Newly-born pulsars as sources of ultrahigh energy cosmic rays

0.43 MB·English

by  Ke Fang

Draftversion January26,2012
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
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
ABSTRACT
2 Newly-born pulsars offer favorable sites for the injection of heavy nuclei, and for their further
1 accelerationtoultrahighenergies. Onceacceleratedinthepulsarwind,nucleihavetoescapefromthe
0 surrounding supernova envelope. We examine this escape analytically and numerically, and discuss
2
the pulsar source scenario in light of the latest ultrahigh energy cosmic ray (UHECR) data. Our

n calculations show that, at early times, when protons can be accelerated to energies E >1020eV, the
a youngsupernovashelltends to preventtheir escape. Incontrast,becauseof their highercharge,iron-
J peakednucleiarestillacceleratedtothehighestobservedenergiesatlatertimes,whentheenvelopehas

5 becomethinenoughtoallowtheirescape. Ultrahighenergyironnucleiescapenewly-bornpulsarswith
2 millisecond periods and dipole magnetic fields of ∼1012−13G, embedded in core-collapse supernovæ.
Duetotheproductionofsecondarynucleons,theenvelopecrossingleadstoatransitionofcomposition

] from light to heavy elements at a few EeV, as observed by the Auger Observatory. The escape also
E
results in a softer spectral slope than that initially injected via unipolar induction, which allows for
H
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
h
would reproduce satisfactorily the current UHECR data. Possible signatures of this scenario are also
p
discussed.
-
o
r
t 1. INTRODUCTION natural explanation if the sources were transient, such
s
as gamma-ray bursts or newly-born pulsars. The de-
a
[ 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
1 Letessier-Selvon& Stanev 2011for recentreviews). The degree deflection over 100 Mpc) between charged parti-
v measurementofafluxsuppressionatthehighestenergies cles and the photons propagating in geodesics, so that
7 (Abbasi et al. 2008; Abraham et al. 2010b), reminiscent the sources should already be extinguished when cos-
9 ofthe “GZKcut-off”(Greisen1966;Zatsepin & Kuzmin mic rays are detected on Earth. Even in this case, for
1
1966) produced by the interaction of particles with the proton dominated compositions and intergalactic mag-
5
cosmic microwavebackground(CMB) photons for prop- netic fields of reasonable strengths, the UHECR arrival
.
1 agations over intergalactic scales, has appeased the de- directions are expected to trace the large scale struc-
0 bate concerning the extragalactic provenance of UHE- tures where the transient sources are distributed, with a
2 CRs. ThisfeaturenotonlysuggeststhatUHECRswould possible bias (Kalli et al. 2011). The precise role of ex-
1 originate outside our Galaxy, but also that the sources tragalactic magnetic fields in UHECR propagation may
v: of the highest energy particles should be located within be clarified in the future through extensive Faraday ro-
i ∼100Mpc distance,inourlocalUniverse. However,the tation surveys (see, e.g., Beck et al. 2007) and indirect
X sourcesremainamysteryandresultsfromtheAugerOb- measurements of gamma-ray halos around blazars (e.g.,
r servatory on the arrival directions and chemical compo- Neronov & Semikoz 2009).
a sition of UHECRs make the picture even more puzzling.
Thecompositionmeasurementsatthehighestenergies
Hints of anisotropies in the sky distribution of cosmic oftheAugerObservatoryaresurprising. Abraham et al.
rays above 60 EeV were reported by the Auger Obser- (2010a) reporta trendfrom a protondominated 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 modelscanbeextrapolatedtotheseenergies. Thistrend
highestenergyevents. Thismightbeexplainedbystrong is not confirmed by the HiRes experiment (Abbasi et al.
deflectionsthatcosmicrayscouldexperienceinpresence 2005)norbythepreliminarydataoftheTelescopeArray
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 Northernhemisphere(whileAugerobservestheSouthern
hemisphere). Onemaynotehoweverthatbothresultsre-
main consistent with those of Auger within quoted sta-
1Department of Astronomy & Astrophysics, Kavli Institute
tistical and systematic errors.
for Cosmological Physics, The University of Chicago, Chicago,
Illinois60637, USA.
2TheoreticalAstrophysics,CaliforniaInstituteofTechnology, From a propagation point of view, heavier nuclei
1200ECaliforniaBlvd.,M/C350-17,Pasadena,CA91125,USA are favored compared to light elements for a given en-
2
ergy as they can travel hundreds of megaparsecs be- is consistent with a GZK cutoff (Abbasi et al. 2008;
forelosingtheirenergybyphoto-disintegrationprocesses Abraham et al.2010b). Adecadeago,thechemicalcom-
on the cosmic backgrounds due to their lower energy position was also barely detectable at the highest ener-
perbaryon(e.g.,Stecker & Salamon1999;Bertone et al. gies while recent results suggesta puzzling trend toward
2002; Allard et al. 2005, 2008; Hooper et al. 2005). Nu- heaviernuclei. Anewinvestigationofthepulsarscenario
clei of charge Z can also be in principle accelerated to asUHECRsourcesistimely,inthelightofthedatathat
an energy typically Z times larger than protons in a has been recently acquired.
given electromagnetic configuration. Propagation mod-
Inthispaper,weexaminethekeymechanismsinvolved
els 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 effects
higher maximum energies for heavier elements (present
ofthe dense supernovaenvelopethat surroundsthe neu-
in a slightly higher abundance than Galactic) have been
tronstar,and that acceleratedparticles have to traverse
shown to reproduce the composition trends observed by
on their way to the interstellar medium. We perform
Auger (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-
tionsoftheenvelopecrossingandpredicttheout-coming
vide a plausible source for the injection of these specific
features that particles should bear after the escape. It
compositions. 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 could
their escape from the acceleration site remains open.
satisfactorily explain the latest UHECR observations.
Heavy nuclei dominated injection models are quite
The layoutofthis paper is the following. In Section2,
rare in the astrophysical literature of candidate sources.
wereview andupdate the discussions relatedto the pro-
A direct injection of large proportions of heavy nu-
duction of UHE heavy nuclei in newly-born pulsars. In
clei into an acceleration region requires either an ini-
Section 3, we describe the supernova envelope modeling
tial metal-rich region, or an efficient nucleosynthesis in
used to develop our analytical estimates and to perform
the 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 the
Pruet et al. 2002). Active galactic nuclei (AGN), which
supernovæ envelopes are presented in Section 3. In Sec-
aretheotherpopularsitesforUHECRaccelerationmod-
tion4wediscuss theimplications ofthe newly-bornpul-
els, are observedto have a solar to super-solarmetallici-
sar model in view of the available UHECR observations.
ties, but with a lowproportionof nucleiheavierthanni-
There,wearguehowareasonablysmallfractionofextra-
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
novæcould reproduce satisfactorilythe currentUHECR
surfaceandearlyconditionsthatarepropitiousforheavy
data,anddiscussobservablesignaturesthatcouldprobe
nuclei 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 & Ostriker
1969), due to their important rotational and magnetic
2. UHEHEAVYNUCLEIPRODUCTION
energy reservoirs. Galactic pulsars have been suggested
INNEWLY-BORNPULSARS
asthesourcesofcosmicraysaroundthekneeregionupto
the ankle (Karakula et al. 1974; Bednarek & Protheroe In this section, we review and discuss some key points
1997, 2002; Giller & Lipski 2002; Bednarek & Bartosik relatedto the productionof UHE heavynuclei innewly-
2004). Blasi et al.(2000)proposedthatironnucleiaccel- bornfast-spinningneutronstars. Ournumericalapplica-
erated in the fastest spinning young neutron stars could tionsfocusonisolatedrotation-poweredpulsarsofradius
explain the observed cosmic rays above the ankle in a R∗,10 ≡ R∗/10km, angular velocity Ω4 ≡ Ω/104s−1,
Galactic source scenario. They assumed that the strip- principal moment of inertia I ≡ I/1045 gcm2, and
45
pingofheavynucleifromthesurfaceofthestarisaplau- magnetic dipole moment µ ≡ µ/1030.5cgs with µ =
30.5
sible seeding and derived a spectrum based on the spin
BR3/2 = 1030.5 cgs(B/6 × 1012 G)R3 , with B the
down of young pulsars (J ∝ E−1). Arons (2003) stud- ∗ ∗,10
surface dipole field strength. We show in Section 3 that
ied the birth of extragalactic magnetars as the source
such parameters would enable the escape of UHE nuclei
of ultrahigh energy protons, developing the acceleration
from the surrounding supernova envelope.
mechanism in detail and assuming that the magnetar
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) proposals
Rapidly rotating neutron star magnetospheres
for the origin of UHECRs were elaborated to explain
are promising particle acceleration sites (see, e.g.,
the absence of the GZK cut-off in the observed spec-
Shapiro & Teukolsky 1983 and references therein). In
trum reported by AGASA (Takeda et al. 1998) with-
the out-flowing relativistic plasma, the combination
out invoking the so-called top-down models (see, e.g.,
of the fast star rotation and its strong magnetic field
Bhattacharjee & Sigl 2000). An increase in the expo-
can induce, in principle, potential differences of order
sure at the ultrahigh energies by the HiRes and Auger
Φ = Ω2µ/c2. Provided that particles of charge Z can
Observatories have shown that the UHECR spectrum
experience a fraction η of that potential, they can
3
be accelerated to the energy (Blasi et al. 2000; Arons however likely to prevent the acceleration of particles
2003): to the highest energies both in the polar cap and the
outer gap. Venkatesan et al. (1997) and Arons (2003)
E(Ω)=ZeΦη =3×1020Z26η1Ω24µ30.5 eV (1) discussed that particles accelerated in the wind region
withr≫R withR the radiusofthelightcylinder,do
where η ≡η/0.1 and Z ≡Z/26 for iron nuclei. L L
1 26 not suffer curvature radiative losses.
Energy losses by gravitational waves and electromag-
In the next paragraphs, we follow the arguments of
netic radiation lead to the spin-down of the pulsar (see
Arons (2003) to calculate the radius at which particle
Shapiro & Teukolsky 1983 and references therein)3, and
acceleration is most likely to occur. We also take into
thus to the production of particles of lower and lower
account the effects of curvature radiation of pions that
energies as time goes. Under the assumption that
was not previously considered, though it could be more
the Goldreich-Juliancharge 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-
wards (Michel 1991). In regions of the wind where
dNi = 9 c2I E−1 1+ E −1 , (2) the rest mass density is not dominated by electron and
dE 4Zeµ E positron pairs, the plasma can be considered as force-
(cid:18) g(cid:19) free. Insuchregions,andforthecaseofalignedrotators,
where Eg is the critical gravitational energy at which Contopoulos & Kazanas (2002) calculated that charged
gravitational 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 negligible
highest energies when the magnetic field of the star be- when r ≫ r = γ R . The intial Lorentz factor
min,lin L L
comesµ&1033cgs. Magnetarsarethusaffectedbythese of the particles entering the wind, γ , can take values
L
losses. For pulsars with milder fields that are the main between 10−103 depending on the magnetospheric pa-
concernofthispaper,gravitationalwavelossesarenegli- rameters. Beyond r ≫ r , particles flow out nearly
min,lin
gible, and Eg ≫1020eV. In this case, the injected spec- radially (they “surf-ride” the fields) and the wind acts
trum reads (Blasi et al. 2000): like a linear accelerator: the Lorentz factor of the out-
flowing plasma increases linearly as γ ∼r/R .
dN w L
i =5×1023I (Z µ E )−1eV−1, (3)
45 26 30.5 20 Arons (2003) extended the work of
dE
Contopoulos & Kazanas (2002) to oblique rotators
Thespin-downtimeatwhichparticlesofenergyE 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), for
wave losses are negligible, reads (Arons 2003): r > rdiss ∼ 2κ±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
t (E)= −1 (4)
spin 8µ2Ω2 E pairs. Calculations of pair creation in ultra-magnetized
i (cid:18)3×10(cid:19)20eV Z η I neutron stars suggest κ± ∼10−100 (Baring & Harding
∼3×107 26 1 45 s. (5) 2001). Arons (2003) discussed that surf-riding acceler-
(cid:18) E (cid:19) µ30.5 ation can still occur in these more general cases. He
argues further that magnetic dissipation via Alfv´en
where E is the maximum acceleration energy corre-
i wave emission beyond r would lead to an even more
sponding to the initial angular velocity Ω. The spin- diss
i efficient surf-riding process, the waves acting as strong
down time at which particles of energy E can be accel-
pondermotive forces on the ions. The Lorentz factor of
erateddoesnotdependontheinitialrotationvelocityof
the ions (of mass m) would then reach values as high
the neutron star Ω, for E ≪E. i
i i as γ = ZeηΦ/(mc2) > γ for r > r . The results
i i w diss
obtained for the unipolar induction toy-model described
2.2. Acceleration sites in Section 2.1 can then be applied.
The curvature radius of a surf-riding ion at distance
Variousauthorshavediscussedparticleaccelerationin-
r ≫ r reads (Arons 2003): ρ = 2ργ2, where
sidethelightcylinderofpulsarsandmagnetars(see,e.g., min,lin c l w
Harding & Lai 2006 for a review). Possible sites include ρl ∼ ηr is the Larmor radius of the particle.4 One
the polar cap region, just above the magnetic pole of can calculate that, to avoid photon curvature radiation
thestar(e.g.,Sturrock1971;Harding & Muslimov2001,
2002), the “slot gap” region along the last open field
4 The complete expression of the curvature radiation given by
line between the polar cap and the light cylinder (Arons Arons (2003) is ρc = 2ρlγw2/cos(Ω,µ). The angle between the
1983),andintheoutergapregionclosetothelightcylin- rotation axis and the magnetic dipole moment needs to satisfy
der (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 etal. 1999). In the model of Arons (2003), it
windssuggestthatthespin-downratemaybefasterthanobtained isspecifically this currentof ions that istapped into the windfor
inthestandard“vacuum dipole”model(Bucciantini etal.2006). acceleration.
4
losses, the acceleration of particles at E ≡ E/1021eV elements, it is possible that heavy nuclei get injected in
21
needs 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
r =E1/2 (6) of the stellar material via Kelvin-Helmholtz instabilities
min,c A46m4c4ηΩ4
(cid:18) p (cid:19) or oblique shocks was also proposed (Zhang et al. 2003;
∼6×106E1/2Z1/6A−2/3η−1/6Ω−2/3cm. (7) Wang et al. 2008). This mechanism requires however
21 26 56 1 4 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 between
is more constraining; it reads (Herpay & Patk´os 2008):
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 =6×10−141021 eVA−561 χ s , (8) envelope in that region has a metallicity high enough to
mix large amounts of heavy nuclei in the wind.
where χ ≡ E2~/(ρ A2m3c5). We present here only the
c p The nucleosynthesis of heavy elements by r-process
case of charged pions π+, as this process dominates the in the neutrino-driven wind at the very early phase of
case of the emission of π− and π0 (Herpay et al. 2008). the proto-magnetar formation has also been discussed
Onecanreadilyseethatχ∼13E2 A−2η−1Ω (R /r)3 ≪ by Metzger et al. (2011a,b). These authors find that
21 56 1 4 L
1 and thus, t ≫ t , for sufficiently large r ≫ R in the production rate of nuclei with A & 56 can be
c,π acc L
the wind. Numerically, for the same parameters as in important during the first 1 to ∼a few 100 s, when
Eq. (6), the acceleration above E21 ≡ E/1021eV needs the electron fraction Ye could be fairly low, the wind
totakeplaceatr >rmin,c,π ∼2×107cmtoavoidenergy expansion time τexp . 103 s, and the entropy S .
losses through curvature radiation of charged pions. 100k nucleon−1,asisrequiredforasuccessfulr-process
b
(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-
impacttheaccelerationofUHECRs. Theearlyneutrino-
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 10−100s (Pons et al. 1999), and the
tized wind case, as the evolution of S and τ is mostly
windshouldthenbecomerelativisticandnonradiatively exp
ruled by thermal ingredients (and the rotation speed) in
dissipative. Afewdaysafterthesupernovaexplosion,the
the times considered. However, we will see in the next
temperature of the soft thermal photons from the sur-
section that the supernova envelope at t ∼ 10−100 s
face of the neutron star drops to T . 107 K and photo-
is too dense to allow the escape of particles, whatever
disintegration on this background radiation can also be
their mass number. At later times, as the wind cools
neglected,eveninsidethelightcylinder(Protheroe et al.
and becomes relativistic, the neutrino heating efficiency
1998; 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 our
sible acceleration sites thus lie close to the equato- framework.
rial plane of the star, at a distance R > r ≡
a min
max(rmin,lin,rmin,c,rmin,c,π)∼3×107−9Ω−41 cm,assum- 3. UHECRESCAPEFROMSUPERNOVAENVELOPES
ing γ .103. The fact that r &r implies that the
L min diss
unipolar induction toy-modelcould apply, and that par- Particles accelerated in the pulsar wind further need
ticles could reachultrahigh energies within this range of to escape from the pulsar wind nebula itself, and then
distances. fromthe surroundingyoung supernovaenvelope. 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
strongly magnetized proto-magnetar scenario discussed
One can mention three channels via which heavy ions
by Metzger et al. (2011a) —see Appendix A for further
couldbe seededinthe neutronstarwind. Note thatsce-
discussions.
narios of pulsar winds loaded with heavy nuclei give a
satisfactory explanation to some observations. For in- Theescapeofacceleratedionsfromthemagnetarwind
stance, the morphological features of the Crab Nebula nebulawasdiscussedbyArons(2003). InSection2.2,we
could be the signature of resonant scattering of pairs of arguedthatatdistancesr≫R ,thecurvatureradiusof
L
electrons and positrons by heavy nuclei (Hoshino et al. the ions reads: ρ ∼ 2ηr3/R ≫ r. Hence, particles are
c L
1992; Gallant & Arons 1994). not coupled to the magnetic field lines and can escape
the wind beyond r .
The classical argument that applies best in min
our scenario is that iron nuclei can be stripped In supernova envelopes, magnetic fields are of order
off the neutron star surface, as has been sug- a few mG at most (see, e.g., Reynolds et al. 2011 for a
gested by Ruderman & Sutherland (1975) and review). The Larmor radius of the ions is thus much
Arons & Scharlemann (1979). Strong electric fields largerthanthesizeoftheenvelopeandtheirtrajectories
combined with bombardment by particles can extract can be treated rectilinearly. We give in the following
ions from the polar cap regions, where the co-rotation section, estimates of the density profile and composition
charge is positive provided that Ω·B < 0. The surface of young supernova envelopes, that we use to study the
of a neutron star being composed mainly of iron-peaked escape of UHECRs analytically and numerically.
5
3.1. Supernova envelopes sity of the envelope, the detailed density profile is not
crucial to our calculations.
AsdiscussedforinstancebyChevalier(2005),rotation-
Under the assumption of adiabatic, spherically
powered pulsars can originate in various types of core-
symmetric flows, the numerical calculations of
collapsesupernovæ: inTypeIIsupernoværesultingfrom
Matzner & McKee (1999) show that the density of
red supergiant stars with most of their hydrogen enve-
a Type II supernova inthe dense centralregioncantake
lope intact (SNIIP), or with most of their hydrogen lost
values as high as:
(SNIIL and IIb), or in Type Ib or Type Ic supernovæ
(SNIb/c) that stem from stars with all their hydrogen ρ (t)∼10−16M5/2 E−3/2t−3 gcm−3 . (12)
lost. See also Maeda et al. 2007, Woosley 2010, and SNII ej,10 ej,52 yr
Piro & Ott 2011, Kasen & Bildsten 2010, for supernovæ
Most type II supernovæ eject a mass of order M
associated with magnetars. Chevalier (2005) finds that, ej,10
(Woosley & Weaver1995). This dense,relativelyflatre-
of the remnants with central rotation-powered pulsars,
gion extends to radius R ∼ 2(E /M )1/2t and is sur-
the pulsar properties do not appear to be related to the b ej ej
roundedby a steep outer power-lawprofile. The column
supernova category.
density that the acceleratedparticles have to traverseto
Within a few days after the explosion, the supernova escape is then:
enters a free expansion phase with velocity distribution
v =r/t, thatlasts severalhundreds ofyears. A straight- ySNII(t)=ρSNIIRb ∼4Me2j,10Ee−j,152t−yr2 gcm−2. (13)
forwardwayto model the evolutionofthe density of the
ejectaistoassumethattheejectedmassMej willexpand ForTypeIb/c/bcsupernovæ,onecanapplythemodel
spherically in time with a mean velocity vej over a shell of Matzner & McKee (1999) for the explosion of a star
of radius RSN = vejt. The ejected velocity, Eej, relates with a radiative envelope, which yields:
to the supernova explosion energy and the ejected mass
through: ρ (t)=7×10−17 v −1.06×
SNIb/c 0.01c
1/2
vej =2 MEej ∼109Ee1j/,522Me−j,11/02cms−1 , (9) Me1j.,927Ee−j,05(cid:16).297t−yr3 g(cid:17)cm−3 , (14)
(cid:18) ej(cid:19)
out to radius R , beyond which the density decreases
b
where we defined Mej,10 ≡ Mej/10M⊙ and Eej,52 = steeply. We have assumed in this estimate an explosion
Eej/1052ergs. Mostcore-collapsesupernovæareinferred energyofEej,52andanejectamassofMej,2=Mej/2M⊙,
to have explosion energy E ∼ 1051ergs. However, for which are derived from the observation of such objects
ej
the 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 ∼ 1052ergs will be transfered reads
within a fraction of a year to the surrounding ejecta
Rb
(see Kasen & Bildsten 2010). Depending on the radia- y (t)= ρ dr ∼9M2 E−1 t−2 gcm−2 .
tion conversionefficiency of this energy,the surrounding SNIb/c SNIb/c ej,2 ej,52 yr
Z0
supernovacouldbecomeultraluminous. Someultralumi- (15)
nousSNIb/candSNIIhaveindeedbeendetectedwithan
Equations (11), (13), and (15) agree within factors of
explosion energy & 1052ergs (e.g., Nomoto et al. 2001;
a few. It is thus reasonable to consider Eqs. (10) and
Woosley 2010; Piro & Ott 2011; Barkov & Komissarov
(11) as representative of the envelope mean density and
2010).
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 M would enhance the column den-
ej
ρ (t)= Mej ∼2×10−16M5/2 E−3/2t−3gcm−3 , sity. The effects of suchcases onparticle escape are also
SN (4/3)πv3t3 ej,10 ej,52 yr discussed throughout the paper.
ej
(10) Onecanfurther note that ifthe pulsarwindshreds its
where tyr ≡ t/1yr, which is the timescale to reach a surrounding supernova envelope, as discussed in Arons
pulsar spin that enables the acceleration of iron up to (2003)forthemagnetarcase,disruptedfragmentswould
∼ 1020.5 eV (see Eq. 4). The column density integrated expand in the interstellar medium. In this case, one can
over RSN as a function of time reads weightthe initial supernovadensityby C−2/3, C ≡δρ/ρ
y (t)=ρ R ∼2M2 E−1 t−2 gcm−2. (11) being a factor measuring the clumpiness of the envelope
SN SN SN ej,10 ej,52 yr
(Murase et al. 2009). A high C wouldease the escape of
UHECRs from the envelope. However, the values of C
More detailed modelings show that the density evo-
remaindifficulttoevaluate,asnoobservationalevidence
lution of the ejecta is expected to depend on the type
of such phenomena has been detected.
of supernova. Yet, we demonstrate in what follows that
Eq. (11) above provides a good estimate for the evolu-
tion of the integrated column density of various types of Thecompositionofthesupernovaejectadependsupon
supernovaenvelopes. Indeed, wewillsee inthe nextsec- the type, progenitor mass, and the final interior mass
tion 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 at
interactionssolelydependontheintegratedcolumnden- least some pulsars arise from massive star progenitors
6
(Muno et al. 2006). But, as mentioned before, rotation- The condition of escape from the supernova envelope
powered pulsars and magnetars have been invoked for can thus be written as t /t <1, yielding
dyn had
a wide variety of supernova types. The composition of
a type Ib supernova is roughly 50% helium and 50% t>tesc,p ∼1.2×107Mej,10Ee−j,15/22s for proton,(18)
C/O: e.g., the Woosley (2010) progenitor is ∼50% he-
t>t ∼3.8×107M E−1/2s for iron, (19)
lium, ∼43% carbon and ∼7% oxygen. Type Ic super- esc,Fe ej,10 ej,52
novae (more numerous than Ib supernovae) are com-
where we assumed a supernova density profile following
posedalmostentirelyofC/Oandheavierelements: e.g.,
Eq. (10). Cosmic rays at ultrahigh energy will escape
Mazzali et al. (2010) argued that SN 2007gr was com-
only if they are produced at late times t & 1 yr, when
posed of roughly 75% C, 15% O, 8% 56Ni, and 2% S.
the envelope density has decreased. Because nuclei of
Type II supernovae have a range of ejecta, ranging from
charge Z at a given energy E are produced at a time
roughly60%H,30%He,and10%C/Otoexplosionsvery t (E) ∝ Z (Eq. 4), one has t /t ∝ Z−2. The
spin dyn had
similar to type Ib supernova with small amounts of H.
escape condition from the baryonic envelopes at a fixed
WewilldiscussinSection3.4howtheescapedUHECR E should consequently be eased for heavier nuclei.
spectrumvariesbetweenpurehydrogenandpurehelium
StillassumingthesupernovadensityprofileofEq.(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
particles should not be able to escape the envelope:
3.2. Analytical estimates
1/2 −1
In accord with the discussion at the beginning of Sec- 8µ2Ω2 3M σξ
E =E 1+ i ej (20)
tainodn(31.11,)wperowviidllecaonrseiadseornianbtleheesftoilmloawtiengofththaet Eevqosl.u(t1io0n) cut,Z i  9 Ic3 4πmbve2j! 
ofthedensityofthe supernovaenvelopesurroundingthe  σ 1/2
neutron star. ∼7.5×1018Z1η1I45µ−301.5Me−j,110Ee1j/,522 σp eV(21)
Successful escape of UHECRs from the envelope will (cid:16) σ (cid:17) 1/2
occur if the shell crossing time tdyn is shorter than the ∼1.2×1020Z26η1I45µ−301.5Me−j,110Ee1j/,522 σFe eV(22)
coolingtimebyhadronic,t ,andphoto-hadronic,t ,
had Nγ (cid:16) (cid:17)
interactions. wherethefirstnumericalapplicationcorrespondstopro-
tons and the second to iron nuclei. Note that under the
TheaccelerationofaparticletotheenergyE happens crude approximation that σ ∝ A2/3, E ∝ Z/A1/3.
cut,Z
atatime after pulsarbirth: t(E)≃t (E). We canas-
spin For E ≪E, E does not depend on Ω.
cut,Z i cut,Z i
sumethatthethicknessofthesupernovashelltotraverse
at a time t is given by R ≃tv . Indeed, from the val- This trend is illustrated in Figure 1, where the main
SN ej
uesgiveninSection2.2,the accelerationsite Ra ≪RSN, timescalesatplayaredisplayed: tdyn andthad asa func-
as soon as t & 100 s. The crossing time for UHECRs tion of particle energy E, for various pulsar parameters
traveling 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-
R v lope at higher energies, as they can reach these energies
SN ej
tdyn(E)≃ c ≃ c tspin(E) . (16) at later times. Lower magnetic fields (µ . 1031) lead to
longer t at a fixed E (Eq. 4), while high pulsations
spin
As the expansion time scale of the envelope is tex = (Ω&104s) lead to higher acceleration energies (Eq. 1).
R /v ≃ t < t , one can neglect the evolution
SN ej spin dyn
of the envelope density during the escape of a particle. When iron nuclei are injected, secondary particles are
produced by hadronic interactions for times t < t .
esc,Fe
The timescale for hadronic interaction losses can be
Thesesecondariesofmassandchargenumbers(A,Z)can
expressed as:
escapethe envelopeonly attimes t>t , wheret
esc,Z esc,Z
is defined as the time at which t /t = 1. Hence,
t (E)=m {cρ [t (E)]σ(E)ξ(E)}−1 , (17) dyn had
had b SN spin secondariesthat will escape from the envelope have nec-
essarily been produced between t < t < t , i.e.,
where m is the mass of the dominant target ion com- esc,Z esc,Fe
b the lightest secondaries will escape first. This translates
posing the envelope. The parameters ξ(E) and σ(E)
in terms of the energy range of the primary iron to:
are the elasticity and the cross-sectionof the interaction
E <E <E (t ), where we can further express
at energy E. For our analytical estimates, we evalu- cut,Fe Fe esc,Z
E (t ) = (26/Z)E . The main fragment among
ate their values roughly from the hadronic interaction Fe esc,Z cut,Z
secondary particles will thus emerge from the envelope
model EPOS (Werner et al. 2006). We assume that the
between energies E′ .E .E′ , with
cross-sections of the hadronic interactions do not vary low,Z cut,Z
fsotrropnrgoltyoanb-porvoetEon>in1te0r1a8cetVionasndansedtσthFeem=t1o.2σ5pb=fo1r3i0romnb- El′ow,Z≡5A6Ecut,Fe (23)
protoninteractions. The number ofnucleons carriedout
ateachinteractioncanvaryfrom1toA−1,withalarge ∼2.1×1018Aη1I45µ−301.5Me−j,110Ee1j/,522eV ,
spread in values. For demonstrative purposes, we take
26A
an average value of ξ =0.4 for both p−p and p−Fe in- E′ ≡ E (24)
cut,Z 56Z cut,Z
teractions. These calculations are done accurately using
EPOS in our numerical calculations in the next section. ∼3.5×1018Aη I µ−1 M−1 E1/2 eV.
1 45 30.5 ej,10 ej,52
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
(dashedlines)andenergylosstimebyhadronicinteractionsthad (solidlines)aredisplayedasafunctionofparticleenergyE,forpureiron
(red) and pure proton (blue) injections. The timescales are calculated for various pulsar initial rotation velocities Ωi =103,104s−1 and
magneticdipolemoments µ=1030 to1031.5cgs,aslabeled. Theother pulsarparametersaresettoI=1045gcm2,η=0.1.
The numerical estimates are calculated for secondary time by photo-disintegrationof order:
protons. Peaksofthe varioussecondaryelements should
appear in the escaped cosmic-ray spectrum at their re-
spective energies. A tail due to lower energy secondary tAγ,th=[cξAγ(∆ǫAγ/ǫ¯Aγ)σAγUth/ǫγ]−1 (25)
nucleons (E <E′ ) followingapproximatelya power- 3/8
lawwithindex∼lo−w,1Z/2shouldalsobeproducedtogether ∼105A−0.21 Eej,52 M−9/8t9/4s (26)
withthemainfragment,downtoPeVenergies. Theam- 56 ηγ2,1ηt2h! ej,10 yr
plitude of this tail around ∼ E′ is about a fraction
low,Z
of the number of the main fragment.
where ∆ǫ /ǫ¯ ∼ 0.4A0.21, σ ∼ 8×10−26A cm−2
From Eq. (16) and (17), one can derive: Aγ Aγ 56 Aγ 56
(Murase et al. 2008), and we take for the elastic-
tdyn/thad ∼ 3 × 1010t−22Me2j,10Ee−j,152 at t2 ≡ t/100s, ity of the Aγ interaction: ξAγ = 1/A (which
for (A,Z) = (90,40). We assumed a cross-section is a crude approximation). This estimate of
σ = 1.5b for nuclei-proton interactions and an elas- the cooling time is valid for cosmic-ray energy
90
ticity of ξ = 0.4, at energies E ∼ 1020 eV (in the target E ∼4×1017(η η )−1/4E1/8 M−3/8t3/4eV, and
A,peak γ,1 th ej,52 ej,10 yr
rest-mass frame). This demonstrates that nuclei with
isaboutoneorderofmagnitudelargerforE &E ,
A & 56 that could be injected at times t ∼ 10−100 s A A,peak
as the photo-disintegration cross-section lowers. At the
if a successful r-process occurred in the neutrino-driven highest energies (E ∼ 1020eV), photo-disintegration
wind (Section 2.3), cannot survive the crossing of the A
couldthus play arole onthe escape ofcosmic raysif the
supernova envelope.
radiation and thermalization efficiencies are higher than
η η &10−2. Therateofwindenergygoingtoradiation
γ th
Ultrahigh energy ions could also experience photo- is evaluated to be of order 10% (e.g., Kasen & Bildsten
disintegrationin the radiationfields generatedatthe in- 2010), but the thermalization fraction of these photons,
terfacebetweenthepulsarwindandthesupernovashell. 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-type
if the supernova explosion is driven by the pulsar wind,
structures could lead to a leaking of the high energy
asexpectedformillisecondrotators. Afractionη ofthe
γ photons, and the thermalization fraction could be as
wind energy ∼ (1/2)IΩ2 can be converted to radiative
i low as . 10%. A higher acceleration efficiency η would
energy via internal shocks and another fraction η of
th alsoenable particles to reachthe highest energiesby the
thisradiationthenthermalizesdependingontheopacity
time the radiation field intensity has become negligible.
ofthemedium. Thisthermalcomponentpeaksatenergy
Given these uncertainties, and for simplicity, we will
ǫγ =kT ∼0.4(ηγ,1ηth)1/4Ee−j,15/28Me3j/,180ty−r3/4eV, with en- assume in this paper that the radiation field can be
ergy density U ∼ 0.5η η E−1/2M3/2t−3ergcm−3, neglectedfor the escape of UHECRs fromsupernova en-
th γ,1 th ej,52 ej yr velopes,thebaryonicbackgroundplayingthemajorrole.
where η ≡η /0.1. This backgroundleads to a cooling
γ,1 γ
To summarize, the conditions for successful accelera-
8
tion and escape above 1020eV can be written as:
BΩ2 &(1012.4G)×(104s−1)2Z−1η−1R−3
i 26 1 ∗,10 (27)
(B .1012.8GZ26A5−61/3η1I45Me−j,110Ee1j/,522R∗−,310
Highervaluesofthemagneticfieldwouldallowhigherac-
celeration energies, but would require lower ejecta mass
and higher explosion energies. Note that 10M⊙ can
be viewed as an upper bound for the ejecta mass for
type II supernovæ (Woosley et al. 2002). One might
also advocate that the presence of clumps could lower
the overall densities and allow the escape of particles at
E > 1020 eV. All in all, the parameter space allowed
for successfulaccelerationandescape appearsto be nar-
row, but we will see in Section 4.1 that the low rate of
sourcesrequiredtoaccountfortheobservedUHECRflux
would compensate for this issue. A higher acceleration
efficiency η would also broaden the allowed parameter
space.
3.3. Numerical Setup
Fig. 2.—UHECRspectrumbefore(dash)andafter(solid)escape
As discussed in the previous section, the hadronic in- fromahydrogen supernovaenvelopewithMej,10 andEej,52,with
teraction between UHECRs and the baryonic envelopes pηu=re0p.1r,otΩon=i1n0je4c.0tiso−n1.,Tahnedpµu=lsa1r03p0a.r5acmgse.ters are I =1045gcm2,
is the determinant factor that would affect the injected
UHECR spectrum. pulsars embedded in pure hydrogen, helium and carbon
supernovæ.
The interactionswith the baryonicenvelopeswerecal-
culated by Monte-Carlo for injected nuclei and their
secondaries. As in Kotera et al. (2009), we used the 3.4. Numerical Results
hadronic interaction model EPOS (Werner et al. 2006)
andthefragmentationmodelofCampi & Hu¨fner(1981), 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 using
CONEX (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, the
interaction products can be derived from the nuclei-
3.4.1. Pure Hydrogen supernova envelope
proton interaction case by a superposition law. In the
targetrestframe,theproductsoftheinteractionbetween
Figure 2 presents the injected (in dash line) and es-
aprojectileofmassnumberandenergy(A ,E )and
proj proj caped (in solid line) spectra of pure proton injection by
atargetnucleusofmassnumberA areroughlyequiv-
targ a pulsar with initial angular speed Ω = 104s−1 and
alent to A times the products of the interaction be- i
targ magnetic dipole moment µ = 1030.5cgs. The injected
tween a projectile with (A ,E /A ) and a target
proj proj targ spectrum follows the characteristic −1 spectral index in
proton. The exact cross-sections are nonetheless com-
Eq. (2). As predicted in Eq. (21) UHE protons above
puted with EPOS.
∼10EeVfailtoescapethesupernovaenvelope,sincethe
In the simulations, we modeled pulsars with initial region is still very dense at the time they are produced.
angular velocity Ωi ∼ 103.0−4.2s−1 and magnetic mo- Below a few EeVs protons are free to escape. Protons
ment µ ∼ 1030−33cgs, corresponding to a surface mag- with energy in between can partially escape with signifi-
netic dipole field B ∼ 2×1012−15G. Notice that there cant flux suppression. EPOS shows that for one 10EeV
is an upper limit (∼ 104.2s−1) on the initial angular primary proton, the peak of interaction products lies at
speed (Haensel et al. 1999). For each set of parame- 1014eV; the chance of resulting a secondaryprotonwith
ters, 107 cosmic rays are injected following a power-law E ≥ 1017eV is less than 0.01. Therefore we can barely
energy spectrum as in Eq. (2) with minimum injection see the secondary protons in our energy window of sim-
energy E = 1017eV, and the maximum acceleration ulation.
min
energy E calculated in Eq. (1). Above E, the spec-
i i The spectra of pure iron injection by pulsars with
trum cuts-off exponentially. Nuclei with initial energy Ω =104s−1andµ=1030.5,1031.5cgsareshowninFig3.
E are injected at a radius Ra = 1010 cm (correspond- Ini the top plot (µ = 1030.5cgs, Ω = 104s−1), primary
ing to ∼ 3rmin for Ωi = 104s−1, see discussion in Sec- iron nuclei with energy up to E = 1.2 × 1020eV
tion 2.2) at the time t (E), and propagate through a cut,Fe
spin can escape without significant loss. As discussed in our
supernova envelope of total ejected mass 10M⊙ (2M⊙ analytical estimates, most secondaries should originate
in Type Ib/c supernova case) expanding at a constant
from primary iron nuclei with energy between E =
rate vej = 109Ee1j/,522Me−j,11/02cms−1. The evolution of the 1.2×1020eVand56×Ecut,p=4.2×1020eV,correcuspt,Foend-
ejecta density is assumedto follow Eq. (10). We studied ing to the iron cutoff and iron mass number times the
9
When the magnetic field is 10 times stronger (µ =
1031.5cgs,Ω = 104s−1, bottom plot of Fig 3), the pul-
i
sar spins faster and the cutoff for primary and secon-
daries are lowered by 10 times (see Eq. 24). Hence,
the µ = 1031.5cgs case presents a similar shape as the
µ =1030.5cgs 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 ≪E the gravitational wave losses are negligible and
g
t is independent on the initial rotation speed Ω for
spin i
E ≪ E. A pulsar with higher initial angular veloc-
i
ity can inject UHECRs with greater maximum energy.
However a minimum spin period ∼0.4ms is allowed for
neutron stars (Haensel et al. 1999) corresponding to an
upper limit (∼ 104.2s−1) on the initial angular speed.
Magnetic dipole moments µ greater than 1032cgs would
make the spin-down process too fast to allow UHECR
escape. On the other hand pulsarswith µ<1030cgs are
not energetic enough to accelerate particles to ultrahigh
energy (see Eq. 27). To determine the best escaping re-
gionwe ran a parameter scanwith 15×15 sets of (Ω,µ)
and the results are presented in Figure 4.
Wedefine the cut-offenergyE asthe energythe ra-
cut
tiobetweentheescapedandinjectedparticlesislessthan
10%. Itcorrespondsapproximatelytothehighestenergy
ofescapedcosmicraysE definedinEq.(21). InFig-
cut,Z
ure4,the contoursrepresentE reachedafterescaping
cut
hydrogensupernova envelopes with M and E for
ej,10 ej,52
pulsarswithdipolemomentµandinitialangularvelocity
Ω. In the proton case (top), protons with energy above
i
1020eV cannot escape the supernova envelope in our
model. In the ironcontours(bottom), the parameter re-
gion with (µ≈1030.00−30.72cgs)×(Ω ≈103.95−4.20s−1)
i
allowsthe escape ofironnucleiwith energygreaterthan
1020eV. Thisparameterscanisbasedonasupernovaen-
velope with density profile described inEq. (10). Higher
values of explosion energy and lower ejecta mass could
lead to a broader enclosed parameter region that allows
the escape, as E scales with M−1 E1/2 (Eq. 20).
cut,Z ej,10 ej,52
Our results agree with the theoretical prediction from
Fig.1inBlasi et al.(2000),exceptthatwehaveasmaller
parameterareathat allowsescape. This comes from our
Fig.3.— UHECR spectrum before (dash) and after (solid and
dashdotted)escapefromhydrogensupernovaenvelopewithMej,10 assumption that only η ∼ 10% of the induced potential
and Eej,52, with pure iron injection. The pulsar parameters are turns into UHECR energy.
I = 1045gcm2, η = 0.1, Ω = 104s−1, and µ = 1030.5cgs (top),
and µ=1031.5cgs (bottom). Different compositions are listed as 3.4.2. Helium-Carbon/Hydrogen-Helium supernova
inthelegendbox.
envelopes
cutoffofsecondaryprotons. InagreementwithEq.(24), ResultsinFigure5arefromsimulationswithanejecta
secondaries lie between (1.0−5.0)×1018eV for proton, mass of Mej = 10M⊙, explosion energy Eej = 1052ergs,
2.0×1018−1.3×1019eV for helium, 7.9×1018−4.0× and a composition of pure 4He (top) and pure 12C (bot-
1019eV for CNO, (1.3−7.1)×1019eV for Mg-like ele- tom). As discussed in Sec. 3.3, realistic envelopes for
ments and2.0×1019−1.1×1020eV for Si-likeelements, SNII and SNIb/c are more complicated and could be
withthepeakpositionsscaledtothemassnumberofthe evaluated by a combination of Fig. 3 and 5. Spectra of
elements and the bump width being almost the same in UHECRs escaped from envelopes abundant in heavier
logarithmic coordinates. The significant tail of protons elementsmaintainfeaturesfromthatwith apurehydro-
below 1EeV comes from the products of the hadronic gen envelope. For instance, in case of a pure helium
interactions. On average, each interaction of a 500EeV envelope (top plot in Fig. 5), the spectrum preserves
iron nucleus results in one EeV proton among its prod- the ‘original’ secondary peaks at 6.3× 1017eV for hy-
ucts. Thestrongsignalsfromsecondarynucleicontribute drogen,2.5×1018eV for helium, 8.0×1018eV for CNO,
to a steeper overall spectrum (in solid black) which fol- 1.3×1019eV for Mg-like elements and 2.5×1019eV for
lows ∼E−2 at 1018.5−1020eV. Si-like elements. These peaks are similar to the ones in
10
Fig. 4.—Parameterspacewithcut-offenergy(Ecut,Z)contours, Fig.5.—UHECRspectrumafterescapefromasupernovaenve-
for a hydrogen supernova envelope with Mej,10 and Eej,52, and lopewithMej,10andEej,52,withcomposition(top): 100%4Heand
pulsarparameters I45 andη1. Thesolidlines referto cut-offpar- (bottom): 100% 12C. The pulsar parameters are I = 1045gcm2,
ticleenergies aftertheescape. Upisprotoninjectionanddownis η=0.1,Ω=104s−1,andµ=1030.5cgs.
ironinjection. Noticethatcurrentneutronstarmodelssuggestan
upper limitof rotational speed at Ωi ≤ 104.2s−1. Note also that tribute to an increment of primaries around 4×1019eV
Ecut,Z scaleswithMe−j,110Ee1j/,522 (Eq.20). for helium envelopes and 1.3 × 1019eV 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. IMPLICATIONSFORTHESCENARIOOFUHECR
due to the 4 times heavier interactant.
PRODUCTIONINNEWLY-BORNPULSARS
The case of heavy envelopes can generate multiple
ThesuccessofaUHECRsourcescenarioliesinitsabil-
peaks to the left of the original peaks due to multiple
ity to reproduce these observations: i) the energy spec-
products. According to the superposition law, the num-
berofproductsscaleswithAN afterN interactionswith trum, ii) the composition, iii) the anisotropy,and iv) on
the fact that it requires a rate of sources consistentwith
envelope baryons of mass number A. So the later gener-
the population studies inferred from other astronomical
ations (tertiaries and so forth) whose energy are mostly
observations.
below 1018eV are far more numerous than the earlier
generations(primariesandsecondaries). This bringsthe As we discuss in this section, the results obtained in
low end of the original peaks up to be a second, or even this paper suggest that all four points could be rea-
thirdadditionalpeaksforallcompositions;theyalsocon- sonably achieved in the extragalactic rotation-powered