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Nuclear Physics B (Proc. Suppl.) 28B (1992) 85-89 North-Holland
PROCEEDINGS SUPPLEMENTS
COSMIC RAY ACCELERATION ABOVE 102°eV AND COSMIC STRINGS
J.:l. Quenby, K. Naidu
Astrophgsics Group, Blacketg Laboratory Imperial College o.f Science, Technologll and Medicine Prince Coneort Road, London SW7 2BZ, UK
R. Lieu
University o.f Bet&e/el/ Cali]ornia, USA
Assuming that active galactic nuclei relativistic jets provide the most favourable sites for Cosmic Ray acceleration to the highest energies, the problem of predict|rig the u_r.per cut off to the spectrum is re-exunined. Numerical simulations of shock acceleration are performed for parallel and near perpendicular shocki and a relativistic spec~- up of acceleration in comparison with that predicted by standard theory involving the effective diffusion coefficient normal to the shock surface is confirmed, both for relativistic parallel shocks and any near perpendicular shock where the de HolTman-Teller fi'ame is moving rclativistically. However, likely diffusive escape times derived from AGN radio estimates of the field limit the m~x'.,~um cx:t off to about i0~°eV total energy. The predictions of Bhattacharjee [1] that the decay of Cosmic Strings could produce measurable cosmic ray fluxes above 102°eV are briefly discussed. 'Cusp evaporation' will not work but some unknown mechanism for dissipating 10 -3 of string energy into energetic pvxticles is susceptible to experimental investigation above 102°eV, provided we are confident in understanding conventional acceleration.
1. INTI~ODUCTION
Topological defects, f~r example cosmic strings
Kibble [2], provide a popular mechanism for ex-
plaining the da ta on the large-scale, gravitational
f luctuations from the structure of the universe.
Since particle energies resulting from the even-
tual decay of these defects extend to the GUT
scale of 10ZSGeV, it is temptiug to sear ch for cos-
mic str ing signatures in the very highest, energy
cosmic rays. In this case, the top end at ~he cos-
mi_c ray spectrum due to conventional, electrody-
namic acceleration is the unwanted background
and needs to be well understood. An ideal situa-
tion would be a firm prediction tha t conventional
acceleration ceases at i02°eV, the present upper
limit of ~xperir~ent, ~llowing detectors built for
higher energy recording to search for early uni-
verse signatures or to put upper limits on pos-
sible processes. Our twofold aim is therefore to
re-examine our underscsnding of particle accel-
eration by electrodynmuic means and to review
the available predictions relative to Topological
defects decay. Others have discussed the limita-
tions of isolated pulsars ~ galactic accelerators
and the problems of trying to make DC electric
field acceleration in AGN jets. We confine our-
selves to the most promising mechanism for ob-
taining the highest energy particles, namely dif-
fusive shock acceleration ~t the large-scale, ter-
0920.-.5632/92/$05.00 © 1992- Elsevier Science Publishers B.V. All rights reserved.
86 JJ. Qucnby et al. / Cosmic ray acceleratiw~ above 1020 eV and cosmic strings
ruination shocks of the relativistic AGN jets.
2. NUMERICAL SIMULATIC~N OF
DIFFUSIVE S~OCK ACCELERATION
Although non-linear effects can be important
in shock acceleration of Cosmic rays, the test
particle model that we adopt here is applica-
ble if either the total number of cosmic rays in-
jected is a small fraction of the energy density in
the plasma or alternatively at the upper cut off,
where the escape time is sufficiently short com-
pared with the acceleration time that there is
not enough energy ill the particles in resonance
with the appropri.ate scattering wavelengths to
modify this part of the turbulence spectrum. Be- cause non-linear effects are neglected, a strong shock compression ratio of 4 is adopted. Follow- ing Quenby and Lieu [3] the scattering is taken
to be isotropic with a parallel mean free path A =
41r~ tot cyclotron radius r 9, as based upon in-
terplanetary Cosmic ray propagation studies. A
guiding centre approximation is adopted so th,~J. a particle has a probability of moving Z along a field line at pitch angle 0 proportional to exp
( z ) For an inclined shock, field-normal A(cos e ) . " , t
angle ~bl, parallel diffusion only dominates over
perpendicular diffusion if cot2qJ > 1. We
find that for the above values of A, the guid- ing centre approximation applies for ~_<88 °, but
if field line wand~,-ing is a major contribution
to perpendicular diffusion as in interplanetary
space, our approach is limited to q~_<68 °. Particles are injected 40 mean free paths
upstream ~f ;he shock in the half hemisphere
centred on the field line and directed toward
the shock according to the weighting factor
cos0sin0. The upstream injection is in the
-pstream plasma frame, but on reaching the
shock a relativistic transformation to the E_. -- 0
frame is made. On reaching the shock, particles are transmitted or reflected according to con-
servation of the first adiabatic invariant in the
E_. = 0 or de Hoffman-Teller frame. Newman
et al [4] demonstrate approximate conservation even in a highly turbulent model. Transmitted particles are relativistically transformed to the
downstream plasma frame until a scattering
returns them to the shock. The standard analytical solution for test par-
ticle acceleration when the flow speeds Vl (up-
~t~.~,n) and V2 (downstream) are non relativistic
but the particle speed v ~ c yields a differential
number spectrum n(p)o~p -a , where p is particle
momentum, o - (r -4- 2) / ( r - I) and r = VI/V2
is the compression ratio. An acceleration time
constant.
r(P) = -3
for diffusion coefficient K - ~ cos2~ neglecting
K .L. Quenby and Lieu [4] have investigated the
case Vl - 0.96c and V2 - 0.32c for a plane par-
allel shock and show by the above Monte Carlo
method that average, single cycle shock crossing energy gain is A p / p -- 10.5, corresponding to
approximately a .?2 energy enhancement where
72 --- ( I - V2/c 2) and V is the relativistic dif-
ference between V] and V2. The numerical value
of the time constant found in the Monte Carlo
was a factor 13.5 less than given by the above ex-
pression for t-(p). Also the power law exponent
Q -- i.2, rather than cr - 5/2 expected non-
relativistically. Other workers have noticed this
spectral flattcning [5] and accel,~ration speed up
[6] for parall,~! shocks.
The ac~.,lal occurrence of such high speed up-
stream flows in AGN jets as adopted by Quenby
and Lieu [3] is not certain, except perhaps near
the core [7]. However spectral ~attening occurs
.l.,l. Quenby et al. / Cosmic ray acceleration above 102° eV and cosmic strings 87
for slower flows provided the shock inclination angle is sufficiently high that the upstream fluid speed measured in the de Hoffman-Tel!er fre~me, -1.5 U = Uah sec Oup for shock velocity Uah and where ~ -!.6 ~'up is measured in the upstream frame, tends to
. - -! .7 the velocity of light [8]. We have repeated the ] Monte Carlo for a range of upstream de Hoffman- ~ -!.8
f~ Teller frame flow speed, s and ~ angles for a corn- -1.9 pression ratio of 4. Fif~ure 1 shows the differen- -2 tied number spectral index for ~bl = 88 °, and
-2.1 ~2 = 89.50 against U, and figure 2, shows the 0
ratio of the numeri,cedly measured acceleration time to that given by equation 1 for the same shock parameters, against U. For these param- eters, which just satisfy our least stringent con- dition of the dominance of parallel diffusion, the spectral index reduces from the non- relativistic limit of 2 to 1.5 while the acceleration time re- duces from the equation 1 value to nearly one
tenth of the value as U varies from 0.I to 0.9 in 0.8 units of c. F o r ~ = 68 °, the speed up is not so dramatic, rea~iii~g only 0.35 of the equation 1 0.6 value, but coafirming the trend that renders the de Hoffman-Teller frame upstream speed as the 0.4 crucial parameter.
0.2
3. MINIMUM ACCELERATION TIMES AND DIFFUSIVE LOSS TIMES FOR AGN'S, 1020 - 1021eV
The terminating shocks of AGN jets are as- trophysical sites where the acceleration speed- up will apply. Minimum energy approach to an interpretation of synchrotron radiation leads to typical field strengths ~:-the most of 4.10 -4 gauss o~.'er di.qtances of 3 kpc in these radio lobes al-
though the dimension may be up to 20 kpc [9]. Data from the lar&e radio galaxy CygA [I0], which may not be directly relevant to our local supercluster, are consistent with fields of 4.10 -e
- 1 . 4 ' ' ' , ' ' ' , ' . ' i , • . I . • .
g
' ' I | ° ' ' I , • , I I ' • ] I ' '
02. 0.4 0.6 0.8 U~tream flow speed
Fig. 1. Differential number spectral index versus Up- s tream flow velocity in the de Hoffmann-Teller frame (in
units of c).
0 o 0 1 ~ " " o : 4 " - 0 ~ e " 0 1 s " "
Upstream flow speed
Fig. 2. Ratio of experimental to theoretical time to accel- erate particle (~) versus Upstream flow velocity in the de
Hoffmmm-Teller frame (in units of c).
gauss over 300 kpc. Using the expression for mean free path, ~ccei-
eration time and relativistic speed-up relevant to a Vl ~- 0.96c, V2 = 0.32c shock as discussed in section 2, we find a~ acceleration time of 1.810 s Yr for the 410 -4 gauss hot-spot. This is far below the photo-pion loss time of 3107 Yr at I021eV.
88 J..I. Quenby et al. I Cosmic ray acceleration above 1020 eV and cosngc stn'ngs
Diffusive escape will be by the shortest route
parallel to flux tubes of B__. Allowing this distance
to be R-10 kpc and an escape time at 1021eV, Te
given by Te - (3/4)R2/~ 2, we find Te - 2.410 s Yr for a 'hot spot', Te -" 2.1104 Yr for an outer lobe with R -300 kpc.
Clearly with the above escape times, an en- ergy of 1021eV cannot be obtained and in fact, escape and energy gain times are equal only a'~ 3.51019eV in an outer lobe and at 1.2102°eV in a
hot spot. However, in both these cases, the mean free path is roughly equal to the dimensions of the system, so it is likely that only particle en- ergies per nucleon of 1019eV are accelerated in AGN jets [3]. To account for 102°eV total energy,
an enhancement of heavy element composition is required.
4. COSMIC STRING CONTRIBUTION AT THE HIGHEST ENERGIES
Topological defects are traps for the supermas- sive gauge and higgs bosons of GUTs (X parti-
cles) which if free have extremely short lifetimes. Symmetry breaking below temperatures equiv-
alent to 1016GeV leaves the symmetry inside a
cosmic string, a popular example of a defect, dif-
ferent to the symmetry outside. Cosmic strings have thicknesses ,,-10 -2s - 10-3°cm, masses per
unit length ~1022gcm-1 and evolve due to a ten-
sion within the string as the universe expands. Intersection and chopping off of loops maintains
the typical scale of the long string network as the
horizon scale and the total cnergy density as a
small and constant fraction of the total energy
density of the universe. The closed loops formed
by the chopping process loose energy by gravi-
tational radiation or fast evaporation into ener-
getic particles.To investigate the implications for
cosmic ray origin, Bhattacharjee [1] defines the
average rate of primary loop formatio~ as
dn! 1
where n 3 is the number density of loops at forma-
tion time t! and ~ is the number of sub-horizon- sized loops formed per horizon sized volume per
Hubble time at t ! . The typical loop length L! is
Ls = o, ts = Ms (3) where c~ is a numerical constant < 1, M! is
the total energy of the loop and p is the en-
ergy per unit length of the string. (Note: 'Natu-
ral units' are used for equations 2, 3 and 4 and
in dimensionless numbers given in this section,
c = h = Mpav/'G = k B = 1 Loops collapse or self-intersect at a time t given by
ts = + -1 (4)
in a half-period of oscillation. Collapse of a loop
or its break-up into a large number of pairs will
lead to particle production because of the micro- physical interaction of the overlapping regions. Let f be the fraction of total primary loop en- ergy going into high energy particles.
Bhattacharjee [1] uses QCD to go from the X
particle decay into quarks and leptons and via
ideas of hadronic jet production taking into ac-
count the effects of the microwave background
on the resulting cosmic ray spectrum to pre-
dict the 10 Is - 102~eV flux. Air shower exper-
iments at 102°eV limit the product of ~fc~/3rl to
_<1.710 -9 where rl is mass per unit length. Now
~/3~0.57 from numerical simulation of string be- haviour, but may be expected to be -~ I on gen-
eral grounds of energy conservation in the con-
text of a scaling solution to large-scale fluctua-
tions. ~~10 -e if strings have enough gravity to
explain large scale structure. Hence experiment puts a limit l_< 10 -3.
If the decay into cosmic rays is by cusp-
radiation, that is the formation of cusp-like
.I.J. Quenby et al. / Cosmic ray acceleration above 10 :° eV and cosmic strings 89
kinks in the oscillating loops with overlapping
regions,the best one can obtain is to reduce loop mass per unit length to allow r / = Gp = 10 -15.
[11] Here the predicted flux is still 10 -4 of the
observed cosmic ray flux at 1019eV and the
gravitational effect on the large scale structure
has been completely lost. Moreover, gravita-
tional radiation is likely to reduce the cusp
radiation efficiency by a back reaction which
reduces the speed of movement to less than . .
c. [12] While the discussion in this section" is
explicitly about cosmic strings, there is a clear
general principle. To have enough exotic matter
left over from the Big Bang to provide sufficient
gravitational potential to explain the large
scale clustering of galaxies and to maintain a
fluctuations spectrum which does not depend on
red shift epoch, it is likely that one needs about
I0 -3 of the mass energy of the exotic matter to
be available for cosmic ray production in order
to provide measurable fluxes > 102]eV. Only an
ill-understood cosmic string collapse mechanism
fulfills this role at the moment. Because known
electro-dynamic acceleration processes do not
seem to work above 102°eV total energy, detec-
tors with sensitivities better than 10-15m-2s - I
(~_ 102°eV) may provide new limits on early
universe phase transitions and their signatures
in the current epoch.
Pub. World $c/,-.nfi~c,(1991), 882. [2] T.W.B.Kibble, J.PhyJ. Ag, (1976), 1387. [3] J.J. Quenby and R. Lieu, Nature 842, (1989) 654. [4] P.L. Newman, X. Monssas, J.J. Quenby, J.F. Valdes-
Galicia and Z. Theodossiou-Eksterinidi, Astrou and Asfrophy. 255, (1992) 443.
[5] J.G. Kirk and P. Schneider, Astrophys. J. S22, (1987), 2s6.
[6] D.C. Ell;son, F.C. Jones and S.P. Reynolds, Astro. ~av. ~. 8eo, (199o), 702.
[7] R.C. Walker, J.M. Benson and S.C. Urwin, A:tro. phys.J. 816, (1987), 546.
[8] K.R. Ballard and A.F. Heavens, MNRA$ (1992), accepted for pub.
[9] R.A. Liang, MNRA$, 105 (1981), 261. [10] P.J.Har6rave and M. Ryle MNRA$ :175,(1976),481. [11] P. Bhattacharjee, Phys. Rew.D 40 (1989), 3968. [12] J.M. Quashnock and T. Piran, Phys. Rew D (1991),
sub. for pub.
Acknowledgements
Discussions with Ray Rivers, Paul Shellard
and Neil Turock are gratefully acknowledged.
R e f e r e n c e s
[1] P. Bhattacharjee, Astrophysical aspects of the most energetic cosmic rays, eds Nagaro and Takahara,
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