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Stars are formed from the condensation of large clouds of gas to form protostars, these are stars that have not begun thermonuclear fusion. The clouds from which these protostars are formed are largely hydrogen, however, there may also be other elements present which originate from material generated in previous stars that shed their contents into space. Stars produce their energy through the thermonuclear fusion of atoms, initially this will largely be through the thermonuclear fusion of hydrogen. Different processes occur at different stages of a stars life which depend on composition, temperature and pressure/density. These factors are influenced by the size of the star to start with, and so some stars will never have certain processes occurring within them. Below is a table which shows various processes, the fuel required, temperature for the process to occur and the minimum mass (relative to the sun) required to reach the appropriate temperature and pressures.
As the protostar contracts under gravity, the pressure and temperature at the core increases. When the temperature is greater than about 1 x 107 K thermonuclear fusion can start. The first process to start will be hydrogen burning through what is known as the proton-proton chain. There are three processes typically described known as proton-proton I (PP I), PP II and PP III.

PP I is the main process, in the Sun it is about 85%, PP II contributes to about 15% and PP III to about 0.02% of the proton-proton chain processes.

The processes are shown below:

The overall reaction from the PP I process is the conversion of four 1H into one 4He. The difference in atomic mass is about 0.028 mass units, which by using Einstein's equation E = mc2 can be converted into the equivalent energy released.

In the presence of heavier elements generated in previous stars , particularly carbon and nitrogen, a catalytic sequence can occur known as the CNO cycle. However, because the coulombic energy barriers are much higher (about 6 or 7) than for the PP chain it requires a higher temperature to significantly get going (about 1.5 x 107K). Above about 1.8 x 107 K the CNO cycle can predominate if there is sufficient carbon and/or nitrogen. In the Sun it accounts for about 15% of the energy conversion (coulombic forces are charge-charge interactions, in this case the energy barrier is from the repulsive proton-proton force).

The CNO cycle is shown below. Notice that one carbon and four hydrogen nuclei go in and one carbon and one helium comes out, the carbon can then re-cycle, hence the catalytic nature of this process.

As the helium is produced it becomes the major component of the core, helium concentrates at the centre of the core, which leaves a shell of hydrogen round the core burning. The temperatures and pressures at this stage are not high enough in low mass stars for helium fusion to occur. The star enters the red giant phase of its life, the outer layers expand and drop in temperature. The core contracts, and in stars greater than two or three solar masses there is a gradual onset of He fusion. In stars less than this there is a point where what is known as the Helium Flash occurs, which is a rapid onset of the He fusion process. At this point the red giant has a hydrogen burning shell with a helium burning core. Helium fusion is also known as the triple alpha process, it requires three He to generate one C.
At the temperatures required for this reaction to occur (108 K) other reactions can occur:

12C + 4He converted to 16O + Gamma ray

16O + 4He converted to 20Ne + Gamma ray

20Ne + 4He converted to 24Mg + Gamma ray

This generates a core consisting of C,O and Ne.

In higher mass stars when the temperatures reach 5-8 x 108 K carbon burning can take place

The sequence can continue to form 32S, 36Ar and 40Ca. The sequence beyond 40Ca is a bit more complicated with positrons and neutrinos involved, but leads to 44Sc and 48Ti.

Through a combination of the above processes a large star can synthesise the elements up to iron. Beyond this requires the force of a supernova explosion. These stars will have started off with more than 8 solar masses when they were first formed.

At various points in the history of a star, material can get dredged up from deeper layers an brought to the surface. For example, this has happened in stars known as carbon rich stars where significantly raised levels of carbon can be seen spectroscopically in the outer layers. In this case it happens to stars greater than about two solar masses during the AGB phase.

The H and He generated can then go on to take part in other reactions which form different nuclei.

In stars less than 4 solar mass units (mass at formation) they never reach temperatures and pressures high enough to proceed beyond helium burning. Eventually these stars burn out their cores and now have a helium burning shell surrounded by a hydrogen burning shell. At this point the star is moving into the second red giant phase on the Asymptotic Giant Branch (AGB) phase. The outer layers are shed in high winds and eventually what is left is the bright central star in a planetary nebula.

Stars larger than this undergo further contraction of the core after He and C depletion and the temperature rises further. When temperatures greater than 109 K are reached the alpha process can occur. The first step is the endothermic reaction of neon with a gamma photon to create 16O and a very highly energetic 4He. This particle has enough energy to break the coulombic barrier of neon nuclei and a series of other combinations is then possible as shown. The subsequent reactions after the initial reaction are exothermic.

PP I process
PP II process
PP III process
The CNO cycle
The triple alpha process
Carbon burning
The alpha process