Implications

The advantage of surveying external galaxies with a significant supernova rate is that we can translate our results into estimates of abundances and rates. We scale our rates using the observed supernova rate of $R_{SN}=0.15$ year$^{-1}$ ( $0.05 < R_{SN} < 0.35$ at 90% confidence). As we discussed in PaperI, this is significantly higher than standard star formation rate estimates for these galaxies, but the SN rate is directly proportional to the massive star formation rate rather than an indirect indicator, and similar discrepancies, although not as dramatic, have been noted in other contexts (e.g., Horiuchi et al.2011). In this section we first outline how we will estimate rates, and then we discuss the constraints on analogs of $\eta$Car and the implications of our sample of luminous dusty stars.

We are comparing a sample of $N_{SN}=3$ supernovae observed over $t_{SN}=20$ years to a sample of $N_c$ candidate stars which are detectable by our selection procedures for a time $t_d$. In PaperI we used DUSTY to model the detection of expanding dusty shells and found that a good estimate for the detection time period was

$$t_d = t_w + 66 \left( { 100~\hbox{km~s}^{-1} \over v_e } \r... ...82} \left( { M_e \over M_\odot } \right)^{0.043}~\hbox{years}$$ (5)

for shells with masses in the range $-1 \leq \log M_e/M_\odot \leq 1$ around stars of luminosity $5.5 \leq \log L_*/L_\odot \leq 6.5$ where $t_w$ is the duration of the ``wind'' phase and the second term is an estimate of how long the shell will be detected after the heavy mass loss phase ends. The principle uncertainty lies in the choice of the velocity, $v_e$. If the rate of events in the sample is $R_e$, then we expect to find $N_e = R_e t_d$ candidates.

The transient rate in a sample of galaxies is less interesting than comparing the rate to the supernova rate. Let $f_e$ be the fraction of massive ( $M_{ZAMS}>8\,M_\odot$) stars that create the transients, where $f_e = (M_C/8M_\odot)^{-1.35}$ if we assume a Salpeter IMF (Kennicutt1998), that all stars more massive than $8M_\odot$ become supernovae and that all stars more massive than $M_C$ cause the transients. If each star undergoes an average of $N_e$ eruptions, then the rate of transients is related to the rate of supernovae by $R_e = N_e f_e R_{SN} = F_e R_{SN}$. The interesting quantity to constrain is $F_e = N_e f_e$ rather than $R_e$. Poisson statistics provide constraints on the rates, where $P(D\vert R) \propto (R t)^N \exp(-R t)$ for $N$ events observed over a time period $t$. This means that the probability of the rates given the data is

$$P(R_{SN},R_e\vert D) \propto P(R_{SN}) P(R_e) (R_{SN} t_{SN})^3 (R_e t_d)^{N_c} \exp(-R_{SN}t_{SN}-R_e t_d)$$ (6)

where $P(R_{SN})$ and $P(R_e)$ are priors on the rates which we will assume to be uniform and we have set $N_{SN}=3$. If we now change variables to compute $F_e$ and marginalize over the unknown supernova rate, we find that the probability distribution for the ratio of the rates is

$$P(F_e \vert D) \propto F_e^{N_c} \left( F_e t_d + t_{SN}\right)^{-5-N_c}$$ (7)

with the standard normalization that $\int P(F_e\vert D) dF_e \equiv 1$. For our estimates of $F_e$ we present either 90% confidence upper limits or the value corresponding to the median probability and symmetric 90% probability confidence regions. Note that the probability distribution really just depends on the product $F_e t_d$, so the results for any given estimate of $t_d$ are easily rescaled.