From Special Relativity to Feynman Diagrams.pdf

And neutrons with a consequent reduction of rest mass

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and neutrons) with a consequent reduction of rest mass which is released in the form of energy (radiation). The fact that the solar energy, or more generally, the energy of a star, could not originate from chemical reactions, can be inferred from the astronomical observation that the mean life of a typical star, like the sun, is of the order of 10 9 10 10 years. If the energy released by the sun were of chemical origin, one can calculate that the mean life of the sun would not exceed 10 5 10 6 years. It is only through the conversion of mass into energy, explained by the theory of special relativity, that the lifetime of stars can be fully explained in relation to their energy emission.
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2.1 Relativistic Energy and Momentum 51 Without entering into a detailed description of the sequences of nuclear processes taking place in the core of a burning star (which also depend on the mass of the star), we limit ourselves to give a qualitative description of the essential phenomenon. We recall that after the formation of a star, an enormous gravitational pressure is generated in its interior, so that the internal temperature increases to typical values of 10 6 10 7 K . At such temperatures nuclear fusion reactions begin to take place, since the average kinetic energy of nucleons is large enough to overcome the repulsive (electrostatic) potential barrier separating them. At sufficiently short distances, the interaction between nucleons is dominated by the attractive nuclear force and nucleon bound states can form. The fundamental reaction essentially involves four protons which give rise, after intermediate processes, to a nucleus of Helium, 4 2 He , together with two positrons and neutrinos : 4 × 1 1 H 4 2 He + 2 e + + 2 ν e . (2.30) where e + denotes the positron (the anti-particle of an electron) and ν e the (electronic) neutrino, their masses being respectively: m e + = m e 0 . 5 MeV , m ν e 0 . (Note that ionized hydrogen, that is protons, comprise most of the actual content of a star.) Thereaction( 2.30 )istheaforementionednuclearfusiontakingplaceintheinterior of a typical star. To evaluate the mass reduction involved in this reaction we use the value of the mass of a 4 He nucleus, and obtain: M = 0 . 0283 u = 0 . 0283 × 931 . 494 MeV / c 2 26 . 36 MeV / c 2 . This implies that every time a nucleus of 4 He is formed out four protons, an amount of energy of about 26 . 36 MeV is released. Consider now the fusion of 1 Kg of ionized hydrogen. Since 1 mole of 1 1 H , weighting about 1 gr , contains N A 6 . 023 × 10 23 (Avogadro’s number) particles, there will be a total of 1 . 5 × 10 26 reactions described by ( 2.30 ), resulting in an energy release of: 6 E ( 1 Kg ) = 26 . 36 × 1 . 5 × 10 26 MeV 3 . 97 × 10 27 MeV 6 . 35 × 10 14 J . On the other hand, we know that a star like the sun fuses H 1 1 at a rate of about 5 . 64 · 10 11 Kg s 1 , the total energy released every second by our star amounts approx- imately to: E t = 6 . 35 × 10 14 × 5 . 64 × 10 11 3 . 58 × 10 26 J s 1 , This implies a reduction of the solar mass at a rate of: 6
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