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Newly born pulsars


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Newly born pulsars

  1. 1. Draft version January 26, 2012 Preprint typeset using L TEX style emulateapj v. 5/2/11 A NEWLY-BORN PULSARS AS SOURCES OF ULTRAHIGH ENERGY COSMIC RAYS Ke Fang1 , Kumiko Kotera 1,2 and Angela V. Olinto1 Draft version January 26, 2012 ABSTRACTarXiv:1201.5197v1 [astro-ph.HE] 25 Jan 2012 Newly-born pulsars offer favorable sites for the injection of heavy nuclei, and for their further acceleration to ultrahigh energies. Once accelerated in the pulsar wind, nuclei have to escape from the surrounding supernova envelope. We examine this escape analytically and numerically, and discuss the pulsar source scenario in light of the latest ultrahigh energy cosmic ray (UHECR) data. Our calculations show that, at early times, when protons can be accelerated to energies E > 1020 eV, the young supernova shell tends to prevent their escape. In contrast, because of their higher charge, iron- peaked nuclei are still accelerated to the highest observed energies at later times, when the envelope has become thin enough to allow their escape. Ultrahigh energy iron nuclei escape newly-born pulsars with millisecond periods and dipole magnetic fields of ∼ 1012−13 G, embedded in core-collapse supernovæ. Due to the production of secondary nucleons, the envelope crossing leads to a transition of composition from light to heavy elements at a few EeV, as observed by the Auger Observatory. The escape also results in a softer spectral slope than that initially injected via unipolar induction, which allows for a good fit to the observed UHECR spectrum. We conclude that the acceleration of iron-peaked elements in a reasonably small fraction ( 0.01%) of extragalactic rotation-powered young pulsars would reproduce satisfactorily the current UHECR data. Possible signatures of this scenario are also discussed. 1. INTRODUCTION natural explanation if the sources were transient, such as gamma-ray bursts or newly-born pulsars. The de- The origin of the highest energy cosmic rays flection in the extragalactic magnetic fields should in- still remains a mystery (see Kotera & Olinto 2011; deed induce important time delays (∼ 104 yr for one Letessier-Selvon & Stanev 2011 for recent reviews). The degree deflection over 100 Mpc) between charged parti- measurement of a flux suppression at the highest energies cles and the photons propagating in geodesics, so that (Abbasi et al. 2008; Abraham et al. 2010b), reminiscent the sources should already be extinguished when cos- of the “GZK cut-off” (Greisen 1966; Zatsepin & Kuzmin mic rays are detected on Earth. Even in this case, for 1966) produced by the interaction of particles with the proton dominated compositions and intergalactic mag- cosmic microwave background (CMB) photons for prop- netic fields of reasonable strengths, the UHECR arrival agations over intergalactic scales, has appeased the de- directions are expected to trace the large scale struc- bate concerning the extragalactic provenance of UHE- tures where the transient sources are distributed, with a CRs. This feature not only suggests that UHECRs would possible bias (Kalli et al. 2011). The precise role of ex- originate outside our Galaxy, but also that the sources tragalactic magnetic fields in UHECR propagation may of the highest energy particles should be located within be clarified in the future through extensive Faraday ro- ∼ 100 Mpc distance, in our local Universe. However, the tation surveys (see, e.g., Beck et al. 2007) and indirect sources remain a mystery and results from the Auger Ob- measurements of gamma-ray halos around blazars (e.g., servatory on the arrival directions and chemical compo- Neronov & Semikoz 2009). sition of UHECRs make the picture even more puzzling. The composition measurements at the highest energies Hints of anisotropies in the sky distribution of cosmic of the Auger Observatory are surprising. Abraham et al. rays above 60 EeV were reported by the Auger Obser- (2010a) report a trend from a proton dominated compo- vatory, but most of the anisotropy signal seems to issue sition at a few EeV toward an iron dominated compo- from a clustering of events over a few tens of degrees sition at around 40 EeV (continuing up to 60 EeV, see around the region of Centaurus A (Abreu et al. 2010). Abreu et al. 2011b), assuming that hadronic interaction No powerful sources are observed in the direction of the models can be extrapolated to these energies. This trend highest energy events. This might be explained by strong is not confirmed by the HiRes experiment (Abbasi et al. deflections that cosmic rays could experience in presence 2005) nor by the preliminary data of the Telescope Array of particularly intense extragalactic magnetic fields or if (Tameda et al. 2011), who report light primaries in the they were heavy nuclei. This absence might also find a Northern hemisphere (while Auger observes the Southern hemisphere). One may note however that both results re- 1 Department of Astronomy & Astrophysics, Kavli Institute main consistent with those of Auger within quoted sta- for Cosmological Physics, The University of Chicago, Chicago, tistical and systematic errors. Illinois 60637, USA. 2 Theoretical Astrophysics, California Institute of Technology, From a propagation point of view, heavier nuclei 1200 E California Blvd., M/C 350-17, Pasadena, CA 91125, USA are favored compared to light elements for a given en-
  2. 2. 2ergy as they can travel hundreds of megaparsecs be- is consistent with a GZK cutoff (Abbasi et al. 2008;fore losing their energy by photo-disintegration processes Abraham et al. 2010b). A decade ago, the chemical com-on the cosmic backgrounds due to their lower energy position was also barely detectable at the highest ener-per baryon (e.g., Stecker & Salamon 1999; Bertone et al. gies while recent results suggest a puzzling trend toward2002; Allard et al. 2005, 2008; Hooper et al. 2005). Nu- heavier nuclei. A new investigation of the pulsar scenarioclei of charge Z can also be in principle accelerated to as UHECR sources is timely, in the light of the data thatan energy typically Z times larger than protons in a has been recently acquired.given electromagnetic configuration. Propagation mod- In this paper, we examine the key mechanisms involvedels where a heavy composition arises at the highest in the production of UHECRs by newly-born pulsars,energies due to a combination of a low proton maxi- and discuss their implications, considering the latest ob-mum acceleration energy (around 10 EeV) and Z times servational results. We focus in particular on the effectshigher maximum energies for heavier elements (present of the dense supernova envelope that surrounds the neu-in a slightly higher abundance than Galactic) have been tron star, and that accelerated particles have to traverseshown to reproduce the composition trends observed by on their way to the interstellar medium. We performAuger (Allard et al. 2008; Aloisio et al. 2009). However, detailed analytical and numerical Monte-Carlo calcula-these works focus on the propagation, and do not pro- tions of the envelope crossing and predict the out-comingvide a plausible source for the injection of these specific features that particles should bear after the escape. Itcompositions. The problem of finding powerful sources is found that a small fraction of extragalactic rotation-that inject mainly these low abundance elements and of powered young pulsars embedded in supernovæ couldtheir escape from the acceleration site remains open. satisfactorily explain the latest UHECR observations. Heavy nuclei dominated injection models are quite The layout of this paper is the following. In Section 2,rare in the astrophysical literature of candidate sources. we review and update the discussions related to the pro-A direct injection of large proportions of heavy nu- duction of UHE heavy nuclei in newly-born pulsars. Inclei into an acceleration region requires either an ini- Section 3, we describe the supernova envelope modelingtial metal-rich region, or an efficient nucleosynthesis in used to develop our analytical estimates and to performthe accelerating outflow. These requirements are hardly our numerical simulations of the escape of UHECRs.met by fireball-type gamma-ray bursts (Lemoine 2002; Our main results on the escape of UHECRs from thePruet et al. 2002). Active galactic nuclei (AGN), which supernovæ envelopes are presented in Section 3. In Sec-are the other popular sites for UHECR acceleration mod- tion 4 we discuss the implications of the newly-born pul-els, are observed to have a solar to super-solar metallici- sar model in view of the available UHECR observations.ties, but with a low proportion of nuclei heavier than ni- There, we argue how a reasonably small fraction of extra-trogen (e.g., Groves et al. 2006; Mathur & Fields 2009). galactic fast spinning young pulsars embedded in super-Young neutron stars on the other hand possess iron-rich novæ could reproduce satisfactorily the current UHECRsurface and early conditions that are propitious for heavy data, and discuss observable signatures that could probenuclei injection. the pulsar model. Our conclusions are drawn in Sec- Pulsars have been suggested as possible accelerators tion 5.of cosmic-rays since their discovery (Gunn & Ostriker1969), due to their important rotational and magnetic 2. UHE HEAVY NUCLEI PRODUCTIONenergy reservoirs. Galactic pulsars have been suggested IN NEWLY-BORN PULSARSas the sources of cosmic rays around the knee region up tothe ankle (Karakula et al. 1974; Bednarek & Protheroe In this section, we review and discuss some key points1997, 2002; Giller & Lipski 2002; Bednarek & Bartosik related to the production of UHE heavy nuclei in newly-2004). Blasi et al. (2000) proposed that iron nuclei accel- born fast-spinning neutron stars. Our numerical applica-erated in the fastest spinning young neutron stars could tions focus on isolated rotation-powered pulsars of radiusexplain the observed cosmic rays above the ankle in a R∗,10 ≡ R∗ /10 km, angular velocity Ω4 ≡ Ω/104 s−1 ,Galactic source scenario. They assumed that the strip- principal moment of inertia I45 ≡ I/1045 g cm2 , andping of heavy nuclei from the surface of the star is a plau- magnetic dipole moment µ30.5 ≡ µ/1030.5 cgs with µ =sible seeding and derived a spectrum based on the spin BR∗ /2 = 1030.5 cgs (B/6 × 1012 G)R∗,10 , with B the 3 3down of young pulsars (J ∝ E −1 ). Arons (2003) stud-ied the birth of extragalactic magnetars as the source surface dipole field strength. We show in Section 3 thatof ultrahigh energy protons, developing the acceleration such parameters would enable the escape of UHE nucleimechanism in detail and assuming that the magnetar from the surrounding supernova envelope.wind disrupts the supernova envelope to allow the es-cape of accelerated particles. 2.1. Acceleration by unipolar induction The Blasi et al. (2000) and Arons (2003) proposalsfor the origin of UHECRs were elaborated to explain Rapidly rotating neutron star magnetospheresthe absence of the GZK cut-off in the observed spec- are promising particle acceleration sites (see, e.g.,trum reported by AGASA (Takeda et al. 1998) with- Shapiro & Teukolsky 1983 and references therein). Inout invoking the so-called top-down models (see, e.g., the out-flowing relativistic plasma, the combinationBhattacharjee & Sigl 2000). An increase in the expo- of the fast star rotation and its strong magnetic fieldsure at the ultrahigh energies by the HiRes and Auger can induce, in principle, potential differences of orderObservatories have shown that the UHECR spectrum Φ = Ω2 µ/c2 . Provided that particles of charge Z can experience a fraction η of that potential, they can
  3. 3. 3be accelerated to the energy (Blasi et al. 2000; Arons however likely to prevent the acceleration of particles2003): to the highest energies both in the polar cap and the outer gap. Venkatesan et al. (1997) and Arons (2003) E(Ω) = Ze Φ η = 3 × 1020 Z26 η1 Ω2 µ30.5 eV 4 (1) discussed that particles accelerated in the wind regionwhere η1 ≡ η/0.1 and Z26 ≡ Z/26 for iron nuclei. with r ≫ RL with RL the radius of the light cylinder, do not suffer curvature radiative losses. Energy losses by gravitational waves and electromag-netic radiation lead to the spin-down of the pulsar (see In the next paragraphs, we follow the arguments of Arons (2003) to calculate the radius at which particleShapiro & Teukolsky 1983 and references therein)3 , and acceleration is most likely to occur. We also take intothus to the production of particles of lower and lower account the effects of curvature radiation of pions thatenergies as time goes. Under the assumption that was not previously considered, though it could be morethe Goldreich-Julian charge density (Goldreich & Julian constraining than the curvature radiation of photons.1969) is entirely tapped in the outflow for acceleration,and using the expression of the pulsar spin-down rate, Outside the light cylinder, the dipole field struc-one can derive the energy spectrum of the accelerated ture cannot be causally maintained and the field be-particles (Arons 2003): comes mostly azimuthal, with field lines spiraling out- −1 wards (Michel 1991). In regions of the wind where dNi 9 c2 I −1 E the rest mass density is not dominated by electron and = E 1+ , (2) positron pairs, the plasma can be considered as force- dE 4 Zeµ Eg free. In such regions, and for the case of aligned rotators,where Eg is the critical gravitational energy at which Contopoulos & Kazanas (2002) calculated that chargedgravitational wave and electromagnetic losses are equal. particles flow out with a motion along the (nearly az-The gravitational wave losses start dominating at the imuthal) magnetic field lines that becomes negligiblehighest energies when the magnetic field of the star be- when r ≫ rmin,lin = γL RL . The intial Lorentz factorcomes µ 1033 cgs. Magnetars are thus affected by these of the particles entering the wind, γL , can take valueslosses. For pulsars with milder fields that are the main between 10 − 103 depending on the magnetospheric pa-concern of this paper, gravitational wave losses are negli- rameters. Beyond r ≫ rmin,lin, particles flow out nearlygible, and Eg ≫ 1020 eV. In this case, the injected spec- radially (they “surf-ride” the fields) and the wind actstrum reads (Blasi et al. 2000): like a linear accelerator: the Lorentz factor of the out- flowing plasma increases linearly as γw ∼ r/RL . dNi = 5 × 1023 I45 (Z26 µ30.5 E20 )−1 eV−1 , (3) Arons (2003) extended the work of dE Contopoulos & Kazanas (2002) to oblique rotators The spin-down time at which particles of energy E can and to regions in the wind where magnetic dissi-be accelerated in the voltage drop, when gravitational pation occurs (i.e., in non force-free regimes), forwave losses are negligible, reads (Arons 2003): r > rdiss ∼ 2 κ± RL . Here κ± is the ratio between the number density of heavy ions (that we assume equal to 9 Ic3 Ei the Goldreich-Julian density) and of electron-positron tspin (E) = −1 (4) pairs. Calculations of pair creation in ultra-magnetized 8 µ2 Ω 2 i E neutron stars suggest κ± ∼ 10 − 100 (Baring & Harding 3 × 1020 eV Z26 η1 I45 2001). Arons (2003) discussed that surf-riding acceler- ∼ 3 × 107 s. (5) E µ30.5 ation can still occur in these more general cases. He argues further that magnetic dissipation via Alfv´n ewhere Ei is the maximum acceleration energy corre- wave emission beyond rdiss would lead to an even moresponding to the initial angular velocity Ωi . The spin- efficient surf-riding process, the waves acting as strongdown time at which particles of energy E can be accel- pondermotive forces on the ions. The Lorentz factor oferated does not depend on the initial rotation velocity of the ions (of mass mi ) would then reach values as highthe neutron star Ωi , for E ≪ Ei . as γi = ZeηΦ/(mi c2 ) > γw for r > rdiss . The results obtained for the unipolar induction toy-model described 2.2. Acceleration sites in Section 2.1 can then be applied. Various authors have discussed particle acceleration in- The curvature radius of a surf-riding ion at distance 2side the light cylinder of pulsars and magnetars (see, e.g., r ≫ rmin,lin reads (Arons 2003): ρc = 2ρl γw , whereHarding & Lai 2006 for a review). Possible sites include ρl ∼ ηr is the Larmor radius of the particle.4 Onethe polar cap region, just above the magnetic pole of can calculate that, to avoid photon curvature radiationthe star (e.g., Sturrock 1971; Harding & Muslimov 2001,2002), the “slot gap” region along the last open field 4 The complete expression of the curvature radiation given byline between the polar cap and the light cylinder (Arons 2 Arons (2003) is ρc = 2ρl γw / cos(Ω, µ). The angle between the1983), and in the outer gap region close to the light cylin- rotation axis and the magnetic dipole moment needs to satisfyder (e.g., Cheng et al. 1986a,b; Bednarek & Protheroe (Ω, µ) < 90◦ to avoid curvature radiations. In such a configu- ration, one can expect an outflow of ions to form from the po-1997, 2002). Energy losses by curvature radiation are lar cap to the rotational equator, along the last closed field lines (the so-called “return current”, Goldreich & Julian 1969; Michel 3 Numerical simulations of magnetized neutron star relativistic 1975; Contopoulos et al. 1999). In the model of Arons (2003), itwinds suggest that the spin-down rate may be faster than obtained is specifically this current of ions that is tapped into the wind forin the standard “vacuum dipole” model (Bucciantini et al. 2006). acceleration.
  4. 4. 4losses, the acceleration of particles at E21 ≡ E/1021 eV elements, it is possible that heavy nuclei get injected inneeds to take place at radius greater than: the wind by these means. Z e2 1 1/6 Heavy nuclei loading of the pulsar wind by mixing rmin,c = E 1/2 4 6m4 c4 ηΩ4 (6) of the stellar material via Kelvin-Helmholtz instabilities A p or oblique shocks was also proposed (Zhang et al. 2003; 1/2 1/6 ∼ 6 × 106 E21 Z26 A56 −2/3 −1/6 −2/3 η1 Ω4 cm . (7) Wang et al. 2008). This mechanism requires however that a jet goes through the stellar core, a case that is not considered in the present study. Kelvin-Helmholtz The cooling timescale for curvature radiation of pions instabilities might also occur at the interface betweenis more constraining; it reads (Herpay & Patk´s 2008): o the wind nebula and the supernova remnant (Jun 1998; E e0.039/χ van der Swaluw et al. 2004), but it is unlikely that the tc,π = 6 × 10−14 A−1 56 s, (8) envelope in that region has a metallicity high enough to 1021 eV χ mix large amounts of heavy nuclei in the wind.where χ ≡ E 2 /(ρc A2 m3 c5 ). We present here only the p The nucleosynthesis of heavy elements by r-processcase of charged pions π + , as this process dominates the in the neutrino-driven wind at the very early phase ofcase of the emission of π − and π 0 (Herpay et al. 2008). the proto-magnetar formation has also been discussed 2 56 −1One can readily see that χ ∼ 13E21 A−2 η1 Ω4 (RL /r)3 ≪ by Metzger et al. (2011a,b). These authors find that1 and thus, tc,π ≫ tacc , for sufficiently large r ≫ RL in the production rate of nuclei with A 56 can bethe wind. Numerically, for the same parameters as in important during the first 1 to ∼ a few 100 s, whenEq. (6), the acceleration above E21 ≡ E/1021 eV needs the electron fraction Ye could be fairly low, the windto take place at r > rmin,c,π ∼ 2 × 107 cm to avoid energy expansion time τexp 103 s, and the entropy S −1losses through curvature radiation of charged pions. 100 kb nucleon , as is required for a successful r-process (see, e.g., Hoffman et al. 1997). Though these results The radiation fields in the pulsar wind are unlikely to are obtained for the case of a highly magnetized proto-impact the acceleration of UHECRs. The early neutrino- magnetar driving a jet (as in Bucciantini et al. 2007),driven wind should end within the Kelvin-Helmholtz they can be applied in a non-collimated mildly magne-timescale of about 10 − 100 s (Pons et al. 1999), and the tized wind case, as the evolution of S and τexp is mostlywind should then become relativistic and non radiatively ruled by thermal ingredients (and the rotation speed) indissipative. A few days after the supernova explosion, the the times considered. However, we will see in the nexttemperature of the soft thermal photons from the sur- section that the supernova envelope at t ∼ 10 − 100 sface of the neutron star drops to T 107 K and photo- is too dense to allow the escape of particles, whateverdisintegration on this background radiation can also be their mass number. At later times, as the wind coolsneglected, even inside the light cylinder (Protheroe et al. and becomes relativistic, the neutrino heating efficiency1998; Bednarek & Protheroe 2002). drops, shutting off the r-process. It is thus unlikely that In the pulsar wind beyond the light cylinder, pos- this channel can seed heavy nuclei in the wind in oursible acceleration sites thus lie close to the equato- framework.rial plane of the star, at a distance Ra > rmin ≡max(rmin,lin , rmin,c , rmin,c,π ) ∼ 3×107−9 Ω−1 cm, assum- 4 3. UHECR ESCAPE FROM SUPERNOVA ENVELOPESing γL 103 . The fact that rmin rdiss implies that theunipolar induction toy-model could apply, and that par- Particles accelerated in the pulsar wind further needticles could reach ultrahigh energies within this range of to escape from the pulsar wind nebula itself, and thendistances. from the surrounding young supernova envelope. We as- sume in this study that the supernova envelope is not totally disrupted by the wind, and that particles do not 2.3. Heavy nuclei injection escape through a region punctured by a jet, like in a One can mention three channels via which heavy ions strongly magnetized proto-magnetar scenario discussed by Metzger et al. (2011a) —see Appendix A for furthercould be seeded in the neutron star wind. Note that sce- discussions.narios of pulsar winds loaded with heavy nuclei give asatisfactory explanation to some observations. For in- The escape of accelerated ions from the magnetar windstance, the morphological features of the Crab Nebula nebula was discussed by Arons (2003). In Section 2.2, wecould be the signature of resonant scattering of pairs of argued that at distances r ≫ RL , the curvature radius ofelectrons and positrons by heavy nuclei (Hoshino et al. the ions reads: ρc ∼ 2ηr3 /RL ≫ r. Hence, particles are1992; Gallant & Arons 1994). not coupled to the magnetic field lines and can escape The classical argument that applies best in the wind beyond rmin .our scenario is that iron nuclei can be stripped In supernova envelopes, magnetic fields are of orderoff the neutron star surface, as has been sug- a few mG at most (see, e.g., Reynolds et al. 2011 for agested by Ruderman & Sutherland (1975) and review). The Larmor radius of the ions is thus muchArons & Scharlemann (1979). Strong electric fields larger than the size of the envelope and their trajectoriescombined with bombardment by particles can extract can be treated rectilinearly. We give in the followingions from the polar cap regions, where the co-rotation section, estimates of the density profile and compositioncharge is positive provided that Ω · B < 0. The surface of young supernova envelopes, that we use to study theof a neutron star being composed mainly of iron-peaked escape of UHECRs analytically and numerically.
  5. 5. 5 3.1. Supernova envelopes sity of the envelope, the detailed density profile is not crucial to our calculations. As discussed for instance by Chevalier (2005), rotation-powered pulsars can originate in various types of core- Under the assumption of adiabatic, sphericallycollapse supernovæ: in Type II supernovæ resulting from symmetric flows, the numerical calculations ofred supergiant stars with most of their hydrogen enve- Matzner & McKee (1999) show that the density oflope intact (SNIIP), or with most of their hydrogen lost a Type II supernova in the dense central region can take(SNIIL and IIb), or in Type Ib or Type Ic supernovæ values as high as:(SNIb/c) that stem from stars with all their hydrogen 5/2 −3/2lost. See also Maeda et al. 2007, Woosley 2010, and ρSNII (t) ∼ 10−16 Mej,10 Eej,52 t−3 g cm−3 . yr (12)Piro & Ott 2011, Kasen & Bildsten 2010, for supernovæ Most type II supernovæ eject a mass of order Mej,10associated with magnetars. Chevalier (2005) finds that, (Woosley & Weaver 1995). This dense, relatively flat re-of the remnants with central rotation-powered pulsars,the pulsar properties do not appear to be related to the gion extends to radius Rb ∼ 2(Eej /Mej)1/2 t and is sur-supernova category. rounded by a steep outer power-law profile. The column density that the accelerated particles have to traverse to Within a few days after the explosion, the supernova escape is then:enters a free expansion phase with velocity distribution 2 −1v = r/t, that lasts several hundreds of years. A straight- ySNII (t) = ρSNII Rb ∼ 4 Mej,10Eej,52 t−2 g cm−2 . yr (13)forward way to model the evolution of the density of theejecta is to assume that the ejected mass Mej will expand For Type Ib/c/bc supernovæ, one can apply the modelspherically in time with a mean velocity vej over a shell of Matzner & McKee (1999) for the explosion of a starof radius RSN = vej t. The ejected velocity, Eej , relates with a radiative envelope, which yields:to the supernova explosion energy and the ejected massthrough: v −1.06 ρSNIb/c (t) = 7 × 10−17 × 1/2 0.01c Eej 1/2 −1/2 1.97 −0.97 Mej,2 Eej,52 t−3 g cm−3 , (14) vej = 2 ∼ 109 Eej,52 Mej,10 cm s−1 , (9) yr Mej out to radius Rb , beyond which the density decreaseswhere we defined Mej,10 ≡ Mej /10 M⊙ and Eej,52 = steeply. We have assumed in this estimate an explosionEej /1052 ergs. Most core-collapse supernovæ are inferred energy of Eej,52 and an ejecta mass of Mej,2 = Mej /2 M⊙ ,to have explosion energy Eej ∼ 1051 ergs. However, for which are derived from the observation of such objectsthe pulsars with millisecond to sub-millisecond periods (Drout et al. 2010). The corresponding column density,considered here, one can expect that the rotation en- taking into account the velocity distribution v = r/t,ergy of order (1/2)IΩ2 ∼ 1052 ergs will be transfered readswithin a fraction of a year to the surrounding ejecta Rb(see Kasen & Bildsten 2010). Depending on the radia- ySNIb/c (t) = 2 −1 ρSNIb/c dr ∼ 9 Mej,2Eej,52 t−2 g cm−2 . yrtion conversion efficiency of this energy, the surrounding 0supernova could become ultraluminous. Some ultralumi- (15)nous SNIb/c and SNII have indeed been detected with an Equations (11), (13), and (15) agree within factors ofexplosion energy 1052 ergs (e.g., Nomoto et al. 2001; a few. It is thus reasonable to consider Eqs. (10) andWoosley 2010; Piro & Ott 2011; Barkov & Komissarov (11) as representative of the envelope mean density and2010). column density, for types II and Ib/c supernovæ. Equa- The mean density over RSN (t) can then be written: tions (13), and (15) show that higher ejecta energy Eej and lower masses Mej would enhance the column den- Mej 5/2 −3/2 −3 sity. The effects of such cases on particle escape are alsoρSN (t) = 3 ∼ 2×10−16Mej,10 Eej,52 tyr g cm−3 , (4/3)πvej t3 discussed throughout the paper. (10) One can further note that if the pulsar wind shreds itswhere tyr ≡ t/1 yr, which is the timescale to reach a surrounding supernova envelope, as discussed in Aronspulsar spin that enables the acceleration of iron up to (2003) for the magnetar case, disrupted fragments would∼ 1020.5 eV (see Eq. 4). The column density integrated expand in the interstellar medium. In this case, one canover RSN as a function of time reads weight the initial supernova density by C −2/3 , C ≡ δρ/ρ 2 −1 ySN (t) = ρSN RSN ∼ 2 Mej,10Eej,52 t−2 g cm−2 . yr (11) being a factor measuring the clumpiness of the envelope (Murase et al. 2009). A high C would ease the escape of More detailed modelings show that the density evo- UHECRs from the envelope. However, the values of C remain difficult to evaluate, as no observational evidencelution of the ejecta is expected to depend on the typeof supernova. Yet, we demonstrate in what follows that of such phenomena has been detected.Eq. (11) above provides a good estimate for the evolu-tion of the integrated column density of various types of The composition of the supernova ejecta depends uponsupernova envelopes. Indeed, we will see in the next sec- the type, progenitor mass, and the final interior masstion that the escape of UHECRs is determined by their of the supernova. CXO J164710.2-455216’s associa-interactions on the baryonic envelopes. Because these tion with the Westerlund 1 star cluster argues that atinteractions solely depend on the integrated column den- least some pulsars arise from massive star progenitors
  6. 6. 6(Muno et al. 2006). But, as mentioned before, rotation- The condition of escape from the supernova envelopepowered pulsars and magnetars have been invoked for can thus be written as tdyn /thad < 1, yieldinga wide variety of supernova types. The composition of −1/2a type Ib supernova is roughly 50% helium and 50% t > tesc,p ∼ 1.2 × 107 Mej,10 Eej,52 s for proton, (18)C/O: e.g., the Woosley (2010) progenitor is ∼50% he- −1/2lium, ∼43% carbon and ∼7% oxygen. Type Ic super- t > tesc,Fe ∼ 3.8 × 107 Mej,10Eej,52 s for iron, (19)novae (more numerous than Ib supernovae) are com- where we assumed a supernova density profile followingposed almost entirely of C/O and heavier elements: e.g., Eq. (10). Cosmic rays at ultrahigh energy will escapeMazzali et al. (2010) argued that SN 2007gr was com- only if they are produced at late times t 1 yr, whenposed of roughly 75% C, 15% O, 8% 5 6Ni, and 2% S. the envelope density has decreased. Because nuclei ofType II supernovae have a range of ejecta, ranging from charge Z at a given energy E are produced at a timeroughly 60% H, 30% He, and 10% C/O to explosions very tspin (E) ∝ Z (Eq. 4), one has tdyn /thad ∝ Z −2 . Thesimilar to type Ib supernova with small amounts of H. escape condition from the baryonic envelopes at a fixed We will discuss in Section 3.4 how the escaped UHECR E should consequently be eased for heavier nuclei.spectrum varies between pure hydrogen and pure helium Still assuming the supernova density profile of Eq. (10),envelopes (or helium and carbon envelopes). and using the spin-down time given in Eq. (4), one can express the cut-off energy above which injected primary 3.2. Analytical estimates particles should not be able to escape the envelope: 1/2 −1   In accord with the discussion at the beginning of Sec- 8 µ2 Ω2 3Mejσξ ition 3.1, we will consider in the following that Eqs. (10) Ecut,Z = Ei 1 + 2  (20) 9 Ic3 4πmb vejand (11) provide a reasonable estimate of the evolutionof the density of the supernova envelope surrounding the 1/2 σp 1/2neutron star. ∼ 7.5 × 1018 Z1 η1 I45 µ−1 Mej,10 Eej,52 30.5 −1 eV(21) σ Successful escape of UHECRs from the envelope will 1/2 σFe 1/2occur if the shell crossing time tdyn is shorter than the ∼ 1.2 × 1020 Z26 η1 I45 µ−1 Mej,10 Eej,52 30.5 −1 eV (22) σcooling time by hadronic, thad , and photo-hadronic, tN γ ,interactions. where the first numerical application corresponds to pro- tons and the second to iron nuclei. Note that under the The acceleration of a particle to the energy E happens crude approximation that σ ∝ A2/3 , Ecut,Z ∝ Z/A1/3 .at a time after pulsar birth: t(E) ≃ tspin (E). We can as- For Ecut,Z ≪ Ei , Ecut,Z does not depend on Ωi .sume that the thickness of the supernova shell to traverseat a time t is given by RSN ≃ tvej . Indeed, from the val- This trend is illustrated in Figure 1, where the mainues given in Section 2.2, the acceleration site Ra ≪ RSN , timescales at play are displayed: tdyn and thad as a func-as soon as t 100 s. The crossing time for UHECRs tion of particle energy E, for various pulsar parameterstraveling at the speed of light then reads: Ω and µ, and for both pure iron and pure proton injec- tions. As expected, iron particles can escape the enve- RSN vej lope at higher energies, as they can reach these energies tdyn (E) ≃ ≃ tspin (E) . (16) at later times. Lower magnetic fields (µ 1031 ) lead to c c longer tspin at a fixed E (Eq. 4), while high pulsationsAs the expansion time scale of the envelope is tex = (Ω 104 s) lead to higher acceleration energies (Eq. 1).RSN /vej ≃ tspin < tdyn , one can neglect the evolutionof the envelope density during the escape of a particle. When iron nuclei are injected, secondary particles are produced by hadronic interactions for times t < tesc,Fe . The timescale for hadronic interaction losses can be These secondaries of mass and charge numbers (A, Z) canexpressed as: escape the envelope only at times t > tesc,Z , where tesc,Z is defined as the time at which tdyn /thad = 1. Hence, thad (E) = mb {c ρSN [tspin (E)]σ(E)ξ(E)}−1 , (17) secondaries that will escape from the envelope have nec-where mb is the mass of the dominant target ion com- essarily been produced between tesc,Z < t < tesc,Fe , i.e.,posing the envelope. The parameters ξ(E) and σ(E) the lightest secondaries will escape first. This translatesare the elasticity and the cross-section of the interaction in terms of the energy range of the primary iron to:at energy E. For our analytical estimates, we evalu- Ecut,Fe < E < EFe (tesc,Z ), where we can further expressate their values roughly from the hadronic interaction EFe (tesc,Z ) = (26/Z)Ecut,Z . The main fragment amongmodel EPOS (Werner et al. 2006). We assume that the secondary particles will thus emerge from the envelope ′ ′cross-sections of the hadronic interactions do not vary between energies Elow,Z E Ecut,Z , withstrongly above E > 1018 eV and set them to σp = 130 mb A ′for proton-proton interactions and σFe = 1.25 b for iron- Elow,Z ≡ Ecut,Fe (23)proton interactions. The number of nucleons carried out 56 1/2at each interaction can vary from 1 to A − 1, with a large ∼ 2.1 × 1018 Aη1 I45 µ−1 Mej,10 Eej,52 eV , 30.5 −1spread in values. For demonstrative purposes, we take 26 A ′an average value of ξ = 0.4 for both p − p and p−Fe in- Ecut,Z ≡ Ecut,Z (24)teractions. These calculations are done accurately using 56 Z 1/2EPOS in our numerical calculations in the next section. ∼ 3.5 × 1018 Aη1 I45 µ−1 Mej,10 Eej,52 eV. 30.5 −1
  7. 7. 7 Fig. 1.— Timescales at play for the escape of UHECRs from a supernova envelope with Mej,10 and Eej,52 . The crossing time tdyn(dashed lines) and energy loss time by hadronic interactions thad (solid lines) are displayed as a function of particle energy E, for pure iron(red) and pure proton (blue) injections. The timescales are calculated for various pulsar initial rotation velocities Ωi = 103 , 104 s−1 andmagnetic dipole moments µ = 1030 to 1031.5 cgs, as labeled. The other pulsar parameters are set to I = 1045 g cm2 , η = 0.1. The numerical estimates are calculated for secondary time by photo-disintegration of order:protons. Peaks of the various secondary elements shouldappear in the escaped cosmic-ray spectrum at their re-spective energies. A tail due to lower energy secondary tAγ,th = [c ξAγ (∆ǫAγ /¯Aγ )σAγ Uth /ǫγ ]−1 ǫ (25) ′nucleons (E < Elow,Z ) following approximately a power- 3/8law with index ∼ −1/2 should also be produced together Eej,52 −9/8 ∼ 105 A−0.21 56 2 2 Mej,10 t9/4 s (26) yrwith the main fragment, down to PeV energies. The am- ηγ,1 ηth ′plitude of this tail around ∼ Elow,Z is about a fractionof the number of the main fragment. where ∆ǫAγ /¯Aγ ∼ 0.4 A0.21 , σAγ ∼ 8 × 10−26 A56 cm−2 ǫ 56 From Eq. (16) and (17), one can derive: (Murase et al. 2008), and we take for the elastic-tdyn /thad ∼ 3 × 1010 t−2 Mej,10 Eej,52 at t2 ≡ t/100 s, 2 2 −1 ity of the Aγ interaction: ξAγ = 1/A (whichfor (A, Z) = (90, 40). We assumed a cross-section is a crude approximation). This estimate ofσ90 = 1.5 b for nuclei-proton interactions and an elas- the cooling time is valid for cosmic-ray energyticity of ξ = 0.4, at energies E ∼ 1020 eV (in the target 1/8 −3/8 3/4 EA,peak ∼ 4 × 1017 (ηγ,1 ηth )−1/4 Eej,52 Mej,10 tyr eV, andrest-mass frame). This demonstrates that nuclei with is about one order of magnitude larger for EA EA,peak ,A 56 that could be injected at times t ∼ 10 − 100 s as the photo-disintegration cross-section lowers. At theif a successful r-process occurred in the neutrino-driven highest energies (EA ∼ 1020 eV), photo-disintegrationwind (Section 2.3), cannot survive the crossing of the could thus play a role on the escape of cosmic rays if thesupernova envelope. radiation and thermalization efficiencies are higher than ηγ ηth 10−2 . The rate of wind energy going to radiation Ultrahigh energy ions could also experience photo- is evaluated to be of order 10% (e.g., Kasen & Bildstendisintegration in the radiation fields generated at the in- 2010), but the thermalization fraction of these photons,terface between the pulsar wind and the supernova shell. ηth , is not known, due to the uncertainties on the opacities in the internal shock region. Mixing and This radiation field can be expected to be significant Rayleigh-Taylor instabilities effects creating finger-typeif the supernova explosion is driven by the pulsar wind, structures could lead to a leaking of the high energyas expected for millisecond rotators. A fraction ηγ of the photons, and the thermalization fraction could be aswind energy ∼ (1/2)IΩ2 can be converted to radiative i low as 10%. A higher acceleration efficiency η wouldenergy via internal shocks and another fraction ηth of also enable particles to reach the highest energies by thethis radiation then thermalizes depending on the opacity time the radiation field intensity has become negligible.of the medium. This thermal component peaks at energy Given these uncertainties, and for simplicity, we will −1/8 3/8 −3/4ǫγ = kT ∼ 0.4 (ηγ,1 ηth )1/4 Eej,52 Mej,10 tyr eV, with en- assume in this paper that the radiation field can be −1/2 3/2ergy density Uth ∼ 0.5 ηγ,1 ηth Eej,52 Mej t−3 erg cm−3 , neglected for the escape of UHECRs from supernova en- yr velopes, the baryonic background playing the major role.where ηγ,1 ≡ ηγ /0.1. This background leads to a cooling To summarize, the conditions for successful accelera-
  8. 8. 8tion and escape above 1020 eV can be written as: −1 −1 −3 BΩ2 (1012.4 G) × (104 s−1 )2 Z26 η1 R∗,10 i 12.8 −1/3 −1 1/2 −3 (27) B 10 G Z26 A56 η1 I45 Mej,10 Eej,52 R∗,10Higher values of the magnetic field would allow higher ac-celeration energies, but would require lower ejecta massand higher explosion energies. Note that 10 M⊙ canbe viewed as an upper bound for the ejecta mass fortype II supernovæ (Woosley et al. 2002). One mightalso advocate that the presence of clumps could lowerthe overall densities and allow the escape of particles atE > 1020 eV. All in all, the parameter space allowedfor successful acceleration and escape appears to be nar-row, but we will see in Section 4.1 that the low rate ofsources required to account for the observed UHECR fluxwould compensate for this issue. A higher accelerationefficiency η would also broaden the allowed parameterspace. 3.3. Numerical Setup Fig. 2.— UHECR spectrum before (dash) and after (solid) escape As discussed in the previous section, the hadronic in- from a hydrogen supernova envelope with Mej,10 and Eej,52 , with pure proton injection. The pulsar parameters are I = 1045 g cm2 ,teraction between UHECRs and the baryonic envelopes η = 0.1, Ω = 104.0 s−1 , and µ = 1030.5 the determinant factor that would affect the injectedUHECR spectrum. pulsars embedded in pure hydrogen, helium and carbon supernovæ. The interactions with the baryonic envelopes were cal-culated by Monte-Carlo for injected nuclei and theirsecondaries. As in Kotera et al. (2009), we used the 3.4. Numerical Resultshadronic interaction model EPOS (Werner et al. 2006)and the fragmentation model of Campi & H¨fner (1981), u We first assume a pure hydrogen envelope. The re-as implemented in the air shower simulation code sults are presented in Section 3.4.1. Simulations usingCONEX (Bergmann et al. 2007). more supernova envelopes with heavier composition are discussed in Section 3.4.2. In the case of a non-hydrogen baryonic envelope, theinteraction products can be derived from the nuclei-proton interaction case by a superposition law. In the 3.4.1. Pure Hydrogen supernova envelopetarget rest frame, the products of the interaction between Figure 2 presents the injected (in dash line) and es-a projectile of mass number and energy (Aproj , Eproj ) and caped (in solid line) spectra of pure proton injection bya target nucleus of mass number Atarg are roughly equiv-alent to Atarg times the products of the interaction be- a pulsar with initial angular speed Ωi = 104 s−1 andtween a projectile with (Aproj , Eproj /Atarg ) and a target magnetic dipole moment µ = 1030.5 cgs. The injectedproton. The exact cross-sections are nonetheless com- spectrum follows the characteristic −1 spectral index inputed with EPOS. Eq. (2). As predicted in Eq. (21) UHE protons above ∼ 10 EeV fail to escape the supernova envelope, since the In the simulations, we modeled pulsars with initial region is still very dense at the time they are produced.angular velocity Ωi ∼ 103.0−4.2 s−1 and magnetic mo- Below a few EeVs protons are free to escape. Protonsment µ ∼ 1030−33 cgs, corresponding to a surface mag- with energy in between can partially escape with signifi-netic dipole field B ∼ 2 × 1012−15 G. Notice that there cant flux suppression. EPOS shows that for one 10 EeVis an upper limit (∼ 104.2 s−1 ) on the initial angular primary proton, the peak of interaction products lies atspeed (Haensel et al. 1999). For each set of parame- 1014 eV; the chance of resulting a secondary proton withters, 107 cosmic rays are injected following a power-law E ≥ 1017 eV is less than 0.01. Therefore we can barelyenergy spectrum as in Eq. (2) with minimum injection see the secondary protons in our energy window of sim-energy Emin = 1017 eV, and the maximum acceleration Ei calculated in Eq. (1). Above Ei , the spec- The spectra of pure iron injection by pulsars withtrum cuts-off exponentially. Nuclei with initial energy Ωi = 104 s−1 and µ = 1030.5 , 1031.5 cgs are shown in Fig 3.E are injected at a radius Ra = 1010 cm (correspond- In the top plot (µ = 1030.5 cgs, Ω = 104 s−1 ), primarying to ∼ 3 rmin for Ωi = 104 s−1 , see discussion in Sec- iron nuclei with energy up to Ecut,Fe = 1.2 × 1020 eVtion 2.2) at the time tspin (E), and propagate through a can escape without significant loss. As discussed in oursupernova envelope of total ejected mass 10 M⊙ (2 M⊙ analytical estimates, most secondaries should originatein Type Ib/c supernova case) expanding at a constant from primary iron nuclei with energy between Ecut,Fe = 1/2 −1/2rate vej = 109 Eej,52 Mej,10 cm s−1 . The evolution of the 1.2×1020 eV and 56×Ecut,p = 4.2×1020 eV, correspond-ejecta density is assumed to follow Eq. (10). We studied ing to the iron cutoff and iron mass number times the
  9. 9. 9 When the magnetic field is 10 times stronger (µ = 1031.5 cgs, Ωi = 104 s−1 , bottom plot of Fig 3), the pul- sar spins faster and the cutoff for primary and secon- daries are lowered by 10 times (see Eq. 24). Hence, the µ = 1031.5 cgs case presents a similar shape as the µ = 1030.5 cgs case except an overall shift to lower ener- gies by a factor of 10. As pointed out in Section 2.1, at low energies when E ≪ Eg the gravitational wave losses are negligible and tspin is independent on the initial rotation speed Ωi for E ≪ Ei . A pulsar with higher initial angular veloc- ity can inject UHECRs with greater maximum energy. However a minimum spin period ∼ 0.4 ms is allowed for neutron stars (Haensel et al. 1999) corresponding to an upper limit (∼ 104.2 s−1 ) on the initial angular speed. Magnetic dipole moments µ greater than 1032 cgs would make the spin-down process too fast to allow UHECR escape. On the other hand pulsars with µ < 1030 cgs are not energetic enough to accelerate particles to ultrahigh energy (see Eq. 27). To determine the best escaping re- gion we ran a parameter scan with 15 × 15 sets of (Ω, µ) and the results are presented in Figure 4. We define the cut-off energy Ecut as the energy the ra- tio between the escaped and injected particles is less than 10%. It corresponds approximately to the highest energy of escaped cosmic rays Ecut,Z defined in Eq. (21). In Fig- ure 4, the contours represent Ecut reached after escaping hydrogen supernova envelopes with Mej,10 and Eej,52 for pulsars with dipole moment µ and initial angular velocity Ωi . In the proton case (top), protons with energy above 1020 eV cannot escape the supernova envelope in our model. In the iron contours (bottom), the parameter re- gion with (µ ≈ 1030.00−30.72 cgs) × (Ωi ≈ 103.95−4.20 s−1 ) allows the escape of iron nuclei with energy greater than 1020 eV. This parameter scan is based on a supernova en- velope with density profile described in Eq. (10). Higher values of explosion energy and lower ejecta mass could lead to a broader enclosed parameter region that allows −1 1/2 the escape, as Ecut,Z scales with Mej,10 Eej,52 (Eq. 20). Our results agree with the theoretical prediction from Fig.1 in Blasi et al. (2000), except that we have a smaller Fig. 3.— UHECR spectrum before (dash) and after (solid and parameter area that allows escape. This comes from ourdash dotted) escape from hydrogen supernova envelope with Mej,10 assumption that only η ∼ 10% of the induced potentialand Eej,52 , with pure iron injection. The pulsar parameters are turns into UHECR energy.I = 1045 g cm2 , η = 0.1, Ω = 104 s−1 , and µ = 1030.5 cgs (top),and µ = 1031.5 cgs (bottom). Different compositions are listed as 3.4.2. Helium-Carbon/Hydrogen-Helium supernovain the legend box. envelopescutoff of secondary protons. In agreement with Eq. (24), Results in Figure 5 are from simulations with an ejectasecondaries lie between (1.0 − 5.0) × 1018 eV for proton, mass of Mej = 10 M⊙ , explosion energy Eej = 1052 ergs,2.0 × 1018 − 1.3 × 1019 eV for helium, 7.9 × 1018 − 4.0 × and a composition of pure 4 He (top) and pure 12 C (bot-1019 eV for CNO, (1.3 − 7.1) × 1019 eV for Mg-like ele- tom). As discussed in Sec. 3.3, realistic envelopes forments and 2.0 × 1019 − 1.1 × 1020 eV for Si-like elements, SNII and SNIb/c are more complicated and could bewith the peak positions scaled to the mass number of the evaluated by a combination of Fig. 3 and 5. Spectra ofelements and the bump width being almost the same in UHECRs escaped from envelopes abundant in heavierlogarithmic coordinates. The significant tail of protons elements maintain features from that with a pure hydro-below 1 EeV comes from the products of the hadronic gen envelope. For instance, in case of a pure heliuminteractions. On average, each interaction of a 500 EeV envelope (top plot in Fig. 5), the spectrum preservesiron nucleus results in one EeV proton among its prod- the ‘original’ secondary peaks at 6.3 × 1017 eV for hy-ucts. The strong signals from secondary nuclei contribute drogen, 2.5 × 1018 eV for helium, 8.0 × 1018 eV for CNO,to a steeper overall spectrum (in solid black) which fol- 1.3 × 1019 eV for Mg-like elements and 2.5 × 1019 eV forlows ∼ E −2 at 1018.5 − 1020 eV. Si-like elements. These peaks are similar to the ones in
  10. 10. 10 Fig. 4.— Parameter space with cut-off energy (Ecut,Z ) contours, Fig. 5.— UHECR spectrum after escape from a supernova enve-for a hydrogen supernova envelope with Mej,10 and Eej,52 , and lope with Mej,10 and Eej,52 , with composition (top): 100% 4 He andpulsar parameters I45 and η1 . The solid lines refer to cut-off par- (bottom): 100% 12 C. The pulsar parameters are I = 1045 g cm2 ,ticle energies after the escape. Up is proton injection and down is η = 0.1, Ω = 104 s−1 , and µ = 1030.5 cgs.iron injection. Notice that current neutron star models suggest anupper limit of rotational speed at Ωi ≤ 104.2 s−1 . Note also that −1 1/2 tribute to an increment of primaries around 4 × 1019 eVEcut,Z scales with Mej,10 Eej,52 (Eq. 20). for helium envelopes and 1.3 × 1019 eV for carbon en- velopes.a pure hydrogen envelope (see the first plot of Fig. 3),except that they are located at 4 times lower in energy, 4. IMPLICATIONS FOR THE SCENARIO OF UHECRdue to the 4 times heavier interactant. PRODUCTION IN NEWLY-BORN PULSARS The case of heavy envelopes can generate multiplepeaks to the left of the original peaks due to multiple The success of a UHECR source scenario lies in its abil-products. According to the superposition law, the num- ity to reproduce these observations: i) the energy spec-ber of products scales with AN after N interactions with trum, ii) the composition, iii) the anisotropy, and iv) onenvelope baryons of mass number A. So the later gener- the fact that it requires a rate of sources consistent withations (tertiaries and so forth) whose energy are mostly the population studies inferred from other astronomicalbelow 1018 eV are far more numerous than the earlier observations.generations (primaries and secondaries). This brings the As we discuss in this section, the results obtained inlow end of the original peaks up to be a second, or even this paper suggest that all four points could be rea-third additional peaks for all compositions; they also con- sonably achieved in the extragalactic rotation-powered
  11. 11. 11pulsar scenario. Newly-born pulsars are natural candi- the Galaxy, which is consistent with the supernova rate,dates to reproduce points ii) and iii), due to their iron- Lorimer 2008). Among the ‘normal’ pulsar population,peaked surface (if the composition at the highest ener- it is difficult to infer the number of objects that wouldgies proves to be actually heavy, as the measurements satisfy Eq. (27), as the distribution of pulsars accordingof Auger seem to indicate) and their transient nature. to their initial rotation velocities and magnetic field isThough point i) is challenged by the fact that the toy not straightforward (see examples of models discussed inmodel of unipolar induction generates a hard spectrum Giller & Lipski 2002 and Bednarek & Bartosik 2004).that does not fit the observed UHECR spectrum, our Faucher-Gigu`re & Kaspi (2006) find that the birth eresults show that the slope could be naturally softened spin period distribution of pulsars is normal, centeredduring the escape from the supernova envelope (also seen at 300 ms and with standard deviation 150 ms, and thatin Bednarek & Bartosik 2004 for Galactic pulsars). The the initial magnetic field follows a log-normal distribu-range of parameters for the pulsar and its surrounding tion with log(B/G) ∼ 12.65 and σlog B ∼ 0.55. Theysupernova allowed for a successful acceleration and es- stress however that this distribution of birth spin peri-cape at the highest energies is relatively narrow. This ods is not precisely constrained by their method, andpotential issue is however compensated by two advan- considerable deviations from this statistics could be ex-tages. First, the range of values required for the initial pected. Such a distribution would imply that 2% ofparameters of both the pulsar and its supernova are close the ‘normal’ pulsar population could be endowed withto the ones inferred for the youngest isolated pulsars ob- sub-millisecond periods at birth.served nowadays (see e.g., Table 3 of Chevalier 2005).Second, the rate of such objects required to account for Equation (27) further depend on the supernova char-the observed flux of UHECRs is low, of order fs 0.01% acteristics (ejected mass and energy). However, as dis-of the ‘normal’ (as opposed to binary millisecond) pul- cussed in Chevalier (2005), the pulsar properties do notsar birth rate. Point iv) can hence also be deemed as appear to be closely related to the supernova category.reasonably satisfied. This introduces an additional degeneracy on the type and total number of objects meeting the requirements for ac- We will also examine in what follows, the implications celeration and escape. Nevertheless, it is promising thatof our results on the arrival directions of UHECRs in the the range of values required for the initial parameters ofsky, and on possible probes of this source scenario. We both the pulsar and its supernova are close to the onesalso discuss the signatures expected for secondary mes- inferred for the youngest observed isolated pulsars (seesengers such as neutrinos, gamma-rays and gravitational e.g., Table 3 of Chevalier 2005).waves. Hence, fs is a small enough fraction to leave reasonable These implications are first discussed under the as- room for poorer injection efficiencies, and to account forsumption that the currently observed UHECR flux has the narrowness of the parameter range of Eq. (27).an extragalactic origin. The contribution of Galactic pul-sar births is discussed in Section 4.6. One should also keep in mind that both HiRes (Abbasi et al. 2009) and the Pierre Auger Observatory 4.1. Required source density and type of source (Abraham et al. 2010b) report systematic uncertainties of order 20% on the absolute energy scale of the spec- The magnetar birth rate necessary to account for the trum, which should be considered for the evaluation ofobserved flux of UHECRs was estimated in Arons (2003) n. ˙and updated for various cases by Kotera (2011). The The distribution inferred by Faucher-Gigu`re & Kaspi esame calculations can be applied to our case, for rotation- (2006) implies that pulsars with birth periods ∼ 300 ±powered pulsars. 150 ms are about ǫ ∼ 30 times more numerous than the For a population of identical neutron stars with ini- submillisecond ones. Such pulsars could potentially ac-tial rotation velocity Ωi and magnetic dipole momentum celerate iron up to E(P = 100 ms) ≃ 1016 eV (Eq. 1). Forµ, satisfying Eq. (27), one can adapt the normalization extragalactic pulsars with a similar acceleration mecha-found by Kotera (2011) for negligible gravitational wave nism to the case we discuss here (i.e., only a fraction fslosses. An identical neutron star assumption is accept- of the existing population leading to cosmic-ray produc-able, in so far as the allowed parameter range of sources tion), the amplitude of the injected spectrum at thesefor particle acceleration and escape is fairly narrow lower energies is well below the observed one (even if 30 −1(Eq. 27). A birth rate of n ∼ 10−8 µ31 Z26 Mpc−3 yr−1 ˙ times more numerous, the hard ∼ E −(1−1.5) power-law,is required to produce the observed UHECR flux, in the below the peak due to the secondary protons, only over-absence of source evolution history. When the emissivity takes the observed ∼ E −3 spectrum closer to ankle ener-of UHECR sources is assumed to follow the star forma- gies). At these low energies, the diffusion of cosmic-raystion history, the pulsar birth rate at z = 0 is of order: in the intergalactic magnetic fields would further preventnSFR ∼ 0.8 n ∼ 0.8 × 10−8 µ31 Mpc−3 yr−1 . This cal- ˙ ˙ them from reaching us, if the sources are located at tensculation assumes that the total Goldreich-Julian charge of megaparsec distances. On the other hand, a Galacticdensity is tapped in the wind for UHECR acceleration population of these more numerous slower pulsar births(Eq. 2). A lower efficiency would result in a lower energy may give important contributions to the cosmic ray spec-flux per source, and thus in higher required densities. trum below the ankle (see, e.g., Bednarek & Bartosik The above rates correspond to a fraction fs 0.01% 2004).of the birth rate of ‘normal’ pulsars, which is of or-der 1.6 × 10−4 Mpc−3 yr−1 (or one per 60 years in
  12. 12. 12 4.2. Propagated escaped energy spectrum stems from the abundant production of secondary nucle- ons, helium and intermediate nuclei at low energies. The cosmic ray spectrum observed by the Pierre AugerObservatory can be described as a broken power-law, After propagation and interactions in the intergalac-E −λ , with spectral index λ ∼ 3.3 below the break (called tic medium, the injection of particles at the source with index ∼ 2 is expected to provide a good fit to the ob-“ankle”) around 1018.6 eV, and λ ∼ 2.6 above, followed served UHECR spectrum. Our escaped composition canby a flux suppression above ∼ 1019.5 eV (Abraham et al. be identified with the mixed composition introduced by2010b). Allard et al. (2008) (see also Aloisio et al. 2011) that One issue of the model advanced by Blasi et al. (2000) contains 30% of iron and assumes a maximum protonand Arons (2003) for the acceleration of UHECRs in pul- energy of Ep,max ∼ 1019 eV. Allard et al. (2008) calcu-sars and magnetars is the hardness of the produced spec- lates that an injection index of order 2.0 − 2.1 is requiredtrum, that hardly fits the observations described above, to adjust the observed UHECR spectrum after propaga-even after propagation. These models were introduced tion through the intergalactic medium. If one assumesin the “AGASA era”, to account for the absence of GZK that the source emissivity in UHECRs has evolved ac-cutoff in the observed spectrum (Takeda et al. 1998). cording to the star formation rate, the required injectionThey aimed at producing a hard spectrum (of spectral index at the source is of ∼ 1.2 (Kotera et al. 2010).index −1, see Eq. 3) to fit the highest energy end of The bumps and irregularities apparent in the escapedthe spectrum, beyond E > 6 × 1019 eV, and do not spectra (Figs. 3, 5) should be attenuated by the prop-fit the slope at lower energies. The latest experiments agation, a possible distribution of neutron star charac-report however that a suppression reminiscent of the teristics (essentially a distribution of the dipole momentGZK cut-off is present at the highest energy end of the µ, initial spin Ωi ), and especially the envelope chemicalUHECR spectrum (Abbasi et al. 2008; Abraham et al. composition.2010b). Hence, a hard spectrum need no longer be advo-cated to explain the measurements, and now constitutes The flux of particles with energy below the anklea disadvantage. should not overwhelm other (possibly Galactic) compo- nents. Our calculations show indeed that the escaped Kotera (2011) proposed to alleviate this issue by intro- spectrum should become harder below E ∼ 1018 eV, withducing a distribution of initial parameters of magnetars a slope of order −1.5 due to the tail of secondary protons.among their population (see also Giller & Lipski (2002) The flux of these lower energy particles should also be di-for the Galactic pulsars case). Such a distribution re- luted by the large dispersion of their arrival times, aftersults in a distribution of the maximum acceleration en- propagation in the intergalactic and Galactic magneticergy, and adequate values can be found to soften the fields.integrated spectrum and fit the observations. The samecalculation can be applied to the case of rotation-powered The injection of a pure proton composition by neutronpulsars. stars is likely only viable in models where the envelope column density is thinner by many orders of magnitudes Note also that in order to have the monoenergetic- compared to classical supernovæ at early times. In thistype acceleration spectrum given in Eq. (2), the wake- situation, the resulting UHECR observable quantitiesfield acceleration which is based on the ponderomotive are similar to what has been discussed until now: aforce requires the magnetic field to be coherent over the hard spectrum injection should be expected after escapeacceleration region. However, much smaller coherence (spectral index −1), that could be reconciled with thescales can be naturally expected, leading to a stochastic observed spectrum by invoking a distribution of neutronacceleration, that could also produce a E −2 spectrum. star characteristics, as in Kotera (2011).Such cases have been studied in different contexts bye.g., Chen et al. (2002); Chang et al. (2009). One probe of this scenario (both in the proton or iron- The results of Bednarek & Bartosik (2004) already rich injection cases) would be a sharp cut-off of the en-show that the injection of iron nuclei and their escape ergy spectrum at energies above Ecut,Fe (or Ecut,p forthrough the pulsar nebula can lead to a softer spectrum pure proton injection). A mild recovery is indeed ex-due to the production of secondary nuclei. This feature pected if the maximum acceleration energy were E >was however not deeply discussed and highlighted, as the 1020.5 eV, as the observed cut-off in the spectrum wouldAGASA energy spectrum available at that time highly then be due to the GZK effect.differed from the current observations. Besides, the cal-culations of Bednarek & Bartosik (2004) are based onsimplified hadronic interaction cross-sections, and on the 4.3. UHECR compositionraw assumption that one interaction leads to the frag-mentation of the primary nucleus in two nuclei with dif- Recent measurements by the Pierre Auger Observa-ferent mass numbers. tory indicate that the cosmic ray composition transi- tions from being dominated by protons below the an- Our detailed analysis demonstrates that, within the kle (∼ 1018.6 eV) to being dominated by heavier nucleirange of pulsar and supernova envelope parameters given with average masses similar to Si or Fe at ∼ 1019 eVin Eq. (27) and Fig. 4, the injection of heavy nuclei and (Abraham et al. 2010a). The instruments located in thetheir escape from the envelope naturally enables the soft- Northern hemisphere, HiRes and Telescope Array, seemening of the energy spectrum to indices of order ∼ 1.5−2 to observe a light composition up to the highest ener-(Figs. 3, 5). As explained in Section 3.4, this softening gies, though the results of the former remain consistent
  13. 13. 13with those of Auger within errors. We caution further- Nevertheless, the distribution of UHECR events couldmore that the composition measured by these experi- follow the large scale structures, where neutron starsments concern energy bins below ∼ 4 × 1019 eV, due to should be concentrated. In particular, these objectsthe lack of statistics at the highest energy end. should be frequently found in star forming galaxies. Such Neutron stars are one of the most likely places to inject distributions would be apparent only if the deflectionsheavy nuclei abundantly, as we discussed in Section 2.3. experienced by particles in the Galactic and intergalac- tic magnetic fields are small. Moreover, anisotropic sig-It is interesting to notice that the escaped compositionresulting from such an injection indicates a transition natures would only be distinguishable for ensembles offrom light to heavy nuclei around the energy observed particles with the highest energies. Above E ∼ EGZK ≡ 6 × 1019 eV, the horizon that particles can travel with-by Auger (Figs. 3, 5). out losing their energy is limited to a few hundreds of Note that Bednarek & Bartosik (2004) found a sim- megaparsecs and the distribution of sources in that localilar transition, but did not devote much discussion on Universe appears anisotropic.that feature. That finding was not necessarily appealing Neutron stars can be considered as transient UHECRduring the AGASA era, when the composition was be- sources. Cosmic rays with energy above EGZK canlieved to be light at the highest energies. To accountfor the continuation of the flux above GZK energies, indeed only be produced during the first ∆ts ∼Bednarek & Bartosik (2004) added to their Galactic pul- 4 Z26 η1 I45 µ−1 yr after the birth of the neutron star. 30.5sar population, a pure proton extragalactic component, This implies that if secondary messengers such as neu-which lightens their overall composition at the highest trinos, gamma-rays, or gravitational waves were pro-energies. duced at the same time as UHECRs, they would not be As mentioned in the previous section, our escaped com- observed in temporal coincidence with the latter. Theposition is similar to the low Ep,max mixed composi- time delay experienced by UHECRs in the intergalactiction introduced by Allard et al. (2008) and Aloisio et al. magnetic field is indeed of order ∼ 104 yrs for one degree(2011). The resulting composition after propagation in deflection over 100 Mpc, which is much longer than thethe intergalactic medium when such a composition is in- duration of the UHECR production.jected is shown to conserve the transition between light Transient sources could lead to bursts of events in theto heavy elements around 1019 eV (Allard et al. 2008). sky if the number of event per source is important, and The injection of a mixed composition with 10% of the arrival times of particles is not diluted by the disper-iron would remain consistent with such a transition. In- sion induced by magnetic deflections (Kalli et al. 2011).deed, Figs. 3, 5 show that the rate of secondary protons For extremely high energy protons and low intergalaticis more than 10 times higher than the rate of injected fields, the total dispersion time due to magnetic deflec-iron. Injected protons would cut-off below Ecut,p and tion, Σt , can be shorter than the detector exposure time,would not overwhelm the escaped iron flux at the high- Texp, the number of sources contributing to the observedest energies. UHECR flux inside a radius of l = 100 Mpc is of order Ns = (4π/3)l3 nTexp ∼ 0.4, using the pulsar birth rate ˙ One can note that, depending on the detailed transi- inferred in the previous section, and Texp = 10 yrs. Thetion from extragalactic to Galactic component, the com- number of events that can be detected from each sourceposition found here may induce an anisotropy signal at is: Nev = EUHECR Aexp /(EGZK 4πl2 ) ∼ 2 × 103 , wherelower energies as was discussed in Lemoine & Waxman we assumed a detector exposure of Aexp = 3000 km2,(2009). With such a dominant heavy composition at as for the Pierre Auger Observatory, and the cosmic-rayultrahigh energies ( 1019.7 eV), one expects that any energy output per source EUHECR ∼ 5 × 1051 erg in ouranisotropy signal at the highest energies would have a milli-second pulsar scenario. It is likely however thatsimilar structure around 2 EeV where the composition cosmic rays arriving from most directions in the sky ex-is proton dominated, about two times stronger. Such perience a significant dispersion in their arrival time, duean anisotropy is not observed by the Auger observatory to magnetic fields: Σt > Texp. This should be the case for(Abreu et al. 2011a), which may question the composi- iron nuclei, unless one assumes unrealistically low mag-tion of the mild anisotropy found at the highest ener- netic fields. In that case, the number of events detectedgies or imply a more complex composition structure both from one source would be reduced by a factor Texp /Σt .for the extragalactic as well as the Galactic component The reader can refer to Kotera & Lemoine (2008) andaround EeV. Kalli et al. (2011) for detailed discussions on the depen- The injection of a pure proton composition is not ruled dence of Σt on magnetic field parameters.out either in our scenario, but is only favored under strin- A direct identification of the source could be possiblegent conditions on the early envelope density. if a pulsar was born inside our Galaxy, or close enough to allow X-ray or gamma-ray observations. The dispersion 4.4. Distribution of events in the sky of arrival times inside our Galaxy σGal reads: The radio, X-ray and gamma-ray signals of rotation-powered pulsars and magnetars are too weak to allow 2 2 l Bturbtheir detection beyond our Local Group. For this reason, σGal ∼ 2.5 Z 2 ×a direct spatial coincidence between a neutron star and 10 kpc 4 µGUHECR arrival directions is not expected to be observed, λturb E −2if the source is not born inside our Local Group. yr. (28) 50 pc EGZK