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What is a quark?
If you want to understand a quark star, you first need to know a little bit about elementary particles and quantum theory. An atom is composed of several different particles that each have unique properties like mass or charge. The center of an atom, the nucleus, is made of particles called protons and neutrons, and determines exactly which element the atom is. These protons and neutrons have similar masses, but while the proton has a positive electric charge, the neutron has no net electric charge. These charges work in the exact same way as magnets, since they both operate under the electromagnetic force. It's possible to think of a proton like a very small magnet, that feels an attractive force with particles of the opposite charge and a repulsive force with particles of the same charge. The electromagnetic force between two objects is carried by a third particle called virtual photons, which have no mass.
Protons and neutrons themselves are made up of even smaller particles called quarks, with one proton or neutron being composed of three different quarks. Each of these quarks feels the effects of the strong force, which is actually quite different from the electromagnetic force. Unlike electromagnetic charge, the strong force acts on quarks through color charge. This misleading name actually has nothing to do with the color of quarks, but more to do with the color theory we all learned in elementary school.
Instead of just two charges, the strong force has three charges: red, blue, and green. There are also corresponding anti-charges called anti-red, anti-blue, and anti-green. In a proton or neutron, three quarks can only come together if there is no net color charge, such as when a red, blue and green quark come together. The same can also happen for the anti-charges. You can even end up with a quark ending up with its corresponding anti-charge to form a particle, like a red and an anti-red. The strong force between quarks is carried through a particle called the gluon, which does have mass.
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Stable stars have something called hydrostatic equilibrium. In the center of a star, fusion reactions take place that want to blow the star apart in a massive explosion, while at the same time the force of gravity is trying to collapse the star. When there is a balance between these two forces, a star reaches hydrostatic equilibrium.
Stars reach the end of their lives on the main sequence when they run out of fuel in their cores, ending this hydrostatic equilibrium. In the most massive stars, the force of gravity will collapse it down to a single point, creating a black hole. In somewhat less massive stars, the force of gravity will cause electrons to collide with protons to create neutrons. This neutron star is basically an enormous nucleus made mostly out of neutrons. Rather than fusion, the force of gravity is counteracted by something called neutron degeneracy pressure, which is caused by the tendency of neutrons to not occupy the same position.
At first it was believed that if a star has enough mass to overcome neutron degeneracy pressure, it would form a black hole. In the 1970s, astronomers have come up with the idea that the force of gravity could overcome the strong force and actually rip neutrons apart, breaking them down to their constituent quarks. You could think of this star as a single enormous neutron or proton that is held together by gravity rather than the strong force. It would be denser than a normal neutron star, and some of the strongest evidence supporting quark stars is the discovery of neutron stars that appear to be too dense.
So far, astronomers have discovered two stars that are candidates to be quark stars. RX J1856.5 3752 and 3C58 are both neutron stars that appear to be too small and too cool according to the laws of physics as we know them. Observations were made in 2002 with the Chandra X-Ray Observatory. Another star, XTE J1739-285 could also possibly be a quark star.
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Strange Exotic States and Compact Stars - Cornell University Library
Neutron Star/Quark Star: http://chandra.harvard.edu/photo/2002/0211/0211_illustration.jpg
Atoms to Quarks: http://web.rollins.edu/~jsiry/atoms_to_quarks.gif