The closest a neutron star can get to a black hole is determined by the black hole's event horizon, which is the boundary beyond which nothing, including light, can escape its gravitational pull. The distance between the neutron star and the black hole would depend on several factors, including the mass and spin of the black hole, as well as the velocity and trajectory of the neutron star.
If a neutron star is on a direct collision course with a black hole, it will eventually cross the event horizon and be consumed by the black hole's immense gravitational forces. This point of no return is known as the "tidal disruption radius" or the "Roche limit." Inside this radius, the tidal forces become so strong that they can tear the neutron star apart.
The exact value of the tidal disruption radius depends on the mass and size of the black hole, as well as the structural properties of the neutron star. Typically, for a non-rotating black hole, the tidal disruption radius is around three times the event horizon radius. For rotating black holes, the situation can be more complex due to the effects of frame-dragging.
It's important to note that when a neutron star gets very close to a black hole, it experiences extreme tidal forces and gravitational interactions, leading to the emission of gravitational waves and intense radiation. This scenario is known as a "tidal disruption event" and can be a significant source of observable phenomena for astronomers studying these cosmic events.