11institutetext: Dipartimento di Matematica e Fisica, Università degli Studi
Roma Tre, via della Vasca Navale 84, 00100, Roma, Italy 22institutetext: INAF,
Observatory of Rome, Via Frascati 33, 00077 Monte Porzio Catone (RM), Italy
33institutetext: School of Mathematical and Physical Sciences, Macquarie
University, Sydney, NSW, Australia 44institutetext: Astronomy, Astrophysics
and Astrophotonics Research Centre, Macquarie University, Sydney, NSW,
Australia 55institutetext: Istituto Nazionale di Fisica Nucleare, section of
Perugia, Via A. Pascoli snc, 06123 Perugia, Italy 66institutetext: Institute
of Astronomy, K.U.Leuven, Celestijnenlaan 200D bus 2401, B-3001 Leuven,
Belgium

# Understanding dust production and mass loss on the AGB phase using post-AGB
stars in the Magellanic Clouds

S. Tosi 1122 F. Dell’Agli 22 D. Kamath 3344 P. Ventura 2255 H. Van Winckel 66
E. Marini 22

(Received September 15, 1996; accepted March 16, 1997)

###### Abstract

Context. The asymptotic giant branch (AGB) phase of evolution in low- and
intermediate-mass stars is governed by poorly understood physical mechanisms,
such as convection, mixing, dust production and mass loss, which play a
crucial role on determining the internal structure and the evolution of these
stars. The spectra of post-asymptotic giant branch (post-AGB) stars hold
critical chemical fingerprints that serve as exquisite tracers of the
evolution, nucleosynthesis, and dust production during the AGB phase.

Aims. We aim to understand the variation of the surface chemistry that occurs
during the AGB phase by analysing results from observations of single post-AGB
stars in the Magellanic Clouds. We also aim at reconstruct dust formation
processes, that are active in the circumstellar envelope of AGB stars,
occurring towards the end of the AGB phase and during the subsequent course of
evolution when contraction to the post-AGB has begun.

Methods. We study likely single post-AGB sources in the Magellanic Clouds that
exhibit a double-peaked (shell-type) spectral energy distribution (SED). We
interpret their SED by comparing with results from radiative transfer
calculations, to derive the luminosity and the dust content of the individual
sources. Additionally, we compare the observationally derived stellar
parameters and the photospheric chemical abundances of the target sample with
results from stellar evolution modelling of AGB and post-AGB stars. This
allows for the characterization of the individual sources in terms of initial
mass and formation epoch of the progenitors. The theoretically derived dust
mineralogy and optical depth is used to assess when dust formation ceases and
determine the propagation velocity of the dust-gas system during the post-AGB
evolution.

Results. We find that amongst our target sample of 13 likely single post-AGB
stars with shell-type SED, 8 objects are carbon stars descending from $\sim
1-2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitors. 5 of the 13 objects
are of lower mass, descending from ${\rm M}<1\leavevmode\nobreak\ {\rm
M}_{\odot}$ stars. Based on the dust mineralogy, we find that these 5 stars
are surrounded by silicate dust, and thus failed to become carbon stars. The
dust optical depth and the luminosity of the stars are correlated, owing to
the faster evolutionary time-scale brighter stars, which makes the dusty layer
to be closer to the central object. From our detailed analysis of the SEDs, we
deduce that the dust currently observed around post-AGB stars was released
after the onset of the central star contraction and an increase in the
effective temperature to $\sim 3500-4000$ K.

###### Key Words.:

stars: AGB and post-AGB – stars: abundances – stars: evolution – stars: winds
and outflows

##  1 Introduction

After the core helium burning phase, single stars of initial mass below $\sim
8\leavevmode\nobreak\ {\rm M_{\odot}}$, i.e., low- and intermediate-mass
stars, evolve through the asymptotic giant branch (AGB). When the mass of the
envelope drops below a few hundredths of solar mass the external regions of
the star contract, thus giving the start to the post-AGB evolution (Iben &
Renzini, 1983). During this phase the stars evolve at constant luminosity and
readjust on a configuration which becomes more and more compact as the time
elapses, with the effective temperature increasing from $\sim 3000$ K to $\sim
30000$ K. The post-AGB phase is then followed by the PN evolution, before the
start of the white dwarf cooling (Iben & Renzini, 1983).

The AGB phase is characterized by the occurrence of two physical phenomena
able to alter the surface chemistry of these stars: third dredge-up (TDU, Iben
1974) and hot bottom burning (HBB, Sackmann & Boothroyd 1992). These
mechanisms are still not fully understood from first principles and the
information derived from the analysis and the characterization of post-AGB
stars proves an exceptionally valuable tool for this purpose. The main reason
is twofold: their chemical composition represents the final outcome of AGB
evolution and associated internal enrichment processes. Another reason is that
the surface abundances of post-AGB stars can be determined accurately given
that their spectra are dominated by atomic transitions.

Furthermore, given the peculiar morphology of the SED of post-AGB stars (Woods
et al., 2011), the study of the infrared (IR) excess allows a reliable
determination of the dust properties, in terms of mineralogy, optical
thickness and current distance from the central star: this kind of analysis
provides important information on the dust formation process in the wind of
AGBs during the final evolutionary phases, and more generally on the
efficiency of dust formation in the circumstellar envelope of evolved giants.
This step is of paramount importance to evaluate the role that AGB stars play
as dust manufacturers: indeed it was shown that a few stars evolving through
the late AGB phases are responsible for most of the current dust formation
rate in the dwarf irregular galaxies like IC1613 (Dell’Agli et al., 2016),
IC10 (Dell’Agli et al., 2018), Sextans A (Dell’Agli et al., 2019).

In the present work we use results from observations of post-AGB stars to
characterize the individual sources in terms of mass and age of the progenitor
stars. This step offers the possibility of testing the current AGB evolution
theories, particularly when the surface chemistry of the stars is known from
the observations, and can be compared with the theoretical predictions.
Furthermore, the analysis of the IR excess is used to shed new light on the
dust formation process during the very late AGB phases and possibly after the
beginning of the contraction to the post-AGB. This investigation can draw
information on the time-scale of the AGB to post-AGB transition, and to
investigate the dynamics of the wind moving away from the star after dust
production ceases.

This study benefits from the rich dataset of optically visible post-AGB stars
observed in the Magellanic Clouds (Kamath et al. 2014, 2015, hereafter K14 and
K15). For the single stars in these samples the availability of the optical
and infrared data allowed the reconstruction of the spectral energy
distribution (SED) and the determination of the IR excess, which prove
essential for an estimate of the luminosity and for the determination of the
mineralogy and amount of dust in the surroundings of the star.

The stars are interpreted and characterized under the light of updated AGB
evolutionary sequences, extended on purpose until and past the post-AGB phase.
These tracks also include the description of the dust formation process,
following the schematization proposed by the Heidelberg group (Ferrarotti &
Gail, 2006), on the wake of previous studies on dust production by evolved
stars proposed by our team (Ventura et al., 2012, 2014), which have been used
to interpret the evolved stellar populations of the Magellanic Clouds
(Dell’Agli et al., 2014a, b, 2015a, 2015b) and of Local Group galaxies
(Dell’Agli et al., 2016, 2018, 2019), and to explore the potentialities in
this field offered by the recent launch of the James Webb Space Telescope
(Marini et al., 2020, 2021).

The paper is structured as follows: Section 2 includes the description of the
codes used to model the evolution of the star and the dust formation process
during the AGB and the early post-AGB phases, and the methodology followed to
interpret the optical and infrared observations of the selected sources, to
derive the luminosities and the properties of the dust in their surroundings;
the results of the SED fitting analysis are given in section 3; in section 4
we characterize the sources individually, in order to find out the mass and
age of the progenitors; section 5 presents results regarding the mass loss
suffered by the stars during the late AGB phases, in agreement with the
expectations from stellar evolution modelling; in section 6 we discuss the
properties of the wind and of the physics of the AGB - post-AGB transition,
based on the results obtained in the previous sections. Finally, the
conclusions are given in section 7.

##  2 Numerical, chemical and physical input

The characterization of the stars considered in the present study is based on
results from stellar evolution and dust formation modelling, which are used in
synergy with SED fitting analysis, applied to all the sources investigated. In
the following we describe the most relevant physical ingredients of the codes
used to model the evolution of the star, to describe the dust formation
process in the outflow, and to build synthetic SEDs, which are compared with
the observational data set available.

###  2.1 AGB and post-AGB modelling

The evolutionary sequences were calculated by means of the stellar evolution
code ATON, extensively described in Ventura et al. (1998). The code performs a
full integration of the stellar structure equations, from the centre of the
star to the photosphere. The temperature gradient within regions unstable to
convection is found via the Full Spectrum of Turbulence model (Canuto &
Mazzitelli, 1991). The formal borders of the convective regions are found via
the classic Schwartzschild criterion; however, convective eddies are allowed
to overshoot into radiatively stable regions, by imposing an exponential decay
of velocities beyond the formal border. This approach, which is consistent
with results from numerical simulations (Freytag et al., 1996), was also used
e.g. by Herwig (2000). In the convective regions (including the overshoot
zones) nuclear burning and mixing of chemicals are self-consistently coupled
by means of the diffusive approach proposed by Cloutman & Eoll (1976). Mass
loss during the oxygen-rich phases is described by using the formulation by
Blöcker (1995); for carbon stars we use the treament proposed by the Berlin
group (Wachter et al., 2002, 2008).

The metallicity of the stars considered here span the $-1\leq[{\rm Fe}/{\rm
H}]\leq-0.4$ range. We divided the sources in two groups, according to the
metallicity, and used the $Z=2\times 10^{-3}$ and $Z=4\times 10^{-3}$
evolutionary tracks that we calculated for the study by Kamath et al. (2021),
to study the stars of lower and higher metallicity, respectively. To
investigate the highest luminosity objects we extended the set of models by
Kamath et al. (2021) to include $2\leavevmode\nobreak\ {\rm M}_{\odot}$ and
$2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ evolutionary sequences of
metallicity $Z=2\times 10^{-3}$, which were calculated on purpose for the
present work. Furthermore, the study of the faintest stars required the
calculation of $0.65\leavevmode\nobreak\ {\rm M}_{\odot}$ and
$0.7\leavevmode\nobreak\ {\rm M}_{\odot}$ models of the same metallicity.

###  2.2 The description of dust formation in the wind of evolved stars

For some selected points along the evolutionary tracks we use the physical
parameters of the star, namely mass, luminosity, effective temperature, mass
loss rate and surface chemical composition, to determine the dust production
rate. To this aim we rely on the schematization proposed by the Heidelberg
group (Ferrarotti & Gail, 2006), according to which the dust forms in a
stationary wind, expanding radially from the photosphere of the star.

The wind velocity $v$ is found via the equation of momentum conservation,
which is

${d{\rm v}\over dr}=-{G{\rm M}_{*}\over r^{2}}(1-\Gamma)$ $None$

where $r$ is the radial distance from the center of the star, ${\rm M}_{*}$ is
the (current) mass of the star and $\Gamma={k{\rm L}_{*}/4\pi cG{\rm M}_{*}}$.
In the expression for $\Gamma$, which represents the relative weight of the
effects of radiation pressure and gravity, ${\rm L}_{*}$ is the luminosity of
the star, whereas $k$ is the extinction coefficient, which reflects the
scattering and absorption processes of the photons released from the stellar
photosphere by dust particles.

Therefore, the growth of dust grains and the dynamics of the wind are self-
consistently coupled, as the calculation of $k$ requires the determination of
the size of the dust particles of the different species formed.

The description of the wind is completed by the radial profiles of the
density, which for mass conservation reasons is expressed by the equation

$\rho={\dot{\rm M}\over{4\pi r^{2}{\rm v}}}$ $None$

where $\dot{\rm M}$ is the mass loss rate, whereas v is the wind velocity at
distance $r$ from the centre of the star.

The relevant equations and the details of the numerical treatment are
extensively described, e.g., in Ventura et al. (2012, 2014).

The knowledge of the radial stratification of the thermodynamic variables and
of the grain sizes of the various dust species allows the calculation of the
optical depth, which is found via the following expression:

$\tau_{10}=\pi\int_{r_{in}}^{r_{out}}{n_{d}Q_{10}a^{3}dr}$ $None$

where $n_{d}$ and $a$ are the number density and the size of the dust grains,
$Q_{10}$ is the extinction coefficient at $10\leavevmode\nobreak\ \mu$m,
$r_{in}$ and $r_{out}$ are the distances of the inner and outer borders of the
region of the circumstellar envelope populated by dust particles. The overall
optical depth must be calculated by summing all the contributions from the
individual dust species, the largest contribution being given by those with
the highest extinction coefficients, i.e. solid carbon or silicate dust.

###  2.3 SED Fitting

To build synthetic SED for the comparison with the observational data points
we use the radiation transfer code DUSTY, described in Nenkova et al. (1999).
The code calculates the SED emerging from the star, by accounting for the
reprocessing of the radiation released from the photosphere by a dusty region,
isotropically distributed around the source.

To describe the input radiation entering the dusty layer we use the Kurucz-
Castelli atmosphere models (Castelli & Kurucz, 2003), available for a wide
range of metallicities and effective temperatures.

The effects of dust reprocessing are taken into account by specifying the
optical depth $\tau_{10}$ (see Eq. 3 above for the definition) and the dust
temperature in the inner border of the dusty region, ${\rm T}_{\rm d}$.

For what regards the dust mineralogy, we consider the possibilities that the
dust is composed either by $100\%$ silicates or it is pure carbon dust. In the
latter case, for the higher metallicity sources, we further explore the
possibility that some fraction of silicon carbide is present.

The size of the dust particles is chosen on the basis of the results from dust
formation modelling, described in section 2.2. One of the outcomes of this
modelling is the asymptotic size attained by the dust grains of the individual
species, which is used as input for the dusty code. Typically, we adopt
$0.12\leavevmode\nobreak\ \mu$m for solid carbon, $0.07\leavevmode\nobreak\
\mu$m for SiC, and $0.08\leavevmode\nobreak\ \mu$m for silicates.

The optical constants required as input by DUSTY and considered in the present
analysis are the following: Ossenkopf et al. (1992) for silicates; Zubko et
al. (1996) for amorphous carbon; Pegourie (1988) for silicon carbide; we will
consider the possibility that in the low temperature domain the silicates
assume a crystalline form, in which case we use the optical constants by
Jaeger et al. (1994).

##  3 The analysis of SEDs of the target sample of post-AGBs

Figure 1: Optical and IR data (black points) of SMC and LMC sources classified
as post-AGBs with a shell-like structure by K14 and K15, that we interpret as
surrounded by carbonaceous dust in the present investigation. The grey line in
the middle panel of the bottom line refer to IR spectroscopy results from Volk
et al. (2011). The red lines indicate the best-fit model obtained by the DUSTY
code. The derived stellar and dust parameters for each source are shown in the
different panels.

K14 and K15 assembled results from optical and infrared photometry of evolved
stars in the Magellanic Clouds, and reconstructed the shape of the SED of the
different sources over an extended spectral region. In this work we focus on
13 single stars that were classified as shell-type post-AGBs by K14 and K15.
This choice allows us to use the code DUSTY, described in section 2.3, in the
attempt of providing an interpretation of the data of the different sources.

Our approach is to confront results from synthetic SED modelling with the
photometric data set, to derive the most relevant quantities related to the
individual stars, namely luminosity, dust mineralogy, optical depth and dust
temperature of the internal border of the dusty region. For only one of the
sources considered, namely J004441.03, we could compare the synthetic SED with
the IR spectrum (Volk et al., 2011).

To this aim we start from the effective temperatures and metallicities
reported in K14 and K15, or, when available, from high-resolution spectroscopy
analysis. These information are required to calculate the input spectrum
released from the photosphere of the star, and entering the dusty zone. We
adjust the input parameters required to produce the synthetic SED, until
satisfactory agreement with the observations is reached. This procedure leads
to the derivation of the stellar luminosity and of the dust quantities,
according to the following steps:

  1. 1.

The dust mineralogy can be reliably assessed, because the shape of the SED of
carbon stars differs significantly from that of stars surrounded by silicates
in the $20-30\leavevmode\nobreak\ \mu$m spectral region: indeed the presence
of silicates causes a strongly negative slope of the flux with $\lambda$ at
wavelengths above $\sim 20\leavevmode\nobreak\ \mu$m, whereas in the case of
carbon stars the SED is still decreasing in the same spectral region, but it
is significantly flatter.

  2. 2.

The optical depth of the dust surrounding the individual sources can be
accurately derived, because the dust layer is sufficiently far from the
surface of the star, that the infrared excess in the $8-30\leavevmode\nobreak\
\mu$m spectral region is indeed dependent on $\tau_{10}$ only.

  3. 3.

The peculiar morphology of the SED of post-AGB stars allows the determination
of the dust temperature ${\rm T}_{\rm d}$ of the internal border of the dust
zone, which turns extremely sensitive to the depth and the location of the
minimum in the SED, usually found in the $3-6\leavevmode\nobreak\ \mu$m
spectral region.

  4. 4.

The morphology of the SED in the optical part of the spectrum is used to
derive the values of the interstellar reddening, for which we find consistency
with the results published in K14 and K15.

  5. 5.

The luminosity of the star can be estimated by shifting the synthetic SED
until matching the observed SED profile in the near infrared region.

Once the mineralogy of the dust is determined, the dust temperature can be
derived with an accuracy of the order of $10-20$K. In the cases where WISE,
IRAC and MIPS photometry are available, the optical depth can be fixed with a
$10-15\%$ precision. Finally, the luminosity of the stars is established
mostly from the near-IR part of the spectrum with an uncertainty within
$10\%$.

Figure 2: Same as Fig. 1, reporting the sources that we interpret as surrounded by silicate dust. The symbols used are the same as in Fig. 1. The fraction of silicate dust in the crystalline form is given in each panel. Table 1: Physical and dust properties of the LMC and SMC sources classified as shell-type post AGB stars by K14 and K15. The quantities listed in the various columns are the following: 1 - source ID; 2, 3: metallicity and effective temperatures derived spectroscopically by K14 and K15; 4-7: luminosity, optical depth at $10\leavevmode\nobreak\ \mu$m, dust temperature and distance separating the central star from the inner border of the dusty region, found via SED fitting; 8: mass of the progenitor at the beginning of the AGB phase, deduced via comparison of the derived luminosity with results from post-AGB evolution modelling.  ID |  $[$Fe$/$H$]$ |  ${\rm T}_{\rm eff}$[K] | ${\rm L}/{\rm L}_{\odot}$ | $\tau_{10}$ |  ${\rm T}_{d}$[K] | ${\rm R}_{\rm in}/{\rm R}_{\odot}$ | ${\rm M}/{\rm M}_{\odot}$
---|---|---|---|---|---|---|---
J052220.98 | -0.50 | 5750 | 4500 | 0.015 | 220 | $2.63\times 10^{5}$ | 0.9
J053250.69 | -1.10 | 6000 | 5200 | 0.006 | 250 | $2.06\times 10^{5}$ | 1.00
J004114.10 | -1.04 | 5750 | 5800 | 0.007 | 280 | $1.65\times 10^{5}$ | 1.00
J050632.10 | -0.40 | 7600 | 6000 | 0.003 | 240 | $2.74\times 10^{5}$ | 1.25
J003643.94 | -0.63 | 7500 | 6500 | 0.002 | 250 | $2.56\times 10^{5}$ | 1.50
J051848.84 | -1.00 | 6000 | 6700 | 0.008 | 240 | $2.62\times 10^{5}$ | 1.25
J004441.03 | -1.07 | 6000 | 8500 | 0.002 | 320 | $6.64\times 10^{4}$ | 2.00
J005803.08 | -1.03 | 6500 | 12000 | 0.025 | 280 | $1.20\times 10^{5}$ | 2.50
J045119.94 | -0.40 | 8250 | 5000 | 0.006 | 270 | $2.97\times 10^{4}$ | 0.80
J052241.52 | -0.50 | 8250 | 5000 | 0.017 | 400 | $2.26\times 10^{4}$ | 0.80
J052740.75 | -0.50 | 8250 | 5200 | 0.024 | 470 | $1.22\times 10^{4}$ | 0.90
J050221.17 | -0.60 | 5250 | 4500 | 0.022 | 180 | $5.45\times 10^{4}$ | 0.80
J051906.86 | -1.30 | 5500 | 3800 | 0.015 | 180 | $5.13\times 10^{4}$ | 0.65

Fig. 1 and 2 show the analysis done for each of the sources considered,
divided according to whether they are interpreted as surrounded by carbon dust
and silicates, respectively. We note that the error bars of the photometric
data of the post-AGBs in SMC/LMC (from K14 and K15) are small and hence we do
not display them in our SED fits. In the cases where multi-epoch observations
for the same band are available, we show all the results in a vertical
sequence, in conjunction with the central wavelength of the corresponding
filter.

Tab. 1 reports the metallicities, the effective temperatures, and the
quantities derived from SED fitting, namely the luminosity and the dust
properties, i.e. the mineralogy, the optical depth, the dust temperature and
the distance from the centre of the star of the inner border of the dust
layer. The last column gives the estimated mass of the progenitors at the
beginning of the AGB phase (see next section).

We find that the dust in the surroundings of the stars that we classify as
oxygen-rich sources is composed by a mixture of amorphous and crystalline
silicates, with the latter component accounting for $\sim 50\%$ of the total
dust. The presence of a significant fraction of dust in the crystalline form
is required to reproduce the strongly negative slope of the SED in the
$\lambda>20\leavevmode\nobreak\ \mu$m spectral region, as determined by the
combination of the WISE4 and MIPS data. Use of a pure amorphous dust in the
SED fitting procedure does not allow to reproduce the shape of the SED,
independently of the optical constants adopted.

We note that this conclusion relies on the assumptions upon which the DUSTY
code is based, namely that the dusty zone is spherically distributed and is
characterized by a single dust component and temperature. Taking into account
all these factors would not change the general conclusions drawn regarding the
progenitors of the individual sources and the amount of dust in their
surroundings, but would likely reflect into the relative fractions of
amorphous and crystalline silicates in the case of M-type sources.

The presence of crystalline silicate dust has been so far detected in AGB
sources with optically thick dust envelopes in the Galaxy (Sylvester et al.,
1999). Furthermore, emission bands associated to crystalline silicates in
evolved stars in the LMC were studied by Jones et al. (2014), down to (gas)
mass-loss rates of $\sim 10^{-6}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr,
fairly consistent with the mass-loss rates that low-luminosity, oxygen-rich
stars attain at the tip of the AGB (see section 6.5 below). The explanation
proposed here holds as far as no change in the structure of silicate dust
occurs as the dust moves away while the star contract to the post-AGB phase.

##  4 The progenitors of the post-AGB stars sample

We characterize the sources discussed in the previous section to infer the
mass and formation epoch of the progenitors. To this aim we consider the
position of the stars on the HR diagram, based on the values of effective
temperature and luminosity listed in Tab. 1, and the evolutionary tracks of
model stars of different mass, with the metallicity similar to those reported
in the col. 2 of Tab. 1. This approach can be safely used in the case of post-
AGBs, given the peculiar morphology of the tracks, which run practically
horizontal on the HR plane.

We show in Fig. 3 the position of the different sources on the HR diagram,
overimposed to the evolutionary tracks of $0.65-2.5\leavevmode\nobreak\ {\rm
M}_{\odot}$ stars111The masses considered here refer to the beginning of the
AGB phase, thus they do not take into account possible mass loss during the
RGB ascending. We separate metal-poor sources, with $[$Fe$/$H$]\sim-1$, from
the higher metallicity counterparts. The former, reported in the left panel of
Fig. 3, are compared with tracks of metallicity $Z=2\times 10^{-3}$, whereas
to interpret the latter group we use evolutionary sequences with metallicity
$Z=4\times 10^{-3}$.

In the following we present the interpretation of the stars, discussing
separately the carbon stars and the oxygen-rich objects, grouped according to
the estimated age and metallicity. When available, results from high-
resolution spectroscopy, which allowed the determination of the surface
chemical composition, will be compared with the expectations from stellar
evolution modelling, thus providing a further test of the interpretation
proposed here.

Figure 3: Post-AGB evolutionary tracks of model stars of metallicity
$Z=2\times 10^{-3}$ (left panel) and $Z=4\times 10^{-3}$ (right). The green
track in the right panel refers to a $0.9\leavevmode\nobreak\ {\rm M}_{\odot}$
model star, which experienced a late TP, shortly after the beginning of the
contraction towards the post-AGB. Coloured points indicate the observationally
derived position of the sources considered in the present investigation,
according to the effective temperatures given in K14 and K15 and the
luminosities derived in section 3. Red squares and blue triangles refer to the
sources surrounded by carbonaceous dust and silicates, respectively.

###  4.1 Carbon stars

####  4.1.1 J053250.69 and J004114.10: low-mass, metal-poor carbon stars

Figure 4: Left: Time variation of the surface C$/$O ratio of a
$1\leavevmode\nobreak\ {\rm M}_{\odot}$, metal-poor model star during the AGB
phase, until the post-AGB evolution. Red square and line indicate the measured
value and the error bar from van Aarle et al. (2013). Right: The time
evolution of the surface carbon of two model stars of mass
$1\leavevmode\nobreak\ {\rm M}_{\odot}$ (solid, black line) and
$1.1\leavevmode\nobreak\ {\rm M}_{\odot}$ (green, dotted line), compared with
results from van Aarle et al. (2013).

The lowest luminosity stars among the metal-poor objects surrounded by carbon
dust are J053250.69 and J004114.10. From their position on the HR diagram,
shown in the left panel of Fig. 3, we deduce that they descend from stars with
mass equal to or slightly greater than $1\leavevmode\nobreak\ {\rm
M}_{\odot}$. As far as J053250.69 is concerned, the presence of carbon dust is
consistent with the results from van Aarle et al. (2013), that indicate that
the surface chemistry is enriched in carbon. The surface C$/$O and
$[$C$/$Fe$]$ are in agreement with results from stellar evolution modelling:
this is shown in Fig. 4, where the values given in van Aarle et al. (2013) are
compared with the AGB time variation of the C$/$O ratio and of the surface
carbon content of a $1\leavevmode\nobreak\ {\rm M}_{\odot}$ model star. The
expected increase in the surface $[$O$/$Fe$]$ (not shown in Fig. 4), of the
order of $0.1$ dex, is $0.1-0.2$ dex below the lower limit given in van Aarle
et al. (2013).

Regarding J004114.10, there are no results from spectroscopy. According to the
estimated luminosity we expect that the progenitor is $\sim
0.1\leavevmode\nobreak\ {\rm M}_{\odot}$ more massive than for J053250.69. In
this case we expect a higher carbon enrichment, with a final $[$C$/$Fe$]\sim
1.5$ (see right panel of Fig. 4).

Based on these results, and considering a $0.1-0.2\leavevmode\nobreak\ {\rm
M}_{\odot}$ mass loss during the red-giant branch (RGB) ascending, we expect
that these two sources descend from metal-poor progenitors of mass in the
$1.2-1.3\leavevmode\nobreak\ {\rm M}_{\odot}$ range, born $\sim 3$ Gyr ago.

Figure 5: Time variation of the surface carbon (red line) and oxygen (blue)
content during the AGB evolution of a $1.25\leavevmode\nobreak\ {\rm
M}_{\odot}$ model star of metallicity $Z=0.002$. The red and blue points on
the right side of the plane (with the corresponding error bars) indicate
results for J051848.84 from De Smedt et al. (2015).

####  4.1.2 J051848.84

For this source we derive a luminosity ${\rm L}\sim 6700\leavevmode\nobreak\
{\rm L}_{\odot}$, consistent with the conclusions from De Smedt et al. (2015).
This luminosity, as shown in the left panel of Fig. 3, corresponds to a $\sim
1.25\leavevmode\nobreak\ {\rm M}_{\odot}$ star. The conclusion drawn in
section 3, that the star is surrounded by carbon dust, is consistent with the
discussion in De Smedt et al. (2015), that this source is enriched in carbon,
with $[$C$/$Fe$]=1.2$. The s-process enrichment claimed by De Smedt et al.
(2015) is a further hint of the action of repeated TDU events. The observed
surface 12C and 16O are in agreement with results from stellar evolution
modelling, as shown in Fig. 5, where the results from De Smedt et al. (2015)
are compared with the time evolution of the surface carbon and oxygen of a
$1.25\leavevmode\nobreak\ {\rm M}_{\odot}$ model star. Assuming a
$0.1\leavevmode\nobreak\ {\rm M}_{\odot}$ mass loss during the RGB, we deduce
that the progenitor of J051848.84 was a $1.3-1.4{\rm M}_{\odot}$ star, formed
$\sim 2.5$ Gyr ago.

Figure 6: Time variation of the surface carbon (red lines) and oxygen (blue)
content during the AGB evolution of two $Z=0.002$ model stars of mass
$2\leavevmode\nobreak\ {\rm M}_{\odot}$ (solid lines) and
$2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ (dotted). The red and blue points on
the right side of the plane (with the corresponding error bars) indicate
results for J004441.03 from De Smedt et al. (2015)

####  4.1.3 J004441.03

This source, with an estimated luminosity of $8500\leavevmode\nobreak\ {\rm
L}_{\odot}$, is among the brightest objects in the sample considered in the
present investigation. We concluded in the previous section that it is
surrounded by carbon dust: this is consistent both with the results from De
Smedt et al. (2015), which indicate significant carbon and s-process
enrichment, and with the interpretation based on the position on the HR
diagram (see Fig. 3), which suggests a $\sim 2\leavevmode\nobreak\ {\rm
M}_{\odot}$ progenitor. According to this understanding this is the first
source, among those discussed so far, which did not experience the helium
flash, rather core helium burning occurred in quiescent conditions of thermal
stability. Stars of mass in the $2-3\leavevmode\nobreak\ {\rm M}_{\odot}$
range are expected to experience a series of TDU events that trigger a
significant surface carbon enrichment: Dell’Agli et al. (2015a, b) concluded
that the sources in the LMC exhibiting the largest infrared excesses are the
progeny of $2-3\leavevmode\nobreak\ {\rm M}_{\odot}$ stars.

This interpretation finds a confirmation in the results given in De Smedt et
al. (2015), who find an extremely large surface carbon mass fraction
$[$C$/$Fe$]\sim 1.7$. The results shown in Fig.6 confirm the robustness of the
present analysis, as the final surface content of both carbon and oxygen are
fully consistent with the values given by De Smedt et al. (2015). The
estimated age of J004441.03 is $\sim 1$ Gyr.

####  4.1.4 J005803.08

This is the brightest star in the sample, with an estimated luminosity $\sim
12000\leavevmode\nobreak\ {\rm L}_{\odot}$. The comparison with results from
post-AGB evolution modelling, shown in the left panel of Fig. 3, suggests that
the progenitor of this source is a $2.5\leavevmode\nobreak\ {\rm M}_{\odot}$
star. The expected variation of the surface chemistry of such a star, in terms
of the surface 12C and 16O mass fractions, is shown in Fig. 6, with dotted
lines.

Similarly to the star examined previously, we expect that this source
experienced several TDU events, that caused a significant increase in the
surface carbon, with a final $[$C$/$Fe$]>2$. Some oxygen enrichment is also
expected (see Fig. 6), as also the increase in the s-process abundance.

J005803.08 is highest mass among the sources analysed here, which formed most
recently, with an estimated age slightly older than half Gyr.

Figure 7: Time variation of the surface carbon (red lines) and oxygen (blue)
content during the AGB evolution of two $Z=0.004$ model stars of mass
$1.25\leavevmode\nobreak\ {\rm M}_{\odot}$ (solid lines) and
$1.5\leavevmode\nobreak\ {\rm M}_{\odot}$ (dotted). The red and blue points in
the middle of the plane (with the corresponding error bars) indicate results
for J050632.10 from van Aarle et al. (2013)

####  4.1.5 J050632.10 and J003643.94

From SED fitting (see Fig. 1) we find that both these sources with sub-solar
chemical composition are surrounded by carbon dust. This is consistent with
the estimated luminosities, in the $6000-6500\leavevmode\nobreak\ {\rm
L}_{\odot}$ range, which based on post-AGB modelling correspond to progenitors
of mass slightly above the solar mass (see right panel of Fig. 3), which are
expected to experience a sufficiently large number of TDU episodes to reach
the carbon star stage.

van Aarle et al. (2013) presented results from spectroscopy for J050632.10,
which confirmed carbon enrichment, with $[$C$/$Fe$]$ slightly above unity. As
shown in Fig. 7, this is consistent with the expected carbon enrichment of
$1.25\leavevmode\nobreak\ {\rm M}_{\odot}$ stars of metallicity $Z=0.004$,
which is the derived mass of the progenitor of this source, based on the
luminosity. On the other hand, the expected oxygen enrichment is smaller than
indicated by van Aarle et al. (2013), even in the hypothesis that the gas from
which the star formed was significantly $\alpha-$enhanced, with $[\alpha/{\rm
Fe}]=+0.4$.

J003643.94 is slightly brighter than J050632.10 (see right panel of Fig. 3),
thus we deduce a higher mass progenitor, with ${\rm M}\sim
1.5\leavevmode\nobreak\ {\rm M}_{\odot}$. The results shown in Fig. 7, where
the variation of the surface 12C and 16O for the $1.25\leavevmode\nobreak\
{\rm M}_{\odot}$ and $1.5\leavevmode\nobreak\ {\rm M}_{\odot}$ cases are
reported, indicate that the final surface abundances of carbon and oxygen
should be pretty similar to those found for J050632.10

According to the present interpretation the formation of J003643.94 occurred
around 2 Gyr ago, whereas J050632.10 is older, with an estimated age slightly
above 3 Gyr.

####  4.1.6 J052220.98: a (too) faint carbon star

The position of J052220.98 on the HR diagram reported in the right panel of
Fig. 3 suggests a $\sim 0.8\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitor,
an interpretation that however is at odds with the analysis done in section 3,
where we found that this source is surrounded by carbon dust. Indeed, as
discussed earlier in this section, we expect that carbon stars of metallicity
similar to J052220.98 evolve to the post-AGB phase with luminosities above
$\sim 5000\leavevmode\nobreak\ {\rm L}_{\odot}$, which translates into ${\rm
M}>0.9\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitors.

A possible explanation for this apparently anomalous behaviour is that
J052220.98 descends from a low mass progenitor, which experienced an
extraordinary deep TDU event during the late evolutionary phases: the
consequent large carbon enrichment would have turned the star into a carbon
star. While this possibility cannot be ruled out, we are more predisposed to
consider the hypothesis that this source experienced a late TP shortly after
the beginning of the contraction towards the post-AGB. The idea that helium
ignition might occur after the stars leave the AGB was proposed by Iben et al.
(1983), then further investigated by Blöcker (2001), who studied different
cases, distinguished by the time after the beginning of the overall
contraction when the late TP is ignited. In particular, in the LTP case
discussed by Blöcker (2001), the evolutionary tracks are characterized by a
first excursion to the blue, followed by the expansion to the red, before the
standard post-AGB contraction begins. An example of such a behaviour is
reported in the right panel of Fig. 3, in which the green line refers to the
evolution of a $0.9\leavevmode\nobreak\ {\rm M}_{\odot}$ model star, that
underwent helium ignition immediately after the start of the overall
contraction.

During the occurrence of the loop the luminosity of the star is fainter than
expected on the basis of the classic core mass - luminosity relationship. We
therefore suggest that J052220.98 descends from a $\sim
0.9-1\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitor, which reached the
C-star stage during the AGB, and is now evolving through a loop similar to the
one shown in the figure, after having experienced a late TP.

###  4.2 Oxygen-rich stars

Five out of the sources investigated here show evidence of silicate dust in
their surroundings, based on the analysis presented in section 3. These stars
are indicated with blue triangles in Fig. 3.

A first possibility to explain these objects is that they descend from ${\rm
M}>3\leavevmode\nobreak\ {\rm M}_{\odot}$ stars, which experienced HBB during
the AGB phase; the latter mechanism prevents the surface carbon enrichment,
owing to the proton-capture activity experienced at the base of the convective
envelope, that depletes the surface 12C. This option can be safely ruled out
in the present case, as such massive AGBs are expected to enter the post-AGB
phase with luminosities not below $20000\leavevmode\nobreak\ {\rm L}_{\odot}$,
thus significantly higher than those of the stars considered here, as clear in
Fig.3.

An alternative explanation is that these sources are the progeny of low-mass
stars, that failed to reach the C-star stage. For a given metallicity a
luminosity threshold exists, separating carbon stars from their lower mass
(and fainter) counterparts, that evolve as oxygen-rich objects: indeed the
latter stars at the beginning of the AGB phase are composed by a $\sim
0.5\leavevmode\nobreak\ {\rm M}_{\odot}$ degenerate core and by a convective
envelope of a few tenths of solar masses, that is lost via stellar winds
before the number of TDUs required to reach the carbon star stage is
experienced. The luminosity threshold is sensitive to the metallicity, as the
achievement of the C-star stage is easier the lower the metallicity, owing to
the smaller amount of oxygen in the envelope of the star. In the $Z=0.002$
case we find that carbon stars should have ${\rm L}>4700\leavevmode\nobreak\
{\rm L}_{\odot}$, whereas for $Z=0.004$ the threshold luminosity is $\sim
5200\leavevmode\nobreak\ {\rm L}_{\odot}$.

The luminosities of the five sources characterized by the presence of silicate
dust in their surroundings are within the limits given above, as shown in Fig.
3. In particular J051906.86 and J050221.17 are significantly fainter than the
threshold luminosities discussed above, thus they are interpreted as the
progeny of low-mass stars, with mass at the beginning of the AGB phase of
$\sim 0.7\leavevmode\nobreak\ {\rm M}_{\odot}$ (J051906.86) and $\sim
0.8\leavevmode\nobreak\ {\rm M}_{\odot}$ (J050221.17), which, by assuming
typical RGB mass loss $0.1-0.2\leavevmode\nobreak\ {\rm M}_{\odot}$,
correspond to ages in the $8-10$ Gyr range. These are the oldest objects in
the sample investigated here.

On the other hand J052740.75, J045119.94 and J052241.52 are slightly brighter,
with luminosities close to the afore discussed threshold, holding for the
$Z=0.004$ population. These luminosities correspond to $\sim
0.85\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitors, that formed between 6
and 8 Gyr ago. We might expect to observe some carbon and s-process enrichment
at their surface, as it is likely that they experienced some TDU events before
the entire envelope was lost.

Figure 8: Mass-loss rate at the tip of the AGB for stars of different mass,
reported on the abscissa, and metallicity $Z=0.002$ (magenta points) and
$Z=0.004$ (green diamonds). The values reported in the plane were calculated
by stellar evolution modelling.

Figure 9: The optical depth of the different sources obtained from SED fitting
in section 3 as a function of the luminosity of the stars (left panel) and of
the distance of the internal border of the dusty region from the center of the
star (right). Red squares and blue triangles indicates stars surrounded by
carbonaceous dust and silicates, respectively. Crossed red squares refer to
stars exhibiting evidence of carbon dust, despite their luminosities are
consistent with an M-type nature.

##  5 Dust production during the late AGB phases

During the AGB evolution the stars gradually lose the external convective
mantle. While the central regions contract, the external layers become less
and less bound on the gravitational side to the compact core and expand, and
the mass loss rate is consequently enhanced. This general rule holds for low-
mass objects that evolve as M-type stars, such as those discussed in section
4.2, and even more for carbon stars: indeed in the latter case the expansion
of the structure is further favoured by the rise in the surface carbon, which
is accompanied by a significant increase in the molecular opacities (Marigo,
2002). Massive AGB stars that experience HBB follow a different behaviour, as
in this case the largest mass loss rates are experienced during the phases
when HBB is strongest (Ventura et al., 2015), which happens before a
significant fraction of the envelope is lost. However this is not relevant in
the present context, as we showed in section 4 that the post-AGB stars
discussed here descend from ${\rm M}<3\leavevmode\nobreak\ {\rm M}_{\odot}$
progenitors.

The above arguments indicate that the final AGB phases are the most relevant
to understand dust production from stars. This is because most of the mass is
lost during the last interpulse phases and the dust production rate, tightly
correlated with $\dot{\rm M}$, increases rapidly towards the end of the AGB
evolution. Understanding dust production during the tip of the AGB (TAGB),
i.e. the phase preceding the beginning of the contraction of the star towards
the post-AGB, is therefore crucial to evaluate the dust yields from single
stars and the overall contribution from these objects to dust production in
the host system. Indeed the total dust production rate of some galaxies is
mostly determined by a low fraction of AGB stars, which are currently evolving
through the very final AGB phases (Dell’Agli et al., 2016, 2018, 2019). In
addition, the comprehension of the TAGB turns crucial to interpret the
distribution of stars of galaxies in the observational colour-colour and
colour-magnitude planes, because it is while evolving thorugh the TAGB that
the stars are expected to exhibit the largest infrared excesses, thus the
reddest IR colours (Marini et al., 2021; Dell’Agli et al., 2021).

Fig. 8 shows the mass-loss rate at the TAGB, $\dot{\rm M}^{\rm TAGB}$,
calculated via stellar evolution modelling, as a function of the mass of the
star at the beginning of the AGB phase. $\dot{\rm M}^{\rm TAGB}$ increases
with stellar mass, as higher mass stars reach higher luminosities and radii.
It is clear in the figure the steep rise in $\dot{\rm M}^{\rm TAGB}$ in
correspondence of the threshold mass at which transition from M-type to
C-stars takes place, which is connected to the afore discussed increase in the
surface molecular opacities, taking place when C-rich molecules form.

In the following section we will critically evaluate the mass loss rates
reported in Fig. 8, by comparing the theoretical expectations regarding the
dust in the surroundings of post-AGB stars with the results discussed in
section 3 and 4.

On this regard we will consider the characterization of the individual sources
presented in section 4, which allows the derivation of the mass of the
progenitors and, based on the results shown in Fig. 8, of $\dot{\rm M}^{\rm
TAGB}$. These rates, properly scaled with the effective temperature, lead to
the determination of the variation of the mass loss rates experienced by the
stars during the contraction to the post-AGB.

The comparison with the results deduced in section 3, particularly of the
derived values of $\tau_{10}$, allows to test the $\dot{\rm M}^{\rm TAGB}$
values reported in Fig. 8 and, at the same time, to identify the evolutionary
phase when the dust responsible for the currently observed IR excess was
released. To this scope we will concentrate on some specific points
distributed along the post-AGB evolutionary tracks, each identified by the
value of the effective temperature. For each of these points we will derive
$\tau_{10}^{\rm onset}$, the optical depth that characterizes the SED of the
star if it was observed during the evolutionary stage considered. The values
of $\tau_{10}^{\rm onset}$ are calculated on the basis of stellar evolution +
dust formation modelling, and the application of Eq. 3.

We note that the possibility that dust production continues after the
beginning of the contraction to the post-AGB was already considered, e.g. by
van Hoof et al. (1997), who explored both the possibilities that dust
formation ceased after the AGB, or that it carries on during the post-AGB
phase. In the following we will show that for all the sources considered the
observations are non consistent with the possibility that the dust was
released after the effective temperatures raises above 4000 K. Therefore, here
we do not explore dust formation during the post-AGB phase through which the
sources are evolving nowadays, rather we consider the possibilities that
formation of dust was halted either at the tip of the AGB or during the early
phases after the beginning of the contraction to the post-AGB.

The second step will consist in the determination of $\tau_{10}^{\rm now}$,
the optical depth that characterizes the current SED of the star, if the dust
was released during the stages for which $\tau_{10}^{\rm onset}$ was
calculated. To this aim we consider the general expression given by Eq. 3. In
the computation of the integral on the right-hand side most of the
contribution comes from the regions close to the inner border of the dusty
layer, as $n_{d}$ decreases with distance: therefore, $\tau_{10}$ is mainly
determined by the thermodynamic conditions, particularly the density, at the
distance ${\rm R}_{in}$ from the centre of the star. Assuming that $n_{d}$
scales as $r^{-2}$, and that $Q_{10}$ and $a$ do not change as the dust moves
outwards, we find that the optical depth found via Eq. 3 scales as ${\rm
R}_{in}^{-1}$.

For what attains the tip of the AGB or the different post-AGB phases taken
into account, we consider that both solid carbon dust and silicates form at a
distance of $\sim 10\leavevmode\nobreak\ {\rm R}_{*}$ (Dell’Agli, 2012), thus
$\tau_{10}^{\rm onset}\sim(10\leavevmode\nobreak\ {\rm R}_{*})^{-1}$.
Regarding the dust presently surrounding the star, located at a distance ${\rm
R}_{\rm in}$, we have $\tau_{10}^{\rm now}\sim{\rm R}_{\rm in}^{-1}$.
Therefore, the relation we are looking for is

$\tau_{10}^{\rm now}\sim\tau_{10}^{\rm onset}\times(10\leavevmode\nobreak\
{\rm R}_{*}/{\rm R}_{\rm in})$ $None$

The comparison between the values of the optical depth derived via Eq. 4 and
those found in section 3 will allow testing the different hypothesis
considered. The methodology followed to understand how dust formation and mass
loss work during the final AGB phases and during the beginning of the post-AGB
contraction will be also based on the analysis of the derived expansion
velocities with which the dust layer would travel from the evolutionary stage
when it was released until now, in turn calculated based on the derived
distance of the dusty region and the time elapsed, found via the modelling of
the AGB and post-AGB phases. Our reference point is that the wind velocity
during the post-AGB phase is generally assumed to be comparable with the
velocity in the AGB phase, with values that typically span the $10-30$ km$/$s
range (He et al., 2014; Klochkova, Panchuk, & Tavolzhanskaya, 2015).

Whenever the results from this analysis prove not satisfactory, we will model
the post-AGB evolution by adopting different mass-loss rates, in the attempt
of understanding when the dust responsible for the observed infrared excess of
the various sources was released, and to derive the post-AGB mass loss that
would determine evolutionary time-scales consistent with the dynamical
conditions expected.

##  6 Dust formation and wind propagation across the AGB-post-AGB frontier

A summary of the results derived in section 3 from the interpretation of the
optical and infrared data of the stars considered are presented in Fig. 9,
which shows $\tau_{10}$ of the individual sources as a function of the
luminosity (left panel) and of the distance of the inner border of the dust
layer from the centre of the star (right). Different trends can be detected in
the figure, that correlate $\tau_{10}$ with the stellar luminosity and dust
mineralogy, for stars of different metallicity.

For what concerns metal-poor carbon stars (full, red squares in the figure),
we note: a) the correlation between luminosity and optical depth (left panel),
the brighter stars exhibiting larger IR excesses; b) luminosity and ${\rm
R}_{\rm in}$ are anti-correlated, as the inner border of the dusty layer is
closer to the surface the brighter the star.

The metallicity of the stars is also playing a role in the present context:
the two higher metallicity carbon stars, indicated with open squares in Fig.
9, are characterized by optical depths significantly smaller than the lower
metallicity counterparts with similar luminosities.

The oxygen rich stars behave differently from carbon stars. They are all
characterized by similar luminosities (see Fig. 3), and optical depths
extending over a factor $\sim 2$ with respect to the average value
$\tau_{10}=0.02$. Inspection of the right panel of Fig. 9 shows that the dusty
zone for these sources is closer than found for carbon stars.

To explain these trends, and to have a more exhaustive view of dust production
and propagation during the phases from the tip of the AGB until presently, we
apply the methodology described in the previous section, to study in detail a
few objects, selected among those discussed in section 3, diverse in
luminosity and dust composition. The sources selected are J005803.08,
J053250.69 and J050221.17. The first object, discussed in section 4.1.4, is a
bright carbon star, which experienced several TDU episodes, responsible for
the significant carbon enrichment in the surface regions. J053250.69,
discussed in section 4.1.1, is representative of faint, low-mass carbon stars,
that reached the C-star stage during the last thermal pulses. J050221.17 is a
low-mass, oxygen-rich star, which failed to become carbon star, because the
number of TDU events experienced (if any) was not sufficient to reach the
C$/$O$>1$ condition. We also discuss two carbon stars with sub-solar chemical
composition, and J052220.98, the low-luminosity carbon star, for which a
possible characterization was presented in section 4.1.6.

###  6.1 Bright post-AGBs with carbon dust

According to the discussion in section 4.1.4, J005803.08 is the brightest
source investigated, with luminosity $\sim 12000\leavevmode\nobreak\ {\rm
L}_{\odot}$, that indicates a $\sim 2.5\leavevmode\nobreak\ {\rm M}_{\odot}$
progenitor. This is close to the threshold limit required to start HBB
(Ventura et al., 2013), which destroys the surface carbon and prevents the
C-star stage. As the luminosity of the star increases with the initial mass
(see e.g. Fig. 3), we argue that J005803.08 is among the brightest post-AGBs
with a carbon star chemistry.

The AGB evolution of these stars was studied in details by Dell’Agli et al.
(2015a) and more recently by Marini et al. (2021), who outlined the large
quantities of carbon accumulated in the surface regions and the intense dust
production taking place, particularly during the very final AGB phases.
Results from stellar evolution and dust formation modelling suggest that at
the tip of AGB this class of stars experiences mass loss rates slightly in
excess of $10^{-4}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr and efficient dust
production, with rates of the order of a few $10^{-6}\leavevmode\nobreak\ {\rm
M}_{\odot}/$yr; a thick dust shell composed mainly of solid carbon is expected
to surround the star while evolving at the TAGB, with optical depths
$\tau_{10}\sim 5$ (Dell’Agli et al., 2021). At solar or sub-solar chemistries
SiC grains might offer a significant contribution to the overall extinction of
the radiation, but this is not the case for J005803.08, owing to the low
metallicity.

Figure 10: Time variation of the effective temperature of a
$2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ model star during the transition
from the AGB to the post-AGB phase. Times are counted since the beginning of
contraction. Black line indicates the standard result, obtained by adopting
the prescription from Wachter et al. (2002, 2008) for mass loss. Coloured
tracks refer to results obtained by assuming that $\dot{M}$ decreases as ${\rm
T}_{\rm eff}^{-\alpha}$, with $\alpha=2$ (green), 3 (blue) and 4 (red). The
orange, dashed line indicates the effective temperature of J005803.08 given in
K14 and K15.

Figure 11: Solid lines and full points indicate the values of the optical
depth $\tau_{10}^{\rm onset}$ that characterize dust production at the tip of
the AGB and during different phases along the contraction sequence of a
$2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ model star (see text for details).
The colors used correspond to different treatments of mass-loss, following the
same colour coding as in Fig. 10. Dotted lines and open symbols indicate the
values of the optical depth $\tau_{10}^{\rm now}$, that would characterize the
dust responsible for the infrared excess observed nowadays, at the effective
temperature of J005803.08 (6500 K), under the assumption that it was released
while the star was evolving through the evolutionary stages with the effective
temperatures reported on the abscissa. The dashed, orange line represents the
optical depth obtained for J005803.08 from SED fitting.

Fig. 10 shows the time evolution of the effective temperature of a
$2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ model star of metallicity similar to
J005803.08, starting from the latest AGB phases through the contraction to the
post-AGB. The standard evolution, represented by the black line in the figure,
was obtained with the input physics specified in section 2 for the AGB and the
post-AGB phases. In this case the time elapsed from the tip of the AGB until
the current evolutionary stage, at $T_{\rm eff}=6500$ K (see Tab.1), is $\sim
5600$ yr.

For the tip of the AGB and for 5 further evolutionary phases selected during
the post-AGB, corresponding to effective temperatures in the
$3000\leavevmode\nobreak\ {\rm K}<{\rm T}_{\rm eff}<5000\leavevmode\nobreak\
{\rm K}$ range, equally spaced by $500$ K, we calculated the optical depth
$\tau_{10}^{\rm onset}$. The results are shown in Fig. 11. $\tau_{10}^{\rm
onset}$ decreases as the star contracts (and the effective temperature
increases), since the mass loss rate diminishes, thus the wind is less dense
(see Eq. 2), so lower amounts of gas molecules are available to condense into
dust.

We followed the approach described in the previous section to calculate the
corresponding $\tau_{10}^{\rm now}$ for each of the phases considered. We
limited this analysis to the phases with ${\rm T}_{\rm eff}\leq 4000$ K,
because dust production during later phases is too small to account for the IR
excess observed (see Fig. 11). To calculate $\tau_{10}^{\rm now}$ via Eq. 4 we
used the distance of the inner border of the dusty zone derived in section
4.1.4 for J005803.08, i.e. ${\rm R}_{\rm in}=1.2\times
10^{5}\leavevmode\nobreak\ {\rm R}_{\odot}$, and the radius and optical depth
corresponding to each evolutionary phase. The results, indicated with black,
open squares in Fig. 11, are connected with a dotted line.

We are predisposed to rule out the possibility that the dust observed in the
surroundings of J005803.08 was released while the star was evolving at the tip
of the AGB, for the following reasons: a) the expected $\tau_{10}$ is a factor
$\sim 7$ higher than that derived from SED fitting; b) if we estimate the
average expansion velocity with which the dust would have travelled away from
the star, on the basis of ${\rm R}_{\rm in}$ and of the time elapsed since the
tip of the AGB to the current evolutionary status, we obtain $v\sim 0.5$
km$/$s, which is much lower than observed (He et al., 2014; Klochkova,
Panchuk, & Tavolzhanskaya, 2015).

In the context of the results obtained with the input given in section 2
(black squares in Fig. 11), the only possibility consistent with the analysis
done in section 3 is that the dust observed now was released after the
beginning of the contraction to the post-AGB, when the effective temperature
of the star is $\sim 3000$ K. On the other hand we can rule out this
possibility too, because the time elapsed since the evolutionary phase
considered is $\sim 3600$ yr, which, combined with the distance of the dusty
layer, leads to an average expansion velocity below 1 km$/$s, still too small.
As a further motivation to disregard this possibility, we consider that if the
winds and the dust move with a $\sim 0.8$ km$/$s velocity, the dust produced
at the tip of the AGB would now be at a distance $\sim 2{\rm R}_{\rm in}$, and
would determine the formation of a bump in the SED of the star in the
$25-30\leavevmode\nobreak\ \mu$m region, which is not observed.

As we could not obtain any satisfactory overall explanation of the
observations of J005803.08, we explored different mass-loss rate laws, by
changing the slope of $\dot{\rm M}$ with the effective temperature with
respect to the treatment by Wachter et al. (2002, 2008), according to which
$\dot{\rm M}\sim{\rm T}_{\rm eff}^{-7}$. We calculated new evolutionary
sequences for the $2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ model star,
starting from the tip of AGB, where we left the mass loss rate $\dot{\rm
M}^{\rm TAGB}$ unchanged, then extended to the post AGB, during which we
assumed $\dot{\rm M}=\dot{\rm M}^{\rm TAGB}\times({\rm T}_{\rm eff}/{\rm
T}_{\rm eff}^{\rm TAGB})^{-\alpha}$, with $\alpha=2,3,4$. The results of these
runs are reported in Fig. 10, that shows the time variation of the effective
temperature, and in Fig. 11, reporting the corresponding optical depths, with
different colour coding. The evolutionary time scales get shorter the higher
the mass loss rate, as the rate of the contraction process during the AGB to
post-AGB transition is determined by the rate with which the residual envelope
is lost (Iben & Renzini, 1983). The impact of the treatment of mass loss on
the time scales of the AGB to post-AGB transition was studied by van Hoof et
al. (1997), who showed that the rate at which the effective temperatures of
post-AGB stars increase gets significantly shorter when it is assumed that the
mass-loss rate keeps high after the contraction phase begins.

Consistency between the expected and the observed infrared excess (thus
optical depth) is found in the $\alpha=2$ and $\alpha=3$ cases, if we assume
that the dust observed now was released when the star contracted until the
evolutionary phase characterized by effective temperatures in the $3500-4000$
K range. In these cases the derived average expansion velocity of the wind is
in the $5-20$ km$/$s range, thus consistent with the observational evidence
(He et al., 2014; Klochkova, Panchuk, & Tavolzhanskaya, 2015). Furthermore,
these velocities are sufficiently fast that the dust released at the tip of
AGB would in the meantime reach regions $5-20\leavevmode\nobreak\ {\rm R_{\rm
in}}$ away from the central star, thus causing no effects on the SED in the
$\lambda<50\leavevmode\nobreak\ \mu$m spectral region.

In summary, the interpretation of the observations of J005803.08 indicate that
mass loss rates of bright carbon stars evolving at the tip of the AGB slightly
in excess of $10^{-4}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr are consistent
with the infrared excess observed, provided that we assume that the dust was
released when the effective temperatures reached $\sim 3500-4000$ K, and that
the rate of mass loss during the transition to the post-AGB scales as ${\rm
T}_{\rm eff}^{-2}$.

Figure 12: Variation with the effective temperature of the optical depth
characterizing various evolutionary phases of a $1\leavevmode\nobreak\ {\rm
M}_{\odot}$ model star, from the tip of the AGB (first point on the plane) to
the post-AGB. Full squares connected with solid lines indicate the values of
$\tau_{10}^{\rm onset}$ obtained with the physical ingredients discussed in
section 2 to model the evolution of the star, whereas open squares refer to
the expected optical depths that would be observed in the present epoch,
$\tau_{10}^{\rm now}$. Full and open diamonds refer to the results obtained by
assuming a different scaling of mass loss with the effective temperature
during the contraction to the post-AGB (see text for details). Green pentagons
indicate the optical depths obtained by artificially increasing the mass loss
rate at the tip of AGB by a factor 3. The orange, dashed line indicates the
optical depth of J053250.69, obtained by SED fitting.

###  6.2 Low-mass carbon stars

J053250.69 is the faintest among the stars that we interpret as surrounded by
carbon dust, shown in Fig. 3, with the sole exception of J052220.98, which
will be discussed in a separate section. In section 4.1.1 we interpreted this
object as descending from a low mass progenitor, with mass at the beginning of
the AGB phase of the order of $1\leavevmode\nobreak\ {\rm M}_{\odot}$. This
mass is close to the lower threshold required to reach the C-star stage. As
shown in Fig. 4, which reports the time variation of the surface C$/$O and
$[$C$/$Fe$]$ of a $1\leavevmode\nobreak\ {\rm M}_{\odot}$ model star with the
same metallicity of J053250.69, the C-star stage is reached during the last 2
TPs. The expected mass loss rate at the tip of the AGB is $\dot{\rm M}^{\rm
TAGB}\sim 10^{-5}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr (see Fig. 8).

By studying this source we are therefore exploring the low luminosity tail of
the carbon star population, in such a way that the study of this star, coupled
with the analysis of J005803.08 done previously, allows a thorough view of the
late AGB and post-AGB evolution of carbon stars.

In analogy with the analysis of J005803.08, we considered the tip of the AGB
and different phases during the transition to the post-AGB of a $\sim
1\leavevmode\nobreak\ {\rm M}_{\odot}$ model star, characterized by specific
values of the effective temperature, selected in the $3500-5000$ K range,
spaced by 500 K. We used the same methods described earlier to calculate
$\tau_{10}^{\rm onset}$ for these stages. This analysis was applied both to
the standard evolution of the $1\leavevmode\nobreak\ {\rm M}_{\odot}$ model
star used to study J053250.69, with the mass loss description by Wachter et
al. (2002, 2008), and to the case where the post-AGB mass loss rate is assumed
to change as $\dot{\rm M}=\dot{\rm M}^{\rm TAGB}\times({\rm T}_{\rm eff}/{\rm
T}_{\rm eff}^{\rm TAGB})^{-2}$.

The results of this exploration are reported in Fig. 12, that shows with full
symbols the values assumed by the optical depth characterizing the SED of the
star as it evolves from the tip of the AGB (first point on the left side of
the figure) to the post-AGB. The corresponding values assumed by the optical
depth in the present epoch, under the hypothesis that the dust was released
during the evolutionary phases considered, are indicated with open symbols.
The results obtained by standard evolution modelling and those based on the
$\dot{\rm M}\sim{\rm T}_{\rm eff}^{-2}$ law are indicated in Fig. 12 with
black squares and green diamonds, respectively.

We see in Fig. 12 that the optical depths obtained are significantly smaller
than those derived for J053250.69, the differences exceeding a factor 10 in
all cases. This conclusion holds independently of the treatment of mass loss
adopted to describe the contraction to the post-AGB.

A further issue is related to the estimated velocities with which the dusty
layer moved away from the star since it was released, until presently.
According to the standard description of the transition from the tip of the
AGB to the post-AGB, the expected time transcribed from the tip of the AGB to
the current phase would be $\sim 15000$ yr; this result, combined with the
distance of the dusty region derived in section 3, i.e. $2\times
10^{5}\leavevmode\nobreak\ {\rm R}_{\odot}$, corresponds to an average
expansion velocity below 1 km$/$s. Even if we assume that the dust was
released during a later evolutionary phase, say when the effective temperature
was $4000$ K (that is however highly inconsistent with the estimated optical
depth), the average expansion velocity would not exceed 1 km$/$s.

The conclusion drawn from this preliminary analysis is that the mass loss rate
at the tip of the AGB of low mass carbon stars is underestimated. A higher
$\dot{\rm M}^{\rm TAGB}$ is required not only to obtain higher optical depths,
compatible with the infrared excess observed, but also to shorten the
contraction time scales, so that the derived velocities are consistent with
the observations (He et al., 2014; Klochkova, Panchuk, & Tavolzhanskaya,
2015).

Based on these arguments we calculated additional evolutionary sequences
starting from the tip of the AGB, through the contraction to the post-AGB,
until reaching the effective temperature of J053250.69. In these computations
$\dot{\rm M}^{\rm TAGB}$ was artificially enhanced with respect to the $\sim
10^{-5}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr value given above, and the
mass loss rate during the following contraction phases was changed
accordingly, either by assuming the recipe by Wachter et al. (2002, 2008), or
by adopting the $\dot{\rm M}\sim{\rm T}_{\rm eff}^{-2}$ relationship.

The results obtained by adopting a factor 3 increase in $\dot{\rm M}^{\rm
TAGB}$ and $\dot{\rm M}\sim{\rm T}_{\rm eff}^{-2}$ for the following phases
are indicated with green pentagons in Fig. 12. Consistency between the derived
and the expected optical depth is obtained when assuming that the dust
responsible for the currently observed infrared excess was released when the
effective temperature of the star was $3500-4000$ K. In this case the time
interval is reduced to $500-1000$ yr, thus the average expansion velocity of
the wind is between 10 and 15 km$/$s.

We restrict the attention to the $\alpha=-2$ case. Indeed if we use the
scaling of mass loss with effective temperature described in section 2, the
only possibility to obtain an infrared excess comparable to the observed one
is that the dust was released while the star was at the tip of the AGB.
However, in that case the dust would have travelled for $\sim 5000$ yr, thus
the average expansion velocity would be of the order of 1 km$/$s.

Figure 13: Variation with the effective temperature of the optical depth
characterizing various evolutionary phases of a $1\leavevmode\nobreak\ {\rm
M}_{\odot}$ model star, from the tip of the AGB (first point on the plane) to
the post-AGB. Full squares connected with solid lines indicate the values of
$\tau_{10}^{\rm onset}$ obtained with the physical ingredients discussed in
section 2 to model the evolution of the star, whereas open squares refer to
the expected optical depths that would be observed in the present epoch,
$\tau_{10}^{\rm now}$. The orange, dashed line indicates the optical depth of
J050221.17, obtained by SED fitting.

###  6.3 Carbon stars with sub-solar chemistry

The carbon stars sample studied here encompass J050632.10 and J003643.94, two
sub-solar metallicity stars with similar luminosities, around
$6000\leavevmode\nobreak\ {\rm L}_{\odot}$ (see Tab. 1 and right panel of Fig.
3). In section 4.1.5 we suggested that these stars descend from $\sim
1-1.5\leavevmode\nobreak\ {\rm M}_{\odot}$ progenitors, which became carbon
stars.

Both stars, indicated with open, red squares in Fig. 9, fall off the
$\tau_{10}$ vs luminosity trend defined by low-metallicity carbon stars, as
their optical depths are $\sim 2$ times smaller; also, the current distances
of the dusty region from the central star are the largest among the stars in
the whole sample.

These differences can be at least partly explained by the fact that these two
stars are the hottest among the carbon stars considered, as reported in Tab.
1. This indicates that they are evolving through a more advanced post-AGB
phase in comparison with the carbon stars of lower metallicity. This allowed
the dusty layer to travel further away and reach larger distances.

However, the previous arguments alone are not sufficient to account for the
low IR excess observed in the SED of J050632.10 and J003643.94. Indeed, as
shown in the right panel of Fig. 9, the distances of the dusty regions around
these two sources are found to be $\sim 0.05-0.1$ dex higher than in the other
carbon stars: by applying Eq. 4, we find that the corresponding differences in
$\tau_{10}^{\rm now}$ are not expected to exceed $20-30\%$. To recover the
factor $\sim 2$ difference in the derived optical depth we must assume that
dust formation during the late evolutionary phases was significantly less
efficient.

The most plausible explanation of the lower dust formation that J050632.10 and
J003643.94 experienced during the late AGB phases is the smaller number of
gaseous carbon molecules available to condense into solid particles in the
surface regions, in turn related to the small carbon excess with respet to
oxygen. Indeed, the latter is the relevant quantity for the formation of solid
carbon dust, given the high stability of CO molecules (Ferrarotti & Gail,
2006). The carbon excess is intrinsically higher in metal poor stars $Z\sim
0.004$, given the lower initial oxygen content, while it is smaller in stars
with sub-solar chemistry, as the two sources investigated here. This quantity
is also sensitive to any possible oxygen enrichment connected to the TDU
events experienced.

The latter point deserves particular attention. The results shown in Fig. 7
indicate that negligible oxygen enrichment is expected to have taken place in
the surface regions of these two stars. On the other hand van Aarle et al.
(2013) claim significant oxygen enrichment for J050632.10, which implies a low
carbon excess, and consequently poor dust production. We conclude that the low
IR excess characterizing the two stars considered is related to the
metallicity and to the oxygen enrichment, witnessed by results from high-
resolution spectroscopy.

There are some implications related to these results, which will deserve
further investigations in the future. First, the observations of J050632.10,
similarly to other post-AGB stars in the Milky Way (Kamath et al., 2021),
outline a significant enrichment in the surface oxygen, which is not expected
based on the current modelling of the AGB phases. For what regards the
description of mass loss of carbon stars, the formalism must include some
sensitivity to the surface chemical composition, as the efficiency with which
the winds of carbon stars are radiatively driven by dust is highly sensitive
to the carbon excess with respects to oxygen. Finally, in the description of
carbon dust formation, it is recommended that the number density of seeds over
which the dust grains grow, as suggested by Nanni et al. (2013, 2014), is
taken as dependent on the carbon excess, instead of assuming a plain
proportionality to the hydrogen density, as done in most of the descriptions
currently used.

###  6.4 A faint, dusty C-star

In the left panel of Fig. 9 we note the anomalous position of J052220.98,
which falls off the general trend defined by C-stars: despite the low
luminosity, the smallest among the carbon stars in the sample, the optical
depth derived in section 3 is similar to the brightest carbon stars
considered.

In section 4.1.6 we discussed the possible origin of this object, and proposed
that it experienced a late TP, which is the reason for the relatively low
luminosity. If this understanding proves correct, we could hardly apply the
dust formation modelling adopted in the previous cases so far discussed, given
the poor knowledge of the mass loss mechanism characterizing the tranition to
the post-AGB during and after the star experiences a late TP. We speculate
that the large infrared excess observed in the SED of J052220.98 is caused by
a fast removal of the residual envelope, which favoured large mass loss rates,
with enhanced dust production.

###  6.5 Oxygen-rich, faint post-AGB stars

We consider J050221.17 as representative of low-mass stars that failed to
reach the C-star stage. As discussed in section 4.2, this source descends from
a low-mass progenitor, whose mass at the beginning of the AGB phase was around
$0.8\leavevmode\nobreak\ {\rm M}_{\odot}$. Results from stellar evolution
modelling indicate that such low-mass stars with the same metallicity as
J050221.17 experience between 4 and 5 thermal pulses before their external
envelope, initially of mass $\sim 0.3\leavevmode\nobreak\ {\rm M}_{\odot}$, is
lost, and the contraction to the post-AGB phase starts. The surface chemistry
is substantially unchanged since the beginning of the AGB evolution. Similarly
to the carbon stars discussed earlier, the largest rates of mass loss, which
in this particular case are of the order of $10^{-6}\leavevmode\nobreak\ {\rm
M}_{\odot}/$yr, are reached at the tip of the AGB.

The dust formed in the wind of these stars is mainly composed by silicates,
with alumina dust contributing for less than $10\%$ (Dell’Agli et al., 2014a).
Following the method described in section 2, we find that the optical depth at
the tip of AGB is $\tau_{10}=0.5$.

We modelled dust formation in the wind for different points taken along the
evolutionary track, following the same criterion, based on the values of the
effective temperature, chosen for carbon stars: in this specific case,
considering that the tip effective temperature is $\sim 3600$ K, we considered
${\rm T}_{\rm eff}=4000,4500,5000$ K. In analogy with the study of carbon
stars presented earlier in this section, we use the radii of the star during
each of the phases considered, in combination with the distance of the dust
layer, derived for J050221.17 in section 3, namely ${\rm R}_{\rm
in}=5.45\times 10^{4}\leavevmode\nobreak\ {\rm R}_{\odot}$, to calculate the
current optical depth, via Eq. 4. The results are shown in Fig. 13, where we
report the values of $\tau_{10}^{\rm now}$ as a function of the effective
temperature.

The expected duration of the evolution of the star between the tip of the AGB
and nowadays is $\sim 60000$ yr. If we consider the current distance of the
dusty region from the central star ${\rm R}_{\rm in}$, we find that the dust
particles should move extremely slowly since they are released, with average
velocities below 0.1 km$/$s. This is main reason why we believe extremely
unlikely that the dust responsible for the infrared excess currently observed
was released while the star was evolving at the tip of AGB. A further
motivation to disregard this possibility is that the derived $\tau_{10}^{\rm
now}$ is significantly higher than derived in section 3.

Indeed the results shown in Fig. 13 indicate that consistency between the
expected and the derived $\tau_{10}$ are found under the hypothesis that the
dust was released when the effective temperature was in the $4000-4500$ K
range. In the latter case we should assume that the time elapsed since the
dust was released is substantially shorter, which would correspond to average
velocities $\sim 0.5-1$ km$/$s. These velocities are lower than found during
the study of the wind of carbon stars, which according to our understanding is
the reason why in the case of M-type post-AGBs the distance between the dusty
cloud and the central object is smaller than for C-rich stars, as clear in the
right panel of Fig. 9. The generally lower extinction coefficients of
silicates with respect to carbonaceous dust species is the main reason for the
lower expansion velocity with which the dust moved away from O-rich stars than
for the C-rich counterparts. Furthermore, M-type stars are the faintest
objects in the sample studied here, thus the effects of the radiation pressure
on the dust particles, which is proportional to the luminosity of the star,
are lower.

###  6.6 An overall view of dust surrounding post-AGB stars

The results presented in this section allow us to interpret the findings
reported in Fig. 9. The increasing trend of the optical depth with luminosity
(hence mass) traced by metal-poor carbon stars is mainly due to the fact that
brighter stars evolve faster during the AGB-post-AGB transition, thus the
dusty region is closer to the stellar surface. The higher carbon dust
production rates of the stars of higher mass is also playing a role in this
context.

Oxygen-rich stars that failed to reach the C-star stage cover a narrower range
of luminosities than the C-rich counterparts, thus no significant trends can
be detected. The dust production rate during the late evolutionary phases is
significantly smaller than for carbon stars, thus the acceleration experienced
by the outflow is lower, and the current distances of the dusty regions from
the photosphere are shorter.

The SEDs of carbon-rich post-AGBs with sub-solar chemistry ($Z\sim 0.004$)
present smaller infrared excesses with respect to the lower-metallicity
counterparts, owing to the scarcity of gaseous carbon molecules available to
form dust, in turn related to the large surface oxygen abundances.

##  7 Conclusions

We study a sample of single stars identified as post-AGB sources observed in
the Magellanic Clouds, with the goal of providing a characterization in terms
of mass and formation epoch of the progenitors, and of the mineralogy of the
dust responsible for the observed infrared excess. The analysis is based on
the combination of optical and near infrared data, that allow to trace the
morphology of the spectral energy distribution.

The comparison between the observations and synthetic SED modelling obtained
via the radiative transfer code DUSTY allowed to identify a majority of stars
surrounded by carbonaceous dust, and five objects with silicates in the
surroundings. The determination of the luminosity, still obtained within the
SED fitting process, led to the derivation of the initial mass of the
progenitors, based on the tight link between core mass and luminosity that
characterize post-AGB stars. Overall, we find that the sub-sample selected is
composed by stars descending from progenitors with mass in the
$0.7-2.5\leavevmode\nobreak\ {\rm M}_{\odot}$ range, formed in epochs ranging
from 500 Myr to 8 Gyr ago.

For some of the stars considered, the interpretation proposed here finds a
valid confirmation based on the results from high-resolution spectroscopy,
which nicely agree with the expectations from stellar evolution modelling. The
only exception is a star with luminosity $\sim 4500\leavevmode\nobreak\ {\rm
L}_{\odot}$, inconsistent with the presence of carbon dust in the
surroundings, for which we propose an alternative explanation, based on the
occurrence of a late thermal pulse. Overall, the present analysis indicates
that the results from the present generation of AGB models are consistent with
the post-AGB observations, as far as the general pattern of the surface
chemistry with luminosity is concerned. This agreement demands further
investigation before it can be confirmed, given the small number of sources
investigated.

The analysis of the infrared excess observed, combined with results from
evolutionary modelling of the late AGB and post-AGB phases, proves a valuable
tool to understand the physical behaviour of the stars before and during the
AGB - post-AGB transition, as well as to study the dynamics of the outflow,
moving away from the central stars dust currently observed was released.

The general conclusion drawn in this context is that the dust nowadays
observed around post-AGB stars was released after contraction to post-AGB
began, and the effective temperatures increased to $\sim 4000$ K. The infrared
excess observed is primarily determined by the time scale of the post-AGB
evolution, which proves crucial to determine the current location of the dusty
region, and the dust production rate experienced when the stars evolve through
the tip of the AGB and the early post-AGB.

In the case of carbon stars the infrared excess increases with the luminosity
of the stars, as bright stars evolve faster and are expected to experience a
more efficient formation of dust. Detailed comparison with stellar evolution
modelling indicates that the mass loss and dust formation rates of $M\geq
2\leavevmode\nobreak\ {\rm M}_{\odot}$ stars are correctly described, whereas
in the low mass domain a factor $\sim 3$ increase is required. A further
indication from the present study, connected with the small infrared excess
characterizing the SED of a few low-mass stars of sub-solar chemical
composition, is the need of a mass loss prescription that is sensitive to the
surface chemistry, particularly to the carbon excess with respect to oxygen.

The oxygen-rich stars in the sample are interpreted as the progeny of low-mass
stars, that failed to reach the C-star stage. The dust in their surroundings
is mostly composed by silicates in a crystalline form. In this case
satisfactory agreement is found with the theoretical expectations, which
indicate mass loss rates at the tip of the AGB of the order of
$10^{-6}\leavevmode\nobreak\ {\rm M}_{\odot}/$yr. The dusty shells of M-type
stars are found to be on the average closer to the central star with respect
to carbon stars, which we interpret as a result of the lower radiation
pressure acting on silicates than on carbon dust, which results in slower
winds.

As a next step in our research, we intend to extend our study to the Galactic
sample of post-AGB stars presented in Kamath et al. (2022). The upcoming
release of the Gaia DR3 astrometric data will be instrumental in further
capitalising on the Galactic sample of objects. Additionally, as future work,
we will aim to obtain robust observational evidence to further constrain the
dust chemical composition of our target sample. In this regard, high-
resolution IR and mid-IR spectra from James Webb Space Telescope (JWST),
covering the $\sim$ 5 $-$ 28 $\mu$m wavelength regime, will be particularly
useful since the spectra of post-AGB stars show interesting features such as
atomic fine-structure and hydrogen lines, PAHs, crystalline and amorphous
silicates, SiC, MgS, the enigmatic 21$\mu$m feature, alumina,
C2H${}_{\rm{}_{2}}$, SiS, SiO, TiO and H2O (Sloan et al., 2016; Gielen et al.,
2011). The intermingling of these observations with our newly developed models
will allow us to further develop our description of dust formation in the
outflows from post-AGB stars.

###### Acknowledgements.

DK acknowledges the support of the Australian Research Council (ARC) Discovery
Early Career Research Award (DECRA) grant (DE190100813). This research was
supported in part by the Australian Research Council Centre of Excellence for
All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number
CE170100013. HVW acknowledges support from the Research Council of the KU
Leuven under grant number C14/17/082. EM acknowledges support from the INAF
research project “LBT - Supporto Arizona Italia”.

## References

  * Andersen et al. (2003) Andersen, A. C., Höfner, S., & Gautschy-Loidl, R. 2003, A&A, 400, 981
  * Aringer et al. (2009) Aringer B., Girardi L., Nowotny W., Marigo P., Lederer M. T., 2009, A&A, 503, 913
  * Begemann et al. (1994) Begemann, B., Dorschner, J., Henning, T., et al. 1994, ApJ, 423, L71
  * Bensby et al. (2014) Bensby T., Feltzing S., Oey M. S. 2014, A&A, 562, A71
  * Bertola et al. (1995) Bertola F., Bressan A., Burstein D., et al. 1995, ApJ, 438, 680
  * Bertschinger & Chevalier (1985) Bertschinger, E., & Chevalier, R. A. 1985, ApJ, 167, 190
  * Bladh et al. (2019) Bladh S., Eriksson K., Marigo P., Liljegren S., Aringer B. 2019, A&A, 623, A119
  * Blöcker (1995) Blöcker T., 1995, A&A, 297, 727
  * Blöcker & Schöenberner (1991) Blöcker T., Schöenberner D. 1991, A&A, 244, L43
  * Blöcker (2001) Blöcker, T. 2001, Ap&SS, 275, 1
  * Schönberner & Blöcker (1996) Schönberner, D. & Blöcker, T. 1996, Ap&SS, 245, 201
  * Boothroyd & Sackmann (1988) Boothroyd, A. I., & Sackmann, I.-J. 1988, ApJ, 328, 671
  * Bouchet et al. (2015) Bouchet P., García-Marín M., Lagage P.-O., et al. 2015, PASP, 127, 612
  * Bowen (1988) Bowen, G. H., 1988, ApJ, 329, 299
  * Boyer et al. (2013) Boyer M. L., Girardi L., Marigo P., et al. 2013, ApJ, 774, 83
  * Brunner et al. (2018) Brunner M., Maercker M., Mecina M., Khouri T., Kerschbaum F. 2018, A&A, 614, A17
  * Cameron & Fowler (1971) Cameron A. G. W., Fowler W. A., 1971, ApJ, 164, 111
  * Canuto (1992) Canuto, V. M. 1992, ApJ, 392, 218
  * Canuto & Mazzitelli (1991) Canuto V. M. C., Mazzitelli I., 1991, ApJ, 370, 295
  * Casagrande et al. (2011) Casagrande L., Schönrich R., Asplund M., et al. 2011, A&A, 530, A138
  * Castelli & Kurucz (2003) Castelli, F. & Kurucz, R. L. 2003, Modelling of Stellar Atmospheres, 210, A20
  * Chen et al. (2019) Chen Y. Q., Zhao G., Zhao J. K., et al. 2019, AJ, 158, 249
  * Cioni & Habing (2003) Cioni M.-R. L., Habing H. J. 2003, A&A, 402, 133
  * Cloutman & Eoll (1976) Cloutman, L. D. & Eoll, J. G. 1976, ApJ, 206, 548.
  * Cox (1980) Cox, J. P. 1980, Theory of Stellar Pulsation (Princeton University Press, Princeton) 165
  * Cox (1969) Cox, A. N.,& Stewart, J. N. 1969, Academia Nauk, Scientific Information 15, 1
  * Cristallo et al. (2009) Cristallo, S., Straniero, O., Gallino, R., et al. 2009, ApJ, 696, 797
  * Cristallo et al. (2011) Cristallo S., Piersanti L., Straniero O., et al. 2011, ApJS, 197, 17
  * Cristallo et al. (2015) Cristallo S., Straniero O., Piersanti L., et al. 2015, ApJS, 219, 40
  * D’Ercole et al. (2008) D’Ercole A., Vesperini E., D’Antona F., et al. 2008, MNRAS, 391, 825
  * Ek et al. (2019) Ek, M., Hunt, A. C., Lugaro, M., et al. 2019, Nature Astronomy, 4, 273
  * Dell’Agli (2012) Dell’Agli, F., 2012, Master Thesis, University of Rome ”Roma Tre”
  * Dell’Agli et al. (2014a) Dell’Agli F., García-Hernández D. A., Rossi C., et al. 2014a, MNRAS, 441, 1115
  * Dell’Agli et al. (2014b) Dell’Agli F., Ventura P., Garcia Hernandez D. A., et al. 2014b, MNRAS, 442, L38
  * Dell’Agli et al. (2015a) Dell’Agli F., Ventura P., Schneider R., et al. 2015a, MNRAS, 447, 2992
  * Dell’Agli et al. (2015b) Dell’Agli F., García-Hernández D. A., Ventura P., et al. 2015b, MNRAS, 454, 4235
  * Dell’Agli et al. (2016) Dell’Agli F., Di Criscienzo M., Boyer M. L., et al. 2016, MNRAS, 460, 4230
  * Dell’Agli et al. (2018) Dell’Agli F., Di Criscienzo M., Ventura P., et al. 2018, MNRAS, 479, 5035
  * Dell’Agli et al. (2019) Dell’Agli F., Di Criscienzo M., García-Hernández D. A., et al. 2019, MNRAS, 482, 4733
  * Dell’Agli et al. (2021) Dell’Agli, F., Marini, E., D’Antona, F., et al. 2021, MNRAS, 502, L35
  * De Smedt et al. (2015) De Smedt, K., Van Winckel, H., Kamath, D., et al. 2015, A&A, 583, A56
  * Di Criscienzo et al. (2013) Di Criscienzo M., Dell’Agli F., Ventura P., et al. 2013, MNRAS, 433, 313
  * Di Criscienzo et al. (2016) Di Criscienzo M., Ventura P., García-Hernández D. A., et al. 2016, MNRAS, 462, 395
  * Draine (2003) Draine B. T., 2003, ApJ, 598, 1017
  * Draine & Lee (1984) Draine, B. T., & Lee, H. M. 1984, ApJ, 285, 89
  * Eriksson et al. (2014) Eriksson, K., Nowotny, W., Höfner, S., et al. 2014, A&A, 566, A95
  * Ferrarotti & Gail (2001) Ferrarotti, A. S., & Gail, H.-P. 2001, A&A, 371, 133
  * Ferrarotti & Gail (2002) Ferrarotti, A. S., & Gail, H.-P. 2002, A&A, 382, 256
  * Ferrarotti & Gail (2006) Ferrarotti A. S., Gail H.-P. 2006, A&A, 447, 553
  * Fleischer et al. (1992) Fleischer, A. J., Gauger, A., & Sedlmayr, E. 1992, A&A, 266, 339
  * Freytag et al. (1996) Freytag, B., Ludwig, H.-G., & Steffen, M. 1996, A&A, 313, 497
  * Frost et al. (1998) Frost, C. A., Cannon, R. C., Lattanzio, J. C., et al. 1998, A&A, 332, L17
  * Garcia-Berro & Iben (1994) Garcia-Berro E., Iben, I. 1994, ApJ, 434, 306
  * García-Hernández et al. (2016) García-Hernández, D. A., Ventura, P., Delgado-Inglada, G., et al. 2016, MNRAS, 458, L118
  * Gielen et al. (2011) Gielen, C., Cami, J., Bouwman, J., et al. 2011, A&A, 536, A54
  * Ginolfi et al. (2018) Ginolfi, M., Graziani, L., Schneider, R., et al. 2018, MNRAS, 473, 4538
  * Gioannini et al. (2017) Gioannini, L., Matteucci, F., & Calura, F. 2017, MNRAS, 471, 4615
  * Gładkowski et al. (2019) Gładkowski, M., Szczerba, R., Sloan, G. C., et al. 2019, A&A, 626, A92
  * Goebel & Moseley (1985) Goebel, J. H. & Moseley, S. H. 1985, ApJ, 290, L35
  * Gong et al. (2017) Gong M., Ostriker E. C., Wolfire M. G. 2017, ApJ, 843, 38
  * Gruendl et al. (2008) Gruendl, R. A., Chu, Y.-H., Seale, J. P., et al. 2008, ApJ, 688, L9
  * Grevesse & Sauval (1998) Grevesse N., Sauval A. J 1998, SSrv, 8 5, 161
  * Groenewegen & de Jong (1993) Groenewegen, M. A. T., & de Jong, T. 1993, A&A, 267, 410
  * Groenewegen et al. (2007) Groenewegen, M. A. T., Wood, P. R., Sloan, G. C., et al. 2007, MNRAS, 376, 313
  * Groenewegen et al. (2009) Groenewegen, M. A. T., Sloan, G. C., Soszynski, I., Petersen, E. A. 2009, A&A, 506, 1277
  * Groenewegen & Sloan (2018) Groenewegen, M. A. T., & Sloan, G. C. 2018, A&A, 609, A114
  * Gullieuszik et al. (2012) Gullieuszik, M., Groenewegen, M. A. T., Cioni, M.-R. L., et al. 2012, A&A, 537, A105
  * Hanner (1988) Hanner, M. 1988, Infrared Observations of Comets Halley and Wilson and Properties of the Grains
  * Harris & Zaritsky (2009) Harris, J., & Zaritsky, D. 2009, AJ, 138, 1243
  * He et al. (2014) He J. H., Szczerba R., Hasegawa T. I., Schmidt M. R., 2014, ApJS, 210, 26.
  * Herwig (2000) Herwig, F. 2000, A&A, 360, 952
  * Herwig (2005) Herwig, F. 2005, ARA&A, 43, 435
  * Hollenbach et al. (1971) Hollenbach, D., Salpeter, E. E., 1971, ApJ, 163, 155
  * Hony et al. (2002) Hony, S., Waters, L. B. F. M., & Tielens, A. G. G. M. 2002, A&A, 390, 533
  * Iben (1974) Iben, I. Jr. 1974, ARA&A, 12, 215
  * Iben (1975) Iben, I. 1975, ApJ, 196, 525
  * Iben & Renzini (1983) Iben, I. & Renzini, A. 1983, ARA&A, 21, 271
  * Iben et al. (1983) Iben, I., Kaler, J. B., Truran, J. W., et al. 1983, ApJ, 264, 605
  * Iglesias & Rogers (1996) Iglesias, C. A., Rogers, F. J. 1996, ApJ, 464, 943
  * Jaeger et al. (1994) Jaeger, C., Mutschke, H., Begemann, B., Dorschner, J., Henning, Th. 1994, A&A, 292, 641
  * Johansson et al. (2012) Johansson J., Thomas D., Maraston C., 2012, MNRAS, 421, 1908
  * Jones et al. (2014) Jones, O. C., Kemper, F., Srinivasan, S., et al. 2014, MNRAS, 440, 631
  * Jones et al. (2017) Jones O. C., Meixner M., Justtanont K., Glasse, A. 2017, ApJ, 841, 15
  * Jørgensen et al. (2000) Jørgensen, U. G., Hron, J., & Loidl, R. 2000, A&A, 356, 253
  * Kamath et al. (2012) Kamath, D., Karakas, A. I., & Wood, P. R. 2012, ApJ, 746, 20
  * Kamath et al. (2014) Kamath, D., Wood, P. R., & Van Winckel, H. 2014, MNRAS, 439, 2211, K14
  * Kamath et al. (2015) Kamath, D., Wood, P. R., & Van Winckel, H. 2015, MNRAS, 454, 1468, K15
  * Kamath et al. (2021) Kamath, D., Dell’Agli, F., Ventura, P., et al. 2021, arXiv:2112.05535
  * Kamath et al. (2022) Kamath, D., Van Winckel, H., Ventura, P., et al. 2022, ApJ, 927, L13
  * Karakas et al. (2002) Karakas, A. I., Lattanzio, J. C., & Pols, O. R. 2002, PASA, 19, 515
  * Karakas (2010) Karakas A. I. 2010, MNRAS, 403, 1413
  * Karakas & Lattanzio (2014) Karakas A. I., Lattanzio J. C. 2014b, PASA, 31, e030
  * Karakas & Lugaro (2016) Karakas A. I., Lugaro M. 2016, ApJ, 825, 26
  * Karakas et al. (2018) Karakas A. I., Lugaro M., Carlos M., et al. 2018, MNRAS, 477, 421
  * Kato et al. (2007) Kato, D., Nagashima, C., Nagayama, T., et al. 2007, PASJ, 59, 615
  * Kemper et al. (2010) Kemper, F., Woods, P. M., Antoniou, V., et al. 2010, PASP, 122, 683
  * Klochkova, Panchuk, & Tavolzhanskaya (2015) Klochkova V. G., Panchuk V. E., Tavolzhanskaya N. S., 2015, AstL, 41, 14
  * Kraemer et al. (2017) Kraemer, K. E., Sloan, G. C., Wood, P. R., et al. 2017, ApJ, 834, 185
  * Kubryk et al. (2015) Kubryk M., Prantzos N., Athanassoula E., 2015, A&A, 580, A126
  * Laor & Draine (1993) Laor, A., & Draine, B. T. 1993, ApJ, 402, 441
  * Lattanzio (1987) Lattanzio, J. C. 1987, ApJ, 313, L15
  * Lattanzio (1993) Lattanzio, J. C. 1993, Planetary Nebulae, 235
  * Lewis et al. (2013) Lewis K. M., Lugaro M., Gibson B. K., et al. 2013, ApJ, 768, L19
  * Lugaro et al. (2014) Lugaro M., Tagliente G., Karakas A. I., et al. 2014, ApJ, 780, 95
  * Lugaro et al. (2018) Lugaro M., Karakas A. I., Pető M., et al. 2018, Geochim. Cosmochim. Acta., 221, 6
  * Maercker et al. (2018) Maercker M., Khouri T., De Beck E., Brunner M., Mecina M., Jaldehag O. 2018, A&A, 620, A106
  * Marigo et al. (1999) Marigo, P., Girardi, L., & Bressan, A. 1999, A&A, 344, 123
  * Marigo (2002) Marigo, P. 2002, A&A, 387, 507
  * Marigo & Aringer (2009) Marigo P., Aringer B., 2009, A&A, 508, 1538
  * Marigo et al. (2020) Marigo, P., Cummings, J. D., Curtis, J. L., et al. 2020, Nature Astronomy, doi:10.1038/s41550-020-1132-1
  * Marini et al. (2019) Marini, E., Dell’Agli, F., Di Criscienzo, M., et al. 2019, ApJ, 871, L16
  * Marini et al. (2020) Marini, E., Dell’Agli, F., Di Criscienzo, M., et al. 2020, MNRAS, 493, 2996
  * Marini et al. (2021) Marini, E., Dell’Agli, F., Groenewegen, M. A. T., et al. 2021, A&A, 647, A69
  * Mathis (1990) Mathis J. S., 1990, ARA&A, 28, 37
  * Matteucci (2012) Matteucci F., 2012, Chemical Evolution of Galaxies (Berlin Heidelberg: Springer-Verlag)
  * Mattsson et al. (2008) Mattsson, L., Wahlin, R., Höfner, S., et al. 2008, A&A, 484, L5
  * Mattsson et al. (2010) Mattsson, L., Wahlin, R., & Höfner, S. 2010, A&A, 509, A14
  * Mattsson & Höfner (2011) Mattsson, L. & Höfner, S. 2011, A&A, 533, A42
  * Mattsson et al. (2012) Mattsson, L., Wahlin, R., & Höfner, S. 2012, A&A, 546, C1
  * Matsuura et al. (2006) Matsuura, M., Wood, P. R., Sloan, G. C., et al. 2006, MNRAS, 371, 415
  * Meixner et al. (2006) Meixner, M., Gordon, K. D., Indebetouw, R., et al. 2006, AJ, 132, 2268
  * Messenger et al. (2013) Messenger, S. J., Speck, A., & Volk, K. 2013, ApJ, 764, 142
  * Minchev et al. (2013) Minchev I., Chiappini C., Martig M., 2013, A&A, 558, A9
  * Miller Bertolami (2016) Miller Bertolami, M. M. 2016, A&A, 588, A25
  * Nanni et al. (2013) Nanni A., Bressan A., Marigo P., et al. 2013, MNRAS, 434, 2390
  * Nanni et al. (2014) Nanni A., Bressan A., Marigo P., et al. 2014, MNRAS, 438, 2328
  * Nanni et al. (2016) Nanni A., Marigo P., Groenewegen M. A. T., et al. 2016, MNRAS, 462, 1215
  * Nanni et al. (2018) Nanni A., Marigo P., Girardi L., et al. 2018, MNRAS, 473, 5492
  * Nanni (2019a) Nanni, A. 2019a, MNRAS, 482, 4726
  * Nanni et al. (2019b) Nanni A., Groenewegen M. A. T., Aringer B., et al. 2019b, MNRAS, 487, 502
  * Nanni et al. (2019c) Nanni A., Groenewegen M. A. T. Aringer, B., et al. 2019, IAU Symposium, 478
  * Nanni et al. (2020) Nanni, A., Burgarella, D., Theulé, P., et al. 2020, A&A, 641, A168
  * Nanni et al. (2020b) Nanni A., Marigo P., Groenewegen M. A. T., et al. 2020, IAU General Assembly, 405
  * Nenkova et al. (1999) Nenkova, M., Ivezic, Z., & Elitzur, M. 1999, Thermal Emission Spectroscopy and Analysis of Dust, Disks, and Regoliths, 20
  * Ossenkopf et al. (1992) Ossenkopf, V., Henning, Th., Mathis, J. S. M. 1992, A&A, 261, 567
  * Paczyński (1970) Paczyński, B. 1970, Acta Astron., 20, 287
  * Pastorelli et al. (2019) Pastorelli, G., Marigo, P., Girardi, L., et al. 2019, MNRAS, 485, 5666
  * Pastorelli et al. (2020) Pastorelli, G., Marigo, P., Girardi, L., et al. 2020, MNRAS, 498, 3283
  * Pegourie (1988) Pegourie, B. 1988, A&A, 194, 335
  * Pérez-Mesa et al. (2017) Pérez-Mesa, V., Zamora, O., García-Hernández, D. A., et al. 2017, A&A, 606, A20
  * Rau et al. (2017) Rau G., Hron J., Paladini C., Aringer B., Eriksson K., Marigo P., Nowotny W., Grellmann R. 2017, A&A, 600, A92
  * Rau et al. (2019) Rau G., Ohnaka K., Wittkowski M., Airapetian V., Carpenter K. 2019, ApJ, 882, 37
  * Renzini & Voli (1981) Renzini, A., & Voli, M. 1981, A&A, 500, 221
  * Renzini (1989) Renzini, A. 1989, Planetary Nebulae, 131, 391
  * Rieke et al. (2015) Rieke G. H., Wright G. S., Böker T., et al. 2015, PASP, 127, 584
  * Romano et al. (2001) Romano D., Matteucci F., Ventura P., et al. 2001, A&A, 374, 646
  * Romano et al. (2010) Romano D., Karakas A. I., Tosi M., et al. 2010, A&A, 522, A32
  * Romano et al. (2017) Romano D., Matteucci F., Zhang Z.-Y., et al. 2017, MNRAS, 470, 401
  * Romano et al. (2019) Romano D., Matteucci F., Zhang Z.-Y., et al. 2019, MNRAS, 490, 2838
  * Sackmann & Boothroyd (1992) Sackmann I.-J., Boothroyd A. I., 1992, ApJL, 392, L71
  * Sandin & Mattsson (2020) Sandin, C. & Mattsson, L. 2020, arXiv:2006.11296
  * Schneider et al. (2014) Schneider R., Valiante R., Ventura P., et al. 2014, MNRAS, 442, 1440
  * Schneider et al. (2016) Schneider, R., Hunt, L., & Valiante, R. 2016, MNRAS, 457, 1842
  * Schultheis et al. (2019) Schultheis M., Rich R. M., Origlia L., et al. 2019, A&A, 627, A152
  * Schwarzschild & Härm (1965) Schwarzschild, M. & Härm, R. 1965, ApJ, 142, 855. doi:10.1086/148358
  * Sharp & Huebner (1990) Sharp, C. M., & Huebner, W. F. 1990, ApJS, 72, 417
  * Siess (2007) Siess, L. 2007, A&A, 476, 893
  * Skrutskie et al. (2006) Skrutskie, M. F., Cutri, R. M., Stiening, R., et al. 2006, AJ, 131, 1163
  * Sloan et al. (2014) Sloan, G. C., Lagadec, E., Zijlstra, A. A., et al. 2014, ApJ, 791, 28
  * Sloan et al. (2016) Sloan, G. C., Kraemer, K. E., McDonald, I., et al. 2016, ApJ, 826, 44
  * Speck et al. (2009) Speck, A. K., Corman, A. B., Wakeman, K., et al. 2009, ApJ, 691, 1202
  * Spitoni et al. (2015) Spitoni E., Romano D., Matteucci F., et al. 2015, ApJ, 802, 129
  * Srinivasan et al. (2009) Srinivasan S., Meixner M., Leitherer C., et al. 2009, AJ, 137, 4810
  * Srinivasan et al. (2010) Srinivasan S., Sargent B. A., Matsuura, M., et al. 2010, AJ, 137, 4810
  * Srinivasan et al. (2011) Srinivasan, S., Sargent, B. A., Meixner, M. 2011, A&A, 532, A54
  * Srinivasan et al. (2016) Srinivasan S., Boyer M. L., Kemper F., et al. 2016, MNRAS, 457, 2814
  * Sylvester et al. (1999) Sylvester, R. J., Kemper, F., Barlow, M. J., et al. 1999, A&A, 352, 587
  * Thorsbro et al. (2020) Thorsbro B., Ryde N., Rich R. M., et al. 2020, arXiv e-prints, arXiv:2003.11085
  * Valiante et al. (2009) Valiante R., Schneider R., Bianchi S., et al. 2009, MNRAS, 397, 1661
  * Valiante et al. (2017) Valiante R., Gioannini L., Schneider R., Matteucci F., Dell’Agli F., Di Criscienzo M. 2017, Memorie della Societá Astronomica Italiana, 88, 420
  * van Hoof et al. (1997) van Hoof, P. A. M., Oudmaijer, R. D., & Waters, L. B. F. M. 1997, MNRAS, 289, 371
  * van Loon et al. (1999) van Loon, J. T., Groenewegen, M. A. T., de Koter, A., et al. 1999, A&A, 351, 559
  * van Aarle et al. (2013) van Aarle, E., Van Winckel, H., De Smedt, K., et al. 2013, A&A, 554, A106
  * Vassiliadis & Wood (1993) Vassiliadis E., Wood P. R. 1993, ApJ, 413, 641
  * Ventura et al. (1998) Ventura, P., Zeppieri, A., Mazzitelli, I., D’Antona, F., 1998, A&A, 334, 953
  * Ventura et al. (1999) Ventura, P., D’Antona, F., & Mazzitelli, I. 1999, ApJ, 524, L111
  * Ventura et al. (2000) Ventura P., D’Antona F., Mazzitelli I. 2000, A&A, 363, 605
  * Ventura et al. (2001) Ventura P., D’Antona F., Mazzitelli I., et al. 2001, ApJ, 550, L65
  * Ventura & D’Antona (2005a) Ventura P., D’Antona F. 2005a, A&A, 431, 279
  * Ventura & D’Antona (2005b) Ventura, P., D’Antona, F. 2005b, A&A, 439, 1075
  * Ventura (2010) Ventura, P. 2010, Light Elements in the Universe, 147
  * Ventura et al. (2012) Ventura P., Di Criscienzo M., Schneider R., et al. 2012, MNRAS, 420, 1442
  * Ventura et al. (2013) Ventura P., Di Criscienzo M., Carini R., D’Antona F., 2013, MNRAS, 431, 3642
  * Ventura et al. (2014) Ventura P., Dell’Agli F., Schneider R., et al. 2014, MNRAS, 439, 977
  * Ventura et al. (2015) Ventura P., Karakas A. I., Dell’Agli F., et al. 2015, MNRAS, 450, 3181
  * Ventura et al. (2016) Ventura P., Karakas A. I., Dell’Agli F., et al. 2016, MNRAS, 457, 1456
  * Ventura et al. (2018) Ventura P., Karakas A., Dell’Agli F., García-Hernández D. A.,Guzman-Ramirez L. 2018, MNRAS, 475, 2282
  * Ventura & Marigo (2009) Ventura, P., & Marigo, P. 2009, MNRAS, 399, L54
  * Ventura & Marigo (2010) Ventura, P., & Marigo, P. 2010, MNRAS, 408, 2476
  * Volk et al. (2002) Volk, K., Kwok, S., Hrivnak, B. J., et al. 2002, ApJ, 567, 412
  * Volk et al. (2020) Volk, K., Sloan, G. C., & Kraemer, K. E. 2020, Ap&SS, 365, 88
  * Volk et al. (2011) Volk K., Hrivnak B. J., Matsuura M., Bernard-Salas J., Szczerba R., Sloan G. C., Kraemer K. E., et al., 2011, ApJ, 735, 127.
  * Wachter et al. (2002) Wachter A., Schröder K. P., Winters J. M., Arndt T. U., Sedlmayr E., 2002, A&A, 384, 452
  * Wachter et al. (2008) Wachter A., Winters J. M., Schröder K. P., Sedlmayr E., 2008, A&A, 486, 497
  * Weiss & Ferguson (2009) Weiss, A., & Ferguson, J. W. 2009, A&A, 508, 1343
  * Wegg et al. (2019) Wegg C., Rojas-Arriagada A., Schultheis M., et al. 2019, A&A, 632, A121
  * Winters et al. (2000) Winters, J. M., Le Bertre, T., Jeong, K. S., Helling, C., & Sedlmayr, E. 2000, A&A, 361, 641
  * Woods et al. (2011) Woods, P. M., Oliveira, J. M., Kemper, F., et al. 2011, MNRAS, 411, 1597
  * Xu et al. (2016) Xu, Y., Lin, Y., Zhang, J., et al. 2016, ApJ, 825, 111
  * Zamora et al. (2014) Zamora, O., García-Hernández, D. A., Plez, B., et al. 2014, A&A, 564, L4
  * Zaritsky et al. (2004) Zaritsky, D., Harris, J., Thompson, I. B., et al. 2004, AJ, 128, 1606
  * Zhukovska et al. (2008) Zhukovska S., Gail H.-P., Trieloff M. 2008, A&A, 479, 453
  * Zijlstra et al. (2006) Zijlstra, A. A., Matsuura, M., Wood, P. R., et al. 2006, MNRAS, 370, 1961
  * Zinner (2008) Zinner, E. 2008, Organic Matter in Space, 251, 341
  * Zubko et al. (1996) Zubko, V. G., Mennella, V., Colangeli, L., et al. 1996, MNRAS, 282, 132