The EDIBLES Survey VII. A survey of C 2 and C 3 in interstellar clouds

Context. Small linear carbon chain radicals like C 2 and C 3 act both as the building blocks and dissociation fragments of larger carbonaceous species. Their rotational excitation traces the temperature and density of local environments. However, these homo-nuclear di-and triatomic species are only accessible through their electronic and vibrational features because they lack a permanent dipole moment, and high signal to noise ratio data are necessary as the result of their generally low abundances in the interstellar medium (ISM). Aims. In order to improve our understanding of small carbon species in the ISM, we carried out a sensitive survey of C 2 and C 3 using the ESO Di ff use Interstellar Bands Large Exploration Survey (EDIBLES) data set. We also expanded our searches to C 4 , C 5 , and 13 C 12 C isotopologue in the most molecule-rich sightlines. Methods. We fit synthetic spectra generated following a physical excitation model to the C 2 (2-0) Phillips band to obtain the C 2 column density ( N ) as well as the kinetic temperature ( T kin ) and number density ( n ) of the host cloud. The C 3 molecule was measured through its ˜ A − ˜ X (000-000) electronic origin band system. We simulated the excitation of this band with a double-temperature Boltzmann distribution. Results. We present the largest combined survey of C 2 and C 3 to date in which the individual transitions can be resolved. In total we detected C 2 in 51 velocity components along 40 sightlines, and C 3 in 31 velocity components along 27 sightlines. Further analysis confirms the two molecules are detected in the same velocity components. We find a very good correlation between N (C 2 ) and N (C 3 ) with Pearson correlation coe ffi cient r = 0 . 93 and an average N (C 2 ) / N (C 3 ) ratio of 15.5 ± 1.4. A comparison with the behaviour of the C 2 DIBs shows that there are no clear di ff erences among sightlines with and without detection of C 2 and C 3 . This is in direct contrast to the better-studied non-C 2 DIBs who have reduced strengths in molecule-rich environments, consistent with the idea that the C 2 DIBs are indeed a distinguishable DIB family. We also identify for the first time the Q (2), Q (3), and Q (4) transitions of the 13 C 12 C (2-0) Phillips band in the stacked average spectrum of molecule-rich sightlines, and estimate the isotopic ratio of carbon 12 C / 13 C as 79 ± 8, consistent with literature results. At this stage it is not possible yet to identify these transitions in individual sightlines. Our search for the C 4 and C 5 optical bands was unsuccessful; even in stacked spectra no unambiguous identification could be made.


Introduction
Many carbon chain species have been detected via submillimetre and/or radio observations.However, due to the lack of a permanent dipole moment, bare carbon chains can only be accessed through their electronic and vibrational features.The simplest bare carbon molecule, interstellar C 2 , was first detected towards the highly reddened star Cyg OB2 12 through its A-X (1-0) Phillips band in the near-IR (Souza & Lutz 1977).This was followed by more extensive observations of other bands and towards more targets (e.g.Hobbs & Campbell 1982;van Dishoeck & de Zeeuw 1984;Sonnentrucker et al. 2007;Wehres et al. 2010).The population on the C 2 rotational excitation ladder is determined simultaneously by collisional and radiative processes (van Dishoeck & Black 1982), which makes C 2 a powerful diagnostic tool for deriving local environmental factors, such as the kinetic temperature and density within the interstellar cloud.
An emission band around 4051 Å was first noted in the spectrum of comet Tebbutt (Huggins 1881), and Douglas (1951) subsequently assigned it to C 3 .Observations of this band in diffuse interstellar environments have been difficult due to the much lower column density of C 3 ; the first tentative detection was made in 1995 in the highly reddened sightline towards HD 147889 (Haffner & Meyer 1995).C 3 molecules were also observed with Herschel in the ν 2 bending mode towards Sagittarius B2 and star-forming cores (Cernicharo et al. 2000;Mookerjea et al. 2010).To date, only a handful of measurements of the C 3 4051 Å band are available in the literature, and the data are mostly sightlines known to be molecule-rich (Maier et al. 2001;Roueff et al. 2002;Galazutdinov et al. 2002;Oka et al. 2003;Ádámkovics et al. 2003;Schmidt et al. 2014).In addition to the C 3 origin band, Schmidt et al. (2014) also reported electronic transitions involving vibrationally excited C 3 in its Ã state.High-resolution spectra are required to accurately measure the C 3 column density.Once these measurements are available, individual transitions can be resolved, allowing the rotational excitation to be characterised, and the contribution from the higher J-levels to be determined (see e.g.Oka et al. 2003).
Small carbon molecules and chains are considered building blocks of more complex carbon-bearing compounds (Wakelam et al. 2009), and column density ratios of different carbon chains help us constrain the chemistry that drives their formation and destruction.The destruction process may also involve the photo-dissociation of (larger) polycyclic aromatic hydrocarbons (PAHs; Hrodmarsson et al. 2022).C 4 and C 5 molecules may be abundant in protoplanetary nebulae and circumstellar shells around evolved stars (Bernath et al. 1989;Cernicharo et al. 2002;Hargreaves et al. 2014), yet attempts to search for their spectral fingerprint in diffuse and translucent environments have been unsuccessful so far (Maier et al. 2002(Maier et al. , 2004)).Additionally, the isotopic ratio of carbon may trace Galactic chemical evolution (Milam et al. 2005).Hamano et al. (2019) report a marginal detection of the Q(3) transition of the 13 C 12 C (0,0) Phillips band towards Cyg OB2 12 (see also Gredel et al. 2001), and Giesen et al. (2020) reported the first detection of 13 CCC and C 13 CC through their ro-vibrational transitions at 1.9 THz.The measurement of an isotopologue also provides insight into local peculiarities since the isotope exchange rate that sets the isotopic fraction depends on environmental parameters such as temperature (see e.g.Rocha & Linnartz 2021;Visser et al. 2009).
Since their first discovery in the 1920s (Heger 1922), more than 500 diffuse interstellar bands (DIBs) have been identified in the near-UV, optical, and near-IR regions (Hobbs et al. 2008(Hobbs et al. , 2009;;Geballe et al. 2011;Fan et al. 2019).Despite this centurylong history, the carriers of DIBs remain unknown, except for the five near-IR DIBs that were recently attributed to C + 60 (Campbell et al. 2015;Walker et al. 2015;Cordiner et al. 2019;Ebenbichler et al. 2022).High-resolution observations reveal substructures within several DIBs that suggest they have molecular origins (Sarre et al. 1995;Kerr et al. 1998;Cami et al. 2004).
Molecules of different types have been proposed as DIB carriers -from complex structures, such as (charged) fullerenes and PAHs (see e.g.Foing & Ehrenfreund 1994;Gredel et al. 2011;Salama et al. 2011;Salama & Ehrenfreund 2014, and their citations), to smaller molecules, such as carbon chains of different sizes (Motylewski et al. 2000;Oka et al. 2013;Zack & Maier 2014; see Herbig (1995) for a review) -despite the fact that none of their absorptions fully overlap with DIB features.
C 2 and C 3 molecules are present in the dense molecular regions of diffuse and translucent cloud environments.Their detection provides physical information on interstellar medium (ISM) environmental conditions, such as the kinetic temperature inferred from their rotational excitation.The column density ratios between C 2 and the 'first-generation molecules' H 2 and CH can trace the average ISM conditions of the sightline (Thorburn et al. 2003;Welty et al. 2013), similar to the use of the molecular hydrogen fraction, f H 2 , but presumably tracing deeper regions in the interstellar clouds (e.g.Fan et al. 2017).This information in turn can be compared to the behaviour of DIBs and provides constraints on their carriers (e.g.Cami et al. 1997;Sonnentrucker et al. 1997).This may eventually lead to the identification of the C 2 DIBs, whose equivalent widths become larger when the sightline is more abundant in C 2 (Thorburn et al. 2003).
In this paper we present a survey of C 2 and C 3 signals in the EDIBLES data.This dataset is briefly introduced in Sect. 2. We present our measuring methods and results for C 2 in Sect. 3 and for C 3 in Sect. 4. These results are discussed in Sect.5, and we link these results to the behaviour of DIBs in Sect.6. Section 7 describes our efforts to search for the 13 C 12 C, C 4 , and C 5 transitions.We summarise our conclusions in Sect.8.

Observations and spectral data
The ESO Diffuse Interstellar Bands Large Exploration Survey (EDIBLES) is a sensitive spectroscopic survey of a large sample of DIB sightlines towards early-type stars using the UV-visual echelle spectrograph (UVES; Dekker et al. 2000;Smoker et al. 2009) mounted on the Very Large Telescope (VLT).EDIBLES contains observations towards 123 sightlines with high-spectral resolution (R∼80,000 in the blue arm and ∼100,000 in the red arm), high sensitivity (signal-to-noise ratio S/N ∼400-1,000 in the optical region), and covering a wide wavelength range (305-1042 nm).The program is described in Cox et al. (2017), and we also refer the reader to earlier publications that used EDIBLES data (i.e.Lallement et al. 2018;Elyajouri et al. 2018;Bacalla et al. 2019;MacIsaac et al. 2022) for more details.
In this paper, we focus on the C 2 (2-0) Phillips system around 8757 Å, and the C 3 Ã − X origin band around 4051 Å.All spectra used in this work were obtained with the 437+860 setting of UVES.For sightlines with multiple observations, we interpolated, normalised, and co-added all available data into a single and high-quality spectrum for the final analysis.
Figure 1 shows spectral segments around the target transitions of the C 2 (2-0) band and the fitted model of three representative sightlines.
3.1.Line list and modelling of the C 2 (2-0) Phillips band We adopted the wavelengths and oscillator strengths ( f j j values) of the C 2 (2-0) Phillips band from Sonnentrucker et al. (2007) for J up to 16 (Table 1).The R(2) and R( 10) transitions at 8753.949 and 8753.578Å were excluded due to a nearby CCD artefact.van Dishoeck & Black (1982) provided a comprehensive model of the rotational excitation of C 2 by considering both radiative and collisional processes.The relative population of C 2 for different J-levels probes the kinetic temperature T kin as well as the number density n.Snow & McCall (2006) turned this model into an online C 2 Calculator1 that computes the relative population for different J-levels at given the kinematic temperature T kin and the H + H 2 number density n.We generated a finely sampled grid from this calculator, for T kin between 10 and 100 K with an interval of 1 K, and n between 10 and 1000 cm −3 with an interval of 10 cm −3 .The parameter grid is then interpolated using a 2D cubic spline.
We followed the routine described in Elyajouri et al. ( 2018) to generate synthetic C 2 spectra and assumed Gaussian optical depth profiles for all C 2 transitions (see also Kokkin et al. 2007).With the knowledge on the relative population on different Jlevels, the total column density N, the Gaussian broadening parameter b, and the velocity offset v off (relative to the barycentric frame), we were then able to calculate the opacity spectrum for a velocity component.For each of the sightlines, opacity spectra for all velocity components where C 2 transitions can be discerned are co-added and then converted to flux units.This synthetic spectrum was finally convolved with a Gaussian kernel with a full width at half maximum (FWHM) of 2.8 km • s −1 that simulates the instrumental profile.

C 2 measurements and results
Under typical ISM conditions, the Q(2), Q(4), and Q( 6) are the strongest transitions of the C 2 (A-X) band.We first visually examined the presence of these absorption lines to screen the tar-get sightlines for a more detailed model-fitting process.We paid close attention to the asymmetries and multiple peaks in the line profiles and compared them to the velocity components identified in the Na i 3302 Å doublet.Such efforts helped us determine whether the C 2 lines should be measured in multiple velocity components, each of which is characterised by its own set of parameters (i.e. the kinetic temperature T kin , the number density n, the total column density N, the broadening factor b, and the velocity offset v off relative to the barycentric frame).
The rotational transitions of the C 2 (2-0) Phillips band span a ∼100 Å wavelength range.Since they take up a very limited fraction of the window, we chose to focus on spectral segments with the target C 2 transitions without having to fit a global continuum.These segments are 1.8 Å or ∼60 km • s −1 on each side of the target lines, and are combined when a segment overlaps with another.We normalised these segments with third-order polynomial continua.Telluric lines were identified from the HITRAN database (Rothman 2021) in this process.In most cases, the telluric lines could be corrected by fitting them with a Gaussian profile so their presence would not affect the normalisation.If that was not possible, the spectral segment was excluded from the fitting process.
The synthetic spectrum was fitted to the normalised data using a customised Python script based on the Sherpa package (Freeman et al. 2001).The parameters in the model were optimised using a Levenberg-Marquardt χ2 minimisation.The 1σ errors were estimated by independently tuning each parameter from its optimal value until the sum of χ 2 reached χ 2 min + 1.The C 2 (2-0) Phillips band is detected at the 3σ level in 37 sightlines, and tentatively in another 3 sightlines 2 .Nine of these sightlines -HD 41117, HD 61827, HD 112272, HD 147084, HD 149404, HD 161056, HD 183143, HD 186745, and HD 186841 -contain two velocity components in which the target C 2 band is detected, while three components are detected towards HD 167971.Figure 1 demonstrates spectral data and best-fit results of three representative sightlines and Appendix A contains a gallery of all 40 sightlines.The fitted parameters and their 1σ uncertainties are tabulated in Table C.1.In the table we also include literature values for targets that have been studied previously.For the sightlines in common with our study, both the kinetic temperatures and densities we derive are in good agreement with existing literature values, and typically with smaller uncertainties.

Temperatures and densities of C 2 -bearing clouds
We supplemented our derived values for T kin and n in 40 sightlines with literature values for an additional 36 targets (see the bottom part of Table C.1) to study the distribution function of these parameters.
Fig. 2 shows the distribution of the derived kinetic temperatures and densities using results from our study (red) as well as including literature values (blue).The temperature distribution peaks in the 30-40 K range, with an unweighted average kinetic temperature (using all values) of 40±16 K; we also note that the median value is 40 K.Only four values are higher than 60 K.The density distribution is similarly strongly peaked in the 100-200 cm −3 range.The unweighted average and standard deviation   4) line is still below the 3σ detection limit.We considered multiple rotational lines simultaneously when deciding about possible identifications.The C 2 molecules are detected in two velocity components in HD 161056 (bottom).The combined model (red) is plotted against the spectral data, while synthetic spectra for the two velocity components (blue and orange) are plotted separately with an offset.We focus on the spectral segments around the C 2 transitions, and the wavelength grid is hence discrete.Rotational lines being considered in each sightline may differ due to telluric interference and/or data quality.These rotational lines are labelled with vertical bars.The best-fit parameters for the kinetic temperature, T kin , number density, n, and the column density of the velocity component, N, are listed in each panel.
is 247 ±146 cm −3 , but the median value is 202.8 cm −3 and only six values are above 500 cm −3 .As can be ascertained from Table C.1, these values typically have large uncertainties.

C 3 models and measurements
The strongest C 3 transitions in the optical region lie around 4051 Å and are from the origin 000-000 band of the Ã1 Π u − X1 Σ + g system.This band is characterised by a piled-up Q-branch in the centre, while the rotational lines from the P-and R-branches can be resolved in our data (Fig. 3).This allows us to determine the rotational excitation and thus the total column density of C 3 in a more accurate way by including the contributions from the higher J-levels whose transitions are too weak to be detected.Some transitions involving vibrationally excited levels in the upper Ã state have recently been detected in diffuse inter- stellar environments (Schmidt et al. 2014).These transitions are also within the EDIBLES wavelength range, but in wavelength ranges where we lack sufficient S/N to detect these weak lines.

Line list of the C 3 Ã − X band
Wavelengths of the C 3 Ã − X origin band transitions are reported in several works (e.g.Gausset et al. 1965;McCall et al. 2003;Tanabashi et al. 2005;Martin-Drumel et al. 2023).Zhang et al. (2005) report the presence of at least two perturbing energy levels that are close to the upper Ã 1 Π u state.These perturbing states would change the observation in two ways, where a) the effective f j j values of the original transitions are smaller than theoretically predicted, and b) some of these perturbed lines are strong enough to be detected in the spectrum.Such interference, especially when the lower J-levels are involved, should be considered when simulating the C 3 spectrum (Schmidt et al. 2014;Haddad et al. 2014).
Table 2. Line list used to simulate the C 3 Ã − X 000-000 band.
Rotational 20  Table 2 lists the 44 transitions we considered to simulate the C 3 4051 Å band, including the original Q-branch lines for J up to 30, the original P-and R-branch lines for J up to 20, and eight strong perturbed lines reported in Schmidt et al. (2014).The wavelengths of the target transitions are adopted from Table 3 of Schmidt et al. (2014).For the rotational lines that are not included in that source, we used the values from Tanabashi et al. (2005) after removing their correction of 0.04 cm −1 on the wavenumbers.This treatment provides better agreement on the lower J-level transitions between Tanabashi et al. (2005) and Schmidt et al. (2014) (as well as Gausset et al. (1965) and Mc-Call et al. (2003)).We adopted the more widely used oscillator strength of the C 3 000-000 band as f 000 = 0.016 (e.g.Maier et al. 2001;Ádámkovics et al. 2003;Oka et al. 2003;Schmidt et al. 2014).However, a slightly smaller value of f 000 = 0.0146 is used in some studies (e.g.Roueff et al. 2002;Galazutdinov et al. 2002) due to the uncertainties in the Frank-Condon factor (see e.g.Perić-Radić et al. 1977;Jungen & Merer 1980).The C 3 column densities would be 10% larger if this smaller f 000 value is used.Without the perturbed lines, the f j j values of the original transitions can be calculated as g 000-000 band in three representative sightlines.Top panel: HD 147889, in addition to harbouring the largest N(C 2 ), also harbours the largest N(C 3 ) in our data sample.The transitions are all labelled, including those to the original and the perturbed upper energy levels (the latter are highlighted with asterisks; refer also to Sect.4.1 and Table 2).Bottom panels: The C 3 band is tentatively detected towards HD 23180, mainly through its piled-up Q branch in the middle.The sightline of HD 161056 contains two velocity components of C 3 .The synthetic spectra for the two components are plotted with a positive offset in different colours. (1) Due to the presence of perturbed lines, the effective f j j values are different from the theoretical calculations.We used the experimentally determined f j j values from Schmidt et al. (2014), except for transitions not listed there.In that case, we used the theoretical results of Eq. 1.These transitions are mostly from the higher J-levels with limited population, and the interference from the perturbed lines becomes very subtle.

Excitation and modelling of C 3
Under typical diffuse interstellar conditions, C 3 molecules have a relatively short lifetime, and collisional processes may not significantly change the population of the higher J-levels (Roueff et al. 2002;Schmidt et al. 2014).Observations suggest that the rotational excitation of C 3 is characterised by two temperatures with the transition point around J = 14.The excitation temperature of lower J-levels, T low , is governed more by collisions and is typically a few tens of kelvin, whilst the temperature for the higher J-levels, T high , is governed more by radiative processes and is usually a few hundred kelvin (see e.g.Ádámkovics et al. 2003;Maier et al. 2001;Welty et al. 2013).The much higher excitation for the higher J-levels leads to a shallow decrease in their population, and in total the J > 14 levels may contribute ∼15 % of the total column density (Ádámkovics et al. 2003).
To simulate the rotational excitation of the C 3 molecules and obtain the relative population of different J-levels, we composed a two-temperature Boltzmann distribution model with the following steps: 1. Take the J = 14 population as reference, by defining {N}={N low (J=0), N low (J=2), ..., N low (J=12), N(J=14), N high (J=16), ..., N high (J=100)}.5. Scale {N J } so it adds up to unity.
Despite the approximation, this model provides results that agree well with the literature (see the next subsection).The rotational energies of the different J-levels required in the Boltzmann distribution are calculated using the molecular constants in Tanabashi et al. (2005) as follows: (2) The column density of C 3 is typically one order of magnitude lower than C 2 and for the much weaker C 3 absorption strengths optically thin conditions can be assumed (Fig. 3).Becker et al. (1979) report the radiative lifetime of C 3 as approximately 200 ns, and natural broadening can hence be ignored.We thus used simple Gaussian profiles in flux units to simulate the rotational lines of the C 3 4051 Å band, and rotational lines from the same velocity component would share the same FWHM and velocity offset (v off , relative to barycentric frame) parameters.We also set a lower boundary of 3.0 km • s −1 for the FWHM, so the Gaussian profiles have at least the instrumental width, in addition to physical processes that broaden the C 3 transitions.

Measurements and results
As for C 2 , we first visually examined the target C 3 transitions, especially the more strongly piled-up Q branch contour.The C 3 Ã − X 000-000 band spans a narrow wavelength window of ∼6 Å, and we used a cubic spline continuum to normalise the entire wavelength region.The compact distribution of the transitions hampers the identification of possible extra velocity components.We thus referred to the C 2 measurements and the Na i 3302 Å doublet of the same sightline to determine the number of components needed for the model.Each of these components is characterised by its own set of T low , T high , column density N, Gaussian width factor FWHM, and v off , which is relative to the barycentric frame.We followed the same routine as C 2 for parameter optimisation and error estimation (see Sect. 3.2).
The target C 3 band is detected at the 3σ level in 19 sightlines, and tentatively in another 8 sightlines3 .These include 4 sightlines where C 3 is measured in two velocity components (HD 61827, HD 149404, HD 161056, and HD 186841).In Table 3 we summarise the C 3 measurements and compare to available literature values, and we find good agreement in most of the cases, especially with results from high-resolution data.The C 2 and C 3 detection are clearly linked; all sightlines with a C 3 detection (Table 3) also exhibit C 2 absorption.The best-fit models are visualised for three representative targets in Fig. 3 and the full gallery is available in Appendix B.
The temperature parameters may be less well determined when N(C 3 ) is small, especially for T high .Assuming average excitation conditions where T low = 50 K and T high = 250 K, the higher J levels harbour ∼25% of the total population.This leaves less than 5% of the total equivalent width for each of the J-levels considered in our model, which is further divided into the P, Q, and R transitions.These transitions are then very weak and provide limited constraints on minimising the χ 2 during the fitting process.

C 2 versus C 3
In this section we compare the velocity offsets and column densities of C 2 and C 3 obtained in our survey (Fig. 4).Since there may be multiple velocity components in some sightlines, each data point in the plot represents an individual ISM cloud.We find the velocity offsets of C 2 and C 3 agree with each other (within uncertainties), which suggests the two molecules are detected in the same velocity components, and validates further comparison of their column densities.The column densities of C 2 and C 3 show a tight correlation among the 30 velocity components from 26 sightlines with r = 0.93 (Fig. 4, lower panel).We find the best-fit line to have a relatively small intercept.The slope of the best-fit line, indicating the N(C 2 )/N(C 3 ) ratio, is 15.5±1.4.While it is widely agreed that N(C 3 ) should be one magnitude lower than N(C 2 ), it used to

This Work b
Literature c Sightline a  b The Gaussian broadening factor FWHM has a lower boundary of 3.00 km • s −1 to reflect instrumental profile; the fitting boundaries of the excitation temperature for the higher J-levels T high are between 100 and 1,000 K; the velocity offset v off is relative to the barycentric frame.be difficult to determine the N(C 2 )/N(C 3 ) ratio with good accuracy due to the required high quality data.In a pioneering survey by Oka et al. (2003), the authors also found a good correlation between N(C 2 ) and N(C 3 ), and estimated a N(C 2 )/N(C 3 ) ratio of ∼40.However, since the individual rotational lines could not be resolved in their medium resolution data (R∼38,000), Oka et al. (2003) focused on the piled-up Q branch and assumed it contributed half of the total intensity of the band.This approximation may underestimate contributions from the higher J-levels and hence N(C 3 ) (Ádámkovics et al. 2003).
The N(C 2 )/N(C 3 ) ratio of 15.5 in this work is slightly larger than the value of ∼10 reported by Schmidt et al. (2014), and is in fact close to the prediction for a typical diffuse interstellar cloud with A V ∼2.0 mag in the model presented by Schmidt et al. (2014).This is approximately the extinction of our targets (Cox et al. 2017).However, the N(C 2 )/N(C 3 ) ratio can be quite different in circumstellar environments, presumably related to the much higher density and/or enhanced radiation field.For example, literature reports column densities of both C 2 and C 3 to be ∼10 15 cm −2 towards the well-studied carbon star IRC+10216 (see e.g.Hinkle et al. 1988;Bakker et al. 1997;Cernicharo et al. 2000;Hargreaves et al. 2014), and N(C 2 )/N(C 3 ) ∼1.
In diffuse and translucent clouds the C 2 molecules may be formed bottom-up along the reaction C + + CH → C + 2 + H.The product then turns into C 2 molecules via multiple channels that consist of series of abstractions of H atoms and dissociative recombinations (e.g.Federman & Huntress 1989;Federman et al. 1994;Sonnentrucker et al. 2007).Oka et al. (2003) suggested that C 3 molecules are formed via dissociative recombination of C 3 H + (Oka et al. 2003), while Bergeat & Loison (2001) advocate for non-negligible contribution from neutral-neutral reactions between C and C 2 H 2 .It is also possible for the two molecules to be formed from detachment of larger molecules in a top-down manner.In either case, there must be a strong chemical link between C 2 and C 3 since their column densities are tightly correlated.Thorburn et al. (2003) introduced the C 2 DIBs as 'a class of weak, narrow bands whose normalised equivalent widths W(X)/W(6196) are well correlated, specifically with N(C 2 )/E B−V via power laws'.The C 2 DIBs demonstrate different behaviour compared to the 'regular' non-C 2 DIBs, in particular in dense environments (see e.g.Fan et al. 2017).Contrary to their name, the C 2 DIBs may correlate better with E B−V than with N(C 2 ) (Elyajouri et al. 2018).The strong C 2 DIBs λλ4963 and 4984 are detected in sightlines exposed to intense radiation fields and with low f H2 values (see e.g.Fan et al. 2017Fan et al. , 2019)), suggesting that their carriers are not necessarily directly chemically linked to C 2 or other molecules.

C 2 , DIBs, and C 2 DIBs
In previous sections we carefully examined all 123 EDI-BLES targets and detected the C 2 (2-0) Phillips band and the C 3 Ã − X origin band in a large number of sightlines.The detection of C 2 and C 3 molecules usually suggests dense and molecular environments along the sightlines (see also Appendix C).We refer to the 41 sightlines that show C 2 features as 'C 2 sightlines' and the rest as 'non-C 2 sightlines'.This section discusses how C 2 and non-C 2 DIBs behave in these sightlines.Since DIB measurements are not the primary focus of this work, we adopted the DIB equivalent widths (W(DIBs)) from Fan et al. (2017).Readers are referred to the original paper for details on how the measurements were obtained.
Figure 5 compares the general behaviour of four DIBs, in terms of W(DIBs) -E B-V correlations, between the C 2 and non-C 2 sightlines.The comparisons are made for two non-C 2 DIBs λλ5780 and 6196, and two of the well-recognised C 2 DIBs λλ4963 and 4984.Both non-C 2 DIBs have reduced strengths in the C 2 sightlines, which was described as the 'skin effect' in early DIB observations (e.g.Wampler 1966;Strom et al. 1975;Meyer & Ulrich 1984;Herbig 1995).This observational fact suggests that the inner parts of the clouds have a limited contribution to the column density of the general DIB material (Snow & Cohen 1974).As a more quantitative analysis, Fan et al. (2017) reported 'lambda-shaped' behaviour of non-C 2 DIBs where their normalised strengths (W(DIB)/E B-V ) decrease as a function of f H2 for f H2 > ∼ 0.3.This is also approximately the range of the f H2 values for the C 2 sightlines we identified (Appendix C).
In direct contrast, the C 2 DIBs λλ4963 and 4984 showed undifferentiated behaviour among C 2 and non-C 2 sightlines.This is consistent with the observations that W(C 2 DIBs)/E B-V remains relatively constant (with large scatter) in sightlines with large f H2 values (Fan et al. 2017).The C 2 DIBs are originally defined by an enhanced ratio of W(C 2 DIBs)/W(6196) in the C 2 sightlines (Thorburn et al. 2003).This increment is more likely the result of decreased W(6196) rather than increased W(C 2 DIBs), and provides evidence that the C 2 DIBs may trace denser environments than the non-C 2 DIBs.However, since the C 2 DIBs are not enhanced in the C 2 sightlines, their carriers may not reside in the deep layers of the ISM clouds where C 2 and other secondgeneration molecules are found (e.g.CN and CO; see Fan et al. 2022).

One step further: Search for more complicated carbon chains
The positions of the transitions from the 13 C 12 C (0-0) and (1-0) Phillips A 1 Π-X 1 Σ + bands were first provided by Amiot & Verges (1983) using Fourier spectroscopy.Hamano et al. (2019) recently reported the first tentative detection of 13 C 12 C in the ISM using these data.This marginal detection is for the Q(3) transition of the (0-0) band and is made towards the heavily reddened sightline of Cyg OB2 12.To the best of our knowledge, no dedicated laboratory studies are yet available on the 13 C 12 C (2-0) bands.We calculated the required transition wavelengths and f j j values using the well-known isotopic dependence of the Dunham vibration-rotation coefficients (Watson 1980).
For this purpose, we followed the same approach as described in Ram et al. (2014) and Rousselot et al. (2012) and utilised the molecular parameters of the main 12 C 2 isotopologue (Chen et al. 2015) and the isotopic relationships (Ram et al. 2014;Rousselot et al. 2012;Watson 1980): Here the primed and unprimed quantities are equilibrium spectroscopic constants for 13 C 12 C and 12 C 2 , respectively (as defined in Chen et al. 2015) and ρ = µ main /µ rare = 0.98052, with the µ Table 4. Mass-scaled spectroscopic constants of 13 C 12 C in the (2-0) Phillips system.
Constants a This work b Band origin 11354.65876B v 1.74133 1.741417 1.51402 D v ×10 6  6.46215 6.530 6.06544 q v ×10 4 -1.77837q D ×10 9 4.31457 All values are in cm −1 .a B v and D v are rotational and centrifugal distortion parameters, while q v and q D are Λ-type doubling constants.v define vibrational quantum numbers; see Chen et al. (2015) for details.b Calculated from 12 C 2 (Chen et al. 2015) and isotopic relations (see the main text).c Laboratory spectroscopic data; see Amiot & Verges (1983).
being the reduced mass.The resulting primed (equilibrium) constants of 13 C 12 C (Eq. 3) were then used as input into standard spectroscopic formulas (see e.g.pages 3- 4 Chen et al. 2015) to introduce the proper vibrational dependence into the relevant mass-scaled parameters.Their final values are gathered in Table 4.The list with calculated line positions was subsequently generated using the final spectroscopic constants in Table 4 and PGOPHER (see Table 5; Western 2017).
We employed the methodology described in Ram et al. (2014) to determine the f j j values for the 13 C 12 C (2-0) Phillips bands.Briefly, the potential energy curves for each electronic state are represented by Rydberg-Klein-Rees (RKR) classical turning points (Le Roy 2014).These are calculated using the RKR1 program (Le Roy 2014) and the mass-scaled spectroscopic constants of the upper (A 1 Π) and lower (X 1 Σ + ) 13 C 12 C states  (Eq.3).With the corresponding ab initio electronic transition dipole moment function reported by Yurchenko et al. (2018), Einstein A-coefficients (A J ′ →J ′′ ) were then determined using the LEVEL code (Le Roy 2017), and finally converted into the f j j values following the usual expression (Ram et al. 2014): Here, ν is the transition wavenumber (in cm −1 ) associated with the lower (J ′′ ) and upper (J ′ ) ro-vibronic J states.Unlike the regular C 2 molecule, 13 C 12 C has additional transitions from odd J-levels due to its hetero-nuclear nature.Our simulation suggests that the Q(1) -Q( 4) transitions are the strongest under typical excitation conditions with T exc ∼ 60 K. Their calculated wavelengths and f j j values are summarised in Table 5.We also included the calculated (following the same approach described above) f j j values of the regular 12 C 2 molecule for comparison.As expected, they differ by less than 1.2%.We also note that our calculated f j j values for the 12 C 2 molecule reproduce to within 0.4% those reported by Sonnentrucker et al. (2007), hence further supporting the reliability of our approach.We focused on the sightlines of HD 63804, HD 147889, and HD 169454 because of their large C 2 column densities and good S/N in the target region.These factors make them plausible candidates to search for the 13 C 12 C transitions, yet we cannot make robust detection on the target 13 C 12 C transitions in any of these sightlines (Fig. 6, upper panel).To further increase the sensitivity of our data, we continued by performing a S/N-weighted spectral stacking for the three sightlines.The measured S/N of the stacked spectrum reaches ∼ 3,000, and the Q(2), Q(3), and Q(4) transitions are detected in the stacked spectrum (Fig. 6, lower panel).This detection validates our theoretical prediction on the wavelengths, which is based solely on isotopic relations (Rousselot et al. 2012;Ram et al. 2014).
The stacking process, however, averages different ISM conditions of the three sightlines, and we cannot obtain detailed excitation information for 13 C 12 C in the stacked data.Because of this, the best approach to even calculating the column density of 13 C 12 C requires analysis of a wide range of probable excitation conditions especially regarding the excitation temperature.We compiled a series of models with T exc between 40 and 80 K (with an interval of 5 K), and obtained the relative strengths of the Q(1) to Q(4) transitions from PGOPHER.We assumed a Gaussian profile with instrumental widths for each of the transitions.The best fit with the minimal χ 2 occurs when T exc = 55 K with N( 13 C 12 C) = 2.30 × 10 12 cm −2 .The standard deviation of N( 13 C 12 C) among models of different T exc is 0.31 × 10 12 cm −2 (∼ 13%), representing the uncertainty due to the unknown excitation conditions.
The average N(C 2 ) is 8.92 × 10 13 cm −2 for the three sightlines 4 .The N(C 2 )/N( 13 C 12 C) ratio is then ∼ 39 ± 4, or a 12 C/ 13 C ratio of ∼ 79 ± 8.This is very close to the same ratio of 72±26 observed towards the circumstellar shell of the post-asymptoticgiant-branch star HD 56126 (Bakker & Lambert 1998).The 12 C/ 13 C ratio can also be obtained from other carbon-bearing molecules.For examples, Milam et al. (2005) reported ∼68 by considering CO, H 2 CO, and CN, and Stahl et al. (2008) find 76±2 using CH + measurements of the local ISM.Again the findings of this work are well aligned with the above literature results.
The 12 C/ 13 C ratio traces the large-scale Galactic chemical evolution as well as local peculiarities of the ISM conditions (e.g.Milam et al. 2005;Furuya et al. 2011;Liszt & Ziurys 2012).Rocha & Linnartz (2021) discussed the temperature dependence of the isotopic fractionation of carbon.Their calculation suggests that a smaller 12 C/ 13 C ratio can be expected in sightlines characterised by lower kinetic temperature, since the forward isotopeexchange reaction 13 C + 12 C 2 → 13 C 12 C + 12 C is exothermic by 26.4 K and the reverse endothermic process becomes largely inefficient at low temperatures.We note in passing that the 13 C + + 12 C 2 reaction, despite not being considered by Rocha & Linnartz (2021) in the context of dense molecular clouds, is also expected to play a key contribution to the overall carbon isotopic fractionation, particularly under the diffuse ISM environments (Loison et al. 2020;Colzi et al. 2020).Compared to T kin = 20 K, the 12 C/ 13 C ratio can be a factor of 2 greater when T kin = 100 K.The identification in stacked data cannot assign the 12 C/ 13 C ratio to specific ISM conditions, but such dependences may be examined in future studies.

C 4 and C 5
The C 4 and C 5 molecules are the simplest bare carbon chains after C 2 and C 3 .In the optical region, the strongest C 4 band occurs at 3789 Å and is between the 3 Σ − u and 3 Σ − g states.The laboratory spectrum of this band is characterised by the piled-up R-branch to the blue and more expanding P-branch contour to the red (Linnartz et al. 2000;Fig. 7, left panel).The strongest C 5 optical band is at 5109 Å and Forney et al. (1996) assigned this band to the 1 Π u − X 1 Σ + g system.However, Hanrath & Peyerimhoff (2001) noted a discrepancy of ∼0.5 eV in the transition energy and suggested the 5109 Å band to be associated with formally forbidden 1 Σ − u and 1 ∆ u states.Nevertheless the most prominent feature of this band is the piled-up Q-branch in the middle while the contours of the P-and Q-branches are much shallower (Fig. 7, right panel).
Similar to our search for 13 C 12 C, we searched for these optical bands of C 4 and C 5 in candidate sightlines characterised by large N(C 3 ) and high S/N in the wavelength region (namely, HD 147889, HD 169454, and HD 210121 for the C 4 3789 Å band and HD 154368, HD 169454, and HD 203532 for the C 5 5109 Å band).However, neither of the target transitions can be identified in individual sightlines or their stacked spectra (Fig. 7).We instead estimated the upper limits of N(C 4 ) and N(C 5 ) by considering the most prominent feature in their band profile (i.e. the R-branch contour for C 4 3789 Å band and the Q-branch contour for C 5 5109 Å band) while assuming it contributes 50% of the total absorption.The results are summarised in Table 6.
The C 5 molecules have been detected in earlier work via their IR transitions in the carbon-rich circumstellar shell around IRC +10216.Bernath et al. (1989) suggested N(C 5 ) might be ∼ 1/10 of N(C 3 ), yet a more recent observation provided a much smaller N(C 5 )/N(C 3 ) ratio of 1/68 (Hargreaves et al. 2014).Theoretical models also predict N(C 4 ) and N(C 5 ) to be one to two orders of magnitude smaller than N(C 3 ) (e.g.Terzieva & Herbst 1998;Schmidt et al. 2014), and the much smaller f j j values of the C 4 and C 5 optical bands make their detection even more difficult.These f j j values are also inconsistent in the literature.For example, Mühlhäuser et al. (2000) reported f j j ∼0.0001 for the C 4 3789 Å band, which is merely one-sixth of the value used in Table 6 (see also Jungen & Xu 2003;Pacchioni & Koutecký 1988).Under the best case scenario where N(C 4 ) and N(C 5 ) is ∼ 1/10 of N(C 3 ) and their transitions have the largest f j j values reported in the literature, the S/N of the most molecule-rich sightlines should be boosted by a factor of 2-10 to increase the chance for a successful C 4 or C 5 detection.

Summary
We conducted an extensive survey of C 2 and C 3 signatures in the EDIBLES dataset, using the C 2 A 1 Π u ← X 1 Σ + g (2-0) Phillips band and the C 3 Ã 1 Π u − X1 Σ + g 000-000 electronic origin band.Our data have high S/N and can resolve individual ro-vibronic transitions of the above bands, which allowed us to characterise the rotational excitation and obtain more accurate column densities.The results and conclusions we have reached are as follows: 1. We detect the C 2 (2-0) Phillips band in 51 velocity components along 40 sightlines; 48 velocity components have detections at the 3σ level or better, and three sightlines have tentative detections.The data were interpreted following the excitation model of van Dishoeck & Black (1982), and we were able to obtain information about the kinetic temperature and density in addition to the column density.2. We detect the C 3 Ã − X 000-000 band in 31 velocity components along 27 sightlines; 23 velocity components in 19 sightlines have detections at the 3σ level or better, and eight sightlines have tentative detections of C 3 .Due to the lack of a detailed excitation model, we assumed a double-temperature Boltzmann distribution and report the excitation temperatures and column density.3. The C 2 and C 3 transitions have consistent velocity offsets within their uncertainties, suggesting that the two species are detected in the same velocity components.We find a good correlation between N(C 2 ) and N(C 3 ), namely r = 0.93.The average N(C 2 )/N(C 3 ) ratio in our survey is 15.5±1.4,which is lower than a previously reported value of ∼ 40 (where individual C 3 transitions were not resolved) and slightly higher than the value of ∼10 reported by Schmidt et al. (2014).4. We confirm that the C 2 /C 3 detections occur in ζ-type sightlines characterised by large f H2 values and low W(5780)/W(5797) ratios.We compared the behaviour of DIBs in C 2 and non-C 2 sightlines.Unlike regular DIBs, the C 2 DIBs exhibit a similar behaviour in the two types of sightlines.They may trace denser regions in the ISM clouds than the regular DIBs, yet not as deep as where C 2 molecules are found.5. We searched for the optical bands of C 4 , C 5 , and the singly substituted isotopologue of C 2 ( 13 C 12 C) in sightlines with high C 2 /C 3 column densities.We identified the 13 C 12 C (2-0) Phillips band in a stacked spectrum via its Q(2), Q(3), and (tentatively) Q( 4) transitions.The estimated 12 C/ 13 C ratio is 79±8.For C 4 and C 5 , only upper limits to the column density can be derived, as in earlier unsuccessful surveys.
The high spectral resolution and high S/N offered by the ED-IBLES dataset thus result in accurate determinations of the physical parameters that describe the denser parts of diffuse clouds, and provides us with insight into the chemistry in these regions.  Σ − u − 3 Σ − g 3789 Å band (left) and the C 5 5109 Å band, possibly from the 1 Π u − X 1 Σ + g system (right).Their lab spectra are shown at the bottom as reference (red).The sightlines are selected by their measured N(C 3 ) and the S/N of the wavelength region, which is indicated by the dashed blue lines.We cannot find the target transition in either individual sightlines nor their stacked spectrum.Upper limits on N(C 4 ) and N(C 5 ) can be estimated from their most prominent feature, and are around 10 13 cm −2 and 10 12 cm −2 , respectively.

Fig. 1 .
Fig. 1.Spectral segments from three representative sightlines for the C 2 (A-X) Phillips (2-0) band.HD 147889 (top) has the highest C 2 column density in our data sample.HD 23016 (middle) shows a tentative detection, as the strongest Q(4) line is still below the 3σ detection limit.We considered multiple rotational lines simultaneously when deciding about possible identifications.The C 2 molecules are detected in two velocity components in HD 161056 (bottom).The combined model (red) is plotted against the spectral data, while synthetic spectra for the two velocity components (blue and orange) are plotted separately with an offset.We focus on the spectral segments around the C 2 transitions, and the wavelength grid is hence discrete.Rotational lines being considered in each sightline may differ due to telluric interference and/or data quality.These rotational lines are labelled with vertical bars.The best-fit parameters for the kinetic temperature, T kin , number density, n, and the column density of the velocity component, N, are listed in each panel.

Fig. 2 .
Fig. 2. Histograms of the kinetic temperatures (top) and densities (bottom) in C 2 -bearing clouds.In both panels, the red histogram represents the values determined in this study; the blue histogram also includes literature values (see TableC.1).The single data point with an error bar indicates the (unweighted) average of all values and the standard deviation.
f j j values are taken from Schmidt et al. (2014) unless otherwise specified.a Wavelengths adopted from Tanabashi et al. (2005) and f j j values based on theoretical calculations.b Perturbed transition.

Fig. 4 .
Fig. 4. Comparison between the C 2 and C 3 measurements.Each point represents a velocity component for which both molecules are detected.Upper panel: The velocity offsets of C 2 and C 3 (relative to the barycentric frame) are well aligned, suggesting the two molecules are detected in the same velocity component.Lower panel: N(C 2 ) and N(C 3 ) have a good correlation, with r = 0.93.The dashed blue line represents the best linear fit from orthogonal distance regression.The 1σ confidence interval of the fit is shown as the shadow.The red triangles represent the 3σ upper limits of N(C 3 ) in velocity components without C 3 detection.These upper limits are not considered in the linear regression.

Fig. 5 .
Fig. 5. Behaviour of non-C 2 DIBs λλ5780 and 6196 (left panels), and C 2 DIBs λλ4963 and 4984 (right panels) in C 2 and non-C 2 sightlines.The non-C 2 DIBs show the 'skin effect' and have reduced strengths in the C 2 sightlines.However, the behaviour of the C 2 DIBs is rather similar in the two types of sightlines.They may trace deeper regions of a cloud than the non-C 2 DIBs, but not as deep as the C 2 molecules.The DIB equivalent width data are gathered from Fan et al. (2017).

Table 5 .
Calculated line list of the 13 C 12 C (2-0) Phillips band.molecular constants of 12 C 2 (Chen et al. 2015) and isotopic relations.See the main text for more details.c f j j values of 12 C 2 from Sonnentrucker et al. (2007).

Fig. 6 .
Fig. 6.Search for 13 C 12 C (2-0) Phillips transitions.Top panel: Spectral segments of HD 63804, HD 147889, and HD 169454 around the target wavelength region, and simulated data (red) for 13 C 12 C with T exc = 55 K. From left to right, the four transitions are Q(1), Q(2), Q(3), and Q(4).Unfortunately, we cannot identify these features in any individual sightline.Bottom panel: Stacked (average) data of the three sightlines.We have a 3σ detection of the Q(2) and Q(3) transitions, and a marginal detection of the Q(4) transition.The estimated column density of 13 C 12 C is 2.30 ± 0.31 × 10 12 cm −2 in the stacked spectrum, corresponding to a 12 C/ 13 C ratio of 79±8.The dashed blue lines around the spectra represent 1σ uncertainties.

Fig. 7 .
Fig. 7. Spectral data around the C 43 Σ − u − 3 Σ − g 3789 Å band (left) and the C 5 5109 Å band, possibly from the 1 Π u − X 1 Σ + g system (right).Their lab spectra are shown at the bottom as reference (red).The sightlines are selected by their measured N(C 3 ) and the S/N of the wavelength region, which is indicated by the dashed blue lines.We cannot find the target transition in either individual sightlines nor their stacked spectrum.Upper limits on N(C 4 ) and N(C 5 ) can be estimated from their most prominent feature, and are around 10 13 cm −2 and 10 12 cm −2 , respectively.

Tkin
Fig. A.2. Same as Fig. A.1 but for HD 24398, HD 27778, HD 41117, and HD 45314.For HD 41117, synthetic spectra for individual velocity components are shown in orange and blue.

Fig. A. 3 .
Fig. A.3.Same as Fig. A.1 but for HD 54239, HD 61827, HD 63804, and HD 73882.For HD 61827, synthetic spectra for individual velocity components are shown in orange and blue.

TkinFig. A. 4 .
Fig. A.4. Same as Fig. A.1 but for HD 80558, HD 112272, HD 147084, and HD 147683.For HD 112272 and HD 147084, synthetic spectra for individual velocity components are shown in orange and blue.

FigFig
Fig. A.8. Same as Fig. A.1 but for HD 167971, HD 169454, HD 170740, and HD 170938.For HD 167971, synthetic spectra for individual velocity components are shown in orange, blue, and green.

Fig
Fig. A.10. Same as Fig. A.1 but for HD 185859, HD 186745, HD 186841, and HD 203532.For HD 186745 and HD HD186841, synthetic spectra for individual velocity components are shown in orange and blue.
aAir wavelength.b Not included in the measurements due to adjacent CCD artefact.

Table 3 .
Best-fit parameters of the C 3 models in 27 sightlines.
12cm −2 ) a Best-fit results are listed in separate entries for sightline harbouring multiple velocity components.

Table 6 .
Upper limits of N(C 4 ) and N(C 5 ).Assuming the most prominent feature has Gaussian profile and contributes to half of the total absorption.