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Following a supernova explosion a star may turn into a white dwarf, neutron star, black hole, or just a stellar dust & gas leftover.
Excluding the latter case, why and how does the star's core matter stays collapsed, after such an event where matter is burst and scattered in space?
In order to "blow something up" you need to release more energy than its binding energy and have a way of trapping that energy so it can't escape in another way.
At the centre of a core collapse supernovae is a 10 km radius, $1.4 M_{\odot}$ ball of (almost) neutrons. Its gravitational binding energy is $\sim GM^2/R = 5\times 10^{46}$ J.
This is almost exactly how much energy is released by the collapse of the core from a much larger size (i.e. the energy of the supernova is gravitational to begin with) and since some of that energy goes into dissociating iron nuclei and making neutrons (both endothermic processes) and most of the rest escapes in the form of neutrinos, then there can't be enough energy to unbind the core. Only a tiny fraction (1%) of this energy is transferred to the envelope of the original star, which since it has a much larger radius (by at least 5 orders of magnitude), is enough to overcome its gravitational binding energy and blast it into space.
The case of a type Ia supernova (an exploding white dwarf) is quite different. Here the energy source is not gravitational collapse, but from a thermonuclear detonation of all the carbon and oxygen that make up the white dwarf, to form iron peak elements. This exothermic process rapidly releases enough energy to unbind the original star (e.g. see here) and it is completely destroyed.
What's missing from the above explanations is what is really going on that causes any kind of explosion at all.
I'm going to steal from xkcd to help with this:
And here's an article from the Max Planck Institute that talks in depth about the nature of the neutrino aspect:
https://www.mpg.de/11368641/neutrinos-supernovae
Ultimately, when the star is in it's dying moments, it starts emitting neutrinos. A lot of neutrinos... with a lot of energy. Now, I'm sure you're thinking "what would that do... they don't weigh much of anything". But this is literally like being buried in a football stadium with ants... there are so many neutrinos packing so much energy that they literally cause the outer matter of the star to be blown outwards with large enough energy to carry it away from the gravity well of the remaining matter.
Ah... but how does any matter remain? Because close to the center, the gravity well is deepest, and also close to the center any particle (nucleus/neutron) is being bombarded just about equally in all directions by neutrinos... so the total momentum effectively cancels to zero. Some of the matter is moved a bit... but falls back into the very deep gravity well.
I'm sure it would be a sight to behold... for that brief moment before you were vaporized by neutrinos (and all the other energy) at least.
Found the answer on NASA site
The collapse happens so quickly that it creates enormous shock waves that cause the outer part of the star to explode!
This means the core survives the blast somehow
After a supernova explosion, the event might leave a compact object as a neutron star or a blackhole. The object can still accrete materials such as from fall back accretion or its companion star. If the object is a neutron star, it might further collapse into a blackhole.
Note that massive stars in the 50-150 solar mass range can explode in a supernova end leave no core whatsoever, because of a thing called "pair instability".
In a star, there are two opposing forces which usually balance each other Gravity is a force which induces collapse, while radiation pressure from the fusion reactions within resists the tendency to collapse. Small, sun-like stars, when they have used up most of their hydrogen fuel, will start "burning" helium and become red giants. When the helium runs out they will puff off their outer layers in a nova and collapse to form a white dwarf about the size of Earth. These white dwarfs are amazingly dense and heavy, because most of the mass of the original star has been compressed into a comparatively tiny volume. Further collapse is resisted by a force called electron degeneracy pressure.
Stars much larger than the sun will go on fusing elements beyond helium, building up layers of successively heavier elements until they reach iron. Fusion of elements beyond iron requires an input of energy rather than producing any, and the nuclear fires go out, so deprived of support from radiation pressure the outer layers of the star collapse, producing a supernova explosion. Electron degeneracy pressure is not enough to prevent a more drastic collapse than occurs with much smaller stars. According to the mass of the collapsing star, this will either result in the formation of a neutron star, which is like a gigantic atomic nucleus of incredible density about 6 miles across but containing a mass equivalent to several of our suns, or it will collapse further to form a black hole singularity in which matter enters a state not fully understood by science. Our sun, by the way, is 860,000 miles in diameter..
