7
COSMIC RAYS FROM PULSARS G. S. Saakyan, D. M. Sedrakyan, E. V. Chubaryan, R. M. Avakyan, and G. P. Alodzhants A new mechanism is suggested in this paper for the generation of cosmic rays in the magnetosphere of pulsars. Equations are derived for the total number of particles in the magnetosphere, the rate of decrease of this number with time, and the second derivative of the period with respect to time. On the basis of the observed flux of cosmic rays, the total number of pulsars in the galaxy is estimated, as well as their distribution with respect to the periods (it is assumed that cosmic rays are generated principally by pulsars). The idea has been advanced in a number of papers [1-3] that cosmic radiation is gener- ated in pulsars. It is suggested in the papers [2, 3] that high-energy cosmic rays are form- ed by means of the acceleration of particles beyond the light cylinder in the wave field of the magnetic-dipole radiation of an oblique rotator. A different mechanism is suggested in this paper for the generation of cosmic radiation particles in the magnetosphere of pulsars. i. The Magnetosphere of Pulsars The parameters of the magnetosphere of pulsars have been investigated in the paper [4]. In the presence of a strong magnetic field a ring-shaped plasma magnetosphere with the fol- lowing parameters: the interior radius of the ring is equal to ={2GM I . "~ k 3~ ~ / (I) where M is the mass of the star and ~ is the angular velocity of the rotation, is formed near a rotating baryon star in the vicinity of its magnetic equator. There is no matter in the distance range R < r < rl. Here R is the radius of the star. The plasma extends as far as the light cylinder, i.e., to a distance where c is the velocity of light. order C r 2 = --, (2) ~2 The effective thickness of the magnetosphere is of the 7.42.106 wl/2 z o = -- ' 6 , (3) where T = I06T6 is the temperature of the plasma. It is assumed that the axis of rotation coincides with the direction of the magnetic moment. The radial distribution of the density of particles in the plasma is determined by the diffusion equation [4] On(r,at t)_v{xr~n2(r' /)[ ;nZ(r,n "~(r, t)t) k ?• (4) Erevan State University. Translated from Astrofizika, Vol. Ii, No. 4, pp. 679-687, October'December, 1975. Original article submitted July 12, 1974. This material is protected by copyright registered in the name of Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part oft his publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $Z50. 44

Cosmic rays from pulsars

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COSMIC RAYS FROM PULSARS

G. S. Saakyan, D. M. Sedrakyan, E. V. Chubaryan, R . M. Avakyan, and G. P. Alodzhants

A new mechanism is suggested in this paper for the generation of cosmic rays in the magnetosphere of pulsars. Equations are derived for the total number of particles in the magnetosphere, the rate of decrease of this number with time, and the second derivative of the period with respect to time. On the basis of the observed flux of cosmic rays, the total number of pulsars in the galaxy is estimated, as well as their distribution with respect to the periods (it is assumed that cosmic rays are generated principally by pulsars).

The idea has been advanced in a number of papers [1-3] that cosmic radiation is gener- ated in pulsars. It is suggested in the papers [2, 3] that high-energy cosmic rays are form- ed by means of the acceleration of particles beyond the light cylinder in the wave field of the magnetic-dipole radiation of an oblique rotator.

A different mechanism is suggested in this paper for the generation of cosmic radiation particles in the magnetosphere of pulsars.

i. The Magnetosphere of Pulsars

The parameters of the magnetosphere of pulsars have been investigated in the paper [4]. In the presence of a strong magnetic field a ring-shaped plasma magnetosphere with the fol- lowing parameters: the interior radius of the ring is equal to

={2GM I . "~ k 3 ~ ~ / (I)

where M is the mass of the star and ~ is the angular velocity of the rotation, is formed near a rotating baryon star in the vicinity of its magnetic equator. There is no matter in the distance range R < r < rl. Here R is the radius of the star. The plasma extends as far as the light cylinder, i.e., to a distance

where c is the velocity of light. order

C r 2 = - - , (2) ~2

The effective thickness of the magnetosphere is of the

7 . 4 2 . 1 0 6 wl/2 z o = - - ' 6 , ( 3 )

where T = I06T6 is the temperature of the plasma. It is assumed that the axis of rotation coincides with the direction of the magnetic moment.

The radial distribution of the density of particles in the plasma is determined by the diffusion equation [4]

On(r,at t)_v{xr~n2(r ' / ) [ ;nZ(r,n "~(r, t)t) k ?• (4)

Erevan State University. Translated from Astrofizika, Vol. Ii, No. 4, pp. 679-687, October'December, 1975. Original article submitted July 12, 1974.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, New York, N. Y. 10011. No part o f t his publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available from the publisher for $Z50 .

44

Page 2: Cosmic rays from pulsars

Here the following notation is introduced:

c~- 4 c% '~ ( 2~rn ) ~12 A 3 ~2 \ k T

[ (~T)'~"2 i _ A = In (4r.n)ir;e 3

F m9.% Gram F 3

where n is the density of particles in a hydrogen plasma, ~ is the magnetic moment of the dipole field, m is the mass of the proton, r is the distance from the axis of rotation, k is Boltzmann's constant, and e is the charge of the electron. Equation (4) allows a self- similar solution of the form

n (r, t) = s , r , . (5) t

Substituting (5) into (4) for f(r), we obtain the following equation:

( m 9 ~ GmM 1 Z . ) f , /4m~2 2.5GraM l ) f @ 1 0 ' (6) f " + f ' I k T ' k r r ~ r \ ' k T k T r ~ 2or'

where t h e pr ime s i g n d e n o t e s d i f f e r e n t i a t i o n w i t h r e s p e c t to r . An a p p r o x i m a t e s o l u t i o n o f this equation with the boundary conditions

has the form [4] f(rx) = /0"2) -- 0

f (r ) =

d) d b 1-- 7 ~ for r l ~ r ~ - ~ r 2

4.8 b --r~r~ (1 -- ~r2/'~U2 for ~-8 r2 ~< r .< r.,, i7)

where

The total number of particles in the magnetosphere is equal to

N ( O = -

Taking account of (3) and (7), we find

0 3 r~

; i 2~ e--ZZ/ZX dz rf (r) dr. t ,

- - o o rl

= i0 -'~ where ~3o p, and p = 2v/~ is the rotation period of the pulsar. We rewrite Eq. (8) in the form

N ( - ) = N~ 1 -k =/to" (9)

Here the notation t = to + T is introduce; T = 0 corresponds to the time of the pulsar~s observation and

No (p) 2.35-10 ~~ 4~z [0 , �9 6 P3o P

(lO)

449

Page 3: Cosmic rays from pulsars

represents the number of particles at the instant x = 0. It is obvious from (9) that to is the time during which the number of particles in the magnetosphere decreases by a factor of two.

2. Some Relationships among the Parameters of a Pulsar

We postulate that the observed slowing down of the rotation rate of pulsars is caused by the escape of particles at the light cylinder. In agreement with this postulate one can write

a L ldN, ( 1 1 )

d~ d~

where L is the angular momentum of the star, I is the momentum of a single particle leaving the star at the light cylinder,

and c is the energy of the particle.

l = r e - - , (12)

It follows from the last two equations that

4~I dp ~p2 dN, (13)

p dr dv

where I is the moment of inertia of the baryon star. It is easy to note that the relation- ship (13) also follows from the law of conservation of energy. If now we calculate from (9) the value of dN/dT at the instant T = 0 and substitute it into (13), we obtain the equa- tion which defines to:

t~ + 12 =~1 Nat~ 4=~I \ o

Here p ~ d p / d T , and Noto and c are functions of the rotation periods of the pulsars. The dependence of Noto on p is determined by Eq. (!0). Knowing also the dependence of e on p and taking the values of p and ~ at the present time from the observational data, one can determine with the help of Eq. (14) the dependence of to on the parameters of the star.

Let us now determine the dependence'of e on p.

The plasma is strongly magnetized right up to the light cylinder itself and rotates in a rigid manner along with the star. The injection of particles into outer, space occurs at the light cylinder at distances very close to r= = c/a, when the velocities of the particles are relativistic. In this aase the radius of curvature on the trajectory of a particle in the magnetic field increases strongly, and when it begins to exceed the radius of the mag- netosphere r2, the particle, moving along a diverging spiral, leaves the magnetosphere.

Evidently, this occurs when the Larmor frequency becomes equal to the angular velocity of the rotation ~,

cZeB . . . . . !2,

where g is the energy and Ze is the charge of the particle. leaving the magnetosphere is equal to

(15)

Thus the energy of a particle

4~2eFZ 1 21.1 ~aoZ

c ~ p2 p2

Substituting (16) into (14) and taking account of (i0), we find for to

1 M~. 2,~3 T6,3o to ---- 1 . 1 2 ' 1 0 a I~; 2 \ - P - - / o j "

(16)

(17)

Thus, knowingthe mass of the star, its moment of inertia (see Table I), magnetic moment, and the temperature of the magnetosphere, and using observational data for p and ~, one can deter- mine the total number of particles in the magnetosphere and the time for the particles in it to decrease by a factor of two.

4 5 0

Page 4: Cosmic rays from pulsars

TABLE i. Range of the Most Important Param- eters of Baryon Stars Which MayBe Pulsars [ l l , 121

p {0) M/M. ' (L ) [ I. 10- 44 I , ---' (g/cma) ] (g" cm~) [ N-10-57 . . . . . . . �9 10 ,. (see -1 )

8.69.1013

1.37-1014 1.78.1014 2.27.1014

2.88.1014 3.62.1014 4.56.1014 5.77.1014 7.39.1014 9.59.1014 1.27.10 t5

1.75.1015 2.1015

0.118 O. 145

O. 172 O. 208

0.265 O. 372 0.518

O. 672 O. 845 1 . 0 5

1 . 2 6

1 . 4 1

1 , 4 6

160

66 43 25 18

15 14 14

13 13

13

13

12

5.50

4.4

3.3 2.2

1.6 2.6

3.8

5.2 6.9 8.7

10

t l 11

0.15

0.16 0.17 0.22

0.32 0. 472

0.65 0.85

1.1 1.3 1.7

1.9 1.9

0.59

2.2 5.7

10

19

23 31

44 57 74

89 100

110

Note. M, R, and I are the mass, radius, and moment of inertia of the star, N is the total number of baryons in it, p(0) is the central

density, and ~max = (GM/R3) I/2 is the maximum value of the angular velocity of rotation.

For the pulsar in the Crab Nebula, taking M = 0.5 M =,, I = i04" g.cm 2, a = 200 sec -I, = 1031G'cm 3, andT6=l, we find No = 2.43"10 ~2 and to = 250 yr.

The value obtained for to is approximately four times smaller than the known age of pulsar 0531. This indicates that, notwithstanding the comparative youth of this pulsar, its magnetosphere has already had time to reach a quasisteady state. On the other hand, the flux of particles per unit time is of the order of No/to % 3.11"i032sec-~. It is obvious from Eq. (16) that the energy removed by a single proton and electron is of the order of 2"10 ~ ergs, and the energy losses produced by the corpuscular radiation of the pulsar are of the order of 6.1037 ergs/sec, which agrees in order of magnitude with the total observed x-ray radiation of the Crab Nebula [5]. It is possible to calculate the value for this same pulsar of the second derivative of the period with respect to the time. According to (9) and (13),

, = - ,,o I i T(;)o ,: + T (;)o 1' at the instant T = 0, whence we obtain p~ 9.10 -23 sec -I for the pulsar which we are discuss- ing. This value exceeds the observed value [6, 7] by a factor of nine, which one can assume to be satisfactory agreement, since the values of the parameters M, ~, T, and I may differ somewhat from the values which we selected. Eq. (18) can give important information about the parameters of pulsars if one is successful in measuring p accurately in addition to ~.

3. Cosmic Radiation from Pulsars.

The primary spectrum of cosmic radiation is approximated by the equation [8, 9]

7 ( E ) d E _ t A E - Z 6 d E for 0 . 0 1 6 < E < a 2 8 0 e r g s ( 1 9 )

- [ (1 .236 :! 0 .25) . 10 -2 E -3"-•176 dE ' ~ r i 2 8 0 ~ E ~ 6.4.105 ergs.

Here J(E)dE is the flux of particles calculated per unit solid angle (cm-~'sec -! sr-1). We find from the matching condition at E = 1280 ergs that A = 1.69-10 -4 . The number of par- ticles with energy E > Eo is denoted by J:(Eo).

According at Eq. (16), the particles ejected by the pulsar having the maximum period of 3.67 seconds possess the least energy. Their energy is c = 1.6 ergs. Employing (19) for the value of J1 at Eo = 1.6 ergs, we find

451

Page 5: Cosmic rays from pulsars

6

Fig.

�9 �9 �9 �9

-,1..6. -1 2 -0.8 -0.4 0 0 0.4

IgP

I. Plot of the dependence of the logarithm of the age of pulsars on the logarithm of the period.

TABLE 2. Galaxy

Period, p (see)

0.2 0.4

Number of Pulsars in the

i N'o t of p _ _ u!sars with period < p " theory'

in galaxy [ in vo lume observation of radius .... 2,kp, e ,,

135 4 8 6.61.102 169 27

0.6 5.50.104 0.8 2.19-105 1,0 6.35,105 1.2 1.45.106 1.6 4.79-106 2.0 1.20-107 3.0 6.31-107 4.0 I 6 6 1G s

! 1.41-t02 50 I 5.61.102 69

1.63-10 ~ 70 3.72,10 ~ 81 1.22.10 s 98 3.08.10 ~ I01 1.61-10 ~ 104 4.27.10 ~ 105

J1 (1 .6e rgs ) = 5 -10 - s c m . Z s e c . l s r _ ~

However, according to recent experimental data [i0],

J , (1.6 ergs) = 1 . 3 4 - 1 0 - s era_ 2 sec_lsr?1.

Let us estimate the total number of pulsars in our galaxy assuming that the observed flux of cosmic rays is caused only by their corpuscular radiation. We will also assume that the distribution of the pulsars at our epoch with respect to the periods is steady. If tg is the time interval which elapses after the establishment of this steady-state distribution, P(p)dp is the number of pulsars in the range (p, p + dp), and V is the volume of the galaxy, then

c.t dN g--~ P(p} dp ~ 4~VJ (E)dE. (20)

Substituting the value of dN/dT from Eq. (13) into (20), we find

P(p) dp -- 4=Vel~ p J ( E ) dE. ( 2 1 ) lc3tg p

The slowing-down time p~p for the majority of old pulsars is of the order of 106-107 yr; therefore, one can assume p/p to be constant in estimating the total number of pulsars in the galaxy. Then

452

Page 6: Cosmic rays from pulsars

0 s

Pl (P,) ---- j P (P) dp~ Po

4r, el~Vlc% (-P) J~(Eo), (22)

where P1(po) is the total number of pulsars with periods less than po, and po = 3.67 sec. 2

Substituting the values into (22) for the volume of the galaxy, V = ~Rgh = 7~3"i06~ cm ~ (R r = 12.5 kpc is the radius and h = 5 kpc is the thickness of the galaxy) and the intensity of cosmic radiation J1(Eo), we get

Pg(Po) ~-2.2"10~ lb._g_0 (._~_]. (23) \ p /

Here (p/~) i s measured i n u n i t s o f 10 s y r , and 144 = 10 -4~ I g.'cm =. I f we t a k e >30 = I44 = 1 and t_ = 109 y r f o r a l l p u l s a r s , t h e n t h e e x p e c t e d number i n the g a l a x y i s o f t h e o r d e r o f 106-1~ r . I t ' i s known t h a t t he m a j o r i t y o f p u l s a r s a r e l o c a t e d a t a d i s t a n c e o f R = Q,5 kpe , Based on t h e e s t i m a t e o f (22 ) , t he number o f such p u l s a r s i s a p p r o x i m a t e l y equa l to

P = P~(Po) ~ / = 1.6 10 " ~(po). (24)

We obtain P~ 10a-10 ~ for those same values of the parameters ;30, I~4, and tg. If our estimates are true, then we must assume that less than 10% of all the pulsars occurring within a sphere of radius 0.5 kpc have been detected up to the present time.

Finally, using the relationship (21), it is possible to find the expected period distri- bution of pulsars corresponding to the observed flux of cosmic radiation.

The slowing-down time p/~ can be derived as a function of the period p on the basis of existing observational data. A plot of log p/~ versus log p is given in Fig. i, which can be approximated by the equation

P 3.15.1014 1.1>.-l~.,, (25) P

Errors as large as 50% produced by the large scatter in the values of p/p are not ex- cluded in this equation. Substituting (19) and (25) into (21) and having expressed E in terms of p by Eq. (16), we obtain the following period distribution of pulsars:

4.53- 0.9871g P ( p ) = 34.9.p for p <~ 0.128 sec,

p(p) = 2.97.p~.3~ 0.~87,,...I, for p ~ 0.128 sec. (26)

The number of pulsars in the galaxy having period less than a specified value is given in Table 2 on the assumption that the entire flux of primary cosmic radiation is produced only by pulsars. We do not assume that the proposed mechanism is the only one possible to explain cosmic radiation~ The possibility of the existence of other mechanisms for its generation is not excluded. This refers, in particular, to the cosmic radiation of energy E < 10 13 eV.

The presence of other mechanisms for the generation of cosmic radiation obviously results in a decrease in the expected number of pulsars in the galaxy. In this sense, the numbers cited in the second column of Table 2 give an upper limit to the number of pulsars. One can evidently also conclude from the data of this table that a significant part of the slow pulsars has still not been observed.

LITERATURE CITED

I. T. Gold, Nature, 22!, 25 (1969). 2. J. E. Gunn and J. P. Ostriker, Phys. Rev. Lettr., 22, 728 (1969) 3. C. F. Kennel, G. Schmidt, and T. Wilcox, Phys. Rev. Lett., 31__, 1364 (1973). 4. R. M. Avakyan, A. P. Avetusyan, G. P. Alodzhants, G. S. Saakyan, D. M. Sedrakyan, and

E. V. Chubaryan, Astrofizika, 11, 27 (1975). 5. P. Morrison, Ann. Rev. Astron. Astrophys., ~, 325 (1967). 6. C. Pepa!iolis, N. P. Carleton, and P. Horowitz, Nature, 228, 445 (1970). 7. I. G. Duthie and P. Murdin, Astrophys. J., 163, 1 (1971).

453

Page 7: Cosmic rays from pulsars

8. A. Ramakrishnan, Elementary Particles and Cosmic Rays, Pergamon (1963). 9. S. Khayakova, The Physics of Cosmic Rays [Russian translation], Mir, Moscow (1973).

i0. M. J. Ryan, J. F. Ormes, and V. K. Balsubrahmanyan, Phys. Rev. Lett., 28, 985 (1972). Ii. G. S. Saakyan, Equilibrium Configurations of Degenerate Gaseous Masses [in Russian],

Nauka, Moscow (1972). 12. G. G. Arutyunyan, D. M. Sedrakyan, and E. V. Chubaryan, Astron. Zh., 48, 496 (1971).

GRAVITATION FIELD OF A FLAT UNIFORM DISK

R. M. Avakyan and I. Horsky

The gravitation field produced by a flat uniform disk is studied. Inside the configuration components are found of the metric tensor, as well as the pres- sure distribution. Matching with the exterior solution of Taub is carried out. An approximate analytic solution is found for low central pressures, and a physical interpretation given of a solution which is constant on the outside.

Problems with spherically symmetric distributions of matter have been investigated by a number of scientists. Their detailed analysis can be found in [i, 2]. The rotation problem has also been essentially solved recently. In [3, 4] the exterior solution was analyzed for a rotating configuration, and interior numerical solutions were also obtained.

It would also be of some interest to consider configuration of planar symmetry [5]. In the present article gravitation field produced by a flat uniform disk is analyzed.

i. In the static case the space--time metric can be written as

d g = e~(~)~ ~ - - e ~(~) ( d Z + dg ~) -- d . 2 ( 1 . 1 )

(the Oz axis being perpendicular to the disk, and the xy plane being in the middle of the disk).

The functions v(z), 1(z), as well as the distribution of the energy density p(z) and of pressure P(z) in the interior of the configuration can be found by solving the Einstein and hydrodynamic equations:

8~k ?," 4- 3 >. '~ . . . . . ( 1 . 2 ) 4 d ["

),'~ X%' 8v.k -~ - - P , ( 1 . 3 )

4 2 d

I r

2 4

x" x'e + X ' v ' 8~k P , (1.4)

2 4 4 d

P" - ( P -~. p), ( 1 . 5 ) 2

Erevan State University. Purkine University, Brno (Czechoslovakia). Translated from Astrofizika, Vol. Ii, No. 4, pp. 689-697, October-December, 1975. Original article sub- mitted December 9, 1974.

This material is protected by copyright registered in the name o f Plenum Publishing Corporation, 227 West 1 7th Street, N e w York, N. Y. l 10011. No part o f this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic,

mechanical, photocopying, microfilming, recording or otherwise, wi thout written permission o f the publisher. A copy o f this article is available f rom the publisher for $7.50.

454