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Astron. Astrophys. 330, 389-398 (1998)

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2. Errors inherent in the theory

2.1. Basic concept

We consider the acceleration of particles in spherical shocks with the concept of first order Fermi acceleration (see, e.g., Drury 1983). In this concept energetic particles are cycling back and forth across the shock region, gaining energy each time they turn back; since the two sides of a shock are a permanently compressing system the particles gain energy. In a spherical shock the particles also lose energy from adiabatic expansion. Furthermore, in a system where the unperturbed magnetic field is perpendicular to the shock normal, particles can modify their energy by drifts, here dominated by curvature drifts; the particles move sideways in a curved magnetic field, and experience an electric field from the motion through a magnetic field; the component of the motion parallel to the perceived electric field leads to an energy change. It has to be noted that for plane parallel shocks the drifts have been shown by Jokipii (1982) to be equivalent in their effect to the Lorentz transformation; here we emphasize that we use the curvature and gradient drifts only, which give an additional effect. A key ingredient in such an approach is the time spent by a particle on either side of the shock. Observations as well as stability arguments lead us to the notion, that the time spent on either side of the shock is given by a transport coefficient [FORMULA] given by fast convective motion (see Biermann 1993a, 1997b). The construction of this transport coefficient is the major step in the argument, and is based on the concept of the smallest dominant scale, a scale either in real space or in phase space. The resulting expression for the spectrum of the energetic particles is the same as in, e.g., Drury (1983), except for an additional term for energy gains from drifts (Jokipii 1987).

For the expansion of a spherical shock into a stellar wind we adopt the basic magnetic configuration of a Parker spiral (Parker 1958, Jokipii et al. 1977), where the magnetic field in the equatorial plane is an Archimedian spiral with [FORMULA] dominating over [FORMULA], decreases with [FORMULA] towards the pole and outwards with radius r, and becomes mostly radial in the pole region. We write for the wind velocity [FORMULA], for the shock velocity [FORMULA], and for the downstream gas velocity in the shock frame [FORMULA].

2.2. Assumptions and Systematic Uncertainties

The assumptions adopted are inspired by Prandtl's mixing length approach (Prandtl 1925, 1949); all use the key proposition that the smallest dominant scale, either in geometric length, or in velocity space, gives the natural transport coefficient. In this sense the assumptions are derived from a basic principle.

Our basic, argument 1, based on observational evidence as well as theoretical arguments, is that for a cosmic ray mediated shock the convective random walk of energetic particles perpendicular to the unperturbed magnetic field can be described by a diffusive process with a downstream diffusion coefficient [FORMULA] which is given by the thickness of the shocked layer and the velocity difference across the shock, and is independent of energy.

The upstream diffusion coefficient can be derived in a similar way, but with a larger scale based on the same column density as in the downstream layer. This leads to the second critical, argument 2, namely that the upstream length scale is just [FORMULA] times larger.

It must be remembered that there is a lot of convective turbulence which increases the curvature: The characteristic scale of the turbulence is [FORMULA] for strong shocks, again, as an example, in the case of the wind-SN, and thus the curvature is [FORMULA] maximum. Half the maximum of the curvature allows for the net balance of gains and losses for the energy gain due to drifts (argument 3), and so we obtain then for the curvature [FORMULA] which is twice the curvature without any turbulence; this increases the curvature term for the spectral range below the knee.

The diffusion tensor component [FORMULA] can be derived similar to our heuristic derivation of the radial diffusion term [FORMULA], again by using the smallest dominant scales. The characteristic velocity of particles in [FORMULA] is given by the erratic part of the drifting, corresponding to spatial elements of different magnetic field direction, and this is on average the value of the drift velocity [FORMULA], possibly modified by the locally increased values of the magnetic field strength, and the characteristic length is the distance to the symmetry axis [FORMULA] (argument 4); this is the smallest dominant scale as soon as the thickness of the shocked layer is larger than the distance to the symmetry axis, i.e. [FORMULA].

Rapid convection also gives a competing diffusion in the [FORMULA] -direction, independent of particle energy; this will begin to dominate as soon as the energy dependent [FORMULA] -diffusion coefficient reaches this maximum at a critical energy. As long as the [FORMULA] -diffusion coefficient is smaller, it will dominate particle transport in [FORMULA] and the upper limit derived here is irrelevant. When the [FORMULA] -diffusion coefficient reaches and passes this maximum given by the fast convection, then the particle in its drift will no longer see an increased curvature due to the convective turbulence due to averaging and the part of drift acceleration due to increased curvature is eliminated. Again, a detailed consideration of gains and losses of the drift energy gains leads to the spectrum of particles beyond the knee. The critical energy derived in this way is the same as that derived from a phase-space argument near the poles.

All these arguments are inspired by Prandtl's mixing length approach; all use the key proposition that the smallest dominant scale, either in geometric length, or in velocity space, gives the diffusive transport discussed. We assume this to be true even for the anisotropic transport parallel and perpendicular to the shock.

We emphasize the analogy to simple limiting scaling arguments such as a) the estimate of the temperature gradient in the lower hydrogen convection zone on the Sun, which is followed by nature to a very good approximation (Strömgren 1953, p. 65), and b) the estimate of the Kolmogorov turbulence spectrum (Rickett 1990, Goldstein et al. 1995), which appears to be also followed by nature in many sites over many orders of magnitude in length scale. Whether the cosmic rays follow also such a limiting scaling argument, as regards their spectrum, to such an accuracy remains to be seen. This paper is a step to verify the straight forward prediction for 28 chemical elements individually.

In addition, we i) use the simplified notion of a purely spherical shock; ii) ignore the modifications of the shock introduced by the cosmic rays themselves, except in the conceptual derivation of the initial argument, where the cosmic rays are critical for the instability; and iii) use a test particle approach.

We have to emphasize very strongly that these uncertainties mean that the spectral indices derived for the powerlaw region of the various components of the cosmic rays correspond to a limiting argument: If things were really as simple - and they are likely to be much more complicated - then the spectrum derived and any comparison with data has to be taken with considerable caution.

On the other hand, the very simplicity of the proposed concept makes it easier to test and this is what we propose to do.

2.3. The error budget

2.3.1. Below the knee

As discussed in paper CR III, the simplifications which we did make in treating the flow field of the expansion of a supernova explosion into the interstellar medium lead to an uncertainty of [FORMULA] in spectral index.

For wind-supernovae we can estimate one uncertainty, which arises from the finite wind speed of Wolf Rayet stars, or those massive stars with strong winds which explode as supernovae. These wind speeds can go up to several thousand km/sec, while the supernova shock is variously estimated to [FORMULA] km/sec to twice that much. As a limiting argument we use that the ratio of the wind speed to the supernova shock speed is [FORMULA] ; this gives a steepening of the derived spectral index of the particle distribution by 0.04. This uncertainty also may correspond to curvature of the spectrum, since there is a time-evolution as the shock progresses out through the stellar wind: As more energy of the shock is dissipated and more mass of the stellar wind snowplowed, the shock slows down; then those particles already accelerated keep their flatter spectrum (see Eq. 2.44 of Drury 1983), while those particles freshly injected and accelerated will have a steeper spectrum. Thus, in the range [FORMULA] we obtain a spectral index in the range [FORMULA]. Therefore we ascribe to the spectral index derived here an uncertainty of [FORMULA], which describes both the uncertainty in an assumed powerlaw, and the possible curvature. This way of writing makes it clear that we do not expect the distribution of observed spectral indices to follow a gaussian distribution, but rather to be biased towards faster winds, which steepen the spectrum.

2.3.2. The knee

In the Fermi-acceleration process there is energy gain and energy loss in each cycle which an energetic particle crosses the shock region; one part of this energy gain is due to drifts. At a certain rigidity ([FORMULA]) the drift contribution is reduced, and so the slope of the spectrum changes. This critical energy is given by


In a Parker-wind the product of radius r and magnetic field strength [FORMULA] is a constant with radius. Z is the charge of the particle and e is the electromagnetic charge unit. [FORMULA] is the advance speed of the shock caused by the supernova explosion; [FORMULA].

This implies that the chemical composition at the knee changes so, that the gyroradius of the particles at the spectral break is the same, implying that the different nuclei break off in order of their charge Z, considered as particles of a certain energy (and not as energy per nucleon). In an all-particle spectrum in energy per particle, this introduces a considerable smearing.

There is one additional cosmic ray component from that latitude region near the pole of the magnetic field structure in the wind, where the magnetic field is predominantly radial rather than tangential. This region we call the polar cap. Thus the spectrum is harder in the polar cap region, because we are close to the standard parallel shock configuration, for which the particle spectrum is well approximated by [FORMULA] at the source. Because of spatial limitations most of the hemisphere has to dominate again above the knee, although with a fraction of the hemisphere that decreases with particle energy. This introduces a weak progressive steepening of the spectrum with energy. The superposition of such spectra for different chemical elements, including the polar cap contribution, has been tested (see paper CR IV). The results of these checks suggest that the polar cap may be the source for the flattening of the cosmic ray spectrum as one approaches the knee feature.

We note that we are using a limiting argument to derive the spectrum below the knee, and again use a limiting argument (see below) for the spectrum above the knee. Close to the knee, such an argument breaks down on either side, and so a softening of the knee feature is to be expected. On top of such a softened knee feature the polar cap is an additional component.

The expression for the particle energy at the knee also suggests by the clearly observed break of the spectrum that the actual values of the combination [FORMULA] must be limited in range for all supernovae that contribute appreciably in this energy range. We have speculated on possible reasons for such a behaviour elsewhere (paper CR I, and in Biermann 1995a). We note that the recent results from the Tibet array suggest that the knee may in fact be rather smooth (Amenomori et al. 1996), in apparent contrast to the earlier description of the Akeno data (Nagano et al. 1984; see also Stanev et al. 1993).

2.3.3. Beyond the knee

Beyond the knee, the drift contribution to the cyclical energy gain of individual particles is reduced, and so we obtain a steeper spectrum with an error which we write as [FORMULA]. As noted above the use of limiting arguments to derive the spectrum on either side of the knee implies that the knee itself may be quite soft, and thus curvature is to be expected.

2.3.4. The ultimate cutoff

The maximum energy particles can reach depends linearly on the magnetic field


If stars that explode as wind-supernovae were to vary widely in their magnetic field strength, then this maximum energy would also vary from star to star, and as a result the sum of all contribution would appear as strongly curved downwards.

2.4. The predictions

The proposal is that three sites of origin account for the cosmic rays observed, i) supernova explosions into the interstellar medium, ISM-SN, ii) supernova explosions into the stellar wind of the predecessor star, wind-SN, and iii) radio galaxy hot spots for the extragalactic component. Here the cosmic rays attributed to supernova-shocks in stellar winds, wind-SN, produce an important contribution at all energies up to [FORMULA] GeV.

Particle energies go up to 100 Z TeV for ISM-SN, and to 100 Z PeV with a bend at 600 Z TeV for wind-SN. Radiogalaxy hot spots may go up to near 1000 EeV at the source. These numerical values are estimates with uncertainties of surely larger than a factor of 2, since they derive from an estimated strength of the magnetic field, and estimated values of the effective shock velocity (see above).

The spectra [FORMULA] are predicted to be


for ISM-SN (paper CR III), and


for wind-SN below the knee, [FORMULA] for wind-SN above the knee, and [FORMULA] at injection for radiogalaxy hot spots. The polar cap of the wind-SN contributes an [FORMULA] component (allowing for leakage from the Galaxy), which, however, contributes significantly only near and below the knee, if at all.

The chemical abundances are near normal for the injection from ISM-SN, and are strongly enriched for the contributions from wind-SN.

This means that the sources for cosmic ray particles and the sources for the enrichment of the interstellar medium are the same, and hence it is no surprise that the isotopic ratios are similar for galactic cosmic ray sources and the solar system (DuVernois et al. 1996a, 1996b).

That spallation-produced secondaries require a strongly enriched original composition has been confirmed by a calculation of the formation of light elements in the early Galaxy and a comparison with observed abundances (Ramaty et al. 1997); we consider this to be an important check on the picture proposed here.

At the knee the spectrum bends downwards at a given rigidity, and so the heavier elements bend downwards at higher energy per particle. Thus beyond the knee the heavy elements dominate all the way to the switchover to the extragalactic component, which is, once again, mostly Hydrogen and Helium, corresponding to what is expected to contribute from the interstellar medium of a radiogalaxy, as well as from any intergalactic contribution mixed in (Biermann 1993c). This continuous mix in the chemical composition at the knee already renders the overall knee feature in a spectrum in energy per particle unavoidably quite smooth, a tendency which can only partially be offset by the possible polar cap contribution, since that component also is strongest at a given rigidity (for details see the discussion in paper CR IV). This is confirmed by Amenomori et al. (1996). They determined a spectral index of [FORMULA] below the knee and [FORMULA] above the knee with the slope changing continuously between [FORMULA] eV and [FORMULA] eV.

We note that uncertainties of the spectrum derive from a) the time evolution of any acceleration process as the shock races outward, b) the match between ISM-SN and wind-SN, c) the mixing of different stellar sources with possibly different magnetic properties, and d) the differences in propagation in any model which uses different source populations. These uncertainties translate into a distribution of powerlaw indices of the spectra, to curvature of the spectra, to a smearing of the knee feature, and to a smoothing of the cutoffs. Obviously, this is in addition to the underlying uncertainty associated with the concept of the smallest dominant scale itself.

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Online publication: January 8, 1998