Black Hole Disk Accretion in Supernovae

THE ASTROPHYSICAL JOURNAL, 489 : 227È233, 1997 November 1 ( 1997. The American Astronomical Society. All rights reserved. Printed in U.S.A.

BLACK HOLE DISK ACCRETION IN SUPERNOVAE SHIN MINESHIGE AND HIDEKO NOMURA Department of Astronomy, Faculty of Science, Kyoto University, Sakyo-ku, Kyoto 606-01, Japan ; minesige=kusastro.kyoto-u.ac.jp, nomura=kusastro.kyoto-u.ac.jp

MASAHITO HIROSE Theoretical Physics, Astronomical Observatory of Japan, Mitaka 181, Japan ; hirose=yso.mtk.nao.ac.jp

AND KENÏICHI NOMOTO AND TOMOHARU SUZUKI Department of Astronomy and Research Center for the Early Universe, School of Science, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan ; nomoto=astron.s.u-tokyo.ac.jp, suzuki=astron.s.u-tokyo.ac.jp Received 1997 February 25 ; accepted 1997 June 11

ABSTRACT Massive stars in a certain mass range may form low-mass black holes after supernova explosions. In such massive stars, fallback of D0.1 M materials onto a black hole is expected because of a deep gravi_ tational potential or a reverse shock propagating back from the outer composition interface. We study hydrodynamical disk accretion onto a newborn low-mass black hole in a supernova using the smoothed particle hydrodynamics method. If the progenitor was rotating before the explosion, the fallback material should have a certain amount of angular momentum with respect to the black hole, thus forming an accretion disk. The disk material will eventually accrete toward the central object because of viscosity at a supercritical accretion rate, M0 /M0 [ 106, for the Ðrst several tens of days. (Here, M0 is the Eddington luminosity divided by c2.) We crit then expect that such an accretion disk is optically crit thick and advection dominated ; that is, the disk is so hot that the produced energy and photons are advected inward rather than being radiated away. Thus, the disk luminosity is much less than the Eddington luminosity. The disk becomes hot and dense ; for M0 /M0 D 106, for example, T D 109(a /0.01)~1@4 K and o D crit 103(a /0.01)~1 g cm~3 (with a being the viscosity parameter) in the vicinity of vis the black hole. Dependvis vis ing on the material mixing, some interesting nucleosynthesis processes via rapid proton and alphaparticle captures are expected even for reasonable viscosity magnitudes (a D 0.01), and some of them vis hole. could be ejected in a disk wind or a jet without being swallowed by the black Subject headings : accretion, accretion disks È black hole physics È supernovae : general 1.

INTRODUCTION

tion depends on Mmax and thus on the equation of state of NS & Bethe (1994) have argued that nuclear matter. Brown Mmax can be as small as D1.5 M and main-sequence stars NS D18È30 M can leave low-mass _ with black holes behind _ supernova explosions. Brown, Bruenn, & Wheeler (1992), motivated by the lack of clear observational indications of neutron star activity in SN 1987A, have suggested that SN 1987A may have formed a black hole. The observations of SN 1987A are thus critical to clarify whether SN 1987A has formed a neutron star or a black hole. Currently, the bolometric luminosity of SN 1987A is D1036 ergs s~1, which can be explained by the energy deposition from the 44Ti decay (Suntze† 1997 ; Kumagai et al. 1993 ; Nomoto et al. 1994 ; Chugai et al. 1997 ; see, however, Bouchet 1997). However, a possible energy supply from a buried pulsar cannot be ruled out. Therefore, it is necessary to examine the observational consequences of possible events associated with the formation of a neutron star or a black hole. In this regard, the important event is a possible fallback of material onto a newborn compact object. Such a fallback would be induced by a deep gravitational potential of relatively massive stars (Woosley & Weaver 1995 ; Nomoto et al. 1993) or by a reverse shock propagating inward from the outer composition interface (Chevalier 1989). The gases that are falling back would form a spherical accretion Ñow or a rotating viscous disk, depending on the angular momentum of the accreting gas. Spherical accretion Ñow in a supernova was examined by

Stars more massive than D8 M on the main sequence _ end of their evolution. undergo gravitational collapse at the Depending on the stellar mass and binarity, the collapsing star forms either a black hole or a neutron star and gives rise to a Type II, Type Ib, Type Ic, or optically dark supernova (see Nomoto et al. 1993 for a review on dark supernovae and the black hole formation). Because of the complicated neutrino processes, however, we do not yet understand what is the exact mechanism that transforms collapse into explosion and which stars form neutron stars or black holes. The following suggestions have been made concerning black hole formation : 1. A progenitor with a mass exceeding a certain limit will experience a prompt formation of a high-mass black hole with little mass ejection. Tsujimoto et al. (1997) have estimated this limit to be 50 ^ 10 M from a comparison _ and observations of between the theoretical [O/Fe] ratio metal-poor stars. No optically bright supernovae are expected from such a collapse. 2. Progenitors with masses smaller than the above limit but greater than 20È40 M might undergo a delayed forma_ hole because the mass of a tion of a low-mass black newborn hot neutron star exceeds the maximum mass of the cold neutron star, Mmax. In this case, most materials in the heavy element mantleNSare ejected, and the events would be observed as Type II or Type Ib supernovae (Nomoto et al. 1993 ; Brown & Bethe 1994 ; Woosley & Weaver 1995). Which star undergoes such a low-mass black hole forma227

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MINESHIGE ET AL.

Chevalier (1989), Bethe (1990), Houck & Chevalier (1991), and Brown & Weingartner (1994). These papers pointed out the importance of photon trapping (see Blondin 1986). A supercritical accretion (with a mass Ñow rate exceeding L /c2) onto a newborn compact object is plausible. If the E newborn object is a neutron star, a considerable amount of energy generated on its solid surface should be somehow carried away, presumably by neutrinos. The expected luminosity is several times 1038 ergs s~1 for the neutron star case, while it is much less, D1034È1035 ergs s~1, for the black hole case in which there is no solid surface (Park 1990). Note, however, that the luminosity of D1038 ergs s~1 given above corresponds to the Eddington limit for the electron scattering opacity. In SN 1987A, the accreting gases contain a large amount of iron so that the line opacity may be much larger than the electron scattering opacity. For the latter case, the expected luminosity could be as low as D1035ergs s~1 (Pinto 1997). If most of the infalling gas originates from the progenitor rotating with an angular velocity exceeding that of the solar envelope (see, e.g., Shigeyama & Nomoto 1990), the formation of a gaseous disk is highly feasible. Disk accretion was considered by Meyer & Meyer-Hofmeister (1989) and Mineshige, Nomoto, & Shigeyama (1993). If disk accretion occurs, however, it is generally believed that the disk luminosity should be, at least, of the order of the Eddington luminosity (D1038 ergs s~1), even for the black hole case, because of the high conversion efficiency from gravitational energy to radiation energy in disk accretion. If this is the case, such a high luminosity should have been observed in SN 1987A. In the present study, we Ðrst perform smoothed particle hydrodynamics (SPH) simulations to see how ambient materials with angular momentum accrete toward the central object (° 2). The rate of mass accretion onto a compact object can be estimated from the simulations. We then discuss the expected disk structure in ° 3, in which we demonstrate that photon trapping can also occur in disk accretion, which largely reduces the radiation energy output. This type of disk is already known : the so-called slim disk (or optically thick, advection-dominated disk). Finally, we discuss possible nucleosynthesis within such a disk.

2.

FORMATION OF A VISCOUS DISK IN A SUPERNOVA

2.1. Physical Assumptions We here study the accretion process of ambient gases around a compact object with a mass of D1È3 M . Since a _ kick velocity imparted to the compact object is D100È400 km s~1, which is less than the free-fall velocity (D1000 km s~1), Chevalier (1989) suggested that the compact object will end up comoving with ambient matter traveling at the same speed. We thus assume that the compact object is at rest with respect to its ambient matter (see Brown & Weingartner 1994). For simplicity, we follow only the hydrodynamical accretion process of a He core and neglect the e†ects of radiation on gases and nuclear reactions within the disk (possible nuclear reactions will be discussed later). Three representative mass distributions are considered. In a thin shell model, we assume

Vol. 489

o\

7

d d o for r [ ¹ r ¹ r ] ; 0 0 2 0 2 0

otherwise ,

(1)

where r \ 5 ] 1010 cm is the place at which a reverse 0 shock appears (see Woosley 1988), d is the width of the shell, and o is determined so as to give a total mass of 0.1 M as 0 _ o \ 0.1 M /(4nr2 d). This model may not be realistic, but 0 _ 0 it is useful in visualizing how ambient gases accrete to form a disk. In spherical models, we assume either a uniform density distribution with o\

G

o for r ¹ r ; 0 0 0 otherwise

(2)

or a central mass condensation with o \ o (r/r )~2 . (3) 0 0 Here, again, r \ 5 ] 1010 cm, and o is determined to give 0 r equal to 0.1 M , i.e., 0 o \ 0.3 M /(4nr3 ) the mass inside 0 _ 0 ) for the_latter. 0 for the former case and o \ 0.1 M /(4nr3 0 _ We Ðx the initial sound velocity to be c 0\ 3.45 ] 107 cm s s~1 (see Chevalier 1989), which corresponds to the initial temperature of T D 107 K. Note that the gas pressure and radiation pressure are about the same order at r for the 0 parameter set given above (the Ñow is radiation pressureÈ dominated inside r ). In the present study, we assume the mass of the central0 object to be M \ 1.4 M so that a _ hole forsmall amount of mass accretion may0induce black mation (see Brown & Bethe 1994). Gravity due to the central object exceeds gas and radiation pressure so that rapid accretion is initiated, even if we assumed no initial velocities for ambient materials. There are good reasons to believe that the initial gas cloud is rotating (see Mineshige et al. 1993). It is thus important to see how the initial angular momentum a†ects accretion processes. In the present study we compare two cases with di†erent initial angular frequencies, )/) \ 0.0 and 1.0 0 velocity at with ) \ 10~3 rad s~1 (for which the rotational 0 r is v \ 5 ] 107cm s~1). For simplicity, we assume 0 0angular frequencies (independent of the initial posiuniform tion of the gas). The angular velocity is thus proportional to the radius, r, and the angular momentum is proportional to r2. Note that the equilibrium rotational velocity at r is (GM /r )1@2 D 6.1 ] 107(M /1.4 M )1@2 cm s~1 Z r 0) . 0 cloud is not rotation 0 _ 0 0 The 0initial supported so that rapid infall of the cloud is expected. 2.2. Basic Equations and Numerical Procedures We use the SPH method (see a review by Monaghan 1992) to solve the hydrodynamic equations. The hydrodynamic equations for the ith particle are

A B A B

1 dv i \ [ $P ] g ] a (4) i i o dt i de P i\[ (5) $ Æ v ]Q i dt o i P \ (c [ 1)o e (6) i i i where ¿ , P , g , a , Q , e , and c denote the velocity, pressure, i i viscosity, i i i viscous and shock heating, intergravity, iartiÐcial nal (heat) energy, and speciÐc heat ratio, respectively. In the

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BLACK HOLE DISK ACCRETION IN SUPERNOVAE

present study, we assume c \ 4/3, since the infalling gas is likely to be radiation pressureÈdominated (Chevalier 1989). We include viscous heating but omit radiative cooling in the energy equation. This is a good approximation when the disk is advection dominated, as we will demonstrate later (see the next section). The kernel of each SPH particle has a proÐle

1 W \ ] ij nh3

C A BA B D A B

g

r 3 ij [ 2 h r 3 2 [ ij h

r 2 ij ] 4 h

0

r for 0 ¹ ij \ 1 h r for 1 ¹ ij \ 2 h r for ij º 2 , (7) h

where r \ o r [ r o and h is the smoothing length of SPH ij which i is kept j particles, constant as h \ 2.4 ] 109 cm. Physical quantities at each location are then calculated using this kernel. For example, the local density at r is given by o(r) \

P

o(r@)W (r, r@)dr@ ,

(8)

which is, in the SPH forms, n o(r ) \ ; mW , (9) i ij j/1 at the position of the ith particle (r ). Here m is the mass of i number of particles, one SPH particle, and n is the total with n \ 104 and m \ 0.1M /n \ 10~5M in the present _ _ calculations. We prescribe the kinematic viscosity as I ol 4 k \ a oc h , 2 SPH s

(10)

where a represents the viscosity term in the SPH scheme SPHconfused with the viscosity parameter, a , in the (not to be vis of the standard Shakura-Sunyaev model) and I is a constant order of unity. We assigned a \ 1.0 in the present study to assure numerical stability.SPH Since kinematic viscosity is related to the viscosity parameter through l D a c H vis s (with H being the half-thickness of the disk), our deÐnition leads to a D (h/H)a D 2.4 ] 109/H, which is of the vis SPH order of unity. The basic equations are integrated by the second-order predictor-corrector method. To test the code, we calculated the steady disk structure, conÐrming that the calculated surface density distribution reproduces the analytical result well, including the factor [1 [ (r /r)1@2] that represents the in at r . We also solved inner boundary condition imposed in and compared the viscous di†usion process with our code the result with the analytical solution, Ðnding agreement to within a few percent. The initial mass distributions were given in the previous subsection. In the shell models, we assign d \ h, i.e., we put one SPH particle in the radial direction for a randomly chosen set of angles (both h and /). The SPH code has a Ðnite spatial resolution, of the order of h D 2.4 ] 109 cm in the present simulations, and cannot resolve the disk structure in the vicinity of the central object (e.g., within several Schwarzschild radii from the center). Fortunately, however, the inner disk structure may well be described by the theory of geometrically thin accretion disks, since the e†ects of dynamical infall onto the disk should be negligible in such a

229

compact region. We thus Ðrst discuss the disk formation process and evaluate the mass accretion rate by SPH simulations and then discuss the inner disk structure by using the estimated accretion rates. It is essential in the present study to allow the total number of SPH particles to change because of the addition of new particles from outside (which represents the mass input to the disk) and because of the removal of particles that enter the innermost region (which corresponds to the mass accretion onto the central object). Numerically, we remove all the particles within R \ h \ 2.4 ] 109 cm sink from the disk center at each time step. The number of removed particles per unit time corresponds to a mass Ñow rate at r \ R . To check how the results depend on the sink value of R , we calculated the same models with R \ h sink sink and 5h and found that the mass Ñow rate changes only by a small amount (less than a factor of 2). We thus conclude that the precise value of R will not alter our order-ofmagnitude estimates of M0 . sink 2.3. Numerical Results of Disk Formation Figure 1 is a plot of the time evolution of SPH particle distributions for the thin-shell models without (Fig. 1a) and with (Fig. 1b) the initial angular momentum. In each panel, the horizontal axis is taken to be parallel to the rotation axis of the initial gas cloud. When the ambient gas has no angular momentum () \ 0.0), it accretes toward the center with a free-fall velocity (see Fig. 1a), whereas if the gas has initial angular momentum () \ ) ), it Ðrst falls onto the equatorial plane, forming a rotating0 gas disk, and then accretes inward via viscosity (see Fig. 1b). Since the total disk angular momentum is conserved (except for a small amount of angular momentum loss because of the removal of accreting gases at R ), most of the original angular momentum sink in the disk material should be transported outward and carried by a small amount of gas rotating at large radii. That is, some amount of gas is left there. Figure 2 illustrates the time variation of the total accreted mass (the mass that has entered the innermost region) in the upper panel and that of mass accretion rates through the sink radius in the lower panel, respectively. Since it takes D(r3/GM )1@2 D 800 s for the shell material to fall to the 0 0 plane, mass accretion is initiated at t D 800 equatorial s D 3 ] 10~5 yr. Figures 3 and 4 illustrate the total accreted mass and the mass accretion rate as functions of time for the uniform spherical model and for the centrally condensed spherical model, respectively. Since the mass was initially distributed down to a small radius, mass accretion immediately starts. Although the disk shape is not as clear as in the shell models, mass accretion rates look similar between two cases with a di†erent initial mass distribution. Note that each line has two di†erent slopes. Since a viscous disk does not form for the nonrotating case, the later gradual decline seen in M0 of the rotating case represents accretion via the viscous di†usion process. The disk viscosity thus maintains continuous mass accretion Ñow onto the central hole at later times. The derived mass accretion rate is roughly D 9.4 ] 1023(a t/yr)~1.01 g s~1 , shell vis M0 D 2.8 ] 1022(a t/yr)~1.35 g s~1 , uni vis M0 D 4.4 ] 1021(a t/yr)~1.43 g s~1 , con vis

M0

(11)

t=0.2

1

0

0

0

z

1

-1

-1

-1

0

1

-1

-1

x

0

1

-1

x

t=0.6

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0

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0

z

1

-1

-1

0

1

1

t=1.0

1

-1

0 x

t=0.8

z

z

t=0.4

1

z

z

t=0.0

-1

-1

0

1

-1

x

x

0

1

x

FIG. 1a t=0.2

1

0

0

0

z

1

-1

-1

-1

0

1

-1

-1

x

0

1

-1

x

t=0.6

1

0

0

0

z

1

-1

-1

0 x

1

1

t=1.0

1

-1

0 x

t=0.8

z

z

t=0.4

1

z

z

t=0.0

-1

-1

0 x

1

-1

0

1

x

FIG. 1b FIG. 1.ÈCollapse of thin shells toward a central object with a mass of M \ 1.4 M . (a) Free-fall collapse of a thin shell with no angular frequency. (b) _ is normalized by L \ 1.22 ] 1011 cm and the initial shell radius is Disk accretion of an initially rotating shell with angular frequency of ) . The0length scale 0.41 in this unit. The elapsed times are 0.0, 0.2, 0.4, 0.6, 0.8, and 1.0 (in0units of T \ 3.14 ] 103 s) from the upper left-hand to the lower right-hand in both panels, respectively. There exists a marked di†erence in (a) and (b) ; the formation of a rotating disk and subsequent disk accretion are clear in the latter.

BLACK HOLE DISK ACCRETION IN SUPERNOVAE

FIG. 2.ÈTotal mass that has entered the innermost region with the sink radius, R \ 2.4 ] 109 cm (upper) and the mass accretion rate across this radius forsink thin-shell models. The units of mass and mass-Ñow rate are 1.0 M and 1.0 M yr~1 D 6.3 ] 1025 g s~1, respectively. The initial angular _ frequencies are_)/) \ 0.0 and 1.0. It is evident that mass accretion is decelerated when the0disk material is rotating.

FIG. 4.ÈSame as Fig. 2, but for the centrally condensed sphere model

for the shell, uniform sphere, and centrally concentrated sphere models, respectively, all with ) \ ) . Note that the 0 accretion rates have the power-law dependence on time as M0 P (t/t )~a with t corresponding to the viscous timescale 0 a positive 0 constant (generally 1 \ a \ 2). This is and a being expected by the self-similar solutions for a viscous disk under the assumption of constant total angular momentum (see Mineshige et al. 1993). The scalings with a in equation (11) were introduced by using the relation t Pvisa~1. 0 vis becomes For smaller a , the mass accretion timescale vis longer so that more material remains in the disk, which causes larger M0 at t ? (r3/GM )1@2 D 1000 s. Obviously, the radial accretion occurs 0more0 quickly in the nonrotating case, unless mass is somehow supplied continuously from outside the system. We denote the Eddington accretion rate by

A

B

M L 4nGM 0 0 \ 2.8 ] 1017(gs~1) , (12) 4 E\ 1.4 M c2 ci _ es where L is the Eddington luminosity, i is the opacity (we took i E^ 0.2 cm2 g~1). The normalizedesaccretion rates are es m5 D 3.4 ] 106(a t/yr)~1.01 , shell vis m5 D 1.0 ] 105(a t/yr)~1.35 , uni vis m5 D 1.6 ] 104(a t/yr)~1.43 , (13) con vis for the shell, uniform sphere, and centrally concentrated sphere models, respectively. Therefore, the mass accretion rate far exceeds the critical rate. For comparison, the mass accretion rate due to spherical accretion from an expanding envelope is estimated as M0

crit

FIG. 3.ÈSame as Fig. 2, but for the uniform sphere model

231

232

MINESHIGE ET AL. pressure, hydrostatic balance leads to

(Chevalier 1989) M0 D 1.2 ] 1022(t/yr)~15@8 g s~1 ,

(14)

and the normalized accretion rate is m5 4 M0 /M0 3.

crit

\ 4 ] 104(t/yr)~15@8 .

(15)

CONSEQUENCE OF HYPERCRITICAL DISK ACCRETION ONTO A BLACK HOLE

3.1. Hypercritical Accretion We have studied, via SPH simulations, how a mass of D0.1 M with an initial angular momentum accretes _ toward a central black hole. The result is a hypercritical disk accretion. This is not surprising, if we compare it with the case of X-ray binaries. Some X-ray binaries (XBs) shine steadily at nearly the Eddington luminosity on a long timescale. How much mass is contained in these disks ? Roughly, the disk mass is M D nR2& with the disk size R D 1011 disk & D 103 g cm~1 (see Mineshige & cm and surface density Kusunose 1993). This gives only M D 3 ] 1025 disk amount of g D 1.5 ] 10~8 M . Even with such a small _ mass, the disk can radiate nearly at the Eddington luminosity. In the case of XBs, of course, a constant mass supply at a rate of M0 D 10~8 M yr~1 is essential. If the mass _ supply stops, the disk luminosity will diminish on the viscous timescale. In fact, the viscous timescale (for the case of XBs) is

A B A B A

A B B A B

R3 1@2 T a ~1 1 vir D 1(yr) vis q D vis a GM T 0.1 0 vis M ~1@2 R 1@2 T ~1 0 . (16) ] M 1011 cm 104 K _ Here, T is the virial temperature. Hence, the disk mass is vir of M0 q D 10~8 to 10~7M for a \ 0.01È0.1. of the order vis that if we put a mass _ of D0.1 vis M on a It is thus reasonable _ dynamical timescale,

A B

Vol. 489

A

B A B

R3 1@2 R 3@2 M ~1@2 0 D 3000(s) , (17) q D dyn GM 1011 cm M 0 _ the averaged mass accretion rate far exceeds the critical rate. The hypercritical accretion is hence inevitable, once an envelope with a mass exceeding 10~7 M falls back to _ The consethe central object after a supernova explosion. quence of such a hypercritical accretion is advectiondominated Ñow (Abramowicz et al. 1988), in which photon trapping occurs. Note that some discussion of hypercritical disk accretion is made by Chevalier (1996), who discussed the disk structure using the self-similar solution for advection-dominated disks (Narayan & Yi 1994). 3.2. Optically T hick, Advection-dominated Disks When M0 exceeds the critical rate, a disk becomes advection dominated and optically thick, as long as the shearviscous tensor depends on the radiation pressure. The detailed structure of such an optically thick, advectiondominated Ñow (the so-called slim disk) was obtained by Abramowicz et al. (1988). Typical density and temperature of such a disk can be estimated, in order of magnitudes, as follows. First, the half-thickness of the disk, which is roughly a pressure scale-height, becomes comparable to (but, by a factor of a few, smaller than) the disk radius : H [ r. Since the radiation pressure dominates over the gas

A B A B

p 1@2 r3 1@2 rad . (18) o GM 0 We henceforth treat the aspect ratio, H/r( [ 1), as a parameter. The disk temperature can be derived after some algebra : HD

T4D

3GM 0 &(Hr) , 2ar2

(19)

where a is the radiation constant and & is surface density. Second, from the steady state relation we Ðnd

AB

H ~2 M0 M0 1 . \ 3nl 3na (GM r)1@2 r vis 0 Inserting this expression into equation (20), we let &\

A

B A B AB A B A B A B A B A B

M0 1@4 GM 1@8 0 2na a r5 vis a ~1@4 D 1.5 ] 109(K) vis 0.01

TD

(20)

H ~1@4 r

m5 1@4 M ~1@4 0 106 M _

r ~5@8 H/r ~1@4 . (21) 5R 1/2 g Note that the temperature maximum occurs around r D 5R (Abramowicz et al. 1988). Similarly, the average density is g ]

AB A B A BA B A B A B

H ~3 & M0 1 D 2H 6na (GM r3)1@2 r vis 0 a ~1 m5 M ~1 0 D 5.7 ] 102(g cm3) vis 0.01 106 M _ r ~3@2 H/r ~3 . (22) ] 5R 1/2 g If the density distribution in the z-direction (vertical to the disk plane) has a Gaussian proÐle, o(z) \ o exp [[z2/ c (2H2)], where o is the density on the equatorial plane c (z \ 0), we Ðnd o6 \ o /(2n)1@2. For a D 0.01, m5 D 106, vis such high temM D 1 M , and r Dc 5R , we expect 0 _ g peratures and densities as T D 109 K and o D 103 g cm~3. c To summarize, in the disk formed after a supernova explosion, very high densities and temperatures can be achieved just as in massive, main-sequence stars. Efficient nucleosynthesis is thus expected (see the next subsection). In such a disk, photon trapping occurs. The timescale of photon di†usion in the z-direction is o6 4

q

dif

D

Hi & es , c

(23)

while the viscous timescale (in advection-dominated Ñows) is nr2& q D . vis M0 The trapping radius, inside which q

(24)

[ q , is vis H M 2@3 M0 i H es D 2.5 ] 105(cm)m5 . (25) R \ tr r M 2nc r _ For m5 Z 106, we Ðnd R ? 1010 cm. Photon trapping is tr very efficient over almost all of the disk, as in spherical

AB

dif

A BA B

No. 1, 1997

BLACK HOLE DISK ACCRETION IN SUPERNOVAE

accretion. If so, hard radiation from the innermost part of the disk cannot be observed, and we expect rather soft radiation, optical to UV, originating from the outer, cool parts. Note that the trapping radius does not depend on a , since vis both the photon di†usion timescale and the viscosity timescale are proportional to &, thus in proportion to a~1. vis If a neutron star is formed after a supernova explosion, the energy produced by accretion should eventually be liberated, presumably by neutrinos. Still, large energy output, of the order of the Eddington rate, is expected for this case (Brown & Weingartner 1994). In the case of a black hole, on the other hand, both the internal energy of gas and the radiation energy can be advected toward and swallowed by the central black hole. Much less energy than that produced by accretion is likely to be observed. The fact that the luminosity of SN 1987A has decreased below the Eddington luminosity does not preclude the presence of disk accretion onto a newborn black hole. This indicates either that there was no fallback material at all or that the compact object that was left behind after the explosion was a black hole. This conclusion originally derived for the spherical accretion cases can apply, even if the accreting gas initially has angular momentum. 3.3. Possible Nucleosynthesis The possibilities of nucleosynthesis in accretion disks have been discussed by several authors, initiated by Chakrabarti, Jin, & Arnett (1987) and Jin, Arnett, & Chakrabarti (1989). According to recent analyses by Arai & Hashimoto (1992, 1995) and Arai, Hashimoto, & Fujimoto (1996) based on a realistic disk model (Paczynski & Abramowicz 1982 ; Abramowicz et al. 1988), very small a vis

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(\10~6) is required to produce a considerable amount of heavy elements. This is because small a yields slower acvis cretion speed, thereby increasing local density and temperature. Unless we specify very small a , we cannot vis achieve sufficiently the high density (Z103 g cm~2) needed to produce heavy elements. The required a values are vis much less than the widely accepted values, a Z 0.01, vis which were estimated from the simulations of eruptive variables (such as dwarf novae and X-ray transients ; see, e.g., Mineshige & Kusunose 1993). Unlike disks in the usual binary sources, where mass accretion rates do not largely exceed the critical rate, a hypercritical accretion is expected in the present case. Note that, roughly, o P M0 . That is, if M0 can largely exceed the critical value, we can easily attain a sufficiently large local density to promote nucleosynthesis for even a relatively large a vis (D0.01). The disk becomes hot and dense ; at D100 days after the fallback, we estimated M0 /M0 D 106, which yields crit /0.01)~1 g cm~3. If T D 109(a /0.01)~1@4K and o D 103(a vis vis mixed down to some hydrogen and helium have been deeper layers and accreted (see, e.g., Hachisu et al. 1990), interesting nucleosynthesis processes via rapid proton and alpha-particle captures on heavy elements would take place. The elements produced in this way might be advected inward and swallowed by the central black hole, but some of them could be ejected in a disk wind or a jet. This work has been supported in part by the Grant-inAid for ScientiÐc Research, 05242102 (K. N.), 06233101 (K. N., S. M.), 08640329 (S. M.), and COE research, 07CE2002 (K. N., T. S.), of the Ministry of Education, Science, Sports and Culture in Japan.

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