The surprising Crab pulsar and its nebula: A review

The pulsar emits radiation across the electromagnetic spectrum with a period of $P_{Crab}=33.6$ ms, which is slowing down by ${\dot{P}}_{Crab}=4.2\times {10}^{-13}$ [Manchester2005, Abdo2013]. The corresponding loss of rotational energy is $\dot{E}\approx 5\times {10}^{38}$ erg s${}^{-1}$, assuming a moment of inertia of the neutron star of $I\approx {10}^{45}$ g cm${}^{-2}$. Only $\sim 1$% of this energy is emitted in electromagnetic radiation. The radiation is thought to be collimated in beams of light, which sweep the field of view of the Earth, creating the observed pulsations. The phase-averaged spectral energy distribution (SED) is shown in Fig. 2. It is composed of three main components: the first one in the radio band, a second energetically dominant component peaking in the X-ray band and a third one emerging above energies of $\sim 100$ MeV.

The phase profile of the emission is shown in Fig. 3. As for the majority of pulsars detected in gamma rays by the Fermi Large Area Telescope (LAT), the radiation as a function of phase $\varphi$ is concentrated in two peaks [Abdo2013]. The pulses arrive approximately simultaneously across the electromagnetic spectrum, only small shifts between energy bands are seen ($\Delta \varphi \lesssim 0.01$; \citeasnounOosterbroek2008 and \citeasnounAbdo2010). The main pulse $P1$ is located at $\varphi \approx 1.0$ and the inter-pulse $P2$ at $\varphi \approx 0.4$. A faint precursor to $P1$ is detected at radio frequencies, and at frequencies above the optical bridge emission is found between $P1$ and $P2$. A peculiarity of the pulse profile of the Crab is that it is unusually irregular on short time scales. Giant radio pulses with a flux 1000 times the average are seen randomly during $P1$ and $P2$ [Lundgren1995, Cordes2004, Popov2007]. Optical emission was seen to increase during such events [Shearer2003, Strader2013], but no such correlation has been found so far with higher energies [Bilous2011, Bilous2012, Aliu2012]. Pulsar glitches, during which for a limited time the spin-down frequency increases by up to $\sim {10}^{-5}$ Hz, are observed $\sim 1$ per year [LyneA.G.1993, Espinoza2011, Wang2012].

The pulsar emission is found to be polarized. The position angle $PA$ of the linearly polarized component varies with pulse phase. The polarization properties in radio depend on frequency. Generally, the polarization degree and angle vary smoothly, with no abrupt changes during the main pulses [Moffett1999]. The Crab is one of the few pulsars for which polarization has also been detected at optical wavelength [Sowikowska2009, Moran2013]. The optical polarization angle swings from $PA\approx {40}^{\circ }$ to $PA\approx {170}^{\circ }$ during P1, decreases again during the bridge emission and swings from $PA\approx {90}^{\circ }$ to $PA\approx {180}^{\circ }$ during P2.

The appearance of the nebula in the sky is approximately ellipsoidal with a major axis of $\sim 7$ arc minutes and a minor axis of $\sim 4.6$ arc minutes. This corresponds to a projected length of $\sim 4.1$ pc and $\sim 2.7$ pc, respectively. As one observes the nebula at higher frequencies, a toroidal structure becomes increasingly apparent. Images of the inner region in X-rays and optical are shown in Fig. 4. A torus surrounding the pulsar and a jet emerging perpendicular to it are apparent. It is striking that there is no bright emission in the region within $\sim 10$ arc seconds of the pulsar [Hester1995, Hester2002, Mori2004a, Temim2006]. The pulsar wind is apparently radiationless (or “cold”), until interaction with the ambient medium happens. The first interaction is commonly thought to occur at the “inner ring” observed in the X-ray image [Weisskopf2000].

As can be seen in Fig. 4, the inner nebula is a highly dynamical place. Thin arcs of increased emission, so called “wisps”, are observed to move outwards from the inner ring into the body of the nebula [Scargle1969]. Wisps can be seen in radio, optical and X-rays, however their positions do not always coincide between frequencies. Typically, their inferred velocity is between 0.03 c and 0.5 c [Hester2002, SchweizerThomas2013], with indications for radio wisps to be slightly slower [Bietenholz2001, Bietenholz2004]. Additionally, several other structures on scales of a few arc seconds are seen close to the inner ring and along the jet. Most prominently the “Sprite” is seen as a fuzzy region at the base of the jet in the optical images, and the “inner knot”²²2There is some confusion with the nomenclature used in the literature. The “inner knot” is sometimes also referred to as “knot 1”, and the “Sprite” sometimes referred to as “Anvil”. is detected only $\sim 0.6$ arc second south east of the pulsar (the inner knot is too close to the pulsar to be visible in Fig. 4, see e.g. \citeasnounTziamtzis2009 or \citeasnounMoran2013). Sprite, wisps and the inner knot are known to be variable down to time scales of a few hours [Hester2002, Melatos2005].

Spectrally, the nebula shows a trend to softer spectra with increasing distance from the pulsar. This is interpreted as the radiative cooling of the highest energy electrons as they are convected and diffuse away from the inner nebula. Interestingly however, the spectrum in the inner region of the nebula is rather uniform. The torus, the jet and the small scale structures mentioned in the previous paragraph show no strong spectral variations among them from infrared to X-ray energies [Veron-Cetty1993, Willingale2001, Temim2006, Mori2004a, Tziamtzis2009]. At radio frequencies the picture is more complex. As radio emission originates from a larger volume in the nebula, it is generally more difficult to disentangle line of sight effects. However, also at radio frequencies the spectrum is harder in the inner nebula [Green2004a, Arendt2011].

The emission from the nebula is found to be linearly polarized from radio to hard X-rays [Wilson1974, Weisskopf1978, Bietenholz1991, Hester2008, Dean2008, Forot2008, Aumont2010]. Generally, the polarization angle $PA\approx {125}^{\circ }$ is aligned with the symmetry axis of the nebula, indicating that the magnetic fields are predominantly azimuthal. A detailed study was recently performed in optical by \citeasnounMoran2013. They find that the polarization of the wisps, the inner knot and the torus are all within a few degrees of the aforementioned $PA$ and have a high degree of polarization $PD\sim 50\%$. While these structures are found to be variable in flux, their PA and PD show no significant variations.

The broad band SED of the nebula is shown in Fig. 2. One can see that the energy output of the nebula is a factor $\gtrsim 10$ larger than the one for the pulsar (in addition, the isotropic pulsar flux is likely even lower due to the beaming of its emission). In the nebula, the peak of the emission is located in the UV. The emission from radio to X-rays is known to be due to synchrotron emission thanks to polarization measurements. At high energies ($\gtrsim 400$ MeV), a second emission component emerges due to inverse Compton emission of the same electrons. This component is energetically subordinate with respect to the synchrotron emission by a factor $\gtrsim 100$.

A pulsar can be seen in first order as a rotating, perfectly conducting sphere, with a dipole magnetic field. The magnetic moment is generally not aligned with the rotational axis (the so called “oblique rotator”). Compression of the magnetic field of the progenitor star onto a neutron star of $\sim 12$ km diameter results in magnetic fields of the order of $B\approx 3.8\times {10}^{12}\left(\frac{P}{P_{Crab}}{)}^{1/2}\right(\frac{\dot{P}}{{\dot{P}}_{Crab}}{)}^{1/2}$ G at the equatorial surface of the neutron star. The rotation of this field induces an electric potential of the order of $\Delta \Phi \approx 4\times {10}^{16}\left(\frac{\dot{P}}{{\dot{P}}_{Crab}}{)}^{1/2}\right(\frac{P}{P_{Crab}}{)}^{-3/2}$ V between the poles and the equator of the star. After obtaining vacuum solutions of the oblique rotator [Deutsch1955], it was realized that such strong potentials would remove particles from the neutron star surface and fill the magnetosphere with plasma. The magnetosphere becomes charge separated to compensate the electric potential [Goldreich1969]. This makes the calculation of the magnetosphere properties, as magnetic field structure and electric densities and currents, intrinsically non linear, and dependent on the microphysics of the neutron star surface and the magnetospheric plasma. We therefore still do not have a fully self-consistent description of the magnetosphere of pulsars.

Over the past decade however, there have been significant progress in our understanding of the magnetosphere using the force free electrodynamics (FFE) approximation. The FFE approximation can be derived from magnetohydrodynamics (MHD) in the limit of inertia-free plasma, where the transport of energy and momentum is purely electromagnetical [Komissarov2002, Blandford2002]. As in MHD, plasma is abundant and electric fields $E$ parallel to the magnetic fields $B$ are shortened out ($\overrightarrow{E}\cdot \overrightarrow{B}=0$). There is no particle acceleration or resistivity. This approach is supported by the fact that modeling of pulsar wind nebulae suggest that electrons are injected at high rates from the magnetosphere into the nebula ${\dot{N}}_{+-}>{10}^5\times {\dot{N}}_{GJ}$, where ${\dot{N}}_{GJ}=7.6\times {10}^{33}\left(\frac{P}{P_{Crab}}{)}^{-1/2}\right(\frac{\dot{P}}{{\dot{P}}_{Crab}}{)}^{1/2}$ s${}^{-1}$ is the “Goldreich-Julian” current [Bucciantini2011]. Charge densities in the magnetosphere are therefore expected to significantly higher than the densities needed to compensate the electric potential due to pair creation within the magnetosphere (see \citeasnounArons2012 and references therein). Further supporting the FFE approximation is that only $\lesssim 10\%$ of the spin-down energy of pulsars is typically radiated in the form of pulsed emission, suggesting a low resistivity within the magnetosphere. Several simulations of magnetospheres in the FFE approximation have been made over the last years [Spitkovsky2006, Kalapotharakos2009, Bai2010]. The charge density and magnetic field structure in the plane of the angular moementum and magnetic moment vectors are shown in Fig. 6. The charge separation of the magnetosphere is apparent. A current sheet emerges in the equatorial plane, at the boundary between closed and open field lines³³3Field lines which return to the pulsar within the light cylinder are usually referred to as “closed”, others as “open”. and crosses the light cylinder.

The FFE approximation cannot be exact, as there must be some resistivity in the magnetosphere. The latter depends on the properties of the co-rotating plasma, and has not been derived from first principles to date. However, ad-hoc introduction of resistivity into FFE and MHD simulations have been performed over the last years [Li2012, Kalapotharakos2012a, Kalapotharakos2012b, Tchekhovskoy2013]. The basic magnetosphere properties, as e.g. the presence of the equatorial current sheet, are in agreement with the FFE approach. It was shown by \citeasnounLi2012 that the total spin-down luminosity and the potential drop over the open field lines transitions smoothly between the FFE approximation and the vacuum solution with increasing resistivity. The real magnetosphere of pulsars is therefore likely somewhere in between these two descriptions.

Neither MHD nor FFE simulations address the microphysics of the acceleration of particles within the magnetosphere. As currents flow out of the light cylinder on the open field lines, charge starved regions can emerge behind them (so called “gaps”). An electric potential drop can build up in these regions until it is discharged by electron pair cascades and particles are accelerated. A polar gap is thought to emerge up to a distance of $\lesssim {10}^6$ cm above the magnetic poles [Sturrock1971, Ruderman1975, Holloway1975, Daugherty1982, Baring2004]. Gaps location have also been proposed out in the magnetosphere, along the last open field lines (slot gap [Arons1979, Muslimov2004]; annular gap [Qiao2004, Du2012]), reaching out to the light cylinder (outer gap [Cheng1976, Cheng1986, Romani1995, Cheng2004]). Alternatively, particles might also be accelerated in reconnecting magnetic fields at the equatorial current sheet [Li2012, Tchekhovskoy2013]. In either case, the particles loose their energy after leaving the acceleration region due to interaction with the magnetic and photon fields of the magnetosphere, and secondary particle cascades might be induced. As the direction of the particles is bound closely to the magnetic field lines, co-rotating cones of light are emitted around both magnetic poles. As one or both of them pass our line of sight, the observed electromagnetic pulsations are produced.

It was shown at the early days of pulsar studies, that the magnetic field lines become asymptotically radial beyond the light cylinder in the FFE approximation [Ingraham1973, Michel1974]. This justified studying the fields and flows outside of the light cylinder in the “split-monopole” approximation, where the field lines from the pulsar are thought to be a monopole which inverses its field direction at the equator. An analytic solution to the obliquely rotating split-monopole was found by \citeasnounBogovalov1999. Generally, plasma that flows outside of the magnetosphere on the open field lines begins to trail the pulsar rotation, creating a helical pattern. A current sheet emerges which undulates within an angle $\alpha$ around the equatorial plane (the so called “striped wind”). The magnetic field lines become predominantly toroidal, and reverse their direction at the current sheet (see Fig. 5). Qualitative agreement with the split-monopole solutions has recently been found in FFE and MHD simulations out to a distance of $\lesssim 10R_{LC}$ [Kalapotharakos2012, Tchekhovskoy2013].

The Poynting flux per solid angle of the split monopole solution scales as $P\propto si{n}^2\theta$, where $\theta$ is the angle with respect to the rotation axis of the pulsar. As most of the energy is emitted around the equatorial plane, the striped wind plays an important role in the dynamics of the nebula. The wind is thought to be magnetically dominated at its launch, with $\sigma >>1$, where $\sigma$ is the ratio of the magnetic to the kinetic energy of the flow. However, as we will discuss in the next section a low-sigma flow of $\sigma \sim 0.003$ was inferred at the termination of the wind from 1D MHD modeling of the synchrotron nebula. The emerging question how magnetic energy is transferred to particle energy in the flow is referred to as the “$\sigma$-problem”. In principle, at low latitudes the alternating magnetic fields of the striped wind make it susceptible to magnetic reconnection due to instabilities in the current flow [Coroniti1990]. However, it was realized that due to the relativistic motion of the wind, the time scales of these processes are likely too long [Lyubarsky2001, Kirk2003]. Recently, a new paradigm is emerging were magnetic energy is thought to be dissipated predominantly after the wind termination, as we will discuss in the next section.

As the cold pulsar wind can not be observed before it termination, its properties can only be inferred indirectly by observing the synchrotron nebula. Spherical symmetric MHD solution downstream of the termination surface were derived by \citeasnounKennel1984, predicting the magnetic field distribution in the nebula. The key parameter of this model is the magnetization of the wind just before its termination. In particular the distance of the termination surface from the pulsar in the Crab could only be reproduced if the magnetization at this point is low $\sigma \sim 0.003$. The magnetic field strength is then expected to increase behind the shock and have values between 100-300 $\mu$G in the synchrotron nebula [DeJager1992, AtoyanA.M.1996, Hillas1998, Meyer2010, Martin2012, Torres2013]. This field is lower than the one expected in the case of equipartition between particle energy and magnetic fields, indicating that $\sim 1/30$ of the internal energy in the nebula is in magnetic form [Hillas1998]. Similar magnetic field values are inferred by interpreting the hardening of the integrated spectrum of the nebula between the radio and the optical band due to electron cooling [Marsden1984].

It is typically assume that particles are re-accelerated at the wind termination surface. However, it is unlikely that diffusive shock acceleration of particles takes place, as the emerging shock is expected to be relativistic and oblique [Sironi2011]. Instead, the turbulence downstream of the shock might trigger magnetic reconnection, which can lead to non-thermal particle acceleration [Lyubarsky2003]. Particle in cell (PIC) simulations show that, independent of the wind magnetization, the energy of the alternating field components of the striped wind is dissipated into kinetic energy downstream of the wind termination [Sironi2011a]. Assuming that electrons are accelerated into a power-law spectrum at the termination surface one dimensional MHD models reproduces the spectral energy distribution of the nebula qualitatively, with the exception of the radio band [AtoyanA.M.1996, Volpi2008]. As the life time of radio emitting electrons is much longer than the age of the nebula, these particles might have been injected in the past. Radio emission might therefore come from a separate electron population, which could have been injected during the high spin-down phase of the pulsar following the super novae explosion [AtoyanA.M.1996]. Alternatively, radio emitting electrons could be continuously accelerated throughout the nebula due to magnetic reconnection in MHD turbulence [Nodes2004]. Magnetic reconnection downstream of the wind termination can also reproduce the spectrum of radio emitting electrons, if electrons are injected at a rate of ${\dot{N}}_{+-}\sim {10}^8\times {\dot{N}}_{GJ}$ into the pulsar wind and the magnetization is high with $\sigma >30$ [Sironi2011a]. The implications of high electron multiplicities therefore deserve further study.

While the one dimensional models of the Crab laid out our general picture, it was clear form the beginning that the asymmetry of the nebula needs to be taken into account for a more quantitative description of the nebula. The dynamics of the flow at and beyond the wind termination have therefore been studied with multidimensional MHD simulations over the past decade [Komissarov2004, Bucciantini2006, Porth2013]. The properties of the pulsar wind at its termination and the particle energy spectrum after the wind termination are free parameters of these simulations. Tracing the maximum particle energy in the flow makes it possible to compute synchrotron, inverse Compton and polarization maps of the nebula, which can be compared to observations [delZanna2006, Volpi2008, Volpi2009]. Axially symmetric models with an appropriately chosen azimuthal dependence of the wind qualitatively reproduce the Crab morphology: a torus emerges, and the flow is bend backwards due to the “hoop-stress” in the downstream, creating a jet (see Fig. 7). They explain the emergence of wisps and their dynamics. Their increased brightness is thought to be predominately due to Doppler beaming towards our line of sight [Camus2009]. The simulations predicted variability of a few percent per year in the overall X-ray emission of the nebula [Volpi2008], which has recently been discovered [Wilson-Hodge2011, Kouzu2013]. This is remarkable in high-energy astrophysics, where predictions and measurements are seldom done on this level of accuracy.

As the early 1D MHD solution, the 2D simulations show that a low-sigma wind with $\sigma \sim 0.02$ matches the observed morphology, and the location the shock at the observed $R_{WT}$. Recently, the first 3D simulations were performed [Mizuno2011, Porth2013, Porth2013b]. Interestingly, these simulations show that the morphology of the Crab can also be obtained at high magnetization ($\sigma >1$). As was pointed out by \citeasnounBegelman1998, 1D and 2D MHD solutions prevent the growth of turbulence due to current instabilities. Due to this turbulence, magnetic fields are randomized and remain predominantly axial only close to the wind termination (see Fig. 7), and pressure and magnetic field gradients in the nebula are reduced when compared to 1D or 2D models. The increased turbulence induces magnetic dissipation in the downstream, alleviating the $\sigma$-problem [Lyutikov2010]. While it might be too early for a final verdict, it appears that the origin of the latter lay in the simplifications made in 1D and 2D models.

The success of the MHD simulations in modeling many of the nebula’s properties shows that the general picture we have outlined is likely a good approximation of reality. However, several observational findings can not be explained or have not been studied yet. A quantitative comparison of the morphology obtained in the simulations to the observed one has not been performed to date. In particular, the inner ring does not seem to be found consistently in the simulations. At odds with the expectation from Doppler boosting, the Chandra observations show that the inner ring has similar brightness at the front and the receding side. We also do not have an explanation why the ring appears to be composed of a series of knots, which are highly variable in time [Weisskopf2000, Porth2013b]. The observation that the wisps at different wavebands are generally not aligned has not been studied in the simulations. If Doppler beaming is predominantly responsible for their enhanced brightness, one would naively expect radio, optical and X-ray wisps to be co-spatial. In addition, 2D MHD simulations with low magnetization overestimate the observed inverse Compton emission [Volpi2008]. It will be interesting to see if this discrepancy is reduced in 3D models of high-magnetization.

The high-energy end of the SED of the Crab pulsar is shown in Fig. 8. The best fit parametrizations of the Fermi-LAT measurement by a power-law function with an exponential cutoff ($b=1$) and a sub-exponential cutoff ($b<1$) are also displayed. The latter parametrization is preferred statistically at a $\gtrsim 3\sigma$ confidence level, indicating a slower than exponential fall of the spectrum. This is not unusual for LAT detected pulsars [Abdo2013]. However, unexpectedly VHE gamma-ray measurements showed an enhanced emission above an extrapolation of both of these parametrizations [Aliu2008, Aliu2011, Aleksic2011, Aleksic2012] ⁴⁴4Statistically marginal experimental discrepancies can be seen between different measurements in Fig. 8. In particular, the MAGIC mono data have larger fluxes compared to the other measurements. However, within systematic errors these differences are not significant.. Considering systematic errors, the Fermi-LAT and VHE-gamma-ray data can be described by one power-law function above $E\approx 4$ GeV [Aliu2011]. No indication of a spectral cutoff has been found up to photon energies of $\sim 400$ GeV.

The phase distribution of the VHE gamma-ray events is shown in Fig. 3. The peaks of the gamma-ray pulses coincide with the radio ones. Compared to the HE-gamma-ray pulsations, the VHE ones are narrower. This narrowing of the pulse width with increasing energy can also be seen in the HE and VHE data independently [Abdo2010, Aleksic2012]. Interestingly, a hardening of the spectrum of $P2$ with respect to $P1$ is observed in similar form in two components of the SED: the relative flux of $P2$ increases with respect to $P1$ from radio to hard X-rays, and from HE gamma-rays to the VHE gamma-rays.

While radio emission from pulsars is usually attributed to coherent emission close to the polar gap, at higher frequencies the emission has to come from outer parts of the atmosphere. In particular the gamma-ray emission is expected to come from distance of $0.1-1R_{LC}$ [Romani1995, Qiao2004, Muslimov2004]. This is inferred from the phasograms and spectra of the population of gamma-ray detected pulsars [Abdo2013]. For the Crab pulsar, this expectation is confirmed in a model-independent way by the detection of emission beyond $100$ GeV. To avoid absorption of gamma rays due to pair production on the pulsars magnetic fields, the photons have to be emitted at a distance $\sim {10}^7$ cm $\approx 0.1R_{LC}$ from the neutron star, a factor $\gtrsim 10$ above the expected location of the polar gap (Eq. 1 in \citeasnounBaring2004).

The emission mechanism at the high-energy end of the SED of pulsars is expected to be dominated by curvature radiation of particles streaming along magnetic field lines. However, the high energy of the pulsations makes this unlikely for the Crab. Radiative losses due to curvature radiation would produce a cutoff in the spectrum above gamma-ray energies of ${ϵ}_{cut}=150$ GeV $(E/B{)}^{3/4}\sqrt{R/R_{LC}}$, where $E$ is the accelerating electric field [Aliu2011]. The latter is typically assumed to be $E\lesssim 0.1B$ in pulsar gaps. The absence of a cutoff in the spectrum up to energies $>100$ GeV therefore implies $R>R_{LC}$. As particles stream along magnetic field lines only within the light cylinder, curvature radiation is likely not responsible for the VHE pulsations.

It was proposed by \citeasnounLyutikov2012 that the emission at the high-energy end of the SED is dominated by Inverse Compton scattering. Indeed, the pulsed emission can be explained by up-scattering of photons in particle cascades induced by outer gaps [Aleksic2011, Aleksic2012] or annular gaps [Du2012]. Alternatively, Inverse Compton emission can also occur in the striped wind [Bogovalov2000, Kirk2002]. As proposed by \citeasnounAharonian2012 and \citeasnounPetri2012, the VHE emission may result from the up-scatter of pulsed X-ray photons. If this is the case, the VHE observations are the first direct observational signature of the cold pulsar wind. In order to fit the observed pulse profile and spectrum of the VHE emission, the wind needs to be abruptly accelerated at $20-50R_{LC}$. However, while the gamma-ray emission can be explained in this picture, the observed narrowing of the VHE pulses with increasing energy would not be expected.

The gamma-ray flares are observed in the energy range between the synchrotron and inverse Compton component of the spectral energy distribution. The average photon flux of the synchrotron component is $F_{ns,100}=(6.1\pm 0.2)\times {10}^{-7}$ cm${}^{-2}$ s${}^{-1}$ and the one of the inverse Compton component is $F_{ns,100}=(1.1\pm 0.1)\times {10}^{-7}$ cm${}^{-2}$ s${}^{-1}$ above 100 MeV. For comparison, photon flux of the pulsar above this energy is $F_{p,100}=(20.4\pm 0.1)\times {10}^{-7}$ cm${}^{-2}$ s${}^{-1}$. The emission of the inverse Compton component of the nebula and the pulsar flux are found to be constant in time within measurement accuracies. However, in the synchrotron nebula is variable on all time scales which can be resolved. The power density spectrum as a function of frequency of the flux variations is compatible with a power-law $PDS\propto {\nu }^{-0.9}$ [Buehler2012]. The flux can remain below the detection threshold of the Fermi-LAT for a month, with flux upper limits well below the average flux value [Abdo2010]. On the other hand, the flux can rapidly increase within a few hours during flares. Whether or not the flares are distinct events or the high-energy end of a continuous spectrum in variability is unclear. Time scales down to $\approx 10$ hours can be resolved in the PDS with no sign of a break in the spectrum. However, the flares are far outlayers in the flux distribution and the spectrum during the brightest flares clearly shows the emergence of a new spectral component in the SED.

As of September 2013, six flares have been reported [Buehler2012, Ojha2012, Striani2013, Mayer2013]. Due to the statistical nature of the flux variations, the definition of a flare is somewhat arbitrary. In all flares reported to date the synchrotron component of the nebula had a peak flux $F_{ns,100}>35\times {10}^{-7}$ cm${}^{-2}$ s${}^{-1}$. Equally, the definition of a flare duration is not straight forward; typically, the flux is increased with respect to the monthly average flux for $\sim 1$ week. The SED around the peak of the flares is shown in Fig. 9. One can see that the spectral behavior differs strongly between flares. During the flare in February 2009, e.g. the flux increased with no measurable spectral changes with respect to the average nebula flux, while during the flare in April 2011 a new spectrum component with a rising flux was observed. Generally, all spectra show significant emission up to $\sim 1$ GeV.

The flare of April 2011 gave us the most detailed look into the flare phenomenon to date [Striani2011, Buehler2012]. The light curve of this flare is shown in Fig 10. Its high flux allowed flux measurements down to time scales of $\sim 20$ minutes. The flux doubled within $t_d\lesssim 8$ hours at the rising edges of the two main bursts during the flare. The spectral evolution is shown in Fig. 11. Interestingly, the apparently complex evolution can be parametrized in a simple way: the emerging spectral component is well characterized by a power-law spectrum with an exponential cutoff. The spectral index $\gamma =1.27\pm 0.12$ remains constant within errors during the flare, whereas the cutoff energy $E_C$ and the total energy flux of the synchrotron component above 100 MeV vary as $L_{ns,100}\propto E_C^{3.42\pm 0.86}$. At the maximum of the flare, the cutoff energy is ${ϵ}_{C,max}=375\pm 26$ MeV and the total isotropic luminosity in the synchrotron component is $L_{max,100}\approx 4\times {10}^{36}$ erg s${}^{-1}$, approximately 1% of the spin-down power of the pulsar.

The angular resolution of current HE instruments is $>18$ arc minutes, not enough to determine the emission region of the flares within the nebula. From the beginning, it was clear that in order to pinpoint the emission region, correlated variability at radio, optical or X-rays is needed, making use of the $<1$ arc second angular resolution achieved in these wavebands. However, to date, despite extensive efforts a detection of the flares outside of the HE gamma-ray band remains elusive. Strictly simultaneous observations were obtained during the September 2010, April 2011 and March 2013 flares [Vittorini2011, Morii2011, Lobanov2011, Weisskopf2013, Aliu2014]. Particularly dense observations with the Chandra and Hubble Space Telescopes and the Keck and VLA Observatories were carried out during the April 2011 flare. No increased emission was detected from radio to X-rays for any structure of the nebula above the usual levels. This finding was very unexpected. The inferred flux upper limits show that the SED of the HE gamma-ray flare steeply drops with decreasing frequency, as was already suggested by the hard spectral index of the flaring component in HE gamma-rays during the flare.

No pulsations are found in the gamma-ray emission of the flares. The pulsar properties remain unchanged during the outbursts, no change in the spin-down period or flux were found in radio, X-rays or gamma-rays [Abdo2011, Morii2011, Buehler2012]. The time scale of the recurrence of pulsar glitches is similar to the recurrence of the gamma-ray flares, however, there is no obvious correlation in time between these two events.

To date, the Crab flares remain mysterious. We do not know what causes them and where they come from within the nebula. Several ideas have been proposed, but no definite answers can be given today. Any explanation will have to encompass all the presented observations, so far theoretical models have addressed different aspects of the problem. The rapid variability implies that, unless there is ultrarelativistic beaming, the flare emission comes from a small region within the nebula $R_f\lesssim c\cdot t_d\approx {10}^{-4}$ pc, small even when compared to nebula sub-structures as the Sprite, wisps or the inner knot with projected scales greater ${10}^{-2}$ pc.

A puzzling observation is that flare emission is detected up to photon energies of $\approx 1$ GeV. Synchrotron emission appears to be the only radiation process which is efficient enough to account for the flare emission in the nebula environment [Abdo2011]. However, particles accelerated in MHD flows can only emit synchrotron emission up to a maximum energy ${ϵ}_{max}=160$ MeV [Guilbert1983, Uzdensky2011]. Therefore, either MHD conditions are not valid in the flaring region, or the emission is relativistically boosted towards our line of sight ⁵⁵5Another alternative is that the flare emission is “jitter radiation”, which could e.g. be emitted in the striped wind if the length scale of magnetic turbulence is much smaller than the distance between stripes [Teraki2013]. Both scenarios are possibly interrelated: a breakdown of the MHD conditions occurs in magnetic reconnection events and beaming of particles naturally occurs in the reconnection layer [Zweibel2009, Uzdensky2011, Cerutti2012, Sturrock2012].

That magnetic reconnection process is responsible for the particle acceleration is also indicated by the fact that the proposed alternatives have severe difficulties: diffusive shock acceleration generally does not produce electron spectra which are hard enough to explain the emission during the April 2011 flare and is expected to be inefficient at the termination of the pulsar wind [Ellison2004, Sironi2011, Sironi2013], and the proposed acceleration due to absorption of ion cyclotron waves is expected to act on longer time scales [Amato2006]. Magnetic reconnection was studied with PIC simulation in the context of the flares by \citeasnounCerutti2013 and \citeasnounBaty2013. Cerutti et al. show that emission beyond ${ϵ}_{max}$ is possible in such events. The variations in motion of the accelerated particle beams can explain the observed flux variations with a PDS compatible with the measured one. The spectrum of the April 2011 flare can be qualitatively reproduced, including its dynamical evolution and the observed correlation between cutoff energy and luminosity. The gamma-ray flares are therefore likely connected to explosive reconnection events triggered by current instabilities.

Regions with increased magnetic dissipation are the preferred emission sites, as e.g. downstream of the wind termination, near the inner ring [Bednarek2011, Lyutikov2012, Bykov2012]. In particular, at high latitudes the flow is expected to remain relativistic in the downstream and can be beamed into our line of sight. This produces increased emission from the so called “arch shock” of the wind termination, which was associated to the inner knot [Komissarov2011]. Another possibility is that the flares originate from the base of the jet. As pointed out by \citeasnounLyubarsky2012, turbulence and increased magnetic dissipation are expected in this region, which might be associated to the Sprite. However, no observational signatures related to the flares have been found for any of these regions to date.

The location of the emission region might in principle be different between flares. A distribution of flares throughout the nebula has been proposed, where the beamed gamma-ray emission from different regions randomly crosses our line of sight [Yuan2011, Clausen-Brown2012]. Such a process can produce PDS of flux variations compatible with the observed one, linking the flare phenomenon to the flux variations on longer time scales. The flares might therefore be responsible for a significant part of the magnetic dissipation in the nebula [Komissarov2012].