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When the two protons stick together, one of them is converted into a neutron. Actually what is going on here is that protons are made up of elementary particles called quarks. Quarks typically come in sets of three. Protons have two up quarks and one down quark. Neutrons are made of two down quarks and an up quark. When a proton is converted into a neutron, one of the up quarks is converted into a down quark.
The product of this fusion is called a deuteron, a proton bound to a neutron. Also, when the fusion event happens, a neutrino and a positron are ejected, as well as a photon. We can write this as a sort of equation:
P + P = D + n + e+ + g
Here, the Greek symbol n (nu) stands for the neutrino, the g (gamma) stands for the photon, and the e+ stands for the positron. A positron is just like an electron, except that it is made of antimatter instead of normal matter. In collisions like this, physical quantities like mass, energy, charge, spin, matter/antimatter all must be conserved. Since the neutron is neutral (has no charge) a particle with a positive charge, the positron, must be emitted. Since the positron is an antimatter particle, a matter particle, the neutrino, must be emitted.
The conservation of energy means there is also something else very important happening here. If you add up the masses of the deuteron, positon and neutrino, the total mass is less than the mass of the original two protons. This matter was converted to pure light energy. The amount of light energy can be found by relating mass to energy using Einstein's famous equation:
E = mc2
The c in this equation is the speed of light, which is a very big number. c2 is a huge number. This means that transforming a small amount of matter creates a large amount of energy, as indicated in the animation below. In fact, to produce the energy we calculate coming from the sun, fusion in the core fuses 600 billion tons of hydrogen into helium every second.