Table Of Content

Neutrino-cooled Accretion Disks around Spinning Black Holes Wen-xin Chen∗ and Andrei M. Beloborodov∗ ∗PhysicsDepartmentandColumbiaAstrophysicsLaboratory,ColumbiaUniversity,538West 6 120thStreet,NewYork,NY10027 0 0 2 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 7 neutrinofluxeswhichmaydepositenoughenergyabovethedisktogenerateaGRBjet. 5 1 1 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 0 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