Stochastic Chemical Evolution of Radioactive Isotopes with a Monte Carlo
  Approach
  Short-lived radionuclides (SLRs) with mean-lives $\tau$ of a few to hundreds
Myr provide unique opportunities to probe recent nucleosynthesis events in the
interstellar medium, and the physical conditions in which the Sun formed. Here
we quantify the uncertainty in the predicted evolution of SLRs within a parcel
of interstellar gas given the stochastic nature of stellar enrichment events.
We assume that an enrichment progenitor is formed at every time interval
$\gamma$. For each progenitor, we randomly sample the delay time between its
formation and its enrichment event, based on several delay-time distribution
(DTD) functions that cover a wide range of astrophysical sites. For each set of
$\tau$, $\gamma$, and DTD function, we follow the abundances of SLRs for 15
Gyr, and repeat this process thousands of times to derive their probability
distributions. For $\tau/\gamma\gtrsim2$, the distributions depend on the DTD
function and we provide tabulated values and analytical expressions to quantify
the spread. The relative abundance uncertainty reaches a maximum of $\sim$ 60%
for $\tau/\gamma=1$. For $\tau/\gamma\lesssim1$, we provide the probability for
the SLR abundance to carry the signature of only one enrichment event, which is
greater than 50% when $\tau/\gamma\lesssim0.3$. For
$0.3\lesssim\tau/\gamma\lesssim 2$, a small number of events contributed to the
SLR abundance. This case needs to be investigated with a separate statistical
method. We find that an isolation time for the birth of the Sun of roughly
$9-13$ Myr is consistent with the observed abundances of $^{60}$Fe, $^{107}$Pd,
and $^{182}$Hf in the early Solar System, when assuming $\tau/\gamma\sim3$ for
these isotopes.
