What is the closest distance for an undiscovered old neutron star and how long would it take to get here? To help people scared of neutron stars This is to help people who are scared of neutron stars. My questions are


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*What is the faintest a neutron star could be

*How close could it be and not yet be detected

*How long would it take to get here if it was traveling as fast as the fastest known stars.


I am using these figures at present


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*Absolute magnitude 23

*a third of a light year for a 10" telescope limiting magnitude 13, and around 13 light years for the GAIA star survey

*Over 80 years for 10" telescope easy visibility and over 8,000 years for the GAIA star survey


They are scared because of an absurd YouTube video with over a million views, by someone going by the name of TopMan 2.0. He names a particular neutron star. It's utterly absurd, it's an estimated 424 light years away and he claims it will hit us 75 years from now based on a spanish blogger who in turn got it from a National Geographic fictional over the top absurd "documentary" which didn't name any star as it was a made up scenario acted out by actors. 
Children as young as 7 are posting comments to his video, saying they are afraid they are going to die. I tried talking to him, but he is not going to add anything to the video to say that it is fake. He must be getting a fair bit of ad revenue from it. Even when I explain how absurd it is the people who contact me are still scared of neutron stars.
This is part of my work trying to help people scared of these absurd fake doomsdays. Often they are getting help from therapists for the extreme fear and anxiety generated by these videos. Ordinary folk who flunked physics, and see one of these videos on the web, and then the fear overwhelms them and they get such severe anxiety they no longer think clearly and panic many times a day.
Background and calculation here.
Anyway, it would help them to know that we could detect a faint neutron star if there was one close by. I thought members here might be interested in the question. The main uncertainty is for 1 how faint a neutron star can be, I'm basing it on Rob Jeffries' answer here, assuming temperature of Sirius, but I'm not sure if that is the faintest it could get, if it is a billions of years old neutron star in our stellar neighbourhood.
 A: There are all sorts of things to consider here and I doubt there can be a definitive answer.
First: how many neutron stars are there - or more pertinently, what is the density in the solar neighbourhood.
There are about 1000 stars within 15pc of the Sun down to about $0.2M_{\odot}$. Most of these are main sequence stars that are less massive (and more long-lived) than the Sun, with odd exceptions like Sirius and Arcturus. About 10% are white dwarfs that have evolved from objects with initial masses of $1-8M_{\odot}$. If we assume there are 900 stars within 15pc that were born with $M\leq 1M_{\odot}$, then we can integrate an assumed initial mass  function and further assume that all stars with $8\leq M/M_{\odot}<25$ have already ended their lives as neutron stars. The lower limit is fairly solid; the upper limit is much more uncertain, but because of the steepness of the initial mass function ($N(M) \propto M^{-2.3}$) it doesn't really change the numbers of neutron stars much (but does change the [small] numbers of black holes!).
Thus the fraction of stars that end up as white dwarfs would be
$$f_{\rm WD} \sim \frac{\int^{8}_{1} M^{-2.3}\ dM}{\int^{25}_{0.2} M^{-2.3}\ dM} = 0.11,$$
which ignores the negligible contribution of even higher mass stars to the total. This is in reasonable agreement with observation, but will be slightly overestimated because not all stars more massive than $1M_{\odot}$ have died.
Armed with some confidence that this calculation works for white dwarfs, we can do the same calculation for neutron stars.
$$f_{\rm NS} \sim \frac{\int^{25}_{8} M^{-2.3}\ dM}{\int^{25}_{0.2} M^{-2.3}\ dM} = 0.006.$$
i.e. If there are 1000 total stars within 15pc, there should be 6 neutron stars.
This is likely to be an overestimate because a large fraction of neutron stars are created in supernovae and obtain a large momentum kick that can give them velocities of hundreds of km/s. That means they should be under-represented in the Galactic disc and some will have been ejected from the Galaxy. This is probably a factor of $\sim 2$ effect.
See also https://astronomy.stackexchange.com/questions/16678/how-far-away-is-the-nearest-compact-star-remnant-likely-to-be?noredirect=1&lq=1 for a similar calculation, where i used some slightly different assumptions and numbers (which gives a flavour of the uncertainties involved).
Second: Could we actually see these nearby neutron stars? Now, the age distribution is likely to be reasonably uniform over the age of the Galaxy or perhaps even weighted to older ages. Neutron stars lose their original "birth heat" on timescales of thousands to millions of years, mostly by the emission of neutron stars. By the time they get to a million years old they can only be kept hot by accretion from the interstellar medium or perhaps by some sort of Ohmic heating driven by the decay of their magnetic fields or frictional processes associated with their spindown and decoupling between superfluids and "normal" fluids in the crust and core. 
Of these reheating mechanisms, accretion from the interstellar medium is probably not important for neutron stars close to the Sun, because our local ~100 pc is a local bubble of hot and relatively sparse interstellar gas. But the other processes are very uncertain and until we start detecting the thermal radiation from old neutron stars, we just don't know how luminous they will be. 
In Position of Neutron Stars in H R diagrams I gave an estimate of $M_{v} \sim 23$ for the absolute visual magnitude of the neutron star surface has cooled to 10,000K or so, but I would say this could be uncertain by a factor of a few either way, which results in orders of magnitude uncertainties in their luminosities and so about $\pm 5$ magnitudes on $M_v$!
If the absolute magnitude was $<20$, then there is a chance that Gaia might detect one of those $\sim$ few old neutron stars within 15 pc. It would have a large parallax, probably a large proper motion, and a calculation of its luminosity compared with its temperature would quickly reveal it was much smaller than a hot white dwarf. On the other hand, if $M_V>20$ then there is virtually no chance that the Gaia survey would spot it, because that is about its apparent magnitude sensitivity limit. So unless one of those few nearby neutron stars was closer than a few pc it would just be too faint to see. The Large Synoptic Survey Telescope, due to start operation in 2023, should survey the sky to much fainter limits and really does stand a chance of detecting a population of these objects.
Third: I should point you to another possibility that I addressed (and dismissed) in https://astronomy.stackexchange.com/questions/16578/will-gaia-detect-inactive-neutron-stars/16699#16699 That is that Gaia might see the gravitational lensing of background stars by a foreground and nearby neutron star. In the answer referred to, I showed that this is possible but rather unlikely.
In conclusion there is little reassurance to be given. The likelihood of a neutron star disrupting the solar system is very low and since they are orders of magnitude less common than "ordinary stars" (which would do just as much damage!) and none of these ordinary stars show much likelihood of coming nearer than 10,000 au to us in the the forseeable millions of years (e.g. Bailer Jones 2018) it would be unfortunate in the extreme to be struck by something that is much rarer in the next 100 years. Conspiracy theorists and other wackos should focus on the very much more real threats of global warming and anti-biotic-resistant bacteria, rather than claiming that something we see $>400$ light years away can reach us in 75 years...
