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In 1987, light from supernova SN1987A was detected by telescopes on Earth. The supernova occurred in the Large Magellanic Cloud, a distance d = 1.5 ? 1021 m away, making it the closest in centuries. Observations of this event tell us a remarkable amount about elementary particles. a. Both light and neutrinos were produced in the core of the supernova. Neutrinos are elementary particles which interact extremely weakly with ordinary matter. Detectors on Earth saw a few dozen of these neutrinos, in a burst which occurred about T = 3 hours before the light arrived. i. One explanation of these observations is that the neutrinos’ speed v was faster than the speed of light c, violating special relativity. If this is the case, find v c in m/s. ii. Another explanation is that the light was slowed down by the gas in the solar system, while the neutrinos always moved at speed c. Suppose the solar system has a uniform index of refraction n within a radius D = 1013 m. What would n have to be to explain the time delay? Neither of these explanations seem plausible; the modern accepted explanation is that the light was trapped for some time inside the supernova, while the neutrinos were able to leave immediately. Therefore, for the rest of this problem you should assume special relativity holds. The results listed on the previous page may be helpful. The neutrinos did not all arrive at once. The first arrived with an energy of about E1 = 40 MeV, and the last arrived about t = 10 s later with an energy of about E2 = 20 MeV. b. One explanation of these observations is that neutrinos have a small mass m, so that when they have energy E mc2, their speed v is slightly slower than the speed of light. i. Find an approximate expression for c v, to leading nontrivial order in mc2/E. ii. Using the information above, numerically estimate the neutrino mass m, in units of eV/c2. c. Another explanation is that the neutrinos did not travel in straight lines, but rather were deflected by the intergalactic magnetic field. Suppose this field is uniform, B = 1013 T, and directed perpendicular to the line joining Earth and the supernova, and that neutrinos have charge q = e. i. If a neutrino has momentum p, then in the presence of the magnetic field, it travels in a circle of radius r = p/(qB) d, and its path to the Earth has a total length `. Find an approximate expression for ` d, to leading nontrivial order in d/r. ii. Using the information above, and assuming the neutrino mass is very small so that the e↵ect in part b is negligible, numerically estimate .