ON THE POLAR CAP CASCADE PAIR MULTIPLICITY OF YOUNG PULSARS

The Astrophysil Journl, 81:144 (26pp), 215 September 1 215. The Amerin Astronomil Soiety. All rights reserved. doi:1.188/4-637x/81/2/144 ON THE POLAR CAP CASCADE PAIR MULTIPLICITY OF YOUNG PULSARS A. N. Timokhin 1,2 nd A. K. Hrding 1 1 Astrophysis Siene Division, NASA/Goddrd Spe Flight Center, Greenbelt, MD 2771, USA; ndrey.timokhin@ns.gov 2 University of Mrylnd, College Prk (UMDCP/CRESST), College Prk, MD 2742, USA Reeived 215 April 8; epted 215 July 3; published 215 September 8 ABSTRACT We study the effiieny of pir prodution in polr ps of young pulsrs under vriety of onditions to estimte the mximum possible multipliity of pir plsm in pulsr mgnetospheres. We develop semi-nlyti model for lultion of sde multipliity whih llows effiient explortion of the prmeter spe nd orroborte it with diret numeril simultions. Pir retion proesses re onsidered seprtely from prtile elertion in order to ssess different ftors ffeting sde effiieny, with elertion of primry prtiles desribed by reent selfonsistent non-sttionry model of pir sdes. We rgue tht the most effiient sdes operte in the urvture rdition/synhrotron regime, the mximum multipliity of pir plsm in pulsr mgnetospheres is few 1 5. The multipliity of pir plsm in mgnetospheres of young energeti pulsrs wekly depends on the strength of the mgneti field nd the rdius of urvture of mgneti field lines nd hs stronger dependene on pulsr inlintion ngle. This result questions ssumptions bout very high pir plsm multipliity in theories of pulsr wind nebule. Key words: elertion of prtiles plsms pulsrs: generl strs: neutron 1. INTRODUCTION The ide tht prodution of eletron positron pirs in mgnetospheres of rottion-powered pulsrs is intimtely onneted with their tivity hd been proposed by Sturrok (1971) only few yers fter the disovery of pulsrs (Hewish et l. 1968). Sine then it hs beome n integrl prt of the stndrd pulsr model nd tody, there is little doubt tht n tive rottionlly powered pulsr produes eletron positron plsm. Although the pulsr emission mehnism(s) is still not yet identified, there is strong empiril evidene tht pulsrs stop emitting rdio wves when pir formtion eses the threshold for pir formtion roughly orresponds to the deth line in pulsr prmeter spe, wht ws lredy noted by Sturrok (1971). Furthermore, the nrrow peks in mny pulsr high-energy light urves (Abdo et l. 21) require pervsive sreening of the whole mgnetosphere by pir plsm, exept in nrrow elertor gps (e.g., Wtters et l. 29; Pierbttist et l. 215). Understnding pir plsm genertion in pulsr mgnetospheres is therefore of ruil importne for developing pulsr emission models. In the stndrd pulsr model, initilly proposed by Goldreih & Julin (1969) nd Sturrok (1971), the mgnetosphere is filled with dense pir plsm whih sreens the elerting eletri field everywhere exept some smll zones whih re responsible for prtile elertion nd emission. Pir plsm is primrily reted vi onversion of γ rys in the strong mgneti field ner the polr ps (PCs). Pir prodution in the PCs is ornerstone of the stndrd model without dense plsm produed t the PCs, t the bse of open mgneti field lines, the mgnetosphere would hve lrge volumes with unsreened eletri field, s pir retion in e.g., outer gps (Cheng et l. 1976) nnot sreen the eletri field over the rest of the mgnetosphere. Chrge strvtion (Arons & Shrlemnn 1979) or vuum gps (Rudermn & Sutherlnd 1975) t the polr p, when the number density of hrged prtiles is not enough to sreen the eletri field, leds to formtion of elerting zone(s). Some hrged prtiles enter this zone, re elerted to very high energies, nd emit gmm rys tht re bsorbed in the ultrstrong mgneti field, reting eletron positron pirs. The pirs, being reltivisti, n lso emit pir produing photons nd so the vlnhe develops until photons emitted by the lst genertion of pirs n no longer produe pirs nd espe the mgnetosphere. The pir plsm reted by pulsrs flows out of the mgnetosphere long open mgneti field lines nd provides the rditing prtiles for the surrounding Pulsr Wind Nebule (PWNe). Models of PWNe depend (t lest) on the density of the plsm, wht produes the observed synhrotron nd inverse Compton emission. Estimtes of the pir multipliity (the number of pirs produed by eh primry elerted prtile) needed to ount for the emission from the Crb pulsr wind nebul (PWN) rnge from bout 1 5 1 6 (de Jger et l. 1996) up to 1 7 (e.g., Buintini et l. 211); for the Vel PWN the multipliity is estimted to be bout 1 5 (de Jger 27). PWNe therefore give the most ompelling evidene for pir prodution nd pir sdes in t lest young energeti pulsr mgnetospheres. Although PWNe re observed only round young pulsrs (< few times 1 4 yers), evidene for pirs, t lest for high plsm densities lrger thn those provided by primry prtiles, n lso be found in older pulsrs. Synhrotron bsorption models for the elipse in the double pulsr system PSR J737 339 (Lyutikov 24; Arons et l. 25) require pir multipliity of round 1 6 for the reyled 22 ms pulsr in tht system. The sde proess in pulsr PCs hs been the subjet of extensive numeril s well s nlytil studies (e.g., Dugherty & Hrding 1982; Gurevih & Istomin 1985; Zhng & Hrding 2; Hibshmn & Arons 21, 21b; Medin & Li 21). The pir plsm multipliity obtined in these studies ws signifintly lower thn estimtes of pir plsm multipliity in PWNe, s it did not exeed few 1 4. Most of those works onsidered pir retion together with the prtile elertion, whih mkes these nlyses dependent on the elertion model onsidered. These studies lso ssumed 1

The Astrophysil Journl, 81:144 (26pp), 215 September 1 stedy, time-independent elertion of the primry prtiles. However, reent studies by Timokhin (21) nd Timokhin & Arons (213) hve found tht pulsr polr p pir sdes re not time-stedy in the generl se of rbitrry urrent, prtiulrly those required by globl mgnetosphere models (e.g., Contopoulos et l. 1999; Spitkovsky 26; Timokhin 26; Klpothrkos & Contopoulos 29). In this pper we study the question of wht the mximum pir multipliity hievble in pulsr polr p sdes is nd under whih irumstnes it is hieved. In ontrst to previous pir sde studies, we tke multistep pproh. We onsider the physil proesses in pir sdes nd prtile elertion models seprtely in order to lerly set prt different ftors influening the effiieny of pir sdes. We first ssess how eh of the mirosopi proesses ffets the finl multipliity nd the pulsr prmeter rnges tht result in the lrgest possible pir multipliity. Then, we employ the most reent model of non stedy-stte prtile elertion in pulsr PCs nd derive simple nlytil estimte for the mximum energy of prtiles elerted in non-sttionry sde. One of the most importnt results of our study is strong upper limit on pir plsm multipliity in pulsrs. We limit ourselves to the se of sdes t the PCs of young 3 pulsrs s from previous theoretil studies of polr p sdes, suh pulsrs re expeted to be the most effiient pir produers. We rely on results of previous sde studies in our hoie of the speifi sde proess, nmely sdes initited by urvture rdition (CR) of primry prtiles. The pln of the pper is s follows. In Setion 2 we briefly disuss the effiieny of sdes in generl nd give n overview of the most effiient sde proess in pulsr PCs. In subsequent setions we onsider in detil ll physil proesses in suh sdes. Setions 3 7, the lrgest prt of this pper, re devoted to development of simple semi-nlytil model for estimtion of pir prodution effiieny in PCs of young pulsrs. This model llows effiient explortion of pulsr prmeter spe nd helps to lrify the min ftors ffeting sde multipliity. Then, in Setion 8 we use diret numeril simultions of PC sdes in Crb-like pulsr to show tht preditions of our semi-nlytil model re indeed orret. We disuss unertinties of urrent pulsr models in Setion 1 nd summrize nd disuss our findings nd their implitions in the Disussion. 2. PHYSICS OF POLAR CAP CASCADES: AN OVERVIEW An eletron positron sde n be thought of s proess of splitting the energy of primry prtiles into the energies of seondry prtiles. The mximum multipliity, the number of seondry prtiles for eh primry prtile, of n idel sde initited by single primry prtile with energy ò p would be p kmx 2. ( 1) g,es 3 Pulsrs with strong polr p sdes should hve potentil drop over the polr p well in exess of the pir formtion threshold s well s lrge mgneti fields B 1 11 G nd short rottionl periods, so tht prtile elertion hppens over short distne nd sde develops in the region with strong mgneti field. The best single prmeter seleting pulsrs with suh properties is the smll hrteristi ge t = P 2P. A detiled disussion of the pulsr prmeter rnge where pproximtions used in this pper re formlly pplible is given t the end of Setion 6.2. where g,es is the mximum energy of photons esping from the sde (or the minimum energy of pir produing photons). Not ll of the primry prtile energy goes into pir produing photons, nd pirs reted in the sde do not rdite ll of their energy into the next genertion of pir produing photons. Hene, kmx is only the theoretil upper limit on the multipliity of rel sde. The totl pir yield of the sde is ombintion of four ftors: () number of primry prtiles, (b) initil energy of primry prtile the higher the energy the more pirs n be produed, () threshold for pir formtion the lower the threshold the higher the multipliity, (d) effiieny of splitting the energy of primry prtiles into pirs the higher the frtion of prtile energy going into pir reting photons, s opposed to the finl kineti energy of the prtiles nd photons below the pir formtion threshold, the higher the multipliity. In young, fst rotting pulsrs, the eletri field in the polr p elertion zones is strong nd primry prtiles n be elerted up to very high Lorentz ftors, γ 1 7. At these energies the most effetive rdition proess is CR. CR effiieny grows rpidly with the prtile energy nd for young pulsrs beomes the dominnt emission mehnism for primry prtiles. For seondry prtiles, whih re substntilly less energeti thn the primry ones, the primry wy to rete pir produing photons is vi synhrotron rdition. In Setion 9 we rgue tht lthough nother possible emission mehnism for seondry prtiles Resonnt Inverse Compton Sttering (RICS) of soft X-ry photons emitted by the NS n generte pir produing photons in pulsrs with mgneti field 3 111 G, tht hnnel never beomes the dominnt one for pir prodution in young pulsrs, t best resulting in pir multipliity omprble to the one of the synhrotron hnnel. Hene, in young pulsrs tht re expeted to hve the highest multipliity pir plsms, the polr p sdes should operte primrily in the CR-synhrotron regime ll known studies of sdes in pulsr polr p gree on this point. In this pper we study in detil CR-synhrotron sdes. The resulting multipliities will be good estimtes for wide rnge of pulsr prmeters, however, s we onsider only synhrotron rdition of seondry prtiles, for pulsrs with mgneti field 3 112 G our nlysis might underestimte the sde multipliity by ftor of 2, see Setion 9. Figure 1 gives shemti overview of eletron positron sde development in polr p regions of young pulsrs. Shown re the first two genertions in sde initited by primry eletron. Primry eletrons emit CR photons (lmost) tngent to the mgneti field lines; primry eletrons nd CR photons re genertion prtiles in our nottion. Mgneti field lines re urved nd the ngle between the photon momentum nd the mgneti field grows s the photon propgtes further from the emission point. When this ngle beomes lrge enough, photons re bsorbed nd eh photon retes n eletron positron pir genertion 1 eletron nd positron. The pir momentum is direted long the momentum of the prent photon nd t the moment of retion, the prtiles hve non-zero momentum perpendiulr to the mgneti field. They rdite this perpendiulr momentum lmost instntneously vi synhrotron rdition nd then move long mgneti field lines. Although these seondry prtiles re reltivisti, their energy is muh lower thn tht of the primry eletron nd their urvture photons nnot rete 2

The Astrophysil Journl, 81:144 (26pp), 215 September 1 pirs. After the emission of synhrotron photons, seondry prtiles (genertion 1 nd higher) no longer ontribute to sde development. Genertion 1 photons (synhrotron photons produed by the genertion 1 prtiles) re lso emitted (lmost) tngent to the mgneti field line s the seondry prtiles re reltivisti nd propgte some distne before quiring the neessry ngle to the mgneti field nd reting genertion 2 pirs. These pirs in their turn rdite their perpendiulr momentum vi synhrotron rdition, emitting genertion 2 photons. The sde initited by single CR photon stops t genertion where the energy of synhrotron photons flls below g,es. Only primry prtiles emit pir produing photons s they move long the field lines; ll seondry prtiles emit pir produing photons just fter their retion. The sde development n be thus divided into two prts: (i) primry prtiles emit CR photons s they move long mgneti field lines nd (ii) eh CR photon gives rise to synhrotron sde, when synhrotron photons rete suessive genertion of pirs whih emit the next genertion synhrotron photons t the moment of retion. This division goes between genertion nd ll subsequent sde genertions. In the following setions we nlyze ll four ftors regulting the yield of CR-synhrotron sdes listed t the beginning of this setion (in reverse order, from d to ). In Setions 3 7 we develop simple semi-nlytil models for lultion of multipliity of strong PCs sdes initited by primry eletron, whih we then ompre with detiled numeril omputtion desribed in Setion 8. We strt with gmm-ry bsorption in strong mgneti field in Setion 3; then we disuss the effiieny of the synhrotron sde in Setion 4. CR is onsidered in Setion 5 nd in 5.3 we disuss the multipliity of CR-synhrotron sde initited by single prtile of given energy. This overs items nd d from the list of ftors ffeting sdes effiieny. In Setion 6 we ddress item b from the list energy of primry prtiles inititing the sde. Setions 7 nd 8 re devoted to the totl multipliity of CR-synhrotron sde when prtile elertion is tken into ount in Setion 7 we present results for sde multipliities from semi-nlytil model nd in Setion 8 we present results of detiled numeril simultions of sdes in Crb-like pulsr. In Setion 9 we disuss the role of RICS in PCs sdes nd rgue tht onsidering only CR-synhrotron sdes gives us n dequte estimte of the pir multipliity in young pulsrs. Finlly, in Setion 1, we ddress item the men flux of primry prtiles nd the totl yield of CR-synhrotron sde in n energeti pulsr. We rgue tht this is the most importnt ftor regulting pir yield in energeti pulsrs. Despite the unertinty in determining the men flux of primry prtiles, we n set rther strit upper limit on pir multipliity in pulsrs. 3. PHOTON ABSORPTION IN THE MAGNETIC FIELD 3.1. Opity for g B Pir Prodution The opity for single photon pir prodution in strong mgneti field is (Erber 1966) f 4 B ( g, y) =.23 b sin y exp - ( 2) 3 where b º BB q is the lol mgneti field strength B normlized to the ritil quntum mgneti field Bq = e f 2 = 4.41 1 13 G, ψ is the ngle between the photon momentum nd the lol mgneti field, f = e 2» 1 137 is the fine struture onstnt, nd = m = 3.86 1-11 m is the redued Compton wvelength. The prmeter χ is defined s 1 b sin y, ( 3) 2 º g where g is the photon energy in units of m e 2. For onveniene from here on, ll prtile nd photon energies will be quoted in terms of m e 2. The optil depths for pir retion by high energy photon in strong mgneti field fter propgting distne l is l ( g ) ò B ( g ) t, l =, y() x dx, () 4 where integrtion is long the photon s trjetory. Expression (2) is urte if the mgneti field is smll ompred to the ritil field B q, b <.2 suffies, nd if g sin y > 2 so tht the reted pir is in high Lndu level; pir prodution into low Lndu levels nd for higher mgneti fields hs been disussed in Dugherty & Hrding (1983) nd Hrding et l. (1997). For most pulsrs, the mgneti field in polr p regions is smller tht B q /3. In this pper we study sdes in pulsrs with norml mgneti fields nd so we neglet high-field effets. Expression (2) beomes inurte when pirs re reted t low Lndu levels, ner the pir formtion threshold, it overestimtes the opity nd even formlly llows pir formtion below the kinemti threshold g sin y = 2. However, pirs reted t low Lndu levels will not give rise to strong sdes, s their perpendiulr energy will be too low to emit pir-produing synhrotron photons. Hene, for the se of strong sdes, urte tretment of pir formtion ner the kinemti threshold is not neessry. Throughout this pper we use Erber s pproximtion (2) in ll our nlytil lultions but introdue ut off t the kinemti threshold g sin y = 2 s desribed t the end of this setion, thus tking into ount the esstion of pir formtion below the the kinemti threshold. If the men free pth (mfp) of pir-produing photon l γ is omprble to or lrger thn the hrteristi sle of the mgneti field vrition L B, this photon will not initite strong sde with the sme emission proesses by whih it ws produed. The reson for this is s follows. The opity for γb pir prodution exponentilly depends on the mgneti field strength nd photon energy vi χ. The energy of the next genertion photon will be smller thn tht of the primry one, nd, beuse the primry photon hs lredy trveled the distne over whih the mgneti field hs substntilly delined, the mgneti field long the next genertion photon s trjetory will be substntilly weker thn tht long the primry photon s trjetory. 4 The next genertion photon s mfp will be muh lrger thn thn tht of the primry photon, nd, even if this seondry photon will be bsorbed, it will be the 4 In the polr p, photons re emitted by ultrreltivisti prtiles moving long mgneti field lines. At emission points photons re lmost tngent to field lines nd the differenes in initil photon pith ngles for different genertions n be negleted. 3

The Astrophysil Journl, 81:144 (26pp), 215 September 1 lst sde genertion. Hene, in strong sde, for ll but the lst genertion photons, l g L B. A resonble estimte for L B would be the distne of the order of the NS rdius R ns s ny globl NS mgneti field deys with the distne s ( r R ns ) - d, δ 3. Very lolized, Sun spot like mgneti fields, re, in our opinion, of no importne for the generl pulsr se s the probbility of suh spot to lie t the polr p should be rther low, i.e., most pulsrs should be ble to produe plsm in more or less regulr mgneti field. A dipole field, δ = 3, is often onsidered s resonble ssumption for generl pulsr model. A pure dipole field, however, seems to be too idelized of n pproximtion, s even if the NS field is pure dipole, it will be slightly disturbed by the urrents flowing in the mgnetosphere. In generl, ner dipole mgneti fields with different urvtures of mgneti field lines should be exmined in sde models. We onsider strong sdes with lrge multipliities, where, s rgued bove, photons propgte distnes muh shorter thn the hrteristi sle of the mgneti field vrition, so we ssume tht in the region where most of the pirs re produed the mgneti field is onstnt. The rdius of urvture of mgneti field lines r is not smller thn L B, nd s l g L B the ngle ψ is lwys smll, the pproximtion sin y» y is very urte. For photons emitted tngent to the mgneti field line, dx = r dy. In our pproximtion both b nd r re onstnts. From Eqution (3) we hve y = 2 g b, nd substituting it into Eqution (4) we n express the optil depth τ to pir prodution s n integrl over χ (, ( l) ) 4 t (, l) = r g y g At exp d, ( 5) 2 b ò - 3 g where A º.92f» 1.74 18m- 1 t. Integrting Eqution (5) over χ by prts two times we n get n expression for τ in terms of elementry funtions nd the exponentil integrl funtion Ei: r t( ) = At 2 g b 2 4 4 8 1 e 2 3 9 Ei 4 - - 3 - -. ( 6) 3 Ei(z) is widely used speil funtion, defined s ò Ei() z =- exp( -t) t dt. There re effiient numeril -z lgorithms for its lultion implemented in mny numeril librries nd sientifi softwre tools; using Eqution (6) for lultion of the optil depths will result in muh more effiient numeril odes thn diret integrtion of Eqution (5). As the optil depth to pir formtion grows exponentilly with χ (nd distne), for nlytil estimtes it is resonble to ssume tht ll photons re bsorbed when they hve trveled the distne l γ suh tht τ(l γ ) = 1. We denote the vlue of χ when the optil depth rehes 1 s : : t( ) = 1. ( 7) In ll our omputtions we will use s the vlue of the prmeter χ t the point of the photon s bsorption. is solution of the nonliner Eqution (7) with τ given by Eqution (6). Beuse of the exponentil dependene of τ on Figure 1. Shemti representtion of eletron positron sde in the polr p of young pulsr, see the text for desription. 1/χ it is to be expeted tht 1 should hve lose to liner dependene on logrithms of g, b, nd r. We solved Eqution (7) numerilly for different vlues of g, b, nd r nd, indeed, the inverse quntity 1 depends lmost linerly on log g, log b, nd log r in wide rnge of these prmeters. Mking modest size tble of 1 vlues one n lter use pieewise interpoltion to find prtiulr vlue of quite urtely. In Figure 2 we plot ontours of 1 s funtions of log( g ) nd log( B) for three different vlues for the rdius of urvture of the mgneti field lines. We wnt to point out tht 1 differs from the vlue 1 = 15 used by Rudermn & Sutherlnd (1975), espeilly for higher energy photons. Although this differene is only ftor of few, s we will point out lter, the sde effiieny in eh genertion depends on the orresponding vlue of, nd in strong sde with severl genertions, the estimte for the finl multipliity will be substntilly ffeted by the vlue of. Erber s expression is not pplible for pirs produed t low Lndu levels s it overestimtes the opity nd, formlly, solutions for obtined from Eqution (7) llows pir retion even for g sin y < 2. In our nlytil tretment we introdue limit on to orret for the kinemti threshold. For photons bove kinemti threshold from Eqution (3) it follows tht > b, ( 8) 4

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 2. Contour plot of 1 s funtion of the logrithms of the mgneti field strength B in Guss, nd photon energy g normlized to the eletron rest energy, for three vlues of the rdius of urvture of mgneti field lines r = 1 6, 17, 18 m. 1 vlues shown on this plot re lulted from Eqution (7) nd re not orreted for the kinemti threshold (see text). Figure 3. Contour plot of 1 orreted for kinemti threshold ording to Eqution (9). These vlues of 1 re used in ll lultions within semi-nlyti model. Nottions re the sme s in Figure 2. the point of bsorption y through we get (nonliner) eqution for g,es In ll our nlytil lultion we get from Eqution (7) nd then use the mximum vlue of nd b: = mx (, b). (9 ) g,es = 2 In Figure 3 we plot ontours of 1 whih inorporte orretions to due to the kinemti threshold ording to Eqution (9). It is ler from the plot tht this orretion ffets only ses with high mgneti field nd low prtile energies. r ( g,es, b, r ). ses R ns b (1) The nonlinerity in this eqution is due to dependene of on g,es, b, nd r. Using n interpoltion tble for 1 this eqution is very esy to solve numerilly for ll resonble vlues of physil prmeters involved. In Figure 4 we plot energy of esping photons, log g,es s funtion of the rdius of urvture of mgneti field lines r nd mgneti field strength B for ses = 1. For smller vlues of ses the whole plot would move to the left. This figure shows (n obvious) trend tht for higher mgneti field nd smller rdii of urvture, the energy of esping photons is lower, whih llows for more sde genertions nd lrger multipliity. The brek in ontour lines round log B 12.4 is due to the kinemti threshold. 3.2. Energy of Esping Photons As disussed bove, photons esping the sde re those whose mfp lg is lrger thn the distne of signifint mgneti field ttenution LB. The forml riteri we use for lulting the energy of esping photons g,es is lg ( g,es ) = ses R ns; ses is dimensionless prmeter quntifying the esping distne in units of R ns. The photon mfp is lg = r y nd expressing the ngle between the photon momentum nd the mgneti field t 5

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 4. Energy of esping photons: ontours of log g,es s funtion of logrithms of the rdius of urvture of mgneti field lines r in m nd mgneti field strength B in Guss for ses = 1. For given geometry of the mgneti field more urte nlytil expression for the energy of esping photons n be derived using n ext expression for ψ long photon s trvel pth by expnding the integrl in Eqution (4) round its upper endpoint, s ws done by Hibshmn & Arons (21b). However here we try to explore different possible mgneti field onfigurtion exploring prmeter spe in r nd our simple estimte is urte to ftor of few. 4. SYNCHROTRON CASCADE In this setion we disuss the synhrotron sde, where most of the eletron positron pirs re reted. The synhrotron sde is the prt of the whole sde tht is initited by genertion photons. In the synhrotron sde eh genertion s primry photon is divided into mny (lower energy) next genertion s pir-produing photons by synhrotron rdition of freshly reted pirs. 4.1. Frtion of Prent Photon Energy Remining in the Csde A high energy photon, when bsorbed in the mgneti field, produes n eletron nd positron; the totl energy of these prtiles is equl to the energy of the photon. Prtile moment just fter prodution hve pith ngles equl to y, when pir prodution tkes ple well bove threshold. When pirs re reted t high Lndu levels, s is the se in strong sdes, reltivisti prtiles hve non-zero pith ngles nd they rdite their perpendiulr energy vi synhrotron rdition; in superstrong mgneti fields, this hppens lmost instntneously. The omponent of prtile momentum prllel to the mgneti field is unffeted by synhrotron rdition nd so the finl Lorentz ftor of the prtile,f will be 2-1 2,F 1,I 1,I 2 1 2 = - b» + y - 11 ( ) ( ) ( ) where b º v is prtile veloity long the mgneti field line nd,i is the initil Lorentz ftor of the prtile right Figure 5. Frtion of the prent photon energy rdited s synhrotron photons by freshly reted pirs: ontours of zsyn s funtion of logrithms of the prent photon energy g nd mgneti field strength B in Guss for r = 17 m. fter retion. If the photon bsorption hppens t < 1 whih is indeed the se ner pulsr PCs, see Setion 3.1 with high degree of ury we n ssume tht the energy of the photon is eqully divided between the eletron nd positron (e.g., Dugherty & Hrding 1983). Expressing y in Eqution (11) through, y = 2 g b, nd the initil prtile energy through the photon energy g,,i = g 2 we get,f s funtion of nd b,f 2 1 2 - g 2 1 = + b. ( 12 ) The frtion z syn of photon energy rdited s synhrotron photons, whih is going into subsequent pir retion, is z syn 2(,I -,F) = g 2-1 2 = 1-1 +. ( 13) b Beuse of the kinemti threshold (8) the minimum vlue of this frtion is zsyn b.292 =, i.e., formlly 5 t lest ;3% of bsorbed photon energy will go into synhrotron rdition of reted pirs. In Figure 5 we plot the frtion of pir-produing photon energy rdited s synhrotron photons by freshly reted pirs given by Eqution (13). Contours of zsyn re plotted s funtions of the photon energy g nd mgneti field strength B, the rdius of urvture ws ssumed to be r = 17 m. The dependene of zsyn on r (vi ) is very wek, nd Figure 5 is good representtion of how z syn depends on g nd B for ny r of interest. It is evident from Figure 5 tht for higher mgneti field strengths, B 3 112 G, nd lower energies of prent photons, progressively smller frtion of the prent photon 5 As we mentioned bove, the physis of ner-threshold pir formtion is more omplited nd our simplified tretment is less urte in this regime. 6

The Astrophysil Journl, 81:144 (26pp), 215 September 1 energy nd mgneti field strength, the rdius of urvture ws ssumed to be r = 17 m. Two ler trends re visible on this plot: the lower the energy of the primry photon the lrger the number of seondry synhrotron photons produed t eh onversion event, nd the higher the mgneti field the smller is the number of synhrotron photons. The first one is generl trend of emission proesses when higher energy prtiles emit less photons whih, however, hve lrger energies. The seond trend is due to the suppression of the synhrotron sde disussed bove, in Setion 4.1. 4.3. Multipliity of Synhrotron Csde The genertion i + 1 sde photon is synhrotron photon emitted t the event of pir retion by photon of genertion i. Expressing b, y, nd () i g through nd g from Eqution (14) we get for the hrteristi energy of the next genertion photon Figure 6. Number of synhrotron photons with the hrteristi energy g,syn emitted t eh g e + e - onversion: ontours of n syn s funtion of logrithms of the prent photon energy g nd mgneti field strength B in Guss for r = 17 m. energy goes into synhrotron photons; the rest remins in the kineti energy of the reted pirs moving long mgneti field lines. The portion of the prent photon energy energy left in kineti energy of pirs does not go into prodution of next genertion pirs but is lost from the synhrotron sde. 6 The reson for this is tht for higher mgneti field strengths pirs re reted when the photon hs smller pith ngle y, so tht smller frtion of the photon energy goes into perpendiulr pir energy, nd hene, smller frtion of the photon energy is emitted nd remins in the sde. 4.2. Number of Seondry Photons At eh pir retion event the prent photon is effetively trnsformed into n eletron positron pir nd lower energy synhrotron photons. Those photons beome prent photons for the next sde genertion or espe the mgnetosphere, terminting the sde. The hrteristi energy of synhrotron photons emitted by newly reted prtile in terms of quntities used in this pper is given by 3 = b y 2 2 g,syn,i. ( 14) The number of synhrotron photons with the hrteristi energy g,syn these photons rry most of the energy of synhrotron rdition emitted t eh event of onversion of prent photon with energy g into n eletron positron pir is n syn z syn g g,syn = 4 3 z syn. ( 15) In Eqution (15) both z syn nd re funtions of B, g, nd r. In Figure 6 we plot ontours of n syn s funtions of the photon 6 The kineti energy of pirs might be tpped by RICS sde brnhes, see Setion 9. () i g ( i 1) g + 3 i = g, ( 16) 4 ( i+ 1) ( ) g where nd re energies of i th nd ( i + 1) th genertion photons. The photon energy degrdes with eh suessive genertion of the sde. This degrdtion elertes s the sde proeeds through genertions beuse with the derese of the photon energy inreses, see Figure 2. The photon mfp in onstnt mgneti field goes s lg µ g. For photon energies 1 4 g, when 1, the mfp of suessive genertions inreses by t lest n order of mgnitude in eh suessive genertion; therefore the rudeness of our pproximtion for estimting the energy of esping photons, Eqution (1), ould ffet only the lst sde genertion. The rpid energy degrdtion results in rther smll number of genertions in polr p sdes s the energy of pir-produing photons rpidly rehes the threshold energy. On the other hnd, the number of emitted synhrotron photons inreses with the derese of the energy of the prent photon, see Figure 6; onsequently synhrotron sdes n hve quite lrge multipliities despite smll number of genertions. Eh sde photon retes 2 prtiles t the moment of ( i+ 1) g bsorption, nd prtiles re produed until g,es. The totl number of prtiles generted in synhrotron sdes initited by primry photon n be lulted by summtion over ll genertions of the number of pirs produed in eh sde genertion. The lgorithm for this lultion is shown in Appendix A, Algorithm 1. We will use this lgorithm for lultion of the polr p sde multipliity fter we disuss CR, the rdition mehnism responsible for generting the primry photons for synhrotron sdes in young energeti pulsrs. Finlly we wish to point out how the mgneti field strength ffets the multipliity of the synhrotron sde. From disussions presented in Setions 3.2 nd 4.1 it is evident tht, for the sme energy of the primry photon, higher mgneti field strength results in (i) reduing the finl energy of esping photons, therefore inresing sde effiieny, nd t the sme time (ii) in suppression of sde effiieny by foring smller frtion of the photon energy to go into perpendiulr energy of the reted pirs nd so short-iruiting the sde. 7

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Therefore, there should be sweet spot in mgneti field strength where the synhrotron sde is most effiient. 5. CR In this setion we disuss how primry prtiles emit photons whih lunh the synhrotron sde. As we disussed bove the most effiient proess for supplying the primry (genertion ) photons in young energeti pulsrs is CR. 5.1. Frtion of the Primry Prtile Energy going into the Csde Ultrreltivisti prtiles moving long urved mgneti field lines emit eletromgneti rdition with power (Jkson 1975) P 2 e2 3 m CR = 5 e 2 2 4, ( 17) r where P CR is normlized to m e 2 s, is the prtile s energy normlized to m e 2 ; we do not distinguish between eletrons nd positrons. Sine we re treting the pir genertion problem seprtely from the problem of prtile elertion, we onsider sdes produed by prtiles injeted in region with sreened eletri field, so tht prtiles re not elerted nd only lose energy to CR. The prtile s energy dereses with time ording to the eqution of motion d =-P CR. ( 18) dt Solving Eqution (18) we get for the prtile energy fter it trvels the distne s from the injetion point ( ) 3-1 3 () s = 1 + 3H s ( 19) 2 r (see lso Hrding 1981), where s is normlized to R ns, is the initil prtile energy; onstnt H is defined s H = ( 2 3) Rnsre» 1.88 1-7m2, where re = e2 me2 is the lssil eletron rdius. The frtion z CR of the initil prtile energy lost to CR fter prtile hs trveled distne s CR is ( scr) z CR ( scr) = 1 -. ( 2) If the energy of these CR photons goes into retion of eletron positron pirs, zcr gives the effiieny of the CR prt of the full sde. The eletri field in the elertion zone trnsforms eletromgneti energy into prtile s kineti energy, whih is then rdited s pir produing photons. Only the photon s energy n be divided in hunks rried by lrge number of pirs. The sde will hve high effiieny if (i) primry prtiles hve high energy, (ii) emit most of their energy s photons, nd (iii) injet these photons in the region where the synhrotron sde n work effetively, i.e., in region lose to the NS whih is smller thn the hrteristi sle of mgneti field vrition L B. In Figure 7 we plot the frtion of the primry prtile energy emitted s CR photons fter the prtile hs trveled Figure 7. Frtion of the primry prtile energy emitted s CR photons over the distne scr = 1: ontours of zcr s funtion of logrithms of the initil prtile energy nd the rdius of urvture of mgneti field lines r in m. distne scr = 1. Shown re ontours of zcr ( scr) s funtion of the initil prtile energy nd the rdius of urvture of mgneti field lines r. For smller vlues of s CR the whole plot would move to the right. CR is most effiient in trnsferring prtile energy into the sde in the prmeter spe orresponding to the lower right tringulr region of Figure 7. Going from the upper left (smller, lrger r ) to the lower right (lrger, smller r ) on this plot, not only the prtile energy inreses but lso the frtion of the energy whih n go into the sde. For ertin rnge of nd r the energy put into the sde by the primry prtile grows fster thn the energy of tht prtile, i.e., the frtion of prtile s energy going into the sde inreses stronger thn linerly with the energy of the prtile. For more or less regulr globl mgneti fields, with r 17 m, the trnsition between effetive nd ineffetive CR sdes ours t prtile energies ~ 1 7, nd the effiieny is very sensitive to the prtile energy. In this prmeter rnge even modest inrese of the primry prtile energy n result in lrge boost of the sde multipliity. 5.2. Energy of CR Photons nd Critil Prtile Energy The hrteristi energy of CR photons emitted by prtile with the energy is (Jkson 1975) 3 3,r 5.8 13-1 3 g =» r,7,7, ( 21) 2 r where r,7 º r 17 m nd,7 º 17. The number of CR photons emitted by the prtile while trveling distne ds normlized to R ns is dn ds R P CR ns CR g,r ( 22) Eh CR photon bove the pir formtion threshold will be primry photon for the synhrotron sde disussed in 8

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 8. Critil prtile energy bove whih it n emit pir-produing photons vi urvture rdition: ontours of log,th s funtion of logrithms of the rdius of urvture of mgneti field lines r in m nd mgneti field strength B in Guss for ses = 1. Setion 4. The ritil energy t whih primry prtiles n produe pir-reting CR photons n be lulted by equting g,r given by Eqution (21) to the espe photon energy g,es from Eqution (1). In Figure 8 we plot the ritil prtile energy whih ould initite pir prodution with CR photons,th for ses = 1. Shown re ontours of log,th s funtion of the rdius of urvture of mgneti field lines r nd mgneti field strength B. Primry prtiles should hve energies 1 6 to be ble to initite pir prodution vi CR. 5.3. Multipliity of CR-synhrotron Csde The multipliity of the the CR-synhrotron sde the totl number of prtiles produed by single primry eletron or positron elerted in the gp n be omputed by multiplying the number of CR photons n CR, Eqution (22), by the number of prtiles produed in the synhrotron sde initited by these photons n syn, Eqution (15), nd integrting it over the distne where CR n initite sde scr CR syn ò k = n syn dn ds CR ds. ( 23) The tul lgorithm we use to ompute the totl multipliity is the Algorithm 2 from Appendix A. Integrtion in Eqution (23) is done ssuming onstnt vlues for B nd r, s disussed in Setion 3.1. In Figure 9 we plot kcr syn s funtion of the initil prtile energy log nd mgneti field strength log B for three different rdii of urvture of mgneti field lines r = 16, 17, 18m ssuming scr = ses = 1. It is evident from these plots tht in dipolr mgneti field, with r» 18 m, the mximum hievble multipliity is k CR syn few 14 ~ even in sdes initited by extremely energeti primry prtiles. If the rdius of urvture is n order of mgnitude less, rther high multipliity kcr syn 15 ould be hieved in polr p sdes for mgneti field strength B 1 12 G nd prtile energies 1 7, prmeters quite relisti for young pulsrs. For strongly non-dipolr mgneti field, with r» 16 m the multipliity n be nother order of mgnitude higher k CR syn few 16 ~. The properties of CR do not depend on the strength of the mgneti field, therefore the effet of the mgneti field strength on the multipliity of CR-synhrotron sde is due to the synhrotron sde nd how mny CR photons pir produe. As disussed t the end of Setion 4.3 there should be n optimum mgneti field strength where the multipliity is the highest; the multipliity dereses both for higher (due to lrger energy left in prtile motion long mgneti field lines) s well s lower mgneti field (due to inrese of the energy of esping photons). This trend is lerly visible on ll plots of Figure 9. The highest sde multipliity is hieved for mgneti field round B 1 12 G. The vlue of this optimum mgneti field grows slightly with inrese of r, but it stys round ~ few 112 G even for dipolr mgneti fields. This is noteworthy in view of the ft tht B 1 12 G is the typil vlue of mgneti field strength for norml pulsrs. For ny given energy of the primry prtile the derese of the sde multipliity towrds stronger mgneti fields is fster thn for weker fields. The dependene of sde multipliity on the initil energy of the primry prtile is nonliner. Let us onsider wht hppens when the initil prtile energy goes from the highest to the lowest vlue (horizontl diretion in plots of Figure 9). For the highest vlues of multipliity dereses uniformly, but then it drops by n order of mgnitude in rther smll rnge of (for B ~ 112 G it hppens round ~ 16.3 for r = 16 m, ~ 16.8 for r = 17 m, nd ~ 17.4 for r = 18 m). After bout hlf dede of vlues the multipliity drops gin to 1, when no prtiles n be produed (for B 112 ~ G it hppens round ~ 15.7 for r = 16 m, ~ 16.3 for r = 17 m, nd ~ 17 for r = 18 m). The first effet is due to the derese of the effiieny of CR, disussed in Setion 5.1. For lower initil energies of the primry prtile there is less energy vilble to rete pirs, not only beuse the prtile energy is smller, but lso due to smller effiieny of CR in produing photons the primry prtile keeps most of its energy, depositing only smll frtion of it in the sde zone. The drop in the effiieny of CR z CR, where it beomes less thn 1% (see Figure 7) mnifests in rpid derese of sde multipliity by n order of mgnitude on ll plots of Figure 9 for mgneti field strengths where the mximum multipliity is hieved. This drop in kcr syn is most prominent for B ~ 112 G nd is less pronouned for both higher nd lower mgneti field strengths due to lower effiieny of the sde disussed bove. The seond drop in k CR syn, towrds 1, is due to the threshold in pir formtion for those prtile energies CR photons hve too low n energy to initite sde. The blue region on the plots of Figure 9 show the prmeter spe where no prtiles n be produed by the CR-synhrotron sde. It does not men, however, tht no pirs n be produed in the polr p sdes for suh primry prtile energies. Insted of CR, the primry photons for synhrotron sde will be produed by inverse Compton sttering (ICS) of therml photons emitted by the NS, however, those primry photons will hve muh lower energies nd multipliities of suh sdes will be quite low (see Hrding & Muslimov 22). 9

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 9. Multipliity of CR-synhrotron sde: ontours of log k CR syn s funtion of logrithms of the primry prtile energy nd mgneti field strength B in Guss for three vlues of the rdius of urvture of mgneti field lines r = 16, 17, 18m. Assumed vlues for hrteristi lengths: s = s = 1. CR es Figure 1. Effiieny of CR-synhrotron sde s given by Eqution (24): ontours of zcr syn s funtion of logrithms of the primry prtile energy nd mgneti field strength B in Guss for three vlues of the rdius of urvture of mgneti field lines r = 16, 17, 18m. Assumed vlues for hrteristi lengths: s = s = 1. CR es We find it lso instrutive to ompre the multipliity of the CR-synhrotron sde with the theoretil upper limit on sde multipliity k mx, given by Eqution (1) in Setion 2. The rtio z CR syn k = k CR syn mx ( 24) n be onsidered s the effiieny of splitting the energy of the primry prtile into pirs. In Figure 1 we plot zcr syn for the sme vlues of prmeters s kcr syn in Figure 9. Despite lower multipliity for smller vlues of B, the sde effiieny is higher, i.e., more of the initil energy of the primry prtile goes into pir formtion s opposed to the energy of esping photons nd kineti energy of pirs nd primry prtiles. This trend is disussed in Setions 4.1 nd 5.1. It is interesting to note tht for B few 111 G the sde n be quite effiient in splitting notieble frtion of primry prtile energy into pirs. For mgneti field 1 12 G the frtion of the primry prtile s energy going into pir prodution sturtes t 1%; this is the limiting effiieny of the highest possible multipliity sde in typil pulsr. The dependene of z on is similr to the dependene of k on. CR syn CR syn 6. PARTICLE ACCELERATION 6.1. Overview of Prtile Aelertion Regimes In this setion we will get n estimte for the energy of primry sde prtiles. In the following disussion we will rely on results of self-onsistent modeling of pir sdes by Timokhin (21) [T1] nd Timokhin & Arons (213) [TA13]. First, we give brief overview of how prtile elertion proeeds ording to these simultions. 1

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 11. Shemti representtion of gp formtion nd evolution for sdes in Rudermn Sutherlnd regime with jm jgj >. See the text for explntion. Whether nd how the pir formtion long given mgneti field lines ours depends on the rtio jm jgj of the urrent density required to support the twist of mgneti field lines in the pulsr mgnetosphere (e.g., Timokhin 26; Bi & Spitkovsky 21), jm º ( 4p) B, to the lol GJ urrent density, jgj º hgj, where h GJ = BP is the GJ hrge density. For the Rudermn & Sutherlnd (1975) sde model, where prtiles nnot be extrted from the NS surfe, effetive prtile elertion nd pir formtion is possible for lmost ll vlues of jm jgj (T1). In the spe hrge limited flow regime, first disussed by Arons & Shrlemnn (1979), pir formtion is not possible if < jm jgj < 1, but is possible for ll other vlues of jm jgj (TA13). Pir formtion is lwys non-sttionry: n tive phse when prtiles re elerted to ultrreltivisti energies nd give rise to eletron positron sdes burst of pir formtion is followed by quiet phse when reently generted dense eletron positron plsm sreens the eletri field everywhere. As pir plsm leves the tive region, it flows into the mgnetosphere, nd lter into the pulsr wind. When the density of the pir plsm drops below the minimum density neessry for sreening of the eletri field gp ppers hrge strved region where the eletri field is very strong, of the order of the vuum eletri field. To sreen the eletri field, the plsm density must be high enough to provide both the GJ hrge density h GJ nd the imposed urrent density j m. The trnsition between the region(s) still filled with plsm (herefter we ll it plsm til ) nd the gp is shrp plsm still pble of sreening the eletri field moves in bulk, its density is lose to the ritil density. The plsm density drops bruptly t the gp boundry; within the gp the prtile number density is muh smller thn the ritil density (see Figures 22 nd 23 in Setion 1). The motion of the boundry between the plsm filled region nd the gp sets the gp s growth rte. Some prtiles enter the gp nd re elerted to ultrreltivisti energies. The gp grows until the energy of prtiles elerted there beomes suffiient to strt eletron positron sdes nd the yle repets. The gp is not sttionry; in lmost ll ses it moves s whole fter its growth hs been terminted by newly reted pirs its upper nd low boundries re moving in the sme diretion. The gp n move, keeping its size for long time, or it might dispper rther quikly. The detils of gp behvior where the gp ppers, wht diretion it moves, how fst it disppers depend on the rtio jm jgj. However, despite these differenes, the wy in whih the highest energy prtiles re elerted is very similr in ny regime whih llows pir formtion studied in T1 nd TA13. Nmely, the size of the hrge strved region grows s the til of pir plsm moves, the bulk veloity of the til v sets the rte of the gp expnsion. Prtiles entering the gp from the til re elerted in lrger nd lrger gp, until they re ble to produe pir-produing photons. The ple where these photons re bsorbed nd produe pirs is the other boundry of the gp. 6.2. Energy of Primry Prtiles In this setion we obtin quntittive estimte for the mximum energy of elerted prtiles using s n exmple the se of the Rudermn Sutherlnd (RS) sde, when prtiles nnot be supplied from the surfe of the NS. As we mentioned before, gp formtion nd prtile elertion in the spe hrge limited flow regime, when pir retion is llowed, is very similr to the RS se nd estimtes for prtile energies obtined in this setion re pplible for the spe hrge limited flow with jm jgj > 1 nd j m j GJ <. The GJ hrge density is positive nd we onsider the se when the rtio jm jgj >. In Figure 11 we show shemti piture of how prtiles re elerted in this sde. On the top of eh figure we show the eletri field in the elerting region nd on the bottom shemti representtion of plsm motion in nd round the gp; plot () orresponds to the time when the eletri field sreening hs just strted, plot (b) shows welldeveloped gp moving into the mgnetosphere. These shemti plots illustrte results of tul simultions of RS sdes shown in Figures 3, 4, 11, nd 13 in T1. At the beginning of the burst of pir formtion, the gp ppers t the NS surfe nd its upper boundry is the til of plsm left from the previous burst of pir formtion, where the prtile number density is still high enough to sreen the eletri field (region I in Figure 11). Eletrons nd positrons in this til re trpped in eletrostti osilltions nd the bulk veloity of this til v is sub-reltivisti, but for lrge urrent densities (round or greter thn j GJ ) it is quite lose to. Eletrons from this til whih get to the gp boundry re pulled into the gp nd re elerted towrd the NS. As the til moves, the gp grows; the urrent nd hrge density in the gp is due to the flux of eletrons from the til nd so it remins onstnt within the gp. The gp growth is stopped when eletrons reh n energy high enough to produe pir-reting photons. This first-genertion of pirs strt sreening the eletri field eletrons move towrd the NS nd positrons re elerted towrd the mgnetosphere nd strt produing pir 11

The Astrophysil Journl, 81:144 (26pp), 215 September 1 reting photons s well (region II in Figure 11()). In numeril simultions (T1) the first-genertion positrons moving towrd the mgnetosphere hve pproximtely the sme energies s the primry eletrons whih initited the dishrge. Beuse those positrons re ultrreltivisti they prtilly o-move with the photons nd so new pirs re injeted lose to their prent prtiles mking blob of pir plsm moving into the mgnetosphere (region II in Figure 11(b)). 7 This blob is the lower boundry of the elerting gp, nd the gp exists until this blob thes up with the til from the previous pir formtion yle. For lrge urrent densities this n tke while s v is lose to. Plsm leking from the blob forms the new til (region IIb on Figure 11(b)). In the dishrge desribed bove primry prtiles re moving in both diretions nd initite sdes towrd the NS (eletrons) nd the mgnetosphere (positrons). As the dishrges hppen lose to the NS surfe, the sde n fully develop only in the diretion of the mgnetosphere prtiles moving towrd NS slm onto the str s surfe before they n produe lot of pirs. For RS dishrges the primry, genertion, prtiles inititing the full sde in Figure 1 re positrons in region II in Figure 11(b). As we mentioned bove, the energy of those positrons is very lose to the energy of the primry eletrons nd here, for the ske of simpliity, we provide estimtes only for the energy of primry eletrons. 8 The evolution of the eletri field in ny given point x nd moment of time t is given by (see, e.g., Eqution (1) in TA13) E ( x, t) =-4 p( j( x, t) - jm ) º-4 p ȷ, ( 25) t j is the tul urrent density long given mgneti field line nd j m is the urrent density imposed by the mgnetosphere. The differene ȷ º j - j m in the gp remins onstnt. When the upper boundry of the gp moves with the onstnt speed v this eqution n be integrted to get the eletri field in the gp x - x E( x, t) = E( x, t) + 4p ȷ + 4 p ȷ ( t - t). ( 26) v Where E ( x, t) is the eletri field within the gp t the moment t t the point x. If we ssume tht the gp boundry is t x t the moment t, then E ( x, t) =. Eletrons enter the gp from bove nd re quikly elerted by the strong eletri field nd move with reltivisti speed prtilly from the moment they leve the plsm til. If prtile enters the gp t t (in point x ) its oordinte is x = x - ( t - t), substituting x into Eqution (26) the eletri field seen by tht prtile is given by 4p E( xp) = ȷ 1 + ( x - x) º 4phGJ xj l ( 27) v In the lst step in Eqution (27) we denote with l º x - x the distne trveled by the prtile in the gp nd introdue x j 7 See lso Figure 22 in Setion 1 where we show snpshot from numeril simultions of the sde orresponding to the stge shown in Figure 11(b). 8 The reson for both kinds of primry prtiles, eletrons nd positrons, quiring lmost the sme energies is tht the potentil drop experiened by eh of them is regulted by the proess of pir formtion, rther thn by the detils of their elertion. We did nlytil estimtes for the finl energies of the first-genertion positron bsed on the model presented in this setion; the differene between energies of the primry eletrons nd first-genertion positrons in the frme of the model is bout 2%. defined s ȷ x j º 1 +, ( 28) j v GJ where j GJ is the GJ urrent density in n ligned rottor B jgj º hgj = P. ( 29 ) xj is ftor whih shows how stronger/weker the eletri field in the gp is ompred to the sitution of stti vuum gp in n (nti-)ligned rottor, like the one onsidered by Rudermn & Sutherlnd (1975). v onst is good pproximtion to the numeril results nd so xj onst.in sdes long mgneti field lines where j m is lose to the lol vlue of j GJ in n ligned rottor x j ~ 2, for the sme sitution in pulsr with inlintion ngle of 6, x j ~ 1. For energy losses dominted by CR, free elertion is good pproximtion for B 111 G (see Appendix B). If rdition losses re negligible, the prtile s eqution of motion is dp dt =-ee, ( 3) where p = mg is prtile s momentum. In terms of the distne trveled by the prtile in the gp l = ( t - t ) with E given by Eqution (27), prtile eqution of motion n be written s dp 4pe = h GJ x j l. ( 31) dl Integrting this eqution nd expressing j GJ through pulsr prmeters we get for prtile energy 2p B B P l 2 = x j. ( 32) q The distne the primry prtile trvels in the region of unsreened eletri field l gp the size of the gp s seen by the moving prtile is the sum of the distne the prtile trvels before emitting pir-produing photons terminting the gp e l,gp nd the distne these protons trvel until the bsorption point l g,gp e l = l + l g,gp. ( 33) gp,gp For ny given prtile, the lrger the distne l e the prtile trvels to the emission point, the higher the prtile energy nd the energy of CR photons it emits, nd so the smller is the distne trveled by the photon until the bsorption point l g. The distne the prtile trvels in the gp l gp is the minimum e vlue of l = l + l g beuse one the first pirs re injeted the vlnhe of pir retion will led to sreening of the eletri field. The photon men free pth l g n be estimted from Eqution (3) s l 2 r g =. ( 34) b Photon energy g depends on the prtile energy whih depends on l e ording to Eqution (32), nd so l g is funtion of l. e e l gp n be found by minimizing l = l + l g g 12

The Astrophysil Journl, 81:144 (26pp), 215 September 1 using l e s n independent vrible: l gp is the vlue of l whih stisfies dl dl e e =, l g,gp nd l,gp re the vlues of l g nd l e where l rehes its miniml vlue. Using Eqution (33) we n write n eqution for l g,gp dl g,gp =-1. ( 35) e dl,gp If the photon energy depends on l e s µ ( l e g ), then Eqution (35) is redued to e,gp l = l g,gp, ( 36) e where l g,gp is expressed in terms of l,gp using Eqution (34). l gp is then given by l 1 e = + l ( ) gp,gp. 37 The finl energy of the primry prtile is given by Eqution (32) with l = l gp. Plese note tht beuse the gp moves, the tul size of the gp (see Figure 11()) is v h 1 gp = + l gp. ( 38 ) The energy of the CR photons depends on the prtile energy s 3 (Eqution (21)), the prtile energy depends on l s l 2 (Eqution (32)), hene, µ ( l e ) 6 g nd = 6. Substituting expression for (Eqution (32)) into the expression for CR photon energy g (Eqution (21)), the ltter into Eqution (34), nd the resulting expression for l g into Eqution (36) with = 6 fter lgebri trnsformtions we e get the following expression for l,gp e,gp l 4 Bq 2 3 = p3 1 7 1 7-3 7 x r 2 7 P 3 7 B - 4 7. ( 39 ) The size of the gp s seen by the moving prtile ording to Eqution (37) is j 4 1 7-3 7 2 7 3 7-4 7 j,7 12 l 2 1 x r P B m, ( 4) gp where B12 º B 112 G nd r,7 º r 17 m. Substituting l = l gp into Eqution (32) we get for the finl energy of prtiles elerted in the gp 9, 49 1 7 pb q 2 7 1 7 = xj r 4 7 P- B 3 1 7-1 7 18 5 17 2 7 1 7 4 7 P 1 7 1 7 x r - - B 12. ( 41) j The dependene of the primry prtiles energy on pulsr period P, filling ftor xj nd the strength of the mgneti field B is very wek, the only substntil dependene is on the rdius of urvture of mgneti field lines. The reson for this is the strong dependene of CR photon energy on the energy of 3 emitting prtiles g µ, Eqution (21). Chnges in the threshold energy of pir produing photons whih n stop the 9 Our expression for the energy of primry prtiles hs the sme dependene on r, P, nd B s the expression for the potentil drop in the gp derived by Rudermn & Sutherlnd (1975), their Eqution (23). This is to be expeted s in both ses prtiles re elerted by the eletri field whih grows linerly with the distne nd the size of the gp is regulted by bsorption on urvture photons in mgneti field. The differene is in the presene of ftor xj nd different numeril ftor.,7 Figure 12. Primry prtile energy: ontours of log, s funtion of logrithms of the rdius of urvture of mgneti field lines r in m nd mgneti field strength B in Guss. We used the following vlues for gp prmeters P = 33 ms, x j = 2 nd = 17. gp growth use only modest vrition of the energy of primry prtiles, whih expliitly depends on P, x j, nd B, but not on r, Eqution (32). The energy of pir produing photons sets the energy of elerted prtiles, nd in pulsrs with strong elerting eletri field the gp will be smller thn in pulsrs with weker elerting field. In Figure 12 we plot the energy of prtiles, elerted in the gp of pulsr with P = 33 ms s funtion of the rdius of urvture of mgneti field lines r nd mgneti field strength B, ssuming x j = 2 nd = 17. The vlue» 17orresponds to of CR photons emitted by reltivisti prtiles with = 2.5 17 in mgneti field B = 112 G with r = 17 m, this is good estimte for in Eqution (41) for young pulsrs s the dependene on is very wek. This plot lerly illustrtes the dependene of, on B nd r weker mgneti field nd/or lrger rdius of urvture requires lrger photon energies for terminting gp growth, nd so the energy of the primry prtiles is lrger. We derived Eqution (41) under the following ssumptions: (i) prtiles re elerted freely, i.e., rdition retion n be negleted, (ii) the length of the gp is muh smller tht the polr p rdius, so tht one dimensionl pproximtion n be used, (iii) the mgneti field is B.2Bq» 8.8 112 G so tht the opity to gb pir retion is desribed by Eqution (2). Constrints on the pulsr prmeters (i) nd (ii) re derived in Appendix B nd Appendix C orrespondingly. Plotted on the PṖ digrm, Figure 13, these restritions selet the rnge of pulsr period nd period derivtives shown s yellow region where ll three ssumptions re vlid. In this figure, the one-dimensionl pproximtion (ii) is vlid to the left of the solid line, given by Eqution (59), the pproximtion (i) of free elertion bove the dot-dshed line, given by Eqution (54), nd pulsrs with B <.2B q re below the dotted line. We see tht most of young norml pulsrs, inluding gmm-ry pulsrs from the Fermi seond pulsr tlog, fll in this rnge. 13

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 13. PṖ digrm with the yellow re showing the rnge of prmeters where pproximtion for prtile elertion used in this pper is pplible, see text for desription. Pulsrs from ATNF tlog Mnhester et l. (25), http://www.tnf.siro.u/reserh/pulsr/psrt, re shown by blk dots, γ-ry pulsrs from the seond Fermi tlog (Abdo et l. 213) by red dots. 7. CASCADE MULTIPLICITY PER PRIMARY PARTICLE: SEMI-ANALYTICAL MODEL We now ombine the results of Setion 5.3 onerning the sde multipliity for fixed energy of the primry prtile, nd of Setion 6.2 onerning the energy of primry prtiles elerted in the gp with given prmeters P, x j, strength B, nd r. The multipliity of CR-synhrotron sdes depends on the energy of the primry prtile, mgneti field B, nd rdius of urvture of mgneti field lines r. The only signifint dependene of the energy of elerted prtiles in young pulsrs is on the rdius of urvture of mgneti field lines r ; the dependene on P, x j, nd B is very wek, see Eqution (41). Therefore, when prtile elertion is tken into ount, the overll sde multipliity κ n depend substntilly only on B nd r. In CR-synhrotron sde hnges in nd r hnge κ in opposite diretions for higher multipliity is higher, for lrger r multipliity is lower. But the energy of the primry prtiles elerted in PCs of young pulsrs is higher for lrger rdii of urvture of mgneti field lines r. Hene, inresing r lowers the sde multipliity for fixed, but t the sme time inreses, whih prtilly ompenstes the derese of sde multipliity. Therefore, the finl sde multipliity should hve rther wek dependene on r, whih leves the mgneti field strength B the only prmeter signifintly ffeting the multipliity of strong sdes in PCs of young pulsrs. In Figure 14 we show the finl multipliity s funtion of mgneti field strength nd rdius of urvture of mgneti field lines. We present plots for three sets of prmeters P nd xj whih differ by n order of mgnitude. As expeted, both pulsr period s well s the filling prmeter xj hve very smll effet on the finl multipliity. For the rnge of r nd B in Figure 14, the CR-synhrotron sde hs the highest effiieny primry prtiles lose most of their energy s CR photons within the distne scr = Rns where the sde is possible nd higher multipliities nnot be hieved. Aording to Figure 14 the sde multipliity sles with r roughly s k µ ( r ), with 1 2. The intervl [ 16, 18] m represents the most resonble vlues of r for ny globl mgneti field onfigurtion in the pulsr polr p. For very different mgneti field onfigurtions highly multipolr field with r ~ 16 m versus dipole field with r ~ 18 m the multipliity differs by less thn n order of mgnitude. The dependene of κ on the mgneti field is stronger, with the mximum rehed ner B 1 12 G. It is remrkble ft tht the multipliity of the most effiient sdes is sensitive mostly to the strength of the mgneti field. The multipliity is not very sensitive to r nd for typil pulsr mgneti field of 1 12 G is round 1 5. For fixed B the totl pir yield the totl number of prtiles injeted into the mgnetosphere depends then only on the flux of primry prtiles. This is true for pulsrs where the elerting potentil is regulted by pir prodution nd where sde operte in CR-synhrotron regime hve high effiieny. 8. CASCADE MULTIPLICITY PER PRIMARY PARTICLE: NUMERICAL SIMULATIONS In Setions 3 7 we developed semi-nlytil model of strong polr p sdes. In order to verify the key ssumptions nd onlusions of this model we hve performed numeril simultions of time-dependent polr p pir sdes. Beuse suh numeril simultions re quite time onsuming we limited ourselves to the se of young pulsr with prmeters similr to the Crb pulsr (P = 33 ms) nd mgneti field strength B nd rdius of urvture of mgneti field lines r hving vlues resulting in high multipliity. Our gol ws to show tht the min ssumptions nd onlusions of our nlysis of polr p sdes re relisti. More extensive self-onsistent numeril studies of the polr p sdes will be done in subsequent ppers. As outlined in Setion 6.1, prtiles re quikly elerted in the gp whih is muh smller thn the typil distne over whih the full sde develops. The primry pir produing prtiles re moving most of the time in the region with sreened eletri field. If the primry prtile energies re known, the full sde n be modeled using trditionl Monte-Crlo tehniques (Dugherty & Hrding 1982); to obtin the energies of primry prtiles inititing the CRsynhrotron sde self-onsistent model of the sde (Timokhin 21; Timokhin & Arons 213) is neessry. We hve performed numeril simultions of time-dependent polr p pir sdes in two-step proess. In the first step, we use hybrid Prtile-in-Cell/Monte Crlo (PIC/MC) ode PAMINA (PIC And Monte-Crlo ode for sdes IN Astrophysis) to simulte the initil self-onsistent eletri field genertion, prtile elertion nd eletri field sreening ner the NS to obtin the distribution funtions of the eletrons nd positrons in the elertion/sreening region. This ode inludes only CR of the prtiles nd first genertion of pirs needed to sreen the gp, nd so does not follow the full synhrotron sde. The detils of this ode re desribed in Timokhin (21), Timokhin & Arons (213). In the seond step, we use nother ode to simulte the full pir sde in the pulsr dipole field bove the PC, inluding both CR of primry prtiles nd synhrotron rdition of pirs. 14

The Astrophysil Journl, 81:144 (26pp), 215 September 1 This ode, bsed on the lultion desribed in detil in Hrding & Muslimov (211) [HM11], is Monte-Crlo simultion of the eletron positron pir sde generted bove PC by elerted prtiles in the region of sreened eletri field. Although HM11 inluded stedy prtile elertion omponent, this omponent is not used in the present lultion. We therefore ssume tht the prtiles nd the further pirs they rete do not undergo ny elertion. This ode tkes s input the distribution funtions of elerted prtiles output by the time-dependent PAMINA ode tht re moving wy from the NS surfe nd simultes the ombined sde from ll of the prtiles. Although our setup is pble of lulting full sdes generted by primry prtiles with rbitrry distribution funtions, in the simultions desribed in this setion we used monohromti injetion of primry prtiles. On the one hnd, the energy distribution of the most energeti primry prtiles whih produe the bulk of the pirs in mny ses is lose to monohromti (see, e.g., Setion 1, Figure 23), nd on the other hnd this enbles us to ompre numeril simultions with preditions of our semi-nlytil theory. The energy of the primries ws lulted from the self-onsistent model however. The MC ode first follows the primry prtile in disrete steps long the mgneti field line t mgneti oltitude θ, strting from the lotion x nd prtile energy t time t pek t the pek of the pir prodution yle in PAMINA ode, omputing its CR. The steps Dx re set to the minimum of frtion.1 of NS rdius nd the distne over whih the prtile would lose 1% of its energy to CR. Dividing the CR spetrum t eh step into logrithmi energy intervls, representtive photon from eh energy intervl is followed through the urved mgneti field until its point of pir prodution (determined s rndom frtion of the men-free pth). The number of CR photons in eh energy bin, n CR,is determined by the energy loss rte nd verge energy in tht bin. The pirs produed by the photon, or the esping photon number, is then weighted by n CR. The reted pir is ssumed to hve the sme diretion nd hlf the energy of the prent photon. Although the CR photons re rdited prllel to the mgneti field, they must quire finite ngle to the field before produing pir, so the reted pirs hve finite pith ngles t birth. Eh member of the pir emits sequene of ylotron nd/or synhrotron photons, strting from its initil Lndu stte until it rehes the ground stte, ssuming the position of the prtile remins fixed (given the very rpid rdition rte). As desribed in HM11, when the pir Lndu stte is lrger thn 2, the symptoti form of the quntum synhrotron rte (Sokolov & Ternov 1968) is used to determine the photon emission energy nd finl Lndu stte. When the Lndu stte is below 2, the full QED ylotron trnsition rte (Hrding & Preee 1987) is used. At lrge distnes bove the NS surfe, when the mgneti field drops below.2b q,we short-ut the individul emission sequene nd use n expression for the spetrum of synhrotron emission for n eletron tht loses ll of its perpendiulr energy (Tdemru 1973). Eh emitted photon is then propgted through the mgneti field from its emission point until it pir produes or espes. The next genertion of pirs re then followed through their synhrotron/ylotron emission sequene. By use of reursive routine tht is lled upon the emission of eh photon, we n follow n rbitrry number of pir genertions. The sde ontinues until ll photons from eh brnh hve esped. As eh member of eh reted pir ompletes its synhrotron emission, its ground-stte energy, position, nd genertion number re stored in pir tble. As eh photon either pir produes or espes, its energy, genertion, nd position of pir retion or espe re stored in tbles for bsorbed nd esping photons. The photons nd pirs from ll elerted prtiles re summed together to produe the omplete sde portrit t tht time step. The NS mgneti field in the MC ode desribed bove is distorted dipole with n zimuthl (f) omponent whih is offset from the enter of the NS. The mgneti field is given by B B R ns 3 = r 1 er os[ q( 1 + )] + eq 2 sin [ q( 1 + )] 1 - ef eq ( + sin qos q) sin ( f- f), ( 42) 2 where B is the surfe mgneti field strength t the mgneti pole, r is the rdil oordinte, = e os( f - f) is the prmeter hrterizing the distortion of polr field lines, nd f is the mgneti zimuthl ngle defining the meridionl plne of the offset PC. The prmeter ε sets the mgnitude nd the prmeter f sets the diretion of symmetry of this zimuthl omponent. Setting e = gives pure dipole field struture, while non-zero vlue of ε produes n effetive offset of the PC from the dipole xis in the diretion speified by f. For non-zero ε, the rdius of urvture of the mgneti field lines is smller thn dipole in the diretion of the offset nd lrger thn dipole in the diretion opposite to the diretion of offset. This prtiulr prmetri form for the mgneti field ws used in simultions of sttionry sdes in HM11, it ws hosen to ount for distortion of the shpe of the PC used by urrents flowing in pulsr mgnetosphere. Suh zimuthl symmetries in the ner-surfe mgneti field is used by the sweepbk of the field lines ner the light ylinder due to retrdtion (e.g., Dyks & Hrding 24) nd urrents (e.g., Timokhin 26; Bi & Spitkovsky 21; Klpothrkos et l. 214) or, dditionlly, by symmetri urrents in the NS. We did not perform systemti study of ll prmeter spe with our numeril simultions, whih will be done elsewhere, with our numeril simultions we test ssumptions nd preditions of our semi-nlyti sde model. Any form of mgneti field with djustble rdius of urvture of mgneti field lines would serve our purposes, but using the mgneti field given by Eqution (42) llows omprison with the most reent simultions in the frme of the previous-genertion sde models HM11. We explored ses of pir sdes both for pure dipole fields nd for zimuthlly distorted fields. Multipliities obtined from the numeril simultions gree resonbly well with the semi-nlyti model, within ftor of few. As n exmple we desribe in detil results of prtiulr simultions with pulsr prmeters yielding high multipliity for sde t the pek of the pir retion yle. The mgneti field is B = 1 12 G nd is modertely distorted, with the offset e =.4 resulting in the rdius of urvture of mgneti field lines ner the NS r = 8.8 16 m. The initil 15

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 14. Multipliity of polr p sdes: ontours of log k s funtion of logrithms of urvture of mgneti field lines r in m nd mgneti field strength B in Guss for three sets of the gp prmeters ( P [ ms ], x ): ( 33, 2), ( 33,.25), ( 33, 2). In ll ses we used = 17for lultion of the energy of primry prtiles. j energy of primry prtiles from PAMINA simultions is = 2.3 17. In Figures 15 nd 16 we show set of five plots whih we ll sde portrits. These plots illustrte different spets of the sde development by showing moments of the photon or prtile distribution funtion f ( x, E). The top pnel shows the number of prtiles produed in eh energy nd distne bin dn ( x, E) = f ( x, E) dx de, ( 43) s 2D olor mp. The number of prtiles is olor oded in logrithmi sle ording to the olor br on the right. The middle left plot shows the energy distribution E f ( xi, E) of prtiles produed t four distnes xi, i = 1... 4; different line styles orrespond to different distnes ording to the left plot legend. These spetr re essentilly ross-setions of the mp of dn( x, E) (multiplied by prtile energy) long four lines shown in the plot for dn( x, E). The bottom left plot shows the energy distribution of ll prtiles produed in the sde (dn( x, E) integrted long x diretion nd multiplied by E) ò x mx Ef( E) = E dxf x, E. ( 44) ( ) In this nd the following plots, olored lines show ontributions of different sde genertions; lines re olor-oded ording to the right plot legend. The middle right plot shows the differentil pir prodution rte number of prtiles produed in distne bin (dn( x, E) integrted long the E diretion) ò Emx dn () x = dx f ( x, E) de. ( 45) The bottom right plot shows the umultive pir prodution rte totl number of prtiles produed up to the distne x x Emx N( x) = dx f x, E de. ( 46) ò ò ( ) Figure 15 shows the sde portrit of the pirs (we do not differentite between eletrons nd positrons). The sde extends to bout ~ R, where it dies out ompletely s the 6 ns mgneti field strength nd the elerted prtile energy derese. The pir number grows very quikly, within few tenths of NS rdius, nd then sturtes t k» 7.4 1 4. The mjority of pirs re produed t distnes < R ns (see plot for N ( x) ), whih supports the ssumption bout the length of the sde zone being ~ R ns mde in Setion 3.1. The number of pir genertions is highest t R ns, up to six in this se. 1 The ourrene of most of the sde genertions t distnes up to ~ R ns is the evidene tht the photon mfp in strong sde is indeed smll, l g R ns, so tht the sde initited by ny given prtile goes through severl genertions within the distne ~ R ns. The lrge extent of the sde zone reltive to l g is due to ontinuous injetion of pir-produing CR photons (see the blue line in the plot of dn(x); genertion pirs re produed by CR photons). The lrgest ontribution to pir multipliity in this se omes from genertion 1, pirs reted by the first synhrotron photons (see plots for dn(x) nd N ( x)). In our simultions for different pulsr prmeters, ontributions of genertion 1 nd 2 to the pir multipliity sometimes beome omprble; for weker sdes the reltive ontribution of genertion is higher thn in this se, however we did not see genertions 3 nd higher produing the mjority of pirs. The number of sde genertions is not very lrge in ny of our simultions (severl t most), but in eh genertions high numbers of pir produing photons re emitted whih results in high multipliity. The energy of reted pirs derese with distne (see plots for dn( x, E) nd E f ( xi, E)), mostly beuse of energy losses of primry prtiles whih results in lower energy CR photons. The pir spetrum extends down to few m 2, sine the sde is very effiient t onverting initil pir energy into more photons (nd pirs). Degrdtion of pir energies through sde genertions disussed in Setion 4.3 is lerly visible on the plot for E f ( E) the mximum pir energy systemtilly dereses with sde genertions t ll distnes. 1 There re too few pirs produed in the 6th genertion to show in the plot; urves orresponding to this genertion re below the lower limit of ll plots in Figure 15; this genertion shows in the portrit for photons, Figure 16. 16

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 15. Csde portrit of eletron positron pirs. Top pnel: number of prtiles produed in eh energy nd distne bin dn( x, E) olor oded in logrithmi sle ording to the olorbr on the right. Middle left: E f ( xi, E) energy distribution of prtiles produed t four distnes xi, i = 1... 4; different line styles orrespond to different distnes ording to the left plot legend. Bottom left: E f ( E) energy distribution of ll prtiles produed in the sde. Middle right: dn (x) differentil pir prodution rte number of prtiles produed in distne bin. Bottom right: N ( x) totl number of prtiles produed up to the distne x. Color lines in plots for E f ( E), dn(x), nd N ( x) show ontributions of different sde genertions, lines re olor-oded ording to the right plot legend. Thik blk lines show ontributions of ll sde genertions. x is the distne from the NS normlized to NS rdius R ns nd E is prtile energy normlized to m e 2. Prtile number density is normlized to n GJ. Prmeters of this simultion: pulsr period P = 33 ms; the mgneti field in the PC hs B = 112 G nd 8.8 16 r = m; initil energy of primry prtiles = 2.3 17. Figure 16 show the portrit of photons esping the sde. Photon genertion is CR while the higher genertions ( 1) re synhrotron/ylotron rdition. Although the highest energy photons re produed nerest the NS surfe, these photons re bsorbed by pir prodution ttenution so tht the spetr t the lowest ltitudes show shrp utoffs ner 1 m 2. This utoff is lerly visible in the plot for Ef ( xi, E) for x1 =.15Rns shown by dshed line. In the plot for dn( x, E) the utoff is evident s shrp horizontl boundry of the olored region for x.16r ns. The esping photon energies inrese with distne from the NS surfe, s the mgnetosphere beomes more trnsprent. The highest esping photon energies re produed ner the end of the sde, t round 4 NS rdii. The highest energy photons esping the sde re CR photons (blue line in the plot for Ef(E)). Synhrotron rdition is emitted by pirs right fter their retion, so tht the pir formtion nd synhrotron rdition end t the sme distne in the plot for dn(x), the number of photons in eh genertion drops t lrge distnes in ordne with the drop of number of injeted pirs shown on similr plot in 17

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 16. Csde portrit of esping photons for the sme sde s in Figure 15. Nottions nd normliztions re the sme s in Figure 15. Figure 15. Above ~ 4R ns the emitted CR photon energies drop s the primry prtiles ontinue to lose energy. The spetrum of esping photons lso brodens t the lower end beuse, s the mgneti field dereses, so does the ylotron energy whih sets the lower limit of the synhrotron spetrum. Thus the lowest nd the highest energy esping photons re produed t the lrgest distnes from then NS, f. spetr t different x i in the plot for Ef ( xi, E). The bulk of high energy emission from the PC sde omes from synhrotron rdition of pirs in genertions 1, 2, nd 3. Quntittively, our semi-nlyti theory ompres with this prtiulr numeril simultion s follows. For pulsr prmeters used in this simultion, the energy of primry prtiles, ording to Eqution (41) should be» 3.6 17, whih is» 1.6 times lrger thn the result obtined from numeril,num simultions with PAMINA ode» 2.3 17. For the multipliity of the CR-synhrotron sde strted by primry prtiles with monohromti energies,num, the semi-nlyti model gives k CR syn 1.56 15», from Eqution (23), whih is lso bout 2 times higher thn the vlue obtined in numeril simultions k num» 7.4 14. The ombined model from Setion 7, whih uses the nlyti model for prtile elertion s n input for the semi-nlyti model of CR-synhrotron sde, predits for the multipliity k» 2.9 15, see Figure 14, whih is» 4 times lrger thn the multipliity from numeril simultions. The disrepny with the semi-nlyti model for severl other numeril simultions we performed with different prmeters is of the sme order. We ttribute this disrepny mostly to the pproximtion of onstnt mgneti field in the numeril simultions, where B nd r (whih is derived from B) depend on the distne ording to Eqution (42), photon bsorption 18

The Astrophysil Journl, 81:144 (26pp), 215 September 1 dereses nd beomes less effiient with the distne. We think tht for suh simple model the greement with the numeril simultions is resonble nd the model n be used for estimtes of multipliities in young energeti pulsrs. 9. RICS Another emission mehnism for reltivisti prtiles besides urvture nd synhrotron rdition is ICS. In strong mgneti fields typil for pulsr PCs, ICS n our in the resonnt regime, when the photon energy in the eletron s rest frme is equl to the ylotron energy. The ross-setion for sttering of suh photons is gretly enhned ompred to tht of non-mgneti sttering. It hs been noted tht resonnt ICS (RICS) with the soft therml photons from the NS surfe is importnt for high-energy emission from pulsr PCs (Sturner 1995; Zhng & Hrding 2) nd n be importnt in the development of polr p sdes beuse for quite wide rnge of pulsr prmeters, sttered photons n be bove pir-formtion threshold (e.g., Sturner et l. 1995; Zhng & Hrding 2). In this setion we rgue tht lthough RICS n ply role in the development of polr p sdes, it never beomes the dominnt soure for pir multipliity nd, therefore, onsidering only CR-synhrotron sdes provides dequte estimtes for pir multipliity in strong sdes of norml pulsrs. A detiled study of the role of RICS in polr p sdes will be presented in subsequent pper. First, let us onsider the effiieny of RICS in trnsforming prtile kineti energy into rdition. The distne over whih prtile loses most of its energy to RICS is given by (Dermer 199; Sturner 1995; Zhng & Hrding 2): l RICS =-.61 T - B - ln- 1 - exp - 2 6 1 12 2 134B T 1 - m 1 12 6( s) m ( 47) where T 6 is the temperture of the NS surfe in units of 1 6 K, B 12 is the mgneti field strength in units of 1 12 G, nd ms = os qs, where q s is the ngle between the moment of the sttering photon nd prtile in the lb frme. If the NS is young, its surfe temperture omes mostly from ooling nd should be bout 1 6 K. It emits X-ry photons neessry for RICS from the whole surfe, nd for prtiles t distne omprble to the NS rdius the rnge of ms is quite lrge. For older NS, with full surfe tempertures below few 1 5 K only the polr p region, heted by the bkflow of elerted prtiles up to ~ few 1 6 K (e.g., Hrding & Muslimov 21, 22), n emit enough photons for RICS to beome importnt. In the ltter se when the prtile rehes distne omprble to the width of the polr p rp 1.4 14 P m, where P is pulsr period in seond, the rnge of ms gets very smll nd photons quikly get out of resonne. So, for young NSs RICS is n importnt rdition proess if l RICS ~ Rns; for old NSs this ondition hnges to l RICS ~ rp. In Figure 17 we plot lrics s funtion of prtile energy nd mgneti field B, for T = 1 6 K nd m s =.5. It is evident from this plot tht primry prtiles, with > 1 6, lose negligible mount of their energy vi RICS, nd CR is the dominnt emission mehnism for genertion of strong sdes. For old NSs, RICS of photons from heted PCs Figure 17. Distne (in m) over whih prtile loses its energy vi resonnt inverse Compton sttering. Contours of log l RICS re plotted s funtion of logrithms of prtile energy nd mgneti field B in Guss for NS surfe temperture T = 1 6 K nd m =.5. might be n importnt emission mehnism only for very nrrow energy rnge of low energeti prtiles, in one of the lter sde genertion the prmeter spe between two l RICS = 14 ontours is rther smll nd for most vlues of B the sttered photons will be below pir formtion threshold. Hene, RICS n be ompletely negleted in strong polr p sdes of old NSs. For hot NSs RICS beomes n importnt emission mehnism for wide rnge of moderte prtile energies < 1 4 the energy rnge between ontours of 6 l RICS = 1 in Figure 17 is quite wide. In the ltter se, the digrm for physil proesses in strong polr p sde n hve the generl form shown in Figure 18, with RICS photons in some ses rrying non-negligible energy strting from sde genertion 1. The extension of the semi-nlytil nlysis of CR-synhrotron sde developed in Setions 3 7 to the whole sde tking into ount multiple sde brnhes is not strightforwrd nd we postpone it to future publitions. Here we will mke rough estimtes of the ontribution of RICS sde brnhes to the multipliity of the whole sde. Let us now disuss the effiieny of RICS sde brnhes in splitting the vilble energy into pirs. The energy of photons fter sttering by prtiles of energy in the RICS regime is (e.g., Zhng & Hrding 2) s g,rics = 2,F b, ( 48) where,f, given by Eqution (12), is the kineti energy of prtile moving long mgneti field line the finl energy of freshly reted pirs fter they emit ll perpendiulr to B energy vi synhrotron rdition. Using Equtions (48), (12) together with Eqution (21) for the energy of CR photons nd Eqution (41) for the energy of the primry prtile, we get n upper limit on the energy of the genertion 1 RICS photons the highest energy RICS photons. In Figure 19 we plot the rtio of ( 1) the energy of genertion 1 RICS photons g,rics to the energy of photons esping from the sde zone es s funtion of 19

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 18. Digrm showing the generl hin of physil proesses in strong polr p sde. Csde genertions re shown on the left numbers onneted by double rrows. In eh genertion prtiles e (eletrons nd/or positrons) produe photons ( g CR vi urvture rdition, g syn vi synhrotron rdition, g RICS vi Resonnt inverse Compton sttering), whih re turned into pirs of the next sde genertion. The CR-synhrotron sdes studied in detil in this pper re shown by solid rrow, dshed rrows show RICS initited brnhes whih re disussed only in Setion 9. ( i 1) + g,syn Figure 2. Rtio of the hrteristi energies of synhrotron nd RICS ( i + 1) g,rics photons emitted by freshly reted pirs (( i + 1) th genertion sde i i photons): ontours of log ( +,syn 1 ) ( + 1) g g,rics re plotted s funtion of () i logrithms of the prent photon energy g nd mgneti field strength B in Guss for r = 17 m. For stronger mgneti fields, however, RICS photons do ontribute to pir multipliity. The hrteristi energy of RICS photons in terms of the energy of the previous genertion photon n be obtined by substituting,f into Eqution (48) 2-1 2 = g b 1 +. ( 49) b ( i + 1) i g,rics ( 1) g Figure 19. Rtio of of the hrteristi energy of RICS photons,rics emitted by the first genertion pirs to the energy of esping photons: ontours of log ( 1) g,rics g,es re plotted s funtion of of logrithms of the rdius of urvture of mgneti field lines r in m nd mgneti field strength B in Guss. We used the following vlues for gp prmeters P = 33 ms, x j = 2 nd = 17. the rdius of urvture of mgneti field lines nd the strength of the mgneti field B. It is esy to see tht for mgneti fields weker thn few 1 11 G, even the highest energy RICS photons re not pble of produing eletron positron pirs. This eqution desribes the energy degrdtion in eh sde genertion for the RICS proess. In Figure 2 we plot the rtio of the hrteristi energies of synhrotron nd RICS photons (given by Equtions (49) nd (16) orrespondingly) produed by pirs reted by the sme prent photon s funtion of the () i prent photon energy g nd mgneti field strength B. The energy of RICS photons re lwys smller thn the energy of synhrotron photons, nd for B < 112 G signifintly so. Beuse of the muh fster energy degrdtion in the RICS proess, sde brnhes initited by RICS photons re in generl shorter thn those initited by synhrotron photons. In lter sde genertions RICS photons freely espe the sde zone while synhrotron photons emitted by the sme prtiles re bsorbed, still splitting the energy of the prent photons into pirs. Therefore, RICS sde brnhes should be in generl less effiient in splitting the energy thn synhrotron ones. Finlly let us ddress the question of how muh energy is going into RICS brnhes of the sde. The energy powering RICS brnhes is the kineti energy of pirs moving long mgneti field lines, while synhrotron brnhes re powered by the perpendiulr energy of freshly reted pirs. From Equtions (12) nd (13) we get for the rtio of the 2

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 21. Rtio of perpendiulr to prllel to B energy of freshly reted pirs: ontours of ^ s funtion of logrithms of the prent photon energy g nd mgneti field strength B in Guss for r = 17 m. perpendiulr to prllel pir energies ^ 2-1 2 = 1 + - 1. ( 5) b As > b (see Eqution (8)), the minimum vlue for this rtio is 2-1.414. In Figure 21 we plot ^ s funtion of the energy of the prent photon nd mgneti field strength B. For most of the prmeter spe in eh pir retion event, the frtion of energy going into the RICS sde brnh is smller thn those going into the synhrotron brnh. Even in the se when more energy is left in the pirs prllel motion, the energy vilble to the RICS brnh is only 1.5 times lrger thn the energy vilble for synhrotron brnh. In this se, however, the vlue of is lose to b, i.e., pir formtion ours ner the kinemti threshold nd is not very effiient (see Setion 3.1). Hene, sdes where RICS brnhes hve more vilble energy should not be very effiient. Summrizing the bove rguments, RICS brnhes re less effiient in splitting photon energy into pirs nd the energy vilble to those brnhes is in the best se omprble to the energy vilble for synhrotron brnhes. The effet of RICS brnhes re strongest for higher vlues of the mgneti field, few 1 12 G, when pir formtion hppens lose to kinemti threshold nd the sde multipliities re lower thn for weker mgneti fields. The finl multipliity of the totl sdes should be less thn twie tht of the pure CRsynhrotron sde in the best se nd so the pure CRsynhrotron sdes studied in this pper provide good estimtes for the multipliity of strong polr p sdes. 1. FLUX OF PRIMARY PARTICLES AND PAIR YIELD As we hve shown bove, for the most effiient polr p sdes the totl pir yield for given vlue of the mgneti field B depends mostly on the flux of primry prtiles. Aording to self-onsistent models of polr p elertion zones (T1, TA13), prtile elertion is intermittent nd the pttern of plsm flow nd elertion effiieny depends on the rtio of the urrent density imposed by the mgnetosphere to the GJ urrent density jm jgj s well s the boundry onditions t the NS surfe whether prtiles n be extrted from the surfe or not. There re essentilly three qulittively different regimes of plsm flow tht determine the flux of primry pir-produing prtiles: (i) in spe hrge limited flow with < jm jgj < 1 prtiles re elerted up to very low energies, ~ few, nd no pirs re produed, (ii) in spe hrge limited flow with jm jgj > 1 signifint flux of primry prtiles of the order of ~ jgj e is elerted through the gp while the gp is moving towrd the NS, (iii) in ll other ses i.e., for ny urrent density in Rudermn Sutherlnd model nd for jm jgj < in spe hrge limited flow regime in eh burst of pir formtion blob of primry prtiles is produed during formtion of the gp, nd then no signifint mount of primry prtiles is reted until the formtion of the next gp. In Figure 22 we plot s n exmple for se (iii) phse portrit nd densities of plsm nd photons in the phse spe for Rudermn Sutherlnd sde with jm jgj = 1. The proess of gp formtion for this se is desribed t the beginning of Setion 6.2. When the gp is formed, only very few prtiles lek from the plsm til in this se eletrons, visible s line-like feture in the top two pnels of Figure 22 re elerted in the gp, the flux of these prtiles is muh less thn ~ jgj e nd their ontribution to the pir prodution n be negleted. The blob of primry prtiles whih ws reted during the gp formtion hs density of few n GJ in its densest prts nd size omprble to the gp s height. Those prtiles in this se positrons in the re surrounded by dotted line on the plot for positron phse spe density in Figure 22 rete the vst mjority of pirs. The density of primry prtiles in the blob is high but no new primry prtiles re reted until the next gp is formed, the duty yle of the sde is smll. The resulting flux of primry prtiles verged over the period of gp formtion is rther low. In se (ii), in spe-hrge limited flow with super GJ urrent density, the sitution is qulittively different in tht during the lifetime of the gp, nd not only during the time the gp is forming, s in se (iii), signifint onstnt flux of prtiles is going through the elertion zone. The duty yle of suh sde should be substntilly higher thn in ny other types of sdes. As n exmple of suh sde we plot in Figure 23 phse portrit nd the densities of plsm nd photons in phse spe for sde in spe-hrge limited flow regime with jm jgj = 1.5. The gp is formed t some distne from the NS nd moves towrd it. When the first prtiles re formed the proess of eletri field sreening proeeds in similr wy to the se of Rudermn Sutherlnd sdes; the blob of ultrreltivisti prtiles is reted nd prtiles leking from it rete til of mildly reltivisti plsm sreening the eletri field behind the blob; this blog moves towrd the NS reting pirs. However, onstnt flux of prtiles extrted from the NS surfe in this se eletrons, visible s line-like feture in the top two pnels of Figure 23 re elerted in the gp. The density of these eletrons is high (for the se shown in Figure 23 it is 1.25 n GJ ) nd these prtiles, nd not the prtiles from the blob, produe most of the pirs. 21

The Astrophysil Journl, 81:144 (26pp), 215 September 1 Figure 23. Snpshot of the phse spe for sde in Spe Chrge Limited Flow regime. Csde n pulsr prmeters: jm jgj = 1.5B = 112 G, P = 33 ms, x j =.25. Types of plots nd nottions re the sme s in Figure 22. Figure 22. Snpshot of the phse spe for Rudermn Sutherlnd sde t the end of dishrge. Csde n pulsr prmeters: jm jgj = 1B = 112 G, P = 33 ms, x j = 1. Horizontl xis prtile positron x, vertil xis prtile momentum normlized to me; the vertil xis is logrithmi exept for the region round zero momentum (-5 < p < 5), where the sle is liner. The top pnel shows phse spe portrit of the sde: eh dot represents numeril prtile (every 1th prtile is plotted); blue dots eletrons, red dots positrons, blk dots photons. Three plots beneth show prtile number density in phse spe: p- vs. x-eletrons, p+ vs. x-positrons, pg vs. x-photons. Prtile number density is olor-oded ording to the olor mp on the right in units of n GJ. Prtiles whih produe most of the pirs re positrons inside the re surrounded by dotted line mrked s bulk of pirproduing prtiles. On the top of the plots we show the sketh of the struture of the elertion zone from Figure 11(b). between the beginning of suessive bursts of pir retion, then the pir multipliity s disussed in previous setions should be multiplied by the ttenution ftor t fk = tive (51) Tsde to get the verge multipliity of pir sdes. The existing self-onsistent simultions of prtile elertion (T1, TA13) re inonlusive bout the sde repetition rte. The numeril resolution in these simultions ws indequte for tht purpose beuse of opious pir formtion the Debye length of plsm t some point beme smller thn the ell size nd the formtion of the plsm til ould not be simulted urtely; the repetition rte, however, is determined by the mildly reltivisti plsm in the plsm til, s the next burst of pir formtion strts only when plsm leves the polr p region. Future numeril simultions would ddress Beuse of the intermitteny of pir formtion, the resulting multipliity must be djusted by the reltive frtion of time during whih primry prtiles re produed. If ttive is the time of tive prtile elertion nd Tsde is the time 22