Neutrino-cooled Accretion Disks around Spinning Black Holes

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by  Wen-xin Chen

Neutrino-cooled Accretion Disks around
Spinning Black Holes
Wen-xin Chen∗ and Andrei M. Beloborodov∗
∗PhysicsDepartmentandColumbiaAstrophysicsLaboratory,ColumbiaUniversity,538West
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120thStreet,NewYork,NY10027
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Abstract. We calculate the structureof accretiondiskarounda spinningblack holefor accretion
n rates M˙ =0.01−10M⊙ s−1. The model is fully relativistic and treats accurately the disk micro-
a physics including neutrino emissivity, opacity, electron degeneracy,and nuclear composition.We
J find that the accretion flow always regulates itself to a mildly degenerate state with the proton-

8 to-nucleon ratioYe <∼0.1 and becomes very neutron-rich.The disk has a well defined “ignition”
radius where neutrino flux raises dramatically, cooling becomes efficient, andY suddenly drops.
e
1 Wealsocalculateothercharacteristicradiiofthedisk,includingthen -opaqueandn -trappingradii,
v andshowtheirdependenceonM˙.Accretiondisksaroundfast-rotatingblackholesproduceintense
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neutrinofluxeswhichmaydepositenoughenergyabovethedisktogenerateaGRBjet.
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0 Most GRB models assume a short-lived accretion disk as a primary source of energy. Such
6 diskshavehugeaccretionratesM˙ =0.01−10M⊙ s−1 andtheircoolingmechanismisneutrino
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emission[1,2,3].Specialfeaturesofneutrino-cooleddisksare:
/
h
p —Themaincoolingisduetoreactions p+e−→n+n andn+e+→ p+n¯.
- —Electrondegeneracyaffectsthediskstructure.
o
r —Thediskcanbeopaquetoneutrinoswhichaffectsthecoolingrate.
t
s —Nuclearcompositionchangeswithradius,consumingenergy.
a
—Neutrinocoolingandelectrondegeneracyleadtoveryhighneutronrichness.
:
v
i The disk model may be formulated following the classical vertically-integrated approach of
X f
Shakura & Sunyaev and approximating the azimuthal motion in the disk, u , as Keplerian
r
rotation. A small radial velocity is superimposedon this rotation. It is approximately given by
a
ur =a B(H/r)2ruf whereH ishalf-thicknessofthedisk(foundfromhydrostaticbalance),Bis
anumericalfactordeterminedbyKerrmetric,anda =0.1−0.01.
Themainequationsoftheaccretiondiskareasfollows.
d(UH) U+Pd(r H)
Energyconservation: F+−F−=cur − . (1)
(cid:20) dr r dr (cid:21)
1 r dY d
Leptonconservation: (N˙n¯−N˙n )=cur e + (nn −nn¯) . (2)
H (cid:20)m dr dr (cid:21)
p
r
Chargeconservation: ne−−ne+ =Yem . (3)
p
1−Y
Chemicalbalance: m e−m n =kTln e +(mn−mp)c2. (4)
(cid:18) Y (cid:19)
e
HereF+ istheviscouslydissipatedenergyperunitarea,F− istheneutrinoenergyflux,N˙n and
N˙n¯ arethenumberfluxesofneutrinosandantineutrinos,r isthemassdensity,T istemperature,
Yeistheproton-to-baryonratio,m eandm n arethechemicalpotentialsofelectronsandneutrinos,
ne− andne+ arenumberdensitiesofelectronsandpositrons.TheenergydensityU andpressure
P include contributions of baryons, electrons, positrons, radiation, and neutrinos. They are
calculatedbyintegratingthecorrespondingdistributionfunctions(Fermi-Diracdistributionsfor
e± andn ,n¯ ifthediskisopaquetoneutrinos).
Notethatweincludetheadvectiontermsinthelawsofenergyandleptonconservation(right-
handsideofeqs.1and2).Theydescribethemostimportanteffectsofadvection.Othereffects
ofheatstoragein thedisk—in particular, thedynamicaleffectsoftheradialpressuregradient
onur anduf —arerelativelysmallandneglectedhere,i.e.thestandarda -diskmodelisusedto
calculateuf andur.
The chemical balance and m n are used only in the n -opaque region; then the cooling rate is
calculatedasF− =cUn /tn +cUn¯/tn¯.Inthen -transparentregion,F− equals2H(U˙e−p+U˙e+n)
whereU˙e−p andU˙e+n aretheratesofenergyreleasebyreactionse−+p→n+n ande++n→
p+n¯.Theabundanceofa -particlesisdeterminedbythenuclearstatisticalequilibrium.
We have solved the set of disk equations numerically starting from an outer region where
no neutrino cooling takes place and assuming that the infalling matter is initially made of a
particles(heavierelementsaredecomposedinto a -particlesasthematterapproachesthe black
hole).AschematicpictureoftheaccretiondiskanditscharacteristicradiiareshowninFigure1.
FIGURE1. Schematicpictureofthediskwithcharacteristicradiiindicated.
Theouterregionr>50r (r =2GM/c2isthegravitationalradiusoftheblackhole)isdom-
g g
inated by a particles. Neutrino cooling is negligible in this region and the viscouslydissipated
heatis storedin thediskandadvectedinward.Thedestructionofa particlesatra ∼50rg con-
sumes ∼7 MeV per baryon, which makes the disk somewhat thinner. At the "ignition" radius
rign (slightly smaller than ra ), kT reaches ∼ MeV and neutrino cooling becomes significant,
furtherreducingdiskscale-heightH/r.Withdecreasingradiusthediskbecomesopaquetoneu-
trinos(firstton andthenton¯).IntheinnermostregionofdiskswithM˙ >2M⊙s−1theneutrinos
aretrappedandadvectedtowardtheblackhole.Figure2showsthefoundcharacteristicradiias
functionsofM˙ fordiskswitha =0.1aroundblackholeswithspinparametersa=0and0.95.
Wefindtwogeneralpropertiesofthedisk:
1. — It regulates itself to a mildly degenerate state with m = 1 ∼ 3kT. The reason of this
e
regulationisthenegativefeedbackofdegeneracyandneutronizationonthecoolingrate:higher
FIGURE 2. Characteristic radii in the accretion disk as a function of M˙ for a black hole of mass
M =3M⊙ and spin parameter a=0.95 (left) and 0 (right). Viscosity parameter a =0.1 is assumed.
NeutrinocoolingdoesnotignitewhenM˙ isbelow0.02M⊙s−1 forKerrand0.1M⊙s−1 forSchwarzschild
black holes. Neutrino trapping takes place for Kerr black holes when M˙ >2M⊙ s−1. The disk extends
downtothemarginallystableorbit,whichis3r fora=0and≈r fora=0.95.
g g
degeneracym e/kT →fewerpositrons(n+/n−∼e−m e/kT)→lowerequilibriumrateofneutrino
emission→lowercoolingrate→highertemperature→lowerdegeneracy.
2. — In a broad range of relevant accretion rates the disk is found to reach Y ∼ 0.1. It
e
correspondstoaveryhighneutron-to-protonratio∼10.
Figure 3 shows examples of the disk structure around a Kerr black hole with a=0.95 with
accretion rate 0.2M⊙ s−1, for three different values of viscosity parameter a . The transitions
atthecharacteristicradiimanifestthemselvesinthenon-monotonicbehaviorofthedegeneracy
parameterh =m /kT,F−/F+,Y ,andH/r.InparticularthesharpmaximumofF−/F+appears
e e
attheignitionradius.Thediskisquitethinatr<r andcooledlocally,F−≈F+.
ign
ThehighneutronrichnesshasimportantimplicationsfortheglobalpictureofGRBexplosion.
Whentheneutron-richmaterialisejectedinarelativisticjetanddevelopsalargeLorentzfactor,
theneutronsdecayatdistance∼1016 cmandaffecttheobservedexplosion.
Disks around rapidly rotating black holes produce a high neutrino flux. It deposits energy
abovethedisk(vianeutrinoannihilationn +n¯ →e++e−)andcandriveanoutflowwithpower
L>1051 erg/s.Adetailedstudyofn n¯ annihilationaroundaKerrblackholeisinpreparation.
ACKNOWLEDGMENTS
Thisworkwas supportedbyNASA grant NAG5-13382.
REFERENCES
1. R.Popham,S.E.Woosley,&C.Fryer,Astrophys.J.,518,356(1999).
2. A.M.Beloborodov,Astrophys.J.,588,931(2003).
3. K.Kohri,R.Narayan,&T.Piran,Astrophys.J.,629,341(2005).
18 1013
a =0.1 a =0.1
2T/mc)e 111246 aa ==00..0031 3m) 11001112 aa ==00..0031
ature (k 1 80 sity (g/c 1100190
mper 6 Den 108
e 4
T 2 107
0 106
1 10 100 1000 1 10 100 1000
r/r r/r
g g
5.5 1.4
a =0.1 a =0.1
4 .55 aa ==00..0031 1.2 aa ==00..0031
4 1
hgeneracy 23 ..355 -+F/F 00..68
e
D 2 0.4
1.5
0.2
1
0.5 0
1 10 100 1000 1 10 100 1000
r/r r/r
g g
0.8 102
a =0.1 a =0.1
0.7 aa ==00..0031 h 101 aa ==00..0031
0.6 ept 100
D
0.5 cal 10-1
H/R 00..34 no Opti 10-2
0.2 eutri 10-3
N
0.1 10-4
0 10-5
1 10 100 1000 1 10 100 1000
r/r r/r
g g
0.55
a =0.1 a =0.1
0.5 a =0.03 1 a =0.03
Ye 0.45 a =0.01 a =0.01
n mass fraction 00 000..23...23455 ee neucleon Xf 000...468
oto 0.15 Fr
Pr 0.1 0.2
0.05
0 0
1 10 100 1000 1 10 100 1000
r/r r/r
g g
FIGURE3. NumericalsolutionforaccretiondiskwithM˙ =0.2M⊙s−1 aroundablackholewithmass
M=3M⊙ andspinparametera=0.95.Threemodelsareshownwithviscosityparametera =0.1,0.03
and0.01.Thedegeneracyparameterh isdefinedasm /kT,X isthemassfractionoffreenucleons.
e free