Electron-Positron Pair Production by Gamma Rays in an Anisotropic Flux Of

Electron-positron pair production by gamma rays in an anisotropic flux of soft photons, and application to pulsar polar caps Guillaume Voisin, Fabrice Mottez, Silvano Bonazzola

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Guillaume Voisin, Fabrice Mottez, Silvano Bonazzola. Electron-positron pair production by gamma rays in an anisotropic flux of soft photons, and application to pulsar polar caps. Monthly Noticesof the Royal Astronomical Society, Oxford University Press (OUP): Policy P - Oxford Open Option A, In press, pp.1 - 19. hal-01614371

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Electron-positron pair production by gamma rays in an anisotropic ﬂux of soft photons, and application to pulsar polar caps

Guillaume Voisin,1? Fabrice Mottez,1,2 Silvano Bonazzola1 1LUTH, Observatoire de Paris, PSL Research University, 5 place Jules Janssen, 92190 Meudon, France 2LUTH, CNRS, 5 place Jules Janssen, 92190 Meudon, France

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT Electron-positron pair production by collision of photons is investigated in view of application to pulsar physics. We compute the absorption rate of individual gamma- ray photons by an arbitrary anisotropic distribution of softer photons, and the energy and angular spectrum of the outgoing leptons. We work analytically within the approximation that 1 mc2/E > /E, with E and the gamma-ray and soft-photon maximum energy and mc2 the electron mass energy. We give results at leading order in these small parameters. For practical purposes, we provide expressions in the form of Laurent series which give correct reaction rates in the isotropic case within an average error of ∼ 7%. We apply this formalism to gamma rays flying downward or upward from a hot neutron star thermally radiating at a uniform temperature of 106K. Other temperatures can be easily deduced using the relevant scaling laws. We find differences in absorption between these two extreme directions of almost two orders of magnitude, much larger than our error estimate. The magnetosphere appears completely opaque to downward gamma rays while there are up to ∼ 10% chances of absorbing an upward gamma ray. We provide energy and angular spectra for both upward and downward gamma rays. Energy spectra show a typical double peak, with larger separation at larger gamma-ray energies. Angular spectra are very narrow, with an opening angle ranging from 10−3 to 10−7 radians with increasing gamma-ray energies. Key words: (stars:) pulsars: general – radiative transfer – relativistic processes – stars: neutron – X-rays: general – gamma-rays: general

1 INTRODUCTION of gamma rays by the extragalactic background light. Nu- merical integration was needed to obtain practical results. In Electron-positron pair creation by collision of two photons, contexts such as active galactic nuclei or X-ray binaries, var- also called Breit-Wheeler process, is important in a series of ious formulations and approximations were developed. Ap- astrophysical questions (Ruffini et al. 2010). Among them is proximated analytical expressions were given in Bonometto the filling of recycled pulsar magnetospheres with plasmas. & Rees(1971) and Agaronyan et al.(1983) in the case of an The cross-section of two-photon-pair creation has been isotropic soft-photon background distribution and averaging derived in Berestetskii et al.(1982). This is a function of over outgoing angles of the produced leptons. The expres- the four-momentum of both electrons. In pulsar magneto- sion of Agaronyan et al.(1983) also applies for a bi-isotropic spheres, there is generally a huge reservoir of low-energy photon distribution (both strong and weak photon distri- photons and a small number of high-energy photons. In or- butions are isotropic) without angle averaging over leptons. der to decrease computational cost compared to pairwise In these papers, the authors provide the energy spectrum calculations, the cross-section is integrated over the distri- of the outgoing leptons. An exact expressions in the case of bution of the low-energy photons. The exact formula for the bi-isotropic photon distribution is derived in Boettcher & reaction rate on an isotropic soft-photon background was Schlickeiser(1997), as well as a comparison to the previous first derived in Nikishov(1962) to estimate the absorption approximations that favors the approach in Agaronyan et al. (1983) for its better accuracy.

? E-mail: [email protected] (GV) The standard picture of a pulsar magnetosphere as-

c 2017 The Authors 2 G. Voisin et al. sumes that its inner part is filled with plasma and corotates P- P+ P- P+ with the neutron star with angular velocity Ω∗. The primary plasma is made of matter lifted from the neutron-star surface by electric fields Goldreich & Julian(1969). These + particles have highly relativistic energies; their motion in the neutron-star magnetic field generates synchrotron and curvature gamma-ray photons. In addition to primary particles, Ks Kw Ks Kw Sturrock(1971) has shown that electron-positron pairs are created in or near the acceleration regions of the magnetosphere. This provides plasma capable of screening the elec- Figure 1. Reaction of electron-positron pair creation from a pair tric field component parallel to the magnetic field. There are of photons represented to first order by Feynman diagrams. Pho- tons have 4-momenta K and K while electron and positron two processes of pair creation : two-photon process, and one- s w have respectively P− and P+. photon in the presence of a strong magnetic field. The one- photon process is the most efficient with young and standard pulsars, of which magnetic field is in the range B ∼ 106 − 108 dependent absorption of gamma rays interacting with the T(Burns & Harding 1984). The photon-photon pair-creation diffuse extragalactic background light (Nikishov 1962; Gould process can become more important with high-temperature & Schréder 1966). This effect drastically limits the horizon polar caps, and when the magnetic field is below 106 T as of the gamma-ray universe, and this has been taken into in recycled pulsar magnetospheres. Anisotropy of the soft- account in the science case of high-energy gamma ray obser- photon sources is prone to be important as they are expected vatories (Vassiliev 2000). to come either from the star (hot spots) or from synchrotron In this paper, we revisit the computation of the two- radiation in magnetospheric gaps. That is the main reason photon pair-creation rate with the aim of dealing with arbi- of our present investigation. trarily anisotropic soft-photon background distribution. In Many detailed studies of pair-creation cascades in pul- addition, we give formulas for angle and energy spectra in sar magnetospheres are based only on the one photon pro- order to be able to determine in which state pairs are cre- cess. This is for instance the case in the recent studies in ated. After an introduction to the two-photon pair creation Timokhin & Harding(2015). Others take the two reactions equations in section2, the integral over the low-energy pho- into account (Chen & Beloborodov 2014; Harding et al. tons is defined in section3. Practical expressions for spectra 2002). are derived in section ??, and applications to the cosmic mi- In numerical simulations of pulsars, the pair-creation crowave background and to a hot neutron star are developed rate is generally estimated with simple proxies. For instance, in section 5.2. in Chen & Beloborodov(2014), a mean free path l = 0.2R∗ is used for the one-photon process, and l = 2R∗ for the two-photon process. The rate of creation of pairs is not 2 THE TWO-PHOTON-QUANTUM- explicited as a function of the electron (or positron) mo- ELECTRODYNAMICS mentum, neither of the local photon background. Instead, REACTION pair creations are supposed to be abundant enough to sup- ply electric charges and current densities. The authors write When not specified, we use a unit system where the speed that this approximation is somehow similar to the force-free of light c = 1. approximation. In Harding et al.(2002), both one-photon and two-photon processes are taken into account, and the 2.1 General formalism two-photon process is controlled by a mean free path derived from Zhang & Qiao(1998), where anisotropy is par- Any quantum-electrodynamics reaction from an initial tially taken into account : the energy integral has a lower quantum state |ii to an outgoing state |oi can be repre- limit that depends on the angular size of the hot cap pro- sented as the decomposition on a final states basis {| fk i} viding the soft-photon background. Besides, these authors of the evolved state Sˆ|ii, Sˆ being the evolution operator, do not provide spectra for the created pairs although the X | i h | ˆ| i| i energy distribution of the outgoing particles are important o = fk S i fk (1) for the dynamics of pair cascades. A more complete model k needs an integration over every local surface element with a From that starting point, if one is able to derive the ap- threshold that depends on the location of each elementary propriate evolution operator, one can then determine the emitter. This is what the results of the present paper allow probability of transition from a given state to any state of to do within some approximation, together with angular and the final basis. We are interested in the reaction which yields energy spectra of the outgoing pairs. an electron e− and a positron e+ from the encounter of two Pair creation by two photons is also important in photons. Common applications take place in a frame where high energy gamma-ray astrophysics. Many papers about one is ”strong”, that is high-energy, and the other is ”weak”. gamma-ray bursts and active galactic nuclei refer to Svens- Hence we call them γs and γw , and son(1987) and the integrated mean free path in this paper γ + γ → e− + e (2) is also based on Nikishov(1962). Actually, spectra of TeV s w + radiation observed from distant (beyond 100 Mpc) extra- The state of a free photon can be decomposed on a galactic objects suffer essential deformation during the pas- plane-wave basis parametrized by four-momentum and po- sage through the intergalactic medium, caused by energy- larisation. The common assumption is that the effective

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 3

state of a photon is very well approximated by one plane which turns into wave at the time of the encounter. Such a state is not phys- 0 0 2 ical in itself, because it cannot be normalized i.e. it does Ks Kw (1 − cos ξ) ≥ 2m (10) 2 not belong to the L space, or more physically because the where ξ is the angle between the two photons. Heinsenberg inequality imposes to the wave function to be A few important properties of dw can be evidenced by entirely spread through space as a consequence of the per- taking a look at cross-section(6) averaged over every possi- fect determination of momentum. Though, this assumption ble direction of the outgoing lepton (Berestetskii et al. 1982). should be valid over a local four-volume of space-time V δt As a result, the averaged cross-section depends only on the ˆ where the interaction through operator S takes place. kinematic invariant τ = t/(4m2) the ratio between the CM Equivalently, free electrons and positrons live on a ba- energy and the threshold energy. Without loss of generality sis of plane-wave spinors parametrized by a four-momentum in the present discussion, we can assume that the reaction and a spin. From now on, the leptons are characterized by 0 takes place in the CM frame, such that the elementary cur- their charges and their four-momenta P+ = (P+, p~+) and P− = rent j = 2/V. The ultra-relativistic limit (Berestetskii et al. 0 0 ~ (P−, p~−) and photons by their four-momenta Ks = (Ks , ks ), 1982) shows that the cross-section vanishes like log τ/τ. This 0 ~ and Kw = (Kw , kw ). We consider that their distributions are kind of decrease with energy is a common feature of quantum averaged over spin and polarization respectively. Following mechanical cross-sections. Moreover, one can numerically es- Berestetskii et al.(1982), the Lorentz-invariant cross-section timate the CM energy√ corresponding to the maximum of the equations are derived in terms of kinematic invariants (also reaction rate to be t ' 1.4(2m). called Mandelstam variables), defined as Concerning the angular dependency, leptons are cre- 2 2 ated almost isotropically when the reaction is near thresh- s = (P− − K ) = (P − K ) , (3) s + w old while their momenta become aligned with those of the 2 2 t = (Kw + Ks ) = (P+ + P−) , progenitor photons when going to higher energies (see e.g. 2 2 Berestetskii et al.(1982)). In the observer’s frame this trans- u = ((P− − Kw ) = (P+ − Ks ) . lates in a larger angular dispersion for reactions close to The conservation of four-momentum writes threshold. Equations (3-8) fully describe the interaction for a given s + t + u = 2m2, (4) pair of photons; but in a pulsar’s magnetosphere, there is where m is the mass of the electron. a huge amount of photon pairs. In a simulation, it is not The probability dw per unit time of making a pair is possible to compute dw for each pair; we need a statistical approach and a kind of ”collective” reaction rate dW. We dw = d2σ × j, (5) define it in the next section. where dσ is the Lorentz-invariant cross-section !2 m2 m2 m2 d2σ = −ds8πr2 × + + (6) 2.2 The pair reactions that count in a pulsar e t2 s − m2 u − m2  magnetosphere  ! ! m2 m2 1 s − m2 u − m2 +  − + , In a pulsar magnetosphere, the weak photons Kw are mostly 2  2 2 2 s − m u − m 4 u − m s − m # caused by the black-body radiation of the neutron star, or 1 possibly by synchrotron from secondary pair cascades. Their where re is the classical radius of the electron , ds is the 0 0 energies range in the X-ray domain. The strong photons are differential of the kinematic invariant s at P− and Ks fixed, caused by the synchro-curvature radiation of energetic par- ds = 2d(p~− · k~s ) = 2 p~− k~s sin(p~−, k~s )d(p~−, k~s ) (7) ticles (electrons, positrons, and possibly ions). Their energies are in the gamma-ray domain. They are more scarce and j is the elementary two-particle flux of the reaction, than weak photons. Let’s follow a ”rare” high-energy, strong 1 Ks · Kw photon taken from a phase-space distribution . We as- j = , (8) f s 0 0 V Ks Kw sume that it flies through an abundant stream of low-energy, weak photons with a distribution f . Strong photons negli- 2 w and V is the interaction volume previously defined . Only gibly interact with other strong photons because they are the current j is frame-dependent. In particular it reads j = not abundant, and because the reaction would likely be 2/V in the center of mass (CM) of the reaction. far above threshold in Eq. (10), and therefore inefficient. Let us notice that a reaction is possible only if the en- Weak photons do not interact with other weak photons since ergy of the two photons exceeds the mass energy of the elec- their energies are under the reaction threshold. Thus, only tron and of the positron. One shows that the kinematic in- weak/strong interactions remain, but weak photons are so variant t (3) is equal to the square of the energy in the CM. numerous that a reaction negligibly changes their distribu- This allows to define the frame-invariant criterion tion. Because strong photons are less abundant, pair cre- √ t ≥ 2m, (9) ations can change their distribution. Hence, for the simulation of a pulsar’s magnetosphere, we need to compute the probability of interaction of a strong photon on the back- 1 e2 −15 In international units re = 2 ' 2.8179 ·10 meters, with ground distribution fw of weak photons. Indeed, it does not 4π0 mc 0 the electric permittivity of vacuum. matter which weak photon is annihilated but we want to up- 2 include a factor c in the definition of j when it is not assumed date f s as well as the lepton distributions with the outcome that c = 1. of the reactions. With our representation of the involved

MNRAS 000,1–19 (2017) 4 G. Voisin et al.

4 4 particles, this amounts to compute the probability dW of Equivalently, L− is the subset of R × R parametrized by creating a lepton of four-momentum P from a photon Ks, Kw with the following constraints: − − dW = dWK →P . (11) P+ = Kw (P− Ks )(a), s 2 2 4 kP+ k = m c (b), For example, one could think of high-energy synchrotron  k k2  Kw = 0 (c), (16) or curvature photon emitted above the polar cap of a pul-  0  Kw ≥ 0 (d), sar and flying through a stream of thermal photons emitted   P0 ≥ mc2 (e). by the crust. Let us notice that the probabilities of making  +  a positron or an electron are the same, and that a four- Condition (a) expresses the conservation of four-momentum.  0 momentum has four components but only three are inde- Conditions (b) and (e) come from P+ ∈ Πm, and conditions pendent since kPk2 = m2c4. These three free parameters can (c) and (d) come from Kw ∈ Π0. We can compute the num- be parametrized by one direction (two parameters) and the ber of degrees of freedom in L−. The set L− is a subset of energy of the particle. Πm × Π0 of dimension 8. The condition (a) on quadrivectors substracts 4 degrees of freedom. The conditions (b) and (c) both substract 1 degree of freedom. We are left with a set L− of dimension 2. 3 PROBABILITY OF REACTION FOR A Some of the conditions in Eq (16) are already incorpo- GIVEN PHOTON DISTRIBUTION rated in the solution of our problem. Condition (c) is already Quite naturally, the desired probability is the sum over all implicitly met in Eq. (14). Condition (a) is also implicitly met by the set of variables used. Only (b) is not straightfor- the possible reactions involving a photon Ks from the back- ward, since P is not directly part of the variables of inte- ground, that would produce an electron at P− (respectively + gration. One can still convert it into a condition on the three a positron at P+), other four-vectors by putting (a) into (b) and using (c), the X three following equalities being equivalent: dWfw (Ks, P−) = dw(Kw , Ks, P)× Nw × Ns, 2 2 4 L−={(Kw,P+)/Ks →P− } kP+ k = m c 2 2 4 (12) kKw − (P− − Ks )k = m c (17) 0 0 0 ~ 0 where Nw and Ns are the number of photons of four- Kw (Ks − P−) − ks − p~− Kw cos ξ = Ks · P−, (18) momentums Kw and Ks respectively within the interaction ξ = angle(k~s − p~−, k~w ) (19) volume V. In spite of greater simplicity in the CM frame, ~ 0 ~ we must use Eq. (6) in the laboratory frame, because the where kw = Kw . The limit case where ks = p~−, for which CM frame would be different for each of the summed pairs cos ξ is not defined, is physically impossible because Eq. (18) 0 of photons. Since low energy photons are parametrized con- would imply Kw < 0, in contradiction with condition (d). tinuously, we must change our sum for an integral, which With some algebra, we can show that Ks · P− ≥ 0 and that 0 yields the condition |cos ξ| ≤ 1 imposes Kw > min, where ~ 3 3 ~ K · P− Nw → fw (x~w , kw )d ~xd kw , = s (20) Z min 0 0 c K · K k~ − p~− + K − P dW (K , P ) = N s w f (~x, k~ )d6Ω, s s − fw s − s 0 0 w w L− V Ks Kw 0 0 More precisely K w ([−1,cos ξ0[) = [ min,+∞[ and Kw (cos ξ > 6 2 3 3 ~ where d Ω = d σd ~xd kw . (13) cos ξ0) < 0. We can distinguish three regimes of approximation: We assume that strong photons are spread out in space q such that their density does not vary on the interaction vol- 0 2 2 Ks >> p− : min ∼ m + p− − p− cos θ, (21) ume V such that their local density is ns = Ns/V. Conse- r quently the differential probability of interaction per unit 0 0 1 − cos θ K ∼ p− : ∼ K , time reads s min s 2 0 Ks << p− : min ∼ p−. (22)

dWfw (Ks, P−) = nsWk , (14) For further approximations, we consider that the weak- Z cK · K photon distribution has a cut-oﬀ at = max < m/4. W = d2σ s w f (~x, k~ )d3k~ , k 0 0 w w w L− Ks Kw k m/4 > max = 128keV. (23) where the volume element dV = d3~x. Because the weak distribution function is in the most extreme case composed of thermal X-rays typically in the range 1 − 10 keV for a pulsar, this approximation is reasonable. 3.1 The domain of integration

Let us precisely deﬁne the domain L− of integration. We 4 2 4 0 4 GENERAL SOLUTION note Πα = {P ∈ R : kPk = α c , P ≥ |α|} such that Πm is the set of lepton four-momenta (m being the mass of the 4.1 Energy spectrum electron) and Π0 is the set of photon four-momenta. Then, The probability of interaction depends on the integral Wk L−(P−, Ks ) = {(P+, Kw ) ∈ Πm × Π0 : Kw − P+ = P− − Ks }. (15) deﬁned in Eq. (14). In this section, Wk is directly expressed

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 5

as a multiple integral with explicit boundaries. The results fw depends on the angle φw , which allows a direct integra- exposed in this section can be used directly for applications. tion defining the function ( see also (A50)) The path followed to compute them are described in ap- Z 2π 2 pendixA. A summary of the notations and useful relations Fw (r, µ) = fw r, φw , z(r , µ) dφw , (36) φ is given in appendixB.The new expression of Wk involves w =0 new variables that appear both in the integrand and in the where z(r2, µ) is defined in equation (A47) by boundaries of the integral. First, new notations are intro- 2 ! duced, for shorter formulas, k r max z(r2, µ) = − 4µ . (37) 4 µkmax k Ks ≡ k,~k = k~z , (24) The integral Wk in (28) is approximated by P ≡ P0, p~ , (25) Z p Z µ 4 Z R ≡ 2 2 X ai Kw ,~x = (x, y, z) , (26) W = c2π dp dµ 2Fw (r, µ)rdr. (38) ~k µi θ ≡ angle(~k, p~). (27) p1 µ1 i=1 r=0 Here, the boundaries of the integration domain are left arbi- With the new notations related to P and to K , the inte- − s trarily. The reaction probability integrated other every out- gration set L (P , K ) can be rewritten L (p,cos θ, k). Let − − s − going momenta can be computed as well. In this case the µ Ω be the set of angular components of the electron P−, we integral is taken from µmin to 1 and p ranges between k/2 rewrite Wk as and a maximum pmax defined such as µmin(p = pmax) = 1. Z Z 2 We find d σ Ks · Kw 3 W = c dΩ fw (k~w )d k~w . (28) s ~k dΩ 0 0 Ω L− Ks Kw k m2 pmax = 1 + 1 − (39) We wish to compute the probability of making a pair of 2 kmax * + which the electron P− is in a volume of phase space defined . / by The spectrum of outgoing lepton energy is readily obtained as , - k/2 < p1 < p < p2 < k dW Z 1 4 Z R ~k X ai (cos θ, φ) ∈ Ω = [C1,C2] × [0,2π] with Cmin ≤ C1 < C2 < 1 (max (p, k − p)) = c2π dµ 2Fw (r, p, µ)rdr. dp µi µmin i=1 r=0 where p , p ,C and C can be set arbitrarily as long as the 1 2 1 2 (40) above inequalities are correct. After the computations exposed in sectionA, Wk is transformed into a multiple integral with explicit boundaries. Before showing it, a new set of 4.2 Angular spectrum variables is introduced. The parameter µ parametrizes cos θ, It is also possible to compute the angular spectrum of the outgoing leptons. The problem has to be split in two, 2 ! 2(k − p) m whether one consider the higher-energy particle (p > k/2) or cos θ = 1 − µmax − . (29) kp 2p2 the lower-energy particle (p < k/2). For the higher-energy particle, one takes equation (38) It varies in an interval µ ∈ [µ ,1] where µ = 1 corresponds min and changes variable µ to c = 1 − cos θ using equation (29). to cos θ = c and µ is such that cos θ = 1, θ min min One then obtains 1 km m 0 dW Z p2 4 Z R µmin = (30) ~k X ai 4 p(k − p) max = 2πc dp 2Fw (r, p,cθ )rdr (41) dc ci θ p1 i=1 r=0 We define the dimensionless coefficients ai (p), where 2 2 2 m k − 2kp + 2p dµ kp a (p) = − , (31) 0 1 2 ai = ai = ai , (42) 8kp (k − p) dcθ 2(k − p)max m4 k2 − 4kp + 2p2 and the domain of integration has the following limits a (p) = − , (32) 2 3 16maxkp (p − k) pmin = k/2 ≤ p1 ≤ p2 ≤ pmax (43) m6(3k − 2p) a (p) = , (33) with 3 2 3 2 32maxp (k − p) r m2 c m2 8 1 + 1 − − θ km maxk 22 a (p) = − , (34) maxk max 4 3 4 2 pmax = , (44) 64maxp (k − p) m 2max/m + cθ k/m

and obtained by inverting eq. (30). The limits for cθ are given p by R = 2max µ(1 − µ). (35) 0 ≤ cθ ≤ cθ max (45) In the following we do not write the p dependance of the ai coeﬃcients except when otherwise stated. It is convenient to with 2 ! express the weak-photon three-momentum in cylindrical co- max m cθ max = 2 − , (46) ordinates ~x = (r, φw , z). Then, only the distribution function k k2

MNRAS 000,1–19 (2017) 6 G. Voisin et al.

for which pmax = k/2 + (1). where one checks that rξ2 < rξ1 . We need the intersection

In virtue of 37, Fw now depends explicitly on p, hence [rξ2 ,rξ1 ] ∩ [0, Rmax] which implies solving for Rmax(µξ1,2 ) =

the dependence in (41). rξ1,2 , which gives us For the lower-energy lepton, we need first to establish 2 the kinematic relation between its outgoing angle defined 1 sin ξ1,2 µξ = (57) 0 − 0 1,2 2 1 + cos ξ by cos θ = 1 cθ and the higher-energy lepton variables. All 1,2 primed quantities refer to the lower-energy lepton. Taking where one checks that µξ2 > µξ1 . We can rewrite the energy the strong-photon direction along the z axis we have the spectrum (40) as relation 0 0 pz 1 − c = . (47) dW~ θ p0 k (max (p, k − p)) = (58) dp Using the conservation of momentum (16)a) one can express Z Z 0 min max µmin, µξ1 ,1 rξ1 cθ to leading order 2 2πcre dµ dr + 2 µmin max 0,rξ 1 − p˜ m˜ − p˜ * 2 0 1−p˜ p˜ 2 . 1 − c = + (1) (48) Z min µ ,1 Z R 4 θ 1/2 ξ2 X ai m˜ 2 2 , dµ dr 2F (r, µ)r. 1 − p˜ − − m˜ 2 i w 2p ˜ µ , µ , ,r µ min max min ξ1 1 max 0 ξ2 i=1 /+ where every quantities tilded quantity is expressed in unit Concerning the angular spectrum, nothing more needs - of k, a˜ = a/k. Since there is no dependence on cθ one can to be done for lower-energy leptons, and for higher-energy directly deduce that leptons we proceed similarly as for the energy spectrum dW~ dW~ dp above. Starting from (41) one needs to replace µ by its ex- k = k , (49) dc0 dp dc0 pression as a function of cθ and p in R. This allows us to θ θ define the p analogs of µξ1,2 by which can be expressed using q 1 + r¯2 max + 1 − r¯2 dc0 2m ˜ 2 m˜ 4 + 2(1 − p˜)2p˜(3p ˜ − 1) + m˜ 2(1 − p˜(p˜(5 − 2p ˜) + 2) ξ1,2 cθ k ξ1,2 θ pξ = k , (59) = 1,2 k 2 max p 3/2 2 + cθ + r¯ d k(1 − p˜)2 m˜ 4 − 4m ˜ 2p˜ + 4(1 − p˜)2p˜2 max ξ1,2 cθ k

(50) where one shows that pξ1 < pξ2 . The spectrum for higher- energy leptons is then obtained from (41) 0 after numerical inversion of (48). One finds that cθ is a monotonously increasing function of p and that dW~ k = (60) 2 dc 0 m c (k/2) = 4 , (51) Z min max p , p , p Z r θ k2 min ξ1 max ξ1 s c2π dp dr + 2 2 pmin max 0,rξ 0 max m * 2 c (p ) = 1 + 1 − . (52) . θ max 2 Z Z 4 0 m kmax min pξ2, pmax R X a * + , dp dr i 2F (r, µ)r. . / i w p , p , p ,r µ min max min ξ1 max max 0 ξ2 i=1 , - + 4.3 With conic boundary conditions / The weak-photon distribution (36) is defined by - We consider the case where the soft photon distribution is X (n,m) n m defined everywhere between two cones of axis ~k and of half- Fw (,¯ Cξ ) = Fw ¯ Cξ . (61) n,m apertures 0 ≤ ξ1 < ξ2 ≤ π, and the distribution Fw (36) is given as a function of the coordinates (¯ = /max,Cξ = Integrations over r in (58) and (60) yield expressions of cos ξ), where we deduce from (A48) and (A45) the type 2 r¯ P4 ai R r2 ¯ = + µ, (53) i 2Fw (r, µ)rdr = 4µ i=1 µ r1 ! m r 2 P4 P Pm − l l−i+1 n−l 2 2µ 2max i= ai n,m ( 2) µ r¯¯(r¯, µ) C = 1 − , (54) l=0 l r1 ξ ¯ f g (62) where r¯ = r/max. We are now looking for the appropriate boundary con- where ditions to apply to integral (38). Using the fact that ¯(r¯ , µ)n−l+1−¯(r¯ , µ)n−l+1 r 2 1 − − 2 n−l 2 2µ n−l+1 if n l , 1 π z(r , µ) r¯¯ (r¯, µ) = ¯(r¯ , µ) . tan − ξ = (55) r1  2µ log 2 if n − l = 1 2 r f g  ¯(r¯1, µ)  where z(r2, µ) is defined in eq. (37), one finds the new bound-  (63)  aries in r by inverting this relation. The resulting r bound- To obtain the final spectrum (58) (resp. (60)), integra- aries are given by tion over µ (resp. over p) is possible analytically : the first ! 1 1 line of (63) is a rational fraction that can be integrated rξ1,2 = 2max µ + , (56) tan ξ1,2 sin ξ1,2 throught partial fraction decomposition and the second

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 7 R line yields expressions of the type xk log(polynomial(x))dx ξ = π/2 one obtains from (67) an estimate of ζ for the peak (where k is integral) which values are given in most relevant of reactions textbooks such as (Gradshteyn et al. 2000). However, the kkbT resulting expressions may be lengthy and a direct numerical ζ = ' 0.8 (68) m2 integration might sometimes be more efficient. which is the right order of magnitude. One could argue that at such energies reactions would occur more face-on, meaning ξ < π/2 which is consistent with the higher peak position 5 APPLICATIONS found on figure2. We now proceed to the computation of pair energy spectra. 5.1 Isotropic black-body background distribution Figure3 shows the pair-creation spectra for different values of . Those spectra are directly computed using equa- Here we propose to check our approximation eq. (38) ζ tion (38) and expressed as a function of which allows the against the exact isotropic case described in Nikishov(1962); p/k same scaling law as in equation (66), with the momentum Agaronyan et al.(1983); Boettcher & Schlickeiser(1997). p of one of the created leptons, and normalize each spectrum We assume a high-energy photon hitting on a thermal soft to unity such that the obtained spectral shape are universal photon background given by i.e. do not depend on the temperature of the black-body or 2 1 on the absolute value of k, but only on ζ. The shape and fbb() = (64) (~c2π)3 e/kBT − 1 evolution of the spectra with the strong-photon energy is consistent with Agaronyan et al.(1983). In this paper, the where T is the temperature of the body and k the Boltz- B authors consider spectra resulting from the reaction of two mann constant. We choose a cutoff = 20T (see Eq. max isotropic monoenergetic photon distributions with energies 23) such that the neglected part of the black-body spectrum and k that are symmetrical with respect to (k + )/2. Here, (64) represents less than ∼ e−20 ∼ 10−9 the total amount every spectrum is symmetrical with respect to p/k = 0.5 as of background photons. We perform a Chebyshev interpo- result of neglecting (/k) terms. Besides the shape of these lation (see e.g. Grandclément & Novak(2009)) of 2 f () bb spectra is very reminiscing of pair-creation in the photon- on the 25 first Chebyshev polynomials achieving a relative plus-magnetic-field process that is well-known in the field of accuracy better than one thousandth everywhere and better neutron-star magnetospheres (Daugherty & Harding 1983). than 10−6 for 0 ≤ /T ≤ 10, energies between which most The analogy is not fortuitous since the latter process can in of the photons are. This then allows us to derive the coeffi- principle be seen as the interaction of a strong photon with cients of the Laurent serie describing f with poles of order bb an assembly of magnetic-field photons. We see on figure3 one and two. Then, we produce the spectra of figure3 and that each spectrum is made of two peaks that move apart 2. and become narrower and weaker as the reaction occurs far- On the top panel of figure2 we plot the total probability ther above threshold. Notice that the narrowing is relative of absorbing a strong-photon of energy k as a function of to the momentum span and not absolute. kk T The separation of the peaks at higher energies results ζ = b . (65) (mc2)2 from the fact that the cross-section favors alignment of in- going and outgoing particles in the center-of-mass frame if This parametrization by ζ makes the temperature depen- the energy is much larger than the threshold energy. It fol- dency simple lows that a Lorentz boost to the observer’s frame along this !3 axis results in a low-energy and a high-energy particle. The k T W ∝ B . (66) intensity of the peaks of course depends on the background ~k 2 mc distribution, but also on the cross-section which decays as Here we choose to take T = 2.7K which allows to reproduce log(τ)/τ (see section 2.1). The latter dependency explains the result of Gould & Schréder(1966) (dashed line) concern- the above-threshold decrease of the peak intensity and the ing absorption on the cosmic microwave background. The former explains the below-threshold decrease, as shown on lower panel of figure2 shows the ratio between our formula the lower panel of figure3. One notices that spectra are not and the exact formula of Nikishov(1962). It shows that our smooth in their center, which is naturally explained by our result is fifty percent off at ζ < 1 and asymptotically tends approximations that ensure continuity at the center but not to the correct value for large ζ, the difference between the continuity of derivatives. two curves is ∼ 10% around the maximum of the curve located at ζ ∼ 2. On average on the range plotted on fig.2, our formula overestimates the reaction rate by 7%. 5.2 Above a hot neutron star A toy model can help us understand the shape of this In this section, we consider a homogeneously hot neutron curve. The peak of a black-body spectrum is roughly at star at temperature T and two kinds of photons : the down bb ' 5kbT. The cross-section peaks when the center-of-mass photons and the up photons. Down photons are going radi- energy is 1.4(2m), so if one approximates the black-body ally toward the center of the star while up photons are going spectrum to its peak one gets in the opposite direction, away from the star. This configura- 2 tion aims at approximating a pulsar magnetic pole. Indeed, bbk(1 − cos ξ) ' 3.9m . (67) in a pulsar magnetosphere high-energy photons are expected For an isotropic distribution of soft photons, collisions take to be mostly created by curvature radiation of electrons and place at every angle ξ ∈ [0, π]. Taking the intermediate value positrons flowing along magnetic field lines that can be con-

MNRAS 000,1–19 (2017) 8 G. Voisin et al.

10-20 Nikishov 10-21 VMB

10-22

10-23

10-24

Proba / meter strong photon 10-25 10-1 100 101 102 103 104 105 ζ =(kk T)/(mc2 )2 1.2 B 1.1 1.0 0.9 0.8 0.7

VMB/Nikishov 0.6 0.5 0.4 10-1 100 101 102 103 104 105 2 2 ζ =(kkB T)/(mc )

Figure 2. Comparison of absorption of high-energy photons on a black-body background with Nikishov’s formula (dashed line) and with our’s (VMB, plain line). The scaling (66) is that of a black-body at Tbb = 2.7 K (see formula (66)) to give an estimate of the effect of the cosmic-microwave background. In this case the energy of the strong photons ranges between k ∼ 100 TeV and k ∼ 108 TeV. The bottom panel shows the ratio between the two theories . The ratio of probabilities averaged over k is about 1.07. The peak of our curve occurs around 2.6m2/T while Nikishov’s is around 1.9m2/T . The ratio between the two curves at the position of our peak is approximately 1.01. sidered radial at low altitudes above the poles. Note that ton with respect to the star, and in particular the up pho- we do not consider only a hot cap here but the full star, as tons. means of geometrical simplification. In this configuration, the distribution of soft photons is still given by eq. (64) except that it is now zero when the The case of pair production from photon-photon colli- angle ξ between the soft and the strong photon is beyond sions in pulsar magnetospheres was studied by authors such the horizon of the star as seen from the strong photon (see as Zhang & Qiao(1998); Harding et al.(2002). In these pa- figure4). For a photon going upward, the horizon is defined pers, the authors generalize the formula of Nikishov(1962) by with a minimum energy threshold for the background dis- −1 tribution corrected by a factor (1 − cos θc ) where θc is the R∗ sin ξ < = sin ξhorizon (69) maximum viewing angle on the hot polar cap of the star. R∗ + h In other words, they consider an isotropic black-body distri- where R is the radius of the star (typically 10km) and h is bution where only photons within the viewing angle of the ∗ the height above its surface. Consequently, we use eqs. (58) cap contribute, however with a threshold energy that corre- and (60) with angles ξ = 0, ξ = ξ for a up photon and sponds to the largest incidence angle only since the threshold 1 2 horizon ξ = π − ξ , ξ = π for a down photon. does not depend on the location of the emitter on the cap. 1 horizon 2 Figure5 shows the probability of reaction per unit Therefore, this approximation overestimates the threshold length (we will sometimes say ”reaction rate”) as a func- which generally translates in underestimating the reaction tion of ζ at various heights h above the cap (left panel), rate. This has little consequences when the viewing angle and as a function of h at various ζ (right panel). As in the is wide, which is the case very close to the cap. However, previous subsection, the temperature dependance is T3 for one expects a faster decrease as one goes away from the cap a given value of ζ. All the figures in this section are made and the factor (1 − cos θ )−1 grows larger. As an example, c with a fiducial temperature of 106K. With this value the the authors of Zhang & Qiao(1998) compute a maximum conversion from ζ to k is : k ' 5.9 · 103ζmc2. At the lowest reaction probability of 5.7 · 10−5 m−1 at a viewing angle of altitude we computed, h = 10−3 R , the peak of the reaction 90◦ when we get 6.7 · 10−5m−1 (see peak of the down-photon ∗ rate is around ζ = 1.6 for down photons and about an order h = 10−3 curve on figure5 for an estimate), but they obtain of magnitude higher for up photons ζ ' 16. This is a direct only 6.3 · 10−6 m−1 at 45◦ when we still have a probability of consequence of the threshold eq. (67) given the less favorable 4.3 · 10−5 m−1 (see their equation 9, for T = 106 Kelvins)). incidence angles of up photons. Another point is that the po- Besides, an interest of our formalism is that it can in sition of the maximum shifts to lower ζ as height increases principle deal with any other orientation of the strong pho- for down photons, but to larger ζs for up photons. As can

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 9

1.0

0.8 1 ζ =2 10− · 0.6 ζ =1 ζ =1 101 · 0.4 ζ =1 102 · ζ =1 103 0.2 ·

0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Probability / momentum unit length (unit of the peak) Lepton momentum / Strong photon momentum -29

r 10 e t e m

) c -30 m

10 f o

t i n u (

m -31

u 10 t n e m o m

/ -32

y 10 t i l i b a b o r -33 P 10 10-1 100 101 102 103 2 2 ζ =(kkB T)/(mc ) (dimensonless)

Figure 3. Spectra of outgoing leptons (electron or positron) for different strong-photon momenta k on a black-body background at temperature Tbb (top panel). m is the mass of the electron, the speed of light and the Boltzmann constant are taken to be unity. The amplitudes are normalized to the amplitude of the peaks of each spectrum, and these amplitudes are reported on the lower panel. As in figure2 these amplitudes are normalized to correspond to the cosmic-microwave background. The most intense peaks arise around a 2 momentum k such that its reaction with a background photon at Tbb is at threshold, i.e. kTbb ∼ m . The more above threshold, the more separated, narrow and low the peaks are. The separation of the peaks can be understood as a mere relativistic-frame effect, by analogy with a two-photon collision. be seen on the right-hand-side panel, the reaction probabil- tons. Because soft photons are arriving ”from behind”, there ity per unit length is fairly stable (within a factor of two) is always an inner cone inside which the reactions are below until ∼ 1R∗, after which it decays very sharply. The decay is threshold, and an outer cone limited by the angle beyond sharper as ζ increases for down photons and smoother for up which the cross-section is too small if ζ 1 or the view- photons, which explains the crossing between some curves ing angle if the strong-photon energy is small enough. The on the right-hand-side panel. lower the energy of the strong photon the wider the outer A qualitative reasoning explains these behaviors. For a cone and the most sensitive to viewing angle the reaction rate is. That explains why, contrary to down photons, the low-energy down photon (i.e. ζ . 1), most of the soft photons most likely to react are in a narrow cone almost face-on reaction rate decays slower with altitude when ζ is larger on with the strong photon. The aperture of the cone defines the figure5. With this reasoning, one also understands why the limit beyond which the reaction is below threshold. When energy of the reaction-rate peak (left panel) is quite stable the strong photon is higher, the almost face-on soft pho- at low altitudes and becomes smaller for down photons at tons are the last to disappear because of the shrinking of high altitudes (& 1R∗) or larger for up photons. the viewing angle. As energy rises, this cone becomes wider Figure6 shows the optical depth of strong photons as since soft photons provoking a near-threshold reaction are a function of ζ through 10R∗ from the surface. The optical located at a wider angle according to formula (67). Inside depth is defined by the first cone also appears a co-axial cone with a narrower Z 10R∗ aperture inside which photons are not contributing signif- τζ (10R∗) = Wζ (h)dh. (70) icantly anymore, since reactions are too far above thresh- 0 old (and therefore the cross-section is too small) because Because of the effects mentioned above the peak for down of small incidence angles. At large strong-photon energies photons is slightly shifted downward at ζ ' 1.4 while upward (ζ 1), the soft photons close to the outer cone are the for up photons at ζ ' 30. The corresponding typical Lorentz first to disappear when the viewing angle shrinks because of factors of the created particles are 8 · 103 and 2 · 105 respec- a larger height. This explains the faster decay of the reac- tively. The peak optical depths are respectively τ ' 8.8 and tion probability with h for larger ζs of down-photon curves τ ' 0.23 at a temperature of 106 Kelvins. One concludes that on figure5. The same kind of reasoning applies for up pho- at this temperature more than three out of four up photons

MNRAS 000,1–19 (2017) 10 G. Voisin et al.

of viewing angle. The same thing applies for the position ξhorizon of the most energetic peak pp (and the least energetic at k − pp). Down-photon peaks are wide wp ∼ 0.45 for ζ . 2 π - ξhorizon and then sharply narrow while their position smoothly goes from pp/k ∼ 0.8 to pp/k . 1 at large ζs. On the contrary, up photons are very sensitive to altitude, which is explained by the fact that the higher above the star, the narrower the viewing angle and therefore the incidence angle, and the more energetic up photons need to be for the reaction to be at or above threshold. As a consequence up-photons peaks are very centered at low values of ζs, with pp/k ∼ 0.6, and h are even more centered at higher altitudes. With ζ rising, the energy distribution becomes increasingly asymmetric as − pp/k → 1 , although it takes a larger ζ at higher altitude. Similarly, peak widths are growing with ζ until a maximum wp ∼ 0.45 at a ζ all the more large that altitude is high, after which wp drops sharply. This sharp change of slope happens because the two peaks separate (see comment of ﬁgure7 ). Figure8 shows the normalized angular spectra for both up and down photons, and both higher-energy (p > k/2) and lower-energy (p < k/2) outgoing leptons at various val- R* ues of ζ. It is remarkable that apart from their amplitudes O (not visible on this normalized plot), these spectra do not change much with height apart at large and very unlikely

Figure 4. A neutron star of center O and radius R∗ above which angles, and therefore we limit ourselves to only one height. a up photon and a down photon are represented by radial arrows These spectra are monotonously decreasing as the angle be- of opposite directions. Both photons are represented at an height comes larger, and the larger ζ the larger the outgoing an- h above the surface of the star. From this height, they can interact gle. Lower-energy leptons have larger outgoing angles than with soft photons coming from the surface of the star within a their higher-energy counterpart and are not created below cone of aperture ξhorizon, eq. (69), represented by dashed lines. a minimum angle defined in equation (51). For a given ζ, The incidence angle between the strong photon and soft photons leptons created from down photons are always going out at therefore lies between 0 and ξhorizon for the up photon (purple larger angles, and the difference is growing at larger angles upward arrow), and between π−ξ and π for the down photon horizon of the spectrum. In a pulsar magnetosphere the outgoing (blue downward arrow). may be important because the pairs will radiate more or less synchrotron radiation depending on their momentum at the peak energy escape the magnetosphere if no other perpendicular to the local magnetic field. We see here that reaction or source of soft photons opacifies it. The mag- the angles with respect to the progenitor strong photon are netosphere may become opaque if the star is hotter than overall very small, which is expected from relativistic col- ∼ 1.6 · 106K, temperature for which the maximum optical limation. If one assumes that strong photons are produced depth reaches 1 owing to the T3 dependence of the reaction though curvature radiation along the magnetic-field lines, rate. Down photons with ζ between ∼ 0.25 and ∼ 64 have then the angle distributions presented on figure8 matter optical depths larger than unity and therefore are absorbed only if the mean free path is much shorter than the radius before they hit the star except if they are emitted at very of curvature of the field line. This is not the case with the low altitudes h R∗. The maximum optical depth of down parameters presented in this section, and would probably photons is below one, namely the magnetosphere is trans- require an extra source of photons. parent, for a temperature below 0.5 · 106K. Let’s notice that our approximation of a uniformly hot star obviously leads to overestimating the optical depth on distances larger than 6 DISCUSSION the size of an actual hotspot. Figure7 shows the energy spectra of the created leptons Recent simulations of aligned millisecond-pulsar magneto- (left panel) and the evolution of the position and widths of spheres indicate that significant pair production may occur the peaks as a function of ζ at various heights (right panel). near the so-called separatrix gap and y point (see Cerutti The spectra have the same double-peaked structure as in the & Beloborodov(2016) and references therein) near the light isotropic case (figure3 ) but evolve differently depending on cylinder. This implies that the source of pairs be photon- the orientation of the strong photon. The general principle photon collisions. However, in the most detailed modeling is the same : the more above threshold the more separated of pair creation realized by Chen & Beloborodov(2014), peaks, with the consequence that they narrow when they photon-photon pairs are created with a constant and uni- get close to the limits of p/k ∈ [0,1]. For down photons, form mean free path of 2R∗. If one assumes that the source the width of the peaks wp (in unit of k) has very little de- of soft photons is only provided by the star, this assump- pendence on altitudes which is due to the fact that for the tion seems reasonable close to the star, h < 2R∗, but greatly range of ζ . 20 visible on this plot (right panel), the ef- underestimated beyond owing to the exponential cutoff of ficient soft photons are mostly face-on and suffer no effect the reaction rate with altitude (figure5). This issue can be

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 11

10-4 10-4 3 Up, h = 1 10− R Up, ζ =1.6 · ∗ Up, h = 1R Up, ζ =1.6 101 ∗ · Up, h = 5R Up, ζ =1.6 103 ∗ · 3 1 Down, h = 10− R Down, ζ =1.6 10− ∗ · -5 Down, h = 1R -5 Down, ζ =1.6 10 ∗ 10 Down, h = 101 R Down, ζ =1.6 102 ∗ · Down, h = 102 R ∗ ) ) K K

6 6 0 0 1 1 = -6 = -6 T T

10 10

r r o o F F ( (

1 1 − − s s r r e e t t e e m m

10-7 10-7

10-8 10-8

10-1 100 101 102 103 104 10-3 10-2 10-1 100 101 102 ζ =(kk T)/(mc2 )2 Height (in R ) B ∗

2 2 Figure 5. Probability of reaction per meter of a strong photon of momentum k as a function of ζ = (kkBT )/mc ) and height h above a star of radius R∗. Up-triangle markers represent photons going radially up from the star. Down-triangle markers represent photons going down to the star. The probability scales like T 3, according to equation (66), and is here represented using a fiducial T = 106K. Left-hand-side panel : probability as a function of ζ at various heights. Right-hand-side panel : probability as a function of height h at various ζ. overcome if another source of soft photons can be found, & Erber 1975; Daugherty & Harding 1983) resulting for example from synchrotron radiation near the ! light cylinder. Moreover, in these simulations, the direction 9 B sin θ 4 −1 ~ω B sin θ W ~ ' 4.3·10 exp − m with χ = , of strong photons relative to the soft-photon sources is not γB χ1 Bc 3 χ 2mc2 Bc taken into account, which can have an effect of several orders (71) of magnitude on reaction rates with a strong dependence on strong-photon energies (see figure6). The energy separation where ~ω is the energy of the gamma photon, B the intensity of the two outgoing leptons (figure7) may also have an im- of the local magnetic field, and θ the angle between the di- portant impact on the subsequent synchrotron radiated by rection of the magnetic field and the direction of the photon. the pair. Indeed, the synchrotron peak frequency scales like The photon-magnetic-field optical depth heavily depends on γ2, where γ is the Lorentz factor of the particle around the the magnetic-field geometry: typically, a photon in the pul- magnetic field. Therefore, a typical situation in which the sar magnetosphere is emitted parallel to the local magnetic higher-energy lepton takes 10 times more energy than the field due to relativistic beaming, thus starting with a reac- other (dotted line on figure5) results in two synchrotron tion rate which increases along the propagation as WγB~ = 0 θ peaks radiated two orders of magnitude apart. This situa- increases. The upper limit of the reaction rate of the photon tion is reached at values of ζ for which the optical depth on can be estimated by considering sin θ = 1 everywhere, al- figure6 is still high i.e. more than half the peak value. Notice though this is bound to largely overestimate the probability that we implicitly assume here that both particles share the of creating a pair. same angle with respect to the local magnetic field, which is Figure9 shows a comparison between the reaction rate justified by small outgoing angles shown on figure8. of equation (71) for a range of values of B sin θ versus the photon-photon reaction rates computed at various altitudes The photon-photon mechanism competes with the of figure5. Considering that the range of probable photon photon-magnetic-field mechanism γ + B~ → e+e− for the cre- energies lies below 100 GeV one sees that the photon-photon ation of pairs in pulsar magnetospheres. For our present dis- mechanism can dominate near the surface of millisecond 5 cussion, we focus on magnetic fields smaller than the critical pulsars where B sin θ . 10 Teslas, and clearly dominates 9 magnetic field Bc = 4.4 · 10 Teslas. Additionally the photon for down photons at altitudes h ≥ 10R∗ where, assuming −3 energies produced by curvature or synchrocurvature radia- a dipolar magnetic field where B ∝ (R∗ + h) , one has tions in pulsar magnetospheres cannot exceed ∼ 100 GeV B sin θ . 100 Teslas for millisecond pulsars. For up photons, owing to radiation reaction (see e.g. Viganòet al.(2014)). it is less clear which mechanism dominates without taking With these two limits, the photon-magnetic-field reaction into account a particular magnetic geometry. It should also rate can be computed with the asymptotic expression (Tsai be noted that, if a comparable soft-photon density can be

MNRAS 000,1–19 (2017) 12 G. Voisin et al.

101 Down Up

0 ) 10 K

6 0 1

r o F (

h t p

e -1

d 10

l a c i t p O

10-2

10-1 100 101 102 103 104 2 2 ζ =(kkB T)/(mc )

Figure 6. Reaction rate in equation (28) integrated from 0 to 10R∗ for up and down photons as a function ζ. The amplitudes are valid for a homogeneously hot star of temperature T = 106K, and can be converted to other temperatures using the T 3 scaling law (66). For photons going down toward the star, the peak is at ζ ' 1.4 with an amplitude of ' 8.8. For photons going up away from the star, the peak is at ζ ' 30 with an amplitude of ' 0.23. achieved in the outer magnetosphere as is necessary in some case the strong photon) located at inﬁnity and are therefore recent simulations (e.g. Chen & Beloborodov(2014)), the inapplicable in the present case where we consider reactions −3 photon-photon mechanism would be largely dominating the between 10 R∗ and 100R∗. Instead, one would need to com- photon-magnetic-ﬁeld mechanism in this region of the mag- pute numerically the geodesics followed by the soft photons netosphere. between the surface and the strong photon.

We have neglected two effects possibly important in our The second neglected effect is the effect of a strong mag- application to a hot neutron star, section 5.2: general rel- netic field on the cross section for photon-photon pair cre- ativity and the effect of the strong magnetic field on the ation. It has been worked out by Kozlenkov & Mitrofanov pair-production cross section. The former effect, general rel- (1986). This cross section shows a sawtooth behavior at en- ativity, redshifts the spectrum of soft photons, and enlarges ergies corresponding to the quantified Landau levels of the the visible horizon of the star. Indeed, light bending due to outgoing leptons. Unfortunately, this cross section is very the gravitational field of the star curves the trajectories of unwieldy for practical calculations. However, its effect is low soft photons in such a way that part of the surface beyond or moderate in magnetic fields much lower than the criti- the geometrical horizon of the star becomes “visible” by the cal field Bc , which fortunately corresponds to the range of strong photon. Simple analytical formulas for the effective parameters where photon-photon pair creation can domi- soft-photon distribution with general-relativistic effects have nate over photon-magnetic-field pair production (see above been given by Beloborodov(2002); Turolla & Nobili(2013). and figure9), and therefore the range of interest of our for- However, these expression are valid for an observer (in our malism. As mentioned before, the other important domain

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 13

0 Down 10 1.0 Up

3 p ,h =1.0 10− 0.8 p · pp ,h =1.5 =5 0 101 x pp ,h . a · m ) =1 0 10 3 p wp ,h . − d p ·

/ 0.6 w

wp ,h =1.5 , W k d

( 1 w w / / =5 0 10 p p p wp ,h . p p · d 3 / pp ,h =1.0 10−

W ·

d -1 0.4 10 pp ,h =1.5 3 w ,h =1.0 10− p · wp ,h =1.5

0.2

p /k 0.0 p 0.0 0.2 0.4 0.6 0.8 1.0 100 101 102 103 104 105 p/k 2 2 ζ =kkB T/(mc )

Figure 7. Left-hand-side panel : example of two normalized spectra of energy of created particles. These spectra are normalized to the amplitude of the largest peak, and the energy in abscissa is normalized to the energy of the incident strong photon k. In both cases, k corresponds to ζ = 10 at an altitude h = 0.5R∗, and the only difference resides in the up or down orientation of the strong photon. As in the isotropic case3, spectra are generally made of two peaks more or less thin and separated. The width at half maximum of peaks wp is defined in the two possible cases : if one side of a peak never reaches its half before rising again to another peak in which case the width is taken to be half of the double-peak width, or if the peak is well defined on both sides in which case the definition is straightforward. The position of the most energetic peak p p /k is defined as well. Right-hand-side panel : Evolution with ζ of positions p p /k of the higher-energy peak (curves on the higher part of the plot), and widths at half maximum wp (curves on the lower part of the plot) for up and down photons at various heights h (in units of R∗). Positions are ranging from 0.5 at low ζs which corresponds to a perfectly centered peak or to a null spectrum when a reaction is below threshold (lowest energies of up photons), to ' 0.98 at large ζs. The horizontal dotted line shows the positions at which the ratios between the two peaks is 10. Widths at half maximum are rising to ∼ 0.45 until the two peaks separate and drop sharply to ' 0.029. of application of our formalism is the outer magnetosphere its large validity ( 1) comes if the reaction is far above 2 where the magnetic field is also much smaller than Bc , in- threshold i.e. kmax/m 1. cluding in the case of young pulsars such as the Crab. In section 5.1, we compare our formalism with the exact formula that can be found in the literature (Nikishov(1962), or Agaronyan et al.(1983) eq. 4 and 5 for a more detailed formulation), and show that our approximated formulation gives results accurate at ∼ 7% on average, with ∼ 10% near 7 CONCLUSION the peak and asymptotically tend to the exact value at high We propose a formalism to analyze photon-photon pair energies. However, the difference can be as large as ∼ 50% at creations with an arbitrarily anisotropic soft-photon back- low energies. We show pair spectra that are consistent with ground. This formalism allows to calculate energy and angle those of Agaronyan et al.(1983) in the isotropic case. spectra of outgoing pairs, as given by formulas (58) and (60) In section 5.2 we show that the differences created by respectively. the strong anisotropy of radiation near a hot neutron star Calculations are carried using two approximations : the are much more important than a few percent, potentially first being that the strong photon is much more energetic reaching several orders of magnitude depending on energy, than the soft-photon cutoff energy max, and the second that direction of the strong photon, and altitude above the star. the outgoing higher-energy lepton of momentum p~ be very We consider two directions for strong photons : radially to- aligned with the progenitor strong photon of momentum ~k ward the star (down photons) and away from the star (up in the sense that ~k − p~ / ~k − p~ 1,(A22), where per- ⊥ k photons). In both cases reaction rates are stable until 1R∗, pendicular and parallel components are taken with respect before undergoing an exponential cutoff. However, the peak to ~k. This latter approximation is the most stringent one. of strong-photon absorption occurs at an energy ∼ 10 times Indeed, one can show that the inequality itself (< 1) is al- larger for up photons. Energy pair spectra show two peaks ways true within the frame of our first approximation, but symmetric with respect to k/2, similarly to the isotropic

MNRAS 000,1–19 (2017) 14 G. Voisin et al.

100

10-1 x a m ) Θ d /

W -2 0

d 10 ζ =10 ( / ζ =101 Θ

d 2

/ ζ =10

W ζ =103 d ζ =104 -3 10 ζ =100 ζ =101 ζ =102 ζ =103 ζ =104 10-4 10-7 10-6 10-5 10-4 10-3 Angle (rad)

Figure 8. Angular spectra for various values of ζ = k kb T and both up and down orientations (respectively up-triangle and down- (mc2)2 triangle markers) normalized to their maximum values. Solid lines correspond to angular spectra of the higher energy leptons (p > k/2) and dashed-line spectra to their lower-energy counterparts (p < k/2). All these spectra correspond to a height above the star h = 0.001R∗ which is representative for all other heights. Indeed their amplitudes signiﬁcantly change with height but their shapes (and therefore the shown normalized spectra) barely change.

GeV 1 1 2 3 4 3 10− 3 3 10 3 10 3 10 3 10 · · · · · 3 Up, h = 1 10− R 0 · ∗ 10 Up, h = 1R ∗ Up, h = 5R -1 ∗ 10 3 Down, h = 10− R ∗

) -2 Down, h = 1R

K 10 ∗ 1 6 5 0 5 0 5 0 5 0 Down, h = 10 R ...... 0 5 5 4 4 3 3 2 2 ∗ 0 0 0 0 0 0 0 0 1

-3 1 1 1 1 1 1 1 1 2

= Down, h = 10 R 10 ======

T ∗ θ θ θ θ θ θ θ θ

n n n n n n n n

i i i i i i i i r

s s s s s s s s o

B B B B B B B B

F -4

( 10

1 − s r -5 e

t 10 e m

10-6

10-7

10-8 10-1 100 101 102 103 104 2 2 ζ =(kkB T)/(mc )

Figure 9. Comparison of reaction rates of the γγ → e+e− process (solid lines) versus the γ + B~ → e+e− process (dashed lines). The photon-photon reaction rates are identical to those on the left-hand-side panel of figure5, but a temperature of T = 106 K is taken giving the photon energies in GeV reported on the upper horizontal axis corresponding to the ζ values on the lower axis. The photon-magnetic- field reaction rates are given in the χ 1 approximation (see text) within which they depend only on the the photon energy and the magnetic intensity perpendicular to the photon direction B sin θ (in Teslas), where θ is the angle between the local magnetic field and the photon direction.

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 15

~ case. These peaks separate as the energy of the reaction k β~ rises. We show that such a difference in energy between the two outgoing leptons can importantly affect the synchrotron ~p emission of the pairs for a large range of strong-photon en- ~z ergy compared to a simple model in which both components θ of a pair take away the same energy. These findings are meant to contribute to a better modeling of pair creation from photon-photon collisions in pulsar magnetospheres. Recent millisecond-pulsar-magnetosphere simulations gave an important role to this pair-production mechanism (Cerutti & Beloborodov 2016). However, the current state of modeling leaves an important uncertainty on ~x, ~y the amount of soft photons needed to sustain such pair dis- charges. The results of this work provide means to estimate Figure A1. Coordinate system. In our approximation, ~k is a the mean free path on a soft-photon background resulting quasi symmetry axis. from a homogeneously hot neutron star. Moreover it provides formulas to obtain results with virtually any soft pho- ViganòD., Torres D. F., Hirotani K., Pessah M. E., 2014, Monthly ton distribution, in particular resulting from secondary syn- Notices of the Royal Astronomical Society, 447, 1164 chrotron close to the light cylinder. The possibility to gen- Zhang B., Qiao G. J., 1998, Astronomy and Astrophysics, 338, 62 erate energy spectra allows to differentiate between the two components of a pair and therefore to differentiate their synchrotron emissions. APPENDIX A: DERIVATION OF THE GENERAL RESULT We show the that domain of integration can be approxi- REFERENCES mated by an hyperboloid of revolution. Then, we compute the integral W over this surface assuming that the distri- Agaronyan F. A., Atoyan A. M., Nagapetyan A. M., 1983, Astro- k physics, 19, 187 bution function of weak photons is given by a polynomial Beloborodov A. M., 2002, The Astrophysical Journal Letters, 566, (e.g. Taylor expansion). A variety of notations and relations L85 is used, we summarize them in appendixB. Berestetskii V. B., Lifshitz E. M., Pitaevski L. P., 1982, Quantum electrodynamics. Oxford University Press, Oxford, Royaume- Uni A1 Parametrization of L− by the Boettcher M., Schlickeiser R., 1997, Astronomy and Astrophysics, three-momentum of the weak photons 325, 866 Bonometto S., Rees M. J., 1971, Monthly Notices of the Royal The spectrum of pair creation is the density of probability Astronomical Society, 152, 21 of making a pair as a function of the energy of one of the Burns M. L., Harding A. K., 1984, Astrophysical Journal, 285, particles. By definition, it is symmetric with respect to half 747 of the total energy k + ' k: if one of the particles has Cerutti B., Beloborodov A. M., 2016, Space Science Reviews, pp an energy p then the other has k − p as a result of energy 1–26 conservation. Therefore we consider only the upper half of Chen A. Y., Beloborodov A. M., 2014, Astrophysical Journal Let- the spectrum, for p > k/2 and ters, 795, L22 dW dW Daugherty J. K., Harding A. K., 1983, The Astrophysical Journal, (p) = (k − p) (A1) 273, 761 dp dp Goldreich P., Julian W. H., 1969, Astrophysical Journal, 157, 869 Therefore, we are left with the very helpful ordering Gould R. J., Schréder G., 1966, Physical Review Letters, 16, 252 Gradshteyn I. S., Ryzhik I. M., Jeffrey A., Zwillinger D., 2000, k & p m,max, (A2) Table of Integrals, Series, and Products Grandclément P., Novak J., 2009, Living Reviews in Relativity, which allows to write : ! 12, 1 m2 m2 Harding A. K., Muslimov A. G., Zhang B., 2002, The Astrophys- P0 = p + + . (A3) 2p 2 ical Journal, 576, 366 p Kozlenkov A. A., Mitrofanov I. G., 1986, Zh. Eksp. Teor. Fiz, 91, Further, we learn from the angle-averaged cross-section 1978 (Berestetskii et al. 1982) that when the reaction is way above Nikishov A. I., 1962, Soviet physics JETP, 14 threshold, one of the particles of the pair takes most of the Ruffini R., Vereshchagin G., Xue S.-S., 2010, Physics Reports, energy while the other takes almost nothing (section2), 487, 1 which reinforces our assumption. In the following calcula- Sturrock P. A., 1971, Astrophysical Journal, 164, 529

Svensson R., 1987, MNRAS, 227, 403 tion we note (n) a development up to a bounded function Timokhin A. N., Harding A. K., 2015, Astrophysical Journal, 810, of !n !n 144 m n m n ∼ ∼ max ∼ max . (A4) Tsai W.-y., Erber T., 1975, Physical Review D, 12, 1132 k p k p Turolla R., Nobili L., 2013, The Astrophysical Journal, 768, 147 Vassiliev V. V., 2000, Astroparticle Physics, 12, 217 This leads to the conclusion that p~ is almost aligned with

MNRAS 000,1–19 (2017) 16 G. Voisin et al.

~ 3 k. Indeed, we can show that min < max implies that mo- It can be shown that, provided that max < 8 m and for any mentum can be conserved only if cos θ > Cmin, where relevant θ or p, k + C = max (P0 − ) (A5) α > β⊥ (A21) min kp max 1 q Similarly, provided that max < 4 m (see Eq. (23)), max 2 2 0 0 2 − k + p − 2kP + 2kmax − 2P max + max. kp β⊥ < 1 (A22) This is approximated as βk 2 The smaller with respect to m the more effective m max(k − p) max Cmin = 1 + − 2 + (3) . (A6) these constraints will be. (Notice that the functions are 2p2 kp monotonous on the appropriate range.) Moreover, the max- Therefore we set ~k as the main axis of our coordinate system imum value of 1 − cos θ is the limiting factor for max , and (Figure A1), parallel to the unit vector ~z of the direct triad therefore these limits are less stringent if one√ considers cre- 3 (~x,~y,~z). For a weak photon of energy : ation of particles at smaller angles. Besides, kmax is the higher bound of the energy of the two photons in the center (k − p) m2 1 − cos θ ≤ 2 − + (3) (A7) of mass frame, and given our condition k m, max close kp 2p2 to m leads to an energy way above threshold in Eq. (10), By squaring relevantly the mass-shell constrain 16b) one ob- and therefore very unlikely to happen (section2), although tains the following quadratric constrain : it depends on the angle of incidence of the weak photon on the strong one as well. For these reasons, we should consider 0 2 2 P − k = ~x · ~k − p~ + K · P (A8) that the higher limit for max is a ”smooth” one meaning that most photons of the weak distribution should actually which can be rewritten as not be close to max, even when max is close to the limit ~x(α21 − β)~x − 2Aβ~ · ~x = A2, (A9) m/4, except if one has a very peculiar photon distribution. This discussion a posteriori justifies condition 23. The proper where vectors associated to the proper values in Eq. (A16) are A = K·, P (A10) ~v1 = (0,0,1), β~ = ~k − p~, (A11) ~v2 = (0,1,v2z ), − 0 α = k P , (A12) 2βx βy ~v = (1, ,v ), 3 2 2 3z and β is defined by βy − βx ) 2 ~x β~x = β~ · ~x . (A13) with 2βy βz We find v = << 1, (A23) 2z 2 2 2 βz − βy βx 0 0 2 β β β β = 2βx βy βy 0 . (A14) x z y 1 v3z = 2βx βz − 4 << 1. (A24) 2β β 2β β β2 β2 − β2 β2 − β2 .* x z y z z /+  z y  z x . /   Let’s rewrite Eq. (A9) in a dimensionless form, The above components are negligible in virtue of Eqs. (A21) , -   and (A22). Therefore, any vector parallel to the z axis has α2 β β~ ~x( 1 − )~x − 2 · ~x = 1. (A15) its image parallel to the z axis, and any vector perpendicular 2 2 A A A to the z axis has its image roughly perpendicular to the z The three proper values of this quadratic form are axis. 2 2 2 2 2 2 We can simplify the orthogonal proper values in Eq α β α βy α β ( − x , − , − z ). (A16) (A16), which are now both equal to : A2 A2 A2 A2 A2 A2 α2 The geometrical type of this quadratic form is determined (A25) by the signs of its proper values. For this we express the A2 different quantities using the approximation defined in Eq. It can be shown that (A2), ! α < βk, (A26) 1 k kp A = m2 + (1 − cos θ) + (A17)(1) βk − α 2 p m2 = (1) . (A27) m k − p max = 2m2 µ This implies that the parallel proper value (the third one in m m ! Eq. (A16)) is negative. Because the parallel proper value is k p m α = m − − + (2) (A18) negative while the two orthogonal values are positive, the m m 2p q 2 2 p βx + βy = β⊥ = p 2(1 − cos θ) + (3) (A19) 3 The energy of one photon in the center of mass of two photons, ! √ k one at energy k and another at energy , is k(1 − cosω) where β = β = p − cos θ (A20) z k p ω is the angle between the two photons 3-momenta

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 17 quadratic form in Eq.(A9) describes a paraboloid of revolu- ~z tion. The above remarks and Eqs. (A21) and (A22) allow to simplify the quadratic form in Eq. (A9). We are left with ~kw 2 2 β α2 α2 β α2 z 2 2 2 2 z k~ − − (z+z0) + (y +z ) = 1−z − , (A28) s ,k~ A2 A2 A2 0 A2 A2 w * + * + where , - , - Aβk L0 z = . (A29) 0 2 2 ~x, ~y βk − α

β2 2 Dividing everything by z − α we get our ﬁnal, although min A2 A2 not fully standard, form of Eq. (A9): zmin

(z + z )2 x2 + y2 0 − = 1 − δ2, (A30) 2 L2 z0 where the characteristic orthogonal radius L and the dis- placement δ are

2 A2 βk L2 = , (A31) 2 2 2 α βk − α 2 2 Figure A2. Representation of the mass shell (16)b) within ap- βk − α δ2 = . (A32) proximation (23). 2 βk Equation A28 describes a paraboloid of revolution of axis ~z, Further, one may show that z = min ~x 4 which corre- i.e. parallel to the strong photon 3-momentum ~. We notice δ k sponds to the physical idea that the smallest weak photon that L2/z2 = (1) meaning that the hyperboloid is very 0 that can produce a pair is the one that hits the strong pho- steep around the parallel axis. Besides, the small displace- ton head-on. How does it compare to determined in the ment , 2 , is responsible for a shift of the bottom of min δ δ = (1) previous section ? With the notations used in this section, the paraboloid under the plane of zero parallel momemtum. This corresponds to reactions with head-on weak photons A min = . (A37) that are in general of smaller energies, as shown on ﬁgure β~ + α A2.

We must remeber that the inclusion of L− into the Using the approximation in Eq. ((A22)), β~ = βk + paraboloid defined in Eq. (A30) is derived from the condi- β⊥/βk , we get tion in Eq. (16b). It must be completed with the condition in Eq. (16e) that reads min = zδ + β⊥/βk . (A38) ~x · β~ + A ≥ 0. (A33) This is consistent with the definition of min in Eq. (20) as From a geometrical point of view, this means that the rel- the minimum energy allowed in L−. evant photons are those with momenta above the plane of normal vector β~ of equation ~x · β~ = −A. Using relation (A22), this approximates to the plane orthogonal to the parallel di- A2 Integration rection, ~z, at position : We now compute the integral Wk defined in Eq. (14). Within A the frame of our approximations, Eq. (A30) shows that the zmin = − ' −2min (A34) ~ βk probability of making a pair is symmetric around k. This leads to a first angular integration of φ that yields a 2π fac- As a consequence, only the upper sheet of the hyperboloid 3 tor. The differential element d k~w = dxdydz is constrainted defined in Eq. (A30) corresponds to the physical mass shell. by Eq. (A30) that defines L−. Thus we write z = z(x, y, p) Indeed this hyperboloid crosses the parallel axis ~z at abscissa ± zδ such that : p 4 Then one shows easily from (A30) that x2 y2 has a minimum z± = −z 1 ± 1 − δ2 (A35) + δ 0 L/z0 2 2 2 2 for zm = −z0 = −z0 L/z0 + L /z ( L /z = (1)). 1+L2/z2 0 0 0 This relation takes into account the fact that z0 |zmin|. A βk A Hence zm ' − while zδ = . Using the fact that α . βk Because δ << 1, α α βk one gets that zm . zδ , which means zm is slightly under the p 1 bottom of the hyperboloid. Since x2 + z2 can be easily shown to z = −z 1 − 1 − δ2 ' − z δ2 (A36) δ 0 2 0 be a growing function of z, its smallest value can only be zδ .

MNRAS 000,1–19 (2017) 18 G. Voisin et al. through the constrain L−(p,cos θ, k) and We note ! ! ∂z ∂z k2 m2 2 µ dz = dp. (A39) = − max , − ∂p ∂p left 2(k p)p kp k ! ! We can now write (28) as follows ∂z r2 = 1 + . ∂p 42 µ2 Z C2 Z p2 right max W = c 2π d cos θ dp × ~k We need the absolute value of ∂z/∂p, and one can show from C1 p1 (A49) that it is always negative, so that we shall always take Z d2σ K · K ∂z s w f (k~ ) dxd(A40)y, the opposite of Eq. (A49) and remove the absolute value in 0 0 w w L (x,y) dΩ K K ∂p − s w the following developments. The distribution function fw is where is the projection of onto the plane. the only element that depends on φw . Moreover, the integra- L−(x, y) L− (x, y) We need to expand the different quantities appearing in tion over the hyperboloid L− leads to get rid of z through Eq. (A40). Let us start with an explicit projection of the up- (A47). For further developments, we explicitly keep track of per hyperboloid on the plane (~x,~y). This projection is a disc the fact that L− = L−(p,cos θ) = L−(p, µ) of radius Z 2π p F (r, p, µ) = f r, φ , z(r2, µ) dφ . (A50) − w w w w R = 2max µ(1 µ), (A41) φw =0 where µ is defined in Eq. (29). For the change of variable We can separate the integration of (28) in several parts. The θ → µ, we need to switch the integration on p with the parts with a dependance on r are to be found in the Jacobian integration on cos θ in (A40), with |∂z/∂p| (see Eq. (A49)) of which we take only the rightmost factor, the current in Eq. (A44), the distribution function, (k − p) max d cos θ = 2 dµ (A42) and the differential element rdr. Parts that depend only on p k µ or p are the differential cross section in Eq. (A43), the Moreover, the shape of the domain naturally suggests to two first factors in the Jacobian ∂z in Eq. (A49) , and the use polar coordinates in the plane (~x,~y), with radius r = ∂p q Jacobian in Eq. (A42). The dependance of the integrated 2 2 x + y , angle φw and dxdy = rdrdφw . The differential distribution function Fw is not known a priori. We obtain cross-section is Z p2 Z µ2 2 2 2 ∂ cos θ d σ ∂z d σ r 2 pm2 2 2 W = c2π dp dµ × (A51) − e m m − ~k = 2 2 + − ∂µ dΩ ∂p Ω 4 kmax µ 4max pµ 4max (k p)µ p1 µ1 left d " Z R m2 m2 1 p 1 k−p ∂z Ks · Kw − − − .(A43) Fw (r, p, µ)r dr. 4max pµ 4max (k−p)µ 4 k−p 4 p 0 0 r=0 ∂p right Ks Kw

The elementary current in Eq. (8) is The boundary conditions on the p integral, (pi )i=1,2 must K · K be understood as pi = max (pi , k − pi ) in virtue of symmetry s w = 1 − cos ξ + β /β , (A44) 0 0 ⊥ k (A1). At lowest order, one can shows that the r part of the Ks Kw integrand is merely equal to 2Fw (r, p, µ)rdr and that the µ where part can be reduced after a partial fraction decomposition cos ξ = 1 − 2µ max + (1) . (A45) to 4 q X ai (p) 2 2 2 , (A52) The expression of = x + y + z needs as well to be µi i=1 developed as a function of x, y and p, which implies to write a clear expression for z(p). From Eq. (A30), where the dimensionless coefficients ai (p) are given by Eq. s (31). Then Eq. (A51) can be formally reduced to Eq. (38). 2 2 x + y 2 z = −z0 + z0 1 + − δ . (A46) L2 One can show that under approximations in Eq. (23), APPENDIX B: FORMULA COMPENDIUM 2 2 x +y < 1/16 and should be in practice much smaller. There- L2 The cosine of the angle θ between the strong photon ~k and fore, the outgoing lepton p~ is parametrized below by ! z x2 + y2 z ' 0 − δ2 (A47) cos θ = 1 − c (B1) 2 L2 and the following parametrization by µ can lead to signifi- This allows to make explicit, cant simplifications 1 2 2 2 2 = x + y + 4µ max , (A48) k − p m2 4µmax c = 2 max µ − (B2) p k 2p2 as well as the derivative of z : . ! ! ∂z k2 m2 2 µ r2 The following quantities are used are intermediates in = − max 1 + . (A49) 2 2 the derivation of the hyperboloid of integration, ∂p 2(k − p)p kp k 4max µ

MNRAS 000,1–19 (2017) Photon-photon pairs above a pulsar polar cap 19

A = K · P (B3) β~ = ~k − p~ (B4) α = k − P0 (B5) 2 βx 0 0 2 β = 2βx βy βy 0 (B6) 2 * 2βx βz 2βy βz βz + . / and can be explicited to relevant order (see (A4) ) as , - − A = m2 1 k + k p c + (1) = 2m2 k p max µ 2 p m2 m m k p m α = m m − m − + (2) q 2p √ (B7) 2 2 | βx | ∼ βy ∼ βx + βy = β⊥ = p 2c + (3) β = β = p k − 1 + c z k p . The characteristics of the hyperboloid (A30), are then related to the previous quantities by

Aβk k − k z0 = 2 2 = 2(k−p) (k p + pc) = 2 + (1) βk −α β2 2 2 A2 k pk m2 L = 2 2 2 = 4(k−p) 2c + 2 = µkmax α βk −α p 2 2 β −α 2 2 k p m max δ = 2 = k−p+2c p 2c + 2 = 4µ k + (2) βk p (B8) , where c = 1 − cosθ. From this one ﬁnds the derivative of z : ! ! ∂z k2 m2 2 µ r2 = − max 1 + (B9) 2 2 ∂p 2(k − p)p kp k 4max µ

This paper has been typeset from a TEX/LATEX ﬁle prepared by the author.

MNRAS 000,1–19 (2017)