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,