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SOLUTIONS TO EXERCISE 17
(a) If dark matter is, as current theories suggest, not made of protons and neutrons, then we can approximate the total mass of protons and neutrons in the observable universe by the total mass of stars. There are about 100 billion galaxies in the observable universe, and about 100 billion stars per galaxy. A typical star (such as our Sun) has a mass of 2 * 10^30 kg. Thus, the mass of all the stars in the observable universe is M = 10^11 * 10^11 * 2 * 10^30 = 2 * 10^52 kg. Protons and neutrons have nearly identical mass, 1.67 * 10^(-27) kg. Electrons have a much smaller mass (1836 times smaller than a proton, in fact), so we can leave them out of the calculation without adding much error. Thus, after all neutrons decay into protons, there will be about (2 * 10^52 kg) --------------------------- = 1.2 * 10^79 protons. (1.67 * 10^(-27) kg/proton) (b) After every half-life, half of all the protons will be gone. Doing some simple calculations, we see a pattern: After 10 half-lives, there will be about 1/(10^3) as many protons as before. After 20 half-lives, there will be about 1/(10^6) as many protons as before. After 30 half-lives, there will be about 1/(10^9) as many protons as before. So let's keep counting: After 40 half-lives, about 1/(10^12) the number of protons will remain. After 50 " , about 1/(10^15) " After 60 " , about 1/(10^18) " After 100 " , about 1/(10^30) " After 200 " , about 1/(10^60) " After 260 half-lives, about 1/(10^78) the number of protons will remain. We are assuming that the half-life of a proton is 10^33 years. So, after (260 * 10^33) years, there will be about (1.2 * 10^79)/(10^78) = 12 protons left. They will decay in about 4 more half-lives. So all the protons in our current observable universe will have decayed away after (260 + 4) * (10^33 yr) = 264,000,000,000,000,000,000,000,000,000,000,000 years. |