As is true with most scientific discoveries, the discovery and understanding of nuclear fusion processes in the Sun required many different stages involving many different scientists.
First: Nuclear fusion is the process by which light elements are fused together to form heavier elements. For instance, the Sun–like all main-sequence stars–generates energy by fusing hydrogen into helium. During these reactions, a minute amount of the initial matter is transmuted into energy in accordance to what is arguably the most famous equation in physics:

This equation arose out of Albert Einstein’s Special Theory of Relativity (1905), which established the matter-energy equivalence principle, much to the shock and chagrin of the physics community, which had previously regarded them as separate.
In 1920, British physicist Francis Aston (1877–1945) determined that four hydrogen atoms were slightly more massive than one helium atom, which suggested that hydrogen could be combined to form helium while some the initial material could be transmuted into energy.
Sir Arthur Eddington (1882–1944), the foremost relativity expert at the time (apart, of course, from Einstein), then proposed that the Sun could have generated its energy through the proton-proton chain that fuses hydrogen into helium. By the mid 1920s, scientists had realized that the Sun’s age had to have been between 20 – 100 million years old. Gravitational contraction, the mechanism which had been presumed to have been responsible for the Sun’s energy generation, had been dismissed because the outer layers of the Sun would have had to have extended out to Earth around the time of its formation. In 1926, Eddington published his theory in the now famous publication The Internal Constitution of the Stars.
Eddington proposed nuclear fusion as the mysterious mechanism because such reactions produce prodigious amount of energy. Also, the Sun’s mass, which had been established by Henry Cavendish in the late 18th century, was known to have been sufficient to sustain such reactions over immense time scales. Even though we now know that the Sun must be around 4.6 billion years old, the fusion reaction rate and the available core fuel material would enable nuclear fusion to power the Sun not only for those 4.6 billion years, but also for billions of more years in the future.
In 1939 Hans Bethe (1906–2005) published the paper Energy Production in Stars, in which he showed that the Carbon-Nitrogen-Oxygen reactions were of vital importance to powering the stars. Although CNO reactions account for about 1.7% of the Sun’s energy generation, this cycle is prominent in stars which are more than 1.5 times as massive as the Sun.
In the late 1960s astrophysicists Raymond Davis Jr (1914–2006) and John N Bahcall (1934–2005) developed the Homestake Experiment, involving the capture of solar neutrinos in a vast underground reservoir of perchloroethyelne. Although their detections were significantly fewer than expected, hence the now-resolved Solar Neutrino Problem, it did provide the first proof that nuclear fusion reactions were occuring in the Sun. Even though the energy released in the solar core requires many thousands of years to migrate outward, almost all the neutrinos escape the solar core at once. Hence, to detect them is to determine that core reactions are still continuing.

neutrino detector in operation. Image: Department of Energy.
Astronomers have every faith now that nuclear fusion processes power the stars.
However, there are no fusion reactions occuring in the core of neutron stars. Like white dwarfs, neutron stars are stellar remnants, not active stars. Although nuclear “burning” can occur on the surfaces of these remnants if they accrete matter away from a stellar companion, neutron stars are not powered by fusion. Neutron degeneracy pressure counterbalances the intense gravitational contraction within a neutron star in order to sustain its shape.





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