A Study of the Dusty Disks and Shells around Post-RGB Stars in the LMC

: A new class of dusty post-Red Giant Branch (post-RGB) stars has recently been identiﬁed in the Magellanic Clouds. Their spectral energy distributions (SEDs) suggest that their mass-ejecta are similar to dusty post-Asymptotic Giant Branch (post-AGB) stars. We 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 in the disks and shells and set rough constraints on the dust grain compositions and sizes. The shells were signiﬁcantly more massive than the disks. Our models suggest that circumstellar disks, when present, are geometrically thick with a substantial opening angle, which is 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 tip of the RGB for the post-RGB sources in our sample. Amorphous silicate emission features at 10 and 18 µ m are seen in the model spectra of several post-RGBs. A surprising result is that the ejected dust in certain post-RGB sources appears to be carbon-rich, thus, providing independent support for the hypothesis of binary interactions leading to the formation of dusty post-RGB objects.

After mass-loss has depleted most of the stellar envelope, the stars evolve to higher temperatures through the post-AGB phase at almost constant luminosity. The largely spherical (AGB) envelopes are transformed into aspherical bipolar/multipolar morphologies (Sahai & Trauger [7], Sahai, Morris & Villar [8]). Several mechanisms may explain the formation of the jet engines that produce such outflows and produce bipolar/multipolar structure in PPNe (Phillips & Ramos-Larios [9] and references therein), the most popular amongst these being common-envelope evolution (CEE) in close binary systems.
Binarity may also be responsible for the rapid and unexpected evolution of the Boomerang Nebula, which shows bipolar morphology similar to post-AGB PPNe (Wegner & Glass [10]; Sahai & Nyman [11]). However, the luminosity of the central star in the Boomerang is much lower (L ∼ 300 L ) than is possible for a post-AGB star. Sahai et al. [12] showed that the Boomerang is most likely a post-RGB star and that a merger with a binary companion may have triggered its extreme mass loss (∼10 −3 M yr −1 ) at a very high ejection velocity (165 kms −1 ), over a relatively short period (3500 yr) via CEE.
Kamath et al. [13][14][15] identified a group of stars in the LMC/SMC that are of too low a luminosity (<2500 L ) to be post-AGB, and rather these appear to be a new class of post-RGB objects. Further, they determined the properties of this class of objects. These objects show large mid-IR excess.
Optical spectroscopy allowed determination of their stellar parameters (T eff , log g, [Fe/H] and E[B-V]). Using optical and Spitzer photometry, they integrated the spectralenergy-distributions (SEDs) to constrain the bolometric flux of the stars. The distance modulus for the LMC/18.54 mag and SMC/18.93 mag (Keller & Wood [16]) enabled reliable luminosity estimates. The Boomerang, with its mid-infrared excess and low luminosity, appears to be a Milky Way analog.

Sample Selection
We selected a sample of eight post-RGB stars in the LMC with mid and far-IR excess from Kamath et al. [14], hereafter KWVW15. Our sample consists of an equal number of "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 emission lies beyond 10 µm. The "disk" sources show near-IR excess indicative of hot dust and the peak of the dust SED lies around 10 µm or sometimes even bluer. Further, on a color-color plot, the "shell" sources have [3.6] Additionally, to ensure that the objects are indeed post-RGB (and not post-AGB), we selected objects with L < 1000 L .
In order to investigate if the properties of the ejecta (e.g., mass, mass-loss rate, temperature, disk-to-shell mass ratio) are different between post-RGB and post-AGB stars, we also selected a sample of post-AGB LMC stars from KWVW15, equally divided between "shell" and "disk" types. Our post-AGB stars are common between KWVW15 and van Aarle et al. [17], the only exception being J051906.86-694153. 9. This star showed the BaII line at 4554.03 Å (KWVW15). The presence of BaII indicates a s-process enriched post-AGB object.
For many of the sources, KWVW15 obtained vastly different values for the observed and photospheric luminosities 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.
The photometry of the objects in this study was compiled by KWVW15 and is available online from the Vizier database (https://vizier.u-strasbg.fr/viz-bin/VizieR, accessed on 31 January 2022) The photometric magnitudes were corrected for the combined effects of Galactic and LMC reddening using a mean E(B−V) = 0.08 (Keller & Wood [16]). 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. [18]. In the mid-infrared, we applied the extinction law by Gao et al. [19]. The observed SEDs extend from the U-band to 24 µm.
When the Sil-Ow grain composition did not provide a good fit, we tested different grain compositions. We used the Mathis, Rumpl and Nordsieck, MRN [25] 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. 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 that affect the SED, our strategy for exploring the parameter space is as follows. We first attempted to fit the SED using a single shell (one-component model), varying the dust temperature on the inner shell boundary (T d ), the relative shell thickness (Y = ratio of the outer to the inner shell radius), the optical depth (τ) at 0.55 µm and the grain-size distribution for a choice of grain composition.
If systematic discrepancies remained between the fit and the data, we then attempted a two-component fit, in which we added an inner component, representative of a hot, compact disk and varied 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. [26,27]. 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 presented in Tables 1 and 2. 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 Figure 1.
A correctly illuminated model of the outer shell is then 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. Illustration showing circumstellar dust geometry with an inner disk and an outer shell. R in and R out are the inner and outer radius of the disk (shell). A disk with an opening angle θ d intercepts Sin(θ d /2) (referred to as disk-fraction in this paper) of starlight. Table 1. Important parameters derived from the best-fit post-RGB models.
In some post-AGB stars, the SEDs may be a result of an interaction between a slowmoving cool 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. [27]). Thus, a fit to the SED of the post-AGB stars J050632.10-714229.8, J051906.86-694153.9 and J045623.21-692749.0 was obtained assuming a pair of nested shells, i.e., a warm inner shell covering 4π solid angle and a cool outer shell.
We arrived at the best-fit models, Figures 2 and 3 based on visual inspection of the observed and modeled data. In doing so, we gave more importance to matching the photometry at longer wavelengths (λ > ∼ 2 µm) as 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 sources where visual inspection could not discern between models, we used the reduced chi-square, to decide the best-fit model; 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).

Derived Parameters
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 to the LMC is d = 50 kpc. We estimated the luminosity for each model as L = 4πd 2 F bol .
For objects obeying a r −2 density distribution, the dust mass in the circumstellar component is given by, M d = 4πR 2 in Y(τ 100 /κ 100 ) and the total mass (gas+dust), M gd is approximately equal to the gas mass, M d δ for δ >> 1 (see Sarkar & Sahai [28]). Here, R in is the inner radius of the dust shell inferred from the output of the DUSTY code, Y is the shell relative thickness specified in the DUSTY input (R out /R in ), τ 100 is the shell optical depth at 100 µm, κ 100 is the dust mass absorption coefficient at 100 µm and δ is the gas-to-dust ratio. As in Sarkar & Sahai [28], we assume κ 100 = 34 cm 2 g −1 and δ = 200.
The dust mass 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 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. [29]; Nanni et al. [30]. van Loon et al. [29] used a value of 500 for the LMC. Roman-Duval et al. [31] 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 [32]). A higher value of δ would imply a proportionate increase in the masses of the ejecta (dust and gas) that we derive for our sample of stars.
The dust mass absorption coefficient (κ) is poorly constrained. This may be different for warm ( > ∼ 300 K) and cold dust ( < ∼ 300 K) (e.g., Demyk [33]). κ 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. We uniformly used κ at 100 µm for the cool and warm dust.
The derived parameters from our best-fit models are summarized in Tables 1 and 2 (Table 1) are shown (horizontal lines) on a plot of theoretically estimated dust masses in the ejecta of common envelope systems versus initial stellar mass, taken from Figure 2 of Lü et al. [34]. FGB refers to the first red giant branch. FG2006 refers to results by Ferrarotti & Gail [35], showing the dust masses produced in the dust-driven outflows of AGB stars.

Results
We modeled the SEDs of a select sample of post-RGB and post-AGB objects (eight in each class) in the LMC. Our main conclusions are listed below: • We found that the published classification of these objects as "shell" or "disk" sources was not robust. While shells were present in all sources, our modeling showed that, additionally, the presence of a disk was (a) required in some "shell" sources ( , implying disks with large opening angles (∼41 • ± 6 • ) and hence were geometrically thick structures. The large opening angles appeared to be roughly consistent with the gas density of the ejected envelope as seen in numerical simulations of CEE, relatively soon after CEE occurred (e.g., see Figure 1 of García-Segura et al. [36]). • We derived the total dust mass in the disks and shells and set rough constraints on the dust grain composition and sizes. The shells are significantly more massive than the disks. The dust mass in the disks of post-RGB (post-AGB) sources lies in the range of 5 × 10 −11 -1 × 10 −7 M (5 × 10 −11 -4 × 10 −6 M ) and in the shells from 3 × 10 −7 -1 × 10 −4 M (4 × 10 −7 -3 × 10 −4 M ). • We found evidence that, for some post-RGB sources, the ejected matter may be carbonrich, even though it is expected to be oxygen-rich. For the post-RGB star, J055102.44-685639.1, our modeling revealed that the disk is optically thick (τ = 1.0) and amC-Hn grains provided a reasonable fit to the SED in the near and mid-infrared. The cool outer shell has a combination of warm silicates and silicon carbide. While amorphous carbon has been observed in the dusty disk around C-rich post-AGB stars (e.g., HR4049, Acke et al. [37], 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) dredgeup-events that occur at the centers of AGB stars. 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. • We found systematic discrepancies between the model and observed SEDs in the 5.8-8 µm 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, e.g., PAH molecules have been found in the circumstellar environment of the oxygen-rich red giant, HD233517 (Jura et al. [38]). The presence of PAHs provides independent support for the hypothesis of binary interaction leading to the formation of post-RGB objects. • The amorphous silicate emission features at 10 µm and 18 µm are visible in the model spectra of several of our objects. • Comparison of our model results with predictions of dust mass in the circumstellar ejecta of binary systems wherein the primary is a red giant suggests that CE ejection occurred near or at the tip of the RGB.

Discussion
Our study shows that post-RGB stars have large mid-infrared excesses resulting from the presence of dusty disks and shells. The extreme mass loss required for the infrared excess may be a consequence of CE interaction in binary systems as seen, e.g., in certain AGB stars in the LMC (Dell'Agli et al. [39]). These may be the result of standard CE ejection due to a rapid plunge in of the companion (Paczynski [40]) or a slow inspiral phase (Ivanova et al. [41], Podsiadlowski et al. [42], Ivanova et al., 2013 [43], Clayton et al. [44]).
An alternate scenario to these, proposed recently by Glanz & Perets [45], is that, during the inspiral phase, the CE expands to a very large radius, thus, allowing dust condensation and the formation of a dust-driven wind that evaporates the CE. Some fraction of the ejected mass may fall back and interact with the binary leading to the formation of circumbinary disks (Kashi & Soker [46]).
Dust formation in CE ejecta wherein the primary is a RGB star was studied by Lü et al. [34] and Iaconi et al. [47]. Lü et al. [34] were pessimistic about the difficulty of observing 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 showed that dust is relatively easily detected in post-RGB sources between radii of few ×10 14 and 10 18 cm.
Author Contributions: Conceptualization, R.S.; methodology, R.S. and G.S.; software, G.S.; validation, R.S. and G.S.; formal analysis, G.S. and R.S.; writing-original draft preparation, G.S.; writing-review and editing, R.S and G.S. All authors have read and agreed to the published version of the manuscript.