Does nuclear fission also happen inside the Sun?

Fission is the splitting of atoms, either by radioactive decay or by collisional impact. Certainly radioactive decay occurs because the sun contains many radioactive isotopes including thorium, uranium etc. Basically fission happens irrespective of any environmental constraints because it is an intrinsic property of radioactive nuclides.

Although the energy produced by fission is comparable to what is produced by fusion, the core of the sun is dominated by hydrogen and at temperatures where hydrogen fusion is possible, so that the dominant source of energy per cubic meter is in fusion rather then the fission of very low abundance radioisotopes. Fission is not a significant source of energy so long as the temperatures and densities are high enough for fusion to occur.

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All answers are provided by Dr. Sten Odenwald (Raytheon STX) for the
NASA IMAGE/POETRY Education and Public Outreach program.