Having spent a long happy life on the Main Sequence, like all things, the death of a star is inevitable, however it is often a catalyst for creating new stars, planets or even life!
The death of a star can take a number of forms, all depending on it's mass.
As the amount of Hydrogen fuel in the core goes down so does the fusion rate and the and the amount of energy generated. Since the star was in thermal equilibrium (energy generated = energy radiated = star stable) a drop in fusion causes lower energy (temperature) in the core and a drop in pressure. This decrease in pressure causes the star to contract slightly, and with the contraction an increase in core pressure. This will cause the temperature to rise and a new hydrogen burning shell in a region previously too cold for fusion. In this new hydrogen burning shell the fusion rate increases as does temperature. This is the stars last gasp as almost all the hydrogen is gone by now. As this happens the star becomes a little brighter and a little cooler and makes a jump on the HR Diagram.
The hydrogen burning shell generates more energy than the core did (it has access to a much larger volume of the star's mass) and the star increases sharply in luminosity and expands in size to become a red giant (or super giant depending on its mass). Even though the star is brighter and produces more energy, its pressure has increased such that its surface area has become very large, and the surface temperature of the star drops into the K and M spectral type regions.
This process can take several million years, after which different mass stars will do their own thing.
A Sun-like star is classified as a star below 1.5 times the mass of the Sun. These will shed the outer layers of via pulsations and strong stellar winds. Without these opaque outer layers the remaining core of the star shines very brightly and is very hot. The ultraviolet radiation emitted by this core ionises the ejected outer layers of the star which radiate as a planetary nebula.
Eventually the star will cool down to a point where it cannot radiate enough ultraviolet radiation to ionize the expanding gas cloud. The star becomes a white dwarf, and the gas cloud recombines becoming invisible.
A white dwarf's mass is around the same mass as when it was on the main sequence, however its volume has shrunk to that resembling the size of the Earth, and as such it is very dense.
Over the next few million years the star will cool to a temperature at which it is no longer visible and become a cold black dwarf. Since no white dwarf can be older than the Universe, even the oldest white dwarfs still radiate at a few thousand kelvins, and no black dwarfs are thought to exist yet.
Like smaller stars, large stars of up to 3 solar masses will burn their fuel and collapse. Unlike small stars who can regain an equilibrium, the huge stars collapse in on themselves under gravitational pressure. Since the stars mass is greater than the Chandrasekhar limit, the outer layers of the star collapse in on the core, the forces holding atomic nuclei apart in the innermost layer of the core suddenly give way. The core implodes under its own mass, and no further fusion process can ignite or prevent collapse.
Under these great pressures a process known as photo disintegration, gamma rays decompose iron into helium nuclei and free neutrons, which absorb energy, and electrons and protons merge via electron capture, producing neutrons and electron neutrinos which escape. During the collapse a new "neutron rich" core is created from the newly created neutrons which is 6000 times the temperature of the Sun's core.
After the supernova, all that is left is a neutron star - a very small, dense and hot mass of neutrons and sometimes a supernova remnant. The gravitational field on the surface is about 2x1011 times stronger than on Earth. Such a strong gravitational field acts as a gravitational lens and bends the radiation emitted by the star such that parts of the normally invisible rear surface become visible.
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