Is there any information about the future of the neutron star? What I am interested what happens to it as the time passes. Say a billion years. Would it still be a neutron star? Would it collapse to black hole? Would it simply dissipate?
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A newly formed neutron star cools (via neutrino emission) to an effectively "zero-temperature" configuration in a matter of a few minutes.
What is meant by that is that it contracts to a radius where the separation between its neutrons is so small that neutron degeneracy pressure and the pressure associated with mutual repulsion between neutrons (from the strong nuclear force) becomes large enough to support the star, and since neither of these depend on temperature, the star can continue to cool off without any change in radius. In other words, the radius it has when it is at an interior temperature of $10^{10}$K (and a surface temperature of about $10^{8}$K), a few minutes after formation, and the radius it has for any cooler temperature are essentially the same. Hence neutron stars do not contract, become denser and collapse into black holes as they cool.
A neutron star cools for the first million years or so by various neutrino emission mechanisms in the interior. Thereafter, at a surface temperature of about 100,000 degrees, the dominant cooling is radiation from the surface, which then proceeds on timescales of only millions of years. The neutron star is then thought to get colder and colder (NB no old, cold neutron stars have ever been observed, but they are so small and so faint at low temperatures that they could never be found with current technology, but there should be about a billion of them out there in our Galaxy).
The future of the neutron star is therefore to be a fading cinder. The vast majority of the billion or so neutron stars thought to exist in our galaxy are probably at temperatures of around 1000-10,000 degrees, and probably maintained at that temperature by the accretion of small amounts of gas from the interstellar medium. They may also be reheated by Ohmic dissipation of their magnetic fields or viscous dissipation of their rotational kinetic energy.
Now, it is possible, that if a neutron star accreted enough material, then at some point in the future it could exceed the maximum possible mass for a neutron star and become a black hole. But most neutron stars appear to be born at masses well below this limit and it would likely take many galaxy lifetimes to accrete enough material for them to make that transition. Neutron stars born into binary systems may experience more rapid accretion. This accretion is observed (in X-rays) and could result in black hole formation on shorter timescales.
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$\begingroup$ Thank you for the answer. So it means, if there is no accretion, they will eventually continue loosing mass and cooling... until they vanish completely? $\endgroup$– Amiga500Sep 15, 2017 at 7:09
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1$\begingroup$ I don't fully understand your use of terminology. You say it cools quasi-immediately to "effectively zero temperature", and that then cools further to a surface temperature of 100,000K, so there's some (understandable) mismatch there. $\endgroup$ Sep 15, 2017 at 16:14
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1$\begingroup$ @EmilioPisanty Zero-temperature means that the gas can be treated as completely degenerate. This actually happens when the neutron star temperature drops below about 10 BILLION degrees. i.e. The difference in it's radius between when it has a temperature of $10^{10}$K and when it has cooled to any colder temperature (even absolute zero) is negligible. $\endgroup$– ProfRobSep 15, 2017 at 17:31
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1$\begingroup$ @EmilioPisanty I use zero temperature as shorthand for completely degenerate. Once degenerate, a change in T has no structural effect. Neutrino losses go as $T^8$, photon losses go as $T^4$, so completely dominate at low temperatures. The notion of a timescale is a bit arbitrary. Obviously at colder temperatures, the NS has much less thermal energy to get rid of, so that is why the cooling timescales can be comparable even though the losses are smaller. $\endgroup$– ProfRobSep 15, 2017 at 20:01
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1$\begingroup$ Sure, I get the shorthand but it's nontrivial to grasp even for a technical audience, so it could use a bit more explanation just to ease the reader in. +1 anyways, thanks for sharing. $\endgroup$ Sep 15, 2017 at 20:15