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 To cite this version: 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 HAL Id: hal-01614371 https://hal.archives-ouvertes.fr/hal-01614371 Submitted on 10 Oct 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. MNRAS 000,1{19 (2017) Preprint 10 October 2017 Compiled using MNRAS LATEX style file v3.0 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 ap- proximation 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 pri- mary 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 cur- vature 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 magneto- sphere. 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 & Schr´eder 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 de- rived 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 jii to an outgoing state joi can be repre- limit that depends on the angular size of the hot cap pro- sented as the decomposition on a final states basis fj fk ig viding the soft-photon background. Besides, these authors of the evolved state Sˆjii, Sˆ being the evolution operator, do not provide spectra for the created pairs although the X j i h j ˆj ij 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.