Ultrahigh-Energy Cosmic Rays from the "En Caul" Birth of Magnetars

0.31 MB·

by  Anthony L. Piro

ACCEPTEDFORPUBLICATIONINTHEASTROPHYSICALJOURNAL
PreprinttypesetusingLATEXstyleemulateapjv.5/2/11
ULTRAHIGH-ENERGYCOSMICRAYSFROMTHE“ENCAUL”BIRTH1OFMAGNETARS
ANTHONYL.PIRO2ANDJUNAA.KOLLMEIER2,3
AcceptedforpublicationinTheAstrophysicalJournal
ABSTRACT
6 Rapidly-spinningmagnetarscanpotentiallyformbythe accretioninducedcollapseofa white dwarforby
1 neutronstarmergersiftheequationofstateofnucleardensitymatterissuchthattwolowmassneutronstars
0 cansometimesformamassiveneutronstar ratherthanablackhole. Ineithercase, thenewlybornmagnetar
2 is an attractive site for producingultrahigh-energycosmic rays (particles with individualenergiesexceeding
1018eV;UHECRs). Theshort-periodspinandstrongmagneticfieldareabletoaccelerateparticlesuptothe
y
appropriateenergies,andthecompositionofmaterialonandaroundthemagnetarmaynaturallyexplainrecent
a
M inferencesofheavyelementsinUHECRs. Weexplorewhetherthesmallamountofnataldebrissurrounding
thesemagnetarsallowstheUHECRstoeasilyescape. We alsoinvestigatetheimpactontheUHECRsofthe
uniqueenvironmentaroundthemagnetar,whichconsistsofabubbleofrelativisticparticlesandmagneticfield
0
withinthedebris. RatesandenergeticsofUHECRsareconsistentwithsuchanorigineventhoughtheratesof
1
eventsthatproducerapidly-spinningmagnetarsremainveryuncertain.Thelowejectamassalsohelpslimitthe

high-energyneutrinobackgroundassociated with this scenario to be below currentIceCube constraints over
]
E mostofthe magnetarparameterspace. A uniquepredictionisthatUHECRs maybe generatedin old stellar
H environmentswithoutstrongstarformationincontrasttowhatwouldbeexpectedforotherUHECRscenarios,
suchasactivegalacticnucleiorlonggamma-raybursts.
.
h
Subjectheadings:astroparticlephysics—cosmicrays—stars: magnetars
p
-
o
r 1. INTRODUCTION ∼1043.5−1044ergMpc−3yr−1 (Katzetal. 2009), which
t mustbesatisfiedbythecombinedrateandenergyproduction
s The origin of ultrahigh-energy cosmic rays (UHE-
a CRs), particles that can have a kinetic energy as large of any model. Most recently, the Auger Observatoryresults
[ point to a composition of UHECRs that may be dominated
as a thrown baseball, remains one of the outstand-

2 ing questions in astrophysics (e.g, Kotera&Olinto 2011; byheaviernucleiatthehighestenergies(Aabetal.2014a,b).
If confirmed, this would pose a problem for any model that
v Letessier-Selvon&Stanev2011). Theirmysteryhassparked
producesUHECRssimplyasprotons.
5 awiderangeofmodelsfortheaccelerationsitesofUHECRs,
At the same time, motivated by different physicalconsid-
2 with varying levels of success in explaining their energetics
erations,therehasbeenincreasingfocusonthepossibilityof
6 and spectrum. This includes, but is not limited to, galaxy
high-energyeventsthatmayresultinaquickly-spinningmag-
2 clusters(Inoueetal.2007;Koteraetal.2009),activegalactic
netar with a small amount of surrounding material. These
0 nuclei(AGNs;Dermeretal.2009;Takami&Horiuchi2011),
mainly come in three varieties, the first of which is the ac-
. classical gamma-ray bursts (GRBs; Waxman 2004, 2006;
1 cretion induced collapse (AIC) of a white dwarf (WD) to a
Dermer 2010), low-luminosity GRBs (Muraseetal. 2006),
0 neutron star (NS). In this case, either through the merger of
magneticallydominatedjetsinthemillisecondprotomagnetar
6
twocarbon-oxygenWDs,oraccretionontoanoxygen-neon-
1 model for GRBs (Metzgeretal. 2011), engine-drivensuper-
magnesiumWD,theWDelectroncaptures,losesitspressure
: novae(Wangetal.2007;Chakrabortietal.2011),AGNflares
v (Farrar&Gruzinov2009),andnewly-born,fast-rotatingpul- support and collapses to an NS (Canal&Schatzman 1976;
i Nomoto&Kondo1991). Thelargeangularmomentumasso-
X sars(Fangetal.2012;Koteraetal.2015).
ciated with such an eventmay naturally lead to short-period
Observationally, there are many clues about UHECRs to
r spinsandastrongmagneticfield(Duncan&Thompson1992;
a help constrain these models. The flux suppression at the
Price&Rosswog 2006; Zrake&MacFadyen 2013). In the
highestenergies(Abbasietal.2008;Abrahametal. 2008)is
second case, the merger or grazing collision of two NSs
consistentwithUHECRsinteractingwithcosmicmicrowave
may in some circumstances(dependingon their masses and
background photons or other sources of extragalactic light,
the maximum mass allowed by the equation of state of nu-
knownastheGZKeffect(Greisen1966;Zatsepin&Kuz’min
clear density matter; Hebeleretal. 2013) produce a stable,
1966). This strongly suggests an extragalactic origin,
but also limits their source to be in our local universe high-massNSratherthanablackhole(e.g.,O¨zeletal.2010;
within ∼ 100Mpc. The energy injected into UHECRs is Giacomazzo&Perna2013;Kiziltanetal.2013). Inthethird
case, the massive NS may be present, but only for a lim-
ited time due to support by thermal pressure and/or dif-
1Thebirth“encaul”referstotherarecircumstanceinwhichanewborn ferentialrotation(Lehneretal. 2012;Paschalidisetal. 2012;
emergesinafullyintactamnioticsac. Birthofthisnatureisconsidered Kaplanetal. 2014). In any case, there is considerable in-
asignofgoodfortuneinmanycultures. Here, werefertothenewborn
terest in the potential transient signature that these events
magnetar similarly surrounded bya small amount ofnatal material and
similarlyfortunateasacosmic-rayaccelerator. may make (Piro&Kulkarni 2013; Metzger&Piro 2014;
2Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA Piro&Thompson 2014) as well as the exciting possibility
91101,USA;[email protected] thattheymaybeobservedaselectromagneticcounterpartsto
3School of Natural Sciences, Institute for Advanced Study, Einstein
gravitationalwavesources.
Drive,Princeton,NJ08540,USA
2
Here we propose that such events may also be ideal sites spinning near its centrifugalbreak-up limits, torques from a
for producing UHECRs. The fast spin and high magnetic neutrino-drivenwind(Thompsonetal.2004)orgravitational
field of a newly born magnetar provide a natural location wavelosses(Piro&Ott2011) likelyspinitdownduringthe
foracceleratingUHECRstothecorrectenergies. Mostcriti- firstmillisecondstospinperiodsofP =2π/Ω ∼2−5ms,
i i
cally,thelowamountofnatalmaterialsurroundingthemag- andthusweconsiderthisthe“initial”periodforpurposesof
netars should, in principle, make it easier for UHECRs to ourcalculations.
escape the magnetar and seed the local universe. In this
work, we examine this possibility to determine whether this 2.1. EnergyInputbytheMagnetar
provides a viable site for UHECR production. In particu- Theinitialrotationalenergyofthemagnetarwithmoment
lar, we investigate whether the unique environment around ofinertiaI is
themagnetar,whichincludesabubbleofrelativisticparticles
andmagneticfieldinflatedbythe magnetar(Metzger&Piro E = 1IΩ2 =4.9×1051I Pi −2erg, (1)
2014; Siegel&Ciolfi 2015a,b), impedes and/or imprints it- rot 2 i 45 2ms
selfontheUHECRsinanyway. Furthermore,weinvestigate (cid:18) (cid:19)
whether such a scenario will have implications for the rates whereI45 = I/1045gcm2. The spin downof the magnetar
andenvironmentswheretheUHECRsmaycomefrom. withmagneticmomentµ=R∗3B∗injectsenergywithatime-
Ourmodeldiffersinseveralcriticalrespectsfromprevious dependentrateof
studiesthatfocusedonfast-spinningpulsars(e.g.Arons2003;
Fangetal.2012). Inparticular,theseworkshavefocusedon L (t)= µ2Ω(t)4 = µ2Ω4i 1+ t −2, (2)
a core-collapse scenario as a precursor to the UHECR en- sd c3 c3 t
(cid:18) sd(cid:19)
gine. As a result, a largeamountof ejecta materialthat sur-
wherethespindowntimescaleis
roundsthepulsar(thepulsarwindnebula;usuallyassumedto
be ∼ 5 −10M⊙ in line with estimates from the Crab neb- Ic3 P 2
ula) in comparison to the “en caul” birth we consider with t = =22.8µ−2I i min. (3)
only ∼ 0.1M⊙ of surroundingmaterial. A large amountof sd 2µ2Ω2i 33 45(cid:18)2ms(cid:19)
ejectagenericallymakesitmoredifficultforUHECRstoes-
Theejectaaroundthemagnetarisinflatedbythisenergyinput
cape the core-collapse environment as shown by Fangetal.
withatime-dependentvelocityof
(2012). Another important difference is that we focus on
muchearlier timesafterthe magnetarisborn(∼daysrather dR t 1/2
than∼monthstoyears),whenthenebulaaroundthepulsaris vej(t)= dtej = 2Me−j1 Lsd(t′)dt′+ve2j,i , (4)
denserinelectron/positronpairs. Thedensityofchargedpar- (cid:20) Z0 (cid:21)
ticlesisthenbetterdescribedinasaturatedregime(Svensson whichcan be simply integratedto find R (t). At late times
ej
1987) ratherthan simply givenby the Goldreich-Julianden- thevelocityoftheejectacangethigh
sity (Goldreich&Julian 1969). We discuss the impact of
thesedifferencesonourmodelbelow. v (t≫t )≈(2E /M )1/2
ej sd rot ej
In Section 2, we present a simplified model for following −1
the time-dependent features of the environment around the ≈0.7cI1/2M−1/2 Pi , (5)
newlycreatedmagnetar.InSection3,weusethesemodelsto 45 −2 2ms
(cid:18) (cid:19)
calculateourmainresultsforwhatenergyparticlesmaycome
fromthemagnetarandhowtheseparticlespropagatethrough whereM−2 = Mej/10−2M⊙. To simplifyouranalysis, we
focusonvaluesofE andM wherev .c.
the environmentof the magnetar. In Section 4, we estimate rot ej ej
the expectedrates of such eventsin comparisonto measure-
2.2. EvolutionofthePerimagnetarEnvironment
ments of UHECRs. We also discuss various constraints on
our scenario, including the rate of expected associated elec- InordertoelucidatetheUHECRpropagation,wedividethe
tromagnetictransients,galacticandintergalacticpropagation perimagnetarenvironmentinto two primary regions: (1) the
issues, andthe high-energyneutrinobackground. Finally,in ejectashelland(2)thenebula(orbubble)ofrelativisticparti-
Section5,wesummarizeourresultsanddiscusspotentialfu- clesaroundthemagnetar(seethediagraminMetzger&Piro
turework. 2014). In each of these regions, both the particle density
and radiation content must be followed since these can in-
2. THEMAGNETARANDSURROUNDINGREGION teract with UHECRs and alter their fate. In this simplified
To followthe environmentaroundthe newly bornmagne- picture, we do not explicitly follow the dynamicsof the ter-
tar, we construct a toy model that replicates the key phys- mination shock between the magnetar wind and the nebula.
ical features. For more detailed discussions and calcula- Nevertheless, this shock is important for converting the ini-
tions, with a particular focus on the electromagnetic signa- tialPoynting-fluxdominatedoutflowofthemagnetarintoki-
tures, the reader should consult Metzger&Piro (2014) and netic energy, as has been extensively studied in past work
Siegel&Ciolfi(2015a,b). (e.g.,Lyubarsky&Kirk2001;Arons2008;Lyubarsky2010;
The general picture we are considering is that, whether Granotetal. 2011; Porthetal. 2013). Since the presence of
born from an AIC or from an NS merger, the initial state is theterminationshockdoesnotimpacttheshelldynamicsbe-
a magnetar with surface dipole field B∗ ∼ 1014 − 1015G yond providing a potential site for particle acceleration, we
surrounded by Mej ∼ 0.01M⊙ of ejecta with a velocity mostlyignoreitwhensolvingforthepropertiesofthenebula.
of v ∼ 0.1c (Dessartetal. 2009; Metzgeretal. 2009a,b; For the ejecta region, the density simply evolves as the
ej,i
Leeetal. 2009; Ferna´ndez&Metzger 2013). The specific ejectaexpandsas
values can of course change and we will explore the depen-
3M
dencies on them. Although the magnetar may be initially ρ (t)= ej . (6)
ej 4πR (t)3
ej
3
The composition is thought to be primarily iron-rich ele- of the energy budget of the nebula in the calculations by
ments due to the higher electron fraction produced by neu- Metzger&Piro (2014), but this only represents a small cor-
trino irradiation from the magnetar (Metzgeretal. 2009b; rectionsinceη .0.1.
B
Ferna´ndez&Metzger 2013). We estimate the number den- Nextwe consideris the radiation contentin both the neb-
sity ofnucleiin the ejecta as n ≈ ρ /56m , where m is ula and surrounding ejecta. These are highly coupled. The
ej ej p p
theprotonmass. non-thermal radiation in the nebula is dominated by syn-
The second region is the relativistic bubble interior to the chrotronandinverse-Comptonemission(Metzgeretal.2014;
ejecta. This is inflated by the magnetar spin down luminos- Muraseetal.2015). Thisisthenpartiallyabsorbedandther-
ity, whichcanbe partitionedinto a combinationofe± pairs, malizedbybound-freereactionsbytheiron-peakelementsin
magneticfield,radiation,and,importantlyforthestudyhere, the ejecta. In the Appendix of Metzger&Piro (2014), we
UHECRs. The high density of radiation and magnetic field show that this interplay can be accurately followed in each
within the nebula gives rise to copious e± pairs, with the region by simply considering two coupled differentialequa-
strength of the pair creation described by the “compactness tions. Thenon-thermalenergyofthenebula,E ,issimply
nth
parameter” givenbytheenergyinjectionfromspindownminusadiabatic
andabsorptivelossesasafunctionoftime
3E σ
ℓ= nth Th (7)
4πRe2jmec2 dEnth =Lsd−Labs− Enth . (11)
dt t+t
i
where E is the energy in non-thermal radiation within
nth
the nebula (which we describe in more detail below) and Whatisabsorbedbytheejectafromthenon-thermalreservoir
σ is the Thomson cross section. When ℓ & 1, the num- isconvertedtoheatandthereforetheevolutionofthethermal
Th
ber density of these pairs can be estimated by balancing the radiationoftheejecta,Eth,evolvesas
rate of pair creation and annihilation (Metzgeretal. 2014;
dE E
Metzger&Piro2014),resultingin th =L −L − th . (12)
abs rad
dt t+t
i
1/2
n±(t)= 4YLsd(t) , (8) Theseequationshaveanumberoftermsthatwenowdescribe
πR (t)3σ m c3 indetail. Therateofabsorptionofnon-thermalradiationis
(cid:20) ej Th e (cid:21)
where σTh = 6.65× 10−25cm is the Thomson cross sec- L = fabsEnth, (13)
tion and Y ≈ 0.1 is the pair multiplicity factor for a satu- abs t
diff,neb
rated cascade (essentially the fraction of the magnetar spin
wheref is the fractionof non-thermalradiationabsorbed
down that goes into pairs; Svensson 1987). In our case, abs
bytheejectaandthediffusiontimeacrossthenebulais
ℓ & 1 for the times we consider as we show explicitly be-
low. This density is in contrast to other work on UHECRs
tdiff,neb =(σThRejn±+1)Rej/c. (14)
frommagnetarsinacore-collapsescenario(Fangetal.2012;
Lemoineetal. 2015) that use a Goldreich-Julian density in- The term L appearsin both Equations(11) and (12) as it
abs
stead (Goldreich&Julian 1969). Another difference is that mediatesthetransportofenergybetweenthesetworeservoirs
Equation (8) assumes that the e± are fairly sub-relativistic. ofradiation. Therateofradiativeenergylossfromtheejecta
Thisis reasonablebecause evenif the pairsare createdwith issimply
a high Lorentz factor, the ratio of the Compton cooling
E
timescaletCtothetimethenebulahasbeenexpandingtis Lrad = th , (15)
t
diff,ej
tC = πmecRe3j ∼ πmecve3jt where
t σ E t σ L
Th nth Th sd
−2 κ M R
∼0.1µ233I4−51/2M−−23/2 1dtay , (9) tdiff,ej = 4eπsRe2ejj +1! cej, (16)
(cid:18) (cid:19)
andthusthepairscoolrapidlyinthedensenebula. isthephotondiffusiontimethroughtheejectaandκ isthe
es
Inadditiontoe± pairs,thenebulawillbefilledwithmag- electron scattering opacity of the ejecta. The term L is
rad
netic field. As mentioned above, the general picture is that importantforestimatingthethermalradiationobservedfrom
themagnetaroutflowisPoynting-fluxdominatedinsideofthe the heated ejecta. The timescale t = R /v is the ini-
i ej,i ej,i
terminationshock,butdissipationneartheshockresultsina tial expansion timescale of the ejecta. It is included to pro-
muchsmaller mean field in the rest of the nebula. Since the ducewell-behavedsolutionsatearlytimesandtheresultsare
exact conversion factor is not known, especially for the ex- largelyinsensitive to its value. Finally, the last term in each
treme case of magnetars we consider here, we quantify this expression accounts for the adiabatic expansion of the rela-
uncertainty with a factor η that is the fraction of the mag- tivisticgas,andhencehasthefamiliarscalingofE /tat
B (n)th
netarspindownthatgoesintothismagneticfield. Thisthen latetimes.
gives From the above description, given specific properties for
the magnetar and ejecta, one can solve for the evolution in
Bneb(t)≈ R6η(Bt)3 tLsd(t′)dt′ 1/2, (10) enveoj,lunti±o,n.EAntnheaxnadmEpltehstooluitnivoensitsigparteestehnetepdoisnsiFbilgeuUreHsE1CaRnds
(cid:18) ej Z0 (cid:19) 2 for Pi = 3ms, B∗ = 5×1014G, and Mej = 0.01M⊙.
as an estimate of the time dependent mean magnetic field In this case we use f = 1, which generally reproduce the
in the nebula. We note that this magnetic field was left out featuresof more detailed treatments (Metzger&Piro 2014).
4
Starting with Figure 1, the thermal light curve L shows
rad
the characteristic supernova-like shape, which peaks on a
∼ 1day timescale, potentially being the best electromag-
netic signature for identifying such events (note the non-
thermal X-rays have a similar luminosity and time scale as
well, Metzger&Piro 2014). The observed L is well be-
rad
lowL becauseofthelargeopticaldepthinpairshindering
sd
thetransferofenergyfromthenebulatotheejecta,ascanbe
seen from τ± & τej near the light curve peak. Here we use
τ± = σThRejn± and τej = κesMej/4πRe2j. In the bottom
panel, we plot nej and n±, which will be important for un-
derstandingUHECRpropagationinsubsequentsections. We
alsoplotthecompactnessℓinFigure1,demonstratingthatit
isgreaterthanunityformostoftheevolution,justifyingour
treatmentofthee± pairsinthesaturatedregime.
InFigure2,wesummarizemoreimportantquantitiesfrom
the evolution of this model. The top panel shows both the
temperatureoftheejectaalongwiththeeffectivetemperature
ofthethermalopticalsignature.Thesebecomeroughlyequal
once τ . 1. The mean magnetic field in the nebula B ,
ej neb
calculated from Equation (10), will be important for under-
standing the acceleration of UHECRs as well as potentially
coolingthem via synchrotronradiation. In the bottompanel
wesummariesthecriticaldensitiesineachregime,alongwith
FIG.1.—Exampleevolutionoftheejectaandnebulaaroundanewlyborn,
quicklyrotatingmagnetarusingPi=3ms,B∗=5×1014G,andMej= theradiusoftheexpandingejectaRej.
0.01M⊙.ThetoppanelshowstheenergyinputbythespindownLsd(red,
dashedline)andthethermal radiation leavingtheejecta Lrad (blue, solid 3. UHECRENERGIESANDPROPAGATION
line). Thebottompanelshowstheelectronscatteringopticaldepthsforthe
pairsinthenebula(green,dashedline)andtheejecta(blue,solidline),τ± ESTIMATES
andτej,respectively,aswellasthecompactnessℓ(red,dot-dashedline).The In Section 2, we presented the basic physical model and
horizontal,blackdottedlinesimplyindicateswherethesequantitiesbecome
unity. demonstratedthatwecanfollowtheevolutionoftheencaul
magnetar in a straightforward fashion. Here, we estimate
theenergyoftheindividualUHECRparticlesandshowhow
the perimagnetarenvironmentaffectsthe ultimatedestiny of
UHECRsproducedfromthischannel.
3.1. MaximumEnergyofUHECRs
We first summarize the basic energetics to show that, in-
deed,themagnetarsurfacecaneasilyaccelerateCRstoultra-
high energies. Relativistic magnetic rotators have voltage
dropsacrosstheirmagneticfieldof
µΩ(t)2
Φ(t)= . (17)
c2
A particle of charge Z that is stripped off the surface of the
magnetar(bysomecombinationofstrongelectricfields,bom-
bardmentofparticles, andboilingbystellarheat)canbeac-
celeratedtoanenergy
−2
P
E =ZeΦ=3.3×1021Zµ eV. (18)
surf 33
2ms
(cid:18) (cid:19)
This shows that in practice it is reasonable to expect nuclei
acceleratedtosufficientenergiestoproduceUHECRs, espe-
cially for large charges(Z > 1). The problemis that when
thisaccelerationoccurswithinthelightcylinderofthemag-
FIG.2.— Further properties oftheejecta andnebula from the modelin netar R = Ω/c (which is much less than R ), then cur-
Figure1.ThetoppanelshowsthetemperatureoftheejectaTej(blue,dashed vature rLadCiation losses preventacceleration to seujch high en-
line),theeffectivetemperatureoftheejectaTeft(solid,greenline),andthe
meanmagneticfieldinthenebulaBnebusingηB =0.1(dot-dashed,purple ergies. For this reason, both Arons (2003) and Fangetal.
line). Thebottompanelshowsthedensitypairsinthenebulaandtheejecta, (2012) consider acceleration in the wind at radii ≫ R to
LC
n±(red,dot-dashedline),thedensityofionsintheejectanej(green,dashed avoidthisproblem.
lliinnees),ianntdhethbeorttaodmiuspaonfetlhiesejujescttaacRoeinjc(ibdleunec,esofolirdthliinsep).arTtihcuelcaornmveordgeeln.ceof Due to this difficulty, we instead consider the energyions
canbeacceleratedtojustfromconfinementbymagneticfields
5
withinthenebula(Lemoineetal.2015),whichgives
E (t)≈ZeB (t)R (t). (19)
conf neb ej
Themagneticfieldthatacceleratesparticlescanalsolimitthe
energyof the UHECRs due to synchrotronlosses. Equating
the synchrotron cooling time with the gyration time of the
particlesresultsin
3 (Am c2)2
E (t)= p . (20)
synch 2(Ze)3/2B (t)1/2
neb
Withthisinmind,weconsiderthehighestpossibleenergyfor
UHECRsatanygiventimetobetheminimumbetweenE
conf
andE .
synch
ThismaximumenergyisshownintheupperpanelofFig-
ure3 forUHECRswitheitherprotonorFe composition. At
earlytimesthenebulaisdenseenoughinmagneticfieldsthat
synchrotron cooling dominates and the energy is ≈ E .
sync
Then, as the nebula expands and the magnetar spins down,
synchrotronlossesdecreaseuntiltheparticleenergyisinstead
≈ E . The maximum energy for UHECRs from a given
conf
magnetariswherethesetwoenergiesareroughlyequal.Also
note that heavier nuclei generally reach higher energies be-
causetheirlargerchargepromotesmoreacceleration.
Eventhoughparticlesmaybeacceleratedtosuchhighen- FIG.3.—IntheupperpanelwesummarizethemaximumenergytheUHE-
CRscanbeacceleratedtoasafunctionoftimefromanewlybornmagnetar,
ergies,interactionwiththeenvironmentaroundthemagnetar usingPi =3ms,B∗ =5×1014G,andMej =0.01M⊙forthisexam-
maypreventtheUHECRsfromleavingunhindered. Wenow ple.ThesolidcirclesshowwheretheUHECRscanbeginescapingfromthe
consider each of these, both in the ejecta and the nebula, in ejecta. Thebottompanelsummariestheevolutionofrelevanttimescalesim-
moredetail. portantforunderstandingtheescapeofUHECRsfromtheejectasurrounding
themagnetar. Thetimescaletcross = Rej/c(green,solidline)showshow
longittakesaUHECRisescapetheejecta. Thismustbeshorterthanthe
3.2. InteractionswiththeEjecta hadroninteractiontimescaleforescapetooccur.Thesetimescalesareshown
forbothaproton-composition(blue,dashedline)andiron-composition(red,
For the ejecta, we use a hadron interaction model to esti-
dottedline)UHECRs.Alsoplottedisthepulsarspindowntimescale(purple,
mate thetimescale for sappingthe UHECRs of theirenergy. dot-dashedline),whichshowswhentheenergyoutputofthepulsarbegins
Thistimescaleisgivenby decreasing.
t (E)=[cn σ (E)ξ(E)]−1, (21)
ej ej ej
With this picture in mind, we nextestimate the maximum
whereσ isthecrosssectionforhadronicinteractions,andξ
ej energypossibleforiron-compositionUHECRsasa function
istheelasticity.Alsonotethatallthesetermswillhaveenergy
dependenciesin detail. The crosssectionforiron-protonin- ofthe initialspin periodPi andthe magneticfield B∗. This
is done by considering the highest possible energy for the
teractionsis1.25barn,whichapplieswhenZ =1. Sincethe
UHECR past the time when they start escaping the ejecta
crosssectionshouldscaleasroughlyZ2/3,foriron-ironinter- (past the solid circles from Figure 3). The results are sum-
actionsweestimatethecrosssectionas≈11barn. Although marized in the contour plot shown in Figure 4 where we
ξcanchangesignificantlydependingonthenumberofnucle- use Mej = 0.01M⊙. For these calculations we assume
ons,wetakeittobeoforderunitygiventheuncertaintiesin f = 1, which gives light curve solution that qualitatively
the cross sections which are degeneratewith this. The rele- matchMetzger&Piro(2014)andallowustocoverawidepa-
vant timescale to compare these to is the crossing timescale rameterspaceofmodels. Inthefuture,amoredetailedtreat-
fortheejecta. mentofthethermalizationshouldbeconducted.Thisdemon-
strates that to producethe highest possible energyUHECRs
t (t)=R (t)/c. (22)
cross ej requiresaspinperiodofP ≈ 2−4msandarathermodest
i
Whent <t ,theUHECRscanescapetheejectawithout (foramagnetar)magneticfieldofB ≈1−4×1014G(note
cross ej
significantinteraction(andviceversa). thatalthoughweconsiderP .2msunlikelyforthereasons
i
InFigure3,wepresentanexamplecalculationoftherele- mentionedabove,weincludethisparameterspaceinourplots
vanttimescalesforassessingtheinteractionofUHECRswith forcompleteness). The reasonwhythe highestfield magne-
the ejecta alongwith maximumpossibleenergyfor the gen- tarsaredisfavoredforproducinghighenergyparticlesisthat
eratedUHECRsasafunctionoftime. We alsocomparedif- theyspin downtoo fast. Eventhoughtheyproducehighen-
ferentcompositionsof UHECRs, bothprotons(blue, dashed ergyparticles,theseallinteractwiththeejectaatearlytimes
lines)andiron(red,dottedlines). Thisdemonstratesthatpro- beforeitcanbecomesufficientlydiffuse.
tonssatisfyt <t earlierduringtheevolutionandbegin When we consider more massive ejecta the maximumen-
cross ej
escapingearlier(everythingtotherightofthebluecirclees- ergies for the UHECRs decrease by slightly (∼ 10% for
capes). Nevertheless,comparingwiththeenergytheprotons 0.05M⊙) and the “sweet spot” for most efficiently acceler-
would haveat that time, protonsdo not reachas high of en- atingUHECRsshifttoslightlylowermagneticfields. Thisis
ergiesastheiron. Thus, if allotherthingsare equal, a large becausealowermagneticfieldspinsdoesthemagnetarmore
chargeisfavoredforproducingthehighestenergyUHECRs. slowly so that the time scale for the energy injection better
6
FIG.4.— Contours show the maximum possible energy for iron- FIG.5.—Evolutionofrelevanttimescalesimportantforunderstandingthe
compositionUHECRsasafunctionoftheinitialspinperiodPiandmagnetic escape ofUHECRsfrom theejecta surrounding the newly bornmagnetar,
fieldB∗ofthemagnetar,usinganejectamassof0.01M⊙.Eachcontouris usingPi = 3ms,B∗ = 5×1014G,andMej = 0.01M⊙ forthisex-
labeledinunitsofeV. ample. Thenebulacrossingtimetcross andthespindowntimetsd arethe
sameasinFigure3.Thetwoadditionaltimescalesarethatforphotodisinte-
matchesthe longer diffusiontime throughthe more massive grationwithinthenebulatnthandwiththeejectatth,whichareestimatedin
Equations(24)and(25),respectively.
ejecta. With very high ejecta masses, as in the more asso-
ciated with core-collapse(e.g.,Fangetal. 2012), theshiftto
lowerenergiescaninsomecasesmakeitdifficulttoproduce
UHECRsatall. Overallthough,the ejectamassimpactsthe and thus strongly increases as the ejecta expands and cools.
energeticslessthanB∗ andPiforourmodel,andthuswefo- Nevertheless,duringthefirst∼1−5daysoftheevolutionwe
cusonthesetwoparametersformostoftherestofourstudy. generallyfindthattth ≪tcrossandthustheUHECRswillbe
subject to strong photo-hadronicinteractions. This does not
3.3. Photo-hadronicinteractions prevent iron UHECRs, but these estimates imply that lower
massnucleiwillalsonecessarilybepresent.Wesavedetailed
AfractionoftheX-rayscomingoutofthenebulathermal-
calculationsofthecompositionoftheUHECRsasafunction
ize with the ejecta. This generates a bath of thermal pho-
ofenergyforfuturework. Wedonotethatoncet & t ,
tons that can lead to photodisintegrationsof the iron nuclei th cross
itisstill possiblethatE & 1020eV andthusUHECRsfrom
when they attempt to pass through. To estimate when this
iron-likenucleiwillsurvive.
shouldoccur,welargelyfollowthemethodologypresentedin
Anotherpotentiallocationforstrongphoto-hadronicinter-
Koteraetal.(2015).
actionsiswithinthenebulaitself,whichisfullofnon-thermal
For ejecta with temperature T , the photon energy peaks
ej X-rays. Herethehighenergyoftheindividualphotonsmake
at roughly ǫ ≈ 2.8k T ≈ 2.4T eV, where T =
th B ej 4 4 it easier to exceed the Lorentz threshold. Using the total
T /104K (see Figure 2). In contrast, the threshold en-
ej energy in non-thermal X-rays, E , we thus estimate the
ergy of the photodisintegration process for iron-like nuclei nth
timescaleforinteractionwiththeseas
is ǫ ≈ 18.3A−0.21MeV (Pugetetal. 1976; Muraseetal.
A 56 −1
2n0u0cl9e)i,.wThheerreesAu5lt6in=gthAre/s5h6olddeLnootreesnttzhefaacttoomrfiocrmphaosstoodfistihne- tnth ∼ σǫAc43πERnt3h , (25)
tegrationis nth ej!
γ ≈ ǫA ≈7.7×106A−0.21T−1, (23) whereǫnthistheenergyofthephoton.Thus,athigherphoton
thres ǫ 56 4 energiesthistimescaleislongersincethepower-lawdistribu-
th
tionofphotonsisdecreasing. Thisspectrumcontinuesupto
oranenergyofE ≈ 4×1017eVforiron-likenuclei. Thus, roughlythethresholdforpaircreation∼2m c2 ∼MeV.
e
thisthresholdisalwayseasilymetinourscenario.
We summarize the main argumentsof this section in Fig-
BesidesexceedingthisLorentzfactor,thethephotodisinte-
ure 5, where we plot the relevanttimescales for an example
gration must be sufficiently fast in comparison to the cross-
model. This demonstratesthat where t < t photodis-
th cross
ing timescale (Muraseetal. 2014). Using a cross section
integrationsintheejecta arerelevantandthisprocessgener-
of σA ≈ 8×10−26A56cm2 for a Giant Dipole Resonance ally dominatesoverthe photodisintegrationswithin the neb-
(Pugetetal.1976;Muraseetal.2008),thistimescaleis
ula. ThislikelyleadstoamixedcompositionofUHECRsup
σ acT4 −1 toafewdays.Afterthistimescale,theUHECRscanleavethe
t ∼ A th ∼21A−1T−3s, (24) magnetarlessimpactedbytheenvironment.
th ǫ 56 5
(cid:18) th (cid:19)
7
4. GLOBALENERGETICS,RATES,AND Wenowshowthatindeedtheorder-of-magnitudeconsistency
CONSTRAINTS argumentsaboveagreenicelywithmorecarefulcalculations.
To understand how representative this value of f is re-
WeshowedinSection3thattheindividualencaulmagnetar cr
quiresaslightlymoredetailedmodeloftheUHECRproduc-
isapotentialsiteforUHECRgeneration.Wenowinvestigate
tion rate. To perform this, we first assume that the spec-
theexpectedenergeticsandratesassociatedwiththepopula-
trum of UHECRs accelerated at any given time follows a
tionofsuchsystemsandtheassociateduncertaintiestherein.
dN/dE ∝ 1/E scaling, similar to previous discussions
about pulsar magnetospheres (Arons 2003). This assumes
4.1. OrderofMagnitudeEstimates
a Goldreich-Julian injection (Goldreich&Julian 1969) and
TheenergygenerationrateofUHECRsabove1019.3eVis has mainly been chosen to illustrate the energy constraints
estimatedtobeintherange≈1043.5−1044ergMpc−3yr−1 on our model. In general, the spectrum could be different
(Waxman 1995; Berezinskyetal. 2006; Katzetal. 2009; depending on the acceleration type (for example, shock ac-
Murase&Takami 2009). In detail, such estimates depend celeration,Lemoineetal.2015),andinaddition,otherfactor
ontheassumedenergyspectrumandvolumeoverwhichthe suchasthedetailsofthesourcepopulationandthemagnetic
UHECRs can be observed, which may even be impacted by field within the nebula may also soften the spectrum (e.g.,
the (uncertain) composition. Nevertheless, such rough rates Fangetal.2013)Next,weassumethatafractionη . 1of
cr
are helpful for constraining potential mechanisms. In the themagnetarspindowncanbeusedforUHECRacceleration.
followingdiscussionswe use 1043.75ergMpc−3yr−1 as the Energyconservationthenrequires
characteristicratetocomparetoforgeneratingUHECRs.
Forthemagnetarmodel,themaximumenergyavailableper Econf dN˙
EdE ≈η L , (27)
eventisErot,butwedonotexpectthisentireenergyreservoir dE cr sd
tobeputintoUHECRs. Assumingafractionf doesmakeit ZEmin
cr
intoUHECRs,therequiredrate,estimatedfromEquation(1) where Emin is the smallest energy for accelerated particles
andthevolumetricestimateabove,issimply andcanbeignoredaslongasEmin ≪ Econf. Equation(27)
canbesolvedtofind
P 2
R≈1.1×10−8I4−51 2mis fc−r1Mpc−3yr−1. (26) dN˙ ≈ ηcrLsd 1. (28)
(cid:18) (cid:19) dE E E
conf
For AICs, it is helpful to compare to the Type Ia super-
ThisdescribestheUHECRsproducedwithinthenebula,but
novaratesincebothtypesofeventsareproducedfromWDs.
as we’ve discussed previously, they mightnot all escape the
The Lick Observatory Supernova Search found a rate of
nebula. This is especially true at the highest energies due
R = (3.01±0.062)×10−5Mpc−3yr−1 (Lietal. 2011),
Ia to synchrotron loses. To capture this physics, we only inte-
and thus, R/RIa ≈ 4 × 10−4fc−r1. This can be compared gratedN˙/dE uptoE whenE < E . Thetime-
totheAICratesthathavebeenproposedintheliterature,al- synch synch conf
dependentenergyoutputisthen
though these are known to be very uncertain. Using popu-
lationsynthesis,Yungelson&Livio(1998)findAICratesof η L (t), E >E
cr sd synch conf
8 × 10−7 − 8 × 10−5yr−1 for the Milky Way, depending Lcr(t)= η L (t) Esynch(t) , E <E (29)
onassumptionsaboutthecommon-envelopephaseandmass ( cr sd Econf(t) synch conf
transfer, which corresponds to volumetric rate of R ∼ (cid:16) (cid:17)
AIC Thisisthentimeintegratedovertheevolutionofthespinning
3×10−9 −6×10−7Mpc−3yr−1. Similar constraints are
down magnetar to get the total energy output in UHECRs.
alsoobtainedfromobservedabundancesofneutron-richiso-
This is summarized in Figure 6, where we plot contours of
topes (Hartmannetal. 1985; Fryeretal. 1999). This would
thefractionf of E thatgoesinto UHECRs. Thiscalcu-
givearelativerateR /R ∼1×10−4−2×10−2,which cr rot
AIC Ia lationuses η = 0.1, which is notdissimilar to the fraction
isconsistentwithourmodelforvaluesoff ∼0.01−0.1. cr
cr ofpulsarpowerfoundtoexperiencethefullpotentialdropby
TheotherscenarioformakingUHECRsfromencaulmag-
Chen&Beloborodov(2014),andtheseresultscanbesimply
netars we are considering is if some NS mergers produce
scaled to other values of η because the dependence is just
massive NSs rather than BHs. Indeed, there is now evi- linear. Thisshowsthatf cirsgreaterthan≈ 0.01overasig-
dence that some NSs could have masses well above 2M⊙ nificantfractionofthepacrrameterspace.
(vanKerkwijketal.2011;Romanietal.2012), whichfavors
AfurthercomparisonisshowninFigure7. Assumingthat
an equation of state that makes such a scenario more likely.
BothpopulationsynthesisandjusttheobservedGalacticNS agivenB∗ andPi arerepresentativeofthemajorityofmag-
netarsthatproduceUHECRs,weestimatewhatrateofevents
binariesfavoraratethatiscomparabletoorgreaterthanthe
areneededtoproducetheobservedUHECRrate, inunitsof
AIC rate. For example, Kimetal. (2005) estimate a Galac-
theTypeIasupernovarate. Asdiscussedabove,typicalAIC
tic rate of ∼ (0.1 − 3) × 10−4yr−1 or a volume rate of
andneutronstar mergerratesare. 0.01inthese units, thus
∼ 4 × 10−8 − 2 × 10−6Mpc−3yr−1. Again, this rate is thisscenariofavorsthelowerleftquadrantofFigure7.
consistent with what is needed for UHECRs for values of
f ∼ 0.01, taxing neitherthe NSmergerrate northe avail- 4.3. RateandCounterpartswithintheGZKvolume
cr
ableenergyreservoiroftheencaulmagnetarpopulation.
UHECRs propagating through intergalactic space interact
withthecosmicmicrowavebackground(CMB)orevenother
4.2. DetailedEnergyandRateConstraints
sourcesofextragalacticbackgroundlight(EBL),whichlimits
Theconsistencyargumentsaboveimplyafractionofrota- howfarwecandetectthem.ThisisknownastheGZKeffect
tionalenergyf ≈ 0.01goesintoUHECRs. Theseratesare (Greisen1966;Zatsepin&Kuz’min1966). Forbothprotons
cr
souncertain,however,itisreasonableto askwhethersucha andFe-likenuclei, thetypicalscale is similarand onthe or-
valuecanbeunderstoodfromindependentlinesofargument. der of D ∼ 170Mpc (Metzgeretal. 2011). Using the
GZK
8
astherateofeventswithintheGZKvolume. Suchadistance
scaleisinfactinterestingforotherastrophysicalinterests.For
example, this is similar to the horizonexpected for detected
NS mergers by ground-based interferometric detectors such
asAdvancedLIGO(Abadieetal.2010). Thiswilleithercon-
strainorprovideadirectmeasurementofNSmergingevents
withintheGZKvolume.
Inaddition,theapparentbolometricmagnitudeoftheelec-
tromagnetic transient of such an UHECR producing event
wouldbe≈15.5(Metzger&Piro2014)at200Mpc. Inprin-
ciple, these events could have been detected by wide-field,
transient surveys like the Palomar Transient Factory (PTF;
Rauetal. 2009) and the Panoramic Survey Telescope and
Rapid Response System (Pan-STARRS; Kaiseretal. 2002).
Unfortunately, given their particularly short timescales of
∼ 1day and blue color, it is possible they have so far been
missed. Also, note that the events best suited for producing
UHECRs will have smaller magnetic fields and correspond-
inglyfainterelectromagneticcounterparts.
In the future, this will change with surveys that
have particularly rapid cadences, such as the Zwicky
Transient Facility (ZTF; Lawetal. 2009), the All-
Sky Automated Survey for Supernovae (ASAS-SN;
Shappeeetal.2014),ortheLargeSynopticSurveyTelescope
FIG.6.—Thefractionfcr oftherotational energyErot agivenmagne- (LSSTScienceCollaborationetal. 2009, depending on the
tar(oroftheentirepopulationofmagnetarsgivenacharacteristicmagnetic
fieldandinitial spin)thatcanputintoUHECRswithE > 2×1019eV, finalcadence). Infact, suchaneventmayhavealreadybeen
assumingafraction,ηcr =0.1,ofthespindownenergygoesintoUHECR observedinGRB080503(Perleyetal.2009),whichshowed
acceleration. a rebrightening in both optical and X-rays. Thus, perhaps
another way to constrain the rate of these events is through
GRBs (although the X-ray/optical signature is expected to
bemoreisotropicthanthebeamedgamma-rays). Associated
radio, either from the plerion associated with the magnetar
(Piro&Kulkarni 2013) or interaction with the interstellar
medium(Nakar&Piran2011) mayalsoprovideconstraints.
Either way, it appears that these events may be observable
by independent means on multiple different fronts if they
areindeedmakingUHECRs(forexample,seeMetzgeretal.
2015; Horeshetal. 2016). Within the next ∼ 3−4yrs we
willhaveamuchbetterempiricalhandleontheratesofsuch
eventsandtheirroleinUHECRacceleration.
4.4. IntergalacticPropagation
UHECRs with largechargescan be significantlydeflected
by the intergalactic magnetic field, so it is a good check to
considerthiseffectinthecontextofourmodel. Asaparticle
withchargeZetravelsoveramagneticfieldwithcorrelation
length λ, it typically gets deflected by an angle α ∼ d/r ,
L
where r = E/ZeB is the Larmor radius and B is
L IGM IGM
theintergalacticmediummagneticfield.Travelingoveradis-
tanceofD,thetypicaltotaldeflectionwillbe
1/2
D λ
θ ∼ . (31)
λ r
FIG.7.—AssumingthatagivenB∗andPiarerepresentativeofmostfast- (cid:18) (cid:19) L
spinningmagnetarsgenerated,thecontoursshowtherateofeventsneededto Thisdeflectionneedstobelessthan∼1radwithintheGZK
explaintherateofUHECRsobservedinunitsoftheTypeIasupernovarate,
takentobe3.01×10−5Mpc−3yr−1(Lietal.2011). volume, otherwise the path length of an UHECR will be in-
creasedbytoomuch. ThiswoulddecreasetheeffectiveGZK
volumetothepointthatitmaycauseproblemswithgettinga
sufficientrateofevents. Thislimitsthemagneticfieldtobe
requiredratesestimatedinEquation(26),wethenestimate
−1/2
D
P 2 f −1 D 3 BIGM .8×10−9Z−1E20 GZK
RGZK ≈23I4−51(cid:18)2mis(cid:19) (cid:18)0.c0r1(cid:19) (cid:18)170GMZKpc(cid:19) yr−1, (cid:18)17λ0Mpc−(cid:19)1/2
(30) × G, (32)
1Mpc
(cid:18) (cid:19)
9
whereE =E/1020eV.
20
Another consideration is that the intergalactic magnetic
field should provide enough smoothing such that at least
one source within the GZK volume is providing UHECRs
at any given time, otherwise there would be large angular
anisotropiesobservedintheUHECRrate.Forthistobesatis-
fied,thediffusiontime,whichisroughlyτ ∼ θ2D/cshould
be longer than R−1 , which we derived in Equation (30)
GZK
above.Thisyields
−1 1/2
P f
B &7×10−14Z−1E I1/2 i cr
IGM 20 45 2ms 0.01
(cid:18) (cid:19) (cid:18) (cid:19)
D −5/2 λ −1/2
× GZK G, (33)
170Mpc 1Mpc
(cid:18) (cid:19) (cid:18) (cid:19)
asalowerlimitfortheintergalacticmagneticfield. Interest-
ingly,theseupperandlowerlimitsshowthatinprinciplemea-
suringthepropertiesoftheintergalacticmagneticfieldcould
helpconstrainthepropertiesofthemagnetarsgeneratingthe
UHECRs, albeit the wide range of the constraints probably
makethisdifficultinpractice.
4.5. GalacticPropagation
FIG.8.— Contours show the background flux of high-energy neutri-
One is finally left wondering, “What if an AIC occurred nos expected from Equation (38) in units of GeVcm−2s−1sr−1. The
rate R is assumed to be sufficient to produce UHECRs with a rate
intheMilkyWay?”Similartotheestimatesmadeabove,the of 1043.75ergMpc−3yr−1, and this assumes a volume of high-energy
propagationtimeacrossthediskoftheGalaxyis
neutrinos out to a distance of 4000Mpc. The IceCube upper limit
is resolved as function of energy, but is generally in the range of ∼
D 2 λ 10−8GeVcm−2s−1sr−1ormuchhigher.
τ ∼58Z2E−2
20 15kpc 0.1kpc
(cid:18) (cid:19) (cid:18) (cid:19) TheopticaldepthofUHECRstohadroninteractions,which
B 2 weestimatedinSection3.2inthecontextofpreventingUHE-
× 3×1I0S−M6G yr. (34) CRsfromescaping,is∼ tej/tcross. Thusweestimatetheef-
(cid:18) (cid:19) ficiencyofpionproductionas
ForZ = 26,thisimpliesτ ∼ 4×104yrandthusforheavy f ∼min(t /t ,1). (36)
π ej cross
nucleithediffusiontimeislongerthanthetimescalebetween
AICorNSmergereventsintheGalaxy. Soanon-zerofrac- Furthermore, we assume that 0.2 of the energy of a given
tion of UHECRs we measure today could have originated UHECR can be convertedto the energy of a pion, and 0.25
fromaneventintheMilkyWay. Thedeflectionangleis ofthepionenergythengoesintotheneutrino(aswasdonein
Fangetal. 2016). This givesan overallconversionfactor of
θ∼5×10−2ZE2−01 15Dkpc 1/2 fνIn∼te0g.r0a5tinfrgomovaergitvheeneUnHtirEeCURHtEoCaRneeuntreirngoy. spectrum, we
(cid:18) (cid:19) can derive the total energy of neutrinos that are generated.
λ 1/2 B Note that in this particular case, f can be taken outside of
× 0.1kpc 3×1I0S−M6G rad, (35) theenergyintegralsince weareusπinga simplisticmodelfor
(cid:18) (cid:19) (cid:18) (cid:19) the hadronic interactions, which may not be true in a more
which for Z = 26 this implies deflection angles of ∼ 50◦, detailed treatment. This greatly simplifies the evaluation of
whichisobviouslynotinthesmallangleapproximationfrom theintegral,andtotalneutrinoluminositycomesoutto
whichthisformulawasderived!Theconclusionisthateither
L (t)∼f f (t)η L (t). (37)
Z ∼1andthereisverylittlecontributionfromtheGalaxyor ν ν π cr sd
thatZ =26andalthoughthereissomecontributionfromthe Note that here we are using the entire luminosity that goes
Galaxy,associatingthemwithlocationsintheMilkyWaywill intoUHECRs,whichisrepresentedbythetermη L ,rather
cr sd
bedifficultduetotheirlargedeflectionsfromtheirorigins. than just the fraction that escape the magnetar environment,
which is given in Equation (29). This is because it is this
4.6. ConstraintsfromHigh-EnergyNeutrinos luminosity of UHECRs that is important for generating the
high-energy neutrinos. To relate this to the observed high-
HadronicinteractionsbetweentheUHECRsandtheejecta
energyneutrinoflux,weintegrateL overalltimeforasingle
surroundingthe magnetar will lead to charged pion produc- ν
eventandoveradistanceD,resultingin
tion,whichsubsequentlydecayintohigh-energyneutrinosvia
π± →e±+νe(ν¯e)+νµ+ν¯µ. Althoughadetailedstudyof DR ∞
thisprocessisoutsidethescopeofthispaper,wemakeasim- Fν = Lν(t)dt, (38)
4π
pleestimateoftheexpectedneutrinobackgroundtocheckfor Z0
consistency with constraints from the IceCube Observatory whereR is thevolumetricrate ofevents. We donotinclude
(TheIceCubeCollaboration2011;Aartsenetal.2013). additionalneutrinosproducedbysecondarynucleiandpions,
10
since our ejecta is so much less massive than in supernova- Using a simple model of the nebula and ejecta surrounding
relatedscenarios(Muraseetal.2009). the newly born magnetar, we considered both the accelera-
Thecalculationofthishigh-energyneutrinobackgroundis tion of UHECRs and their ability to escape their production
summarized in Figure 8. At each location in this plot we site, which each imprintfeatureson the emergentUHECRs.
assume that B∗ and Pi are representative of all UHECR- We also exploredthe requiredrates for this scenario as well
producing magnetars. Furthermore, we set R to be the rate as the propagationof these particlesthoughthe intergalactic
neededtomatchtheobservedUHECRrate(assummarizedin medium or even the interstellar medium, especially for the
Figure7). FinallyweuseD =4000Mpc,essentiallyassum- casewherethenucleiareheavy. Perhapsmostexcitingly,the
ingaconstantrateofUHECRproductionovertheentirehis- volumeoverwhichtheseUHECRscanpropagatetoEarthis
toryoftheUniverse. Amoredetailedtreatmentthatincludes similar to the distancesthat these eventscan be detectedvia
thetime-dependentcosmicstarformationrateandconnecting electromagneticandgravitationalwavesignaturessothatitis
thistotherateofencaulmagnetarbirthsovertimeisoutside likelythisscenariowillbetestedwithinthenext∼3−5yrs.
thescopeofthiswork. Lookingtothefuture,akeyissueforUHECRswillbecon-
TheIceCubeupperlimitsareresolvedasafunctionofen- clusivelymeasuringwhetherthecompositionisindeedheavy
ergyandreachaminimumof∼ 10−8GeVcm−2s−1sr−1 at and how it is changing as a function of particle energy. A
a neutrino energy of ∼ 1014eV. Furthermore, going up to heavy composition is difficult to reconcile with AGN and
energies of ∼ 1018GeV − 1020GeV, the upper limits are mostGRBscenarios,butaniron-richcompositionisnaturally
∼ 3 × 10−8 − 3 × 10−7GeVcm−2s−1sr−1. Comparing expected for a NS source such as young pulsars (Fangetal.
to Figure8, thisis aboveour expectedbackgroundrate over 2012) or the magnetars studied here. A protomagnetar jet
mostoftheparameterspace,andnotethatweareintegrating couldalsogiveheavynuclei(Metzgeretal.2011)andinfact
over neutrinosof all energiesand thus likely overestimating could reach elements such as zirconium, tellurium, or plat-
thepotentialbackground.Thereasonweareabletomeetthe inumatthehighestenergies,themeasurementofwhichwould
neutrinoconstraintsisthatthelowamountofejectanaturally bestrongsupportforthismodel.
gives t /t ≪ 1 over most of the time when UHECRs Animportantextrinsicdifferencebetweentheencaulmag-
ej cross
are produced in the greatest amounts. Thus, for the same netar scenario and most other proposed scenarios for UHE-
reason we are able to efficiently produce UHECRs, we are CRs is the galaxy environment. AIC and NS mergers are
alsoabletonotoverproducehigh-energyneutrinos. Wenote expected to occur in both old and young stellar environ-
thattheremayalsobephoto-hadronicinteractionswithinthe ments, and in fact this is well-known from short GRBs
nebulathatmayalsoproduceneutrinos(Lemoineetal.2015; (Fong&Berger2013). Thisisverydifferentthanyoungpul-
Fangetal. 2016). Since this process is stronger for protons sars or the long GRBs associated with protomagnetar jets
thanheavynuclei,andourmainfocushereisontheheavynu- thatwouldbeinyoung,starformingregions. Thusevidence
cleiasUHECRs, we save sucha calculationforfuturework thatUHECRsarebeingproducedinoldstellarenvironments
where the spectrum of UHECRs and the non-thermalradia- wouldbestrongsupportforthemodelpresentedhere.
tioniscomputedinmoredetail.
We thank Andrew MacFadyen for important discussions
5. DISCUSSIONANDCONCLUSIONS
whenweweredevelopinginitialideasforthiswork.Wethank
InthisstudyweconsideredtheproductionofUHECRsby John Beacom for suggesting a comparison with the high-
quicklyspinningmagnetarsfollowingtheAICofWDsorthe energyneutrinobackground. We alsothankJonathanArons,
mergerofNSs–processesthatsetupthegenericinitialcon- KumikoKotera,BrianMetzger,KohtaMurase,E.SterlPhin-
ditionswehavetermed“encaulbirth”asthenewlybornmag- ney,andToddThompsonfortheirhelpfulfeedbackonprevi-
netar hasa small shroudof ejecta in its immediate environs. ousdrafts.
REFERENCES
Aab,A.,Abreu,P.,Aglietta,M.,etal.2014a,Phys.Rev.D,90,122005 —.2013,JCAP,3,010
—.2014b,Phys.Rev.D,90,122006 Farrar,G.R.,&Gruzinov,A.2009,ApJ,693,329
Aartsen,M.G.,Abbasi,R.,Abdou,Y.,etal.2013,ApJ,779,132 Ferna´ndez,R.,&Metzger,B.D.2013,MNRAS,435,502
Abadie,J.,Abbott,B.P.,Abbott,R.,etal.2010,ClassicalandQuantum Fong,W.,&Berger,E.2013,ApJ,776,18
Gravity,27,173001 Fryer,C.,Benz,W.,Herant,M.,&Colgate,S.A.1999,ApJ,516,892
Abbasi,R.U.,Abu-Zayyad,T.,Allen,M.,etal.2008,PhysicalReview Giacomazzo,B.,&Perna,R.2013,ApJ,771,L26
Letters,100,101101 Goldreich,P.,&Julian,W.H.1969,ApJ,157,869
Abraham,J.,Abreu,P.,Aglietta,M.,etal.2008,PhysicalReviewLetters, Granot,J.,Komissarov,S.S.,&Spitkovsky,A.2011,MNRAS,411,1323
101,061101 Greisen,K.1966,PhysicalReviewLetters,16,748
Arons,J.2003,ApJ,589,871 Hartmann,D.,Woosley,S.E.,&ElEid,M.F.1985,ApJ,297,837
Arons,J.2008,inAmericanInstituteofPhysicsConferenceSeries,Vol. Hebeler,K.,Lattimer,J.M.,Pethick,C.J.,&Schwenk,A.2013,ApJ,773,
983,40YearsofPulsars:MillisecondPulsars,MagnetarsandMore,ed. 11
C.Bassa,Z.Wang,A.Cumming,&V.M.Kaspi,200–206 Horesh,A.,Hotokezaka,K.,Piran,T.,Nakar,E.,&Hancock,P.2016,
Berezinsky,V.,Gazizov,A.,&Grigorieva,S.2006,Phys.Rev.D,74, ArXive-prints
043005 Inoue,S.,Sigl,G.,Miniati,F.,&Armengaud,E.2007,ArXivAstrophysics
Canal,R.,&Schatzman,E.1976,A&A,46,229 e-prints
Chakraborti,S.,Ray,A.,Soderberg,A.M.,Loeb,A.,&Chandra,P.2011, Kaiser,N.,Aussel,H.,Burke,B.E.,etal.2002,inSocietyofPhoto-Optical
NatureCommunications,2,175 InstrumentationEngineers(SPIE)ConferenceSeries,Vol.4836,Survey
Chen,A.Y.,&Beloborodov,A.M.2014,ApJ,795,L22 andOtherTelescopeTechnologiesandDiscoveries,ed.J.A.Tyson&
Dermer,C.2010,ArXive-prints S.Wolff,154–164
Dermer,C.D.,Razzaque,S.,Finke,J.D.,&Atoyan,A.2009,NewJournal Kaplan,J.D.,Ott,C.D.,O’Connor,E.P.,etal.2014,ApJ,790,19
ofPhysics,11,065016 Katz,B.,Budnik,R.,&Waxman,E.2009,JCAP,3,20
Dessart,L.,Ott,C.D.,Burrows,A.,Rosswog,S.,&Livne,E.2009,ApJ, Kim,C.,Kalogera,V.,Lorimer,D.R.,Ihm,M.,&Belczynski,K.2005,in
690,1681 AstronomicalSocietyofthePacificConferenceSeries,Vol.328,Binary
Duncan,R.C.,&Thompson,C.1992,ApJ,392,L9 RadioPulsars,ed.F.A.Rasio&I.H.Stairs,83
Fang,K.,Kotera,K.,Murase,K.,&Olinto,A.V.2016,JCAP,4,010 Kiziltan,B.,Kottas,A.,DeYoreo,M.,&Thorsett,S.E.2013,ApJ,778,66
Fang,K.,Kotera,K.,&Olinto,A.V.2012,ApJ,750,118 Kotera,K.,Allard,D.,Murase,K.,etal.2009,ApJ,707,370