Pulsars lose their rotational energy by driving ultrarelativistic winds of electrons, positrons and, possibly, ions. If a pulsar is surrounded by a supernova remnant, its winds are thought to terminate at a collisionless shock front, the location of which is determined by the balance between the wind ram pressure and the total pressure in the nebula caused by accumulation of the energy injected in the nebula over the time of the pulsar (Rees and Gunn, 1974). This condition, as well as an assumption that the energy density of magnetic field is approximately half of the overall pressure (so-called "equipartition condition"), gives quite accurate estimates of the position of the shock and the average nebular magnetic field, in particular for the most prominent pulsar-driven nebula or plerion (filled supernova remnant), the Crab Nebula.
The MHD wind model developed by Kennel and Coroniti (1984) satisfactorily explains many basic features of the Crab Nebula. In this model the pulsar wind is terminated by a standing reverse shock which accelerates electrons up to 1016 eV and randomises their pitch angles. This leads to formation of a bright synchrotron source in the region downstream of the shock. A synchrotron nebulae is formed when the ultrarelativistic, kinetic energy dominated wind is confined in a slowly (nonrelativistically) expanding shell of the supernova remnant. Thus, the major fraction of the rotational energy of the pulsar is eventually released in nonthermal synchrotron radiation which extends to the hard X-ray or even Y-ray energy domain. Inverse Compton scattering of the 2.7 K CMBR and (in the Crab Nebula) the synchrotron radiation of the plerion by the same ultrarelativis-tic electrons lead to the formation of more energetic Y-radiation extending to multi-TeV energies. Although in many cases the energy release in IC Y-rays may constitute only a small fraction of the synchrotron luminosity of the plerion, the high energy Y-radiation provides a unique channel of information about the basic parameters of the nebula.
Broad-band nonthermal radiation of the Crab Nebula
The Crab Nebula is a unique cosmic laboratory with an unprecedentedly broad spectrum of the observed nonthermal radiation which extends over 21 (!) decades of frequencies – from radio wavelengths to very high energy Y-rays (see Fig. 2.7). This emission is dominated by two major mechanisms connected with interactions of relativistic electrons with the magnetic and photon fields of the nebula. While the synchrotron component is responsible for the radiation from radio to relatively low energy Y-rays (E < 1 GeV), the inverse Compton (IC) scattering of electrons is thought to be the most probable mechanism for TeV Y-rays.
Fig. 6.14 Adaptively smoothed Chandra HETG-ACIS-S image of the central 200" X 200" region of the Crab Nebula.
The typical energies of electrons responsible for production of synchrotron photons in different energy bands of the Crab spectrum are indicated in Fig. 2.7. Note that although the conclusion about highest energy electrons is still based on a model (although well justified) assumption about the synchrotron origin of the hard X-rays/low-energy Y-rays, the detection of Y-rays well above 1013 eV is the first unambiguous evidence of effective acceleration of particles beyond 1014 eV.
It is commonly accepted that the synchrotron nebula is powered by the relativistic wind of electrons generated at the pulsar and terminated by a standing reverse shock wave at a distance
(Rees and Gunn, 1974). Relativistic MHD models, even in their simplified form (e.g. ignoring the axisymmetric structure of the wind and its interaction with the optical filaments), successfully describe the general characteristics of the synchrotron nebula, and predict realistic distributions of relativistic electrons and the magnetic field in the downstream region behind the shock (Kennel and Coroniti, 1984).
While the synchrotron and IC mechanisms seem to provide a reasonable explanation of the overall nonthermal radiation of the Crab Nebula (see Fig. 2.7), one cannot exclude possible deviations in different frequency domains from the simplified picture of the outer nebula described by the spherically symmetric MHD models. Indeed, the imaging of the Crab Nebula by the Hubble Space Telescope (Hester et al. 1995) revealed a very rich and complex structure of the inner part of the nebula, on scales down to 0.2". It consists of features like wisps, jets, knots, etc., and exhibits cylindrical symmetry. The recent imaging of the Crab Nebula in X-rays by Chandra (Weisskopf et al., 2000) with subarcsecond resolution confirms the striking richness of the inner nebula (Fig. 6.14). In particular, these observations revealed, for the first time, an X-ray inner ring within the X-ray torus (Aschenbach and Brinkmann, 1975), as well X-ray hot spots along the inner ring.
The distinct axisymmetrical structure of the inner nebula is strong evidence that the most of the rotational energy of the pulsar is released in the form of a wind which flows along the pulsar equator. But, after 30 years of extensive theoretical efforts, astrophysicists are still far from a full understanding of the important details of the physics of interaction of the pulsar wind with the synchrotron nebula. This prominent source remains a great challenge for future theoretical work.
Synchrotron and IC radiation
To calculate the nonthermal radiation of the Crab Nebula, one has to specify the spatial distribution of the magnetic field, the acceleration site(s) and the injection spectrum of the relativistic electrons, as well as the character of their propagation in the nebula. The calculations presented below are based on the MHD model of Kennel and Coroniti (1984). Although this model assumes a spherical geometry, which significantly deviates from the picture shown in Fig. 6.14, it provides a quite accurate estimates for IC Y-ray fluxes integrated over the size of the nebula.
The MHD model of Kennel and Coroniti (1984) provides all necessary parameters, in particular it allows self-consistent calculations of the spatial and spectral distribution of high energy electrons, freshly accelerated at the wind termination shock and injected into the nebula (wind electrons).
Meanwhile, the radio emission of the nebula requires an additional low energy (E < 100 GeV) component of electrons (radio electrons) accumulated, most probably, during the whole history of the Crab Nebula. Although the origin and site(s) of this "relic" component are not yet established, it can be easily incorporated into the calculations of the broad-band synchrotron spectrum with a minimum number of assumptions based on radio observations.
Fig. 2.7 demonstrates the good agreement of calculations with the observed spectrum of the Crab Nebula up to hard X-rays, and a reasonable explanation of the Y-ray fluxes up to 1 GeV by the synchrotron radiation. The best fit is reached by the following combination of the spectra of the radio and wind electrons:
and
with
The presentation
of the spectrum of the wind electrons nwe(E) in such a form provides a the necessary (and natural in the framework of MHD model) flattening of the spectrum below E* ~ 200 GeV. The transition from the hard to steep power-laws in the total (radio + wind) electron spectrum at energies around 100 GeV accounts for the sharp steepening of the spectrum at IR/optical wavelengths. Therefore detailed spectroscopic measurements in this energy region would allow us to specify more precisely the values of Ec and E* which define the degree of "smoothness" of transition from radio to wind electrons. Independent information about the electrons in this transition region is contained in the 10 to 100 GeV Y-rays produced by the same electrons upscattering the ambient low-frequency radiation. Meanwhile, determination of the high energy cutoff
is contingent on measurements of Y-rays in the 1 MeV to 1 GeV energy band.
The existence of ultra-relativistic electrons in the synchrotron nebula provides production of detectable TeV Y-ray fluxes through IC scattering (Gould, 1965; De Jager and Harding, 1992; and Atoyan, 1995; De Jager et al., 1996; Atoyan, 1996; Hillas et al., 1998). Since the target radiation fields which play a major role in the production of IC Y-rays in the Crab Nebula (synchrotron, thermal infrared, and 2.7 K CMBR) are well known, the flux of IC Y-rays can be calculated with good accuracy. For the given flux of synchrotron X-rays, the number of TeV electrons strongly depends on the nebular magnetic field. Therefore the IC Y-ray fluxes are very sensitive to the average magnetic field:
Thus the comparison of the predicted and observed TeV Y-ray fluxes allows determination of the magnetic field in the central r ~ 0.5 pc region (where the bulk of X-rays and TeV Y-rays are produced) with accuracy
Although the statistical significance of Y-ray observations of the Crab Nebula is very high, the systematic uncertainties in the flux estimates remain rather large,
Even with the present uncertainties, the TeV observations favour the magnetic field in the X-ray production region to be 
which is in agreement with the estimated equipartition field (Marsden et al., 1984).
Fig. 6.15 The radial distribution of the magnetic field in the Crab Nebula expected within the framework of the MHD model of Kennel and Coroniti (1984) for different values of a-parameter. r = rs corresponds to the distance of the wind termination shock from the pulsar.
The shape of the spectrum of IC Y-rays does not depend much on the basic parameters of the nebula, in particular, it is almost independent of the parameter a (the ratio of the electromagnetic energy flux to the particle energy flux at the shock) which in the framework of MHD model defines the spatial distribution of the magnetic field
Although the calculations in Fig. 2.7 correspond to
the measured synchrotron fluxes can be equally well fitted with a anywhere between 0.001 and 0.01. Indeed, since at distances
where the bulk of synchrotron and IC photons are produced, the magnetic field B(r) calculated with the MHD model of Kennel and Coroniti (1984) depends rather weakly on a (Fig. 6.15), only minor changes in the assumed injection spectrum of electrons are required to provide the same synchrotron spectrum for different a within 0.001-0.01. Consequently, as far as the target photon fields for IC Y-rays are fixed, there is no room for strong dependence of the calculated IC fluxes on a.
In Fig. 2.7 are shown the synchrotron and IC components of radiation, calculated in the framework of spherically symmetric MHD model, which describes the flux of the Crab Nebula over the whole range of observed frequencies fairly well. At the same time, in some specific energy intervals, in particular in the 1-10 MeV, 1-10 GeV, and perhaps also at > 10 TeV bands the observed energy spectra cannot be easily explained within the simplified synchrotron-Compton model. Below we discuss possible ways to account for these features.
Second High Energy Synchrotron Component
The spectral measurements of the unpulsed radiation of the Crab by COMPTEL revealed an unexpected flattening of the spectrum at energies 1-10 MeV (van der Meulen et al., 1998) which follows the well established steepening of the spectrum above 100 keV. Since such sharp feature could hardly be attributed to peculiarities in the injection spectrum of shock accelerated electrons, a more natural interpretation of this spectral feature can be given assuming the existence of an additional radiation component. Explanation of this radiation excess in terms of nuclear Y-ray line emission is not supported by observations (van der Meulen et al., 1998), and more importantly, it contradicts the total luminosity of the Crab Nebula since only
of the energy losses of nonrelativistic protons and nuclei is re leased in prompt Y-ray lines, while the main part goes to heating of the ambient gas, and thus should show up in the form of thermal radiation with an unacceptably high luminosity
While remaining in the framework of the hypothesis of the synchrotron origin for the radiation up to 1 GeV, the steepening above 100 keV implies an exponential cutoff in the injection spectrum of the wind electrons at energies smaller than
used in Fig. 2.7 . If so, the flat spectrum observed by COMPTEL requires a second population of high energy electrons. In Fig. 6.16 a possible fit of the observed fluxes up to 1 GeV by two-component synchrotron emission is presented. The first component is attributed to the same wind electrons as in Fig. 2.7 but with 
For the second component a very hard acceleration spectrum, for example of Maxwellian type,
with 
has been assumed.
Fig. 6.16 Synchrotron and IC radiation components produced by the first (solid) and second (dashed) populations of electrons (see text). The heavy solid line shows the totalflux. The hatched region corresponds to
which generally describes the reported fluxes from 
Although here
actually the mean energy of the second population
is larger that the highest energy particles
effectively present in the first population. Note that the required acceleration power in the second electron population is only
less than 1% of the first (main) population,
This explains the small contribution of the second electron population in the total IC Y-ray fluxes up to energies
(see Fig. 6.16).
The possible sites of acceleration of the second electron population could be the peculiar compact regions such as wisps, knots, etc. Since the equipartition magnetic field in these regions is estimated to be as high as few mG (in that case E2 is reduced to
the highest energy electrons could not escape the acceleration sites due to severe synchrotron losses. Although these variable structures, with typical size 0.2" (Hester, 1995), are not resolvable by low-energy Y-ray instruments, the detection of variability of the 1-100 MeV emission would be direct proof for the synchrotron origin of the "MeV bump".
Bremsstrahlung and
gamma rays?
The second feature seen in Fig. 6.16 is the deficit in the predicted IC fluxes compared to the reported fluxes at GeV energies. The expected IC Y-ray flux in this energy region is estimated to be, within 20 per cent accuracy,
For the equipartition magnetic field of 
this flux is a factor of 5 below that measured by EGRET (Nolan et al., 1993, Fiero et al., 1998). In order to explain the measured fluxes in this energy region, an additional component of GeV radiation was suggested in De Jager et al. (1996), which the authors in their best-fit "synchrotron+IC" model call a second IC power-law component. To increase the IC flux one has to suppose that the magnetic field in the radio nebula (where the low energy Y-rays are produced) is
However this value does not agree with the well defined flux of TeV radiation. Also, \ such a low magnetic field would imply that the energy in radio electrons is
while the energy in the magnetic field is a factor of 30 smaller. This would make the confinement of electrons in the nebula rather problematic. Thus, if the high Y-ray flux above 1 GeV reported by EGRET originates outside the pulsar, i.e. in the nebula, one may need to invoke additional radiation mechanism.
Bremsstrahlung gamma-rays. For a mean gas density in the nebula of 
the flux of the bremsstrahlung Y-rays cannot exceed 15 % of the flux of IC Y-rays. In fact, in the Crab Nebula the gas is concentrated mainly in dense filaments where
(Davidson and Fesen, 1985).
In the case of a uniform distribution of relativistic electrons throughout the nebula the effective gas density is defined by the mean density of the nebula,
However, if electrons are trapped, at least partially, in the regions of high density, i.e. if they propagate slower inside the filaments than outside, then
The fluxes of bremsstrahlung Y-rays calculated for
are shown in Fig. 6.17 . The contribution of the "amplified" bremsstrahlung flux not only could explain the measured GeV Y-ray fluxes, but also would significantly modify the spectrum at very high energies.Indeed, in the energy range between 100 GeV and 10 TeV the superposition of the IC and the "amplified" bremsstrahlung components results in an almost power-law spectrum with an index
in contrast to the curved IC Y-ray spectrum alone, which is hard at
but becomes significantly steeper at higher energies
Fig. 6.17 The contributions of different Y-ray production mechanisms to the total nonthermal radiation of the Crab Nebula. The Synchrotron and IC components are the same as in Fig. 6.16. The bremsstrahlung and
Y-ray fluxes are calculated for
-decay gamma-rays. Interactions of the nucleonic component of accelerated particles (which may in principle acquire a significant part of the power of relativistic wind at the reverse shock; see Arons, 1996) with the ambient gas lead to the production of Y-rays through secondary
However, since the average gas density in the nebula is low, the contribution of this mechanism to the Y-radiation could be detectable only in the case of partial confinement of relativistic particles in the filaments, so that
If so, the
rays may show up (on top of the steep IC spectrum) at TeV energies and beyond (Atoyan, 1996; Bednarek and Protheroe, 1997). In this regard, the detection of up to 50 TeV Y-rays from the Crab as reported by the CANGAROO (Tan-imori et al., 1998a) and HEGRA groups (Horns et al., 2003) may have significant implications concerning the content of the wind and propagation/interaction of accelerated particles in the filaments. Although the reported fluxes do not contradict, within the uncertainties in the nebular magnetic field, an IC origin of the radiation, the estimated differential power-law spectra from 1 TeV to 50 TeV with power law index close to 2.5-2.7 seems to be significantly harder than the predicted IC spectrum 
In Fig. 6.17 the spectrum of
Y-rays is shown, calculated for a power-law differential spectrum of accelerated protons with
exponential cutoff at
and significant flattening below
,as is expected from wind acceleration models. For
used in
Fig. 6.17, the
fluxes shown correspond to a total energy in accelerated protons of
a quite acceptable amount from the point of view of the energy budget of the Crab. It is interesting to note that for the chosen parameters the superposition of 3 components of
in the power-law spectrum with
over the entire energy range from 100 GeV to 100 TeV. This spectrum significantly differs from the pure IC spectrum, and provides a better fit to the reported data, the compilation of which is presented in Fig. 6.16 and 6.17 by the hatched zone. However, given the large uncertainties in the reported Y-ray fluxes, one cannot at present make a strong statement about the role of the bremsstrahlung and n0 signatures in the Crab spectrum.
Fig. 6.18 Predicted brightness distributions of high energy Y-rays in the Crab Nebula at different energies. The heavy solid curve corresponds to the brightness distribution at E = 100 GeV expected in the case of enhanced bremsstrahlung contribution. All other curves are for the pure IC Y-rays .
The objectives of future gamma ray studies
The multiwavelength observations of the Crab Nebula have already provided important information about the nonthermal energy in the form of magnetic fields and relativistic electrons. However, many details remain still unresolved which need to be addressed by future observations. Importantly, the fluxes of the source exceed, by at least one order of magnitude, the sensitivities of the current or planned instruments at practically all frequencies of the observed spectrum. This ensures further significant progress in understanding the complex processes in the interaction of the relativistic pulsar wind with the nebula. Below we outline some issued to be addressed by new observations in different energy bands.
Probing electrons and B-fields in the synchrotron nebula. The most informative frequency band to probe the acceleration site(s) and the character of propagation of the ultrarelativistic electrons in the nebula is the X-ray domain. The recent studies of of the spatial and spectral structure of the X-ray emission of the Crab Nebula by Chandra (Weisskopf et al., 2000) provide essential material for comprehensive modelling of synchrotron radiation in the inner part of the nebula. Since the cooling time of electrons responsible for the X-radiation, tsy « 50(B/10-4 G)-3/2(e/1 keV)-1/2 yr, is very short compared to the age of the source, the total flux of synchrotron X-rays is a perfect calorimetric measure of the (quasi)-continuous acceleration rate of ultrarelativistic electrons, WW ~ Lx, and thus also the power of the kinetic energy dominated wind. At the same time, since the luminosity of synchrotron X-rays emitted in this regime is almost independent of the magnetic field, the X-ray measurements do not tell us much about the strength and spatial distribution of the magnetic field. Such information is contained in TeV Y-rays produced by the same electrons, because the fraction of energy of electrons released in IC Y-rays is determined by the ratio of energy density of magnetic field (<x B2) to the energy density of target photon fields.
Although the limited angular resolution of Y-ray detectors does not allow, at least presently, mapping of the source on subarcmin scales, the measurements of integral fluxes of IC Y-rays at different energies, being coupled with synchrotron radiation in the relevant energy bands, could compensate for this disadvantage. Such an analysis is possible due to the relationship between the energies of synchrotron and IC photons produced by the same electrons. Thus, the spatial distribution of synchrotron radiation in different bands provides important (although not completely model- independent) information about the regions of production of Y-rays at different energies. The brightness distributions of Y-rays shown in Fig. 6.18 are calculated within the framework of the spherically symmetric MHD model of Kennel and Coroniti (1984). Although more realistic spatial distributions of the TeV electrons seen in Chandra images would modify these symmetric distributions, the results presented in Fig.6.18 quite correctly describe the average extensions of Y-ray production regions in different energy bands. For example, since the TeV Y-rays are produced by IC scattering of the electrons responsible for the observed keV X-rays, the estimate of the magnetic field based on keV/TeV data relates to the central r — 0.5 pc region of the Crab Nebula. A model-independent estimate of the magnetic field in the outer parts of the (optical) nebula can be provided by measurements of Y-ray fluxes at E — 100 GeV. Similarly, the fluxes of E > 10 TeV Y-rays, combined with hard X-ray data (E > 100keV), could allow determination of the magnetic field in the vicinity of the wind shock front at r — 0.1 pc. Remarkably, as the target photon fields for IC Y-rays are well known, the accuracy of the estimate of the magnetic field could be better than 25%, provided that the Y-ray fluxes are measured with an accuracy of better than 50%.
While at energies between 1 and 10 TeV IC scattering dominates over all other possible radiation mechanisms, at energies below 1 TeV and above 10 TeV other processes connected with interaction of the relativistic particles with the nebular gas might contribute to the production of Y-rays as much as the IC does. Therefore determination of the magnetic field based on Y-ray fluxes in these energy regions requires separation of the IC contribution from the possible contamination due to Y-rays of other origin. The shape of the spectrum of IC Y-rays does not vary much with the basic parameters of the nebula. While at GeV energies the IC spectrum is very hard, with a power-law index of rY « 1.5, in the VHE region the spectrum gradually steepens from T7 « 2 at E – 100 GeV to T7 « 2.5 at E – 1 TeV, and rY « 2.7 at E — 10 TeV. Confirmation of this spectral shape is one of the important issues to be addressed by new observations, in particular by the stereoscopic systems of imaging atmospheric Cherenkov telescopes which provide good spectrometry with energy resolution < 20 per cent and high localisation precision for point VHE sources with subarcmin accuracy. Although this still is not sufficient for adequate study of the angular structure of the VHE Y-ray production region in the Crab, in combination with expected large photon statistics it could provide an answer as to whether the observed Y-rays are produced in nebula, or should they be attributed to the pulsar or unshocked wind.
Interaction of accelerated particles with filaments. Although the uncertainties in the reported fluxes at energies 1-10 GeV do not allow us to draw definite conclusions about the conflict between the observations and the predicted IC Y-ray fluxes, the confirmation of high EGRET fluxes by future observations, e.g. by GLAST, would require more effective mechanism(s) responsible for Y-ray production in this energy region. The bremsstrahlung of radio electrons seems to be an intriguing possibility. In particular, it implies an effective confinement of the relativistic particles in dense filaments to provide sufficiently high effective gas density ("seen" by relativistic particles) neff > 50 cm-3. Another possible explanation of the EGRET excess flux is IC radiation of the unshocked pulsar wind, provided that the Lorentz factor of the wind is smaller by an order of magnitude than the "nominal" value, r — 106, required by the model of Kennel and Coroniti (1984).
The enhanced rate of interactions of relativistic particles with the gas due to their confinement in high density regions, opens the possibility of probing the content of the pulsar wind by searching for n0 Y-rays in the spectrum of the Crab at very high energies. Indeed, for the effective gas density neff > 50cm-3, the total energy of accelerated protons is estimated as Wp < 2 x 1048 erg. This is quite a reasonable amount for the energy budget of the Crab, which however would be sufficient for significant modification (hardening) of the resulting "IC+n0" spectrum. If future accurate spectro-metric observations of the Crab show that the power-law Y-ray spectrum measured at TeV energies extends, without noticeable steepening, well beyond 10 TeV, it would be strong evidence of acceleration of a nucleonic component of cosmic rays in the Crab up to energies of 1015 eV, as well as a high concentration of relativistic particles in dense gas regions. Interestingly, since this will result in enhanced bremsstrahlung Y-rays as well, the total "IC+bremsstrahlung+n0" radiation spectrum is expected to be almost a single power-law for over three decades in energy, from 100 GeV to 100 TeV.
Origin of hard (MeV) synchrotron radiation. The steepening in the synchrotron spectrum above 100 keV followed by hard MeV radiation ("MeV bump") can be naturally interpreted as the superposition of two different radiation components, both, most probably, of the synchrotron origin. If the first component is attributed to the diffuse emission of the synchrotron nebula, the second component could originate in one or a few compact structures, e.g. in knots or in the jet, where the strong magnetic field (B > 10-3 G) could create favourable conditions for both effective acceleration of highest energy electrons and production of synchrotron radiation up to — 100 MeV. The limited angular resolution of hard X-ray/soft Y-ray detectors does not allow direct identification of these compact structures. On the other hand, since the cooling time of electrons producing MeV synchrotron radiation does not exceed the light crossing time of these compact structures, the detailed study of the spectrum and flux variability of radiation in this energy domain by INTEGRAL and GLAST missions, could compensate, to some extent, for the lack of spatial information.
Indirect information about the site(s) of the synchrotron "MeV bump" is also contained in low energy X-rays. Indeed, even in the case of an extremely hard injection spectrum (e.g. monoenergetic or Maxwellian type) of electrons responsible for the "MeV bump", the energy flux of this component at 1-10 keV range cannot be less than 10-10 erg/cm2s, so it perhaps could be resolved by Chandra even if this flux is due to the superposition of a large number of subarcsecond structures. Note, however, that the detection of these structures in X-rays will not be sufficient to identify them as the sites of "MeV bump". The important criteria for such identification would be the energy spectrum of this component which is expected to be essentially harder than the X-ray spectrum of the surrounding diffuse synchrotron nebula.
