I have been searching for the number of type II (core-collapse) supernovae per unit of stellar mass formed.
It is my understanding that a star must have an initial mass of at least 8 times and no more than 40-50 times the mass of the Sun in order to end its life as a supernova. I can't seem to find the answer in any papers.
About 1 type II supernova for every 100 solar masses of stars formed.
It is a simple calculation once you have decided on the form of your initial stellar mass function (IMF) and the initial mass limits you think are appropriate for stars that end their lives as type II supernovae.
If we call the number of star born per unit mass $\Phi(m)$, then the answer you are looking for is that the number of supernovae per unit mass is: $$\frac{N_{\rm sn}}{M} = \frac{ \int^{m_2}_{m_1}\ \Phi\ \ dm}{\int^{m_{\rm hi}}_{m_{\rm lo}}\ m\ \Phi\ \ dm}\ , $$ where $m_1$ and $m_2$ are the lower and upper mass limits for supernovae progranitors and $m_{\rm lo}$ and $m_{\rm hi}$ are the lower and upper mass limits for the IMF.
We could do a toy calculation assuming the Salpeter IMF with $\Phi = A(m/M_{\odot})^{-2.3}$ for $m>1 M_{\odot}$, where $A$ is just a normalisation constant, and a flatter $\Phi = A (m/M_\odot)^{-1.3}$ for $1.0 > m/M_{\odot} > 0.08$. Other assumed IMFs are possible, but this should do a reasonable job and is similar to that proposed by a number of authors - e.g. Kroupa (2001). We can use $m_1 = 8M_\odot$, $m_2= 40M_\odot$, $m_{\rm lo}=0.08M_\odot$ (the brown dwarf limit) and $m_{\rm hi} = 100 M_{\odot}$. The lower limits are reasonably well understood, the upper limits less so, but their exact value has little influence on the answer because of the steepness of the Salpeter IMF. $$ \frac{N_{\rm sn}}{M} = \frac{ \int^{40}_{8}\ m^{-2.3}\ \ dm}{\int^{100}_{1}\ m^{-1.3} \ dm + \int^{1}_{0.08}\ m^{-0.3} \ dm }\ = \frac{1}{81}\ . $$
i.e. There should be about 1 type II supernova for every 100 solar masses of stars that are formed.
This is in good agreement with observational estimates of the current Milky Way type II supernova rate (2 per century - Hartmann et al. 2006) divided by the current Milky Way star formation rate (about 2 solar masses per year Elia et al. 2023).
