Particle acceleration and radiation reaction in strong spherical electromagnetic waves
MMNRAS , 1–15 (2021) Preprint 25 février 2021 Compiled using MNRAS L A TEX style file v3.0
Particle acceleration and radiation reaction in strong sphericalelectromagnetic waves
J. Pétri ? Université de Strasbourg, CNRS, Observatoire astronomique de Strasbourg, UMR 7550, F-67000 Strasbourg, France.
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Strongly magnetized and fast rotating neutron stars are known to be efficient particle accelerators within theirmagnetosphere and wind. They are suspected to accelerate leptons, protons and maybe ions to extreme relativisticregimes where the radiation reaction significantly feeds back to their motion. In the vicinity of neutron stars, magneticfield strengths are close to the critical value of B c ∼ , T and particle Lorentz factors of the order γ ∼ areexpected. In this paper, we investigate the acceleration and radiation reaction feedback in the pulsar wind zone wherea large amplitude low frequency electromagnetic wave is launched starting from the light-cylinder. We design a semi-analytical code solving exactly the particle equation of motion including radiation reaction in the Landau-Lifshitsapproximation for a null-like electromagnetic wave of arbitrary strength parameter and elliptical polarization. Underconventional pulsar conditions, asymptotic Lorentz factor as high as 10 − are reached at large distances from theneutron star. However, we demonstrate that in the wind zone, within the spherical wave approximation, radiationreaction feedback remains negligible. Key words: magnetic fields - methods : analytical - stars : neutron - stars : rotation - pulsars : general
Strong magnetic fields dragged by fast rotation induce hugeelectric fields able to accelerate charged particles to ultra-relativistic speeds. Such conditions are met around stronglymagnetized and fast spinning neutron stars known as pul-sars and magnetars. These compacts astrophysical objectsare indeed suspected to fill the interstellar and intergalacticmedium with the most energetic particles in the universe andmaybe also to produce part of the ultra high energy cosmicrays. These ideas where for instance explored by Gunn & Os-triker (1969) by using a vacuum wave and then improved byKegel (1971) assuming a refractive index different from va-cuum. These ideas were also revisited by Thielheim (1990).It is still unclear where and how efficient such accelerationmechanisms are around neutron stars. However, three mainregions have been identified : the inner magnetosphere, thatis the corotating quasi-static zone (Goldreich & Julian 1969),the wind zone (Coroniti 1990; Michel 1994) where a low fre-quency large amplitude electromagnetic wave is launched andthe termination shock of the pulsar wind (Pétri & Lyubarsky2007). Alternatively, magnetized relativistic outflows can alsoproduce high energy particles via the Fermi process, diffusiveshock acceleration, shock drift acceleration or magnetic re-connection, see for instance the review by Matthews et al.(2020).In this work, we focus on particle acceleration by largeamplitude electromagnetic waves. Relativistic acceleration of ? . E-mail: [email protected] charged particles with mass m and charge q in a plane electro-magnetic wave reveals efficient when the strength parameterdefined by a = ω B /ω becomes much larger than one. Here ω B is the particle cyclotron frequency and ω the wave frequency.The strength parameter a gives a first guess for the energygained by a particle starting from rest when accelerating inthe electromagnetic field during one period of the wave. Toorders of magnitude, the particle momentum divided by itsmass is γ β ≈ a where β is the normalized velocity with res-pect to the speed of light and γ the associated Lorentz factor.As typical values for this strength parameter a , we remem-ber that for visible light, taking a wavelength of λ = 1 µ mand a flux of 1 W / m corresponding to a magnetic field of2 · 10 − T, it amounts to a ≈ − (cid:28) . (1)Such optical waves are therefore unable to accelerates par-ticles to even mildly relativistic speeds. For current statetechnology with laser power of 10 W / m , it becomes si-gnificantly larger than one and up to values about a ≈ . (2)Mildly relativistic regimes are reachable by current state-of-the-art technology. It is even expected to be soon possibleto study radiation reaction effects during electron accelera-tion phases and to test the Lorentz-Abraham-Dirac (LAD)prescription for the charged particle equation of motion sub-ject to radiation reaction. The correction term brought to theLorentz force introduced by Abraham (1902, 1904) and rein-vestigated by Lorentz (1916) was eventually formulated in © 2021 The Authors a r X i v : . [ a s t r o - ph . H E ] F e b J. Pétri the relativistic regime by Dirac (1938). This so-called LADequation is still awaiting for experimental support and ve-rification. It is known to be subject to run-away solutionsthat must be discarded. There exist an extensive literatureon this topic, see for instance Rohrlich (2007) for a summaryor also alternative radiation reaction contributions like theone deduced by Eliezer (1948). In the astrophysical contextof strongly magnetised rotating neutron stars, for instancefor the archetypal Crab pulsar the strength parameter canreach extremely large values as high as a = 10 (cid:29) r L = c/ω ), about 10 but still extremely high. Pulsarsare therefore excellent candidates to push particles to ultra-relativistic energies by producing an electromagnetic kick ona very short time scale. The strength parameter at the light-cylinder, where the wave emerges has decreased by severalorders of magnitude but remains significantly larger than onedepending on the period P and its derivative ˙ Pa L ≈ q Bm ω (cid:16) Rr L (cid:17) ≈ , (cid:18) ˙ P − (cid:19) / (cid:16) P (cid:17) − / (4)thus still very large, a L (cid:29)
1. In the present investigation, westudy particle motion starting from this remote region up tovery large distances, r (cid:29) r L where a plane wave is a verygood approximation. Quantitative accurate results will bederived thanks to exact analytical solutions of the particle4-velocity in a plane electromagnetic wave in the so-calledLandau-Lifshits approximation derived by Piazza (2008) andretrieved by Hadad et al. (2010) in a different form. Howe-ver, the Landau-Lifshits approximation being a first orderexpansion of the Lorentz-Abraham-Dirac equation, it is notthe only possible choice, see for instance the review by Ham-mond (2010) and also the exposition of alternative theoriesby Burton & Noble (2014).Gunn & Ostriker (1969) were among the first to reco-gnize the potential of pulsars to produce high energy cosmicrays. Following simple arguments, they found an estimate forthe maximum energy that was refined by Ostriker & Gunn(1969). Laue & Thielheim (1986) studied the accelerationof protons and electrons for an perpendicular rotator in theLandau-Lifshits approximation, showing their orbit and Lo-rentz factor evolution with time. They also give detailed mapsof maximum energy depending on the initial longitude of theparticle. See also Leinemann (1988) for similar ideas. Kegelet al. (1995) studied acceleration and radiation of chargedparticles in strong electromagnetic waves using exact ana-lytical solutions for linearly and circularly polarized waves.They also looked for cold plasma effects. Strongly magneti-zed rotating neutron stars are believed to be efficient cosmicray accelerators (Thielheim 1991) but a clear picture of whatkind of particles and to which energies they can be accelera-ted is still lacking. Thielheim (1993) performed a careful ana-lysis of particle acceleration in a spherical wave field producedby a rotating dipole. This work was continued by Thielheim(1994) who also computed some plasma configurations. In thesame vain, Michel & Li (1999) studied particle motion in aplane wave and in the Deutsch field. Tolan (1992) presentedapproximated analytical solutions for a test particle evolvingin a rotating magnetic dipole and around a pulsar. He showed that radiation reaction and gravitation are negligible compa-red to the geometrical effect of a decaying spherical wave inthe pulsar wave zone.The propagation of strong electromagnetic waves in denseplasmas, being uniform or showing gradients, was performedback to the 70s by Max & Perkins (1971) where conditionsfor the transmission of a plane wave are given. The radiationdamping of a strong linearly polarized wave launched by apulsar and due to electron-positron pair synchro-Comptonradiation was explored by Asseo et al. (1978). They showedthat for conditions prevailing in the Crab pulsar, the wavefades away within only several wavelengths.All these works focused on the large scale acceleration. Onthe opposite side, Ferrari & Trussoni (1974) computed par-ticle acceleration and radiation in the near field region, veryclose to the neutron star surface, showing significant radiationreaction braking in this near zone. However, as the strengthparameter a decreases due to the dipolar nature of the ma-gnetic field, acceleration and radiation reaction become lessand less effective.Earlier works already worried about the effect of radiationreaction on particle acceleration. For instance Heintzmann &Grewing (1972) studied particle acceleration and radiationreaction in plane and spherical waves, see also Grewing et al.(1975) for radiation effects in pulsar fields. Grewing et al.(1973) then showed that the presence of a longitudinal ma-gnetic field significantly reduces the maximum Lorentz factorof the accelerated particles. Surprisingly, radiation reactionis able to increase the asymptotic Lorentz factor of the char-ged particle when interacting for instance with an intenselaser pulse as shown by Fradkin (1979). Synchrotron radia-tion spectra are also modified because of the decaying orbitof electrons in an uniform magnetic field as shown by Nelson& Wasserman (1991).Radiation reaction is usually treated as a perturbation ofthe Lorentz force and called the Landau-Lifshits approxi-mation. Therefore Finkbeiner et al. (1990) checked the va-lidity of this approximation in pulsar vacuum fields whichrequires a classical description of the emitting particles, theradiation field and the smallness of the radiation reactionforce compared to the Lorentz force in the particle instanta-neous rest frame. This justifies the approach of integratingthe particle equation of motion in the Landau-Lifshits limitfor highly relativistic particles as performed by Finkbeineret al. (1989) starting from the neutron star surface. In strongelectromagnetic fields, the quantum nature of the particlesalso emerges, leading to an additional equation for the evo-lution of the particle spin as implemented numerically by Liet al. (2020).Finding accurate and exact analytical solutions to the par-ticle equation of motion is crucial in ultra-strong electroma-gnetic fields as shown by Pétri (2020b) in the Lorentz forcelimit. Some applications to neutron stars have been exploredby Tomczak & Pétri (2020). Different approaches exist totackle the problem of finding exact and efficient implementa-tions of the Lorentz force equation, see for instance Gordonet al. (2017) and Gordon & Hafizi (2021) who also discussthe possibility to add radiation reaction.In this paper we study particle acceleration and radiationreaction in the wind zone, approximating the field locally bya plane wave with decreasing amplitude in order to mimic aspherical wave. Our integration of the test particle equation MNRAS , 1–15 (2021) article acceleration and radiation reaction of motion relies on exact analytical solutions of the Landau-Lifshits equation for either time-dependent elliptically pola-rized plane waves or constant null like electromagnetic fields.These solutions are recalled in Sec. 2 and serve as a buildingblock for our algorithm. As a first step towards a more generalalgorithm able to integrate semi-analytically any field confi-guration, we also try an algorithm based on locally constantelectromagnetic field solutions. Both numerical schemes arethen tested in plane polarized waves for the Lorentz forcein Sec. 3 and with radiation reaction in the Landau-Lifshitslimit in Sec. 4, showing the good agreement of the locallyconstant approximation with the analytical solution. We thendiscuss our new results about acceleration efficiency and fi-nal Lorentz factor of particles in a spherical waves in Sec. 5including radiation reaction. The limitation of our presentstudy focusing on null-like electromagnetic fields is discussedin Sec. 6. Conclusions are drawn in Sec. 7. Our aim is to solve the particle acceleration and radiationdamping problem by time dependent numerical simulations,sticking as close as possible to known exact analytical solu-tions. We start from the linearised Lorentz-Abraham-Diracequation leading to first order to the Landau-Lifshits pres-cription (Landau & Lifchitz 1989) such that du i dτ = qm F ik u k + q τ m ∂ ‘ F ik u k u ‘ + q τ m (cid:18) F ik F k‘ u ‘ + ( F ‘m u m ) ( F ‘k u k ) u i c (cid:19) . (5) q and m are the particle charge and rest mass, u i its 4-velocity, τ its proper time, F ik the electromagnetic or Fa-raday tensor, c the speed of light and τ the light crossingtime across the electron classical radius r e (within a factorunity) τ = q π ε m c = 23 r e c ≈ ,
26 · 10 − s . (6)Fortunately, there exist some exact analytical solutions tothis equation (5) in either a constant electromagnetic field orfor an elliptically polarized plane wave depending on the twoelectromagnetic invariants I = E /c − B (7a) I = E · B /c (7b)where E and B are the electric and magnetic field respec-tively as measured by some inertial observer. The two im-portant parameters defining the family of solutions are thestrength parameter a and the radiation reaction efficiency ω τ according to the following definitions a = ω B ω (8a) b = ω τ (8b) ω B = q Bm = q Em c . (8c)Introducing the weighted and normalized electromagneticfield tensor by ˜ F ik = q F ik /m ω ∝ a and a normalized time ˜ τ = ω τ , the Landau-Lifshits equation (5) is rewritten wi-thout dimensions as d ˜ u i d ˜ τ = ˜ F ik ˜ u k + b ˜ ∂ ‘ ˜ F ik ˜ u k ˜ u ‘ + b (cid:0) ˜ F ik ˜ F k‘ ˜ u ‘ + ( ˜ F ‘m ˜ u m ) ( ˜ F ‘k ˜ u k ) ˜ u i (cid:1) . (9)The ordering of the right-hand side terms are γ a, γ a b, γ a b, γ a b , γ being the particle Lorentzfactor. For our application to neutron stars, a (cid:29) γ (cid:29)
1, therefore the last term dominates the radiationreaction. This last term is γ a b = γ ω B τ times theLorentz force. Therefore radiation reaction force becomesdominant in the regime where γ ω B τ (cid:38) γ (cid:38) , (cid:16) B (cid:17) − / . (10)For the remainder of this paper, we use a Cartesian coordi-nate system labelled by ( x, y, z ) and the corresponding Car-tesian basis ( e x , e y , e z ). Moreover, the plane wave propagatesin the x direction, with a frequency ω , has a wave-vector k and a polarization electric vector E in the yOz plane. Thusby construction B x = E x = 0, E · B = 0 and E = c B . Aswe remind in the next section, exact analytical solutions havebeen found for those waves. An exact analytical solution of the Landau-Lifshits equa-tion has been given by Piazza (2008) and Hadad et al. (2010).For completeness, as our algorithm heavily relies on this so-lution, we recall it by adopting slightly different notationscompared to Hadad et al. (2010).Let us assume a plane electromagnetic wave in vacuumwith wave number k and frequency (more properly calledpulsation) ω such that the vector potential A i is given by thereal part < of a complex potential f ( ξ ) ε i A i = A < [ f ( ξ ) ε i ] (11)with A the potential amplitude. f ( ξ ) is an arbitrary func-tion of the phase given by ξ = k i x i = ω t − k x , the four-position vector is x i = ( c t, x, y, z ), the four-wavenumber k i =( ω/c, k, ,
0) and the space like polarization vector ε i . Thestrength of the wave is given in term of the parameter a defined by a = q A m c . (12)Note that it can be positive or negative depending on the par-ticle charge. The solution for the 4-velocity is then expressedby introducing several functions as ψ ( ξ ) = Z ξ ˆA ( y ) · ˆA ( y ) dy (13a) τ ( ξ ) = ξ k · u − τ a Z ξ ψ ( y ) dy (13b) χ i = Z ξ ˆ A i ( y ) ψ ( y ) dy (13c) k · u = γ ω (1 − β ) (13d)where the prime in ˆ A i ( y ) denotes the derivative with respect MNRAS000
0) and the space like polarization vector ε i . Thestrength of the wave is given in term of the parameter a defined by a = q A m c . (12)Note that it can be positive or negative depending on the par-ticle charge. The solution for the 4-velocity is then expressedby introducing several functions as ψ ( ξ ) = Z ξ ˆA ( y ) · ˆA ( y ) dy (13a) τ ( ξ ) = ξ k · u − τ a Z ξ ψ ( y ) dy (13b) χ i = Z ξ ˆ A i ( y ) ψ ( y ) dy (13c) k · u = γ ω (1 − β ) (13d)where the prime in ˆ A i ( y ) denotes the derivative with respect MNRAS000 , 1–15 (2021)
J. Pétri to the argument y . Therefore, the full solution for an arbitrarywave is u i k · u = u i k · u + a c k · u (cid:2) − ( ˆ A − ˆ A ) i + k i k · u (cid:18) ( ˆA − ˆA ) · u − a c ( ˆA − ˆA ) (cid:19)(cid:21) + τ c (cid:2) − a ( ˆ A − ˆ A ) i + a χ i + k i k · u (cid:0) a ( ˆA − ˆA ) · u − a c ψ − a c ( ˆA − ˆA ) · ( ˆA − ˆA ) − a χ · u + a c ( ˆA − ˆA ) · χ (cid:1)(cid:3) + τ c k i (cid:20) − a ( ˆA − ˆA ) a ( ˆA − ˆA ) · χ + a ψ − a χ (cid:21) . For charged particles immersed in the neutron star electro-magnetic field, outside the light-cylinder, the field convergesto a elliptically polarized plane wave depending on the cola-titude θ . It is linearly polarized at the equator θ = π/ θ = 0 and θ = π , showingany kind of elliptic polarization between the poles and theequator. Therefore, in order to keep the discussion as generalas possible, we focus on elliptically polarized waves with awave vector k i = ( ω/c, k, ,
0) and being a linear superposi-tion of a left-handed and right-handed elliptically polarizedwave with characteristics ε i c = (0 , , , − i ) / √ f ± ( ξ ) = √ e ± i ( ξ − ξ ) (14b) A i ± = A (0 , , cos( ξ − ξ ) , ± sin( ξ − ξ )) (14c) E i ± = − ω A (0 , , − sin( ξ − ξ ) , ± cos( ξ − ξ )) (14d) B i ± = k A (0 , , cos( ξ − ξ ) , ± sin( ξ − ξ )) = k A i ± . (14e) ξ is the initial phase of the wave and the sign ± refers to aleft or right handed elliptical polarization. The full vector po-tential is a sum of left and right-handed elliptically polarizedwaves such that A i = α A i + +(1 − α ) A i − = (0 , , cos( ξ − ξ ) , (2 α −
1) sin( ξ − ξ ))(15)where α ∈ [0 ,
1] with α = 1 / α = 0 or α = 1for circularly polarized waves with opposite handedness.For spatially varying waves like spherical waves emitted byrotating neutron stars, we need to integrate the 4-velocity u i to deduce the phase dependence of the 4-position. Notingthat u i = dx i /dτ = ( k · u ) dx i /dξ we find that dx i dξ = u i k · u (16)which can also be integrating analytically for elliptically po-larized waves starting from the expression (14a).In the special case of particle propagation in an ellipticallypolarized wave, deviation from the Lorentz force motion setsin when a b (cid:38) Unfortunately, the most general electromagnetic field is notnull-like (meaning I = I = 0). The simplest generalization leading to a tractable analytical solution is for constant fields.Following the procedure described by Heintzmann & Schrüfer(1973), we introduce the electromagnetic tensor eigensystemsolution such that eigenvalues λ i (possibly complex values)satisfy λ i = I ± p I + 4 I I = 0, then at least two eigenvalues vanish. For a null-like field meaning I = I = 0, all eigenvalues vanish andsolutions are given in the previous paragraph. If all eigen-values λ i are non zero then the associated eigenvectors ψ i are null-like, ψ i · ψ i = 0 because of the antisymmetry of theelectromagnetic tensor. Moreover they are explicitly given by( ψ i ) k = (cid:18) λ i E + c I λ i c , λ i E + c I B + λ i E ∧ B (cid:19) . (18)These eigenstates form a complete basis for the four dimen-sional velocity space. The 4-velocity is then adequately pro-jected onto this basis according to u k ( τ ) = X i =1 k ( τ ) f i ( ψ i ) k e λ i τ . (19)The f i are the components of the 4-velocity in the ψ i ba-sis. They are deduced from the initial conditions u k (0) = P i =1 f i ψ i and the damping factor is k ( τ ) = X i = j f i f j ( ψ i · ψ j ) e ( λ i + λ j ) τ ! − / . (20)In the absence of radiation reaction, this damping factorequals unity. In the most general electromagnetic field confi-guration, both invariants are non-vanishing, there are fourdistinct eigenvalues, two real and two complex conjugatedand the eigenvectors form a full basis for the velocity spacejustifying the above projection scheme. Such configurationsare met around rotating magnetized neutron stars, from themagnetosphere, inside the light-cylinder (the static zone)through the light-cylinder, the transition zone down to thewave zone, outside the light cylinder. Therefore the solu-tion (19) is the most appropriate building block to constructnumerical schemes integrating particle trajectories aroundstrongly magnetized neutron stars. However, in the presentstudy, we focus only on plane waves for which analytical so-lutions exist, allowing detailed quantitative comparisons anderror estimates between the algorithm proposed here and theexpected values.If some eigenvalues vanish, expression (18) cannot be ap-plied straightforwardly. Special care is required in these limi-ting cases. Of particular interest is the case when I = I = 0.Then all the eigenvalues vanish, the field is null- (or light-like)and the eigensystem must be solved separately as shown inthe previous paragraph.If I = 0 and I = 0 two eigenvalues vanish and the othertwo are either real and opposite or purely complex and oppo-site depending on the sign of I . If I >
0, the electric field E dominates, the solutions being real and given by λ , = ±√ I representing a pure electric accelerating solution. If I < B dominates and λ , = ± i √− I representingoscillatory solutions, a simple magnetic gyration in the ap-propriate electric drift frame. In this case of perpendicular MNRAS , 1–15 (2021) article acceleration and radiation reaction electric and magnetic fields, two eigenvalues vanish and for anon vanishing magnetic field B = the eigenvectors of thetwo dimensional null space are (cid:16) ωc , ωc E ∧ B B + µ B (cid:17) (21)with ( ω, µ ) ∈ R two arbitrary and uncorrelated reals gene-rating the two dimensional null space.In the special case of a zero magnetic field the above ex-pression (21) fails and a separate treatment is required. Theeigenvalues are real and given by λ i = (0 , , − E/c, + E/c ) . (22)The associated eigenvectors are( ψ ) k = (0 , k ) (23a)( ψ ) k = (0 , k ) (23b)( ψ ) k = ( − E, E ) (23c)( ψ ) k = (+ E, E ) (23d)with k · E = k · E = 0 and k ∧ k ∝ E . The spatialvectors k and k span the spatial plan orthogonal to theelectric field E .In this paper, we are interested in null-like fields with I = I = 0, corresponding to electromagnetic waves launched bya rotating magnetic dipole, as seen at large distances r (cid:29) r L , well outside the light-cylinder. In this special case, alleigenvalues vanish λ i = 0 and the solution for ellipticallypolarized waves applies. The aforementioned formal solutions depend on severalphysical parameters that have been reduced to two norma-lized quantities, namely the strength a and the damping b parameters. In order to quantitatively find the exact solu-tion, we need to impose the initial conditions given by theinitial phase of the wave ξ and the initial velocity of theparticle u injected at phase ξ . We will only consider initialvelocities aligned with the wave propagation direction suchthat u i = Γ c (1 , β , ,
0) where Γ is the initial Lorentz fac-tor and β the normalized spatial velocity.Note that the strength parameter is a Lorentz invariantbecause the electromagnetic field F (meaning E or B ) bet-ween two frames of relative velocity β transforms accordingto F = D F with the Doppler factor D = 1 / Γ (1 − e x · β ).The frequency is Doppler shifted based on ω = D ω (24)rendering the ratio B/ω constant and equal to B /ω forframe velocities aligned with the wave propagation direction.Therefore a = a is indeed a relativistic invariant. When theparticle has an initial velocity such as u it suffices to trans-form to the instantaneous particle rest frame at the initialtime, to compute the solution in this frame with a particleat rest and finally to transform position and velocity back tothe observer frame. The Lorentz factor γ in the rest frame isrelated to the Lorentz factor γ measured by the observer by γ = γ Γ (1 − β · β ) (25)which simplifies for ultra-relativistic particles and β aligned with β to γ ≈ γ r − β β . (26)The period as measured by the observer also suffers from thetime dilation effect. The Doppler effect for the wave frequencycombined with the Lorentz transform for the time intervalbetween two periods c ∆ t = γ ( c ∆ t + β ∆ x ) (27)corresponding to ξ = 2 π shows that the period is changed to ω T circ = 2 π D γ [1 + ( β + 1) a ] (28)for a circularly polarized wave and to ω T lin = 2 π D γ [1 + 34 ( β + 1) a ] (29)for a linearly polarized wave (to be compared with particlesstarting at rest, see Pétri (2020b)). We will check this pointof view in the numerical tests discussed in Sec. 3.If the initial phase ξ of the wave at the particle injectionpoint does not vanish, the acceleration process is not optimalin the sense that the highest Lorentz factor γ will be less than γ max = 1 + 2 a . For instance for the motion without radia-tion reaction, the particle is insensitive to the initial phaseof a circularly polarized wave but sensitive to an ellipticallypolarized wave, the worst case being a linearly polarized wavewith α = 1 /
2. For those waves the maximum Lorentz factoris γ max ( ξ ) = 1 + a (1 + k cos ξ k ) therefore a factor 4less for ξ = π/ ξ = 0 when a (cid:29)
1. This will also be checked in our subsequent tests.
In this section, we perform some tests of the constant fieldapproximation for plane waves and compare our results withthe exact analytical solutions detailed in the previous section.We distinguish cases with particles initially at rest from caseswith particles initially moving at relativistic speed catchingup the wave or moving in opposite direction to the wave. Wethen close the test section by a discussion of the impact of theinitial phase of the wave on the acceleration efficiency. Thephase is indeed another important parameter controlling themaximum energy reached by the particle.
Let us assume that particles are injected at rest in an elec-tromagnetic wave with an initial phase equal to zero ξ = 0at the particle location. The maximum Lorentz factor is thenalways given by γ max = 1 + 2 a whatever the polarizationof the wave. In order to check the integration of the par-ticle equation of motion in a constant electromagnetic field,we compare the exact analytical solution with the constantfield integrator. Several examples are shown without radia-tion reaction, a strength parameter up to a = 10 and cir-cular or linear polarization modes such that α = { , . } .Fig. 1 shows the periodic variation of the Lorentz factor fora circularly polarized wave and a strength parameter log a = MNRAS000
Let us assume that particles are injected at rest in an elec-tromagnetic wave with an initial phase equal to zero ξ = 0at the particle location. The maximum Lorentz factor is thenalways given by γ max = 1 + 2 a whatever the polarizationof the wave. In order to check the integration of the par-ticle equation of motion in a constant electromagnetic field,we compare the exact analytical solution with the constantfield integrator. Several examples are shown without radia-tion reaction, a strength parameter up to a = 10 and cir-cular or linear polarization modes such that α = { , . } .Fig. 1 shows the periodic variation of the Lorentz factor fora circularly polarized wave and a strength parameter log a = MNRAS000 , 1–15 (2021)
J. Pétri
Figure 1.
Evolution of the Lorentz factor of a particle initially atrest and for a circularly polarized wave with log a = { , , , } .Solid lines represent the exact analytical solutions and dottedpoints the constant field approximation. Figure 2.
Same as Fig. 1 but for a linearly polarized wave. { , , , } . The time is normalised with respect to the periodfor a circularly polarised wave T circ given by Eq. (28) with β = 0. The numerical solution marked as symbols perfectlyoverlaps with the analytical solution in solid line.Fig. 2 shows the equivalent results for a linearly polari-zed wave and time normalisation according to T lin given byEq. (29) with β = 0. Here also, the match is perfect.In this section, we saw that the particle gained energy fromthe wave but at the end of a cycle, i.e. after a phase variation ξ of 2 π , the particle returned to a state at rest, losing its kineticenergy due to the "braking" of the wave. The process is fullyreversible in time for the Lorentz force. This is typical of awave-particle interaction. We will see that when dissipation isadded to the equation of motion, like for instance radiationreaction, the particle does not return to rest but keeps aminimal kinetic energy. The process is no longer fully timereversible. If the particle enters the wave with an initial relativisticvelocity, the situation changes from the evolution found pre-viously. A Lorentz boost in the rest frame of the particle doesnot affect the nature of the wave, it remains null-like but the
Figure 3.
Evolution of the Lorentz factor of a particle injectedwith a relativistic speed and for a circularly polarized wave with a = 10 . The legend shows log( γ ) with the convention that a nega-tive value means a velocity vector pointing in a direction oppositeto the wave propagation. Figure 4.
Same as Fig. 3 but for a linearly polarized wave. wave frequency is Doppler shifted to a new frequency ω ac-cording to Eq. (24). The periodicity in the particle Lorentzfactor also changes to Eq. (28) or to Eq. (29) depending onthe wave polarization. Some examples for a circularly pola-rized wave are shown in Fig. 3 and for a linearly polarizedwave in Fig. 4. The initial Lorentz factor is γ and shown inthe legends as a logarithmic log γ with the convention thata negative value means a velocity vector pointing in a di-rection opposite to the wave propagation. The constant fieldapproximation, in dotted points, agrees with the exact ana-lytical solution, in solid lines. The phase ξ when the particle enters the wave also affectsits subsequent trajectory. The impact of this initial phase isscrutinised by varying ξ in multiples of π/ ξ ∈{ , π/ , π/ , π/ } . Some results are shown for a circularlypolarized wave in Fig. 5 by fixing the strength parameterto a = 10 . As expected, for such waves, the trajectory isindependent of the initial phase because only the ( E , k ) planerotates without varying the strength of E or B with ξ . Themaximum Lorentz factor is always γ max = 1+2 a ≈ . MNRAS , 1–15 (2021) article acceleration and radiation reaction Figure 5.
Evolution of the Lorentz factor of a particle injected atdifferent initial phases for a circularly polarized wave with a = 10 . Figure 6.
Same as Fig. 5 but for a linearly polarized wave.
For linearly or elliptically polarized waves, the initial phaseimpacts the trajectory and the maximum Lorentz factor asshown for instance in Fig. 6 for a linearly polarized wavewith a = 10 . Acceleration is most effective when injectionhappens at ξ = 0 and a factor 4 less efficient if injectionoccurs at ξ = π/ In conditions prevailing around rotating magnetized neu-tron stars, the electromagnetic field strength and the particleLorentz factors are so large that radiation reaction efficientlyslows down the particle by lowering its kinetic energy, conver-ting it into radiation. In this section we redo the same analysisas in the previous section except that we add the radiationreaction without removing any term in the Landau-Lifshitsapproximation.The strength of radiation damping is controlled by the nor-malised parameter b defined in Eq. (8). Typical values for Figure 7.
Evolution of the Lorentz factor of a particle initially atrest with radiation reaction set to log b = −
20 and for a circularlypolarized wave with log a = { , , , } . neutron stars are b = 4 · 10 − (cid:16) P (cid:17) − (30)where P = 2 π/ Ω is the pulsar period in second. It is stron-gest for millisecond pulsars reaching values of b ≈ − for a 2 ms pulsar. For the simulations presented below, we uselog b = −
20. The perturbation in the Lorentz force also in-cludes terms involving γ and a as explained in the paragraphafter Eq. (9). When the particle starts at rest, the radiation reaction va-nishes. Whatever the strength and damping parameters a and b , the particle evolves initially only according to the Lorentzforce. The trajectories are therefore identical to the previouscases without radiation reaction. Only when the Lorentz fac-tor reaches high enough values for the perturbation to becometo the same order of magnitude as the Lorentz part will theparticle deviate from its dissipationless motion. This is seenin Fig. 7 showing the particle Lorentz factor evolving in a cir-cularly polarized wave for log a = { , , , } . By inspectionof Fig. 8, we deduce that the behaviour in a linearly polarizedwave is very similar, only the largest strength parameters lea-ding to the largest Lorentz factors will perturb the Lorentzforce. Indeed, only the case a = 10 leads to the radiationdominated motion in the regime a b = 10 (cid:29)
1. All othercases a well approximated by the Lorentz force motion, ex-cept for a = 10 where we observe a slight increase in theperiodic variation in γ with time, see the plots in green pointin Fig. 7 and Fig. 8.Radiation reaction drastically inflates the typical time scaleof Lorentz factor variation as can be checked in Fig. 9 sho-wing an increase by 10 orders of magnitude in the case of a = 10 for circularly as well as for linearly polarized waves,respectively in solid lines and dashed lines with and withoutradiation reaction (resp. LL in blue and LF in orange). Themaximum Lorentz factor also increases significantly when ra-diation reaction is included, see Fig. 10. In the aforementio-ned case, there is an increase by 4 to 5 orders of magnitude.Finally, Fig. 11 summarizes the spatial evolution of this Lo-rentz factor, demonstrating the stretching effect of radiation MNRAS000
1. All othercases a well approximated by the Lorentz force motion, ex-cept for a = 10 where we observe a slight increase in theperiodic variation in γ with time, see the plots in green pointin Fig. 7 and Fig. 8.Radiation reaction drastically inflates the typical time scaleof Lorentz factor variation as can be checked in Fig. 9 sho-wing an increase by 10 orders of magnitude in the case of a = 10 for circularly as well as for linearly polarized waves,respectively in solid lines and dashed lines with and withoutradiation reaction (resp. LL in blue and LF in orange). Themaximum Lorentz factor also increases significantly when ra-diation reaction is included, see Fig. 10. In the aforementio-ned case, there is an increase by 4 to 5 orders of magnitude.Finally, Fig. 11 summarizes the spatial evolution of this Lo-rentz factor, demonstrating the stretching effect of radiation MNRAS000 , 1–15 (2021)
J. Pétri
Figure 8.
Same as Fig. 7 but for a linearly polarized wave.
Figure 9.
Phase evolution of the inertial frame clock normalizedtime ω t of a particle initially at rest with and without radiationreaction (resp. LL and LF) for a circularly (solid line) and linearly(dashed line) polarized wave with log a = 3 and log b = − reaction. The achievable energy is much higher but it requiresmore time or space to attain its asymptotic value.We conclude that when the radiation reaction is taken intoaccount, i.e., in the presence of deceleration induced by ra-diative friction, the particle energy becomes greater than wi-thout taking it into account. This statement appears counter-intuitive but it is not related to the well-known runaway so-lutions of the LAD equation because these motions do notshow any exponential grow of the Lorentz factor as would bethe case for a runaway solution. Indeed, the Landau & Lif-shits prescription is free of these parasitic solutions becauseit is a second order in time equation of motion. Thereforethe non-physical self-accelerating solutions are absent in theLandau-Lifshits equation. The reason leading to more effi-cient acceleration in case of radiation reaction is related tothe precise time evolution of the particle in the electroma-gnetic wave.Gunn & Ostriker (1971) showed indeed that, for any ini-tial conditions, a particle evolving in a plane electromagneticwave with radiation reaction (in the Landau-Lifshits pres-cription) slowly increases its energy with time, a kind of "ra-diative pumping" as they said. The radiation reaction can beinterpreted as a friction causing a delay in the particle res-ponse to the field, inducing a lag between its velocity and the Figure 10.
Same as Fig 9 but for the phase evolution of theLorentz factor.
Figure 11.
Evolution of the Lorentz factor with distance accor-ding to Fig. 9 and Fig. 10. accelerating electric field, causing the slowly in time increasein kinetic energy, typically γ ∝ t / as they showed. The factthat radiation reaction can decrease or increase the Lorentzfactor in plane waves has also been noticed by Heintzmann& Grewing (1972). Such pumping is not effective in spheri-cal waves because the kinetic energy increase occurs mainlyduring the phase locked motion at the beginning of the ac-celeration process and requires many cycles with constantstrength parameter. When the particle starts at a relativistic speed, for thesame simulation runs as in the previous section, the maxi-mum Lorentz factor reached by the particle is not sufficient tosignificantly perturb the Lorentz force if the particle catchesup the wave. We therefore do not observe any difference bet-ween radiation reaction and solely Lorentz force evolutionwhen inspecting Fig. 12 for a circularly polarized wave orFig. 13 for a linearly polarized wave, in the cases markedwith a positive log γ , meaning particles moving in the samedirection as the wave. This is due to the fact that the effec-tive damping parameter b = ω τ as measured in the particlerest frame decreases by several orders of magnitude due to MNRAS , 1–15 (2021) article acceleration and radiation reaction Figure 12.
Evolution of the Lorentz factor of a particle injectedwith a relativistic speed and for a circularly polarized wave with a = 10 . Figure 13.
Same as Fig. 12 but for a linearly polarized wave.
Doppler shifting of the wave frequency ω = D ω (cid:29) ω . Tothe contrary, for a head on collision between the particle andthe wave, the apparent wave frequency is blue shifted due tothe Doppler effect, and the effective damping parameter b increases by several orders of magnitude. Radiation reactionbecomes significant and the particle trajectory is affected bythe perturbing force. This is clearly seen for negative log γ (meaning particle moving in opposite direction to the wavepropagation i.e. a head-on collision) in Fig. 12 and Fig. 13where the Lorentz factor slowly drifts to larger and largervalues. We already saw that a circularly polarized wave is insensi-tive to the initial phase due to its symmetry of rotation aboutan axis parallel to the wave vector k . This holds true for ra-diation reaction as demonstrated in Fig. 14 and as expecteddue to this symmetry property. However, as for the Lorentzforce, the Landau-Lifshits equation remains also sensitive tothe initial phase for the linearly polarized wave as seen inFig. 15.We recall that in all the above exposed examples, the exactanalytical solutions for the 4-velocity are known and served Figure 14.
Evolution of the Lorentz factor of a particle injected atdifferent initial phases for a circularly polarized wave with a = 10 . Figure 15.
Same as Fig. 14 but for a linearly polarized wave. as a check for our algorithm. We found that exact analyticalsolutions for the constant field approximation gives sensiblythe same results. Therefore, it demonstrates that the constantfield approximation can serve as a building block for verygeneral null-like electromagnetic fields.After these extensive tests of our numerical algorithm fornull-like electromagnetic fields, we apply it in the contextof neutron star vacuum magnetospheres outside the light-cylinder where spherical waves are launched, described locallyas plane waves to a good approximation.
Since the work of Deutsch (1955), we know that a rotatingmagnetic dipole launches a large amplitude low frequencyelectromagnetic wave at large distances r (cid:29) r L , that is ap-proximated by a spherical wave of definite polarization de-pending on the colatitude θ and decreasing with distance like r L /r . Indeed, along, the rotation axis, the wave is a circularlypolarized wave whereas along the equator, it is completely li-nearly polarized. In between these two limits, the wave showsany degree of elliptical polarization, left handed or right han-ded.In this last section, we apply our algorithm to a real as-trophysical context of particle acceleration and radiation inthe wave zone outside the light-cylinder of a neutron star. As MNRAS000
Since the work of Deutsch (1955), we know that a rotatingmagnetic dipole launches a large amplitude low frequencyelectromagnetic wave at large distances r (cid:29) r L , that is ap-proximated by a spherical wave of definite polarization de-pending on the colatitude θ and decreasing with distance like r L /r . Indeed, along, the rotation axis, the wave is a circularlypolarized wave whereas along the equator, it is completely li-nearly polarized. In between these two limits, the wave showsany degree of elliptical polarization, left handed or right han-ded.In this last section, we apply our algorithm to a real as-trophysical context of particle acceleration and radiation inthe wave zone outside the light-cylinder of a neutron star. As MNRAS000 , 1–15 (2021) J. Pétri a typical value of the magnetic field strength at this light-cylinder, we choose B L ≈ r equal to the light-cylinder if not otherwise speci-fied, r = r L . Moreover, we employ the spherical wave ap-proximation meaning a decrease in the field amplitude like( E, B ) ∝ r L /r and where the wave field components E , B and propagation direction n are mutually orthogonal. Anykind of polarization can be considered, linear, left/right cir-cular and elliptical polarization. Let us assume that particles are injected at rest at a dis-tance r = r L from the neutron star centre and an arbitrarycolatitude θ with respect to the rotation axis. Because thewave amplitude decrease with distance, particles do not reachthe maximum energy γ max of a plane wave. The actual maxi-mum energy is much less and does not scale as (1 + 2 a ) anymore as we will proof.Indeed, Fig. 16 shows the acceleration efficiency for cir-cularly, elliptically and linearly polarized waves respectivelyin solid, dotted and dashed line, with α = { , . , . } wi-thout radiation reaction and strength parameter log a = { , , , } . The maximum Lorentz factor found from theseruns scales roughly as a . in all cases, the weakest valuesbeing obtained for a linear polarization, Fig. 17. Followingthe arguments exposed by Michel & Li (1999), the par-ticle reaches its maximum energy after travelling a distance r c ≈ π a where a c is the strength parameter at the dis-tance r c . But at these distances, the strength parameter hasdecreased to a value a c = a r L /r c . Solving for the distance,we get r c ≈ π / a / r L . A good guess of this final Lorentzfactor is given by γ fin ≈ a ≈ a/π ) / (31)which is in agreement with the fitted exponent of 0 . ≈ / α = 0 .
5) compared to thecircular or elliptic polarization with α = { , . } . Therefore,around a rotating magnet, acceleration is most effective alongthe rotation axis and weakest around the rotational equator.Fig. 18 shows the same results as in Fig. 16 but with ra-diation reaction fixed to log b = −
20. Because the perturba-tion scales as a b and a decreases with distance, radiationfeedback does not produce any significant perturbation tothe particle motion, except in the efficient acceleration zonearound the light-cylinder for the case a = 10 . We concludethat radiation reaction does not impact the particle motionin the wave zone of a pulsar in this vacuum case.The final Lorentz factor is also relatively insensitive to theinitial position r of the particle at rest. Fig. 19 indeed showsthe Lorentz factor dependence on distance for particles evol-ving in a spherical wave with a = 10 , for several polarizationstates and several initial positions log( r /r L ) = { , , , } .For linear polarization as long as particles are injected at ra-dii shorter than the distance where the asymptotic energyis attained, in the example about r c /r L ≈ , the maxi-mum Lorentz factor is noticeably the same. For circular po- Figure 16.
Evolution of the Lorentz factor for a circularly po-larized wave ( α = 0) in solid line, an elliptically polarized wave( α = 0 .
2) in dotted line and linearly polarized wave ( α = 0 .
5) indashed line. The strength parameter is log a = { , , , } . Figure 17.
Final Lorentz factor for a circularly ( α = 0), ellipti-cally ( α = 0 .
2) and linearly ( α = 0 .
5) polarized wave. The law inEq. (31) is shown in red solid line.
Figure 18.
Same as Fig. 16 but with radiation reaction fixed tolog b = − , 1–15 (2021) article acceleration and radiation reaction Figure 19.
Evolution of the Lorentz factor with distance for acircularly ( α = 0), elliptically ( α = 0 .
2) and linearly ( α = 0 . r /r L = { , , , } and with a =10 . Figure 20.
Maximum Lorentz factor, extracted from Fig. 19, fora circularly ( α = 0), elliptically ( α = 0 .
2) and linearly ( α = 0 . r /r L ) = { , , , } and a = 10 . larization and elliptic polarization, this maximum γ slightlydecreases with r , not even by a factor two for a distance in-crease of three orders of magnitude, see Fig. 20. The energyacquired by a particle therefore only depends on the wavecharacteristic, that is its polarization α and strength para-meter a for injections at distances r < r c . At distance r > r c ,particle acceleration efficiency sharply decreases. Around a neutron star, particles entering the waves areinjected already at high Lorentz factors from the magnetos-phere, within the light-cylinder. We do not expect them tobe picked up at rest by the wave, see the discussion belowin Sec. 6. Therefore we imposed initial conditions where par-ticles catch up the wave at relativistic speed. We already sawthat particles reach Lorentz factors well above γ max impo-sed by the strength parameter a if the particle catches upthe wave without radiation reaction. This scaling with a is Figure 21.
Evolution of the Lorentz factor for a linearly po-larized wave ( α = 0 .
5) with a = 10 , initial Lorentz factorlog γ = { , , , } and varying initial phases, ξ = 0 in dashedlines, ξ = π/ ξ = π/ typical of a coherent wave/particle interaction in the phase-locking stage.Fig. 21 shows an example of linear polarization with a =10 , injection factors log γ = { , , , } and varying initialphase ξ , ξ = 0 in dashed lines, ξ = π/ ξ = π/ ξ = 0. Indeed at initial highLorentz factors, the particle does not feel any electromagneticfield because it almost exactly catches up the wave in its nodewhere B = E = . For γ = 10 only after having travelleda distance 10 r L will the particle start to accelerate. If theinitial phase differs from zero like for instance ξ = π/ ξ = π/
2, then the linear polarization results resemble thecircular polarization evolution shown in Fig. 22 because theparticle accelerates right at the injection place r = r L . Forcircular polarization, because of the symmetry of the field, themotion remains insensitive to the initial phase of the wave,only the initial Lorentz factor matters.Simulations including radiation reaction terms in the equa-tion of motion according to the Landau-Lifshits prescriptiondo not alleviate the conclusions drawn above. We indeed che-cked by inspection of the linear and circular polarization re-sults that the discrepancies are irrelevant. The phase at which the particle enters the wave is also arbi-trary. Its value ξ can vary from injection points to injectionpoints. We already showed some examples in the previous pa-ragraphs. Let us summarize our findings for particles startingat rest at r = r L and different polarization states.Fig. 23 shows a particle entering the wave with a = 10 and different initial phases ξ = { , π/ , π/ , π/ } for cir-cularly, elliptically and linearly polarized waves respectivelyin solid, dotted and dashed line. All things being equal, li-near polarization is always the least efficient configuration toenergize charged particles. If radiation reaction is included,we checked that nothing changes significantly again.The radiation feedback never perturbs the motion of a MNRAS000
2, then the linear polarization results resemble thecircular polarization evolution shown in Fig. 22 because theparticle accelerates right at the injection place r = r L . Forcircular polarization, because of the symmetry of the field, themotion remains insensitive to the initial phase of the wave,only the initial Lorentz factor matters.Simulations including radiation reaction terms in the equa-tion of motion according to the Landau-Lifshits prescriptiondo not alleviate the conclusions drawn above. We indeed che-cked by inspection of the linear and circular polarization re-sults that the discrepancies are irrelevant. The phase at which the particle enters the wave is also arbi-trary. Its value ξ can vary from injection points to injectionpoints. We already showed some examples in the previous pa-ragraphs. Let us summarize our findings for particles startingat rest at r = r L and different polarization states.Fig. 23 shows a particle entering the wave with a = 10 and different initial phases ξ = { , π/ , π/ , π/ } for cir-cularly, elliptically and linearly polarized waves respectivelyin solid, dotted and dashed line. All things being equal, li-near polarization is always the least efficient configuration toenergize charged particles. If radiation reaction is included,we checked that nothing changes significantly again.The radiation feedback never perturbs the motion of a MNRAS000 , 1–15 (2021) J. Pétri
Figure 22.
Evolution of the Lorentz factor for a circularly polari-zed wave with a = 10 , initial Lorentz factor log γ = { , , , } .The curves are insensitive to the initial phase ξ . Figure 23.
Evolution of the Lorentz factor for circularly, ellipti-cally and linearly polarized waves respectively in solid, dotted anddashed line, with a = 10 and several initial phases ξ . charged particle in a spherical wave on the pulsar wind zone.To a very good approximation, this perturbation as imple-mented in the Landau-Lifshits prescription, is irrelevant insuch a case. A last configuration of interest concerns the "col-lision" between the pulsar large amplitude low frequency va-cuum wave with an incoming charged particle. We refer tothis process as a head on collision and investigate it in thefollowing closing paragraph. Particles catching up the spherical wave is less efficientthan particles hitting this wave in head on "collision". Wetherefore also investigated the propagation of particles travel-ling towards the neutron star, permeating its electromagneticfield. So let us consider a particle coming from infinity. In ourruns, it means particles starting at sufficiently large distanceswhere the electromagnetic field has sufficiently decreased tobecome negligible for the particles to follow straight lines.The particle moves at a relativistic speed with initial Lorentzfactor γ , in the negative e x direction. Concretely, we alsofixed the large distance to r/r L = 10 at time t = 0. The par-ticle travels towards the star, feeling an outgoing wave with Figure 24.
Evolution of the Lorentz factor for a linearly polarizedwave with a = 10 , initial Lorentz factor log γ = i with i ∈ [1 .. ξ = 0 in dashed lines, ξ = π/ ξ = π/ Figure 25.
Same as Fig. 24 but for circular polarization. an increasing strength parameter a . At some distance r min ,the electromagnetic field overcomes the particle inertia andturns it back into the positive e x direction. The minimal dis-tance of approach r min depends on the initial particle energy.As an example, we injected particles with initial Lorentzfactors log γ = i with i ∈ [1 ..
8] in a wave of strength a =10 . Fig. 24 shows the evolution of the Lorentz factor forcounter-propagating particles and several initial phases ξ forlinear polarization. Fig. 25 shows the equivalent evolutionfor circular polarization. The gain in energy after bouncingback is irrelevant and independent of the initial phase whenentering the wave. It is about a factor 2.5 for all runs.A simple picture helps to understand the small gain inenergy. Let us assume a particle moving in vacuum in the ne-gative e x direction with Lorentz factor γ . At x = 0, it entersa region x < E direc-ted along the positive e y direction and B directed along thepositive e z direction. For such constant fields, exact analyticalsolutions are known and given for instance by Pétri (2020b).The particle is deflected by the magnetic part meanwhile ac-celerated by the electric part. The particle comes out of theelectromagnetic field back to the vacuum region x > e x direction. It can be shown that MNRAS , 1–15 (2021) article acceleration and radiation reaction Figure 26.
Minimal distance of approach and Lorentz factor fora circularly and linearly polarized wave with a = 10 and initialLorentz factor log γ = i with i ∈ [1 .. ξ for linear polarization . the final Lorentz factor after escape is related to the initialspeed β by γ fin = (1 + 3 β ) γ = γ + 3 p γ − ≈ γ . (32)Therefore we find a factor 4 not to different from the factor 2.5in view of the simple picture we used. This conclusion holdsirrespective of the sign of the charge.Fig. 26 shows the minimal distance of approach r min depen-ding on the initial Lorentz factor γ and polarization state.A good fit is given bylog (cid:16) r min r L (cid:17) ≈ (8 . circ / (8 . lin − log γ (33)the constant value depends on the polarization state, circu-lar or linear. This minimum distance can be estimated bynoting that the particle turns back whenever its Larmor ra-dius r B = γ c/ω B is comparable to the wavelength of order r L . In such a situation, the particle performs a half turn inan approximately constant electromagnetic field. Equallingboth values leads to γ ≈ ω B ω = a = a r L r min . (34)In other words, the product γ r min remains constant andequal to a r L that is approximately 10 . The energy gainin this head on collision remains therefore also too weak toaccount for any acceleration process. We considered exclusively waves with zero electromagne-tic invariants which seems far from reality around a neu-tron star. However a plane wave solution represents an ex-cellent approximation to the electromagnetic field felt by anultra-relativistic particle in its rest frame (Ritus 1985). The-refore the zero electromagnetic invariants assumption is auseful simple case to compute approximate solutions in theultra-relativistic regime. For instance in low density laserplasma simulations, the field is that of a plane wave, the-refore zero invariants apply to high accuracy if the plasmacurrent feedback is neglected. Moreover, if particles move atultra-relativistic speeds, as in high intensity laser experiments or around neutron stars, in their rest frame the two electro-magnetic invariants I and I nearly vanish. Indeed, theirnormalized magnitude defined by E · B E + c B = E · B E + c B ∝ γ (cid:28) E − c B E + c B = E − c B E + c B ∝ γ (cid:28) /γ where γ is the particle Lorentz factor inthe observer frame. This approximation breaks down only invery special configurations, for instance when particle velo-city, electric field and magnetic field are all collinear. Thisapproximation called "locally constant crossed field approxi-mation (LCFA)" is extensively used in the computation ofQED effects in laser experiments.Nevertheless, we emphasize that in a pulsar magnetohydro-dynamical (MHD) wind, the electromagnetic invariants arenot exactly equal to zero. The solutions given in the previoussections can only barely represent the more realistic situationfor a relativistically magnetized outflow. For instance, in idealMHD where the plasma possesses an infinite conductivity, theelectric field vanishes in the plasma rest frame and the windstructure is well approximated by the split monopole solu-tion of Bogovalov (1999). More generally speaking, particleacceleration in relativistic magnetized outflows is central tothe explanation of gamma-ray bursts (GRB). The composi-tion in neutrons and protons and their dynamics impacts onthe observational appearance of the GRBs as shown by De-rishev et al. (1999). Moreover, relativistic jets in blazars canefficiently convert bulk kinetic energy into radiation as foundby Stern & Poutanen (2008). Similar problems and outcomesare discussed by Beskin (2018) who summarizes the historyof pulsar theory development. He points out the importanceof the outflow mass load and its particle content made es-sentially of electrons and positrons with high pair multiplici-ties to understand the dynamics of the pulsar wind. Prokofevet al. (2015) showed that in an MHD wind with non vani-shing invariants, not only acceleration, but also decelerationof particles is possible. Nokhrina & Beskin (2017) also consi-dered the case of particle injection with arbitrary energies.These ideas are now also supported by PIC simulations seenby Philippov & Spitkovsky (2018), Sironi & Cerutti (2017) orCerutti et al. (2020). Clearly, the variety of solutions is muchricher for non null fields and required a deeper investigation.Our particle injection scheme from an arbitrary point inthe wave assumes their birth at that point. A more carefulanalysis would require an investigation of their entire trajec-tory, starting from the vicinity of the stellar surface wherethe electromagnetic field resembles more to a quasi staticdipole magnetic field and quadrupole electric field. Physi-cally, particles are injected via electron/positron pair creationthrough magnetic photon absorption (Erber 1966) or photon-photon interaction through the Breit-Wheeler process (Breit& Wheeler 1934). This injection mechanism associated withpair production has been widely discussed in the literature.Magnetic photon absorption occurs mainly around the polarcaps (Sturrock 1971; Ruderman & Sutherland 1975; Al’beret al. 1975; Fawley et al. 1977) where the magnetic field isstrong enough to disintegrate a high energetic photon. Ou-ter gaps (Cheng et al. 1986) are the privileged sites for thephoton-photon interaction although the pulsar striped windbecomes a serious alternative (Lyubarskii 1996). Cheng & MNRAS000
Minimal distance of approach and Lorentz factor fora circularly and linearly polarized wave with a = 10 and initialLorentz factor log γ = i with i ∈ [1 .. ξ for linear polarization . the final Lorentz factor after escape is related to the initialspeed β by γ fin = (1 + 3 β ) γ = γ + 3 p γ − ≈ γ . (32)Therefore we find a factor 4 not to different from the factor 2.5in view of the simple picture we used. This conclusion holdsirrespective of the sign of the charge.Fig. 26 shows the minimal distance of approach r min depen-ding on the initial Lorentz factor γ and polarization state.A good fit is given bylog (cid:16) r min r L (cid:17) ≈ (8 . circ / (8 . lin − log γ (33)the constant value depends on the polarization state, circu-lar or linear. This minimum distance can be estimated bynoting that the particle turns back whenever its Larmor ra-dius r B = γ c/ω B is comparable to the wavelength of order r L . In such a situation, the particle performs a half turn inan approximately constant electromagnetic field. Equallingboth values leads to γ ≈ ω B ω = a = a r L r min . (34)In other words, the product γ r min remains constant andequal to a r L that is approximately 10 . The energy gainin this head on collision remains therefore also too weak toaccount for any acceleration process. We considered exclusively waves with zero electromagne-tic invariants which seems far from reality around a neu-tron star. However a plane wave solution represents an ex-cellent approximation to the electromagnetic field felt by anultra-relativistic particle in its rest frame (Ritus 1985). The-refore the zero electromagnetic invariants assumption is auseful simple case to compute approximate solutions in theultra-relativistic regime. For instance in low density laserplasma simulations, the field is that of a plane wave, the-refore zero invariants apply to high accuracy if the plasmacurrent feedback is neglected. Moreover, if particles move atultra-relativistic speeds, as in high intensity laser experiments or around neutron stars, in their rest frame the two electro-magnetic invariants I and I nearly vanish. Indeed, theirnormalized magnitude defined by E · B E + c B = E · B E + c B ∝ γ (cid:28) E − c B E + c B = E − c B E + c B ∝ γ (cid:28) /γ where γ is the particle Lorentz factor inthe observer frame. This approximation breaks down only invery special configurations, for instance when particle velo-city, electric field and magnetic field are all collinear. Thisapproximation called "locally constant crossed field approxi-mation (LCFA)" is extensively used in the computation ofQED effects in laser experiments.Nevertheless, we emphasize that in a pulsar magnetohydro-dynamical (MHD) wind, the electromagnetic invariants arenot exactly equal to zero. The solutions given in the previoussections can only barely represent the more realistic situationfor a relativistically magnetized outflow. For instance, in idealMHD where the plasma possesses an infinite conductivity, theelectric field vanishes in the plasma rest frame and the windstructure is well approximated by the split monopole solu-tion of Bogovalov (1999). More generally speaking, particleacceleration in relativistic magnetized outflows is central tothe explanation of gamma-ray bursts (GRB). The composi-tion in neutrons and protons and their dynamics impacts onthe observational appearance of the GRBs as shown by De-rishev et al. (1999). Moreover, relativistic jets in blazars canefficiently convert bulk kinetic energy into radiation as foundby Stern & Poutanen (2008). Similar problems and outcomesare discussed by Beskin (2018) who summarizes the historyof pulsar theory development. He points out the importanceof the outflow mass load and its particle content made es-sentially of electrons and positrons with high pair multiplici-ties to understand the dynamics of the pulsar wind. Prokofevet al. (2015) showed that in an MHD wind with non vani-shing invariants, not only acceleration, but also decelerationof particles is possible. Nokhrina & Beskin (2017) also consi-dered the case of particle injection with arbitrary energies.These ideas are now also supported by PIC simulations seenby Philippov & Spitkovsky (2018), Sironi & Cerutti (2017) orCerutti et al. (2020). Clearly, the variety of solutions is muchricher for non null fields and required a deeper investigation.Our particle injection scheme from an arbitrary point inthe wave assumes their birth at that point. A more carefulanalysis would require an investigation of their entire trajec-tory, starting from the vicinity of the stellar surface wherethe electromagnetic field resembles more to a quasi staticdipole magnetic field and quadrupole electric field. Physi-cally, particles are injected via electron/positron pair creationthrough magnetic photon absorption (Erber 1966) or photon-photon interaction through the Breit-Wheeler process (Breit& Wheeler 1934). This injection mechanism associated withpair production has been widely discussed in the literature.Magnetic photon absorption occurs mainly around the polarcaps (Sturrock 1971; Ruderman & Sutherland 1975; Al’beret al. 1975; Fawley et al. 1977) where the magnetic field isstrong enough to disintegrate a high energetic photon. Ou-ter gaps (Cheng et al. 1986) are the privileged sites for thephoton-photon interaction although the pulsar striped windbecomes a serious alternative (Lyubarskii 1996). Cheng & MNRAS000 , 1–15 (2021) J. Pétri
Ruderman (1980) envisaged even an ion outflow from the po-lar caps. Traditionally, the acceleration process starts at thebirth place and goes on smoothly up to the light-cylinder orfurther. Nevertheless, in some circumstances, Beskin & Rafi-kov (2000) found an efficient and abrupt acceleration phasein a narrow band around the light-cylinder. The injectionproblem is crucial for the outcome of kinetic pulsar magne-tosphere simulations. Particles can be extracted right at thesurface (Wada & Shibata 2011) or everywhere within thelight-cylinder as done by Chen & Beloborodov (2014) for anaxisymmetric magnetosphere. Both injection schemes lead tovery different stationary states. The role of the particle in-jection rate was studied by Kalapotharakos et al. (2018), seealso Brambilla et al. (2018). On a more fundamental side,Timokhin & Harding (2019) performed a careful analysis ofthe pair production efficiency, updating their previous workpresented in Timokhin & Harding (2015).The places where particles enter the wave and their asso-ciated kinetic energy at injection into this wave determinesthe large scale motion towards the termination shock. Thewhole story of particle production, propagation, radiationand mixing into the interstellar medium requires a carefulbottom-up analysis encompassing the smallest and the lar-gest time and spatial scales. This preliminary work was onlyintended to explore the propagation and radiation part in thelarge amplitude low frequency electromagnetic wave.
Neutron stars are believed to be efficient particle accele-rators. However, this acceleration process must be quanti-fied depending on the magnetosphere model, being vacuum,force-free or dissipative as well as on radiation feedback. Mo-reover, realistic physical parameters are required in orderto avoid artificial down-scaling of the problem. In this pa-per we proposed a new approach to tackle those difficulttasks. First we designed an algorithm to solve analyticallyand semi-analytically for the particle equation of motion inthe Landau-Lifshits approximation checking it on known so-lutions. Next we applied it to spherical waves as those laun-ched by a rotating neutron star. We found that the acce-leration efficiency depends on the wave polarization state,strength parameter and on the particle injection conditions,that is its initial speed when entering the wave and the waveinitial phase. Because the spherical wave amplitude decreasesoutside the light-cylinder, we found no evidence of significantradiation damping in the wave zone except in the immediatevicinity of the light-cylinder.We plan to extend our analysis to waves possessing an elec-tromagnetic field component along the direction of propaga-tion in order to apply it to the exact solution of a magneticdipole rotating in vacuum and known as Deutsch solution. Insuch configurations, the light-like electromagnetic field ap-proximation fails and the constant electromagnetic field ap-proximation must be used to treat the most general geometry.The full 3D nature of the problem could then also be incor-porated in order to study particle velocities deviating fromthe wave propagation direction.Last but not least, the plasma content of the magnetos-phere must be taken into account for the most realistic andself-consistent electromagnetic field/particle/radiation inter- action. We plan to study test particle motion in those dissipa-tive magnetospheres as found for instance by Pétri (2020a).
ACKNOWLEDGEMENTS
I am grateful to the referee for helpful comments and sug-gestions. This work has been supported by the CEFIPRAgrant IFC/F5904-B/2018 and ANR-20-CE31-0010.
DATA AVAILABILITY
The data underlying this article will be shared on reaso-nable request to the corresponding author.
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