n in the case of isotropic diffusion.
4.13. Induction acceleration mechanisms
4.13.1. The discussion on the problem of induction acceleration mechanisms
The hypothesis that the CR particles can be accelerated up to very high energies by an electromagnetic mechanism of induction type was first set forth by Swann (1933) as long as more than 70 years ago. He proposed the betatron acceleration mechanism and qualitatively developed this model later in the work (Swann, 1960), in which he studied charged particle acceleration up to the CR energies; he treated particle acceleration as being owed to electromagnetic induction associated with alternating magnetic fields of the stars (the betatron mechanism). For the sake of brevity axial symmetry is considered, in which the magnetic field H and the relevant vector-potential U are generated by circular currents around the axis z. If U is independent of r (i.e. dU/dr = 0) and the magnetic field component
Hz varies as ^ r-1 the particle will move along a circular orbit. As the field increases for 106 sec from 0 to 2000 Gs inside a circle of radius ro = 109 cm, the particle energy increases up to 3 x 1014 eV. A decrease of the field, however, will result in a particle's deceleration. It has been shown that even if dU/dz ± 0, a 'deep' trap of the curve U (r) permits the existence of a stable circular orbit on which the particle may gain energy for a long period. Special selection of the function U (r, t) will result in that the trap will shift with time outwards from the axis and disappear at a certain distance from the axis. At that moment the accelerated particles, without being decelerated, will be ejected from the trap along a rapidly unwinding spiral.
A similar concept was developed by Terletsky (1959) who examined the possibility of particle acceleration by the electromagnetic field generated in the vicinities of a rotating body when its rotation axis and the magnetic moment do not coincide as a result of unipolar induction. Swann (1960) noted in the discussion of paper Terletsky (1959) that he had considered this problem many years earlier and concluded that the results of the calculations were dubious owing to an uncertainty of the motion state of the medium surrounding such rotating astronomical body (star or planet). The fact is that if the medium co-rotates with a body, the electromotive forces will not be induces in such medium. The ions may have been accelerated at great distances from the body, but the magnetic field at those distances should be frozen into the plasma.
Therefore only a certain transient region is of interest in this case. It was noted, in numerous works that the induction mechanisms could not be effective because of the high conductivity of the cosmic plasma. In connection with that Swann (1960) noted, that the shielding electrical currents in rarified plasma could not be significant and that in any case their density was smaller than Nec, which gives
~5 X10 A cm at N = 1cm , but such currents could not hamper the particle's energy gain in the induction acceleration mechanism.
4.13.2. Charged particle acceleration up to very high CR energies by rotating magnetized neutron star
Gunn and Ostriker (1969) suggested following induction mechanism of charged particle acceleration by fast rotating neutron star provided maximal energy of accelerated protons up to 1021 eV. For determinate it was considered pulsar in the Crab remnant. Its main parameters are supposed as following: the magnetic moment (perpendicular to the axes of rotation) ¡u = 4.17 x1030Gs.cm3, moment of inertia I = 1.39x1045 g.cm2, quadruple inertia moment Iq = 6.12x1041g.cm2, the initial angle velocity Qo = 1.03 x104sec—1. Here the values for I and Qo are taken according to model of Hartle and Thorne (1968) for neutron star with the mass 1.4 MS and with the strength of magnetic field on the star's surface Hp ~ 1012 Gs . The value of Iq corresponds to ellipsoidality ~ 10—4. Such star will emit quadruple gravitation radiation with frequency 2Q and magnetic dipole radiation with frequency Q. The equation of particle moving in this case can be integrated analytically with the solution t — Tm
3c3i 45c5/ Tm — 2 2 ~ 180 days; Tg —-- = 0.16 days;
fal g 4GI2Q.4C
The gravitation radiation will be predominate during the time
In this period parameters will be change in the following way:
>( + tTg) Hr = Hro(1 + tjtg)4; Lmd = Lmdo(1 + t/Tg
Lmdo = 4.8x 1045erg/sec; Lgqo = 2.7x 1048erg/sec; Hro = 1.7x 1011Gs.
In Eq. 4.13.4 Hr is the strength of magnetic field in the radiation zone; the magnetic field on the surface Hp is supposed to be constant. At the moment of time t = 915 years after Supernova explosion the star radiated magnetic dipole and gravitation quadrupole radiation with the power
Lmd(t = 915years)= 5.5x1038erg/sec; Lgq(t = 915years) = 1.0x 1038erg/sec. (4.13.6)
For the all time of Crab remnant living it gives ~ 5 x 1050 erg total energy in the low-frequency electromagnetic radiation.
In paper of Gunn and Ostriker (1969) it was not considered more complicated problem on the interactions in the local wave zone r < ric, but investigated in details the region r > rc where electromagnetic wave can be considered as spherical. The equation of moving of the test particle with charge Ze and mass mac in such wave will be dv„ Ze
dT macc where is the tensor of electro-magnetic field, and vM is the velocity vector of accelerated particle. Let us determine the wave in the point (0,0,ro) where particle are injected; then
[H f [(0,1,0)J Ze ric = c/Q; 0)l = ZeHr/macC (4.13.9)
are the radius of the wave zone and gyro-frequency, correspondingly. The equations of particle moving are as following:
For particles which start from the rest the equations of particle moving will be
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