Astroparticle Physics

ELSEVIER Astroparticle Physics 7 (1997) 183-202

Kinetic theory of cosmic rays and gamma rays in supernova remnants. I. Uniform interstellar medium

E.G. Berezhko a, H.J. Vijlk b** a Institute of Cosmophysical Research and Aeronomy, Lenin Ave. 31, 677891 Yakutsk, Russia

b Max-Planck-Institutfr Kerphysik, Posrfach 103980, D-69029 Heidelberg, Germany

Received 16 December 1996; accepted 1 March 1997

Abstract

Kinetic models of particle acceleration in supernova remnants (SNRs) are used to determine the cosmic ray (CR) nucleon and, for the first time, also the associated y-ray spectrum during SN shock propagation in a uniform interstellar medium. SNR evolution is followed numerically taking into account the backreaction of accelerated CRs on the overall dynamics. The high energy CRs also produce TO-decay y-rays. The model for SNRs includes injection of suprathennal particles at the shock front and heating of the thermal plasma due to the dissipation of Alfven waves in the precursor region. It is shown that the CRs are accelerated with very high efficiency. About 50% of the explosion energy is absorbed by CRs at maximum during the SNR evolution even for relatively low injection rates. The maximum energy achieved by accelerated CR protons is about lOI eV for a Bohm-limit diffusion coefficient. The main flux of high energy y-rays is produced during the early Sedov phase and decreases thereafter. The results are compared with earlier models based on the hydrodynamic approximation for CR transport and test particle estimates. For a moderate ambient gas density of 0.3 H-atoms cmm3, corresponding to a warm interstellar medium, a magnetic field strength of 5 pG, and an explosion energy of 105’ erg, the integral TeV y-ray flux from a SNR at a distance of 1 kpc exceeds lo-” photons cm-*s-I at peak luminosity. This is higher than previously estimated peak fluxes by factors of the order of 7, and is primarily due to the large shock compression during the sweep-up phase. Spatially, SNRs without a central compact object are shell sources also in y-rays throughout all their evolutionary phases. Given that such SNRs exist close-by they should be observable in particular with sensitive ground based instruments. @ 1997 Elsevier Science B.V.

1. Introduction

It is widely believed that the main part of the nucleonic cosmic rays (CR) in the Galaxy is generated in

Supernova Remnants ( SNRs). This old idea, which for a long time was based on energy arguments alone, has been greatly supported by the development of diffusive shock acceleration theory (see, for example, the reviews [ l-31). At the same time there are as yet no firm observational arguments that the nuclear component of CRs is indeed produced in SNRs, even though recent y-ray data [4] for energies above 100 MeV from the Compton

* Corresponding author.

0927-6505/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PII SO927-6505(97)00016-9

184 E.G. Berezhko. H.J. Vdk/Astroparticle Physics 7 (1997) 183-202

Gamma Ray Observatory (CGRO) are consistent with the theory: if the nucleonic component of the CRs is strongly enhanced inside SNRs then, through inelastic nuclear collisions leading to neutral pion production and

subsequent decay, y-rays will be noticeably produced. We shall not address here the question of y-ray production by electron synchrotron emission, Bremsstrahlung,

or the inverse Compton (IC) effect on ambient low energy photons. Especially at very high energies exceeding - 10 GeV, IC y-rays appear to dominate the emission from plerions like the Crab nebula (e.g. [ 5,6] ), where presumably a relativistic wind of electron-positron pairs from the pulsar is dissipated in a circumstellar

termination shock deep inside the SNR shell. The IC effect associated with acceleration of ultrarelativistic

electrons inside the extended, turbulent SNR shell, or at its leading shock (along with the ions considered here), may also contribute quantitatively to the y-ray luminosity of a SNR as simple estimates suggest [ 7,8]. Therefore in any specific source this leptonic contribution to the y-ray flux needs to be estimated before the

hadronic y-ray emission can be compared with the theoretical models presented below. The To-decay y-ray emission of SNRs in the energy regions beyond ey = 100 MeV has been studied for the

case of a uniform interstellar medium (ISM) [ 9,10,1 I]. The studies 19,101 were based on CR acceleration models [ 12-141 in the framework of CR hydrodynamics, and were combined with parametrizing assumptions for the particle momentum distribution taken from approximate test particle models [ 15,161. In [ 12,131 the hydrodynamics was further simplified by breaking the remnant up into several spherical shells whose bulk characteristics and dynamical evolution were described by coupled equations, essentially derived from the basic conservation laws (‘simplified models’). The closure parameters used for the hydrodynamic description were

the same in all these studies. In particular in [ lo], the predicted y-ray fluxes were also discussed in the context

of an analysis of the various backgrounds and of current instrumental sensitivities. An important conclusion

was that the most favourable energy range for detecting #-decay y-rays from SNRs is probably from 1 to 10 TeV.

Here we shall concentrate on the acceleration theory and the resulting y-ray emission. We expect that the y-ray fluxes derived in these hydrodynamic models are rather realistic, insofar as they rather correctly give

the overall CR production efficiency which is the most important factor for y-ray production. At the same time several important features of nonlinear shock modification by the CR backreaction can be studied only

in a detailed kinetic treatment which must be done numerically [ 17-191. The same is obviously true for the selfconsistent determination of the CR distribution function and the resulting y-ray spectrum. This theory is

developed here to study CR and y-ray production in SNRs in the case of an arbitrary circumstellar mass

distribution around the progenitor star. In doing this we assume the scattering of energetic particles to be governed by the Bohm limit for the

diffusion coefficient in the local mean magnetic field. The argument for Bohm diffusion as a limiting case stems from a consideration of the quasilinear evolution of the unstable scattering MHD waves, excited by the accelerated particles themselves. For a parallel shock the steady state wave amplitudes 6B have been discussed in the test particle limit [20] as well as in the nonlinear case [21]. Both suggest that SB/B can formally exceed unity for strong shocks whose magnetosonic Mach number exceeds values of the order of 10. Under these circumstances the quasilinear approximation itself breaks down. Since SNR shocks have Mach numbers of the order of 100 or more during much of their active lifetime, the asymptotic limit 6B/B -+ 1 (Bohm diffusion) should indeed be reached at least within factors of a few. At late phases of the SNR evolution this must become invalid again. But then acceleration is in any case inefficient (see Fig. 2). Therefore the use of the Bohm limit may in practice somewhat overestimate the upper cutoff energies and the acceleration efficiency at a given injection rate. In principle (for quasiparallel shocks), it gives upper limits for these quantities. However, at the late phases the spectral slope of the freshly accelerated particles is in a lowest order test particle approximation in any case independent of the diffusion coefficient and given only by the shock compression ratio. Thus one need not worry about the small inaccuracy introduced by the continued use of the Bohm diffusion coefficient. In the nonlinear regime of strong shocks indeed all dynamical variables are strongly coupled with one another. But just in this case the system should naturally drive itself towards Bohm limit.

E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202 185

In the present paper, we apply the theory to the case of SNR evolution in a uniform medium, corresponding to the limit of negligible influence of the progenitor star on its environment. In this sense we extend the above mentioned earlier studies to the kinetic level. In a companion paper (referred to hereafter as paper II) we study CR acceleration and y-ray production in SNRs evolving in a circumstellar medium that can be strongly nonuniform due to the action of a wind from the progenitor star.

The various aspects of the model are described in Section 2. Section 3 contains the results; they are discussed in Section 4. A summary and the conclusions are given in Section 5.

2. Model

2.1. CR acceleration and SNR evolution

The mechanical energy Es,, released in a SN explosion is during the early phase of evolution in the form of kinetic energy of an expanding shell of ejected mass. The motion of these ejecta produces a strong shock wave in the ambient medium, and the shock radius R, increases with velocity Vs. Diffusive propagation of energetic particles in the collisionless scattering medium allows them to intersect the shock front many times before they are finally left behind the shock. Each pair of subsequent intersections increases the particle energy. In plane geometry and for a steady state shock this regular or diffusive shock acceleration process [22-241 creates a power law-type CR momentum spectrum. Due to their large energy content the CRs can in turn dynamically modify the shock structure.

The description of CR acceleration by a SN shock wave in spherical symmetry is based on the diffusive transport equation for the CR distribution function f( r. p, t) [ 25,261:

5 = v(Kvf) - wcvf + FPg + Q,

where r, t, and p denote the radial coordinate, the time, and particle momentum, respectively, and K is the CR diffusion coefficient. In addition,

WC = w for r < Rs, wc = w+ca for r > R,,

where w is the radial mechanical velocity of the scattering medium (the thermal gas), c, is the speed of forward Alfven waves generated in the upstream region by the anisotropy of the accelerating CRs. In the downstream plasma the propagation directions of the scattering waves are assumed to be isotropized (e.g. [ 121) .

The phenomenological source term

Q = Q&r - R,) (2)

describes particle injection at the subshock from the thermal gas into the acceleration process. The gas subshock, situated at r = Rsr is treated as a discontinuity. We restrict our consideration to the most simple case of monoenergetic injection, described by the source current density in phase space

Qo = Ninjul -S(P - Pinj> 9 4TPkj

where u = V’ - w, pinj is the momentum of injected particles, V, is the shock speed, and NinjUi is the number of injected particles per unit time and unit surface of the shock. The subscripts 1 (2) refer to points just ahead (behind) the subshock.

Conceptually, injection of ions at shocks is a well understood process, as shown by (one-dimensional) plasma simulations [ 27,281, Monte Carlo calculations [ 291 and analytical theory [ 301. In addition the analysis

186 E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202

of observational data obtained in solar wind shocks allows the empirical derivation of an injection rate [ 311 in favourable cases, although typical solar wind shocks have moderate Mach numbers in comparison with

SNR shocks. Nevertheless, the connection between injection theories and a dynamical model of a spherically symmetric SNR is yet to be established. In this situation it is useful to explore the dependence of the CR production on the injection rate. In order to do that one can parameterize the injection rate (or the number

density Ninj) and the injection momentum pinj, and study the sensitivity of the CR acceleration efficiency to these parameters. We adopt here a simple injection model, in which a small fraction 7 of the incoming protons

is instantly injected at the gas subshock with a speed A > 1 times the postshock gas sound speed c,z (e.g.

1321):

Ninj = wa lm, pinj = Amen,

where m is the particle (proton) mass. For simplicity, we shall always use h = 2.

(4)

It is important to note that in a spherical shock, due to the influence of geometrical factors, the accelerated CRs are unable to modify completely the front of the nonstationary, expanding shock for a strongly energy dependent CR diffusion coefficient [ 17-19,33,34] : in the upstream region accelerated particles occupy a volume increasing with energy and time. In contrast to the plane wave case, the acceleration process can not produce particle energies in excess of some cutoff energy where this volume becomes too large and where the acceleration

process is unable to fill it with a CR number density sufficiently high to smooth out the shock entirely. This is at least true for a SNR in a uniform medium. For a nonuniform circumstellar medium the same property need

not hold although it is a likely one as long as spherical symmetry can be assumed. A further consequence of the existence of a subshock is a quasistationary CR pressure since in this case injection is never interrupted as it would be expected to be for a smooth transition. The subsequent formalism for a spherical SNR assumes this

existence of a sufficiently strong subshock also in the case of a nonuniform circumstellar medium. It is necessary to solve Eq. (1) in the downstream (r < R,) and upstream (r > R,) regions in turn and then

to match the solutions at the subshock. The boundary condition which allows this is obtained through a spatial

integration of Eq. (1) over the upstream volume, including the subshock [ 17,191. Technically this is achieved by multiplying Eq. ( 1) by 4?rr2dr and integrating each term over the interval R, - 0 < r < 0;). The result can

be written in the form [ 191:

where

(5)

is the effective velocity of the upstream scattering medium, which is ‘felt’ by particles with momentum p, and fR(p, t) = f(r = R,,p, t) is the CR momentum distribution at the shock front.

We assume that the Bohm diffusion coefficient is a good approximation for strong shocks, characterized by very strong wave generation as shown in [ 3.51. For simplicity the CR diffusion coefficient

K(P) = k'Bc/3 (7)

is used, where pe is the gyroradius of a particle with momentum p in the magnetic field B, and c is the speed of light. It differs from the Bohm diffusion coefficient in the non-relativistic energy region, but this difference is absolutely unimportant because of the very high acceleration rate at p < mc. In the compression region we use a modified Bohm limit in the form K = ~~p~/p, where the subscript s corresponds to the current shock

E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202 187

position r = R,. For want of a deeper theory the additional factor ps/p was taken to prevent the instability of the precursor [ 36-381. Such a nonresonant precursor instability should ultimately lead to more irreversible heating and thus entropy increase of the thermal gas, and therefore diminish the smoothing of the shock by the CRs, while maximizing energetic particle scattering [ 391. As long as the shock remains far from becoming totally smoothened by the CRs, the modified Bohm limit for K appears as a reasonable way to take the effects of the instability into account. Such a form for K can also qualitatively describe the expected decrease of the CR scattering mean free path in a medium of varying density.

The thermal gas that carries the scattering fluctuations is described by the gas dynamic equations

ap, F + (wV)Pg + Y,(VW)Pg = cu,(l - y,)c,VP,,

where p, yr and Pg denote the mass density, specific heat ratio, and the pressure of gas, respectively, and

(11)

is the CR pressure. These gas dynamic equations include the CR backreaction via the term -VPc. They also describe the gas heating due to AlfvCn wave dissipation in the upstream region [ 39,211; it is given by the parameter ff, = 1 at r > R, and LY, = 0 at r < R,.

AlfvCn wave dissipation as an additional gas heating mechanism strongly influences the structure of the modified shock in the case of large Mach number M >> m . It substantially restricts the growth of the shock compression ratio CT = &PI which in absence of the Alfven wave dissipation has been found to reach extremely high values u x M3i4 in the case of large Mach numbers [ 18,191.

Eqs. ( 1 )-( 11) should be solved with the initial (t = 0) conditions:

f (PI = 0, P=PO(r,L)r Pe = PH(r, b), w = wO(r,tw), (12)

which describe the ambient CR and gas distribution (in the general case modified by a wind from the progenitor star emitted during the time period t,,,).

The result of the SN explosion of a massive star, many days after the explosion, is freely expanding gas with a velocity field ZJ = r/t. The density profile of the ejecta is describable by a similarity solution [40-421

Ft-3, Pej =

u < 4,

Ft-3(u/u,)-k, u > ur,

where

F _ 1 [3(k-3)Mej15'* 4Tk [10(k-5)Es,]3/2’ Or= ’

k > 0,

(13)

and Mej is the total ejected mass. For SNRs the value of the parameter k typically lies between 8 and 12. The pressure in the expanding ejecta is negligible. Interaction with the ambient material modifies the ejecta

density distribution and leads to a reverse shock in the ejecta material, ultimately heating it to very high temperatures (e.g. [ 141). We shall ignore this reverse shock here and describe the ejecta dynamics in an approximate manner, assuming that the modified ejecta consist of two parts: a thin shell (or piston) moving

188 E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202

with some speed V, and a freely expanding part which is described by the distribution ( 13). The piston includes the decelerated tail of the distribution (13) with initial velocities u > R,,/t:

m

M, = 41rt3 s

dou*pej(u, t) 9 (14)

R,,lf

where R, is the piston radius. The dynamics of the piston is quite satisfactorily described in the framework of the simplified thin-shell approximation [43], in which the thickness of the shell is neglected. Accordingly, the

motion of the piston is given by the equation

d(MpV,) =~TR;[P,(~=R~-O)-P~(~=R~+O)-P~(R~+O)]. dt

(15)

The CR pressure P,( r = R, - 0) is produced by particles that have penetrated through the piston. We include this possibility by the following boundary condition at the piston surface

a.f K- = -4, for r = Rp + 0, (16)

where 4 = F~/4n-p*, and FD is the density of the CR diffusive flux through the piston. It can be written

parametrically as

where K,, = ~~p$/p~ is the CR diffusion coefficient in the piston material, and pp and lp are the mean density and thickness of the piston respectively. To derive this expression the thin-piston condition 1, = SR,, S < 1 was employed. For the numerical calculation we shall use the value S = 0.1.

Behind the piston (r < Rp ) the CR distribution function obeys the approximate equation

(17)

which can be derived from Eq. ( 1) taking into account that the spatial distribution of CRs is nearly uniform in the volume V = 47rRi/3 due to the large diffusion coefficient there. The last term in this equation describes

the diffusive CR flux through the piston surface S = 47rRi.

It was shown that in the case of a uniform ISM the CR penetration through the piston is unimportant for SN shock evolution and CR acceleration [ 18,191. In the early free expansion phase this process is insignificant because the diffusion coefficient K~ is too small due to the large density pp. In the intermediate Sedov phase, when the dominant part of CRs is produced, the piston size is small in comparison with the shock size ( Rp < R,). Therefore CR penetration into the region r < R, is also dynamically unimportant in this stage of SNR evolution. In the case of an intensive presupernova wind a relatively large volume around the progenitor is occupied by a low density bubble. The main amount of CRs and y-rays is produced when the SN shock interacts with the dense surrounding shell. Piston and shock sizes are comparable during the most effective CR production phase and one can expect that CR penetration through the piston is more important in comparison with the case of a uniform ISM (see paper II).

In general the consideration of a realistic velocity distribution (13) for the ejecta should be important during the initial stage of SNR evolution when the amount of swept-up ambient material is much less than the ejecta mass. During this period the actual piston speed is much higher than the often used expression for the initial piston speed VO = dw. Higher shock speed leads to much more intensive CR and y-ray production.

Radiative gas cooling is not included in our model. This process becomes important at the late phase of SNR evolution [9], when CR acceleration is inefficient due to the low shock velocity.

E.G. Berezhko. H.J. Wk/Astroparticle Physics 7 (1997) 183-202 189

The results to be discussed in the next section have been obtained by numerical methods. For detailed descriptions we refer the reader to earlier publications [ 17,191.

2.2. Gamma-ray production

Energetic CR protons generate y-rays in inelastic collisions with gas nuclei which produce neutral pions that subsequent decay. The resulting integral y-ray emissivity of a SNR, in units of photons per unit volume and second with energies greater than +, can be written as

00

Qr(+) =47r J

dpp2cr,,Z,ncN,f(r,p,t), (18)

PV

where

ff 2 0.01876~ mb

pP = 38.5 + 0.461n mc (19)

is the inelastic p-p cross-section [ 441, Z; denotes the so-called spectrum-weighted moment of the inclusive cross-section, pr is the momentum of a CR particle with a kinetic energy E = ey, Ng = p/m is the gas number density, (Y = - 1 - d In n/d Ins is the power law index of the integral CR energy spectrum, and n = 4rp2f is the CR differential number density. We use the approximation for the spectrum-weighted moment [lo], limiting Zy” from above by the value 0.2,

Zy” = min(O.2, 101.49-2.73a+0.530*} (20)

The integral y-ray flux at distance d from the source is given by

m

Fy(Ey) = 4?i- J

drr2Q,( s,)/4n-d2. (21)

0

3. Results

Detailed theoretical investigations of CR acceleration and SNR evolution in a uniform ISM have revealed important features of this process. For a wide range of the injection rate the CR acceleration efficiency is very high and almost independent thereof [ 18,191. We shall often use here the particular values of the injection parameter r) = 10B3 and 71 = 10e4 which correspond to rather moderate and quite low injection rates, respectively. We also restrict our considerations to typical values of the SN parameters: explosion energy Es,, = 10sl erg, ejecta mass Mej = lOMa, initial piston speed V,a = 20000 km/s, and the value k = 8.6 which corresponds to the ejecta of SN 1978A 1451.

The scaling parameters of the spherical shock are the sweep-up radius

and the sweep-up time to = R&6, where Vo = d-. 2E M We will consider the case of the so-called warm ISM which is characterized by a hydrogen number density No = 0.3 cmF3 and a temperature T = lo4 K. The ISM density is pa = 1.4Nam (we assume 10% of helium nuclei in the ISM). The values of the scaling parameters are then Rc = 6.14 PC and to = 1893 yr.

In order to study the influence of the assumed ISM magnetic field Ba on the CR and y-ray production, we present in addition the results of calculations performed for different values of Ba. We consider the following

190 E.G. Berezhko. H.J. Wk/Astroparticle Physics 7 (1997) 183-202

Energy components of the SNR

0.01 0.1 1 10 0.1 I IO 100

t/t, t/t,

Fig. 1. Time dependence of ejecta (E,,), gas kinetic (Ek) and internal ( Eg) energies, and CR internal (kinetic) energy EC, normalised to the explosion energy Esn, for the injection rates T,I = 10T3 (a,b) and 7 = 10e4 (c,d), ISM magnetic field Bo = 5 pG (a,c,d) and Bo = 30 pG (b), the ejecta velocity distribution ( 13) (a.b,c) and the uniformly moving ejecta (d). expanding in a uniform ISM with number density NO = 0.3 cmm3; to = 1893 yr.

cases: (a) 7 = lop3 (hereafter referred to as high injection, to contrast it to the value 17 = 10m4 below), BO = 5 ,uG; (b) 71 = 10-3, Bo = 30 ,uG (hereafter referred to as high field) ; (c) 77 = 10d4 (hereafter referred to as low injection), Bo = 5 PG. In all these cases we assume the initial velocity distribution of the ejecta in the form (13) with k = 8.6. Besides that we also consider the simpler case (d) r) = 10b4, Bo = 5 ,uG, with all the ejecta initially moving with same the velocity V,IJ = Vi = dm.

3.1. SNR energy components

In Fig. I the different SNR energy components, the ejecta energy

&It

4 = M, Vi/2 + 27rmt3 s

duu4pej(u,t), 0

the internal and kinetic gas energies E, and Ek, respectively,

Em = 677 J drr* [Pg(r, t) - ~@<r>] , RP

(22)

(23)

E.G. Berezhko, H.J. Kilk/Astroparticie Physics 7 (1997) 183-202 191

cm

Ek = 27~ drr2pw2, (24)

RP

and the CR (kinetic) internal energy

E~=16~mc2~drr2~dpp2(~~p,mc)2+I-1)f(r,p,r) (25)

0 0

are presented as a function of time. In all cases the CR energy E, initially increases with time, reaches a peak value at t = 3 + lOto depending on the value of Bo, and then slowly decreases with time due to adiabatic cooling in the SNR interior. Even for low injection the peak value of EC is quite large. With Bc = 5 ,X G it exceeds 50% of the explosion energy. The acceleration efficiency increases with increasing injection rate. But the dependence of EC on 7 is not very strong. The peak value of E, increases by less than 15% if the injection rate is increased by a factor of ten, as one can see by comparison of Figs. la and lc.

The CR acceleration efficiency depends also on the value of the ambient magnetic field Bo in the ISM. A high field Ba increases the Alfvtn wave dissipation effect which significantly restricts the shock modification and the acceleration efficiency [ 191. One can see formally from Figs. la, b that an artificial increase of B. from 5 to 30 ,uG leads to a decrease of the peak value of E, from 0.57E,, to 0.43E,,.

3.2. CR proton distribution

Since K 0; B;‘, the ambient magnetic field also influences the value of the CR cutoff momentum. This cutoff pm is determined by the expression 1341

mc R,V, Pm =Alcs(mc>’ (26)

whereA=qo[2+2b+e+d-(y-l)/v].Theparameterd=(rVw)za/[(cT-l)V,] describes theeffect of particle adiabatic cooling in the downstream region, and the dimensionless parameters

v=dlnR,/dlnt, b = dlnfR/dlnt, e =dlnK,/dlnt

describe the time variation of the shock radius, of the CR distribution function, and of the CR diffusion coefficient, respectively, whereas qo = 3a/ (u - 1) is given in terms of the shock compression ratio u = p2/po.

The actual values of pm can be estimated from Fig. 2 where the CR proton distribution function at the shock front fR(p, t) = f( r = R,,p, t) is plotted for five different phases of the SNR evolution. In the initial stage t < to the SN shock expands with a constant’speed which is about V, = 22 x lo3 km/s in the case of the ejecta velocity distribution ( 13), and with V, M 3500 km/s in the case (d) where initially the ejecta all move with 40. In the early free expansion phase (t < to) when the shock speed is almost constant (Y = 1 ), d = - l/2 and b = 0 [ 341. Taking e = 0 and qo = 4 we have in the case BO = 5 ,uG, pm = 6 x 106( t/to)mc for V, = 22 x lo3 km/s, and pm = 2 x 105(t/to)mc for V, = 3500 km/s. One can see from Fig. 2 that the numerical values for the cutoff momentum are in good agreement with formula (26) in the initial stage when the shock speed is indeed almost constant (which is true for t/to < 10e2 in the case V, = 22 x lo3 km/s, and for t/to < 0.1 in the case V, = 3500 km/s).

For low injection nonlinear effects produced by accelerated CRs during the initial stage of evolution (I < 0.1 to) are small and the CR spectrum is almost a pure power law fR o( pm4 (see Fig. 2~). Later, at t > 0.1 to, the shock modification becomes noticeable. If we approximate the CR distribution function by a power law fR cx p-4, the power law index q will be close to

192 E.G. Berezhko, H.J. Vd’lk/Astroparticle Physics 7 (1997) 183-202

Proton distribution function at the shock front

p/me p/me

Fig. 2. CR distribution function fR(p) in arbitrary units (ax) at the shock front as a function of momentum p at five different times for the same four cases as in Fig. 1.

for p < mc, and for p > mc it will lie in the interval q1 < q < 4, where

q* = A!?- u’ - 1’

(27)

(28)

ai = (ur - c,)/u:! = a,(1 - l/Mar), (T’ = (V, - ca)/u2 = (+(I - l/M&), as = p~/p~ is the subshock compression ratio, and M, = u/c, is the Alfvenic Mach number.

For ~7 = 10m3 the shock modification becomes even more essential already in the very early phase r = O.OOlto

(see Figs. 2a, b).

3.3. Shock compression ratios, spectral index

The compression ratios a, and u and the lower bound q1 to the index q are represented in Fig. 3 as functions of time. For low injection the maximum of the shock modification occurs for t = 2ra (see Figs. 3c, d), when the overall shock compression ratio reaches the value u = 6. For t > 2t0 the shock deceleration leads to a decrease of the CR acceleration efficiency. As a consequence the shock modification decreases with time and u approaches the subshock compression ratio.

For ~7 = 10e3 the shock modification is larger and reaches its maximum much earlier (see Figs. 3a, b). For Bo = 5 ,uG the shock compression ratio reaches its peak value u = 13 at t = O.Olto, while in the high field case the peak compression ratio u = 7.5 is much smaller due to the AlfvCn wave dissipation effect.

E.G. Berezhko, H.J. Viilk/Astroparticle Physics 7 (1997) 183-202 193

Compression ratios and lower index limit

(a) 1 (b)

Fig. 3. Total shock compression ratio U, subshock compression ratio (T S, and the lower bound qr to power law index as a function of time for the same four cases as in Fig. 1.

In all cases the lower index bound q, increases with time at 1 > 2to which shows that the acceleration process

produces CRs with a progressively steeper spectrum. A particularly rapid increase of q, takes place for high fields due to the decrease of the Alfvenic Mach number.

3.4. Spatially integrated momentum spectrum

The spatially integrated (overall) momentum spectrum of the SNR

M

WA t) = 1613~~ s

drr2f( r,p, t) (29)

0

represented in Fig. 4, is close to a pure power law N cx p -* in the entire momentum range at low injection

(Figs. 4a, b) . The amplitude of the spectrum in the initial phase of SNR evolution, t < to, is very sensitive to the initial ejecta velocity distribution. During this phase the ejecta described by the formula (13) cause much

higher speeds V, than would the mean ejecta speed VP,. The CR cutoff momentum is also considerably larger

in this case, since pm 0; V,. For T,I = 10e3 the overall CR spectrum produced by the modified shock has much less power law character

(Figs. 4a, b) . The high energy spectrum, p > 103mc, is harder than in the low injection case. The power law index in this energy range is about 1.7. The low energy part (p < mc) of the overall CR spectrum on the

other hand, is relatively steeper.

194 E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202

Overall proton momentum spectrum

w, Cc) 1 I-\,

1 1/1/ :-c.-.,. vt, (a!

,- ’ 10-J -..._..._ 10-2 _ .._ _ 1 \ ’ .‘\ ,0-z 0.1 1 1 ‘\. \

, _:_:l._, J \. \

01 / :.. ‘.

: ; :.._. ‘., \ t , : ‘.., ‘._ i ---- ,0 _-__

‘\ ; ; ; .,.._ ‘., \ 10 - ,’ 1 102 -:: 1 .,

‘,.. ,, ’ \

~-.~ \-.--,! I \_, ‘. . . . . \ \ ,’ ; ! / i ‘\ ‘.,, -.\ ‘\

,’ .I -..\ ..,,. x.

: ., ., ‘\ . . . . . ‘,, \ : , :i -.,. ‘..., ( , , ,: ! I i;

.

, I:

‘.._,

, !: j

._.. ..\‘\ ‘\ ‘.(\‘\ . . ‘..

; !:: , !;I

.‘\ ; .\

‘\

’ : ; . ‘... .\ t

, ; ; \ \

I ‘.._ ., ‘\. ‘\ ‘. ‘.., \, i

‘1. ; ! :

‘. ‘. ‘., , ‘..

I ,; : ‘. t ‘. I , : 1 ‘5

.r

!’ : \ ,. 1

-2 0 2 4 6 -4 -2 0 2 4 6

b9(P/m4 l09(P/d

Fig. 4. Spatially integrated (overall) CR momentum spectrum N(p) in arbitrary units, at five different times, for the same four cases as in Fig. 1.

3.5. Integral y-ray spectrum

The main properties of y-ray production in SNRs are determined by these aspects of the CR acceleration process. Fig. 5 shows the expected integral y-ray llux spectrum F,( ey) for energies Ed > 1 GeV, normalized

to the distance d = 1 kpc. Five different stages of SNR evolution are exhibited. For low injection rate (Figs.

5 c,d) the y-ray spectrum is close to a pure power law Fy 0: E;’ in energy up to some cutoff energy

am*(t) x 0. le,( t), where c,,,(t) = pm (t)c is the cutoff energy of the corresponding CR spectrum (Fig. 4). The amplitude and the cutoff energy of the y-ray spectrum are considerably larger during the initial phase for the case of the initial ejecta velocity distribution (13). During the Sedov phase (t > to), the y-ray production is quite insensitive to the assumed initial ejecta velocity distribution (compare Figs. 5c and d).

During the sweep-up epoch 0.01 < t/to < 1 of strong shock modification for high injection (cf. Figs. 3a, b), the y-ray spectra are significantly harder at high energies ey > 10” eV.

3.6. y-ray jhx evolution

The integral flux of y-rays with energy ey = 1 TeV is plotted in Fig. 6 as a function of time. The peak value

of the y-ray flux, reached somewhat after 10 is not very sensitive to the injection rate and the ISM magnetic field strength whereas the time profile F,(t) depends rather strongly on Bu and 77. For t < to the CR and the y-ray production are proportional to the injection rate. At this stage of SNR evolution the y-ray production is especially sensitive to the assumed ejecta velocity distribution with the result that the y-ray production is much more efficient in the case where the ejecta are distributed according to Eq. (13) compared to the case of a unique ejecta velocity. In the Sedov phase the y-ray flux is almost independent of the injection rate and

E.G. Berezhko, H.J. Viilk/Astroparticle Physics 7 (1997) 183-202

Integral gamma-ray flux at distance d=i kpc

195

Fig. 5. Integral y-ray spectrum, normalized to a distance of 1 kpc, at five different stages of the SNR evolution, as a function of energy ev, for the same four cases as in Fig. 1.

Integral gommo-ray flux ot distance d=l kpc

‘0 I I’, I I

’ -3 -2 -1 0 1

w/t,)

Fig. 6. Time dependence of integral flux F, of y-cays with energy 9 = lOI* eV, normalized to a distance of 1 kpc, for 7 = lO-3 and Bo = 30 /ALG (full line), TJ = lop3 and Bo = 5 /.LG (dashed line), 71 = 10m4 and Bo = 5 /LG (dash-dotted line). The dotted line corresponds to the case of single-velocity ejecta for 7) = 10m4 and BO = 5 PG.

196 E.G. Berezhko, H.J. ViitklAstroparticle Physics 7 (1997) 183-202

of the initial ejecta velocity distribution, but quite sensitive to the ISM magnetic field strength Ba. This can be demonstrated by choosing Ba = 30 ,uG when the expected y-ray flux decreases much more rapidly with time

for t > 5~ in comparison with the case Bo = 5 PG. The reason for this important result is that for a large ISM magnetic field the effective shock compression ratio u’ decreases not only due to the decrease of (7 but also

due to the decrease of the Alfven Mach number. At the late stage t > 5f0, the Alfvtn Mach number M, = b/c,

is comparatively small, M, < 5, and the CR acceleration efficiency at relativistic energies becomes very low. For this reason the y-ray flux now decreases with time at all energies. The shock continues to accelerate CRs

but with a very steep spectrum. Freshly accelerated CRs still dominate in the overall CR spectrum for E < lo’*

eV and the CR spectrum (as well as the y-ray spectrum) becomes more and more steep. At the same time the shock produces a negligible amount of fresh particles at the higher energies E > lOI eV in comparison with

the previously produced number of CRs. At a late stage of SNR evolution CRs mainly lose their energy due to adiabatic cooling. This is less important

for the highest energy particles because they occupy the whole SNR volume including the precursor almost

uniformly due to their large diffusion coefficient. Particles with lower energy are situated deeper inside the SNR

and therefore loose energy more rapidly. Due to this reason the CR and consequently also the y-ray spectra, become even harder at the late stage in the energy range eu > 1 TeV (see Fig. 5b). There is an additional reason for this hardening of the y-ray spectrum. Intermediate energy CRs around E N lOI eV are produced inefficiently at these late stages but are still confined inside the remnant. Therefore the spatial overlap of the distribution of CRs with the thermal gas density distribution (which is strongly peaked near the subshock) decreases with time in comparison with both low and very high energy CRs, leading to a relative depression

of the y-ray flux F,, near .sy N lOI eV.

4. Discussion

In order to explain the main aspects of the evolution of the y-ray flux and to compare with previous models it is useful to represent this flux in a simple form. This can be done by means of approximately performing the integration in expression (21) over the volume occupied by CRs with the aid of the spatial dependencies

of gas density and CR

distribution of shocked 7a, b), we can write

distribution shown in Fig. 7 for the case v = 10m3, B = 5pG. In the case when the

gas behind the shock front is close to uniform, i.e. in the free expansion phase (Figs.

F, 0: pN(> ~1,

where p is the average gas density in the volume occupied by CRs which is mainly the volume between the

piston, at R, =0.95R,, and the shock, and where

N(> P) = dp’N(p’) 1 P

is the total number of CRs in the SNR with momenta greater than p x lOe,/‘c. In the case of a strongly modified shock the overall CR spectrum has a power law form N(p) cc p-7 with y < 2 in the relativistic

energy range. Therefore we have

NC> P) = N(P)P _[l-($-y],

where pm is the cutoff momentum in the overall spectrum N(p). It is useful in this case to also represent the CR energy content in the form

197 E.G. Berezhko, H.J. Viilk/As~roparficle Physics 7 (1997) 183-202

Fig. 7. Radial dependencies of the gas density p(r, t) (normalized to its upstream value po), and of the CR distribution function f(p, r) multiplied by p4 ( in arbitrary units). The radial distance r is given in units of the shock radius Rx(t). Figs. 7a, b, and c correspond to f/t0 = 0.1, 1, and 10, respectively, where 10 denotes the sweep-up time. The CR distribution is shown for four different particle energies Et (in units of eV) as indicated in Fig. 7a. For r < ro the piston has a radius R, x 0.95R, and an assumed thickness IP = 0.1 R,,, and sepemtes freely expanding ejecta in 0 < r < 0.9Rp from shocked interstellar material at Rp < r < R,. The artificial density step at r = R, - 1, is a quantitatively unimportant consequence of the assumed finite value for I,. For t < ro the piston becomes decelerated strongly so that ultimately Rp << R,.

E c

= 1112c3 N(mc) 2-y [(E,“-11.

For the momenta me < p < pm and for y < 2, we can write the approximate expression

1-Y mc(2 - y)

Pm(Y - 1)

which shows that the time dependence of the y-ray flux has the form

Fy 0: pECp;-’ 0: jSE,( R,v,)y-2, (30)

if we take into account that according to relation (26) we have pm c( R,V,. In the opposite case, when the CR space distribution is close to uniform, i.e. during the Sedov phase (Fig.

7c), the corresponding relation has the form

Fy c( M,ec( R,K>y-2,

198 E.G. Berezhka, H.J. Wk/Astroparricie Physics 7 (1997) 183-202

where e, is the CR energy density, and M, is the mass of the shocked gas in which the CRs are confined. In the Sedov phase R,V, 0: t-‘/’ and the factor (R,V,)y-* can be considered as a constant. Therefore the time

dependence of the expected y-ray flux at energies .s,. > 10 GeV can be written in the form

Fy cx Mgec. (31)

During the free expansion phase both factors p and EC are increasing functions of time (see Figs. 1, 3)

which leads to the growth of the y-ray flux (Fig. 6). To the extent that ii = ape the average gas number density

reaches large values at the end of the free expansion phase (see Fig. 3). The total CR energy EC reaches its maximum value somewhat later (see Fig. 1). The third factor (R,V,)Y-* decreases in the free expansion phase and is almost constant during the Sedov phase. Relation (30) explains the fact that the y-ray flux begins to

drop with time earlier than does the CR energy.

Both the peak value and the time dependence of the expected y-ray flux F, are not insignificantly different

from the hydrodynamic prediction [9,10]. Regarding the peak value it is best to compare the kinetic and the

hydrodynamic result [lo] around t = to, where the downstream gas density is roughly uniform between piston and shock. Specifically the case 77 = 10e3 and Bo = ~,uG (corresponding to the dashed curve in Fig. 6) lends itself for an approximate comparison with the total y-ray flux Fi” resulting from Fig. 1 of Ref. [lo]. For

n = 0.3 cme3, ESN = 10” erg, d = 1 kpc, LY = 2.0, as well as 0 = 0.5, Eq. (9) of Ref. [lo] gives a maximum

value Ego cu 2.7 . lo-” photons cm -* s-’ for E > 1 TeV, since (QY/Ec)a=2.0 = 1.89(Qy/Ec)o=2.t. At t = to,

F,? is roughly a factor of 3 lower, cf. Fig. 1 of Ref. [ lo]. For the same values of n, ESN, d, a, and t, our F,,

corresponding to case (a) of Fig. 6, equals F,( E > 1 TeV) N 7. lo-” photons cm-* s-‘, i.e. a factor of about 8 larger than F’O. This factor can be approximately explained by an enhanced CR energy EC = 3.3 Ei” and an

enhanced compiession ratio (+ N (9/4)a” ’ m the kinetic relative to the hydrodynamic case [ lo], resulting in

(E,(T) /( Ez”~lo) pv 6.8. Thus, ultimately it is the combination of harder spectrum, increased CR energy, and

higher compression ratio that leads to the higher y-ray flux in the kinetic picture. Regarding the time dependence, the kinetic model predicts a y-ray flux which is rising much earlier during

the free expansion phase together with EC because the CR acceleration efficiency is considerably higher due to the much larger shock speed described by Eqs. (13)-( 15) (see Fig. 6). The simplified models [lo] did not take the power-law velocity distribution (13) of the ejecta into account. Their F, rises very late but sharply during sweep-up (compare their Fig. la) to a level which remains about constant throughout the Sedov phase.

Interestingly the same behaviour occurs in Dorti’s model [9] (compare his Fig. 7c), although his detailed gas dynamical calculation [ 141 shows the very high early shock speed implied by Eqs. ( 13)-( 15). His rise of

EC during the sweep-up phase is even slower than in our case (d) (compare his Fig. 3~). The CR energy of Dorfi’s mode1 [9] falls off similarly to the kinetic case in the Sedov phase, whereas in the simplified models

[ 101 EC stays high for a much longer period. The peak value of the y-ray flux is in the kinetic treatment reached after a few to and then falls off roughly

linearly with time, by a factor of about 10 (and more so in the high field case) over a time period 10 times larger than the time of maximum. One reason for this decrease is the damping of the scattering AlfvCn waves which get their energy from the CR pressure gradient. This influences in particular the late time behaviour of EC_ Quantitatively the damping effect requires a good knowledge of the spatial structure in the CR precursor which may explain the difference for E, between [ lo] on the one hand, and [9] and our present treatment on the other.

As far as the maximum y-ray flux is concerned there is not only a quantitative but also a qualitative difference between the hydrodynamic and the kinetic models. This concerns the spatial overlap between CR intensity and gas density as a function of particle energy discussed earlier. Whereas the very high energy CR particles near the upper cutoff are distributed more or less uniformly across the SNR, this is not true for lower energy CRs which produce the y-ray spectrum at y-energies less than a few TeV. They are remaining more and more inside the SNR where the gas density p is low. Therefore the overlap integral ( 19) decreases as time proceeds.

E.G. Berezhko, H.J. Wk/Astroparticle Physics 7 (1997) 183-202 199

Hydrodynamic models can only calculate the CR energy density e, which is an integral over all particle energies. Therefore they cannot reproduce any differential overlap effect. As a result their F,(t) remains roughly constant throughout the Sedov phase. Quantitatively the peak of the y-ray flux (and its quasi-constant level during the Sedov phase) for the hydrodynamic models can be somewhat larger than in our present theory. These models

predict very strong modifications of the SNR shock by the CR pressure gradient and for certain parameters

even a complete smoothing during some phases of the evolution. In this case about all internal energy behind the shock front is contained in the CRs. However, as discussed in Section 1, the kinetic treatment shows that for a spherical SNR the CRs cannot modify the shock completely and therefore the CR production efficiency

remains lower. The hydrodynamic treatment obviously has difficulties to produce CRs efficiently in the early sweep-up phase with its limited dynamical range in particle momentum, while not being overefficient in the

Sedov phase. The present, detailed theory allows the absorption of more than ten percent of the explosion energy into CRs

for t < to (see Fig. l), and a correspondingly high y-ray flux at these early times (Fig. 6). This may have interesting consequences for SNRs approaching the Sedov phase of the same type as Tycho’s SNR, even though unfortunately that particular object is probably not a strong y-ray source due to its low ambient gas density and

explosion energy [ lo]. Formula (3 1) gives a simple explanation of the behaviour of the y-ray flux Fy at different times during the

Sedov phase. If the shock continues the acceleration of CRs, then e, 0; pay: and h4, = M,, cc Ri (M,y, is

the mass of ambient gas swept up by the shock), resulting in Fy 0: RzVf which is constant during the Sedov phase. This explains the nearly constant y-ray flux predicted by the hydrodynamic models [ 9,101.

Intermediate and highest energy CRs are not effectively produced in the late Sedov phase any more. Previously produced intermediate energy CRs remain confined in the shock interior. This means that a constant gas mass M, < M,, is interacting with these particles, while e, decreases with time due to adiabatic cooling. Therefore

the production of y-rays with energies around lo’* eV drops with time. The highest energy CRs occupy the whole SNR volume almost completely and uniformly, i.e. M, = M,, cc

R:. Therefore the intensity of the highest energy y-rays decreases more slowly compared to those of the intermediate energy band.

It is important to note the kinetic theory prediction that sufficiently high values of the y-ray flux are expected only during a rather limited period of SNR evolution. For example the y-ray flux F,( 1 TeV) > lo-” cm-*s-r

according to Fig. 6 can be observed only during the period 0.5 + SOto after explosion for a standard magnetic field value Bo = 5 ,uG. For higher fields this period becomes shorter. This fact reduces the possibility to detect high energy y-ray emission from SNRs of unknown age.

For a given set of SN parameters the y-ray flux should roughly linearly scale with the ISM number density NO. At different NO it will have almost the same dependence on t/to. In other words, a SN explosion in a

denser medium produces a higher y-ray flux but during a shorter period of time (to 0: Ni”3). One can expect

also that the peak value of the y-ray flux and its time profile F,(t/fo) are almost independent of the value of

the ejected mass Mej. However the absolute time scale to increases proportional to M::".

5. Summary and conclusions

Our numerical calculations, based on CR kinetic theory and on a phenomenological injection model, demon- strate that CR nucleons can be very efficiently accelerated in SNRs. In a uniform ISM considered here, about 50 percent of the explosion energy is transformed into CR internal energy during the main (adiabatic) phase of SNR evolution.

The peak value of the CR energy content is reached when the SN shock has swept up an amount of gas roughly equal to several times the ejected mass. Thereafter the CR energy content decreases slowly with

200 E.G. Berezhko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202

time due to adiabatic cooling and wave dissipation. This leaves room for a release of 10 + 20 percent of the explosion energy in the form of CRs upon the ultimate dissolution of the SNR into the ISM, consistent with the observational requirement (see [ 441) .

At least in core collapse SNRs the extremely high maximum ejecta velocities lead to correspondingly high shock speeds at early times and thus to strong shock modifications due to the high acceleration efficiency during the sweep-up phase.

The CR differential energy spectrum is always close to a power law with index 2 although shock modification

and late-time shock weakening both tend to steepen the spectrum at nonrelativistic particle energies and to harden it significantly to an index value below 2 in the relativistic region. The maximum energy of accelerated particles is about lOI eV if we assume the diffusion coefficient to be as small as the Bohm limit.

The high energy y-ray production by inelastic collisions of CR nucleons with gas nuclei is already strong during the sweep-up phase due to the strong gas compression and a roughly 10 percent absorption of the explosion energy by CRs. This makes nearby (d N 1 kpc) SNRs in a sufficiently dense ISM already good

candidates for y-ray detection when they are quite young in an evolutionary sense. For an explosion with 105* erg at 1 kpc distance into an ISM of density No = 0.3 H-atoms cmm3 and magnetic field strength 5 ,uG, the maximum TeV y-ray flux of several lo-” photons cmv2 s-l is reached at the end of this phase i.e. at a time

N 2000 yr after the explosion. On the other hand, at least in the most interesting TeV energy region, the further evolution of the y-ray flux is characterized by a roughly linear decrease by a factor of ten if the system age has reached ten times the sweep-up time to. The rate of flux decrease is higher for larger Alfvtn velocity e.g. due to a larger ISM magnetic field strength which decreases the CR acceleration efficiency but leaves the adiabatic

cooling unchanged to first approximation. The peak y-ray flux is proportional to No; to scales as Nc1’3. The integral y-ray spectra which we obtain are close to a power law with index 1. In part like the CR

energy spectra, they harden above y-ray energies er > 10’ ’ eV due to shock modification and late-time shock weakening that result in power law indices smaller than 1. They exhibit a smooth cutoff at ay N lOI3 eV,

roughly ten times lower than the CR cutoff. Previous estimates, based on a hydrodynamic approximation for the CRs, gave significantly lower maxima

of the CR spectrum in the early Sedov phase, and in particular underestimated the particle acceleration and

even more the y-ray fluxes during the sweep-up phase. They also did not show the subsequent decrease of

the y-ray flux during the Sedov phase. By their very nature hydrodynamic theories could only make resonable assumptions about the form of the CR and y-ray spectra and their cutoffs during SNR evolution, using test particle calculations and interpretations of observations of CR secondary nuclei in the ISM.

Regarding the y-ray spectra there is an additional effect that influences their shape: at times after the maximum of the flux particles of intermediate energies, E = 1012 + lOI eV, are no more very strongly produced

in comparison to lower energy particles. Yet, in contrast to the highest energy CRs with E > lOI3 eV, they tend to remain confined in the SNR interior towards which the gas density p(r) strongly decreases. This decrease of spatial overlap between CR and gas density distribution reduces the y-ray production and leads to a relative

depression of the y-ray flux F, near ey N 1012 eV that increases with time. The sum of all these kinetic and configurational effects leads to the temporal decrease of Fy starting with the Sedov phase.

The limiting flux F1 detectable in the TeV-range by modern ground-based instruments, like upcoming systems of imaging atmospheric Cherenkov telescopes, is about Fl = lo-l2 photons cmd2 s-’ (e.g. [ lo] ). According to our results we expect a y-ray flux at 10” eV that exceeds FI = 10-l’ photons cm-* s-l from a SNR at distance d = 1 kpc in a warm uniform ISM to exist for times t, = lo4 f lo5 yr depending of the magnitude of the AlfvCn velocity. Assuming crudely that SNRs are distributed uniformly in the Galactic disk we can attempt to estimate the number N,,, of SNRs of this kind that are observable with limiting sensitivity at any given time:

E.G. Berethko, H.J. Vdk/Astroparticle Physics 7 (1997) 183-202 201

where Ye,, is the Galactic SN rate, f denotes the filling factor of the ‘warm’ ISM, and Rgal is the radius of the Galactic disk. Taking vsn = l/30 yr-‘, f = 0.5, and R,l = 15 kpc, we obtain Nsn = 7 + 74.

For imaging Cherenkov telescopes this number increases if the threshold decreases below lOI eV. Indepen- dently of the type of detectors their threshold energies Erh play a role for the observation of an object with a spectral cutoff.

It is well known that the Whipple telescope [46] with E,h x l/3 TeV, the CANGAROO telescope [ 471 with E,h = 1 TeV, the AIROBICC array with & % 25 TeV [ 481, as well as the high-threshold detector arrays CASA-MIA [ 491, CYGNUS [ 501, and JANZOS [ 5 1 ] with Eth N 10’ TeV have not detected positive signals from any of the candidate sources observed. In part this may be due to high threshold energies above the cutoff predicted here. In part this may be also due to the extended rather than point-like nature of these sources which renders the rejection of CR background events and thus a positive identification of a y-ray source difficult. In addition, there are great uncertainties in the astronomical determination of the ambient medium parameters like the ISM density No and its spatial distribution, as is for example the case for the SNR y-Cygni [4,52]. Projection effects in an optically thin medium are prominent amongst these uncertainties. As a consequence we will have to await deeper observations, also with stereoscopic systems (cf. [53] ) that should be able to overcome the technical difficulties presented to the low-threshold imaging atmospheric Cherenkov technique by extended sources. Also there is a need to improve the astronomical multi-wave analysis of individual candidate SNRS.

However we should take these somewhat complex but clearly negative observational results as a serious reason to reevaluate the various theoretical aspects of particle acceleration in astronomical objects. One of these aspects concerns the characteristics of the circumstellar medium around SN progenitors. As a next step we should therefore abandon the approximation of a uniform ISM around a point explosion site. The majority of SNRs in our Galaxy appears to have massive star progenitors with more or less well-developed stellar wind bubbles around them. The companion paper II investigates such cases.

AcknowIedgments

We thank L.T. Ksenofontov for his help in the preparation of the manuscript. EGB acknowledges the hospitality of the Max-Planck-Institut fur Kernphysik where the main part of this work was carried out. Much of the final work was done during a short visit of HJV at the Institute of Cosmophysical Research and Aeronomy. He thanks the Yakutsk institute for its hospitality. The work has in part been supported by the Verbundforschung Astronomie und Astrophysik of the German BMBF under contract number 05 2HD66A, Teilprojekt:E.

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