Millimetre observations of the S-type AGB star $\chi$ Cygni: variability of the emission of the inner envelope

$\chi$ Cygni was observed with NOEMA using 12 antennas in the extended A- and B-configurations in February and March 2023, midway between minimum and maximum light (Figure 1) and in February 2024, closer to minimum light. Observations were using two individual frequency setups covering a total frequency range of $\sim$32 GHz in the 1.3 mm atmospheric window. The low frequency setup covers frequency ranges of 213.8 GHz to 221.6 GHz in LSB and 229.4 GHz to 237.2 GHz in USB and the high-frequency setup covers 242.8 GHz to 250.6 GHz in LSB and 258.4 GHz to 266.2 GHz in USB. The high-frequency setup is identical to the one previously used to observe RS Cnc (Winters et al., 2022) whereas the low-frequency setup was shifted upwards by 1 GHz. We used the new 250 kHz mode of the PolyFiX correlator to cover the full 32 GHz bandpass with a continuous spectral resolution of 250 kHz, corresponding to intrinsic velocity resolutions between 0.35 km s^-1 and 0.28 km s^-1. All data were finally re-binned to a spectral resolution of 0.5 km s^-1. Phase and amplitude calibration was done on the two quasars J1948+359 and 2013+370, which were observed every $\sim$20 min. Pointing and focus of the telescopes was checked about every hour, and corrected when necessary. The bandpass was calibrated on different strong quasars (3C279, 2013+370, 2200+420). The absolute flux scale was fixed on MWC349, whose flux is regularly monitored and calibrated on planets whose fluxes are known. The accuracy of the absolute flux calibration at 1.3 mm is estimated to be better than 15%. The data were calibrated and imaged within the GILDAS¹¹1https://www.iram.fr/IRAMFR/GILDAS suite of software packages using CLIC for the NOEMA data calibration and $uv$ table creation and the MAPPING package for merging, subsequent self-calibration and imaging of the data sets. Continuum data were extracted for each sideband of the two frequency setups individually by filtering out spectral lines and phase self-calibration was performed on the corresponding frequency averaged data. The gain tables containing the self-calibration solutions were then applied to the spectral line $uv$ tables using the SELFCAL procedures provided in MAPPING. The resulting data sets were imaged applying robust weighting with a threshold of 0.1 to increase the spatial resolution by typically about a factor 2. The resulting dirty maps were then CLEANed using the Högbom algorithm (Högbom, 1974). The final beam characteristics and sensitivities of the individual and combined data sets from A- and B-configuration are listed in Table 1.

The low frequency setup was observed in A- and B-configurations in 2023, each during 4 hours. The corresponding data have been merged, resulting in a maximum recoverable scale of $\sim$2 arcsec. The baselines ranged from 23 m to 1649 m. The high frequency setup was observed in the A-configuration during 4 hours in 2023 and in B-configuration during 2.6 hours in 2024, with baselines ranging from 65 to 1612 m. The associated $uv$ coverages are illustrated in Figure 2 and details are given in Table 2.

Evidence for major differences between the 2023 and 2024 observations of the emissions of the H¹²CN(3-2) line is illustrated in Figure 3. It prevented merging the associated A- and B-configuration data for this line. We comment on this in more detail in Sub-section 2.2 below.

We studied carefully the impact of the lack of short spacing by imaging disc sources of different sizes and different radial dependence of the brightness. We found that reliable imaging is obtained for angular distances $R$$<$1.5-2 arcsec for the merged A+B data, and $R$$<$0.6-0.8 arcsec for the A-configuration data.

In what follows, unless otherwise stated, we present line emission results obtained after subtraction of the contribution of continuum emission. Table 1 summarises some properties of strong line emissions. Some other clearly detected line emissions, not considered in the present article, include: SO(6₅-5₄), SO(7₆-6₅), H₂O at 232.68670 GHz, SiS$\nu$=0,12-11), SiS($\nu$=0,13-12), AlF($\nu$=0, $J$=7-6, $F$=15/2-17/2), TiO($\nu$=1,7-6), ²⁹Si¹⁷O($\nu$=0,6-5) and PN($N$=5-4,$J$=6-5). Information on these lines is given in Table 4 and their spectra are shown in Figure 19.

We use coordinates centred on the continuum emission (Figure 4), $x$ pointing east, $y$ pointing north and $z$ pointing away from Earth. Its position in 2023 is found to be the same for both low frequency and high frequency setups and each of the lower and upper sidebands: RA=19h 50min 33.87986s and Dec=32^o 54’ 49.6350”. The 2024 data have been re-centred to take into account the proper motion of $\chi$ Cygni between 2023 and 2024. The flux density of the continuum emission is measured to be 54 mJy, within $\pm$8 mJy depending on frequency. The angular resolution does not allow for resolving the stellar disc. We checked that the 2023 and 2024 measurements agree within 0.8 mJy, well within uncertainty.

The angular distance to the centre of the star is calculated as $R$$=$$\sqrt{x^{2}+y^{2}}$. Position angles, $\omega$, are measured counter-clockwise from north. The Doppler velocity, $V_{z}$, is relative to the adopted systemic velocity of 10 km s^-1 in the local standard of rest (LSR) frame.

Figure 3 compares the dependence on baseline length of the absolute values of the visibility for the emission of three molecular lines, Si¹⁶O(5-4), Si¹⁶O(6-5) and H¹²CN(3-2) for A- and B-configurations separately and for two Doppler velocity channels. The justification for such a choice will become apparent later (see Section 3). For the time being, we simply note the spectacular singularity of the H¹²CN(3-2) emission, giving evidence for a major variability between the 2023 and 2024 observations at Doppler velocities close to the systemic velocity of the star. This is in contrast to the Si¹⁶O(5-4) and Si¹⁶O(6-5) emissions, which show very similar distributions at short baselines, independently from the year of observation. However, we also note some smaller differences on these two lines, preventing us from ascertaining that their emission did not vary between 2023 and 2024. Similar small differences are also present for the other line emissions reported in the present article, however none being as spectacular as for H¹²CN(3-2) at $V_{z}$=0. The variability of the SiO($\nu$=1,$J$=6-5) line emission is studied in Sub-section 3.8 and is reliably understood as caused by a maser: its interpretation does not pose a particular problem. In contrast, the interpretation of the variability of the H¹²CN(3-2) line is challenging: while it is natural to think of masers as a possible cause, it is very unusual to observe such strong maser emission from a vibrational ground state, it only rarely occurred in the case of carbon-stars. We discuss this issue in Sub-sections 3.2 and 5.1.

The observations reported below, using the upgraded NOEMA array, probe the inner CSE of $\chi$ Cygni with a higher angular resolution than previously possible, revealing new details of its morpho-kinematics. In the present section, we limit the study to the interior of a circle of 1 arcsec radius centred on the star, within which we have established that reliable imaging has been achieved. In addition to the dominant CO and SiO molecular line emissions, we report the emissions of the HCN and CS lines, which, as being emitted by carbon-bearing molecules, are of specific interest to the study of an S-type star. As a general rule, unless specified otherwise, we limit the study to 2023 data using the A-configuration: as B-configuration data were obtained from two different observing periods, a possible variability makes their inclusion less straightforward. However, we have repeated the analysis presented in the present section using the merged A+B-configuration data and we have made sure that all our conclusions are robust in this respect. Altogether, we cover the following molecular lines: ¹²CO(2-1), ¹³CO(2-1), Si¹⁶O(5-4), Si¹⁶O(6-5), Si¹⁷O(6-5), H¹²CN(3-2), H¹³CN(3-2) and ¹²CS(5-4).

As will become apparent in what follows, the observations reveal different properties in the very close, unresolved neighbourhood of the star and farther away. We therefore distinguish between two complementary regions within the circle $R$$<$1 arcsec, S₁ and S₂ (Figure 5a). S₁ corresponds approximately to twice the average beam size, namely 0.5$\times$0.3 arcsec² at a position angle of 32^o. S₂ is its complement in the circle $R$$<$1 arcsec. Moreover, we distinguish between the central Doppler velocity interval, $|V_{z}|$$<$5 km s^-1, and larger red-shifted Doppler velocities, 5$<$$V_{z}$$<$12 km s^-1, for which the effect of absorption is expected to be smaller than in the blue-shifted hemisphereAs will become apparent in Section 3.2, the observations reveal different properties in the very close, unresolved neighbourhood of the star and farther away.

Table 3 lists, for each line, the intensities measured in regions S₁ and S₂ in each Doppler velocity interval separately. Figure 5b,c displays Doppler velocity spectra integrated over regions S₁ and S₂ separately and Figure 5d,e displays radial distributions integrated over the low and high Doppler velocity intervals separately. The left panel shows a clear absorption peak on the blue-shifted side, revealing progressive acceleration up to a terminal velocity of $\sim$8-10 km s^-1. In S₁, all line emissions peak near systemic velocity, the H¹²CN(3-2) line standing out as particularly narrow: its FWHM is only 4.0$\pm$0.2 km s^-1, meaning a $\sigma$ of $\sim$1.7$\pm$0.1 km s^-1 for a bin size of 0.5 km s^-1; we study this feature in Sub-section 3.2. In S₂, the ¹²CO(2-1) spectrum shows, in addition to the central peak, enhancements near terminal velocity, affected by absorption on the blue-shifted side, typical of an extended spherical CSE having reached terminal velocity at short distances from the star. A similar red-shifted enhancement is also visible on other lines; as they cover much shorter radial distances, its relative intensity is significantly smaller. In the higher Doppler velocity interval, Figure 5e shows that all line emissions have similar radial dependences: we study this $V_{z}$ interval in Sub-section 3.3. In the low $|V_{z}|$ interval, the radial distributions displayed in Figure 5d give again evidence for the H¹²CN line standing out as particularly narrow, possibly revealing the dominance of a gravitationally bound static layer. It may also suggest a similar trend for the Si¹⁷O(6-5) and the ¹³CO(2-1) lines, however with a much lower signal-to-noise ratio.

The observations reveal different properties in the very close, unresolved neighbourhood of the star (S₁) and farther away (S₂) The H¹²CN(3-2) intensity is nearly an order of magnitude larger than the SiO intensities in S₁ and close to them in S₂; relative to ¹²CO(2-1) emission, H¹³CN(3-2) emission is over three times larger in S₂ than in S₁. Comparing the relative values taken by the line emission intensities in the regions of the data-cube listed in Table 3 provides useful information on the relative abundances of the associated molecules and on the mechanism of excitation of the upper states. However, a general concern, abundantly discussed in the published literature, is the impact of absorption on the line emission of densely populated molecules, such as ¹²C¹⁶O and Si¹⁶O. We compare in Sub-section 3.4 the Si¹⁶O(5-4) and Si¹⁶O(6-5) emissions and in Sub-section 3.5 the emissions of the ¹²C and ¹³C isotopologues of the CO(2-1) and HCN(3-2) molecular lines and the emissions of the ¹⁶O and ¹⁷O isotopologues of the SiO(6-5) molecular line.

Using A-configuration data, maps of the line intensities integrated over the $|V_{z}|$$<$5 km s^-1 interval are displayed in Figure 6 for each line separately. In contrast to ¹²CO(2-1) emission, which was shown by Castro-Carrizo et al. (2010) to extend up to $\sim$40 arcsec and which we study in Section 4, the Si¹⁶O line emission was shown earlier by Lucas et al. (1992) and Ramstedt et al. (2009) to be confined to short distances from the star, the latter authors quoting a half-maximum radius of 1.6 arcsec. Similarly, Schöier et al. (2013) quote a half-maximum radius of 1.8$\pm$0.8 arcsec for H¹²CN emission. The confinement of SiO molecules is a general feature of M- and S-types AGB stars, commonly interpreted as the result of their condensation on dust grains and of photo-dissociation. We see from the intensity maps displayed in Figure 6 and from the spectra displayed in Figure 5b that all lines give evidence for the presence of a slowly expanding reservoir of gas surrounding the star and covering a radial range similar to, or smaller than that covered by Si¹⁶O emission.

As was shown in the preceding sub-section by the radial distributions displayed in Figure 5d, we observe an outstanding confinement of H¹²CN(3-2) emission in the immediate neighbourhood of the star. We further illustrate its singularity by displaying in Figure 7 a map and Doppler velocity distributions of the ratio of its emission to those of the Si¹⁶O(6-5), H¹³CN(3-2) and ¹²CS(5-4) lines, all using A-configuration data. Duari & Hatchell (2000) had previously inferred from the intensity of emission of high excitation transitions that H¹²CN molecules were formed within 200-300 mas from the centre of the star, but their single-dish observations could not reveal that the emission of low excitation lines would be confined to such short distances. The fact that all high-frequency emissions are simultaneously observed in the same sideband, implying in particular a common calibration, leaves little room for an explanation in terms of instrumental or data processing flaws. As neither H¹³CN(3-2) nor ¹²CS(5-4) emission displays such a spectacular confinement near the photosphere as H¹²CN(3-2) emission does, the singularity of the latter is not a straightforward result of HCN being a carbon-bearing molecule, enhanced by the large C/O ratio of the star. In summary, we can state with confidence that in the 2023 A-configuration data, the interval of low Doppler velocities, $|V_{z}|$$<$5 km s^-1, hosts an H¹²CN(3-2) emission component that has no significant H¹³CN(3-2) counterpart, has both a narrow $V_{z}$ width and a narrow spatial extension, and suggests the presence of an unresolved, gravitationally-bound, static layer.

In contrast, the H¹²CN(3-2) line emission observed in 2024 using B-configuration, as was already apparent from Figure 3, shows quite a different picture. Comparing the H¹²CN(3-2) emissions observed in S₁ and S₂ in the $|V_{z}|$$<$5 km s^-1 interval, we find that their ratio is $\sim$1.5 in 2023 and $\sim$0.2 in 2024 while their sum increases by only 7%. This suggests that the total emission has simply been smeared around the star between 2023 and 2024 but that the total emission has stayed approximately constant, as illustrated in the lower-left panel of Figure 3 by comparing Doppler velocity spectra observed in 2023 and 2024. The beam associated with the B-configuration 2024 observations covers region S₁, implying that some emission having its source within this region is detected in S₂, causing a decrease of the detected S₁/S₂ emission ratio. To tell apart this instrumental effect from the actual expansion of the emission, we smeared the radial distribution observed in 2023 using the A-configuration to predict what it would produce using the B-configuration. Comparing the result with the actual 2024 observation, we estimate that the radial extension of the H¹²CN(3-2) emission has increased by $\sim$0.1 arcsec between the two years. If such an increase were caused by an isotropic expansion of the gas volume, it would mean an expansion velocity at the scale of $\sim$80 km s^-1, conflicting with the observed narrow width of the line profile, $\sim$$\pm$2 km s^-1.

Such singular behaviour could not have been revealed by earlier single-dish observations. The ratio between the H¹²CN(3-2) and Si¹⁶O(5-4) line intensities measured by Bieging et al. (2000) in 1998 on $\chi$ Cygni using the Heinrich Hertz Submillimeter Telescope was 12.3/10.8=1.14, and these authors quote an average value of 1.2$\pm$0.5 for S stars. This result, being obtained with a single-dish telescope, covers a broad angular range, at variance with the present observations, making the comparison difficult. The H¹²CN(3-2)/Si¹⁶O(5-4) emission ratios that we measure over the whole Doppler velocity interval, $|V_{z}|$$<$20 km s^-1, vary between 1.78 and 1.55, the larger value corresponding to a circle $R$$<$1 arcsec in the A-configuration and the lower value to a circle $R$$<$2 arcsec in the B-configuration (namely in 2023 for SiO and in 2024 for HCN). These are, in all cases, well above the 1998 value of 1.14 measured by Bieging et al. Taken at face value, this suggests that the singular component of HCN emission that we observe is absent from the 1998 observation and changed in morphology, but not in total intensity, between 2023 and 2024. Indeed, the lower row of Figure 3 shows that in the high velocity channel, $V_{z}$$=$7 km s^-1, the H¹²CN(3-2) and Si¹⁶O(5-4) emissions are very close from each other, but in the low velocity channel, $V_{z}$$=$0, the two lines display very different distributions. This may suggest that H¹²CN(3-2) and Si¹⁶O(5-4) emissions would have a similar morphology if one could ignore the enhanced H¹²CN(3-2) emission confined to low Doppler velocities. It raises the question of the evolution of the situation in the years to come: will the enhanced H¹²CN(3-2) emission disappear, will it stay at its 2024 level, will it evolve to yet an other morphology? To answer such a question, new A-configuration observations need to be made in 2025; B-configuration observations would improve the $uv$ coverage but they should be made close in time to allow for reliable merging. In the present situation, we are left with a major question that we are unable to answer with sufficient confidence: does the confined H¹²CN(3-2) enhanced emission observed in 2023-2024 reveal the presence close to the photosphere of an enhanced molecular abundance or simply an enhanced excitation of the line, due to masing and/or shock dynamics?

Intensity maps integrated over the high Doppler velocity interval, 5$<$$V_{z}$$<$12 km s^-1, are displayed in Figure 8. With the exception of the ¹³CO(2-1) line emission, which is in the noise, all other line emissions are enhanced in the north-western quadrant, including the H¹²CN(3-2) line emission which also hosts the high Doppler velocity tail of the narrow component discussed in the preceding sub-section. In order to better illustrate the morpho-kinematics at stake in the north-western quadrant, 270^∘$<$$\omega$$<$360^∘, we display in Figure 9, for each line emission separately, $V_{z}$ vs $R$ maps. They show patterns reminiscent of that observed in EP Aquarii by Nhung et al. (2024), suggesting an interpretation in terms of a localised and episodic mass ejection, expected to be associated with shock waves near maximal pulsation amplitude, and having enhanced intensity and radial velocity over a particular convective cell. In such a case, the shock waves push away the gas to larger distances from the star with radial velocities larger than the normal expansion velocity: a depression of emission develops in their wake, the size of which increases with time. The enhancement of emission seen in the high Doppler velocity interval, 5$<$$V_{z}$$<$12 km s^-1, is associated with the gas having been pushed outward and a depression of emission is seen at lower values of $V_{z}$. However, in the present case, the situation is not as clear as it was in the case of EP Aqr because the data are of lesser sensitivity and lower angular resolution. The present data, by themselves, while being consistent with such a scenario, cannot exclude other possible interpretations. Higher values of the angular resolution, of the sensitivity and of the maximal recoverable scale would allow for more accuracy and better confidence in the reliability of this interpretation. If its validity were confirmed, it would suggest that the resulting mass ejection reaches between 1 and 2 arcsec from the centre of the star, implying that it occurred 5 to 10 decades ago over the red-shifted north-western quadrant with a radial velocity having reached $\sim$12 km s^-1. The $V_{z}$ vs $R$ maps displayed in Figure 9 show that the enhancement of mass ejection is not limited to the red-shifted hemisphere but also extends over the blue-shifted hemisphere where, however, the impact of absorption is stronger. This is consistent with the suggested interpretation, the pulsation shock waves covering the whole solid-angle and being simply enhanced over particular convective cells. Radiation pressure on dust formed in the gas originally compressed by the shock wave would accelerate the gas and thereby further increase the density in the affected region (e.g. Freytag & Höfner, 2023).

Exploring the light curve over the past 50 years reveals important variations of the pulsation amplitude but does not reveal obvious singular episodes of exceptionally large ones. The magnitude reached at maximal light varies between 3 and 6, spanning a factor $\sim$16 in intensity, and has a mean value of $\sim$4.5 and an rms value of $\sim$0.5 (a factor $\pm$1.6) with an approximately Gaussian distribution. The highest maximum occurred in 2006. Similar remarks can be drawn when exploring the light curve over the past century (McAdam & Greaves, 2000). Probably worth mentioning is the presence of a small wiggle on the light curve at the time of the 2023 observations (Figure 1). Such a wiggle is visible at mid-rise of each pulsation, however taking very variable forms; such is the case, in particular, after 1975, when Hinkle et al. (1982) claim evidence for the sudden formation and subsequent progressive decrease of an 800 K layer within some 200-300 mas away from the centre of the star. Similar features were first reported by Le Bertre (1992), particularly clearly on C-star AFGL 1085, and commented upon by Winters et al. (1994). They have now been observed on other S- and C-stars, but are not well understood (Merchan-Benitez et al., 2023).

In the LTE approximation, the unabsorbed emissivity $\varepsilon$ is proportional to $A_{ji}(2J+1)$e${}^{-Eup/T}/T$, where $A_{ji}$ is the Einstein coefficient, $J$ the angular momentum of the upper state, $E_{up}$ its energy and $T$ the temperature. The ratio between Si¹⁶O(5-4) and Si¹⁶O(6-5) emissions, when neglecting opacity (more precisely when neglecting the difference between the effects of opacity on each of the two line emissions) and assuming level populations predicted by LTE, is therefore expected to be 0.48e^12.5/T where 0.48 is the ratio of the Einstein coefficients multiplied by (2$J$+1) and 12.5 K the difference between the values of $E_{up}$ (see Table 1). The measured values listed in Table 3, using 2023 A-configuration data, give mean emission ratios of 0.86 in S₁ and 0.92 in S₂, corresponding to temperatures of 19 and 21 K. Using instead A+B-configuration data, which however mixes 2023 and 2024 observations, these values become 0.76 in S₁ and 0.68 in S₂, corresponding to temperatures of 27 and 36 K. Such temperatures are lower than expected: according to Justtanont et al. (2010), they are only reached at distances exceeding 700 to 1000 au from the centre of the star. Assuming that the line emissions probe distances at arcsec scale, the temperature should exceed 100 K, implying that the emission ratio should be about half the measured value. In the same approximation, the ratio of $\varepsilon$ to the optical depth $\tau$ is proportional to $f^{3}$/(e${}^{\Delta{E}/T}-1$), where $f$ is the frequency of the transition and $\Delta{E}$ its energy, proportional to $f$. For $T$$>>$$\Delta{E}$, as the Einstein coefficients are approximately proportional to $f^{3}$, we expect the optical depth to be proportional to $f$e${}^{-Eu/T}/T^{2}$. Both $f$ and $T$ (the temperature probed by the transition) are larger for Si¹⁶O(6-5) than for Si¹⁶O(5-4) and their effects partly compensate each other; but, if absorption were much stronger for Si¹⁶O(6-5) than for Si¹⁶O(5-4), the observed discrepancy could be the simple result of a difference of opacity between the two lines.

In order to get an independent evaluation of the emission ratio, we analysed archived ALMA/ACA observations of the SiO(5-4) and SiO(6-5) line emissions of $\chi$ Cygni from Project 2021.2.00147.S (PI M. Saberi). The Doppler velocity spectra are displayed in Figure 10. The ACA measurement covers the whole emission while the NOEMA measurement is over $R$¡1.5 arcsec only. As the extension of the SiO(5-4) emission is larger than that of the SiO(6-5) emission, the SiO(5-4)/SiO(6-5) emission ratio is expected to be smaller when measured by NOEMA than when measured by ACA. Indeed, it is $\sim$0.7 instead of 0.86, probably consistent within uncertainties.

Such large emission ratios have been measured earlier for other stars, such as, for example, 0.77 for R Dor (De Beck & Olofsson, 2018). The importance of accounting properly for radiative transfer when interpreting line emission ratios has been underscored by several authors, such as Bieging et al. (2000) or Ramstedt et al. (2014). Over 30 years ago, Bujarrabal & Alacolea (1991) had already remarked, from their IRAM 30 m single dish observations of Si¹⁶O lines from $\chi$ Cygni, that emission ratios such as Si¹⁶O(2-1)/Si¹⁶O(3-2) deviated significantly from LTE expectation and they were blaming it on the large opacity of the Si¹⁶O lines. However, Lucas et al. (1992), observing that the half-power radius of $\chi$ Cygni stays between 0.6 and 0.7 arcsec across the whole SiO(2-1) line profile, argued that such behaviour could not be accounted for by standard dynamics, even in the presence of large opacities. A situation very similar to the present case of $\chi$ Cygni was studied by Winters et al. (2022) in the case of RS Cnc, an MS-type, post-third-dredge-up semi-regular Long Period Variable. In that case, the Si¹⁶O(5-4)/Si¹⁶O(6-5) emission ratio, averaged over a projected distance from the centre of the star of $\sim$1 arcsec, is $\sim$0.65, closer to the LTE value than for $\chi$ Cygni, but still implying temperatures of $\sim$40-50 K, lower than expected. The authors attempted an LTE description, blaming the low temperature on a large optical depth, causing the line emission to explore preferentially the external layers of the CSE.

In such a context, it is instructive to obtain an estimate of the opacity of the SiO lines, which we do by producing a simple LTE model of the morpho-kinematics and integrating the radiative transfer equation over a spherical CSE. We find that standard radial distributions of the velocity, molecular abundance and temperature give good fits to the data. Precisely we use as reference parameters (Justtanont et al., 2010; Van de Sande et al., 2018) a mass-loss rate of 2.4 M_⊙ yr^-1, a terminal velocity of 8 km s^-1, an SiO/H abundance ratio of 10^-5, meaning a density of 1.5 SiO/cm³ at a distance of 165 au from the centre of the star, multiplied by a Gaussian having a $e$-folding radius of 215 au, a radial velocity increasing from 2 km s^-1 at 17 au to 2+6(1$-$0.1/$r$)^0.8 at a distance $r$, a radial dependence of the temperature of the form $T$(K)=100(210/$r$[au])^0.78. However, when varying these within reasonable limits, we find that the emission ratio SiO(5-4)/SiO(6-5) remains between 0.5 and 0.6: absorption, even when strong, does not much affect the emission ratio and cannot explain the large values that are observed. A systematic study of such emission ratios, including observations of other lines, would help, not only for $\chi$ Cygni but also for RS Cnc, R Dor and other AGB stars, to better tell apart the respective contributions of opacity and of level population and to understand what precisely is causing them to deviate from the LTE regime. In particular, non-LTE models have been mostly used to simulate single-dish observations and a detailed comparison with currently available interferometer data would help with telling apart different contributions; this is, however, beyond the scope of the present article.

The availability of observations of CO(2-1) and HCN(3-2) line emissions for both ¹²C and ¹³C isotopologues should provide an estimate of the value taken by the ¹²C/¹³C abundance ratio. Indeed, as both the Einstein coefficients and upper state energies take very similar values for both isotopes, we might expect the emission ratio to be a good approximation of the abundance ratio: the differences between the upper state energies are less than 1 K and the ratios of the Einstein coefficients are 1.14 for CO and 1.09 for HCN. Similarly, the Si¹⁶O(6-5)/Si¹⁷O(6-5) emission ratio, with a ratio of Einstein coefficients of 1.12 and a difference of upper states energies of only 1.7 K, should provide an estimate of the ¹⁶O/¹⁷O abundance ratio; and the ²⁸SiO(5-4)/²⁹SiO(5-4) emission ratio, with a ratio of Einstein coefficients of 1.04 and a difference of upper states energies of only 0.4 K, should provide an estimate of the ²⁸Si/²⁹Si abundance ratio. However, opacity is expected to lower the measured emission of the ¹²C, ¹⁶O and ²⁸Si isotopologues, namely to decrease the measured values of the ¹²C/¹³C, ¹⁶O/¹⁷O and ²⁸Si/²⁹Si abundance ratios compared to their true values. In contrast with the case discussed in the preceding sub-section, we now expect a significant impact of the effect of opacity as the optical depth is larger for the ¹²C, ¹⁶O and ²⁸Si than for the ¹³C, ¹⁷O and ²⁹Si isotopologues. This has been discussed in the published literature in the case of the CO(1-0), CO(2-1) and CO(3-2) line emissions from $\chi$ Cygni observed using single-dish telescopes (De Beck et al., 2010; Wallerstein et al., 2011; Ramstedt et al., 2014) with a ¹²C/¹³C abundance ratio between 33 and 40, about twice as large as the measured line emission ratio.

For each of the HCN(3-2), CO(2-1), SiO(6-5) and SiO(5-4) line emissions, Figure 11 displays the dependence on $R$ of the relevant line emission ratios in the low Doppler velocity interval (in the case of the SiO(5-4) line emission, we show A+B merged data as the line was observed exclusively in 2023). They share a common feature: close to the star, in the low Doppler velocity interval, the emission ratios take different values than elsewhere. In the case of the HCN line, it is not surprising, it was already clear from the analysis presented in Sub-section 3.2 and in particular in the right panel of Figure 7. However, in the case of the CO and SiO lines, it is unexpected. In particular, in both the CO(2-1) and SiO(6-5) cases the emission ratios are abruptly decreasing below $R$$\sim$0.3 arcsec and it is tempting to blame high opacities; but we see from Figure 5 that the decrease of the ratios close to the star is in part the result of an increase of the rare isotope emissions, rather than a decrease of the ¹²C and ¹⁶O isotopologue emissions as high opacities produces. In contrast with elsewhere, both ¹³CO(2-1) and Si¹⁷O(6-5) emission ratios are observed with a reasonable signal to noise ratio in the low Doppler velocity intervals of region S₁ where ¹²CO(2-1)/¹³CO(2-1)=12$\pm$2 and Si¹⁶O(6-5)/Si¹⁷O(6-5)=9$\pm$1 using data of the A+B-configuration, both lower than expected. Using data of the A-configuration only does not make much difference: the Si¹⁶O(6-5)/Si¹⁷O(6-5) ratio becomes 7$\pm$1. In the case of the ²⁸SiO(5-4)/²⁹SiO(5-4) ratio, the decrease close to the star is more progressive: its value is $\sim$1.9$\pm$0.3 in region S₁ and increases to $\sim$3$\pm$0.5 elsewhere within $R$$<$1 arcsec. In the case of HCN(3-2) emission it is instead in the high Doppler velocity interval that the rare isotopologue is observed with a reasonable signal-to-noise ratio, giving a surprisingly low emission ratio H¹²CN(3-2)/H¹³CN(3-2)=5.0$\pm$0.5 using 2023 data of the A-configuration.

As already mentioned, single-dish measurements of the ¹²C/¹³C emission ratio are available for CO only and are at least half the associated abundance ratio of 33-40, significantly larger than the values reported here for the CO emission ratio in region S₁ at low Doppler velocities and of the HCN emission ratio in region S₁+S₂ at large Doppler velocities. Very low abundance ratios are typical of J-type carbon-stars, such as Y CVn or RY Dra with values of 2 and 2.5, respectively (Ramstedt et al., 2014). Known processes that may produce such low ratios are either of nuclear or chemical origin. Nuclear processes can be of the “cool bottom processing” type (Boothroyd & Sackmann, 1999), particularly efficient in low mass Red Giants; in regions where the CNO cycle is operative, ¹³C can be formed by the decay of the $\beta$-unstable isotope ¹³N produced in the reaction ¹²C(p,$\gamma$)¹³N, and then dredged-up to the stellar surface (Sengupta et al., 2013). A typical example of a chemical process is the fractionation of CO molecules by interaction with carbon ions, ¹³C⁺+¹²CO$\rightarrow$¹²C⁺+¹³CO (Mamon et al., 1988).

In the case of the ¹⁶O/¹⁷O abundance ratio, using observations of CO near-infrared overtone band transitions, Hinkle et al. (2016) measured ¹⁶O/¹⁷O=328$\pm$100 for $\chi$ Cygni, Smith & Lambert (1990) ¹⁶O/¹⁷O$\sim$710 for the MS-type star RS Cnc, Harris et al. (1985) 625$\leq$¹⁶O/¹⁷O$\leq$3000 for a sample of five MS- and three S-type stars; from the emission ratio of CO(1-0) and CO(2-1) lines, Kahane et al. (1992) measured 242$\leq$¹⁶O/¹⁷O$\leq$840 for a sample of five carbon-rich envelopes. However, De Beck & Olofsson (2018) measured an emission ratio Si¹⁶O(6-5)/Si¹⁷O(6-5) of only 66 for the CSE of R Dor and Winters et al. (2022) measured only $\sim$50 for the CSE of RS Cnc, remarking that the larger optical depth of the Si¹⁶O(6-5) line causes this ratio to be smaller than the ¹⁶O/¹⁷O abundance ratio.

Finally, in the case of the ²⁸Si/²⁹Si abundance ratio Schöier et al. (2011) measure 8$\pm$2 using Herschel/HIFI .

In summary, we can state with confidence that in the regions where the rare isotopologue emissions are measured with a reasonable signal-to-noise ratio, we observe unexpectedly low emission ratios: Si¹⁶O(6-5)/Si¹⁷O(6-5)$\sim$7-9, ¹²CO(2-1)/¹³CO(2-1)$\sim$13 and ²⁸SiO(5-4)/²⁹SiO(5-4)$\sim$2 in the low Doppler velocity interval of region S₁ and H¹²CN(3-2)/H¹³CN(3-2)$\sim$5 in the high Doppler velocity interval within $R$$<$1 arcsec. Moreover we observe a decrease of the Si¹⁶O(6-5)/Si¹⁷O(6-5) and ¹²CO(2-1)/¹³CO(2-1) emission ratios close to the star, caused in part by an increase of the rare isotopologue emissions. In other regions, the lower values of the signal-to-noise ratios prevent a reliable analysis but estimates using B-configuration or merged A+B data suggest that the isotopic emission ratios are consistent, within large uncertainties, with the ratios measured using single-dish telescopes.

The presence of high Doppler velocity wings confined within a few stellar radii from the centre of the star is a common feature of millimetre observations of M- and S-types AGB stars (Darriulat et al., 2024), sometimes very clear, as for example in $o$ Cet (Nhung et al., 2022), EP Aqr (Nhung et al., 2024) or RS Cnc (Winters et al., 2022), sometimes much less as in the cases of L₂ Pup (Hoai et al., 2022a; Homan et al., 2017). The observed effective line broadening is commonly understood as revealing the simultaneous presence of out-flowing and in-falling gas in the inner layers of the CSE where shocks associated with pulsation and convection are dominant. However, a detailed description is still lacking, in particular concerning the dependence on stellar phase and the different time scales of variability.

In the present case of $\chi$ Cygni, the presence of a high Doppler velocity mass ejection in the north-western quadrant complicates the analysis. Figure 12 displays Doppler velocity spectra integrated over three intervals of position angles and four intervals of $R$ for three of the line emissions discussed in the preceding sections. As CO(2-1) emission shows no significant line broadening, consistent with the dominance of large radial distances explored by this line, it is not displayed in the figure. On the blue-shifted side, where absorption complicates the analysis, H¹²CN(3-2) shows no clear line broadening while Si¹⁶O(6-5) and Si¹⁶O(5-4) show some with $|V_{z}|$ increasing from 12-13 to 18-20 km s^-1, confined within 0.4 arcsec. On the red-shifted side, H¹²CN(3-2) is completely dominated by the mass ejection, which is also strong in Si¹⁶O(5-4) and Si¹⁶O(6-5). In summary, absorption and mass ejection make it difficult to reveal high Doppler velocity wings in the spectrum of $\chi$ Cygni; if they are present, which one cannot assert with certitude, they are weak.

The emission of the vibrationally-excited line of Si¹⁶O($\nu$=1,$J$=6-5) was observed in the extended A-configuration in February 2023. Figure 13 illustrates the result. The Doppler velocity spectrum displays a profile characteristic of maser emission, with a narrow peak at $\sim$$-$1.5 km s^-1. This is indeed the only remarkable feature visible on the channel maps, which are otherwise dominated by the convolution with the beam of the stellar disc emission, elongated in the NE-SW direction. Another enhancement of emission is observed in the 3$<$$V_{z}$$<$8 km s^-1 interval of Doppler velocity. The high value of the energy of the $J$=6 upper level, 1812 K, implies that the line emission probes the very close neighbourhood of the photosphere. However, the angular resolution is insufficient to map reliably the maser emission and possibly reveal multiple emitting spots simultaneously present in the close neighbourhood of the star, as commonly observed using very long baseline observations on several other stars (e.g. Cotton et al. 2004) and as well established in the case of SiO maser lines observed from $\chi$ Cygni. Small differences between the spectra observed on February 18 and February 21 may reveal variability, as can be expected from such maser emission.

Observation of the same line emission using the B-configuration, also illustrated in Figure 13, was made in February 2024. It shows that both the identified maser and the enhanced emission between 3 and 8 km s^-1 have disappeared, suggesting that the latter feature was also caused by masers. While the broader beam of the B-configuration would smear the image of point-like maser enhancements, this has no impact on the Doppler velocity spectrum and cannot account for the disappearance.

As mentioned in the introduction, SiO masers in $\chi$ Cygni have been observed on many occasions, however usually in lower rotational transitions, mostly $J$=1-0 and 2-1. To our knowledge the observation of a maser in the $J$=6-5 transition is presented here for the first time. Observations reported in the available literature cover a broad range of vibrational excitations, from $\nu$=0 (ground state) in many cases and up to $\nu$=6 for $\chi$ Cygni (Rizzo et al., 2021) and give evidence for strong day-to-day variability, more moderate for Miras than for semi-regular stars (Gómez-Garrido et al., 2020) but, still, in excess of 10% in the case of the $\nu$=1,2, $J$=2-1 transitions for $\chi$ Cygni (Rizzo et al., 2021).

The presence in 2023 of a strong H¹²CN(3-2) line emission confined within some 200 mas from the centre of the star challenges interpretation. As we already remarked, its expansion to larger radii in 2024, at the scale of $\sim$100 mas in a year, meaning $\sim$80 km s^-1, contrasts with the narrow width of the line profile, $\sim$$\pm$2 km s^-1. An explanation in terms of maser emission would account for both the confinement of the emission and the exclusivity of the H¹²CN(3-2) line but would suggest variability in intensity at a same location, not variability in location at a same intensity as is observed. Moreover, as remarked by Bieging (2001), ground vibrational state HCN masers have only been observed in optically bright carbon stars with relatively low mass-loss rates. These include, in particular, a H¹³CN(1-0) maser from Y CVn (Izumiura et al., 1987) and a H¹²CN(1-0) maser from six low mass loss rate carbon stars (Izumiura et al., 1995). Olofsson et al. (1998), from an extensive study of a large sample of AGB stars, in particular W Ori and X TrA, suggest that the H¹²CN(1-0) maser emission observed in low mass-loss-rate ($<$$\sim$5$\times$10^-7 M_⊙ yr^-1) carbon stars is due to radiative excitation with radial amplification. However, Bieging (2001) remarks that an alternative interpretation may be the heating effects of shock waves that are driven periodically through the extended stellar atmosphere by the large-amplitude pulsations of the stars, causing HCN molecules to be successively destroyed in the shock and reformed in the post-shock gas: shocks might then produce a “chemical pump” similar to that discussed by Schilke et al. (2000) for IRC+10216, the basic idea being that the HCN molecules are formed in random vibrational states, population inversion being caused by wide differences in the radiative decay rates of the different vibrational levels. The idea that a shock wave might produce masers travelling outward is of course attractive, but too speculative.

Yet, the observation of unexpectedly low ¹²CO/¹³CO, ²⁸Si/²⁹Si and ¹⁶O/¹⁷O emission ratios in the close neighbourhood of the star at small Doppler velocities, reported in Sub-section 3.5, adds to the difficulty to propose a sensible explanation of what happens in this region. In particular, the contrast observed at low Doppler velocities between the low value of the ¹²CO/¹³CO and the high value of the H¹²CN/H¹³CN emission ratios challenges interpretation, suggesting that two very different mechanisms may have successively dominated the dynamics in this region. It is useful, in this context, to recall the basic assumptions, explicitly spelled out by Willacy & Cherchneff (1998), underlying shock chemistry models. Stellar pulsations cause large amplitude compressional waves to propagate outward and to steepen into shocks outside the stellar photosphere. These periodic shocks travel through the inner layers of the CSE and alter dramatically the temperature and the density of the gas. Layers of gas are then accelerated upwards but then decelerate and eventually fall back to their initial position under the influence of stellar gravity, leading to the formation of a stationary layer close to the stellar photosphere. The shock is assumed to form at stellar radius and its strength to be damped as it travels outwards. Using such a model, Duari & Hatchell (2000) and Cherchneff (2006) have studied a C/O=0.95 star hit by shocks formed at stellar radius, with velocities of 32 and 25 km s^-1, respectively. Figure 18 compares the radial dependence of the molecular abundances obtained by Cherchneff (2006), normalised at a distance of 5 stellar radii, for CO, SiO, CS and HCN molecules separately: the latter decreases by a factor $\sim$15 between 1.5 and 3.5 stellar radii while the others are nearly constant. Such a dramatic effect is however not commented by the author. According to the description of the relevant shock chemistry described in the paper, it is probably related to HCN formation via the chain of reactions H+C₂$\rightarrow$CH+C, CH+N$\rightarrow$CN+H, CN+H₂$\rightarrow$HCN+H, which relies on the formation of CH molecules in the shock front: the CH formation reaction has a very high activation barrier of over 3$\times$10⁴ K, which prevents its occurrence in other regions and CH molecules do not contribute to the formation of CO and SiO molecules, and contribute only little to the formation of CS via CN+S$\rightarrow$CS+N, dominated instead by C+SO$\rightarrow$CS+O.

While offering an interesting illustration of the mechanisms at stake, such a model cannot be expected to provide a reliable description of reality. Indeed, Gobrecht et al. (2016) and Marigo et al. (2016), using a similar model with different parameters find instead a deep depression of HCN abundance in the first few stellar radii. We simply retain from these arguments that the HCN abundance in the first few stellar radii, at variance with the abundance of the other molecules, seems to be extremely sensitive to the physical properties of the shock and to its amount of damping when moving outwards: within the limits of our current understanding of the shock dynamics at stake, an order of magnitude enhancement of HCN abundance within a few stellar radii, as predicted by Cherchneff (2006) and Duari & Hatchell (2000), is quite plausible even if the very speculative nature of the above scenario cannot be understated. Moreover, the complexity of the dynamics at stake may also be expected to eventually offer favourable conditions for chemical pumping schemes causing significant maser emission.

While the scenario developed in the previous sub-section rests on the important role played by strong shocks in the CSE of $\chi$ Cygni, the very large value taken by the H¹²CN(3-2)/H¹³CN(3-2) emission ratio in the inner layer may be related to the recent occurrence of third-dredge-up episodes. Indeed, both shocks and third-dredge-ups have been shown to play a dominant role, but the relation between the two, if any, is unclear. Indeed, for now several decades, as described in the introduction, both shocks and third-dredge-ups have been commonly accepted to play a dominant role, but the present observations raise questions about their precise respective contributions.

An important argument in proposing the observations reported in the present article was the S-type nature of $\chi$ Cygni, with a large C/O ratio of 0.95-0.98 (Duari & Hatchell, 2000; Justtanont et al., 2010). Indeed, we know from Uttenthaler (2013) and (Uttenthaler et al., 2019) that Tc-rich and Tc-poor stars populate different regions of the period-luminosity diagram, suggesting that third-dredge-ups, bringing new elements to the star surface, play a significant role in the evolution of the star. The large value of the ¹²C/¹³C ratio implies that $\chi$ Cygni is indeed in an advanced stage of evolution, and the presence of technetium in its spectrum is evidence for a third dredge-up event to have occurred recently. However, the observed confinement of the H¹²CN(3-2) emission, contrasting with the absence of a similar feature in ¹²CS and H¹³CN production, is not in itself evidence for approaching the carbon-rich regime. We find it difficult to identify precisely the respective contributions of shock waves and third dredge-ups to the dynamics of the CSE. There seems to be no clear relation between the two. Indeed, the simulations performed by Duari & Hatchell (2000) and Cherchneff (2006) show that the production of such molecules by shock chemistry is essentially independent from the C/O ratio, and very much so over the M- to S-type transition. The pulsation periods are not much larger than the values which they take for M-stars and the mass-loss rates have similar distributions for M-, S- and C-stars (Ramstedt et al., 2009). In general, the episodic and inhomogeneous nature of the wind morpho-kinematics, and particularly the occurrence of brief episodes of mass-loss, are also observed in Tc-poor stars such as R Leo (Hoai et al., 2023), R Dor (Nhung et al., 2019) or W Hya (Hoai et al., 2022b) and cannot be interpreted as signalling the approach of the carbon-star regime.