Compton driven beam formation and magnetisation via plasma microinstabilities
aa r X i v : . [ phy s i c s . p l a s m - ph ] F e b Under consideration for publication in J. Plasma Phys. Compton driven beam formation andmagnetisation via plasma microinstabilities
Bertrand Martinez † , Thomas Grismayer and Luis O. Silva GoLP/Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade deLisboa, 1049-001 Lisbon, Portugal(Received xx; revised xx; accepted xx)
Compton scattering of gamma rays propagating in a pair plasma can drive the formationof a relativistic electron positron beam. This process is scrutinised theoretically andnumerically via particle-in-cell simulations. In addition, we determine in which conditionsthe beam can prompt a beam-plasma instability and convert its kinetic energy intomagnetic energy. We argue that such conditions can be met at the photosphere radius ofbright Gamma Ray Bursts.
1. Introduction
The interaction of gamma-rays with a pair plasma is a fundamental problem inastrophysics. For instance, it is present in the context of Gamma-ray Bursts (GRBs).The latter involve the explosion of a stellar mass object, which energy is expelled inthe form of a relativistic ejecta. As this ejecta propagates, its interaction with theambient medium generates a strong emission which is detected by satellites as well asground-based observatories. Recent efforts proved observations can cover a large rangeof frequencies including the radio, optical, x-ray and gamma-ray bands, up to TeVenergies (Acciari et al. ∼ cm . This ejecta is a relativistic plasma in expansion. Most of its energy comes fromphotons, but it also contains pairs, an unknown fraction of baryons and may even bemagnetised. At this early stage of expansion it is so dense that photons cannot escape(their mean free path is larger than unity). At the photospheric radius ∼ cm , itsdensity has decreased enough to enable a fraction of gamma-rays to propagate in theambient medium (Cavallo & Rees 1978). One of the pending questions related to GRBsis how does the ejecta dissipates its energy above such a large distance of ∼ cm ? † Email address for correspondence: [email protected]
B. Martinez, T. Grismayer and L. O. Silva
This question has been addressed in two ways by researchers, as detailed in this recentreview on GRBs (Kumar & Zhang 2015).Firstly, they brought forward models that describe the overall evolution of the ejecta.The most used is the hot fireball model which assumes the energy of the ejecta isdissipated at the photosphere radius and in internal/external shocks (Narayan et al. et al. et al. et al.
30 keV and and he observes electron acceleration in a plasma wakefield. According tothe author, this wakefield is excited by an electrostatic force acting to restore the chargeseparation induced by Compton deflections. Contrary to this work, our investigationfocuses on a regime where the Compton cross-section is beamed, above
10 MeV . Inaddition, we consider a range where Compton prevails over the Bethe-Heitler pairproduction ( γe → e + e − e ), for energies below
100 MeV . Formulas for all these threeprocesses can be found in Lightman (1982). For this gamma-ray range −
100 MeV , oneexpects Compton scattering to scatter electrons mainly forward such that they form arelativistic beam (Del Gaudio et al. a ).In light of these previous results, we investigate the interaction of gamma-rays (10-100MeV) with a background pair plasma. In section 2, with the support of a 1D theoreticalmodel and Particle-In-Cell (PIC) simulations, we evidence how a gamma-ray beam (10-100 MeV) propagating in a pair plasma can drive the formation of a relativistic anddense electron positron beam via Compton scattering. In section 3, we specify underwhich conditions the pair beam can trigger a beam-plasma instability, thus generating asmall scale magnetic field which extent is discussed. Section 4 confirms the robustness ofthis process for various photon sources. Finally, we argue in section 5 that these conditionscan be met at the photospheric radius of GRBs.
2. Pair beam formation
Let us consider a semi-infinite gamma-ray beam of density n ω with a monoenergeticdistribution. We denote by ǫ = ~ ω/mc the photon energy normalized by the electron rest ompton driven beam formation and magnetisation x direction in an infinite cold pair plasma atrest density n p associated to an angular frequency ω p . In the frame of this study, we focuson the energy range ≪ ǫ < /α f = 137 where Compton scattering prevails over thetwo photon Breit-Wheeler process, and pair creation in the Coulomb field of an electronor positron (Lightman 1982). The variable α f denotes the fine structure constant.The formation process of the pair beam relies on the beaming of Compton cross-section for high-energy photons. Let us consider an electron at rest experiencing Comptondeflection from a gamma-ray of energy ǫ . dσ kn /dΩ and σ kn ( ǫ ) are the angular-differentialand total Compton cross-sections (Klein & Nishina 1928), and θ is the polar angleassociated to Ω , the photon angle after one scattering. The Lorentz factor of the electronafter the deflection γ can be deduced from an energy and momentum balance as γ/ǫ =1 + 1 /ǫ − / [1 + ǫ (1 − cos θ )] . We introduce φ the angle between the deflected electronmomentum and the incident photon direction and obtain cotan( φ ) = (1 + ǫ ) tan( θ/ .Averaging those quantities over the Compton angular cross-section, we obtain for ǫ ≫ , h γ i Ω ≃ ǫ and h sin φ i Ω ≃ /ǫ . In the frame of our problem, this proves that there is asimultaneous beaming of photons, electrons and positrons centered on the direction ofthe incident gamma-ray.We assume the longitudinal momentum of the pair beam ( p x ), can be approximated byits average over the Compton cross-section p x ≃ p h p x i Ω , and we define the transversemomentum spread induced by Compton scattering as ∆p ⊥ = p h p ⊥ i Ω . For ǫ ≫ : p x mc ≃ ǫ and ∆p ⊥ mc ≃ r ǫ ǫ . (2.1)Equation (2.1) shows that the deflected electron energy can be as high as the photon oneand that the transverse momentum spread of the pair beam is typically a few percentsof its longitudinal momentum ∆p ⊥ /p x ≃ − , for ≪ ǫ < /α f .The density of photons decreases as they are scattered at a frequency τ − ω = 2 n p cσ kn .It thus follows the equation dn ω /dt = − n ω /τ ω . The solution reads n ω /n ω = exp( − t/τ ω ) .In the high energy limit ǫ ≫ where σ kn ( ǫ ) ≃ πr e ln( ǫ ) /ǫ , the photon density can beapproximated as constant n ω /n ω ≃ . Indeed ω p τ ω ∝ r − / e n − / p ǫ/ ln( ǫ ) & for anyplasma density n p cm − , a conservative upper bound for astrophysical systems.To derive the equation for the pair beam density n b at the photon front, we simplifythe modeling by focusing on a thin layer of length c/ω p , over which no collective plasmaprocesses are expected. In order to determine the density for all positions in the beam atany time t, a conservation argument can be used. The density increases at a frequency / (2 τ ω ) via Compton scattering, and is subject to dilution since scattered electrons movewith β x < and dephase with the photon front. Based on these two ideas, we obtainedthe following equation dn b dt = n ω τ ω − n b τ d . (2.2)where ω p τ d = 1 / (1 − β x ) is the typical dilution time, and β x ≃ p h β x i Ω . In the highenergy limit ǫ ≫ , we obtain ω p τ d ≃ ǫ ln ǫ and n b n p = √ π ln ( ǫ ) (cid:18) n ω r e n p (cid:19) / (cid:20) − exp (cid:18) − tτ d (cid:19)(cid:21) . (2.3)Equation (2.3) proves that the normalized pair beam density depends on n ω in unitsof p n p /r e . It also establishes that the front density of the pair beam is linearly loadedwith the propagation distance for early times t τ d . In this limit, dilution effects arenegligible and n b /n p ≃ σ kn ( ǫ ) n ω ct . For t ≫ τ d , the beam cannot exceed the maximum B. Martinez, T. Grismayer and L. O. Silva
Figure 1.
2D electron density profile at three instants (a1) ω p t = 0 (b1) ω p t = 90 (c1) ω p t = 130 . (a2-b2-c2) 1D projections along propagation direction x for the same instants(green thick curve) and theoretical pair beam density (black thin curve). The gamma-ray beampropagates from left to right and its front is marked by the dashed black line. (d1-d2) f ( p x ) and f ( p ⊥ ) distributions of electrons scattered by Compton at the beam front for time ω p t = 130 .For the plasma density n p = 1 cm − , one has c/ω p = 5 . × cm . density √ π ln ( ǫ ) n ω r / e /n / p . Since we expect collective processes for such time scales,we underline that this limit is an upper bound of the maximum achievable density.We stress that this model is valid whatever the photon and plasma densities. Therelevance for laboratory or astrophysical environments will be discussed in section 3.It remains applicable as long as collective plasma effects do not play a significant role.The growth rate of the beam-plasma instability will be inferred in section 2. In terms ofphoton energies, it is strictly limited to the range ≪ ǫ < /α f = 137 . However, we willdiscuss in the third section the reasons we believe it could be extended to a larger rangeof . ǫ . .We have reported how a pair beam can be created when gamma-rays propagatein a background pair plasma via Compton scattering. This process is corroboratedwith theory and is now confronted to simulation results. We have run 2D Particle-In-Cell (PIC) simulations with the code Osiris (Fonseca et al. et al. b ), similarly to earlierwork (Haugbølle et al. ) with a uniform density n p = 1 cm − , as displayedin Figs. 1(a1-a2). In this pair plasma, we propagate a monoenergetic ( ǫ = 100 ) gamma-ray beam along the x direction from left to right. It presents a uniform density n ω =10 cm − with a front indicated by the black dashed line in Figs. 1(b2-c2). Periodicboundary conditions are assumed in the transverse direction y for all species and fields.The domain extends on (160 c/ω p ) with cells. The cell dimensions are δx = δy =0 . c/ω p and the time step is δt = δx/ . We initialize / / particles per cell forelectrons, positrons and photons. ompton driven beam formation and magnetisation Figure 2.
Pair beam density n b /n p versus the photon density n ω in unit of p n p /r e for plasmadensities n p = 1 cm − (X) and n p = 10 cm − (O) at two times in the simulations. ω p τ d ≃ is the beam dilution time scale. The process of pair beam formation is illustrated in Fig. 1. Figs. 1(a1-b1-c1) exhibit2D profiles of the electron density at three instants ω p t = 0 , and . Figs. 1(a2-b2-c2) are projections of theses densities along x the propagation direction. Based on thesefigures, we can witness the formation and propagation of an electron beam (positrons aresuperimposed by symmetry). We found excellent agreement of the simulations with thetheoretical estimates (black lines) of Eq. (2.3). The beam has a triangular shape sinceit is loaded linearly with the propagation distance. The momenta distributions f ( p x ) and f ( p ⊥ ) of deflected electrons are exemplified in Figs. 1(d1-d2) at time ω p t = 130 .The distribution f ( p x ) exposes a peak at p x /mc ≃
100 = ǫ , exactly as our theoreticalestimate in Eq. (2.1). The transverse momentum profile f ( p ⊥ ) includes both directions p y and p z . The two distributions are centered and characterized by a ≃ mc standarddeviation, in agreement with Eq. (2.1). This is conducted in the range of validity ofthe theoretical model with periodic conditions in the transverse direction, and does notevidence collective plasma processes with a significant impact on the beam formation.Another set of 2D simulations was run to assert the relevance of this process of pairbeam formation. Our goal is to check the range of validity of the scaling inferred for itspeak density in Eq. (2.3). We consider two sets of simulations. The first set is characterisedby a pair plasma of density n p = 10 cm − , with photon densities n ω = 10 - cm − .The second presents a pair plasma density of n p = 1 cm − , with photon densities n ω = 10 - cm − . Connections with laboratory or astrophysical parameters willbe discussed in section 5. Given the symmetry of our problem, we reduce the domain sizeto (24 c/ω p ) and follow the photon beam in a moving window. The domain has cells with dimensions δx = δy = 0 . c/ω p and a time step δt = δx/ .Figure 2 represents the pair beam density as a function of the incident photon densityfor two instants, t/τ d = 0 . and . . The normalization by the factor ( n p /r e ) / comesfrom Eq. (2.3). It enables to reveal a general scaling of the normalized pair beam density n b /n p , whatever the initial plasma density n p . We reported the two sets of simulations,with pair plasma densities n p = 10 cm − (circles) and n p = 1 cm − (crosses). Despitesuch a wide gap of initial conditions, simulations evidence that the normalized pair beamdensity at a given time t/τ d remains exactly the same as long as n ω ( r e /n p ) / is constant. B. Martinez, T. Grismayer and L. O. Silva
This scaling matches exactly with the prediction of Eq. (2.3), illustrated as a black linein Fig. 2.We have generalised the validity of the results obtained in Fig. 2, to account for baryonloading. Firstly, we ran all the simulations presented before with a fraction of protondensity up to . n p in the pair plasma. We noted that this does not change significantlythe pair beam density over the whole range of photon and plasma densities we consider.This verification was done as astrophysical pair plasma are usually expected to contain afraction of protons. Secondly, we ran one 3D simulation, confirming our previous findings.We performed it for a plasma density of n p = 1 cm − and photon density of n ω =10 cm − . The domain size is (24 c/ω p ) and the photon beam is followed in a movingwindow. The domain has cells of dimensions δx = δy = δz = 0 . c/ω p and thetime step is δt = δx/ . We initialize / / particles per cell for electrons, positrons andphotons. The transverse boundary conditions for particles and fields are all periodic. Forthis 3D simulation, the evolution of the peak beam density is the same as for the 2Dsimulation with the same parameters. While we will discuss this in detail in section 3,we add that the growth rate of the beam plasma instability in this 3D simulation is alsothe same as in the corresponding 2D simulation.
3. Onset of a beam-plasma instability
The normalized pair beam density n b /n p increases with time and is expected totrigger beam-plasma instabilities. Two of them can compete and lead to an exponentialgrowth of magnetic fields (Bret et al. ω − p Γ CFI = ( n b /n p γ b ) / . Secondly, the OBlique filamentationInstability (OBI), with growth rate ω − p Γ OBI = √ n b /n p γ b ) / / / . We can predictthat our prevailing modes will be oblique since we consider Lorentz factors ≪ γ b . /α f and densities n b /n p . Above some critical angle in k space, such modes are expected tobe damped like filamentation instability modes due to thermal effects (Bret et al. et al. α th = n b /n p ≃ γ b ( p ⊥ /mcγ b ) . Using the estimates from Eq. (2.1),we get n b /n p ≃ / (6 ln ǫ ) and ω − p Γ ≃ (7 √ / / [ ǫ log( ǫ )] − / . (3.1)We have neglected dilution effects, assuming that the time required to reach the densitythreshold α th remains much lower than the typical dilution time scale τ d . This conditioncan be recast as n ω ( r e /n p ) / ln ǫ ≫ . (3.2)It is worthwhile to stress that provided this condition (3.2) is met, the pair beam willalways be magnetized.In addition, it seems meaningful to evaluate the energy conversion efficiency fromthe incident photons to magnetic fields. We define η = E B z / E ω , the ratio betweenthe magnetic energy E B z ≃ R c B z / π d x and the incident photon energy. For a semi-infinite beam (and 1D), the ratio η is also the ratio of the energy densities. For thesake of simplicity, we estimate an upper bound for the magnetic field energy E B z . R c | B z | / π d x . Using Parseval theorem, it can be expressed as E B z . R c | B z | / π d k .The value of the saturated magnetic field can be approximated by equating the bouncefrequency of trapped particles to the growth rate of the instability (Yang et al. ompton driven beam formation and magnetisation eB z ( k ) /mω p ≃ ( γ b /p x )( ω − p Γ ) ω p /kc . The energy conversion efficiencyis then deduced by introducing the incident photon energy density ǫn ω η . π √ ! / n p n ω (cid:0) ǫ ln ( ǫ ) (cid:1) / . (3.3)It is important to underline this inequality only expresses an upper bound for theconversion efficiency. From Eq. (3.3), η can be maximized by rising the ratio n p /n ω .However this factor cannot be higher than a certain value due to our assumptions. Thederivation of η relies on a fast growth of the pair beam density, before dilution effects takeplace. As previously stated, this implies n ω ln ( ǫ ) p r e /n p & which can be rephrasedas n ω [cm − ] & × n / p [cm − ] . (3.4)This upper limit for n p /n ω therefore bounds the conversion efficiency to η ≪ − n / p [cm − ] . This restriction seems unreasonably low but it implies that thetotal photon energy is barely affected by this loss. As a consequence, the generationof magnetic fields can in principle take place over long distances. For a perfectlycollimated photon source emerging from the photosphere ( ≃ cm ), this length canbe as high as the Compton mean free path L ω [cm] = cτ ω ≃ n − p [cm − ] . In fact,the typical opening angle of the GRB ejecta is in the range θ ◦ , as estimatedin Racusin et al. (2009). With this more realistic assumption, our estimates are onlyvalid on distances over which the photon density satisfies the condition (3.4). Given agamma-ray density between cm − and cm − that we inferred at the exit ofthe photosphere of GRBs (see section 5) and assuming this photon density decreases as /r , the condition (3.4) is fulfilled for distances ranging from a few photospheric radii ∼ cm up to larger values of × cm . We did not verified this estimate due tothe computational cost of simulations with L [cm] > n − / p [cm − ] .
4. Generalisation for various photon distributions
We have considered idealized photon distributions. We now examine more realisticones. Particle-In-Cell simulations previously presented confirmed that the pair beamcan set off a beam plasma instability. We investigate a simulation with plasma density n p = 10 cm − and photon density n ω = 10 cm − . Figure 3 reports the energy of the B z field, denoted by E B z and normalized by the incident photon energy E ω (blue solidcurve).We first focus on the monoenergetic photon distribution (blue solid curve). Thegrowth rate for all modes k is ω − p Γ = 7 . × − . The B z field profile is displayedduring the linear phase of the instability (see the inset) and shows the dominant modeis oblique: k = ( k x , k y ) ≃ (1 . , . ω p /c . Its growth rate is ω − p Γ = 0 . close to ω − p Γ ≃ . , the theoretical prediction from Eq. (3.1). The criteria given is actuallyfulfilled: n ω ( r e /n p ) / ln ( ǫ ) ≃ ≫ . The theoretical upper bound of the energyconversion efficiency given by Eq. (3.3) is η . . × − , which is the order of thesimulation result η = 5 × − . We also checked that the pair beam density rises slowly( +30% ) on the instability time scale. This legitimates a posteriori the use of estimatesfrom linear theory.The aforementioned discussions are focused on a monoenergetic gamma-ray beam. Weemploy the δ notation for a Dirac delta function, and write it as f = f x = δ ( p x − mc ) .We intend to extend these results for more complex distributions. Firstly, we consider B. Martinez, T. Grismayer and L. O. Silva
Figure 3.
Magnetic field energy E B z , normalized by the incident photon energy E ω .Monoenergetic photon distribution (blue solid curve), Maxwellian transverse profile (orangedashed curve), front density gradient (red dotted curve), and synchrotron distribution (greendot-dashed curve). The growth rate of the field is determined from the slope of the black dashedcurves. Inset: typical B z field profile during the linear phase. a photon distribution f = f x f y with f x = δ ( p x − mc ) and f y a Maxwellianof temperature mc . Since the Compton cross-section is beamed for gamma-rays,the pair beam presents a similar transverse momentum spread as the photon beam,which is known to lower the growth rate of the OBI (Bret et al. ω − p Γ = 7 . × − → × − , see theorange dashed curve. Secondly, photons have a monoenergetic distribution f = f x with f x = δ ( p x − mc ) but their density profile has an exponential shape of scalelength
100 c /ω p , followed by a flat profile of density n ω . Since the pair beam densityis proportional to the photon density, its profile first presents a density gradient andthen the triangular shape observed in Fig. 1(b2). As a result, the growth of the OBIis delayed in time, slightly lower but noticeable ( ω − p Γ ≃ . × − ). This time delayis comparable to the gradient length, as can be seen on the red dotted curve in Fig. 3.Thirdly, we model a synchrotron distribution for the incident photon beam. We setup this energy distribution as a longitudinal momentum distribution for the photons f = f x = F ( η, χ ) (Klepikov 1954). It has a peak at an energy of ~ ω/mc = 100 . Photonswith an energy of -
100 MeV are Compton scattered and contribute to increase thepair beam density compared to the monoenergetic case. Indeed, the Klein-Nishina cross-section is a decreasing function of photon energy. As a result, the growth of the B z fieldenergy is faster: ω − p Γ = 7 . × − → . × − , as seen on the green dot-dashed curvein Fig. 3.
5. Relevance for astrophysics and laboratory environments
In previous sections, we have identified a fundamental plasma physics process. We haveexplained how gamma-rays propagating in a pair plasma can induce the formation of arelativistic pair beam via Compton scattering. The beam density increases with time butit has some transverse momentum spread. We stressed that due to this, a beam-plasmainstability can only be triggered if the photon density is high enough n ω [cm − ] & ompton driven beam formation and magnetisation Figure 4.
Relevance of the pair beam formation and magnetisation in the frame of thelaboratory and GRB-CBM interaction at the photospheric radius ∼ cm . Above the straightline (area I): one expects only pair beam formation. Below the line (area II), one also awaitsthe beam-plasma instability. The full dot represents an experimental gamma-ray source by NonLinear Inverse Compton (NLIC) (Poder et al. et al. et al. × n / p [cm − ] , see Eq.(3.4). We now discuss whether this condition can be fulfilledat the photospheric radius of a GRB.Figure 4 covers a large range of pair plasma density n p and photon densities n ω .It reports whether the condition n ω [cm − ] & × n / p [cm − ] is fulfilled or not.The latter limit is plotted by the oblique line in Fig. 4 and enables to distinguish tworegimes of interaction. The first one is represented by all the area above the oblique lineand is denoted I. In this case, the photons are expected to form a relativistic electronpositron beam via Compton scattering. However, the pair beam density is too low and itstransverse momentum spread is too high to enable the onset of the instability as detailedin Sec. 3. The second interaction regime is depicted by the area below the oblique lineand is denoted II. In this regime, the pair beam density grows high enough to allow theinstability to develop.For the photospheric radius of a GRB, we evaluated a range of gamma-ray densities andbackground pair plasma densities. Although the pair plasma density at the photosphereradius of a GRB cannot be directly measured, its order of magnitude can be safelyassumed to be in the range . −
100 cm − (Panaitescu & Kumar 2003). The gamma-ray density (10-100 MeV) is determined to be n ω ∈ (10 , ) cm − with estimatesfrom the fireball model. From these orders of magnitudes, we observe in Fig. 4 that theestimated conditions at the photospheric radius can fulfill the condition in Eq. (3.4). Inparticular, one can therefore conclude that the instability can be expected for the GRBswhenever the photon density at the photosphere is above cm − , and whenever thebackground plasma density lies in the range . −
100 cm − . This corresponds to GRBswith an isotropic equivalent luminosity above × erg / s .To illustrate this limit, we detail the estimates to infer such gamma-ray density at thephotosphere radius, following Kumar & Zhang (2015). Let us consider a compact objectwith a radius ∼ cm and an isotropic equivalent luminosity L in erg / s . For simplicitywe provide the estimate in the compact object frame, thus neglecting the red shift of its0 B. Martinez, T. Grismayer and L. O. Silva host galaxy. The fireball emerges from the compact object and experiences an adiabaticexpansion. Its temperature at the photospheric radius ∼ cm can be inferred to be [1 . ×L / . The notation L is defined as L = L / (10 erg / s) . We assume typicalGRBs have an isotropic equivalent luminosity in the range L ∈ (10 − ) erg / s , inline with data recently gathered (Abbott et al.
70 keV up to . However,the isotropic equivalent luminosity is defined for photons in the standard energy band up to
10 MeV . We then deduce the fraction of photons in the energy range -
100 MeV , where our results are valid, by assuming a black body energy distribution.To be more precise, this ratio of photons (10-100 MeV) becomes & if the GRBluminosity is & × erg / s . The final step is to deduce a gamma-ray density fromthis total luminosity in gamma-rays. To this purpose, we assumed the GRB ejecta isa spherically expanding shell and its thickness is the GRB duration, typically rangingbetween . to
100 s .It is worth to question whether this beam formation and magnetisation can take placein the laboratory. Despite the ongoing worldwide efforts, no large size and confinedpair plasmas have been formed in a laboratory environment yet. One can still mentionthat pair jets of high density n p ∼ cm − can be generated by fast electronsgoing through a mm-sized high-Z targets. The fast electrons can be generated by anintense laser (Chen et al. et al. et al. et al. n ω ∼ cm − of the experimental photon source recentlyobtained by Non Linear Inverse Compton scattering (Poder et al. et al. et al. n ω ∼ cm − with a distribution peaked at 100 MeV. One can see inFig. 4 that even this high gamma-ray density is not enough to witness the onset of theinstability in a laboratory frame. Our results indicate that exploration of this scenario inthe laboratory is unlikely in the short term due to the combined requirements of plasmaand gamma-ray densities.
6. Conclusions
To summarize, we reported a study of gamma-ray propagation in a background pairplasma. We showed that it can lead to the formation of a relativistic pair beam, thanksto the beaming of the Compton cross-section for photons above 10 MeV. We developedtheoretical arguments that have been explored with PIC simulations. We showed thepair beam can achieve a relativistic Lorentz factor with a density comparable to thebackground plasma. In addition, we quantified the transverse momentum spread ofthe beam, which is induced by the Compton cross-section. We demonstrated that thepair beam, as it propagates, can convert its kinetic energy to magnetic energy via theoblique instability, although limited by its transverse momentum spread. The conversionefficiency of this process is low. We extrapolated from PIC simulation results that itcould lead to the generation of magnetic fields on distances larger than a parsec (up to ≃ cm ) for a perfectly collimated GRB ejecta, and on lower distances, from cm to × cm for a less collimated one. We showed that these results can be used toaddress the energy dissipation of gamma-rays at the photospheric radius of GRBs. We ompton driven beam formation and magnetisation et al. a ). The second may be tounderstand the impact of this beam formation and magnetization on global GRB modelssuch as the hot fireball and others. Acknowledgments
The authors acknowledge fruitful discussions with Dr. Del Gaudio and Dr. Schoef-fler. This work was supported by the European Research Council (ERC-2015-AdGGrant 695088), FCT (Portugal) grants PD/BD/114323/2016 in the framework of theAdvanced Program in Plasma Science and Engineering (APPLAuSE, FCT grant No.PD/00505/2012). We acknowledge PRACE for awarding access to resource MareNostrumbased in Spain. Simulations were performed at MareNostrum (Spain).
REFERENCESAbbott, B. P., Abbott, R., Abbott, T. D., Acernese, F., Ackley, K., Adams, C.,Adams, T., Addesso, P., Adhikari, R. X. & V. B. Adya, et al.
The Astrophysical Journal (2), L13.
Acciari, V. A., Ansoldi, S., Antonelli, L. A., Engels, A. Arbet, Baack, D.,Babić, A., Banerjee, B., Barres de Almeida, U., Barrio, J. A. & González,J. Becerra, et al.
Nature (7783), 459–463.
Bret, A., Firpo, M.-C. & Deutsch, C.
Physical Review Letters , 115002. Bret, A., Gremillet, L. & Bénisti, D.
Physical Review E , 036402. Cavallo, G. & Rees, M. J.
Monthly Notices of the Royal Astronomical Society (3), 359–365.
Chen, H., Link, A., Sentoku, Y., Audebert, P., Fiuza, F., Hazi, A., Heeter, R. F.,Hill, M., Hobbs, L. & A. J. Kemp, et al.
Physics of Plasmas (5), 056705. Cole, J. M., Behm, K. T., Gerstmayr, E., Blackburn, T. G., Wood, J. C., Baird,C. D., Duff, M. J., Harvey, C., Ilderton, A. & Joglekar, et al.
Physical Review X , 011020. Del Gaudio, F., Fonseca, R. A., Silva, L. O. & Grismayer, T. a Plasma wakesdriven by photon bursts via compton scattering.
Physical Review Letters , 265001.
Del Gaudio, F., Grismayer, T., Fonseca, R. A. & Silva, L. O. b Compton scatteringin particle-in-cell codes.
Journal of Plasma Physics (5), 905860516. Fonseca, R. A., Silva, L. O., Tsung, F. S., Decyk, V. K., Lu, W., Ren, C.,Mori, W. B., Deng, S., Lee, S. & Katsouleas, T., et al. ‚ Computational Science — ICCS 2002 (ed. Peter M. A. Sloot, Alfons G. Hoekstra,C. J. Kenneth Tan & Jack J. Dongarra), pp. 342–351. Berlin, Heidelberg: Springer BerlinHeidelberg. B. Martinez, T. Grismayer and L. O. Silva
Frederiksen, J.T.
TheAstrophysical Journal (1), L5–L8.
Gruzinov, A. & Mészáros, P.
The Astrophysical Journal (1), L21–L24.
Haugbølle, T., Frederiksen, J.T. & Åke, N.
Physics of Plasmas (6), 062904. Klein, O. & Nishina, Y.
Nature (3072), 398–399.
Klepikov, N P
Journalof Experimental and Theoretical Physics , 19. Kumar, P. & Zhang, B.
PhysicsReports , 1 – 109.
Levinson, A. & Cerutti, B.
Astronomy & Astrophysics , A184.
Liang, E., Clarke, T., Henderson, A., Fu, W., Lo, W., Taylor, D., Chaguine, P.,Zhou, S., Hua, Y. & Cen, X., et al. W.cm − Laser Irradiating Solid Targets.
Scientific Reports , 13968. Lightman, A. P.
TheAstrophysical Journal , 842–858.
Lyutikov, M
New Journal of Physics (7), 119–119. Martins, S. F., Fonseca, R. A., Silva, L. O. & Mori, W. B.
The Astrophysical Journal (2), L189–L193.
Medvedev, M.V. & Loeb, A.
The Astrophysical Journal (2), 697–706.
Mehlhaff, J M, Werner, G R, Uzdensky, D A & Begelman, M C
Monthly Notices of the Royal Astronomical Society (1), 799–820.
Mészáros, P. & Rees, M. J.
The Astrophysical Journal , 278.
Mészáros, P. & Rees, M. J.
The Astrophysical Journal (1), 232–237.
Narayan, R., Paczynski, B. & Piran, T.
The Astrophysical Journal , L83.
Panaitescu, A. & Kumar, P.
AIP Conference Proceedings (1), 305–312.
Pelletier, G., Gremillet, L., Vanthieghem, A. & Lemoine, M.
Physical Review E ,013205.
Piran, T.
Review of Modern Physics , 1143–1210. Poder, K., Tamburini, M., Sarri, G., Di Piazza, A., Kuschel, S., Baird, C. D., Behm,K., Bohlen, S., Cole, J. M. & Corvan, D. J., et al.
PhysicalReview X , 031004. Racusin, J. L., Liang, E. W., Burrows, D. N., Falcone, A., Sakamoto, T., Zhang,B. B., Zhang, B., Evans, P. & Osborne, J.
The Astrophysical Journal (1), 43–74.
Rees, M. J. & Mészáros, P.
The Astrophysical Journal , L93.
Sampath, A., Davoine, X., Corde, S., Gremillet, L., Gilljohann, M., Sangal, M.,Keitel, C. H., Ariniello, R., Cary, J. & Ekerfelt, H., et al.
Physical Review Letters , 064801.
Sarri, G., Poder, K., Cole, J., Schumaker, W., Di Piazza, A., Reville, B., Doria,D., Dromey, B., Gizzi, L., Green, A., Grittani, G., Kar, S., Keitel, C. H.,Krushelnick, K., Kushel, S., Mangles, S., Najmudin, Z., Thomas, A. G. R., ompton driven beam formation and magnetisation Vargas, M. & Zepf, M.
Nature Communications . Silva, L. O., Fonseca, R. A., Tonge, J. W., Mori, W. B. & Dawson, J. M.
Physics ofPlasmas (6), 2458–2461. Spitkovsky, A.
The Astrophysical Journal (1), L5–L8.
Uzdensky, D. A.
SpaceScience Reviews (1), 45–71.
Xu, T., Shen, B., Xu, J., Li, S., Yu, Y., Li, J., Lu, X., Wang, C., Wang, X. & Liang,X., et al.
Physics of Plasmas (3), 033109. Yang, T.Y.B., Arons, J. & Langdon, A. B.
Physics of Plasmas (9), 3059–3077. Zhang, B. & Yan, H.
The Astrophysical Journal726