Astroparticle Physics

76 5 Acceleration Mechanismsln(1 + P)γ ≈ln(1 + ε) ≈ P ε . (5.37)Experimentally one finds that the spectral index up to energiesof 10 15 eV is γ = 1.7. For higher energies the primarycosmic-ray-particle spectrum steepens with γ = 2.5.8 Problems1. Work out the kinetic energy of electrons accelerated ina betatron for the classical (v ≪ c) and the relativisticcase (B = 1Tesla,R = 0.2 m). See also Problem 1.2.2. A star of 10 solar masses undergoes a supernova explosion.Assume that 50% of its mass is ejected and theother half ends up in a pulsar of 10 km radius. What isthe Fermi energy of the electrons in the pulsar? What isthe consequence of it?3. It is assumed that active galactic nuclei are powered byblack holes. What is the energy gain of a proton fallinginto a one-million-solar-mass black hole down to theevent horizon?4. If the Sun were to collapse to a neutron star (R NS =50 km), what would be the rotational energy of such asolar remnant (M ⊙ = 2 × 10 30 kg, R ⊙ = 7 × 10 8 m,ω ⊙ = 3 × 10 −6 s −1 )? Compare this rotational energy tothe energy which a main-sequence star like the Sun canliberate through nuclear fusion!5. In a betatron the change of the magnetic flux φ =∫B dA = πR 2 B induces an electric field,∫E ds =−˙φ ,in which particles can be accelerated,E =−˙φ2πR =−1 2 RḂ.The momentum increase is given byṗ =−eE = 1 eRḂ . (5.38)2What kind of guiding field would be required to keep thecharged particles on a stable orbit?