The sun is mostly composed of hydrogen and helium, which are the most abundant elements in the universe and were formed shortly after the Big Bang. The sun also contains a small fraction of heavier elements such as carbon, oxygen and iron. These elements did not exist early in the history of the universe. They were formed inside stars, where temperature and pressure are so high that atoms merge to form bigger ones through powerful fusion reactions. Throughout their life or when they explode at the end of their life, stars disseminate these elements in space. The space between stars is thus far from empty and contains atoms, molecules and ions, although at very low densities. When they assembled, both the sun and the earth included some of these elements formed in previous stars. In fact, we are all made of this stardust!
There are two main forces at work in nuclear fusion: the electromagnetic force and the strong nuclear force. The repulsive electromagnetic force between positively-charge nuclei is long-range but relatively weak, while the attractive strong nuclear force is short-range but strong. When two nuclei are far enough apart, the repulsive electromagnetic force dominates, holding the nuclei apart. As the two nuclei get closer, the electromagnetic repulsion gets stronger and it gets harder and harder to push the nuclei together. When the two nuclei get close enough, the attractive short-range nuclear force dominates and the two nuclei stick together to form a new nucleus. For this reason, it takes a lot of pressure to push nuclei close enough that they fuse together.
In principle, any two nuclei can be fused into a single nucleus. However, it is the easiest to fuse (and the most energy is released from) nuclei that have little electromagnetic repulsion because they have little electric charge. The nuclei with the least electric charge are the lighter elements, such as hydrogen and helium. In stars, most of the fusion taking place is hydrogen fusing with itself or with other light elements. Since gravity is what provides the pressure to ignite nuclear fusion in stars, and since gravity is caused by mass, all you need is a big enough mass of hydrogen in order to end up with burning stars. There is very little oxygen in stars. The oxygen that is there was created by hydrogen fusing repeatedly until it made the oxygen.
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