A&A 395, 943-953 (2002)
E. G. Berezhko 1 - L. T. Ksenofontov 1 - H. J. Völk 2
1 - Institute of Cosmophysical Research and Aeronomy, 31 Lenin Av., 677891 Yakutsk, Russia
2 - Max Planck Institut für Kernphysik, Postfach 103980, 69029 Heidelberg, Germany
Received 8 April 2002 / Accepted 23 August 2002
The nonlinear kinetic model of cosmic ray (CR) acceleration in supernova remnants (SNRs) is used to describe the properties of the remnant of SN 1006. It is shown, that the theory fits the existing data in a satisfactory way within a set of parameters which is consistent with the idea that SN 1006 is a typical source of Galactic CR nucleons, although not necessarily of CR electrons. The adjusted parameters are those that are not very well determined by present theory or not directly amenable to astronomical observations. The calculated expansion law and the radio-, X-ray and -ray emissions produced by the accelerated CRs in SN 1006 agree quite well with the observations. A rather large interior magnetic field G is required to give a good fit for the radio and X-ray synchrotron emission. In the predicted TeV -ray flux from SN 1006, the -decay -rays, generated by the nuclear CR component, dominate over the inverse Compton (IC) -rays, generated by the CR electrons in the cosmic microwave background. The predicted source morphology in high energy -rays roughly corresponds to that of the synchrotron emission. The predicted integral -ray flux extends up to energies 100 TeV if CR diffusion is as strong as the Bohm limit. Only if the interior magnetic field is much lower in the SNR, G, then the observed -ray emission can be due to the accelerated electron component alone. In this case, not plausible physically in our view, the lowest permissible value of the electron to proton ratio is rather high, and the maximum individual energy and total energy content of accelerated nucleons so small, that SN 1006 can not be considered as a typical source of the nuclear Galactic CRs.
Key words: ISM: cosmic rays - acceleration of particles - stars: supernovae: individual: SN 1006 - radio continuum: ISM - X-rays: ISM - gamma rays: theory
In the last years significant efforts have been made to obtain direct observational evidence whether the Galactic cosmic rays (CRs) - relativistic nucleons and electrons - are indeed generated in supernova remnants (SNRs). The expected -decay -ray emission, produced in nearby SNRs by the accelerated protons in their collisions with thermal gas nuclei, is marginally high enough to be detectable by the present generation of imaging atmospheric Cherenkov telescopes (e.g. Drury et al. 1994; Naito & Takahara 1994; Berezhko & Völk 1997). Positive results of such observations would constitute a necessary condition for a dominant role of SNRs in the production of the Galactic CRs and of their energy spectrum up to the knee energy 1015 eV.
If SNRs accelerate nucleons at least to 100 TeV/n as it is required for Galactic CRs (e.g Berezhko & Ksenofontov 1999), then in smooth extension of the expected hadronically generated -decay -ray peak at 67.5 MeV (e.g. Stecker 1971; Dermer 1986) one expects the nucleonic -ray spectrum to have an almost single power-law form to about 1 TeV, before turning over above 10 TeV. At the same time the leptonic Inverse Compton (IC) -ray spectrum may be quenched at a lower cut-off energy due to electron synchrotron losses (as in our model). Then -ray observations within the range from 10 MeV to 100 TeV would be useful in discriminating between the nucleonic and leptonic components.
Recent observations of nonthermal X-rays and -rays indicate that at least CR electrons are accelerated in SNRs. SN 1006 is one of the SNRs for which there is evidence that electrons reach energies of about 100 TeV (Koyama et al. 1995; Tanimori et al. 1998, 2001). It is also one of the three shell type SNRs in which TeV -ray emission has been detected up to now. However, the interpretation of these data is not unique. Depending on the assumed values for the unknown physical parameters of SN 1006 (mainly the value of the magnetic field, the electron to proton ratio, and to some extent also the nucleon injection rate), the observed high-energy -ray emission can be predominantly either inverse Compton radiation due to CR electrons scattering on the microwave background radiation (as predicted by Mastichiadis 1996; Pohl 1996; Mastichiadis & de Jager 1996; and Yoshida & Yanagita 1997 from the X-ray synchrotron emission, and through the interpretation of the X-ray and -ray data by Aharonian 1999; Aharonian & Atoyan 1999; Berezhko et al. 1999), or -decay emission due to hadronic collisions of CRs with gas nuclei (Aharonian 1999; Aharonian & Atoyan 1999; Berezhko et al. 1999; Berezhko et al. 2001).
The thermal and nonthermal X-ray emission has recently been rediscussed by Allen et al. (2001) who draw phenomenological conclusions on the -ray emission to be dominated by IC radiation as well as on the production and the characteristics of the nonthermal nucleonic particle component which they estimate to have a total energy of 1050 ergs.
In contrast, our starting point is the overall SNR dynamics and the acceleration theory of the nonthermal component. Thus we accept the X-ray results and the quantitative distinction between thermal X-ray emission and the nonthermal synchrotron components, including the constraints on the external density derived from them. We calculate the outer shock size, speed and compression ratio as well as the nonthermal quantities as solutions of the nonlinear, time-dependent kinetic equations for electrons and protons as functions of space, time and particle momentum, rather than phenomenologically assuming forms of the electron and proton momentum distribution functions.
In this way we also obtain full morphological information. We determine the unknown parameters like the upstream magnetic field strength and the electron to proton ratio, and constrain the nucleon injection rate which cannot yet be accuratetely calculated from known theory, by comparing with the observations. For this purpose we use the selfconsistent kinetic model of diffusive acceleration of CRs in SNRs (Berezhko et al. 1996; Berezhko & Völk 1997) and investigate the -ray emission from SN 1006. A preliminary version of our results is given in Berezhko et al. (2001).
In contrast to a previous study (Berezhko et al. 1999) we restrict ourselves to the so-called Bohm limit for CR diffusion near the shock, assuming efficient and strong excitation of magnetohydrodynamic waves by the accelerating particles themselves. This is consistent with the large magnetic field strengths which we infer for this remnant.
Our considerations show that, together with a renormalization due to the lack of spherical symmetry of the nucleon injection, the existing SNR data are consistent with very efficient acceleration of CR nuclei at the SN shock wave which converts a significant fraction of the initial SNR energy content into CR energy as required for a typical source of the Galactic CR nuclei. The relative amount of energetic electrons is rather small, and may require additional sources to SNe type Ia like SN 1006. Therefore the observed -ray emission of SN 1006 can indeed be of hadronic origin, and we consider this as the physically most plausible solution from the point of view of acceleration theory.
Nevertheless, the existing observations do not strongly exclude a solution in which nuclear CRs play no important role and all nonthermal emissions are of leptonic origin. Therefore we analyze whether the existing data can also be fitted by an essentially different set of parameters. We demonstrate that there is an alternative possibility to fit the overall data, albeit with somewhat lower quality.
It implies a (physically not plausible) much lower injection rate and acceleration efficiency of protons, a rather large electron to proton ratio, and a lower magnetic field value compared with the case of efficient CR nucleon acceleration. If we would assume SN 1006 to be a typical representative, in this second case we could not consider the SNRs as the source population of the Galactic CRs due to the low proton acceleration efficiency. To discriminate empirically between these rather different scenarios from our results -ray measurements above 10 TeV are required: in this very high energy range we expect a measurable -ray flux only in the case of efficient nucleonic CR production. We also confirm the earlier result (Aharonian 1999; Aharonian & Atoyan 1999; Berezhko et al. 1999; Berezhko et al. 2001) that the IC -ray emission should cover the SNR almost uniformly, whereas the -decay emission should be peaked behind the shock front and limited to two polar caps where the interstellar magnetic field is parallel to the shock normal.
Since SN 1006 is a type Ia supernova (SN) we suggest that its evolution takes place in a uniform interstellar medium (ISM), not modified by a wind from the progenitor star. The general picture is then well-known.
A SN explosion ultimately ejects a shell of matter with total energy and mass . During an initial period the shell material has a broad distribution in velocity v. The fastest part of these ejecta is described by a power law (e.g. Jones et al. 1981; Chevalier 1982). The interaction of the ejecta with the ISM creates a strong shock there which accelerates particles.
Our nonlinear model (Berezhko et al. 1996; Berezhko & Völk 1997) is based on a fully time-dependent solution of the CR transport equation together with the gas dynamic equations in spherical symmetry. It yields at any given phase of the SNR evolution the complete spatial distribution of gas and accelerated CRs and their spectrum. This makes it possible to calculate any kind of emission, and its morphology, produced by the accelerated particles.
The CR diffusion coefficient is taken as the Bohm limit
The number of suprathermal protons injected into the acceleration process is described by a dimensionless injection parameter which is a fixed fraction of the ISM particles entering the shock front. For simplicity it is assumed that the injected particles have a velocity four times higher than the postshock sound speed. Unfortunately there is no complete selfconsistent theory of a collisionless shock transition which can predict the value of the injection rate and its dependence on the shock parameters. For the case of a purely parallel plane shock hybrid simulations predict a quite high ion injection (e.g. Scholer et al. 1992; Bennet & Ellison 1995) which corresponds to the value of our injection parameter. Such a high injection is consistent with analytical models (Malkov & Völk 1995, 1996; Malkov 1998) and confirmed by measurements near the Earth's bow shock (Trattner & Scholer 1994). We note however that in our spherically symmetric model these results can only be used with some important modification. In reality we must consider the evolution of the large scale SN shock which expands into the ISM with its magnetic field. In the case of SN 1006, at the current evolutionary phase, the shock has a size of several parsecs. On such a scale the unshocked interstellar magnetic field can be considered as uniform since its random component is characterized by a much larger main scale of about 100 pc. Then our spherical shock is quasi-parallel in the polar regions and quasi-perpendicular in the equatorial region. This magnetic field essentially suppresses the leakage of suprathermal particles from the downstream region back upstream when the shock is more and more oblique (Ellison et al. 1995; Malkov & Völk 1995). Applied to the spherical shock in the uniform external magnetic field it would mean that only small regions near the poles, covering less than 10% of the shock surface, allow a sufficiently high injection that ultimately leads to the transformation of an essential part (more than a few percent) of the shock energy into CR energy, whereas the main part of the shock is an inefficient CR accelerator. If one takes into account the Alfvén wave excitation due to CR streaming (which becomes efficient already at a very low injection rate ) the local injection rate has to be averaged over the fluctuating magnetic field directions and is lower than for the purely parallel case by a factor of hundred. Therefore we adopt here the value for the injection parameter (Völk et al. 2002). The detailed choice in the next section is made in order to achieve a nonlinear shock modification that is consistent with the observed radio spectral index.
According to our above estimate a substantial part of the shock still efficiently injects and accelerates CRs. This fraction is effectively increased relative to the above percentage due to broadening of the injection region by the strong wave field as well as by CR diffusion perpendicular to the mean magnetic field. In addition, the overall conservation equations ensure an approximately spherical character of the overall dynamics. Therefore, we assume the spherically symmetric approach for the nonlinear particle acceleration process to be approximately valid in those shock regions where injection is efficient. To take the effective injection fraction into account, we then need to introduce a renormalization factor for the CR acceleration efficiency and for all the effects which it produces in the SNR. A rough estimate performed for the simple case of spherical shock expended in the uniform outer magnetic field (Völk et al. 2002) gives .
Note that such a picture is consistent with the observed structure of SN 1006: the intense radio and X-ray emissions come from two bright rims with radially oriented magnetic field (Reynolds & Gilmore 1993). They may be the polar regions of a quasispherical shock in an outer magnetic field. The -ray observations may indicate a similar asymmetry (Tanimori et al. 1998) which would be well explained by a corresponding concentration of the accelerated nucleons towards the polar regions.
We assume that electrons are also injected into the acceleration process,
still at nonrelativistic energies below
electron injection mechanism is not very well known (e.g. Malkov & Drury
2001) for simplicity we consider their acceleration starting from
the same momentum as that of protons. At relativistic energies (for
protons) these electrons have exactly the same dynamics as the protons.
Therefore, neglecting synchrotron losses,
their distribution function at any given time has the form
The electron distribution function
deviates only at sufficiently large momenta from
due to synchrotron losses,
which are taken into account by supplementing the ordinary diffusive
transport equation with a loss term:
The solution of the dynamic equations at each instant of time yields the CR spectrum and the spatial distributions of CRs and gas. This allows us to calculate the expected flux of -rays from -decay due to hadronic (p-p) collisions of CRs with the gas nuclei. Following the work of Dermer (1986) and its later improvement by Naito & Takahara (1994) we use here the isobar model at the protons kinetic energies GeV and the scaling model at GeV with a linear connection between 3 and 7 GeV. This model agrees very well with the simpler approach, introduced by Drury et al. (1994) (see also Berezhko & Völk 1997, 2000; Berezhko et al. 1999) at high energies GeV, except in the cutoff region, where the scaling model yields an essentially smoother turnover of the -ray spectrum at somewhat lower energies.
The choice of
allows us to determine in addition the electron
distribution function and
to calculate the associated emission. The expected synchrotron
flux at distance dfrom the SNR is given by the expression (e.g. Berezinskii et al.
Since in the shock region where particle injection and acceleration is efficient the upstream magnetic field is assumed to be almost completely randomized due to intense Alfvén wave generation. For such a strongly turbulent upstream field, consistent with Bohm diffusion, the postshock field strength B2 is given by by , where . We shall approximately equate with the gas compression ratio at the shock that We suggest also that in such a young SNR like SN 1006 the downstream magnetic field in a relatively thin region between the shock and the ejecta is approximately uniform and we shall take .
The relativistic electrons produce -ray emission due to
IC scattering of background photons. It is not
difficult to show that due to the hard spectrum of
accelerated electrons only the 2.7 K cosmic microwave background (CMB) is
important in the case considered. The expected differential flux of IC
-rays as a function of their energy
represented in the form:
Energetic electrons produce also nonthermal bremstrahlung (NB) emission, interacting with both ambient electrons and protons. We use standard formulae for the cross section to calculate the NB -ray flux (e.g. Baring et al. 1999). As it was already demonstrated by these last authors, in young SNRs the primary energetic electron number density due to electrons directly accelerated from the suprathermal postshock distribution by far exceeds that of secondaries which were produced via the decay of charged pions created in p-p collisions. Hence the contribution of secondaries to the NB, IC, and synchrotron emission can be neglected.
Since SN 1006 is a type Ia SN we use typical SN Ia parameters in our calculations: ejected mass , k=7, and a uniform ambient ISM.
The ISM density is the most relevant parameter, especially for the -ray production (e.g. Berezhko et al. 1999). Here we use the most appropriate value of the ambient number density of ISM hydrogen cm-3 as well as the distance d=1.8 kpc, consistent with X-ray and optical imagery of SN 1006 (Winkler & Long 1997, see also Allen et al. 2001 and references therein).
The value T0=104 K is used for the ISM temperature. Note that SNR and CR dynamics are not sensitive to T0, because the shock structure is mainly determined by the Alfvénic Mach number.
The other important parameter is the ISM magnetic field B0. Unfortunately there is no direct way to determine its value from the observations. Since it influences the synchrotron spectrum in an essential way, we use an upstream magnetic field value B0=20 G. It is required to provide the observed shape of the synchrotron spectrum in the radio and the X-ray bands.
The gas dynamics problem is characterized by the following length,
time, and velocity scales:
shock expansion law during the free expansion phase (t<t0) is then
|Figure 1: a) Shock radius and shock speed ; b) total shock () and subshock ( ) compression ratios; c) ejecta ( ), CR (), gas thermal ( ), and gas kinetic ( ) energies as a function of time. Scale values are R0=3.2 pc, V0=14 675 km s-1, t0=212 years. The dotted vertical line marks the current epoch. The observed size and speed of the shock (Moffett et al. 1993), are shown as well.|
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The calculations together with the experimental data are shown in Figs. 1-5. For the assumed SN distance and ISM density an explosion energy erg fits the observed SNR size and its expansion rate (Moffet et al. 1993). The shown uncertainties of and are calculated as the uncertainties of the angular size and expansion rate times the distance d=1.8 kpc.
According to Fig. 1a SN 1006 is indeed already in the adiabatic phase. The assumed injection rate leads to a significant modification of the shock which at the current epoch, t=995 yr, has a total compression ratio and a subshock compression ratio (Fig. 1b).
The different SNR energy components, the ejecta energy
the gas kinetic (
), and thermal (
and the CR energy ,
are presented in Fig. 1c as
functions of time. The acceleration process is characterized by a high
efficiency in spherical symmetry: at the current time
53% of the explosion energy has been already transferred to CRs, and the
CR energy content
continues to increase to a maximum of
about 60% in the later Sedov phase (Fig. 1c), when particles start
to leave the source. As usually predicted by the model, the CR
acceleration efficiency is
significantly higher than required for the
average replenishment of the Galactic CRs by SNRs, corresponding to
As discussed before, this
discrepancy can be attributed to the physical conditions at the shock
surface which influence the injection efficiency. The magnetic field
geometry is the most important factor: at the quasiperpendicular portion
of the shock ion injection (and subsequent acceleration) is presumably
depressed compared with the quasiparallel portion. Therefore the number of
nuclear CRs, calculated within the spherically-symmetrical approximation,
should be renormalized by this depression factor (see Völk et al.
2002 for details). Assuming SN 1006 to be an average Galactic CR
source, the renormalizing factor should be
that according to our calculation the actual energy of accelerated protons
in SN 1006 at the present epoch is
|Figure 2: The overall (= spatially integrated) CR spectrum as function of momentum. Solid and dashed lines correspond to protons and electrons, respectively. Thick and thin lines correspond to efficient and inefficient proton acceleration, respectively.|
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The shape of the overall electron spectrum
that of the proton spectrum N(p) at high momenta
due to the synchrotron losses in the downstream region
with magnetic field
G which is assumed
uniform in this region (
to expression (4) the synchrotron losses should become important
for electron momenta greater than
In detail the momentum dependence of the spatially integrated electron spectrum from Eq. (13) is somewhat more complicated at high energies. This can be understood as follows:
The shock constantly produces the electron spectrum
up to the maximum momentum
which is much larger than .
spectrum is not modified by synchrotron losses within the downstream
region of thickness
|Figure 3: Synchrotron emission flux as a function of frequency for the same two cases as in Fig. 2. Solid and dashed lines correspond to efficient and inefficient proton acceleration, respectively. The observed X-ray (in black color: Hamilton et al. 1986; in red color: Allen et al. 1999) and radio emissions (in black color: Reynolds 1996) are shown.|
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The maximum electron momentum
can be estimated
by equating the synchrotron loss time (4) and the acceleration
The parameters and G provide good agreement between the calculated and the measured synchrotron emission in the radio- and X-ray ranges (Fig. 3). The steepening of the electron spectrum at high energies due to synchrotron losses naturally yields a fit to the X-ray data with their soft spectrum. Note that two kinds of X-ray data are presented in Fig. 3. The Hamilton et al. (1986) data represent the total (i.e. thermal plus nonthermal) X-ray emission in a relatively wide frequency range, where the nonthermal synchrotron emission contribution dominates at frequencies Hz. At these highest frequencies the Allen et al. (1999) RXTE nonthermal data are also presented (in red color), lying slightly below the Hamilton et al. data points. Clearly our nonthermal model spectra must lie below the Hamilton et al. spectral points at Hz. Such a smooth spectral behavior is achieved in a 20 G upstream field.
Since the total number of accelerated protons has to be reduced by the factor , the same number of accelerated electrons then corresponds to a renormalized parameter . Then the total energy of accelerated electrons in SN 1006 is erg.
The ratio of accelerated electrons to protons thus comes out lower by a factor of about 6 than the canonical ratio of 0.01, generally deduced from observations of the Galactic CRs. This is an interesting consequence of our model: if SN 1006 is typical for the nuclear Galactic CR source population, then other sources like young Pulsars or Pulsar Nebulae must also significantly contribute to the Galactic CR electron population (e.g. Aharonian et al. 1995; Pohl & Esposito 1998). An additional fraction might come from secondary interstellar electrons/positrons due to the decay of charged pions that are produced in inelastic collisions of CR nuclei with nuclei of the interstellar gas. Since the majority of Galactic Supernovae is not of the type Ia to which SN 1006 belongs, but rather occurs as a consequence of core collapse events, there is the alternative possibility that core collapse Supernovae produce a higher electron to proton ratio. Such an increase might have to do with the different circumstellar magnetic field structure of massive stars, but this is indeed only a possibility for which we have no proof at present.
The radio data are fitted by a power law spectrum , whose index (Allen et al. 2001) is noticeably larger than 0.5, corresponding to the electron spectrum produced by an unmodified shock with compression ratio . In our case the shock is essentially modified by the backreaction of the accelerated protons (see Fig. 1b): its compression ratio . At the same time low energy electrons with momenta ( GeV), which produce synchrotron emission at GHz, are accelerated at the subshock which has the compression ratio . Therefore these electrons have a steeper spectrum that leads to the expected radio spectrum , fitting the experimental data very well (Fig. 3). The fact that the observed value of the radio power law index exceeds the value 0.5 can be considered as an indication that the shock is essentially modified. A relatively high upstream magnetic field strength B0=20 G, compared with typical ISM values B0=5 G, and a corresponding downstream value G are required to have radio emitting electron energies in the steep part of their spectrum GeV on the one hand, and to give a smooth rollover of the synchrotron spectrum at frequencies Hz on the other. According to model calculations by Lucek & Bell (2000), the existing ISM field can indeed be significantly amplified near a strong shock by CR streaming.
Such a strongly nonlinear behavior has three consequences in that (i) it
leads to the high magnetic field strength, required to understand the
measured nonthermal synchrotron spectrum, (ii) it is consistent with our
assumption of Bohm diffusion, and (iii) it suggests a value
for the injection parameter that we use (see also Sect. 2).
|Figure 4: IC (dashed lines), -decay (solid lines) and NB (dash-dotted lines) integral -ray energy fluxes as a function of -ray energy for the same cases as in Fig. 2. High energy -ray data (Tanimori et al. 1998), the EGRET upper limit as given by Mastichiadis & de Jager (1996), and the JANZOS upper limit (Allen et al. 1995) are shown.|
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The calculated integral IC and -decay -ray energy fluxes are presented in Figs. 4 and 5 together with the available experimental data.
In Fig. 4 we present also the integral NB -ray fluxes; they play no important role for SN 1006. Note that the number of accelerated protons, which produce -decay -rays, has been reduced by the factor .
According to the calculation, the hadronic -ray production exceeds the electron contribution by a factor of about 7 at energies TeV, and dominates at TeV (Fig. 4). The calculation is in good agreement with the TeV-measurements reported by the CANGAROO collaboration (Tanimori et al. 1998), and it does not contradict the EGRET upper limit cm- 2 s-1 at GeV (cf. Mastichiadis & de Jager 1996). This is also confirmed by Fig. 5, where we compare our calculations with the revised CANGAROO data (Tanimori et al. 2001) plus an extended set of EGRET upper limits (Naito et al. 1999).
The differential -decay -ray flux
has the expected peak at photon
MeV (see also Naito & Takahara 1994).
This feature may be used for the identification of a hadronic contribution
to the observed -ray flux at the particle energies <10 GeV
mainly producing it. However, two considerations weaken this possibility.
of all, for reasons of instrumental sensitivity, we expect only nearby
SNRs to be detectable in the foreseeable future. Such objects have
diameters approaching 1 degree, and SN 1006 is an example. The diffuse
Galactic -ray background is large at these energies due to the
steep spectrum of the Galactic CRs that produce it, as pointed out by
Drury et al. (1994). Therefore, we expect SNRs to be best
observable at much higher -ray energies
GeV, where the
hard source spectra typically dominate over the background emission.
Secondly, from a strictly empirical point of view, such a nuclear CR
component that produces the -decay bump need not extend far beyond
the GeV region. For the existence, in a SNR, of a nuclear CR component
approaching the energy region of the knee in the Galactic spectrum, the
detection of a 67.5 MeV -ray feature is therefore neither a
necessary nor a sufficient condition.
|Figure 5: Differential -decay ( solid lines) and IC ( dashed lines) -ray energy fluxes as a function of -ray energy for the same cases as in Fig. 2. The recent differential high energy -ray energy flux data (Tanimori et al. 2001) and EGRET upper limits (Naito et al. 1999) are also shown.|
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On the other hand, due to the high magnetic field strength which we deduce for this remnant, the -ray spectra produced by the electronic and hadronic CR components have rather similar shapes in the energy interval 0.1 GeV 10 TeV due to electron synchrotron losses. As a consequence the best observational possibility to discriminate between leptonic and hadronic contributions at high energies GeV is to measure the -ray spectrum at energies higher than 10 TeV, where we expect these two spectra to be essentially different. Therefore the clear detection of a substantial flux at energies TeV would provide direct evidence for its hadronic origin.
On the other hand one may dismiss the relevance of a deviation of the observed radio spectrum from the form , and of a likely thermal contribution to the soft X-ray flux around 1017 Hz, and try to reproduce all the observed emissions by effects of electrons alone (e.g. Pohl 1996; Mastichiadis & de Jager 1996; Yoshida & Yanagita 1997; Aharonian & Atoyan 1999). The number of injected and accelerated protons and all the effects which they can produce in SN 1006 is suggested to be negligibly small in this extreme case. For example, the proton contribution to the TeV -ray emission should be at least an order of magnitude lower compared with the previous case. At such low proton numbers the nonlinear shock modification due to their backreaction is negligible, and shock acceleration takes place in the test particle limit.
In order to make clear how different the required set of relevant physical parameters compared to the case of efficient CR nucleon acceleration is, we performed a calculation which corresponds to inefficient CR acceleration. We use a proton injection rate which yields an upper limit for the proton acceleration efficiency that is consistent with their low contribution to the -ray production. This low CR nucleon production efficiency also implies that the effect of magnetic field amplification due to CR streaming is small. Therefore we have used a typical upstream ISM magnetic field strength B0=4 G. We note that such a low value was even assumed to be required in the compressed downstream region - G - to fit the experimental data (Tanimori et al. 2001). All other ISM and SN parameters are the same as in the previous case of efficient CR acceleration. Since during the early evolutionary phase the global SNR dynamics is rather insensitive to the number of accelerated CRs, the observed SNR size and its expansion rate are equally well reproduced in this case.
In this test particle regime the electron spectrum, produced by a strong
unmodified shock, can be analytically approximated by (Berezhko
|Figure 6: Observed total radio flux of SN 1006 (Reynolds 1996) as a function of frequency with model spectra superimposed. Solid and dashed curves correspond in this figure to the high and low proton injection/acceleration efficiency, respectively.|
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The results for this inefficient proton injection/acceleration case are also presented in Figs. 2-7. One can see from Fig. 2 that, due to the lower magnetic field value, one needs about five times more accelerated electrons compared with the previous case to fit the radio and X-ray data (see Fig. 3).
In the inefficient case the -ray production is dominated by the electron contribution (Fig. 4). Since synchrotron losses are not important, the predicted IC spectrum is essentially harder for -ray energies between about 109 and 1011 eV than in the efficient proton acceleration model. The differential -ray CANGAROO spectrum appears in better agreement with the -decay emission of the efficient model than with the inefficient model IC prediction (Fig. 5).
In Fig. 6 the calculated synchrotron fluxes are compared with the experimental data in the radio range. One can see that the steeper spectrum which corresponds to the efficient CR acceleration case gives a better fit to the radio data than the spectrum corresponding to the case of inefficient proton injection, even though the experimental accuracy does not very clearly distinguish between these to variants. Note that is the average value of the power-law index within the frequency range shown in Fig. 6. In fact, due to the concave shape of the electron spectrum, the index slightly decreases with increasing frequency, with and at the lowest and largest frequency, respectively.
|Figure 7: Radial dependence of the -ray brightness for the -ray energy TeV. Thick and thin curves correspond to the high and low proton injection/acceleration efficiency, respectively.|
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A similar situation is encountered in the X-ray range. An essentially steeper electron spectrum in the case of efficient CR accelerations much better fits the nonthermal X-ray data by lying below the total X-ray flux near the frequency which corresponds to the photon energy (Fig. 3).
At given number of protons and magnetic field strength an electron to proton ratio is required in the inefficient case, in order to reproduce the observed emission.
Nucleonic CRs absorb in this case only
It is important to note that the -ray brightness of the remnant
The -ray brightness calculated for the two different cases is presented in Fig. 7 for the -ray energy TeV. One can see the essentially different radial dependence for the two cases. The -rays of hadronic origin are expected to be concentrated in the postshock region, whereas the IC -ray brightness, calculated for quite a large level of turbulence near the contact discontinuity (see Berezhko & Völk 2000 for details), is almost uniformly distributed across the visible disk of the remnant. As it is seen from Fig. 7, the IC high energy emission of the inefficient case produces a noticeable halo around the shock - for SN1006 its size is about 0.1 degrees - that can in principle be used to discriminate between the -ray emissions of hadronic and leptonic origin.
The reported -ray flux was detected from the same outer part of SN 1006 which shows the radio-emission. As it was already pointed out (Aharonian 1999; Aharonian & Atoyan 1999; Berezhko et al. 1999; Berezhko et al. 2001), this can be considered as an observational argument favoring a strong role for the nuclear CR component.
The nonlinear kinetic model for CR acceleration in SNRs has been applied to SN 1006 in order to explain its observed properties. We have used stellar ejecta parameters , k=7, distance d=1.8 kpc, and ISM number density cm-3 from X-ray and optical imaginary of SN 1006. For these parameters an explosion energy erg is required to fit the observed size and expansion speed which are determined by the ratio .
The number of accelerated electrons required to fit the radio and X-ray emission of SN 1006, and correspondingly the role of accelerated protons in -ray production, depends essentially on the magnetic field value B0.
It was demonstrated that for low magnetic field B0=4 G all the observed emissions can be dominated by the electron contribution. Protons are then assumed to be injected into the acceleration much less efficiently than electrons. For this test particle case the lowest permitted value of the electron to proton ratio is . It exceeds the canonical value 0.01 observed in situ in the neighborhood of the Solar System for the Galactic CRs. The maximum energy of accelerated CRs and their total energy content in this case are only eV and erg respectively. These numbers are too low for such SNRs to be considered as the main sources of the nucleonic Galactic CRs.
If CRs in SN 1006 are produced due to the diffusive shock acceleration process, then even in the case of inefficient proton injection quite a large but plausible downstream magnetic field G is required to fit the data. It is several times larger than assumed in a simple estimate by Tanimori et al. (2001), because the shock produces in this case an electron spectrum which is significantly harder than the spectrum assumed in that estimate.
The existing SNR data are better approximated if a significantly larger upstream magnetic field value B0=20 G and a physically much more plausible, efficient nucleon injection rate are assumed. Such an ion injection rate is estimated from injection theory, and consistent with the observed radio spectral index. The required magnetic field strength, that is significantly higher than the rms value G in the ISM, might be the result of non-linear amplification near the SN shock by the CR acceleration process itself.
The results for this case of efficient proton acceleration are then as follows:
We find that after adjustment of the predictions of the nonlinear spherically-symmetric model by a renormalization of the number of accelerated nuclear CRs to take account of the large area of quasiperpendicular shock regions in a SNR, good consistency with all observational data can be achieved, including the reported TeV -ray flux. The -decay -ray flux produced by the nuclear CR component exceeds the flux of IC -rays generated by the electronic CR component at all energies above about 100 MeV. The theory did not make use of any knowledge to be derived from -ray measurements. Therefore the reported TeV flux from SN 1006 supports the idea that the nuclear CR component is indeed produced in SNRs. The -decay -ray flux comes from two polar caps of the remnant.
The maximum energy of accelerated protons eV and their total energy content erg, reproduced in this case, are consistent with the requirements for the Galactic CR sources. The electron to proton ratio of , on the other hand, is lower than the canonical value 0.01.
Comparing the case of efficient proton acceleration with the inefficient proton acceleration case, we see that the expected -decay -ray flux extends up to almost 100 TeV, whereas the IC -ray flux reaches less than about 10 TeV. Therefore the detection of -ray emission above 10 TeV would imply evidence for a hadronic origin.
We therefore conclude that the analysis of SN 1006 on the basis of overall SNR dynamics and nonlinear diffusive shock acceleration theory results in a picture where the nuclear component is strongly accelerated, consistent with all data for this SNR.
The ratio of accelerated electrons to protons comes out lower by a factor of about 6 than the canonical ratio of 0.01, generally deduced from observations of the Galactic CRs. If SN 1006 is typical for the nuclear Galactic CR source population, then other sources like young Pulsars or Pulsar Nebulae must also significantly contribute to the Galactic CR electron population. The Crab Nebula is a point in case. Since the majority of Galactic Supernovae is not of the type Ia to which SN 1006 belongs, but rather occurs as a consequence of core collapse events, there is the alternative possibility that core collapse Supernovae produce a higher electron to proton ratio.
Phenomenological studies of the -ray emission from SN 1006 on the basis of the observed synchrotron emission have preferred a dominance of IC emission from electrons in the -ray part of the emission spectrum. This scenario can not explain the apparent bipolar morphology inferred from the existing -ray measurements and yields a much less convincing approximation to the radio and X-ray synchrotron spectrum. In addition, as we have shown, the IC emission should reach at best about 10 TeV. Apart from future detailed determinations of the -ray morphology one therefore needs to precisely measure the -ray flux at energies between 100 GeV and 100 TeV, and preferably even from 10 MeV upwards. The detailed spectra and in particular the existence of -rays with very high energies above 10 TeV should allow a confirmation, or a rejection, of CR nucleon production in SN 1006 with an acceleration efficiency that is consistent with the requirements on the Galactic CR energy budget.
This work has been supported in part by the Russian Foundation for Basic Research (grants 00-02-17728, 99-02-16325). EGB and LTK acknowledge the hospitality of the Max-Planck-Institut für Kernphysik, where part of this work was carried out.