Table Of ContentSupernova neutrinos, from back of the envelope to supercomputer
7
0 Christian Y. Cardallab∗
0
2 aPhysics Division, Oak Ridge National Laboratory,
n Oak Ridge, TN 37831-6354,United States of America
a
b
J Department of Physics and Astronomy, University of Tennessee,
9 Knoxville, TN 37996-1200,United States of America
2
While an understanding of supernova explosions will require sophisticated large-scale simulations, it is never-
1
thelesspossibletooutlinethemostbasicfeaturesoftheneutrinoemissionresultingfromstellarcorecollapsewith
v
a pedestrian account that, through reliance upon broadly accessible physical ideas, remains simple and largely
1
self-contained.
3
8
1
0
7 1. BACK OF THE ENVELOPE tionstoaneutronstar,’toparaphraseBaadeand
0 Zwicky, involves the emission of another electri-
/ Because the central features of a system can
h cally neutral particle ‘discovered’ (theoretically,
often be elucidated with simple models, and be-
p if not experimentally) in the 1930s: the neu-
- cause neutrino emission is—from nature’s point trino. Once conditions for their production are
o
of view of raw energetics—the central feature of
r reached, the weakness of neutrino interactions
t core-collapse supernovae, one may hope to un-
s implies that their emission is the most efficient
a derstandthebasicfeaturesofsupernovaneutrino means of cooling. Neutrino emission is responsi-
: emission quickly and easily.2
v ble for the degeneracy of the precollapse stellar
Xi Shortlyafterthediscoveryoftheneutroninthe core of mass M ≈ 1.5M⊙ (the Chandrasekhar
early 1930s Baade and Zwicky declared: “With
mass); the core’s eventual instability, at least in
ar all reserve we advance the view that supernovae part; and the energy loss of about 1.7×1053erg
represent the transitions from ordinary stars to
that ultimately allows collapse to proceed to ra-
neutron stars, which in their final stages consist dius Rcold ≈11km as a cold neutron star.4
of extremely closely packed neutrons” [9]. For
the ‘core-collapse’varieties of supernovae (Types willprovetobeaparticularlysharpmachete:
Ib/Ic/II) this basic picture prevails today with
strong theoretical and observational support. 2EN +ER=−EG, (1)
The story3 of how an ‘ordinary star transi- whereEN andER arethecontributions toastar’sinter-
nalenergyinnonrelativisticandultrarelativisticparticles
respectively, and
∗Oak Ridge National Laboratory is managed by UT-
Battelle, LLC, for the DoE under contract DE-AC05- 3GM2
00OR22725. EG=−5 R (2)
2Aself-containedbutsimplederivationwillbecarriedon
in these footnotes. In terms of nuclear physics, weak in- isthetotalgravitationalenergyofastarofuniformdensity
teractions,andhydrodynamicsrelevanttostellarcollapse (assumed here throughout) with mass M and radius R.
and bounce, as well as spatial neutrino diffusion during (In this presentation G is Newton’s constant; moreover,
post-bounce deleptonization, this presentation exhibits a ¯h=c=k=1.)
cavalierdisregardfordetailthatbordersoncriminalneg- 4The assumption that the stellar core burns all the way
ligence. For responsible treatments of these aspects that to56Fe—thetopofthebindingenergycurve,fromwhich
26
neverthelessretainasomewhatanalyticflavor(thoughre- no further energy can be extracted by fusion—allows an
lyingtoanextentonnumericalsimulationsorexperimen- iron core temperature TFe 0.9 MeV to be estimated
≈
talinformation),seeforinstanceRefs. [1,2,3,4,5,6,7,8]. as follows. Fusion requires tunneling through a Coulomb
3Inslashingthroughthisproblem,thevirialtheorem[10] barrier. Take the uncertainty principle ∆x∆p 1
≈
1
2
as a measure of the potency of tunneling, with ∆p nantthatresultstherefrom. (Takinginhomogeneous stel-
2µZ1Z2α/∆x as a momentum scale corresponding t≈o larstructureintoaccountyieldsM ≈1.22M⊙(Ye,0/0.46)2
[12].)
pthe energy height of the Coulomb barrier (here µ is the
Next consider the role of neutrinos in the ultimate in-
reducedmassofcollidingnucleiofprotonnumbersZ1and
stabilityofthecore. AsitapproachestheChandrasekhar
Z2, and α 1/137 is the fine structure constant). Take
≈ mass,oneproximatecauseofthecore’sinstability—which
alsoT Z1Z2α/∆x as thecharacteristic thermal energy
≈ in factwill beused here to define the onset of collapse—
perparticleneededtobringthecollidingnucleiwithin∆x
of each other. Elimination of ∆x yields T = 2Z2Z2α2µ. iselectroncapture, whichsapspressuresupportfromthe
1 2 core. Electroncapturebecomespossiblewhentheelectron
Burningtends to proceedbysuccessive capture ofαpar-
Fermienergy
ticles (which have modest Z, and are tightly bound and
tµh=eremfoNre(4a)b(5u2n)d/a(4nt+),5s2o)atpakpirnogprZia1te=to2a4p,uZta2ti=ve2fi,naalnαd (ǫF)e=π1/3 3 2/3 MYe 1/3 1 (5)
capture to5266Fe gives TFe ≈0.9MeV (mN ≈939 MeV is (cid:16)2(cid:17) (cid:16) mN (cid:17) R
agenericnucleonmass).
rises above the energy threshold required to convert a
Iftheelectronsarenotalreadydegenerateatthispoint,
proton to a neutron. Crudely considering a nucleus of
they must soon become so, thanks to neutrino emission.
proton number Z and mass number A as comprising de-
A temperature TFe ≈ 0.9 MeV is almost twice the elec- generate Fermi gases of nonrelativistic protons and neu-
tprootnenmtiaaslsofmseim, isloarinmtahgeniatubdseenpceosoitfroannsewleiclltrboenthcheremmaicllayl trons contained in a volume of radius rA = rNA1/3
produced. But these e+ will annihilate with e− into νν¯ (where rN ≈ 1.2 fm), the typical emitted neutrino en-
pairs that escape freely; hence we expect cooling by νν¯ ergyǫ¯νe isnotthetypical energyofanelectroncaptured
from its (ambient) degenerate Fermi sea, because the en-
emission back down towards T ≈ me ≈ 0.5 MeV. In re- ergy required to lift a bound proton from its Fermi sea
sponse to the resulting loss of pressure support the core
(within the nucleus) to the top of the neutron Fermi sea
contracts to higher density, with the loss of gravitational
potential energyincreasingthetemperature andbringing (within the nucleus) must first be paid. Hence ¯ǫνe ≈
aboutfurtherpairemission,andsoon. Continuedincrease (ǫF)e−∆−[(ǫF)n;A,Z−(ǫF)p;A,Z](where(ǫF)n;A,Z and
in density with temperature regulated to 0.5 1 MeV (ǫF)n;A,Z aretheFermienergiesoftheneutronsandpro-
∼ − tons boundinthenucleus), or
eventually results in electron degeneracy, which becomes
the dominant source of pressuresupport. (Iron core cen- 3 3π2 1/3(A Z)2/3 Z2/3
strtealllatremevpoelruattiounrecsoodfes∼[101.]5.)−1 MeV are in fact seen in ǫ¯νe≈(ǫF)e−∆−4(cid:18) 2 (cid:19) m−NrN2 A−2/3 (6)
The consequences of degeneracy are dire. As the mass
ainngddsteangsei,tytohfethveeliorcointycoorfeitnhceredaseegednuerriantgetheleecfitnraolnbsuarpn-- ≈(ǫF)e−∆−34(cid:18)3π22(cid:19)1/3(1−Yme)N2/r3N2−Ye2/3, (7)
proaches the speed of light, at which point they can pro-
vide no further pressure support. To estimate the core’s where ∆=mn mp me 0.8MeV and itisassumed
− − ≈
maximum mass, take EN = 0 and ER = (EF)e in Eq. that all baryons are locked up in nuclei characterized by
(1),where the same Z and A. Electron capture can proceed when
theright-handsideispositive;using(ǫF)e asgiveninEq.
3π1/3 3 2/3 MYe 4/3 1 (5) yields R0 810 km at the onset of electron capture
(EF)e= 4 (cid:16)2(cid:17) (cid:16) mN (cid:17) R (3) for M = 1.5M≈⊙ and Ye,0 = 0.46, corresponding to a
density of ρ0 1.3 109 g/cm3. Incidentally, we are
is the internal energy of a uniform core supported by a now in a posit≈ion to×compute the entropy per bayon of
degenerateFermigasofrelativisticelectrons;theelectron the precollapse core: s0 0.85 1.5 (corresponding to
fractionYe istheratioofthe netnumber density ofelec- T0 0.5 1MeV)receives≈contrib−utionsfromthenucleiof
trons ne− −ne+ (or, by charge neutrality, the number mas≈snum−berA=56,therelativisticdegenerateelectrons,
densityofprotonsnp)tothetotalbaryonnumberdensity andphotons:
n. NotethecancellationofradiusthatthenresultsinEq.
(s1ta),ratrcileesatroinsduipcpatoirotnittshealftbsoymreetlahtiinvgisgtoicesdehgaeynweirraecwyhpernesa- sA = 1 5+ln mA mAT0 3/2
sure alone. The resulting expression nevertheless yields A(cid:26)2 (cid:20) ρ0 (cid:16) 2π (cid:17) (cid:21)(cid:27)
anestimateofthemassofthisextremeconfiguration,the 0.31 0.33, (8)
≈ −
so-calledChandYrea,s0ekh2ar mass: se− = π2Ye,0(ǫTF0)e ≈0.52−1.04, (9)
M ≈1.5M⊙(cid:16)0.46(cid:17) , (4) sγ = 4π2mNT03 0.02 0.14, (10)
45 ρ0 ≈ −
whereYe,0 0.46istheratioZ/Aoftheprotonandmass
nmuamssbeorfstohfe≈52c66oFree.uTphoinsmcoalylabpseet,aaknednaosfathneecstoimmpaatectorfetmhe- Awhaernedm(AǫF≈)em=NA(3iπs2tρh0eYme,a0s/smoNfa)1n/u3cle≈us4o.f4mMasesVnuismbtheer
3
initial electron Fermi energy. (As was the case with T0,
thesevaluesofρ0ands0areinreasonableagreementwith
the central ironcore values produced bystellar evolution
codes[11].)
Having characterized the precollapse core and deter-
minedthat itis doomedto collapse—andsupposing that
the degeneracy pressure of nonrelativistic nucleons will
haltcollapse—whatmightweexpectthefinalradiusRcold
of the compact remnant to be, and why do we expect it
toconsist of neutrons rather than amixtureof neutrons,
protons,andelectrons? Awarethatweakinteractionscan
interchange neutrons and protons, we provisionally take
our star of “extremely closely packed [nucleons]” to con-
sist of nonrelativistic degenerate Fermi gases of neutrons
andprotonsandarelativisticdegenerateFermigasofelec-
trons. HenceEN =(EF)n+(EF)p inEq. (1),where Figure1. Inthecrudemodelofgravitationalcollapse
presented here, the neutrino diffusion time scale τ
ν
9 3π2 1/3 [M(1 Ye)]5/3 exceeds the free-fall time scale τG when the core ra-
(EF)n = 20(cid:18) 2 (cid:19) m8−/3R2 , (11) diushas shrunktoabout 180 km,which corresponds
N to a trapped lepton fraction of Ye,trap ≈0.33.
9 3π2 1/3 (MYe)5/3
(EF)p = 20(cid:18) 2 (cid:19) m8N/3R2 , (12) Basic characterizations of the precollapse core
and ER =(EF)e of Eq. (3). A second constraint is pro- and the cold neutron star to which it collapses
vided by chemical (‘β’ or ‘weak’) equilibrium, which re- may be sufficient to estimate the total energy
quires loss to neutrinos, but additional features of this
µn=µp+µe, (13) emission—which flavors are emitted, at what en-
ergies, over what time scales—depend crucially
wherethe chemical potentials µi include particle masses. ontheflowofelectronleptonnumberandthegen-
Thatis,µi=mi+(ǫF)i atzerotemperature,with
eration of entropy. ‘Neutronization’ via electron
3 3π2 1/3 [M(1 Ye)]2/3 captureentailstheflowofelectronleptonnumber
(ǫF)n = 4(cid:18) 2 (cid:19) m5−/3R2 , (14) from degenerate electrons to electron neutrinos,
N
and the thermalenergyassociatedwith finite en-
(ǫF)p = 43(cid:18)3π22(cid:19)1/3 (Mm5Y/3e)R22/3, (15) tarnodpyanrteisnuelutstriinnothseotfhaelrlmflaavloerms.i5ssionofneutrinos
N
The salient point about the flow of electron
and(ǫF)e given byEq. (5). Taking M =1.5M⊙, simul-
taneous solution of Eqs. (1) and (13) as specified above numberduringcollapseisthatwhileelectronneu-
yieldsRcold=11kmandYe,cold=1.0×10−2. Hencethe trinos initially escape freely, they eventually be-
label ‘neutron star’ is well deserved! The single equation come trapped, as Fig. 1 illustrates.6 A roughly
(EF)n=−EG/2yields
1.5M⊙ 1/3 ofenergyequalinmagnitudetohalfthefinalneutronstar
Rcold=11km (16) gravitational energy,
(cid:16) M (cid:17)
s0wi.ti1ty4hnfmY≈e−3n0e.d2ge5lerifcvmtee−dd3.frios(mTwhifetrhorminesueaxltfpianecrgtiombreaonrfytaotlwnmoneouafmstubhreeermvdaeenluntes- Eν ≈1.7×1053erg(cid:16)1.5MM⊙(cid:17)2(cid:16)11Rkm(cid:17), (17)
of the size of atomic nuclei, which also feature densely mustbelosttoneutrinos(andabitofittotheexplosion)
packed nucleons. A neutron star is not unlike an atomic beforetheneutronstarcanreachitscoldfinalstate.
nucleusofastronomicalproportions!) 5In a sense these two themes are complementary, for de-
Notethatneutrinoemissionisnecessarynotonlytoget generacy entails lowentropy ingeneral, andinparticular
collapsestarted, but alsoforittoproceed tocompletion. anetelectronnumberpreventstheriseofapositronpop-
This is another consequence of the virial theorem: from ulationthatwouldallowνν¯pairemissionofallflavorsby
Eq. (1) we see that the total energy of the precollapse e+e− annihilation.
coreis(EF)e+EG≈0,whilethetotalenergyofthecold 6That neutrino trapping will occur is shown by a com-
neutron star is (EF)n+EG ≈ EG/2. Hence an amount parisonofthecollapseandneutrinodiffusiontimescales,
4
estimated 5.0×1050 erg are releasedin νe before
trapping;the gradualriseintotalluminosityand
cutoff at trapping,as wellas the typicalneutrino
energyduringinfall,aredepictedintheleftpanel
of Fig. 2 at negative (i.e. pre-bounce) times.7
baryons remain locked in heavy nuclei through trapping
is given in the next footnote). In accordance with the
above remarks, a neutral current scattering cross section
pernucleon(G2/2)¯ǫ2 issuggested bydimensionalanal-
F νe
ysis. Appliedtoanentirenucleus yieldsnotjustafactor
Acorrespondingtoanincoherentsumoverscatteringsoff
individualnucleons,butafactorA2 correspondingtocol-
lective scattering off the entire collection of nucleons—a
coherent sum in the scattering amplitude. This is be-
Figure 2. Crudely estimated neutrino luminosities cause the neutrino wavelength is larger than the size of
(thick) and characteristic energies (thin). Note the the nucleus (as can easily be verified from Eq. (7) and
changeintimeandluminosity(butnotenergy)scales rA =rNA1/3 with rN ≈1.2fm). With the nucleus then
between theleft and right panels. The left panelis a treated as a point particle, the cross section is partially
close-up of infall and bounce at t=0. analogous to Rutherford scattering of a charged particle
fromapointnucleusofprotonnumberZ,withourfactor
ofA2associatedwith‘weakchargenumber’Acorrespond-
ingtothefactorZ2 intheRutherfordscatteringformula.
butbeforeexaminingtheseawordonweakinteractionsis Ascollapsebegins,
inorder. For energies much lowerthan the masses ofthe
W+,−andZ0bosons(whoseexchangeareresponsiblefor R 3/2 1.5M⊙ 1/2
‘charged-’and‘neutral-current’interactionsrespectively), τG≈0.11s 810km M , (20)
theeffective weakinteractionLagrangianis (cid:16) (cid:17) (cid:16) (cid:17)
L=−(4GF/√2)(Jcc)†µ(Jcc)µ−(GF/√2)(Jnc)µ(Jnc)µ, baneednτtνaeke≈nt0osdineficene¯ǫνtehe≈on0seatsogficvoelnlapbsye.EWq.h(il7e)nheaustrhineores
twhheeroevetrhaellFsecramleicoofnwsteaanktiGntFer=act1i.o1n7×am1p0−lit1u1dMeseV(β−d2esceatys iτnνietiaalslyResdcaepcreeaesaessilyd,utrhinegdeccorlelaapseseinimτGplyantdhaitncarteassoemine
beingtheprototype),and(Jcc)µand(Jnc)µarea‘charged pointneutrinoswillbetrapped(inadditiontotheexplicit
current’ (which changes electric charge by one unit) and dependence of τνe on R in Eq. (19), see Eq. (5) for the
‘neutral current’ constructed of quark and lepton fields increasein(ǫF)e andhenceǫ¯νe asRdecreases).
(e.g. Appendix B of Ref. [13]). In this presentation the In order to estimate the trapped lepton fraction, note
factors 8G2 and G2/2 are taken as crude estimates of thataslongastheνe fromelectroncaptureescapefreely,
F F
leading factors in charged and neutral current processes theelectronfractionchanges accordingto
respectively,withneutrinoenergyfactorssuppliedtogive
thecorrectdimensionalityforratesorcrosssections. dYe ≈ −YeΓeA→Aν dt (21)
Now to the collapseand neutrino diffusiontimescales. 3πR 1/2
Assumingthatthecollapseattendingthecore’sapproach ≈ 8YeG2F¯ǫ5ν(cid:16)GM(cid:17) dR. (22)
totheChandrasekharmassiscatastrophic, suchthatthe
core is essentially in free-fall, the instantaneous contrac- The factor of Ye gives the fraction of baryons that are
tiontimescalewhenthecoreisatdensity ρis protons; the charged current capture rate per proton on
4πR3 1/2 nesutcimleiatoefdminassacncuomrdbaenrceA,wΓitehA→prAevνio≈us8rGem2Faǫ¯r5νk,sh(aisnbceoenn-
τG≈(Gρ)−1/2=(cid:18)3GM(cid:19) . (18) strasttoneutrinoscattering,thereisnocoherenceonthe
nucleusasawhole);anddt≈−dτGhasbeentaken. Equa-
Ontheotherhandthetrappedelectronneutrinodiffusion tion(22),withEqs. (7)and(5),isanordinarydifferential
timescaleis equation for Ye whose solutionis shown inFig. 1. Using
this solution in connection with Eqs. (18) and (19), the
τνe ≈ 3λRνe2 ≈ 3R2ρmσNνAA→Aν ≈ 89πAGm2FN¯ǫ2νe MR, (19) tRTrh0aip=spie8sd1i0llleukpsmttor,naatnefrddacgAtriao=pnh5ii6sc.atlhTlyehveinarleFuseiugol.tf1iYsfeYorwe,htMrearpe=τ1G.05.=3M3τ.⊙ν,.
Awh2e(rGe2Fλ/ν2e)¯ǫi2νsethisetnheeutcrrionsosmseecatnionfrefeorpacothh,erσenνtA→neAuνtra≈l 7taTinheedtobtyalinetneegrrgaytinregleadsEeνdein=νeǫ¯νeb(eMfo/remtNra)p(dpYin≈eg/diRs)odbR-
current scattering off a heavy nucleus of mass number A from the initial radius to the trapping radius. The lu-
(the dominant source of opacity; the argument that the minosity during infallis Lνe =dEνe/dτG, and is plotted
5
The other point to make about collapse is of a soft landing cushioned by abundant degrees
that the entropy per baryon changes only mod- of freedom capable of gently absorbing the ki-
estly, which conditions what happens on dynam- netic energy of infall, a hard ‘bounce’ can be ex-
ical time scales at the end of implosion.8 Instead pectedwhennucleondegeneracypressurekicksin
at supranuclear densities.9 This is accompanied
eainrsgFoybigits.aig2nievadesnainbfyuthnEecqtpi.orne(7vo)io.futIsinmfaoeloltton=fott−ehτiisGst.uhsTeehdse.olnuetuiotnrinYoe(eRn)- b2)y,ainbvuorlvstinogfaνneeemneisrsgiyonlo(ssseoeftahbeolueftt5p.9an×e1l0o5f1Feirgg.
8Toseethattheentropyperbaryonchangesonlymodestly andthesuddenreleaseoftrappedelectronlepton
duringcollapse,consider thefirstlawofthermodynamics number.10 Moreover,anadditional6.8×1051erg
intheform
Tds=dq−Xi µidYi. (23) aaYnnLdd=νt¯heYeec,ofitmrraspet,tintehrtomecsahelcesomonidbcaetcleo(rm‘mβe’svoasrnm‘iwsahleleasaksb’)eece−aq,uuseile+ib,drYpiu,Lmn≈., ν0e,
Heredq=d(e/n)+pd(1/n)(whereeistheinternalenergy 9The estimates of post-bounce neutrino emission pre-
density,nisthebaryonnumberdensity,andpisthepres- sented below involve reasoning from the approximate
sure);butitisequal,asthelabeldqsuggests,totheheat global states ofaneutron star inchemical equilibriumat
perbaryonexchanged withthe environment. Thesecond finite temperature. In addition to thermal photons and
termrepresentstheentropyperbaryongeneratedbyout- νµν¯µ and ντν¯τ pairs, there must be allowance for arbi-
of-equilibrium ‘chemical’ changes to the core (µi are the trary degeneracy in νeν¯e pairs, relativistic e+e− pairs,
chemical potentials including rest masses of the species and nonrelativistic neutrons and protons. For relativis-
i, and dYi are the changes in the numbers per baryon). tic pairs of arbitrary degeneracy there are simple ex-
There are arguments for the smallness of ds both before pressions for the differences of number densities and the
andaftertrapping. sums of energy densities of particles and antiparticles, as
Beforeneutrinotrapping both energyand leptonnum- well as the entropy [6,3]. For the nonrelativistic nucle-
ber are lost via electron capture, and the modesty of ons the needed quantities can be derived from the Lan-
entropy increase results from a substantial cancellation dau potential Ω(T,V,µ) = pV = E TS µN. This
− − −
between these contributions. The energy per baryon can be written in terms of a polylogarithm function as
lost to electron capture neutrinos that freely escape is Ω(T,V,µ) = V(m3T5/2π3)1/2Li5/2[−exp(µ/T)], whose
tdrqon≈ca¯ǫpνetudrYee.inMvoelavnewshdilYeeth=e cdhYepmi=cal cdhaYnnge(thofe efilercst- sdylemdbboylicandyersievlaft-riveesspeacntdinngusmymerbicoallicevmaaluthateimonatciacsnpbaechkaagne-
equality arisingfrom charge conservation−and the second (thoughtheSommerfeldexpansionseemsmorereliablefor
from baryon number conservation), so that iµidYi = µte/rTize>d1b0y−tw20o).paTrhaemgeltoebrasl(ntheueternontrostpayrpsteartebsaaryreonchsaraancd-
(µe+µp µn)dYe. Hence P
− the lepton fraction YL = Ye+Yνe) and seven unknowns
Tds≈(ǫ¯νe+µn−µp−µe)dYe. (24) b(Re,soTlv,eYde,arµeet,hµeνvei,rµianl,thaenodreµmp)o.fTEhqe. s(e1v);encheeqmuiactailoneqsutio-
It turns out that the leading factor cancels exactly with librium,Eq. (13)withµνe addedtotheleft-handside;the
the prescription for electron capture neutrino energy of totalentropyperbaryon;andfour‘particlenumberequa-
Eq. (7). In a more careful calculation the cancellation tions’definingYe,Yνe =YL−Ye,Yp=Ye,andYn=1−Ye
wouldnot be exact, but itwouldstill besubstantial. An intermsofthechemicalpotentials, temperature,etc.
importantconsequenceofthesmallchangeinentropybe- 10Aneutronstarstate(asdescribedinthepreviousfoot-
foretrapping isthat the baryons remaininnuclei (which note) with s = 1.5 and YL = Ye,trap has a total energy
featurethecoherentscatteringcrosssection∝A2): using Etotmuchlowerthan−Eνe,infall(theenergylosttoνees-
A = 1, R = 180 km, and T = 1 5 MeV in Eq. (8) capeduringinfall),soitcannotbethepost-bouncestate.
−
shows that s 4 6 would be required for the nuclei to (As noted earlier, the precollapse initial condition sup-
≈ −
disassembleintofreenucleons beforetrapping. ported by purely relativistic energy is characterized by
Itcanalsobearguedthatthechangeinsbetweentrap- Etot = 0, which follows from Eq. (1).) Examining a
ping and core bounce will be modest, this time because sequence of states withhigher s inabid toreach Etot =
the terms of Eq. (23) tend to vanish individually. First, −Eνe,infall eventually yields a ‘burst state’ with s ≈ 5.9
dq 0 since neutrinos can no longer carry away energy. whoseradiusR 303kmisequaltoitselectronneutrino
≈ ≈
Second,withneutrinotrapping, trapping radius. (As before, the neutrino trapping radii
are given by equating a diffusion time scale τν = 3R2/λ
Xi µidYi=(µe+µp−µn−µνe)dYe+µνedYL, (25) λwνiteh≈a(dnyNnσaNmi+calnntiσmne)−s1c,alwehτedreynσN=≈(GGρ)F−ǫ¯12ν/e2/.2 iHseforer
neutral current scattering on nucleons and σn ≈8GFǫ¯2νe
where YL = Ye+Yνe is the trapped lepton fraction (in is for charged current absorption on neutrons. For later
(annνaelo−gynwν¯eit)h/nt)h.e BdeefitwnieteionntroafpYpeinggivaenndebaorulinerc,e,Yνwehi≡le aunsed, nλoνtµe,ν¯tµh,νaτt,νλ¯τν¯e≈≈((nnNNσσNN)−+1n.)pσHp)e−n1ce, wahsetrheeσcpo≈reσrne-,
6
or so is lost from the core before the dust set-
tles after a dynamical time scale of about 24 ms
post-bounce;11 in the left panelof Fig. 2 this full
amountisattributedtoneutrinoemissionequally
divided among all flavors.12
Theneutrinoemissionencounteredthusfarac-
counts for only about 10% of the total energy to
ultimatelybelost,withtherest—about1.5×1053
erg—temporarilystoredasheat,thankstotheen-
tropygeneratedby bounce andits aftermath. As
the hotnascentneutronstar shrinks(seeFig. 3),
this energy is lost to neutrino emission of all fla-
vors (albeit with a modest hierarchy of luminosi-
ties and average energies) over a period of many Figure 3. As the nascent neutron star shrinks in
seconds, as depicted in Fig. 2—a recognizable if radius in response to neutrino emission during the
veryroughcaricatureoflightcurvesproducedby post-bounce cooling phase, the total energy Etot of
detailed models [14,15,16,17].13 the core and entropy per baryon s decrease mono-
tonically even as the temperature T increases before
bounds to this radius, νe with ǫ¯νe ≈ 3µe/4 ≈ 7.1 MeV makingitsfinalplunge. Startingat0.1spost-bounce,
escape freely in a burst in which a lepton fraction dYL diamonds mark time intervals of 0.2 s; starting at 1
and energy Eνe,burst ≈ ¯ǫνe(M/mN)dYL are lost during s, squares mark time intervals of 2 s.
a timeτdyn ≈24 ms that characterizes this ‘burststate.’
In order to find dYL, a sequence of states is searched by
decreasing YL while adjusting s to stay just below the 2. SUPERCOMPUTER
(varying)νe trappingradius. Allsuchstatesarefoundto
becharacterizedbyEtot<−Eνe,infall−Eνe,burst,allthe That some basic features of supernova neu-
waydowntoYνe ≈0. Theconclusionisthatinthiscrude trinoemissioncanbemotivatedbyastick-figure-
model the deleptonization accompanying the νe burst is
essentiallycomplete,withEνe,burst≈¯ǫνe(M/mN)Ye,trap quality model provides a baseline degree of in-
evaluated attheinitial‘burststate.’ stant gratification. This is in contrast to the ex-
11Attheendofthesequence ofstates withdeclininglep-
plosionmanifest inan expandingsupernova rem-
ton number searched as described in the preceding foot-
nant’s kinetic energy, which, as a 1% subsidiary
note,westilldonothaveastatewithEtot=−Eνe,infall−
Eνe,burst; further energy loss is needed. But it is found detailfromnature’sperspective,hasbeenrequir-
at the end of this searched sequence that the population ingdecadesofdetailedstudybynumerouspeople
of thermal neutrinos of all flavors has begun to make a to understand [18].14
nontrivial contribution to Etot, so their emission is now
a viable means of prompt energy loss so long as the star
is outside the trappingradius ofthese flavors. Hence our 5.3 to 0. The luminosities Lνi and average energies ǫ¯νi
‘post-bouncestate’afterthedustsettlesisthatcharacter- for the various species are found by equating two differ-
izedbyYL=Ye andthehighestentropyperbaryonthat ent expressions forthe neutrinoluminosities necessaryto
keeps the star within the smallest trapping radius (that allow a transition between two successive neutron star
of νµ,τ and their antiparticles, which as described in the states. Defining dEtot to be the difference in Etot be-
previous footnote have the largest mean freepath). This tween two successive states in the sequence, and assum-
isthestatewiths 5.3andR 130kmatthefarright ing that the star contrives to release an equal amount
≈ ≈
of Fig. 3. The additional energy loss referred to in the of energy in each flavor, one expression is the ratio of
maintext isthe differencebetween Etot ofthisstateand dEtot/6totheneutrinodiffusiontimescaleτν (givenfor
−
1−2ESoνem,ienffarlalc−tiEonνeo,bfutrhsti.s would actually be transferred on tinhethvearisoeuqsuesnpceec.iesAinseacopnrdevieoxupsrefsosoiotnnotfeo)r alutmthinaotsiptoyinits
a dynamical time scale to layers exterior to the core—in Lνi = 4πR2(7π2/960)(ǫ¯νi/3)4, a neutrino version of the
an incipient ‘prompt’ explosion, it was (rashly) hoped in Stefan-Boltzmann law with Teff,i ≈ ¯ǫνi/3. Once the ǫ¯νi
bygone days—by processes that cannot be discussed in andLνi aredetermined,thesequenceisrelatedtoatime
detail here, such as the direct momentum impulse of the coordinate by defining dt ≡ −dEtot/( iLνi) for each
reboundingcore,‘pdV’work,shockheating, andsoon. stepofthesequence.
P
13The sequence of neutron star states in Fig. 3 is that 14Suchisthe(probablyunavoidable)resultofouranthro-
characterized by YL = Ye and s stepped down from pocentricchauvinism. Fordecades, andnotyetsureeven
7
But stick figures do not ultimately satisfy, of 2. G. M. Fuller, W. A. Fowler, M. J. Newman,
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15
tivelyillustrativethanpredictivelyrobust. This 5. G. E. Brown, H. A. Bethe, G. Baym, Nucl.
is one indication of the ultimate necessity of de- Phys. A 375 (1982) 481–532.
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fulnessofthisproblemarestilljustoverthehori- Universe, Vol. 69 of Frontiers in Physics,
zon, but efforts to meet this problem in its full Addison-Wesley, 1990.
measure are already underway [19,20]. 14. R.Buras,M.Rampp,H.-T.Janka,K.Kifoni-
dis,Astron.Astrophys.447(2006)1049–1092.
15. M. Liebend¨orfer, O. E. B. Messer, A. Mez-
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