Nonthermal emission of supernova remnant SN 1006 revisited: theoretical model and the H.E.S.S. results
The properties of the Galactic supernova remnant (SNR) SN 1006 are theoretically re-analyzed in the light of the recent H.E.S.S. results. Nonlinear kinetic theory is used to determine the momentum spectrum of cosmic rays (CRs) in space and time in the supernova remnant SN 1006. The physical parameters of the model – proton injection rate, electron-to-proton ratio and downstream magnetic field strength – are determined through a fit of the result to the observed spatially-integrated synchrotron emission properties. The only remaining unknown astronomical parameter, the circumstellar gas number density, is determined by a normalization of the amplitude of the gamma-ray flux to the observed amplitude. The bipolar morphology of both nonthermal X-ray and -ray emissions is explained by the preferential injection of suprathermal nuclei and subsequent magnetic field amplification in the quasi-parallel regions of the outer supernova shock. The above parameters provide an improved fit to all existing nonthermal emission data, including the TeV emission spectrum recently detected by H.E.S.S., with the circumstellar hydrogen gas number density close to values derived from observations of thermal X-rays. The hadronic and leptonic -ray emissions are of comparable strength. The overall energy of accelerated CRs at the present epoch is of the order of 5 % of the total hydrodynamic explosion energy, and is predicted to rise with time by a factor of . The relevance of CR escape from the SNR for the spectrum of the -ray emission is demonstrated. The sum of the results suggests that SN 1006 is a CR source with a high efficiency of nuclear CR production, as required for the Galactic CR sources, both in flux as well as in cutoff energy.
Subject headings:acceleration of particles — cosmic rays — gamma-rays: general radiation mechanisms:non-thermal — shock waves — supernovae: individual(SN 1006) —
The Galactic cosmic rays (CRs), with proton energies below a few eV, are generally believed to be accelerated in shell-type supernova remnants (SNRs). To establish that the Galactic SNRs are indeed the main sources of the Galactic CRs one needs at least a handful of SNRs with clearly determined astronomical parameters, like the type of supernova explosion, the SNR age, the distance, and the properties of the circumstellar medium, together with the nonthermal emission characteristics from which the energetic particle spectra can be deduced with a theoretical model. The most suitable objects are type Ia supernovae which have rather old and low-mass progenitors, not associated with star forming regions, and thus supposedly explode into a uniform circumstellar medium – at least into one which is not modified by mass loss in the form of a stellar wind. The type of explosion also restricts the total explosion energy rather tightly and fixes the ejecta mass to a value close to the Chandrasekhar mass.
One such object is the historical remnant SN 1006: the distance was determined using optical measurements with relatively high precision (Winkler et al., 2003) and all other astronomical parameters are quite well known (e.g. Cassam-Chenaï et al., 2008). A special characteristic of SN 1006 is the fact that it lies about 550 pc above the Galactic plane (Bocchino et al., 2011) in a very low-density environment. This unusual property suggests that the ambient interstellar medium (ISM) is also free of density inhomgeneities from cooling instabilities (e.g. Inoue et al., 2012).
It has been shown earlier (Berezhko et al., 2009), in the following referred to as BKV09, that a nonlinear kinetic theory of CR acceleration in SNRs (Berezhko et al., 1996; Berezhko & Völk, 1997) 111 Similar approaches were recently developed and applied to study the properties of other SNRs (Zirakashvili & Aharonian, 2010; Zirakashvili & Ptuskin, 2011). was consistent with all observational data available at the time. This included the H.E.S.S. energy flux density erg cm s of the Very High Energy (VHE: GeV) -ray emission from one of the detected polar caps, integrated over the observed energy interval , where TeV and TeV (Naumann-Godo et al., 2008), as well as the fluxes of the nonthermal emission in the radio and X-ray regions measured with Chandra (Allen et al., 2004) and Suzaku (Bamba et al., 2008). Note that the above H.E.S.S. energy flux density was attributed in BKV09 to the whole remnant, whereas in fact it represents about half of the total flux. This is the reason why the values of the ambient ISM density and of the SN explosion energy, estimated below, are correspondingly larger in comparison with BKV09.
Compared with BKV09, in the following the physical parameters of SN 1006 – mainly the ambient gas density and the energy production efficiency – are re-estimated, based on the more recent measurements of the -ray spectrum and of the emission morphology (Acero et al., 2010). In addition, the time profile of the acceleration efficieny is presented and the radial dependence of the -ray brightness components is given. Finally the radial profiles of the gas density and of the CR spectral energy density per unit logarithmic bandwidth are compared in a discussion of the energetic particle escape from the remnant.
2. Summary of assumptions and earlier results
As a type Ia supernova SN 1006 presumably ejects roughly a Chandrasekhar mass . Since the gas density is observed to vary only mildly across the SNR (Acero et al., 2007), it appears reasonable to assume the circumstellar gas density and magnetic field to be roughly uniform. The ISM mass density , is characterized by the hydrogen number density . As the most reliable distance estimate the value kpc is adopted (Winkler et al., 2003).
As in BKV09 a nonlinear kinetic theory of CR acceleration in SNRs (Berezhko et al., 1996; Berezhko & Völk, 1997) is applied. With one exception this theory includes equations for the most important physical processes which influence CR acceleration and SNR dynamics: shock modification by CR backreaction, MHD wave damping and thus gas heating within the shock transition, and synchrotron losses of CR electrons under the assumption of a strongly amplified magnetic field within the remnant .
The values of three scalar parameters in the governing equations (proton injection rate , electron to proton ratio and the upstream magnetic field strength ) can be determined from a fit of the solutions that contain these parameters to the observed spatially integrated synchrotron emission data at the present epoch. Both, and , are assumed to be independent of time. The parameter values for SN 1006, evaluated in this way, agree very well with the Chandra measurements of the X-ray synchrotron filaments and were obtained in BKV09 by the analysis of the radio data compiled by Allen et al. (2004) (see also Allen et al., 2001, 2008) and of the most accurate X-ray data of Chandra (Allen et al., 2004) and Suzaku (Bamba et al., 2008).
In order to explain the detailed -ray spectrum, values of the hydrodynamic supernova explosion energy erg and erg are taken to fit the observed shock size pc and shock speed km s (Moffett et al., 1993; Katsuda et al., 2009) at the current epoch yr for the ISM hydrogen number densities and , respectively. These densities are consistent with the observed level of the VHE emission, as shown below. The best-fit value of the upstream magnetic field strength G is quite insensitive to . The resulting current total shock compression ratios for and become now and 4.9, respectively, whereas the subshock compression ratios remain both close to . Note, that the explosion energy value erg is somewhat lower compared with erg that one would expect extrapolating the value erg obtained earlier (Ksenofontov et al., 2005) for according to the expected dependence . However this difference is well within the range determined by the uncertenties of the observed values and taking into account the relation .
Since the properties of the accelerated CR nuclear and electron spectra and their dependence on the relevant physical parameters, as well as the dynamical properties of the system were described in detail in BKV09, they will not be discussed here once more.
3.1. Acceleration efficiency
In Fig. 1 the time dependence of the fractional energy contained in accelerated CRs during the SNR evolution is presented. Note that the value of is reduced by a factor of (see below) compared with the value calculated within the spherically symmetric model. According to Fig. 1, and 0.065 for and , respectively. This is lower than the value which is required on average for each SNR for their population to be the main source of CRs in the Galaxy. The reason is that SN 1006 is a quite young object in an evolutionary sense: according to Fig. 1 is expected to approach this canonical value in the subsequent evolution. On the other hand, the requirement for an average efficiency of is based on an assumed Galactic average value erg which is about one half of our value erg. Therefore the calculated efficiency fulfills the average requirement even for the present epoch.
3.2. Overall nonthermal spectra
Fig. 2 illustrates the consistency of the synchrotron and -ray spectra, calculated with the best set of parameters ( as well as for , and as well as for ), with the observed spatially integrated spectra. The H.E.S.S. data (Acero et al., 2010) for the NE and SW limbs, respectively, have been multiplied by a factor of 2, in order facilitate comparison with the full deduced -ray flux. As can be seen from Fig. 2, the calculated synchrotron spectrum fits the observations both in the radio and the X-ray ranges (Allen et al., 2004; Bamba et al., 2008) very well. Note that the Chandra flux (Allen et al., 2004) is from a small region of the bright NE limb with minimal contributions from thermal X-rays; this X-ray flux was normalized to the overall Suzaku flux (Bamba et al., 2008) at energies keV (for details, see BKV09).
The only important parameter which can not be determined from the analysis of the synchrotron emission data is the external gas number density : Fig. 2 shows that the spectrum of synchrotron emission is almost non-sensitive to the ambient gas density. Consequently, numerical solutions have been calculated for the pair of values and which appear to bracket the density range consistent with the H.E.S.S. -ray measurements. It should be noted that this estimate for is by a factor of larger than the estimate of BKV09. The reason is that in the analysis of BKV09 the total calculated -ray energy flux was compared with the energy flux from the NE rim. However, the latter flux corresponded only to about 50% of the total observed energy flux, as it had been reported for the H.E.S.S. instrument by Naumann-Godo et al. (2008). A detailed comparison of the -ray spectrum with the calculated spectrum will be made in the context of Fig. 4.
3.3. Gamma-ray morphology
The -ray morphology, as found in the H.E.S.S. measurements (Acero et al., 2010), is consistent with the prediction of a polar cap geometry on account of the strongly preferred injection of nuclear particles in the quasi-parallel regions of the shock (Völk et al., 2003). Such a geometry has also been found experimentally from an analysis of the synchrotron morphology in hard X-rays by Rothenflug et al. (2004) and Cassam-Chenaï et al. (2008). This means that the -ray emission calculated in the spherically symmetric model must be renormalized (reduced) by a factor , as in Ksenofontov et al. (2005) and BKV09. This renormalization factor is applied here. The total -ray flux from the source is the sum of the fluxes from these two regions.
This morphology is also a key argument for the existence of an energetically dominant nuclear CR component in SN 1006, because only such a component can amplify the magnetic field to the observed degree. The accelerated electrons are unable to amplify the magnetic field to the required level (BKV09).
The question, whether these bright NE and SW regions of SN 1006 represent quasi-parallel portions of the SN shock or not, is still debated (Petruk et al., 2009; Schneiter et al., 2010; Morlino et al., 2010). On the other hand Bocchino et al. (2011) have recently argued from the radio morphology that the radio limbs – morphologically similar to the X-ray and -ray limbs – are polar caps and that electrons are accelerated with quasi-parallel injection efficieny. Like the aforementioned theoretical arguments and X-ray analyses, this argues against a scenario, where the magnetic field is perpendicular to the shock normal (Fulbright & Reynolds, 1990). Bocchino et al. (2011) also concluded that the remaining asymetries (converging limbs and different surface brightness), clearly visible also in the H.E.S.S. -ray data (Acero et al., 2010), could be explained by a gradient of the ambient ISM magnetic field strength.
Additional arguments which confirm that the NE and SW regions correspond to quasi-parallel shock regions are given by Reynoso et al. (2011). Their analysis of the polarization of radio emission led them to the conclusion that the magnetic field in SN 1006 is radial at the NE and SW lobes but tangential in the SE of the radio shell. In addition they established the maximum fractional polarization in the SE which implies that the magnetic field is highly ordered there. On the other hand the low fractional polarization in the lobes suggests a considerable randomization of the magnetic field. The fact that a turbulent magnetic field coexists with the brightest synchrotron emission in the SW strongly supports efficient acceleration of the CR nuclear component, followed by magnetic field amplification which provides its own directional randomization.
3.4. Relative contributions to the gamma-ray flux
Fig. 3 shows the -decay, the inverse Compton (IC), and the total (-decay plus IC) -ray energy spectra of the remnant, calculated for and , respectively. It can be seen that the -ray spectrum produced by the nuclear CRs is rather close to the IC emission spectrum produced by CR electrons alone, especially in the case of . Even for the case which is close to the minimum density for the remnant, as argued by Acero et al. (2007), the total predicted -ray flux is still quite similar in spectral shape to the total flux for (and essentially also for ), as shown in Fig.6 of BKV09. Unless the actual gas density would be small compared to , it would therefore be very difficult to observationally distinguish a hypothetical dominant IC scenario from the mixed -ray spectrum discussed here. Therefore, in the VHE range – except perhaps in the last decade of energy, near the cutoff, see subsection 3.6 – the observed -ray spectrum alone is not able to discriminate between the hadronic -decay and the leptonic IC -ray components. However, it was already shown by Ksenofontov et al. (2005) that such a low total VHE emission flux, with a highly depressed IC -ray flux due to the synchrotron losses of high energy electrons, is only possible if the nuclear CR component is efficiently produced with accompanying strong magnetic field amplification.
3.5. Radial profile of gamma-ray brightness
A possibility for an experimental discrimination between -decay and IC -rays is given by the measurement of the radial profile of the -ray brightness. It was shown early on (Berezhko et al., 2002) that the radial profile of the TeV -decay -ray emission has a peak near the remnant rim with a width of about 20% of the remnant radius. On the other hand, the peak of the IC -ray emission was predicted to be much narrower, actually narrower by an order of magnitude. This latter characteristic is physically identical to the filamentary structure detected subsequently with Chandra in keV X-rays (Bamba et al., 2003; Long et al., 2003) (and for Cas A by Vink & Laming (2003)), because the keV synchrotron emission and the TeV -ray IC emission are produced by the same electrons, with energies approaching 100 TeV.
In Fig. 4 radial profiles of the -ray brightness are shown as functions of radial projection distance corresponding to energies TeV, calculated for . It is seen that the profile of the IC component is indeed dramatically thinner than the profile of the -decay component. The profile of the total -ray emission, which is the sum of these two components, has a width at half maximum of about that is smaller than that observed by H.E.S.S. The total radial emission profile, smoothed to the H.E.S.S. point spread function, agrees with the H.E.S.S. observations for but exceeds the H.E.S.S. values considerably at low radial distances . The reason for this discrepancy is simple. As it has been said in subsection 3.3, only those nuclear CRs that occupy two cones with opening angle 20, with symmetry axes going from the remnant center toward the NE and SW directions, respectively, should be taken into account to model the hadronic emission of the actual remnant due to the strong angular dependence of the nuclear injection rate. The radial profiles of the -ray emission lobes, corresponding to the NE-SW directions produced by these CRs, are well consistent with the H.E.S.S. data (Fig. 4). In this case the line of sight at does not intersect the outer spherical region of the remnant which contains most of the accelerated CRs that provide the brightness depression at .
A discrimination between these -decay and IC -ray components would require a -ray instrument with an angular resolution that is an order of magnitude higher than that of H.E.S.S., or of any other existing -ray instrument. It is, however to be noted that the measured radial H.E.S.S. profile, which gives a width of the measured peaks of 20% of the remnant radius (Acero et al., 2010), is clear evidence that the nuclear CR component is accelerated efficiently. Indeed, if the energy content in CR nuclei were small due to inefficient proton acceleration one would have to conclude that the magnetic field in the remnant is not amplified to any substantial degree. In such an inefficient scenario electrons with energies of tens of TeV, which produce the TeV -ray IC emission, would be extremely smoothly distributed across the remnant – almost uniformly in the spherically symmetric case (Berezhko et al., 2002, 2003). Therefore the expected radial width of the -ray emission region would be considerably larger than that expected from efficiently accelerated protons.
3.6. External density
In order to exhibit the most detailed comparison of the calculated -ray spectra with the observations, in Fig. 5 the theoretical spectra for two ambient densities are presented together with the observed H.E.S.S. spectra for each of the polar caps on an expanded energy scale. For ease of comparison, the theoretical spectra are divided by a factor of two, which would correspond to the emission from a single polar cap region for an ideal dipolar morphology.
The H.E.S.S. spectra from the two regions are compatible with decreasing power-law distributions , with and somewhat different fluxes , in the detected energy range (Acero et al., 2010). As Fig. 2, Fig. 3 and Fig. 5 show, the H.E.S.S. data are also compatible with the more complex total theoretical -ray spectral energy flux density over this range that has a maximum at . Above 10 GeV the theoretical spectral energy flux can be approximated analytically by a power law with an exponential cutoff:
with cutoff energies TeV and 37 TeV for and , respectively. These two analytical curves represent the best fit to the H.E.S.S. data. At the same time they coincide within 4%, respectively, with the theoretical curves presented in Fig. 5.
The quality of fit to the H.E.S.S. data is characterized by the values and 2.2 for the NE and SW part of the remnant respectively. It is comparable with the quality of fit by the pure power law distributions , with and different fluxes (Acero et al., 2010), which have, according to our calculation, a and 2.5 for the two regions, respectively.
In order to discriminate between those different spectral forms the observational energy range should be extended both below the present H.E.S.S. threshold energy and above the present maximum energy detected by H.E.S.S.
According to Fig. 5 the H.E.S.S. data are consistent with an ISM number density in the interval . There is also some indication that the actual number density is closer to the lower end of this interval, say . However this is a weak conclusion, given the approximations made in the theoretical model in terms of the reduction factor (see above). It should be noted nevertheless that the lower end of this interval is also preferred from the point of view of supernova explosion theory: the corresponding hydrodynamic explosion energy erg is close to the upper end of the typical range of type Ia SN explosion energies that vary by a factor of about two (Gamezo et al., 2005; Blinnikov et al., 2006). In addition, from their X-ray measurements also Acero et al. (2007) concluded that a value is probably representative for the ambient density around SN 1006.
A solid observational conclusion is that the total emission from the NE limb exceeds that from the SW limb (Acero et al., 2010). Given that the hadronic -ray flux is proportional to , the result in Fig.5 suggests that this excess is due to a gas density difference cm between the two -ray emission regions across the SNR, i.e. a relative difference of . Given that the density difference between the thermally emitting NW and SE regions is of the order of cm (Acero et al., 2007) such a minor density difference is indeed quite plausible. However, it does not explain the deviations from full dipolar symmetry in terms of a noticeable convergence of the limbs and thus should simply complement the gradient in the magnetic field strength claimed by Bocchino et al. (2011) from radio observations.
It should be noted that the cutoff of the theoretical -ray spectrum (Fig. 5) begins already at eV. This value is unexpectedly low since the overall proton power-law spectrum extends up to a proton energy eV (BKV09). The reason is the progressive decrease of overlap between the radial gas density profile and the radial profile of the proton distribution function at energies eV. As Fig. 6 shows, the gas number density has a peak at the subshock position and decreases towards the remnant center by a factor of 2 at a distance from the subshock position. (The thickness of the thermal gas precursor is about ; therefore it is hardly visible in Fig. 6.) The corresponding radial dependence of the proton distribution in momentum , multiplied by , i.e. of the spectral energy density , with energies eV, has a similar behaviour; it shows a relatively small particle number in the upstream region . At higher energies eV the thickness of the radial distribution increases with energy, so that for eV its overlap with the gas density profile is by a factor of two lower than for eV. This leads to a lower cutoff of the -ray spectrum compared with what one would expect in the simple case of energy-independent overlap of CR protons with the gas distribution.
This spread of CRs into the upstream region , which for the highest energies is faster than the shock expansion , represents the diffusive escape of CRs from the expanding SNR and was predicted by Berezhko & Krymsky (1988). Such a CR escape is expected in particular in the Sedov phase when the maximum energy of CRs, accelerated in the given evolutionary phase, decreases with time, because the value of is shock-size limited (see Berezhko, 1996, for details) rather than time limited. At any given phase (where is the sweep-up time) efficient CR acceleration at the SN shock takes place only for energies , whereas for CRs with energies , produced during earlier times, the acceleration process becomes inefficient and these CRs expand into the upstream region. Within the SNR model discussed in this paper CR escape is relatively slow because the model assumes Bohm diffusion in the amplified magnetic field everywhere upstream for any CR energy. In reality this is expected to be true only for CRs with energies in the vicinity of the shock, where these CRs produce significant magnetic field amplification. At large distances upstream of the shock, , or/and for higher CR energies , CR diffusion is much faster than Bohm diffusion. Therefore in actual SNRs CR escape is expected to be faster and more intense than the present model predicts (see Ptuskin & Zirakashvili, 2003; Drury, 2011, for more detailed considerations). Since SN 1006 is only at the very beginning of the Sedov phase this underestimate of the magnitude of escape is however not very critical.
Based on a nonlinear kinetic model for CR acceleration in SNRs the physical properties of the remnant SN 1006 were examined in a detail which corresponds to that of the available observations. Since the relevant astronomical parameters as well as the synchrotron spectrum of SN 1006 are measured with high accuracy, the values of the relevant physical parameters of the model can be estimated with similar accuracy for this SNR: proton injection rate , electron-to-proton ratio and downstream magnetic field strength G.
As a result the flux of TeV emission detected by H.E.S.S. is predicted to agree with an ISM hydrogen number density , consistent with the expectation from X-ray measurements. The corresponding hydrodynamic SN explosion energy erg is somewhat above but fairly close to the upper end erg of the typical range of type Ia SN explosion energies. The efficiency of CR production is predicted to be between 5 and 6 percent up to the present epoch and expected to approach of the order of 10 precent during the further evolution. Normalized to a standard value erg, already the present efficiency is about 10 percent.
The magnetic field amplification properties of this SNR can be understood as the result of azimuthal variations of the nuclear ion injection rate over the projected SNR circumference in terms of injection at quasi-parallel shocks only and the corresponding acceleration. This predicts the dipolar -ray emission morphology and is compatible with a polar cap-type X-ray synchrotron morphology. The -ray morphology is the key argument for the existence of an energetically dominant nuclear CR component in SN 1006. The magnetic field amplification allows the proton spectrum to extend to energies eV. The difference between the -ray emission from the NE polar region and that from the SW polar region can be attributed to a small density difference between those regions.
It is shown that the high-energy tail of the -ray spectrum is affected by CR escape from the SNR interior, consistent with the comparatively low -ray cutoff near eV.
It is also concluded that the radial profile of the TeV -ray emission measured by H.E.S.S. is evidence that the nuclear CR component is indeed efficiently produced. In the opposite case of inefficient nuclear CR production the magnetic field would not be expected to be amplified and the radial profile of the IC-dominated -ray emission would be expected to be significantly smoother than observed.
The sum of all these results suggests the conclusion that SN 1006 is a CR source with a high efficiency of nuclear CR production, required for the Galactic CR sources, both in flux as well as in cutoff energy.
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