Common Envelope Mass Ejection in Evolved Stars: Modeling the Dust Emission from post-RGB stars in the LMC

Geetanjali Sarkar
Department of Physics,
Indian Institute of Technology, Kanpur

U.P., India

Raghvendra Sahai
Jet Propulsion Laboratory

Pasadena, CA, USA

Common Envelope (CE) systems are the result of Roche lobe overflow in interacting binaries.
The subsequent evolution of the CE,
its ejection and the formation of dust in its ejecta
while the primary is on the Red Giant Branch, gives rise to a recently-identified evolutionary class –
dusty post-RGB stars. Their spectral energy distributions (SEDs) suggest that their mass-ejecta are similar to dusty post-Asymptotic Giant Branch (post-AGB) stars. We have modeled the SEDs of a select sample of post-RGB and post-AGB stars in the Large Magellanic Cloud (LMC),  quantified the total
dust mass (and gas mass assuming gas-to-dust ratio) in the disks and shells and
set constraints on the dust grain compositions and sizes.  We find that the shell masses in the post-RGBs are generally less than those in post-AGBs, with the caveat that a substantial amount of mass in both types of objects may lie in cold, extended shells. Our models suggest that  circumstellar
disks, when present, are geometrically thick, flared structures with a substantial opening angle,
consistent with numerical simulations of CE evolution (CEE). Comparison of our model dust mass values with the predictions of dust production during
CEE on the RGB suggest that CEE occurred near or at the top of the RGB for the post-RGB sources in our sample. A surprising result is that the ejected dust in some post-RGB sources appears to be carbon-rich, providing independent support for the hypothesis of binary interaction leading to the formation of dusty post-RGB objects.

§ INTRODUCTION

The demise of most stars in the Universe (i.e., in the 1–8  range) has traditionally been thought to occur as a
result of heavy mass-loss (with rates as high as $10^{-4}$ ) on the Asymptotic Giant
Branch (AGB), when the stars are very luminous ($L\sim5000-10,000$ ) and cool ($T_{eff} < 3000$ K). This heavy
mass-loss is believed to occur via a dusty, spherical, radiatively-driven wind, typically expanding at
$\sim5-20$ . After the mass-loss has depleted most of the stellar envelope, the
stars evolve to higher temperatures through the post-AGB phase at almost constant
luminosity. A major unsolved problem in this evolution is that post-AGB stars are surrounded either by a disk
(dpAGB objects[disk-prominent post-AGB objects: Sahai et al. 2011]), and/or an expanding dusty shell (pre-planetary
nebulae or PPNe) that is generally highly aspherical – objects with round morphology are absent during this evolutionary phase! HST imaging surveys have clearly revealed the aspherical morphologies of pre-planetary nebulae (PPNs)
(e.g. Sahai et al. 2007; Ueta et al. 2000).
The current consensus is that these dramatic changes in the geometry and kinematics of
mass-loss during the AGB-to-PPN transition are a result of strong interaction with a binary
companion (e.g., Sahai 2018).

Intriguingly, binarity may also be responsible for the rapid and unexpected evolution of RGB stars to the
PPN stage (post-RGB stars).  The Boomerang Nebula in our Galaxy (also the coldest object in the Universe: Sahai & Nyman 1997) is the only
known representative of a post-RGB star
that has evolved to the PPN phase (Sahai et al., 2017). The Boomerang's luminosity is much lower
($L\sim300$ ) than possible for a post-AGB object, and Sahai et al., 2017 show that merger with a binary companion most likely triggered
extreme mass loss ($\sim$10$^{-3}$ M$_\odot$yr$^{-1}$) at a very high ejection velocity (165 ),
over a relatively short period (3500 yr) via common-envelope evolution (CEE).

A class of post-RGB objects similar to the Boomerang (i.e., with $L\lesssim1000$ ) have recently been identified in the Large and Small Magellanic Clouds
(LMC and SMC), using optical spectroscopy and $Spitzer$ photometry (Kamath et al. 2014, 2015, 2016). The circumstellar dust in the ejecta heated by the central star results in these objects having large mid-IR excesses. The optical spectroscopy allows a determination of stellar parameters ($T_{\rm eff}$, log $g$, [Fe/H] and E[B–V]) and integration of the spectral-energy-distribution (SED) constrains the bolometric flux. The known distance to the LMC ($\sim$ 50 kpc) then allows
reliable luminosity estimates for these stars; the low luminosities ($<$ 2500 L$_\odot$)
indicate that these stars have not yet reached the AGB phase. The post-RGB stars are thus separated from post-AGB ones by their low luminosities.
And they can be distinguished from Young Stellar Objects (YSOs) by the value of log $g$, which is generally lower in post-RGB stars by about 1.2.

In this paper, we investigate a
sample of  this new class of post-RGB stars in the LMC, believed to be rapidly evolving to
the PN stage. We also study a sample of post-AGB LMC stars discovered by
Kamath et al., 2015 for comparison, in order to investigate if the properties of the ejecta (e.g., mass, mass-loss rate, temperature, disk-to-shell mass ratio) are different in these two classes. In order to understand the influence of a binary companion on the mass loss history and rapid evolution
of a post-RGB star to the PPN stage, we model the circumstellar dust distribution of these objects, using their
near to far-infrared SEDs.  We chose objects classified as “shell" or “disk" sources from
Kamath et al. (2015: hereafter KWVW15)[According to KWVW15, “Shell" sources are those that show a double-peaked SED in the near- to mid-IR region, whereas the “disk" sources
do not show two distinct flux peaks but display a near-IR excess]. The presence of a disk and/or shell and the ratio of the shell
to disk mass (when both can be identified) should help in understanding the physical processes contributing to the
early evolution of these stars to the PN stage.

The plan of this paper is as follows.  <ref> describes the selection of sources for our modeling study,   <ref> describes our modeling methodology, and  <ref> and  <ref> provide details of the SED modeling for each of our post-RGB and post-AGB objects. In  <ref>, we provide a discussion of our modeling results, and  <ref> gives our conclusions.

§ TARGET SELECTION

The post-RGB and post-AGB stars in this study are taken from KWVW15 – these are listed in Tables <ref>, <ref>, <ref> and <ref> together with their photometry. They
are divided roughly equally between “shell” and “disk”
sources. KWVW15 introduced this classification based on a visual
inspection of their dust SEDs. The shell sources show far-IR excess and
the peak of the dust distribution lies beyond 10 $\micron$.  The disk sources
show near-IR excess indicative of hot dust and the peak of the
dust SED lies around 10 $\micron$ or sometimes even bluer. Further, on a
colour-colour plot, the shell sources have [8]$-$[24] $>$ 4.0 and [3.6]$-$[4.5] $<$ 0.5.
disk sources, on the other hand, have [3.6]$-$[4.5] $>$ 0.5 or is some cases [3.6]$-$[4.5] $<$ 0.5
but in combination with [8]-[24] $<$ 3.0.

In order to ensure that for the post-RGBs in our study, CEE occured on the RGB, we selected objects with
L $<$ 1000 L$_{\odot}$. These post-RGB candidates have the largest 24 $\micron$-excess,
implying the presence of the largest amount of dust, using the 8$-$to$-24\,\micron$ color
(from IRAC4 and MIPS1 band photometry).  The post-AGB stars are common between KWVW15
and van Aarle et al. (2011), the only exception being J051906.86-694153.9. This star showed
the BaII line at 4554.03Å, (KWVW15).  The presence of BaII indicates a
s-process enriched post-AGB object.

§ MODELLING THE SPECTRAL ENERGY DISTRIBUTION: METHODOLOGY

We have modeled the SEDs of the selected sources using the one-dimensional radiative transfer code, DUSTY [Ivezić et al. , 2012]. Optical properties for six different grain types are included
in the DUSTY code. These are `warm' and `cold' silicates (Sil-Ow and Sil-Oc) from Ossenkopf et al. (1992),
silicate and graphite grains (Sil-DL and grf-DL) from Draine and Lee (1984), amorphous carbon (amC-Hn) from Hanner (1988) and silicon carbide (SiC-Pg) from Pégourié (1988).
For our models, the dust grain composition was specified to be “warm" (Sil-Ow) silicates. Only when Sil-Ow grain composition did not provide a satisfactory fit, we tried different
grain compositions.  We used the Mathis, Rumpl, Nordsieck
(MRN; Mathis et al., 1977)
grain size (a) distribution function, n(a) $\propto$ a$^{-q}$ for a$_{\rm min}$ $\leq$ a $\leq$ a$_{\rm max}$. DUSTY allows one to use standard MRN parameters (q = 3.5, a$_{\rm min}$ = 0.005$\micron$ and a$_{\rm max}$ = 0.25$\micron$) or modified MRN parameters. Whenever required, we used a modified MRN distribution by altering a$_{\rm min}$ and a$_{\rm max}$. The dust density was assumed to be proportional to r$^{-2}$, where r is the radial distance from the star.

Since there are multiple input parameters associated with the dusty circumstellar
environment that affect the SED, our strategy for exploring the parameter space is as
follows. We first attempt to fit the SED using a single shell (one-component model),
varying the dust temperature on the inner shell boundary (T$_{\rm d}$), the relative
shell thickness (Y = ratio of the outer to the inner shell radius), the optical
depth ($\tau$) at 0.55$\micron$, and the grain-size distribution. These model
numbers are given a suffix `s'. If systematic discrepancies
remain between the fit and the data, we then attempt a two-component fit, in which we add
an inner component, representative of a hot, compact disk, and vary its T$_{\rm d}$, Y, $\tau$,
and dust-grain properties. Specific wavelength ranges of the SED are relatively more sensitive to the shell and disk, and help us to constrain their properties in a non-degenerate manner.

§.§ Photometric magnitudes, extinction and stellar model atmospheres

The photometry of the objects in this study has been compiled by Kamath et al., 2015
and is available online from the Vizier database (https://vizier.u-strasbg.fr/viz-bin/VizieR).
The Vizier database was further searched to compile the error bars
and upper(lower) limits if any on the photometric magnitudes.
The photometric magnitudes were corrected for the combined effects of Galactic and
LMC reddening using a mean E(B$-$V) = 0.08 [Keller & Wood, 2006]. From the
ultraviolet to the near-infrared, we used the LMC's average extinction curve and
R$_{v}$ = 3.41 as derived by Gordon et al. , 2003. In the mid-infrared,
we applied the extinction law by Gao et al. , 2013. Kamath et al., 2015 have derived the
effective temperatures (T$_{\rm eff}$), gravities (log $g$) and metallicities [Fe/H] of the
objects used in the study. For the purpose of modeling, we used the Kurucz stellar atmosphere models
closest to these parameters from http://kurucz.harvard.edu/grids.html.

§.§ Estimating the circumstellar mass

The DUSTY code outputs the SED, normalized to the bolometric flux, F$_{\rm bol}$. We determined F$_{\rm bol}$ by
scaling the model SED to match the de-reddened SED of our sources.
The luminosity and total mass (M$_{\rm gd}$) in the  circumstellar component (gas+dust) was computed for each model.
The distance (d) to the LMC is 50 kpc.
We estimated the luminosity for each model as $L = 4\pi$d$^{2}$F$_{\rm bol}$.
For objects obeying a r$^{-2}$ density distribution, the total mass in the circumstellar component (gas+dust) is estimated using [Sarkar & Sahai, 2006]:

M$_{\rm gd}$ = 4$\pi$R${^{2}_{\rm in}}$Y($\tau_{100}/\kappa_{100})\delta$

where, R$_{\rm in}$ is the inner radius of the dust shell, Y is the shell relative thickness specified in the DUSTY input (R$_{\rm out}$/R$_{\rm in}$), $\tau_{100}$
is the shell optical depth at 100 $\micron$, $\kappa_{100}$ is the dust mass absorption coefficient at
100 $\micron$ and $\delta$ is the gas-to-dust ratio. We assume $\delta=200$, and $\kappa_{100}=34$ cm$^{2}$g$^{-1}$,
as in Sarkar & Sahai, 2006.

The inner radius (R$_{\rm in}$) of the modeled dust
shell/disk is inferred from the output of the DUSTY code.
The dust mass absorption coefficient ($\kappa$) is poorly constrained. It may be different for warm ($\gsim$ 300K)
cold dust ($\lsim$ 300K) (e.g. Demyk 2018). $\kappa$ also differs with wavelength. Its value at near-IR wavelengths where the bulk of
the emission from an inner disk would occur is also not known. For the cool and warm dust, we have used
$\kappa$ at 100$\micron$ [Sarkar & Sahai, 2006]. M$_{\rm gd}$ also scales in proportion to Y and $\delta$ (discussed below).

§.§.§ Thickness of the circumstellar component

In general, the total mass in the circumstellar shell is a lower limit to the actual mass in the shell because the dust temperature decreases with radius – thus the outer regions of the shell emit dominantly at wavelengths longward of the maximum wavelength for which we have data, $\sim$24 . To understand this, we look at the post-RGB objects, J051920.18-722522.1 (shell-source) and
J045755.05-681649.2 (disk-source).
Increasing the thickness of the cool outer shell (Y) by a factor of 10 in these objects, increased the total shell mass by the same factor (Table <ref>) but did not cause any noticeable change in the fit to the observed SEDs (Figs. <ref> and <ref>).

§.§ Gas-to-dust ratio

The derived mass-loss rate depends on the assumed gas-to-dust ratio. Considering that RGB stars are much less luminous
that their AGB counterparts, the gas-to-dust ratio may deviate from the typical value of 200 for the post-AGBs.
The dust composition around the post-RGBs may be different from that around the post-AGBs. The gas-to-dust ratio may also depend on the metallicity
of the galaxy, e.g. van Loon et al., 1999, Nanni et al., 2019. van Loon et al., 1999  use a value of 500 for the LMC.
Roman-Duval et al. (2014) found gas-to-dust ratios of
380$^{+250}_{-130}$ in the LMC.
Determining the gas-to-dust ratio as a function of fundamental stellar parameters
(e.g., luminosity and metallicity) and evolutionary phase is still a distant goal (Sahai 2009). A higher value of $\delta$ would imply
a proportionate increase in the masses of the ejecta that we derive  for our sample of stars.

§.§ Expansion velocity and mass loss rate

Typical expansion velocities of the dust shells around post-AGB and PNe range from 10$-$30 [Nyman et al., 1992]. In contrast,
Sahai & Nyman, 1997 found that the Boomerang Nebula's cold outer shell was extending at 165 . For the
high luminosity post-AGB stars the radiative pressure from the central star would influence the expansion velocity. In
RGB stars the luminosity is much lower and hence one would expect a different mechanism to drive the velocity of the
cold shell. Such a mechanism may be expulsion due to common envelope evolution in a binary system.
We estimate the age of the cold dust shell in the post-RGBs assuming a typical expansion velocity of 50 . The thickness of
the model shell provides an ejection time-scale, and together with the model shell mass (M$_{\rm gd}$), an estimate
for the mass loss rate (dM/dt).

In the Boomerang nebula, the FWHM of the emission from the central waist/disk region is
4.3 [Sahai et al., 2017], which may suggest an expansion velocity of 2.2 . Jura & Kahane, 1999 suggest
FWHM $<$ 5 from molecular gas observations around evolved red giant stars.  Hence,
for the warm inner disk, we assumed an expansion velocity of 2 to derive
the mass-loss rates.

In contrast, for the post-AGBs we assumed an expansion velocity of 15 for the cool outer shell and
2 for the inner disk. In the case of nested shells, e.g. in the post-AGB “shell" source, J050632.10-714229.8  ( <ref>), we assumed an expansion velocity of 50 for the inner shell.

§ MODELS OF POST-RGB OBJECTS

The detailed modeling of the post-RGBs objects listed in Table <ref> is discussed below. Our procedure for determining the best-fit model is explained in detail for J043919.30-685733.4, together with a discussion of additional models demonstrating the effect of varying different input parameters. For the remaining sources, the figures corresponding to additional model SEDs which check for the robustness of the fits by
varying the different input parameters (Table <ref>) are given in the Appendix. For ease
of reference, the figures in the appendix are numbered in accordance with the post-RGB object number in
Table <ref>. The derived model parameters are given
in Table <ref>, and important physical parameters of the adopted best-fit models (flagged with
$\dagger$ symbol) are summarized in Table <ref>.

§.§ J043919.30-685733.4

This object is classified as a post-RGB shell source by Kamath et al., 2015, with T$_{\rm eff}$ = 6313K, log $g$ = 1.5
and [Fe/H] = $-$2.0. We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 6250K, log $g$ = 1.5 and [Fe/H] = $-$ 2.0.
In Fig. <ref>, we
show the observed and model SEDs obtained using varying shell thickness, Y = 8, 100 and 500. We
assumed the standard MRN grain size distribution. With T$_{\rm d}$ = 550 K and $\tau$=1.5,
we find a reasonable fit to the SED, except in the mid-IR region ($\sim$ 5 $-$ 12 $\micron$).
We find a lower limit on the shell thickness of $Y\sim100$, from the WISE 22 $\micron$ and MIPS 24 $\micron$ fluxes.

Taking (Y = 100) as the lower limit to the shell thickness, we investigated whether variation in the
value of T$_{\rm d}$  can improve the fit in the mid-IR region.
Models with T$_{\rm d}$ = 145, 175 and 500K (Fig. <ref>) were computed using a standard MRN grain size distribution, and $\tau$=0.8.
We find that the discrepancy
in the near-IR persists, except for the model with T$_{\rm d}$ = 500 K.  However, in this
case, the model shows a large silicate feature which is not observed – although using larger size grains in
the shell may suppress this emission feature, the
far-IR flux too falls much more rapidly than that indicated by the MIPS 24 $\micron$ data. Since the shell is optically thin
at all wavelengths longward of $\sim1\micron$, the model SED is not sensitive to the shell geometry.

We conclude that a single-shell cannot fit the SED adequately, and resolving these discrepancies requires a two-component model, i.e.
a warm inner component and a cool outer shell.
The inner component is likely to have a disk structure, as in the Boomerang Nebula.

We approximated the inner disk to be an axially symmetric wedge-shaped fraction ($\sim$ 0.35) of a sphere[“disk fraction" in Tables <ref> and <ref>] (Sahai et al., 2003). Assuming the
fraction that is not covered by the disk, reasonable fit to the far-IR flux distribution ($\lambda\gtrsim10\micron$) could be obtained
for T$_{\rm d}$ = 130K, thickness (Y = 20), $\tau$=0.95 and standard MRN grain size distribution (blue curve in Fig. <ref> and <ref>).
Y = 20 is a lower limit for the shell thickness.  For a warm inner disk with thickness (Y =1.4) and an r$^{-2}$ density, alternate model fits were obtained
assuming standard MRN grain size distributions (model # 1: T$_{\rm d}$ = 1000 K, yellow curve, Fig. <ref>) and modified
MRN (a$_{\rm min}$ = 1 $\micron$ and a$_{\rm max}$ = 25 $\micron$) grain size distributions (model # 2: T$_{\rm d}$ = 800K, yellow curve,
Fig. <ref>) in the disk.

The sensitivity of the SED to different a$_{\rm min}$ and a$_{\rm max}$ grain sizes in the
inner disk is shown in Fig. <ref>.  A smaller a$_{\rm min}$
(by a factor of 10) would cause a deficit in the near-IR range.
Similarly, decreasing a$_{\max}$ (by a factor of 5) causes a deficit between 5 and 10 $\micron$. Increasing a$_{\rm max}$ however, does not appear to effect the SED.

For both models # 1 and # 2 (Figs. <ref> and <ref>), there is a mismatch for the IRAC4 (8$\micron$) flux.
We convolved the model flux with the IRAC4 response curve and estimated the magnitude at 8 $\micron$ for
models # 1 (14.96 mag) and # 2 (14.76 mag). In comparison, the observed magnitude at 8 $\micron$ (corrected for extinction) is significantly lower (13.84).
The IRAC4 bandpass covers a strong PAH emission feature which is not included in the DUSTY code. If we were to include the
excitation of PAHs, it is possible that a better fit would be obtained in this region.

For both models  # 1 and # 2, we show the sensitivity of the SEDs to the thickness (i.e., Y=outer-radius/inner-radius) of the inner disk in Figs. <ref> and
<ref>. Increasing the thickness causes a mismatch in the mid-IR and with WISE band 3 (W3/12$\micron$) flux.

In our two-component models (models # 1 and # 2), it was assumed that the shell is illuminated
by the radiation from the central star and captures $\sim$ 0.65 of the bolometric flux of the star. The disk
was treated as a separate entity illuminated by the central star
and capturing ($\sim$ 0.35) of the bolometric flux from the star. The output was the sum
of the modeled fluxes in each instance.

However, in the solid angle covered by the disk, the radiation illuminating the shell is not the direct starlight, but the attenuated, scattered and emitted stellar radiation from the disk. In the solid angle not covered by the
disk, the shell is illuminated by the central star radiation in addition to the scattered and thermal output
from the disk. We therefore constructed the correct illuminating spectrum for the shell in each of the two solid angles accounting for these effects. The DUSTY code
was run for each component of the shell with modified illumination and the outputs were then added in the appropriate
fractions – we will hereafter refer to this class of models as “correctly-illuminated"
models (for this and other sources), and they are given a suffix `c'  (see Table <ref>, model # 1,c & # 2,c and Figs. <ref> & <ref>).

The fit in the far-IR is identical for the two models; we compared the model and observed
SEDs in the near and mid-IR region to choose our best-fit model. The model SED is closer to
the observed J,H,K fluxes in the case of model # 1,c, and we adopt this as the best-fit for J043919.30-685733.4.
Our estimated luminosity (L = 116 L$_{\odot}$) is in close agreement with the observed luminosity of the object
(L = 106 L$_{\odot}$, KWVW15).
The total mass accumulated in the shell is 5.2 $\times$ 10$^{-3}$ M$_{\odot}$ over a relatively short period of
$\sim$ 200 $-$ 300 years at a mass loss rate of 7.1$\times$ 10$^{-6}$ $-$ 1.1$\times$ 10$^{-5}$ M$_{\odot}$ yr$^{-1}$ (Table <ref>). The inner disk is very young ($\sim$ 2 years)
with a mass of $\sim$ 2.19 $\times$ 10$^{-8}$ M$_{\odot}$ and a mass loss rate of 1.18$\times10^{-8}$ M$_{\odot}$yr$^{-1}$.

§.§ J051347.57-704450.5

This object is classified as a post-RGB shell source by KWVW15, with T$_{\rm eff}$ = 4500 K, log $g$ = 1.0 and
[Fe/H] = $-$0.5. We used a Kurucz stellar atmosphere model with these parameters for
modeling the SED of the object. A fit to the optical and IR fluxes could be obtained using
T$_{\rm d}$ = 250K, modified MRN grain size distribution (a$_{\rm min}$ = 0.1$\micron$,
a$_{\rm max}$ = 0.25$\micron$), $\tau$ = 0.35 and Y = 3.0 (Fig. <ref>). In (Figs. <ref> $-$ <ref>), we show
the effects of varying the temperature at the inner boundary of the
dust shell (T$_{\rm d}$), the optical depth, ($\tau$), grain sizes (a$_{\rm min}$, a$_{\rm max}$) and shell
thickness (Y). We find that the GSC 2.2 R-band flux is much below the modeled SED.
Three R-band magnitudes are available for the object from Vizier, R = 16.74 $\pm$ 0.44
(GSC 2.2 Catalogue/STScI 2001), R1 = 15.76 and R2 = 16.29 (USNO-B 1.0 Catalog; Monet et al. 2003). We plotted the corresponding extinction corrected fluxes in the SED. The USNO-B 1.0 R-band fluxes are closer in agreement with the modeled SED. Mismatch between the modeled SED and the I-band flux is seen in several of our objects and is discussed in  <ref>. Since a single shell model provided satisfactory fits to the optical and IR fluxes, we did not try a two-component model
for this star.

The luminosity of the object is estimated to be 776 L$_{\odot}$. KWVW15 integrated the flux under the observed SED
and estimated a luminosity of 712 L$_{\odot}$. They derived a photospheric luminosity of 840 L$_{\odot}$.
From our best-fit model, the total mass accumulated in the shell is 4 $\times10^{-5}$ M$_{\odot}$
over a period of 17.8 years at a mass loss rate of 2.24 $\times10^{-6}$ M$_{\odot}$yr$^{-1}$.

§.§ J051920.18-722522.1

This object is classified as a post-RGB shell source by KWVW15, with T$_{\rm eff}$ = 4500 K, log $g$ = 1.5 and
[Fe/H] = $-$1.5. We used a Kurucz stellar atmosphere model with these parameters.
Figs. <ref> $-$ <ref> show the observed and modeled SED of the object along with effects of varying the temperature at the inner boundary
of the dust shell (T$_{\rm d}$), the optical depth ($\tau$), grain sizes and shell thickness (Y).
The R and I-band fluxes are well below the modeled SED ( <ref>).
A fit to the U,B,V and far-IR fluxes could be obtained using T$_{\rm d}$ = 200K, MRN grain size distribution
(a$_{\min}$ = 0.005$\micron$, a$_{\max}$ = 0.25$\micron$), $\tau$=0.75 and Y = 20 (model # 1,s).
The fit at MIPS band 1 (24 $\micron$, Fig. <ref>) suggests that Y = 20 may be taken as a lower limit
for the shell thickness. However, the modeled flux
remained well below the observed flux in the mid-IR region. To account for this deficit, we assumed a two-component
model (model # 2) with a warm (T$_{\rm d}$ = 500 K), inner disk, in addition to a cool outer shell at 110 K. This disk
was approximated as a wedge-shaped fraction ($\sim$ 0.4) of a sphere with $\tau$=0.4, thickness (Y) = 2.0 and modified
MRN grain size distribution (a$_{\rm min}$ = 0.3$\micron$ and a$_{\rm max}$ = 20$\micron$). The cool outer shell is
characterised by $\tau$=1$\micron$, thickness (Y) = 20 and MRN grain size distribution.
We constructed the correctly illuminated spectrum (model # 2,c) for the shell in the two solid angles (Fig. <ref>).

From the correctly illuminated model # 2,c, we estimated luminosity,
L = 582 L$_{\odot}$. Integrating the flux under the observed
SED, KWVW15 estimated L = 496 L$_{\odot}$.
The total mass in the shell is estimated to be 3.44 $\times10^{-2}$ M$_{\odot}$ accumulated over $\sim$ 650 $-$ 800 years at a
mass loss rate of $\sim$ 2.1$\times10^{-5}$ $-$ 2.6$\times10^{-5}$M$_{\odot}$yr$^{-1}$. The inner disk is much younger (17.3 years) with a
mass of 2.59$\times10^{-5}$ M$_{\odot}$ and a mass loss rate of 1.5$\times10^{-6}$M$_{\odot}$yr$^{-1}$.

§.§ J053930.60-702248.5

This object is classified as a post-RGB shell source by KWVW15, with T$_{\rm eff}$ = 4315 K, log $g$ = 0.5 and
[Fe/H] = $-$0.9. We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 4250 K , log $g$ = 0.5 , [Fe/H] = $-$1.0
for modeling the SED of this object. A fit to the
optical and IR fluxes could be obtained using T$_{\rm d}$ = 300K, MRN grain size distribution
(a$_{\rm min}$ = 0.005$\micron$, a$_{\rm max}$ = 0.25$\micron$), $\tau$=0.7 and Y = 10 (Fig. <ref>). Figs. <ref> $-$ <ref>, show the effects of varying the temperature at the inner boundary of the dust shell (T$_{\rm d}$), the optical depth, ($\tau$), grain sizes (a$_{\rm min}$, a$_{\rm max}$) and shell thickness (Y).
The fit at MIPS band 1 (24 $\micron$, Fig. <ref>) suggests that Y = 10 may be taken as a lower limit for the shell thickness.  Since a single shell model provided satisfactory fits to the optical and IR fluxes, we did not try a two-component model for this star.

Our estimated luminosity (295 L$_{\odot}$) is in good agreement with the observed (248 L$_{\odot}$) and photospheric luminosity (294 L$_{\odot}$) values of KWVW15. From our best
fit model, the total mass accumulated in the shell is 5.8 $\times10^{-5}$ M$_{\odot}$ over a period of 27.2 years at a mass loss rate of 2.13 $\times10^{-6}$ M$_{\odot}$yr$^{-1}$.

§.§ J045555.15-712112.3

This object is classified as a post-RGB disk source by Kamath et al., 2015, with T$_{\rm eff}$ = 10000 K, log $g$ = 2.0 and [Fe/H] = $-$0.5. We used a Kurucz stellar atmosphere model with these parameters.
Fig. <ref> shows a one-component model fit to the observed SED (model # 1,s)
using Sil-Ow grains, T$_{\rm d}$ = 550 K, a$_{\min}$ = 0.005 , a$_{\max}$ = 1.0 , $\tau$=2.3 and Y = 20. The effects of varying the temperature at the inner boundary of the dust shell (T$_{\rm d}$), the optical depth ($\tau$), grain sizes and shell thickness (Y) are shown in Figs. <ref> $-$ <ref>.
Our model fit (# 1,s) is deficit in the near-IR region and additionally, there is a mismatch with the IRAC2 (4.5$\micron$) and W2 (4.6$\micron$) fluxes and IRAC3 (5.8 $\micron$) flux.

For a better fit in the near-IR region, we assumed a two-component model (model # 2) with a warm (T$_{\rm d}$ = 1200 K), thin inner disk, in addition to the cool outer shell at 550K. The disk composed of Sil-Ow grains was approximated as a wedge-shaped fraction ($\sim$ 0.1) of a sphere with $\tau$=0.5, thickness (Y) = 10 and grain size distribution, a$_{\rm min}$ = 0.005 $\micron$ and a$_{\rm max}$ = 0.1 $\micron$.  We constructed the correctly illuminated spectrum for the shell in the two solid angles (# 2,c, Fig. <ref>). Besides the continuing
mismatch at IRAC2 (4.5$\micron$), W2 (4.6$\micron$) and IRAC3 (5.8 $\micron$), the fit remained
unsatisfactory in the near-IR.

A better fit in the near-IR was obtained using an alternate two-component model (model # 3). We used graphite grains (grf-DL) for the warm inner disk (T$_{\rm eff}$ = 800K) and a combination of Sil-Ow (0.8) and grf-DL (0.2) for the cool outer shell (T$_{\rm eff}$ = 500K). The inner disk was approximated to be a wedge-shaped fraction (0.1) of a sphere with $\tau$=0.7, thickness (Y) = 5 and MRN grain size distribution (a$_{\rm min}$ = 0.005 $\micron$ and a$_{\rm max}$ = 0.25 $\micron$). The cooler shell
is modeled using $\tau$=1.8, thickness (Y) = 2 and MRN grain size distribution. Fig. <ref> shows the correctly illuminated two-component model (# 3,c).

We explored models which may give a better fit at 4.6$\micron$.
The two-component model # 4 is shown in Fig. <ref>. The warm inner disk
(T$_{\rm eff}$ = 1000K) composed of Sil-Ow grains was approximated to be a wedge-shaped fraction (0.4) of a sphere with $\tau$=1.0, Y =10 and MRN grain size distribution. The cool outer shell
(T$_{\rm eff}$ = 300K) has Sil-Ow grains, $\tau$=2.5 and Y =2 and MRN grain size distribution. The model provides a good fit at 4.6$\micron$ but is discrepant with the observed values in the near-IR and
at IRAC3 (5.8$\micron$) and IRAC4 (8$\micron$).

We investigated the possibility that a stellar companion dominates the emission at and around 4.6$\micron$ in models # 2 and # 3, with suitable adjusments in the disk and the shell parameters, by fitting a blackbody curve to the flux at 4.6$\micron$. We find that a blackbody with
T$_{\rm eff}$ = 800K, and L = 39 L$_{\odot}$ would account for the flux at 4.6$\micron$. Since a main-sequence companion star with such a low T$_{\rm eff}$ would have a very low luminosity (i.e., much lower than 39 L$_{\odot}$), we conclude models including a stellar companion cannot explain the discrepancy at 4.6$\micron$.

We conclude that the relatively low flux at 4.6$\micron$ likely results from the presence of a deep absorption feature at 4.6$\micron$. Such a feature has been observed in YSOs (Lacy et al. 1984), and is believed to be due to solid phase CO and cyano group molecules. J045555.15-712112.3  was classified originally as a possible YSO by Gruendl & Chu (2009). KWVW15 re-classified it (and many other objects also originally classified as YSOs), using an automated spectral typing pipeline (STP) that focusses on the
Balmer-line region for the determination of T$_{\rm eff}$, for objects with T$_{eff}>8000$K (Kamath et al. 2014).
However, an examination of the optical spectrum (downloaded from VizieR) reveals a lack of reliable discernible spectral features in this region. It is therefore plausible that for this case, the STP fit did not provide a good estimate of T${\rm eff}$ and hence log $g$. A higher log $g$
value would support the classification of J045555.15-712112.3 as a possible YSO.

Nevertheless, if the object is indeed a post-RGB source, we adopt model # 3,c as the best-fit to the observed SED (Fig. <ref>).  Given the carbon-rich circumstellar dust
in this model, the mismatch at IRAC3 (5.8$\micron$) may be attributed to PAH emission, indicating that J045555.15-712112.3 is a CH giant (see  <ref>).
From this model, we estimated a luminosity of 621L$_{\odot}$ for J045555.15-712112.3. KWVW15 integrated the flux under the observed SED and estimated a luminosity of 454 L$_{\odot}$. In contrast, they derived a photospheric luminosity of 191 L$_{\odot}$. They attributed this difference to either a non-spherically symmetric dust envelope or
substantial flux contribution from another object coincident with J045555.15-712112.3 in the sky.
Based on our best-fit model # 3,c, the total mass accumulated
in the shell is 8.73$\times10^{-5}$M$_{\odot}$ over $\sim$ 1.4 $-$ 4.7 years at a mass loss rate of 4.94$\times10^{-6}$ $-$ 1.71$\times10^{-5}$M$_{\odot}$ yr$^{-1}$.
The mass accumulated in the disk is 2.67$\times10^{-6}$M$_{\odot}$ over 66.9 years at a mass loss rate of 3.99$\times10^{-8}$M$_{\odot}$ yr$^{-1}$.

§.§ J045755.05-681649.2

This object is classified as a post-RGB disk source by KWVW15, with T$_{\rm eff}$ = 4927 K, log $g$ = 1.5 and
[Fe/H] = $-$0.3. We used a Kurucz stellar atmosphere model with Teff = 5000K, log $g$ = 1.5 and [Fe/H] = $-$0.3. We first modeled the SED of the object, assuming a single
dust shell surrounding the central star (model # 1,s, Fig. <ref>).
A fit to the optical and far-IR fluxes could be obtained using T$_{\rm d}$ = 500K,
MRN grain size distribution (a$_{\min}$ = 0.005 , a$_{\max}$ = 0.25 ),
$\tau$=1.0 and Y = 20 (model # 1,s). The I-band flux is well below the modeled SED.
Additionally, there is a mismatch at W2 (4.6$\micron$)
and IRAC4 (8$\micron$). These discrepancies have been discussed further in ( <ref>). The mismatch at IRAC4 may be attributed to PAH emission, similar to
J043919.30-685733.4.  Figs. <ref> $-$ <ref>
show the observed and modeled SED of the object along with effects of varying the temperature at the inner boundary
of the dust shell (T$_{\rm d}$), the optical depth ($\tau$), and shell thickness (Y).

To obtain a better fit in the mid-IR region, we tried a two-component model (model # 2), i.e. a warm inner disk, approximated as an
axially symmetric wedge-shaped fraction (0.4) of a sphere and a cool outer shell. Assuming the fraction that is not
covered by the disk, reasonable fit to the far-IR flux distribution ($\lambda\gtrsim10\micron$) could be obtained for
for T$_{\rm d}$ = 400K, thickness (Y = 30), $\tau$=0.9 and modified MRN grain size distribution (a$_{\rm min}$ = 0.1 $\micron$,
a$_{\rm max}$ = 1.0 $\micron$). For the warm inner disk, T$_{\rm d}$ = 1300 K, Y = 2, $\tau$=0.5 and modified MRN grain
size distribution (a$_{\rm min}$ = 0.005 $\micron$, a$_{\rm max}$ = 2.0 $\micron$) was used.
We constructed the correctly illuminated model for the shell in the two solid angles
(# 2,c). The output modeled flux distribution is shown in Fig. <ref>.

Our two-component model (# 2,c) provides a better fit in the optical (U,B,V photometric bands) and the
mid-IR region (Fig. <ref>) . Hence, we adopt this as the best-fit model for J045755.05-681649.2
From our model ( Table <ref>) the luminosity is estimated to be L = 217 L$_{\odot}$. The total mass accumulated in the shell
is $\sim$ 5.73 $\times$ 10$^{-5}$ M$_{\odot}$ over a short period of $\sim$ 30 $-$ 40 years at a mass
loss rate of $\sim$ 7$\times$ 10$^{-7}$ $-$ 9 $\times$ 10$^{-7}$ M$_{\odot}$ yr$^{-1}$.
The inner disk is much younger ($\sim$ 1.9 yr) with a mass of $\sim$ 9.64$\times$10$^{-9}$ M$_{\odot}$.
and a mass loss rate of $\sim$ 5$\times10^{-9}$ M$_{\odot}$ yr$^{-1}$.
KWVW15 integrated the flux under the
observed SED and estimated a luminosity of 190 L$_{\odot}$. They derived a photospheric luminosity of
215 L$_{\odot}$.

§.§ J050257.89-665306.3

This object is classified as a post-RGB disk source by KWVW15, with T$_{\rm eff}$ = 4586 K, log $g$ = 0.5 and
[Fe/H] = $-$0.5. We used a Kurucz stellar atmosphere model with Teff = 4500K, log $g$ = 0.5 and [Fe/H] = $-$0.5. A single shell model could not provide a fit to the near and mid-IR fluxes (model # 1,s; Fig. <ref>). Hence, we tried two-component models with a warm inner disk and a cool outer shell. A fit to the optical
and IR flux could be obtained under two scenarios: a two-component model in which the disk and shell are composed of Sil-Ow grains (model # 2) and another two-component model in which the disk is composed of amC-Hn grains and shell is composed of Sil-Ow grains (model # 3). The discrepancies at I-band and IRAC4 (8$\micron$)
been discussed in  <ref>.

In model # 2, the inner disk (T$_{\rm d}$ = 1200K) is composed of Sil-Ow grains with modified
MRN grain size distribution (a$_{\rm min}$ = 0.3$\micron$ and a$_{\rm max}$ = 5.0$\micron$),
$\tau$=0.5 and shell thickness (Y) = 3.0. The outer shell (T$_{\rm d}$ = 240K) is composed of
Sil-Ow grains with modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$ and
a$_{\rm max}$ = 1.0$\micron$), $\tau$=1.2 and shell thicknes (Y) = 10.0. The disk was approximated as
an axially symmetric wedge-shaped fraction (0.4) of a sphere. The correctly illuminated model (# 2,c) is presented in Fig. <ref>.

In model # 3, the inner disk (T$_{\rm d}$ = 1000K) is composed of amC-Hn grains with modified
MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$ and a$_{\rm max}$ = 0.5$\micron$).
$\tau$=0.6 and shell thickness (Y) = 2.0. The outer shell (T$_{\rm d}$ = 250K) is composed of
Sil-Ow grains with MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$ and a$_{\rm max}$ = 0.25$\micron$), $\tau$=1.7 and shell thicknes (Y) = 10.0. The disk was approximated as an
axially symmetric wedge-shaped fraction (0.3) of a sphere. The correctly illuminated model (# 3,c) is presented in Fig. <ref>.

Model # 2 is in better agreement with the optical (U,B) fluxes and
hence we adopted this as the best-fit. From our correctly illuminated model # 2,c (Table <ref>),
the luminosity is estimated to be L = 303 L$_{\odot}$.
The total mass accumulated in the shell is 2.68$\times10^{-4}$M$_{\odot}$ over a period of $\sim$ 30 $-$ 38 years at a mass loss rate of $\sim$ 4$\times10^{-6}$M$_{\odot}$ yr$^{-1}$. The inner disk is much younger ($\sim$ 4.3yr) with a mass of $\sim$ 5.77$\times10^{-8}$M$_{\odot}$ and a mass loss rate of $\sim$ 1.34$\times10^{-8}$M$_{\odot}$ yr$^{-1}$. KWVW15 integrated the flux under the observed SED and estimated a luminosity of 267 L$_{\odot}$. They derived a photospheric luminosity of 177 L$_{\odot}$.

§.§ J055102.44-685639.1

This object is classified as a post-RGB disk source by KWVW15, with T$_{\rm eff}$ = 7625 K, log $g$ = 1.0 and
[Fe/H] = $-$0.5. We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 7500K, log $g$ = 1.0 and [Fe/H] = $-$0.5.
The near-IR flux distribution of this object suggests the presence of a warm inner disk. A single shell
model (model # 1,s) could not provide a fit to the optical and near-IR fluxes. We therefore tried two component models with a warm inner disk and cool outer shell made up of Sil-Ow grains.
In these models, # 2a and # 2b,
the inner disk was approximated as a fraction (0.4 and 0.3 respectively) of a sphere.
However, we could not obtain a simultaneous fit to the optical and near-IR fluxes. The model fits # 1,s, # 2a and # 2b are shown in Fig. <ref>.

The near-IR flux of the object indicates a dust temperature in excess of 1500K. This would exceed the sublimation temperature of silicate grains. We therefore tried amorphous carbon grains (amC-Hn) for the inner disk (model # 3). In the DUSTY input, the dust sublimation temperature was changed to 2500 K.  Amorphous carbon has a higher sublimation temperature than silicate grains. While amorphous carbon has been observed in the dusty disk around C-rich post-AGB stars (e.g. HR4049, Acke et al. 2013), we do not expect to see it in the circumstellar environment of a post-RGB star because such dust is believed to form when the C/O ratio is $>$1 in the star's atmosphere, following formation of C via 3-$\alpha$ nucleosynthesis and (the third) dredge-up – events that occur at the centers of AGB stars. However, in the case of J055102.44-685639.1, we find that the disk is optically thick ($\tau$=1.0) and amC-Hn grains provide a reasonable fit to the SED in the near and mid-infrared (Fig. <ref>).
The cool outer shell has a combination of warm silicates (Sil-Ow/0.4) and  silicon carbide (SiC-Pg/0.6). The carbon-rich circumstellar chemistry may be explained if the post-RGB star is a CH giant in a binary system that formed when the post-RGB progenitor accreted carbon-rich matter from a more massive AGB companion (now a WD) before undergoing CE ejection. Such a scenario may also account for the presence of PAH molecules in the circumstellar environment of some post-RGB stars (see  <ref>).

In model # 3, the disk was approximated as an axially symmetric wedge-shaped fraction (0.3) of a sphere with amC-Hn grains, T${\rm d}$ = 2000 K, Y = 7.0, $\tau$=1.0 and modified MRN grain
size distribution (a$_{\rm min}$ = 0.005$\micron$, a$_{\rm max}$ = 0.05$\micron$). For the outer shell, we used a combination of Sil-Ow/0.4 and SiC-Pg/0.6 grains, T${\rm d}$ = 350 K, Y = 3.0, $\tau$=12.0 and modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$, a$_{\rm max}$ = 0.07$\micron$). At the large optical depth of the outer shell, the SiC feature goes into absorption (Fig. <ref>).
Since the inner disk is optically thick, we cannot apply the “correctly illuminated
model" formalism to this object.

From the best-fit model # 3 (Table <ref>), we estimated a luminosity of 621L$_{\odot}$.
The total mass accumulated in the shell
is 3.05 $\times$ 10$^{-3}$ M$_{\odot}$ over a period of $\sim$ 12.5 years at a mass
loss rate of $\sim$ 2.43 $\times$ 10$^{-4}$ M$_{\odot}$ yr$^{-1}$.
The inner disk is  younger ($\sim$ 8.8 yr) with a mass of $\sim$ 1.99$\times$10$^{-8}$ M$_{\odot}$ and a mass loss rate of $\sim$ 2.27$\times10^{-9}$ M$_{\odot}$ yr$^{-1}$. KWVW15 estimated 452L$_{\odot}$ and 815L$_{\odot}$ respectively for the observed and photospheric luminosity of the object.

§ MODELS OF POST-AGB OBJECTS

The detailed modeling of the post-AGBs objects listed in Table <ref> is discussed below.
The derived model parameters from the adopted best-fit models
(flagged with $\dagger$ symbol)
and the physical
parameters corresponding to these models are given
in Table <ref>; and important physical parameters of the adopted best-fit models are summarized in Table <ref>. We check
for the robustness of the fits by varying the different input model parameters (see
Table <ref>). These additional models are shown in the Appendix.

§.§ J050632.10-714229.8

This object is classified as a post-AGB shell source by KWVW15, with T$_{\rm eff}$ = 7614 K,
log $g$ = 0.5 and [Fe/H] = $-$0.4. However, based on high resolution spectra, [van Aarle et al., 2013] obtained a
much lower T$_{\rm eff}$ (= 6750K), log $g$ = 0.5 and a lower [Fe/H] (= $-$1.0). We prefer the
[van Aarle et al., 2013] values based on their high resolution data. Hence, we used a Kurucz stellar atmosphere model
with T$_{\rm eff}$ = 7000K, log $g$ = 0.5 and [Fe/H] = $-$1.0. Fig. <ref> shows a one-component model fit to observed SED using Sil-Ow grains, standard MRN grain size distribution, T$_{\rm d}$ = 200 K, $\tau$=0.6 and Y = 20 (model # 1,s).
This model is deficient in the mid-IR region, particularly at
IRAC3 (5.8$\micron$) and IRAC4 (8$\micron$).

Since J050632.10-714229.8 is a post-AGB star, the SED may
be the result of an interaction between a slow-moving cool outer shell ejected during the previous AGB
phase (typical V$_{\rm exp}$ = 15 kms$^{-1}$) and a fast-moving warm inner shell ejected during the
post-AGB phase (typical V$_{\rm exp}$ = 50 kms$^{-1}$), similar to the case of
IRAS 22036+5306 [Sahai et al., 2006]. Hence, we assumed a pair of nested shells, i.e. a warm inner shell covering 4$\pi$ solid angle and a cool outer shell.

Fig. <ref> shows the fit obtained in the mid-IR using
Sil-Ow (grain type # 1) in the inner shell, T$_{\rm d}$ = 400K, standard MRN grain size distribution, $\tau$=0.4 and Y = 2. It also shows that changing the grains from warm (Sil-Ow) to cold silicates (Sil-Oc) doesn't alter the fit significantly in the
mid-IR, so we use Sil-Ow grains for the inner shell.

Since J050632.10-714229.8 is a post-AGB star, its circumstellar environment may have a mix of carbon and silicate dust grains.
Fig <ref> shows the fits obtained using combinations of
amC-Hn and Sil-Ow grains. We find that
Sil-Ow (# 1) at 400K, a combination of amC-Hn/0.1 and Sil-Ow/0.9 (# 2) at 350K and
a combination of grf-DL/0.3 and Sil-Ow/0.7 (# 3) at 350K
(Fig. <ref>) all provide equally good fits to the
observed SED in the mid-IR. However, neither combination
of grains could provide a satisfactory fit to the flux from 4.6$\micron$ $-$ 8$\micron$.
These discrepancies are further discussed in  <ref>.

In our nested shell model, the cold shell is illuminated by the radiation emerging from the warm inner shell.
Using each of the above best-fit warm inner-shell models, we obtained best-fit models for the cool outer shell
(nested models # 1, # 2 and # 3; Table <ref>). Fig <ref> shows the complete correctly illuminated model SED of
J050632.10-714229.8 for each grain combination.

Our estimated luminosity (5434L$_{\odot}$) is in good agreement with KWVW15's observed
luminosity for the object (4910L$_{\odot}$). They estimated a much higher photospheric luminosity
of 7606L$_{\odot}$. The latter may be in error since it is based on their estimate
of T$_{\rm eff}$ and metallicity from low resolution spectra which is significantly different from that
based on high resolution spectra of [van Aarle et al., 2013]. From our best-fit models, the total mass
accumulated in the outer shell, shell age and mass loss rate respectively are 7.69$\times10^{-2}$M$_{\odot}$, 1020 yrs, 7.51$\times10^{-5}$M$_{\odot}$yr$^{-1}$ (nested model # 1); 6.94$\times10^{-2}$M$_{\odot}$, 1030 yrs.,
6.73$\times10^{-5}$M$_{\odot}$yr$^{-1}$ (nested model # 2) and 5.91$\times10^{-2}$M$_{\odot}$, 1020 yrs.,  5.81$\times10^{-5}$M$_{\odot}$yr$^{-1}$ (nested model # 3).

§.§ J051848.84-700247.0

This object is classified as a post-AGB shell source by KWVW15, with T$_{\rm eff}$ = 6015 K, log $g$ = 0.0 and [Fe/H] = $-$1.0.
KWVW15 suggest that based on high resolution spectra, log $g$ = 0.5. They cite [van Aarle et al., 2013]. However, we could
not find the object in the latter. Hence, we used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 6000 K, log $g$ = 0.0 and [Fe/H] = $-$1.0.
Based on VizieR, for the R-band magnitude we used 14.80 $\pm$ 0.44 (GSC 2.2 Catalogue/STScI 2001)[We did not use KWVW15's
R-band magnitude (14.31) because we could not find a corresponding reference.] We modeled the SED of the
object using T$_{\rm d}$ = 350K, standard MRN grain size distribution,
$\tau$=3.2 and Y = 20. Figs. <ref> $-$ <ref> show the
effects of varying the temperature at the inner boundary of the dust shell (T$_{\rm d}$), the
optical depth ($\tau$), and shell thickness (Y). Since a single shell model provided
satisfactory fits to the
optical and IR fluxes, we did not try a two-component model for this object.

We estimate a luminosity of 6210L$_{\odot}$. KWVW15 obtained vastly different values for
the observed and photospheric luminosity of the object, 4477L$_{\odot}$ and 14112L$_{\odot}$ respectively.
Their observed luminosity is obtained by integrating the flux under the observed SED. In the event of circumstellar
dust and reddening, the former may be significantly less than the actual luminosity. Their derived
photospheric luminosity depends on the extinction correction applied to the observed V-magnitude.
KWVW15's formalism attempts to account for both interstellar and circumstellar reddening.
If the V-magnitude is over-corrected, they would derive a higher luminosity for the object.

From our best-fit model, the total mass accumulated in the shell is 1.76$\times10^{-2}$M$_{\odot}$
over 1100 years at a mass-loss rate of 1.6$\times10^{-5}$M$_{\odot}$yr$^{-1}$.

§.§ J051906.86-694153.9

This object is classified as a post-AGB shell source by KWVW15,
with T$_{\rm eff}$ = 5613K, log $g$ = 0.0 and [Fe/H] = $-$1.3.
We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 5500K, log $g$ = 0.0 and [Fe/H] = $-$1.5.
A single shell model could not simultaneously fit the observed flux distribution from the optical to the far-IR.
Similar to the case of the post-AGB star, J050632.10-714229.8, we assumed a pair of nested shells.
A fit to the optical, mid- and far-IR data was obtained
using amC-Hn grain type, standard MRN grain size distribution, T$_{\rm d}$ = 2000K, $\tau$=0.35 and shell thickness (Y) = 2.0 (Fig. <ref>).  The radiation emerging from this warm shell
illuminates a cool outer shell of
SiC-Pg grains with modified MRN grain size distribution (a$_{\rm min}$ = 2.3$\micron$, a$_{\rm max}$ = 3.0$\micron$), T$_{\rm d}$ = 160K, $\tau$=0.07 and shell thickness (Y) = 2.0.
Fig. <ref> shows the correctly illuminated nested shell model.

We estimate a luminosity of 2018L$_{\odot}$. This is in good agreement with the observed luminosity estimated
by KWVW15 (2052L$_{\odot}$). However, as in the case of the post-AGB star, J051848.84-700247.0, KWVW15 report
vastly different values for the observed and photospheric luminosities of the object, 2052L$_{\odot}$ and 4246L$_{\odot}$ respectively. From our best-fit model, the total mass accumulated in the shell is 1.78$\times10^{-4}$M$_{\odot}$  over a period of 51.4 years at a mass loss rate of 3.46$\times10^{-6}$M$_{\odot}$yr$^{-1}$.

§.§ J053250.69-713925.8

This object is classified as a post-AGB shell source by KWVW15, with T$_{\rm eff}$ = 6073K,
log $g$ = 1.0 and [Fe/H] = $-1.1$. Based on high resolution spectra, [van Aarle et al., 2013] obtained
T$_{\rm eff}$ = 5500K, log $g$ = 0.0 and [Fe/H] = $-$1.0. We prefer the
[van Aarle et al., 2013] values based on their high resolution data. Hence, we used a Kurucz stellar atmosphere model
with T$_{\rm eff}$ = 5500K, log $g$ = 0.0 and [Fe/H] = $-$1.0. A single shell model gave
a reasonable fit to the near, mid and far-IR data (Fig. <ref>). The fit
was obtained for T$_{\rm d}$ = 250K, using a mix of
graphite (grf-DL/0.4) and amorphous carbon (amC-Hn/0.6) grains,
modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$, a$_{\rm max}$ = 0.07$\micron$),
$\tau$=0.7 and shell thickness (Y) = 10.0.

We estimate a luminosity of 4657.0L$_{\odot}$. This is in good agreement with the observed luminosity
estimated by KWVW15 (4223L$_{\odot}$). However, similar to the case of the post-AGBs, J051848.84-700247.0 ( <ref>)
and J051906.86-694153.9 ( <ref>), KWVW15 estimated a much higher photospheric luminosity, 11056L$_{\odot}$.

From our best-fit model, the total mass accumulated in the shell is 3.35$\times10^{-2}$M$_{\odot}$ over
a period of 1710 years at a mass loss rate of 1.96$\times10^{-5}$M$_{\odot}$yr$^{-1}$.

§.§ J045623.21-692749.0

This object is classified as a post-AGB disk source by KWVW15, with T$_{\rm eff}$ = 4500 K, log $g$ = 0.0 and
[Fe/H] = $-$1.0. We used a Kurucz stellar atmosphere model with these parameters. A single shell model (Fig. <ref>)
could not provide a fit to the far-IR data.
An overall fit from the optical to the far-IR
could be obtained under two scenarios: (1) a nested shells model (# 1) as in the case of the post-AGB shell sources J050632.10-714229.8 and J051906.86-694153.9 and (2) a two-component model with a warm inner disk and a cool outer shell (model # 2).

In case of the former (# 1), the inner shell is composed of Sil-Ow grains
with modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$ and a$_{\rm max}$ = 5.0$\micron$),
T$_{\rm d}$ = 1100K, $\tau$=0.6 and thickness (Y) = 3.0. The outer shell is composed of Sil-Ow grains with
standard MRN grain size distribution, T$_{\rm d}$ = 150K,
$\tau$=0.10 and thickness (Y) = 2.0. Fig. <ref> shows the correctly illuminated nested shells model.

In the second scenario (# 2), the warm inner disk (T$_{\rm d}$ = 1100 K) is composed of amC-Hn grains and the cool outer shell (T$_{\rm d}$ = 400 K) is made up of Sil-Ow grains. The amC-Hn disk
was approximated as a wedge-shaped fraction (0.5) of a sphere with $\tau$ = 0.5, thickness (Y)= 2 and modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$ and a$_{\rm max}$ = 2.0$\micron$). The Sil-Ow shell has modified MRN grain size distribution (a$_{\rm min}$ = 0.40$\micron$ and a$_{\rm max}$ = 1.0$\micron$), $\tau$ = 0.22 and shell thickness (Y) = 15. The correctly illuminated model SED is shown in Fig. <ref>.

The estimated luminosity based on the two models is 6598L$_{\odot}$ (# 1) and 6600L$_{\odot}$ (# 2).
KWVW15 gave an observed luminosity estimate of 6864L$_{\odot}$ and and a photospheric luminosity of 7131L$_{\odot}$.

The total mass accumulated in the outer shell (nested shells model # 1) is estimated to be 8.25$\times10^{-4}$M$_{\odot}$ in 225 years at
a mass-loss rate of 3.7$\times10^{-6}$M$_{\odot}$ yr$^{-1}$.
For the two-component model (# 2), the total mass accumulated in the shell is estimated to be 2.95$\times10^{-4}$M$_{\odot}$ in $\sim$ 78 years at
a mass-loss rate of 1.6$\times10^{-6}$M$_{\odot}$ yr$^{-1}$. The total mass in the disk is 5.63$\times10^{-7}$M$_{\odot}$ in 14.9 years at a
mass-loss rate of 3.8$\times10^{-8}$M$_{\odot}$yr$^{-1}$.

§.§ J051418.09-691234.9

This object has been classified as a post-AGB disk source (KWVW15), a population II Cepheid variable (Soszyński et al. 2008) and a RV Tauri star (van Aarle et al. 2011). KWVW15 estimated T$_{\rm eff}$ = 6112 K, log $g$ = 0.5 and
[Fe/H] = $-$1.6. We adopted Kurucz stellar atmosphere model with T$_{\rm eff}$ = 6000 K, log $g$ = 0.5 and [Fe/H] = $-$1.5.
For the R magnitude we used 13.78[We did not use KWVW15's R-band magnitude (13.96) because we could not find a corresponding reference.] from the NOMAD catalog (Zacharias et al. 2005).

A reasonable fit
to the near, mid and far-IR data (Fig. <ref>) is obtained assuming an inner disk (T$_{\rm d}$ = 1100K) composed of amorphous carbon (amC-Hn) with modified MRN grain size distribution (a$_{\rm min}$ = 0.005$\micron$, a$_{\rm max}$ = 2.0$\micron$), $\tau$=5.0 and thickness (Y) = 15.0. The disk is assumed to be a wedge-shaped fraction (0.25) of a sphere. The outer shell (T$_{\rm d}$ = 600K) is composed of silicate (Sil-Ow/0.4) and graphite (grf-DL/0.6) grains with standard MRN grain size distribution, $\tau$=0.4 and thickness (Y) = 30.0.
Since the inner disk is optically thick, similar to the post-RGB star, J055102.44-685639.1
( <ref>) we cannot apply the “correctly illuminated model" formalism to this object.

The variable nature of the star may explain the
difference between the
WISE (ALLWISE) and IRAC data at their bands 1 and 2 ($\sim$3.5$\micron$ and 4.5$\micron$) as also the mismatch between model and observed I-band data. ALLWISE photometry at bands 1(3.4$\micron$) and 2(4.6$\micron$) is generally an improvement over the WISE data at these wavelengths ( <ref>).
From our best-fit model, the luminosity of the object is 7763L$_{\odot}$. KWVW15 obtained 6667L$_{\odot}$ for the observed luminosity and 4703L$_{\odot}$ for the photospheric luminosity.
The total mass accumulated in the shell is estimated to be 1.21$\times10^{-3}$M$_{\odot}$ over 800 years at a mass-loss rate of 1.51$\times10^{-6}$M$_{\odot}$ yr$^{-1}$. The total mass in the disk is 4.67$\times10^{-5}$M$_{\odot}$ over 220 years at a
mass-loss rate of 2.12$\times10^{-7}$M$_{\odot}$yr$^{-1}$.

§.§ J055122.52-695351.4

This object has been classified as a post-AGB disk source (KWVW15), a
Population II Cepheid variable (Soszyński et al. 2008) and a RV Tauri star (van Aarle et al. 2011).
KWVW15 estimated T$_{\rm eff}$ = 6237 K, log $g$ = 1.5 and
[Fe/H] = $-$2.5 for the star.  We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 6250 K, log $g$ = 1.5 and [Fe/H] = $-$2.5. The difference between the
WISE (ALLWISE) and IRAC fluxes at bands 1 and 2 ($\sim$3.5$\micron$ and 4.5$\micron$) maybe
attributed to the variable nature of the object.

A reasonable fit to the near, mid and far-IR data could be obtained assuming a single shell model (Fig. <ref>) with a mix of graphite (grf-DL/0.7) and amorphous carbon (amC-Hn/0.3) grains, modified MRN grain size distribution (a$_{\rm min}$ = 0.05$\micron$, a$_{\rm max}$ = 0.3$\micron$), T$_{\rm d}$ = 450K, $\tau$=0.78 and thickness (Y) = 6.

We obtained a luminosity of 4116L$_{\odot}$ for the object in close agreement with the observed (3780L$_{\odot}$) and photospheric (3657L$_{\odot}$) luminosity estimated by KWVW15.
The total mass accumulated in the shell is estimated to be
1.18$\times10^{-3}$M$_{\odot}$ in 243 years at a mass-loss rate of 4.86$\times10^{-6}$M$_{\odot}$yr$^{-1}$.

§.§ J052519.48-705410.0

This object has been classified as a post-AGB disk source (KWVW15), a
Population II Cepheid variable (Soszyński et al. 2008) and a RV Tauri star (van Aarle et al. 2011). KWVW15 estimated T$_{\rm eff}$= 8117, log $g$ = 1.0 and
[Fe/H]= $-$0.5 . We used a Kurucz stellar atmosphere model with T$_{\rm eff}$ = 8000K , log $g$ = 1.0 and [Fe/H] = $-$0.5. The difference between the
WISE (ALLWISE) and IRAC fluxes at bands 1 and 2 ($\sim$3.5$\micron$ and 4.6$\micron$) maybe
attributed to the variable nature of the object.

Single shell models # 1  and # 2 (Fig <ref>) could not
provide simultaneous fit to the near, mid and far-IR data. Using Sil-Ow grains a
fit could be obtained to the near and far-IR fluxes (# 1) while
single shell grf-DL model (# 2) provided a fit to the near-IR and IRAC (3.6$\micron$,
4.5$\micron$ and 5.8$\micron$) fluxes.

Hence we tried two-component model fits to the observed SED. Fig <ref> shows the "correctly illuminated" best-fit two-component model for J052519.48-705410.0 (model # 3,c).
The warm inner disk (T$_{\rm d}$ = 800K) is approximated as a wedge-shaped fraction (0.35) of a sphere and
is made up of Sil-Ow grains with standard MRN grain size distribution, $\tau$=1.0 and thickness (Y) = 20. The outer
shell (T$_{\rm d}$ = 625K)  is made up of grf-DL grains with modified MRN grain size distribution (a$_{\rm min}$ =0.005$\micron$, a$_{\rm max}$ = 1.5$\micron$), $\tau$=0.7 and thickness (Y) = 2.0.

From our best-fit model # 3,c, we estimated a luminosity of 3804L$_{\odot}$. KWVW15 obtained
3219L$_{\odot}$ and 4943L$_{\odot}$ respectively for the observed and photospheric luminosities.
The total mass in the disk and shell was estimated to be 7.21$\times10^{-5}$M$_{\odot}$ and 7.26$\times10^{-5}$M$_{\odot}$ respectively.
The age of the disk is estimated to be
947 years while the shell is $\sim$2$-$4 years old. The mass loss rates for the disk and shell respectively are
2.26$\times10^{-5}$M$_{\odot}$yr$^{-1}$ and 7.61$\times10^{-8}$M$_{\odot}$yr$^{-1}$.

§ DISCUSSION

§.§ Common envelope evolution and ejecta mass

Common envelope (CE) evolution can occur in a close binary system where the primary is a giant star,
as a result of the formation of a common-envelope due to mass transfer from the primary.
The giant overfills the Roche lobes of both stars, engulfing the giant core and the companion.
If enough orbital energy is deposited to the CE via dynamical friction, the whole envelope is ejected on a
dynamical timescale. The CE ejecta then expand rapidly. Some fraction of the ejected mass may reside in an expanding, dense, equatorial structure and/or a bound circumbinary disk.

Lü et al. (2013) discuss dust formation in the CE ejecta of binary systems wherein the giant is
on the RGB (they call it the first giant branch or FGB) and the companion is a 1  degenerate star. Since RGB stars are expected to be oxygen-rich
following the first dredge-up, they use only olivine, pyroxene, and quartz-type silicates and
iron grains for discussion. They find that silicate and iron-rich dust in the CE ejecta
can form over a very wide range of radial distances ($10^{14}-10^{18}$ cm) from the RGB star (Fig. 5, Lü et al. 2013) which
easily encompasses our R$_{\rm in}$ values for the cold dust shells from our best-fit models (Table <ref>). In Lü et al.'s models,
the mass of dust produced in the CE ejecta of giants with masses 1$-$7 M$_{\odot}$ ranges from about 10$^{-9}$M$_{\odot}$ to 10$^{-2}$M$_{\odot}$. The lower end of this mass range corresponds to CEE occurring at the base of the RGB, and the model parameter $\gamma=0.2$ (see eqn. 2 of  Lü et al. (2013) for a definition of $\gamma$). Higher values of $\gamma$ produce a more rapid in the temperature of the expanding ejecta, result in CE ejecta with relatively higher densities, and thus more efficient dust formation.  Thus, the upper end of the ejecta mass range corresponds to CEE occurring anywhere between the base and the top of the RGB with $\gamma\ge0.3$, or at the top of the RGB with $\gamma\ge0.2$ (Fig. 2, Lü et al. 2013).

For our sample of post-RGB objects, the values of total mass (M$_{\rm gd}$) in the outer shell range from about 6$\times10^{-5}$ M$_{\odot}$ to  3$\times10^{-2}$ M$_{\odot}$
(Table <ref>) – inferred from our model dust masses that lie in the range 3$\times10^{-7}$ M$_{\odot}$–$1.5\times10^{-4}$ M$_{\odot}$.
We show our model dust mass values (M$_{\rm gd}$(outer shell)/$\delta$) overplotted on Fig. 2 of Lü et al. (2013) (Fig. <ref>) – since our derived  masses are likely lower limits to the total dust mass, its most likely that either $\gamma\ge0.3$ and/or CEE has occurred near or at the top of the RGB for our post-RGB sources. Lü et al. (2013) were pessimistic about the difficulty to observe
the dust produced in CE ejecta, since “the distance of dust formation
in the CE ejecta is between $\sim10^{14}$ and $\sim10^{18}$ cm and is relatively
far away from the FGB star"; however our study clearly shows that dust is relatively easily detected in our sample of post-RGB sources between few$\times10^{14}$ and $\sim10^{18}$ cm.

§.§ “Disk" and “Shell" source classification

KWVW15 classified the post-RGB and post-AGB sources into two groups – “disk" and “shell" (Van Winckel 2003), based on their SEDs.
The “disk" sources are inferred to have a stable compact circumbinary disk and are confirmed binaries (with orbital periods, P$_{orb}\sim100-2000$ d), whereas the “shell" sources have not resulted (so far) in any clear detected binary orbit (Hrivnak et al. 2011), suggesting that these are either
single-stars or have companions with relatively very wide orbits.

However, our modeling shows that the KWVW15 “shell"/“disk" classification is not robust. All “disk" sources in our sample require the presence of “shells". In addition, we find the presence of a disk in some “shell" sources (J043919.30-685733.4 and J051920.18-722522.1), and we exclude the presence of a disk in some “disk" sources (J045623.21-692749.0 and J045623.21-692749.0).

We find that the disk fractions are surprisingly large (typically 0.3–0.4), implying that these are geometrically thick, flared structures with a substantial opening angle. The large opening angles appear to be roughly consistent with the gas density of the ejected envelope as seen in numerical simulations of CEE, relatively soon after CEE occurs (e.g., see Fig. 1 of García-Segura et al. 2018). It appears unlikely that the “disk" structures in our models represent the stable compact circumbinary disks envisaged by Van Winckel (2003), although it is possible that the latter reside within the former. The inner rims of circumbinary disks may be puffed up, as suggested by the modeling of near and mid-IR interferometric imaging of the disk structure in the post-AGB star IRAS 08544-4431 (Deroo et al. 2007), and may mimic the physical effect of absorbing a substantial part of the direct stellar radiation within a relatively large solid angle.

§.§.§ Source Luminosities

We summarize our derived model stellar luminosities, together with those provided by KWVW15 in Tables <ref> (for post-RGB sources) and <ref> (for post-AGB sources). We find that our values are in reasonable agreement with KWVW15's observed values. For a few objects where KWVW15's observed and photospheric values are very discrepant, our model values provide a more accurate estimate of the stellar luminosities.

§.§.§ Ages of the dusty disks and shells

A comparison of the ages of the disk and shells can be potentially useful for constraining details of the physical mechanism responsible for the transition of RGB stars directly (i.e., without ascending the AGB) towards the PN phase, which has been argued to be CEE. For example, in the Boomerang Nebula, the age of the central disk structure is estimated to be 1925 yr, less than the age of the shell of $\gtrsim3840$ yr, which implies that the disk was formed after the (inferred) companion entered the CE, resulting in the ejection of the primary red giant star's envelope and the subsequent merging of the companion with the primary's core. In general, we find that the disk age is generally lower than the shell age for the objects in our study.
Our estimate for the age depends on (i) the assumed expansion velocities for the disk and the shell respectively (V$_{\rm exp}$) and (ii) on the thickness of the shell/disk (Y). Further, our derived Y
value is likely a lower limit to the estimated shell thickness. We illustrate this point with the post-RGB objects, J045755.05-681649.2 (Y =30) and J051920.18-722522.1 (Y =20).  In Figs. <ref> and <ref> we plot the correctly illuminated two-component models for the objects with the thickness of the outer shell increased by a factor of 10, i.e. Y = 300 and 200 respectively. No noticeable change is observed in the fits. The derived model parameters corresponding to these fits are given in <ref>.
In the absence of data beyond 24$\micron$, it is quite plausible that the shell is even more extended (i.e. the actual value of Y is much larger than in our model), further increasing the age difference
between the shell and the disk. For example, in the post-AGB object, IRAS 22036+5306, where the modeled SED included data up to $\sim$200  the outer radius of the shell is a factor $\sim40$ larger than the inner radius [Sahai et al., 2006].

There are however two exceptions, the post-RGB object, J045555.15-712112.3 and the post-AGB object, J052519.48-705410.0.  In these two, the disk is estimated to be older than the shell.
For J045555.15-712112.3, a change in the value of Y for the outer shell, impacts the overall model fit to the observed SED. The nature of this object is ambiguous; it may be a YSO ( <ref>).
The post-AGB star, J052519.48-705410.0 has been classified as a Cepheid (Soszyński, 2008) and a RV Tauri variable (van Aarle et al. 2011). The variable nature of this object is also evidenced by the difference between the IRAC and WISE (ALLWISE) fluxes ( <ref>). The disk being older
than the shell is consistent with the RV Tauri classification of the object (Waters et al. 1993).
In RV Tauri systems, the hot dust is believed to be related to the presence of a companion and maybe the remnant of a phase of high mass loss in the past or rapid mass transfer during Roche lobe overflow.
The outer shell is usually related to more recent mass-loss history (Van Winckel et al. 1999; Waters et al. 1993).

§.§.§ RV Tauri post-AGB stars

The post-AGB disk stars, J051418.09-691234.9, J052519.48-705410.0, J055122.52-695351.4 are
known RV Tauri variables (van Aarle et al. 2011). Galactic RV Tauri variables with infrared excess are
almost always associated with circumstellar disks (e.g. Lloyd Evans 1995, 1997; Van Winckel et al. 1999;
Gielen et al. 2007). The RV Tauri stars in the LMC may also be expected to show similar morphology (Manick et al. 2018). However, our model SEDs show that although J052519.48-705410.0 and J055122.52-695351.4 harbour a warm inner disk in addition to the outer shell, the observed SED of the RV Tauri star, J055122.52-695351.4 did not require a circumstellar disk, as it could be fitted by a single shell model (Fig. <ref>).

§.§ Systematic discrepancies in SED Modeling

In this section, we discuss some systematic discrepancies between the model and observed SEDs and their implications. Some of these may be due to the presence of dust emission features – JWST/MRS spectroscopy will be needed to test this hypothesis. Other discrepancies may be related to systematic uncertainties in the photometry.

§.§.§ PAH emission

We find systematic discrepancies between the model and observed SEDs in the 5.8–8$\micron$ region for some of our sources – J043919.30-685733.4, J045555.15-712112, J045755.05-681649.2, J050257.89-665306.3 and J050632.10-714229.8.
These discrepancies may be explained by the presence of
PAHs in the dust disks/shells of our sources.

We discuss two possibilities that may account for the presence of PAH molecules in oxygen-rich post-RGB stars. Both of
these have been discussed by Jura et al. (2006) for the case of HD233517, an evolved oxygen-rich red giant showing
PAH features in the circumstellar environment. In the first scenario, since the dusty post-RGB stars in our sample are most likely in close binary systems, the companion of the post-RGB star which may have been more massive than the latter on the MS, may have evolved to become a C-rich AGB star and transferred C-rich material to the post-RGB star progenitor, making it carbon-rich[this mechanism is believed to be responsible for dwarf carbon stars and CH stars]; then, the post-RGB progenitor, when it evolved to become a red giant underwent a CE interaction with the companion (then a WD), ejecting C-rich material that made PAHs. Another possible scenario is that when the more massive star of the binary pair became a red giant, it engulfed the secondary, and the mass transfer and resulting ejection of circumstellar matter via CEE that followed harbored the required physical conditions for the formation of PAHs. For example, Fischer-Tropsch
catalysis on the surface of iron grains (Willacy 2004) converted much of the carbon initially contained in CO into hydrocarbons, and subsequently, analogous to the atmospheric chemistry in Jupiter (Wong et al. 2003), these
simple hydrocarbons were converted into PAHs.

PAH emission is believed to be the result of excitation of the PAH molecules and/or photodissociation of the CO molecule in dense torii. The latter is believed to account for the presence of PAH emission features in O-rich circumstellar environments of post-AGB stars, PPNe  and PNe (Beintema et al. 1996; Waters et al. 1998; Matsuura et al. 2004; Cerrigone et al. 2009; Guzman-Ramirez et al. 2011).  It is widely accepted that UV flux is required for both these mechanisms. However, UV flux cannot account for the excitation of PAH molecules in the circumstellar environment of post-RGB stars, because of their low T$_{eff}$. Li & Draine (2002) argue that UV flux is not imperative for the excitation of PAH molecules – depending on the sizes of the PAH molecules, these may be excited by photons covering a wide range of wavelengths. In particular, it is possible for PAH molecules to be excited by photons longwards of UV wavelengths – in fact, PAH emission may be observed even if the effective temperature of the exciting star is as low as 3000K.

§.§.§ Silicate emission feature

The $\sim$10  amorphous silicate emission feature is visible in the model spectra of several of our objects.
We illustrate the effects of choice of silicate grains
(Sil-Ow, Sil-Oc), dust temperature at the inner boundary of the dust shells (T$_{\rm d}$) and maximum grain size on the
silicate emission feature, using the post-AGB (shell) source, J050632.10-714229.8 as an example. Changing the grains from “warm" (Sil-Ow) to “cold" (Sil-Oc) silicates doesn't do much to alter the fit in the
mid-IR (Fig <ref>). The silicate emission feature is also not very sensitive to changing T$_{\rm d}$ (Fig <ref>).
Incorporating larger Sil-Ow grains in the inner shell has the effect of broadening the wings of the
silicate emission feature while reducing its intensity. Fig <ref> shows the fits
for different values of a$_{\rm max}$.

§.§.§ R and I band data

For several of our post-RGB and post-AGB sources, the model SEDs do not provide good matches to the R- and I-band data.
The R magnitudes, taken from the GSC 2.2 catalog (as in KWVW15), often have large error bars, e.g. the post-RGB sources, J051347.57-704450.5 and J051920.18-722522.1. The modeled SEDs do not agree with the extinction corrected Rmag even after taking into account the large error bars. For the post-RGB source J051347.57-704450.5, in addition to the GSC 2.2 data, we plotted the  extinction corrected USNO-B 1.0 R1 and R2 fluxes (Fig. <ref>). We find that the latter are in better agreement with the modeled SED.

KWVW15 compiled I magnitudes from the MCPS Catalog for the LMC (Zaritsky et al. 2004). In
Tables <ref> and  <ref> we compare the I-band
magnitudes from Zaritsky et al. 2004 with data from the DENIS catalog (Cioni et al. 2000) and
USNO-B 1.0 (Monet et al. 2003). The Zaritsky et al. (2004) data is in good agreement with the DENIS data with the exception of the post-RGB source, J050257.89-665306.3 ($\Delta$Imag $\sim$ 1 mag). The corrected DENIS Imag for this object is in good agreement with our modeled SED.
(Fig. <ref>). In most cases, the USNO-B 1.0 magnitudes are brighter than the Zaritsky et al. (2004) data and would show better agreement with our modeled SEDs.

§.§.§ WISE and ALLWISE photometry

The WISE magnitudes from the All-Sky Data Release (Cutri et al. 2012) and the ALLWISE magnitudes (Cutri et al. 2013) are listed in tables <ref> and <ref>.
A mismatch between modeled and observed SEDs at WISE band 2 (W2/4.6$\micron$) was observed for the post-RGB sources, J043919.30-685733.4, J045555.15-712112.3 and J045755.05-681649.2. For these sources, we additionally plotted the ALLWISE (W2$^{\prime}$/4.6$\micron$) magnitudes.
Because of processing improvements in compiling the photometric data and the addition of a second coverage of the sky in the short wavelength bands, ALLWISE achieves better sensitivity in the W1 and W2 bands. The latter data shows good agreement with our modeled SEDs  for J043919.30-685733.4 (Figs. <ref>, <ref> and J045755.05-681649.2 <ref>). In J045555.15-712112.3, the 4.6$\micron$ flux may be due to an additional inner shell ( <ref>). Similarly, we notice a mismatch between modeled and
observed SEDs for the post-AGB “disk” sources in the W1 and W2 fluxes. For all the post-AGB sources, we additionally plotted the ALLWISE magnitudes. Whenever the mismatch between WISE and ALLWISE data significantly
exceeds the error in the observed photometry, it may be due to source variability. Three of the
post-AGB disk sources, J051418.09-691234.9, J052519.48-705410.0, J055122.52-695351.4 are known
RV Tauri variables ( <ref>).

§ CONCLUSIONS

We have modeled the SEDs of a select sample of post-RGB and post-AGB objects (8 in each class) in the LMC,  drawn from a large list of such sources classified as possessing “disks" or “shells" based on the shape of their SEDs. A circumstellar disk suggests binarity. Close binary systems undergoing CEE can result in the
the premature evolution of an RGB star into a PNe. We have used the 1-D radiative transfer code, DUSTY, approximating disk structures as wedge-shaped fractions of a sphere (disk-fractions). We have derived the total dust mass (and using an assumed gas-to-dust ratio, the total mass of gas) in the disks and  shells, and set constraints on the dust grain composition and sizes. Our main conclusions are listed below:

* The masses of ejecta in the disks of post-RGBs (post-AGBs) lie in the range $\sim10^{-8}-10^{-5}$  ($\sim10^{-8}-5\times10^{-5}$ ) and
in the shells from $\sim5\times10^{-5}-10^{-2}$  ($\sim10^{-4}-8\times10^{-2}$ ). On average, the shell masses in
post-RGBs are less than those in post-AGBs, with the caveat that a substantial amount of mass in both types of objects may lie in an extended cold shell, that is detectable only at wavelength longwards of the MIPS 24$\micron$ band.

* The disk fractions are surprisingly large (typically 0.3–0.4), implying that the disks are geometrically thick, flared structures with a substantial opening angle. The large opening angles appear to be roughly consistent with the latitudinal  variation of gas density in the ejected envelope as seen in numerical simulations of CEE, relatively soon after CEE occurs.

* We find evidence that for some post-RGB sources the ejected matter may be carbon-rich, even though it is expected to be oxygen-rich. For J055102.44-685639.1, the inner disk is composed of amorphous carbon grains and the outer shell has a mix of silicate and silicon carbide grains. In addition, for J043919.30-685733.4, J045555.15-712112, J045755.05-681649.2 and J050257.89-665306.3, PAH emission may account for the discrepancy between model and observed SEDs. The presence of PAHs provides independent support for the hypothesis of binary interaction leading to the formation of post-RGB objects.

* We find that the published classification of these objects as “shell" or “disk" sources is not robust. Our modeling shows
the presence of a disk is (a) required in some “shell" sources (the post-RGB sources: J043919.30-685733.4
and J051920.18-722522.1), and (b) not required in some “disk" sources (the post-AGB sources: J045623.21-692749.0 and J055122.52-695351.4).

* Comparison of our model dust mass values with the predictions of models of CEE on the RGB that produce dust suggest that CEE occurred near or at the top of the RGB for the post-RGB sources in our sample.

G.S. would like to acknowledge financial support from the Department of Science and Technology (DST), Government of India, through a grant numbered SR/WOS-A/PM-93/2017. R.S.'s contribution to the research described here was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA, and funded in part by NASA via ADAP awards, and multiple HST GO awards from the Space Telescope Science Institute.

[Acke et al. , 2013]Acke, B., Degroote, P., Lombaert, R., de Vries, B.L. et al.  2013, A&A, 551, 76
[Beintema et al. , 1996] Beintema, D. A., van den Ancker, M. E., Molster, F. J. et al.  1996, A&A, 315, L369
[Cutri , 2012] Cutri, R. M., Wright, E. L., Conrow, T. et al. 2012, Explanatory Supplement to the WISE All-Sky Data Release Products. Tech. rep.
[Cutri , 2013]Cutri, R. M., Wright, E. L., Conrow, T. et al. 2013, Explanatory
Supplement to the ALLWISE Data Release Products. Tech. rep.
the WISE All-Sky Data Release Products. Tech. rep.
[Cerrigone et al. , 2009] Cerrigone, L., Hora, J. L., Umana, G. & Trigilio, C.  2009, , 703, 585
[Cioni et al. , 2000] Cioni, M. -R., Loup, C., Habing, H. J. et al. 2000, , 144, 235
[Demyk et al. , 2018] Demyk, K., Meny, C., Leroux, H. et al. 2017, A&A, 606, 50
[Deroo et al., 2007] Deroo, P., Acke, B., Verhoelst, T., et al. 2007, , 474, L45
[Draine & Lee , 1984] Draine, B. T. & Lee, H. M. 1984, , 285, 89
[Gao et al. , 2013] Gao, Jian, Jiang, B. W., Li, Aigen & Xue, M. Y. 2013, , 776, 7
[Gordon et al. , 2003] Gordon, K. D., Clayton, G. C., Misselt, K. A., Landolt, A. U. & Wolff, M. J. 2003, , 594, 279
[Gruendl & Chu , 2009] Gruendl, R. A., Chu, You-Hua, 2009, ApJS, 184, 172
[Guzman-Ramirez et al. , 2011] Guzman-Ramirez, L., Zijlstra, A. A., Níchuimín, R., et al.  2011, , 414, 1667
[García-Segura et al., 2018] García-Segura, G., Ricker, P. M., & Taam, R. E. 2018, , 860, 19
[Gielen et al. , 2007] Gielen, C., Van Winckel, H., Waters, L. B. F. M., Min, M.,
& Dominik, C. 2007, A&A, 475, 629
[Hanner, 1988] Hanner, M.S. 1988, NASA Conf. Pub. 3004, 22
[Ivezić et al. , 2012] Ivezić, Z., Nenkova, M., Heymann, F. & Elitzur, M. 2012, User Manual for DUSTY (V4)
[Jura & Kahane, 1999] Jura, M., Kahane, C. 1999, , 521, 302
[Jura et al., 2006] Jura, M., Bohac, C. J., Sargent, B., Forrest, W. J. et al. 2006,
, 637, L45
[Kamath et al., 2014] Kamath, D., Wood, P. R. & Van Winckel, H. 2014, ,
439, 2211
[Kamath et al., 2015] Kamath, D., Wood, P. R. & Van Winckel, H. 2015, , 454, 1468
[Kamath et al., 2016] Kamath, D., Wood, P. R., Van Winckel, H., & Nie, J.D.  2016, A&A, 586, L5
[Keller & Wood, 2006] Keller, S. C. & Wood, P. R. 2006, , 642, 834
[Lacy et al. , 1984] Lacy, J. H., Baas, F., Allamandola, L. J., Persson, S. E.
et al. 1984, , 276, 533
[Li & Draine, 2002]Li, Aigen and Draine, T. 2002, , 572, 232
[Lloyd Evans , 1995] Lloyd Evans, T. 1995, Ap&SS, 230, 169
[Lloyd Evans , 1997] Lloyd Evans, T.  1997, Ap&SS, 251, 239
[Lü et al., 2013]Lü, G., Zhu, C., Podsiadlowski, P. 2013, , 768, 193
[Manick et al. , 2018] Manick, R., Van Winckel, H., Kamath, D., Sekaran, S. &
Kolenberg, K  2018, A&A, 618, 21
[Mathis et al., 1977] Mathis, J. S., Rumpl, W. & Nordsieck, K. H. 1977, , 217, 425
[Matsuura et al., 2004] Matsuura, M., Zijlstra, A. A., Molster, F. J. et al.  2004 , 604, 791
[Monet et al., 2003] Monet, D. G., Levine, S. E., Canzian, B. 2003, , 125, 984
[Nanni et al., 2019]Nanni, A., Groenewegen, M. A. T., Aringer, B. et al. 2019, 487, 502
[Nyman et al., 1992]Nyman, L. -Å., Booth, R. S., Carlström, U. et al. 1992, , 93, 121
[Ossenkopf et al., 1992] Ossenkopf, V., Henning, Th. & Mathis, J. S. 1992, , 261, 567
[Pégourié, 1988] Pégourié, B. 1988, A&A, 194, 335
[Roman-Duval et al., 2014] Roman-Duval, J., Gordon, K. D., Meixner, M. et al., 2014 , 797, 86
[Sahai & Nyman, 1997] Sahai, R. & Nyman, Lars-Ake, 1997 , 487, L155
[Sahai et al., 2003] Sahai, R., Zijlstra, A., Sánchez Contreras, C., & Morris, M., 2003  , 586, L81
[Sahai et al., 2006] Sahai, R., Young, K., Patel, N. A., Sánchez Contreras, C. and Morris, M. 2006 , 653, 1241
[Sahai et al., 2007] Sahai, R., Morris, M., Sánchez Contreras, C. and Claussen, M. 2007, , 134, 2200
[Sahai, 2009] Sahai, R.  2009, Astro2010: The Astronomy and Astrophysics Decadal Survey, Science White Papers, no. 256
[Sahai et al., 2011]Sahai, R., Claussen, M. J., Schnee, S. et al. 2011, , 739, L3
[Sahai et al., 2017] Sahai, R., Vlemmings, W.H.T. & Nyman, L.A. 2017, , 841, 110
[Sahai , 2018] Sahai, R. 2018, Galaxies, 6, 102
[Sarkar & Sahai, 2006] Sarkar, G. & Sahai, R. 2006, , 644, 1171
[Soszyński et al., 2008] Soszyński et al. 2008, Acta Astronomica, 58, 293
[van Aarle et al., 2011]van Aarle, E., Van Winckel, H., Lloyd Evans, T. et al.  2011, A&A, 530, A90
[van Aarle et al., 2013]van Aarle, E., Van Winckel, H., De Smedt, K. et al.  2013, A&A, 554, 106
[van Loon et al., 1999]van Loon, J.T., Groenewegen M. A. T., de Koter A., Trams N. R., Waters L. B. F. M., Zijlstra, A. A, Whitelock, P. A  1999, A&A, 351, 559
[Van Winckel et al., 1999] Van Winckel, H., Waelkens, C., Fernie, J. D.,
Waters, L. B. F. M. 1999, A&A, 343, 202
[Van Winckel, 2003] Van Winckel, H. 2003, , 41, 391
[Waters et al., 1993]Waters, L. B. F. M., Waelkens, C., Mayor, M.,
Trams, N. R. 1993, A&A, 269
[Waters et al., 1998] Waters, L.B.F.M., Cami, J., de Jong, T., et al.  1998, Nature, 391, 868
[Willacy, 2004] Willacy, K. 2004, ApJ, 600, L87
[Wong, 2003] Wong, A.-S., Yung, Y. L., & Friedson, A. J. 2003, Geophys. Res. Lett., 30, 30
[Zacharias et al., 2005] Zacharias, N., Monet, D. G., Levine, S. E., Urban,
S. E., Gaume, R., Wycoff, G. L. 2004, AAS, 205, 4815
[Zaritsky et al., 2004] Zaritsky, D., Harris, J., Thompson, I. B., Grebel, E. K. 2004, , 128, 1606

Photometric data of post-RGB stars
S.No.     Object     Type     U     B     V     R     I     J     H     K     [3.6]     [4.5]     [5.8]     [8.0]     [24]

1     J043919.30-685733.4     Shell     20.093     19.475     18.805     18.100     17.900     17.206     16.597     16.333     16.068     15.741         13.856     9.285

2     J051347.57-704450.5     Shell     19.909     18.510     17.077     16.740     15.683     14.661     13.933     13.719     13.661     13.728     13.625     13.563     9.284

...     ...     ...     ...     15.76      ...     ...     ...     ...
...     ...     ...

...     ...     ...     ...     16.29     ...     ...     ...     ...
...     ...     ...

3     J051920.18-722522.1     Shell     19.408     18.650     17.426     17.020     16.152     15.154     14.540     14.396     13.977     13.785     13.159     12.057     8.052

4     J053930.60-702248.5     Shell         19.934     18.243     18.600     16.915     15.810     14.981     14.756      14.611     14.686     14.405     14.160     9.451

5     J045555.15-712112.3     disk     18.216     18.455    18.165     17.610         16.790     16.350     16.011     14.232     14.190     11.518     9.974     7.258

6     J045755.05-681649.2     disk     20.118     19.563     18.321         17.205     16.369     15.651     15.470     15.059     14.922     14.557     12.480     9.677

7     J050257.89-665306.3     disk     19.988     19.788     18.518     17.610     17.563     15.979     15.242     14.666     14.387     13.830     13.829     11.652     8.565

8     J055102.44-685639.1     disk     20.108     19.951     18.861     18.210     17.821     16.346     15.598     14.734     13.890     13.383     12.727     10.138     5.961

WISE and ALLWISE magnitudes of post-RGB stars

S.No.     Object     W1     W2     W3     W4     W1$^{\prime}$     W2$^{\prime}$     W3$^{\prime}$     W4$^{\prime}$

1     J043919.30-685733.4     16.416     16.259     13.07$^{\dagger}$     9.380$^{\dagger}$     16.194     15.894     12.938	    8.988$^{\dagger}$

2     J051347.57-704450.5      ...     ...     ...     ...     ...     ...     ...     ...

3     J051920.18-722522.1     14.084     13.807     10.882     8.462     14.069     13.789     10.752     8.297

4     J053930.60-702248.5     14.907     15.130     11.350     9.302     14.804     15.206     11.274     9.015

5     J045555.15-712112.3     14.034     13.774     8.696     6.833     14.035     13.814     8.762     6.896

6     J045755.05-681649.2     15.233     15.460     11.489     9.825$^{\dagger}$     15.144	    15.094     11.253     9.190$^{\dagger}$

7     J050257.89-665306.3     14.174     14.134     10.747     8.472     14.169     4.138     11.201     8.773

8     J055102.44-685639.1     14.223     13.409     9.120     6.076     14.247     13.410	    9.042     6.022

$^{\dagger}$: Magnitudes are upper limits

WISE magnitudes: W1, W2, W3, W4

ALLWISE magnitudes: W1$^{\prime}$, W2$^{\prime}$, W3$^{\prime}$, W4$^{\prime}$

Photometric data of post-AGB stars
S.No.     Object     Type     U     B     V     R     I     J     H     K     [3.6]     [4.5]     [5.8]     [8.0]     [24]

1     J050632.10-714229.8     Shell     14.801     14.717     14.298     13.950     13.780     13.408     13.254     13.168     12.989     12.827     11.441     9.192     5.118

2     J051848.84-700247.0     Shell     18.692     17.142     15.429     14.800     13.852     13.152     12.713     12.568     12.372     11.992     10.889     8.558     4.214

3     J051906.86-694153.9     Shell     ...     16.800     15.470     ...     14.473     13.652     13.048     12.900     12.765     12.827     12.761     12.786     8.289

4     J053250.69-713925.8     Shell     17.985     16.533     15.069     14.240     13.803     13.124     12.842     12.709      12.488     12.280     11.255     8.898     4.733

5     J045623.21-692749.0     disk     17.707     16.211     14.628     14.070     13.123     12.416     11.740     11.430      10.533     10.420     10.068     9.486     7.189

6     J051418.09-691234.9     disk     15.837     14.924     14.552     13.780     13.853     13.089     12.635     12.093      10.144     9.394     8.669     7.835     6.007

7     J055122.52-695351.4     disk     15.870     15.589     15.160     14.460     14.009     13.717     13.317     13.099     11.669     10.678     9.676     8.408     5.844

8     J052519.48-705410.0     disk     16.310     15.450     14.973     14.800     14.276     14.221     13.829     13.459      10.897     10.119     9.502     8.647     6.591

WISE and ALLWISE magnitudes of post-AGB stars

S.No.     Object     W1     W2     W3     W4     W1$^{\prime}$     W2$^{\prime}$     W3$^{\prime}$     W4$^{\prime}$

1     J050632.10-714229.8     13.077     12.960     7.979     5.160     13.072     12.959     7.957	    5.147

2     J051848.84-700247.0     ...           ...        ...       ...         ...         ...         ...        ...

3     J051906.86-694153.9     12.529     12.656     10.844     7.499     12.515     12.747     13.366$^{\dagger}$     9.628$^{\dagger}$

4     J053250.69-713925.8     12.537     12.360    	7.348     4.873     12.555     12.370	    7.327	    4.846

5     J045623.21-692749.0     10.695     10.460     8.752     7.260     10.709     10.504	    8.851     7.743

6     J051418.09-691234.9     10.669     9.584     7.290     6.025     10.554     9.500     7.297     6.111

7     J055122.52-695351.4     12.248     11.009     7.606     5.969     12.029     10.763	    7.578	    5.889

8     J052519.48-705410.0      11.954     10.791     8.312     6.989     11.668	    10.566     8.272     6.856

$^{\dagger}$: Magnitudes are upper limits

WISE magnitudes: W1, W2, W3, W4

ALLWISE magnitudes: W1$^{\prime}$, W2$^{\prime}$, W3$^{\prime}$, W4$^{\prime}$

Models of post-RGB objects

J043919.30-685733.4 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) models with Sil-Ow grains

550     161      8       1.82     1.5     0.005     0.25     1.8$\times10^{-15}$     140.0     1.45$\times10^{-5}$     8.09     1.79$\times10^{-6}$

550     65.6     100     1.79     1.5     0.005     0.25     1.8$\times10^{-15}$     140.0     1.74$\times10^{-4}$     112      1.55$\times10^{-6}$

550     38.4     500     1.79     1.5     0.005     0.25     1.8$\times10^{-15}$     140.0     8.72$\times10^{-4}$     565      1.54$\times10^{-6}$

145     27.2     100     26.3     0.8     0.005     0.25     1.5$\times10^{-15}$     116.0     2.01$\times10^{-2}$     1650     1.22$\times10^{-5}$

175     32       100     16       0.8     0.005     0.25     1.5$\times10^{-15}$     116.0     7.51$\times10^{-3}$     1010     7.45$\times10^{-6}$

500     66.7     100     1.79     0.8     0.005     0.25     1.5$\times10^{-15}$     116.0     9.32$\times10^{-5}$     112      8.30$\times10^{-7}$

12lTwo-component model, # 1: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.35

130 “shell"     41.3     20      29.1     0.95     0.005     0.25     9.81$\times10^{-16}$     76.0     5.87$\times10^{-3}$     351      1.67$\times10^{-5}$

1000 “disk"     764      1.4     0.29     0.5      0.005     0.25     5.19$\times10^{-16}$     40.0     2.19$\times10^{-8}$     1.86     1.18$\times10^{-8}$

12lmodel # 1: varying the thickness of the warm inner disk

1000     383     5.0     0.27     0.5     0.005     0.25     5.19$\times10^{-16}$     40.0     6.67$\times10^{-8}$     17.2     3.89$\times10^{-9}$

1000     271     10      0.27     0.5     0.005     0.25     5.19$\times10^{-16}$     40.0     1.31$\times10^{-7}$     38.3     3.42$\times10^{-9}$

12lTwo-component model, # 2: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.35)

130 “shell"     41.3     20      29.1     0.95     0.005     0.25     9.81$\times10^{-16}$     76.0     5.87$\times10^{-3}$     351     1.67$\times10^{-5}$

800 “disk"     644      1.4     0.14     0.25     1.0     25.0     5.19$\times10^{-16}$     40.0     5.34$\times10^{-7}$     0.89     5.99$\times10^{-7}$

12lmodel # 2: varying the thickness of the warm inner disk

800     338     5.0      0.13     0.25     1.0     25.0     5.19$\times10^{-16}$     40.0     1.66$\times10^{-6}$     8.30     2.00$\times10^{-7}$

800     240     10.0     0.13     0.25     1.0     25.0     5.19$\times10^{-16}$     40.0     3.25$\times10^{-6}$     18.5     1.76$\times10^{-7}$

12l$^{\dagger}$Correctly illuminated model # 1,c: two-component
model assuming the shell to be divided into two parts

$^{\dagger}$130 “shell(low-lat)"    42.8     20     27.7     0.65     0.005     0.25     9.81$\times10^{-16}$     76.0     3.64$\times10^{-3}$     334     1.09$\times10^{-5}$

$^{\dagger}$130 “shell(high-lat)"    43.1     20     18.2     0.65     0.005     0.25     5.19$\times10^{-16}$     40.0     1.56$\times10^{-3}$     219     7.14$\times10^{-6}$

$^{\dagger}$1000 “disk"     764      1.4     0.29     0.5      0.005     0.25     5.19$\times10^{-16}$     40.0     2.19$\times10^{-8}$     1.86     1.18$\times10^{-8}$

12lCorrectly illuminated model # 2,c: two-component
model assuming the shell to be divided into two parts

130 “shell(low-lat)"    43.1     20     27.3     0.60     0.005     0.25     9.81$\times10^{-16}$     76.0     3.25$\times10^{-3}$     329     9.90$\times10^{-6}$

130 “shell(high-lat)"    43.1     20     19.8     0.60     0.005     0.25     5.19$\times10^{-16}$     40.0     1.71$\times10^{-3}$     238     7.18$\times10^{-6}$

800 “disk"     644      1.4     0.14     0.25     1.0     25.0     5.19$\times10^{-16}$     40.0     5.34$\times10^{-7}$     0.89     5.99$\times10^{-7}$

J051347.57-704450.5 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12l$^{\dagger}$Single shell (one-component) model with Sil-Ow grains

$^{\dagger}$250     155     3.0     14.1     0.35     0.1     0.25     1$\times10^{-14}$     776     4.00$\times10^{-5}$     17.8     2.24$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

200     127     3.0     23.8     0.35     0.1     0.25     1$\times10^{-14}$     776.0     1.15$\times10^{-4}$     30.2     3.79$\times10^{-6}$

300     180     3.0      9.5     0.35     0.1     0.25     1$\times10^{-14}$     776.0     1.82$\times10^{-5}$     12     1.51$\times10^{-6}$

12lVarying the shell's optical depth

250     155     3.0     14.0     0.30     0.1     0.25     1$\times10^{-14}$     776.0     3.37$\times10^{-5}$     17.7     1.91$\times10^{-6}$

250     154     3.0     14.2     0.40     0.1     0.25     1$\times10^{-14}$     776.0     4.66$\times10^{-5}$     18     2.59$\times10^{-6}$

12lVarying the shell's grain sizes

250     155     3.0     13.1     0.35     0.05     0.25     1e-14     776.0     4.48$\times10^{-5}$     16.6     2.7$\times10^{-6}$

250     155     3.0     14.4     0.35     0.1     0.35     1e-14     776.0     4.84$\times10^{-5}$     18.2     2.66$\times10^{-6}$

12lVarying the shell's thickness

250     181     2.0     14.3     0.35     0.1     0.25     1$\times10^{-14}$     776.0     2.75$\times10^{-5}$     9.06     3.04$\times10^{-6}$

250     99.9     10.0     13.9     0.35     0.1     0.25     1$\times10^{-14}$     776.0     1.3$\times10^{-4}$     79.2     1.64$\times10^{-6}$

J051920.18-722522.1 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s

200     63.5     20     18.4     0.75     0.005     0.25     7.5$\times10^{-15}$     582.0     1.85$\times10^{-3}$     221     8.34$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

150     49.7     20     38.4     0.75     0.005     0.25     7.5$\times10^{-15}$     582.0     8.04$\times10^{-3}$     462     1.74$\times10^{-5}$

250     75.8     20     10.9     0.75     0.005     0.25     7.5$\times10^{-15}$     582.0     6.47$\times10^{-4}$     131     4.94$\times10^{-6}$

12lVarying the shell's optical depth

200     64.0     20     18.2     0.65     0.005     0.25     7.5$\times10^{-15}$     582.0     1.56$\times10^{-3}$     219     7.14$\times10^{-6}$

200     63.0     20     18.6     0.85     0.005     0.25     7.5$\times10^{-15}$     582.0     2.14$\times10^{-3}$     224     9.55$\times10^{-6}$

12lVarying the shell's grain sizes

200     63.6     20     20.4     0.75     0.05     0.25     7.5$\times10^{-15}$     582.0     1.55$\times10^{-3}$     246     6.3$\times10^{-6}$

200     63.4     20     19.8     0.75     0.005     0.5     7.5$\times10^{-15}$     582.0     2.29$\times10^{-3}$     238     9.6$\times10^{-6}$

12lVarying the shell's thickness

200     102     5     18.6     0.75     0.005     0.25     7.5$\times10^{-15}$     582.0     4.75$\times10^{-4}$     47.3     1.00$\times10^{-5}$

200     50.4     40     18.3     0.75     0.005     0.25     7.5$\times10^{-15}$     582.0     3.67$\times10^{-3}$     453     8.10$\times10^{-6}$

12lTwo-component model, # 2: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.4)

110 “shell"     36.8     20     70.6     1.00     0.005     0.25     4.5$\times10^{-15}$     349.0     3.63$\times10^{-2}$     851     4.27$\times10^{-5}$

500 “disk"     325     2.0     1.09     0.4     0.3     20.0     3$\times10^{-15}$     233.0     2.59$\times10^{-5}$     17.3     1.5$\times10^{-6}$

12lVarying the minimum and maximum grain sizes in the warm inner disk

500     329     2.0     1.15     0.40     0.1     20.0     3$\times10^{-15}$     233.0     1.39$\times10^{-5}$     18.2

500     326     2.0     1.00     0.40     0.5     20.0     3$\times10^{-15}$     233.0     3.01$\times10^{-5}$     15.9     1.89$\times10^{-6}$

500     325     2.0     1.22     0.40     0.3     5.0     3$\times10^{-15}$     233.0     2.46$\times10^{-6}$     19.3     1.27$\times10^{-7}$

12l$^{\dagger}$Correctly illuminated model # 2,c: two-component
model assuming the shell to be divided into two parts

$^{\dagger}$110 “shell(low-lat)"     37.8     20     66.1     0.65     0.005     0.25     4.5$\times10^{-15}$     349.0     2.07$\times10^{-2}$     797     2.60$\times10^{-5}$

$^{\dagger}$110 “shell(high-lat)"     37.8     20     53.7     0.65     0.005     0.25     3$\times10^{-15}$     233.0     1.37$\times10^{-2}$     647     2.11$\times10^{-5}$

$^{\dagger}$500 “disk"     325     2.0     1.09     0.4     0.3     20.0     3$\times10^{-15}$     233.0     2.59$\times10^{-5}$     17.3     1.5$\times10^{-6}$

12lShell thickness increased by a factor of 10 in model # 2,c

110 “shell(high-lat)"     17.6     200     65.9     0.65     0.005     0.25     4.5$\times10^{-15}$     349.0     0.206     8320     2.48$\times10^{-5}$

110 “shell(high-lat)"     17.6     200     53.6     0.65     0.005     0.25     3$\times10^{-15}$     233.0     0.136     6760     2.01$\times10^{-5}$

500 “disk"     325     2.0     1.09     0.4     0.3     20.0     3$\times10^{-15}$     233.0     2.59$\times10^{-5}$     0.691     3.75$\times10^{-5}$

J053930.60-702248.5 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12l$^{\dagger}$Single shell (one-component) model with Sil-Ow grains

$^{\dagger}$300     112     10     4.77     0.70     0.005     0.25     3.8$\times10^{-15}$     295     5.81$\times10^{-5}$     27.2     2.13$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

250     97.5     10     7.08     0.70     0.005     0.25     3.8$\times10^{-15}$     295     1.28$\times10^{-4}$     40.4     3.16$\times10^{-6}$

350     125     10     3.52     0.70     0.005     0.25     3.8$\times10^{-15}$     295     3.16$\times10^{-5}$     20.1     1.57$\times10^{-6}$

12lVarying the shell's optical depth

300     113     10     4.72     0.60     0.005     0.25     3.8$\times10^{-15}$     295     4.88$\times10^{-5}$     27.0     1.81$\times10^{-6}$

300     111     10     4.83     0.80     0.005     0.25     3.8$\times10^{-15}$     295     6.79$\times10^{-5}$     27.5     2.47$\times10^{-6}$

12lVarying the shell's grain sizes

300     112     10     5.27     0.7     0.05     0.25     3.8$\times10^{-15}$     295     4.83$\times10^{-5}$     30.1     1.60$\times10^{-6}$

300     112     10     5.26     0.7     0.005     0.50     3.8$\times10^{-15}$     295     7.53$\times10^{-5}$     30.0     2.51$\times10^{-6}$

12lVarying the shell's thickness

300     145     5     4.83     0.70     0.005     0.25     3.8$\times10^{-15}$     295     2.97$\times10^{-5}$     12.2     2.43$\times10^{-6}$

300     87.8     20     4.76     0.70     0.005     0.25     3.8$\times10^{-15}$     295     1.15$\times10^{-4}$     57.3     2.01$\times10^{-6}$

J045555.15-712112.3 (disk source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s:

550     101     20     4.49     2.3     0.005     1.0     8$\times10^{-15}$     621.0     4.49$\times10^{-4}$     54.0     8.32$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

500     95.3     20     5.33     2.3     0.005     1.0     8$\times10^{-15}$     621.0     6.35$\times10^{-4}$     64.2     9.89$\times10^{-6}$

600     108      20     3.79     2.3     0.005     1.0     8$\times10^{-15}$     621.0     3.20$\times10^{-4}$     45.6     7.02$\times10^{-6}$

12lVarying the shell's optical depth

550     105     20     4.41     2.0     0.005     1.0     8$\times10^{-15}$     621.0     3.78$\times10^{-4}$     53.1     7.11$\times10^{-6}$

550     99.6     20     4.51     2.5     0.005     1.0     8$\times10^{-15}$     621.0     4.94$\times10^{-4}$     54.3     9.09$\times10^{-6}$

12lVarying the shell's grain sizes

550     93.3     20     5.83     2.3     0.005     0.25     8$\times10^{-15}$     621.0     5.70$\times10^{-4}$     70.2     8.11$\times10^{-6}$

550     105      20     3.64     2.3     0.005     2.0     8$\times10^{-15}$     621.0     3.93$\times10^{-4}$     43.8     8.97$\times10^{-6}$

550     105      20     4.36     2.3     0.05     1.0     8$\times10^{-15}$     621.0     3.67$\times10^{-4}$     52.5     6.98$\times10^{-6}$

12lVarying the shell's thickness

550     168     5     4.56     2.3     0.005     1.0     8$\times10^{-15}$     621.0     1.16$\times10^{-4}$     11.6     1.00$\times10^{-5}$

550     130     10     4.51     2.3     0.005     1.0     8$\times10^{-15}$     621.0     2.27$\times10^{-4}$     25.7     8.83$\times10^{-6}$

550     88.1     30     4.46     2.3     0.005     1.0     8$\times10^{-15}$     621.0     6.67$\times10^{-4}$     82.0     8.13$\times10^{-6}$

12lTwo-component model, # 2: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.1)

550 “shell"     101     20     4.26     2.3     0.005     1.0     7.2$\times10^{-15}$     559.0     4.04$\times10^{-4}$     51.3     7.89$\times10^{-6}$

1200 “disk"     241     10     0.363     0.5     0.005     0.1     8$\times10^{-16}$     62.0     7.19$\times10^{-7}$     51.8     1.39$\times10^{-8}$

12lCorrectly illuminated model # 2,c: two-component model assuming the shell to be divided into two parts

550 “shell(low-lat)"    104     20     4.11     2.10     0.005     1.0     7.2$\times10^{-15}$     559.0     3.45$\times10^{-4}$     49.6     6.96$\times10^{-6}$

550 “shell(high-lat)"    113     20     1.06     2.10     0.005     1.0     8$\times10^{-16}$     62.0     2.30$\times10^{-5}$     12.8     1.80$\times10^{-6}$

1200 “disk"     241     10     0.363     0.5     0.005     0.1     8$\times10^{-16}$     62.0     7.19$\times10^{-7}$     51.8     1.39$\times10^{-8}$

12lTwo-component model, # 3:
cool Sil-Ow/0.8 and grf-DL/0.2 outer shell and a warm grf-DL inner disk (disk fraction=0.1)

500 “shell"    216     2     7.64     1.8     0.005     0.25     7.2$\times10^{-15}$     559     9.6$\times10^{-5}$     4.84     1.98$\times10^{-5}$

800 “disk"    356     5     1.06     0.7     0.005     0.25     8$\times10^{-16}$     62     2.67$\times10^{-6}$     66.9     3.99$\times10^{-8}$

12l$^{\dagger}$Correctly illuminated model # 3,c: two-component model assuming the shell to be divided into two parts

$^{\dagger}$500 “shell(low-lat)"     224     2     7.42     1.60     0.005     0.25     7.2$\times10^{-15}$     559.0     8.06$\times10^{-5}$     4.71     1.71$\times10^{-5}$

$^{\dagger}$500 “shell(high-lat)"    236     2     2.14     1.60     0.005     0.25     8$\times10^{-16}$     62.0     6.71$\times10^{-6}$     1.36     4.94$\times10^{-6}$

$^{\dagger}$800 “disk"    356     5     1.06     0.7     0.005     0.25     8$\times10^{-16}$     62     2.67$\times10^{-6}$     66.9     3.99$\times10^{-8}$

12lTwo-component model, # 4: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.4)

300 “shell"     138     2     12.5     2.5     0.005     0.25     3.6$\times10^{-15}$     279.0     2.84$\times10^{-4}$     7.92     3.59$\times10^{-5}$

1000 “disk"     226     10     0.919     1.0     0.005     0.25     2.4$\times10^{-15}$     186.0     3.08$\times10^{-6}$     131     2.35$\times10^{-8}$

J045755.05-681649.2 (disk source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s:

500     118     20     1.94     1.0     0.005     0.25     2.8$\times10^{-15}$     217.0     2.75$\times10^{-5}$     23.4     1.18$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

450     110     20     2.34     1.0     0.005     0.25     2.8$\times10^{-15}$     217.0     4.00$\times10^{-5}$     28.2     1.42$\times10^{-6}$

550     125     20     1.65     1.0     0.005     0.25     2.8$\times10^{-15}$     217.0     1.98$\times10^{-5}$     19.9     9.98$\times10^{-7}$

12lVarying the shell's optical depth

500     119     20     1.93     0.9     0.005     0.25     2.8$\times10^{-15}$     217.0     2.44$\times10^{-5}$     23.2     1.05$\times10^{-6}$

500     117     20     1.97     1.1     0.005     0.25     2.8$\times10^{-15}$     217.0     3.12$\times10^{-5}$     23.8     1.31$\times10^{-6}$

12lVarying the shell's thickness

500     200      5      1.99     1.0     0.005     0.25     2.8$\times10^{-15}$     217.0     7.20$\times10^{-6}$     5.04     1.43$\times10^{-6}$

500     85.5     50     1.94     1.0     0.005     0.25     2.8$\times10^{-15}$     217.0     6.89$\times10^{-5}$     60.4     1.14$\times10^{-6}$

12lTwo-component model, # 2: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.4)

400 “shell"     91.0     30     2.31     0.9     0.1     1.0     1.68$\times10^{-15}$     130.0     5.84$\times10^{-5}$     42.6     1.37$\times10^{-6}$

1300 “disk"     921     2     0.12     0.5     0.005     2.0     1.12$\times10^{-15}$     87.0     9.64$\times10^{-9}$     1.94     4.98$\times10^{-9}$

12l$^{\dagger}$Correctly illuminated model # 2,c: two-component model assuming the shell to be divided into two parts

$^{\dagger}$400 “shell(low-lat)"    93.3     30     2.20     0.6     0.1     1.0     1.68$\times10^{-15}$     130.0     3.52$\times10^{-5}$     40.5     8.7$\times10^{-7}$

$^{\dagger}$400 “shell(high-lat)"    93.5     30     1.74     0.6     0.1     1.0     1.12$\times10^{-15}$     87.0     2.21$\times10^{-5}$     32.1     6.9$\times10^{-7}$

$^{\dagger}$1300 “disk"     921     2     0.12     0.5     0.005     2.0     1.12$\times10^{-15}$     87.0     9.64$\times10^{-9}$     1.94     4.98$\times10^{-9}$

12lShell thickness increased by a factor of 10 in model # 2,c

400 “shell(low-lat)"     43.0     300     2.19     0.6     0.1     1.0     1.68$\times10^{-15}$     130.0     3.48$\times10^{-4}$     415     8.40$\times10^{-7}$

400 “shell(high-lat)"     43.1     300     1.74     0.6     0.1     1.0     1.12$\times10^{-15}$     87.0     2.21$\times10^{-4}$     331     6.69$\times10^{-7}$

1300 “disk"     921     2     0.12     0.5     0.005     2.0     1.12$\times10^{-15}$     87.0     9.64$\times10^{-9}$     1.94     4.98$\times10^{-9}$

J050257.89-665306.3 (disk source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s:

250     64.9     30     8.14     1.1     0.005     0.25     4.2$\times10^{-15}$     326.0     7.97$\times10^{-4}$     150     5.32$\times10^{-6}$

12lTwo-component model, # 2: cool Si-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.4)

240 “shell"     90.2     10     7.23     1.20     0.005     1.0     2.34$\times10^{-15}$     182.0     3.05$\times10^{-4}$     41.3     7.38$\times10^{-6}$

1200 “disk"     677     3.0     0.136     0.5     0.3     5.0     1.56$\times10^{-15}$     121.0     5.77$\times10^{-8}$     4.32     1.34$\times10^{-8}$

12l$^{\dagger}$Correctly illuminated model # 2,c: two-component model assuming the shell to be divided into two parts

$^{\dagger}$240 “shell(low-lat)"     93.7     10     6.69     0.75     0.005     1.0     2.34$\times10^{-15}$     182.0     1.63$\times10^{-4}$     38.2     4.27$\times10^{-6}$

$^{\dagger}$240 “shell(high-lat)"    93.7     10     5.37     0.75     0.005     1.0     1.56$\times10^{-15}$     121.0     1.05$\times10^{-4}$     30.6     3.43$\times10^{-6}$

$^{\dagger}$1200 “disk"     677     3.0     0.136     0.5     0.3     5.0     1.56$\times10^{-15}$     121.0     5.77$\times10^{-8}$     4.32     1.34$\times10^{-8}$

12lTwo-component model, # 3: cool Sil-Ow outer shell and warm amC-Hn inner disk (disk fraction=0.3)

250 “shell"     90.8     10     6.76     1.70     0.005     0.25     2.59$\times10^{-15}$     201.0     2.83$\times10^{-4}$
38.6     7.34$\times10^{-6}$

1000 “disk"     726     2.0     0.242     0.6     0.005     0.50     1.11$\times10^{-15}$     86.0     2.21$\times10^{-8}$
3.84     5.75$\times10^{-9}$

12lCorrectly illuminated model # 3,c: two-component model assuming the shell to be divided into two parts

250 “shell(low-lat)"     93.2     10     6.35     1.3     0.005     0.25     2.59$\times10^{-15}$     201.0     1.91$\times10^{-4}$     36.2     5.27$\times10^{-6}$

250 “shell(high-lat)"    93.9     10     3.86     1.3     0.005     0.25     1.11$\times10^{-15}$     86.0     7.05$\times10^{-5}$     22.0     3.2$\times10^{-6}$

1000 “disk"     726     2.0     0.242     0.6     0.005     0.50     1.11$\times10^{-15}$     86.0     2.21$\times10^{-8}$
3.84     5.75$\times10^{-9}$

J055102.44-685639.1 (disk source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s:

190     44.0     20     37.5     4.0     0.005     0.25     7$\times10^{-15}$     543     4.1$\times10^{-2}$     452     9.08$\times10^{-5}$

12lTwo-component model, # 2a: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.4)

350 “shell"     1.28     3     7.06     8.0     0.005     0.07     5.4$\times10^{-15}$     419     2.12$\times10^{-3}$     8.95     2.37$\times10{-4}$

1400 “disk"     872     2.0     0.361     0.5     0.005     0.12     3.6$\times10^{-15}$     279.0     1.05$\times10^{-7}$     5.72     1.84$\times10^{-8}$

12lTwo-component model, # 2b: cool Sil-Ow outer shell and warm Sil-Ow inner disk (disk fraction=0.3)

350 “shell"     128     3     7.20     8.00     0.005     0.07     5.6$\times10^{-15}$     435.0     2.2$\times10^{-3}$     9.12     2.41$\times10^{-4}$

1400 “disk"     748     2.0     0.278     1.00     0.005     0.08     2.4$\times10^{-15}$     186.0     2.34$\times10^{-7}$     4.41     5.3$\times10^{-8}$

12l$^{\dagger}$Two-component model, # 3:  cool Sil-Ow/0.4 and SiC-Pg/0.6 outer shell and warm amC-Hn inner disk (disk fraction=0.3)

$^{\dagger}$350 “shell"     118     3     9.89     12.0     0.005     0.07     5.6$\times10^{-15}$     435.0     3.05$\times10^{-3}$     12.5     2.43$\times10^{-4}$

$^{\dagger}$2000 “disk"     813     7.0     0.09     1.0     0.005     0.05     2.4$\times10^{-15}$     186.0     1.99$\times10^{-8}$     8.79     2.27$\times10^{-9}$

12lmodel # 3: varying the dust temperature at the inner radius of the cool outer shell

250     87.8     3     20.0     12.0     0.005     0.07     5.6$\times10^{-15}$     435     1.25$\times10^{-2}$     25.3     4.92$\times10^{-4}$

450     143     3     6.11     12.0     0.005     0.07     5.6$\times10^{-15}$     435     1.17$\times10^{-3}$     7.75     1.5$\times10^{-4}$

12lmodel # 3: varying the thickness of the cool outer shell

350     138     2     10.0     12.0     0.005     0.07     5.6$\times10^{-15}$     435.0     2.09$\time10^{-3}$     6.35     3.29$\times10^{-4}$

350     75.6     10     9.72     12.0     0.005     0.07     5.6$\times10^{-15}$     435.0     9.83$\times10^{-3}$     55.5     1.77$\times10^{-4}$

12lmodel # 3: varying the optical depth of the cool outer shell

350     122     3     9.74     9.00     0.005     0.07     5.6$\times10^{-15}$     435.0     2.22$\times10^{-3}$     12.4     1.8$\times10^{-4}$

350     1152     3     9.99     15.0     0.005     0.07     5.6$\times10^{-15}$     435.0     3.9$\times10^{-3}$     12.7     3.08$\times10^{-4}$

12lmodel # 3: varying the maximum grain size in the cool outer shell

350     118     3     9.05     12.0     0.005     0.05     5.6$\times10^{-15}$     435.0     3.97$\times10^{-3}$     11.5     3.46$\times10^{-4}$

350     119     3     10.8     12.0     0.005     0.09     5.6$\times10^{-15}$     435.0     2.14$\times10^{-3}$     13.7     1.56$\times10^{-4}$

12lmodel # 3: varying the dust temperature at the disk's inner radius

1800     726     7.0     0.12     1.0     0.005     0.05     2.4$\times10^{-15}$     186.0     3.47$\times10^{-8}$     11.6     2.99$\times10^{-9}$

2200     899     7.0     0.07     1.0     0.005     0.05     2.4$\times10^{-15}$     186.0     1.21$\times10^{-8}$     6.85     1.77$\times10^{-9}$

12lmodel # 3: varying the disks' thickness

2000     1120     3.0     0.09     1.00     0.005     0.05     2.4$\times10^{-15}$     186.0     8.8$\times10^{-9}$     2.97     2.96$\times10^{-9}$

2000     542     20.0     0.09     1.00     0.005     0.05     2.4$\times10^{-15}$     186.0     5.63$\times10^{-8}$     27.7     2.03$\times10^{-9}$

12lmodel # 3: varying the disks' optical depth

2000     837     7.0     0.09     0.8     0.005     0.05     2.4$\times10^{-15}$     186     1.55$\times10^{-8}$     8.68     1.79$\times10^{-9}$

2000     791     7.0     0.09     1.2     0.005     0.05     2.4$\times10^{-15}$     186     2.44$\times10^{-8}$     8.88     2.75$\times10^{-9}$

12lmodel # 3: varying the disks' maximum grain size

2000     832     7.0     0.09     1.0     0.005     0.10     2.4$\times10^{-15}$     186     1.62$\times10^{-8}$     8.83     1.83$\times10^{-9}$

2000     838     7.0     0.07     1.0     0.005     0.50     2.4$\times10^{-15}$     186     1.17$\times10^{-8}$     6.93     1.68$\times10^{-9}$

The inner disk is approximated to be an axially symmetric wedge-shaped fraction of a sphere. This fraction is referred to as “disk fraction".

shell: Parameters for the outer shell in two-component models

disk: Parameters for the inner disk in two-component models

shell(low-lat): Correctly illuminated models, part of shell in the solid angle not covered by the disk

shell(high-lat):Correctly illuminated models, part of shell in the solid angle covered by the disk

suffix `s':  Single shell model

suffix `c':  Correctly illuminated two-component model

$^{\dagger}$: Adopted (best-fit) model
1The (input) dust temperature at shell inner radius
2The (output) dust temperature at shell outer radius
3The (input) ratio of the dust shell's outer to inner radius
4The (inferred) inner radius of the dust shell
5The (input) dust shell's optical depth at $0.55\micron$
6The (inferred) circumstellar (gas+dust) mass
7The (inferred) dust shell's age
8The (inferred) mass-loss rate characterizing the dust shell

Models of post-AGB Objects

J050632.10-714229.8 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains, # 1,s

200     62.4     20     81.1     0.60     0.005     0.25     7$\times10^{-14}$     5434     2.87$\times10^{-2}$     3260     8.83$\times10^{-6}$

12lVarying the dust temperature at the shell's inner radius

180     57.1     20     106.0     0.60     0.005     0.25     7$\times10^{-14}$     5434     4.93$\times10^{-2}$     4260     1.16$\times10^{-5}$

220     67.4     20     64.7     0.60     0.005     0.25     7$\times10^{-14}$     5434     1.83$\times10^{-2}$     2600     7.05$\times10^{-6}$

12lNested Shell Models: The warm inner shell

12lgrain type # 1: Sil-Ow; varying the temperature at the inner radius of the warm shell

450     286     2     15.4     0.4     0.005     0.25     7$\times10^{-14}$     5434     6.92$\times10^{-5}$     9.77     7.08$\times10^{-6}$

$^{\dagger}$400     260     2     18.9     0.4     0.005     0.25     7$\times10^{-14}$     5434     1.05$\times10^{-4}$     12.0     8.71$\times10^{-6}$

350     233     2     24.3     0.4     0.005     0.25     7$\times10^{-14}$     5434     1.72$\times10^{-4}$     15.4     1.12$\times10^{-5}$

12lgrain type # 1: Sil-Ow; varying the maximum grain size in the inner shell for T$_{\rm d}$ = 400K

400     263     2     16.9     0.4     0.005     1.0     7$\times10^{-14}$     5434     1.11$\times10^{-4}$     10.7     1.03$\times10^{-5}$

400     265     2     11.6     0.4     0.005     5.0     7$\times10^{-14}$     5434     1.12$\times10^{-4}$     7.34     1.53$\times10^{-5}$

12lgrain type: Sil-Oc

400     261     2     16.4     0.4     0.005     0.25     7$\times10^{-14}$     5434     6.72$\times10^{-5}$     10.4     6.45$\times10^{-6}$

12lgrain type # 2: amC-Hn:0.1, Sil-Ow:0.9

$^{\dagger}$350     239     2     26.5     0.4     0.1     0.5     7$\times10^{-14}$     5434     1.59$\times10^{-4}$     16.8     9.48$\times10^{-6}$

12lgrain type: amC-Hn:0.2, Sil-Ow:0.8

350     241     2     27.3     0.4     0.1     0.5     7$\times10^{-14}$     5434     1.73$\times10^{-4}$     17.3     1.00$\times10^{-5}$

12lgrain type # 3: grf-DL:0.3, Sil-Ow:0.7

$^{\dagger}$350     163     5     23.7     0.4     0.1     1.0     7$\times10^{-14}$     5434     7.5$\times10^{-4}$     60.0     1.25$\times10^{-5}$

12l$^{\dagger}$Nested shells # 1: correctly illuminated cool Sil-Ow outer shell with Sil-Ow (grain type # 1) inner shell, T$_{\rm d}$ = 400K

$^{\dagger}$100     73.9     2     484     0.45     0.005     0.25     7$\times10^{-14}$     5434     7.69$\times10^{-2}$     1020     7.51$\times10^{-5}$

12l$^{\dagger}$Nested shells # 2: correctly illuminated cool Sil-Ow outer shell with amC-Hn:0.1, Sil-Ow:0.9 (grain type # 2) inner shell, T$_{\rm d}$ = 350K

$^{\dagger}$100     74.3     2     488     0.40     0.005     0.25     7$\times10^{-14}$     5434     6.94$\times10^{-2}$     1030     6.73$\times10^{-5}$

12l$^{\dagger}$Nested shells # 3: correctly illuminated cool Sil-Ow outer shell with grf-DL:0.3, Sil-Ow:0.7 (grain type # 3)  inner shell, T$_{\rm d}$ = 350K

$^{\dagger}$100     74.8     2     481     0.35     0.005     0.25     7$\times10^{-14}$     5434     5.91$\times10^{-2}$     1020     5.81$\times10^{-5}$

J051848.84-700247.0 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12l$^{\dagger}$Single shell (one-component) model with Sil-Ow grains

$^{\dagger}$350     77.4     20     27.5     3.2     0.005     0.25     8$\times10^{-14}$     6210.0     1.76$\times10^{-2}$     1100     1.6$\times10^{-5}$

12lVarying the dust temperature at the shell's inner radius

300     69.6     20     37.4     3.2     0.005     0.25     8$\times10^{-14}$     6210.0     3.25$\times10^{-2}$     1500     2.17$\times10^{-5}$

400     84.4     20     21.5     3.2     0.005     0.25     8$\times10^{-14}$     6210.0     1.08$\times10^{-2}$     864     1.25$\times10^{-5}$

12lVarying the shell's optical depth

350     80.9     20     26.7     2.5     0.005     0.25     8$\times10^{-14}$     6210.0     1.3$\times10^{-2}$     1070     1.21$\times10^{-5}$

350     74.6     20     28.3     4.0     0.005     0.25     8$\times10^{-14}$     6210.0     2.33$\times10^{-2}$     1140     2.05$\times10^{-5}$

12lVarying the shell's thickness

350     126     5     28.1     3.2     0.005     0.25     8$\times10^{-14}$     6210.0     4.59$\times10^{-3}$     237     1.93$\times10^{-5}$

J051906.86-694153.9 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lNested Shell Models: The warm inner shell: amC-Hn grain-type

$^{\dagger}$2000     1480     2     0.231     0.35     0.005     0.25     2.6$\times10^{-14}$     2018.0     9.58$\times10^{-9}$     0.146     6.54$\times10^{-8}$

12lVarying the dust temperature at the warm shell's inner boundary

1700     1260     2     0.355     0.35     0.005     0.25     2.6$\times10^{-14}$     2018.0     2.27$\times10^{-8}$     0.225     1.01$\times10^{-7}$

2300     1700     2     0.160     0.35     0.005     0.25     2.6$\times10^{-14}$     2018.0     4.62$\times10^{-9}$     0.102     4.54$\times10^{-8}$

12lVarying the optical depth of the warm inner shell

2000     1510     2     0.224     0.20     0.005     0.25     2.6$\times10^{-14}$     2018.0     5.16$\times10^{-9}$     0.142     3.63$\times10^{-8}$

2000     1460     2     0.237     0.50     0.005     0.25     2.6$\times10^{-14}$     2018.0     1.44$\times10^{-8}$     0.150     9.58$\times10^{-8}$

12lVarying the thickness of the warm inner shell

2000     945     7     0.225     0.35     0.005     0.25     2.6$\times10^{-14}$     2018.0     3.18$\times10^{-8}$     0.856     3.72$\times10^{-8}$

12l$^{\dagger}$Nested shells: correctly illuminated cool SiC-Pg outer shell with warm amC-Hn inner shell

$^{\dagger}$160     125     2     24.3     0.07     2.3     3.0     2.6$\times10^{-14}$     2018.0     1.78$\times10^{-4}$     51.4     3.46$\times10^{-6}$

12l Nested shells: varying temperature at the inner boundary of
the cool outer shell

130     104     2.0     43.9     0.07     2.3     3.0     2.6$\times10^{-14}$     2018.0     5.8$\times10^{-4}$     92.8     6.25$\times10^{-6}$

200     152     2.0     13.8     0.07     2.3     3.0     2.6$\times10^{-14}$     2018.0     5.72$\times10^{-5}$     29.1     1.96$\times10^{-6}$

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lNested shells: varying optical depth of the cool outer shell

160     125     2.0     24.0     0.05     2.3     3.0     2.6$\times10^{-14}$     2018.0     1.24$\times10^{-4}$     50.8     2.44$\times10^{-6}$

160     125     2.0     24.8     0.1     2.3     3.0     2.6$\times10^{-14}$     2018.0     2.63$\times10^{-4}$     52.3     5.03$\times10^{-6}$

12lNested shells: varying the minimum and maximum grain sizes in the
cool outer shell

160     127     2.0     35.7     0.07     1.0     3.0     2.6$\times10^{-14}$     2018.0     2.23$\times10^{-4}$     75.4     2.96$\times10^{-6}$

160     122     2.0     21.5     0.07     2.3     5.0     2.6$\times10^{-14}$     2018.0     2.30$\times10^{-4}$     45.5     5.07$\times10^{-6}$

12lNested shells: varying the thickness of the cool outer shell

160     78.7     10.0     23.9     0.07     2.3     3.0     2.6$\times10^{-14}$     2018.0     8.62$\times10^{-4}$     455     1.89$\times10^{-6}$

J053250.69-713925.8 (shell source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12l$^{\dagger}$Single shell (one-component) model with Sil-Ow grains

$^{\dagger}$250     82.4     10     90.1     0.7     0.005     0.07     6$\times10^{-14}$     4657.0     3.35$\times10^{-2}$     1710     1.96$\times10^{-5}$

12lVarying the dust temperature at the shell's inner radius

200     66.9     10     154     0.7     0.005     0.07     6$\times10^{-14}$     4657.0     9.74$\times10^{-2}$     2920     3.34$\times10^{-5}$

300     98.7     10     58     0.7     0.005     0.07     6$\times10^{-14}$     4657.0     1.39$\times10^{-2}$     1100     1.26$\times10^{-5}$

12lVarying the shell's optical depth

250     85.3     10     90.1     0.50     0.005     0.07     6$\times10^{-14}$     4657.0     2.4$\times10^{-2}$     1710     1.40$\times10^{-5}$

250     79.8     10     90.8     0.90     0.005     0.07     6$\times10^{-14}$     4657.0     4.38$\times10^{-2}$     1730     2.54$\times10^{-5}$

12lVarying the maximum grain sizes in the shell

250     82.2     10     83.9     0.70     0.005     0.03     6$\times10^{-14}$     4657.0     3.69$\times10^{-2}$     1600     2.31$\times10^{-5}$

250     84.9     10     94.2     0.70     0.005     0.2     6$\times10^{-14}$     4657.0     2.57$\times10^{-2}$     1790     1.43$\times10^{-5}$

12lVarying the shell thickness

250     162     2     90.8     0.70     0.005     0.07     6$\times10^{-14}$     4657.0     6.81$\times10^{-3}$     1920     3.55$\times10^{-5}$

250     63     20     90.1     0.70     0.005     0.07     6$\times10^{-14}$     4657.0     6.71$\times10^{-2}$     3620     1.85$\times10^{-5}$

J045623.21-692749.0 (disk source)

T$_{\rm d}$(in)a     T$_{\rm d}$(out)b     Yc     R$_{\rm in}$d     $\tau$e     a$_{\rm min}$     a$_{\rm max}$     $F_{\rm bol}$     L     M$_{\rm gd}$f     Ageg     dM/dth

(K)     (K)          (10$^{14}$cm)          ($\micron$)     ($\micron$)     ()     ()     ()     (yr)     ()

12lSingle shell (one-component) model with Sil-Ow grains

1100     163     50     1.29     0.6     0.005     5.0     8.5$\times10^{-14}$     6598     5.23$\times10^{-5}$     1340     3.91$\times10^{-7}$

12lNested Shell Models: The warm inner shell (model # 1) : Sil-Ow grain type