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

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 (RGB), 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 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 structures with a substantial opening angle, consistent with numerical simulations of CE evolution (CEE). Comparison of our model dust masses with the predictions of dust production during CEE on the RGB suggest that CEE occurred near or at the tip of the RGB for our post-RGB sources. A surprising result is that some post-RGB stars harbor carbon-rich dust, believed to form when C/O $>$ 1, e.g. following triple-alpha nucleosynthesis and third dredge-up events in AGB stars. This anomaly strengthens the hypothesis that dusty post-RGBs are born in binary systems.


INTRODUCTION
The demise of most stars in the Universe (i.e., in the 1-8 M range) has traditionally been thought to occur as a result of heavy mass-loss on the Asymptotic Giant Branch (AGB), with rates in the range 10 −8 −10 −4 M yr −1 (Höfner & Olofsson 2018), when the stars are very luminous (L ∼ 5000 − 10, 000 L ) and cool (T ef f < 3000 K). This heavy mass-loss is believed to occur via a dusty, spherical, radiatively-driven wind, typically expanding at ∼ 5 − 20 km s −1 . 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 1 ), 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, see e.g. Sahai (2018) and references therein.
Intriguingly, binarity may also be responsible for the rapid and unexpected evolution of Red Giant Branch (RGB) stars to the PPN stage (post-RGB stars). The Boomerang Nebula -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), in our Galaxy. The Boomerang's luminosity is much lower (L ∼ 300 L ) 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 (∼10 −3 M yr −1 ) at a very high ejection velocity (165 km s −1 ), 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 1000 L ) have recently been identified in the Large and Small Magellanic Clouds (LMC and SMC), using optical spectroscopy and Spitzer photometry (Kamath et al. 2014(Kamath et al. , 2015(Kamath et al. , 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 eff , log g, [F e/H] and E(B − V )) and integration of the spectral-energy-distribution (SED) constrains the bolometric flux. The known distance to the LMC (∼ 50 kpc) then allows reliable luminosity estimates for these stars; the low luminosities (< 2500 L ) 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. Further, the spectroscopically estimated log g values are compared with theoretical values that a star with similar luminosity and T ef f would have in the post-RGB and pre-Main Sequence phase. Stars in the latter phase would be typically 15-20 times more massive and hence differ in log g by ∼ 1.2 (Kamath et al. 2016). However, for stars with poor quality optical spectra, the estimated log g and hence subsequent classification may not be accurate; e.g. J051845.47-690321.8, a post-AGB star was wrongly classified as a Young Stellar Object (YSO): Kamath & Van Winckel (2019). In addition, since the post-AGB/RGB stars are an old to intermediate age population, they are expected to be more metal poor than the YSOs which would belong to the young LMC population with a mean metallicity of [F e/H] −0.5 (Kamath et al. 2015).
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): hereafter KWVW15 2 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 KWVW15. 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. § 2 describes the selection of sources for our modeling study, § 3 describes our modeling methodology, and § 4 and § 5 provide details of the SED modeling for each of our post-RGB and post-AGB objects. In § 6, we provide a discussion of our modeling results, and § 7 gives our conclusions.

TARGET SELECTION
The post-RGB and post-AGB stars in this study are taken from KWVW15 -these are listed in Table 1 together with their photometry. The photometry 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. KWVW15 use the average of epoch 1 and 2 (Meixner et al. 2006) Spitzer MIPS 24 µm photometry. Although these values are close enough and the use of an average is justified, we use the photometric data from both the epochs with their corresponding error bars from the NASA/IPAC Infrared Science Archive (https://irsa.ipac.caltech.edu).
A total of eight post-RGB and post-AGB stars have been analysed. These are divided 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 SED lies beyond 10 µm. The disk sources show near-IR excess indicative of hot dust and the peak of the SED lies around 10 µm or sometimes shortwards. Further, on a color-color plot, see In order to ensure that for the post-RGBs in our study, CEE occured on the RGB, we selected objects with L < 1000 L . 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 & Lee (1984), amorphous carbon (amC-Hn) from Hanner (1988) and silicon carbide (SiC-Pg) from Pégourié (1988). For stars on the RGB, the circumstellar envelope is expected to be oxygen-rich. The change from oxygen-rich to carbon-rich chemistry may occur as a result of specific binary interactions (see § 6.3.1) and/or once the star has undergone thermal pulses on the AGB. Silicates are expected to be the main dust species in oxygen-rich outflows, see e.g. Gail et al. (2009). Further, as the name suggests, 'warm silicates' are expected to dominate in regions of high dust temperature and/or in fast moving circumstellar outflows (Ossenkopf et al. 1992). In contrast, 'cold silicates' are expected in the cool highly obscured circumstellar environment around OH/IR stars and in the interstellar medium (ISM). Hence, for our models, the dust grain composition was chosen 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) ∝ a −q for a min ≤ a ≤ a max . DUSTY allows one to use standard MRN parameters (q = 3.5, a min = 0.005 µm and a max = 0.25 µm) or modified MRN parameters. The MRN grain size distribution, for a mixture of silicate and graphite grains, reproduces the Milky Way extinction curve. The Magellanic Cloud extinction curves also obey the same power law for the grain size distribution albeit for different silicate and graphite abundance ratios (Pei 1992). Whenever required, we used a modified MRN distribution by altering a min and a max . The dust density was assumed to be proportional to r −2 , where r is the radial distance from the star.
There are multiple input parameters associated with the dusty circumstellar environment: the dust temperature at the inner shell boundary, T d (in); the relative shell thickness (ratio of outer, R out to inner radius, R in ), Y = R out /R in ; the optical depth at 0.55 µm, τ ; choice of grain composition and the grain-size distribution.
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 above input parameters. 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 d , Y, τ , 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.
For the two-component fit (inner disk + outer shell), we approximated the inner disk by a spherical shell that intercepts a fraction of the direct starlight e.g., as in (Sahai et al. 2003(Sahai et al. , 2006. Such a shell is thus roughly equivalent to an axially-symmetric wedge-shaped fraction of a sphere; this fraction is hereafter referred to as the "disk fraction" and is listed in Tables 2 and 3. Thus, a disk with an opening angle of θ d is approximated by a shell that intercepts a fraction, sin(θ d /2) of the radiation emitted within a 4π solid angle, and the corresponding "disk-fraction" is sin(θ d /2). An illustration of the circumstellar geometry is provided in Fig. 1.
A "correctly illuminated" model of the outer shell is constructed assuming the shell to be divided into two parts. The fraction of the shell that lies in the shadow of the disk (= the disk-fraction) is illuminated by star light attenuated by the disk, together with the sum of the scattered and thermal emission from the disk within that fraction. The remaining fraction of the shell is illuminated by direct starlight plus the remaining fraction of the sum of scattered and thermal emission from the disk. The DUSTY code is run separately for each of the two parts of the outer shell and the outputs are added proportionately to obtain the final SED. These correctly illuminated model numbers are given a suffix 'c'.
In some post-AGB stars, the SEDs may be a result of an interaction between a slow-moving cold outer shell ejected during the previous AGB phase and a fast-moving warm inner shell ejected during the post-AGB phase similar to the case of IRAS 22036+5306 (Sahai et al. 2006). A fit to the SED is then obtained assuming a pair of 'nested shells', i.e., a warm inner shell covering 4π solid angle and a cold outer shell. In the correctly illuminated nested shells model, the cold shell is illuminated by the radiation emerging from the warm inner shell.

Chi-square statistic
We arrive at the best-fit models based on visual inspection of the observed and modeled data. In doing so, we have given more importance to matching the photometry at longer wavelengths (λ > ∼ 2 µm) because these are much less affected by the relatively uncertain intervening interstellar absorption along the line-of-sight to each object, and potential stellar variability in the optical and near-infrared. For the sake of completeness, we also calculated the reduced chi-square statistic for each of the models. The reduced chi-square is given by: where, O i is the observed flux, M i is the model flux, σ i is the error in the observed flux and N−p−1 is the number of degrees of freedom, with N equal to the number of observed datapoints and p equal to the number of free parameters (= 5 for single shell models and 10 for two-component models).
While estimating χ 2 we omitted the WISE photometric data and used the ALLWISE data due to its improved photometric sensitivity at bands 1 and 2 ( § 6.3.4). For our dataset then, the maximum number of degrees of freedom is 12 for single shell models and 7 for two-component models. Using Pearson's chi-square distribution table (https://www.statology.org/chi-square-distribution-table), we find that the critical chi-square values for 5% significance level and degrees of freedom 7 and 12 correspond to 14.1 and 21.0 respectively. The critical values decrease with decrease in the degrees of freedom. For the majority of our stars, the calculated chi-square statistic lies well above the critical value for both single shell and two-component models. Whenever possible, the model with the least χ 2 value has been chosen as the best-fit for a particular source. However, the chi-square statistic cannot be unambiguously used to decide between our models. Given the large number of free parameters, p vis-á-vis the number of observed data points, a much larger dataset is required for the use of the reduced chi-square method to unambiguously decide the goodness of fit.

Photometric magnitudes, extinction and stellar model atmospheres
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). The DUSTY code requires the spectral shape of the source illuminating the dusty shell. Kamath et al. (2015) have derived the effective temperatures (T ef f ), gravities (log g) and metallicities [F e/H] of the objects used in the study. Hence, 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 bol . We determined F bol by scaling the model SED to match the de-reddened SED of our sources. The luminosity and dust mass (M d ) in the circumstellar component was computed for each model. The distance (d) to the LMC was adopted to be 50 kpc. We estimated the luminosity for each model as: For objects obeying a r −2 density distribution, the dust mass in the circumstellar component is given by: From this, we inferred the total mass (gas+dust), M gd = M d δ (Sarkar & Sahai 2006) where, τ 100 is the shell optical depth at 100 µm, κ 100 is the dust mass absorption coefficient, κ, at 100 µm and δ is the gas-to-dust ratio. We assume κ 100 = 34 cm 2 g −1 , as in Sarkar & Sahai (2006). Although this equation is valid for any wavelength, we selected the latter to lie in the far-infrared range, where the SED is dominated by optically-thin thermal emission and therefore the optical depth is better constrained by the model than that at short wavelengths. We note that κ is poorly constrained in general. It may be different for warm ( > ∼ 300 K) and cold dust ( < ∼ 300 K), e.g. (Demyk et al. 2018). We have therefore chosen to use the same value of κ 100 in Eq. 3 for the cool shell and warm disk. The inner radius of the dust shell, R in scales as L 1/2 where L is the luminosity (Ivezić et al. 2012). DUSTY computes R in for 10 4 L ; we use our estimated luminosity (Eq. 2) for each source, to infer R in for the dust shell/disk in it. M d also scales in proportion to Y (discussed below).

Thickness of the circumstellar component
The estimated dust mass scales in proportion to Y (Eq. 3) and may therefore be taken as a a lower limit to the actual mass in the shell. 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, Y of the cold outer shell by a factor of 10 in these objects, increased the total shell mass by the same factor (Table 2) but did not cause any noticeable change in the fit to the observed SEDs (Figs. 4.2 and 7.4). While R in is constrained by T d which is an input parameter, we cannot constrain R out in our SED modeling. This is 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, ∼24 µm. In this paper, we adopt the minimum Y value needed to fit the 24 µm photometry.

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 than 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 gasto-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). In this paper, we adopt δ = 200. A higher value of δ would imply a proportionate increase in the derived masses of the ejecta (gas+dust).

Expansion velocity and mass loss rate
We have no reliable way of estimating the expansion velocity of the cool shells in post-RGB objects. For (the only example of) a post-RGB object where this information is available -the Boomerang Nebula -the cold outer shell is expanding at 165 km s −1 . Typical expansion velocities of the shells around post-AGB and PNe are lower, ∼ 10 − 30 km s −1 (Nyman et al. 1992), although collimated outflows in pre-planetary nebulae can have expansion velocities comparable to that in the Boomerang's cool shell e.g., (Sánchez Contreras & Sahai 2012;Bujarrabal et al. 2001) We have therefore assumed an expansion velocity of 50 km s −1 for the cool shells for the post-RGB sources in order to estimate their ages. The thickness of the model shell provides an ejection time-scale, and together with the model shell mass (M gd ), an estimate for the mass loss rate (dM/dt). Our estimate of dM/dt is sensitive to both the extent of the dust shell and the assumed expansion velocity (it is possible that the dust shell is much larger, as in the Boomerang Nebula.) For the warm inner disk in these objects, the expansion velocity is likely to be much lower. For example, in the Boomerang nebula, the FWHM of the emission from the central waist/disk region is 4.3 km s −1 (Sahai et al. 2017), which may suggest an expansion velocity of 2.2 km s −1 . Jura & Kahane (1999) suggest FWHM < 5 km s −1 from molecular gas observations around evolved red giant stars. We have therefore assumed an expansion velocity of 2 km s −1 to derive the ages and mass-loss rates for the warm inner disks in post-RGB.
In the post-AGB phase, the cold outer shell is most likely a remnant of the previous AGB phase. Since typical AGB expansion velocity is 15 km s −1 (Olofsson 1993), we adopt this value for the cool post-AGB shells. For the inner disk, we use an expansion velocity of 2 km s −1 , similar to the post-RGBs. The SED of some post-AGB sources such as J050632.10-714229.8 ( § 5.1) indicate the presence of 'nested shells'. Following the original AGB phase, a post-AGB wind may develop. Radiation pressure from the hot central post-AGB star on the less dense post-AGB wind may accelerate the dust to velocities of upto 150 km s −1 (Szczerba & Marten 1993). Since, our post-AGB sources are relatively cool (T ef f < 10000 K), we assume a smaller value of 50 km s −1 for the expansion velocity of the warm inner post-AGB shells.

MODELS OF POST-RGB OBJECTS
The detailed modeling of the post-RGBs objects listed in Table 1 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. The physical parameters and derived model parameters corresponding to each of the post-RGB models are given in Table 2. We check for the robustness of the fits by varying the different input model parameters. All model fits are shown in their respective Figure Sets. The adopted best-fit models are flagged with † symbol. Corresponding to each figure set, a sample figure displaying the best-fit model is shown. Important physical parameters of the adopted best-fits are summarized in Table 4. 4.1. J043919. 30-685733.4 This object is classified as a post-RGB shell source by Kamath et al. (2015), with T ef f = 6313K, log g = 1.5 and [F e/H] = −2.0. We used a Kurucz stellar atmosphere model with T ef f = 6250 K, log g = 1.5 and [F e/H] = − 2.0. In Fig. 2.1 we show the observed and model SEDs obtained by varying Y = 8, 100 and 500. We assumed the standard MRN grain size distribution. With T d (in) = 550 K and τ = 1.5, we find a reasonable fit to the SED, except in the mid-IR region (∼ 5 − 12 µm). We find a lower limit on Y ∼ 100, from the MIPS 24 µm flux.
Taking Y = 100 as the lower limit to the shell thickness, we investigated whether variation in the value of T d can improve the fit in the mid-IR region. Models with T d (in) = 145, 175 and 500 K, Fig. 2.2 were computed using a standard MRN grain size distribution, and τ = 0.8. We find that the discrepancy in the near-IR persists, except for the model with T d (in) = 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 µm data. Since the shell is optically thin at all wavelengths longward of ∼ 1µm, 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.
Following the two-component approach outlined in Sec. 3, we found that a good fit to the SED could be obtained with a disk fraction ∼ 0.35, together with a cold outer shell that dominates the emission in λ 10µm region. This cold shell has T d (in) = 130 K, Y = 20, τ = 0.95 and standard MRN grain size distribution (blue curve in Fig. 2.3 and 2.5). Y = 20 is a lower limit for the shell thickness. For the warm inner disk with Y = 1.4 and an r −2 density, alternate model fits were obtained assuming standard MRN grain size distribution, model # 1: T d (in) = 1000 K, yellow curve, Fig. 2.3 and modified MRN grain size distribution: a min = 1 µm and a max = 25 µm, model # 2: T d (in) = 800 K, yellow curve, Fig. 2.5) in the disk.
The sensitivity of the SED to different a min and a max grain sizes in the inner disk is shown in Fig. 2.7. A smaller a 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 µm. Increasing a max however, does not appear to effect the SED.
For both models # 1 and # 2 (Figs. 2.3 and 2.5), there is a mismatch with the IRAC4 (8 µm) flux. We convolved the model flux with the IRAC4 response curve and estimated the magnitude at 8 µm for models # 1 (14.96 mag) and # 2 (14.76 mag). In comparison, the observed magnitude at 8 µm (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.
We show the sensitivity of the SEDs to varying the Y parameter for models # 1 and # 2 in Figs. 2.4 and 2.6. Increasing the thickness causes a mismatch in the mid-IR and with WISE band 3 (W3/12 µm) flux.
In the two-component models (models # 1 and # 2), it was assumed that the shell is illuminated by the radiation from the central star and captures ∼ 0.65 of the bolometric flux of the star. The disk was treated as a separate entity illuminated by the central star and capturing (∼ 0.35) of the bolometric flux from the star. The output was the sum of the modeled fluxes in each instance. Finally, we constructed the correctly illuminated models 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. This choice is re-affirmed by a comparison of reduced chi-square in the two cases, model # 1,c (18.1) and model # 2,c (30.4). In the light of the above discussion, we omitted the IRAC4 (8 µm) flux and the W4 flux while calculating the reduced chi-square. The latter is an upper limit. Our estimated luminosity, L = 116 L from model # 1,c is in close agreement with the observed luminosity of the object, L = 106 L (KWVW15). The total mass accumulated in the shell is 5.2 × 10 −3 M over a relatively short period of ∼ 200 − 300 years at a mass loss rate of 7.1×10 −6 − 1.1 × 10 −5 M yr −1 ( Table 2). The inner disk is very young (∼ 2 years) with a mass of ∼ 2.19 × 10 −8 M and a mass loss rate of 1.18×10 −8 M yr −1 . 4.2. J051347. 57-704450.5 This object is classified as a post-RGB shell source by KWVW15, with T ef f = 4500 K, log g = 1.0 and [F e/H] = −0.5. We used a Kurucz stellar atmosphere model with these parameters for modeling the SED of the object. Our adopted best-fit model corresponds to T d (in) = 250 K, modified MRN grain size distribution: a min = 0.1 µm, a max = 0.25 µm, τ = 0.40 and Y = 3.0 ( Fig. 3.1; χ 2 = 69.2). In Figs. 3.2 − 3.5, we show the effects of varying T d (in), τ , grain sizes and 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 ± 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 § 6.3.3. 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 . In comparison, KWVW15 estimated a luminosity of 712 L by integrating the flux under the observed SED and a photospheric luminosity of 840 L . From our best-fit model, the total mass accumulated in the shell is 4.66 ×10 −5 M over a period of 18 years at a mass loss rate of 2.59 ×10 −6 M yr −1 . 4.3. J051920. 18-722522.1 This object is classified as a post-RGB shell source by KWVW15, with T ef f = 4500 K, log g = 1.5 and [F e/H] = −1.5. We used a Kurucz stellar atmosphere model with these parameters. A fit to the U,B,V and far-IR fluxes could be obtained using T d (in) = 200 K, MRN grain size distribution, τ = 0.75 and Y = 20 (model # 1,s; χ 2 = 89.1). Figs. 4.3 − 4.6 show the effects of varying T d (in), τ , grain sizes and Y in model # 1,s. For all cases, the R and I-band fluxes remain well below the modeled SED ( § 6.3.3). To account for this deficit, we assumed a two-component model (model # 2) with a warm (T d (in) = 500 K), inner disk, in addition to a cold outer shell at 110 K. This disk was approximated as a wedge-shaped fraction (∼ 0.4) of a sphere with τ = 0.4, Y = 2.0 and modified MRN grain size distribution: a min = 0.5 µm and a max = 20 µm. The cold outer shell is characterised by τ = 1 µm, Y = 20 and MRN grain size distribution. We constructed the correctly illuminated spectrum (model # 2,c; χ 2 = 17.5) for the shell in the two solid angles ( Fig. 4.1). Increasing the thickness of the cold outer shell by a factor of 10, does not cause any appreciable change in the fit to the SED ( Fig. 4.2, § 3.3.1).
Based on visual inspection and a comparison of reduced chi-square values, we adopted model # 2,c as the best-fit. From this, we estimated luminosity, L = 582 L . In comparison, KWVW15 estimated a luminosity of 496 L by integrating the flux under the observed SED. The total mass in the shell is estimated to be 3.41 ×10 −2 M accumulated over ∼ 645 − 792 years at a mass loss rate of ∼ 2.1 × 10 −5 − 2.6 × 10 −5 M yr −1 . The inner disk is much younger (15.9 years) with a mass of 3.01×10 −5 M and a mass loss rate of 1. , τ , grain sizes and Y . We find that the reduced chi-square is least for a modified MRN grain size distribution, a min = 0.005 µm and a max = 0.50 µm ( Table 2, Fig. 5.4). Visual inspection of the model plot in this case reveals that this is due to better agreement with ALLWISE W4 data. However, the latter model shows a mismatch with the IRAC4 (8µm) flux. Hence, we adopt the model with standard MRN grain size distribution as the best-fit. For the same reason, we also reject the model with τ = 0.8 ( Fig. 5.3) even though the model has a lower chi-square value than our adopted best-fit. 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 ) is in good agreement with the observed (248 L ) and photospheric luminosity (294 L ) values of KWVW15. From our best fit model, the total mass accumulated in the shell is 5.8 ×10 −5 M over a period of 27.2 years at a mass loss rate of 2.13 ×10 −6 M yr −1 . 4.5. J045555. 15-712112.3 This object is classified as a post-RGB disk source by Kamath et al. (2015), with T ef f = 10000 K, log g = 2.0 and [F e/H] = −0.5. We used a Kurucz stellar atmosphere model with these parameters. Fig. 6.1 shows a one-component model fit to the observed SED (model # 1,s) using Sil-Ow grains, T d (in) = 550 K, a min = 0.005 µm, a max = 1.0 µm, τ = 2.3 and Y = 20. The effects of varying T d (in), τ , grain sizes and Y are shown in Figs. 6.5 − 6.8. Our model fit (# 1,s) is deficit in the near-IR region and additionally, there is a mismatch with the IRAC2 (4.5 µm), W2 (4.6 µm) and IRAC3 (5.8 µm) fluxes.
For a better fit in the near-IR region, we assumed a two-component model (model # 2) with a warm, T d (in) = 1200 K, thin inner disk, in addition to the cold outer shell at 550 K. The disk composed of Sil-Ow grains was approximated as a wedge-shaped fraction (∼ 0.1) of a sphere with τ = 0.5, Y = 10 and grain size distribution: a min = 0.005 µm and a max = 0.1 µm. We constructed the correctly illuminated spectrum for the shell in the two solid angles (# 2,c, Fig. 6.2). We tried an alternate two-component model (model # 3) using graphite grains (grf-DL) for the warm inner disk (T ef f = 800 K) and a combination of Sil-Ow (0.8) and grf-DL (0.2) for the cold outer shell (T ef f = 500 K). The inner disk was approximated to be a wedge-shaped fraction (0.1) of a sphere with τ = 0.7, Y = 5 and MRN grain size distribution: a min = 0.005 µm and a max = 0.25 µm. The cold shell is modeled using τ = 1.8, Y = 2 and MRN grain size distribution. Fig. 6.3 shows the correctly illuminated two-component model (# 3,c). Besides the continuing mismatch at IRAC2 (4.5 µm), W2 (4.6 µm) and IRAC3 (5.8 µm), the fit remained unsatisfactory in the near-IR.
Further, we investigated the possibility that a stellar companion dominates the emission at and around 4.6 µm in models # 2 and # 3, with suitable adjustments in the disk and the shell parameters, by fitting a blackbody curve to the flux at 4.6 µm. We find that a blackbody with T ef f = 800 K, and L = 39 L would account for the flux at 4.6 µm. Since a main-sequence companion star with such a low T ef f would have a very low luminosity (i.e., much lower than 39 L ), we conclude models including a stellar companion cannot explain the discrepancy at 4.6 µm.
The relatively low flux at 4.6 µm may result from the presence of a deep absorption feature at 4.6 µm. 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 ef f , for objects with T ef f > 8000K (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 ef f and hence log g. A higher log g value would support the classification of J045555.15-712112.3 as a possible YSO.
In the scenario that the object is indeed a post-RGB source as classified by KWVW15 and not a YSO, we explored models which may give a better fit at 4.6 µm. The two-component model # 4 is shown in Fig. 6.4. The warm inner disk, T d (in) = 1000 K, composed of Sil-Ow grains was approximated to be a wedge-shaped fraction (0.4) of a sphere with τ = 1.0, Y = 10 and MRN grain size distribution. The cold outer shell, T d (in) = 300 K has Sil-Ow grains, τ = 2.5 and Y = 2 and MRN grain size distribution. The model provides a good fit at 4.6 µm but is discrepant with the observed values in the near-IR and at IRAC3 (5.8 µm) and IRAC4 (8µm). The discrepancy at IRAC bands 3 and 4 may be due to PAH emissions and has been discussed further in Sec. 6.3.1. Since the inner disk is optically thick, we cannot apply the "correctly illuminated model" formalism to this object.
We calculated the reduced chi-square for each of the models, model # 1,s/χ 2 = 619.8, model # 2,c/χ 2 = 1125.5 and model # 3,c/χ 2 = 1318.3, model # 4/χ 2 = 116.6. Based on the above discussion, if the object is indeed a post-RGB source, we adopt model # 4 as the best-fit to the observed SED.
From this model, we estimated a luminosity of 465L for J045555.15-712112.3. KWVW15 estimated a luminosity of 454 L by integrating the flux under the observed SED and a photospheric luminosity of 191 L . They attributed this difference in luminosity estimates to either a non-spherically symmetric dust envelope or substantial flux contribution from another object coincident with J045555.15-712112.3 in the sky. The total mass accumulated in the shell is 2.84×10 −4 M over ∼ 7.9 years at a mass loss rate of 3.59×10 −5 M yr −1 . The mass accumulated in the disk is 3.08×10 −6 M over 131 years at a mass loss rate of 2.35×10 −8 M yr −1 .

J045755.05-681649.2
This object is classified as a post-RGB disk source by KWVW15, with T ef f = 4927K, log g = 1.5 and [F e/H] = −0.3. We used a Kurucz stellar atmosphere model with T ef f = 5000 K, log g = 1.5 and [F e/H] = −0.3. We first modeled the SED of the object, assuming a single dust shell surrounding the central star (model # 1,s, Fig. 7.1). A fit to the optical and far-IR fluxes could be obtained using T d (in) = 500 K, MRN grain size distribution (a min = 0.005 µm, a max = 0.25 µm), τ = 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 µm) and IRAC4 (8 µm). These discrepancies have been discussed further in ( § 6). The mismatch at IRAC4 may be attributed to PAH emission, similar to J043919.30-685733.4. Figs. 7.5 − 7.7 show the observed and modeled SED of the object along with effects of varying T d (in), τ , and 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 cold outer shell. Assuming the fraction that is not covered by the disk, reasonable fit to the far-IR flux distribution (λ 10µm) could be obtained for for T d (in) = 400 K, Y = 30, τ = 0.9 and modified MRN grain size distribution: a min = 0.1 µm, a max = 1.0 µm. For the warm inner disk, T d (in) = 1300 K, Y = 2, τ = 0.5 and modified MRN grain size distribution: a min = 0.005 µm, a max = 2.0 µm 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. 7.2.
Since, both the one and two-component models could not provide a fit at IRAC4 (8 µm), we omitted the flux at this point for the reduced chi-square calculation. Further, the ALLWISE W4 flux is an upper limit and is also omitted from the chi-square calculation. The reduced chi-square is least for the single shell model with T d (in) = 550K, χ 2 = 20.7 (Table 2). This value is comparable with χ 2 = 24.7 for the two-component model # 2,c. Since the two fits appear similar, we overplotted the two models ( Fig. 7.3). On visual inspection, we find that the latter provides a better fit in the region of the mid-IR IRAC bands, 3µm − 6µm with a lower value for the residual sum (98.8) than in the former case (186.4). Hence, we adopt the two-component model # 2,c as the best-fit for J045755.05-681649.2. Increasing the thickness of the outer shell by a factor of 10 in model # 2,c does not cause any appreciable change in the fit to the SED ( Fig. 7.4, § 3.3.1).
From our best-fit model, the luminosity is estimated to be L = 217 L . In comparison, KWVW15 estimated a luminosity of 190 L by integrating the flux under the observed SED and a photospheric luminosity of 215 L . The total mass accumulated in the shell is ∼ 5.73 × 10 −5 M over a short period of ∼ 30 − 40 years at a mass loss rate of ∼ 7 ×10 −7 − 9 × 10 −7 M yr −1 . The inner disk is much younger, ∼ 1.9 yr. with a mass of ∼ 9.64 × 10 −9 M . and a mass loss rate of ∼ 5 × 10 −9 M yr −1 .

J050257.89-665306.3
This object is classified as a post-RGB disk source by KWVW15, with T ef f = 4586K, log g = 0.5 and [F e/H] = −0.5. We used a Kurucz stellar atmosphere model with Teff = 4500 K, log g = 0.5 and [F e/H] = −0.5. A single shell model could not provide a fit to the near and mid-IR fluxes (model # 1,s; Fig. 8.1, χ 2 = 141.77). Hence, we tried two-component models with a warm inner disk and a cold 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 µm) have been discussed in § 6. In calculating the reduced chi-square, we have omitted the flux at 8µm.
In model # 2, the inner disk (T d (in) = 1200 K) is composed of Sil-Ow grains with modified MRN grain size distribution: a min = 0.3 µm and a max = 5.0 µm, τ = 0.5 and Y = 3.0. The outer shell (T d (in) = 240 K) is composed of Sil-Ow grains with modified MRN grain size distribution: a min = 0.005 µm and a max = 1.0 µm, τ = 1.2 and 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, χ 2 = 48.4) is presented in Fig. 8.2.
In model # 3, the inner disk (T d (in) = 1000 K) is composed of amC-Hn grains with modified MRN grain size distribution: a min = 0.005 µm and a max = 0.5 µm. τ = 0.6 and Y = 2.0. The outer shell (T d (in) = 250 K) is composed of Sil-Ow grains with MRN grain size distribution: a min = 0.005 µm and a max = 0.25 µm, τ = 1.7 and 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, χ 2 = 43.9) is presented in Fig. 8.3.
The reduced chi-square is comparable for models # 2,c and # 3,c. On visual inspection we find that model # 2,c shows better agreement with the optical (U,B) fluxes ( Fig. 8.4). Hence we adopt the two-component model # 2,c as the best fit for J050257.89-665306.3.
From our correctly illuminated model # 2,c (Table 2), the luminosity is estimated to be L = 303 L . In comparison, KWVW15 estimated a luminosity of 267 L by integrating the flux under the observed SED and a photospheric luminosity of 177 L . The total mass accumulated in the shell is 2.24×10 −4 M over a period of ∼ 30 − 38 years at a mass loss rate of ∼ 3×10 −6 M yr −1 . The inner disk is much younger (∼ 4.3yr) with a mass of ∼ 5.77×10 −8 M and a mass loss rate of ∼ 1.3×10 −8 M yr −1 .

J055102.44-685639.1
This object is classified as a post-RGB disk source by KWVW15, with T ef f = 7625 K, log g = 1.0 and [F e/H] = −0.5. We used a Kurucz stellar atmosphere model with T ef f = 7500 K, log g = 1.0 and [F e/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, χ 2 = 600.2) could not provide a fit to the optical and near-IR fluxes. We therefore tried two component models with a warm inner disk and cold 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. 9.1.
The near-IR flux of the object indicates a dust temperature in excess of 1500 K. This would exceed the sublimation temperature of silicate grains. We therefore tried amorphous carbon grains (amC-Hn) for the inner disk (model # 3, χ 2 = 220.0). 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-α nucleosynthesis and (the third) dredge-up -events that occur at the centers of AGB stars. In the case of J055102.44-685639.1, we find that the disk is optically thick (τ = 1.0) and amC-Hn grains provide a reasonable fit to the SED in the near and mid-infrared ( Fig. 9.2). The cold outer shell has a combination of warm silicates and silicon carbide. 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 Common Envelope (CE) ejection. Such a scenario may also account for the presence of PAH molecules in the circumstellar environment of some post-RGB stars (see § 6.3.1).
Our choice of a two-component model (# 3) as the best fit is re-affirmed by a lower value of the reduced chi-square in this case, χ 2 = 147.3. The disk was approximated as an axially symmetric wedge-shaped fraction (0.3) of a sphere with amC-Hn grains, Td = 2000 K, Y = 7.0, τ = 1.0 and modified MRN grain size distribution: a min = 0.005 µm, a max = 0.05µm. The outer shell (Sil-Ow/0.4 and SiC-Pg/0.6) has T d = 350 K, Y = 3.0, τ = 12.0 and modified MRN grain size distribution: a min = 0.005µm, a max = 0.07 µm. At the large optical depth of the outer shell, the SiC feature goes into absorption ( Fig. 9.2). Further, 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 2), we estimated a luminosity of 621L . In comparison, KWVW15 estimated 452L and 815L respectively for the observed and photospheric luminosity of the object. The total mass accumulated in the shell is 3.05 × 10 −3 M over a period of ∼ 12.5 years at a mass loss rate of ∼ 2.43×10 −4 M yr −1 . The inner disk is younger, ∼ 8.8 yr. with a mass of ∼ 1.99×10 −8 M and a mass loss rate of ∼ 2.27 × 10 −9 M yr −1 .

MODELS OF POST-AGB OBJECTS
The detailed modeling of the post-AGB objects listed in Table 1 is discussed below. The physical parameters and derived model parameters corresponding to the post-AGB models are given in Table 3. We check for the robustness of the fits by varying the different input model parameters. All model fits are shown in their respective Figure Sets. The adopted best-fit models are flagged with † symbol. Corresponding to each figure set, a sample figure displaying the best-fit model is shown. Important physical parameters of the adopted best-fits are summarized in Table 4.  Fig. 10.1 shows a one-component model fit to observed SED using Sil-Ow grains, standard MRN grain size distribution, T d (in) = 200 K, τ = 0.6 and Y = 20 (model # 1,s). This model is deficient in the mid-IR region, particularly at IRAC3 (5.8 µm) and IRAC4 (8 µm).
Since J050632.10-714229.8 is a post-AGB star, we modeled the SED using the 'nested shells' approach outlined in Sec. 3. Fig. 10.2 shows the fit obtained in the mid-IR using Sil-Ow (grain type # 1) in the inner shell, T d (in) = 400 K, standard MRN grain size distribution, τ = 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. The circumstellar environment of a post-AGB star may have a mix of carbon and silicate dust grains. Fig.  10.3 shows the fits obtained using different grain combinations. We find that Sil-Ow (# 1) at 400 K, a combination of amC-Hn/0.1 and Sil-Ow/0.9 (# 2) at 350 K and a combination of grf-DL/0.3 and Sil-Ow/0.7 (# 3) at 350 K ( Fig.  10.4) all provide 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 µm − 8 µm. These discrepancies are further discussed in § 6. Using each of the above best-fit warm inner-shell models, we obtained best-fit models for the cold outer shell (nested models # 1; χ 2 = 181.6; # 2; χ 2 = 164.2 and # 3; χ 2 = 154.7; Table 3). Fig. 10.5 shows the complete correctly illuminated model SED of J050632.10-714229.8 for each grain combination.
The reduced chi-square is least for model # 3. Hence, we adopt this as the best-fit model. Our estimated luminosity (5434L ) is in good agreement with KWVW15's observed luminosity for the object (4910L ). They estimated a much higher photospheric luminosity of 7606L . The latter may be in error since it is based on their estimate of T ef f and metallicity from low resolution spectra which is significantly different from that based on high resolution spectra of van Aarle et al. (2013). From model # 3, the total mass accumulated in the outer shell, the shell age and mass loss rate respectively are 5.91×10 −2 M , 1020 yrs., 5.81×10 −5 M yr −1 . show the effects of varying T d , τ and 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 . KWVW15 obtained vastly different values for the observed and photospheric luminosity of the object, 4477L and 14112L 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 1.76×10 −2 M over 1100 years at a mass-loss rate of 1.6×10 −5 M yr −1 .

J051906.86-694153.9
This object is classified as a post-AGB shell source by KWVW15, with T ef f = 5613K, log g = 0.0 and [F e/H] = −1.3. The WISE and ALLWISE W3 and W4 data are discrepant for this source, with ALLWISE reporting upper limits significantly lower than the WISE magnitudes. Given the good agreement between the Spitzer MIPS 24 µm and the WISE W4 fluxes, we have chosen to ignore the ALLWISE W3 and W4 data for this source. We used a Kurucz stellar atmosphere model with T ef f = 5500 K, log g = 0.0 and [F e/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 d (in) = 2000 K, τ = 0.35 and Y = 2.0 ( Fig. 12.1). The radiation emerging from this warm shell illuminates a cold outer shell of SiC-Pg grains with modified MRN grain size distribution: a min = 2.3 µm, a max = 3.0 µm, T d (in) = 160 K, τ = 0.07 and Y = 2.0. Fig. 12.2 (χ 2 = 21.3) shows the correctly illuminated nested shell model. In calculating the reduced chi-square for this source, we used the WISE W3 and W4 fluxes.
We estimate a luminosity of 2018L . This is in good agreement with the observed luminosity estimated by KWVW15 (2052L ). 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 and 4246L respectively. From our best-fit model, the total mass accumulated in the shell is 1.  2013) values based on their high resolution data. Hence, we used a Kurucz stellar atmosphere model with T ef f = 5500 K, log g = 0.0 and [F e/H] = −1.0. A single shell model gave a reasonable fit to the near, mid and far-IR data ( Fig. 13.1, χ 2 = 214.4). The fit was obtained for T d (in) = 260 K, using a mix of silicate (Sil-Ow/0.2) and graphite (grf-DL/0.8) grains, modified MRN grain size distribution: a min = 0.005 µm, a max = 0.50 µm, τ = 0.9 and Y = 2. The presence of graphite dust indicates that it is a carbon-rich post-AGB star. Hence, the mismatch at IRAC3 (5.8 µm) and IRAC4 (8 µm) may be attributed to PAH emissions.
We estimate a luminosity of 4657.0L . This is in good agreement with the observed luminosity estimated by KWVW15 (4223L ). However, similar to the case of the post-AGBs, J051848.84-700247.0 ( § 5.2) and J051906.86-694153.9 ( § 5.3), KWVW15 estimated a much higher photospheric luminosity, 11056L . From our best-fit model, the total mass accumulated in the shell is 6.45×10 −3 M over a period of 144 years at a mass loss rate of 4.48×10 −5 M yr −1 .
This object is classified as a post-AGB disk source by KWVW15, with T ef f = 4500 K, log g = 0.0 and [F e/H] = −1.0. We used a Kurucz stellar atmosphere model with these parameters. A single shell model (Fig. 14.3) 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 cold 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 min = 0.005 µm and a max = 5.0 µm, T d (in) = 1100 K, τ = 0.6 and Y = 3.0. The outer shell is composed of Sil-Ow grains with standard MRN grain size distribution, T d (in) = 150 K, τ = 0.10 and Y = 2.0. Based on the reduced chi-square values, we adopt the nested shells model (# 1) as the best-fit. The estimated luminosity based on this model is 6598L . KWVW15 gave an observed luminosity estimate of 6864L and and a photospheric luminosity of 7131L . The total mass accumulated in the outer shell is estimated to be 8.25×10 −4 M in 225 years at a mass-loss rate of 3.7×10 −6 M yr −1 . 5.6. 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 ef f = 6112 K, log g = 0.5 and [F e/H] = −1.6. We adopted Kurucz stellar atmosphere model with T ef f = 6000 K, log g = 0.5 and [F e/H] = −1.5. For the R magnitude we used 13.78 4 from the NOMAD catalog (Zacharias et al. 2005).
A reasonable fit to the near, mid and far-IR data ( Fig. 15.1, χ 2 = 98.0) is obtained assuming an inner disk (T d (in) = 1100 K) composed of amorphous carbon (amC-Hn) with modified MRN grain size distribution: a min = 0.005 µm, a max = 2.0 µm, τ = 5.0 and Y = 15.0. The disk is assumed to be a wedge-shaped fraction (0.25) of a sphere. The outer shell (T d (in) = 600 K) is composed of silicate (Sil-Ow/0.4) and graphite (grf-DL/0.6) grains with standard MRN grain size distribution, τ = 0.4 and Y = 30.0. Since the inner disk is optically thick, similar to the post-RGB star, J055102.44-685639.1 ( § 4.8) 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 (∼3.5 µm and 4.5 µm) as also the mismatch between model and observed I-band data. ALLWISE photometry at bands 1(3.4 µm) and 2(4.6 µm) is generally an improvement over the WISE data at these wavelengths ( § 6.3.4). From our best-fit model, the luminosity of the object is 7763L . KWVW15 obtained 6667L for the observed luminosity and 4703L for the photospheric luminosity. The total mass accumulated in the shell is estimated to be 1.21×10 −3 M over 800 years at a mass-loss rate of 1.51×10 −6 M yr −1 . The total mass in the disk is 4.67×10 −5 M over 220 years at a mass-loss rate of 2.12×10 −7 M yr −1 . 5.7. 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 ef f = 6237 K, log g = 1.5 and [F e/H] = −2.5 for the star. We used a Kurucz stellar atmosphere model with T ef f = 6250 K, log g = 1.5 and [F e/H] = −2.5. The difference between the WISE (ALLWISE) and IRAC fluxes at bands 1 and 2 (∼3.5 µm and 4.5 µm) may be attributed to the variable nature of the object.
We obtained a luminosity of 4116L for the object in close agreement with the observed (3780L ) and photospheric (3657L ) luminosity estimated by KWVW15. The total mass accumulated in the shell is estimated to be 1. There is a significant difference between the WISE (ALLWISE) and IRAC fluxes at bands 1 and 2 (∼3.5 µm and 4.6 µm) which may (at least partly) be due to the photometric variability. We have chosen to fit the IRAC data, and have excluded the WISE(ALLWISE) fluxes at 3.6 µm, 4.6 µm and 12 µm for estimating the reduced chi-square.
Single shell models # 1 and # 2 ( Fig. 17.1) 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 (model # 1) while single shell grf-DL model (model # 2) provided a fit to the near-IR and IRAC (3.6 µm, 4.5 µm and 5.8 µm) fluxes.
Hence we tried two-component model fits to the observed SED. Fig. 17.2 shows the best-fit two-component model for J052519.48-705410.0 (model # 3, χ 2 = 35.9). The warm inner disk (T d (in) = 800 K) is approximated as a wedgeshaped fraction (0.35) of a sphere and is made up of Sil-Ow grains with standard MRN grain size distribution, τ = 1.0 and Y = 20. The outer shell (T d (in) = 600K) is made up of grf-DL grains with modified MRN grain size distribution: a min = 0.005 µm, a max = 0.5 µm, τ = 0.8 and Y = 2.0. Since the inner disk is optically thick, we cannot apply the "correctly illuminated model" formalism to this object.
From our best-fit model # 3, we estimated a luminosity of 3804L . KWVW15 obtained 3219L and 4943L respectively for the observed and photospheric luminosities. The total mass in the disk and shell was estimated to be 3.64×10 −5 M and 1.25×10 −4 M respectively. The age of the disk is estimated to be 451 years while the shell is ∼ 20 years old. The mass loss rates for the disk and shell respectively are 8.08×10 −8 M yr −1 and 6.12×10 −6 M yr −1 .

Common envelope evolution and ejecta mass
When the primary in a binary system overflows its Roche lobe and the resulting mass transfer proceeds too rapidly to be accreted by the compact companion, a CE system results. CEE (Paczynski 1976;Ivanova et al. 2013) may give rise to close binary systems. Two scenarios may exist -a rapid plunge in of the companion or a slow spiral-in phase. In the former, if enough orbital energy is deposited to the CE via dynamical friction, the whole envelope is ejected on a dynamical timescale (Paczynski 1976). The second case provides an alternate route for the envelope ejection (Ivanova et al. 2002;Podsiadlowski et al. 2010) over 'several dynamical time scales' (Ivanova et al. 2013;Clayton et al. 2017). Recently, a third scenario has been proposed by Glanz & Perets (2018) wherein dust driven winds similar to those observed in AGB stars may lead to the ejection of the CE.
Following the ejection of the CE, dust formation may occur in the expanding gas (Lü et al. 2013;Iaconi et al. 2020) which may explain the presence of circumstellar dust shells in these systems. Some fraction of the ejected mass may also fall back and interact with the binary leading to the formation of circumbinary disks (Kashi & Soker 2011). 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 M 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 in values for the cold dust shells from our best-fit models (Table 2). In Lü et al.'s models, the mass of dust produced in the CE ejecta of giants with masses 1−7 M ranges from about 10 −9 M to 10 −2 M . The lower end of this mass range corresponds to CEE occurring at the base of the RGB and the model parameter γ = 0.2 (see eqn. 2 of Lü et al. (2013) for a definition of γ). Higher values of γ 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 tip of the RGB with γ ≥ 0.3, or at the tip of the RGB with γ ≥ 0.2 (Fig. 2, Lü et al. (2013)).
For our sample of post-RGB objects, the values of total mass, M gd in the outer shell range from about 4×10 −5 M to 3×10 −2 M (Table 4) -inferred from our model dust masses, M d that lie in the range 2×10 −7 M -1.5 × 10 −4 M . We compare our dust mass estimates with those of Lü et al. (2013) in Fig. 18. Since our derived masses are likely lower limits to the total dust mass, its most likely that either γ ≥ 0.3 and/or CEE has occurred near or at the tip 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 ∼ 10 14 and ∼ 10 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×10 14 and ∼ 10 18 cm. Dust formation in CE ejecta wherein the primary is a RGB star has also been studied by Iaconi et al. (2020) using a 3D SPH hydrodynamic simulation. Their range of dust masses obtained were comparable to that of Lü et al. (2013). 6.2. "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 ∼ 100 − 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 disks with large opening angles (∼ 41 • ± 6 • ) and hence geometrically thick structures. 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 6 (for post-RGB sources) and 6 (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 3840 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 with the exception of the post-RGB object, J045555.15-712112.3 and the post-AGB object, J052519.48-705410.0. Our estimate for the age depends on (i) the assumed expansion velocities for the disk and the shell respectively (V exp ) and (ii) on the thickness of the shell/disk (Y ).
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. 4.2 and 7.4, 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 2. In the absence of data beyond 24 µm, 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) For example, in the post-AGB object, IRAS 22036+5306, where the modeled SED included data up to ∼200 µm the outer radius of the shell is a factor ∼ 40 larger than the inner radius (Sahai et al. 2006). A thicker outer shell in J045555.15-712112.3 would translate into a larger value for the shell age.
The post-AGB star, J052519.48-705410.0 has been classified as a Cepheid (Soszyński et al. 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 ( § 5.8). 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 may be 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).

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 µm region for some of our post-RGB sources -J043919.30-685733.4, J045555.15-712112, J045755.05-681649.2 and J050257.89-665306.3.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 5 ; 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 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 ef f . 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 3000 K.

Silicate emission feature
The ∼10 µm 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 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. 10.2). The silicate emission feature is also not very sensitive to changing T d (Fig. 10.7). 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. 10.8 shows the fits for different values of a 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. Set 3). 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 Table 5 we compare the I-band magnitudes from Zaritsky et al. (2004) with data from the DENIS catalog (Cioni et al. 2000

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 Table 1. A mismatch between modeled and observed SEDs at WISE band 2 (W2/4.6 µm) was observed for the post-RGB sources, J043919.30-685733.4, J045555.15-712112.3 and J045755.05-681649.2. Hence, we additionally plotted the ALLWISE magnitudes of the sources from Vizier. 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 bands 1 and 2. The latter data shows good agreement with our modeled SEDs for J043919.30-685733.4 (Figs. 2.8, 2.9), J045755.05-681649.2 (Fig. 7.2) and J045555.15-712112.3 (Fig. 6.4). Similarly, we notice a mismatch between modeled and observed SEDs for the post-AGB "disk" sources in the WISE band 1 and 2 fluxes. When 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 ( § 6.2.3).

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: 1. The masses of ejecta in the disks of post-RGBs (disks/inner shells of post-AGBs) lie in the range ∼ 9 × 10 −9 − 3 × 10 −5 M (∼ 9×10 −9 −7×10 −4 M ) and in the outer shells from ∼ 4×10 −5 −3×10 −2 M (∼ 1×10 −4 −6×10 −2 M ). 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 µm band.
2. The disk fractions are surprisingly large (typically 0.3-0.4), implying that the disks are geometrically thick structures with a substantial opening angle (∼ 41 • ± 6 • ). 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.
3. 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. 4. 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).
5. 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 tip of the RGB for the post-RGB sources in our sample.
We thank our anonymous referees for their meticulous reviews of our paper, which have helped us improve it. 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.

Note-
The inner disk is approximated as an axially symmetric wedge-shaped fraction of a sphere. This fraction is referred to as "disk fraction" The acronymns for the dust grain compositions refer to 'warm' and 'cold' silicates (Sil-Ow and Sil-Oc) from Ossenkopf et al. (1992), graphite grains (grf-DL) from Draine & Lee (1984), amorphous carbon (amC-Hn) from Hanner (1988) Table 3. Models of post-AGB objects

Note-
The inner disk is approximated as an axially symmetric wedge-shaped fraction of a sphere. This fraction is referred to as "disk fraction". The acronymns for the dust grain compositions refer to 'warm' and 'cold' silicates (Sil-Ow and Sil-Oc) from Ossenkopf et al. (1992), graphite grains (grf-DL) from Draine & Lee (1984), amorphous carbon (amC-Hn) from Hanner (1988)

Note-
The acronymns for the dust grain compositions refer to 'warm' silicates (Sil-Ow) from Ossenkopf et al. (1992), graphite grains (grf-DL) from Draine & Lee (1984), amorphous carbon (amC-Hn) from Hanner (1988) (2013). FGB refers to the first red giant branch. FG2006 refers to results by Ferrarotti & Gail (2006), showing the dust masses produced in the dust-driven outflows of AGB stars. M d scales in proportion to the thickness of the circumstellar component. Determination of the latter is constrained by the long wavelength limit of our data (24 µm). Our estimated M d values may be treated as lower limits to the actual dust mass in the shells.