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The generation of the highest cosmic ray energies
Volume 42A, number 5
1 January 1973
PHYSICS LETTERS
THE GENERATION OF THE HIGHEST COSMIC RAY ENERGIES M. GREWING Institut fir Astrophysik und extraterrestrische Forschung der UniversitiirBonn, W. Germany
and H. HEINTZMANN Znstitutftir Theoretische Physik der Universitiit.Kiiln, W.Germany Received 14 November 1972 The acceleration of charges particles in superstrong electromagnetic numerically, taking into account radiation reaction effects.
The recent discovery of cosmic ray primaries with energies > 1021 eV [l] has furhter increased the requirements for models trying to explain the origin of such particles. We discuss here the acceleration of charged particles to very high energies in intense low frequency electromagnetic waves. It is currently believed that such waves will be generated by pulsars which are taken to be rotating, strongly magnetized neutron stars with the axis of the magnetic moment inclined with respect to the rotation axis of the star. Our discussion is limited to the region outside the “velocity-of-light cylinder” where we may assume vacuum conditions whereas inside the “velocity-oflight cylinder” a dense corotatlng magnetosphere might exist. We do take into account the near field components of a rotating magnetic dipole in addition to the usual wave field components. In studying the particle motion we also take the radiation reaction effects into account. Our discussion is therefore more general than previous discussions in the literature [2-4] . The equation of motion for a charged particle moving in an intense electromagnetic field may be written as -+=(eu+[uX~I)+ux
(1)
fields to energies > 102r eV has been studied
13-f
3 I
log
6 L 9, 5 I I ---__---
-10 /
\
-1 I.
-11-
. I 'O-2
I -1
I log&-l)
I 1
I 2
Fig. 1. The variation of 7, ue , and Us (curves a, b, c respectively) as a function of log(p/p, - 1) for a partrcle moving in a radiation field characterized by f. = 10”. Other initial conditions are given in the text. Curves (a’), (b’), (c’) show the variation of the same quantities neglecting radiation reaction. Curve (d) shows the maximu obtainable energy rmax as function of logp,.
and -&=(e.@+7x
(2)
where we have used the same notation as in a preceding paper [5 ] , i.e. e and b are proportional to the electric and magnetic field respectively, x denotes
345
Volume 42A, number 5
PHYSICS LETTERS
Fig. 2. Dependence of the maximum obtainable energy for the initial conditions chosen (see text) as function of the polar angle at injection e(O). Curves (a), (b), (c) refer to p. = 102, 103, lo4 respectively.
the radiation reaction, * the spatial components of the four-velocity, and 7 the particle’s energy in units of its rest mass energy. Du/d[ is the covariant derivative ofu. The inclusion of the radiation reaction terms in eqs. (1) and (2) has so far prevented us from finding an analytical solution for the particle motion. We did succeed however in solving the set of simultaneous differential equations numerically for a variety of initial conditions. Some of the results are shown in figs. 1 and 2 which both refer to particle motion in the radiation field of a rotating magnetic dipole that is at ninety degrees with respect to the rotation axis of the star. In fig. 1 we have chosen a “standard set” of initial conditions with f. = (eB,/mcS2)p~ = 1015, and a partial starting at rest (up(O) = ue(0) = u+(O) = 0, 7(O)= 1) atp=p, = 100. The polar angle angle at injection was chosen to be e(O) = n/4. Curves labelled (a), (b), (c) show how 7, ue and uv, vary as the particle is accelerated away from the star in essentially radial direction. It should be noted that the initial part of the motion where p R p. is not shown here. For comparison we also have included curves (a’), (b’), (c’) which show the variation of the same quantities in the absence of radiation reaction. Without discussing the differences in detail it is interesting to 346
1 January 1973
note that the influence of radiation reaction is minor. Considering in particular the final energy the particle will reach we see that radiation reaction causes a reduction from 6.1 X lOI to 2.2 X lOI for the initial conditions chosen. The particle would thus still reach an energy in excess of 1021 eV. Also shown in fig. 1 is the variation of the maximum obtainable energy 7max for particles injected at different radii p. (curve d). The numerical calculations were actually carried out to smaller values of p. than shown in the graph but no significant change was found in rrnax. In fig. 2 the same set of initial conditions was chosen as for fig. 1 except that the polar angle at injection e(O) was varied through ninety degrees. The three curves refer to p. = 102, lo3 and lo4 respectively, again showing that particle injection closer to the star is more favorable. The variation of rrnax with 8(O) is particularly interesting for the radiational aspects of this acceleration mechanism which will be discussed in a seperate paper. In summarizing two major conclusions can be drawn from the above results: (1) the radiation reaction terms in eqs. (1) and (2) change the details of the particle motion considerably, the final energy a particle can reach is affected however only mildly. The reduction factors vary betwe,en 1 and 10 depending on the actual set of initial conditions. (2) Under favorable initial conditions evenf, = 1015, which for a neutron star spinning at a rate of 104, would correspond to B * 1016 gauss, leads to a maximum energy of > 1021 eV for protons. This result must be compared with the previous results [2] where f, > 101* was required to obtain such energies.
References
111K. Suga, H. Sakuyama, S. Kawaguchi and T. Hara, Phys. Rev. Lett. 27 (1971) 1604. Phys. Rev. Lett. 22 (1969) 728; J.P. Ostriker and J.E. Gumt, Ap. J. 157 (1969) 1395. 131 M. Grewing and H. H;emt.zmamt, Phys. Rev. Lett. 28 (1972) 381. [41 R.M. Kulsrud, J.P. Ostriker and J.E. Gunn, Phys. Rev. Lett. 28 (1972) 636; R.M. Kulsrud, Ap. J. 174 (1972) L25. 151 M. Grewing and H. Heintzmann, Phys. Lett. 42A (1972) 325.
121J.E. Gunn and J.P. Ostriker,
1 January 1973
PHYSICS LETTERS
THE GENERATION OF THE HIGHEST COSMIC RAY ENERGIES M. GREWING Institut fir Astrophysik und extraterrestrische Forschung der UniversitiirBonn, W. Germany
and H. HEINTZMANN Znstitutftir Theoretische Physik der Universitiit.Kiiln, W.Germany Received 14 November 1972 The acceleration of charges particles in superstrong electromagnetic numerically, taking into account radiation reaction effects.
The recent discovery of cosmic ray primaries with energies > 1021 eV [l] has furhter increased the requirements for models trying to explain the origin of such particles. We discuss here the acceleration of charged particles to very high energies in intense low frequency electromagnetic waves. It is currently believed that such waves will be generated by pulsars which are taken to be rotating, strongly magnetized neutron stars with the axis of the magnetic moment inclined with respect to the rotation axis of the star. Our discussion is limited to the region outside the “velocity-of-light cylinder” where we may assume vacuum conditions whereas inside the “velocity-oflight cylinder” a dense corotatlng magnetosphere might exist. We do take into account the near field components of a rotating magnetic dipole in addition to the usual wave field components. In studying the particle motion we also take the radiation reaction effects into account. Our discussion is therefore more general than previous discussions in the literature [2-4] . The equation of motion for a charged particle moving in an intense electromagnetic field may be written as -+=(eu+[uX~I)+ux
(1)
fields to energies > 102r eV has been studied
13-f
3 I
log
6 L 9, 5 I I ---__---
-10 /
\
-1 I.
-11-
. I 'O-2
I -1
I log&-l)
I 1
I 2
Fig. 1. The variation of 7, ue , and Us (curves a, b, c respectively) as a function of log(p/p, - 1) for a partrcle moving in a radiation field characterized by f. = 10”. Other initial conditions are given in the text. Curves (a’), (b’), (c’) show the variation of the same quantities neglecting radiation reaction. Curve (d) shows the maximu obtainable energy rmax as function of logp,.
and -&=(e.@+7x
(2)
where we have used the same notation as in a preceding paper [5 ] , i.e. e and b are proportional to the electric and magnetic field respectively, x denotes
345
Volume 42A, number 5
PHYSICS LETTERS
Fig. 2. Dependence of the maximum obtainable energy for the initial conditions chosen (see text) as function of the polar angle at injection e(O). Curves (a), (b), (c) refer to p. = 102, 103, lo4 respectively.
the radiation reaction, * the spatial components of the four-velocity, and 7 the particle’s energy in units of its rest mass energy. Du/d[ is the covariant derivative ofu. The inclusion of the radiation reaction terms in eqs. (1) and (2) has so far prevented us from finding an analytical solution for the particle motion. We did succeed however in solving the set of simultaneous differential equations numerically for a variety of initial conditions. Some of the results are shown in figs. 1 and 2 which both refer to particle motion in the radiation field of a rotating magnetic dipole that is at ninety degrees with respect to the rotation axis of the star. In fig. 1 we have chosen a “standard set” of initial conditions with f. = (eB,/mcS2)p~ = 1015, and a partial starting at rest (up(O) = ue(0) = u+(O) = 0, 7(O)= 1) atp=p, = 100. The polar angle angle at injection was chosen to be e(O) = n/4. Curves labelled (a), (b), (c) show how 7, ue and uv, vary as the particle is accelerated away from the star in essentially radial direction. It should be noted that the initial part of the motion where p R p. is not shown here. For comparison we also have included curves (a’), (b’), (c’) which show the variation of the same quantities in the absence of radiation reaction. Without discussing the differences in detail it is interesting to 346
1 January 1973
note that the influence of radiation reaction is minor. Considering in particular the final energy the particle will reach we see that radiation reaction causes a reduction from 6.1 X lOI to 2.2 X lOI for the initial conditions chosen. The particle would thus still reach an energy in excess of 1021 eV. Also shown in fig. 1 is the variation of the maximum obtainable energy 7max for particles injected at different radii p. (curve d). The numerical calculations were actually carried out to smaller values of p. than shown in the graph but no significant change was found in rrnax. In fig. 2 the same set of initial conditions was chosen as for fig. 1 except that the polar angle at injection e(O) was varied through ninety degrees. The three curves refer to p. = 102, lo3 and lo4 respectively, again showing that particle injection closer to the star is more favorable. The variation of rrnax with 8(O) is particularly interesting for the radiational aspects of this acceleration mechanism which will be discussed in a seperate paper. In summarizing two major conclusions can be drawn from the above results: (1) the radiation reaction terms in eqs. (1) and (2) change the details of the particle motion considerably, the final energy a particle can reach is affected however only mildly. The reduction factors vary betwe,en 1 and 10 depending on the actual set of initial conditions. (2) Under favorable initial conditions evenf, = 1015, which for a neutron star spinning at a rate of 104, would correspond to B * 1016 gauss, leads to a maximum energy of > 1021 eV for protons. This result must be compared with the previous results [2] where f, > 101* was required to obtain such energies.
References
111K. Suga, H. Sakuyama, S. Kawaguchi and T. Hara, Phys. Rev. Lett. 27 (1971) 1604. Phys. Rev. Lett. 22 (1969) 728; J.P. Ostriker and J.E. Gumt, Ap. J. 157 (1969) 1395. 131 M. Grewing and H. H;emt.zmamt, Phys. Rev. Lett. 28 (1972) 381. [41 R.M. Kulsrud, J.P. Ostriker and J.E. Gunn, Phys. Rev. Lett. 28 (1972) 636; R.M. Kulsrud, Ap. J. 174 (1972) L25. 151 M. Grewing and H. Heintzmann, Phys. Lett. 42A (1972) 325.
121J.E. Gunn and J.P. Ostriker,