The Crab Nebula ﬂaring activity

The discovery made by AGILE and Fermi of a short time scale ﬂaring activity in the gamma-ray energy emission of the Crab Nebula is a puzzling and unexpected feature, challenging particle acceleration theory. In the present work we propose the shock-induced magnetic reconnection as a viable mechanism to explain the Crab ﬂares. We postulate that the emitting region is located at ∼ 10 15 cm from the central pulsar, well inside the termination shock, which is exactly the emitting region size as estimated by the overall duration of the phenomenon ∼ 1 day. We ﬁnd that this location corresponds to the radial distance at which the shock-induced magnetic reconnection process is able to accelerate the electrons up to a Lorentz factor ∼ 10 9 , as required by the spectral ﬁt of the observed Crab ﬂare spectrum. The main merit of the present analysis is to highlight the relation between the observational constraints to the ﬂare emission and the radius at which the reconnection can trigger the required Lorentz factor. We also discuss different scenarios that can induce the reconnection. We conclude that the existence of a plasma instability affecting the wind itself as the Weibel instability is the privileged scenario in our framework.


Introduction
The Crab Nebula is one of the most detailed studied astrophysical sources. Its large-scale integrated emission was expected to be steady and it was often used to cross-calibrate X-ray and gammaray telescopes and to check their stability over time. The discovery made by AGILE and Fermi [1,2] of a short time scale flaring activity in the energy range 100 MeV -a few GeV has represented a really unexpected feature. This emission is thought to be synchrotron emission by the highest energy particles that can be associated directly with an astronomical source, challenging particle acceleration theory.
Attempts to explain the flaring activity of the Crab Nebula have been proposed. Among the most promising there is the possibility to accelerate particles in magnetic reconnection events inside the nebula, and the gamma-ray emission occurs when they enter a region of enhanced magnetic field [3,4]. Alternatively, these reconnection events can occur in "mini-jets" moving relativistically, and the gamma-rays are Doppler boosted toward the observer [5]. The gamma-ray variability may be also related to the changes in the * Corresponding author.
Lyubarsky [8] discussed the possibility that a shock-driven reconnection at the termination shock may transfer energy from the wind magnetic field to the particles, potentially generating an electron distribution with a spectral index ∼ 1.5 [see e.g. [9]]. Applying his estimates to the Crab Nebula, he showed that the particles are accelerated to a maximum Lorentz factor, suitable to reproduce the observed radio/optical synchrotron spectrum.
In the present analysis we focus on the overall time duration of the flare (∼ 1 day), linking it to emitting region location and to the acceleration mechanism. In particular, we postulate that the emitting region is located at ∼ 10 15 cm from the central pulsar. Then we show that, if similar conditions than the one considered in [8] are fulfilled at ∼ 10 15 cm, hence inside the termination shock, then the electrons can be accelerated up to a Lorentz factor ∼ 10 9 , as required by the spectral fit of the observed Crab flare spectrum. We finally search for a selection criterion among different scenarios that can induce the reconnection. In particular, we discuss an impulsive interaction of the pulsar wind with a shocking material coming out of the pulsar surface or of its magnetosphere, propagating with a supersonic velocity inside the lower dense wind. Then we postulate the existence of a plasma instability affecting the wind itself. We conclude that the second one is the privileged scenario in our framework. The main merit of the present analysis is then to highlight the relation between the observational constraints to the flare emission and the radius at which the reconnection can trigger the required Lorentz factor.

The Crab Nebula flare observations
Four intense gamma-ray flaring episodes from the Crab Nebula have been reported in the gamma-ray energy range 100 MeVa few GeV by AGILE and Fermi/LAT in the period 2007-2011 [1,2,[10][11][12][13]. This activity has been attributed to transient emission in the inner Nebula due to the lack of any variation in the pulsed signal of the Crab pulsar or of any detectable alternative counterpart [14,15]. No global enhancements are seen in other bands [16][17][18][19][20], but high spatial resolution optical and X-ray observation by Hubble Space Telescope (HST) and Chandra detected local enhancement in the "anvil" region [21,22].
The emission can be modelled [1,10,11] as rapid (within 1 day) acceleration followed by synchrotron cooling: the contribution from inverse Compton emission is negligible. The peak of the gamma-ray spectrum reaches a distinct maximum near 500-800 MeV, well above the constraint for the maximum synchrotron photon energy ∼ 150 MeV that can be radiated, assuming equipartition between the electric and the magnetic field [23].
Assuming a bulk Doppler factor ∼ 1 and a local magnetic field B loc ∼ 1 mG (∼ 5 times the average magnetic field of the Nebula), this energy for the synchrotron photons implies that the electrons are accelerated to γ ∼ 10 9 . This modelling assumes that the acceleration process produces a double power-law differential particle energy distribution, with indices p 1 = 2.1 and p 2 = 2.7 and break energy γ br = 10 9 : this reproduces the spectrum of the flaring emission and its non-detection at lower energies, except for the enhancement in the anvil region. Finally, the timescale of the flares sets the dimension of the emitting region to be ∼ 10 15 cm.

The striped pulsar wind
When the magnetic and rotation axes of the pulsar are not parallel, the time-varying electromagnetic field propagates outwards in the form of electromagnetic waves. In the equatorial belt, the magnetic field at a fixed radius alternates in direction at the frequency of rotation, being connected to a different magnetic pole every half-period. The flow in this zone evolves into regions of magnetically-dominated cold plasma, separated by a very narrow, hot, corrugated surface (the current sheet), whose amplitude increases linearly with the distance from the star. The wavelength of these oscillations is at most 2π r L , where r L = c P /2π is the light cylinder radius and P the period of the pulsar (for the Crab pulsar P = 33 ms). Far from the light cylinder, the distance between successive corrugations is small compared to the radius: the current sheet cuts the equatorial plane, and locally it resembles a sequence of concentric, spherical surfaces. This structure is referred to as a striped wind [24].
Lyubarsky [25] showed that, accounting for the pulsar wind acceleration, the distance beyond which the available charge carriers are unable to maintain the necessary current exceeds the radius of the termination shock (r TS = 3 × 10 17 cm [26]), so that only some fraction of the magnetic energy can be converted into particle energy via a magnetic reconnection process in the wind before the plasma reaches this shock front, depending on the reconnection rate (see however [27]). A lower limit may be obtained by assuming that the dissipation keeps the width of the current sheet equal to the particle Larmor radius, which is roughly the same condition as the current velocity being equal to the speed of light [28,29,25]. With this assumption, Lyubarsky [25] estimated the parameters of the flow (see also [8]): a) the maximum distance beyond which the available charge carriers are unable to sustain the current: where ω L = eB/mc is the gyrofrequency at the light cylinder and Ω the pulsar angular velocity; b) the Lorentz factor of the wind: Γ w = 0.5Γ max √ r/r max , where k is the multiplicity coefficient, which is expected to be large (k ∼ 10 3 − 10 4 ) and Γ max = (ω ) /(2kΩ) is the Lorentz factor attained if all the spindown power is converted into kinetic energy of the plasma; c) the wind magnetisation parameter: σ = (Γ max )/(Γ w ) − 1; d) the current sheet width as a fraction of a wavelength 2π r L occupied by two current sheets: ∼ √ r/r max . In the Crab Nebula we have: r max

Acceleration of particles by shock-driven reconnection and the Crab flares
The magnetic reconnection is a direct mechanism to accelerate particles which is naturally able to explain emission frequencies above the synchroton limit, because within the current sheet the magnetic field is almost vanishing and the inductive electric field can efficiently accelerate the electrons (see e.g. [3,4] for an application of this mechanism in the context of the Crab Nebula flares).
In what follows we adopt the analytical model of particle acceleration in a shock-driven reconnection proposed by Lyubarsky [8] that estimates the maximal energy particles can attain when the magnetic field annihilates completely in a pulsar wind as that described in the previous section. The main results of this model have been confirmed by 2D and 3D Particle-In-Cell simulations [9]. Let us consider a region of length l • (in the direction parallel to the equatorial plane) containing two stripes of oppositely directed magnetic field and a current sheet of width 1 δ between them (for a picture of this configuration we refer to Fig. 2 in [8]). Within the current sheet the magnetic field is zero and a resistive electric field is generated, along which the current may flow unbound and particles gain energy from it. If this region is compressed by a shock in a direction orthogonal to the sheet, the reconnection rate is enhanced. The magnetic field dissipates completely when the size of the region transverse to the sheet due to the compression becomes l • → l ∼ δ.
The particle energy distribution within the current sheet in the plasma comoving frame is N(γ ) = K γ −s at 1 γ γ M . Acceleration of relativistic electrons via magnetic reconnection leads to a power-law particle distribution with index s ∼ 1.5 [30,31,9,32].
Thus, the particle density is dominated by low-energy electrons, while the energy density is dominated by high-energy electrons. As in [8], we assume that the power-law index s remains fixed in the compression and only γ M varies, and as a further condition we impose that the sheet width is equal to the maximal gyroradius δ = m e c 2 γ M eB . The resulting maximal Lorentz factor a particle can attain in the plasma comoving frame when the magnetic field dissipates completely is: where σ is the initial magnetisation parameter and • is the current sheet width as a fraction of a wavelength 2π r L occupied by two current sheets. The compression factor k = l • /l necessary for the magnetic field to dissipate completely is k = s−2 • [8]. The continuity equation in the relativistic flow nΓ = const implies that the plasma Lorentz factor at the end of the dissipation stage is Γ d ∼ Γ w /k. The maximal energy in the particle distribution in the laboratory frame is then: (for the complete derivation of Eq. (2) see [8]).
We aim at implementing the shock-induced reconnection scenario sketched above in the context of the flaring activity observed in the Crab Nebula. The first step is to estimate the Lorentz factor that particle can attain if similar conditions are fulfilled in a different region of the wind than the termination shock. Substituting the parameters for the Crab pulsar wind obtained in Section 3 and s = 1.5 in Eq. (2), we find γ max ∼ 10 9 R −3/4 15 . Therefore, if a shock forms in a region well inside the termination shock and compresses the pulsar wind at a distance r ∼ 10 15 cm from the inner pulsar, this simple calculation shows that high energy electrons can be accelerated up to γ max ∼ 10 9 in the laboratory frame. In the vicinity of the current sheet where such an accelerated population of electrons are produced, the shock that triggered the magnetic reconnection process randomises the electric and magnetic fields, and this situation locally will resemble the wind downstream the termination shock, where this mechanism was initially proposed [8]. The high energy electrons interact with the local average magnetic field and do emit synchrotron radiation.
The advantage of this formulation is that the magnetic field where the reconnection mechanism is triggered is larger than that upstream the termination shock, and so also the average local magnetic field downstream the shock is larger. With a local average magnetic field B ∼ 20-100 mG, the resulting synchrotron radiation from electrons accelerated in the laboratory frame to γ max ∼ (8.4-18.5) × 10 8 is hν sync ∼ 800 MeV, in agreement with the spectral fit of AGILE data [see e.g. [10], their Eq. (4)]. This calculation neglects the synchrotron reaction that potentially saturates the maximum Lorentz factor of the electrons to lower values: however, as long as the electrons are accelerated within the current sheet they experience a magnetic field which is much smaller than the local one, reducing radiative losses and allowing for high energy particles [3]. This model adopts for the population of relativistic electrons a power-law energy distribution ∼ γ −1.5 . As a consequence, the synchrotron spectral energy distribution of the emitted photons is hard, suppressing emission at energies much lower than the peak energy (∼ 800 MeV) and keeping the pulsar wind under-luminous in X-rays and lower energies.
The region where the electrons are accelerated is defined by the existence of a specific magnetic configuration (the rotation of the pulsar corrugates the surface where the magnetic field itself vanishes), across which the wind continuously flows. Uzdensky et al. [3] studied the orbit of a pre-accelerated particle in a reconnection layer and found that for a large range of initial values of Lorentz factors and incident angles the orbit of the electron quickly collapses inside the reconnection layer. We thus argue that most of the electrons are indeed trapped inside the current sheet for a timescale sufficient to be accelerated up to the required Lorentz factor. The population of accelerated electrons constitutes an independent component that does not follow the cold wind motion, but moves with the final Lorentz factor attained after the acceleration process. The overall duration of the process is set by the propagation of the shock in a region where the electrons are accelerated to energies ∼ 10 9 , τ ∼ 10 15 /c ∼ 1 day (see Section 5.2).

Physical hints about the shock compressing the pulsar wind
The starting point of the present study is the identification of the size of the emitting region of the Crab flares (i.e. about 10 15 cm) as the distance from the pulsar at which the shockinduced magnetic reconnection mechanism is able to account for the electron Lorentz factor required by the spectral fit of the observed Crab flare spectrum. We take advantage of the simple analytical model by Lyubarsky [8] proposed for the particle acceleration at the termination shock. Nonetheless, the consistent picture we are tracing requires that the origin and the nature of this shock are fixed to some extent. Below we detail some physical hints by exploring two different situations: a shock front produced by an ultrarelativistic, overdense expanding shell in the pulsar wind, or a plasma instability affecting the wind itself.

A collisionless shock compressing the pulsar wind
We explore some of the characteristics of the shocking material to fulfil the observational requirements. We assume that a shell of overdense material of total energy E • is created in the vicinity of the central pulsar, composed by photon and e ± pairs, and loaded with baryons. This scenario resembles the standard fireball model proposed to explain Gamma-Ray Bursts (see e.g. [33]). This fireball 2 accelerates with Γ f ∝ r until it becomes optically thin, the radiation stored inside it escapes and the accelerated baryons continue their expansion at Γ f ∼ const. The shock front, according to our scheme, should compress such a region effectively at a distance from the central pulsar ∼ 10 15 cm. When this interaction starts, a forward and a reverse shock form at the contact discontinuity between the shell and the wind, whose properties in the "strong shock" limit depend from the Lorentz factor of the incoming shell in the wind comoving frame γ f and from the ratio between the particle number densities of the incoming shell and the wind n f /n w [34]. Although we are dealing with a magnetised wind, the shock conditions [34] are still valid by adding the contribution of the magnetic field to the pressure and energy density in the upstream region ( [34] and references therein; see [8] for a complete treatment of the jump conditions for the shock in a striped wind). We also require, since no other emission is observed in the Crab Nebula spectrum, that the emission from the shell producing the shock is negligible with respect to the synchrotron emission from the accelerated electrons in the current sheet: this situation is met if the reverse shock is newtonian (n f /n w > γ 2 f ). In this condition, the Lorentz factor of the shocked wind (relative to the unshocked wind) is ∼ γ f , that relates the (relative) Lorentz factor of the shell to the compression factor.
Several issues are against this formulation. First, Γ f is highly relativistic, at least one order of magnitude larger than the wind Lorentz factor at 10 15 cm, therefore the initial shell must be "radiation dominated" [33]. This implies that most of the fireball energy content is radiated as thermal emission when the fireball becomes transparent, at a temperature T obs = Γ f × 20 keV ∼ 100 MeV [33].
Although the estimate of the flux of this emission is not straightforward, it reasonably exceeds the Crab Nebula emission in that energy range and therefore it should have been detected by a gamma-ray instrument. A second issue is that in an almost pure radiation fireball the transparency is reached too early to accelerate the baryons to such a high Lorentz factor. An alternative possibility is that the fireball is highly magnetised, and that the acceleration is driven by the magnetic pressure instead of the radiation pressure. In this case, however, the baryon acceleration is even less efficient, scaling as Γ ∝ r 1/3 [35]. Finally, it is unclear how such an energy release in the surroundings of the central pulsar can be achieved, especially in absence of any variation in the central pulsar, as glitches [15].

A plasma instability?
The difficulties encountered in the scenario outlined in the previous section lead us to conclude that this hypothesis must be abandoned, at least in its simplest formulation. Thus we investigate alternative mechanisms to trigger the shock-induced magnetic reconnection.
This issue is probably addressable only in a scenario of a strong plasma instability of the pulsar wind, which is not a well established area, especially in view of the very short time scale of the phenomenon. However, we can at least provide very constraining properties of such an instability: i) the shock-induced magnetic reconnection requires necessarily a supersonic compression of the particles and of the magnetic field characterising the wind; ii) the wind is strongly dominated by the magnetic energy, which remarkably overcomes the kinetic contribution and, as a consequence, the Alfven velocity is supersonic in the medium, despite its ultrarelativistic nature. These two points suggest that the shock originates in a compressive instability of the magnetic field, having a magnetosonic nature and propagating with the order of the Alfven speed. These basic considerations lead us to identify the origin of the magnetic instability into an anisotropy in the e ± temperature.
A reliable candidate which satisfies these requirements is the so-called Weibel instability [36]. On a kinetic theory level, it can be recovered by linearly perturbing an anisotropic equilibrium distribution for the particle velocities. 3 For the existence of the instability (which does not rely on the existence of a background magnetic field) the wave vector of the perturbations can be taken in the same direction than the anisotropic velocity component. In order to get an unstable compressive mode, the electron shift must have a non-zero component along the wave vector, that couples the induction equation for the magnetic field growth to the thermodynamics of the plasma, accounting for the growth of the pressure and the energy density. Since the compression of the sheet is radial, we can infer a radial propagation of the instability, associated with a different radial temperature of the wind with respect to the orthogonal one.
If the pulsar wind is well represented by an equilibrium distribution function of the phase space having the form 4 : being (v p , u p ) and (v, u) the velocity of the particles and its variance on a given plane and in the orthogonal direction, respectively, and n the wind particle density, then when u p u the Weibel instability grows as ∼ exp{γ t}, with γ (k) = (ku p ω P )/ ω 2 P + c 2 k 2 (ω P = ne 2 /m e • is the plasma frequency). At the distance from the pulsar r ∼ 10 15 cm, the e ± density is n ∼ 10 −3 cm −3 [25], therefore ω P ∼ 10 3 s −1 . Since we are searching for an instability able to compress every corrugation of the sheet over a region λ s ∼ 2π r L ∼ 10 9 cm, the above dispersion relation should be evaluated in every single region containing a sheet: ω P ∼ c/λ d ck s ∼ c/λ s , 3 The variance of the velocity in a given direction is different from the one in the orthogonal plane. 4 This picture is valid also for any anysotropic background distribution.
where λ D ∼ c/ω P ∼ 10 7 cm is the Debye length of the plasma.
Hence, we get a growth rate of the instability on the sheet width scale of the order γ s ∼ k s u p ∼ k s c ∼ 10 s −1 . We conclude that the Weibel instability is a good candidate to account for the compressive process which triggers the magnetic reconnection inside the pulsar wind. Clearly the emission over a region of about 10 15 cm does not take place simultaneously, but with a phase difference fixed by the light speed. This can be explained by observing that the current sheet is continuously crossed by the particles associated to the background anisotropic momentum distribution and, thus, if in an inner region the condition u p u is met, it propagates to outer regions of the wind with a speed very close to the light one. It is not the temperature anisotropy as a whole, but just the trigger of the instability that propagates.
We have to face a fundamental question concerning the origin of the anisotropic temperature of the background plasma, present in the cold wind component. The anisotropy of a plasma thermal velocity at the equilibrium can be the result of two different and complementary scenarios: i) the plasma is a strongly collisional system and the cross-section responsible for each scattering is an intrinsic anisotropic process; ii) the plasma is a very weak collisional system and it is generated under anisotropic conditions, or its evolution is affected by a direction-dependent mechanism.
In the present study, the nature of the Crab cold wind is clearly non-collisional and, therefore, the Weibel instability may develop under the hypothesis ii). In fact, the mean collision time in the cold e ± plasma is τ c ∼ 10 6 s at the light cylinder and it increases outward like r 2 . The collisionless nature of the plasma comes out by observing that the pairs, after their generation, reach the region r ∼ 10 15 cm in a time smaller by a factor 10 than the collision time τ c . Furthermore, the plasma is generated as whole in a collisionless condition, since the rate of couple creation is such that each pair is created as having de facto a zero scattering crosssection with all the remaining ones. These general considerations allow us to claim the non-collisional nature of this plasma and, hence, to clarify that the development of a Weibel instability is, in principle, possible and, to some extent, even probable: there are no physical reasons for which a fully collisionless plasma, like the Crab cold wind, must have an isotropic equilibrium temperature. Its distribution is indeed sensitive to any anisotropic feature of the generation or propagation environment physics.
The Weibel instability may be quenched in the case of a flow aligned magnetic field [37]. However, this suppression cannot occur if the magnetic field is misaligned with the motion of the system [38]. This result has been proved for symmetric and asymmetric colliding shells [38,39]. This is indeed the case of this plasma configuration, ensuring the development of the inferred unstable mode. In fact, the symmetry of the system and the radial motion of the wind suggest that our compressive mode is mainly radial. Besides, the radial component of the local magnetic field at ∼ 10 15 cm is lower than the toroidal one. Thus, applying the result discussed by Bret [39] to a local non-symmetric profile, we can reliably argue that the Weibel instability is not suppressed by the background magnetic field.
It remains to be settled the nature of the background temperature in the Crab wind, being able to trigger the Weibel instability necessary to support the present scenario for the gamma flaring emission. The detailed description of the morphology of the Crab cold wind and its possible anisotropy profile are out of the scope of the present work since it would require a modelling for different physical aspects, from the nature of the pair formation process to the specific profile of all the most important physical quantities (density, magnetic field, etc) also with respect to transient phenomena. Nonetheless, we notice that the most natural source of anisotropy which can regard the considered system must be searched in the presence of a significant magnetic field within the plasma. Furthermore, since we are interested in a radially propagating instability, it is rather natural (though not mandatory by the problem symmetries) to guess that the highest temperature of the background be along the radial direction, i.e. the variance of the radial velocity of the couples overcomes the one in the perpendicular plane. Finally, we note how the searched background anisotropy is expected to be a transient phenomenon to be searched for in a time dependent process of the Neutron Star, able to pump anisotropically energy in the cold wind. The gamma flare is, in this paradigm, the release of the energy anisotropically stored in the plasma, via a Weibel instability which trigger a magnetic induced reconnection of the equatorial corrugated sheets.
In conclusion, the proposed explanation for the Crab flares has three main merits: i) there is an impressive coincidence between the spatial scale where the electrons can reach the Lorentz factor required to account for the observed flare spectrum in the shockinduced magnetic reconnection scenario and the observed duration of the flare; ii) we excluded a "fireball-like" scenario to trigger the magnetic reconnection; iii) we propose as the most natural candidate the Weibel instability, that is a general property of relativistic plasma and fulfils the spatial and temporal requirements.