Astroparticle Physics

102 6 Primary Cosmic Raysflavours. Based on the 20 recorded neutrino events a totalenergy ofE total = (6 ± 2) × 10 46 Joule (6.42)limits of neutrino massesdifferenceof propagation timeis obtained. It is hard to comprehend this enormous energy.(The world energy consumption is 10 21 Joule per year.) Duringthe 10 seconds lasting neutrino burst Sanduleak radiatedmore energy than the rest of the universe and hundred timesmore than the Sun in its total lifetime of about 10 billionyears.Measurements over the last 40 years have ever tightenedthe limits for neutrino masses. At the time of the supernovaexplosion the mass limit for the electron neutrino from measurementsof the tritium beta decay ( 3 H → 3 He + e − +¯ν e )was about 10 eV. Under the assumption that all supernovaneutrinos are emitted practically at the same time, one wouldexpect that their arrival times at Earth would be subject to acertain spread if the neutrinos had mass. Neutrinos of nonzeromass have different velocities depending on their energy.The expected difference of arrival times t of twoneutrinos with velocities v 1 and v 2 emitted at the same timefrom the supernova ist = r − r = r ( 1− 1 )= r β 2 − β 1. (6.43)v 1 v 2 c β 1 β 2 c β 1 β 2If the recorded electron neutrinos had a rest mass m 0 ,theirenergy would beE = mc 2 = γm 0 c 2 = m 0c 2√1 − β 2 , (6.44)and their velocity( ) 1/2β = 1 − m2 0 c4E 2 ≈ 1 − 1 2m 2 0 c4E 2 , (6.45)since one can safely assume that m 0 c 2 ≪ E. This meansthat the neutrino velocities are very close to the velocity oflight. Obviously, the arrival-time difference t depends onthe velocity difference of the neutrinos. Using (6.43) and(6.45) one getst ≈ r c12m 2 0 c4E 2 1− 1 2β 1 β 2m 2 0 c4E 2 2≈ 1 r E2 m2 0 c4 2 2 − E2 1. (6.46)cE 2 1 E2 2