Superconducting cosmic string loops as cosmic accelerators of cosmic rays

IL NUOVO CIMENTO VOL. 102 B, N. 1 Luglio 1988

Superconducting Cosmic String Loops as Cosmic Accelerators of Cosmic Rays.

Y . H A Y A S H I

Graduate School of Science and Technology, Kobe University - Nada, Kobe 657, Japan

(ricevuto il 29 Aprile 1988)

Summary. - - A superconducting cosmic string loop can carry such a large current - 102~ A which causes a strong magnetic field around it. The motion of the loop yields an electric field which accelerates charged particles. A simple model for the acceleration mechanism is presented. It is shown that particles may get ultra-high-energies -10 ~ eV and their integral energy spectrum is proportional to ~-"~. The expected spectrum for reasonable parameters of the string is consistent with experiments. Constraints on the radius of the loop are also discussed.

PACS 98.70.Sa - Cosmic-ray sources.

1. - I n t r o d u c t i o n .

Primary cosmic rays bring us many messages from the Universe. The air shower experiments show that their energies extend to - 102o eV (1). Such ultra- high-energy hadrons are supposed to be produced by sources different from those producing high-energy particles (< 101SeV). However, the acceleration mechanisms enough to explain ultra-high-energy cosmic rays have not yet been established.

On the other hand, it is probable that cosmic strings appeared at the early stage of the Universe as topological defects accompanied by symmetry breaking in some grand unified theories (GUTs)(2). Astronomically, strings are attractive objects because a large structure of the Universe may be explained by them. In addition, cosmic strings can be superconductive under a certain Higgs structure

(1) A. M. I-IILLAS: Proceedings o f the X V I I ICRC, Vol. 13 (Paris, 1981), p. 69. (2) A. VILENKIN: Phys. Rep., 121, 263 (1985) and references therein.

6 - Il Nuovo Cimento B. 81

82 Y. HAYASHI

and carry a large current (8). Around a superconducting loop with such a large current, a strong magnetic field is produced. If the loop is moving with velocity V in the Universe, an electric field E - Y. B yields and works on charged particles. The highest voltage r that the loop can create is simply estimated in the following way. When the loop, a circle with radius a, carries a current J, then we have

~o V J (1) ~ ~ V B a -

2 '

where B is taken at the centre of the circle. By putting V - c / V 2 and J = Jm~ -- 102o A (the maximum current that the string can carry for 1016 GeV of GUT scale) as typical values, charged particles passed near the moving loop may get the drastic energy ~ - 1022eV.

For above reasons, the superconducting loop can be considered as a source of ultra-high-energy cosmic ray if it exists actually. Therefore, we have to examine the relationship between the ultra-high-energy cosmic rays and the superconducting loop.

The aim of this paper is to consider an acceleration mechanism by a superconducting loop and to investigate conditions which the loop satisfies for being the origin of cosmic rays.

In the next section, the energy given by the loop to the charged particle is roughly estimated. In sect. 3, ultra-high-energy cosmic-ray energy spectra are calculated by a simple model of the moving loop. Furthermore, restrictions on its radius are given from experimental data.

2. - Acceleration mechanism of charged particles.

In this section, we consider an electromagnetic field due to a superconducting cosmic string loop which is a circle with radius a and carries a current J. In the rest frame of the loop (denoted by dashed letters), a vector potential in cylindrical coordinates the origin of which is the centre of the loop has only an azimuthal angle component

(2) AO, ~0J a 1/2 E(k')}

where

(3) k , 2 _ 4ar ' <. 1. (a + r') 2 + z '2

(~) E. WITTEN: Nucl . Phys . B, 249, 557 (1985).

SUPERCONDUCTING COSMIC STRING LOOPS AS COSMIC ACCELERATORS ETC. 83

K(k') and E(k ' ) are the complete elliptic integrals of the first and the second kind. Thus the vector potentials in the rectangular coordinates are

t r r r (4) A'x = - sin 0' A'0, Ay = cos 0 A0 and Az = 0.

Obviously, there is no scalar potential so that the rest loop does not impart energy to charged particles.

Now we assume that the loop is moving in the (x, y)-plane of observer's rest frame and choose the direction of velocity V as x-axis though the loop might be rotat ing around its centre of mass. Then we can obtain vector potentials performing Lorentz transformations for eq. (4):

i - - ! (5) Ax = Ax "/, A.~ - Ay and As = 0,

where ~, = 1/~1 - f12 and c/~ = V. More explicitly, vector potentials can be writ ten a s

(6) ~, ,.oJ'[ / a ,\1~ [[. . E(k')} A . = - s i n ~ 7 - - ~ - I ~ 7 ) l [ t - ~ - ) K ( k ' ) -

and

(7) ~, ,~o J / a ~ '~ f [. E(k ' ) l ,

According to the transformations of coordinates, variables are wl i t ten as follows:

(8) r ''~ = {r(x - Vt)} 2 + y.Z,

(9) k'2 _ 4ar' (a + r ') 2 + z" '

(10) s in0 '= y r ~

and

(11) c o s o ' - y(x - V t ) r '

Here, the origins of x and x' coordinates are coincident with each other at t --- 0. Fur thermore , the following scalar potential exists in this frame as a consequence

84 Y. HAYASHI

of Lorentz transformation:

(12) sinO'(a~l:I(1-~-)K(k')-E(k')}. r = ~.VA'= -~oJVy-~-~--[y ] [[

Next, we try to consider how much energy this potential can give to a charged particle. The work S does on a charged particle is

s j( OA) (13) S = E . d s = - V r .ds. r

Assume that the charged particle exists at (0, Yo, Zo) when t = - ~ (x = - ~ for the centre of the loop) and r = 0 at (0, Y0, Zo) because of k ' = 0. As time goes by, the loop passes through near the particle and leaves. Finally, when the distance between them becomes infinity, r at (0, Yo, Zo) is again zero. Therefore, the scalar potential part in eq. (13) does not contribute, then

J ~A (14) S = - -~-. ds.

To calculate this integral exactly, we have to solve the equations of motion of the particle and this is very difficult. Here, we roughly evaluate S by considering the following ways. Consider that the loop goes near a particle. Since the magnetic fields produced by the loop are not uniform, the motion of the rotation centre equals the motion of the particle on which the force ( - grad B) acts and the particle gets not only x momentum component but also y momentum component. Though the trajectory of such a particle depends on the initial conditions, we approximate that the particle is drifted along the y-axis. The value of S calculated by such an approximation may be the maximum energy which the loop can give the particle.

Furthermore, the time t can be fixed as t = 0, because the time dependence of Ay/3t during the particle moves along y-axis passing through above the loop is

not so strong that we may use (3Ay/3t)lt=o as a typical value. For these reasons, we take the integral path from Y0 to ~ along the y-axis at t = 0 and z = z0 = const. Hence, the work that is the energy ~ given to the particle is thus obtained as

(15) ~ J 3t [t=o

where

.

(16) k '2 = 4alyl (a + y)2 + z~


and

(17) r ' = lYl.

Additionally, as k' is always equal to or smaller than unity,

(18) ( 1 - - ~ - ) K ( k ' ) - E(k') =

= ~ [ ~ 6 k ' 4 + 1 (3" 1/2 2 214-2]

1 ~ ( 2 r - 3)!!~2 k,2r + ...1 ~_~_~ k,t k ' 6 + " ' + 2 [ ( 2 r - 2 ) ! ! J 32- "

Substituting eq. (18) into eq. (15), we get

(19)

4 \Zo] [

Yo+ a 1 [(a + Yofl + zo2] 1/~]

2a a - Yo (a 2 + y~)1/2 [(a - Yofi + z2] 1/2

(for Yo i> 0),

(for Yo < 0).

Since the values in square brackets are of the order of unity, then

~~ 2 (20) ~ 4 ~o "

For example, taking/~ - 1/V2 and J ~ 1020 A, then we obtain

(21) s - 9 . 4 " 1021(eV) (~oo) 2

and, taking ~ ~ l /V2 and J - 1017 A, we obtain

(a)2 (22) ~--9.4" 101S(eV) ~o "

If we want energies near 10 ~~ eV, then Zo- 30 a for eq. (21) and Zo- 0.3 a for eq. (22). Of course various energies can be obtained for various Zo. Therefore we can regard the superconducting string as a source of ultra-high-energy cosmic ray.

3. - Ultra-high-energy cosmic ray and cosmic string loops.

As discussed in sect. 2, cosmic string loops carrying current J, if any, can accelerate charged particles. The energies of particles given by such a loop can

86 Y. HAYASHI

reach (101~ 102~ That is, protons which distribute in the Universe are accelerated by the loop to those energies.

Now, let us estimate the integral spectrum of cosmic ray from a loop. It is assumed that charged particles distribute uniformly with density n and that the loop moves in the same way described before. When the loop passes through, particles whose distance from the (x, z)-plane is smaller than a get energies according to eq. (20). This equation also shows that the same energy is given to particles at the same z0, thus we obtain a differential flux as follows:

(23) d N _ n V a dz _ n V a 2 dz de ~ (r~o VJy) 1/2 ~-~a,

where N is the number of particles accelerated by the loop per unit time. From this equation we get the integral spectrum at ~ times z~.5 as

(24) 1 n V a 2 47: L - - - - ~ 2 (~o V J r ) 1/2 ~1.o =

=2.3- 10Sn /2 r!/2 Jl~ ~L0 (eV)l"5 m-2 s-1 sr-1,

where L is the distance from the earth to the loop. Additionally, we calculate the energy spectrum ~3I(~) from the above expression. Since

~3 I(r = - ~3 d/(> r de '

we obtain the following equation:

(25) (a)2 ~3 I(~) = 1.2- lOSn ~ /33/2 y1/2 j1/2 eL5 (eV)2 m-2 s-1 sr-1.

Equations (24) and (25) can be written in the following forms:

(26) r ~) = Azl.0

and

(27) ~3I(~) =0.5A~ LS.

Here, we consider that cosmic rays above 5.1019 eV mainly come from the cosmic string loop and determine A = 8 . 1 0 .6 of eq. (26) from the data point, ~1.5 i (> ~) _ 4- 1014 (eV 1"~ m -2 s-1 stff) at ~ - 5.1019 eV. This leads to e 1-5 I (> e) ~ 8- 1014 at ~ ~ 1020 eV which is consistent with the experimental value

S U P E R C O N D U C T I N G COSMIC S T R I N G LOOPS AS COSMIC A C C E L E R A T O R S ETC, 87

1030

' - 1025

i E

A 10 z~

10~5 o ~ 1 7 6 1 7 6 8 oooo oooco oo o ~ oo o o i i i i

10 ]~- 1018 1019 10 20 10 21 1022 E(eV)

Fig. 1. - The integral energy spectrum of ultra-high-energy cosmic rays defined by eq. (24) for each set offl and J, denoted by (fl, J), when we set n = 4.2. 106m -s, L = 100 Mpc and a = l p c . A)fl=I/V~ and B) f l=10 -t. 1) J = 3 . 9 . 1 0 2 ~ 2) J = 3 . 9 . 1 0 ~ g A and 3) J = 1017 A. Circles denote experimental data (').

10 z~

~'~ 1019

T

? E _~ ;0 17

A

1015

17 10

~176 o IB,21 . o o o o / /

o o

,

10 ~8 10 ~9 10 2o ~0 2] 1022

E(eV)

Fig. 2. - The integral energy spectrum of ultra-high-energy cosmic rays defined by eq. (24) for the same parameters as fig. 1 except n = 5m -3, Dashed line is eq. (26) with A = 8- 10 -8.

88 Y. HAYASHI

10 26

lo 25

"7 k,

? s

~'~ 10 24-

10 23 1017

o o

10 18 10 19 10 20

E (eV)

Fig. 3. - The energy spectrum of ultra-high-energy cosmic rays defined by eq. (25) for n = 5m -3 and fl = 10 -4 with the same values of J as fig. 1. Filled and open circles denote the experimental data at Akeno (5). Dashed line is eq. (27) with A = 8.10 6.

7.1014 at e -1020 eV. Equat ions (26) and (27) for this value of A are shown in

fig. 1 and 3. Next , let us examine whether reasonable parameters (J, ~, n, a and L) can

reproduce the integral spectrum which is consistent or not with the data. Recently, Ostr iker et al. and Hill et at. (~) pointed out tha t J can reach 3.9.1020 A for M a - 1018GeV or 3.9- 1019A for M ~ - 10~SGeV at the maximum. So, we take the following three J values: J = 1) 3.9.10~~ 2) 3.9.10~9A and 3) 10~7A as a comparison. Fur the rmore , values of velocity of the loop are taken as A) fl = 1/~/-2 and B)f l = 10 .4 (the rotat ion velocity of the solar sys tem in our galaxy). L and a are taken 100 Mpc and a few pc. In addition, two cases are

considered: one is the case that the loop is in a galaxy with n = 4.2- 106m -3 and

the other in a intergalactic space with n = 5 m -a. F igures 1 and 2 show eq. (24) for each variation of parameters sets with n = 4 . 2 . 1 0 6 m -~ and 5 m -3,

respectively. For these values of a and L, these figures indicate tha t eq. (24) is consistent with the data only when ~ = 10 -4, J = 1019 and 1017 A. Figure 3 shows

(4) M. TESHIMA, T. HARA, N. HAYASHIDA, M. HONDA, F. ISHIKAWA, K. KAMATA, T. KIFUNE, M. V. S. RAO and G. TANAHASHI: Proceedings of the XX ICRC, Vol. 1 (Moscow, 1987), p. 404. (5) j . p. OSTRIKER, C. THOMPSON and E. WlTTEN: Phys. Lett. B, 180, 231 (1986); C. T. HILL, D. i . SCHRAMM and T. P. WALKER: Phys. Rev. D, 36, 1007 (1987).


the s p e c t r u m eq. (25) for A) n = 5 m -3 and B) fl = 10 -4 wi th the r ecen t da ta at

A k e n o (5) and it is found t h a t the s p e c t r u m can also be cons is ten t wi th the da t a

for these pa rame te r s .

Now, let us find cons t ra in t s of p a r a m e t e r s due to the exper imenta l data,

~ 1 5 I ( > ~ ) - 4 . 1 0 1 4 at s - 5 . 1019eV. Subs t i tu t ing these values, we obtain the

fol lowing relat ion:

(28) n - j1/2f13/2 rl/2 = 3 .5 .10-14A~ -8.

TABLE I. - Values of a/L fo r various J, fl and n. J's are taken as 1)3.9. 102~ 2) 3.9.1019 A and 3) 1017 A. fl's are taken as A) 1/~v/2 and B) 10 -4. I) is for n = 4.2.106 m -s and II) is f o r n = 5m -3.

I) n = 4.2.106 m -3

1) 2) 3)

A) 7.7.10 -16 1.4.10 -15 6.1.10 -15 B) 6.5" 10 -13 1.2.10 -12 5.1.10 -'2

II) n = 5m -3

1) 2) 3)

A) 7.0.10 -13 1.3.10 -12 5.6.10 -12 B) 5.9.10 -1~ 1.1.10 -9 4.7.10 9

TABLE II. - Constraints f o r a due to that L cannot be larger than the horizon of the Universe (3.1025 m). These values are the largest values of a for each set of parameters of table I.

I) n = 4.2- 106 m -3

1) 2) 3)

A) 2.3. 101~ 4 .1 .101~ 1.8' 1011 B) 1.9" 1018 3.5" 1013 1.5" 1014

II) n = 5m 3

1) 2) 3)

A) 2.1- 1013m 3.8.101~ 1.7- 1014 B) 1.8.1016 3.2.1016 1.4.10 '7

90 Y. HAYASHI

TABLE I I I . - Values of a for each set of paramaters of table I when we set L = 100 Mpc in eq. (28).

I) n=4.2.106m 8

1) 2) 3)

A) 2.4- 109m 4.2.109 1.9- 10 l~ B) 1.9.1012 3.5.1012 1.5.1013

II) n = 5m -3

i) 2) 3)

A) 2.2.1012 m 3.9.10 '2 1.7.10 '3 B) 1.8- 10 '6 3.2.1016 1.4.10 '6

Table I shows the values of a/L for parameters for ~, J and n. The fact that L cannot be larger than the horizon of the Universe (3- 10 ~ m) restricts the radius a listed in table II. It is clear that a can be few pc when n ~ 5 m -8 and ~ - 10 -4. In addition, setting L - 100 Mpc, we obtain the values of radius a shown in table III. From this table, the radius a is a few light years when n - 5 m -'~ and /~ ~ 10 -4.

As we have seen above, cosmic-string loops with suitable radius can be an origin of ultra-high-energy cosmic ray for n - 5 m -3 and ~ ~ 10 -4. In other words, there can be a loop with radius of few pc moving not so fast in the intergalactic space. However, larger loops (a > a few pc) carrying such a large current cannot exist in the Universe.

4. - C o n c l u s i o n s a n d d i s c u s s i o n .

We discussed the acceleration mechanism using the electromagnetic field produced by the moving a superconducting cosmic-string loop carrying large current and put restrictions on the radius of the loop with the help of the observation of ultra-high-energy cosmic ray. Results are the following:

1) ultra-high-energy cosmic-ray energy spectrum due to the loop with suitable parameters can explain the observation,

2) the integral energy spectrum I(> ~) is proportional to -lJ2,

3) the loop might exist in the intergalactic space,

4) the loop moving with large velocity ( - c ) might not exist in the Universe.

There may exist a loop whose radius is of the order of (109 - 1013) m. Such a loop is permitted to be nearer or to be in a galaxy. However, by considering


gravitational radiation, such very small loops have already decayed and cannot exist till now.

The formula for the acceleration energy, eq. (19), might be the maximum voltage because we take the path along the y-axis and t = 0 when we derive eq. (19). So the flux given by eq. (23) may be overestimated. Expected lines shown in fig. 1-3 go down to some extent.

Here, we assume that the loop moves with constant velocity. But giving energy to particles is a kind of friction, so that the loop might stop at last. On the other hand, Vatchaspati and Vilenkin (~) pointed out that gravitational radiation due to asymmetric oscillation can give momentum to the loop, so the loop may continue to move in the Universe.

Though we have calculated eq. (14) as eq. (15), eq. (14) depends on the trajectory of the particle essentially. To examine the condition that eq. (15) is reasonable, we consider the loop' rest frame. In this frame, particles approach the loop with velocity V. If kinetic energy of the particles in the unit volume

~ n m p W / 2 (mp is the mass of the proton) is smaller than the magnetic energy densities at the centre of the loop ~ m - B2/2~o, particles cannot pass above the loop. Equation (15) is not reasonable at least in this case. So, eq. (15) is reasonable for the particles satisfying ~, I>,:,, so that

r T~ 112 TFt~II'2 V (29) B ~< 0 r �9

Then the loop gives energies r smaller than ~, to such particles as follows:

(30) -c, .- V B a ~ (~o mp)l~2 n l / " a V 2 ~ 4.6.10 17 nl~2 a V "2 >1 ~ .

Therefore, the energy spectrum calculated above may be reasonable for ~ ~< ~ and may be overestimated for ~> Er. When we set a-101~m, ~-1019, 101~ 101~ and 109eV for (n,~)=(106,1/V2), (10~,10-4), (5,1/V2) and (5,10-4), respectively. Expected lines in figures may be lowered except the lines for (n, fl, a ) = (4.2.10 ~, l/V2, 101~). So, the constraints considered above may be too strict. In order to calculate the energy E given by the loop to charged particles exactly, we have to solve relativistic equations of motion for the particle and integrate eq. (14) along its path. But to obtain an exact solution for these equations is very difficult. So, we make very rough estimations as described in sect. 2. It will be possible to make evaluations by a numerical calculation.

Because the loop carrying a very large current makes a very strong magnetic field around it, there is a possibility that particles passing through near it might be trapped by the field and form a plasma surrounding it. The plasma may be screening the magnetic fields C). In this case, acceleration considered in this

(~) T. VATCHASPATI and A. VILENKIN: Phys. Rev. D, 31, 3052 (1985). (7) E . M . CHUDNOVSKY, G.B. FIELD, D. N. SPERGEL and A. VILENKIN: Phys. Rev. D, 34, 944 (1986).

92 Y. HAYASHI

paper cannot occur. But if the loop has the radius and the current which produce magnetic fields near the centre of it satisfying ~p > ~m, charged particles may be not t rapped so much and plasma screening effects may be weak.

Throughout this paper, the reduction of the radius by the gravitational radiation is not considered, because the lifetime of the loop ( - 1 0 n s for a -101S m ) is so long compared with the observation time so far ~107s on pr imary cosmic rays that the magnitude of the radius will not reduce so much. Moreover, an energy loss and pile-up of emit ted protons due to collisions with 3 K photons is not considered. If this is considered, restrictions would be changed.

Finally, in this paper it is pointed out that our acceleration mechanism needs

only a large current J , not necessarily Jm~, in a moving loop.

I would like to thank Prof. K. Kobayakawa and Prof. T. Morii for helpful discussions and comments on the paper. I would also like to thank Dr. K. Kitakaze and Dr. H. Nishimura for useful discussions.

�9 RIASSUNTO (*)

Un cappio della stringa cosmica superconduttiva pub trasportare una corrente - 1 0 ~ A cosi grande da causare un forte campo magnetico intorno ad essa. I1 moto del cappio origina un campo elettrico che accelera le particelle cariche. Si presenta un semplice modello per il meccanismo di accelerazione. Si mostra che le particelle possono produrre energie ultra alte - 10 z~ eVe il loro spettro d'energia integrale ~ proporzionale a -1~. Lo spettro previsto per parametri ragionevoli della stringa ~ compatibile con gli esperimenti. Si esaminano anche i vincoli sul raggio del cappio.

(*) Traduzione a cura della Redazione.

C n e p x n p o s o ~ a m x e KOCMNqecKHe cTpynm,~e neTmt, KaK KOCMEIqeCKHe yCKOpNTe.TIll KOCMIfqeCKHX JlyM e~i.

Pe3mme (*). - - CBepxnpoBo~atuaa KOCMrlqeCKa~I cTpyrlna~ neTJm Moz~eT HeCTIt TaKOfi 60J1btliOfi TOK ~ 102~ KOTOpbI~ Bb]3bmaeT CnJlbHoe MarHHTHOe noJle BoKpyr neTJJn. JIBn~eHne neTnn nopoa~aaeT 3aeKTprIqecKoe noJ~e, KOTopoe ycKop~eT 3apa~reHn~ie qacrHU~L YIpeanonaraeTca npoeraa MO~e~ MexaHn3Ma ycKopeaVLq, l"[OKa3bIBaeTC$l, qTO qacTnaS~ MOryT ycKOpnT[,C~ JIO yabTpaBblCOKtlX 3Heprn~i ~ 10 20 aB n ax HHTeFp~JIBHhlI~ 3HepreT~fqecKH~ crleKTp OKa3BIBaeTC~ Ilporlopl~14OHa~ll, HblM e -1/2. O~KnJlaeMhll~ CneKTp Jln~ npaB~OnOaO6nbLX napaMeTpoB cTpya~t cor~acyeTcn c aKcnepnMeHTaMm TaK~Ke o6cy~a~oTca orpaHnaeHn~ Ha paJlHyc ~teT~H.

(*) HepegeOerto pe6a~r4uefi.

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Superconducting cosmic string loops as cosmic accelerators of cosmic rays