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The term core collapse refers to the center region (core) of some astronomical object failing to resist the affects of gravity on the object's surrounding mass, resulting in a runaway gravitational collapse in of the object's core. Often this occurs when some process within the core, that was providing the resistance to support the surrounding material, ceases. Collapse continues until some other mechanism halts it or a black hole is formed. Two common uses of the term within astrophysics are regarding the center of a star and regarding the center of a globular cluster.
A stellar core collapses when there is insufficient fuel for the fusion providing the energy and heat that maintains the required pressure to support the star's current size. A major collapse occurs after a series of short core collapses each raise the temperature to some point at which an additional fusion reaction burns the product of all the fusion so far. When the nucleosynthesis product is iron, further fusion no longer releases energy, and the failing resistance to collapse is not restored. The high density does induce some further fusion (of the iron) but this fusion absorbs energy rather than releasing it, reducing the core temperature, and reducing the supporting pressure. A much larger core collapse results (iron core collapse), releasing a huge amount of gravitational potential energy. If sufficient energy is captured by the outer portion of the star to produce supernova-scale thermal emission (on the order of a foe), it is termed a core collapse supernova. Much of the energy emitted by the core is in the form of neutrinos, of which the vast majority pass through and escape the star, so in even the brightest such supernovae, most of the energy is emitted as neutrinos. If the collapse is into a neutron star, there is a rebound off the neutron degenerate matter, also sending energy outward.
Globular clusters are dynamically unstable: the stars interact much like the molecules of a gas cloud, but there is some "cooling" (kinetic energy sent away from the center) from lighter stars being thrown out of the central region, a consequence of the wide range of stellar masses. Within the central region, binary stars on average lose some of their binding energy during collisions (i.e., during close encounters with other stars). A central region within which a balance between this released energy and that exported gives the globular cluster some long-term stability and in our view from Earth, a region in the center of the cluster shows constant surface brightness, which we term its core. Eventually, when this binding energy release no longer keeps up with the "cooling", a core collapse results, this area of constant surface brightness shrinking, eventually to (from our view) a point. Stars near the center may be generally no more than a few thousand AU separated.