By Dainius Kilda
Throughout their cosmic lives, stars ‘feed’ on the energy produced by nuclear reactions that fuse lighter elements into heavier ones. Nucleosynthesis in the cores of stars calls for infernal temperatures – the inferno starting at around 10 million K. In this heat, matter exists in the state of plasma: a mess of fully ionized atoms, all their electrons unbound from the nuclei. Thermal energy that fusion produces is also what a star needs to sustain the outward pressure in its core to remain stable against the gravitational force. Just like in a normal gas, the pressure in a stellar plasma comes from ions and electrons whizzing and bouncing around at high speeds, and becomes more intense as they are heated.
After about ten billion years of shining glory, a star like our Sun will exhaust its nuclear fuel and become unable to support itself against its own gravity. It will end its life by shedding the outer layers, and the dead core will succumb to a rapid gravitational collapse, as if being crushed under its own weight. Fortunately, the quantum nature of its tiny constituents – electrons – saves the star from impending catastrophe. On quantum scales, electrons behave as particles and waves (known as de Broglie waves) at the same time. When packed closely together, de Broglie waves of electrons begin to overlap and start interacting in a quantum way. Electrons belong to a class of particles called fermions, so their way of interaction needs to comply with the rules of a fermionic society. To see how this works, imagine fermions inside a box where they can only occupy specific quantized energy states – which is exactly what electrons experience in real matter. But fermions are socially awkward beings who avoid each other: two fermions cannot occupy the same quantum state in the same volume simultaneously. This is the Pauli exclusion principle, one of the fundamental principles of quantum mechanics. The very same exclusion principle guides electrons in atoms to form well-organized orbitals, leading to the chemical elements and the periodic table.
Let us try cramming more and more electrons into the box. Once the low-lying energy states are filled, the new electrons are forced into increasingly higher energy levels due to the Pauli exclusion. In real space, the increasing kinetic energy of electrons corresponds to their de Broglie wavelengths becoming shorter, which helps them avoid standing on top of each other. The higher the kinetic energy, the more violently electrons will be jostling against one another. This creates a tremendous pressure on the walls of the box – called degeneracy pressure – resisting further compression at any cost. Fermionic particles can be very protective of their personal space! We can picture this scenario as a social experiment with people elbowing their way through a crowded airport, and the more people we squeeze into the airport, the more intense the elbowing will become.
As the star continues to collapse under its gravity, the degeneracy pressure grows because electrons gain more kinetic energy for their resistance. Eventually, it becomes strong enough to balance the gravitational force and halt the collapse – this takes a pressure that is about 1017 times higher than the atmospheric pressure on the Earth! The stellar core finally settles down to equilibrium as a white dwarf – a hot glowing ball of ultra-dense quantum fluid with a surface temperature of around 105 K, cleverly named ‘degenerate matter’. A white dwarf is nothing but a corpse of a star, which has a mass close to that of the Sun (1030 kg) squeezed within the size of the Earth. It is no surprise that this came as a shock to astronomers when they discovered the first white dwarfs in the early twentieth century. They appeared to possess densities that amounted to having a ton in a cubic centimetre, suggesting a state of matter far beyond the imagination of the day. A.S. Eddington, a prominent English astrophysicist wrote in 1927:
‘We learn about the stars by receiving and interpreting the messages which their light brings to us. The message of the Companion of Sirius when it was decoded ran: “I am composed of material 3,000 times denser than anything you have ever come across; a ton of my material would be a little nugget that you could put in a matchbox”. What reply can one make to such a message? The reply which most of us made in 1914 was—”Shut up. Don’t talk nonsense”‘. 
Unlike a normal gas pressure, degeneracy pressure is an entirely quantum effect oblivious to temperature. A white dwarf thus remains stable forever even as it slowly cools down like an unplugged iron, radiating away its heat during the billions of years of stellar corpse decay.
White dwarfs have another way of defying our common sense. We usually expect that adding mass to an ordinary material will make it bigger. But in a degenerate matter, extra mass causes gravitational compression, which pushes electrons even closer together forcing them to even higher energy states, until their degeneracy pressure can counterbalance the gravity again. As a result, increasing the mass of a white dwarf actually makes it shrink.
This shrinking might make us feel a little uneasy – it goes against our intuition that where increasing mass makes objects bigger. To what extent is it possible to crush things, relying on the Pauli principle to prevent them from breaking apart? There are limits to everything, and degenerate matter is no exception. When a core left by a dying star exceeds 1.4 Solar masses, electron degeneracy pressure is helpless against gravity and can no longer prevent the collapse. The immense compression squeezes electrons into the nuclei where they combine with protons to form neutrons. Luckily again, neutrons are also fermions that obey the Pauli exclusion principle but can tolerate a much smaller amount of personal space. Packed together with much closer separations, they are capable of exerting a far stronger degeneracy pressure and, once again, arrest the collapse before it is too late. The resulting dead star will be composed of neutrons forming a degenerate matter similar to that of electrons in white dwarfs, except with a density and degeneracy pressure reaching levels far more extreme than before. A neutron star may have only a dozen miles across and a surface area just enough to build a big city but hold a few Solar masses squished inside.
We might be lured into believing that nature can continue crushing matter into ever more exotic states as we look at increasingly massive stars leaving behind increasingly massive dead cores. But there is only one more step left to push matter over the brink of a final meltdown. The stellar cores more than three times as massive as the Sun experience gravitational force that no degeneracy pressure or other known effect can withstand. This time, the star is crushed completely, collapsing to form a black hole: that endless source of science fiction stories whose ultra-high density curves spacetime to the point that not even light could escape. The interior of black holes, with all the atoms and elementary particles supposedly destroyed, remains a mystery to contemporary physicists, and it marks the ultimate borderline for extreme densities that one can reasonably perceive as matter.
Exploring matter under extreme conditions has been at the frontiers of modern physics – a discovery of new exotic states often bringing new breakthroughs. Stellar corpses offer us a glimpse of an overwhelmingly dense quantum world, far beyond what we can hope to produce in our terrestrial labs – thus allowing us to push the boundaries of quantum physics.
 Eddington, A. S. (1927). Stars and Atoms. Clarendon Press, Oxford, p. 50.
Fig.1 Star life cycles, cmglee, NASA Goddard Space Flight Center. Licensed under CC BY-SA 4.0 via Wikimedia Commons. Derived from: http://imagine.gsfc.nasa.gov/teachers/lessons/xray_spectra/images/life_cycles.jpg
Fig.2 Sirius A and B artwork, NASA, ESA and G. Bacon (STScI). Public domain image. Source: http://www.spacetelescope.org/images/heic0516b/
Fig.3 Neutron star cross-section, NASA. Public domain image. Source: http://science.nasa.gov/newhome/headlines/sgr_slides/N_sctn2.jpg