[astronomy and astrophysics library] cosmic ray astrophysics || galactic cosmic rays

17. Galactic Cosmic Rays

We have shown in Chap. 16 that the acceleration of galactic cosmic rays below a few 1015 eV Inucl. with power law momentum spectra can be well explained by diffusive shock wave acceleration near galactic supernova remnants, which also, by the overall power requirements (see Chap. 7), are the most probable sources of galactic cosmic rays. Here we address the subsequent propagation of cosmic rays in the Galaxy. We investigate in particular the relation of the cosmic ray moment um spectra near the solar system to the cosmic ray source spectra, and the infiuence of distributed interstellar re-aeeeleration on the momentum speetra.

Another topie of this seetion eoncerns the origin of ultrahigh energy eosmie rays above 1015 eV Inuel. in galaxies (for a review see Axford 1992 [20]). In the papers ofBryant et al. (1992 [78]), Ball et al. (1992 [27]) and Schneider (1993 [478]) the interest in second-order Fermi aeeeleration of this eomponent by eleetromagnetie turbulenee in the interstellar medium has been revived. Observationally it is well established that the interstellar medium is turbulent up to the largest seales, due to the motion of giant gas clouds, supernova explosions, stellar winds and the formation of superbubbles, loops, ehimneys and galaetie fountains. Measurements of the density fiuetuation indieate the presenee of a Kolmogorov-type fiuetuation speetrum up to seales of 100 pe (Riekett 1990 [432]). Together with an interstellar magnetie field strength of c:::: 5 IlG this would allow, aeeording to (16.7.5), the aeceleration up to particle energies of 5 x Z 1017 eV, whieh in the ease of iron nuclei would explain the aeeeleration of eosmie rays up to total energies of 1019 eV. Density fiuetuations are assoeiated with eleetromagnetie fiuetuations, although the exaet relation is quite eomplex (see Wu and Huba 1975 [567], Minter and SpangIer 1997 [361]). We identify the eleetromagnetie fiuctuations with parallel propagating Alfvenie and isotropie fast magnetosonie fiuetuations for which the relevant particle aeeeleration rates have been calculated in Chap. 13.

17.1 Galactic Cosmic Ray Diffusion 2

Beeause of the geometry of our Galaxy as a highly fiattened cylinder we may adopt the model diseussed in Seet. 14.3.3, where the cosmie ray sourees are distributed in a uniform (Q1(X) = q1 = eonst.), axisymmetrie, eylindrieal

R. Schlickeiser, Cosmic Ray Astrophysics© Springer-Verlag Berlin Heidelberg 2002

436 17. Galactic (josmic Rays

disk of radius Ro and half-thickness zo, and the confinement region is a concentric disk with radius R 1 > R o and half-thickness Zl > Zoo At the edges of the confinement region, we apply the free-escape boundary conditions in the form (14.3.18). As before we neglect spatial convection against spatial diffusion, and we assurne the spatial diffusion coefficient to be moment um dependent but independent of position, i.e.

(17.1.1)

with ",(p) a dimensionless function. If the moment um operator (14.3.1b) is also independent of position (i.e. h(x) = 1), the conditions for the scattering time method are fulfilled, and the steady-state cosmic ray distribution is given by (14.3.27),

CXJ CXJ

Mc(R, z,p) = L L A1m(R, z)N1m(p) , (17.1.2) 1=0 m=l

where A1m(R, z) follows from (14.3.23) and (14.3.26). Each function Nlm(p) obeys the leaky-box equation

(17.1.3)

where the spatial eigenvalues Arm according to (14.3.24) enter, and where Op denotes the moment um operator

_ 1 8 (2 8N1m (p) OpNzm(p) = p2 8p P A 2 8p ,

+ P; V1N1m(p) - p2P10ssN1m(P)) -

where we use \7 . V = 3V1 = const.

17.2 Cosmic Ray Propagation in the Leaky-Box Approximation

We first investigate the case wherc no interstellar acceleration occurs A 2 = O. For cosmic ray nud~i with energies above a few hundred MeV jnucleon we can also neglect any continllous moment um los ses and adiabatic losses, so that (17.1.3) simplifies to

(17.2.1)

Under the leaky-box approximation one retains only the first term in the sum (17.1.2) so that (17.2.1) is equivalent to

17.2 Propagation in the Leaky-Box Approximation 437

(17.2.2)

where [ 2 ]-1 Tesc(p) = AQ1/';;(p) (17.2.3)

is the escape time of cosmic rays from the Galaxy.

17.2.1 Secondary /Primary Ratio

We first apply (17.2.2) to a secondary cosmic ray element (e.g. B) that results from the spallation of primary elements (in the case of B predominantly C, N, and 0). The secondary sour ce therefore is given by Qs = O"svnHMcNO ,

where nH denotes the average interstellar gas density and O"s the partial fragment at ion cross-section for the production of B by CNO-nuclei, which is known from accelerator measurements. The transport equation (17.2.2) for boron nuclei reads

(17.2.4)

where we express the catastrophic loss time due to spallation of B as Tc,s =

(vnHO"f,s)-l. Equation (17.2.4) yields the secondary/primary ratio

Ms MCNO O"f.S + X- 1 (p)

in terms of the traversed grammage

(17.2.5a)

(17.2.5b)

which, for the homogenous cosmic ray confinement volume adopted in the leaky-box approximation, agrees with (3.4.1).

The power law decrease of the measured secondary /primary ratios IX p-b with b ~ 0.5 at relativistic particle energies shown in Fig. 3.25 indicates, according to (17.2.5), a corresponding decrease of the cosmic ray escape time Tesc(p) IX p-b and, according to (17.2.3), a corresponding moment um variation of the spatial diffusion coefficient

(17.2.6)

According to (13.2.23), (13.3.77), and (13.4.10) such a moment um variation is possible for Kolmogorov turbulence when the turbulence spectral index q = 2 - b ~ 1.5 is close to the Kraichnan value, regardless of whether interstellar turbulence consists of sI ab AIfvEm waves, isotropie fast mode waves, or a mixt ure of both.

438 17. Galactic Cosmic Rays

17.2.2 Secondary Cosmic Ray Clocks

N ow we consider tvvo isotopes of one secondary cosmic ray element (e.g. beryllium Be): one i,30tope (9Be) is stable, and the other eOBe) decays with lifetime (-yTd ). HowEver, because beryllium is a secondary cosmic ray element produced by spallatitm reactions of (C,N,O)-nuclei in the interstellar medium, their respective sources are

(17.2.7)

where the partial spallation cross-sections for the channels (CNO) --t9Be and (CNO) --t1oBe are known from accelerator measurements.

Expressing the respective catastrophic loss times due to spallation as Tc,9 = (vnHO"f,9)-1 and Tc,lO = (vnHO"f,lO)-l, the transport equation (17.2.2) for the stable 9Be-is l )tope is

Mg [vnHO"f,9 + Te~1(p)l = 0"9vnHMcNo,

while the transport Hquation for the unstable lOBe-isotope reads

The ratio of the two isotopes becomes

0"10 vnHO"f,9 + Te-;2 (p) 0"9 vnHO"f,lO + (-yTd)-l + Te~hp)

(17.2.8)

(17.2.9)

(17.2.10)

All cross-sections anel the decay lifetime appearing in this equation are known, so by measuring thp ratio (17.2.10) at different energies one c:an infer the value of the gas demity in the c:onfinement region nH, and the value and the moment um dependence of the escape time Tesc(p). By combining with inferenc:es drawn on th e traversed grammage (17.2.5) from secondary / primary measurements one CltLn determine the values of nH and Tesp(p) separately.

Table 17.1 summarizes the density and escape time measures from several c:osmic ray isotopes. All measurements are consistent with a cosmic ray escape time of ':::' 107 yr at non-relativistic particle energies. Moreover, the average gas density seen by t he cosmic rays is substantially lower than the Galactic disk average of 1 atom cm -3, wh ich indicates that cosmic rays have spent substantial parts of their lifetime in low-density regions of the interstellar medium like, for example, the halo of our Galaxy.

The estimates in Chap. 6 have shown that the time scale for continuous moment um los ses of relativistic cosmic ray nucleons in the Galaxy are much longer than the cosmic ray escape time of 107 yr. Thus, aposteriori, neglect of these processes in the transport equation (17.2.1) is justified.

17.3 Propagation Induding Interstellar Acceleration 439

Table 17.1. Density and escape time measurements from cosmic ray docks. After Connell et al. (1997 [105])

Isotope p CR. lifetime R.eference

[atoms x cm3] [Myr]

lOBe 0.18(+0.18, -0.11) 17(+24, -8) Garcia-Munoz, Mason, Simpson '77

0.30( +0.12, -0.10) 8.4(+0.4, -2,4) Wiedenbeck, Greiner '80

0.23(+0.13, -0.11) 14(+13,-5) Garcia-Munoz, Simpson, Wefel '81

0.24 ± 0.07 15(+7,-4) Simpson, Garcia-Munoz '88

0.28( +0.14, -0.11) 27(+19, -9) Lukasiak et al. '94

0.23 ± 0.04 18 ± 3 Connell '97

26 Al 0.28(+0.72, -0.19) 9( +20, -6.5) Wiedenbeck '83

0.52( +0.26, -0.20) 13.5(+8.5, -4.5) Lukasiak, McDonald, Webber '94

0.28( +0. 72, -0.19) 15.6(+2.5, -2.6) Connell, Simpson '97

36CI limits limits Wiedenbeck '85

0.39 ± 0.15 11 ±4 This work

I 54 Mn I 0.37( +0.16, -0.11) I 14( +6, -4) I Duvernois '97

N.B.: Adapted from DuVernois 1997 [149]. These are the quoted, published, confinement times--using a different pathlength distribution (PLD) would alter these values somewhat. Mn confinement is for an assumed 54Mn ß- partial half-life of 1 Myr

17.3 Cosmic Ray Propagation Including Interstellar Acceleration

We now investigate the influence of interstellar acceleration (A 2 cF 0), but we again neglect any continuous moment um losses and adiabatic losses. In this case (17.1.3) becomes

According to Chap. 13 (particularly (13.2.42) and (13.3.40)) the moment um diffusion coefficient A2 (p) and the spatial diffusion coefficient KOIi(p) are closely related. Moreover, at relativistic momenta both diffusion coefficients are functions of the cosmic ray moment um per nucleon p = p/A = R/a, where R is the particle rigidity and a == A/IZI the mass to charge number. Therefore it is convenient to work with the particle moment um per nucleon

p=p/A=R/a (17.3.2)

as a variable, and to introduce the particle phase space density


(17.3.3a)

and

(17.3.3b)

which refer to the moment um per nucleon instead of the total momentum. Equation (17.3.1) transforms to

;2! (p2 D(p) dF:;;(P)) - Flm [Tc~P) + Afm~(P)] = - Q(p) , (17.3.4)

where D(p) = A2(p)/A2. Representing thf' interstellar plasma turbulence by a mixt ure of slab

AlfvEm waves and fase magnetosonic waves with Kolmogorov-type power spectrum with spectral index q < 2 we obtain from Chap. 13 that

K . (-) - ~ 1]( q)c ()2-qß -2-q o~P - 3(2-q)(4-q) aE P , (17.3.5)

and

(17.3.6)

where ß = v / c is the cosmic ray particle velocity, E = V/VA. For abbreviation we introduce the parameter

_ 3 (Bo)2 ( C )2-q 1-q ( _1)q-2 1](q) = 2n(q -1) oB eBo kmin cm eV c , (17.3.7)

which is independent of cosmic ray properties. For relativistic particles E c:::: VA/C = const., ß c:::: 1, so that

(17.3.8)

and D( -) D q-2-q p= 10: p, (17.3.9)

with the constants

K1 = ~. 1](q)c 3 (2 - q)(4 - q)

(17.3.10)

3 vi D1 = - ( ) () In (C/VA) 2qq+2c1]q

(17.3.11)

Finally, we introducE the dimensionless moment um variable

(17.3.12)

17.3 Propagation Including Interstellar Acceleration 441

the corresponding phase space density

(17.3.13)

and the two time scales

(17.3.14)

which are characteristic for interstellar acceleration and escape from the Galaxy, respectively. Hence (17.3.4) becomes for relativistic particles

17.3.1 Mathematical Solution

Multiplying by TfQ2~qx2 we can cast the transport equation (17.3.15) into the self-adjoint form

where for convenience we have introduced

and

x= 2-q,

Tf 2~q -Q

Tc

Tf 2(2~q) -Q

Tzm

(17.3.16)

(17.3.17)

(17.3.18)

(17.3.19)

According to Sect. 14.3.4 the self-adjoint equation for general source distribution q(x) is solved in terms of the Green's function G(x, xo) as

firn = TfQX 100 dxo x6q(Xo)G(x, Xo) , (17.3.20)

where the Green's function satisfies the equation

- X4~X_ - [7/'x2 + AX2+X] G = - c5(x - xo) . d [ dG] dx dx

(17.3.21)

17.3.1.1 Construction of the Green's function. With the substitution v = XX / Ao, where Ao is a constant to be fixed later, the homogenous differential equation corresponding to (17.3.21) becomes

(17.3.22)


Making the ansatz g(v) = W(v)exp(-kv)

with the eonstant k, (17.3.22) yields

v~: + [~- 2kV] ~:

(17.3.23)

+ [-(~ + 7jJx~o) + V(k 2 - A~Ö)]W=0.(17.3.24) Obviously, if we eho·Jse the constants to be

(17.3.25)

(17.3.24) reduees to the differential equation of the eonfiuent hypergeometrie functions M(a,b,v) and U(a,b,v) (Abramowitz and Stegun 1972 [3], Chap. 13)

v--+ -- v -- -+-- W-O d2W [3 ] dW [ 3 7jJ] dv2 X dv 2X 2XAl/2 -,

(17.3.26)

and the solution of the homogenous differential equation eorresponding to (17.3.21) beeomes

H(x) = c1H1(x) + C2H2(X) , (17.3.27)

where

[A1/2XX] (3 7jJ 3 2Al~2XX) , H1 = exp --X- M 2X + 2XA1/ 2 ' ,

X (17.3.28)

and

[A 1/2XX] (3 7jJ 3 2Al~2XX) .

H2 = exp - --X U 2X + 2XA1/2 ' ,

X (17.3.29)

The function H 1 (x) fulfills the moment um boundary eondi tion at x = 0, and the function H 2 (x) flllfills the moment um boundary eondition at x = (Xl (see also the diseussion in Seet. 14.3.4.1). Henee we ean eonstruct the Green's funetion as (e.g. Arß:en 1970 [17])

_~ {H1(X)H2(XO) for 0 :::; x:::; Xo } G(x, xo) c= ( ) () V H1 Xo H2 X for Xo :::; x :::; (Xl ,

(17.3.30)

where

[ dH2 dH1 ] 'E = P(xo) H1 dx - H2 dx X=Xo

(17.3.31)

is ealculated from the Wronskian and the function P(xo) = x~-x from the self-adjoint (17.3.16). Using again Abramowitz and Stegun (1972 [3]) for the Wronskian we obtain in terms of gamma functions


(17.3.32)

Colleeting terms, we obtain for the Green's function (17.3.30)

17.3.2 Sources of Cosmic Ray Primaries

For eosmie ray primaries we assurne, on the basis of the diseussion in Chap. 16, that they are injeeted into the interstellar medium from individual supernova remnants with power law souree spectra

q(x) = qox- S - 2 H [x - Xmin] H [xmax - x] , (17.3.34)

at rigidities above some non-relativistie minimum rigidity Rmin and below R max = 1015 V, eorresponding to Xmin = Rmin/(ampc) and X max = 1061a, respectively.

17.3.2.1 Supernova frequency. In spiral galaxies like ours it is estimated that supernova explosions oeeur with a frequeney of 1 event every 50 years per galaxy (e.g. Ostriker and MeKee 1977 [379]). With a volume of our Milky Way (see Chap. 2) of VG = 1f(13 kpe)2 x 0.3 kpe = 2 x 1011 pe3 this yields the supernova rate

(17.3.35)

Aeeording to Ostriker and MeKee (1977 [379]) the maximum radius attained by an individual supernova renmant of initial explosion energy 1051 E 51 , until it adjusts to press ure equilibrium with the ambient interstellar medium of density nH and pressure P4 = (noTI104) K em-3 , is

R - 55Eo.32n-O.16p-O.20 pe max - 51 H 4 . (17.3.36)

During the typieal residenee time of eosmie rays in the Galaxy of Td '::::' 107 yr (eompare Seet. 17.2.2) therefore N s = vSNTd VG = 2 X 105v5o supernova explosions oeeur. If we assurne that these explosions oeeur randomly in the Galaxy, and eaeh of them oeeupies a volume VSN = 41f R'fnax/3, with the maximum radius given by (17.3.36), then the probability that eaeh volume element of our Galaxy is overrun by a supernova explosion during the typieal eosmie ray residenee time is


N. lT V E O.96 Q = ~ = 0 7 50 51 (7 7) VG . n~48 Pg.6 1 .3.3

Most of the interstellar medium is occupied by the coronal phase (compare Chap. 2) with densities nH r:::: 1O-2nc ,2 cm -3, which is in pressure equilibrium with the other gas phases. (Such a small density of the effective cosmic ray residence volume is also suggested by the density estimate in Sect. 17.2.2.) For the probability (17.3.37) we then obtain

v Eo. 96 Q = 64 50 51

. nO.48 pO.6 c,2 4

(17.3.38)

Every point of the ilLterstellar medium is overrun at least six times by a supernova explosion during the typical residence time of cosmic mys. Therefore on these time sc ales we may neglect the "granular" nature of the cosmic ray source distribution in the transport equation (17.3.15), and regard the cosmic ray source term q(x) in (17.3.34) as a quasi-continuous galactic-averaged injection rate.

17.3.2.2 Dispersion of source spectral indices. We noticed in Sect. 6.1 that the observed radio spectral index values a r from individual supernova remnants, in the GaLixy and in the Magellanic Clouds, exhibit a considerable dispersion rYa r:::: 0.1.') around the me an value (ar) = 0.48. Since the radio emission is non-thermal synchrotron radiation from relativistic electrons 8 = 1 + 2ar (see (4.121):, this implies a corresponding dispersion in the source power law spectral index value 8 of cosmic mys in the injection term (17.3.34) that has to be taken into account (Brecher and Burbidge 1972 [75]).

If we represent tbe distribution of radio spectral indices by the Gaussian (Williams and BridlE 1967 [565], Milne 1970 [358])

( ) __ nv [(ar - (ar))2] n v a r -- ~ exp - 2 '

y2wrYa 2rYa (17.3.39)

this yields with 8 = 1 + 2ar and rYs = 2rYa for the corresponding distribution of the cosmic ray momentum distribution

( ) _ V2nv [_(8-(8))2] n 8 - ;:;; exp 2

yWrYs 2rYs (17.3.40)

For the averaged injEction spectrum we obtain

(x-s) = (OO d8 x-s n(8) Jo nv

(17.3.41)


which, with increasing momentum, exhibits a flattening of the source spectrum. For the averaged spectral index we obtain

_ _ In (x- S ) 1 2 8 = - = (8) - -(J" lnx.

lnx 2 s (17.3.42)

This flattening is a consequence of the fact that supernova remnants with the flattest power law spectra dominate in the superposition at large momenta. Hence if we want to account for this dispersion effect we should use, instead of (17.3.34), the modified source distribution

q(x) = q1X-s-2 H [x - Xmin] H [x max - x]

= Q1X-(s)-2 + (1/2)0"; ln(x/x.) H [x - Xmin] H [X max - x] , (17.3.43)

where X* denotes so me reference momentum.

17.3.3 Steady-State Primary Particle Spectrum Below R rnax

We first study the response of (17.3.20) to the single power law injection function (17.3.34) at particle momenta in the range Xmin < X < X max ,

fIrn (Xmin < X < x max ) = Tfqoa x l:::ax dxoxüsG(x, xo)

QoTfaXgo 1-8 = x exp(-Bj2)J(x, X rnin,Xrnax , B) ,

X (17.3.44)

with

J (x, Xrnin, x max , B) == J 1 + J2

= U(a, b, B) 11 dy y(1-s-x)/x (Xmin/X)X

X e- By/ 2 M(a, b, By)

j (xmax/x)X

+ M(a, b, B) 1 dy y(1-s-x)/x

X e- By / 2 U(a, b, By) , (17.3.45)

3 'ljJ b 'ljJ a == 2X + 2X,V/2 = "2 + 2X,V/2 '

3 (2,\1/2) (3/x)-1 r[a] b == X ,go == X3/x r[b] , (17.3.46)

and

X where ( X) 1/x

Xc = 2,\1/2 . (17.3.47)


Using the asymptotic behavior of the confluent hypergeometric functions for small and large argument (Abramowitz and Stegun 1972 [3])

{ I for Izl « 1 M(a,b,z) ~ (r(b)jr(a))za-b eZ for Izl »1 ' (17.3.48a)

U(a b z) ~ {(r(b -l)jr(a)) Z1-b for Izl «1 , 'z-a for Izl »1 ' (17.3.48b)

we can discuss the behavior of the approximate solution (17.3.44)-(17.3.45). Obviously, we have tn consider the four cases X< X max «xc, x «xc< X max ,

Xmin « Xc < x < X max and Xc « Xmin < X < X max . We discuss each in turn.

17.3.3.1 x < X max « xc. In this case the parameter B « 1 for all momentum values x, and the integral J1 with (17.3.48) becomes

') 1 J1 ~ r:.b - 1 B 1- b 1 dy y(1-s-X)/X

r(a) (Xmin/X)X

~Lr -1 ~ ~-1 (b ) ( ) X-3 [( )S-1 ] 8 -- 1 r(a) Xc Xmin

Likewise

r(b 1) j(xmax/x)'" J2 ~ __ -__ B 1- b dy y(1-s-bx)/x r(al 1

X r(b-1) (X )X-3 [ (X )S+2-X] = 8+2-X r(a) Xc 1- X max

Collecting terms in (l7.3.44) we obtain in this case

qOTf (2A1/2 ) (3/X)-1 flm(Xmin < X < X rnax «xc) ~ --1 ( )

8 - X3 /x ~-1 x

1- _mm _ mm [ 3-X (J:" )S-1 8-1 x3-XXS--:-1]

8 + 2 - X x 8 + 2 - X X;;,~-X

qOTf (2A1/2) (3/x)-1

~ 8 - 1 X3/x (~ - 1)

(17.3.49)

. (17.3.50)

(17.3.51)

a very flat distribution due to the dominance of stochastic acceleration.

17.3.3.2 x « Xc < X max • In this case again B « 1 and the integral J1

remains unchanged from (17.3.49). However, the inteßral J2 becomes


(17.3.52)

in terms of the incomplete gamma function r(o, z) and

(17.3.53)

To lowest order in the small parameters (x/xc) « 1 and (xc/x max ) « 1 the distribution function (17.3.44) also approaches the limit (17.3.51), i.e.

1', (2,\1/2)(3/ X)-1 () x-3 qo f X 1-8 X

flm(Xmin «xc «xmax ) '::::' --1 ( ) a x min -S - X3/x 1. _ 1 Xc

x (17.3.54)

17.3.3.3 Xmin « Xc « X « x max • Here the parameter B » 1 for all moment um values x, and the integral h with (17.3.48) becomes

r(b) -b . J2 '::::' r(a)B exp(B/2) J3 , (17.3.55)

where

(17.3.56)

Since B » 1 the main contribution to the integral (17.3.56) comes from small values of t ~ 2/ B « 1, so that ]3 '::::' 2/ B yielding to lowest order in B- 1 « 1 that

2r(b) -b-1 J2 '::::' r(a) B exp(B/2) . (17.3.57)

For the integral J1 we obtain approximately


{1(fc!X)X J1 "':' B-a dy y(1-s-x)/x

(X,nin/X)X

+ T(b) Ba-? 11 dy y[1-s+(a-b-1)xl!x eBY/2} T(a) (xc/x)X

~ B-a {~_ s-l l-s [1- ( . / )S-l] - X X min X mlll Xc S - 1

+ __ B a- beB/ 2 dt (1 _ t)[1-s-x+(a-b)xllxe-Bt/2 T(b) 11-(1/B) }

T(a) 0

~ B-a {~_ 8-1 1-8 [1- ( . / )8-1] - X X min X mlll Xc S - 1

2T(b) B a -b- 1 B/2} + T(a) e . (17.3.58)

Because of the large exponential term, exp(B /2) » 1, the second term in the bracket of (17.3.58) dominates, so that

2T(b) -b-1 J1 "':' h = T(a) B exp(B/2) . (17.3.59)

For the phase space density (17.3.44) we obtain in this case

f ( ) ~ 4qon (2,\1/2/3/ X)-1 x x+3 -(s+2+x) lm Xmin« Xc ~ X «xmax - X3/ X+1 a Xc X ,

(17.3.60) apower law that is steepened by the factor X as compared to the injection distribution (17.3.34).

17.3.3.4 Xc « Xmin « X « Xrn.ax. In this case besides B » 1 we find that By » 1 for all values of y ~ (Xmin/X)x. The integral J2 again is approximated by (17.3.57), whereas the integral J1 simply becomes

J ~ T(b) B-1 •.

1 - T(a) 11

. dy y[1-s- x+(a-b)xl/xeBy/2 (:Lrnin/X)X

T(b) 11-(xmin /x)X = __ B-1'eB/2 dt (1 _ t)[1-s-x+(a-b)xllxe-Bt/2 T(a) 0

~ 2T(b) B- b-1 eB/2 = J - T(a) 2 .

(17.3.61)

Also in this case thf phase space density approaches (17.3.60), i.e.

4 ~ (2,\1/2)(3/ X)-1 f (X // X . // X // X ) ~ qo f axxx+3 X-(s+2+X)

lm c "" mlll "" "" max - 3/x+1 c X

(17.3.62)


17.3.3.5 Interlude. Our analysis in the last subsection has shown that within the moment um range Xmin <::: x <::: X max the steady-state particle distribution for single power law injection approaches the two power law limits

(XXC)X~,3 (17.3.63a)

and

4qoTf (2.V/ 2) (3/x)~1 fZm(x »xc) "-' X3/x+1 aXx2+3 X~(s+2+X), (17.3.63b)

depending on x « Xc and x » xc, respeetively. The rigidity Xc charaeterizes the strength of stochastic acceleration by momentum diffusion. At rigidities below Xc stochastic acceleration is the dominant effect and generates a very flat particle power law spectrum with index X - 3, independent from the injection spectral index. At momenta above Xc stochastic acceleration is negligibly small resulting in a steady-state power law spectrum that is slightly steeper than the injection spectrum. This limit corresponds to (17.2.5) that has been derived neglecting interstellar acceleration completely, and that also exhibits the steepening of the secondary energy spectrum as compared to the primary sour ce spectrum if the traversed grammage X « (J f ~.

According to (17.3.19) and (17.3.47) thc characteristic rigidity Xc

_ (X/2)1/X (Tlm ) 1/2X Xc -

. a Tf (17.3.64)

is determined by the ratio of the escape time Tzm and the acceleration time Tf . Using (17.3.14)~(17.3.15) wc obtain for this ratio

Tzm = 1 D [K (y;, (2l+1)2n2)]~1 rp ( )2 1 1 R2 + 4 2 ' .L f mpc x 1 Zl

(17.3.65)

which, for a highly flattened disk R 1 » Zl, approaches

(17.3.66)

Apart from the normalization factor (mp c)2X, (17.3.66) is the ratio of the cosmic ray escape time TD = 4zU(n2 (2l + 1)2Kd at nucleon rigidity mpc to the cosmic ray acceleration time TF = 1/ D 1 at nucleon energy mpc (compare with the discussion in Seet. 13.2.2.5). According to (17.3.7), (17.3.10) and (17.3.11) we find

X 2 + X VA In ~ 3z1 ( ) ( )

2~x ( ) [ ] 2

(2-X)(4-X) C VA n(2l+1)7](q=2-X) , (17.3.67)


which involves thE· ratio of the escape boundary ZI and the characteristic length r; (see (17.3 .. 7)). Combining (17.3.64) (17.3.67) we obtain

Xc = (X/:!,)I/ X (TD) 1/2X

a'nlpc TF = _1 ._ [ X(2 + X) In (~)] 1/2X (VA) (2-X)/2X

am,c (2-X)(4-X) VA C

[ 3XZI ] I/X

x 2'iT121 + l)r,(q = 2 - X) . (17.3.68)

Because the observed cosmic ray spectrum above 1 GV is certainly steeper than the very fiat distribution (17.3.63a) we have to demand that the value of Xc ::; 10. With rEalistic parameters of the Galaxy in (17.3.68) and (17.3.7) this condition is fulfilled.

17.3.4 Influence of Source Spectral Index Dispersion on the Steady-State Primary Particle Spectrum Below R rnax

We investigate the infiuence of the dispersion in the source spectral indices on steady-state distribution function (17.3.20) by using the modified injection function (17.3.43), instead of (17.3.34),

dxo xosG(x, xo) .

(17.3.69)

Equation (17.3.69) is solved by numerical integration for different parameter values, and thell inserted in (17.1.2) to obtain the full rigidity spectrum at the position of tlH solar system. Figure 17.1 shows the normalized rigidity spectra for three different values of the mean source spectral index (8) = ß, turbulence spectral index q = 1.8, and Rmax = 1015 V. We see that in all three cases slightly curved power laws result below Rmax , and that the rigidity spectrum cuts oe· exponentially at larger rigidities. The cutoff is strongly infiuenced by the va,lue of q as Fig. 17.2 indicates. The closer the value of q lies to 2.0, the more moderate is the cut off.

Significant devia1 ions from a straight power law can be obtained by varying the characteristic rigidity Xc, as Fig. 17.3 indicates. For increasing Xc

(corresponding to increasing TD/TF, see (17.3.68)) the rigidity spectrum become fiatter in accord with (17.3.63).

102

10-2

10-4

X 10-6 "0 ........

10-8 E Z 10-10 "0

* 10-12 lO "-C\i

X 10-14

10-16

10-18

10-20

10-22

10-24

10-6


.......... --, - - - -- ... ,'~"'\ ..

(J= 0.3 q = 1.8

ß= 2.20 ß = 1.96 ß = 1.70

10-5 10-4 10-3 10-2 10-1 1

x/1015V

, .. , .. , .. , .. , .. \ .. \ .

\ .. \ .. \ " \ .. \ .. \ '. \ .. \ \ I', 1\ I', 1\ I \ I \ I' I

Fig. 17.1. Normalized cosmic ray rigidity spectrum for q = 1.8, Rmax = 1015 V and different me an source spectral indices (8) = ß

X "0 ........

E Z "0

* lO "-C\I

X

10-2

-~~-~:'-~::::::\' " 10-4

10-6

10-8

10-10

10-12 Tr:/TF = 1

ß = 1.96 10-14 (J= 0.0

10-16

10-18

10-20 .•••••• q = 1.8 . - - - q = 1.88

10-22 -- q = 1.91

10-24

10-6 10-5 10-4 10-3 10-2 10-1

x/1015V

, , , , , , . , , , .

, , ,

, \ , , , , , , , , , , , , , , , , , , , , , , , , , . , . , ,

\

\ , \

\ \ \ \ I \ I \ I \ \

Fig. 17.2. Normalized cosmic ray rigidity spectrum for Rmax = 1015 V, no source spectral index dispersion and different values of the turbulence spectral index q


><

10-2

10-4

10-6

10-8

R 10-10

~ Z 10-12 "0 * ~ 10-14

N >< 10-16

ß = 1.96 (J= 0.0 q = 1.96

-- ...... ......... __ ....... - , ~ '\ \ ' \ \ \ .. \ .. , · · · \ ..

\ \

\

\ \

10-18 \ \

10-20 ,ri 'F = 1/5 \ ,ri'F = 1 \

10-22 ,;,I'F = 5 ~ \

. . .. . . \ · · · · · · · \ · · · · · · · · \ · · · · · · 10~ ,

~~~-r~ __ rnmw-n~~nm~~nm~~mr~\~mr~ 10-6 10-5 10-4 1 0-3 10-2 10-1 10

x/1015V

Fig. 17.3. Normaliz,~d cosmic ray rigidity spectrum for q = 1.96, Rmax = 1015 V, no sour ce spectral index dispersion, but different values of the characteristic rigidity Xc that reflect the ra'io of escape (TD) to acceleration (TF) time scales at 1 GV

17.3.5 Steady-State Primary Particle Spectrum Above R rnax

Now we investigate ehe response of (17.3.20) to the single power law injection function (17.3.34) at particle momenta above x ~ xmax . With the Green's function (17.3.33), hnd the same notation as in the last section, the steadystate situation bet~'een injection, fragment at ion losses, escape by diffusion and second-order FE"Tmi acceleration yields a cosmic ray spectrum above the highest injection momentum

flm(X ~: xmax ) = Trqoa X l x=ax dxo xÜ8 G(x ~ xo)

Xmin

= foa x e-B / 2 U(a, b, B) , (17.3.70)

with the constant

fo == qogoTf l~,,~ax dxo xüs e-(A'/2x~/x) M (a,b, 2.V~2x~) . (17.3.71)

In the special case of negligi ble fragment at ion los ses (1/J = 0, a = b / 2 = (3 - X)/2X) we can I~xpress the confiuent hypergeometric functions in terms of modified Bessel functions to obtain


with the constant

(17.3.73)

Now, it is essential to realize that the distributions (17.3.70) and (17.3.72) are in normalized particle rigidity x, which is simply energy per nucleon for relativistic particles. However, all measurements in this energy range refer to the total energy E of the cosmic ray particles. Adopting the measured cosmicray composition at lower energies (Swordy et al. 1990 [535]) we calculated the total energy spectrum of all cosmic ray species from their identical rigidity spectrum (17.3.72) for Rmax = 8 X 1015 V and for different values of the parameter X = 2 - q, which characterizes the rigidity dependence of both the escape (by spatial diffusion) time and the moment um diffusion time, by relating for all individual elements their total energy with their rigidity as E = ZR. The resulting all-particle total energy spectra are shown in Fig. 17.4 in comparison with the observed all particle cosmic ray energy spectrum.

6 ;-...

111 ": ->

Q)

Ü . I 5 .::.,., ... ~

.. . (J) .., (J) .

N \- . I '\ E 4-.......... · ...

III .... N W · · *" · · x · · ::J ._-- q 1.667 · 3 · lL.. - 1.8 · --q · · ._----- q 1.88 · · 0'1 · · 0 · · '. · · · 2

12 13 14- 15 16 17 18 19 20

log E (eV)

Fig. 17.4. All-particle cosmic ray total energy spectrum for different turbulence power law spectral indices and Rmax = 8 X 1015 V


One can clearly see tImt for values of q between 5/3 and 1.88, which are compatible with the observed Kolmogorov density turbulence spectrum, we achieve remarkable agreement with the measured energy spectrum of ultrahigh energy cosmic rays up to several '" 1018 eV.

In Fig. 17.5 Wf' show the contributions from a few individual (protons, helium, carbon, m.ygen and iron) elements in the case q = 1.8 to the allparticle energy spectrum. One notices the drastic change in chemical composition from total ellergies below 1016 eV (proton-dominated composition) to lligher energies abü,ve 1016 eV (iron dominated-composition) implied by this model, which can he tested observationally by abundance measurements in this crucial energy range. Figures 17.4 and 17.5 also nicely illustrate how the observed "knee" and "dip" features in the all-particle spectra are reproduced by this in-situ acceleration model.

A very crucial parameter for this in-situ acceleration model is the maximum value Rmax of cosmic ray injection in supernova remnant sources, as Fig. 17.6 demonstr~ll,tes. Only for rather high values of Rmax 2': 8 X 1015 V can we account for theacceleration of > 1016 eV cosmic rays.

6

........ on r-: > q = 1.8

Cl) 5 Ü I "-CI) ..

I --CI) 4 ....

N ____________ .. _ .. ______ ~:~:~:·_·r~~i~\ I

E ........, ---------_ ... ' ... ------ \ '\ \ \' \ " . on

3 ' \ ,... " . N ' \ , , . w He \ \ ' ~ "* ' .

H I' • x \ ' \ :::J C ' , \

" . u.. 0 \ ' . 2 \ ' ~ Fe ' . , I!

Cl Gesamt , t ~

0 ' , , ' . \ ' \ I , . , \ . , , \

1 12 13 14 15 16 17 18 19 20

log E (eV)

Fig. 17.5. Contributi,m of individual elements to the total (= Gesamt) all-particle cosmic ray total energy spectrum for q = 1.8 and Rmax = 8 X 1015 V

17.4 Time-Dependent Calculation 455

6,-----------------------------------~

q = 1.8 ,J) ": 5 >-Q)

C) "-";"

\ .. ..... rf)

";" 4 \ .... rf)

C)l

E \ -l!) \ l"-N

:3 w \ * x

\ ::::l Li:

1 X10 14 V Cl Xo .2 2 Xo = 6x1014 V

Xo = 1 X10 15 V Xo = 8X1015 V

1~~TT~rr~~~~~TTTr~~~~~TT~

12 14 16 18 20

log E (eV)

Fig. 17.6. All-particle cosmic ray total energy spectrum for different values of Rmax

17.4 Time-Dependent Calculation

We now investigate the following hypothesis: if, since the first generation of supernova explosions in our freshly formed galaxy, diffusive shock acceleration at supernova shocks has steadily accelerated cosmic rays up to rigidities of R max = 1015 V, is it conceivable that, since then, over the :s; 1010 yr of the lifetime of our Galaxy, the turbulence in the interstellar medium has accelerated some of these cosmic rays slowly but efficiently to ultrahigh energies by the second-order Fermi mechanism, although the mean escape time from the Galaxy is much shorter?

In Sect. 17.3.5 we demonstrated that in-situ second-order Fermi acceleration in a steady-state situation with injection and escape is capable of explaining the observed all-particle cosmic ray spectrum up to total energies of several 1018 eV. The crucial quest ion is: can the steady-state spectrum (17.3.70) be reached within:S; 1010 yr galactic lifetime?

17.4.1 General Values X#-O

To obtain an answer to this quest ion we have to solve the corresponding (to 17.3.16) time-dependent diffusion-convection equation with steady injection

456 17. Galacti< Cosmic Rays

since time y = 0 für general values of X i= 0:

8f'rn = ~~ [x4-x8flrn] _ [7jJ+ .\xX]frn Oll x 2 8x 8x

+ Qoo;x6 (x - xmax ) H[y] , (17.4.1)

where H denotes the Heaviside's step function, and

(17.4.2)

is the dimensionles" time in units of the acceleration time at Rmax

( ) (Rmax)X Tacc Rmax = Tf -- , mpc

(17.4.3)

and

Tf= (2-X)(4-X) (Bo )2 [WP,i ]l-X (cIVA)2+ x

7r(1 - X) 6B ckmin In (ciVA) 1

(17.4.4)

At rigidities above the maximum injection rigidity x > Xmax it suffices to solve (17.4.1) for 6-function injection at Xmax since, aS (17.3.70) indicates, the variation with rigidlty x is independent of thc specific injcction distribution.

The solution of )7.4.1) for the case X = 0 is well-known (e.g. Kardashev 1962 [256]), see (17.5.2), and Ball et al. (1992 [27]) based their analysis on this limiting case. However, in the light of the interstellar power spectrum with 1.4 'S q 'S 1.9 corresponding to 0.1 'S X 'S 0.6, this case is of limited interest. The genercl solution of (17.4.1) is

fZm(x, y) = Qoo;x.\1/2 (XmaxX)(X-3)/2

x dThv 1'1 [2.\1/2 (xmax x )X/2]

° xsinh(x.\1/2 T )

exp [-7jJT - .\1/2 (X~ax + XX) coth (X.\1/2T) Ix] X ( /) , (17.4.5) sinh X.V 2T

where Iv(z) denotes the modified Bessel function of the first kind, and

v= 13;xxl. (17.4.6)

Figure 17.7 shows the particle spectra above Rmax = 1014 V at different galactic ages as compared to the steady-state solution (17.3.70) for X = 0.2, in which case 1010 yr galactic lifetime corresponds to y = 103 . For timcs y ~ 5 the time-dependent 'lOlution is indistinguishable from the steady-state solution for particle rigid ities x :::; 105 , so that it is justified to use thc thcoretical steady-state spectra for comparison with the observations.

For all values of X within the indicated range the tirne-dependent solutions approach the stcady~state solutions within the galactic lifetime, so that our analysis in Sect. 17.3.5 on the basis of steady-state solutions is justified.

17.4 Time-Dependent Calculation 457

0

-8

r--.. -16 >--0

Q -24 '-----'" Z "'-- -32 r--..

>--Q

-40

'-----'" Z -48 1}=O.2

0> 0 -56

-64

-72 0 2 3 4 5 6

log p/po Fig. 17.7. N ormalized momentum per nucleon spectrum for constant injection since time y = 0 at different later times y calculated from (17.4.5). At late times the curves approach the steady-state solution (17.3.70) shown as a dashed curve. The injection moment um is Po = 1014 eV jc nuc!. From Schlickeiser (1993 [461])

17.4.2 Special Case X = 0

For completeness we list the steady-state Green's function (17.3.33) and the time-dependent solution (17.4.5) in the special case of rigidity-independent escape and acceleration times, i.e. X = O. The steady-state Green's function in this case is

G(x,xo) = ,(17.4.7) (XXO)-3/2 {(XjXO)V1/J+.\+(9/4) 0:<:: x :<:: Xo

2J'lj; + A + (9/4) (x/xo)-Y1/J+'\+(9/4) Xo :<:: x :<:: 00

while the time-dependent solution is

f (CO ) - Qoax In (x/x max ) ( )-3/21Y d -3/2 Im X, Y - XX max TT

47T 1/ 2 J'lj;+A+(9/4) 0

X exp [_('lj;+A+(9/4))T_In2(Xi:max)]. (17.4.8)

With the integral representation (C26) of the modified Bessel function it is straightforward to demonstrate that in the limit y --+ 00 (17.4.8) approaches the steady-state solution (17.3.72) with Xo = xmax .


17.5 Secondary /Primary Ratio Including Interstellar Acceleration

The steady-state Equilibrium distribution of secondary cosmic rays in this case is again given by the solution of the transport equation (14.2.7) where the secondaries SOLirce term

(17.5.1)

is due to the fragmentation of primary cosmic rays with distribution Mp in collisions with interstellar gas. rYp-ts denotes the moment um dependence of the fragment at ion cross-section, n g the interstellar gas density, and v the primary cosmic rar velocity. For relativistic cosmic rays the fragment at ion cross-section is a constant rYo (Letaw et al. 1983 [302]) so that the moment um dependence oft he s,mrce term (17.5.1) is given by the moment um dependence of the primary cosnic ray equilibrium spectrum.

The solution of 1 he secondary transport equation is given by an expression similar to (17.1.2),

CXJ CXJ

Ms(R, z,p) = L L Czm(R, Z)SZm(P) , (17.5.2) z=o m=l

which is quite involved since the expansion coefficients C1m(R, z), according to (14.3.7) and (14.3.8), depend both on the spatial distribution of the interstellar gas den,üty and the cosmic ray primaries (see (17.5.1) above), respectively. Moreo"er, each function SZrn(P) obeys the leaky-box equation

(17.5.3)

where the secondaries spatial eigenvalues Arm' in general, do not equal the primaries spatial eigenvalues.

Analytical (Cow3ik 1980 [106], Lerche and Schlickeiser 1985 [297]) and numerical (Heinbach and Simon 1995 [224]) solutions of the system of equations (17.5.2)-(17.5.3) based on different simplifying assumptions have been obtained. The resulting secondary/primary ratio R(p) = Ms(xs,p)/Mp(xs,p) at the position of tht~ solar system (xs ) decreases according to R(p) cx: p-x in the relativistic momentum range accessible to observations, which accounts for the measured ral ios.

This result can be made plausible by the following argument: under the assumptions made für relatiYistic secondaries (17.5.3) reduces to an equation similar to (17.3.16)

d: [x4-xd~~m] - [7/!sx2 + AsX 2+X ] SZm = -Tfx2axcnorYoP(x) , (17.5.4)

where P(x) denotes Ithe primaries moment um distribution (17.3.44).

17.5 Influence of Interstellar Acceleration 459

According to (17.3.63) the primaries spectrum is approximately apower law P(x » xc) IX X-(s+2+X) for moment um values x » xc. Inserting this power law in (17.5.4) will yield, after repeating the calculations of Sect. 17.3.3 mutatis mutandis, an equilibrium secondary spectrum above Xc as S(x » xc) IX x-(s+2+2X). As a consequence, the ratio of the two becomes

_ S(x) -x R(x» xc) - P(x) IX X , (17.5.5)

as required by observations.

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