Inverse Compton models of TeV emission
The featureless X-ray emission observed by ASCA from the shell of SN 1006 (Koyama et al., 1995) is naturally interpreted as synchrotron emission of
electrons accelerated to energies ~ 100 TeV at the shock front. Motivated by this fact, several theorists (Mastichiadis, 1996; Mastichiadis and De Jager, 1996; Pohl, 1996; Yoshida and Yanagita, 1997) predicted strong TeV emission produced by the same electrons upscattering off the 2.7 K CMBR. Therefore, the detection of TeV emission from SN 1006, claimed soon after these predictions (Tanimori et al., 1998b) strengthened the belief in the electronic origin of TeV radiation. The n0-decay contribution to the observed flux is widely considered to be less important. However, it has been argued that the simple one-zone synchrotron and IC model applied to SN 1006 might have serious theoretical problems. The interpretation of TeV emission in terms of IC mechanism requires several bold conditions, the validity of which requires thorough theoretical studies and new X-ray and Y-ray observations concerning both the spectral and morphological properties of the source:
Constraint on the magnetic field. Since the 2.7 K CMBR serves as the main target photon field for the IC scattering, the flux of Y-rays is determined, for the given X-ray flux
only by the strength of the magnetic field. Therefore, in the framework of the IC origin of TeV Y-rays the ratio of observed fluxes of X-rays and TeV Y-rays provides an estimate of the the strength of the ambient B-field. In the simplified one-zone model which assumes that the synchrotron X-rays and IC Y-rays are produced by same electrons confined in a spatially homogeneous region, Fig 5.1 provides quite accurate estimates for IC fluxes . However, the study of spectral features of the Y-ray emission requires a careful derivation of the spectrum of > 10 TeV electrons which should be controlled by the spectrum of synchrotron X-rays. In the synchrotron-inverse Compton models the spectral fit to the X-ray flux is crucial because the fluxes are produced by electrons in the region of the exponential cutoff E0 between 10 and 100 TeV. For the "power-law with exponential-cutoff" type energy distribution of electrons, the resulting synchrotron radiation depends on the spectral index of electrons re and the parameter
(see Eq.5.10). Strictly speaking, this result is valid for the ^-functional approximation, provided that the deformation of the injection spectrum of electrons due to the energy losses can be ignored
Since the inverse Compton origin of TeV radiation from SN 1006 requires a magnetic field less than 10 ^G, and the age of the accelerator is less than 103 yr, such an approximation is quite acceptable. Moreover, numerical calculations show that this parameter remains a (quasi) invariant even in the general case, although its absolute value could differ somewhat from predictions of the ^-function approximation.
In Fig.5.4 the spectra of synchrotron radiation are shown calculated for the same product
but for 3 different combinations of E0 and B:
I
The power-law index r of electrons is derived from the radio synchrotron spectrum with spectral index
In all calculations the same normalisation to the radio fluxes is used. Therefore different magnetic fields require different total energy in relativistic electrons. At the same time, because the target photon field (2.7 K CMBR) for the IC scattering is fixed, there is a strong dependence of the IC Y-ray fluxes on the magnetic field. In particular, in the case of small magnetic fields,
when the radiative cooling time of 10 to 100 TeV electrons, responsible for the observed X-rays and TeV Y-rays, exceeds the age of the source, we have 
Note that for B < 3^G the radiative losses are dominated by IC on the 2.7 K CMBR.
Fig. 5.4 shows that the interpretation of the observed TeV radiation by the IC mechanism in the framework of a simplified spatially homogeneous (one-zone) model is possible only for an ambient magnetic field in a very narrow range around
Magnetic fields
result in a strong reduction of the IC flux, while the assumption of low magnetic fields,
leads to overproduction of Y-rays.
Maximum electron energy. For a magnetic field in the NE rim 
and the limited age of 103 yr of SN 1006 a question arises as to which mechanism could be efficient enough to accelerate electrons to energies
(in fact, to much higher energies,
taking into account that in a magnetic field as low as 5 ^G only electrons with energy more than 200 TeV could produce synchrotron X-rays up to 8 keV as detected by ASCA). In the model of diffusive shock acceleration, the absolute maximum energy that an electron can achieve is determined by Eq.(5.12), assuming that the cooling time of electrons given by Eq.(5.8) exceeds the age of the source T0:
For
and assuming that the Sedov phase in SN 1006 started several hundred years after the explosion, from Eqs. (5.16) and (5.15) one finds
Thus, even assuming that the acceleration in SN 1006 takes place at maximum possible rate (n = 1), the lower limit on the shock speed exceeds the estimates of vs based on the Sedov solution (Willingale et al., 1996; Winkler and Long, 1997).
Fig. 5.4 The synchrotron and IC fluxes calculated for the homogeneous source without escape of accelerated particles ("one-zone" model). A power-law injection spectrum of electrons with re = 2.15 is assumed in order to fit the radio data. The maximum energy Eo is determined from the condition given by Eq.(5.15) for 3 different values of the magnetic field:
(dashed line),
(solid line), and
(dot-dashed line).
Diffusion effects. The required small magnetic field,
allows the highest energy electrons produced in the shell to effectively escape from the acceleration region. This will give rise to enhanced IC emission outside of the rim, namely in the interior regions of the remnant where the magnetic field could be as low as the typical interstellar B-field. The escape of electrons from the rim is unavoidable because of the diffusive and convective propagation of particles with a characteristic timescale 
The convective escape time is
where u2 is the fluid speed downstream of the shock (in its rest frame), thus at present
for the compression
The width,
of the NE rim in X-rays as measured by ROSAT does not exceed 20 per cent of the angular radius of the remnant of about 17 arcminutes (Willingale et al., 1996) which corresponds to rs = 4.9 dkpc pc. In calculations below a constant
ratio is assumed throughout the evolution of the remnant. The diffusive escape time is
where D(E) is the diffusion coefficient in the shocked region (the rim). In the theory of shock acceleration the diffusion coefficient is generally taken in the form given by Eq.(5.11). The maximum confinement (and therefore the maximum energy) of particles is achieved in the Bohm regime corresponding to n = 1.
The kinetic equation and the solution to this equation for the two-zone model,which treats the overall (i.e. integrated over the volumes) fluxes from the rim (zone 1) and outside (zone 2), allows to study the effects associated with the electron escape. In particular, in the case of small magnetic field the electron escape results in comparable contributions of Y-rays from the rim and outside, as demonstrated in Fig. 5.5. The same could be true also for the synchrotron radiation, unless the magnetic field in the rim does not exceed the average field of the nebula. In fact, the ASCA observations show a narrow X-ray rim with sharp edges. Thus, we may conclude that the magnetic field in the rim is significantly enhanced, which can be naturally explained by strong shock compression. Also, from Fig. 5.5 we may draw the conclusion that the magnetic field in the rim should be within the limits
and n close to 1, i.e. the diffusion in the rim should take place essentially in the Bohm limit. In order to provide maximum electron energy, E0, of order of 30 TeV, the shock speed should exceed 3000 km/s, which implies a large distance to the source,
For this set of parameters, approximately half of the total TeV emission is contributed from the inner parts of the remnant,
The narrow range of the required magnetic field in the rim allows a rather accurate estimate of the total energy of relativistic electrons in SN 1006. For example, for the distance d = 1. 8 kpc, the total electron energy in the rim is 
whereas the energy in the electrons which escaped the rim is
The total energy of the magnetic field in the remnant with
for a distance 1.8 kpc is about 1046 erg. A similar amount of magnetic field energy is expected also in the rim since the amplification of the B-field there is compensated by a smaller volume of the rim. Thus, the inverse Compton origin of TeV Y-radiation requires about 1 percent of the total explosion energy of SN 1006 in relativistic electrons, and implies that the conditions in SN 1006 are far from equipartition between the relativistic electrons and the magnetic field.
Fig. 5.5 Synchrotron (top panel) and IC Y-ray (bottom panel) fluxes calculated in the framework of the 2-zone model for two values of the magnetic field in the rim:
(solid lines) and
(dashed lines). The magnetic field outside of the rim is taken as
The heavy lines show the fluxes from the NE rim, and thin lines correspond to the total fluxes. The distance to the source is assumed to be 1.8 kpc. The maximum electron energy Eo is calculated from Eq. (5.16) for the gyrofactor n =1 for
These two conclusions are intrinsic for the leptonic models of TeV emission of SN 1006. The estimates of energies of relativistic electrons and magnetic fields are very robust and almost independent of the model parameters.
Hadronic origin of TeV emission?
Currently, the nucleonic origin of the observed TeV emission is treated by the community as an inadequate alternative to the IC mechanism, the main argument being the low ambient density of the gas in SN 1006,
(Willingale et al., 1996) as well as the presumed large distance to the source of about 2 kpc (Winkler and Long, 1997). However these arguments are not sufficiently robust to dismiss such an important possibility with far reaching conclusions concerning the origin of the nucleonic component of galactic cosmic rays.
The very fact of the existence of
electrons, as follows from the X-ray data, is evidence for a strong shock in SN 1006, and implies a large compression factor,
or even more, up to 10, as follows from
non-linear studies of shock acceleration in SNRs (e.g. Baring et al., 1999). Therefore it seems not unrealistic to assume a significantly higher density of the gas in the rim region, e.g.
The current estimates of the distance to the source also contain significant uncertainties, and do not exclude a distance of about 1 kpc or even less. To explain the reported flux of TeV emission, for rp = 2 the scaling factor A — 0.83 is needed (Fig. 5.6). This requires for the total energy in accelerated protons
This is only ~ 10 per cent of the total kinetic energy of explosion estimated for SN 1006 as 5 x 1050 erg. The numerical calculations (Berezhko et al., 2002) within a specific nonlinear kinetic model of particle acceleration in SNRs (Berezhko et al., 2002) confirm that indeed the existing TeV data can be explained by accelerated protons.
The estimate of the scaling factor A derived from a comparison of the calculated and observed TeV fluxes depends significantly on the spectrum of the protons. In Fig. 5.6 the n0-decay Y-ray fluxes calculated for 3 spectra of protons with spectral indices, rp =1.8, 2, and 2.1, are shown. The fluxes are normalised to the reported flux at 3 TeV. The corresponding values of the scaling factor are A=0.44, 0.83, and 1.35, respectively. Even in the case of relatively soft spectrum of protons with rp = 2.1, the required scaling factor would be still acceptable if more than 20 per cent of the energy of the supernova explosion could be transformed into accelerated protons. Steeper proton spectra with slopes
do not match the energy budget of the source. Such spectra are excluded also by the EGRET flux upper limit,
The Y-ray spectra in Fig. 5.6 are calculated assuming a maximum proton energy of E0 = 200 TeV. Although the exact value of E0 does not significantly change the requirements to the scaling factor A, it has a noticeable impact on the spectral form of Y-rays above 10 TeV. Therefore only future precise spectroscopic measurements in this energy region can provide an important information about E0. At the same time the existing sub-10 TeV data already tell us that within the framework of nucleonic model of TeV radiation of SN 1006 the cutoff energy E0 cannot be significantly less than 100 TeV. Such large values of E0 could be achieved only under the assumption of a large magnetic field in the shock region, B > 100 ^G (see Eq. 5.16). This assumption does not leave any chance for the IC mechanism to give a noticeable contribution into the observed TeV emission.
Fig. 5.6 The fluxes of
Y-rays calculated for a proton spectrum with Eo =200 TeV, and 3 different power-law indices: rp =1.8, 2 and 2.1. The fluxes are normalised to the reported flux at 3 TeV. The corresponding values for the scaling parameter are A = 0.44, 0.83, and 1.35, respectively.
At the same time, the X-ray emission can be explained by synchrotron radiation for arbitrary magnetic field strengths . In particular, for
the maximum electron energy is determined by the balance between the acceleration rate and the synchrotron energy loss rate, and therefore the position of the corresponding synchrotron cutoff does not depend on the magnetic field and the age of accelerator (see Eq.5.14). The X-ray spectrum of SN 1006 is satisfactorily fitted by Eq.(5.9) with
This gives an interesting estimate of the acceleration rate. Indeed, for the characteristic shock speed
the parameter
the acceleration rate is an order of magnitude slower than the acceleration in the Bohm regime. Then from Eq.(5.16) it follows that in order to accelerate protons to E0 ~ 200 TeV, the magnetic field should be as large as 1 mG. The magnetic field in SNRs can be amplified to such levels through the generation of Alfven waves by accelerated particles themselves (Lucek and Bell, 2000).
Distinct features of electronic and hadronic models
Within the one-zone model for IC Y-radiation from SN 1006, which assumes that all accelerated electrons are confined in the NE rim, the observed X-ray to TeV Y-ray flux ratio requires a very low magnetic field of about 
A more realistic model of Y-ray production that includes the effect of convective and diffusive escape of electrons allows a somewhat larger magnetic field, up to
with preferable values being in a rather narrow range between
Given the short time of particle acceleration
available,
such a small magnetic field would imply a very high shock speed in SN 1006,
in order to provide the acceleration of electrons to energies E0 ~ 30 TeV needed to explain the X-ray and TeV observations.
This model allows definite predictions which could be tested by future observations. A significant, if not dominant, fraction of the IC TeV emission should be produced outside the NE rim. Namely, a Y-ray flux comparable with the flux from NE rim should be expected from the extended inner region of the remnant adjacent to the rim. The size of this region depends on the character of propagation of electrons in the rim. Indeed, even in the case of diffusion of electrons in the Bohm limit in the remnant with a magnetic field of
the characteristic distance of penetration of electrons towards the central region of the remnant can be estimated as
Since the IC production of TeV Y-rays (on the 2.7 K CMBR) with energies less than several TeV takes place in the Thompson regime, and therefore their characteristic energy scales as
the size of the emission region of TeV gamma rays should be as large as the width of the rim, and increases linearly with the energy of the Y-rays. Moreover, if the propagation of electrons in the remnant is much faster than in the Bohm limit
the electrons could fill up practically the entire remnant. In that case the energy dependence of the size of Y-ray emission will be significantly weakened.
Another distinctive feature of the IC origin of TeV radiation is the spectral shape of radiation: (i) very hard, with the photon index r ~ 1.5 below 1 TeV, (ii) flat with r ~ 2 between 1 and 10 TeV, and (iii) very steep above 10 TeV.
The alternative to the IC interpretation is the nucleonic origin of the observed TeV radiation. This interpretation becomes energetically comfortable, which implies no more than 1050 erg in accelerated protons and nuclei, if we assume (i) a small distance to the source of about 1 kpc, and (ii) a significant enhancement of the gas density in the rim due to strong shock compression. This predicts a compact size of the TeV emission comparable with that of the X-ray rim. In the region below 1 TeV the spectrum of Y-rays of nucleonic origin, with a power-law index of
is expected to be steeper than the spectrum of IC Y-rays. However a flatter spectrum of
_ Y-rays cannot be excluded if the protons have an acceleration spectrum as hard as E-15 (see Malkov, 1997).
The shape of the spectrum of n0-decay Y-rays above 1 TeV depends on the characteristic maximum energy of accelerated protons. Since the flux of TeV Y-rays is no longer connected with the X-ray fluxes, the value of the magnetic field in the acceleration region (NE rim) could be as high as 1 mG. Correspondingly, the energies
could be achieved. On the other hand if E0 is significantly less than 100 TeV, one may expect a turnover in the spectrum of Y-rays above several TeV.
Thus, it is possible that both in the low (sub-TeV) and in the high (multi-TeV) energy regions the IC and n0-decay Y-rays have similar spectra. Therefore the spatial rather than spectroscopic measurements of Y-radiation above 100 GeV with future IACT arrays will provide decisive information about the origin of very high energy radiation of SN 1006. The IC models predict significant Y-ray fluxes not only from the rim, but also from the inner parts of the remnant. On the other hand, the n0-decay Y-rays trace the density profile of the gas in the production region, therefore one should expect a rather compact Y-ray source essentially coinciding with the rim.
Because of the small range of magnetic field strength allowed by the IC model of TeV radiation of SN 1006, the energy required for relativistic electrons in this model is predicted with good accuracy:
This is at least two orders of magnitude larger than the energy contained in the magnetic field, even if we ignore the energy contained in accelerated protons. Thus, the interpretation TeV radiation in terms of IC scattering would imply that the conditions in SN 1006 are far from the equipartition regime. Also the acceleration of electrons should proceed in the regime close to the Bohm diffusion limit with n = 1.
The nucleonic model of TeV emission requires a total energy in accelerated protons from 2 x 1050 erg down to 1049 erg, depending on the enhancement factor for the gas density in the rim, the spectral index of the accelerated protons, and the distance to the source. In particular, for a moderate assumption for the spectral index rp = 2, and the gas compression ratio p = 4 (i.e. n = 1.6 cm-3), the energy in protons is estimated to be
In order to accelerate protons to energies E0 ~ 100 TeV the ambient magnetic field should exceed 100 ^G.
Concluding remarks
The synchrotron X-ray emission from SN 1006 is an indication for acceleration of electrons in these objects to energies significantly exceeding 10 TeV. Although the inverse Compton scattering of same electrons unavoidably leads to production of VHE Y-rays, the flux of TeV Y-ray emission from SN 1006 reported by the CANGAROO collaboration exceeds, at least an order of magnitude, the Y-ray flux of inverse Compton origin expected from this supernova remnant, unless one assumes unreasonably small magnetic field in the production region of synchrotron X-rays. The required magnetic of about 5 yU,G seems to be quite unrealistic not only because we generally expect much stronger field in the compressed shocked regions of the shell, but also because it implies acceleration of electrons in the extreme Bohm diffusion regime by the shock with a speed as large as 3500 km/s. Also, the IC model requires a strong departure from the equipartition condition in the region of particle acceleration,
The Y-ray fluxes predicted by hadronic models are also noticeably below the reported TeV flux, unless one assumes that the acceleration of protons proceeds with efficiency significantly exceeding the nominal value of 10 per cent, and that the gas density in the shell is significantly enhanced due to strong shock compression.
Thus, it is clear that both the leptonic and hadronic models face serious difficulties to explain the TeV fluxes reported by the CANGAROO collaboration from SN 1006. Either the CANGAROO result is wrong (the simplest solution from the point of view of a theorist), or something is conceptually wrong with our current understanding of acceleration, propagation and radiation of very high energy particles in supernova remnants (the favoured outcome from the point of view of an observer). On the other hand, there is little doubt that, if the supernova remnants are indeed effective factories of multi-TeV electrons and protons, TeV Y-ray emission should eventually show up, at one flux level or another.
Although the results of this section were explicitly devoted to SN 1006, they, in fact, can be successfully applied to a generic supernova remnant, by considering the acceleration rates of electrons and protons as free parameters. Hopefully, the observations of SNRs with the next generation of detectors with significantly improved sensitivities will comprise a solid observational basis for further phenomenological and theoretical studies. In particular, the extension of measurements both down to GeV energies and beyond 10 TeV will provide the currently missing arguments in favour of (or against) the leptonic or hadronic models. However, key information about the origin of this radiation most probably will be provided by studies of the spatial distribution of TeV emission. The IC models require a very small magnetic field, and therefore relatively broad spatial distribution of TeV radiation. The hadronic models assume strong shock compression of the ambient gas, therefore they predict TeV emission produced essentially in the narrow shell. The forthcoming arrays of imaging Cherenkov telescopes have the potential to address these issues in the near future.
