PeV, and the speeds are in units of $10^3$[km s$^{-1}$]{}. While the ejecta speed in the free expansion SNR phase can have a wide range of values, we use a mean ejecta speed of $\sim 10^4 (M/{\mbox{$M_{\odot}\;$}})^{-1/2}E_{51}^{1/2}$[km s$^{-1}$]{} and take the mean duration of the free expansion phase to be $\sim 200 (M/{\mbox{$M_{\odot}\;$}})^{5/6} n^{-1/2} E_{51}^{-1/2}$yr. In the specific case of a young SN shock propagating through the winds of massive stars in the compact cluster , where $n \sim 0.6$[cm$^{-3}$]{}  [see @munoea06], and assuming $\eta_b \sim$ 0.1, we find ${t_\mathrm{acc}}\approx$ 400 yr for a proton accelerated to $\epsilon_{\rm PeV}\sim$ 40, when $u_{s3} \sim 10$ and $u_{w3}\sim 3$. The high SN shock velocity $u_{s3} \sim 10$ is expected in the free expansion stage if the fast ejecta mass is about one solar mass.

Therefore, for SNRs with ages less than ${t_\mathrm{SNR}}\sim 400$yr, one can get ${t_\mathrm{acc}}< {t_\mathrm{SNR}}$ for 40 PeV protons with standard parameters. In this case, at ${t_\mathrm{SNR}}\sim 400$yr, the SNR radius is $\sim 3-4$pc. Furthermore, the hard spectrum expected from CSFs puts most of the energy into the highest energy protons and one can estimate the power in the highest energy neutrinos produced in inelastic [$p\!-\!p$]{}–collisions by the decay 