© ESO, 2012

1. Introduction

The carbon-rich Mira-type star IRC +10 216 (CW Leo) is located at the tip of the asymptotic giant branch (AGB), where it loses mass at a high rate (~1−4 × 10^-5 M_⊙ yr^-1; Crosas & Menten 1997; Groenewegen et al. 1998; Cernicharo et al. 2000; De Beck et al. 2010). Located at a distance of 120−250 pc (Loup et al. 1993; Crosas & Menten 1997; Groenewegen et al. 1998; Cernicharo et al. 2000), it is the most nearby C-type AGB star. Additionaly, since its very dense circumstellar envelope (CSE) harbours a rich molecular chemistry, it has been deemed a prime carbon-rich AGB astrochemical laboratory. More than 70 molecular species have already been detected (e.g. Cernicharo et al. 2000; He et al. 2008; Tenenbaum et al. 2010), many of which are carbon chains, e.g. cyanopolyynes HC_nN (n = 1, 3, 5, 7, 9, 11) and C_nN (n = 1, 3, 5). Furthermore, several anions have been identified, e.g. C_nH⁻ (n = 4,6,8; Cernicharo et al. 2007; Remijan et al. 2007; Kawaguchi et al. 2007), and C₃N⁻ (Thaddeus et al. 2008), C₅N⁻ (Cernicharo et al. 2008), and CN⁻ (Agúndez et al. 2010). Detections towards IRC +10 216 of acetylenic chain radicals (C_nH), for n = 2 up to n = 8, have been reported by e.g. Guélin et al. (1978, C₄, Cernicharo et al. (1986a,b, 1987b, C₅, Cernicharo et al. (1987a) and Guélin et al. (1987, C₆, Guélin et al. (1997, C₇, and Cernicharo & Guélin (1996, C₈.

The smallest C_nH radical, ethynyl (CCH, or C₂H), was first detected by Tucker et al. (1974) in its N = 1−0 transition in the interstellar medium (ISM) and in the envelope around IRC +10 216. It was shown to be one of the most abundant ISM molecules. The formation of C₂H in the envelope of IRC +10 216 is attributed mainly to photodissociation of C₂H₂ (acetylene), one of the most prominent molecules in carbon-rich AGB stars. Fonfría et al. (2008) modelled C₂H₂ emission in the mid-infrared, which samples the dust-formation region in the inner CSE. The lack of a permanent dipole moment in the linearly symmetric C₂H₂-molecule implies the absence of pure rotational transitions that typically trace the outer parts of the envelope. C₂H, on the other hand, has prominent rotational lines that probe the chemical and physical conditions linked to C₂H₂ in these cold outer layers of the CSE. It has been established that C₂H emission arises from a shell of radicals situated at  ~15′′ from the central star (Guélin et al. 1993). Observations with the IRAM 30 m telescope and Herschel/HIFI show strong emission in several high-N rotational transitions of C₂H, something that is unexpected and challenges our understanding of this molecule in IRC +10 216.

We present and discuss the high-sensitivity, high-resolution data obtained with the instruments on board Herschel (Pilbratt et al. 2010): HIFI (Heterodyne Instrument for the Far Infrared; de Graauw et al. 2010), SPIRE (Spectral and Photometric Imaging Receiver; Griffin et al. 2010), and PACS (Photodetector Array Camera and Spectrometer; Poglitsch et al. 2010), and with the IRAM 30 m Telescope¹ in Sect. 2. The physical model for IRC +10 216’s circumstellar envelope is presented and discussed in Sect. 3. The C₂H molecule and our treatment of it is described in Sect. 4. A summary of our findings is provided in Sect. 5.

Fig. 1 C2H N = 1−0, 2−1, 3−2, 4−3, 6−5, 7−6, 8−7, and 9−8 rotational transitions in IRC +10 216’s envelope as observed with the IRAM 30 m telescope and Herschel/HIFI. The fine structure components are labelled with the respective J-transitions in the top of each panel, while the hyperfine components are indicated with vertical dotted lines. For the sake of clarity, we omitted the components that are too weak to be detected. See Sect. 4 for details on the spectroscopic structure of C2H. Additionally observed features related to other molecules are also identified.

Fig. 2 SPIRE data in the range 475−795 GHz, containing the C2H-lines detected with HIFI. The shaded strips in the panels indicate the spectral ranges of the HIFI data shown in the panels of Fig. 1. The labels mark the principal contributors to the main spectral features. A comparison of emission in the range 609−614 GHz, as obtained with SPIRE and with HIFI, is shown in the inset panel.

2. Observations

2.1. Herschel/HIFI

The HIFI data of IRC +10 216 presented in this paper are part of a spectral line survey carried out with HIFI’s wideband spectrometer (WBS; de Graauw et al. 2010) in May 2010, on three consecutive operational days (ODs) of the Herschel mission. The scan covers the frequency ranges 480−1250 GHz and 1410−1910 GHz, with a spectral resolution of 1.1 MHz. All spectra were measured in dual beam switch (DBS; de Graauw et al. 2010) mode with a 3′ chop throw. This technique allows one to correct for any off-source signals in the spectrum, and to obtain a stable baseline.

Since HIFI is a double-sideband (DSB) heterodyne instrument, the measured spectra contain lines pertaining to both the upper and the lower sideband. The observation and data-reduction strategies disentangle these sidebands with very high accuracy, producing a final single-sideband (SSB) spectrum without ripples, ghost features, or any other instrumental effects, covering the spectral ranges mentioned above.

The two orthogonal receivers of HIFI (horizontal H, and vertical V) were used simultaneously to acquire data for the whole spectral scan, except for band 2a². Since we do not aim to study polarisation of the emission, we averaged the spectra from both polarisations, reducing the noise in the final product. This approach is justified since no significant differences between the H and V spectra are seen for the lines under study.

A detailed description of the observations and of the data reduction of this large survey is given by Cernicharo et al. (2010b) and Cernicharo et al. (in prep.). All C₂H data in this paper are presented in the antenna temperature (${\mathit{T}}_{\mathrm{A}}^{\mathrm{\ast }}$) scale³. Roelfsema et al. (2012) describe the calibration of the instrument and mention uncertainties in the intensity of the order of 10%.

For the current study we concentrate on observations of CO and C₂H. The ten CO transitions covered in the survey range from J = 5−4 up to J = 11−10 and from J = 14−13 up to J = 16−15. The C₂H rotational transitions N = 6−5, 7−6, and 8−7 are covered by the surveys in bands 1a (480−560 GHz), 1b (560−640 GHz), and 2a (640−720 GHz), respectively. The N = 9−8 transition is detected in band 2b (720−800 GHz) of the survey, but with very low S/N; higher-N transitions were not detected. A summary of the presented HIFI data of C₂H is given in Table 1, the spectra are shown in Fig. 1.

2.2. Herschel/SPIRE

In the framework of the Herschel guaranteed time key programme “Mass loss of Evolved StarS” (MESS; Groenewegen et al. 2011) the SPIRE Fourier-transform spectrometer (FTS; Griffin et al. 2010) was used to obtain IRC +10 216’s spectrum on 19 November 2009 (OD 189). The SPIRE FTS measures the Fourier-transform of the source spectrum across two wavelength bands simultaneously. The short wavelength band (SSW) covers the range 194−313   μm, while the long wavelength band (SLW) covers the range 303−671   μm. The total spectrum covers the 446−1575 GHz frequency range, with a final spectral resolution of 2.1 GHz. The quality of the acquired data permits the detection of lines as weak as 1−2 Jy. For the technical background and a description of the data-reduction process we refer to the discussions by Cernicharo et al. (2010a) and Decin et al. (2010b). These authors report uncertainties on the SPIRE FTS absolute fluxes of the order of 15−20% for SSW data, 20−30% for SLW data below 500 μm, and up to 50% for SLW data beyond 500 μm. We refer to Table 2 for a summary of the presented data, and show an instructive comparison between the HIFI and SPIRE data of the C₂H lines N = 6−5 up to N = 9−8 in Fig. 2. Clearly, the lower resolution of the SPIRE spectrum causes many line blends in the spectra of AGB stars.

2.3. Herschel/PACS

Decin et al. (2010b) presented PACS data of IRC +10 216, also obtained in the framework of the MESS programme (Groenewegen et al. 2011). The full data set consists of spectral energy distribution (SED) scans in the wavelength range 52−220 μm, obtained at different spatial pointings on 12 November 2009 (OD 182). We re-reduced this data set, taking into account not only the central spaxels of the detector, but all spaxels containing a contribution to the flux, i.e. 20 out of 25 spaxels in total. The here presented data set therefore reflects the total flux emitted by the observed regions with the assumption that there is no loss between the spaxels. The estimated uncertainty on the line fluxes is of the order of 30%. Owing to instrumental effects, only the range 56−190 μm of the PACS spectrum is usable. This range holds ¹²CO transitions J = 14−13 up to J = 42−41, covering energy levels from  ~350 cm^-1 up to  ~3450 cm^-1. As is the case for the SPIRE data, line blends are present in the PACS spectrum due to the low spectral resolution of 0.08−0.7 GHz.

2.4. IRAM 30 m: line surveys

We combined the Herschel data with data obtained with the IRAM 30 m telescope at Pico Veleta. The C₂H N = 1−0 transition was observed by Kahane et al. (1988); N = 2−1 was observed by Cernicharo et al. (2000). N = 3−2 and N = 4−3 were observed between January and April 2010, using the EMIR receivers as described in detail by Kahane et al. (in prep.). The data-reduction process of these different data sets is described in detail in the listed papers. The observed C₂H transitions are shown in Fig. 1, and details on the observations are listed in Table 1.

3. The envelope model: dust and CO

We assumed a distance d = 150 pc to IRC +10 216, in good correspondence with literature values (Groenewegen et al. 1998; Men’shchikov et al. 2001; Schöier et al. 2007, and references therein). The second assumption in our models is that the effective temperature T_eff = 2330 K, following the model of a large set of mid-IR lines of C₂H₂ and HCN, presented by Fonfría et al. (2008). We determined the luminosity L_⋆, the dust-mass-loss rate Ṁ_dust, and the dust composition from a fit to the SED (Sect. 3.1). The kinetic temperature profile and the gas-mass-loss rate Ṁ were determined from a CO-line emission model (Sect. 3.2). The obtained envelope model will serve as the basis for the C₂H-modelling presented in Sect. 4. We point out that all modelling is performed in the radial dimension only, i.e. in 1D, assuming spherical symmetry throughout the CSE.

3.1. Dust

The dust modelling was performed using MCMax (Min et al. 2009), a Monte Carlo dust radiative transfer code. The best SED fit to the ISO SWS and LWS data, shown in Fig. 3, is based on a stellar luminosity L_⋆ = 11   300 L_⊙. This corresponds well with the results of Men’shchikov et al. (2001)⁴, considering the difference in adopted distance. The combination of L_⋆ and T_eff gives a stellar radius R_⋆ of 20 milli-arcsec, which agrees very well with the values reported by e.g. Ridgway & Keady (1988, 19 mas) and Monnier et al. (2000, 22 mas).

IRC +10 216 is a Mira-type pulsator, with a period of 649 days (Le Bertre 1992). Following Eq. (1) of Men’shchikov et al., we find that L_⋆ varies between L_{⋆,ϕ    =    0} ≈ 15   800   L_⊙ at maximum light (at phase ϕ = 0), and L_{⋆,ϕ    =    0.5} ≈ 6250 L_⊙ at minimum light (ϕ = 0.5). Figure 3 shows the SED-variability corresponding to the L_⋆-variability. Clearly, the spread on the photometric points can be accounted for by the models covering the full L_⋆-range.

The adopted dust composition is given in Table 3. The main constituents are amorphous carbon (aC), silicon carbide (SiC), and magnesium sulfide (MgS), with mass fractions of 53%, 25%, and 22%, respectively. The aC grains are assumed to follow a distribution of hollow spheres (DHS; Min et al. 2003), with size 0.01 μm and a filling factor of 0.8. The population of the SiC and MgS dust grains is represented by a continuous distribution of ellipsoids (CDE; Bohren & Huffman 1983), where the ellipsoids all have the volume of a sphere with radius a_d = 0.1 μm. The CDE and DHS are believed to give a more realistic approximation of the characteristics of circumstellar dust grains than a population of spherical grains (Mie-particles). Assuming DHS for the dominant aC grains was found to provide the best general shape of the SED. All dust species are assumed to be in thermal contact.

The absorption and emission between 7 μm and 10 μm and around 14 μm that is not fitted in our SED model can be explained by molecular bands of e.g. HCN and C₂H₂ (González-Alfonso & Cernicharo 1999; Cernicharo et al. 1999).

Based on an average of the specific densities of the dust components ρ_s = 2.41 g cm^-3, we find a dust mass-loss rate of 4.0 × 10^-8 M_⊙ yr^-1, with the dust density ρ_dust(r), dust temperature T_dust(r) and Q_ext/a_d, with Q_ext the total extinction efficiency, shown in Fig. 4. The inner radius of the dusty envelope, i.e. the dust condensation radius of the first dust species to be formed, is determined at R_inner = 2.7   R_⋆ by taking into account pressure-dependent condensation temperatures for the different dust species (Kama et al. 2009). These dust quantities are used as input for the gas radiative transfer, modelled in Sect. 3.2.

Fig. 3 SED models for IRC +10 216, using Ṁdust = 4.0 × 10-8 M⊙ yr-1, and the dust composition and properties listed in Table 3 and spatial distribution and properties shown in Fig. 4. The SED model is constrained by the ISO SWS and LWS data (full black); photometric points (black diamonds) are taken from Ramstedt et al. (2008) and Ladjal et al. (2010). Dash-dotted grey: model at maximum light, with 15 800 L⊙, dashed grey: best fit to the ISO data, at phase ϕ = 0.24, using 11 300 L⊙, dash-triple-dotted grey: model at minimum light, with 6250 L⊙. The inset is identical, but plotted in log −log  scale.

Fig. 4 Spatial distribution and properties of the dust for our best SED fit to the ISO data at ϕ = 0.24 in Fig. 3: a) dust density ρdust(r), b) dust temperature Tdust(r) , and c) Qext/ad. The dashed lines indicate the inner and outer radii of the dusty envelope.

3.2. CO

To constrain the gas kinetic temperature T_kin(r) and the gas density ρ_gas(r) throughout the envelope, we modelled the emission of 20 rotational transitions of ¹²CO (Sect. 3.2), measured with ground-based telescopes and Herschel/HIFI. The gas radiative transfer was treated with the non-local thermal equilibrium (NLTE) code GASTRoNOoM (Decin et al. 2006, 2010c). To ensure consistency between the gas and dust radiative transfer models, we combined MCMax and GASTRoNOoM by passing on the dust properties (e.g. density and opacities) from the model presented in Sect. 3.1 and Fig. 4 to the gas modelling. The general method behind this will be described in detail by Lombaert et al. (in prep.).

Fig. 5 Overview of the gas kinetic temperature and expansion velocity used in the radiative transfer model calculated with GASTRoNOoM. In the top panel, we indicate the exponents α from the Tkin(r) ∝ rα-power laws used to describe the kinetic temperature, and the radial ranges they apply to. See Sect. 3.2.

The large data set of high-spectral-resolution rotational transitions of CO consists of ten lines observed from the ground and ten lines observed with HIFI, which are listed in Table 4. Since the calibration of ground-based data is at times uncertain (Skinner et al. 1999), large data sets of lines that are observed simultaneously and with the same telescope and/or instrument are of great value. Also, the observational uncertainties of the HIFI data are significantly lower than those of the data obtained with ground-based telescopes (20−40%). Since the HIFI data of CO make up half of the available high-resolution lines in our sample, these will serve as the starting point for the gas modelling.

The adopted CO laboratory data are based on the work of Goorvitch & Chackerian (1994) and Winnewisser et al. (1997), and are summarised in the Cologne Database for Molecular Spectroscopy (CDMS; Müller et al. 2005). Rotational levels J = 0 up to J = 60 are taken into account for both the ground vibrational state and the first excited vibrational state. CO-H₂ collisional rates were adopted from Larsson et al. (2002).

To reproduce the CO lines within the observational uncertainties, we used L_⋆ = 11   300 L_⊙, and a temperature profile⁵ that combines three power laws: T_kin(r) ∝ r^-0.58 for r ≤ 9   R_⋆, T_kin(r) ∝ r^-0.4 for 10   R_⋆ ≤ r ≤ 65   R_⋆, and T_kin(r) ∝ r^-1.2 at larger radii, based on the work by Fonfría et al. (2008) and Decin et al. (2010a). The minimum temperature in the envelope is set to 5 K. The radial profiles of T_kin(r) and ν(r) are shown in Fig. 5. Using a fractional abundance CO/H₂ of 6 × 10^-4 in the inner wind, we find that a gas-mass-loss rate Ṁ of 1.5 × 10^-5 M_⊙ yr^-1 reproduces the CO lines very well. Combining the results from Sect. 3.1 with those from the CO model, we find a gas-to-dust-mass ratio of 375, in the range of typical values for AGB stars (10²−10³; e.g. Ramstedt et al. 2008).

The predicted CO line profiles are shown and compared to the observations in Fig. 6. All lines are reproduced very well in terms of integrated intensity, considering the respective observational uncertainties. The fraction of the predicted and observed velocity-integrated main-beam intensities I_MB,model/I_MB,data varies in the range 74−145% for the ground-based data set, and in the range 89−109% for the HIFI data set. The shapes of all observed lines are also well reproduced, with the exception of the IRAM lines (possibly due to excitation and/or resolution effects).

Fig. 6 Comparison of the observed 12CO emission lines (black histogram), and the line profiles predicted by the GASTRoNOoM-model (red dashed line) with parameters as listed in Table 5. The line transitions J − (J − 1) and the telescopes with which they were observed are indicated in the upper left corner of every panel. The factor IMB,model/IMB,data is given in the upper right corner of every panel.

Fig. 7 Comparison of CO-emission lines measured with PACS (black histogram) and the predicted line profiles (red dotted line). The transitions J − (J − 1) are labelled in the top left corner of each subpanel.

Figure 7 shows the comparison between the observed CO transitions J = 14−13 up to J = 42−41 in the PACS spectrum and the modelled line profiles, convolved to the PACS resolution. Considering the uncertainty on the absolute data calibration and the presence of line blends with e.g. HCN, C₂H₂, SiO, SiS, or H₂O in several of the shown spectral ranges (Decin et al. 2010b), the overall fit of the CO lines is very good, with exception of the lines J = 14−13,   15−14, and 16−15. This significant difference between the model predictions and the data could, to date, not be explained. Both the HIFI data and the higher-J PACS lines are reproduced very well by the model.

Fig. 8 Zoom on the blue wing of a selection of the CO line profiles, with the transitions J − (J − 1) labelled in the top left corner of each panel. We show the comparison between the observed lines and predicted lines with assumed values of νturb = 0.5 km s-1(full grey), 1.0 km s-1(dotted grey), the adopted value of 1.5 km s-1(dashed grey), and 2.0 km s-1(dash-dotted grey).

Skinner et al. (1999) noted the possible presence of a gradient in the turbulent velocity ν_turb, with decreasing values for increasing radial distance from the star. We find no clear evidence in our data set that this decrease in ν_turb is present. The model shown in Fig. 6 uses ν_turb = 1.5 km s^-1, and reproduces the line shapes and intensities to a high degree. In Fig. 8 we compare the predictions of a set of CO-lines using different values for ν_turb in order to assess the influence of this parameter.

Assuming a lower value of 0.5 km s^-1 or 1.0 km s^-1 yields an incomplete reproduction of the emission profile between −44 km s^-1 and −41 km s^-1 for all lines observed with HIFI. Assuming a higher value of 2 km s^-1 leads to the overprediction of the emission in this ν-range for all observed lines, with the strongest effect for the lowest-J lines. The influence of the different values of the turbulent velocity on the emission in the ν-range of −41 to −12 km s^-1 is negligible.

4. C₂H

4.1. Radical spectroscopy

C₂H ($\mathrm{•}\hspace{0.17em}\mathrm{C}\mathrm{C}\mathrm{H}$) is a linear molecule with an open shell configuration, i.e. it is a radical, with an electronic ${}^{\mathrm{2}}\mathrm{\Sigma }^{\mathrm{+}}$ ground-state configuration (Müller et al. 2000). Spin-orbit coupling causes fine structure (FS), while electron-nucleus interaction results in hyperfine structure (HFS). Therefore, to fully describe C₂H in its vibrational ground-state, we need the following coupling scheme: $\begin{array}{ccc}{J}& \mathrm{=}& {N}\mathrm{+}{S}\\ {F}& \mathrm{=}& {J}\mathrm{+}{I}\mathit{,}\end{array}$where N is the rotational angular momentum, not including the electron or quadrupolar angular momentum (S and I, respectively), J is the total rotational angular momentum, and F is the nuclear spin angular momentum.

The strong ΔJ = ΔN FS components have been detected up to N = 9−8. Relative to the strong components, the weaker ΔJ = 0 components decrease rapidly in intensity with increasing N, and they have been detected up to N = 4−3. The HFS has also been (partially) resolved for transitions up to N = 4−3.

Spectroscopic properties of the ground vibrational state of C₂H have most recently been determined by Padovani et al. (2009) and are the basis for the entry in the CDMS.

Our treatment of the radiative transfer (Sect. 4.3) does not deal with the FS and HFS of C₂H, and is limited to the prediction of rotational lines N → N′ with ΔN = 1, according to a ¹Σ approximation. Therefore, all levels treated are described with only one quantum number N. The final line profiles are calculated by splitting the total predicted intensity of the line over the different (hyper)fine components, depending on the relative strength of these components (CDMS), and preserving the total intensity. Note that, strictly speaking, this splitting is only valid under LTE conditions, but that, due to our approach of C₂H as a ¹Σ-molecule, we will apply this scheme throughout this paper.

C₂H has one bending mode ν₂, and two stretching modes, ν₁ and ν₃. The rotational ground-state of the bending mode (ν₂) is situated  ~530 K above the ground state. The $\mathrm{C}\mathrm{C}$ stretch (ν₃) at  ~2650 K is strong. The $\mathrm{C}\mathrm{H}$ stretch (ν₁) at  ~4700 K is weak, and is resonant with the first excited ²Π electronic state at  ~5750 K of C₂H. For each of these vibrational states – ground-state, ν₂, ν₃, and ν₁ – we consider 20 rotational levels, i.e. ranging from N = 0 up to N = 19.

The equilibrium dipole moments of the ground and first excited electronic states have been calculated as μ = 0.769 and 3.004 Debye, respectively (Woon 1995). The dipole moments of the fundamental vibrational states in the ground electronic state are also assumed to be 0.769 Debye. This is usually a good assumption as shown in the case of HCN by Deleon & Muenter (1984). The band dipole moments for rovibrational transitions were calculated from infrared intensities published by Tarroni & Carter (2004): μ(ν₂ = 1 → 0) = 0.110 Debye, μ(ν₃ = 1 → 0) = 0.178 Debye, and μ(ν₁ = 1 → 0) = 0.050 Debye. The influence of the vibrationally excited states on transitions in the vibrational ground state will be discussed in Sect. 4.3.

A more complete treatment of C₂H will likely have to take into account, e.g., overtones of the ν₂-state. These have non-negligible intrinsic strengths (Tarroni & Carter 2004) because of anharmonicity and vibronic coupling with the first excited electronic state. Spectroscopic data for several of these are already available in the CDMS. A multitude of high-lying states may also have to be considered because they have fairly high intrinsic intensities because of the vibronic interaction between the ground and the first excited electronic states.

4.2. Observational diagnostics

The C₂H-emission doublets due to the molecule’s fine structure (Sect. 4.1) are clearly present in all high-resolution spectra shown in Fig 1. For N = 1−0, 2−1, and 3−2, the hyperfine structure is clearly detected in our observations. The double-peaked line profiles are typical for spatially resolved optically thin emission (Olofsson, in Habing & Olofsson 2003), and indicate that the emitting material has reached the full expansion velocity⁶, i.e. ν_∞ = 14.5 km s^-1. This is in accordance with the observed position of the emitting shells (~16′′; Guélin et al. 1999) and the velocity profile derived for this envelope (Sect. 3.2).

To derive information on the regime in which the lines are excited, we constructed a rotational diagram (Schloerb et al. 1983), using $\frac{\mathit{N}}{\mathit{Z}\mathrm{\left(}{\mathit{T}}_{\mathrm{rot}}\mathrm{\right)}}\mathrm{\times }\exp \left(\mathrm{-}\frac{{\mathit{E}}_{\mathit{u}}}{\mathit{k}{\mathit{T}}_{\mathrm{rot}}}\right)\mathrm{=}\frac{\mathrm{3}\mathit{k}\mathrm{\times }{\mathrm{10}}^{\mathrm{36}}}{\mathrm{8}{\mathit{\pi }}^{\mathrm{3}}}\frac{{\mathit{I}}_{\nu \mathit{,}\mathrm{corr}}}{\mathit{\nu S}{\mathit{\mu }}^{\mathrm{2}}}\mathit{,}$(3)where N is the column density, Z the temperature dependent partition function, E_u the energy of the upper level of the transition in cm^-1, k the Boltzmann constant in units of cm^-1 K^-1, T_rot the rotational temperature in K, μ the dipole moment in Debye, and ν the frequency of the transitions in Hz. I_ν,corr is the velocity-integrated antenna temperature, ${\mathit{I}}_{\nu \mathit{,}\mathrm{corr}}\mathrm{=}\frac{\mathrm{1}}{{\mathit{f}}_{\mathrm{bff}}{\mathit{\eta }}_{\mathrm{MB}}}{\mathrm{\int }}_{{\nu }_{\mathrm{LSR}}\mathrm{-}{\nu }_{\mathrm{\infty }}}^{{\nu }_{\mathrm{LSR}}\mathrm{+}{\nu }_{\mathrm{\infty }}}{\mathit{T}}_{\mathrm{A}}^{\mathrm{\ast }}\mathrm{d}\nu$(4)corrected for the beam filling factor (Kramer 1997) ${\mathit{f}}_{\mathrm{bff}}\mathrm{=}\{\begin{array}{c}\\ \mathrm{1}\mathrm{-}\exp \left(\mathrm{-}{\mathrm{[}\sqrt{\ln \mathrm{2}}\mathrm{\times }{\mathit{\theta }}_{\mathrm{source}}\mathit{/}{\mathit{\theta }}_{\mathrm{beam}}\mathrm{]}}^{\mathrm{2}}\right)\mathit{,}& {\mathit{\theta }}_{\mathrm{source}}\mathit{<}{\mathit{\theta }}_{\mathrm{beam}}\\ \mathrm{1}\mathit{,}& {\mathit{\theta }}_{\mathrm{source}}\mathrm{\ge }{\mathit{\theta }}_{\mathrm{beam}}\end{array}$(5)and for the beam efficiency η_MB (see Table 1). We have assumed a uniform emission source with a size of 32′′ in diameter for all C₂H transitions, based on the interferometric observations of the N = 1−0 emission by Guélin et al. (1999).

From Fig. 9 we see that the C₂H emission is characterised by a unique rotational temperature T_rot = 23.3(± 1.3) K, and a source-averaged column density N = 3.84(± 0.09) × 10¹⁵ cm^-2. The unique rotational temperature points to excitation of the lines in one single regime, in accordance with the abundance peak in a confined area that can be characterised with one gas temperature (Guélin et al. 1999).

Fig. 9 Rotational diagram of the measured C2H transitions, based on Eq. (3). Transitions – in the 1Σ approximation – are labelled in grey. The numbers given in parentheses are the uncertainties on the derived column density and rotational temperature.

4.3. Radiative transfer modelling

4.3.1. Collisional rates

For rotational transitions within the vibrational ground state and within the ν₂ = 1 bending state and the ν₁ = 1 and ν₃ = 1 stretching states we adopted the recently published collisional rates of HCN − H₂ by Dumouchel et al. (2010). The lack of collisional rates for C₂H, and the similar molecular mass and size of HCN and C₂H inspire this assumption. The collisional rates are given for 25 different temperatures between 5 and 500 K, which covers the range of temperatures relevant for the C₂H around IRC +10 216. Since ν₃ and ν₁ are at  ~2650 K and  ~4700 K, respectively, collisional pumping to these vibrationally excited levels is unlikely for the low temperatures prevalent in the excitation region of C₂H, i.e. the radical shell at  ~16′′. Hence, for rovibrational transitions between the vibrationally excitated states (ν₂ = 1, ν₃ = 1, ν₁ = 1) and the vibrational ground state we adopted the same rates as for collisionally excited transitions within the vibrational ground state, but scaled down by a factor 10⁴, comparable to what has been done e.g. for H₂O by Deguchi & Nguyen-Q-Rieu (1990); Maercker et al. (2008) and Decin et al. (2010c). We did not consider collisionally excited transitions between the excited vibrational states.

We note that recent calculations by Dumouchel et al. (2010) have shown that the HNC collision rates at low temperatures differ by factors of a few from those of HCN. Since we may expect similar errors for C₂H, there is need for accurate C₂H − H₂ collisional rates as well. The gas density, therefore, is not constrained better than this factor in the models presented in Sects. 3.2 and 4.3.2.

Fig. 10 Fractional abundance of 12CO (black) and C2H (red) relative to H2: model including the density enhancements (full lines), and models without density enhancements, assuming the average interstellar UV field of Draine (1978) scaled by a factor 1 (dotted lines), 0.5 (dashed lines), and 0.25 (dash-dotted lines). See Sect. 4.3.2 for a discussion of this plot.

4.3.2. Constraining the C₂H abundance profile

The calculation of the C₂H/H₂ fractional abundance is based on the chemical model discussed by Agúndez et al. (2010), assuming the average interstellar UV field from Draine (1978), and a smooth envelope structure. The CO and C₂H abundances corresponding to the envelope model presented in Sect. 3.2 are shown in Fig. 10.

The Plateau de Bure Interferometer (PdBI) maps of Guélin et al. (1999) show that the 3mm-lines of the three radicals C₂H, C₄H, and C₆H have their brightness peaks at a radial angular distance of  ~16′′ from the central star. In Fig. 11a the contours of the C₂H map by Guélin et al. (1999) are overlaid on the normalised brightness distribution of the N = 1−0 transition predicted by a (one-dimensional) LTE model based on the abundance profile mentioned above. The extracted PdBI contours correspond to the velocity channel at the systemic velocity of IRC +10 216, representing the brightness distribution of the C₂H-transition in the plane of the sky. From Fig. 11a it is clear that this model leads to a brightness distribution that is concentrated too far inwards to agree with the shown PdBI contours.

Several authors have established that the mass-loss process of IRC +10 216 shows a complex time-dependent behaviour. The images of Mauron & Huggins (1999) and Leão et al. (2006) in dust-scattered stellar and ambient (optical) light, and the CO(J = 1−0) maps of Fong et al. (2003) show enhancements in the dust and the gas density, with quasi-periodic behaviour. Recently, Decin et al. (2011) showed the presence of enhancements out to  ~320′′ based on PACS-photometry, consistent with the images of Mauron & Huggins (1999).

Brown & Millar (2003) presented a chemical envelope model incorporating the dust density enhancements observed by Mauron & Huggins (1999). Cordiner & Millar (2009) added density enhancements in the gas at the positions of the dust density enhancements. They assumed complete dust-gas coupling, based on the work by Dinh-V-Trung & Lim (2008), who compared maps of the molecular shells of HC₃N and HC₅N with the images by Mauron & Huggins (1999). To mimic these density enhancements in our model and evaluate the impact on the excitation and emission distribution of C₂H, we used the approximation of Cordiner & Millar (2009), adding (to our “basic” model, presented in Sect. 3.2) ten shells of 2′′ width, with an intershell spacing of 12′′, starting at 14′′ from the central star. These shells – located at 14′′–16′′, 28′′–30′′, etc. – are assumed to have been formed at gas-mass-loss rates of a factor six times the rate in the regions of “normal density”, the intershell regions. We note that similar episodic mass loss was not added to the “basic” dust model presented in Sect. 3.1. The uncertainties on the data presented in Sect. 3.1 and the photometric variability of IRC +10 216 are too large to constrain the expected small effect of the inclusion of these enhancements.

These density enhancements are included in the chemical model of Agúndez et al. (2010) by combining two regimes. Firstly, a model is run in which the chemical composition of a parcel of gas is followed as it expands in the envelope, considering an augmented extinction (A_V) contribution from the density-enhanced shells located in the outer CSE. The density of the parcel is determined by the non-enhanced mass-loss rate (Ṁ). A second model is run in which the composition of a density-enhanced shell is followed as it expands. The final abundance profile follows the abundance from the first model for the non-enhanced regions and follows the abundance of the second model for the density-enhanced shells. The resulting abundance profiles for ¹²CO and C₂H are compared to those corresponding to a smooth model without enhancements in Fig. 10. The ¹²CO-abundance is affected by the inclusion of density enhancements only in the outer regions of the CSE. The impact on the radiative-transfer results for ¹²CO is negligible, with differences in modelled integrated intensities of at most 3% compared to the model from Sect. 3.2.

We find that the presence of density enhancements significantly alters the abundance profile of C₂H, as was also discussed by Cordiner & Millar (2009). The abundance peak at  ~500 R_⋆ in the smooth model is shifted to  ~800 R_⋆ in the enhanced model, as shown in Fig. 10. This shift indicates that the photochemistry, in this particular case the photodissociation of C₂H₂, is initiated at larger radii than in the smooth model. This is caused by the stronger extinction due to the density enhancements in the outer envelope. At the position of the two innermost density enhancements the fractional C₂H abundance is lower than in the intershell regions. The absolute abundance of C₂H in these regions, however, is higher by a factor 2−3 than in the neighbouring intershell regions. This is a combined effect of (1) the augmented shielding of C₂H₂ from incident interstellar UV radiation (which is no longer true for the outer shells), and (2) chemical effects such as faster reactions with e.g. C₂H₂ to form larger polyynes.

We note that the correspondence between the interferometric observations and the modelled number densities has significantly improved by including density enhancements (Fig. 11b). However, the peak intensity is located at somewhat too small radii compared to the observations. The adopted enhancements in the envelope are linked to Ṁ-values of a factor six times as high as for the intershell regions, corresponding to the values used in the model of Cordiner & Millar (2009). Increasing this factor or the number of shells in our model would increase the total mass of material shielding the C₂H₂ molecules from photodissociation by photons from the ambient UV field, and would hence shift the peak intensity of the predicted emission outwards. Furthermore, the presence of numerous dust arcs out to  ~320′′ is reported by Decin et al. (2011) and their effect will be included in future chemical models.

The “basic” model from Sects. 3.2 and 4.3 and the above introduced model with the density enhancements assume the interstellar UV field of Draine (1978). To assess the effect of a weaker UV field on the photodissociation of C₂H₂, and hence on the spatial extent of C₂H, we tested models assuming interstellar UV fields weaker by a factor of 2 and 4. The corresponding C₂H abundance profiles are shown in Fig. 10, and the modelled N = 1−0 brightness distribution for the latter case is shown in Fig. 11c. Although the correspondence in this figure is very good, the C₄H and C₆H abundances produced by this chemical model cannot account for the coinciding brightness peaks of C₂H, C₄H, and C₆H as reported by Guélin et al. (1999). In contrast, the density-enhanced model does reproduce this cospatial effect.

Fig. 11 Contours (in white) of the brightness distribution of the C2H N = 1−0 transition measured by Guélin et al. (1999) with PdBI, overlayed on the normalised predicted brightness distribution, assuming LTE conditions. The three panels represent the results obtained with three different chemical models. For the map in panel a), we assumed a constant mass loss, and the interstellar UV field of Draine (1978). The map in panel b) shows the brightness distribution when assuming density enhancements in the CSE as described by Cordiner & Millar (2009) and as discussed in Sect. 4.3.2. For panel c), we assumed again a constant mass loss, but an interstellar UV field that is only 25% as strong as that presented by Draine (1978).

4.3.3. Excitation analysis

Assuming the C₂H abundance obtained from the envelope model with density enhancements, we tested the influence of including the vibrational modes by modelling four cases: (1) including only the ground-state (GS) level; (2) including GS and ν₂ levels, (3) including GS, ν₂, and ν₃ levels, and (4) including GS, ν₂, ν₃, and ν₁ levels. An overview of the ratio of the predicted and the observed integrated intensities, I_Model/I_Data, for these four cases is shown in Fig. 12. The comparison of the line predictions for case (1) and case (4) under NLTE conditions is shown in Fig. 13.

Under LTE conditions, I_Model/I_Data for a given transition is the same for all four cases, indicating that the vibrationally excited states are not populated under the prevailing gas kinetic temperatures of  ~20 K in the radical shell. Under NLTE conditions, however, the predicted line intensities are very sensitive to the inclusion of the vibrationally excited states. In the two cases where the ν₃-state is included, the predicted intensity of the N = 1−0 transition is only 9% of the observed value, while transitions with N_up ≥ 4 are more easily excited. This is linked to the involved Einstein-A coefficients. For example, Table 7 gives the Einstein-A coefficients of the transitions involving the N = 2 level of the vibrationally excited states and the N = 3 and N = 1 levels of the ground state. The higher values of the Einstein-A coefficients for transitions to the N = 3 level lead to a more effective population of this level than of the N = 1 level. In particular, when including ν₃ and ν₁, this causes an underprediction of the intensity of the N = 1−0 ground state transition, while this effect is insignificant when only ν₂ is included. A visualisation of this scheme is shown in Fig. 14. This pumping mechanism is not limited to N = 3, but also affects higher levels, explaining the easier excitation of the transitions with N_up ≥ 4 mentioned before.

Fig. 12 Overview of the ratio IModel/IData of integrated intensities for the C2H lines in Fig. 1. The x-axis is labelled according to the upper N-level of the transition, Nup. Different 1Σ-approximations of the C2H molecule are plotted with different symbols: crosses (+ ) for the model including only the ground state (GS), asterisks (∗ ) for the model including GS and the ν2 state, diamonds (⋄ ) for the model including GS, ν2, and ν3, and triangles (△ ) for the model including the GS and all three vibrationally excited states ν2, ν3, and ν1. Black symbols connected with full lines, and red symbols connected with dotted lines represent NLTE and LTE models, respectively.

Fig. 13 Comparison of the measured C2H spectra (black histograms; see Sect. 2 and Fig. 1 for a description and for the identification of additional spectral features) and GASTRoNOoM model predictions under NLTE conditions for the case where (red) only the ground state is included, and (blue) the ground state, and the three vibrational modes are taken into account. These predictions are based on the “enhanced” abundance profile shown in Fig. 10 and the 1Σ-approximation of C2H. The transition N − (N − 1) and the ratio Imodel/Idata are stated in the upper right corner of each panel, according to the colour code of the plots.

4.3.4. Turbulent velocity

As for CO, we tested the influence of ν_turb on the predicted C₂H-emission, considering values ν_turb = 0.5, 1.0, 1.5, and 2.0 km s^-1. We find that the predicted line intensity increases⁷ with 25−55% when ν_turb is increased from 0.5 km s^-1 to 2.0 km s^-1. From the comparison of the predicted and observed shapes of the emission lines, we conclude that values of ν_turb in the range 0.5−1.5 km s^-1 give the best results. This is consistent with our discussion in Sect. 3.2 for the CO-lines, and with values generally used for AGB envelopes (Skinner et al. 1999).

4.4. Vibrationally excited states

The removal of a hydrogen atom from C₂H₂ through photodissociation causes the produced C₂H molecule to be bent, rather than linear (Mordaunt et al. 1998). This means that this formation route of C₂H favours the population of the ν₂  = 1 state over the population of the ground state. However, the Einstein-A coefficients for rovibrational transitions in the band ν₂ = 1 → 0 are in the range 0.01−1 s^-1, and by far exceed the photodissociation rates of C₂H₂, which are of the order of 10^-9 s^-1 (van Dishoeck et al. 2006). Hence, we did not take this ν₂-state population effect into account.

Recently, Tenenbaum et al. (2010) reported on the detection of the C₂H N = 3−2 transition in the ν₂  = 1 state, with the Arizona Radio Observatory Submillimeter Telescope (ARO-SMT). Observed peak intensities are of the order of  ~5−10 mK in antenna temperature, with reported noise levels around 3 mK, and a total velocity-integrated intensity of  ~0.2 K km s^-1. The ν₂ = 1,   N = 3 level is located at  ~560 K and is not populated under LTE conditions, given the gas-kinetic temperature in the radical shell (~20 K). Under NLTE conditions, however, the ν₂-levels are easily populated. The prediction of the ν₂ = 1,   N = 3−2 emission under the different ¹Σ-approximations is shown in Fig. 15. Under the ¹Σ-approximation, and assuming an HPBW of 29′′ at 261 GHz for ARO, we predict a U-shaped profile with peak antenna temperature  ~3 mK. Taking into account the observed fine structure, we hence predict an integrated intensity that is a factor  ~4 lower than the observed value. Considering the substantial uncertainties (the low signal-to-noise ratio of the ARO observations, and the limited C₂H approximation in our model), this agreement is satisfactory.

Fig. 14 Visualisation of the transitions responsible for the depopulation of the N = 1 level in the vibrational ground state (GS), where vibrational states are indicated on the horizontal axis, and rotational levels N on the vertical axis. The colour of the arrows representing the transitions is indicative of the magnitude of the Einstein-A coefficients of these transitions, as given by the colour bar.

Fig. 15 Predicted line profiles for the N = 3−2 transition in the ν2 = 1 vibrationally excited mode, under assumption of NLTE conditions, and different 1Σ-approximations, as it would be observed with the Arizona Radio Observatory Submillimeter Telescope.

5. Summary

We presented new data of CO and C₂H obtained with HIFI, PACS, SPIRE and the IRAM 30 m telescope. High-resolution spectra of CO transitions up to J = 16−15, and of C₂H transitions up to N = 9−8 were presented and modelled. The HIFI data of both CO and C₂H are the first high-frequency-resolution detections of these lines. They were obtained in the framework of a spectral survey of IRC +10 216 over the complete frequency range of the HIFI instrument (Cernicharo et al. 2010b and Cernicharo et al., in prep.).

From an SED fit to ISO data, PACS data, and a set of photometric points, covering the wavelength range 0.1−1000 μm, we obtained a dust-mass-loss rate of 4.0 × 10^-8   M_⊙ yr^-1 and a luminosity of 11300 L_⊙ at a distance of 150 pc. This luminosity value is valid at ϕ = 0.24, the phase at which the ISO data were obtained. The luminosity is then expected to vary between 6250 L_⊙ and 15 800 L_⊙ throughout the star’s pulsational cycle, which has a period  ~649 days.

In order to model IRC +10 216’s wind, we performed the radiative transfer of the dusty component of the CSE consistently with the gas-radiative transfer of CO. The set of 20 high-spectral-resolution CO lines was modelled to constrain the physical parameters of IRC +10 216’s CSE. The kinetic temperature throughout the envelope was described by previous results reported by Fonfría et al. (2008) and Decin et al. (2010a), and is now additionally constrained by the combination of all the high-resolution CO lines. The temperature profile is characterised by T_kin(r) ∝ r^-0.58 for r ≤ 9   R_⋆, T_kin(r) ∝ r^-0.4 for 10   R_⋆ ≤ r ≤ 65   R_⋆, and T_kin(r) ∝ r^-1.2 at larger radii, with an effective temperature T_eff = 2330 K. The derived mass-loss rate is 1.5 × 10^-5 M_⊙ yr^-1. This is consistent with earlier results (e.g. Cernicharo et al. 2000) and gives a gas-to-dust-mass ratio of 375, in line with typical values stated for AGB stars (e.g. Ramstedt et al. 2008). Furthermore, we showed a very good agreement between the predictions for CO lines up to J = 42−41 and the newly calibrated PACS spectrum of IRC +10 216. It is the first time that such a large coverage of rotational transitions of CO is modelled with this level of detail.

We extended our envelope model by including episodic mass loss, based on the model of Cordiner & Millar (2009). This assumption proved very useful in reconciling the modelled C₂H emission with the PdBI map of the N = 1−0 transition of C₂H published by Guélin et al. (1999), and in reproducing the observed line intensities. A decrease of a factor four in the strength of the interstellar UV field also leads to a satisfactory reproduction the PdBI map, but resulted in poorly modelled line intensities. The inclusion of density enhancements in IRC +10 216’s CSE is also supported by observational results based on maps of dust-scattered light (Mauron & Huggins 1999), molecular emisson (Fong et al. 2003), and photometric maps recently obtained with PACS (Decin et al. 2011).

The ground-based observations of C₂H transitions involve rotational levels up to N = 4 with energies up to  ~17 cm^-1, corresponding to temperatures  ~25 K. This temperature is close to the gas kinetic temperature at the position of the radical shell (~20 K). The recent detection of strong C₂H-emission involving levels with energies up to  ~150 K, however, calls for an efficient pumping mechanism to these higher levels. Owing to the spectroscopic complexity of C₂H, with the presence of fine structure and hyperfine structure, we approximated the molecule as a ¹Σ-molecule, exhibiting pure rotational lines, without splitting. We illustrated that the inclusion of the bending and stretching modes of C₂H is crucial in the model calculations, since high-energy levels are much more efficiently (radiatively) populated in this case. At this point, we have not yet included overtones of the vibrational states, nor did we treat the resonance between vibrational levels in the electronic ground state and the first electronically excited A²Π-state. Applying our simplified molecular treatment of C₂H, we can explain the strong intensities of the rotational lines in the vibrational ground state, except for the N = 1−0 transition. We are also able to account for the excitation of the recently observed rotational transition in the ν₂ = 1 state, showing the strength of our approach.

Acknowledgments

The authors wish to thank B. L. de Vries for calculating and providing dust opacities based on optical constants from the literature. E.D.B. acknowledges financial support from the Fund for Scientific Research - Flanders (FWO) under grant number G.0470.07. M.A is supported by a Marie Curie Intra-European Individual Fellowship within the European Community 7th Framework under grant agreement No. 235753. L.D. acknowledges financial support from the FWO. J.C. thanks the Spanish MICINN for funding support under grants AYA2006-14876, AYA2009-07304 and CSD2009-03004. HSPM is very grateful to the Bundesministerium für Bildung und Forschung (BMBF) for financial support aimed at maintaining the Cologne Database for Molecular Spectroscopy, CDMS. This support has been administered by the Deutsches Zentrum für Luft- und Raumfahrt (DLR). M.G. and P.R. acknowledge support from the Belgian Federal Science Policy Office via de PRODEX Programme of ESA. RSz and MSch ackowledge support from grant N203 581040 of National Science Center. The Herschel spacecraft was designed, built, tested, and launched under a contract to ESA managed by the Herschel/Planck Project team by an industrial consortium under the overall responsibility of the prime contractor Thales Alenia Space (Cannes), and including Astrium (Friedrichshafen) responsible for the payload module and for system testing at spacecraft level, Thales Alenia Space (Turin) responsible for the service module, and Astrium (Toulouse) responsible for the telescope, with in excess of a hundred subcontractors. HIFI has been designed and built by a consortium of institutes and university departments from across Europe, Canada and the United States under the leadership of SRON Netherlands Institute for Space Research, Groningen, The Netherlands and with major contributionÂąs from Germany, France and the US. Consortium members are: Canada: CSA, U. Waterloo; France: CESR, LAB, LERMA, IRAM; Germany: KOSMA, MPIfR, MPS; Ireland, NUI Maynooth; Italy: ASI, IFSI-INAF, Osservatorio Astrofisico di Arcetri-INAF; Netherlands: SRON, TUD; Poland: CAMK, CBK; Spain: Observatorio AstronÃşmico Nacional (IGN), Centro de Astrobiología (CSIC-INTA). Sweden: Chalmers University of Technology – MC2, RSS & GARD; Onsala Space Observatory; Swedish National Space Board, Stockholm University – Stockholm Observatory; Switzerland: ETH Zurich, FHNW; USA: Caltech, JPL, NHSC. SPIRE has been developed by a consortium of institutes led by Cardiff University (UK) and including Univ. Lethbridge (Canada); NAOC (China); CEA, LAM (France); IFSI, Univ. Padua (Italy); IAC (Spain); Stockholm Observatory (Sweden); Imperial College London, RAL, UCL-MSSL, UKATC, Univ. Sussex (UK); and Caltech, JPL, NHSC, Univ. Colorado (USA). This development has been supported by national funding agencies: CSA (Canada); NAOC (China); CEA, CNES, CNRS (France); ASI (Italy); MCINN (Spain); SNSB (Sweden); STFC (UK); and NASA (USA). PACS has been designed and built by a consortium of institutes and university departments from across Europe under the leadership of Principal Investigator Albrecht Poglitsch located at Max-Planck-Institute for Extraterrestrial Physics, Garching, Germany. Consortium members are: Austria: UVIE; Belgium: IMEC, KUL, CSL; France: CEA, OAMP; Germany: MPE, MPIA; Italy: IFSI, OAP/OAT, OAA/CAISMI, LENS, SISSA; Spain: IAC.

References

All Tables

All Figures

Fig. 1 C2H N = 1−0, 2−1, 3−2, 4−3, 6−5, 7−6, 8−7, and 9−8 rotational transitions in IRC +10 216’s envelope as observed with the IRAM 30 m telescope and Herschel/HIFI. The fine structure components are labelled with the respective J-transitions in the top of each panel, while the hyperfine components are indicated with vertical dotted lines. For the sake of clarity, we omitted the components that are too weak to be detected. See Sect. 4 for details on the spectroscopic structure of C2H. Additionally observed features related to other molecules are also identified.

In the text

	Fig. 2 SPIRE data in the range 475−795 GHz, containing the C2H-lines detected with HIFI. The shaded strips in the panels indicate the spectral ranges of the HIFI data shown in the panels of Fig. 1. The labels mark the principal contributors to the main spectral features. A comparison of emission in the range 609−614 GHz, as obtained with SPIRE and with HIFI, is shown in the inset panel.
In the text

Fig. 3 SED models for IRC +10 216, using Ṁdust = 4.0 × 10-8 M⊙ yr-1, and the dust composition and properties listed in Table 3 and spatial distribution and properties shown in Fig. 4. The SED model is constrained by the ISO SWS and LWS data (full black); photometric points (black diamonds) are taken from Ramstedt et al. (2008) and Ladjal et al. (2010). Dash-dotted grey: model at maximum light, with 15 800 L⊙, dashed grey: best fit to the ISO data, at phase ϕ = 0.24, using 11 300 L⊙, dash-triple-dotted grey: model at minimum light, with 6250 L⊙. The inset is identical, but plotted in log −log  scale.

In the text

	Fig. 4 Spatial distribution and properties of the dust for our best SED fit to the ISO data at ϕ = 0.24 in Fig. 3: a) dust density ρdust(r), b) dust temperature Tdust(r) , and c) Qext/ad. The dashed lines indicate the inner and outer radii of the dusty envelope.
In the text

	Fig. 5 Overview of the gas kinetic temperature and expansion velocity used in the radiative transfer model calculated with GASTRoNOoM. In the top panel, we indicate the exponents α from the Tkin(r) ∝ rα-power laws used to describe the kinetic temperature, and the radial ranges they apply to. See Sect. 3.2.
In the text

Fig. 6 Comparison of the observed 12CO emission lines (black histogram), and the line profiles predicted by the GASTRoNOoM-model (red dashed line) with parameters as listed in Table 5. The line transitions J − (J − 1) and the telescopes with which they were observed are indicated in the upper left corner of every panel. The factor IMB,model/IMB,data is given in the upper right corner of every panel.

In the text

	Fig. 7 Comparison of CO-emission lines measured with PACS (black histogram) and the predicted line profiles (red dotted line). The transitions J − (J − 1) are labelled in the top left corner of each subpanel.
In the text

	Fig. 8 Zoom on the blue wing of a selection of the CO line profiles, with the transitions J − (J − 1) labelled in the top left corner of each panel. We show the comparison between the observed lines and predicted lines with assumed values of νturb = 0.5 km s-1(full grey), 1.0 km s-1(dotted grey), the adopted value of 1.5 km s-1(dashed grey), and 2.0 km s-1(dash-dotted grey).
In the text

	Fig. 9 Rotational diagram of the measured C2H transitions, based on Eq. (3). Transitions – in the 1Σ approximation – are labelled in grey. The numbers given in parentheses are the uncertainties on the derived column density and rotational temperature.
In the text

	Fig. 10 Fractional abundance of 12CO (black) and C2H (red) relative to H2: model including the density enhancements (full lines), and models without density enhancements, assuming the average interstellar UV field of Draine (1978) scaled by a factor 1 (dotted lines), 0.5 (dashed lines), and 0.25 (dash-dotted lines). See Sect. 4.3.2 for a discussion of this plot.
In the text

Fig. 11 Contours (in white) of the brightness distribution of the C2H N = 1−0 transition measured by Guélin et al. (1999) with PdBI, overlayed on the normalised predicted brightness distribution, assuming LTE conditions. The three panels represent the results obtained with three different chemical models. For the map in panel a), we assumed a constant mass loss, and the interstellar UV field of Draine (1978). The map in panel b) shows the brightness distribution when assuming density enhancements in the CSE as described by Cordiner & Millar (2009) and as discussed in Sect. 4.3.2. For panel c), we assumed again a constant mass loss, but an interstellar UV field that is only 25% as strong as that presented by Draine (1978).

In the text

Fig. 12 Overview of the ratio IModel/IData of integrated intensities for the C2H lines in Fig. 1. The x-axis is labelled according to the upper N-level of the transition, Nup. Different 1Σ-approximations of the C2H molecule are plotted with different symbols: crosses (+ ) for the model including only the ground state (GS), asterisks (∗ ) for the model including GS and the ν2 state, diamonds (⋄ ) for the model including GS, ν2, and ν3, and triangles (△ ) for the model including the GS and all three vibrationally excited states ν2, ν3, and ν1. Black symbols connected with full lines, and red symbols connected with dotted lines represent NLTE and LTE models, respectively.

In the text

Fig. 13 Comparison of the measured C2H spectra (black histograms; see Sect. 2 and Fig. 1 for a description and for the identification of additional spectral features) and GASTRoNOoM model predictions under NLTE conditions for the case where (red) only the ground state is included, and (blue) the ground state, and the three vibrational modes are taken into account. These predictions are based on the “enhanced” abundance profile shown in Fig. 10 and the 1Σ-approximation of C2H. The transition N − (N − 1) and the ratio Imodel/Idata are stated in the upper right corner of each panel, according to the colour code of the plots.

In the text

Fig. 14 Visualisation of the transitions responsible for the depopulation of the N = 1 level in the vibrational ground state (GS), where vibrational states are indicated on the horizontal axis, and rotational levels N on the vertical axis. The colour of the arrows representing the transitions is indicative of the magnitude of the Einstein-A coefficients of these transitions, as given by the colour bar.

In the text

	Fig. 15 Predicted line profiles for the N = 3−2 transition in the ν2 = 1 vibrationally excited mode, under assumption of NLTE conditions, and different 1Σ-approximations, as it would be observed with the Arizona Radio Observatory Submillimeter Telescope.
In the text