Table Of ContentDRAFTVERSIONFEBRUARY4,2008
PreprinttypesetusingLATEXstyleemulateapjv.10/09/06
PROTO-NEUTRONSTARWINDSWITHMAGNETICFIELDSANDROTATION
BRIAND.METZGER1,3,TODDA.THOMPSON2,4,ANDELIOTQUATAERT1
DraftversionFebruary4,2008
ABSTRACT
We solve the one-dimensionalneutrino-heated non-relativistic magnetohydrodynamic(MHD) wind prob-
7 lem for conditions that range from slowly rotating (spin period P&10ms) protoneutron stars (PNSs) with
0 surface field strengths typical of radio pulsars (B.1013G), to "proto-magnetars"with B 1014- 1015G in
≈
0 theirhypothesizedrapidlyrotatinginitialstates(P 1ms). We usetherelativisticaxisymmetricsimulations
2 of Bucciantini et al. (2006) to map our split-mono≈poleresults onto a more physical dipole geometry and to
estimatethespindownofPNSswhentheirwindsarerelativistic. Wethenquantifytheeffectsofrotationand
n
a magneticfieldsonthemassloss,energyloss,andthermodynamicstructureofPNSwinds. Thelatterispartic-
J ularlyimportantinassessingPNSwindsastheastrophysicalsiteforther-process. Wedescribetheevolution
of PNS winds throughthe Kelvin-Helmholtzcoolingepoch, emphasizingthe transition between (1) thermal
4
neutrino-driven,(2)non-relativisticmagnetically-dominated,and(3)relativisticmagnetically-dominatedout-
2 flows. Inthelastofthesestages,thespindownisenhancedrelativetothecanonicalforce-freeratebecauseof
v additionalopenmagneticfluxcausedbyneutrino-drivenmassloss. Wefindthatproto-magnetarswithP 1
2 ms and B&1015 G drive relativistic winds with luminosities, energies, and Lorentz factors (magnetizat≈ion
8 σ 0.1- 1000)consistentwiththoserequiredtoproducelongdurationgamma-rayburstsandhyper-energetic
6 su∼pernovae (SNe). A significant fraction of the rotational energy may be extracted in only a few seconds,
8 sufficientlyrapidlytoaltertheasymptoticenergyoftheSNremnant,itsmorphology,and,potentially,itsnu-
0 cleosynthetic yield. We find that winds from PNSs with somewhat more modest rotation periods ( 2- 10
6 ≈
ms) andwith magnetar-strengthfieldsproduceconditionssignificantlymorefavorableforthe r-processthan
0
windsfromslowlyrotating,non-magnetizedPNSs. Lastly,wearguethatenergyandmomentumdepositionby
/
h convectively-excitedwavesmaybeimportantinPNSwinds. Weshowthatthisfurtherincreasesthelikelihood
p ofsuccessfulr-process,relativelyindependentofthePNSrotationrateandmagneticfieldstrength.
-
o Subjectheadings:stars: neutron—stars: winds,outflows—supernovae: general—gammarays: bursts—
r stars: magneticfields—nuclearreactions,nucleosynthesis,abundances
t
s
a
v: 1. INTRODUCTION ingepochhasbegun,neutrino-drivenwindsfromnon-rotating
i non-magnetic PNSs are unlikely to be energetically impor-
X Ontimescales. 1sfollowingthecorecollapseofa mas- tant on the scale of the kinetic energy of the accompany-
r sive star, neutrino emission from the resulting hot, delep- ing SN (ESN 1051ergs). However, in the presence of a
a tonizing protoneutronstar (PNS) may play an essential role sufficiently str≈ong global magnetic field the dynamics of a
in launching the supernova (SN) shock (Herant et al. 1994; PNS’sneutrino-heatedoutflowaresignificantlyaltered(e.g.,
Burrows, Hayes, & Fryxell 1995; Janka & Müller 1995) or Thompson2003a,b). This point is germanebecause 10%
in generatinglarge-scale anisotropiesthroughhydrodynami- (Kouveliotou et al. 1994; van Paradijs et al. 1995; L∼yne et
calinstabilities(Blondinetal.2003;Schecketal.2006;Bur- al.1998)ofyoungGalacticneutronstarspossesssignificantly
rows et al. 2006a,b). Independent of how the explosion is strongersurfacemagneticfields( 1014- 1015 G)thanthose
initiated at early times, a small fraction of the PNS’s cool- usually inferred from pulsar spin∼-down estimates (“magne-
ing neutrino emission continues heating the surface layers tars”; for a recent review see Woods & Thompson 2004).
of the PNS, driving a persistent thermal wind into the cav- While the precise origin of these large field strengths is un-
ity evacuated by the rapidly-expanding SN shock (Duncan, certain,ithasbeenarguedthattheiramplificationoccursvia
Shapiro,&Wasserman1986;Woosleyetal.1994);thispost- adynamoduringτ (Duncan&Thompson1992;Thompson
KH
explosionneutrino-drivenmass loss persists for the duration &Duncan1993).Theefficiencyofthedynamoisdetermined
ofthePNS’s Kelvin-Helmholtzcoolingepoch,whichlastsa inpartbythecore’sinitialrotationrateΩ =2π/P,andana-
time τKH 10- 100 s (Burrows & Lattimer 1986; Pons et lyticargumentssuggestthattheformation0ofglobal0magnetar-
∼
al.1999). strengthfieldsmightrequireP 1msrotationatbirth.Such
0
∼
Once the SN shock has been launched and the PNS cool- rapidrotationwouldalsoalterthedynamicsofthePNSwind
andprovideareservoirofrotationalenergysignificantonthe
1Astronomy Department and Theoretical Astrophysics Center, 601 scaleoftheaccompanyingSNexplosion:
Campbell Hall, Berkeley, CA 94720; [email protected], E 2 1052M R2 P- 2ergs, (1)
[email protected] rot≃ × 1.4 10 ms
2DepartmentofAstrophysicalSciences,PeytonHall-IvyLane,Princeton where M1.4 is the PNS mass in units of 1.4 M⊙, R10 is the
University,Princeton,NJ08544;[email protected] radiusofthePNSinunitsof10km,andP istheinitialPNS
ms
3Department of Physics, 366 LeConte Hall, University of California, rotationperiodinms.
Berkeley,CA94720
4LymanSpitzerJr.Fellow Previous authors have suggested that if magnetars are in-
2
deed born rapidly rotating, their rotational energy could be is so high (&5 MeV) and the radius of the PNS is so large
efficientlyextractedthroughamagnetizedwind(Usov1992; (R 20- 50km)that,evenforamagnetar-strengthfield,the
ν
∼
Thompson1994;Wheeleretal.2000;Thompson,Chang,& post-explosionoutflowislikelytobepurelythermally-driven.
Quataert 2004, henceforth TCQ). Indeed, a proto-magnetar
(2) By t 1 s the SN shock has propagated to well out-
wind’s energetics, timescale, and potential for highly rel- ∼
side the sonic point of the PNS wind. The PNS will con-
ativistic outflow resemble those of the central engine re-
tractandcooltoapointatwhich,ifthesurfacefieldissuffi-
quired to power long-duration gamma-ray bursts (LGRBs).
cientlystrong,theoutflowbecomesmagneticallydominated.
TCQ argue that an accurate description of magnetized PNS
The strong magnetic field enhances angular momentum and
spin-down must include the effects of neutrino-driven mass
rotational energy loss by forcing outgoing fluid elements to
loss. The significant mass loss accompanying the Kelvin-
effectively corotate with the PNS surface out to the Alfvén
Helmholtz epoch may open the otherwise closed magneto-
radius (R ) at several stellar radii, in analogy with classic
sphere into a “split monopole”-like structure, enhancing the A
workonnon-relativisticstellarwindsandthesolarwind(e.g.,
earlyspin-downrateof the PNS. Inaddition,the massload-
Schatzman1962,Weber&Davis1967,andMestel1968).For
ingofaPNSwind,andhenceitspotentialasymptoticLorentz
finacgtothreΓ,Kieslvlairng-eHlyelcmohnotrlotzlleedpobcyh.theThniesu“trbinaroylounm-lionaodsiintyg”duisr-- s(τuJfficiΩen/tΩl˙y)rcaapnidblye rcootmatpinagraPblNeStso,τtKhHe,sipminp-ldyoinwgnthtiamtemscuaclhe
≡
oftherotationalenergyofthePNScanbeextractedinanon-
sue is particularly important in the LGRB context, where
Γ 10- 1000aretypicallyinferred. relativistic, but magnetically-dominatedwind. The neutrino
∼ luminosity at these relatively early times is still large (e.g.,
Becausemagnetarbirthsarerelativelyfrequent,mostcan- L 1052 ergs s- 1; see Pons et al. 1999, Fig. 14), and, for
notproduceclassicalLGRBs,whichonlyoccurin 0.1- 1% suνffi∼cientlyrapidrotation,masslossissignificantlyenhanced
∼
ofmassivestellardeaths(e.g.,Paczynski2001,Podsiadlowski
bycentrifugalflinging(TCQ).Theoutflowduringthisphase
et al. 2004, Piran 2005); however, a more common obser-
will be collimated about the PNS rotation axis by magnetic
vational signature of magnetar birth may include less ener-
stresses(e.g.,Bucciantinietal.2006;hereafterB06).
geticormildlyrelativisticevents,whichcouldbeobservable
as X-ray transients or unusual SNe. Under some circum- (3)AsthePNScontinuestocool,theluminosityandmass-
stances,theasymmetricenergyinjectionfromproto-magnetar loadingdecreasetothepointatwhichcorotationissustained
winds could produceglobalanisotropiesin the SN remnant, out to nearly the light cylinder (RL ≡2πc/P≃48Pm- 1s km).
as has been detected through polarization measurements of Thespin-downratethenbecomesapproximatelyindependent
˙
someSNe(e.g.,Wangetal.2001,2003). Inaddition,ifsig- ofthemasslossrateMandtheflowbecomesrelativistic,ob-
nificantrotationalenergycanbeextractedsufficientlyrapidly taininghighmagnetizationσ Φ2Ω2/M˙c3, whereΦ isthe
≡ B B
following the launch of the SN shock, the nucleosynthetic totalopenmagneticfluxper4πsteradian(Michel1969). Al-
yield of the SN could be altered (TCQ), which could ex- thoughanultra-relativistic,pulsar-likewindisinevitablesoon
˙
plain nickel-rich, hyper-energetic SNe such as SN 1998bw after τ because M abates as the neutrino luminosity van-
KH
(Galamaetal.1998)orSN2003dh(Hjorthetal.2003;Stanek ishes, relativistic outflow can begin prior to the end of the
etal.2003).Lastly,LGRBsoccurringwithoutassociatedSNe Kelvin-Helmholtz phase. Because this outflow is accompa-
(e.g.,GRBs060505,060614;Fynboetal.2006;Gal-Yamet niedby significantmassloss, it is generallyonlymildly rel-
al. 2006; Della Valle et al. 2006) may be accommodated if ativistic. Althoughtheenergyextractedviarelativisticwinds
aproto-magnetarresultsfromtheaccretion-inducedcollapse isprimarilyconcentratedatlowlatitudes(B06),theconfining
(AIC)ofawhitedwarf(see§4.4). pressureof the overlying,exploding,stellar progenitor(e.g.,
Wheeler et. al 2000; Uzdensky & MacFadyen 2006) or the
Quite apart from the possible impact of PNS winds on
walls of the collimated cavity carved by the preceding non-
the surrounding SN shock, PNS winds themselves have of-
relativistic phase may channel the relativistic outflow into a
ten been considered a promising site for the production of
bipolar,jet-likestructure.
r-process nuclides. However, the conditionsnecessary for a
successfulthirdpeak(A 195)r-processhavenotbeenreal- Theprimaryfocusofthispaperistodelineatethemagnetic
≈
izedindetailedstudiesofnon-rotating,non-magnetizedPNS field strengths and rotation rates requiredto significantly al-
winds (e.g., Qian & Woosley 1996; Cardall & Fuller 1997; ter the characteristics of early PNS evolution by solving the
Otsuki et al. 2000; Thompson et al. 2001). Given the, as one-dimensional(1D)non-relativisticMHD,neutrino-heated
yet,unidentifiedsiteofGalacticr-processenrichmentandthe wind problemfor conditionsthat range fromnormalpulsars
relatively large birthrate of magnetars, it is essential to con- to proto-magnetars. In particular, we quantitatively explore
sider what effects a strong magnetic field and rapid rotation the transition that occurs between stages (1) and (2) above;
might have on nucleosynthesis in PNS winds. Conversely, with these results we analyze some of the immediate conse-
if the nucleosynthetic yield from rotating, magnetized PNS quencesofneutronstarbirth.Wedeferadetailedstudyofthe
windscouldbewell-determinedtheoretically,thebirthperiod transition between stages (2) and (3) to future work, but we
andmagneticfielddistributionofneutronstarscouldperhaps doexplicitlyaddresstheparameterspaceoftheσ=1bound-
be constrained from observations of heavy elemental abun- ary. Relativistic, mass-loaded MHD winds have been stud-
dances. ied recently by B06 in two dimensions for both a monopo-
larandaligned-dipolarfieldstructure,assuminganadiabatic
1.1. StagesofPNSEvolution equationofstate. WhiletheworkofB06iscriticaltounder-
standing the multi-dimensionalcharacter of PNS winds (for
A sufficiently-magnetizedPNSwind goesthroughat least
instance, the degree of collimation and the fraction of open
threedistinctstages ofevolutionfollowingthe launchof the
magneticflux)itdoesnotaddressthe neutrinomicrophysics
SNshock(t=0):
necessary for a directapplication to PNS environments; this
(1) Duringthe earliest phase the PNS surface temperature work and that of B06 are thus complementary in studying
3
PNS spin-downin the presenceof rapid rotationand a large thisprescriptionconservesthemagneticflux4πΦ 4πr2B .
B r
≡
magneticfield. B06showthatthespin-downindipolesimulationscanbeex-
pressed in terms of an equivalentmonopolefield, which de-
pends on the fraction of open magnetic flux. As discussed
1.2. ThisPaper
further in 4.1, we can therefore relate our monopole spin-
§
In 2 we enumeratethe equationsofMHD ( 2.1), discuss downcalculationstothe morerealistic dipolesimulationsof
there§levantmicrophysics( 2.2),andelaborateo§nournumer- B06.
§
ical methods ( 2.3). Section 3 presents the results of our Insteadystate,manipulationofequations(4)and(5)gives
§
calculations and examines the regimes of magnetized PNS theconservedspecificangularmomentum and“theconse-
wind evolution. Section 4 discusses the implications of this quenceofinduction” (e.g.,Lamers&CasLsinelli1999):
work,examiningthetime-evolutionofacooling,magnetized I
rB B
PNS( 4.1),weighingtheimplicationsforLGRBsandhyper- = + =rv - r φ (7)
energe§ticSNe ( 4.2), andconsideringthe viabilityfor third- L Lgas Lmag φ 4πρv
r
§
peak r-process nucleosynthesisin magnetized, rotating PNS =r(v B - v B ) (8)
winds( 4.3).In 4.3.1weconsidertheeffectsthatwaveheat- I φ r r φ
inghave§onther§-processinPNSwindsandin 4.4webriefly The total rate of angular momentum loss from the PNS is
§ ˙ ˙
discuss other contexts in which our calculations may be ap- thereforeJ= M.
plicable, including the accretion-induced collapse of white L
Forconditionsofinterest,photonsaretrappedandadvected
dwarfs, “collapsars,” and merging neutron star binaries. Fi-
withthewind,providingnosignificantenergytransportonthe
nally,section5summarizestheconclusionsofourwork.
timescalesofinterest(Duncan,Shapiro,&Wasserman1986).
Instead,thePNSevolutioniscontrolledbyitsneutrinolumi-
2. PNSWINDMODEL nosity L , which provides heating q˙+ (per unit mass) above
ν ν
thePNSsurface.Includingnetneutrinoheating(q˙ =q˙+- q˙- ;
2.1. MHDEquationsandConservedQuantities ν ν ν
see 2.2)providesasourceterminthewindentropyequation:
§
Making the simplifications of Weber & Davis (1967), we dS
restrictallphysicalquantitiestobesolelyfunctionsofradius T =q˙ν, (9)
dt
r andtimet, andconfineouranalysistotheequatorialplane
sothatthemagneticfieldB= [Br,Bφ]andfluidvelocityv= whered/dt≡∂/∂t+vr(∂/∂r),Sisthewindentropyperunit
[v ,v ]havenoθcomponents.WeemployNewtoniangravity mass, and T is the wind temperature. The asymptotic wind
forraφPNSofmassM actingongasofdensityρ. Wealsoas- entropySa quantifiesthetotalnetheatingaparcelofgasex-
sumethattheplasmaisaperfectconductorwithanisotropic periencesasitiscarriedoutbythePNSwind. Asdiscussed
thermal pressure P. Under these restrictions the time evolu- further in 2.2, when applying equation (9) we assume that
§
tionequationsofnon-relativisticMHDare thewind’scomposition(electronfraction)remainsconstant.
With net heating the Bernoulli integral is not constant
∂ρ 1 ∂
=- (r2ρv ) (2) withradius(insteadystate); insteaditreceBivesanintegrated
∂t r2∂r r contributionfromq˙ :
ν
∂v v2 ∂v 1∂P GM 1 B2 ∂B r
∂tr = rφ - vr ∂rr - ρ ∂r - r2 - 4πρ" rφ +Bφ ∂rφ# (3) M˙∆B=M˙{B(r)- B(r0)}= ρq˙ν∆Ωr′2dr′, (10)
Zr0
∂v 1 B ∂ ∂ where
φ = r (rB )- v (rv ) (4) 1 GM
∂t r 4πρ∂r φ r∂r φ (v2+v2)+h- + Ω, (11)
(cid:20) (cid:21) B≡ 2 r φ r Lmag
∂Bφ = 1 ∂ (r[v B - v B ]) (5) h e+P/ρ is the specific enthalpy, e is the specific internal
∂t r∂r r φ φ r en≡ergy, Ω is the specific magnetic energy, and Ω is the
mag
L
Wehaveneglectedneutrinoradiationpressureinequation(3) stellarrotationrate.
becausetheneutrinoluminosityisalwaysbelowtheneutrino ToassesstheimpactofthePNSwindonitssurroundings,
Eddingtonlimit. Insteadystate,equation(2)givesaradially- itisusefultodefineη,theratiooftherotationalpowerlostby
conservedmassflux: thePNS,E˙ =ΩJ˙,totheasymptoticwindpowerE˙a= aM˙:
rot
M˙ =∆Ωr2ρvr, (6) ΩJ˙ Ω B
where∆Ωistheopeningsolidangleofthewind. Whilethe η≡ E˙a = La , (12)
evolutionequationsconsideredareformallyvalidonlyinthe B
˙ where a is the Bernoulli integral evaluated at large radii.
equatorial plane, quoted values for M will be normalized to B
Magnetized winds are typically assumed to have η 1, but
∆Ω=4π, as if the solutions were valid at all latitudes. As ≈
we find that in some cases η 1 becausemuch of the rota-
discussed in 4.1, this normalization is an overestimate be- ≫
tional energy is used to unbind the wind from the PNS (see
§
causethe PNS’s closedmagneticfluxpreventsmassoutflow
§3.2).
˙
fromneartheequatorandcentrifugalflingingconcentratesM
atrelativelylowlatitudes(B06).
2.2. Microphysics
We take the radial magnetic field structure to be that of a
“split monopole”: B = B (R /r)2, where R is the radius ThelocalheatingandcoolingratesrelevanttotheKelvin-
r ν ν ν
of the PNS and B is the monopole surface field strength; Helmholtz coolingphase have been extensivelyevaluatedin
ν
4
effortstoquantifyunmagnetized,slowly-rotatingPNSwinds heatingandmomentumdepositionin 4.3.1.
§
as an astrophysical site for r-process nucleosynthesis (e.g.,
In this work we include gravitational redshifts, radial
Qian&Woosley1996,Thompsonetal.2001;hereafterT01).
Doppler shifts, and modifications to the “effective solid an-
For our calculations we adopt the heating and cooling rates
gle” (and hence local neutrino flux) presented by the neu-
used in the analytic work of Qian & Woosley (1996, here-
trinosphere in the curved spacetime. The latter effect is de-
after QW) for ν/ν¯ annihilation and for the charged-current
scribed in Salmonson & Wilson (1999), while the Doppler
processes ν +n p+e- and ν¯ +p n+e+. For heating
e ↔ e ↔ and redshifts can be combined into the simple, approximate
and cooling from inelastic neutrino-leptonscattering we use
prescriptionrelevantforallneutrinospecies:
theratesofT01. Thedominantcontributionstothenetheat-
ingratearethecharged-currentprocesses,butscatteringand hǫνni=(φZφD)n+3hǫνni|r=Rν, (13)
annihilation become more significant as the entropy of the
where
wind increases and the thermal pressure becomes radiation- φ α(R )/α(r), φ γ(1- v /c), (14)
Z ν D r
dominated. The scattering and charged-current rates effec- ≡ ≡
tivelyvanishonceT 0.5MeVbecausethee- /e+ pairsan- α(r) 1- (2GM/c2r), γ- 1 1- (v2+v2)/c2, and we
nihilate and the nucl→eons combine into α particles. We ar- ≡ ≡ r φ
tificially take this cutoff into account by setting q˙ = 0 for haveasspumedthatatradiiwherethqeequatorialflowbecomes
ν mildly relativistic, typical neutrinos will primarily be mov-
T <0.5MeV.
ing radially. We emphasize that while we include neutrino
Because the charged-currentinteractions modify the neu- gravitational redshifts in calculating heating rates we calcu-
tronabundanceoftheoutflowtheelectronfractionY should late the wind dynamicsin Newtonian gravity. Includingthe
e
beevolvedinadditiontothetemperatureandpressure. How- effects of the deeper general-relativistic (GR) potential of a
˙
ever,thedynamicsofthewindarenotsensitivetotheprecise SchwarzschildmetriclowersM andincreasesSa,Ya,andthe
e
profileofYeandthus,forsimplicity,wetakeYetobefixedata asymptotic wind speed va (Fuller & Qian 1996; Cardall &
reasonableasymptoticvalueatallradii:Ye=Yea=0.4.Thisis Fuller1997).
agoodapproximationgivenhowrapidlyinradiusYa obtains
e Ourmodel’sequationofstate(EOS)includescontributions
in non-rotating,unmagnetizedcalculations(see T01, Fig. 7)
fromphotonradiation,idealnucleons,andrelativistic,degen-
andhowrelativelyweaklythetotalheatingandcoolingrates
erate electronsandpositrons. Non-relativisticnucleonsgen-
dependonY forY .0.5.
e e erally dominate the EOS near the PNS surface, but within
As in T01 we calculate local neutrino fluxes by consider- a density scale heightaboveR the flow becomesradiation-
ν
ing a single (for all neutrino species) sharp, thermal neutri- dominated. Aswiththeheating/coolingrates,highmagnetic
nosphere at R . To the desired accuracy of our calculations fieldeffectsontheEOSareignored.
ν
thisapproximationis generallygood, evenwhenthe density
scale height(and hence nucleon-absorptionoptical depth) is 2.3. NumericalMethod
extendedbyrapidrotation. Forthispaperweindexstagesof
the PNS thermalevolutionin termsof the anti-electronneu- Inthe steady-state Weber-Daviswind, three criticalpoints
trino luminosity Lν¯e. We scale all other neutrino luminosi- occurintheradialmomentumequationatradiiwheretheout-
ties (Lνe,Lνµ,Lν¯µ,Lντ, and Lν¯τ) as in TCQ: Lνe =Lν¯e/1.3= flow velocity matches the local phase speed of infi˙nitesimal
1n.e0u8trLiνnµo,swpheecrieesµ. dNenootetestheaatcthhoefttohtealotnheeurtfroinuornluemutirninoos/iatyntiis- flu(ideqds.is[t6u]r,ba[7n]c,esa.ndT[h1e1]s)teaardeyfi-sxteadtebeyigthenevraelquueisreMm,eLn,t tahnadt
thenLν ≃4.6Lν¯e. FollowingT01,allfirstenergymomentsat Bthe solution pass smoothly through the slow-magnetosonic,
the neutrinosphere( ǫ E2 / E , where E is the neu- Alfvénic, and fast-magnetosonic points. Physically, we
trinoenergy)wereschaνleid≡whithνliumhinνoisityas ǫ ν L1/4,an- choosethese solutionsover sub-magnetosonic“breezes” be-
ν ν
h i∝ causeweassumetheSNshockorfallbackpressuresatlarge
choring{hǫνei,hǫν¯ei,hǫνµi}at{11,14,23}MeVforLν¯e,51=8, radiiare insufficient to stifle the strong ram-pressuresof the
owfh1er0e51Leν¯er,g51s iss- 1t.heHaingthi-eerleecnterrognynmeuotmrinenotslunmeicneosssiatryyinfournthites wind.However,fallbackatearlytimesisnotwell-understood
becauseitdependssensitivelyonthemechanismforlaunch-
heating calculations( ǫn , ǫn , etc.) are related to the first
h νei h ν¯ei ingtheSNshock(Chevalier1989;Woosley&Weaver1995)
through appropriate integrals over the assumed Fermi-Dirac
and thus this issue deserves further attention. Although the
surfacedistribution. Weshouldnotethattherelationshipbe-
PNS radius, rotation rate, and neutrino luminosity evolve in
tween the neutrinoluminosityand meanneutrinoenergywe
time, for realistic wind conditionsthe timescale requiredfor
haveassumed,whileareasonableapproximation,islikelyto
any of the MHD wavemodesto traverse all critical pointsis
be more complicated. For example, Pons et al. (1999) find
alwaysmuchshorterthanthetimescaleoverwhichthewind
thatthemeanenergyisroughlyconstantforthefirst 10sof
cooling,despitethefactthattheneutrinoluminosityd∼ecreases characteristicsappreciablychange(e.g.,τKHorthespin-down
timescaleτ ). Forthisreasonatime-seriesofsteady-stateso-
monotonically(seetheirFig.18). J
lutions is generally sufficient to accurately model the wind
Theneutrinoheatingandcoolingratesdiscussedabovewill during all phases of the PNS evolution. However, precisely
be modified by the presence of magnetar-strength fields in because all physical solutions must pass through each criti-
theheatingregionduetoquantumeffectsrestrictingtheelec- calpoint,inthetime-independentformulationofthisproblem
tron(positron)phase space (Lai & Qian 1998; Duan & Qian boundaryconditionsmust be placed on the wind solution at
2004).WeneglecttheeffectsthathighBhaveonq˙ anddefer theselocations.Toavoidthisnumericallycomplicatedsingu-
ν
study of these effectsto futurework. In addition, for strong larity structure, we have instead solved the more complete,
surface magnetic field strengths, heating via the dissipation time-dependent version of the problem using the 6th order
of convectively-excitedMHD waves may become important space/3rd order time, “inhomogeneous”2N-RK3 scheme of
(Suzuki&Nagataki2005).Weassesstheimportanceofwave Brandenburg(2001).
5
Ourcodeevolvesthevariables(ρ,T,v ,v ,B ). Thevalue artificiallyplacedonthegridwerecomparedinseveralcases;
r φ φ
ofthePNSmassM,neutrinosphereradiusR ,magneticflux wefoundthatalthoughthevelocitystructurechangesatradii
ν
Φ =B R2, stellar rotation rate Ω, and neutrino luminosity far outside the Alfv´en radius, our imposed boundary condi-
B ν ν
L aretheparametersthatuniquelyidentifyawindsolution. tionhadlittleeffectontheeigenvaluesoftheproblemandthe
ν
We use M = 1.4 M and R =10 km in all of our calcula- correctasymptoticspeedwasobtained(albeitprematurelyin
⊙ ν
tions. Becauseourcodeisintrinsicallynon-conservative,we radius).Sincetheeigenvaluesuniquelydeterminethesteady-
˙
use the constancy of M, , and and equation (10)’s con- state solution, this technique, when necessary, was a useful
straint on as independeLntchecIks on the code’s numerical expedienttoobtainthedesiredtransmagnetosonicsolution.
B
accuracy.Fornumericalstability,anartificialviscosityofthe
formν 2isincludedintheevolutionofeachvariable,where
∇ 3. RESULTS
ν is an appropriately-scaledkinematic viscosity (e.g., Bran-
denburg2001). Figure1summarizesthephysicalregimesofPNSwindsas
We chose the location of the outer boundary, generally at afunctionofLν¯e androtationalperiodP forarepresentative
r 1000km,asacompromisebetweentherun-timetoreach strongly magnetized PNS: Bν =2.5 1014 G, M =1.4 M⊙,
st≈eady-state and the desire to minimize the effects of artifi- and Rν =10 km. A cooling PNS of×fixed surface field will
ciallyforcingthefastpointonthecomputationalgrid(seedis- traverse a path from high to low Lν in this diagram, reach-
cchuossoisoinnginth§e2.3n.u1m).beWreosfpgarciedtphoeirnatdsia(glegnreidrallolyga5r0it0hm- i2c0a0ll0y), itnimgeLscν¯ea,l5e1τ∼Ji0s.l1esastthta=nττKKHH,(tsheeePeNqS. [e2v8o]l)v.esItfothhieghsperinPdodwurn-
to obtain the desired levelof conservationof M˙, , , and ingτKH,butotherwise,thePNSevolvesfromhighertolower
whilesimultaneouslymaintaininglargeenoughaLrtifiIcialvisB- Lν at roughly constant P (see Tables 1 and 2 for represen-
cosity to maintain code stability. With sufficient resolution tative τJ). The regions in Figure 1 correspond to the differ-
andlowenoughviscositythecodeshowsradialconservation entwindphasesoutlinedin 1.1: (1)athermally-drivenwind
§
of all eigenvalues to .1% across the entire grid, although at high Lν and long P; (2) a non-relativistic, magnetically-
wedidnotrequirethislevelofconservationforallsolutions driven wind at high Lν and short P; and (3) a relativistic,
so that we could efficiently explore the parameter space of magnetically-drivenwindatlow Lν andshortP. Inaddition
wind properties. The mass loss rate was the most difficult tothesedifferentregimes,Figure1illustratestherangeofro-
˙
eigenvaluetoconserveyetisaccurate(relativetoitsfullycon- tationperiodsforwhichMisenhancedbycentrifugalflinging
vergedvalue)toatleast 10%forallsolutionspresentedin (roughlyP.2- 3ms;seealsoFig.6)andforwhichτdyn,the
thispaper. ∼ dynamical time at T =0.5 MeV (eq. [35]), and the asymp-
toticwindentropySaarealteredfromtheirnon-rotating,non-
2.3.1. BoundaryConditions magnetizedvalues(whichhasimportantconsequencesforr-
processnucleosynthesis;see 4.3).Inthissection,wepresent
§
The azimuthal speed v at R is set to enforce v = anddiscussthepropertiesofsolutionsforarangeofparame-
φ ν φ,ν
RRνΩan+dvwr,νhBeφre,νΩ/Biνs,thwehsetreellaarsurobtsactrioipntrνatdeetnhoattedseefivnaelusatthioenana-t itlelrusst(rBaνte,dP,iLnν¯Fe)igthuarets1p.anSoemacehoofftthheepnroonp-erretliaetsivoifstwicinredgsiomlues-
ν
gularspeedoftherotatingframeinwhichthesurfaceelectric tionsat Lν¯e =8×1051 ergss- 1, whichcorrespondsto a rela-
field vanishes (MacGregor & Pizzo 1983). In all cases we tively early stage in the PNS coolingevolution, are given in
consider, v R Ω. The temperature at the PNS surface Table 1. Table 2 compares the properties of wind solutions
φ,ν ν
T(R ) (general≃ly 5 MeV) is set by requiring that the net withB =2.5 1015 Gattwodifferentneutrinoluminosities
ν ν
heating rate q˙ν va≈nish; this assumes the inner atmosphereis (Lν¯e =8×1051×and3.5×1051ergss- 1).
in LTE (Burrows & Mazurek 1982). Given our assumption
that matter at the inner grid pointmaintainsLTE, we fix the
densityatR tobe 1012 gcm- 3sothattheneutrinooptical 3.1. Thermally-DrivenWinds
ν
≃
depthτ at the PNS surface reaches 2/3, therebydefining
ν ∼ Thethermally-drivenregioninFigure1correspondstocon-
a neutrinosphere. Forslow rotationwe find thatthe solution
ditions under which the PNS outflow is driven primarily by
outsidetheinnerfewscaleheightsremainsrelativelyinsensi-
neutrinoheating; the magneticfield andthe rotationrate are
tivetoourchoicefortheinnerdensity,althoughwefindthat
˙ unimportantineitheracceleratingorsettingthemasslossrate
Mdependssomewhatsensitivelyonρ(R )asthePNSrotation
ν of the wind. Figure 2 shows the velocity structure of such
rateincreasesapproachingbreak-up.
an effectively non-rotating, non-magnetized (NRNM) solu-
Ifallthreecriticalpointsarecapturedonthenumericalgrid, tionforLν¯e,51=8,Bν =1013G,andΩ=50s- 1(P≃126ms).
the outer boundaryconditionsare notin causal contactwith Notice that the Alfvén radius is relatively close to the PNS
the interior wind and will have no effect on its steady-state surface (R 20 km) and that the sonic point (correspond-
A
≈
eigenvalues. However, as the temperature of the wind de- ing to the fast point in the NRNM limit) is at a much larger
clines,thefastmagnetosonicpointmovestoverylargeradii; radius (R 750 km, approximately the “Parker radius”,
s
in fact, as the sound speed c 0 the fast point formally R GM/(≈va)2,ofanequivalentpolytropicwind,wherevais
s p
→ ≃
approaches infinity (Michel 1969). For this reason, the fast theasymptoticwindspeed). Althoughthemagneticfieldand
pointisdifficulttokeeponthecomputationalgrid. Solutions rotation rate are low enough that they have no effect on the
withoutthefastpointcapturedonthegridaresensitivetothe windenergetics, the AlfvènradiusR is still abovethe PNS
A
outerboundarycondition,withdifferentchoicesaltering,for surface.Angularmomentumlossisthusenhancedbyafactor
instance, the spin-down rate. Therefore, to artificially force of(R /R )2 4overanunmagnetizedwind. Inthisrespect,
A ν
≈
thefastpointonthegridweincreasetheouterradialvelocity PNS winds such as that shown in Figure 2 are analogousto
boundary-conditionuntilthefastpointiscaptured.Otherwise thesolarwind(whichisalso primarilythermally-driven,but
equivalentsolutionswiththe fastpointnaturallylocatedand hasR &R ).
A ⊙
6
Thermally-Driven
10
v
M = v a
. c NRNM
s
M Enhanced ,n = Magnetically-Driven
Sa Reduced R t Reduced
nW .dyn
Ea Enhanced
1
5
,|e
Ln
1 s
=
1
Relativistic
0.1
1 10 100
P(ms)
FIG. 1.— Theregimes ofPNSwindsinthespaceofLν¯e(=Lν¯e,51×1051 ergss- 1)androtation periodPforarepresentative strongly magnetized PNS:
wBνith=L2ν.¯5e,×511∼01140Ga,tRaνt=im1e0tk0m∼,a1ndsMfol=low1.i4ngMt⊙he.AlaumnocnhoopfoltehefieSlNdgsehoomckettroyLisν¯ae,s5s1um∼e0d..1AbycotohleinegnPdNoSftwhiellKevelovlivne-HfreolmmhhoiglthzLpν¯heatsoel(oτwKHLν¯∼ei1n0t-hi1s0d0iasgaraftmer,
explosion). Thesolidline(eq.[18])showstheboundarybetweenPNSwindsthatareprimarilythermally-drivenbyneutrinoheating(highLν;slowrotation)
andwindsthatareprimarilymagneto-centrifugallydriven(lowLν;rapidrotation);forthelatter,thewindpowerE˙aisenhanced(seeeq.[20])andthedynamical
timeτdynisreduced(seeFig.12).Thedottedline(σ=1;eq.[27])showstheboundarybetweennon-relativisticandrelativisticmagnetically-drivenwinds.For
sufficientlyrapidrotation(P.2- 3ms;thedot-dashedline),themass-lossfromthePNS(M˙)isenhancedbecauseofcentrifugalflinging(eq.[24])andthe
asymptoticwindentropySaisreduced(eq.[37])becausemattermovesmorerapidlythroughtheregionofsignificantneutrinoheating.Thedynamicaltimeand
entropyareimportantfornucleosynthesisinthewind(§4.3). ForBν&2.5×1014 Gthethermally-drivenregionshrinks(tolongerPandhigherLν¯e;eq.[18])
andtherelativisticregionexpands(tohigherLν¯e;eq.[27]). ForRν&10km,aswilloccuratearlytimeswhenthePNSisstillcontracting,thewindislikely
tobethermallydriven. ThisfigureillustratesthewiderangeofconditionsunderwhichPNSwindswillbemagnetically-driven,althoughitshouldbecautioned
thatthesurfacedipolefieldBdνipassociatedwiththemonopolefieldBν scalesasBdνip∝BνP(seeeq.[30]),whichmeansthat,forlargeP,thetruedipolefield
appropriatetothisdiagramismuchgreaterthanthemonopolevalueofBν=2.5×1014G.
ManyoftherelevantresultsforNRNMPNSwinds(suchas indeed,fromFigure 2wefindanasymptoticspeedva
NRNM≈
hFineignQ.cW2e),.aiQrfeWLaνpsphroowxǫimνthaa4t,teMda˙sa∝nwalLeyν5¯t/ehi3cahavǫleνl¯yeiaa1sn0s/du3mv(etehrdiefi,ireQdeWqnsu.mfi[5en8rdiac,tabhl]la)yt; p0in.r0oG6xRicmaaatttLehlν¯iyge,5ht1hL=eν8s..aAmTl0eth1wofuoagyuhnwdwittehhafitLnvν¯daNe,RthNoaMutr≃vaNasR0yN.1mMcpst(coLatν¯ilece,5s1si/pn8eea)0dp.3s-
∝ h i are lower than those obtained by T01 primarily because we
the neutrino-drivenmass lossrate is approximatelygivenby
Mb1˙u0Nt5R1wNeiMrtghs≃as-n31o.×rOm1ua0rl-icz4a(aLltciν¯ouen,l5a1lt/oio8wn)e2sr.5fitMnhda⊙nthstha- 1ta,Mt˙woNfhRQeNrMWe∝:Lν¯Le2ν=.5Laνs¯ew,51el×l, hIknainvNeetRiucNseeMndearwgmyin:od˙rE˙seNastRhhNeaMlalos≃wym,(1Np/te2ow)tMitc˙oNnwRiaiNnnMdg(vpraNaovRwiNteMart)i2ios;nheaenlntpicoreet,elynfrtgoiaamls.
ourresultsforM andva wefind
primarilybM˙eNcRaNuMse≃we1.h4a×ve1i0n-c4l(uLdν¯ee,d51n/e8u)2tr.5inMo⊙resd- s1h,iftsin(o1u5r) E˙NaRNMNR≃NM4×104N7R(LNνM¯e,51/8)3.2ergss- 1. (16)
heating rates. Somewhat coincidentally, T01 found a result SincethetimespentatLν¯e,51∼8isonly∼1s(see§4.1),the
totalenergyextractedduringthe Kelvin-Helmholtzepochin
similartoequation(15)fromcalculationsincorporatingGR.
aNRNMwindis 1047- 1048ergs.
Because neutrino heating is so concentratednear the PNS ∼
surface, NRNM winds are barely unboundin comparisonto
3.2. Magnetically-DrivenWinds
thePNSescapespeed(non-relativistically,v (R ) 0.64c);
esc ν
≈
7
by the Michel speed v (B2R4Ω2/M˙)1/3 =σ1/3c (Michel
M ≡ ν ν
1969), exceeds the asymptotic speed obtained if the wind
v were entirely thermally-drivenby neutrino heating (va ).
10-1 f Usingourresultforva ,thewindwillbeintheFMRNRlNimMit
NRNM
formagnetizations
v
y (c) 10-2 r σ&2×10- 4(Lν¯e,51/8)0.9˙≡σFMR. (17)
ocit pUlsyinthgaetqthueatPioNnS(1w5i)ntdoirsemlataegLneν¯eticaanldlyM-d,roivuerncbaelclouwlattihoenscriimti--
el
V v calrotationperiod
10-3 A
v
f PFMR≃15(Lν¯e,51/8)- 1.7B14ms, (18)
whereB =B 10n G. The P=P boundaryin Figure1
ν n FMR
10-4 ismarkedbyas×olidline.Themagnetically-drivenregimeen-
10 100 1000 compasses a large range of PNS parameter space and hence
r (km) generically describes most of a strongly magnetized PNS’s
evolution.Incontrast,thewindfromarelativelyweaklymag-
FIG. 2.— Velocity profile for a thermally-driven wind with Lν¯e,51 =8, netizedPNS will only be dominatedby magneto-centrifugal
Bν=1013G,andΩ=50s- 1(P≈130ms).Thevariablesvr,vφ,vA,andvfare forces late in the cooling epoch. The conditions necessary
theradial,azimuthal,Alfvén,andfastmagnetosonicspeeds,respectively;the for the magnetically-driven phase to dominate the total en-
fast(Alfvén)speedisalsoapproximatelytheadiabaticsound(slow)speedfor
1th0e-r8m,Ea˙laly≃-dr4iv×en10w47inedrsg.sTs-h1is,asnodluτtiJo∼nh8a8s0Ms˙(≈see1T.4a×ble101-).4MTh⊙eMs-i1c,hσel≈sp3ee×d ebregdyisacnudssmedasfsurltohsesrdinur§i4n.g1.the Kelvin-Helmholtz phase will
forthissolutionisvM =σ1/3c≃0.003c,whichislessthanthethermally- Figure 3 shows the velocity structure of a magnetically-
dfireilvdenanadsyrmotpattoiotincrsapteeehdaavcetunaollysigonbitfiacinaendt e(≈ffe0ct.0o6nct)h;eheanccceel,etrhaetiomnagonfetthice driven wind from a PNS with Lν¯e,51 = 8, Bν = 1015 G and
wind. Ω=5000 s- 1 (P 1.3 ms). As the profile of vφ in Figure
≃
3 indicates, the wind corotates to 25 km, which is far in-
≈
side R 46 km because the magnetic field carries a sig-
A
≈
nificant fraction of the angular momentum. In addition, be-
0.7
causethewindismagnetically-driven,thewindspeedatlarge
v
0.6 radiiisalmostanorderofmagnitudelargerthaninaNRNM
f
wind: v =0.54 c v obtains at the outer grid point. The
r M
0.5 sonic point of the≈wind (corresponding to the slow point in
c) v r theFMRlimit) isnowinsideRA, lessthanonestellar radius
city ( 0.4 vA otifofntsheshsouwrfathcea;t wthiitshisinecxrepaescitnegdΩbe,ctahueseloacnatailoynticofcothnesisdoenraic-
elo 0.3 v radiusRs decreasesfromavalueofordertheParkerradiusto
V f avalueindependentofthelocalthermodynamics(seeLamers
0.2 &Cassinelli1999)5:
0.1 v R GM/Ω2 1/3 17P2/3km. (19)
s s,cf≡ ≃ ms
0.0
Forcomparisonwith(cid:0)oursolu(cid:1)tion,Figure3showstheveloc-
10 100 1000
ity structure of an adiabatic wind (γ = 1.15) with approxi-
r (km) mately the same M˙, Ω, B , and surface temperature as our
ν
neutrino-heatedwind.Theadiabaticsolutionagreeswellwith
FIG. 3.—Velocity profileforamagneto-centrifugally-driven windwith ˙
Lν¯e,51=8,Bν=1015G,andΩ=5000s- 1(P≃1.26ms);thevariablesvr,vφ, theneutrino-heate˙dsolutionbecause,althoughMisprimarily
svpAe,evdfs,,arnedspvescatirveetlhy.erTahdiisals,oalzuitmiounthhaal,sAM˙lfv≃én3,.0fa×st,1a0n-d3Msl⊙owsm- 1a,gσne≃tos0o.1n6ic, smeatgbnyetLicνa,lloyn-dcerivMeniws isnpdecbieficeodmtehserveelaloticvietylysitnrudcetpuernedoefntthoef
τJ ≃9 s, and E˙a ≃2.3×1051 ergs s- 1. Comparison plots of vr and vφ thedetailsoftheneutrinomicrophysics. Thisagreementim-
(dashedlines)areforaγ=1.15polytropicwindwithsimilarM˙,Ω,Bν,and pliesthatwecanaccuratelymapour1Dneutrino-heatedcal-
innertemperature.Noticethattheradiusoftheslowpoint(approximatelythe
culationsontomulti-dimensionalpolytropiccalculationsthat
sonicpoint;wherevr=vs)isveryclosetothevalueexpectedinthemagneto-
centrifugallimit:Rs,cf≈19.6km(eq.[19]). employ similar boundary conditions and a similar effective
adiabaticindex(see 4.1).
§
ForaPNSwithagivenneutrinoluminosity,largerrotation ForFMRwinds,theMichelspeedobtainsatlargeradiiand
ratesandmagneticfieldstrengthsleadtoadditionalaccelera- the asymptotic wind power is therefore enhanced relative to
tionintheouter,supersonicportionsofthewind. Iftherota- equation(16):
tionrateandmagneticfield arehighenough,theywilldom-
inate the wind acceleration at large radii. This is the “Fast E˙FaMR=M˙Ba≃(3/2)M˙v2M≃
MagneticRotator”(FMR)limit,usingthestellar-windtermi- 1050B4/3M˙1/3P- 4/3ergss- 1, (20)
nologyof Belcher & MacGregor(1976);see also Lamers& 14 - 3 ms
Cassinelli (1999). An approximate criteria for this limit is 5Thesubscript“cf”,hereandbelow,standsfor“centrifugal”andrelates
thatthemagnetically-drivenasymptoticspeed,givenroughly tothelimitdescribedbyequation(25).
8
100
103
10
..aJ/E s) 102
(
J
= W t
h
1
10
1
0.1
1 10 100 1 10 100
P(ms) P(ms)
FIG. 4.—η=ΩJ˙/E˙a istheratioofspin-downpowerlostbythePNSto
strhotreteanatigsoytnhmsppeBtroνitoic=dw1P0inf1o3drGpLoν¯w(ecer=ors(8se)×q,.11[01012541])G.erTg(sthriissa-n1figgaluen)rd,e2ms.h5oonw×osp1oη0l1ea4smGaagf(udnnieactmitcioofinnedol)df, BfoFνrIGL=ν¯.1e50=.1—38G×Sp1(ic0nr5-od1sosew)r,gns1t0sim-114easGncad(lte4riτamJnog≡nleoΩ)p,/o2Ωl˙e.5as×suraf1af0uc1en4cmtGiaogn(ndoeiaftimrcoofitanetdlido),nsatprneednrigo1td0h1sP5:
and1015 G (asterisk). Forthehighly magnetized solutions (with RA well G(asterisk). ThedecreaseinτJforrapidrotationisduetotheexponential
uorasfdfeidtiht(eoηsuu≃nrfba1ic)ne,d)btnuheteafrwolyrinlaodlwlaonBfdνthheaennedcxethraηigct&hedΩ1r.omtFaootsirotnsolafolwtehlneyerrrgooyttaatetiisoncngaa,plethseentreomrglaaylrlgyies- eansshoacnicaetemdenwtitihntMh˙efeofrfePct.ive2-m3omnosp(oelqe.fi[2e4ld]).BνThsecasluersfaacseBddνiippo∝leBfieνlPdB(sdνeiep
drivenwinds(lowBνandΩ)η≪1becausetherotationalpowerlostbythe eq.[30]), whichimpliesthatthetruedipolefieldappropriate tothisfigure
PNSisinsignificantincomparisontothethermalenergysuppliedbyneutrino canbemuchgreaterthanBν forlargeP. Notethatanapproximateanalytic
heating. expression forτJ inthe magnetically-driven limit (P.PFMR; eq. [18])is
giveninequation(23).
where M˙ = M˙- 3×10- 3 M⊙s- 1 and Ba = E˙a/M˙ = (v3M/va)+ theneutrinoenergy,themagneticfieldbecomesmoreimpor-
(va)2/2≃(3/2)v2MistheBernoulliintegralatlargeradiiinthe tant for unbinding the matter from the PNS. Consequently,
FMRlimit,andiscomposedof2/3magneticand1/3kinetic
only a fractionof the rotationalenergyextractedat the PNS
energy.
surfacereacheslargeradii.
To calculate the angular momentum lost by the PNS,
In the limit of thermally-driven solutions with very low
we note that for any super-Alfvénic outflow, equation (7),
B (even lower than in Fig. 2, such that the Alfvén ra-
ν
equation (8), and conservation of magnetic flux require
dius is interior to the stellar surface), R is the lever arm
ν
that the conserved specific angular momentum obey = for angular momentum loss; thus = ΩR2 and hence η =
oΩuRtfl2A,owwhsepreeedRAmiastcdheefisnethdebyradthiaelpAoslfivtiéonn swpheeerde: thver(RraALd)ia=l aΩn2dR[2ν1M6˙]N)R.NFMo/rEP˙NaRNM1m≈s2th4e(Lrmν¯e,a5l1l/y8-dL)-r0iv.6ePnm- w2sνi(nudssinhgaveeqηs. [151],
BArl(fRvéAn)/p√o4inπtρi(nRAte)r≡msvAo.f ηW=eΩeJ˙s/tiEm˙aa=teΩth2Re2Al/oBcaatidoenfinoefdthine wlahrgicehPeixnplFaiignusrew≫h4y. Pηhdyesiccraelalsye,sthriaspiisdlbyecfoarusseolfuotriosnlosw≪wrioth-
equation(12):
tationratesthe rotationalpowerlost bythe PNSis insignifi-
R2Ω2 (3/2)ηv2, (21) cantincomparisontothethermalenergysuppliedbyneutrino
A ≃ M
heating.
sothat6
R 11η1/2B2/3M˙- 1/3P1/3km. (22) Therateatwhichangularmomentumisextractedfromthe
A|FMR≃ 14 - 3 ms PNSisJ˙=IΩ˙ =ΩR2M˙,whereJ=IΩistheangularmomen-
A
sevFeigraulrseu4rfashceowmsagηnfeotricofiuerlwdsintrdensogltuhtsio(snesewalisthoTLaν¯be,l5e11=).8Rfoe-r teurmtia.ofHtehnecPeN,gSivaenndthIe≃A(l2f/v5é)nMraRd2νiuisstfhroemPNeqSumatoiomne(n2t2o),ftihne-
callthat1/ηrepresentsthefractionoftheextractedrotational spin-downtimeofthePNS(τ Ω/Ω˙)inthenon-relativistic,
J
energyfromthePNSavailabletoenergizethesurroundingen- ≡
magnetically-drivenlimitisfoundtobe
vironment. Inthelimitthatv &v ,wefindthatη 1and
M esc
almostalloftherotationalenergylostbythePNSem≈ergesas τ 440η- 1B- 4/3M˙- 1/3P- 2/3s. (23)
asymptoticwindpower;thislimitisnormallyassumedwhen J|FMR≃ 14 - 3 ms
considering magnetized stellar spin-down. However, as the This result shows explicitly that while increasing the mass-
low Bν solutions in Figure 4 illustrate, winds with short ro- loadingplacesagreaterstrainonthefieldlines(RA∝M˙- 1/3)
tation periods and va <v <v can be magnetically- andhencereducesthenetlossofangularmomentumpergram
drivenandyethaveηNRNM1. TheMprimeasrcyreasonforthisisthat ( =ΩR2 M˙- 2/3), the additionalmass loss carries enough
at high Ω the neutrin≫o heating rate per unit mass is signifi- toLtal angAul∝ar momentum to increase the overall spin-down
cantlyreducedbelowitsNRNMvaluebecausecentrifugally- rate.ThehighmasslossaccompanyingtheKelvin-Helmholtz
acceleratedmatterspendsless time inthe regionwhereneu- epochcanthereforeefficientlyextracttherotationalenergyof
trinoheatingis important. Because the wind absorbsless of thePNS.
Figure 5 shows τ calculated directly from our wind solu-
circ6uTmCsQtantoceosktΩheRηA==v1ra(RssAu)m=pvtiMonanisdnaostsuapmpelidcaηb=le,1.evWeneifinntdhethFatMinRsloimmiet tionsasafunctionoJfPatLν¯e,51=8forseveralmagneticfield
(seeFig.4). strengths(seealsoTable1). Ournumericalresultsagreewell
9
10-1 1012
10-2 1010
M/s) ........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................ 10-3 3cm) 108
.( g/
M (
r
106
10-4
104
10-5
2000 4000 6000 10 100
W (s-1) r(km)
FIG. 6.— Mass loss rate M˙ as a function of the rotation rate Ω at FIG. 7.—Thedensityprofilesintheinner∼100kmformagnetically-
2L.ν¯5e×=180×141G051(deiragmsosn-d1),foarndBν10=1510G13(aGste(crirsoks)s;),aBlsνo=sh1o0w1n4 Gis(Mt˙ri(aΩn)glfeo)r, sd-r1iv(ednowtteindd),s2o0l0u0tiosn- 1s(adtaLsνh¯ee,d51),=an8d,B50ν0=s-1101(5doGt-,dΩas=he7d0).00Trsi-a1ng(sleosl,idc)r,o5s0se0s0,
cLrν¯eeas=es3.w5i×th1in0c5r1eaesrignsgsΩ- 1aannddBBννbe=c2au.5se×ce1n0t1r4ifuGga(dlosuttpepdo,rdtieaxmpoanndd)s.thM˙ehiny-- aAnldfédniaramdoiunsd,sremsparekctitvheelys.loFworpoalilnto,ftthheersaodliuutsiownsh,etrheeTsu=pe0r.s5onMiceVp,oratniodnthoef
dhroowsetavteicr,aMt˙m(oΩs)phneorelo(snegeerFiign.c7re)a.sFeosrwsuitfhficiinecnrtelyaslianrggeBBννb(e&caBuscfe;ethqe.[w26in])d, tmheagdneentiscityacpcreolefirlaetiiosna.lteFroedrsforolumtiothnastwofithatΩhe&rm3a0ll0y0-dsri-v1enthesodluetniosintydusceatloe
corotates pastthesonicpoint. Anapproximate fittothenumericalresults heightinthesubsonicportionofthewind(interiortothetriangle)issignifi-
inthislimitisgivenbyequation(24). cantlylargerduetocentrifugalsupport. Thisenhancesthemasslossrateas
showninFigure6.
withtheanalyticestimatesinequation(23)forwindsthatare
R Ω&c ,wherec isthesoundspeedatthePNSneutri-
intheFMRlimit(i.e.,P.P ;eq.[18]). Thesharpdecline ν s,ν s,ν
FMR
in τ at shortP is dueto the factthatτ M˙- 1/3 andthat M˙ nosphere. We find that the inner sound speed depends only
is enJhanced by centrifugalflinging forJra∝pid rotation. Note, weakly on the neutrino luminosity: cs,ν ≈0.12(Lν¯e,51/8)0.08
c,andthusthatmasslossisenhancedfor
however, that only for B &1015 G is τ τ 10 s for
ν J KH
a millisecondrotator. For Bν =1013 G and∼large P∼the solu- Ω&3600(Lν¯e,51/8)0.08s- 1. (25)
tionsareprimarilythermally-drivenandτ isindependentof
J ˙
P; this occurs because both the Alfvén radius R and mass- Thisregionis denoted“M Enhanced”in Figure1. Equation
˙ A
lossrateM(andhenceτ )areindependentofrotationratefor (25) is in good agreement with our numerically determined
J
thermally-drivenwinds. valueofΩcfdefinedinequation(24).
˙
TheAlfvénradiusandspin-downtimescalculatedinequa- TheenhancementofM impliedbyequations(24)and(25)
˙
tions(22)and(23)dependonthemassloss rateM fromthe does not explicitly depend on the magnetic field strength.
PNS, which itself depends on the PNS’s rotation rate and However,centrifugalsupportofthePNSatmosphereonlyoc-
magneticfieldstrength. Figure6showsourdeterminationof cursifthefieldcansustaincorotationouttothesonicradius
M˙ asafunctionofΩforfieldstrengthsBν =1013,1014,2.5 Rs;otherwiserotationhasmuchlessofaneffectonthemass
Lt1i0νo¯e1n,45,1fo=arnP3d.5.1.02T1-53heGmmsaaatsnsLdν¯laoe,ls5s1so=riant8ce,reinaancserdesaBwsνeitsh=rBa2pν.i5,dtl×hyo1wu0gi1th4hitGrosata×at--t l(qoeusqis.re[rm1a9tee]n)(,tswoefehcitcohhreoitnsaottliuuortnniooinmustptwloieitRshsBlcνoawn>bBBeνcfw,inrwithFteeingre.a6s)R.AT>heRrs,ec-f
ubryacteosmfpoarrtihnegltahregBesνt=m2a.g5net1ic0fi14eGldasntrden1g0t1h5sG,assocluatniobnes.seFeonr Bcf≃2×1014η- 3/4M˙-13/2Pm1/s2G. (26)
×
suffic˙ientlylarge Bν, such that RA&Rs, we find empirically ForBν &Bcf,themasslossfromthePNSnolongerincreases
thatMisgivenby with increasing B (see Fig. 6) because the wind already
ν
˙
M˙ M˙ exp[(Ω/Ω )2] M˙ , (24) co-rotates out to the sonic radius where M is set. In deriv-
˙ ≃ NRNM cf ≡ cf ing Bcf in equation (26), we have used equation (22) for RA
where M is the mass-loss rate for NRNM winds because,undermostconditions,a windthatiscentrifugally-
NRNM
(eq.[15])andΩcf≈2700(Lν¯e,51/8)0.08s- 1. supported will automatically be in the FMR limit, although
the converseis notnecessarily true. We note that even rela-
The enhanced mass loss shown in Figure 6 is due to the
tivelyweakly-magnetizedPNS’sthatarerapidlyrotatingwill
effectofstrongmagneticfieldsandrapidrotationonthesub-
experiencesomedegreeofenhancedmassloss. Forinstance,
sonic, hydrostatic structure of PNS winds. Figures 7 and 8 for Ω=7000 s- 1, even the lowest field strength solution in
show the density and temperature profiles, respectively, for
Figure6(B =1013G)hasamass-lossratealmostanorderof
windswithBν=1015GandLν¯e,51=8atseveralrotationrates. magnitudelνargerthanitsNRNMvalue.
For the most rapidly rotating solution (solid line), Figure 7
showsthatcentrifugalsupportissufficienttoexpandthescale Thenon-relativisticcalculationswehavepresentedhereare
heightofthe hydrostaticatmosphereatsmallradii, resulting onlyapplicableformagnetizationsσ<1;thisisequivalentto
inthemuchhighermasslossratesseeninFigure6. Analyti- requiringR <R c/Ω,theradiusofthelightcylinder.For
A L
≡
cally,weexpectthecentrifugalsupporttobeimportantwhen σ >1 the PNS wind becomes relativistic and its spin-down
10
fields and rapid rotation. Rather, we assume that the PNS
10 has cooled to its final radius and completely established its
globalfieldbyatimet 1sfollowingcore-collapse;wecan
0
∼
thenusethecalculationspresentedinthispapertoinvestigate
thesubsequentevolutionofthePNS.Followingt weassume
0
a simplified PNS cooling evolution (similar to that used in
TCQ,motivatedbyFigure14ofPonsetal.1999):
V) 1
Me t - δ
T( Lν¯e,51(t)=L0 τ :t0<t<τKH
(cid:18) KH(cid:19)
Lν¯e,51(t)=L0exp[- (t- τKH)/τKH]:t>τKH, (28)
where, for definitiveness in what follows, we take L =0.2,
0
0.1 t =1 s, τ =40 s, and δ=1. Thiscoolingevolutionis ap-
0 KH
proximatebecausemagnetar-strengthfieldsandrapidrotation
couldalterτ ortheformofthecoolingprofile(e.g.,δ)by
KH
10 100
affectingtheneutrinoopacityorthedynamicsofthecontrac-
r(km)
tionitself (e.g., Villain et al. 2004; Dessartet al. 2006). For
FIG.8.—Thetemperatureprofilesintheinner∼100kmformagnetically- instance, in 1D collapse calculations with rotation, Thomp-
sd-r1iv(ednowtteindd),s2o0l0u0tiosn- 1s(adtaLsνh¯ee,d51),=an8d,B50ν0=s-1101(5doGt-,dΩash=e7d0).00Trsi-a1ng(sleosl,idc)r,o5s0se0s0, stootna,lQnueuattarienrot,l&umBiunrorsoiwtysa(t20t05)0f.o6usndafttheartbfoournPc∼e i1s ms5,0th%e
anddiamondsmarktheslowpoint,theradiuswhereT =0.5MeV,andthe smallerthaninanon-rotating≃PNS. ∼
Alfénradius,respectively.Noticethatinallbutthemostrapidly-rotatingcase
theT=0.5MeVradiusislocatedbetweentheslowpointandtheAlfvénra- Thedominantuncertaintyinapplyingourresultstomagne-
dius,whichimpliesthatthereissignificantmagneticaccelerationofthewind
tizedPNSevolutionisthatwehaveassumedamonopolefield
attheT =0.5MeVradius.AsshowninFigure12,thissignificantlyreduces
thedynamicaltimeatT=0.5MeV,makingmagnetically-drivenPNSwinds geometry. To relate our results to more realistic dipole sim-
more favorable for r-process nucleosynthesis than thermally-driven winds ulations, we use the recent axisymmetric, relativistic MHD
(see§4.3). simulationsofB06,whosimulateneutronstarspin-downfor
propertieswillchange(see 4.1foradiscussion).Usingequa- σ 0.3- 20. B06showthattheenergyandangularmomen-
˙ § tum≈lossratesfromaligneddipolespin-downcanbedescribed
tion(24)forM(i.e.,assumingB >B ),themagnetizationis
ν cf
accuratelybymonopoleformulaeprovidedthey are normal-
givenby
ized to just the open magneticflux; for instance, we can ac-
σ≃0.05B214Pm- 2s(Lν¯e,51/8)- 2.5exp[- 5.4Pm- 2s(Lν¯e,51/8)- 0.16(]2.7) creunraotremlyalaizpaptliyonooufrBres.ults for τJ (eq. [23]) with a suitable
ν
The σ = 1 boundary is denoted by a dotted line in Figure
To apply the results of B06 we need to estimate the open
1. PNSs with σ >1 and Lν¯e,51 &0.1 will experience a rel- magneticfluxin PNS winds. Inforce-freespin-downcalcu-
ativistic phase accompanied by significant mass loss; this
lations motivated by pulsars it is generally assumed that the
masslosskeepsthewindmildlyrelativistic,incontrasttothe
radius of the last closed magnetic field line (the “Y point”
muchhigherσ spin-downthatwillcommencefollowingτ
KH R ) is coincident with the light cylinder (Contopoulos et
(Lν¯e,51≪0.1). alY. 1999; Gruzinov2005) so that the ratio between the frac-
tionofopenmagneticfluxinthedipoleandmonopolecasesis
4. APPLICATIONSANDDISCUSSION R /R =R /R . Thisassumptionissupportedbyforce-free
ν Y ν L
simulations(Spitkovsky2006;McKinney2006),whichshow
4.1. MagnetizedPNSEvolution that when mass-loading is completely negligible (σ ),
→∞
R R . However, B06 show that rapidly rotating, mildly
Withournumericalresultsinhandthatsampleawiderange Y ≃ L
mass-loaded MHD winds have a larger percentage of open
of PNS wind conditions, we can begin to address the time
magneticfluxthanvacuumorforce-freespin-downwouldim-
evolution of a cooling, magnetized PNS. In the early stages
ply(i.e., R <R ). FromTable4 ofB06we fittheapproxi-
following the launch of SN shock the PNS is likely hot and Y L
matepowerlaw
inflated,witharadiusexceedingthevalueofR =10kmthat
ν R /R 0.31σ0.15 (29)
we have assumed in all of the calculations presented in this Y L≃
paper. This early phase is likely to be thermally-driven for for the range σ [0.298,17.5] and for a fixed rotation pe-
allbutthemosthighly-magnetizedproto-magnetars,and,ifa riodoforderone∈millisecond. Althoughthereareuncertain-
dynamoisatwork,thelarge-scalefielditselfmaystillbeam- tiesinquantitativelyextrapolatingB06’sresults,reachingthe
plifyingduringthisphase(Thompson&Duncan1993). Us- pureforce-freelimitwithR R appearstorequireσ 1.
Y L
ingthecollapsecalculationsofBruenn,DeNisco,&Mezza- Wethereforeconcludethatm≃agnetized,rapidlyrotating≫PNS
cappa(2001)(froma15M⊙progenitorofWoosley&Weaver winds(withσ 10- 3,103 fort <τKH undermostcircum-
1995)T01fitanapproximatefunctionalformtothePNSra- stances)willty∈pic{allyposse}ssexcessopenmagneticflux.
dial contraction: R t- 1/3 such that R (1 s) 15 km and
R (2 s) 12 km. νT∝he recent SN simuνlations≃of Buras et BecausetheresultsofB06forRY/RLcoveronlyarelatively
ν ≃ narrowportionofPNSparameter-spacewemustproceedwith
al.(2003,2006)withBoltzmannneutrinotransportfindasim-
cautioningeneralizingtheirresultstoourcalculations;onthe
ilarneutrinosphereradiusatt 1safterbounce.
∼ otherhand,theirbasicconclusionshowsaweakdependence
Wedonotattempttoaddresstheuncertaintiesinearly-time onσandΩ,andhasasolidtheoreticalexplanation(Mestel&
PNScoolingcalculations,especiallyinthepresenceoflarge Spruit1987).Hence,wehaveattemptedtoapplytheresultsof
