Astroparticle Physics

70 5 Acceleration Mechanismsgravitational collapsepulsar periodsThe gravitational collapse of stars conserves the angularmomentum. Therefore, because of their small size, rotatingneutron stars possess extraordinary short rotational periods.Assuming orbital periods of a normal star of about onemonth like for the Sun, one obtains – if the mass loss duringcontraction can be neglected – pulsar frequencies ω Pulsar of(Θ – moment of inertia)Θ star ω star = Θ pulsar ω pulsar ,ω pulsar =R2 starRpulsar2corresponding to pulsar periods ofω star (5.18)T pulsar = T starR 2 pulsarR 2 star. (5.19)For a stellar size R star = 10 6 km, a pulsar radius R pulsar =20 km, and a rotation period of T star = 1 month one obtainsT pulsar ≈ 1ms. (5.20)The gravitational collapse amplifies the original magneticfield extraordinarily. If one assumes that the magnetic flux,e.g., through the upper hemisphere of a star, is conservedduring the contraction, the magnetic field lines will be tightlysqueezed. One obtains (see Fig. 5.6)∫∫B star · dA star = B pulsar · dA pulsar ,starpulsarFig. 5.6Increase of the magnetic fieldduring the gravitational collapse ofa starB pulsar = B starR 2 starR 2 pulsar. (5.21)For B star = 1000 Gauss magnetic pulsar fields of 2.5 ×10 12 Gauss = 2.5 × 10 8 T are obtained! These theoreticallyexpected extraordinary high magnetic field strengths havebeen experimentally confirmed by measuring quantized energylevels of free electrons in strong magnetic fields (‘Landaulevels’). The rotational axis of pulsars usually does notcoincide with the direction of the magnetic field. It is obviousthat the vector of these high magnetic fields spinningaround the non-aligned axis of rotation will produce strongelectric fields in which particles can be accelerated.For a 30 ms pulsar with rotational velocities ofv = 2πR pulsarT pulsar= 2π × 20 × 103 m3 × 10 −2 s≈ 4 × 10 6 m/s