ANOMALOUS DIFFUSE INTERSTELLAR BANDS IN THE SPECTRUM OF HERSCHEL 36. II. ANALYSIS OF RADIATIVELY EXCITED CH⁺, CH, AND DIFFUSE INTERSTELLAR BANDS

The spectrum of interstellar CH⁺ was discovered by Dunham (1937) as an unidentified absorption line at 4232.6 Å toward ζ Persei, χ² Orionis, 55 Cygni, and χ Aurigae. Douglas & Herzberg (1941) identified the spectrum as due to the $\tilde{A}^1\Pi \leftarrow \tilde{X}^1\Sigma$ (0,0) transition of CH⁺ by laboratory emission spectroscopy. This was the first identification of a molecular ion in interstellar space. Since then, interstellar CH⁺ has been observed through absorption at rest wavelengths (in air) 4232.5 Å (0, 0), 3957.7 Å (1, 0), and 3745.3 Å (2, 0) toward a great many stars, demonstrating the ubiquity of CH⁺ in the diffuse interstellar medium (e.g., Gredel et al. 1993; Crawford 1995; Gredel 1997). Nearly all of the more than 250 detections of CH⁺ absorption lines so far have been of the R(0) transitions of CH⁺ in the ground J = 0 rotational level. The sole exception is the circumstellar absorption toward HD 213985 reported by Bakker et al. (1997). Because of the very fast J = 1 → 0 spontaneous emission (see Section 2.3), the excited J = 1 level is not sufficiently populated for detection in ordinary diffuse clouds. Highly excited CH⁺ lines have been observed in visible emission from the Red Rectangle (Balm & Jura 1992; Hall et al. 1992; Bakker et al. 1997; Hobbs et al. 2004) and in far-infrared emission in the planetary nebula NGC 7027 (Cernicharo et al. 1997), the Orion Bar photon-dominated region (Habart et al. 2010; Naylor et al. 2010), and the disk around Herbig Be star HD 100546 (Thi et al. 2011). Model calculations for the emissions observed in NGC 7027 and the Orion Bar have recently been published (Godard & Cernicharo 2013; Nagy et al. 2013).

The $\tilde{A}^1\Pi \leftarrow \tilde{X}^1\Sigma$ (0,0) and (1,0) bands of CH⁺ from the archive of the Fiberfed Extended Range Optical Spectrograph (FEROS) with a resolution of 48,000 (Kaufer et al. 1999) toward Her 36 and 9 Sgr are compared in Figure 1. Both stars are members of the young cluster NGC 6530. The 3 arcmin separation between the two sight lines corresponds to a linear separation of about 1.3 pc at the roughly 1.5 kpc distance to the cluster. The R(1) and Q(1) absorptions of CH⁺ in the J = 1 excited rotational level, 40.1 K above the ground J = 0 level, are clearly seen for both vibronic bands only toward Her 36. The R(2) and Q(2) lines for the J = 2 level, 120.3 K above the ground level, are not detectable. The non-detection of the J = 1 lines toward 9 Sgr suggests that the excited CH⁺ is found in a small region near Her 36 (Paper I).

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Observed transitions, with oscillator strength f, observed (heliocentric) wavelength in air λ_air, measured equivalent width of the absorption W_λ, Hönl–London factor S, derived CH⁺ column density in the J level N(J), and excitation temperature T_ex, are listed in Table 1. For weak lines, the column densities could be derived from W_λ by using N = (mc²/πe²)(W_λ/fλ²), where f = (8π²m/3he²)νμ_t² is the oscillator strength and μ_t is the transition dipole moment (e.g., Spitzer 1978). The FITS6P profile-fitting program (Welty et al. 2003) was used to fit stronger lines (accounting for any saturation in the absorption) and to fit multiple lines from a given species. A b-value of 1.2–1.6 km s⁻¹ yielded consistent N(0) for the (0,0), (1,0), and (2,0) bands of CH⁺; we adopt N(0) = 9.4 × 10¹³ cm⁻² and N(1) = 1.8 × 10¹³ cm⁻². The excitation temperature T_ex was derived from N(J)/N(0) = (2J + 1)exp (− E_J/kT_ex). Uncertainties are 1σ; upper limits for undetected lines correspond to 3σ.

Table 1. Observed Transitions of CH⁺ and CH toward Her 36, with Oscillator Strength f, Observed Wavelengths (in air) λ_air, Equivalent Widths W_λ, Hönl–London factors S, Derived Column Densities N(J), and Excitation Temperatures T_ex

Molecule	Transition	Line	λair	Wλ	S	N(J)	Tex
			(Å)	(mÅ)		(1013 cm−2)	(K)
CH+	$\tilde{A}^2\Delta \leftarrow \tilde{X}^2\Pi$(0,0) 1	R(2)	4226.982	<0.9	0.4	<0.26	<22.9
	f = 0.00545 2	R(1)	4229.273	7.3 ± 0.3	0.5	1.8 ± 0.1	14.3 ± 0.5
		R(0)	4232.472	40.2 ± 0.3	1	9.9 ± 0.6
		Q(1)	4237.481	6.6 ± 0.3	0.5	1.8 ± 0.1	14.3 ± 0.5
		Q(2)	4239.302	<0.9	0.5	<0.26	<22.9
13CH+	$\tilde{A}^2\Delta \leftarrow \tilde{X}^2\Pi$(0,0) 3	R(0)	4232.200	1.1 ± 0.3	1	0.14 ± 0.02
CH+	$\tilde{A}^2\Delta \leftarrow \tilde{X}^2\Pi$(1,0) 1	R(1)	3955.407	4.4 ± 0.5	0.5	1.9 ± 0.1	15.4 ± 0.4
	f = 0.00331 2	R(0)	3957.618	27.0 ± 0.5	1	8.6 ± 0.4
		Q(1)	3961.994	4.2 ± 0.5	0.5	1.9 ± 0.1	15.4 ± 0.4
13CH+	$\tilde{A}^2\Delta \leftarrow \tilde{X}^2\Pi$(1,0) 1	R(0)	3958.100	1.0 ± 0.4	1	0.19 ± 0.04
CH	$\tilde{A}^2\Delta \leftarrow \tilde{X}^2\Pi$(0,0) 4	R2(1/2)	4300.227	36.0 ± 0.4	1	6.9 ± 0.2
	f = 0.00505 5	R1(3/2)	4303.861	3.4 ± 0.3	1.5	0.30 ± 0.02	6.7 ± 0.1
CH	$\tilde{B}^2\Sigma ^{-} \leftarrow \tilde{X}^2\Pi$(0,0) 6	R2(1/2)	3878.700	5.5 ± 0.5	1/3	3.8 ± 0.2
	f = 0.00320 7	Q2(1/2)	3886.341	12.2 ± 0.6	1	3.3 ± 0.1
		P2(1/2)	3890.148	8.5 ± 0.8	2/3	3.2 ± 0.2

References. (1) Carrington & Ramsay 1982; (2) Larsson & Siegbahn 1983a; (3) Bembenek et al. 1997; (4) Zachwieja 1995; (5) Larsson & Siegbahn 1983b; (6) Ke¸pa et al. 1996; (7) Lien 1984.

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The values of W_λ should be identical for R(1) and Q(1) (which have the same Hönl–London factors), and the values of N(J) and T_ex should be identical for the (0,0) and (1,0) bands. The slight differences in those measured and derived values reflect our measurement uncertainties. We proceed in the following using the excitation temperature of 14.6 K determined from the three CH⁺ bands.

The observed high excitation temperature of T_ex = 14.6 K is exceptional in view of the very short lifetime of CH⁺ in the J = 1 level due to the fast spontaneous emission J = 1 → 0. The energy difference between the J = 1 and 0 levels of CH⁺ has been known since the work by Douglas & Herzberg (1941), but was recently measured directly by Amano (2010) to be 835,137.504 MHz, which is equivalent to 27.86 cm⁻¹ and 40.08 K. The permanent dipole moment of CH⁺ has been calculated by ab initio theory to be μ = 1.656 Debye (Ornellas & Machado 1986), 1.679 Debye (Follmeg et al. 1987), and 1.804 Debye (Sun & Freed 1988). If we use μ = 1.7 Debye, we have the Einstein coefficient A₁₀ = (64π⁴ν³/3hc³)μ²/3 = 0.0065 s⁻¹, corresponding to the lifetime of the J = 1 level of 150 s. The critical density for this decay is estimated to be on the order of n(H) ∼ A₁₀/k_L ∼ 7 × 10⁶ cm⁻³, where k_L ∼ 1 × 10⁻⁹ cm³ s⁻¹ is the Langevin rate constant (Anicich & Huntress 1986). At such a high density, carbon will be primarily in the form of CO and any CH⁺ would be rapidly destroyed by collisions with molecular hydrogen. The number density in the diffuse interstellar medium where CH⁺ abounds is typically on the order of 10² cm⁻³ or less, which is insufficient to maintain CH⁺ in the J = 1 level by collisions. This is the reason why CH⁺ in the J = 1 level has not previously been observed in ordinary sight lines.

The high excitation temperature of 14.6 K must be due to radiative pumping in the environment, most likely by dust emission from the nearby extended infrared source Her 36 SE (Goto et al. 2006). The intensity of radiation must depend on the radial position of the sight line, and therefore what determines the rotational distribution of CH⁺ must be a complicated weighted average of the radiative temperature 〈T_r〉 along the CH⁺ column. If the whole column of CH⁺ is irradiated by the dust emission, then the average radiative temperature 〈T_r〉 due to the dust emission would be equal to the excitation temperature, 〈T_ex〉 = 14.6 K. 〈T_r〉 needs to be higher if only part of the CH⁺ column is irradiated, but we believe that this is not the case since CH⁺ in the J = 2 level is not detected. 〈T_r〉 cannot be much higher than the upper limit on the excitation temperature for J = 2, 〈T_ex〉 < 22.9 K.

The logic used in this subsection to determine the radiative temperature 〈T_r〉 is basically the same as that missed in McKellar's near discovery of the 2.73 K primordial blackbody radiation (McKellar 1941, 1949), but later used by Thaddeus & Clauser (1966) and Meyer & Jura (1984) to determine the temperature of the blackbody radiation from the R(0) and R(1) lines of CN. The cases of CH⁺ and CH are cleaner than that of CN because their J = 1 → 0 and J = 3/2 → 1/2 spontaneous emissions (respectively) are orders of magnitude faster.

The dust emission which populates the J = 1 excited rotational level of CH⁺ and the J = 3/2 excited fine structure level of CH affects the velocity profiles of some DIBs in a more spectacular manner. Six of the DIBs observed toward Her 36 are compared with those in the ordinary, adjacent sight line toward 9 Sgr in Figure 3. For all six DIBs, the velocity profiles toward 9 Sgr (in red) are more or less symmetric with respect to the peak position. The three DIBs λ5780.5, λ5797.1, and λ6613.6 toward Her 36 (in black) shown in the left panel of Figure 3, however, are very asymmetric, with a huge tail extending to the red. We will refer to this Extended Tail toward Red as an ETR. There are also some indications of an ETR for the three DIBs in the right panel of Figure 3, but we have not been able to explain them quantitatively by the mechanism we consider in this paper. They may be due to vibrational hot bands, as discussed in Section 5.3. In any case, the qualitative difference between the variations of the three DIBs in the left panel and those in the right panel of Figure 3 is obvious. The broad deep absorption observed for 9 Sgr near λ6379.3 is a stellar feature and should be ignored.

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We interpret the huge ETR observed for the three DIBs in the left panel of Figure 3 as reflecting the population of many high-J levels of the large polar carriers of the DIBs, due to radiative pumping by dust emission. While this effect is relatively minor for CH⁺ and CH, in which only the lowest excited levels are affected, it can be huge for the DIBs since a great many high-J rotational levels are affected because of the low rotational constants of DIB carriers. For such small rotational constants, the P(J) and R(J) lines are nearly symmetric and linearly spaced by ∼±2BJ (where B is a rotational constant) at low J and the Q(J) lines pile up at the center of the band. As J increases, however, the negative quadratic term (B' − B)J(J + 1) (see the next section) takes over and causes an R-branch band-head and the ETR. The rotational constant in the excited state "B'" is almost always smaller than that in the ground state "B" since an electronic excitation tends to weaken chemical bonds, which makes the molecule larger and the moment of inertia higher.

The pronounced ETRs observed for the three DIBs, λ5780.5, λ5797.1, and λ6613.6 indicate that their carriers are polar molecules with dipole moments. The absence of ETR for λ5849.8, λ6196.0, and λ6379.3 suggests that either the carriers of those DIBs are non-polar or they are large polar molecules with very small B' − B. Either way, the qualitative difference between the variations of λ6613.6 and λ6196.0 eliminates the possibility that they are due to the same molecule, in spite of their near perfect correlation (McCall et al. 2010).

4.2.1. Radiative and Kinetic Temperature, T_r and T_k

4.2.2. Rotational Distribution by Collisions Alone

In the laboratory, where the number density is high, spontaneous emission between rotational levels J → J − 1 is many orders of magnitude slower than collisional processes and is negligible. Then, the detailed balancing between J and J − 1 levels are expressed as

$$\begin{equation} n(J)C_{J-1}^\downarrow = n(J-1)C_{J-1}^\uparrow, \end{equation} \tag{ 1 }$$

where n(J) is the number density of molecules with rotational quantum number J, and $C_{J-1}^\downarrow$ and $C_{J-1}^\uparrow$ are rates for collision-induced rotational transitions (Oka 1973) J → J − 1 and J ← J − 1, respectively. Throughout this paper, we follow the spectroscopists' convention, in which the quantum states of higher energy are on the left and exothermic and endothermic processes are expressed by arrows pointing to the right and left, respectively. The rates satisfy the principle of detailed balancing, that is,

$$\begin{eqnarray} \frac{C_{J-1}^\uparrow }{C_{J-1}^\downarrow } &=& \frac{n(J)_e}{n(J-1)_e} = \frac{g_J}{g_{J-1}}\exp (E_{J-1}-E_J)/kT_k\nonumber\\ &=& \frac{2J+1}{2J-1}\exp (-2hBJ/kT_k), \end{eqnarray} \tag{ 2 }$$

where n(J)_e is the number density in thermal equilibrium, g_J = (2J + 1) is the degeneracy of the level, and E_J = hBJ(J + 1) is the rotational energy of the molecule in the J level (Townes & Schawlow 1955). From Equations (1) and (2), we can derive the thermal Boltzmann number density with collisions alone as

$$\begin{equation} n(J) = n(0)\prod _{m=1}^{J}\frac{C_{m-1}^\uparrow }{C_{m-1}^\downarrow } = \frac{NhB}{kT_k}\ (2J+1)\exp \left[-\frac{hBJ(J+1)}{kT_k}\right], \end{equation} \tag{ 3 }$$

after the normalization n(0) = NhB/kT_k, where N is the total number of molecules.

4.2.3. Rotational Distribution by Collisions and Radiation

When radiation is included along with collisions, the detailed balancing takes the form

$$\begin{equation} n(J)\left(A^J+B_{J-1}^\downarrow \rho +C_{J-1}^\downarrow\right) = n(J-1)\left(B_{J-1}^\uparrow \rho +C_{J-1}^\uparrow\right), \end{equation} \tag{ 4 }$$

where the Einstein A and B coefficients express spontaneous emission and induced radiative effect, respectively (Einstein 1916). Using Einstein's relations $A^J = (8\pi h\nu ^3/c^3)B_{J-1}^\downarrow$, $B_{J-1}^\uparrow /B_{J-1}^\downarrow = (2J+1)/(2J-1)$, and Planck's (1901) formula, $\rho = 8\pi h\nu ^3/(e^{h\nu /kT_r}-1)$, we obtain

$$\begin{eqnarray} &&n(J)\left[A^J\left(1+\frac{1}{e^{h\nu /kT_r}-1}\right) + C_{J-1}^\downarrow \right]\nonumber\\ &&\quad= n(J-1)\left[A^J\frac{2J+1}{2J-1}\frac{1}{e^{h\nu /kT_r}-1}+C_{J-1}^\uparrow \right], \end{eqnarray} \tag{ 5 }$$

instead of Equation (1). A similar equation was used by Goldreich & Kwan (1974) for analyzing the radiation trapping by CO. In the parentheses after A^J on the left-hand side of the equation, the first term represents the number of zero-point photons per phase space, which cause spontaneous emission (Dirac 1927), and the second term represents the number of photons (bosons), which cause induced emission and absorption (Landau & Lifshitz 1969). For the J → J − 1 transition, ν = 2BJ and

$$\begin{equation} A^J = \frac{2^8\pi ^4 B^3\mu ^2}{3hc^3}\frac{J^4}{2J+1}, \end{equation} \tag{ 6 }$$

where μ is the permanent dipole moment.

For collision rates, we use the approximate formulae (Oka & Epp 2004)

$$\begin{eqnarray} C_{J-1}^\downarrow &=& C\sqrt{\frac{2J-1}{2J+1}}\exp (hBJ/kT_k), \nonumber\\ C_{J-1}^\uparrow&=& C\sqrt{\frac{2J+1}{2J-1}}\exp (-hBJ/kT_k), \end{eqnarray} \tag{ 7 }$$

which satisfy the detailed balancing of Equation (2), where C is a collision rate (independent of J) and is proportional to the number density of the environment n. Using Equation (5), together with Equations (6) and (7), we have

$$\begin{eqnarray} n(J) &=& n(0)\prod _{m=1}^{J}\nonumber\\ &&\times\left[\!\frac{\alpha B^3\mu ^2\frac{m^4}{2m-1}\frac{1}{e^{2hBm/kT_r} - 1} + C\sqrt{\frac{2m+1}{2m-1}}e^{-hBm/kT_r}}{\alpha B^3\mu ^2\frac{m^4}{2m+1}\left(\!1+\frac{1}{e^{2hBm/kT_r} - 1}\!\right) + C\sqrt{\frac{2m-1}{2m+1}}e^{hBm/kT_r}}\!\right],\nonumber\\ \end{eqnarray} \tag{ 8 }$$

where α ≡ 2⁸π⁴/3hc³. Unlike Equation (3), n(J) depends on both T_r and T_k, as expected. One obvious shortcoming of this approximate treatment of the collisions is that the effect of collision-induced transitions with |ΔJ| > 1 is not included. Actually, this formalism is a way around dealing with the |ΔJ| > 1 collision-induced transitions for which the rates are not known. We believe that this omission is somewhat justified from the treatment in the previous Section 4.2.2, where such transitions are also not included. Note that Equations (1), (2), and (3) are parallel to Equations (4), (5), and (8).

The number densities n(J) in Equation (8) cannot be expressed analytically as in Equation (3), so we calculate them numerically. They are calculated as a function of the radiative temperatures T_r due to the dust emission and 2.73 K due to the cosmic blackbody radiation, the kinetic temperature T_k, the rotational constant B, the dipole moment μ, and the collision rate C. As examples, calculated population fractions for B = 1000 MHz, μ = 4 Debye, T_k = 100 K, and C = 10⁻⁷ s⁻¹ are given in Figure 4 for T_r = 2.73 K, 14.6 K, and 80 K. As T_r increases, the molecular populations in higher J levels increase rapidly.

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4.2.4. Simulated Spectra

The rotational energy of a linear molecule is given by E_rot/h = BJ(J + 1). We ignore small higher order effects such as centrifugal distortion and λ-doubling. The line positions ν and the Hönl–London factors S are given as

$$\begin{eqnarray} R(J);\hspace{7.22743pt} \nu &=& \nu _0 + 2B^{\prime }(J + 1) + (B^{\prime } - B)J(J + 1)\nonumber\\ S_\parallel &=&\frac{J+1}{2J+1} \quad S_\perp =\frac{J+2}{2(2J+1)} \end{eqnarray} \tag{ 9a }$$

$$\begin{eqnarray} Q(J);\hspace{7.22743pt}\nu &=& \nu _0 + (B^{\prime } - B)J(J + 1) \nonumber\\ S_\parallel &=&0 \quad S_\perp =1/2 \end{eqnarray} \tag{ 9b }$$

$$\begin{eqnarray} P(J);\hspace{7.22743pt} \nu &=& \nu _0 - 2B^{\prime }J + (B^{\prime } - B)J(J + 1) \nonumber\\ S_\parallel &=&\frac{J}{2J+1} \quad S_\perp =\frac{J-1}{2(2J+1)}, \end{eqnarray} \tag{ 9c }$$

where B' is the rotational constant in the upper state of the transition, and S_∥ and S_⊥ are the Hönl–London factors for parallel (Δλ = 0) and perpendicular (Δλ = ±1) transitions, respectively (Herzberg 1989a). Those factors are for transitions with a singlet state λ = 0 in the lower level, but since J is high for the carriers of DIBs, they also give good approximations for levels with small λ. Spectra are calculated by using Equation (9) and the populations n(J) calculated in the previous section. In order to take into account uncertainty broadening due to a short emission lifetime of the upper state or internal conversion by the Douglas effect (Douglas 1966), spectral lines are broadened by Γ = 1/2πΔt. The simulated profiles are then convolved with the FEROS instrumental function (assumed to be a Gaussian with FWHM = 6.25 km s⁻¹).

The most crucial molecular parameter for producing the observed ETR is the difference between rotational constants B' − B. In order to explain the pronounced ETR, we need molecules with large values of β = (B' − B)/B. The experimental values of this constant for carbon molecules (mostly from Maier's group) are listed in Table 2.

Table 2. Variation of the Rotational Constants of Carbon Molecules Upon Electronic Excitation

Molecule	Transition	B	B' − B	β = (B' − B)/B	Reference
		(cm−1)	(cm−1)	(%)
C3	$^1\Pi _u \leftarrow ^1\Sigma _g^+$	0.4305	−0.0183	−4.3	1
ℓ-H2C3	1A1 ← 1A2	0.346783a	−0.01182a	−3.4a	2
C4	$^3\Sigma _u^- \leftarrow ^3\Sigma _g^-$	0.166111	−0.0091	−5.7	3
HC4H+	2Πu ← 2Πg	0.14690	−0.00681	−4.9	4
C5	$^1\Pi _u \leftarrow ^1\Sigma _g^+$	0.0853133	0.0002b	0.2b	5
HC6H+	2Πu ← 2Πg	0.0445943	−0.0008022	−1.8	6
HC8H+	2Πu ← 2Πg	0.0190779	−0.0002106	−1.1	7
HC7N+	2Π ← 2Π	0.0189665	−0.001934	−1.0	8
C24H12		0.011123c	−0.000033c	−0.3c	9

Notes. ^a(B + C)/2 is used instead of B. ^bThe positive change of the rotational constant upon excitation is probably due to a mistake in the analysis (see the text). ^cTheoretical values. References. (1) Gausset et al. 1965; (2) Achkasova et al. 2006; (3) Linnartz et al. 2000; (4) Kuhn et al. 1986; (5) Motylewski et al. 1999; (6) Sinclair et al. 1999; (7) Pfluger et al. 2000; (8) Sinclair et al. 2000; (9) Cossart-Magos & Leach 1990.

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We note that for all molecules except one (C₅), the rotational constant decreases upon electronic excitation, as expected. The reported positive change for C₅ (Motylewski et al. 1999) is probably due to a mistake in the analysis (J. P. Maier 2013, private communication). The large β values of 4.3%, 3.4%, 5.7%, 4.9%, etc., are the essence of the observed pronounced ETR. A small increase on the order of 1% cannot reproduce the pronounced ETR unless T_r is very high. This indicates that the carriers of DIBs with pronounced ETR are perhaps relatively small molecules.

The calculated velocity profiles for λ5780.5, λ5797.1, and λ6613.6 are shown in Figure 5, together with parameters which best mimic the observed ETRs. It has been noted in our simulations that of the seven parameters used in the analysis (T_r, T_k, B, μ, C, β, and Γ), those related to collisions (T_k and C) are not very critical. Even the values of B and μ are not that critical. The most crucial parameters are T_r, β, and Γ (Δt).

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The above analysis has indicated that a large fractional decrease in the rotational constant β (∼5%) and a high radiative temperature (T_r > 50 K) in the environment of the DIB carriers are needed to produce the observed pronounced ETRs. T_r = 14.6 K is not high enough to explain the ETR. The width of an ETR is simply proportional to β, and high T_r extends the rotational quantum number J of the populated levels (as seen in Figure 4), and thus greatly extends the ETR. The requirement of such a high T_r is surprising, but there seems to be no alternative, since the value of β is limited (Table 2). Our calculations are only for linear molecules, but it seems that this requirement holds for other types of molecules with three rotational constants, A, B, and C, as long as the ETRs are ascribed to the differences between the rotational constants of the excited and ground states. Other possible mechanisms, albeit not very realistic ones, are discussed below in Section 5.3.

Although our model calculations for polar linear molecules with high β and T_r give the observed pronounced ETR for the three DIBs as shown in Figure 5, we cannot claim that they are the carriers of the DIBs. In particular, a linear molecule cannot explain the triplet structure of DIB λ6613.6 discovered by Sarre et al. (1995). A perpendicular transition of a linear molecule gives a triplet structure but the central pileup of the Q-branch lines should be stronger than those of the P and R branches by a factor of approximately two due to the Hönl–London factors—which contradicts the observation. The claim by Schulz et al. (2000) that λ6613.6 is due to a parallel band of a long chain cumulane molecule (H₂C_n) does not work either, since the Q-branch pileup is weak for such a molecule.

Kerr et al. (1996) proposed a large planar symmetric top as the carrier of this band. Such a molecule does not have a dipole moment, and thus cannot be the carrier of λ6613.6 (which is sensitive to T_r). In general, non-polar molecules often considered for DIBs (such as C_n, HC_nH, H₂C_nH₂, NC_nN, etc.), symmetric PAHs (such as pyrene, coronene, ovalene, etc.), or C₆₀ and their cations and anions cannot be the carriers of DIBs exhibiting the ETR. Derivatives of PAHs, polar PAHs such as phenanthrene, tetraphen, pentaphene, or recently synthesized olympicene could be the carriers, but most of them probably do not have sufficiently large β to produce the ETR. There are no experimental data for β for coronene, but a theoretical value β = 0.3% (Cossart-Magos & Leach 1990) is an order of magnitude too small to produce the pronounced ETR.

Cami et al. (2004) discovered small variations in the structure of λ6613.6 for different sight lines and interpreted them as due to variations in rotational temperature from 21 K to 25 K. We believe that the rotational temperature is much lower (because of spontaneous emission), and a minute increase of T_r over the 2.73 K blackbody radiation is sufficient to explain the shifts (due to excitation of P and R lines of low J).

A similar difficulty is also encountered for λ5797.1. The high-resolution spectrum of this DIB shows a sharp Q-branch line. Our model calculation requires a large line broadening, which is obviously contradictory. Here we are just content to be able to produce observed huge ETRs using realistic β and T_r, and leave those thorny problems for future studies.

We ascribe the pronounced ETR to small (∼5%) variations in the rotational constants, but there may be two other possible interpretations. It is known that some molecules change their shape from nonlinear in the ground electronic state to linear in the excited state. Some examples are CH₂ ($\tilde{B}^3\Sigma _u^- \leftarrow \tilde{X}^3B_1$), HCO ($\tilde{A}^2\Pi \leftarrow \tilde{X}^1A^{\prime }$), and NO₂ ($\tilde{E}^2\Sigma _u^+ \leftarrow \tilde{X}^2A_1$) (Herzberg 1989b). These molecules are too small to cause DIBs, but if there are larger molecules with the same property, then the change in the rotational constant A by the electronic excitation is 100% and radiative pumping may cause pronounced ETR without having to assume a high T_r. While we have not been able to find relevant experimental data for large molecules of this nature, it is a possibility. Such electronic transitions, however, tend to have many vibronic satellites with high Franck–Condon factors, which are not seen for λ5780.5, λ5797.1, and λ6613.6.

Alternatively, we may interpret the ETR as a result of the excitation of low-frequency vibrational modes (rather than rotational levels in the ground state). The carriers then would not need to be polar molecules. The radiative effect for a vibrational excitation is typically 100 times less than that for rotational excitation, but the collisional excitations are also small and they may compete. The bending frequencies of C₃, 63.4 cm⁻¹ (Schmuttenmaer et al. 1990), and of C₃O₂, 18.3 cm⁻¹ (Fusina et al. 1980), are too high, but there may be larger molecules with lower vibrational frequencies. Such spectra, however, are likely to exhibit blue as well as red wings. Also, its effect will look more like the three examples in the right panel of Figure 3, λ5849.8, λ6196.0, and λ6379.3.

The presence or absence of the ETR allows discrimination between polar and non-polar carriers. The carriers of λ5780.5, λ5797.1, and λ6613.6, which show pronounced ETR, cannot be non-polar molecules. Molecules such as pure carbon chains (C_n) or symmetric hydrocarbon chains (HC_nH, H₂C_nH₂, NC_nN, etc.) and their cations and anions cannot be the carriers of those DIBs. Likewise, symmetric PAHs benzene, pyrene, coronene, ovalene, etc., and their cations and anions (Cossart-Magos & Leach 1990; see also the many PAHs discussed in Sharp et al. 2006) or C₆₀ and their cations and anions cannot be the carriers of DIBs exhibiting strong ETR.

The carriers of DIBs that show the ETR (in the left panel of Figure 3) cannot be the carriers of the DIBs without ETR (in the right panel). In particular, the carriers of λ6196.0 and λ6613.6 cannot be the same molecule, despite the near-perfect correlation of those two DIBs (McCall et al. 2010).

If linear molecules are assumed (as in Section 4), then the number of heavy atoms is likely three to six but not much higher, since the values of β = (B' − B)/B are not high enough for heavier molecules to produce the pronounced ETR (unless the radiative temperature is extremely high). Many linear molecules of this size have already been studied spectroscopically by Maier's group, but cumulene carbene molecules l-H₂C_n (n > 3) may be candidates. Nonlinear molecules are outside the scope of this paper. The high column densities required to produce the strong DIBs are problems (Oka & McCall 2011; Liszt et al. 2012), but this will be a problem for any candidate molecules of this size.

The carriers of λ5849.8, λ6196.0, and λ6379.3, which do not show the ETR as observed in λ5780.5, λ5797.1, and λ6613.6, are most likely non-polar molecules. The possibility that the carriers of those DIBs exist in different locations from other DIBs and are not exposed to the dust emission is perhaps safely neglected. However, our simulations have not excluded the possibility that they may be due to large polar molecules with very small β.

In this paper, we have considered only linear molecules where the ETR is created by large β = (B' − B)/B (∼5%). For nonlinear molecules, there are other possibilities, such as large decreases in the differences of rotational constants A − B or C − B for prolate or oblate tops, respectively. Either way, the relative variations of these constants are not very likely to be much higher than 5%, except for the case mentioned in Section 5.3. In most cases, they are 10 times smaller than needed to produce the pronounced ETR.

Since carriers of DIBs with pronounced ETRs are polar molecules with modest size, they should be observable by radio astronomy. Because of the low density and subthermal rotational distribution in diffuse clouds, the observations need to be by absorption (Liszt et al. 2012), and in the relatively long wavelength region for low-J transitions.