Modelling predicts a molecule-rich disk around the AGB star L2 Puppis

The nearby oxygen-rich AGB star L2 Pup hosts a well-studied nearly edge-on disk. To date, disks around AGB stars have not been chemically studied in detail. By combining a parameterisation commonly used for protoplanetary disks and archival ALMA observations, we retrieved an updated density and temperature structure of this disk. This physical model was then used as input to the first chemical model of an AGB disk. The model shows that the physical structure of the disk has a large impact on its chemistry, with certain species showing large changes in column density relative to a radial outflow, indicating that chemistry could be used as a tracer of disks that cannot be directly imaged. Despite its oxygen-rich nature, the daughter species formed within the disk are surprisingly carbon-rich. Two chemical regimes can be distinguished: cosmic-ray induced chemistry in the midplane and photochemistry induced by the interstellar radiation field in the outer regions. Certain complex organic molecules are formed in the midplane. This occurs via gas-phase chemistry only, as the disk is too warm for dust-gas chemistry. The photochemistry in the outer regions leads to the efficient formation of (long) carbon-chains. The predictions of the model allow us to tentatively put the disk's age $\lesssim 10^5$ yr. Additional observations are necessary to better constrain the physical structure of L2 Pup's disk and are essential to test the predictions made by the chemical model. Our exploratory work paves the way for a more general study of the chemistry of AGB disks.


INTRODUCTION
The asymptotic giant branch (AGB) phase near the end of the lives of stars with an initial mass up to 8 M ⊙ is characterised by vigorous mass loss.AGB stars lose their outer layers via a stellar wind or outflow at a rate between 10 −8 and 10 −4 M ⊙ yr −1 , creating an expanding circumstellar envelope (CSE, Habing & Olofsson 2003;Höfner & Olofsson 2018).CSEs are rich astrochemical environments, with close to 100 different molecules and some 15 types of newly formed dust grains detected so far (Decin 2021).Spherically symmetric outflows appear to be the exception rather than the rule.Asymmetrical structures are ubiquitously observed, ranging from small-scale asymmetries, such as density-enhanced clumps (e.g., Leão et al. 2006;Khouri et al. 2016;Velilla-Prieto et al. 2023), to large-scale structures, such as spirals (e.g., Mauron & Huggins 2006;Maercker et al. 2012) and disks (e.g., Kervella et al. 2016).These large-scale structures are thought to be caused by binary interaction with (sub)stellar companions.Circumbinary disks are expected around younger, lower mass-loss rate AGB stars, evolving into spirals as the mass-loss rate increases and the companion's orbit widens during the AGB phase (Decin et al. 2020).
★ E-mail: mvdsande@strw.leidenuniv.nl The classic spherically symmetric one-dimensional chemical kinetics model of the CSE is hence applicable to only a minority of outflows.The effects of the UV field of a stellar companion and a clumpy substructure on the chemistry throughout the outflow have been included in chemical models of the CSE (Van de Sande et al. 2018; Van de Sande & Millar 2022; Van de Sande et al. 2023).These models show that chemistry can be used to unveil a hidden stellar companion (Siebert et al. 2022) and retrieve its orbital properties (Danilovich et al. 2024).The influence of large-scale density structures on the CSE's chemistry requires specialised chemical models.So far, models have been developed for the density-enhanced shells in the outflow of IRC+10216 (Brown & Millar 2003;Cordiner & Millar 2009;Agúndez et al. 2017), but no general models applicable to a larger set of AGB stars exist.The chemistry of disks around AGB stars has not been studied in detail.
The best-studied AGB disk is that around the oxygen-rich AGB star L 2 Pup, the second-closest AGB star at a distance of 64 pc (van Leeuwen 2007).Optical and infrared observations have shown the clear presence of an almost edge-on compact dusty disk with an inner dust rim at 6 au, spanning to around 15 au.Loops and plumes extending above the disk have also been identified (Ireland et al. 2004; Kervella et al. 2014;Lykou et al. 2015;Ohnaka et al. 2015; Kervella et al. 2015).ALMA observations probed the gaseous disk, which has an inner rim at 2 au, and revealed a tentative planetary companion with a mass of 12 ± 16 M J .This planet is probably not formed within the disk, but is instead rather part of L 2 Pup's (first generation) planetary system (Chen et al. 2016;Kervella et al. 2016).The physical structure of the disk has been proposed to be similar to that of protoplanetary disks (PPDs), albeit with a higher temperature in the midplane as it is assembled from warm stellar outflow material (Homan et al. 2017).
In this paper, we develop a two-dimensional chemical model of L 2 Pup's disk: the first such model of a disk around an AGB star.This exploratory work probes the unique astrochemical regime of a hot and dense disk, irradiated by the interstellar radiation field.By determining the effects of the physical disk structure on the chemistry around L 2 Pup, we can identify the chemical differences with a spherically symmetric O-rich outflow and establish whether the parent species are redistributed into different carriers.Such differences can be used as chemical tracers of disks that are not as easily resolved as the close-by L 2 Pup.Additionally, the evolution of the disk's chemical structure over time can be used to determine the age of the disk.
In Sect.2, we use archival ALMA data to retrieve the disk's density and temperature structure using a parametric model commonly used for PPDs.This physical structure is then used as input to the chemical model in Sect.3, where we also show the chemical modelling results.The physical and chemical modelling results are discussed in Sect.4, followed by our conclusions in Sect. 5.

PHYSICAL MODEL
L 2 Pup was observed by ALMA during Cycle 3 for project 2015.1.00141.S (PI Pierre Kervella) in November 2015.The reduction of the Band 7 observations is described in Kervella et al. (2016), and we use that dataset in the present work.The resulting data have an angular resolution better than 15 mas and a maximum resolvable scale of 200 mas.The  = 3 − 2 lines of 12 CO and 13 CO from the dataset are presented in Homan et al. (2017).The data cubes have a spectral resolution of 0.22 km s −1 and an rms noise level of 2.5 and 2.6 mJy beam −1 , respectively.The channel maps of the lines are presented here in Appendix A, now with beam information.Homan et al. (2017) derived a physical model of L 2 Pup's disk from the 12 CO and 13 CO  = 3 − 2 lines.The model from that work, although provided by the first author, was unable to be reproduced.Given the similarity of the disk to PPDs, we opted to remodel the data using a parametric model commonly used for PPDs.
The parametric model is described in Sect.2.1.Sect.2.2 describes the 3D line radiative transfer code used, Magritte.The parameter space explored is laid out in Sect.2.3, the retrieved physical structure of the disk is presented in Sect.2.4.

Parametric model of a protoplanetary disk
We use the parametrisation of Williams & Best (2014), which assumes a disk surface density profile based on the solution for an axisymmetric, steady-state viscous accretion disk in hydrostatic equilibrium (e.g., Lynden-Bell & Pringle 1974;Hartmann et al. 1998).The surface density of the disk is described by where  0 is a disk reference radius and  describes the radial dependence.Σ 0 gives the global normalisation to the gas mass at  0 , written as with  gas the disk's total gas mass and  in its inner radius.Assuming hydrostatic vertical balance in the disk, the density structure of the disk is described by with   the pressure scale height derived from the midplane temperature.This scale height is given by the ratio between the speed of sound,   , and the Keplerian angular frequency Ω  , or, Here,  * is the stellar mass,   is the Boltzmann constant,  the gravitational constant,   the mass of the hydrogen atom, and  the mean molecular mass.
The midplane temperature profile is parametrised by a power law, with  mid,1 the temperature at 1 au and  the power-law exponent.Similarly, the atmospheric temperature surrounding the disk is parametrised by with  * the stellar temperature and  the power-law exponent.The vertical temperature profile connects Eqs ( 5) and ( 6) by a sine function: where  describes the steepness of the temperature profile and   describes the height at which the disk reaches the atmospheric value.Following Williams & Best (2014), we fix  = 2 and   = 4  ().
For a given stellar mass, the physical structure of the disk is described by eight parameters: four for the density profile, {,  in ,  0 ,  gas }, and four for the temperature profile, { mid,1 ,  atm,1 , ,  }.

Magritte: 3D line radiative transfer
Magritte 1 is an open-source 3D non-LTE line radiative transfer code, mainly used for creating synthetic spectral line observations (De Ceuster et al. 2020a,b, 2022).We initialise the Magritte model by defining the model on a uniform 60 × 60 × 30 au grid.For each point, we define the temperature, velocity, turbulent velocity, H 2 density and CO density.The grid is then resampled to speed up computation by limiting the allowed variation of the density, which we evaluated at the uniform grid positions (Ceulemans et al., in prep.).After model setup, the radiation field is computed assuming non-LTE by taking into account the first seven rotational CO lines.In this way, the energetic state of the CO gas is determined selfconsistently.Afterwards, we create spectral line images for the 12 CO and 13 CO  = 3 − 2 lines.

Parameter space of the physical model
Table 1 shows the parameters of the physical model along with the values which best reproduce the integrated emission profiles.Based on previous observations, we fixed the inner radius of the disk, velocity profile, temperature profile of the disk's atmosphere, the fractional abundance of 12 CO, and the stellar and disk mass.These parameters are listed in the top rows of Table 1.The stellar radius is fixed at 1 au.The inner radius of the disk,  in , is fixed at 2 au (Kervella et al. 2016).The velocity profile is assumed to be Keplerian from the inner radius out to the inner rim of the dusty disk at 6 au.Beyond 6 au, the velocity profile is sub-Keplerian with a power-law exponent of 0.853 (Kervella et al. 2016).The turbulent velocity is fixed to 0.5 km s −1 (Homan et al. 2017).The temperature profile of the atmosphere is fixed at  atm,1 = 2800 K with a power-law exponent of 0.65 (Danilovich et al. 2015).The fractional abundance of 12 CO was taken to be 1 × 10 −4 with respect to H 2 (Homan et al. 2017).The total gas mass of the disk,  gas , is fixed at 2.5 × 10 −4 M ⊙ following Kervella et al. (2015) and assuming a dust-to-gas mass ratio of 0.01.The stellar mass,  * , is fixed to 0.659 × M ⊙ , as derived by Kervella et al. (2016).
The other parameters were allowed to vary and are listed in the bottom rows of Table 1.Their starting points were based on the model of Homan et al. (2017).The midplane temperature,  mid,1 , was varied in increments of 100 K, the power-law of the midplane temperature, , in increments of 0.05, the characteristic disk radius,  0 , in increments of 1 au, the parameter determining the surface density (Eq.1), , in increments of 0.5, and the 13 CO abundance in increments of 1 × 10 −5 w.r.t.H 2 .

Physical structure of the disk
The model calculation was constrained by comparing slices along the disk midplane through the moment 0 (integrated intensity) maps of the observed lines.The moment 0 maps are presented in Kervella et al. (2016) (their Figs. A1 and B1).The slices were chosen so that they go through the emission peaks in the East and West sides of the disk as the disk's position angle is not zero.We choose to use this metric as a radial slice along the major axis of the disk will capture well the radial behaviour of the emission without any complication that may arise from deprojection of the emission from this highly inclined disk.The slice through the moment 0 map of 12 CO is centred on (07 h 13 m 32.s 477, -44 • 38 ′ 17. ′′ 845) at an angle of 9.53 • , that through the 13 CO moment 0 map is centred on (07 h 13 m 32.s 475, -44 • 38 ′ 17. ′′ 843) at an angle of 16.0 • .The slices are different as the 12 CO and 13 CO probe different layers within the disk.
We first modelled the 12 CO line by reproducing the slice through its moment 0 map.Good models were determined by minimising the chi-squared value of the fit to the slice.For these models, we then calculated the corresponding 13 CO model, varying the 13 CO abundance as an additional parameter while keeping the others fixed and calculated the chi-squared value of the fit to the slice through the 13 CO moment 0 map.The 12 CO/ 13 CO ratio is fixed throughout the disk.
Our best-fitting model provides the best reproduction of the 12 CO moment 0 slice, with a chi-squared value of 3.92.A 13 CO abundance of 2 × 10 −5 results in the fourth-best fit in our sampled parameter space to the 13 CO moment 0 slice, with  2 = 1.76.A larger 13 CO abundance of 3×10 −5 results in the third-best model, with  2 = 1.74.However, this would yield a 12 CO/ 13 CO ratio of 3.33.Since this ratio is lower than expected from observations (Ramstedt et al. 2014) and the improvement to reproducing the observations is marginal, we  Kervella et al. (2015).
chose the model with a 13 CO abundance of 2 × 10 −5 as our best model.Note that the model that best reproduces the 13 CO moment 0 slice, with the lowest chi-squared value of 1.65, corresponds to the fourth-best model to the 12 CO moment 0 slice, with  2 = 4.19.
Together with a visual inspection of the moment 0 maps, channel maps, and line profiles, we chose the best-fitting model to the 12 CO moment 0 slice with a 13 CO abundance of 2 × 10 −5 as our final model.Its parameters are listed in Table 1. Figure 1 shows the slices through the moment 0 maps of the 12 CO and 13 CO  = 3 − 2 lines along with the Magritte model results.The negative flux in at the centre of the observed lines is due to line absorption by the molecular gas located in front of the star.The Magritte model's moment 0 and channel maps are shown in Appendix B.
Fig. 2 shows the density and temperature profile of the best-fitting model.We find that L 2 Pup's disk has a surface density profile Σ() ∼  −1 and an  0 of 6 au.This radius corresponds to the radius where the disk's velocity profile changes from Keplerian to sub-Keplerian (Kervella et al. 2016) and where the surface density steepens significantly from a power law (Williams & Cieza 2011).Its midplane is warm with  mid,1 = 900 K, and has a shallow temperature profile characterised by  = 0.2.The 13 CO abundance w.r.t.H 2 is 2 × 10 −5 , yielding a 12 CO/ 13 CO ratio of 5.

CHEMICAL MODEL
The disk chemical model used is the PPD chemical model of Walsh et al. (2015), which includes the full gas-phase chemical network of the UMIST Database for Astrochemistry (UDfA) Rate12 (McElroy et al. 2013) supplemented with a comprehensive dustgas and grain-surface chemistry.The Rate12 gas-phase network includes ion-molecule and neutral-neutral reactions, photoionisation and photodissociation (parametrised to the interstellar radiation field), cosmic-ray-ionisation, and cosmic-ray-induced photoreactions.We include all these processes by default.To this base network, Walsh et al. (2015) added a suite of three-body association reactions and hot neutral-neutral reactions (those with a large activation barrier) from compilations used in combustion chemistry (e.g., Baulch et al. 2005).Also added were a set of collisional dissociation reactions for all species expected to be abundant at the high densi-   1.
ties and temperatures expected in inner disks (i.e., mostly saturated molecules and associated radicals).
The protoplanetary disk chemical code also includes a treatment for self-and mutual-shielding of H 2 , CO, and N 2 (see e.g., Heays et al. 2017).To compute the shielding functions for photodissociation of H 2 , CO, and N 2 , we estimate the visual extinction,   (mag), at each point in the disk by vertically integrating the H 2 column from the disk surface downwards, then scaling the H 2 column to   using the standard ISM scaling (0.5 × (1.59 × 10 21 ) ×   ; Bohlin et al. 1978).This is appropriate because the AGB star is very cool ( eff = 2800 − 3500 K, Kervella et al. 2014;Danilovich et al. 2015), and the only source of UV radiation is the ISM.We then interpolate the shielding functions provided by Heays et al. (2017) over   and column density, assuming conservative fractional abundances for CO and N 2 of 10 −5 w.r.t.H 2 .This shielding factor is then added as a prefactor to the photodissociation rate which is calculated in the usual manner.The chemical model of Walsh et al. (2015) also includes a suite of X-ray reactions and grain surface processes.However, because there is no source of X-rays in this system, X-ray chemistry is effectively inactive and is therefore not included.
The physical model retrieved in Sect. 2 was used as input to the static chemical model.As there are no observational constraints, we assumed that the dust temperature is equal to the gas temperature.
The assumption of thermally coupled dust and gas is reasonable considering the high densities.The dust extinction at each point in the disk was calculated by combining the H 2 column densities of the disk and that in the surrounding tenuous outflow in the vertical direction for each point, assuming the ISM relation between   and H 2 column density.The chemistry is evolved for 10 6 yr, roughly the entire AGB lifetime.
We also calculate a 1D outflow chemical model to compare the predicted disk chemistry with the chemistry in a low-density outflow.The smooth outflow model is the publicly available CSE model of the UDfA Rate22 release (Millar et al. 2024) 2 .It describes a spherically symmetric outflow with a constant mass-loss rate and outflow velocity, with a power-law temperature profile.Following Danilovich et al. (2015), we adopt a mass-loss rate of  = 1.4 × 10 −8 M ⊙ yr −1 , an expansion velocity of  ∞ = 2 km s −1 , and a stellar temperature of  * = 2800 K.We use the same exponent for the temperature power law as in the disk physical model,  = 0.65.The density and temperature throughout the outflow is shown in Fig. C1.
Table C1 lists the O-rich parent species and their initial abundances as derived from a range of observations by Agúndez et al. (2020) and augmented by metal abundances (Van de Sande et al. 2021).Note that Kervella et al. (2016) found a sub-solar metallicity of  = 0.008, which could reduce some of our assumed abundances by a factor ∼ 2 (Karakas & Lugaro 2016).The parent species are used as input to both chemical models.Therefore, we assume that the material captured during the disk generation phase has the same composition as the inner wind as an approximation of the disk's initial composition.We evolve the chemistry for an initial 10 3 yr as our first snapshot to allow the physical conditions to imprint on the chemistry.
The results of the chemical model are presented in the next Sections.We first show the results for the parent species in Sect.3.1, followed by the daughters in Sect.3.2.The formation of complex organic molecules (COMs) and a C-rich reservoir in the outer regions is shown in Sects.3.3 and 3.4, respectively.All abundances are fractional abundances relative to H 2 .

Behaviour of the parent species
Fig. 3 shows the abundance of the parents CO and SO throughout the disk at different time intervals.Their abundance distribution changes over time.The CO abundance decreases in the outer, more tenuous layers due to photodissociation by the interstellar UV radiation field (note that self-and mutual CO self-shielding is included, Sect.3).The abundance of SO decreases in the midplane up to a radius of 10 au after 5 × 10 5 yr, shifting its peak abundance to the outer edge of the disk.This is not due to depletion onto dust as the dust is too warm, but it is caused by reactions with C (liberated predominantly from CO by cosmic rays), which reacts with SO to form CS. SO 2 shows similar behaviour (Fig. C2 in Appendix 3), where reactions with C lead to the formation of SO.The large C/S abundance ratio of ∼ 12 makes these efficient SO and SO 2 destruction pathways.The observed emission of SO and SO 2 (Kervella et al. 2016) suggests that the disk is likely not older than ∼ 10 5 yr; this is further discussed in Sect.4.3.The abundance profiles of all parents at 10 5 yr are shown in Fig. C3.
The radial extent measured at the midplane of the parents at different times within the disk is shown in Fig. 4, together with the size of their molecular envelopes in the outflow model.These are the radii where the initial abundance has decreased by a factor of , corresponding to the -folding radius in the outflow.All parents have a radial extent of ∼ 10 au, which is consistent with the disk's density distribution (Fig. 2).This is smaller than the size of their molecular envelope in the outflow, which is larger by a factor of a few (e.g., for H 2 S or NH 3 ) to two orders of magnitude (for CO and PN).In the disk, the radial extent of most parent species does not change significantly over time.That of the parents SiS, SO 2 , SO and CO 2 decreases by a factor of ∼ 4 − 5 over time; that of PO decreases by a factor of ∼ 2. For SO 2 , SO and CO 2 , this is due to the abundance decreasing in the midplane followed by photodissociation in the outer layers of the disk.The decrease in radial extent for SiS and PO is purely due to photodissociation, these species do not show a gap in the inner midplane (Fig. C3).

Behaviour of daughter species
Fig. 5 shows the abundance of the daughters OH, SiC 2 , CH 3 CN and HCOOH throughout the disk at 10 5 yr.The abundance of OH peaks in the outer regions of the disk, with abundances up to ∼ 3 × 10 −6 .This distribution is due to its formation from the photodissociation of the parent H 2 O and is similar to its behaviour in an outflow, where it forms a molecular shell.
SiC 2 is also present in the outer regions of the disk, but its abundance is largest in the inner disk edge where it reaches ∼ 3 × 10 −6 .This is caused by the unique combination of high density, high temperature, and strong UV radiation field in this region.SiC 2 is formed by the reaction Si + C 2 H 2 , where Si is produced by the photodissociation of the parent SiO.The higher abundance in the inner disk edge is inherited from that of the reactant C 2 H 2 (Fig. C4) and is caused by the temperature dependencies in its formation pathway, which are elaborated on in Sect.4.4.2.
CH 3 CN (methyl cyanide) is not located in the outer edge of the disk, but in the inner disk midplane.Its peak abundance is ∼ 3 × 10 −9 , which is two orders of magnitude larger than its predicted peak abundance in an outflow.This C-bearing molecule is formed via CH 3 + and the parent HCN producing CH 3 CNH + , which then dissociatively recombines by collisions with negatively charged dust grains into CH 3 CN.CH 3 + plays a crucial role in the chemistry of the inner midplane.It is formed via CH 4 + He + , where both reactants have a cosmic ray origin, as CH 4 is formed by the successive addition of hydrogen to C (liberated by cosmic rays from CO). Sect.4.4.1 elaborates on the cosmic-ray induced midplane chemistry.
The abundance of HCOOH (formic acid) is also largest in the midplane.Its peak abundance is ∼ 5 × 10 −9 , three orders of magnitude larger than its peak abundance in an outflow.Its formation is also initiated by cosmic rays: H 3 + (a product of cosmic-ray reactions) reacts with the parent CO to form HCO + .Reaction with the parent H 2 O produces HCOOCH 2 + and HCOOH.Unlike CH 3 CN, its peak abundance is not in the inner regions of the disk but between 5 and 9 au.This is due to the difference in destruction mechanisms: unlike CH 3 CN, HCOOH is efficiently destroyed by collisional dissociation.The rate of this destruction mechanism increases with density, destroying HCOOH in the dense inner midplane.

Formation of complex organic molecules
The COMs CH 3 OH (methanol), CH 3 CN (vinyl cyanide), CH 3 NH 2 (methylamide), C 2 H 5 CN (ethyl cyanide), CH 2 CHCN (acrylonitrile), HCOOH (formic acid), and HCOOCH 3 (methyl formate) are abundantly present in the inner disk and are formed by cosmic-ray induced chemistry in the midplane (Sect.4.4.1).Their abundance distribution at 10 5 yr is shown in Fig. 6.Fig. 7 shows their column density at different times in the disk model, together with their column density in the outflow model.The column density of species in the disk is a disk-averaged column density, calculated via where   is the vertical column density within Δ  and the factor 2 accounts for the symmetry of the model across the midplane.At 10 5 yr, all COMs listed above have a disk-averaged column density larger than 10 10 cm −2 .CH 3 CN has the largest column density of 1.80 × 10 15 cm −2 , followed by CH 3 OH with 3.37 × 10 14 cm −2 and HCOOH with 1.19×10 14 cm −2 .The column densities are at least five orders of magnitude larger than those predicted by the outflow model.The largest differences are seen for CH 3 NH 2 , C 2 H 5 CN, CH 2 CHCN, and HCOOCH 3 , with a difference of at least 8 orders of magnitude, showing that the chemistry in the disk is favouring the formation of larger saturated complex molecules.

Carbon-rich reservoir in the outer regions
Carbon-bearing molecules are abundantly formed in the outer regions of the disk.CO and other C-bearing parents are photodissociated by interstellar UV photons, liberating the reactive C atom.Carbon-chains, e.g., C 2 H 2 and HC 3 N, are abundantly formed in this region (Fig. C4).Longer highly-unsaturated carbon-chains, such as C 10 H 2 , C 8 H, and HC 9 N, are also produced.However, the chemical reaction network is geared towards interstellar chemistry and does not include full saturation of these carbon-chains.Hence, the predicted abundances of these longer chains are likely over-estimated.
We therefore grouped all species with 6 or more C atoms into a "carbon sink" reservoir.Fig. 8 shows the abundance of this sink particle throughout the disk at different times in its evolution.The total abundance of carbon-bearing species is large in the outer disk, with a maximum abundance of ∼ 3 × 10 −5 .This is remarkably large considering the disk is O-rich.Unlike in the outer regions of an AGB outflow, newly liberated C can readily react and form (long) carbon chains thanks to the higher densities in the disk's outer region.

DISCUSSION
The physical structure of the disk has a large influence on the chemistry around L 2 Pup.In Sect.4.1, we compare our newly derived physical structure to PPDs and disks around post-AGB stars.We then compare our results to those of Homan et al. (2017) and suggest future observations to better constrain the density and temperature within the disk.This is important, as the predicted chemistry depends  on the physical structure.In Sect.4.2, we qualitatively compare our predicted abundance profiles to the previous observations of Kervella et al. (2016) as a first-order test of the model.We elaborate on using the predicted SO and SO 2 abundance profiles to age the disk at ∼ 10 5 yr in Sect.4.3.Finally, the chemistry within the disk is discussed in Sect.4.4, where we examine the surprisingly rich C-rich chemistry in the O-rich disk and suggest molecular tracers of the disk structure.

Comparison to PPDs and post-AGB disks
We retrieved that the disk around L 2 Pup has a surface density profile Σ() ∼  −1 and is warm, with a midplane temperature of 900 K and  mid () ∼  −0.2 .The exponent of the surface density distribution, , is on the shallower end of those found for PPDs, for which  ≲ −1 (Andrews 2020).Disks around post-AGB stars have −3 ≤  ≤ −1, although it is difficult to constrain this value with infrared interferometry alone (Hillen et al. 2014(Hillen et al. , 2015;;Corporaal et al. 2023).Our value of  = −1 hence also agrees with post-AGB disks and appears to suggest that the density profile of this disk around an AGB star lies between that of a PPD and post-AGB disk.
The midplane temperature of L 2 Pup's disk is much warmer than in PPDs:  mid,1 = 900 K compared to 100 − 300 K for PPDs.The power-law is also flatter, with  = 0.2 rather than  = 0.5 − 0.7 for PPDs (e.g., Andrews et al. 2009;Williams & Best 2014).The temperatures of post-AGB disks lie within 1400 − 3600 K, which are single blackbody temperatures for the disk as it is, again, difficult to constrain their midplane temperatures using infrared interferometry alone (Kluska et al. 2018;Corporaal et al. 2023).The disk around L 2 Pup again lies between these two ranges: warmer than PPDs, colder than post-AGB disks.This is expected as the gas ejected by the AGB star is warm and its high luminosity can provide a lot of dust heating.In post-AGB stars, the stripped star is much warmer, leading to larger temperatures.

Comparison to Homan et al. (2017)
We used a parametric model commonly used for PPDs to model the same observations used by Homan et al. (2017) as we were unable to reproduce their model from the information provided in that work.We did use the same velocity profile of Kervella et al. (2016) and also adopted certain observational parameters of Homan et al. (2017), as described in Sect.2.3.
We find three main differences between our model and theirs: (i) Homan et al. ( 2017) assumed a three-staged radial temperature profile, which includes an exponential and arctan.This profile includes a feature the authors attribute to the heating of the gas by the dust disk.The parameter space explored to achieve this fit is not explained.We find that a power-law expression (Eq.6) is sufficient to reproduce the observations, indicating that their analytical expression could be overfitted and may have been more complex than needed.
(ii) Homan et al. (2017) retrieved a steep radial density profile at the midplane of (,  = 0) ∼  −3.1 , which is much steeper than our (,  = 0) ∼  −0.4 (Eqs 1, 3, 4).The steepness of their density profile does not agree with observations of either PPDs or post-AGB disks.The large difference in density profile could be due to the differences in temperature structures.
(iii) The 13 CO abundance retrieved by Homan et al. ( 2017) is 1×10 −5 , we find a value of 2×10 −5 w.r.t.H 2 .We used the same 12 CO abundance of 1 × 10 −4 w.r.t.H 2 .This leads to different 12 CO/ 13 CO ratios: 10 by Homan et al. (2017), 5 in this work.While both values lie within the range found by Ramstedt et al. (2014), our value of 5 is on the low side.Optical depth effects might cause such a low 12 CO/ 13 CO ratio.Note that photodissociation effects due to differences in selfshielding between the isotopologues would increase the ratio (Saberi et al. 2019).Additionally, a factor of 2 difference is reasonable within the uncertainties of modelling (Ramstedt et al. 2008).

Refining the physical structure: further observations
Besides CO and 13 CO, the available ALMA data includes rotational transitions of H 2 O, 29 SiO, 30 SiO, SiS, SO, and SO 2 .The isotopologues of SiO were used to constrain the velocity structure of the disk by Kervella et al. (2016).While the other lines shed some light on the chemical structure of the disk (Sects.4.2 and 4.3), further observations are necessary to better constrain the disk's physical structure, given the differences with Homan et al. ( 2017 Prieto et al. 2015).So far, H 2 CO has only been detected around the moderate mass-loss rate O-rich AGB star IK Tau with a column density of 2.8 × 10 14 cm −2 , which is close to our predicted column density (Velilla Prieto et al. 2017).IK Tau likely has a stellar companion shaping its outflow (Coenegrachts et al. 2023).H 2 CO has also been detected in the Rotten Egg Nebula, an AGB star with an Atype companion (Lindqvist et al. 1992), and the Helix Nebula, which might host an unseen dwarf M star (Gruendl et al. 2001;Tenenbaum et al. 2009;Zack & Ziurys 2013).It seems that detections of H 2 CO around evolved stars appear to suggest deviations from spherical symmetry.

Comparison of the chemical model to observations
We qualitatively compare our chemical modelling results with the observations of Kervella et al. (2016).The observed 12 CO emission (Appendix A in Kervella et al. 2016) has a radial extent of ∼ 9 au and a vertical extent of ∼ 3 au.The peak CO abundance predicted by our chemical model lies within ∼ 11 au radially and about 2.5 au vertically (Fig. 3).Keeping in mind that we are comparing abundances with observations, the CO abundance predicted by the chemical model is consistent with the observed 12 CO emission.A standard set of parent species and their abundances was used as input when running the chemical model.The initial CO abundance in the chemical model is a factor 3 larger than that assumed in the physical model, following Homan et al. (2017) (3 × 10 −4 versus 1 × 10 −4 w.r.t.H 2 , Sect.3).As we did not a priori know how the chemical abundances would change over time, nor which abundance (profile) to assume when retrieving the physical structure from the observations, we kept these values when calculating the physical and chemical structures.We find that the results of the chemical model are consistent with its independently derived physical structure.The disk averaged CO abundance at 10 5 yr decreased to 2.86 × 10 −4 w.r.t.H 2 .
Besides the 12 CO and 13 CO  = 3 − 2 lines, Kervella et al. (2016) also presented the SO 2 ( = 0,    ,  = 34 3,31 − 34 2,32 ), SO ( = 0,   = 8 8 −7 7 ), and SiS ( = 1,  = 19−18) lines (their Appendices C -E).The SO 2 and SO emission is more extended vertically (about 6 au and ∼ 3 au, respectively) compared to the vertically compact SiS emission (up to 2 au).The SO abundance of the chemical model is shown in Fig. 3, that of SO 2 and SiS in Fig. C2.At 10 5 yr, the region of highest SO 2 abundance has a vertical extent of ∼ 6.5 au, that of SO about ∼ 5.5 au and of SiS about ∼ 1.5 au.The chemical model hence also qualitatively reproduces the relative vertical extents of SO, SO 2 , and SiS, though a full radiative transfer model is required to check.
Methanol has been detected towards the post-AGB star HD 101584 (Olofsson et al. 2017) and OH 231.8+4.2, a bipolar nebula around an AGB star (Sánchez Contreras et al. 2018).It is detected in the high-velocity outflows of these objects and is thought to originate from post-shock chemistry, where shocks in the outflow desorb the molecule from an icy grain mantle.The retrieved column densities of CH 3 OH are ∼ 10 16 and 1−8×10 14 cm −2 , corresponding to fractional abundances of ∼ 3 × 10 −6 and 10 −8 − 10 −7 w.r.t.H 2 , respectively.Our chemical model predicts a column density of ∼ 10 14 cm −2 around 10 5 yr, with a peak fractional abundance around 3 × 10 −9 w.r.t.H 2 .The strongest CH 3 OH line detected in HD 101584, the 8 −1 -7 0 line at 229.759 GHz, is covered by the 2016.1.00207.S. dataset (PI P. Kervella), with a peak flux around 80 mJy.It is not clearly detected in the archival L 2 Pup data, with an upper limit for the peak flux of around 2−3 mJy.From running a simplified radiative transfer calculation, using the accelerated lambda iteration method code (ALI, e.g., Danilovich et al. 2021, with molecular data from Rabli & Flower 2010), based on the spherically symmetric model of Danilovich et al. (2015) and assuming a Gaussian abundance profile with an -folding radius of 10 15 cm, we determine that for a methanol abundance of 3 × 10 −9 w.r.t.H 2 (the maximum abundance predicted by the chemical model) the 8 −1 -7 0 line would not be detected above the noise for L 2 Pup.The non-detection of CH 3 OH in L 2 Pup could be due to its lower abundance or due to a different excitation mechanism, as methanol is formed in the warm inner midplane of the disk via gas-phase reactions rather than via shock chemistry in a highvelocity outflow, as in the two other objects.Further observations are necessary to confirm the (non)detection of methanol in L 2 Pup's disk.
A radiative transfer analysis of the chemical model predictions lies outside the scope of this paper.Moreover, it is important to note that the predicted chemistry depends on the input physical structure.A parameter study to quantify the impact of different physical structures on the chemistry is computationally intensive and will be addressed in future work.As stated in Sect.4.1.3,further observations are necessary to constrain the retrieved physical model of L 2 Pup's disk.

Age of the disk
After 5 × 10 5 yr, the chemical model predicts a gap in the inner midplane for SO and SO 2 (Figs. 3 and C2), which is about 1 au in height and 9 au radially for both molecules.Such a gap is not seen in the emission maps of SO 2 ( = 0,    ,  = 34 3,31 − 34 2,32 ) and SO ( = 0,   = 8 8 − 7 7 , Appendices C and D in Kervella et al. 2016).The beam size of the observations is 17.7 × 14.5 mas ≈ 1.1 × 0.9 au.The gap predicted by the chemical models should hence be observable, especially as the upper energy level of the SO 2 transition is 582 K. Therefore, under the assumptions of our chemical model of a disk that is formed at a specific time and then evolves in isolation of the outflow, we expect it to be not older than 10 5 yr.Hydrodynamical simulations suggest that while the disk is formed on a timescale of a few 100 yr, some outflowing gas will still be captured by and incorporated into the disk (Chen et al. 2016).If the disk is continually refreshed with new material, its age could be older.
L 2 Pup is currently in the thermally pulsing-AGB (TP-AGB) phase (Kervella et al. 2016), which is a short-lived phase of about 0.5 × 10 5 yr (Rosenfield et al. 2014).Assuming a static model, the disk could therefore both be a remnant from the early-AGB phase or have originated in the TP-AGB phase.The timeframe of 10 5 yr is too short for the massive planet to have been formed within this disk (which would take ∼10 Myr if formed via core accretion, Pollack et al. 1996), and is likely to be a first-generation planet present before the AGB phase.

Chemistry within the AGB disk
The chemistry within the disk around L 2 Pup is different to that in a low-density O-rich outflow both in distribution and composition.The only similarity is the lack of dust-gas chemistry, although this is due to different factors.In the outflow, the density is too low for gas-phase species to efficiently accrete onto the dust.The density in the disk midplane is at least four orders of magnitude larger, but the dust temperature is too high for accreted ices to remain on the dust surface.The dust temperature was taken to be that of the gas, a reasonable assumption given the density of the disk.However, a lower dust temperature could result in the depletion of gas-phase species and the formation of ices.
To compare the chemical composition of the disk to that of the outflow, we use the ratio of their column densities.This column density ratio is weighted by their respective CO column densities, with N the disk-averaged column density of species X (Eq.8) and   the column density of species  in the outflow.This normalises the ratio to the most abundant molecule (after H 2 ), allowing for a better comparison between the chemistries of these different density distributions.The CO column density ratio, NCO / CO , is ∼ 100 for all times in the disk model, decreasing from 126 at 10 3 yr to 98 at 10 6 yr.The chemical composition of the disk is different from that of the outflow: the disk does not simply inherit the initial composition, but also significantly modifies the abundances of parent and daughter species.Unlike in the outflow and despite its O-rich composition, numerous C-bearing molecules are abundantly formed throughout the disk, including COMs.Two main chemical regions can be distinguished: (i) the dense midplane, where chemistry is initiated by cosmic rays, and (ii) the outer edge, where it is initiated by interstellar UV photons.Within the outer region, the hot inner edge displays a different chemistry than the cooler outer regions.The chemistry in the dense midplane differs from that in a PPD because of the higher temperatures in the disk, while the chemistry of the outer regions is different to an AGB outflow due to the higher densities in this region.
Fig. 9 shows the column density ratio for a set of species which have a column density larger than 10 10 cm −2 at some time in the disk and which show a difference of at least an order of magnitude at some time in the disk chemical model.Their abundance profiles are shown in Fig. C4 and the absolute values of their column densities are shown in Fig. C5.These species can be regarded as chemical tracers of the presence of a disk structure.The left panel shows species which have a smaller column density in the disk model compared to the outflow model, the right panel shows those which have a larger column density.The shaded regions group species originating from different types of chemistry: midplane (cosmic ray) chemistry Figure 9. Ratio of the disk column density relative to CO over the outflow column density relative to CO (Eq.9) for species that show a decrease (left panel) and increase in column density relative to the outflow (right panel) of more than an order of magnitude at any time in the disk chemical model.Different markers and colours show the ratio at different times for the disk model.The shaded regions group species originating from different types of chemistry.Blue: midplane (cosmic ray) chemistry, orange: outer edge (photo) chemistry, dark orange: hot inner edge.Boldfaced labels denote species that are formed via cosmic ray and photochemistry.These species can be regarded as chemical tracers of the disk. is highlighted in blue, outer edge (photo) chemistry in orange, with species present in the hot inner edge highlighted in dark orange.Species formed via both cosmic ray and photochemistry are placed in the blue region with a boldface label.Sects.4.4.1 and 4.4.2elaborate on the cosmic-ray induced chemistry in the midplane and in the photochemistry in outer regions, respectively.

Cosmic-ray induced chemistry
Cosmic rays dissociate parent species in the midplane, liberating atoms which participate in a rich chemistry thanks to the high densities in this region.A large role is played by the reactive C, produced mostly by the destruction of the parent CO.Species whose column density is significantly impacted by this chemistry are shown in the blue shaded regions of Fig. 9.
Decrease in column density relative to the outflow Reactions between C and the parents SO and SO 2 form CS, leading to a gap in abundance in the dense midplane and a lower column density relative to the outflow.At 10 5 yr, their column densities are ∼ 5 and ∼ 9 times smaller than in the outflow, increasing to 100 times and ∼ 7 × 10 3 , respectively.CO 2 shows a similar gap in the midplane, which is caused by reactions with Si forming SiO.Its column density is more than 5 orders of magnitude smaller than in the outflow at 10 5 yr.At longer timescales, this difference is reduced to roughly 3 orders of magnitude thanks to its reformation by collisional dissociation of HCOOH (see below).
The decrease in column density of SiS, reaching about an order of magnitude after 5 × 10 5 yr, is caused by reactions with H 3 + , a product of cosmic-ray interactions.The resulting HSiS + reacts with the parent H 2 O, producing SiOH + which goes on to react with the parent NH 3 , funnelling Si away from SiS into SiO.
Increase in column density relative to the outflow Hydrogen is efficiently added to carbon at the high densities and temperatures in the midplane, forming CH 4 .Its column density is about 3 orders of magnitude larger in the disk.CH 4 reacts with N + , formed via cosmic rays, producing HCNH + .This molecule then reacts with H 2 S and NH 3 to form HCN and HNC, with column densities more than 1 and 3 orders of magnitude larger than in the outflow, respectively.The CS abundance increases thanks to the reactions of C with SO 2 and SO, increasing its column density by about 1 order of magnitude.
HCO + is produced by cosmic ray chemistry via H 3 + + CO.Radiative association of HCO + with H 2 O yields HCOOH 2 + , which reacts with H 2 O and NH 3 to form HCOOH (increase in column density ≳ 4 orders of magnitude compared to the outflow).
The CH 3 + cation plays a crucial role in the chemistry of the midplane.It is formed via CH 4 + He + , both products of cosmic rays, forming CH 2 + which reacts with H 2 to CH 3 + .The formation of all detected COMs (and HC 3 N) is linked to this cation.The high densities of the midplane allow for the slow radiative association reactions, dissociative recombination with dust, and collisional dissociation reactions in these pathways to be efficient.The main formation pathways of COMs via reactions with CH 3 + are: • Radiative association between the parent HCN and CH 3 + yields CH 3 CNH + , which dissociatively recombines with negatively charged dust grains to form CH 3 CN (increase in column density of ≳ 3 orders of magnitude at 10 5 yr compared to the outflow).
• NH 3 can radiatively associate with CH 3 + forming CH 3 NH 3 + , which dissociatively recombines with dust to CH 3 NH 2 (column density ≳ 6 orders of magnitude).The association with CH 3 + also forms C 2 H 5 CNH + , which dissociatively recombines with dust to C 2 H 5 CN (≳ 6 orders of magnitude).A sequence of collisional dissociations and reactions with C lead to the formation of CH 2 CCH, which reacts with N to form HC 3 N (≳ 7 orders of magnitude).
• Reactions of CH 3 + with the parent H 2 O gives CH 3 OH 2 + , which produces CH 3 OH by reacting with the parent NH 3 (≳ 3 orders of magnitude).Subsequent reaction with H 2 CO results in H 5 C 2 O 2 + , which dissociatively recombines with dust to HCOOCH 3 (≳ 8 orders of magnitude).
• The formation of methanol initiates the production of H 2 CO, increasing its column density by almost 3 orders of magnitude at 10 5 yr, which is formed by collisional dissociation of CH 2 OH and CH 3 O.Both have their origin in the CH 3 + + H 2 O reaction, which forms both CH 3 OH and CH 3 OH 2 + .CH 2 OH is formed via CH 3 OH + H, CH 3 O is formed after CH 3 OH 2 + reacting with CH 3 OH followed by dissociative recombination with dust and collisional dissociation.
Other species formed by the successive addition of hydrogen to carbon also play an active role in the cosmic-ray induced midplane chemistry.The column density of H 2 CS increases by more than 3 orders of magnitude at 10 5 yr.It is formed via CH 4 + S + (both with a cosmic-ray origin) producing H 3 CS + , which then dissociatively recombines with dust.HNCO is formed by CH 2 + NO and C 2 H 5 CN + OH, with a increased column density of about 3 orders of magnitude at 10 5 yr.The formation of HC 3 N is kickstarted by CH 4 + CH 3 + , forming C 2 H 5 + , which yields C 2 H 4 after reaction with the parents H 2 O and H 2 S. Subsequent reactions with C and N finally produce HC 3 N.

Interstellar UV photochemistry
The chemistry in the more tenuous outer regions of the disk is a rich photochemistry initiated by interstellar UV photons.However, because of the higher densities and temperatures of the disk, the products of this photochemistry are different to those of the photochemistry taking place in the outer regions of the outflow.C 2 H 2 plays an important role in the chemistry of the outer regions of the disk.It is formed by the successive addition of hydrogen to C (produced by photodissociation of mainly the parent CO) into CH, which then reacts with C again to form C 2 .These reactions lead to the formation of C 2 H and finally C 2 H 2 .Species whose column density is significantly impacted by this chemistry are shown in the orange shaded regions of Fig. 9.
Decrease in column density relative to the outflow The daughters OH, CN, HS, NH, and NH 2 are all first-generation daughters formed by photodissociation of the parents H 2 O, HCN, H 2 S, and NH 3 .They are present in a shell in the outer disk.Their main destruction mechanisms are photodissociation and reaction with H 2 to reform the parent species in a photodissociation -hydrogen addition loop, depending on the density.The significant decreases in their column densities relative to the outflow is caused by reactions funnelling them away from their parents.
Reactions with C-bearing species remove OH and CN from their photodissociation -hydrogen addition loops, reducing their column densities by ∼ 5 × 10 3 and 2 orders of magnitude, respectively.OH mainly reacts with C to reform CO and with CH 2 to produce H 2 CO, which is also formed by O + CH 3 .CN is involved in reactions with C 2 H 2 and C 2 H 4 , producing cyanopolyynes and CN-bearing C-chains.
The column densities of HS, NH and NH 2 decrease between one and two orders of magnitude relative to the outflow.S is removed from the photodissociation -hydrogen addition loop by the formation of CS via HS + C and S + CH 2 , and by the formation of H 2 CS via S + CH 3 .NH reacts with O to produce NO, which is then efficiently destroyed by Si forming SiO and NH forming N 2 O, which photodissociates into N 2 .Subsequently, the column density of NO is about two orders lower in the disk after 10 5 yr.NH also efficiently reacts with S to produce NS, which then reacts with O to form NO and SO.
The column density of the cation HCO + is almost 4 orders of magnitude smaller in the disk.Its main formation mechanism is C + + H 2 O.However, unlike in the outflow, C + is also involved in the Cchain chemistry in this region, which reduces the HCO + abundance.
Increase in column density relative to the outflow The column density of C 2 H 2 increases by about 3 orders of magnitude due to photochemistry in the outer regions of the disk.As stated earlier, C 2 H 2 plays an important role in the chemistry in this region.Reactions with SO + , formed by photoionisation of SO and by OH + S + , produce C 2 S, the column density of which increases by two to three orders of magnitude.Reactions with Si, liberated by photodissociation of the parent SiO, produce SiC 2 , whose column density increases by some 5 orders of magnitude at 10 5 yr.SiC is a product of SiC 2 + O, with the latter produced mostly via the photodissociation of the parents H 2 O and SiO.
These four species (C 2 H 2 , C 2 S, SiC 2 , and SiC) show large abundances at the inner edge of the disk, within the first few au (marked by the dark orange region in Fig. 9).This is caused by the strong temperature dependence of the H 2 + C 2 H reaction: its reaction rate depends on temperature as  2.57 (Laufer & Fahr 2004).This has a significant effect at the large temperatures in this region (between 1100 and 1750 K).Even though the Si + C 2 H 2 reaction has a negative temperature dependence ( −0.71 , Canosa et al. 2001), the large increase in C 2 H 2 abundance leads to a larger SiC 2 and SiC abundance in this region.
The formation of SiN and SiH 4 is linked to Si.The SiN column density is about an order of magnitude larger than in the outflow at 10 5 yr.Its production is started by NH 3 + Si + , with the latter produced by the photoionisation of Si or charge exchange with S + .This reaction forms SiNH 2 + which dissociatively recombines with electrons to SiN or recombines to HNSi, which is then photodissociated into SiN.Closer to the midplane, the cation dissociatively recombines with negatively charged dust grains.The SiH 4 column density is more than 5 orders of magnitude larger at 10 5 yr.It is formed by the successive addition of hydrogen to SiH + into SiH 5 + , which then reacts with H 2 O or dissociatively recombines with electrons to SiH 4 .SiH + is formed by Si + H 3 O + , with the latter formed by the addition of H to H 2 O + , a product of the photoionisation of H 2 O.

CONCLUSIONS
We presented an updated model of the density and temperature of the disk around L 2 Pup by fitting archival ALMA 12 CO and 13 CO  = 3 − 2 lines.The disk has a surface density profile Σ() ∼  −1 and is warm, with a midplane temperature of 900 K and  mid () ∼  −0.2 .Unlike Homan et al. (2017), we do not require a three-staged radial temperature profile or an extremely steep vertical density profile (our () ∼  −0.4 vs their ∼  −3.1 .A more simple parametrisation used commonly for PPDs is able to reproduce the observations.
The chemical model presented in this paper is the first chemical model of a disk around an AGB star, specifically of L 2 Pup's disk.The physical structure of the disk has a large impact on its chemistry.All species are constrained to the disk, within ∼ 15 au radially and ∼ 10 au vertically.This is much smaller than the extent of the parents' envelope in the outflow, which ranges from 30 − 3000 au radially from the star and is also where the peak abundances of the daughter species are located.
The chemistry within the disk leads to large abundances of Cbearing molecules, a surprising result given its O-rich composition.Certain COMs are abundantly formed in the midplane.Unlike in PPDs, these are formed purely via gas-phase chemistry as dust-gas chemistry does not play a role at the high temperatures of L 2 Pup's disk.C-chains are readily formed in the outer regions, though caution is needed as the chemical network does not include fully saturated long (more than three C atoms) carbon-chain species.Two chemi-cal regimes can be distinguished: cosmic-ray induced chemistry in the midplane and photoinduced chemistry in the outer regions.The abundances of species are significantly increased or decreased with respect to an O-rich outflow chemical model, allowing us to identify potential and unique tracers of the disk such as H 2 CO, CH 3 CN, and HC 3 N.Moreover, the chemical model can be used to constrain the age of the disk.The appearance of a gap in abundance in the midplane of SO and SO 2 after 5 × 10 5 yr appears to suggest that the disk around L 2 Pup is about 10 5 yr old, as these features were not observed in the data presented by Kervella et al. (2016).
Further observations are necessary to further constrain the physical structure of the disk.In particular, optically thin CO isotopologues are needed to better pin down the disk mass and density structure, while species such as H 2 CO, HC 3 N, CH 3 CN, and C 3 H 2 can be used as empirical temperature probes.Targeting the tracers of the midplane and outer region chemistries will provide an essential test of the chemical model predictions.
Our unique chemical model clearly shows the impact that largescale structures within an AGB outflow can have on its chemistry.This exploratory work paves the way for a more general study of AGB disks, ranging in density and temperature structures.However, note that the 2D model used in this work predicts the chemistry assuming a fixed density structure; coupling chemistry to a hydrodynamical model is necessary to take the effect of dynamical evolution of the chemistry into account.

APPENDIX C: CHEMICAL MODEL RESULTS
This paper has been typeset from a T E X/L A T E X file prepared by the author.
Table C1.Parent species and their initial abundances relative to H 2 for the disk and outflow chemical kinetics models.Abundances are derived from observations, as compiled by Agúndez et al. (2020).Species Initial abundance He 0.17

Figure 1 .
Figure 1.Slice through the moment 0 map of 12 CO (left) 13 CO  = 3 − 2 (right) and along the major axis of the disk (blue) together with the Magritte modelling results (black).The range marked in blue corresponds to the rms error of the moment 0 map of the data.Model parameters are listed in Table1.

Figure 2 .
Figure 2. Density (upper panel) and temperature (lower panel) of the Magritte model that best reproduces the observations of L 2 Pup's disk.The parameters of the physical model are listed in Table1.

Figure 3 .
Figure 3. Fractional abundance of CO (top panels) and SO (bottom panels) throughout the disk as predicted by the chemical model.The colour map shows the logarithm of the abundance w.r.t.H 2 .The different columns show the abundance maps after different times, increasing from left to right.

Figure 4 .
Figure 4. Radial extent of the parent species predicted by the disk and outflow chemical models in au.Their radial extent in the disk measured at the midplane is shown at different times, characterised by different markers and colours.The grey cross shows their extent in a smooth outflow model.

Figure 5 .
Figure 5. Fractional bundance of the daughters OH, SiC 2 , CH 3 CN, and HCOOH throughout the disk as predicted by the chemical model at 10 5 yr.The colour map shows the logarithm of the abundance w.r.t.H 2 .Note that each colourmap has a different dynamic range.

Figure 6 .
Figure 6.Fractional abundances of complex organic molecules (COMs) formed within the disk as predicted by the chemical model after 10 5 yr.The colour map shows the logarithm of the abundance w.r.t.H 2 .Note that each colour map has a different dynamic range.The additional COMs CH 3 CN and HCOOH are shown in Fig. 5.

Figure 7 .
Figure 7. Disk-averaged column densities of COMs produced in the chemical model of the disk at different times, characterised by different markers and colours.The grey cross shows the results from a smooth outflow model.

4. 1
Physical structure of the disk We compare our retrieved physical structure of the disk around L 2 Pup to PPDs and post-AGB disks in Sect.4.1.1.A comparison to the previous model of Homan et al. (2017) is made in Sect.4.1.2.Since our physical model was retrieved using just two molecular emission lines, we describe the observations needed for a further refinement of the model in Sect.4.1.3.

Figure 8 .
Figure 8.Total fractional abundance of all species with 6 or more C atoms (called carbon sinks, Sect.3.4) in the disk as predicted by the chemical model.The colour map shows the logarithm of the abundance w.r.t.H 2 .The different columns show the abundance maps after different times, increasing from left to right.
) (Sect.4.1.2).A better constraint of the physical structure of L 2 Pup's disk will address the differences with theHoman et al. (2017) model and is a crucial input to the chemical model and its predictions.Spatially resolved emission of abundant optically thin CO iso-topologues, such as C 18 O, C 17 O, and 13 CO (Danilovich et al. 2017; De Nutte et al. 2017), can help determine the disk gas mass and its density distribution more accurately as they probe different layers within the disk.Similarly, H 2 CO (e.g., Qi et al. 2013; Pegues et al. 2020), HCN (e.g., Long et al. 2021), HC 3 N, CH 3 CN, and C 3 H 2 (e.g., Bergner et al. 2018; Ilee et al. 2021), are well-known temperature probes in PPDs.Our chemical model predicts column densities of 3.72 × 10 14 , 8.54 × 10 17 , 4.21 × 10 13 , 2.80 × 10 15 , and 3.65 × 10 12 cm −2 , respectively, for these molecules.CH 3 CN and C 3 H 2 have yet not been detected towards O-rich AGB stars: their detection would put strong constraints on our chemical model.HC 3 N has only been detected around OH 231.8+4.2, which has a bipolar outflow with velocities up to ∼ 400 km s −1 (Velilla

Figure A1 .Figure A2 .Figure B1 .Figure B2 .
Figure A1.Channel maps of 12 CO  = 3 − 2 emission toward L 2 Pup.The grey contours show flux levels at 3 and 5 times the rms noise.The beam is shown in white in the bottom left hand corners.

Figure B3 .
Figure B3.Modelled moment 0 map of 12 CO (left) 13 CO  = 3 − 2 (right).The grey contours show flux levels at 3, 5, 10, 20, and 30 times the rms noise.The beam is shown in white in the bottom left hand corners.Model parameters are listed in Table1.

Figure C2 .
Figure C2.Fractional abundance of SO 2 (top row) and SiS (bottom row) throughout the disk as predicted by the chemical model.The colour map shows the logarithm of the abundance w.r.t.H 2 .The different columns show the results after different times, increasing from left to right.

Figure C3 .
Figure C3.Fractional abundances of the parent species throughout the disk as predicted by the chemical model of the parent species after 10 5 yr.The colour map shows the logarithm of the abundance w.r.t.H 2 .Note that each colour map has a different dynamic range.

Figure C4 .
Figure C4.Fractional abundances of some daughter species shown in Fig. 9 throughout the disk as predicted by the chemical model of the parent species after 10 5 yr.The colour map shows the logarithm of the abundance w.r.t.H 2 .Note that each colour map has a different dynamic range.

Figure C5 .
Figure C5.Disk column density for species that show a decrease (left panel) and increase in column density relative to the outflow (right panel) of more than an order of magnitude at any time in the disk chemical model.Different markers and colours show the ratio at different times for the disk model.The shaded regions group species originating from different types of chemistry.Blue: midplane (cosmic ray) chemistry, orange: outer edge (photo) chemistry, dark orange: hot inner edge.Boldfaced labels denote species that are formed via cosmic ray and photochemistry.

Table 1 .
Williams & Best (2014)l for L 2 Pup that best reproduces the data following the parametrisation ofWilliams & Best (2014).The top parameters were fixed based on previous observations.The bottom parameters were obtained from modelling the 12 CO and 13 CO  = 3 − 2 data.Their ranges are described in Sect.2.3.