Rate Limits

One advantage of searching for eruptions in the dust obscured phase is that the process is relatively easy to simulate. We eject $M_e$ of material from a star of luminosity $L_*$ and temperature $T_*$ at velocity $v_e$ over time period $t_e$ and assume it forms dust with total (absorption plus scattering) visual opacity $\kappa_V$ once it is sufficiently distant from the star. We can then use DUSTY (Elitzur & Ivezic2001) to simulate the evolution of the mid-IR luminosities and determine the time $t_d$ during which the source would satisfy our selection criteria. Here we use $\kappa_V=84$ cm$^2~$g$^{-1}$, roughly appropriate for silicate dust, but this is important only to the extent that the ejecta mass can be rescaled as $M_e \propto \kappa_V^{-1}$. The key variable for estimating rates is the expansion velocity $v_e$, because the detection period scales as $t_d \propto v_e^{-1}$. The velocities cited for the supernova impostors (e.g., Smith et al.2011) and the velocity associated with the long axis of $\eta$ Car are high, $v_e \gtrsim 500$ kms$^{-1}$. These velocities are very different from those observed for the older, massive shells in the Galaxy or the shorter axis of $\eta$ Car, where $v_e \lesssim 100$ kms$^{-1}$ (see the discussion of this difference in Kochanek2011a). Here we scale the results to $v_e=100$ kms$^{-1}$ since, for example, it results in our detecting systems with parameters similar to $\eta$ Car at its present age, as observed, and agrees with the expansion velocities of the other massive Galactic shells around luminous stars.

Detection of a shell at late times ($t_d \gg t_e$) is limited by its optical depth and temperature. The shell has total visual optical depth greater than $\tau_V$ for

$$t(\tau_V) = \left( { M_e \kappa_V \over 4 \pi v_e^2 \tau_V}... ...ft( { 100~\hbox{km\,s$^{-1}$} \over v_e }\right)~\hbox{years},$$ (4)

and once $\tau_V <1$ it begins to rapidly fade in the mid-IR. Ignoring Planck factors, the spectral energy ( $\lambda L_\lambda$) peaks at

$$\lambda = { h c \over 4 k T_d } \simeq 2 \left( {L_* \ove... ... v_e \over 100~\hbox{km\,s$^{-1}$} }\right)^{1/2}~\mu\hbox{m},$$ (5)

so the emission peak shifts out of the IRAC bands after several decades, and our survey is primarily limited by the shift of the emission to longer wavelengths rather than the declining optical depth. It is better to search for these sources at 24$\micron$ as has been done in the galaxy (Gvaramadze et al.2010; Wachter et al.2010) but that would require the resolution of JWST (Gardner et al.2006). A reasonable power-law fit to the results ( $-1 \leq \log{M_e/M_\odot} < 1$, $5.5 < \log(L_*/L_\odot) < 6.5$) of the DUSTY models is that the detection period is

$$t_d \simeq t_e + 66 \left( { 100~\hbox{km$s^{-1}$} \over ... ...82} \left( { M_e \over M_\odot }\right)^{0.043}~\hbox{years}.$$ (6)

For $M_e\simeq 10$M$_\odot$ and L $_* \simeq 10^{6.5}$L$_\odot$ like $\eta$ Car, $t_d \simeq t_e + 190 (100~\hbox{km$s^{-1}$}/v_w)$ years where $t_e$ may also be $50$ years or more (see the discussion in Kochanek et al.2012a). For present purposes, we adopt $t_d=200$ years as the period over which our selection criteria would identify an analogue of $\eta$ Car, consistent with the fact that our selection criteria do identify $\eta$ Car.

We can normalize the rate of eruptions to the ccSN rate as

$$R_{erupt} \simeq 0.1 \left( 40 M_\odot \over M_{erupt} \right)^{1.35} N_{erupt} R_{SN} = f_\eta R_{SN},$$ (7)

where $R_{SN}$ is the supernova rate and all stars more massive than $M_{erupt}$ undergo $N_{erupt}$ eruptions. Following the rate arguments in Kochanek (2011a), we can estimate the number of eruptions per massive star needed to explain the massive Galactic shells. If there are $N_{shell} \simeq 10$ massive Galactic shells associated with massive stars ($M>M_{erupt}$), then

$$N_{erupt} \simeq 2 \left( { N_{shell} \over 10 }\right) \l... ...)^{1/2} \left( { v_e \over 100~\hbox{km\,s$^{-1}$} } \right),$$ (8)

where $\tau_V = 0.01$ is the minimum optical depth needed to detect a shell surrounding the star and $R_{SN,MW}\sim 1$/century is the Galaxy's supernova rate. Since the Galactic shells are identified as shells primarily at $24\mu$m, they are easier to find at low optical depths and temperatures than in our extragalactic survey. Thus, the massive Galactic shells imply an eruption rate relative to the supernova rate of $f_\eta \gtrsim 0.2$ since it is unclear whether we possess a complete inventory. Note that with this normalization the rate estimate does not depend on the mass scale $M_{erupt}$.