In the first part of this post you could read about stellar evolution. This text will cover what happens after the unstable giant stages.

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Depending on the mass that the star keeps after losing the outer layers after burning up all the fuel (up to iron), or gravitational collapse didn't provide high enough temperatures and density to start burning heavier elements, stars will end their life as white dwarf (M< 1.4M_☉ ), neutron star (1.4M_☉< 3M_☉) or black hole (M>3M_☉).

Different initial mass leads to different evolution and final stage. Credits: NASA

White Dwarfs

After red giant stadium and losing mass by blowing outer layers, low mass stars - like our Sun, which still have mass up to 1.4M_⊙ will evolve into white dwarfs. When the star burns up its nuclear fuel in hydrostatic equilibrium breaks and the shrinking of the star starts again. If the star can't light up the heavier elements, it will continue to collapse until it is stopped by electron degeneracy pressure. Under the high pressure from the collapse atoms dissolve and matter loses its chemical properties. When dissolved gravitation keeps shrinking the star, compressing electrones and atomic nuclei into a smaller volume - forming the degenerate gas. In this state concentration is extremely high, 10⁸ - 10¹¹ kg/m³. To get the better impression of the extreme conditions here, I will write few densest matters on Earth: mercury ~ 1.36 × 10⁴, gold ~ 1.93 × 10⁴ and platinum ~2.145 × 10⁴.

Free electrons at extreme density and tiny distances between them have relativistic speed. One of the main difference to ideal gas is that pressure of degenerate electron gas is not a function of temperature:

where k'' = constant. Quantum mechanical interaction forces between electrons oppose the gravitational force and stop the collapse. White dwarf stays in hydrostatic equilibrium without any source of energy F_gr=F_QM. Chandrasekhar,brilliant mathematician and astrophysicist, showed that the star evolves into a white dwarf only if it has lower mass than 1.4 solar masses (or progenitor star mass is lower than 8M_⊙). This critical mass M_Ch=1.4M_⊙ is called the Chandrasekhars limit.
Graph showing that more massive white dwarfs have smaller radius. Credit:AlenMcC
Main sequence stars with homogenous structure of ideal gas have larger radius proportional to their mass (R ∝ M^1/3), but this is not the case with the white dwarfs. With more mass gravitational force is stronger, and as a result we have smaller radii.

White dwarfs are hot stars with large density (ranging from 10⁸ kg/m³ on the surface to 10¹¹ kg/m³ in the core), and small radius (R_WD ≥ 0.01R_☉ ≈ 10³ km). Without energy source they cool down, change colour, and emit radiation on longer wavelengths and after the phase of red dwarf they finally become small dark objects we can't see. It is estimated that there are about billion white dwarfs in our galaxy, and we can see only a small fraction in our vicinity.

Neutron Stars

Neutron star is the final phase in the evoultion of a numerous star population with medium mass (1.4M_☉ < M < 3M_☉). These stars go through several stable and unstable red giant phases, producing heavier elements in the core. If iron core is formed it cannot produce nuclear energy further because of negative energy balance (endoenergic fusion). At the temperature of 5 × 10⁹K enormous amount of γ-photons disintegrate iron: Fe⁵⁶+γ → 13He⁴+4n. This photodisintegration results in cooling and immediate implosion - core collapse. Released gravitational potential energy accelerate electrons to ultrarelativistic velocities producing the reaction:

ultrarelativistic electrons dissolve atomic nucleus. Bonding of protons and electrons results in neutron production (which can be packed in significantly smaller volume than atomic nucleus - no Coulomb's repulsion) and neutrinos which leave the star without interactions. Without relativistic electrons to oppose gravity, star continue to collapse even more and concentration in the core reaches 10¹⁵-10¹⁷ kg/m³. At this moment pressure from a degenerate neutron gas oppose gravity and core collapse stops. Compact degenerated core is form while stellar atmosphere is still collapsing, when the layers of atmosphere fall onto the core shockwave is formed and titanic explosion blows the entire atmosphere into surrounding space.

This stellar phenomena that which releases enormous energy is called supernova (this is only one Supernova explosion mechanism). It is very short phase between regular star and compact neutron star that is left as a remnant. Radius of neutron star is about 10km.
Even without energy neutron stars are stable, their gravitational collapse is equalized by quantum-mechanical forces from densely packed neutrons and strong nuclear force between them.

Black Hole

Finally, the most massive stars (M > 3M_☉) cannot fight the catastrophic gravitational collapse, shrinking the massive star into the singularity with infinite density. There are no known physical phenomena to oppose gravity and form the hydrostatic equilibrium like in neutron stars or white dwarfs.

In a strong gravitational field of a collapsing massive star laws of classical physics cease to exist (R → 0, g =GM/R² → ∞). According to Einstein's general theory of relativity, in the gravitational field of a massive collapsing star curvature of space-time is so big that the even light can't escape and the object disappear for the observer.

2D representation of 4D space-time in the region around black hole. Credits: @svemirac (Velibor)

Since the light can't escape the gravitational field, this region of space-time from which we do not have any information is called Black Hole. Depending on the distance from singularity light can bend more or less, or eventually be trapped in the sphere of photons that surrounds the black hole. Schwarzschild provided the first exact solution to critical radius R_g. This is value represent a limit when the star becomes invisible for observation. This radius is called Schwarzschild or gravitational radius:

Boundary region of R_g is Event horizon. Past this point we can't observe anything, and that is the black hole. It is assumed that in the center of the black hole is mass of a collapsed star.

Major contributors to the theory of black holes during 1973-1980 are Y.N. Zel'Dovich and S. Hawking. Their revised theory states that the black hole is not a final phase of evolution of the massive star. Using the uncertainty principle from quantum mechanics they proved that some particles at the boundary of the black hole can escape.

Temperature of the black hole can be calculated using equation below:

where, h - Planck's constant, G - gravitational constant, k - boltzmann constant. We can notice that temperature is inversly proportional to the mass. Black hole with 3M_☉ has temperature just a fraction above the absolute zero, so even if it radiates we cannot detect it because it is weak and submerged into microwave background radiation. By reducing mass over time, temperature of the black hole rises, so does amount of emitted energy, as the result black hole mass loss is accelerated.
Black holes can last very long, but they still have finite life. Black hole will explode and return all the matter from gravitational prison back to the space, where it will continue it cyclic evolution.

Existence of the black holes has never been confirmed by direct observation. Proof of their existence lies in measuring mass of the invisible component in tight binary systems (TBS). If the mass of the invisible component is larger than 3M_☉, and TBS is strong source of X-ray radiation, this object is black hole candidate.
Strong X-ray radiation is emitted when the matter from star in the TBS (composed of BH and another giant star) falls into the black hole, and big spike of x-ray radiation in the form of jets is emitted when the whole star falls into it.

One black hole candidate is in our Galaxy, Cygnus X-1, strong X-ray emitter in TBS with blue supergiant and its invisible companion with mass around 10M_☉.

Fun fact: Black hole representation in the movie Interstellar is the best and most detailed simulation of a black hole up to date. Few scientific papers have been published and PhD dissertation have been written using the data from the movie.

References:

1. Carroll and Ostlie, Introduction to Modern Astrophysics
2. Olga Atancković, Opšta Astrofizika (General Astrophysics)
3. NASA
4. Wikimedia commons