11.6 ± 1.1c	5.1 ± 0.2	68.9 ± 7.6	15.2 ± 1.4
14818	B2Iae	0.28 ± 0.03				81.9 ± 2.7	11.7 ± 0.5b	4.6 ± 0.4	308.0 ± 3.3	91.1 ± 1.9	30.1 ± 0.4	907.5 ± 30.7	127.0 ± 1.2
15558	O5e	0.83 ± 0.08	21.52 ± 0.18	20.89 ± 0.15	0.3 ± 0.2	118.9 ± 4.1	14.0 ± 0.6	5.7 ± 0.5	500.5 ± 5.3	194.9 ± 0.6	60.7 ± 0.7	1097.8 ± 17.7	237.1 ± 1.0
18326	O7V	0.70 ± 0.03				100.7 ± 3.6	13.0 ± 0.5	6.3 ± 1.5	399.4 ± 4.1	145.4 ± 3.3c	45.2 ± 0.6	996.8 ± 14.8	183.0 ± 1.4
19374	B1.5V	0.13 ± 0.03	21.04 ± 0.10a	19.64 ± 0.13a	0.07 ± 0.04	60.5 ± 2.2b	5.5 ± 0.3	0.7 ± 0.2	130.1 ± 2.4	38.0 ± 1.1	13.9 ± 0.5	314.8 ± 9.6	39.8 ± 1.1b
19820	O9IV	0.79 ± 0.03	21.45 ± 0.10a	20.91 ± 0.13a	0.37 ± 0.12	92.2 ± 1.4	14.5 ± 0.4	6.3 ± 0.2	416.0 ± 1.7	124.9 ± 1.4c	46.7 ± 0.3	1052.0 ± 12.0	209.9 ± 0.8
21483	B3III	0.56 ± 0.04	21.00 ± 0.08	20.96 ± 0.13a	0.65 ± 0.11	109.4 ± 3.7b	16.6 ± 0.3b	9.6 ± 0.6	169.9 ± 1.9	89.9 ± 1.1	20.3 ± 0.3	340.2 ± 5.7	92.9 ± 0.9
22951	B0.5V	0.27 ± 0.03	21.04 ± 0.13	20.46 ± 0.14	0.3 ± 0.1	47.8 ± 3.0	8.2 ± 0.4	2.0 ± 0.4	110.3 ± 2.0	45.4 ± 1.6	14.1 ± 0.5	208.0 ± 8.2	47.8 ± 0.7
23180	B1III	0.31 ± 0.03	20.82 ± 0.09	20.60 ± 0.12	0.5 ± 0.1	78.4 ± 2.3b	11.6 ± 0.9b	7.9 ± 0.6	89.5 ± 1.4	69.7 ± 0.4	14.0 ± 0.4	183.3 ± 7.9	50.3 ± 0.4b
281159	B5V	0.85 ± 0.05	21.39 ± 0.30	21.09 ± 0.19	0.50 ± 0.28	111.8 ± 3.9	23.9 ± 0.8	11.7 ± 0.6	310.2 ± 1.8b	110.5 ± 1.3	31.2 ± 0.3	582.2 ± 8.4	146.5 ± 1.4
24398	B1Ib	0.31 ± 0.03	20.80 ± 0.08	20.67 ± 0.14	0.6 ± 0.1	77.1 ± 3.4b	6.4 ± 0.6	3.8 ± 0.2	90.5 ± 4.8	62.1 ± 0.9	17.9 ± 0.3	169.8 ± 7.2	60.8 ± 0.7

Notes.

^aColumn density calculated using surrogates (see Section 3.1). ^bPossible stellar contamination. ^cThe profile of the λ5797.1 DIB corrected from stellar contamination.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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2.3.3. Semi-automated Measuring Procedure

The DIBs in each sight line were measured using a semi-automated routine (arcexam; written by D.E.W.). For each DIB to be measured, the program first displays a small section of the spectrum around that DIB and provides an initial fit to the local continuum. Corresponding spectral segments from the two DIB atlas stars (Hobbs et al. 2008, 2009), and a telluric reference star (10 Lac, uncorrected for telluric absorption) are included to provide guidance regarding the expected DIB widths and profiles (as seen toward the atlas stars), as well as the locations of possible residuals from the correction for telluric absorption. For this study of eight DIBs, spectra of several lightly reddened, low-vsini stars of each spectral type were examined to identify possible blends of each DIB with stellar absorption lines. The user can either accept the program's continuum fit or make adjustments.

Once the continuum is determined, the program determines the extent of the absorption (i.e., the integration limits), then measures the EW as well as the first four moments of the absorption-line profile (central wavelength, width, skewness, and kurtosis). Again, the user can either accept the program's choice of integration limits or choose different ones. Uncertainties on the EWs (1σ) and limits (3σ) for undetected DIBs are based on the fluctuations in the local continuum and the width of the DIB in the two atlas stars (Hobbs et al. 2008, 2009); for these relatively broad features, the uncertainties are often dominated by uncertainties in the placement of the continuum, rather than by photon noise alone (e.g., Sembach & Savage 1992). Depending on such factors as SNR, spectral type, and vsini, the program typically gives acceptable measurements for over 90% of the isolated DIBs without user intervention, and manual fine adjustment on the continuum level and/or integration limits can usually settle the rest. Any possible contamination from telluric residuals, stellar lines, or other DIBs is flagged, and those measurements are excluded from the correlation analyses (except for the λ6283.8 DIB, discussed above). Table 1 lists the star name; the spectral type; the color excess, ${E}_{B-V}$; the column densities (dex) of neutral hydrogen and molecular hydrogen (measured or estimated, see Section 3.1), N(H) and N(H₂); the fraction of molecular hydrogen (by mass), ${f}_{{\rm{H}}2}$; and the measured EWs of the eight DIBs we consider for the 186 sight lines in this survey.

Friedman et al. (2011) studied the correlations between the EWs of eight normal DIBs (λλ5487.7, 5705.1, 5780.5, 5797.1, 6196.0, 6204.5, 6283.8, and 6613.6), N(H), N(H₂), and E_B–V, for 133 sight lines observed with ARCES. The present study has 104 stars and four DIBs in common with those of Friedman et al. (DIBs λλ5780.5, 6196.0, 6283.8, and 6613.6), for which the choice of continuum placement and integration limits are very similar. The fifth DIB in common, λ5797.1, is discussed above. Figure 2 compares the measurements of DIBs λλ5780.5 and 6613.6 obtained in the two studies. Both correlations are essentially perfect (correlation coefficient r = 0.999, slopes are consistent with unity, intercepts are consistent with zero), given the errors in the measurements (e.g., from continuum placement, or subtle blends of adjacent or non-obvious DIBs; see Hobbs et al. 2008 and McCall et al. 2010). The semi-automated program used in this study thus appears to yield results entirely consistent with those obtained by hand, but takes much less time, making the uniform measurement of larger sets of DIBs in even more sight lines quite feasible.

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The E_B–V values were compiled by L.H. from standard sources (see Friedman et al. 2011). The observational data, namely spectral type and observed (B–V), used here for determining the needed stellar properties were taken, as available, from one of the following sources, and generally in order of preference as follows:

• (1)  
  The Bright Star Catalog or its Supplement (BSC);
• (2)  
  The Hipparcos Input Catalog (HIC);
• (3)  
  The various measurements collected for a given star in the SIMBAD database;
• (4)  
  The general literature.

If data were available in neither the BSC nor the HIC, informed choices were sometimes necessary among various distinct data sets available in SIMBAD or the literature. As a last resort in the rare cases in which the preferred source gives a spectral type and/or (B–V) significantly different from the values found in one or more of the subsequent sources, values from the latter were adopted after an investigation of those differences.

Estimated values of intrinsic color (B–V)₀ and of absolute magnitude M_v are required in order to calculate E_B–V and, in the absence of a reliable trigonometric parallax, a stellar distance. The calibrations we adopted for both (B–V)₀ and M_v, in terms of spectral type, are uniformly those of Johnson (1963). Our first requirement in adopting these particular calibrations was methodological consistency, i.e., the need to avoid any systematic errors introduced by modifying our choices of these calibrations during the course of a project that has now spanned 20 years. Over those decades, the various, newer (B–V)₀ calibrations available have generally differed modestly from Johnson's, especially when considering the differences among the measured (B–V) colors and the spectral types that were adopted by different authors to establish the respective calibrations. The corresponding differences between the various, newer M_v calibrations and Johnson's are sometimes somewhat larger. Because the derived stellar distances are of minimal direct interest in this paper, and reliable trigonometric distances are directly available for some of our program stars, we have again consistently used Johnson's calibration where required, in order to maintain internal consistency. We adopt the value 0.03 mag for the typical error of of E_B–V, as a fair representation that covers the major uncertainties: the direct errors of observation of the colors, errors in the spectral type necessary to derive the intrinsic colors of each star, and errors in the calibration to convert from a nominal spectral type to intrinsic colors. The second uncertainty is often likely to be the major contributor, for the star sample we are using.

The integrated molecular fractions of hydrogen for different sight lines are compiled in our database of DIB stars (Friedman et al. 2011). We have attempted to include all stars with H₂ detections from the Copernicus (Savage et al. 1977; Bohlin et al. 1978) and FUSE satellites (for example, Rachford et al. 2002, 2009), and 108 stars in this work have direct measurement of f_H2 = 2N (H₂)/[N(H)+2N(H₂)]. When direct measurements are not available, the EW of the λ5780.5 DIB is used as a surrogate for atomic hydrogen (Herbig 1993: 28 sight lines; Friedman et al. 2011; see Figure 3(A) in this paper), and N(H₂) is estimated based on N(CH) (33 sight lines; e.g., Sheffer et al. 2008; see Figure 3(B) in this paper). The use of surrogates gives us 36 more f_H2 values, for a total of 144.

Concerning the use of surrogates for N(H) and N(H₂), we note there are some sight lines that do not follow the general trends. The outliers more than 3σ away from the best-fit lines are highlighted in red in Figure 3. Most of the sight lines with weaker than "expected" W_λ(5780) versus N(H) appear to have stronger than average radiation fields, while sight lines with stronger than "expected" N(CH) versus N(H₂) may include regions where CH is formed largely by non-thermal effects (e.g., Federman et al. 1985; Zsargó & Federman 2003; Godard et al. 2014). As such cases are very few in number, they have little effect on mean relationships. In particular, the scatter in all figures related to f_H2 in this paper (see Figure 4 and beyond) is not much different for the blue points (indicating use of surrogate f_H2 values) or the black points (using direct f_H2 measurements).

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In this study, we use the fraction of hydrogen in molecular form, f_H2, as an indicator of the "typical" local hydrogen density in the main interstellar clouds in each line of sight (e.g., Cardelli 1994). The integrated f_H2 values for each sight line are plotted versus the normalized DIB EWs, in Figure 4. The color excess, a rough measure of the total amount of interstellar material, ranges from 0.00 to 3.31 mag in our data sample, and we use E_B–V to normalize the EWs of the DIBs as W (DIB)/E_B–V. This represents an attempt to get beyond the general tendency for all interstellar tracers to increase with N(H_tot), so that more local environmental effects in the dominant interstellar clouds might be discernible. Given the large relative uncertainties when E_B–V is small, we omit from the correlation analyses any measurements from sight lines whose reddening is smaller than 0.1 mag. A similar plot, where the EWs of the DIBs are not normalized, is provided in Appendix B. We note that the general behavior of the DIBs does not change upon normalization, although the normalization does greatly reduce scatter due to differences in distance and reddening, especially for f_H2 > ∼0.2 (Figure 12).

Twelve of the sight lines are highlighted in Figure 4 and the following plots because of their well-studied history or interesting behavior. HD 183143 () has often been used as a "standard" sight line for DIB studies (e.g., Herbig 1995; Hobbs et al. 2009). HD 147165 (σ Sco, ) and HD 149757 (ζ Oph, ) represent the defining sight lines of the sigma-zeta effect. The strongest C₂ DIB detections are made in the sight lines toward NGC2024-1 (), HD 204827 (), BD -14 5037 (), and Cyg OB2 #5 (). HD 37903 () at f_H2 = 0.52, HD 73882 () at f_H2 = 0.67, and HD 200775 () at f_H2 = 0.83 are the most significant outliers in Figure 5 (see Section 3.3). For HD 37061 () and perhaps for HD 62542 () (Cardelli & Savage 1988), intense radiation fields may weaken the DIBs. Finally, some of the DIBs toward Herschel 36 () exhibit an anomalously strong extended tail toward the red (ETR), likely due to a strong local IR field (Dahlstrom et al. 2013; Oka et al. 2013).

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The normalized EWs of all five normal DIBs have a lambda-shaped behavior with respect to f_H2 (Figures 4(D) through (H)). This behavior is also found for three additional DIBs λλ5487.7, 5705.1, and 6204.5 from Friedman et al. (2011). By "lambda-shaped behavior," we mean W (DIB)/E_B–V increases with f_H2 at low f_H2, reaches a peak at f_H2 ∼ 0.3, and then declines with increasing f_H2 thereafter. This is a confirmation that W (DIB)/E_B–V for normal DIBs can be weaker in sight lines containing dense cloud regions (e.g., Wampler 1966; Adamson et al. 1991). The three C₂ DIBs, on the other hand, show similar growth when f_H2 is small, but the peak and declining part at larger f_H2 are less marked (Figures 4(A) through (C)). Depending on the fitting range and the particular sight-line sample, the normalized EWs of the C₂ DIBs may increase slightly at large f_H2. Another striking feature of all the plots is the large dispersion in the individual sight-line values of the normalized DIB EWs at a given f_H2, compared to the much smaller errors in the measurements. This cosmic scatter will be discussed in Section 3.4.

In principle, this lambda-shaped behavior could be characterized by three quantities: the slope of the rise in normalized EW for f_H2 < ∼0.15 (representing the growth rate as f_H2 increases in relatively low density gas), the location of the peak of the distribution (generally around f_H2 ∼ 0.3), and the slope of the decline in normalized EW for larger f_H2 (representing the destruction rate at higher densities). In practice, however, precise values for the two slopes have been difficult to determine, due to the limited numbers of sight lines in both the lowest -f_H2 and highest -f_H2 regimes and to the evident steepening of the declines (for the normal DIBs) at the highest f_H2.

The plots in Figure 4 indicate that the slopes at small f_H2 are positive but small for all eight of the DIBs in this study. The λ5797.1 DIB and the ${{\rm{C}}}_{2}$ DIBs exhibit the largest slopes in this regime, while several of the broader DIBs (e.g., the λ6283.8 DIB) increase very little. More dramatic differences among the DIBs are seen in the slopes at higher f_H2. While the slopes for the C₂ DIBs are near zero, the slopes for the 'normal' DIBs become increasingly steep, in some cases reaching values of order −3 (or less) at the highest f_H2.

The lambda-shaped behavior seen for our larger sample of sight lines both confirms and more thoroughly defines the trends seen in several previous studies. Snow & Cohen (1974) had noted that the λ5780.5 and the λ5797.1 DIBs were weaker with respect to E_B–V when dense clouds are present in the sight line (see also Wampler 1966, for the λ4428.2 DIB), corresponding in our case to small W (DIB)/E_B–V at large f_H2. More observations of different regions of the sky (e.g., Strom et al. 1975; Meyer & Ulrich 1984; Adamson et al. 1991) provided further evidence. Jenniskens et al. (1994) and Sonnentrucker et al. (1997) also reported a lambda-shaped behavior for the normalized EWs of DIBs λλ5780.5, 5797.1, 6379.4, 6283.8, and 6613.6, but compared to E_B–V rather than f_H2. A similar lambda-shaped behavior for the normalized EWs of some DIBs (λλ5780.5, 5797.1, and 6353.5) versus the fraction of atomic hydrogen (1-f_H2) was also reported in Cami et al. (1997) for 13 nearby sight lines appearing to have only a single interstellar cloud component (most of which are included here). All previous studies that discussed the lambda-shaped DIB distribution clearly established the sharp rise of the DIB strength with either reddening or f_H2 for f_H2 = ∼0.3. The systematic decrease of the DIB strength with increasing reddening (or f_H2) suggested by those previous studies is seen more clearly in the present work. Our larger data sample clearly establishes the lambda-shaped behavior for those DIBs previously mentioned, as well as for the normal DIB λ6196.0. The steep drop at large f_H2 does not apply for the three C₂ DIBs in this study, however.

Although the EWs of DIBs λλ5780.5 and 5797.1 correlate moderately well (Figure 5(B)), there is considerable scatter, larger than the measurement uncertainties throughout the plot, and the chi-square is large. As described previously, these two DIBs have a similar lambda-shaped behavior with f_H2, but with different slopes at low and high f_H2. The sigma-zeta effect (Sneden et al. 1991; Krełowski et al. 1992) refers to the significant differences observed for the W_λ(5780)/W_λ(5797) ratio (or the ratio of central depths of these two DIBs in some cases), which can range from about 1 to 10 in different sight lines (e.g., Vos et al. 2011; Kos & Zwitter 2013; Welty et al. 2014). Sigma sight lines were originally defined as those where the λ5780.5 DIB is significantly deeper than the λ5797.1 DIB (as toward σ Sco), and vice versa for zeta sight lines (as toward ζ Oph). While there are also differences in far-UV extinction in those two sight lines (flat toward σ Sco, steeper than average toward ζ Oph), the physical relationship between the DIB strengths and the extinction is not understood.

The variation of the W_λ(5780)/W_λ(5797) ratio as a function of f_H2 is shown in Figure 5(A), with the same symbols for significant stars as used in Figure 4. As f_H2 increases from 0 to about 0.2, there is an obvious decline in the average W_λ(5780)/W_λ(5797) ratio. For f_H2 > 0.2, the decline becomes shallower. A similar plot is given in Weselak et al. (2004), where the strengths of the DIBs are represented by central depths (CDs) rather than EWs. In their plot, the CD_5797/CD_5780 ratio undergoes only a slight increase for f_H2 between 0.2 and 0.5, consistent with our finding. In Figure 5(A), there are three noticeable outliers, relative to the well-defined decline of the W_λ(5780)/W_λ(5797) ratio with increasing f_H2: HD 37903 (B1.5V, E_B–V = 0.35 mag., f_H2 = 0.53, highlighted as ), HD 73882 (O8V, E_B–V = 0.70 mag., f_H2 = 0.67, highlighted as ), and HD200775 (B2Ve, E_B–V = 0.63 mag., f_H2 = 0.83, highlighted as ). This last star is the only one in the plots for which f_H2 > 0.8 (though that value is based on the use of surrogates). More detailed discussion concerning these three outliers will be given in Section 4.2.2.

Cosmic scatter is found in all of our f_H2 plots. The dispersion among points at any given f_H2 is always much larger than the typical uncertainties for the individual measurements. The DIB behavior in the previous section thus refers to the general trends in the average normalized EWs of the DIBs, i.e., the orange squares and orange lines in the insets of each panel of Figure 4. Given the high quality of our DIB spectra and the resulting high accuracy of our measurements (Figure 2), this cosmic scatter may reflect different combinations of physical conditions in the various sight lines (and in the responses of the DIBs to those conditions). There is some evidence that the DIBs may be even more sensitive than some of the known atomic and molecular species to changes in the local physical conditions, which can occur over fairly short length scales (Cordiner et al. 2013). Moreover, high-resolution spectra of various atomic and molecular species have revealed complex velocity structure in most sight lines, with significant differences in the properties of the individual velocity components (e.g., Welty et al. 1994, 1996; Crane et al. 1995). However, given the width of the DIBs, it is generally very difficult to actually tie any DIB peculiarity to a single velocity component exhibiting some other peculiar interstellar property. Unfortunately, the individual components cannot be discerned and separated for H, H₂, or the extinction either, and we must resort to measuring those quantities integrated over the sight line. Sight lines with similar f_H2 and E_B–V can thus contain very different ISM environments on the scale of individual clouds. While the most extreme variations will be somewhat obscured in the sight-line averages, some scatter will remain, due to differences in the relative amounts of different kinds of gas in each sight line. This discussion emphasizes the importance of detailed studies of simple sight lines, though identifying such sight lines is extremely difficult with current spectrographs. An alternative is to focus on sight lines that have some outstanding characteristic, so that one cloud may be dominant with regard to DIB behavior, as is the case for Herschel 36 (Dahlstrom et al. 2013; Oka et al. 2013), HD 37903 (this work) or HD 62542 (Snow et al. 2002; D. E. Welty et al. 2017, in preparation).

Pair-wise comparisons between different quantities can identify both general relationships and "discrepant" points that may illuminate the behavior of those quantities under unusual environmental conditions. Tables 2 and 3 present the Pearson correlation coefficients (r_lin for linear units and r_log for logarithmic units, respectively) and the corresponding slopes of the best-fit general trends for pair-wise correlations between N(H), N(H₂), E_B–V, and the eight DIBs measured in this study. While a good correlation between two quantities may suggest that they are related, the slopes can also reveal physical, chemical, or spatial relationships between the various quantities (Section 3 in this work, see also Welty & Hobbs 2001; Sonnentrucker et al. 2007; Sheffer et al. 2008; Welty 2014). No values of N(H₂) or N(H) based on surrogates were used in the correlation analyses. For comparisons involving N(H₂), we only considered sight lines with log[N(H₂)] > 18.5, corresponding to fully shielded ${{\rm{H}}}_{2}$ (Savage et al. 1977). The EWs of DIBs affected by stellar line blending were excluded.

Table 2.  Correlation Coefficients and Slopes for Pair-wise Correlations in Linear Units^a,b

	Wλ(4726)	Wλ(4963)	Wλ(4984)	Wλ(5780)	Wλ(5797)	Wλ(6196)	Wλ(6283)	Wλ(6613)	EB–V	N(H)c	N(H2)c,d
Wλ(4726)	⋯	5.54 ± 0.09	9.75 ± 0.23	0.30 ± 0.01	0.83 ± 0.01	2.58 ± 0.03	0.11 ± 0.01	0.64 ± 0.01	150.4 ± 2.6	1.59 ± 0.34	6.81 ± 1.08
Wλ(4963)	0.96 ± 0.01	⋯	1.81 ± 0.04	0.05 ± 0.01	0.14 ± 0.01	0.42 ± 0.01	0.02 ± 0.01	0.10 ± 0.01	22.6 ± 0.4	0.20 ± 0.05	1.25 ± 0.17
Wλ(4984)	0.91 ± 0.01	0.94 ± 0.01	⋯	0.05 ± 0.01	0.07 ± 0.01	0.17 ± 0.01	0.01 ± 0.01	0.01 ± 0.01	11.7 ± 0.3	0.12 ± 0.03	0.55 ± 0.11
Wλ(5780)	0.81 ± 0.01	0.71 ± 0.01	0.58 ± 0.01	⋯	2.40 ± 0.01	8.71 ± 0.01	0.40 ± 0.01	2.16 ± 0.01	377.0 ± 4.0	5.49 ± 0.93	13.7 ± 2.6
Wλ(5797)	0.90 ± 0.01	0.85 ± 0.01	0.74 ± 0.01	0.93 ± 0.01	⋯	3.22 ± 0.01	0.14 ± 0.01	0.80 ± 0.01	143.1 ± 1.8	1.95 ± 0.34	6.15 ± 0.13
Wλ(6196)	0.83 ± 0.01	0.76 ± 0.01	0.61 ± 0.01	0.97 ± 0.01	0.97 ± 0.01	⋯	0.05 ± 0.01	0.25 ± 0.01	42.2 ± 0.6	0.58 ± 0.10	1.70 ± 0.32
Wλ(6283)	0.75 ± 0.01	0.69 ± 0.01	0.47 ± 0.01	0.96 ± 0.01	0.90 ± 0.01	0.94 ± 0.01	⋯	4.90 ± 0.02	836.0 ± 12.1	10.4 ± 1.9	35.1 ± 5.9
Wλ(6613)	0.83 ± 0.01	0.72 ± 0.01	0.60 ± 0.01	0.96 ± 0.01	0.96 ± 0.01	0.98 ± 0.01	0.92 ± 0.01	⋯	160.9 ± 2.4	2.14 ± 0.54	8.32 ± 1.36
EB–V	0.89 ± 0.01	0.88 ± 0.01	0.76 ± 0.01	0.83 ± 0.01	0.88 ± 0.01	0.86 ± 0.01	0.81 ± 0.01	0.83 ± 0.01	⋯	0.01 ± 0.01	0.04 ± 0.01
N(H)c	0.52 ± 0.06	0.40 ± 0.06	0.39 ± 0.06	0.69 ± 0.05	0.62 ± 0.05	0.58 ± 0.07	0.62 ± 0.06	0.57 ± 0.07	0.64 ± 0.05	⋯	1.01 ± 0.39
N(H2)c,d	0.57 ± 0.05	0.66 ± 0.05	0.53 ± 0.07	0.39 ± 0.07	0.46 ± 0.07	0.45 ± 0.08	0.41 ± 0.06	0.53 ± 0.07	0.71 ± 0.04	0.22 ± 0.08	⋯

Notes.

^aUpper-right section for slopes and lower-left section for correlation coefficients. ^bUsing MC/LINFIT procedure, with no outliers omitted from the fitting. ^cIn the unit of ${10}^{20}\,{\mathrm{cm}}^{-2}$. ^dExclude sight lines with log[N (H₂)] < 18.5.

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We explored two methods for fitting linear trends to the various pair-wise comparisons. The first method employed a Monte Carlo (MC) procedure in which the raw data were modified randomly based on the measured uncertainties (Table 1). The slopes, intercepts, and correlation coefficients (r-values) then were determined using the simple linear fitting program LINFIT,¹⁸ with equal weighting of the data and no rejection of discrepant points. For each correlation, we obtained the average values of r and of the slopes from 1000 Monte Carlo runs, and adopted the standard deviations as errors. The second method employed an iterative fit in which the data are weighted based on the uncertainties in both quantities—using either the IDL FITEXY procedure or a similar procedure (REGRWT)¹⁹ that we have modified to allow for equal weights and/or removal of discrepant points (>2.5σ). While the derived r-values and slopes depend, to some degree, on both the data sample (sight lines included, distributions in values) and the details of the fitting method (e.g., the weighting method and the treatment of "discrepant" points), fairly similar results were often obtained from the two methods for the roughly linear relationships found for both the normal DIB—normal DIB correlations, and the C₂ DIB—C₂ DIB correlations (all of which have r ≥ 0.85). As judged via the weighted rms distances of the data from the fitted lines, the iterative procedures that consider the uncertainties in both quantities (FITEXY and REGRWT) generally yielded better fits to the relationships in log units (Table 3), while the MC/LINFIT procedure generally gave better fits to the relationships in linear units (Table 2). For a number of the comparisons with smaller r-values, FITEXY did not produce good fits to the main trends in the data. In such cases, allowing equal weighting of the data and/or removal of the >2.5σ deviants (typically from one to four sight lines) via REGRWT generally yielded better fits; the slopes from those fits are given in square braces in Table 3.

Because of the wide diversity of characteristics of the sight lines in our sample—e.g., with both very low and very high reddening and/or f_H2—and because of the differences in behavior seen for the normal and the C₂ DIBs, however, some of the other relationships (e.g., normal DIB versus C₂ DIB, or DIB versus E_B–V, N(H), and/or N(H₂)) exhibit significant scatter and/or a degree of bi-modality. Those more complex relationships may reflect (for example) the differences in behavior of different DIBs in sight lines that are comparably reddened, but are dominated either by diffuse atomic gas or by denser, more molecular gas. Such relationships may not be well fitted by simple linear functions y = a + bx of a single variable x (in either linear or log units). Multivariate analyses (e.g., Lan et al. 2015; Ensor et al. 2017) will be required to disentangle the complex dependencies in such cases.

For the normal DIBs, our fits compare well with those obtained by Friedman et al. (2011). For the five normal DIBs in common, our average correlation coefficient with N(H) is r_log = 0.79, compared to their 0.81; both numbers are based on all suitable sight lines (no blends with other DIBs or with stellar lines in the background stars). Pruning the samples to eliminate outliers, Friedman et al. found an average correlation coefficient of 0.88. Comparing this to the small formal errors in the r-values (Tables 2 and 3), it appears that the main uncertainty in a given value for r is not due to the measurement errors for the data, but to the selection of stars used in a given fit (and to the cosmic scatter noted above).

A number of previously examined DIB correlations are confirmed with our somewhat larger sample of stars. DIBs λλ6196.0 and 6613.6, for example, are again found to be nearly perfectly correlated (r_lin = 0.98), as in several previous studies (e.g., Cami et al. 1997; Moutou et al. 1999; Galazutdinov et al. 2002; McCall et al. 2010). Because the amounts of all interstellar constituents generally increase with distance, they are all positively correlated to some degree. Friedman et al. (2011) suggested using r = 0.86 as a threshold to indicate that two ISM quantities could be physically related. By this criterion, all the normal DIBs in this work could be related, as the mutual correlation coefficients, r_lin, are all greater than 0.9.

Good correlations are also found among the three C₂ DIBs (Figures 6(A)–(C)). The correlation coefficients, r_lin, range between 0.96 and 0.91, comparable to the values found for pairs of normal DIBs. The three C₂ DIBs generally do not correlate as well with the normal DIBs, however.²⁰ The average value of r_lin from Table 2 between C₂ DIBs and normal DIBs (15 pairs) is 0.77. The analogous value from Table 3 is r_log = 0.74. These results indicate that the C₂ DIBs represent a distinct class of DIBs well-separated from the normal DIBs. Star-to-star differences in the relative strength of the redward wing of the λ4726.8 DIB suggest that the total feature might not be a pure C₂ DIB, which would tend to degrade its correlation with the other pure C₂ DIBs. However, the good correlations observed among the three C₂ DIBs considered here suggest that any such degradation is minor.

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Another significant difference between the C₂ DIBs and the normal DIBs is seen in their correlations with N(H) and N(H₂). Tables 2 and 3 suggest that the C₂ DIBs are better correlated with N(H₂), while the normal DIBs have better correlations with N(H). The average values of r_lin (and r_log) between the C₂ DIBs and N(H₂) are 0.59 ± 0.04 (0.70 ± 0.04), while the average values between the C₂ DIBs and N(H) are 0.44 ± 0.06 (0.56 ± 0.07). In contrast, r_lin (and r_log) between the normal DIBs and N(H₂) are 0.45 ± 0.02 (0.60 ± 0.08), while the average values for the normal DIBs and N(H) are 0.62 ± 0.04 (0.79 ± 0.04).

The best correlation between a DIB and N(H) is for the λ5780.5 DIB (r_log = 0.86), which is the only correlation that reaches the r = 0.86 threshold.²¹ On the other hand, the best correlation of a C₂ DIB or a normal DIB with N(H₂) is for λ4963.9, where r_log = 0.74, so no DIB exhibits a pure association with N(H₂). While both the normal and the C₂ DIBs are moderately well-correlated with E_B–V, (r_lin = 0.76–0.89), the DIBs λλ4726.8, 4963.9, 5780.5, 5797.1, 6196.0, and 6613.6 reach the level r > 0.85, with the first two being the most closely related to the dust (r_lin = 0.89 and 0.88, respectively).

The slopes of the pair-wise correlations are also reported in Tables 2 and 3, where the abscissa in the correlation is the entity on the right and the ordinate is the entity to the left (e.g., the slope for W_λ(6283) as the abscissa and W_λ(6196) as the ordinate is 0.05 for the linear correlation (Table 2)). The slopes for log-log correlations that do not involve N(H₂) are generally of order unity, consistent with roughly linear relationships. On the other hand, the slopes versus N(H₂) are always smaller than 0.6, possibly due to the very effective accumulation of H₂ after self-shielding is established.

Snow & McCall (2006) proposed a distinction between diffuse atomic, diffuse molecular, and translucent interstellar material, based not on E_B–V or A_V, but on the relative abundances of H and H₂, and the dominant repositories of carbon. In diffuse atomic gas, hydrogen is primarily in neutral atomic form, carbon is primarily singly ionized, and the abundances of even simple diatomic molecules are quite low. In diffuse molecular gas, most of the hydrogen has been converted to H₂, but most of the carbon is still in ${{\rm{C}}}^{+}$ (with small but increasing fractions in C, CO, and other simple molecules). In translucent material, C and/or CO become the dominant carbon species. Photoprocesses, such as photoionization and photodissociation, are significant in both diffuse atomic and diffuse molecular gas, though less so in the latter, due to increased shielding of the ambient UV radiation by dust and, for CO and H₂, by self-shielding via saturated UV absorption lines (Glassgold et al. 1987; Draine & Bertoldi 1996).

Models have suggested that the transition from H to H₂ that occurs when self-shielding becomes effective happens fairly rapidly; measurements of N(H) and N(H₂) in the Galactic ISM indicate that the molecular fraction f_H2 rises dramatically at E_B–V ∼ 0.08 (Savage et al. 1977). In principle, f_H2 thus can be used to distinguish between diffuse atomic and diffuse molecular gas. Large f_H2 will be found when the sight line is dominated by the cores of diffuse molecular clouds, but the actual f_H2 cannot be determined for individual velocity components because the strong low-J lines of H₂ are generally broader than the typical velocity separations between those components (Morton 1975). Based on higher-resolution spectra of H₂ in low-N(H₂) sight lines obtained with IMAPS (Jenkins et al. 2000), on even higher-resolution spectra of other atomic and molecular species that are well-correlated with H₂ (e.g., Welty & Hobbs 2001), and on the b-values inferred from higher-J H₂ lines (e.g., Jensen et al. 2010), it is likely that multiple components are present for most of the sight lines in which H₂ is detected. The highest integrated values of f_H2 in the FUSE survey of highly reddened sight lines are between 0.65 and 0.76 (Rachford et al. 2002, 2009), slightly higher than the highest values found for the typically less-reddened sight lines observed with Copernicus (Savage et al. 1977).

Clues to the behavior and properties of DIBs in these interstellar environments may be obtained from comparisons with the corresponding behavior of known atomic and molecular species. While the roughly quadratic relationships for the trace neutral species Na and K versus ${{\rm{H}}}_{\mathrm{tot}}$ are reasonably consistent with expectations from considerations of ionization balance (including both radiative and grain-assisted recombination), the shallower dependencies for Fe and Ca reflect the increasing depletions of iron and calcium in denser gas (Welty & Hobbs 2001; Welty et al. 2003). Roughly linear relationships with ${{\rm{H}}}_{\mathrm{tot}}$ are found for the dominant ions of little-depleted elements (e.g., O, ${{\rm{P}}}^{+}$, Zn⁺), while somewhat shallower trends are seen for the dominant ions of more severely depleted elements (e.g., Mg⁺, Ti⁺, Fe⁺), again reflecting the increasing depletions of those elements with increasing density (and thus increasing f_H2; e.g., Cardelli 1994; Jensen & Snow 2007a, 2007b; Welty & Crowther 2010)

Because different molecules occupy different locations in the chemical reaction networks and have different photodissociation thresholds, differences in spatial distribution are to be expected (e.g., Federman et al. 1994; Pan et al. 2005). Those differences in distribution can be reflected in the slopes of correlations between those molecules and H₂, with steeper positive slopes for species that are more centrally concentrated in denser parts of the clouds (e.g., Sheffer et al. 2008; D. E. Welty et al. 2017, in preparation). For example, CH, a "first-generation" molecule whose production (in the standard picture of gas-phase chemistry) is initiated by a reaction between ${{\rm{C}}}^{+}$ and H₂, is nearly linearly related to H₂ (e.g., Danks et al. 1983), and is relatively broadly distributed (Pan et al. 2005). On the other hand, "second-" and "third-generation" molecules (e.g., C₂, ${{\rm{C}}}_{3}$, CN, and CO), which require precursor molecules containing heavy atoms for their production, and are thus generally more centrally concentrated in denser regions, exhibit somewhat steeper relationships with H₂ (Sonnentrucker et al. 2007; Sheffer et al. 2008).

There have been a number of indications that the DIBs also depend on local environmental conditions, with some differences in response for individual DIBs. Herbig (1995) found a roughly linear relationship between W_λ(5780) and N(H), with no residual dependence on N(H₂), consistent with that DIB tracing primarily atomic gas and with its carrier being a dominant species (ionization state) in such gas. Friedman et al. (2011) and Lan et al. (2015) found similar relationships with N(H) for a number of additional DIBs as well (though with slightly different slopes and degrees of correlation); Lan et al. also found slight residual correlations or anti-correlations with N(H₂) for a number of the DIBs in their sample. By analogy with the relationships found among the known molecular species, the differences in slopes for the correlations between the EWs of various DIBs and the column densities of atomic hydrogen and of other known species (in log-log plots) may imply differences in the spatial distributions of the DIB carriers, with a possible connection with the widths of the individual DIBs (Welty 2014; Lan et al. 2015). Correlations between the W_λ(5780)/W_λ(5797) ratio and both the N(H)/E_B–V ratio and f_H2 suggest that the λ5797.1 DIB traces somewhat denser regions than the λ5780.5 DIB (Krelowski et al. 1999; Weselak et al. 2004, 2008). While the enhancement of the C₂ DIBs, with respect to the λ6196.0 DIB, in sight lines where the C₂ molecule is more abundant (Thorburn et al. 2003) is at least partially due to the decline of the normalized EW of the λ6196.0 DIB in such sight lines (see Figure 4(F)), our observation that the normalized EWs of the C₂ DIBs do not decline at large f_H2 values nonetheless indicates that they may trace denser regions of the clouds than the normal DIBs. Whether this relative enhancement is due to a more efficient formation of the carriers of the C₂ DIBs in the denser regions is not yet clear.

The DIBs also appear to respond to differences in the local radiation field. Observations that W (DIB)/E_B–V for some DIBs decrease with increasing E_B–V for sight lines probing moderately dense regions provided early evidence that those DIBs are located primarily in the more diffuse outer layers of the clouds (the "skin effect"; Snow & Cohen 1974; Strom et al. 1975; Meyer & Ulrich 1984) and suggested that radiation and shielding effects are important. To varying degrees, the DIBs appear to be weaker in regions where the UV radiation fields are stronger than average (e.g., Orion Trapezium and parts of Sco-Oph; Herbig 1993; Vos et al. 2011). The relatively weak DIBs found in most of the sight lines observed so far in the Large and Small Magellanic Clouds appear to be due to the combined effects of the lower metallicities, typically somewhat enhanced radiation fields, and additional metallicity-related effects (Cox et al. 2006, 2007; Welty et al. 2006). Strong local IR radiation fields can apparently modify the profiles of some of the DIBs (e.g., toward Herschel 36; Dahlstrom et al. 2013), providing clues to the characteristics of the carrier molecules (Oka et al. 2013; Huang & Oka 2015).

In principle, the differences in DIB strength in different environments must reflect the various processes involved in the formation, destruction, and/or modification of the DIB carrier molecules in those environments. Comparisons of the DIB strengths with local physical conditions (e.g., UV/optical/IR radiation fields, densities, fractional ionization, molecular fraction, and metallicity) thus may provide constraints on both the relevant processes and the nature of the carriers. If the carrier molecules generally contain more than five heavy atoms (i.e., other than H; see Huang & Oka 2015), then it seems unlikely that the carriers can be formed "bottom-up" via gas-phase chemistry in the relatively diffuse clouds in which most of the carriers appear to reside, given (for example) the observed trend of decreasing abundances of carbon chain molecules of increasing size in those clouds (Maier et al. 2002). It seems more likely that the carriers are formed "top-down," as fragments of larger molecules or dust grains, which are then able to survive in the diffuse clouds (Duley 2006; Zhen et al. 2014). Other processes (e.g., ionization, hydrogenation or de-hydrogenation, depletion onto grains, and other chemical reactions) may then modify the carriers or remove them from the gas phase to produce changes in the observed DIB strengths.

The radiation field, especially in the UV, is a key environmental factor in the diffuse ISM, given its capability to ionize atoms and to ionize and dissociate molecules. As noted above, the presence of a strong radiation field has been used to explain the anomalously low abundances of molecular hydrogen, trace neutral species, and some DIBs in the significantly reddened Theta Orionis sight lines (Savage et al. 1977; Herbig 1993; Welty & Hobbs 2001). On the other hand, absorption of the incident UV radiation in the outer layers of a cloud, either by dust or by strongly saturated absorption lines, may shield atoms and molecules in the inner parts of the cloud; both mechanisms are significant for the conversion of H to H₂, for example. It has been hypothesized that the carriers of different DIBs may need different amounts of either shielding from or exposure to the ambient radiation field, which then leads to their different behavior when subjected to fields of different strength and/or shape. The nature of the shielding is unclear, however. There are no observations of saturated DIBs, or even any securely confirmed DIBs, in the UV part of the spectrum. Thus, the shielding for DIBs would have to be continuous over the UV spectrum, rather than line shielding as for H₂ or CO (Solomon & Wickramasinghe 1969; van Dishoeck & Black 1988; Warin et al. 1996). Sonnentrucker et al. (1997) developed a simple ionization model for the DIB carriers, based on the analysis of the lambda-shaped distribution, which yielded estimates of DIB ionization potentials (IPs). Those IPs were found to be consistent with those of neutral or singly ionized PAH molecules, which have been proposed to be DIB carriers (and can turn into fullerenes via sequential dehydrogenation and ${{\rm{C}}}_{2}$ fragment losses (Zhen et al. 2014)). The fullerenes are also promising DIB carriers, especially after the identification of ${{{\rm{C}}}_{60}}^{+}$ as the carrier of several DIBs (Campbell et al. 2015). The low abundance of the fullerenes means, however, that they are likely not the only kind of DIB carrier, especially for the strong bands studied here (Omont 2016). The limits on carrier size for the λ5797.1 and λ5780.5 DIBs inferred by Oka et al. (2013) and Huang & Oka (2015) also raise the question of whether PAHs and fullerenes can be the carriers for those DIBs, and more generally, what the dominant ionization state of the DIB carriers might be in any given part of the cloud. It is not certain that the radiation field is the only factor governing the behavior of the DIBs: other parameters, such as density, may contribute as well.

4.2.1. Behavior of Normalized EWs of DIBs versus f_H2

4.2.2. Ratio of W_λ(5780)/W_λ(5797)

In this section, we examine the behavior of some known interstellar species at different f_H2 values, for comparison with the corresponding behavior of the DIBs shown in Figure 4. The normalized column densities of several representative neutral and ionized atomic species (K, Ca, Ca⁺, O, and Ti⁺) and of several diatomic molecules (CH, CH⁺, C₂, CO and CN) versus f_H2 are given in Figure 7.²² These trends with increasing f_H2 reflect the behavior of the column densities of those species versus N(H) and/or N(H₂), as summarized above in Section 4.1. We note that, because even the strongest DIBs in our data sample show no evidence of saturation (Friedman et al. 2011), the DIB EWs are thus proportional to the column densities of their respective carriers, and so are directly comparable to the column densities of these known atomic and molecular species.

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For the trace neutral forms of little-depleted elements (e.g., K in Figure 7(A), and similarly for Na and S, not shown here), the normalized column densities rise monotonically with f_H2 (with some scatter), reflecting both the ionization behavior of those neutrals and the minimal depletions. The three outliers in the lower right are HD37903, HD200775, and HD37903, already noted as being anomalous in the strength of the λ5797.1 DIB. For Ca (Figure 7(C)) and Fe, the normalized column densities exhibit an initial increase, but then decline for f_H2 > 0.2 due to the increasingly severe depletions of calcium and iron in denser gas. Ca⁺ (Figure 7(E)), which is a trace species in very diffuse gas but can become dominant (though also more severely depleted) in higher-density gas (Welty et al. 1996, 2003), also exhibits an initial rise for small f_H2, followed by a sharp decline for f_H2 > 0.2. The normalized column densities of the dominant ions of little-depleted elements (e.g., O in Figure 7(G), and similarly to ${{\rm{P}}}^{+}$ and Zn⁺) are relatively constant for f_H2 < 0.1, but can decline somewhat for higher f_H2, due to their mild depletions in denser gas (and/or, for ${{\rm{P}}}^{+}$ and Zn⁺, to possible underestimation of the column densities from strong absorption lines). The normalized column densities of the dominant ions of typically more severely depleted elements (e.g., Ti⁺ in Figure 7(I) and similarly for Fe⁺) also are roughly constant for small f_H2, but then decline fairly steeply for f_H2 > 0.1 due to the increasingly severe depletions.

The normalized column densities of the neutral diatomic molecules CH, C₂, CO, and CN all rise monotonically with f_H2 (Figure 7(B), (F), (H), and (J)). The increase is steeper for the so-called "second-" and "third-generation" molecules (C₂, CN, and CO), which require other precursor molecules for their formation and thus trace somewhat denser gas than the "first-generation" CH (e.g., Federman et al. 1994; Pan et al. 2005; see Figure 7(B) in this work). The normalized column density of CO exhibits an additional steepening of the trend with f_H2, due to the onset of self-shielding for f_H2 > 0.3 (see Sheffer et al. 2008). The normalized column density of the radical CH⁺, on the other hand, appears to rise with increasing f_H2 (for small f_H2), but then declines for f_H2 > 0.3 (Figure 7(D)), presumably reflecting its rapid destruction in gas of even relatively modest densities, due to collisions with electrons, H atoms, and/or H₂. If the CH⁺ is produced in turbulent dissipation regions (e.g., Godard et al. 2014), then its decline at higher f_H2 could also be the result of a reduced formation rate, as such regions are suppressed at higher densities.

The behavior of most of the known atomic and molecular species thus differs from that of the DIBs considered in this study. While the normalized EWs of the C₂ DIBs show little dependence on f_H2, those of the normal DIBs decline sharply at high f_H2. For the known atomic and molecular species, however, those whose destruction is thought to be dominated by the radiation field—via photoionization (the little-depleted trace neutrals) or by photodissociation (the neutral diatomic molecules)—all exhibit monotonic increases of their normalized column densities with increasing f_H2. However, the trace neutral atomic species are never shielded enough, in our sight line sample, to become dominant. In other words, the reduced photoionization does not significantly reduce the normalized column densities of the dominant ions with respect to the trace neutrals, even at the highest f_H2 probed here. The normalized column densities of the dominant ions of little-depleted atomic species (e.g., O, ${{\rm{P}}}^{+}$) exhibit only modest declines at high f_H2. The species whose normalized column densities do decline significantly at higher f_H2 (i.e., Ca, Ca⁺, Ti⁺, and CH⁺; see Figure 7(C), (E), (I), and (D))—more like the behavior shown by the normal DIBs—all are affected by additional, non-radiative processes that either remove those species from the gas (depletion, collisional destruction) or reduce their formation rates. The turnover point, where the normalized column densities begin to decline, must reflect the interplay between ionization equilibrium (which dominates in more diffuse gas) and depletion or other destructive processes, which become more significant at higher densities and molecular fractions. These comparisons suggest that ionization and shielding do not account entirely for the observed behavior of the DIBs in diffuse atomic and molecular clouds, and that additional destructive processes are likely to significantly affect the abundances of the DIB carriers in the denser gas characterized by higher molecular fractions (f_H2 > 0.3).

The C₂ DIBs defined in Thorburn et al. (2003) may trace the dense parts of interstellar clouds, as their normalized EWs do not decline at larger f_H2, in contrast to the behavior of the normal DIBs. This is clearly seen in Figure 4, as well as in Figures 12 and 13, which give the absolute, unnormalized EWs of the DIBs. The C₂ DIBs exist in the sight lines with highest f_H2 values, which we presume to be the sight lines with highest densities in the clouds.

On the other hand, W_λ(5780) is well-correlated with N(H) (Herbig 1993; Friedman et al. 2011), suggesting that its carrier is in the less-dense cloud envelopes or in the diffuse inter-cloud gas. The decreasing trend of the W_λ(5780)/W_λ(5797) ratio with increasing f_H2 (Figure 5(A)) implies that the carrier of the λ5797.1 DIB exists in a denser region (on average) than that of the λ5780.5 DIB. Thus, there may be spatial differences in the location of different DIB carriers.

In this section, we propose a more detailed sequence for the eight DIBs studied here, according to the behavior of their pair-wise strength ratios as a function of f_H2 (assuming f_H2 is a density indicator). In short, if a decreasing trend for W(DIB_A)/W(DIB_B) is observed with increasing f_H2 (e.g., W_λ(5797)/W_λ(4963) in Figure 8(A)), we conclude that DIB_B favors a higher-density region in the diffuse interstellar cloud than DIB_A, and vice versa. Representative plots are given in Figure 8. We propose this sequence, in increasing order of density of the environment where the carrier of the DIB is found: the λ6283.8 and λ5780.5 DIBs, the λ6196.0 DIB, the λ6613.6 DIB, the λ5797.1 DIB, and the C₂ DIBs. In an onion-like model of interstellar clouds, this proposed sequence would also describe the relative spatial locations of the DIB carriers.

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DIBs from the less-dense regions should have a better correlation with atomic hydrogen (unless they exist in the ionized edges of diffuse clouds), while DIBs found in denser environments should be better correlated with interstellar species that are also found in the interior of clouds (H₂ and CO, for instance). In Tables 2 and 3, following our sequence, the correlation coefficients between W (DIB) and N(H) are generally increasing, while the correlation coefficients between W (DIB) and N(H₂) generally decrease. Moreover, based on the sequence, DIBs in the adjacent portions tend to correlate better. Examples include the moderate correlation between C₂ DIBs and the λ5797.1 DIB, whose carrier is proposed to be found in the denser part of the cloud than any other normal DIB in our sequence. The poor correlations between the C₂ DIBs and the λ5780.5 DIB noted previously (Figure 6) provide another example: those DIBs that are more separated in our sequence are more poorly correlated.

The variation of strength ratio is found in multiple DIB pairs, which have similar behavior to the W_λ(5780)/W_λ(5797) ratio with increasing f_H2. Therefore, the sigma-zeta effect, or the variation in DIB-DIB EW ratio, is not restricted to the λ5780.5 and the λ5797.1 DIBs. This variation becomes more dramatic when the two DIBs being compared are farther apart in the sequence; the most dramatic variation in our DIB sample is found between the C₂ DIBs and the λ6283.8 DIB (as given in Figure 8(F)).

Figure 8 also provides some insights into the well-studied DIBs. For example, Figure 8(C) indicates that the nearly-perfect correlation between the λ6196.0 and the λ6613.6 DIBs breaks down at f_H2 < 0.15 where more than 90% of the gas is atomic. Most of these sight lines have low reddening; a similar result can be seen in the inset of Figure 5 in McCall et al. (2010) as a turnover. Galazutdinov et al. (2002) and McCall et al. (2010) note that these two DIBs have different line shapes and widths, and Krełowski et al. (2016) also found that they have different responses to varying interstellar environments. All these evidences suggest that the λ6196.0 and the λ6613.6 DIBs are likely due to distinct carriers, despite their very good correlation.

Figure 8(E) provides a comparison between the λ5780.5 and the λ6283.8 DIBs, both of which are known for being comparatively robust when a strong radiation field is present in the sight line. The W_λ(6283)/W_λ(5780) ratio varies little with f_H2, with two significant outliers that are Orion Trapezium sight lines (labeled). In those sight lines, the λ5780.5 DIB is weakened by a factor of five compared to N(H) and by two compared to E_B–V, while the λ6283.8 DIB is unaffected compared to E_B–V (see the full version of Table 1). The densities of UV photons in the Trapezium sight lines are greatly enhanced by the proximate O-type stars (Krełowski & Sneden 1995), and the carrier of the λ6283.8 DIB seems to be more robust in such environments.

There have been several sequences of DIBs proposed in the literature. For example, Welty (2014) proposed a sequence based on the slopes between W (DIB) and the column densities of known interstellar species [i.e., N(H), N(H₂), N(CH), N(CH⁺), N(H_tot), E_B–V, N(Na), N(K), and N(Ca)]. For the six DIBs we have in common (DIBs λλ4963.9, 5780.5, 5797.1, 6196.0, 6283.8, and 6613.6), the sequences agree with each other, except for the λ6613.6 DIB. Sonnentrucker et al. (1997) discussed the ionization potentials of the carriers of four DIBs, yielding a sequence consistent with the one proposed above. Additional support for this sequence comes from Lan et al. (2015), who studied the distribution of DIBs using low-resolution SDSS data (York et al. 2000). They proposed a dependence of W (DIB) on the column densities of atomic and molecular hydrogen:

$$\begin{eqnarray*}&&W(\mathrm{DIB})={W}_{21}\ast {\left(\displaystyle \frac{N({\rm{H}})}{{10}^{21}}\right)}^{\alpha }\ast {\left(\displaystyle \frac{N({{\rm{H}}}_{2})}{{10}^{21}}\right)}^{\mu }.\end{eqnarray*}$$

Following the sequence of DIBs we propose, μ shows an increasing trend (see Table 4 and Figure B2 of Lan et al. 2015). Such a trend indicates a growing dependence on H₂ from DIBs λλ5780.5 and 6283.8 to the C₂ DIBs in our sequence. Although these studies are based on different methods, the overall trends are in good agreement with each other, apparently reflecting properties of the carriers of these DIBs.