# How common are neutron stars?

I thought this would be a rather easy question to answer, but it doesn't seem to be addressed anywhere. I haven't even been able to find a paper, article, or page that says it's unknown how common they are. I understand neutron stars, not in the form of pulsars/magnetars would be difficult to detect, but their precursors, 10-29 solar mass stars should be rather easy. Then, couldn't we infer from there? My ultimate question is the ratio of black holes to neutron stars as a result of core collapse.

• "There are thought to be around 100 million neutron stars in the Milky Way, a figure obtained by estimating the number of stars that have undergone supernova explosions." en.wikipedia.org/wiki/Neutron_star – probably_someone Nov 20 '18 at 14:52
• @probably_someone That should be an answer! – user191954 Nov 20 '18 at 17:29

I guess that using the initial mass function (IMF) would yield a rough estimate of the number of neutron stars in a region.

The initial mass function $$\xi (m)$$ describes the inital mass distribution of a population of stars, so if you make an assumption on the mass of stars that will evolve into neutron stars, you would be able to determine an estimate of the number that you are looking for.

For example, you may assume that stars which initial mass lies between $$M_1 = 8~M_{\odot}$$ and $$M_2 = 40~M_{\odot}$$ are very likely to evolve into neutron stars as a result of core collapse. Then, the number $$N$$ of neutron stars that you would find in a region is given by:

$$N = \int^{M_2}_{M_1} \xi (m) dm$$

Where $$\xi (m)$$ is the initial mass function that has been measured in that particular region (please note that studies tend to show that the IMF may be universal).

However, the result will obviously depend on the choice of $$M_1$$ and $$M_2$$, and thus on our knowledge of the mechanisms of formation of neutron stars. Moreover, I doubt that this method is really precise, as it does not take into account every scenario that could lead to the formation or destruction of a neutron star (like the coalescence of two neutron stars that would end up in the creation of a black hole, or the formation of a new neutron star by accretion on a white dwarf in a binary system)

Just my thoughts here, please feel free to comment.

• This would give the estimated number of stars within those mass range. You'd have to make some more assumptions to get the number of neutron stars from this (e.g., SNe rates, avg lifespan, etc). – Kyle Kanos Nov 20 '18 at 17:34
• Given that 8 $M_\odot$ stars have a lifespan of about 55 million years, most that have formed over the past 10 billion years have died already, and the heavier stars even more so. Correcting for this is likely smaller than the other, messier uncertainties. However, I think just using even the Salpeter IMF and these cut-offs likely give a decent order of magnitude estimate. – Anders Sandberg Nov 20 '18 at 18:14