SED Modeling

We fit the SEDs of the 18 self-obscured stars using DUSTY (Elitzur & Ivezic2001; Ivezic & Elitzur1997; Ivezic et al.1999) to model radiation transfer through a spherical dusty medium surrounding a star and Figure7 shows the best fit models. We estimate the properties of a black-body source obscured by a surrounding dusty shell that would produce the best fit to the observed SED (see Figure6 for an example). We considered models with either graphitic or silicate (Draine & Lee1984) dust. We distributed the dust in a shell with a $\rho \propto 1/r^2$ density distribution. The models are defined by the stellar luminosity ($L_*$), stellar temperature ($T_*$), the total (absorption plus scattering) $V$-band optical depth ($\tau_V$), the dust temperature at the inner edge of the dust distribution ($T_d$), and the shell thickness $\zeta=R_{out}/R_{in}$. The exact value of $\zeta$ has little effect on the results, and after a series of experiments with $1<\zeta<10$, we fixed $\zeta=4$ for the final results. We embedded DUSTY inside a Markov Chain Monte Carlo (MCMC) driver to fit each SED by varying $T_*$, $\tau_V$, and $T_d$. We limit $T_*$ to a maximum value of 30,000K to exclude unrealistic temperature regimes.

The parameters of the best fit model determine the radius of the inner edge of the dust distribution ($R_{in}$). The mass of the shell is

$$M_e = \frac{4 \pi R_{in}^2 \tau_V}{\kappa_V}$$ (1)

where we simply scale the mass for a $V$ band dust opacity of $\kappa _V=100\,\kappa _{100}$ cm$^2~$g$^{-1}$ and the result can be rescaled for other choices as $M_e \propto \kappa_V^{-1}$. Despite using a finite width shell, we focus on $R_{in}$ because it is well-constrained while $R_{out}$ (or $\zeta$) is not. We can also estimate an age for the shell as

$$t_e = \frac{R_{in}}{v_e}$$ (2)

where we scale the results to $v_e=100\,v_{e100}$kms$^{-1}$.

For a comparison sample, we followed the same procedures for the SEDs of three well-studied dust obscured stars: $\eta$Car (Humphreys & Davidson1994); the Galactic OH/IR star IRC+10420 (Jones et al.1993; Humphreys et al.1997; Tiffany et al.2010); and M33's VariableA, which had a brief period of high mass loss leading to dust obscuration over the last $\sim 50$years (Humphreys et al.2006; Hubble & Sandage1953; Humphreys et al.1987). We use the same SEDs for these stars as in Khan et al. (2013). In Table3, we report $\chi ^2$, $\tau_V$, $T_d$, $T_*$, $R_{in}$, $L_*$, $M_e$ (Equation1), and $t_e$ (Equation2) for the best fit models for these three sources as well as the newly identified stars. The stellar luminosities required for both dust types are mutually consistent because the optically thick dust shell acts as a calorimeter. However, because the stars are heavily obscured and we have limited optical/near-IR SEDs, the stellar temperatures generally are not well constrained. In some cases, for different dust types, equally good models can be obtained for either a hot ($>25000\,K$, such as a LBV in quiescence) or a relatively cooler ($<10000\,K$, such as a LBV in outburst) star. Indeed, for many of our 18 sources, the best fit is near the fixed upper limit of $T_* = 30000\,K$. To address this issue, we also tabulated the models on a grid of three fixed stellar temperatures, $T_* = 5000\,K, 7500\,K, 20000\,K$, for each dust type. The resulting best fit parameters are reported in Tables 4 and 5.

Figure9 shows the integrated luminosities of the newly identified self-obscured stars described in Section3.2 as a function of $M_e$ for the best fit graphitic models of each source. ObjectX, IRC$+10420$, M33VarA, and $\eta$Car are shown for comparison. Figure10 shows the same quantities, but for various dust models and temperature assumptions. It is apparent from Figure10 and Tables 3, 4 and 5 that the integrated luminosity and ejecta mass estimates are robust to these uncertainties. The exceptions are N2403-4 and N7793-3. Without any optical or near-IR data, many of the models of N7793-3 are unstable so we simply drop it. The only models having a luminosity in significant excess of $10^6\,L_\odot$ are some of the fixed temperature models of N2403-4. These models have a poor goodness of fit and can be ignored.

One check on our selection methods is to examine the distribution of shell radii. Crudely, we can see a shell until it either becomes optically thin or too cold, so the probability distribution of a shell's radius assuming a constant expansion velocity is

$$\frac{dN}{dR_{in}} = \frac{1}{R_{max}} = \hbox{constant}$$ (3)

for $R_{in}<R_{max}$. An ensemble of shells with similar $R_{max}$ should then show this distribution. Figure11 shows the cumulative histogram (excluding N7793-3) of the inner shell radii ($R_{in}$). The curves show the expected distribution where we simply normalized to the point where $F\left(<R_{in}\right)\simeq0.5$. The agreement shows that our sample should be relatively complete up to $R_{max}\simeq10^{16.5}$-$10^{17}$cm which corresponds to a maximum age of

$$t_{max} \simeq 300\,{v_{e100}^{\,\,-1}}\,\,\,\hbox{years}.$$ (4)

Figure12 shows the age ( $t_e=R_{in}/{{v_{e100}^{-1}}}$) of the shells as a function of $M_e$. We also show lines corresponding to optical depths of $\tau_V=1,10,100$. As expected, we see no sources with very low or high optical depths, as we should have trouble finding sources with $\tau _V<1$ due to a lack of mid-IR emission and $\tau _V\gtrsim 100$ due to the dust photosphere being too cold (peak emission in the far-IR). Indeed, most of the dusty stars have $1 < \tau_V < 10$ and none has $\tau_V > 100$. The large $t_e$ estimate for $\eta$Car when scaled by $v_{e100}$ is due to its unusually large ejecta velocities ($\sim 600$kms$^{-1}$ along the long axis (Smith2006; Cox et al.1995) compared to typical LBV shells ($\sim50-100$kms$^{-1}$, Tiffany et al.2010).