Spectrum of cosmic rays, produced in supernova remnants
Nonlinear kinetic theory of cosmic ray (CR) acceleration in supernova remnants is employed to calculate CR spectra. The magnetic field in SNRs is assumed to be significantly amplified by the efficiently accelerating nuclear CR component. It is shown that the calculated CR spectra agree in a satisfactory way with the existing measurements up to the energy eV. The power law spectrum of protons extends up to the energy eV with a subsequent exponential cutoff. It gives a natural explanation for the observed knee in the Galactic CR spectrum. The maximum energy of the accelerated nuclei is proportional to their charge number . Therefore the break in the Galactic CR spectrum is the result of the contribution of progressively heavier species in the overall CR spectrum so that at eV the CR spectrum is dominated by iron group nuclei. It is shown that this component plus a suitably chosen extragalactic CR component can give a consistent description for the entire Galactic CR spectrum.
Subject headings:cosmic rays – shock waves – acceleration of particles – supernova remnants
The main reason why supernova remnants (SNRs) are usually considered as the cosmic ray (CR) source is a simple argument about the energy required to sustain the Galactic cosmic ray (GCR) population against loss by escape, nuclear interactions and ionization energy loss. Supernovae have enough power to drive the GCR acceleration if there exists a mechanism for channeling about 10% of the mechanical energy release into relativistic particles. The high velocity ejecta produced in the supernova (SN) explosion interacts with the ambient medium to produce a strong blast wave. This outer shock may accelerate a small suprathermal fraction of the ambient plasma to high energies.
The only theory of particle acceleration which at present is sufficiently well developed and specific to allow quantitative model calculations is diffusive acceleration applied to the strong outer shock associated with SNRs (e.g. see Berezhko, 2005, for review). Considerable efforts have been made during the last years to empirically confirm the theoretical expectation that the main part of GCRs indeed originates in SNRs. Theoretically, progress in the solution of this problem has been due to the development of a kinetic nonlinear theory of diffusive shock acceleration (Berezhko et al., 1996; Berezhko & Völk, 1997, 2000). This theory attempts to include all the most relevant physical factors, essential for the evolution and CR acceleration in a SNR, at least in its early very energetic stages, and it is able to make quantitative predictions of the expected properties of CRs produced in young SNRs and their nonthermal radiation. The application of the theory to individual SNRs and their known synchrotron emission (Völk, 2003; Berezhko, 2005; Berezhko & Völk, 2006) has demonstrated its capability of explaining the observed SNR properties and in calculating new effects like the extent of magnetic field amplification which leads to the concentration of the highest-energy electrons in a very thin shell just behind the shock. The theory should therefore be able to explain major characteristics of the observed GCR spectrum up to an energy of eV (Berezhko & Ksenofontov, 1999) under the assumption that the ambient magnetic field strength considerably exceeds typical ISM values.
Here we apply kinetic nonlinear theory assuming a time-dependent amplified magnetic field, consistent with multi-wavelength evidence from individual objects, in order to approximately calculate the spectra of CRs produced in Galactic SNRs. It is shown that these spectra are approximately consistent with the existing measurements of GCR spectra up to an energy of eV.
Our nonlinear model is based on a fully time-dependent solution of the CR transport equation together with the gas dynamic equations in spherical symmetry. Since all relevant equations, initial and boundary conditions for this model have already been described in detail elsewhere (e.g. Berezhko & Völk, 2004), we do not present them here and only briefly discuss the most important aspects below.
The basic scenario is that of a point explosion, where the supernova explosion ejects an expanding shell of matter with total energy and mass into the surrounding ISM. Due to the streaming instability CRs efficiently excite large-amplitude magnetic fluctuations upstream of the SN shock (e.g. Bell, 1978). Since these fluctuations scatter CRs extremely strongly, we assume the CR diffusion coefficient to be as small as the Bohm limit , where and are the proton charge and mass, and denote the particle velocity and momentum, is the particle charge number, is the magnetic field strength, and is the speed of light. Note that due to the dependence the shock-produced ion distributions depend on rigidity with the cutoff energy (e.g. Berezhko et al., 1996).
If is the preexisting field in the surrounding ISM, then the Bohm limit implies that the instability growth is restricted by some nonlinear mechanism to the level , where is the wave field. An earlier attempt for a nonlinear description of the magnetic field evolution in a numerical model (Lucek & Bell, 2000; Bell & Lucek, 2001) and more recent instability calculations (Bell, 2004) concluded that a considerable amplification to what we call an effective magnetic field should occur. The Bohm limit is then expected to refer to this amplified field. We adopt this point of view here.
From an analysis of the synchrotron spectrum of SN 1006, Cassiopeia A, Tycho’s SNR (see Berezhko, 2005, for review) such a strong magnetic field amplification can only be produced as a nonlinear effect by a very efficiently accelerated nuclear CR component. The same large effective magnetic field is required by the comparison of our selfconsistent theory with the morphology of the observed X-ray synchrotron emission, in particular, its spatial fine structure.
In fact, for all the thoroughly studied young SNRs, the ratio of magnetic field energy density in the upstream region of the shock precursor to the CR pressure is about the same (Völk et al., 2005): . We note that this amplified magnetic field exceeds the typical ISM value G during much of the early evolution. Our present calculations assume that this is the case for most of the evolution of the SNR and to this extent the results of our calculations below are insensitive to the concrete value of except in the very late SNR evolutionary phase.
The number of suprathermal protons injected into the acceleration process is described by a dimensionless injection parameter which is a fixed fraction of the number of ISM particles entering the shock front. We adopt here a value , which is consistent with the theoretical expectation (Völk et al., 2003) and which is close to the values determined individually for the three objects SN 1006, Tycho’s SNR, and Cas A.
The injection rate of ions heavier than protons can not be calculated with the required precision. Therefore we use ion injection rates which provide ion-to-proton ratios as observed in the GCRs at an energy of TeV. The physical factors (ion injection rate and acceleration efficiency) which determine this ratio were discussed by Berezhko & Ksenofontov (1999).
The overall CR spectrum is formed during the active period of SNR evolution which lasts up to the time when the SN shock becomes too weak to accelerate efficiently a new portion of freshly injected particles. After their release from the parent SNRs the accelerated CRs occupy the confinement volume more or less uniformly with an intensity where is the mean residence time, is rigidity, is momentum, and are total and kinetic particle energy, respectively.
3. Results and Discussion
We use the values erg for the explosion energy and for the ejecta mass which are typical for SNe Ia in a uniform ISM. Note that the main fraction of the core collapse SNe has relatively small initial progenitor star masses between 8 and 15 which therefore do not significantly modify the surrounding ISM through the main sequence wind of the progenitor star (e.g. Abbott, 1982). SNR evolution in this case is similar to that of SNe Ia.
The active phase of the average SNR as CR source was assumed to last until an age of yr (Berezhko et al., 2003).
In Fig.1 we present the calculated intensities of protons (H), Helium, three groups of heavier nuclei, and ”All particles” as a function of kinetic energy, as solutions of the nonlinear equations. Here we have used , with . The results of the recent experiments CAPRICE, ATIC-2 , JACEE and KASCADE, which in our view are the most reliable ones, agree quite well with this theoretical calculation up to the energy eV. One can see that the theory fits the existing data in a satisfactory way up to the energy eV. The main exception – and difficulty – is the Helium spectrum as measured in the recent ATIC-2 balloon experiment which is noticeably harder than the proton spectrum, in contrast to the theoretical expectation.
The second difficulty for the present theory are the calculated very hard overall CR source spectra which, say for protons, have the form for eV. This spectrum is noticably harder than the source spectrum , deduced in the framework of our preferred CR propagation model which includes a selfconsistent halo within a Galactic wind (Ptuskin et al., 1997). Note that this latter model explains well the existing data, except for the observed low GCR anisotropy which is at the moment a common problem for all CR diffusion models without re-acceleration (e.g. Ptuskin et al., 2006), even though the anisotropy is presumably determined by the local structure of the Galactic magnetic field and may deviate significantly from the global characteristics of the propagation model (Ptuskin et al., 1997). However, Ptuskin & Zirakashvili (2003, 2005) have argued that already in the middle Sedov phase nonlinear dissipation will increasingly reduce the field amplification and particle scattering below the Bohm limit. High-energy particles will then increasingly leave the SNR which can only continue to accelerate lower and lower energy particles. Although there is a need for a more detailed analysis of this effect, it should lead to some softening of the overall GCR spectrum as the result of escape at high and continued acceleration at lower energies in older SNRs. In this sense we do not consider the calculated overall source spectra a problem for the arguments put forward here. We will study this question in detail in a subsequent paper.
According to Fig.1 the knee in the observed all-particle GCR spectrum has to be attributed to the maximum energy of protons, produced in SNRs. The steepening of the all particle GCR spectrum above the knee energy eV is a result of the progressively decreasing contribution of light CR nuclei with increasing energy. Such a scenario is confirmed by the KASCADE experiment which shows relatively sharp cutoffs of the spectra of various GCR species at energies eV (Antoni et al., 2005), so that at energy eV the GCR spectrum is expected to be dominated by the contribution from the iron nuclei.
We note that the maximum CR energy in the overall CR spectrum is determined by the CRs produced in the very beginning of the Sedov phase. During the free expansion phase the SN shock produces even higher-energy particles as a result of the higher shock velocities and magnetic field strengths. However, since the mass in the very fast ejecta that produce them is very small, these particles contribute only a very steep part (tail) of the final overall CR spectrum, as was shown earlier (Berezhko & Völk, 2004) for type Ia SNe. For this general reason the same should also be true for wind SNe. As a result these particles play no role for the population of Galactic CRs – in contrast to the assumption of Bell & Lucek (2001) which has been followed by Hillas (2006).
In Fig.2 we present an all-particle spectrum which includes two components: (i) CRs produced in SNRs and (ii) extragalactic CRs, protons plus 10% Helium, presumably produced in Active Galactic Nuclei (Aloisio et al., 2006). The second component has been chosen to have a power-law source spectrum above an energy eV up to energies in excess of eV.
Compared with the source spectrum , the component , observed in the Galaxy, is modified by two factors. At energies eV the shape of is influenced by the energy losses of CRs in intergalactic space as a result of their interaction with the cosmic microwave background that leads to the formation of a ”dip” structure at eV, and to a GZK-cutoff for eV (Aloisio et al., 2006).
For eV the spectrum is determined by the character of CR propagation in intergalactic space. Since we assume the existence of a Galactic Wind, CRs penetrating into the Galaxy from outside are in addition subject to modulation by the wind. We describe this effect by the modulation factor , where the maximum CR energy modulated by the Galactic Wind is about eV (Völk & Zirakashvili, 2004).
Cf. Aloisio et al. (2006), presenting data of the AGASA, Yakutsk and HiRes detectors in Fig.2, we shift the energies by a factor of , 0.85 and 1.2 respectively, so that the measured CR fluxes agree with each other.
According to Fig.2 the calculated GCR spectrum is in reasonable agreement with the existing data. It leads us to the following conclusion: if the observed CR spectrum at energies eV is indeed dominated by the contribution from extragalactic sources [the so-called ”dip-model” of Aloisio et al. (2006)], then we do not need any other Galactic source population except SNRs, as calculated above. However, if the extragalactic sources produce in reality a much harder spectrum (the so-called ”ankle-model”) then their contribution becomes dominant only at energies eV. Therefore, to fit the observed GCR spectrum an additional Galactic source population is required whose contribution is essential in the energy range eV (Hillas, 2006). It could possibly result from CR reacceleration processes (e.g. Berezhko & Ksenofontov, 1999), for example, in the interaction regions of the Galactic Wind induced by the spiral structure in the Galactic Disk (Völk & Zirakashvili, 2004). For further Galactic particle sources, see Hörandel (2007).
We note that these two scenarios for the extragalactic component predict a very different CR chemical composition above about eV. Since within the range eV reacceleration produces a power law tail of the CR spectrum originally produced in SNRs, it is clear that the observed CR spectrum is expected to be dominated by the iron contribution at energies eV. It is very much different from what is expected within the dip-model.
In order to illustrate the CR chemical composition, expected in the latter case, we present in Fig.3 the mean logarithm of the GCR atomic number . At energies TeV increases with energy and for GeV we have . Due to the dependence of the maximum energy of CRs produced in SNRs, , on the charge number and therefore on , CRs become progressively heavier above the proton cutoff energy eV. The mean atomic number of the CRs produced in SNRs goes towards the value as the energy approaches eV. However, already at eV the contribution of extragalactic CRs becomes essential. Therefore reaches its peak value at eV and then diminishes with increasing energy to the value .
The calculated atomic number agrees reasonably well with the existing data up to a particle energy of eV. At higher energies goes down with energy similar to the Yakutsk and HiRes data. Quantitatively the calculated value of agrees well with the HiRes data. Since the existing data demonstrate that above eV CRs become lighter with energy, this could be considered to favor the dip-scenario.
Magnetic field amplification leads to a considerable increase of the maximum energy of the CRs accelerated in SNRs. Calculations performed within nonlinear kinetic theory demonstrate that the expected GCR spectrum, produced in SNRs, fits the existing GCR data in a satisfactory way up to the energy eV. The first knee in the observed all-particle GCR spectrum is attributed to the maximum energy of protons that are produced in SNRs. The steepening of the all-particle GCR spectrum above the knee energy eV is then the result of a progressive depression of the contribution of light CR nuclei with increasing energy. Such a scenario is confirmed by the KASCADE experiment which shows relatively sharp cutoff spectra of GCR species at energies eV (Antoni et al., 2005). A difficulty of the present computation are the resulting very hard overall source spectra from SNRs. We expect however that inclusion of wave dissipation and particle escape from SNRs at later phases of their evolution will rectify this extreme.
If the Extragalactic CR component has a spectrum so that it dominates the observed GCR spectrum already at eV, then GCRs at lower energies are produced in SNRs without significant contribution from other Galactic CR sources. The GCR spectrum up to the energy eV is dominated by the contribution of SNRs, whereas at eV GCRs are predominantly Extragalactic. If the Extragalactic source spectrum is much harder, , then the transition from the Galactic to the Extragalactic component is expected at higher energy eV. Since the expected GCR chemical composition at eV is very different in these two cases, an experimental study of GCR composition could discriminate them. The existing HiRes and Yakutsk data favor the first, so-called dip scenario.
- Abbasi et al. (2005) Abbasi, R. U. et al. 2005, astro-ph/0501317
- Abbott (1982) Abbott, D.C. 1982, ApJ, 263, 723
- Aloisio et al. (2006) Aloisio, R. et al. 2007, Astropart. Phys., 27, 76
- Antoni et al. (2005) Antoni, T. et al. 2005, Astropart. Phys., 24, 1
- Asakimori et al. (2003) Asakimori, K. et al. 2003, ApJ, 502, 278
- Bell (1978) Bell, A.R. 1978, MNRAS, 182, 147
- Bell (2004) Bell, A.R. 2004, MNRAS, 353, 550
- Bell & Lucek (2001) Bell, A.R. & Lucek, S.G. 2001, MNRAS, 321, 433
- Berezhko et al. (1996) Berezhko, E. G., Elshin, V. K., & Ksenofontov, L. T. 1996, JETP, 82, 1
- Berezhko & Völk (1997) Berezhko, E.G. & Völk, H.J. 1997, Astropart. Phys. 7, 183
- Berezhko & Völk (2000) Berezhko, E.G. & Völk, H.J. 2000, A&A, 357, 183
- Berezhko & Ksenofontov (1999) Berezhko, E.G. & Ksenofontov, L.T. 1999, JETPh 89, 391
- Berezhko et al. (2003) Berezhko, E.G. et al. 2003, A&A, 410, 189
- Berezhko & Völk (2004) Berezhko, E.G. & Völk, H.J. 2004, A&A, 427, 525
- Berezhko (2005) Berezhko, E.G. 2005, Adv. Space Res., 35, 1031
- Berezhko & Völk (2006) Berezhko, E. G. & Völk, H. J. 2006, A&A, 451, 981
- Boezio et al. (2003) Boezio, M. et al. 2003, Astropart. Phys., 19, 583
- Egorova et al. (2004) Egorova, V.P. et al. 2004, Nucl. Phys. B (Proc. Suppl.), 136, 3
- Ivanov et al. (2003) Ivanov, A. A. et al. 2003, Nucl. Phys. B (Proc.Suppl.), 122, 226
- Hillas (2006) Hillas, A.M. 2006, J. Phys.: Conf. Ser., 47, 168
- Hörandel (2003) Hörandel, J.R. 2003, J. Phys. G., 29, 2439
- Hörandel (2005) Hörandel, J.R. 2005, astro-ph/0508014
- Hörandel (2007) Hörandel, J.R. 2007, astro-ph/0702370
- Lucek & Bell (2000) Lucek, S. G. & Bell, A. R. 2000, MNRAS, 314, 65
- Panov et al. (2006) Panov, A. D. et al. 2006, astro-ph/0612377
- Ptuskin et al. (1997) Ptuskin, V.S. et al. 1997 A&A, 321, 434
- Ptuskin & Zirakashvili (2003) Ptuskin, V.S. & Zirakashvili, V.N. 2003, A&A, 403, 1
- Ptuskin & Zirakashvili (2005) Ptuskin, V.S. & Zirakashvili, V.N. 2005, A&A, 429, 755
- Ptuskin et al. (2006) Ptuskin, V.S. et al. 2006, ApJ, 642, 902
- Takeda et al. (2003) Takeda, M. et al. 2003, Astropart. Phys. 19, 447
- Völk et al. (2003) Völk, H.J., Berezhko, E. G., & Ksenofontov, L. T. 2003, A&A, 409, 563
- Völk (2003) Völk, H.J. 2003, Proc. 28th ICRC, Invited papers, Vol.8, p.29
- Völk & Zirakashvili (2004) Völk, H.J. & Zirakashvili, V.N. 2004, A&A, 417, 807
- Völk et al. (2005) Völk, H. J., Berezhko, E. G., & Ksenofontov, L. T. 2005, A&A, 433, 229