aa r X i v : . [ a s t r o - ph ] O c t Hyper-accreting black holes
Andrei M. Beloborodov
Physics Department and Columbia Astrophysics Laboratory, Columbia University, 538 West 120thStreet New York, NY 10027, USAAlso at Astro-Space Center of Lebedev Physical Institute, Profsojuznaja 84/32, Moscow 117810,Russia
Abstract.
Hyper-accretion disks are short-lived, powerful sources of neutrinos and magnetized jets.Such disks are plausible sources of gamma-ray bursts. This review describes the disk structure, theneutrino conversion to electron-positron plasma around the disk, and the post-burst evolution.
Keywords:
Accretion – accretion disks – dense matter – gamma rays: bursts
PACS:
INTRODUCTION
Hyper-accretion disks form when a neutron star merges with another compact object,neutron star or a black hole. Recent numerical simulations of mergers [1, 2, 3] are fullyrelativistic and show how most of the mass of the binary system disappears behind theevent horizon in about 10 ms, leaving a rotating debris disk around the black hole. Themass of this centrifugally supported disk is m ∼ . − . ⊙ , similar to what wasfound in previous non-relativistic simulations (e.g. [4, 5]). The ensuing disk accretionis not followed by the merger simulations. It is established on a viscous timescale t visc ∼ . t visc , releasing an energy comparable to mc and emitting copious neutrinos. This diskhas an accretion rate ˙ M ∼ ( M ⊙ / s ) ( m / . ⊙ ) ( t visc / . ) − .Similar neutrino-emitting disks may form during the core-collapse of massive starsif the stellar material has a sufficient angular momentum [8, 9]. These hypotheticalobjects are often called “collapsars.” After the formation of a central black hole of mass M ∼ ⊙ collapsars develop an accretion disk that is fed by the continually infallingstellar material. The high accretion rate ˙ M ∼ . ⊙ s − is sustained for ∼
10 s (the core-collapse timescale). Recent relativistic MHD simulations of this accretion show how theblack hole could accumulate a large magnetic flux and create jets via the Blandford-Znajek process [10].The studies of hyper-accretion disks are greatly stimulated by observations of cos-mological gamma-ray bursts (GRBs, see [11] for a review). Hyper-accretion is ex-pected to produce hyper-jets on a timescale ∼ . −
10 s. If a fraction e jet of the ac-cretion power ˙ Mc is channeled to a relativistic jet, it leads to an explosion with energy E jet ∼ × ( M acc / M ⊙ )( e jet / − ) erg, where M acc is the mass accreted through thedisk. The energy and duration of the jet is consistent with GRB observations.Hyper-accretion disks are markedly different from normal accretion disks in X-rayinaries and AGN. Their optical depth to photon scattering is enormous and radiationis trapped inside the disk, being advected by the matter into the black hole. However,the disk can be efficiently cooled by neutrino emission. Significant neutrino losses canoccur when ˙ M > − M ⊙ s − and make the disk relatively thin and neutron rich.The accreting black hole is expected to have a significant angular momentum, becauseit forms from rotating matter and is further spun up by accretion. The black-hole spinhelps the jet formation through the Blandford-Znajek process. It also affects the black-hole spacetime in such a way that the disk extends to smaller radii and the overallefficiency of accretion significantly increases. For example, the inner radius of the diskaround a maximally-rotating black hole (spin parameter a =
1) is reduced by a factor of6 compared with the Schwarzschild case a =
0. This leads to a higher temperature and ahigher neutrino intensity above the disk, increasing the rate of neutrino annihilation into e ± pairs. Therefore, disks around rapidly spinning black holes can deposit an interestingfraction of their energy into the e ± plasma outside the disk and facilitate the formationof ultra-relativistic jets.The size of a hyper-accretion disk depends on the specific angular momentum of theaccreting matter, l , which is modest in neutron-star mergers and probably even smaller incollapsars. The accretion flows in collapsars are quasi-spherical and may form a special“mini-disk” that is not supported centrifugally and instead accretes on the free-fall time.Larger l leads to standard viscous accretion, which leaves a relict disk carrying the initialangular momentum of the accreted matter. The relict disk gradually spreads to largerradii, and its late evolution may be relevant to the post-burst activity of GRBs. VISCOUS DISKS
As matter spirals into the black hole, it is viscously heated: the gravitational energyis converted to heat. The heat is distributed between nuclear matter, radiation, and e ± pairs, in perfect thermodynamic equilibrium. In particular, the equilibrium e ± populationis maintained. As discussed below, electrons are mildly degenerate in neutrino-cooleddisks, which affects the e ± density. The equilibrium microphysics is determined by onlythree parameters: temperature T , baryon mass density r , and electron fraction Y e (equalto the charged nucleon fraction). Other parameters — e.g. the electron chemical potential m e and density of e ± pairs, n ± — are derived from T , r and Y e . At radii r < ∼ cm,temperature and density are high enough to maintain the nuclear statistical equilibrium,which determines the abundances of all nuclei. Nuclear matter in the disk is dissociatedinto free nucleons n and p inside radius r a = ( − ) r g where r g ≡ GM / c . Thetemperature at this radius is kT < ∼ kT > e ± capture onto nucleons: e − + p → n + n e , e + + n → p + ¯ n e . (1)The escaping neutrinos not only cool down the disk — they also change its electronfraction Y e if the emission rates of n and ¯ n are not exactly equal. The first goal of thedisk modeling is to find T , r , Y e , and self-consistently evaluate the neutrino losses. odeling disk accretion Accretion is quasi-steady in the inner region of the disk where t visc is shorter than thetimescale of ˙ M evolution. Viscosity in accretion disks is caused by MHD turbulence,sustained by the magneto-rotational instability [12]. It creates an effective kinematicviscosity coefficient n , which may be related to the half-thickness of the disk, H , andsound speed, c s : n = a c s H , where a ∼ . − . a prescription. Theequations include radial transport of heat and lepton number.– Local microphysics is calculated exactly: nuclear composition, electron degeneracy,neutrino emissivity and opacity etc., using the equilibrium distribution functions forall species except neutrinos. Neutrinos are modeled separately in the opaque andtransparent zones of the disk, matching at the transition between the two zones.– The model provides only vertically averaged T , r , and Y e (which are approximatelyequal to their values in the midplane of the disk). The diffusion of neutrinos inthe opaque zone is treated in the simplest one-zone approximation (using escapeprobability). This gives a good approximation for the energy losses, however, doesnot give the exact neutrino spectrum emerging from the opaque zone.The vertically-integrated approximation provides no information about the verticalstructure of the disk and its corona. The vertical structure may be eventually understoodwith global 3D time-dependent MHD simulations that include energy losses, althoughthe results of such simulations generally depend on the assumed initial magnetic con-figuration. The behavior of magnetic field on large scales is coupled to the local turbu-lence cascade that extends to scales ≪ H . The microscopic magnetic Prandtl number forhyper-accretion disks has been recently estimated in [20]. The models discussed here areaware of the MHD issues only through the value of a . They are computationally muchcheaper than MHD simulations and allow one to study the disk in a broad range of ˙ M and a .Vertically-integrated disks are described by 1D equations that express conservation ofbaryon number, energy, and momentum (angular and radial) in Kerr spacetime (see [21]for a review). The full set of these equations can be solved [22], and the solutions showthat the deviation from circular Keplerian rotation is small ( < ∼ Thus, the angular velocity of the disk can be approximated by its Keplerian value W K .A small radial velocity is superimposed on this rotation: u r = − a S − c s ( H / r ) , where A strong reduction of W below W K can occur in the limit of a large steady disk with no cooling. Thislimit does not apply to hyper-accretion disks which are transient and have a moderate radius. IGURE 1. Left panel : Contours of the equilibrium Y e ( T , r ) on the T - r plane for n -transparent matter.The electrons become degenerate near the dashed line given by kT deg = ¯ hc ( r / m p ) / = . r / MeV.The Y e contours are calculated assuming that the nuclear matter is dissociated into free nucleons; theyare invalid in the shaded region where matter is dominated by composite nuclei. The “neutronizationline” Y e = . kT n = r / MeV.
Right panel : The equilibrium Y e ( T , r ) for n -opaque matterwith neutrino chemical potential m n =
0. The free-nucleon region is the same as in the left panel. Thecalculation of the equilibrium Y e is now extended into the region of composite nuclei. The neutronizationline Y e = . kT n = . r / MeV. c s = ( P / r ) / is the isothermal sound speed, H is the half-thickness of the disk; S ( r ) is a numerical factor determined by the inner boundary condition [17]. This descriptionof the velocity field in the disk is a good approximation everywhere except in the veryvicinity of the inner boundary where | u r | exceeds c s .In contrast to accretion disks in X-ray binaries and AGN, there is one more conserva-tion law that must be taken into account: conservation of lepton number,1 H ( ˙ N ¯ n − ˙ N n ) = u r (cid:20) r m p dY e dr + ddr ( n n − n ¯ n ) (cid:21) . (2)Here ˙ N n and ˙ N ¯ n are the number fluxes of neutrinos and anti-neutrinos per unit area (fromone face of the disk), n n and n ¯ n are the number densities of neutrinos and anti-neutrinosinside the disk. This equation determines Y e , which is related to the neutron-to-protonratio by Y e = ( n n / n p + ) − and greatly affects the rate of neutrino cooling.In the models solved in [17] and shown below, Y e is calculated using Eq. (2). Note,however, that throughout most of the neutrino-cooled disk, the right side of Eq. (2) issmall compared with each of the two terms on the left side, and Y e is nearly equal to thelocal equilibrium value such that ˙ N ¯ n ≈ ˙ N n . This equilibrium Y e is determined by the localtemperature and density and found for both neutrino-opaque and neutrino-transparentmatter [23, 24, 25]. It is shown in Fig. 1. IGURE 2.
Boundaries of different regions on the r - ˙ M plane for disks around a black hole of mass M = ⊙ and spin parameter a = .
95. Neutrino cooling is inefficient in the shaded region below the" n -cooled" curve and above the “trapped” curve. The shaded region marked “unstable” is excluded: thesteady model is inconsistent in this region because of the gravitational instability. The disk extends downto the marginally stable orbit of radius r ms ≈ r g where r g = GM / c . Left panel : Disks with viscosityparameter a = . Right panel : Disks with viscosity parameter a = .
01. (From [17].)
Overview of disk properties
Hyper-accretion disks have several zones separated by the following characteristicradii:1. Radius r a where 50% of a -particles are decomposed into free nucleons. Thedestruction of a -particles consumes 7 MeV per nucleon, which makes the diskthinner.2. "Ignition" radius r ign where neutrino emission switches on. At this radius, the meanelectron energy becomes comparable to ( m n − m p ) c , enabling the capture reaction e − + p → n + n . Then neutrino cooling due to reactions (1) becomes significant,further reducing the disk thickness H / r .3. Radius r n where the disk becomes opaque to neutrinos and they relax to a thermaldistribution. The disk is still almost transparent to anti-neutrinos at this radius.4. Radius r ¯ n where the disk becomes opaque to anti-neutrinos, so that both n and ¯ n are now in thermal equilibrium with the matter. The disk is still cooled efficientlyat this radius since n and ¯ n diffuse and escape the flow faster than it accretes intothe black hole.5. Radius r tr where neutrino diffusion out of the disk becomes slower than accretion,and neutrinos get trapped and advected into the black hole.The different zones of the disk are shown on the ˙ M − r diagram in Fig. 2. In addition,this figure shows the zone of gravitational instability.hree characteristic accretion rates can be defined: ˙ M ign above which the disk isneutrino-cooled in the inner region, ˙ M opaque above which the disk is opaque to neutrinosin the inner region, and ˙ M trap above which the trapping of neutrinos occurs in the innerregion. The dependence of ˙ M ign , ˙ M opaque , and ˙ M trap on a is well approximated by thefollowing power laws [17],˙ M ign = K ign (cid:16) a . (cid:17) / , ˙ M opaque = K opaque (cid:16) a . (cid:17) , ˙ M trap = K trap (cid:16) a . (cid:17) / . (3)The normalization factors K depend on the black hole spin a . For a = .
95 they are K ign = .
021 M ⊙ s − , K opaque = .
06 M ⊙ s − , K trap = . ⊙ s − and for a = K ign = .
071 M ⊙ s − , K opaque = . ⊙ s − , K trap = . ⊙ s − .To complete this short guide to quasi-steady viscous disks, Figs. 3-4 show T , r , Y e ,and H / r for a model with ˙ M = . ⊙ s − , M = ⊙ , and a = . T e m pe r a t u r e ( k T / m e c ) r/r g a=0.95 a =0.1 a =0.03 a =0.01 10
1 10 100 1000 D en s i t y ( g / c m ) r/r g a=0.95 a =0.1 a =0.03 a =0.01 FIGURE 3.
Disk with ˙ M = . ⊙ s − around a black hole of mass M = ⊙ and spin a = .
95. Threemodels are shown with viscosity a = .
1, 0.03, and 0.01. Radius is measured in units of r g = GM / c =
10 km.
Left panel : Temperature in units of m e c . Right panel:
Mass density. (From [17].) Y e r/r g a=0.95 a =0.1 a =0.03 a =0.01 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 1 10 100 1000 H / R r/r g a=0.95 a =0.1 a =0.03 a =0.01 FIGURE 4.
Electron fraction Y e and thickness of the disk H / r for the same three models as in Fig. 3. r/r g a =0.1 P b /PP e /PP g /PP n /P 0.001 0.01 0.1 1 1 10 100 1000 r/r g a =0.01 P b /PP e /PP g /PP n /P FIGURE 5.
Contributions to total pressure P from baryons, P b , electrons and positrons P e = P e − + P e + ,radiation P g , and neutrinos P n + P ¯ n for the accretion disk with ˙ M = . ⊙ s − around a black hole ofmass M = ⊙ and spin a = . Left panel:
Model with viscosity parameter a = . Right panel:
Model with a = .
01. (From [17].)
The main properties of the neutrino-cooled disk (i.e. at r < r ign ) may be summarizedas follows [17]. ◦ The disk is relatively thin, H / r ∼ . − .
3, especially in the inner region where mostof accretion energy is released. ◦ The n -cooled disk is locally very close to b -equilibrium, ˙ N n ≈ ˙ N ¯ n . In particular, therelation between r , T , and Y e calculated under the equilibrium assumption (Fig. 1) issatisfied with a high accuracy. ◦ Degeneracy of electrons in the disk significantly suppresses the positron density n e + .However, the strong degeneracy limit is not applicable — the disk regulates itselfto a mildly degenerate state with m e / kT = −
3. The reason of this regulation isthe negative feedback of degeneracy on the cooling rate: higher degeneracy m e / kT → fewer electrons (lower Y e ) and positrons ( n e + / n e − ∼ e − m e / kT ) → weaker neutrinoemission → lower cooling rate → higher temperature → lower degeneracy. ◦ Pressure in n -cooled disks is dominated by baryons, P ≈ P b = ( r / m p ) kT , most ofwhich are neutrons (Fig. 5 shows contributions to pressure for two sample models). ◦ All n -cooled disks are very neutron rich in the inner region, with Y e ∼ . NEUTRINO ANNIHILATION AROUND THE DISK
The emitted neutrinos and anti-neutrinos can collide and convert to e ± , thereby deposit-ing energy [26]. The emission of tau and muon neutrinos is negligible [17], so onlyreaction n e + ¯ n e → e − + e + can be considered. Its cross section (assuming center-of-momentum energy ≫ m e c ) is given by s n ¯ n ≈ . × − ( p n · p ¯ n ) ( p n p n ) − cm where p n , p ¯ n are the 4-momenta of n e and ¯ n e , expressed in units of m e c . The cross section ismall and only a small fraction e ∼ − − − of the total neutrino luminosity L con-verts to e ± plasma. Nevertheless, this energy may be sufficient to drive a relativistic jet(or help the formation of a magnetically-dominated jet) since it occurs above the diskwhere the mass density is relatively low, especially near the rotation axis [27].Neutrinos emitted by the disk follow null geodesics in Kerr spacetime. The efficiency e of their annihilation can be calculated numerically by tracing the geodesics, evaluatingthe local energy deposition rate ˙ q n ¯ n [erg s − cm − ] everywhere around the black hole,and then integrating over volume to obtain the net energy deposition rate ˙ E n ¯ n (energy atinfinity per unit time at infinity). The neutrino emission and annihilation is concentratednear the black hole, where accretion is expected to be quasi-steady. ˙ E n ¯ n depends on fourparameters that specify the steady disk model: ˙ M , a , M , and a .The energy deposition rate ˙ E n ¯ n was estimated in [13], approximating geodesics bystraight lines. Fully relativistic calculations were made for several toy models, in par-ticular for disks or tori with uniform temperature or other arbitrary distribution of tem-perature or entropy (see [28] and refs. therein). Recently, the relativistic calculation fora realistic disk around a spinning black hole has been done (Zalamea & Beloborodov,in preparation). ˙ q n ¯ n has been obtained everywhere around the black hole, including itsergosphere, and the dependence of ˙ E n ¯ n on ˙ M , a , M , and a has been determined.Besides tracing the geodesics, this calculation involves a model for neutrino and anti-neutrino spectra emitted by the disk. Using the results of [17], it is straightforward toevaluate the spectra from the transparent zone of the disk. It is more difficult to find thespectrum that emerges from the opaque zone, because the neutrino transport in this zonedepends on the unknown vertical distribution of viscous heating. Various assumptionsmay be made about this distribution [15, 29, 30, 31, 32], including a strong heatingof the magnetic corona above the disk. Note that the corona of a hyper-accretion diskis always in thermodynamic equilibrium, and its temperature T c is determined by itsthermal energy density U c = U g + U e ≈ a r T c where a r = . × − erg cm − K − is the radiation constant. U c is generally smaller than the energy density inside the disk.Therefore, relocating the heating from the disk to its corona cannot significantly increasethe energies of emitted neutrinos.Fortunately, a robust estimate can be obtained for ˙ E n ¯ n in spite of the uncertainty inthe vertical structure of the disk. It is easy to see that all detailed models of neutrinospectrum formation must predict practically the same rate of n ¯ n annihilation above a neutrino-cooled disk. For such a disk, neutrinos carry away a fixed energy flux F − ≈ F + where F + ∼ M W S ( r ) / p is the rate of viscous heating. Therefore, transfer modelsthat predict a higher average energy of emitted neutrinos, E av , must also predict alower number density of the neutrinos above the disk, n ∼ F − / E av c (cid:181) E − . Since theannihilation cross section s n ¯ n (cid:181) E , one finds that the reaction rate ˙ n n ¯ n ∼ c s n ¯ n n n n ¯ n is independent of E av . The energy deposition rate ˙ q n ¯ n ∼ ˙ n n ¯ n E av is proportional to E av .It cannot be changed without a substantial change in temperature (or electron chemicalpotential) of the neutrino source, which would require a huge change in energy densityand therefore is hardly possible.Detailed calculations confirm that ˙ E n ¯ n weakly depends on the details of the verticalstructure and n , ¯ n transfer in the disk. Consider two extreme models for the opaque zone. IGURE 6.
Total energy deposition rate due to n ¯ n annihilation outside the black-hole horizon, ˙ E n ¯ n , asa function of the disk accretion rate, ˙ M . The two characteristic accretion rates ˙ M ign and ˙ M trap depend onthe viscosity parameter a = . a = . M = ⊙ . ˙ E n ¯ n strongly depends on the spin parameter of the black hole; the numerical resultsare shown for two cases: a = a = .
95 (squares). The uncertainty in the vertical structureof the accretion disk leads to a small uncertainty in ˙ E n ¯ n as illustrated by two extreme models: Model A(open symbols) and Model B (filled symbols), see the text for details. The results of both models are wellapproximated by simple Model C that is given by Eq. (4) and also shown in the figure, by solid line for n -opaque disks ( ˙ M > ˙ M opaque ) and by dashed line for n -transparent disks ( ˙ M < ˙ M opaque ). (From Zalamea& Beloborodov, in preparation.) Model A:
Neutrinos n e and ¯ n e are emitted with the same distributions as found inside thedisk (same temperature T and chemical potential m n ). The distribution normalization is,however, reduced compared with the thermal level inside the disk, so that the emergingemission carries away the known energy fluxes F n and F ¯ n that are found in [17]. Model B:
Neutrinos are emitted with a thermal Fermi-Dirac spectrum with chemicalpotential m n = T eff . The effective temperature isdefined by ( / ) s T = F − , where F − = F n + F ¯ n is the energy flux from one side of thedisk and s = a r c / / n e and ¯ n e — the emission of other neutrino species is negligible.The temperature T inside an opaque disk exceeds T eff by factor T / T eff ∼ t / n whenthe neutrino optical depth t n ≫
1. The neutrino chemical potential is modest and, ap-proximately, E av (cid:181) T . Hence, even the extreme Model A gives a moderate enhancementof ˙ E n ¯ n , by the factor T / T eff . Zalamea & Beloborodov (in preparation) calculated ˙ E n ¯ n n both Models A and B. The results of numerical calculations are shown in Fig. 6 anddemonstrate that the difference between the two models is indeed small.It is instructive then to consider Model C : same as Model B except that F − = F + isassumed at all radii. The assumption is clearly incorrect outside the region r tr < r < r ign .Nevertheless, this simplest model gives a good approximation to ˙ E n ¯ n in a broad range˙ M ign < ˙ M < ˙ M trap (Fig. 6). Note that ˙ E n ¯ n in Model C is explicitly independent of a .However, the range of ˙ M where Model C is applicable depends on a (Eq. 3).The scaling of ˙ E n ¯ n with ˙ M is easy to evaluate analytically. The effective surfacetemperature is related to F − ≈ F + by T (cid:181) F − (cid:181) ˙ M . The neutrino number densityabove the disk is proportional to T (cid:181) ˙ M / . The annihilation cross-section s n ¯ n (cid:181) T (assuming kT eff > m e c ). Hence the energy deposition rate ˙ E n ¯ n (cid:181) T (cid:181) ˙ M / , and onecan write ˙ E n ¯ n = ˙ E ( a ) (cid:18) ˙ M M ⊙ s − (cid:19) / , ˙ M ign < ˙ M < ˙ M trap . (4)The normalization factor ˙ E depends on the black hole spin a and must be calculatednumerically. For example, a = .
95 gives ˙ E ≈ erg s − , which implies the efficiency e = ˙ E n ¯ n / L ≈ . ( ˙ M / M ⊙ s − ) / . It is much larger than the corresponding value for anon-rotating black hole, by two orders of magnitude.The strong dependence of ˙ E n ¯ n on a may be seen from the following rough estimate.The neutrino luminosity L peaks at r peak that is a few times the inner radius of the disk, r ms ( a ) — the marginally stable orbit, which is determined by a . The luminosity dependson r ms approximately as r − , and T eff at r peak scales as ( L / r ) / (cid:181) r − / . The energydeposition rate ˙ q n ¯ n scales as T and ˙ E n ¯ n scales as r ˙ q n ¯ n , which yields ˙ E n ¯ n (cid:181) r − / .Then the reduction in r ms by a factor of 3 (as a increases from 0 to 0.95) gives a factorof 60 in ˙ E n ¯ n . This estimate neglects the fact that the gravitational bending of neutrinotrajectories is stronger for smaller r ms . Stronger bending implies a larger average anglebetween neutrinos and leads to an additional enhancement of ˙ E n ¯ n . Therefore, a steeperdependence of ˙ E n ¯ n on r ms is found in numerical simulations. A simple power law˙ E n ¯ n (cid:181) r − . is an excellent approximation to the numerical results for 0 < a < . r g < r ms < r g (Zalamea & Beloborodov, in preparation).Note that ˙ E n ¯ n is defined as the total energy deposition rate outside the event horizon(including the ergosphere). A significant fraction of the created e ± plasma must fall intothe black hole, and only the remaining fraction of ˙ E n ¯ n will add energy to the jet. Thisfraction depends on the plasma dynamics outside the disk, which is affected by magneticfields and is hard to calculate without additional assumptions. The deviation of Model B from Model C at ˙ M ∼ M ign is caused by the overshooting of F − above F + ,which happens just inside of r ign (see Fig. 7 and 16 in [17]). As hot matter accretes into the neutrino-cooled region, its stored heat is quickly emitted with F − / F + reaching ∼ r ≈ r ign /
2. For disks with r ign / ∼ a few r g , this leads to the enhancement of ˙ E n ¯ n by the factor ∼ / compared with Model C thatassumes F − = F + . IGURE 7.
The shaded region shows the range of angular momenta l that form a mini-disk withinsufficient centrifugal support, leading to accretion on the free-fall timescale. l is defined as the angularmomentum of the accretion flow in the equatorial plane; angular momentum decreases toward the polaraxis (shells r = const are assumed to have a uniform angular velocity W ≪ W K at r ≫ r g ). (From Zalamea& Beloborodov, in preparation). LOW-ANGULAR-MOMENTUM DISKS IN COLLAPSARS
The quasi-spherical accretion flows in collapsars create a centrifugally supported diskif the circularization radius of the flow is sufficiently large, r circ ∼ r g . A smallerdisk may not be centrifugally supported and then will accrete on a free-fall timescale[33]. It accretes so fast (super-sonically) that the effects of viscosity can be neglected.A steady model of this “mini-disk” was constructed in [33] and 2D time-dependenthydrodynamical simulations were performed in [34].The mini-disk can be thought of as a caustic in the equatorial plane of a rotatingaccretion flow. It absorbs the feeding infall, and this interaction releases energy, makingthe accretion radiatively efficient. With increasing angular momentum, the size of thedisk grows up to 14 r g c , at which point the centrifugal barrier stops accretion, so that itcan proceed only on a viscous timescale. Thus, the mini-disk model fills the gap betweentwo classical regimes of accretion — spherical ( l < r g c ) and standard accretion disk( l ≫ r g c ) — and is qualitatively different from both.The calculations of [33] were limited to the case of a Schwarzschild black hole.Recently, the model has been extended to the case of a Kerr black hole (Zalamea &Beloborodov, in preparation). Fig. 7 shows the range of angular momenta that leadto mini-disk formation around a black hole of spin 0 < a <
1. The critical angularmomentum for viscous disk formation sets the maximum radius of a mini-disk. Thisradius is ≈ r g = GM / c for a = ≈ r g for a = . r ∼ x ˙ M . M − (cid:18) rr g (cid:19) − / g cm − , U ∼ . r c (cid:18) rr g (cid:19) − . (5)Here x ∼ M . ≡ ˙ M / . ⊙ s − and M ≡ M / ⊙ . Thepostshock matter has m e / kT <
1, and U is dominated by radiation and e ± pairs, whichimplies U ≈ a r T . The postshock temperature is then T ≈ . × r / ( r / r g ) − / K.Disintegration of nuclei in the shock consumes only ∼ − of the energy releasedat r ∼ r g , so the postshock matter can be cooled only by neutrino emission. Neutrinoemission is dominated by two processes: (i) capture reactions (1) provide cooling rate˙ q c ≈ × T r erg cm − s − , and (ii) e ± annihilation e + + e − → n + ¯ n provides˙ q ± ≈ . × T erg cm − s − (see e.g. [35]). This gives˙ q c ≈ × r / (cid:18) rr g (cid:19) − / ergcm s , ˙ q ± ≈ r / (cid:18) rr g (cid:19) − / ergcm s , (6)Approximating the total ˙ q = ˙ q c + ˙ q ± ∼ ˙ q c , one finds˙ q t acc U ∼ − r / (cid:18) rr g (cid:19) M ∼ − x / ˙ M / . (cid:18) rr g (cid:19) − / M − , (7)where t acc ∼ − ( r / r g ) − / M s. The mini-disk is neutrino-cooled if ˙ q t acc / U > ∼ ⊙ s − . Thus, only high- ˙ M mini-disks are sandwiched by radiative shocks that stay near the equatorial plane in theinnermost region of the accretion flow. A large neutrino luminosity, up to ∼ . Mc ,is produced by such disks.For smaller accretion rates, the postshock matter is unable to cool on the free-fall timescale, and the neutrino luminosity from the inner region is suppressed bythe factor ˙ qt acc / U <
1. Then a hot low-angular-momentum bubble must grow aroundthe black hole. Mass flows into the bubble through the shock front that expands to r ≫ r g . Such a bubble is observed in low- ˙ M simulations in [34]. It resembles the bub-ble around viscous disks in the models of [9, 36], except for a slower rotation, lesscentrifugal support, and faster accretion. The shock expansion can be stopped whenit approaches ∼ r g = × M cm. Up to this radius, the postshock temperature, T ≈ × ˙ M / . ( r / r g ) − / K, is high enough to disintegrate nuclei at the density r ∼ × ˙ M . ( r / r g ) − / g cm − . As the shock expands to 40 r g , its energy de-creases to GMm p / r ≈ ( r / r g ) − MeV per nucleon, and a large fraction of this en-ergy is consumed by disintegration (8 MeV per nucleon); therefore, the shock stalls.
PREADING OF VISCOUS DISKS AND NUCLEAR BURNING
Formation of a viscous disk with r circ > r g implies that most of the angular momentumof accreting matter will be stored outside the black hole, in a viscously spreading ring.When matter supply to the disk stops, accretion will proceed from this ring. At any time t , the characteristic size of the ring R ( t ) is where its mass peaks. Alternatively, R can bedefined by J = ( GMR ) / m , where J is the angular momentum carried by the disk and m is its mass. Spreading of merger disks
Immediately after the merger, the characteristic size of the debris disk is R ∼ cm,and its initial mass m may be as large as ∼ . ⊙ . Its viscous evolution starts on atimescale t = ( a W K ) − ( H / r ) − < ∼ . ( a / . ) − s, with accretion rate ˙ M ∼ m / t thatcan exceed 1 M ⊙ s − . The disk is initially hot and n -opaque (cf. Fig. 2); its nuclearmatter is composed of free nucleons n and p .The initial accretion phase lasts ∼ t . Following this stage, the disk mass m ( t ) isreduced and its radius R ( t ) grows to conserve the angular momentum, J = m ( GMR ) / ≈ const , which implies m (cid:181) R − / . Several important changes occur in the disk as it spreadsto R ∼ r g ≈ cm: ◦ Temperature T and electron chemical potential m e in the outer region r ∼ R decreaseto ∼ r ∼ R is not n -cooled anymore: R ( t ) exits the neutrino-cooled region r < r ign on the ˙ M − r diagram (Fig. 2). The viscouslyproduced heat outside r ign is stored and advected by the spreading accretion disk. Thespreading matter is then marginally bound to the black hole, c s ∼ v K . ◦ Electrons become non-degenerated. Pressure is not dominated by neutrons anymore:it is dominated by radiation and e ± pairs, P ≈ P g + P ± ≈ a r T . ◦ Y e freezes. ◦ Nuclear burning occurs: free nucleons n and p recombine into a particles. Thisprocess releases energy of 7 MeV per nucleon, comparable to the binding energy GMm p / r , and unbinds most of the disk matter, ejecting it in a freely expanding wind. All these changes happen as R ( t ) grows from ∼
50 to ∼ r g .A one-zone model of the spreading disk is calculated in a recent work [38]. Let usestimate here one characteristic radius R ⋆ at which t visc = t weak . Here t weak is the timeof conversion n ↔ p through reactions (1) [23]. At R ⋆ , pressure is already becoming The mass of the debris disk is sensitive to the parameters of the binary system before the merger, inparticular to the mass ratio and the spins of the two companions, see e.g. [37] for a review. In addition to nuclear burning and viscous heating, the matter is heated by neutrinos emitted at r ∼ r g (the mass accretion rate by the black hole is still significant when R ( t ) approaches 10 cm). The energydeposited by neutrinos in the advective zone of the spreading disk, r > ∼ r g is comparable to the viscouslydissipated energy in this zone. ominated by radiation and non-degenerate e ± pairs. On the other hand, n and p havenot yet recombined. The timescales t weak and t visc are then given by t weak ≈ (cid:18) kTm e c (cid:19) − s , t visc ≈ a W K ( r ) (cid:18) Hr (cid:19) − . (8)Using the hydrostatic balance P / r = ( H / r ) v with r ≈ m / p r H , one finds kT ≈ . h / M / m / R MeV , m e kT ≈ . Y e h − / M − / m / , (9)where h ≡ H / R , M ≡ M / ⊙ , m ≡ m / g = ( m / .
05 M ⊙ ) , and R ≡ R / cm.(Note that m e / kT depends on m and h only, not R .) This gives t weak t visc ≈ h / a . M − / R / m − / (10) ⇒ R ⋆ ≈ . × h − / a − / . M / m / cm , (11)where a . = a / .
1. The density and temperature of the disk at r = R = R ⋆ are kT ⋆ ≈ . h / a / . M / m − / MeV , (12) r ⋆ ≈ × h − / a / . M − / m − / g cm − . (13)At this temperature and density matter is close to the neutronization line kT n ( r ) = . ( r / g cm − ) / MeV (Fig. 2), T ⋆ T n ( r ⋆ ) ≈ . h / a − / . M / m − / . (14)Hence the equilibrium value of Y e at R ⋆ is Y ⋆ e ≈ .
5. The actual Y e in the spreadingdisk gradually freezes out as R ( t ) passes through R ⋆ and its asymptotic value afterthe transition can differ from Y ⋆ e . The freeze-out Y e ∼ . average valueof Y e . Viscous spreading is a random diffusion process, so different elements of the diskspend different times near R ⋆ , and a longer residence time at R ⋆ gives a higher Y e . Onecan therefore expect a mixture of different Y e in the spreading disk, with a dispersion D Y e / Y e ∼ R ⋆ this mixture is heated by nuclear recombination and ejected ina wind of duration ∼ t visc ( R = cm ) . Subsequent nucleosynthesis in the expandingejecta produces diverse radio-active elements, including some with a long life-time.Their decay can make the ejecta visible to a distant observer [39]. In particular, materialwith Y e ≈ . Ni. Ni decays when the ejecta expand so much thattheir thermal radiation can diffuse out and escape to observer, producing an optical flashsimilar to normal supernovae. preading of collapsar disks
The collapsar disks are continually fed by the infalling stellar matter during a longtime t infall ∼
10 s (and longer, with a decreasing infall rate). The model posits that theangular momentum of the infall, l infall , is sufficiently large to form a viscous disk [8, 9],e.g., the circularization radius of the infall in the numerical model of [9] is r circ < ∼ r g .The accretion timescale at this radius, t visc ∼ × − a − . s is much shorter than t infall .This led [9] and many subsequent works to picture a low-mass disk, m ∼ t visc ˙ M ∼ × − ˙ M that is continually drained into the black hole and re-filled with fresh infallingmatter.The picture of a low-mass viscous disk is, however, implausible. Conservation ofangular momentum requires the following: (1) The disk spreads during t infall to a radius R ∼ × r g ( a / . ) − where the viscous timescale is comparable to t infall . (2) The diskaccumulates mass m that carries angular momentum J = J tot − J acc . Here J tot = M acc l infall is the total angular momentum processed by the collapsar disk, J acc ∼ M acc r g c is theangular momentum accreted by the black hole, and M acc ∼ ˙ Mt infall ∼ M ⊙ is the massaccreted through the disk. Since J tot > J acc for any viscous disk and usually J tot ≫ J acc ,such disks must store J ∼ J tot (unless almost all angular momentum is carried away bya wind). This implies that the disk accumulates the mass, m ≈ M acc l K ( R ) l infall = M acc (cid:18) Rr circ (cid:19) / ∼ M ⊙ . (15)The disk mass may be much smaller than this estimate only if l infall is so small that J tot ≈ J acc , which leaves J ≪ J tot for the disk. This condition leads, however, to theinviscid mini-disk described in the previous section. A low-mass viscous disk couldform only if l infall is fine-tuned toward the boundary between the viscous and mini-diskaccretion regimes (cf. Fig. 7). Spreading of viscous disks in collapsars through a radius ∼ r g is accompaniedby significant changes, similar to the evolution of merger disks described above. Inparticular, matter acquires a positive Bernoulli constant as a result of viscous, nuclear,and neutrino heating. The infalling material of the progenitor star exerts an external rampressure on the disk, and can confine the disk initially, but eventually the disk pressuremust win and its matter will expand with a velocity ∼ × cm/s. The ejected mass ∼ g carries the energy E ∼ erg and will explode the outer parts of the star.To a first approximation, the expansion of the outer disk may be described as a thermalexplosion driven mainly by nuclear burning of n and p into a particles. A fraction ofthe unbound disk matter will turn into Ni and should create a bright supernovae-likeevent. The very small m ∼ .
003 M ⊙ found in the simulations of [9] may be the result of the imposed absorbingboundary condition at r in =
50 km ≈ r g , which is ∼ r ms ∼ r g . The large r in implies an artificially large J acc , which happens to be nearly equal to J tot in themodel, permitting J ≪ J tot . In addition, a (small) fraction of J tot is carried away by the wind. elf-similar spreading at late stages The disk matter that has spread beyond ∼ cm is largely unbound and ejected,however, some matter remains bound and rotating in a remnant disk. Its mass is hardto estimate; it could be as large as ∼ . ⊙ for collapsars and ∼ .
01 M ⊙ for mergers.This remnant is composed of recombined nucleons, and further fusion reactions are not asignificant source of energy (compared with the virial/gravitational energy). The centralsource of neutrinos switches off as ˙ M drops, so neutrino heating is also insignificant.This advective remnant disk will continue to spread viscously to larger radii, graduallydraining its mass into the black hole and possibly losing mass to a wind.If the mass loss through a wind is small, the spreading enters a simple self-similarregime such that J = const , R ( t ) grows as a power-law with time, while ˙ M ( t ) and m ( t ) decrease as power-laws with time. Detailed self-similar models of this type were studied,see e.g. [40, 41]. The advective disk has a scale-height H ∼ r , sound speed c s ∼ v K =( GM / r ) / , and W ∼ W K . Its kinematic viscosity coefficient is n ∼ a c s H ∼ a v K r = a l K ,where l K ( r ) = ( GMr ) / . The disk spreading is a diffusion process described by R ( t ) ∼ n ( R ) t ∼ a ( GMR ) / t ⇒ R ( t ) ∼ a / ( GM ) / t / ∼ R (cid:18) tt (cid:19) / , (16)where subscipt “0” refers to an initial reference moment of time t . The disk mass m ( t ) is then found from the condition J = ( GMR ) / m = const , m ( t ) = J ( GMR ) / ∼ m (cid:18) RR (cid:19) − / ∼ m (cid:18) tt (cid:19) − / , (17)and the accretion rate is given by˙ M ( t ) ∼ mt ∼ ˙ M (cid:18) tt (cid:19) − / . (18)This self-similar solution may not apply if the disk loses mass through a wind and [38]consider solutions that include the wind. In general, advective disks are only marginallybound by the gravitational field of the black hole and their Bernoulli constant can bepositive [42, 43]. This is expected to cause a strong wind. On the other hand, boundsolutions with a negative Bernoulli constant were found for spreading advective disks[41]. The mass loss through a wind then depends on the poorly understood verticaldistribution of viscous heating inside the disk and the behavior of the magnetic fieldabove the disk. CONCLUSIONS
Hyper-accretion disks are formed from matter with a modest angular momentum, withcircularization radius r circ well inside 10 r g ≈ cm. Matter with r circ > ∼ r g is sup-ported by the centrifugal barrier, and its accretion is driven by viscous stresses on aimescale t ∼ . ( a / . ) − s, where a ∼ . − . M exceeds ˙ M ign ∼ . ( a / . ) / M ⊙ s − .Neutrino annihilation above the disk deposits a significant energy that can power GRBexplosions. In contrast to previous expectations, the rate of energy deposition ˙ E n ¯ n isfound to be not sensitive to the details of neutrino transport and the vertical structure ofthe accretion disk. It is given by ˙ E n ¯ n ≈ ˙ E ( ˙ M / M ⊙ s − ) / in a broad range of accretionrates ˙ M ign < ∼ ˙ M < ∼ ˙ M trap (eq. 3). The normalization factor ˙ E is very sensitive to the blackhole spin a . For instance, ˙ E ≈ erg s − is found for a black hole with a = . E for the Schwarzschild case a = Y e ∼ .
1, and free neutrons dominatethe pressure in the disk [17]. The neutron-rich matter may contaminate the jet fromthe accreting black hole and get ejected with a high Lorentz factor. Then the gradualdecay of the ejected neutrons affects the global picture of GRB explosion on scales upto 10 cm, where the GRB blast wave is observed [44, 45, 46].The disk size R grows with time as a result of viscous spreading. Most of the diskmass m ( t ) resides near R ( t ) and its state is not described by the steady model (whichremains valid at radii r < R ). Instead, it is described by a markedly different spreadingsolution. In particular, as the disk spreads to ∼ r g it is heated both viscously andby the nuclear burning of free nucleons into helium. As a result, the disk is disruptedbefore it spreads much beyond 10 cm: the heated flow acquires a positive Bernoulliconstant and gets unbound.Then most of the disk mass m is ejected with a velocity ∼ . ∼ ( m / M ⊙ ) erg. A fraction of the ejected matter acquires Y e ≈ .
5, which favors thesynthesis of Ni as the ejecta expand and their temperature drops. The ensuing gradualdecay of Ni should produce a visible optical flash on a week timescale — a supernova-like event. The flash is expected to be especially bright for collapsars that developmassive spreading accretion disks. A similar (but weaker) flash should be produced bythe spreading disks around a merged binary; the ejected mass can be a few orders ofmagnitude smaller in this case. Ni-rich matter may also be ejected from the inner , geometrically thin, neutrino-cooled disk. This can occur if the inner disk produces a strong wind [47, 48, 49, 50]. Suchwinds are modeled as quasi-steady magnetized outflows, illuminated by neutrinos whichcan heat and de-neutronize the wind material. The details of this plausible mechanismare uncertain because the mass outflow rate and the asymptotic Y e in the wind is hard topredict with confidence — both depend on the assumed MHD behavior of the disk andits corona. In contrast, when matter accretes through ∼ r g (as in the steady-state model), this process isreversed: helium is disintegrated, which leads to cooling . ollowing the main burst, the accretion rate is determined by the amount of matterthat remains bound and rotating around the black hole. ˙ M ( t ) decreases steeply whenthe disk spreads beyond 10 cm and most of its matter is ejected, however, somematter remains bound and continues to accrete. The evolution of ˙ M may be relatedto the observed puzzling features in the afterglow emission of GRBs. The afterglowis likely to be produced by the relativistic blast wave driven by the jet from the centralengine. Its luminosity is determined by the energy and magnetization of the jet as wellas the density profile of the ambient medium at r ∼ − cm. A long-lived jetof luminosity L jet = e jet ˙ Mc would certainly impact the afterglow emission. However,current theories are unable to reliably predict the evolution of ˙ M and e jet . For instance,one could speculate that the jet switches off abruptly as ˙ M decreases below a threshold,which causes the observed steep decay in the afterglow light curves.While the mechanism of the relativistic jet and its evolution with ˙ M ( t ) remain un-certain, the non-relativistic massive ejecta with v ∼ . c is a robust consequence ofviscous-disk accretion. Viscous disks certainly form in merger events. The standard col-lapsar model also assumes the formation of a viscous disk, but this case is less certain.The minimum angular momentum needed to form a disk is ∼ r g c , and collapsars wereproposed as rare events of stellar collapse with l > r g c . Hence, statistically, the accretionflows in collapsars are likely to have small l and their disks can be smaller than ∼ r g .Such mini-disks are not centrifugally supported and accrete faster than viscous disks. Incontrast to the viscous regime, this low-angular momentum accretion leaves no remnantdisk in the end of the core collapse, involves no viscous spreading, and may not ejectmuch mass. However, it still can produce a powerful relativistic jet via the Blandford-Znajek mechanism and/or neutrino heating near the rotation axis. ACKNOWLEDGMENTS
I thank S. Blinnikov for pointing out the work by Imshennik, Nadezhin, & Pinaev [23].This research was supported by NASA
Swift grant.
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