a.i. macfadyen, s.e. woosley and a. heger- supernovae, jets and collapsars

8/3/2019 A.I. MacFadyen, S.E. Woosley and A. Heger- Supernovae, Jets and Collapsars

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THE ASTROPHYSICAL JOURNAL, 550: 410425, 2001 March 20( 2001. The American Astronomical Society. All rights reserved. Printed in U.S.A.

SUPERNOVAE, JETS, AND COLLAPSARS

A. I. MACFADYEN, S. E. WOOSLEY, AND A. HEGERDepartment of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064 ; andrew=ucolick.org, woosley=ucolick.org,

alex=ucolick.org

Received 1999 October 2; accepted 2000 November 29

ABSTRACTWe continue our study of the possible production of supernovae and a variety of high-energy tran-

sients by black hole formation in massive stars endowed with rotation: the collapsar model. The blackhole may form either promptly, since a successful outgoing shock fails to be launched by the collapsediron core (collapsar Type I), or, in a mild explosion, by fallback (collapsar Type II). In the latter case, theinner layers of the star initially move outward but lack adequate momentum to eject all the matterexterior to the young neutron star. Over a period of minutes to hours, D0.15 falls back onto theM

_collapsed remnant, turning it into a black hole and establishing an accretion disk. The accretion rate,D0.0010.01 s~1, is inadequate to produce a jet mediated by neutrino annihilation but is similar toM

_what has been invoked in magnetohydrodynamic (MHD) models for gamma-ray bursts (GRBs). Thisfallback is modeled in detail for two 25 progenitors using two dierent one-dimensional hydrody-M

_namics codes, one Lagrangian and one Eulerian. The production and consequences of jets are thenexplored in both sorts of collapsars. Justication is given for assuming that the jet power is a constanttimes the mass accretion rate, and the consequences ofv\ 0.001 and 0.01 are explored. AdoptingvM0 c2,an initial collimation half-angle of 10, the opening of the jet as it propagates through the exploding staris strongly inuenced not only by the jets kinetic energy but also by its initial pressure and the stellarstructure. Cold jets tend to stay collimated and become even more so, sometimes having an angle ofonly a few degrees when they reach the surface. Jets having higher internal pressure than the stellarmaterial through which they pass, or less initial collimation, spread out and tend to make energetic,asymmetric supernovae accompanied, in helium stars, by weak GRBs. SN 1998bw may have been suchan event, and other events having energies between that of ordinary GRBs and GRB 980425 await dis-covery. In supergiant stars, shock breakout also produces bright X-ray transients that might be a diag-nostic of the model, but even the most powerful jets (equivalent isotropic energy 1054 ergs) will notproduce a GRB in a red supergiant. For such Type II supernovae the limiting Lorentz factor is !B2.Type II collapsars should be more frequent than Type I and may power the most common form ofgamma-ray transient in the universe. However, the GRBs seen by BATSE are, for the most part, toobrief to be Type II collapsars. Those are still attributed to prompt black hole formation. Even therethough, the diverse energies and time structure reect chiey the viewing angle and the variable colli-mation of the jet inside the star, not a highly variable central engine. Indeed, collapsar-induced tran-sients may all have a common total energy in the range 10 511052 ergs.

Subject headings: gamma rays: bursts supernovae: general

1. INTRODUCTION

At the heart of current models for gamma-ray bursts(GRBs), one typically nds a black hole accreting rapidlythrough a disk (e.g., Thompson 1994; Katz 1997; Piran1999; 2000; MacFadyen & Woosley 1999, here-Me sza rosafter MW99; Fryer, Woosley, & Hartmann 1999). Modelsdier in the way the hole comes to exist, in the expectedaccretion rate, and, especially, in the physical process(es)invoked to convert some fraction of the disks energy intorelativistic outow. All agree, however, that some nontrivialfraction, of the accreted mass energy with thev

1, M0 c2, M0

accretion rate through the disk, goes into powering jets. Inaddition, some fraction, of the rotational energy of thev

2,

black hole may also be extracted in a usable([9% MBH

c2)form (Blandford & Znajek 1977; MacDonald et al. 1986;

& Rees 1997 ; 2000 ; Lee, Wijers, &Me sza ros Me sza rosBrown 1999). If these sorts of models are to explain GRBswith equivalent isotropic energies of up to 1054 ergs andD1% beaming factors using stellar mass black holes andaccretion reservoirs, the total efficiency for jet formation,

cannot be much less than 1%. This is the limitv\ v1

] v2

,realized in the most efficient neutrino-powered models

(Popham, Woosley, & Fryer 1999; Janka et al. 1999a;Janka, Ruert, & Eberl 1999b) and is conservative com-pared to what is often invoked for MHD-powered jets frommerging neutron stars. A rst-principles calculation of vremains a formidable task, but an interesting interim step isto assume a constant value of v and explore its conse-quences. That is the strategy adopted in this paper. We willmanufacture jets according to a simple prescription andexplore their interaction with the various stellar environ-ments in which they may be born.

We base our calculations on the collapsar model. Pre-viously (MW99), we explored this model as a possible expla-nation for common GRBs. The key element was collapse ofthe iron core of a rapidly rotating helium star rst to aneutron star, then to a black hole (see also Woosley 1993).The black hole formed within a few seconds of iron corecollapse, about the same time as infalling matter in theequatorial plane was slowed by rotation and piled into adisk. No outward-moving shock was generated prior todisk formation. Henceforth, we shall refer to this modelwhere the black hole forms promptly and makes a jet eitherby neutrino transport or MHD processes as a Type I

410


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SUPERNOVAE, JETS, AND COLLAPSARS 411

collapsar. Additional pioneering studies of the Type I col-lapsar were carried out by Aloy et al. (2000).

It is also quite possible to form a black hole over a longerperiod of time in a supernova that launches an outgoingshock with inadequate strength to eject all the helium andheavy elements outside the neutron star (Woosley &Weaver 1995; Fryer 1999). Fryer estimates that this sort ofbehavior might start around D20 and persist up to 40M

_Depending upon the initial mass of the neutron starM

_.

and the equation of state, the reimplosion of from D0.1 toseveral of the stellar mantle will be adequate to produceM

_a black hole. Excess matter accretes into this hole and, if ithas sufficient angular momentum, forms a disk. We shallrefer to this scenario as the delayed black hole version ofthe collapsar model or a Type II collapsar. Type II isprobably a more frequent event than Type I because itinvolves a more densely populated portion of the stellarmass function.

As we shall see ( 2), the typical accretion rate when mostof the matter falls back in a Type II collapsar is 12 ordersof magnitude less than in Type I, i.e., D0.0010.01 s~1.M

_This is too low for neutrinos to extract disk energy eec-tively and form a jet (Popham et al. 1999; MW99).

However, it is similar to the accretion rate often invoked forMHD models of GRBs, especially in the case of low diskviscosity (e.g., 2000), when D0.1 accretes inMe sza ros M

_10 s. This motivates us to consider the possibility that ener-getic jets will form in a supernova that is already in theprocess of exploding. The accretion timescale is longer thanin the earlier collapsar model, but the total accreted massand its angular momentum are comparable. If the star is ared supergiant, the jet will overtake the supernova shockbefore it reaches the surface, but since the velocity of the jethead is subrelativistic, it may lose its energy input at its basebefore breaking free of the star. In that case, little relativisticmatter would be ejected. For a helium star, however, the jetbreaks out while continuing to receive accretion power at

its base. Its motion may become highly relativistic (seeTable 3 below).

The energy of the jet and the explosion it producesdepend upon the efficiency of MHD processes in extractingaccretion energy from the disk. As noted above, this isuncertain, but it might be as large as D10%. In this paperwe shall make the simple Ansatz that the jet power, at anypoint in time, is an efficiency factor, v, times withM0 c2,vD 0.0010.01 (see 3). Then the energy potentially avail-able for making a jet when only one solar mass accretes isD10511052 ergs. This conservative estimate is still largecompared to the energy of a typical supernova and, if colli-mated into a small fraction of the sky, is also enough energyto make a powerful GRB.

Section 2 describes the initial stellar models and the one-dimensional fallback calculations performed using theKEPLER and PROMETHEUS hydrodynamics codes. Webase our calculations here on two 25 presupernovaM

_models calculated by A. Heger, S. E. Woosley, & N. Langer(2001, in preparation). At sufficiently late times, the calcu-lated accretion rate from fallback agrees well with earlieranalytic estimates by Chevalier (1989). Section 3 furthermotivates our parameterization of the jet kinetic energy uxas and describes some of the factors that enter intovM0 c2estimating v.

Section 4 presents two-dimensional simulations of jetpropagation in both Type I and Type II collapsars. Section4.1 describes how the jet is initiated and introduces three

additional variables (besides jet energy), namely, the jetsstarting radius, opening angle, and pressure (or equivalentlyentropy, or the initial ratio of internal energy to kineticenergy). A ducial opening angle of 10 is employed at 50km. Since special relativity is neglected in our calculations,the initial jet is given a constant speed of 1010 cm s~1 andthe density corresponding to the assumed kinetic energyux. The jets are initiated in two dierent stellar congu-rations corresponding to collapse with high and low disk

viscosity. These calculations show the degree of collimationone might expect at radii much larger than 50 km. Since it isour purpose to explore events that happen on timescaleslonger than can be followed with such a small inner radius(owing to the Courant condition), the results of this sectionprovide guidance for what to use for the jets initiated at alarger radius (109 cm) in 4.2. They also show how the jetproperties might vary for disks of variable viscosity andmass. The calculations here are relevant to both Type I andType II collapsars.

Section 4.2 presents two-dimensional simulations of theresponse of the stellar mantle and envelope to jets assumedto have propagated already to 109 cm. The calculationshere are carried out using as background a supernova that

already launched a weak shock (Type II collapsar) but arequalitatively correct for both types since the jet is so muchmore powerful and travels so much faster than the super-nova shock. A variety of internal energies and two dierentefficiency factors (v\ 0.001 and 0.01) are explored. Jets arefollowed until well after they have broken free of the heliumcore and thus allow meaningful statements to be maderegarding the nal collimation and equivalent isotropicenergy (given the limitations of a Newtonian study). Section5 follows the further propagation of these jets through thehydrogen envelope of a red supergiant to breakout fromstellar surface. A new inner boundary is used at 1012 cm.Finally, in 6 we summarize a variety of high-energy tran-sients that might be produced by collapsars operating in

dierent environments (see also Table 3 below).

2. INITIAL MODELS AND FALLBACK

As has been known for some time (Colgate 1971;Woosley 1988), it is quite possible to produce a fairlymassive black hole in an otherwise successful supernovaexplosion, one that would be about as bright, optically, asone that left a neutron star. Initially, the collapse of the ironcore produces a neutron star and launches a shock, but,especially for more massive helium cores, the shock lacksadequate energy to eject all the matter outside the neutronstar. The gravitational binding energy of the helium coreincreases with mass roughly quadratically, while the explo-sion energy does not (Fryer 1999). Some portion of thematter outside the neutron star initially moves out, thenfalls back.

The rate of this fallback can be increased when the shockwave plows into regions with increasing or3 (Bethe 1990;Woosley & Weaver 1995). So long as the shock remains insonic communication with the origin, its deceleration inthese regions is communicated back to the inner mantle, aportion of which fails to achieve escape velocity. A quanti-tative description of fallback must also include the internalenergy of the ejected zones. At early times, especially a fewseconds, the pressure support provided, e.g., by a piston,plays an important role. Internal energy is converted intoexpansion energy, and zones escape by pushing on thepiston that, based solely upon their kinetic energy budget


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412 MACFADYEN, WOOSLEY, & HEGER Vol. 550

and gravitational potential, would not have. Eventually,however, the expansion becomes supersonic as the soundspeed in the ejecta declines. Matter loses sonic communica-tion with the piston and evolves independently. Carefultreatments of both shock propagation and the inner bound-ary condition are thus essential in any study of fallback.

A. Heger et al. (2001, in preparation) have recently calcu-lated the evolution and simulated the explosion of a grid of15 and 25 main-sequence stars, including the eects ofM

_rotation and mass loss, for several assumptions regardingefficiencies of semiconvection and rotationally inducedmixing. We examine here two of their 25 models. One,M

_with relatively inefficient semiconvective mixing, ended itslife with a low-density hydrogen envelope of 6.57 M

_(model A). The other, with more efficient mixing, had anenvelope of only 1.69 (model B). The helium core massM

_was 8.06 and 8.87 the iron core mass 1.90 and 1.81M

_,

and the stellar radius 8.15] 1013 and 8.65]1013 cmM_

,in models A and B, respectively. The presupernova struc-ture, composition, and angular momentum distribution ofmodel A, which will be used for most of this paper, areshown in Figure 1. Note that A. Heger et al. (2001, inpreparation) assumed rigid rotation on spherical shells,

hence the actual angular momentum in the equatorial planeis 50% higher than the angle-averaged value in Figure 1.Both models had sufficient angular momentum to form aKerr black hole in the center and for the matter in theequatorial plane to form an accretion disk around it. Thatis, there is sufficient angular momentum available for theblack hole to stay at maximum rotation, and the angularmomentum in the equatorial plane is high enough to stop

infall at the last stable orbit (in corotation) around a Kerrblack hole.

For this paper we calculated a new series of supernovaexplosions based upon the models of A. Heger et al. (2001,in preparation), but with a range of kinetic energies at inn-ity (Table 1). The explosion was simulated by a piston thatrst moved inward for 0.45 s until it reached a radius of 500km and then moved outward with an initially highly super-sonic velocity. This velocity was decelerated like a test parti-

cle in the gravitational eld of central mass of timesapistthat of the core. For model A the assumed core below thepiston was 1.97 and for model B was 1.84 TheM

_M_

.initial velocity was chosen so that the piston came to rest at10,000 km (Woosley & Weaver 1995; A. Heger et al. 2001,in preparation). An exception was model A17 in which aterminal radius of 100,000 km was assumed for comparison.

None of these were failed supernovae in the sensedescribed by Woosley (1993) and MW99. While neutrinotransport was not modeled, the motion of the piston gavean outgoing shock that, even for the lowest energy con-sidered, ejected at least the hydrogen envelope of the starwith supernova-like speeds. Each of the models in Table 1would have made a bright Type II supernova without any

additional energy input, though many would have lackedthe characteristic radioactive tail from 56Co decay. Forpiston trajectories that give kinetic energies at innityabove about 1.2] 1051 ergs, calculations by S. Blinnikov(1999, private communication) have shown that the high-mass envelope star (model A) would produce a rathertypical Type IIp supernova. The low-mass envelope gives acurve more like a Type II-L supernova (model B).

TABLE 1

EXPLOSION ENERGY AND FALLBACK

Explosion Energya KEP Fallback PPM Fallback t1@2

b

Run apist

(1051 ergs) (M_

) (M_

) (s)

A01 . . . . . . 0.025 0.255 3.71 3.63 100

A02 . . . . . . 0.05 0.341 3.41 3.35 145

A03 . . . . . . 0.10 0.479 3.03 3.00 265

A04 . . . . . . 0.15 0.595 2.85 2.64 450

A05 . . . . . . 0.20 0.702 2.52 2.23 730

A06 . . . . . . 0.25 0.805 1.96 1.73 1060

A07 . . . . . . 0.30 0.906 1.39 1.22 1140

A08 . . . . . . 0.35 1.007 0.91 0.83 890

A09 . . . . . . 0.40 1.105 0.60 0.60 940

A10 . . . . . . 0.42 1.152 0.53 0.53 1000

A11 . . . . . . 0.44 1.207 0.48 0.47 1060

A12 . . . . . . 0.50 1.326 0.24 0.33 1240

A13 . . . . . . 0.60 1.507 0 0.18 1310

A14 . . . . . . 0.70 1.682 0 9.8 ([2) 970

A15 . . . . . . 0.80 1.850 0 6.2 ([2) 610

A16 . . . . . . 0.95 2.092 0 3.7 ([2) 440A17c . . . . . . 0.025 0.316 3.37 . . . . . .

B01 . . . . . . . 0.20 0.314 5.2 . . . . . .

B02 . . . . . . . 0.30 0.415 4.9 . . . . . .

B03 . . . . . . . 0.40 0.513 4.6 . . . . . .

B04 . . . . . . . 0.60 0.726 3.1 . . . . . .

B05 . . . . . . . 0.70 0.835 2.1 . . . . . .

B06 . . . . . . . 0.75 0.913 1 . . . . . .

B07 . . . . . . . 0.80 0.960 0.06 . . . . . .

B09 . . . . . . . 0.95 1.143 0.05 . . . . . .

B10 . . . . . . . 1.00 1.200 0 . . . . . .

a Final kinetic energy of the ejecta at innity.b Timescale to accrete half of total accreted mass.c Final piston radius of 1010 cm.


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No. 1, 2001 SUPERNOVAE, JETS, AND COLLAPSARS 413

FIG. 1.Presupernova model used for these studies, model A, derivedfrom calculations of A. Heger et al. (2001, in preparation). This model wasevolved with restricted semiconvection, including the eects of rotationand an equatorial rotational velocity on the main sequence of 200 km s~1.(a) Composition of this initially 25 star, now reduced to 14.63 byM

_M_

mass loss. The iron core was removed for this study and replaced by apiston (Fig. 3). (b) Characteristic red supergiant density and temperaturestructures of the presupernova star, especially the cool low-density hydro-gen envelope outside 8.4 (c) Distribution of specic angular momen-M

_.

tum and angular velocity in the presupernova star. The angularmomentum in the equatorial plane is actually 50% higher than this angle-averaged value.

However, especially for energies belowD1.0]1051 ergs,large amounts of material failed to escape and fell back tothe center of the explosion. Fallback masses ranged fromapproximately zero to essentially the entire helium core(Table 1). The accretion was followed in some detail usingtwo quite dierent codes: (1) the same one-dimensionalLagrangian hydrodynamics code, KEPLER (Weaver, Zim-merman, & Woosley 1978), used to simulate the presuper-nova evolution and the explosion; and (2) a one-dimensional version of the Eulerian code, PROMETHEUS(Fryxell, & Arnett 1989, 1991; MW99). PROME-Mu ller,THEUS is an implementation of the high-order Godunovhydrodynamics scheme, piecewise parabolic method(PPM), and we shall henceforth refer to PROMETHEUSas the PPM code.

The eect of various inner boundary conditions on thefallback was then studied. In KEPLER, the inner boundarywas the piston zone, held xed at its nal radius, usually10,000 km. Material that fell back accumulated and came torest. It was distinguished from outgoing material by havingvelocity less than 100 km s~1 and a large positive velocity inthe next zone out.

When the same calculations were repeated using PPM,the results depended sensitively upon the treatment of the

inner boundary condition and the time the calculation waslinked from KEPLER. For a transmitting boundary condi-tion (zero radial gradient of all variables), initiated as theshock left the carbon-oxygen core (10 s after core bounce), ararefaction overtook the outgoing matter and more matterfell back than in the equivalent KEPLER calculation. Forexample, fallback in model A11 was increased from 0.47

in the KEPLER calculation to 1.0 in the PPMM_

M_

calculation, and the explosion energy at 10,000 s wasreduced from 1.21] 1051 to 1.08] 1051 ergs.

It is not physical for the matter to rest on the piston, noris it physical to remove all pressure support from the modelwhile the explosion is still developing. We thus sought acompromise. Figure 2 provides an important detail, regions

in the presupernova star where an outgoing shock wave isexpected to speed up or slow down. Figure 3 shows theradial location of Lagrangian zones as a function of time forthe KEPLER (piston-driven) explosion of model A01.While initially s) all zones are moving rapidly(t[ 60outward, by about 100 s, a bifurcation appears and, by 250s, 2.6 has fallen back onto the piston. This materialM

_would collapse to the origin were the piston removed. Wethus chose to make our link for model A01 to PPM at 100 s(Fig. 4), using a transmitting boundary condition thereafter.This resulted in good agreement between the total fallbackrate calculated with the two codes (Table 1), while the PPM

FIG. 2.Distribution of the quantity or3 in the presupernova model(Fig. 1). The supernova shock will speed up in regions of decreasing or3and slow in regions where or3 increases. The large increase outside 1012 cmoccurs in the hydrogen envelope. The dips at log r \ 9.6 and 10.8 are theedges of the carbon-oxygen and helium cores.


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FIG. 3.Radial history of Lagrangian mass shells in the KEPLER cal-culation of model A01 (Table 1). Dark lines represent mass shells of 0.5

faint lines 0.05 The piston (bottom dark line) moves out, launch-M_

, M_

.ing the shock, and comes to rest, after 19.5 s, at 10 9 cm. The location of thesupernova shock is visible as the outer boundary of the dark concentrationof curved lines. All material on the grid moves outward until 60 s after corebounce, when some begins to fall back and come to rest on the piston.After about 150 s, an accretion shock has developed whose location is anartifact of the stationary piston. In the lower panel the supernova shockcontinues to the surface of the star, but a total of 3.71 eventually fallsM

_back to rest on the piston. The slope of the shock location gives its speed,which decelerates in regions of increasing or3. This model was linked to thePPM code 100 s after material had begun to fall back, but before anyappreciable accumulation on the piston.

calculation more physically described the fallback of matterto the origin at late times.

In addition to the need for hydrodynamical matchingbetween the two codes, the mapping at 100 s also has aphysical basis. During approximately the rst 10 s, aneutrino-driven wind powered by the radiating protoneutron star pushes against the outgoing stellar mantle andprevents its reimplosion. Eventually, however, postshockgas that has failed to achieve escape velocity is pushed backtoward the protoneutron star, both by gravity and by aninwardly directed pressure gradient. This falling materialovercomes the resistance of the low-density neutrino-heatedbubble and penetrates, perhaps by forming ngers, backto the hard surface of the neutron star where it accumulatesin an atmosphere on top of the hard neutron star surface(Fryer, Benz, & Herant 1996; Chevalier 1989). For theaccretion rates relevant for the rst B100 s (Fig. 5), an

FIG. 4.Propagation of the shock shown in Fig. 3 following the remap-ping of the problem into the one-dimensional PPM code at 100 s (dottedline). The solid lines show the velocity (109 cm s~1), log density (g cm~3),and accretion rate s~1) as a function of radius at intervals 50, 100, 150,(M

_200, and 250 s after the remapping. During the rst roughly 20 s, themotion of matter is still inuenced by memory of the KEPLER piston at109 cm and is nearly stationary at small radii. Though the residual supportof this material is articial, only a small amount of mass accretes at thistime, which may be thought of as the interval during which the loss ofpressure at the origin due to black hole formation propagates to 10 9 cm.By 250 s after the remapping (bottom solid line), the accretion rate hasdeclined greatly (Fig. 5).

FIG. 5.Accretion rates for fallback in ve dierent explosions: A01(solid line), A04 (dotted line), A07 (dashed line), A11 (short-dashdotted line),and A14 (long-dashdotted line) (Table 1). Calculations were carried outusing a one-dimensional version of the PROMETHEUS PPM hydrody -namics code. Time is measured from the collapse of the iron core andaccretion rate plotted as a function of the time since the calculation waslinked to the PPM code at 100 s. At late times the accretion rate follows thet~5@3 scaling suggested by Chevalier (1989).


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atmosphere builds up out to several hundred kilometers,at which point neutrino cooling behind the shock balancesthe accretion energy deposition rate and accreting gassettles subsonically onto the neutron star. By 100 s aftercore collapse, enough gas has fallen back for weak explo-sion energies (especially for models A01A03) that theneutron star collapses to a black hole. For intermediateexplosion energies the central object may still be a neutronstar, but the accretion rate is such that the accretion shock

forms interior to our inner boundary at 109 cm so that gasnear the boundary is in free fall and the neutral innerboundary condition is justied. Only for accretion ratesbelow a few times 10~6 s~1, not considered here, wouldM

_the accretion shock move out beyond 109 cm and would theboundary condition have to be modied.

Once a black hole has formed, gas that falls back withsufficient angular momentum will settle into a disk. Whilethis part of the star is not modeled in the current calcu-lations, Popham et. al. (1999) and MW99 found that theinner disk can transport the mass it receives from the col-lapsing star in a steady state. An accretion shock forms nearthe Keplerian support radius of the accreting gas, typicallyseveral hundred kilometers. The upstream collapsing star is

unaware of the accretion disk and collapses in free fallexterior to this region. An exception is outows that candevelop for high-viscosity disks (MW99). These(aZ 0.1)outows are not treated in the present simulations. Since weconsider accretion rates lower by 13 orders of magnitudethan in MW99 and since the MHD processes assumed topower the jets can operate with low disk viscosities (unlikeneutrino annihilation models), the absorbing boundarycondition at 10,000 km is acceptable for these calculations.Future work will address disk outows in detail for the caseof high-viscosity disks, including calculation of the nucleo-synthesis.

The accretion rates obtained from the PPM calculationsare plotted as a function of time in Figure 5. The growth of

the central point mass due to the accretion is shown inMc

Figure 6, and the characteristic time for half the mass to

FIG. 6.Accumulated mass as a function of time for the same calcu-lations shown in Fig. 5.

fallback is given in Table 1. Accretion rates between 10~4and 10~2 s~1 are maintained for hundreds to thou-M

_sands of seconds.

Accretion onto collapsed objects inside of supernovae hasbeen considered previously in a semianalytic fashion byChevalier (1989), who pointed out, for the high accretionrates considered here, that accretion onto a neutron starwould be mediated by neutrino losses (see also Fryer et al.1996; Zampieri et al. 1998). Unless the neutron star mag-

netic eld is unusually strong, the accretion rates we areinterested in are sufficiently high that the neutron star mag-netic eld would be crushed to the surface and sufficientlylow so as not to launch a second supernova shock by neu-trino absorption. Chevalier (1989) gives an approximateformula for the fallback accretion rate based on the parame-ters of SN 1987A and assuming a time sufficiently late thatthe reverse shock has already reached the center:

M0 \ 1.0] 1035t~5@3 g s~1 .

For models with large amounts of fallback, our calcu-lations show that accretion occurs without the mediation ofa strong reverse shock. Indeed, it is not possible for thereverse shock in a red supergiant to make it back to the

center of the star in only a few hundred seconds. Conse-quently, Chevaliers formula becomes applicable muchearlier than the 104 s he estimated for SN 1987A. However,dierent regions of the star have variable entropy, in partic-ular jumps at the boundaries of active burning shells andconvective shells. One large jump is located at the oxygen-burning shell in the presupernova star. As a result, one seesin Figure 5 an early high accretion rate that slows abruptlyafter the silicon shell has fallen in (mass coordinate 2.6 M

_in the presupernova star). After that point, though, duringthe accretion of the carbon-oxygen core, D400 s in modelA04, the accretion rate does start to decline roughly as t~5@3and is in rough quantitative agreement with Chevaliersestimate for SN 1987A.

3. JET ENERGY

From many observations of all sorts of situations inwhich material accretes from a disk into a compact object(star-forming regions, active galactic nuclei, microquasars,SS 433, and planetary nebulae), it seems that jets are aubiquitous phenomenon (e.g., Livio 1999; Pringle 1993). Ineach of these phenomena, the jet typically carries away fromD3% to 30% of the binding energy of the disk and the jetspeed is of the order of the escape velocity. For jets pro-duced near the event horizon of an accreting black hole, therelevant speed is the speed of light. In the case of black holeaccretion, one also has the advantage of knowing thebinding energy of the last stable orbit as a function of Kerr

parameter and black hole mass. The rate at which matterows through that last stable orbit and liberates some ofthis energy in a useful form is just given by the accretionrate. This suggests writing the power of the jet in a simpleparametric form E0

jet\ vM0 c2.

For example, the Blandford-Znajek (1977) process forextracting rotational energy from a black hole with Kerrparameter a4 Jc/GM2 predicts

E0jetB3] 1051a2

A M00.1 M

_s~1

Bergs s~1 ,

or in terms of our efficiency parameter, vB 0.01a2. Thisassumes the development of a nearly equipartition magneticeld, about 1015 G for stellar mass black holes.


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A similar expression that makes the dependence uponmagnetic eld strength explicit is given by MacDonald et al.(1986),

E0jetB 1050a2

A MBH

3 M_

B2A B1015 G

B2ergs s~1 .

To illustrate the uncertainties involved, we consider asteady state disk with accretion rate where HM0 \ 4nrHov,

is the disk scale height and v the radial velocity. The viscoustimescale is approximately where a is theq\ r2/aH2)

K,

disk viscosity parameter and the Keplerian angular)K

velocity. Further, assuming a fraction (d) of the equi-partition magnetic eld energy density (B2/4nD do)

k2 r2),

an advection-dominated disk with HD br, and a diskradius where the energy is mostly generated equal to c timesthe Schwarzschild radius, a similar relation is again foundbetween the jet energy and accretion rate,

E0jetD 0.02

A a0.5

B2A0.5b

B3A3c

B2.5A d0.01

BA0.01a

BM0 c2 .

This formula makes clear the sensitive dependence ofv on a

number of uncertain parameters while indicating thatvD 0.0010.01 is a reasonable choice.Similar powers can also be produced by the magneto-

sphere of the accretion disk and by winds accelerated byviscous dissipation in the disk (MW99; Stone, Pringle, &Begelman 1999). In fact, these powers may dominate theenergy extracted from black hole rotation in most situationsof interest (Blandford & Znajek 1977; Livio, Ogilvie, &Pringle 1999). There one also expects the energy extractionto scale with the energy density in the magnetic eld (timesthe disk area and speed). Using the same assump-Alfve ntions as the previous paragraph, this situation also gives a

jet luminosity that scales as M0 c2.

4. TWO-DIMENSIONAL SIMULATIONS OFJET PROPAGATION

Given a prescription for the jet power and a backgroundstar in which to propagate it, one can then model the inter-action of the jet with the star.

As a matter of convenience, we separate our discussionsand calculations into two regions, one inside 109 cm,another from 109 cm to the surface of model A01 at8.15] 1013 cm. This segregation reects, in part, the diffi-culty of carrying too large a dynamic range on the compu-tational grid. Time steps imposed by the Courant conditionat, e.g.,D 107 cm, are too small to follow the progression ofthe explosion to the surface at D 1014 cm. The split alsoseparates the problem into regions of dierent physics. Inthe inner region, the jet forms, with all the attendant uncer-tain physics that it involves. The outer region, however,contains the bulk of the mass and volume of the star.Occurrences inside 107 cm are discussed in 4.1. Thenumerical models presented there are still subject to theCourant restriction, which makes it difficult to follow fall-back for hundreds or even thousands of seconds, so ourarguments in 4.1 are based upon the stellar and disk struc-ture calculated for Type I collapsars in MW99. Expecteddierences between Type I and Type II are discussed, butwe are mainly interested in this region as providing reason-able inner boundary conditions for the outer region treatedin 4.2.

In the outer region, the jet may be dened by an (angle-dependent) ux at the inner boundary of energy anddensity. Studies of this region will show how jets of dierentproperties aect the star through which they propagate, inparticular how much energy is shared with the star, whetherthe jet suers degradation or additional collimation, andthe degree to which relativistic matter is ejected. Note thatthough the results of 4.2 are discussed within the contextof the collapsar model for GRBs, they are equally appropri-

ate to any model that produces jets with similar properties,for example, jets powered by a highly magnetic, rapidlyspinning neutron star (Usov 1994; Wheeler et al. 2000;Khokhlov et al. 1999).

4.1. Jet Initiation

A variety of processes may operate in the vicinity of arotating black hole to create a jet. Very close to the hole,magnetic elds maintained by currents in the disk andthreading the black hole ergosphere can extract rotationalenergy: the Blandford-Znajek (1977) mechanism. Fartherout, other MHD processes may also extract angularmomentum and energy from the disk to form a jet, e.g., bymagnetocentrifugal forces (Blandford & Payne 1982). See

Koide, Shibata, & Kudoh (1998), Meier (1999), Koide et al.(1999), and Krasnopolsky, Li, & Blandford (1999) for recentdiscussions. These processes tend to produce relatively cold

jets, in the sense that the thermal energy of the jet is notinitially large compared to either the rest mass or the jetkinetic energy. There may be a critical magnetic eld abovewhich these mechanisms are able to provide matter directlywith high Lorentz factor (Meier et al. 1997). Other processeslike neutrino energy deposition (Woosley 1993 ; MW99 ;Janka et al. 1999a, 1999b) or magnetic reconnection(Thompson 1994; Katz 1997) in the disk deposit theirenergy chiey as heat and make hot jets. In these cases theinitial velocity of the jet is small and asymptotic velocity isgiven by the initial ratio of thermal energy to rest mass.

Because the pressure from the deposited energy is initiallyisotropic, the collimation of hot jets then occurs as a conse-quence of the pressure and density gradients of the matter inwhich energy is deposited. Cold MHD jets may thus becollimated both by magnetic elds and by geometry, buthot jets are collimated only by the structure of the mediumin which they expand.

The jets of MW99 are hot jets, focused by the thick accre-tion disk and declining radial pressure gradient in theregion where energy was deposited. In order not to obtain avelocity greatly in excess of the speed of light in their New-tonian code, MW99 were restricted to depositing energy ata rate that, in steady state, gave an internal energy tobaryon loading less than 10 times the rest mass (even thisgave speeds of over 1011 cm s~1 !). These jets easily pen-etrated the helium core in which they were produced whileconcurrently blowing it up, but details of shock breakout,and especially a rst-principles determination of the asymp-totic Lorentz factor of the jet, were obscured by use of ahydrodynamics code that was not special relativistic. Morerecently, Aloy et al. (2000; see also et al. 2000) haveMu llerrecalculated the MW99 conditions using a fully relativisticversion of the PPM code (Aloy et al. 1999) and obtainedsimilar results, i.e., relativistic jet penetration of the heliumcore. In addition, they were able to determine that, at break-out, Lorentz factors of 2030 were achieved in the jet. It isexpected that higher values may be achieved by running the


8/16


calculation longer. These hot jets were all in the context of aType I collapsar.

It is not clear whether the jets made by fallback in a TypeII collapsar will be hot or cold (although jets mediated byneutrino transport are impossible). Given the continuinglimitations of our Newtonian treatment, we can onlyexplore the physics of mildly relativistic warm jets.However, we can vary such things as the kinetic energy andinternal energy loading of the jet, its initial collimation, and

the mass and structure of the accretion disk in which itpropagates in order to understand better how the jet isdynamically focused.

As an example, we consider jets initiated in the two accre-tion disk structures studied by MW99, a high-viscosity disk(model 14A, a\ 0.1; their Fig. 8 at 7.6 s) and a low-viscositydisk (model 14B, a\ 5] 10~4 ; their Fig. 22 at 9.4 s). Themasses of these disks dier by approximately a factor a~1, afew hundred. One might expect a similar dierence in diskmass for constant viscosity, but reducing the accretion rateby a few hundred (Popham et al. 1999). We use the sameconstant energy injection rate for both disk structures toisolate the eect of the disk structure on the jet collimation.This injection rate is therefore not directly determined by

the accretion rate from the disk calculation, but it corre-sponds to v\ 0.1 for the accretion rates ofD0.1 s~1M

_characteristic of the disk calculations (see MW99). The dif-ference between jet behavior initiated in these two modelsqualitatively illustrates the eect we expect for the high

accretion rate in Type I collapsars and the lower accretionrate by 2 orders of magnitude in Type II collapsars.

Jets were initiated in both models at an inner boundaryradius of 50 km, with an opening half-angle of 10, a veloc-ity of 1010 cm s~1, and a total power of 1.8 ] 1051 ergs s~1.The jets were initiated about 10 s after core collapse, 7.6 sfor model 14A and 9.4 s for model 14B at a time when thedisks had fully formed. Gas was still infalling along thepolar axis when the jets were initiated, but the ow was

easily reversed given the energy injection rate employed (forthe appropriate criterion see MW99, 5). At later times

s), as fallback continues the jet has passed the(tZ 100supernova shock and has cleared a path along the polaraxis.

Half of the injected energy was in kinetic energy and theother half in internal energy. The energytorest mass ratiowas thus about 10% so as to avoid superluminal velocitieson the grid. Figures 7 and 8 show the results after 0.6 s of jetpropagation.

Both calculations clearly show the eect of geometricfocusing, but the jet in the lower mass accretion disk ismuch less collimated. Since the subrelativistic gas has com-parable internal and kinetic energies, the sound speed is

near the radial streaming speed, so one expects the jet, in avacuum, to have comparable theta and radial velocities.This is initially the case for the calculation using the lowdisk mass. After 1000 km, the opening half-angle is about20. The pressure gradient of the star is still enough to

FIG. 7.Hydrodynamical focusing of jets initiated in the collapsar disks of MW99 shown h ere for two values of the disk viscosity parameter, a\ 0.1 (left)and 5] 10~4 (right). Each jet was injected at 5] 106 cm with an energy deposition rate of 1.8 ] 1051 ergs s~1, roughly equally in internal and kinetic energy,and an opening half-angle of 10. The gures show the situation 0.6 s after the jets are turned on. The bottom two panels of each gure show the logarithms ofthe pressure (left) and density (right) for the disk before jet initiation. The mass of the disk is approximately 2 orders of magnitude smaller in the high-viscositycase. The top two panels show the logarithm of total energy density ( left) and the velocity (right) 0.6 s after the start of jet injection. The minimum andmaximum values for internal energy are log E(ergs g~1) \ 18.5 and 20.3, for velocity v(cm s~1) \ 5.0] 108 and 1.5] 1010, for density log o(g cm~3) \ 5 and11.5, and for pressure log P(dynes cm~2) \ 23 and 30.1. The color bar indicates the logarithmic interpolation between these two extrema. For example, theyellow-orange region of the density plot represents log o\ 5 ] 0.40(11.5 [ 5) \ 7.6. The jet is clearly less well collimated in the high-viscosity case,presumably because of the smaller mass of the disk and the weaker pressure gradients conning the ow to the polar region. Note also the presence of plumes of high outward velocity at about 45 in the plot for the high-viscosity case. See MW99 for discussion.


9/16


FIG. 8.Further detail of the initial hydrodynamical focusing of the initial jets in the high (left) and low (right) disk viscosity cases (see Fig. 7). Thepressure ranges from 1022 dynes cm~2 (dark blue) to 1027 dynes cm~2 (red), and the region plotted is the inner 400 km (x) by 800 km (y). The longest velocityvectors represent a speed of 1.5] 1010 cm s~1. Both plots are at a time 0.6 s after jet initiation. The lower density in the high-viscosity case results in the jetbeing less well collimated. It expands laterally with approximately the sound speed, which for equal internal and kinetic energies is comparable to its radialspeed. A jet with a higher fraction of internal energy would spread even more, especially if the disk mass were further reduced. The jet in the low-viscositycalculation also propagates in a lower density medium and moves farther before a shock develops.

maintain some mild collimation. In the high disk mass case,though, the opening angle is about half as great, compara-ble to the 10 input angle. If one continued this trend to stillsmaller disks and higher internal energy loading factors, the

jet would fan out still more. Obviously, in the limit of nodisk and energy deposited entirely as a thermal bomb at themiddle, an isotropic explosion would develop. An inter-esting question still to be explored is what would happen inthe absence of a disk, but for o-center energy deposition.

The calculations of MW99 (their Fig. 26) showed that theratio of internal energy to mass density in the jet declinedroughly as r~1@3. That is, the density declined between r~1and r~2, and the radiation entropy, T3/o, was constant. Forconstant mass ux, constant velocity, and constant openingangle (h), the quantity nr2h2ov should be a constant, so thatoP r~2. That the density declined more slowly reects thehydrodynamical collimation and mild deceleration of the

jet. Assuming that the r~1@3 scaling persists for much lowervalues of initial internal energy, one expects the ratio ofinternal energy to kinetic energy to decline by about afactor of 10 going from the region where the jet forms out to109 cm. So for jets that initially had from, say, 1% to 100%of their initial energy in the form of radiation, the ratio ofpressure to kinetic energy, at 109 cm would be in thef

P,

range 0.0010.1. We shall use these values in the nextsection. For much hotter jets, the eects of special relativity(especially the contribution of pressure to momentum)make the above scaling law invalid, and the jets stay muchmore tightly collimated than a nonrelativistic calculationwould suggest (Woosley, MacFadyen, & Heger 1999b;

et al. 2000; Aloy et al. 1999, 2000). For much coolerMu llerjets we shall nd that the answer does not vary much for

(Fig. 10).fP\ 0.01

4.2. Jet Propagation and Supernova Explosion inT ype II Collapsars

The spherically symmetric explosion of model A01, fol-lowed until 100 s after the launch of a weak shock in the

KEPLER code ( 2), was mapped onto the Eulerian grid ofa two-dimensional version of the PPM code. This grid used200 radial zones spaced logarithmically between an innerboundary at 109 cm and the stellar surface at 8.1 ] 1013 cm.Forty angular zones, concentrated near the pole, were usedto simulate one quadrant of the stellar volume, assumingaxial and reection symmetry across the equatorial plane.The angular resolution varied from at the pole to1.25 3.5at the equator. Nine atomic species (C, O, Ne, Mg, Si, Fe,He, n, p) were carried, and the equation of state includedradiation and an ideal gas consisting of electrons and thenine ions. At 100 s, the inner 1.99 of the star wasM

_removed and replaced by a transmitting (zero radial gra-dient of all variables) boundary condition at 109 cm. The1.99 and all subsequently accreted matter contributedM

_to a point-mass term in the gravitational potential that wascalculated using an integral Poisson solver & Stein-(Mu llermetz 1995; MW99). At this time, the weak initial shock wasalready at 1.1] 1010 cm when a simulated jet was turnedon at the inner boundary.

A given jet-powered model was specied by its energyux, mass ux, momentum ux, and theF

e(t), F

o(t), F

u(t),

pressure of the jet, all at 109 cm,Pjet

,

Fe \

E0

jet

(t)

Ajet

, (1)

Fo

\ 2Fe

CfP

A 1cjet

[ 1] 1

B] 1

D~1vjet~2 , (2)

Fu

\ Fo

vjet

, (3)

Pjet

\ 12

fP

Fu

, (4)

with

E0jet

(t) \ vc2

4

5

6

0

0

M0 (t) t\ tp

,

M0 (tp

) maxCA t

tp

B~5@3, 10~6

Dt t

p.


10/16


Here is the time when the accretion rate begins to followtp

the power-law decline, usually several hundred seconds(Fig. 5). The opening half-angle of the jet was taken to be

and the adiabatic index to be The accre-hjet

\ 10 cjet

\ 43

.tion rate was restricted to that matter that fell in throughM0the inner boundary within 45 of the equator. Thus, thedensity of the jet is given by 2vM0 c2/[(4f

P] 1)A

jetvjet3 ],

where is the cross-sectional area of the initial jet,Ajet

1.9] 1017 cm2 for the assumed boundary radius and colli-

mation. The assumed initial velocity of the jet, was 1010vjet,cm s~1.

We considered four cases: (1) v\ 0.001, (2)fP

\ 0.001;v\ 0.001, (3) v\ 0.001, and (4) v\ 0.01,f

P\ 0.01; f

P\ 0.1;

The results of these calculations are summarizedfP

\ 0.01.in Table 2 and Figures 911. Here the model namingfollows the convention JMN, where J indicates thatthe model included a jet but was otherwise based upon theinitial model A01 (Table 1) ; M is the exponent of theefficiency factor, v\ 10~M ; and N is the exponent ofthe pressure factor, The accreted mass, *M inf

P\ 10~N.

FIG. 9.Specic energy density (internal plus kinetic) of the jet andexplosion shown at times of 5, 10, 20, and 27 s after initiation of the jet inmodel J22 (Table 2). The passage of the jet initiates a shock that propa-

gates to lower latitudes, eventually exploding the entire star. The originalsupernova shock can be seen as a pale blue circle at a radius of about2] 1010 cm.

FIG. 10.Pressure in the jet and surrounding star at 5.0 s after theinitiation of the jet in four dierent models (Table 2). For the J3n series,higher pressure clearly leads to greater jet divergence, more mass swept up,and slower propagation. Model J22 had a higher jet energy than the othermodels. The edge of the helium core is at 5 ] 1010 cm o\ 0.6(X

H\ 0.01;

g cm~3), but the presupernova density begins to decrease rapidly below100 g cm~3 outside of 1.4] 1010 cm (green disk).

Table 2, is smaller than the 3.71 computed without a jetM_

(Fig. 5 and Table 1) for model A01 because the jet impeded

the accretion at high latitude and because the accretion wasnot quite complete after 500 s (Fig. 5). The total energyinput by the jet was still v*Mc2, but the energy in Table 2was also reduced by the work done up to 500 s in unbindingthe star and by the internal and kinetic energy that passedinside the inner boundary. The 2.55]1050 ergs due to theinitial shock has been subtracted in Table 2 so that E

totreects only the energy input by the jet.

In all cases a very energetic asymmetric supernovaresulted. Since the integrated mass of the jet in our code wascomparable to that of the stellar material within 10, thetime for jet breakout was approximately the stellar radiusdivided by the jet input speed, or about 8000 s. Since theenergy of the jet engine had declined greatly by that time, as

a result of the declining accretion rate (Fig. 5), the jet thatbroke out was only mildly relativistic (see also 5). Both thelong timescale and the small amount of relativistic matter

TABLE 2

EXPLOSION CHARACTERISTICS AT t \ 400 s AFTER JET INITIATION

*M Etot

Name v fP

(M_

) (1051 ergs) R(h[ 10) R(h[20)

J33 . . . . . . 0.001 0.001 2.76 3.38 0.075 0.037

J32 . . . . . . 0.001 0.01 2.69 3.23 0.102 0.047

J31 . . . . . . 0.001 0.1 2.51 3.00 0.425 0.256

J22 . . . . . . 0.01 0.01 1.72 19.91 0.429 0.230


11/16


FIG. 11.Equivalent isotropic kinetic energy as a function of polarangle for four models having variable energy efficiency factors and internalpressures (Table 2). Except for J32, the calculations are all sampled at 400 safter the initiation of the jet (500 s postbounce). At this point the jets haveexited the helium core and are moving through the hydrogen envelope.Model J32 is shown at two times, once at 400 s and later at 7716 s, as the jetreaches the surface of the star at 8] 1013 cm (dot-dashed line). The colli-mation of model J32 is further improved by its passage through the hydro-gen envelope. Note that the degree of collimation is strongly dependentupon Equivalent isotropic kinetic energy is dened as the integralf

P.

from the center to the surface of the star of its kinetic energy in the solidangle subtended by h and h]*h, divided by the solid angle,2n[cos h[ cos (h]*h)], and multiplied by 4n. The injected energy at thebase of the jet would be a at line out to 10 with a value equal to 66 v*Mc2with *M in Table 1 and 66 \ [1 [ cos (10)]~1. Tick marks along the topaxis give the angular zoning of the two-dimensional code.

are inconsistent with what is seen in common GRBs.However, if the hydrogen envelope had been lost leaving abare helium star, a longer than typical GRB could haveresulted.

Figures 10 and 11 illustrate how the pressure balancebetween the jet and the star through which it propagatesaected its collimation properties. The interaction at latetimes with the hydrogen envelope has a smaller eect on theangular energy distribution that was set chiey by andf

Pthe interaction with the helium core. Model J33 had thelowest internal pressure (note that the actual value of theinitial pressure depends upon the product of v and Thef

P).

nal jet was collimated even more tightly than given by itsinitial injection. That is, a jet initially of 10 half-width willexit the star with an FWHM of less than 2, about 0.06% ofthe sky, though the angular resolution of the code is ques-tionable for such small angles. Meanwhile, the energy atlarger angles was not much greater than that given by theinitial, weak spherically symmetric explosion, 1050.4 ergs.There was little sharing of the jet energy with the star, and,except for the jet, the supernova energy remained low.

This behavior is to be contrasted with models J22 andJ31 where the jet collimation was weaker and much moreenergy was shared with the star. Note that though modelJ22 had about 6 times the total energy of model J31 owingto its larger v, the fraction of energy at large angles in both

these models was signicantly greater than in models J32and J33. That model J22 was not 10 times more energetic(the ratio of the v values) shows the inhibition of the accre-tion by the strong jet. Still, model J22 would be a verypowerful supernova as well as one accompanied by a jet.

As the jet pushes through the star, a shell of relativelyhigh density material is built up at its head. The shell con-tains material from the inner regions of the star that is sweptalong as the jet propagates through the star. The red

regions near the polar axis in Figure 10 correspond to thelocation of the shells. If some of this material is acceleratedto relativistic velocities by the jet, as preliminary relativisticcalculations indicate (Aloy et al. 2000), it may produceobservable features in the spectrum of the burst. More workis needed, however, to determine the detailed compositionof the jet material at large distances from the center of thestar.

The angular factor R(h[ 10) in Table 2 is the ratio ofthe integral of the kinetic energy due to the jet outside 10polar angle (98.5% of the sky) to the total kinetic energy inthe star due to the jet (see Fig. 11). These energies werecomputed by taking the total kinetic energy at 400 s after jetinitiation in both regions and subtracting the kinetic energy

of the initial supernova shock. R(h[ 10) measures theextent to which the jet spread laterally and shared its energywith the rest of the star. The limiting case, R(h[ 10) \ 0,would correspond to a jet that shared none of its energywith the supernova outside an initial 10 polar angle. Thissort of behavior is expected for cold jets with internalpressure small compared to the exploding helium core. Theother extreme, where the jet shared its energy evenly withthe entire star and produced a spherical explosion, wouldcorrespond to R \ cos h\ 0.985. Our hot jets lie some-where between these two limits. The quantity R(h[ 20)was similarly computed for a polar angle of 20. The iso-tropic limit there would be 0.940.

Most of the curves in Figure 11 were evaluated at a time

(500 s) when the central part of the jet had moved welloutside the helium core (initial radius 5.0] 1010 cm,dened by the point where the mass fraction of hydrogen is1%) but had not yet encountered much of the hydrogenenvelope mass. Thus, the energy distribution is also appro-priate for exploding helium stars of mass D9 TheseM

_.

curves then give the angular energy distribution of Type Iband Ic supernovae that would accompany GRBs producedby jets of the specied properties.

For one calculation (model J32), however, the propaga-tion beyond 500 s was considered all the way to the surfaceof the red supergiant at 7820 s note the near(DR/v

jet;

constancy of the jet head speed in the hydrogen envelope).This calculation required a second mapping of the explo-sion onto a new grid. At 500 s a new inner boundary was setup at 1011 cm to alleviate the restrictive time step limitationimposed by the Courant condition at small radii. The massinterior to the new inner boundary was added to the centralpoint mass. A total of 150 radial zones spaced logarithmi-cally between 1011 and 8]1013 cm were used with thesame angular zoning as before. At 500 s the bulk of theaccretion had already taken place (Fig. 6) and the sub-sequent accretion could be approximated using the t~5@3scaling at late times (Fig. 5), s) max [(t/M0 (t) \ M0 (500500)~5@3, 10~4]. The jet was injected as a boundary condi-tion at the new inner boundary, r \ 1011 cm, as before with

and with cm s~1. The value ofhjet

\ 10 vjet

\ 1] 1010 fP


12/16


was reduced to 10~3 to approximate the conversion ofthermal energy to kinetic energy that occurs as the gasexpands adiabatically between 109 cm, where was pre-f

Pviously 0.01, and 1011 cm.

The results of this calculation at the time of jet breakoutare shown in Figure 12. Note the lateral propagation of astrong, high Mach number shock with a large pressure anddensity jump. This shock wraps around the star and, byabout 10,000 s, has ejected even the material in the equato-

rial plane. The jet core spreads signicantly out to about1013 cm, well into the hydrogen envelope, but then is recol-limated by a cocoon of high-pressure, shock-heated gas.Farther out at 5] 1013 cm, it spreads once again as a resultof partial blockage by a plug of high-density material thatthe jet shoves along.

Though these calculations of jet propagation werecarried out against the background of a supernova alreadyexperiencing a weak explosion, the area per unit solid angleof the jet is so large and its speed so great that it rapidlyovertakes and overwhelms the initial shock. Thus, qualit-atively, the results of this section, including Table 2 andFigures 911, would also apply to Type I collapsars.

5. SHOCK WAVE BREAKOUTA strong shock wave breaking out of a star that is experi-

encing mass loss will produce transient electromagneticemission in two ways: (1) as the shock breaks through the

surface and exposes hot material (Colgate 1969, 1974) and(2) as the highest velocity material encounters the circum-stellar wind (Chevalier 1982; Fransson 1984; Leising et al.1994). While the highest energy ejecta in the present modelsmay only have a small solid angle (Fig. 11), there still maybe a very high luminosity in both forms of transients. Thisradiation would not be appreciably beamed.

To explore this possibility, we compute the eects of twostrong, spherically symmetric shocks in model A01 using

the KEPLER code. The motion of the piston at the edge ofthe iron core (see 2) was adjusted to give both models veryhigh kinetic energy at innity, 1.44 ] 1053 and 1.39] 1054ergs. While these were one-dimensional calculations,they should simulate the conditions experienced bystellar regions within solid angles that experience these equivalent isotropic energies (Fig. 11). The advantages ofthe one-dimensional Lagrangian calculations are that veryne zoning can be employed near the surface and radiativediusion can be included in the calculation.

The resulting light curves for the two models are given inFigure 13. Since KEPLER has a very simple radiationtransport scheme (ux-limited radiative diusion) and theopacity is dominantly due to electron scattering, the eec-

tive temperature obtained in these calculations is an under-estimate of the actual (color) temperature, by a factor ofT

c,

23 (Ensman & Burrows 1992). Thus, we expect, for thehigher energy model, a transient with mean photon energy,

FIG. 12.Structure of model J32 as the jet nears the surface 7820 s after core collapse. The total explosion energy at this time is 4.1] 1051 ergs, probablya good approximation to the nal value. The theta velocity radial velocity the logarithm of density and the logarithm of pressure(v

h), (v

r), (log

10P) (log

10P)

are shown with the minimum and maximum values are as follows: [1.5] 109 and 1.5] 109 for [6.7] 107 and 1.0] 1010 for [9 and [6.8 forvh

; vr

;and 2.5 and 10.9 for all in cgs units. The colors indicate the interpolation scale between minimum and maximum (see Fig. 7). Positive islog

10o ; log

10P, v

hmotion away from the polar axis (h\ 0) along an arc of constant radius. The plot shows the expansion of the high-pressure bubble blown by the jetv

hsweeping around the star (red) but also an inner region of collimation (blue, purple, and green). At r \ 7] 1013 cm the x-component of the velocity of theexpansion shock is cm s~1 (compared to a sound speed of only 3 ] 106 cm s~1), while the y-component is cm s~1, resulting inv

x\ 2.0] 109 v

y\ 9.6] 109

an aspect ratio for the bubble between 0.2 and 0.3. The velocity and pressure plots show a collimation of the jet ow near 1 ] 1013 cm, well into the hydrogenenvelope of the star, and evacuation of a low-density column by the jet, but not, however, the plug of high-density material being shoved along by the jet.


13/16


FIG. 13.Shock breakout in spherically symmetric explosions formodel A for the KEPLER calculation of two cases of a 1.44 ] 1053 ergexplosion (dotted line) and a 1.39] 1054 erg explosion (solid line) as afunction of time since the onset of shock breakout. The upper panel givesthe total luminosity and the lower panel the eective temperature, whichprobably underestimates the color temperature, by about a factor ofT

c,

23. The FWHM values of the luminosity curves are 12.5 and 94 s for lowand high energy, respectively. The widths of the temperature curves are 56and 350 s.

keV, lasting for about 10 s. Since these X-rays3kTcD 0.25

would not be beamed, the luminosity would be the value inFigure 12 times an appropriate solid angle (about 1% of thesky), or D1047 ergs s~1. Less energetic but longer lastingtransients would come from regions of the surface thatexperienced a smaller equivalent isotropic energy. This ismuch brighter and harder than the shock breakout tran-sients expected from common Type II supernovae andwould be a diagnostic of our model.

Even more energy may come out, but over a longer time,in the form of X-ray and radio afterglows from circumstellarinteraction. KEPLER is a nonrelativistic hydrodynamicscode, but we can nevertheless estimate the amount of rela-tivistic ejecta following Woosley, Eastman, & Schmidt(1999a). Assuming the near constancy of the quantity!b(or3)1@5 [with b\ v/c and !\ (1 [b2)~1@2] across themildly and strongly relativistic domains (Gnatyk 1985 ;McKee & Colgate 1973), we obtain ! as a function of theexternal mass using the output of KEPLER in the nonrela-tivistic region (Fig. 14). The nest zones in the KEPLERcalculation had mass 1029 g. However, external to about1031 g the density was inuenced by a transition from opti-cally thick to optically thin zones, and outside of 1030 g the

density was aected by a 3 dyne cm~2 surface boundarypressure employed to stabilize the calculation. We thus onlytrust the calculated distribution of or3 out to 1031 g andexpect that there may be relatively minor deviations fromthat power law outside. Of course, we also cannot employzones in our t that in the KEPLER calculation weremoving at superluminal speeds after the conversion of allinternal energy to kinetic energy.

For these reasons, we used only the KEPLER resultsinside 1031 g and used a power-law extrapolation for

FIG. 14.Density structure near the surface of the red supergiant (model A) (left panel) and extrapolated Lorentz factor for the breakout of a shock ofenergy 1.39] 1054 ergs (right panel). Crosses show the product of density times r3 for the hydrogen envelope and zones near the surface. A power lawor3P (*m)0.86 ts the surface of the Kepler model well. Using this density structure and the near constancy of !b(or3)[email protected]] 106 (g1@5), one canextrapolate (dashed line) to obtain the relativistic ! for mass zones outside of 1030.5 g. However, the curve for ! is not valid for external masses less than 1029.3g (see text). The maximum ! for this shock energy is thus !B2.5. The curved dotted line at masses below 1030.7 shows the results one obtains using theactual density structure in KEPLER for optically thin zones rather than the extrapolation ofor3 in the left panel.


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smaller masses. The ts for !b(or3)1@5 gave nearly constantvalues of 7.3] 105 and 3.3] 106 (units g1@5) for the1.44] 1053 and 1.39] 1054 erg explosions, respectively.From these we infer the distribution of! with exterior massgiven in Figure 14 for the high-energy explosion. Much lessrelativistic matter was ejected in the lower energy explosion.

One cannot use the scaling relation for ! indenitely,though. Especially for extended objects like red supergiants,there comes a point when the shock thickness, mediated by

radiation and electron scattering, becomes greater than thedensity scale height. At about the same point, materialbehind the shock will become optically thin and lose energy.For relativistic shocks, both of these conditions occur whenthe Thompson depth of material just behind the shockbecomes unity (Imshennik & Nadyozhin 1989). From theknown dependence of or3 on external mass (Fig. 14),or3 \ 105.31[*M(g)]d with d\ 0.86, one determines theexternal mass beyond which the shock speed cannot beextrapolated,

*M \ 2.0]1029A R

*1014 cm

B2A d0.86

BA0.34 cm2 g~1i

Bg ,

where is the stellar radius. For the much more compactR*

stars considered by Woosley et al. (1999a), small amounts ofmaterial can be accelerated to comparatively large values of!, but here, for red supergiants, there is a limit of!B 2.5.Thus, jet-powered Type II supernovae cannot make GRBsof the common type. The duration of the jet is too short forthe jet itself to acquire large ! outside the star, and therelativistic shock acceleration mechanism fails to yield suffi-ciently energetic material.

However, 10~4 of material with !\ 2.5 is still aboutM_

1050 ergs of kinetic energy. This material will radiate itskinetic energy after encountering roughly its rest mass,which will take a time

q\ 104A10~5 M

_yr~1

M0

BA10 km s~1v

wind

Bs .

For a solid angle 1% of the sky, this again implies D1044ergs s~1, presumably in hard X-rays. In fact, the circumstel-lar interaction of these breakout shocks resembles in manyways the multiwavelength afterglow of GRBs, except thatthere would be no GRB. Since these could be frequentevents and may not be beamed to the same small angle as aGRB, there could be many such orphan afterglows.

6. CONCLUSIONS

It is widely believed that many diverse phenomena ingalactic nuclei (quasars, BL Lac galaxies, Seyfert galaxies,radio galaxies, blazars, etc.) are powered by a commonengine, accretion at a rate of a few solar masses per yearinto a supermassive black hole of several billion M

_(Blandford & Rees 1974; Antonucci 1993). The dierences

relate to the mass and rotation rate of the hole, the environ-ment in which it accretes, and the angle at which oneobserves. We suggest here that a similarly large variety ofenergetic phenomena can also be powered by accretion at amuch higher rate into a small, stellar mass black hole. Itcould be that GRBs are but the tip of the iceberg andthat many other interesting kinds of events await discovery.

In this paper we have drawn attention to the possibilityof two kinds of hyperaccreting black hole scenarios formaking supernovae and high-energy transients. Type I col-lapsars, as explored previously by MW99, have black holesthat form promptly. The black holes in Type II collapsars,on the other hand, form after some delay owing to thefallback from a supernova lacking adequate energy to eject

all of its helium core and heavy elements. Table 3 outlinesthe diverse phenomena resulting from the two collapsartypes occurring in stars with varying radii.

We have shown (Fig. 11) that explosions of nearly con-stant total energy can have highly variable equivalent iso-tropic energy dependent not only upon the angle at whichthey are viewed but also upon how passage through the staracts to collimate (or decollimate) the jet. This should be trueof both Type I and Type II collapsars. For the roughly 10GRBs for which redshifts and GRB energies had been deter-mined as of 1999 July, the inferred energy has an enormousrange: from about 8] 1047 ergs for GRB 980425 (Galamaet al. 1998) to several times 1054 ergs for GRB 990123(Kulkarni et al. 1999). Yet, as MW99 pointed out, the total

explosion energy for both these events may have beenD1052 ergs with the energy collimated into a tight jet forGRB 990123 (see also Fig. 11 and Table 2) and dissipated inproducing a powerful supernova for GRB 980425 (Iwamotoet al. 1998; Woosley et al. 1999a). The collimation proper-ties of the jet may be more important than its total energy indetermining the observed properties of a GRB.

We continue to support the view of MW99 that common,long hard GRBs (mean duration 20 s) seen by BATSE areproduced in Type I collapsars. The GRB commences onlyafter the jet has drilled a hole through the star (510 s) sothat it can expand freely. The jet may be energized either by

TABLE 3

OBSERVABLE PHENOMENA R ESULTING FROM COLLAPSARS

Parameters Type I Type II

B la ck hole fo rm ation tim escale (s) . . . . . . . P ro mpt, q[1 Delayed, 303000M0 (M

_s~1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.1 0.00010.01

qaccretion

a ( s ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B10 303000No H envelope (Wolf-Rayet star) ........ t

engine[ t

bo, GRB ] Type Ib/c SN t

engine[ t

bo, long GRB ] Type Ib/c SN

Small H envelope (Blue supergiant) . . . . . . tengine

\ tbo

, Type II SN; XRT tengine

Z tbo

, long GRB? ] Type II SN

Large H envelope (Red supergiant) .. . . . . tengine

> tbo

, Type II SN; XRT tengine

[ tbo

, Type II SN; XRT

NOTE.All colapsar supernovae are predominantly jet-powered and therefore asymmetric.a A small amount of accretion can continue for a long period after the explosion (see Fig. 5).


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neutrino energy transport from the disk or by MHD pro-cesses. However, we also see the possibility of a number ofother phenomena in Type II collapsars depending upon thecollimation of the jet, the duration and mass of the accre-tion, and the nature of the presupernova star (see alsoTable 3).

Smothered and broadly beamed gamma-ray bursts;GRB 980425.These occur in helium stars in which the jeteither fails to maintain sufficient collimation (e.g., is too

hot compared to the star through which it propagates) orloses its energy input before breaking out of the star s;([10MW99; MacFadyen & Woosley 19981). An energetic super-nova still occurs (SN 1998bw in this case), and a weak GRBis produced, not by the jet itself, but by a strong, mildlyrelativistic shock from breakout (!D 5) interacting with thestellar wind (Woosley et al. 1999a). Kulkarni et al. (1998)and Wieringa, Kulkarni, & Frail (1999) have also arguedthat the radio emission from GRB 980425 was not stronglybeamed and had a total energy not too much greater thanthe GRB ergs), suggesting that GRB 980425 was not([1049a much more powerful GRB observed from the side (as in,e.g., Nakamura 1999). Because these events are so low ingamma-ray energy, many could go undetected by BATSE,

and these could be the most common form of GRB in theuniverse. Because the initial jet may be less eectively colli-mated in GRBs made by supernova fallback, it is temptingto associate these phenomena with delayed black hole for-mation and the stronger GRBs with prompt black holeformation. More study is needed, but we tentatively oermodel J22 in Figure 11 as a possible prototype for SN1998bw. This model at 500 s postbounce would appearessentially as anD1052 erg He core explosion with a small(10) hole along each axis. A hotter jet, lower disk mass,or larger efficiency factor (v) might provide even betteragreement.

L ong gamma-ray bursts ; s.Though typicalqburstZ 100

long, complex bursts observed by BATSE last about 20 s,

there are occasionally much longer bursts. For example,GRB 950509, GRB 960621, GRB 961029, GRB 971207, andGRB 980703 all lasted over 300 s. These long durations maysimply reect the light crossing time of the region where the

jet dissipates its energy (modulo !~2), especially in the exterior shock model for GRBs. However, if the event isdue to internal shocks, the duration depends on the time theengine operates. Such long bursts would imply enduringaccretion on a much longer timescale than one expects inthe Type I collapsar model where the black hole formspromptly. The fallback-powered models discussed in thispaper could maintain a GRB for these long timescales (Fig.6). Indeed, considerable power may still be developed by theGRB days after the initial event with a luminosity decliningroughly as t~5@3 (Fig. 5).

Very energetic supernovae: SN 1997cy.Germany et al.(1999) have called attention to this extremely bright super-nova with an unusual spectrum. The supernova was TypeIIn, and its late-time light curve, which approximately fol-lowed the decay rate of56Co, would require of 56NiZ2 M

_to explain its brightness. Perhaps this was a pair-instabilitysupernova (Woosley & Weaver 1982; A. Heger & S. E.Woosley 2001, in preparation). On the other hand, circum-

1 See also http://www.itp.ucsb.edu/online/gamma

c99/woosley.

stellar interaction could be the source of the energy and theagreement with merely fortuitous. This wouldq

1@2(56Co)

require both a very high explosion energy and a lot of massloss just prior to the supernova. The sort of model describedin 4.2, especially model J22, could provide the large energyin a massive star that would be naturally losing mass at ahigh rate when it died. Large quantities of 56Ni might alsobe made by the wind from the accretion disk (see below),but the presupernova radius of this event was too large and

the jet would have shared its energy with too great a massto make a common GRB. In 5 we placed a limit of![ 2.5on any relativistic ejecta in such events. We thus regard thedetection of a short, hard GRB from the location of SN1997cy as spurious.

Nucleosynthesis: 56Ni and the r-process.An explosionof 1052 ergs focused into 1% of the star (or 1053 ergs into10%) will have approximately the same shock temperatureas a function of radius as an isotropic explosion of 1054 ergs.From the simple expression ergs (Woosley4

3nr3aT4D 1054

& Weaver 1995), we estimate that a shock temperature inexcess of 5]109 K will be reached for radii inside 4] 109cm. The mass inside that radius external to the black hole(assumed mass initially 2 depends on how muchM

_)

expansion (or collapse) the star has already experiencedwhen the jet arrives. Provided the star has not expandedmuch before the jet arrives, an approximate number comesfrom the presupernova model, 3 times the solid angle ofM

_the explosion divided by 4n, orD0.1 Additional 56NiM

_.

will be synthesized by the wind blowing o the accretiondisk (MW99; Stone et al. 1999), and this may be the domi-nant source in supernovae like SN 1998bw. Even for therelatively low accretion rates in Type II collapsars, the tem-perature in the accretion disk is hot enough to photo-disintegrate the accreting matter to nucleons (Popham et al.1999). As this material expands and cools, it will reassemblemostly to 56Ni. Stone et al. (1999) estimate that an appre-ciable fraction of the accreted matter will be ejected in such

a wind.The composition of the jet itself depends upon details of

its acceleration that are hard to calculate. However, itshould originate from a region of high density and tem-perature (Popham et al. 1999). The high density willpromote electron capture and lower The high entropy,Y

e.

low and rapid expansion rate are what is needed for theYe,

r-process (Homan, Woosley, & Qian 1997). The mass ofthe jet, D10~4 (corrected for relativity), is enough toM

_contribute signicantly to the r-process in the Galaxy evenif the event rate was that of supernovae and the jet[1%carried only a fraction of its mass as r-process.

Soft and hard X-ray transients from shock breakout.Focusing a jet of order 1052 ergs into 1%10% of the solidangle of a supernova results in a shock wave of extraordi-nary energy (Fig. 11). As it nears the surface of the star, thisshock is further accelerated by the declining density gra-dient. We have estimated ( 5, Fig. 13) that as the shockerupts from the surface of the star one may have soft X-raytransients of luminosity up to D1049 ergs s~1 times thefraction of the sky to which high-energy material is ejected(typically 0.01). The color temperature at peak would beapproximately 2]106 K (see also Matzner & McKee1999). A 1053 erg shock gave a transient about half as hotand 10 times longer and fainter. The impact of the mildlyrelativistic matter could give an enduring X-ray transientlike the afterglows associated with some GRBs, even though


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the timescale is too long for the X-ray burst to be a commonGRB itself.

Supernova remnants with unusual symmetry.The ex-panding remnant of a supernova exploded by a jet shouldlack spherical symmetry. Along the rotational axis of thepresupernova star matter expands with much greater veloc-ity than along the equator. Dependent upon details of thecircumstellar shock interaction, this could lead to either anelongated nebula characteristic of bipolar outow or a

torus of slower moving, denser debris in the equator. Themassive stars that produce collapsars have a very highabundance of oxygen, so one might seek either toroidal orbipolar supernova remnants (SNRs) that are oxygen rich.Lasker (1980) pointed out that N132D in the LMC mightbe such a remnant. More recent analysis (Sutherland &Dopita 1995; Morse, Winkler, & Kirshner 1995) suggeststhat the remnant may have elliptical structure. AnotherSNR with toroidal structure is E0102[72 in the SMCrecently studied by the Chandra X-Ray Observatory.

Mixing in supernovae: SN 1987A.It is generally agreed(Arnett et al. 1989) that the explosion that gave rise to SN1987A initially produced a neutron star ofD1.4 ThereM

_.

may have beenD0.1 of fallback onto that neutron starM_

(Woosley 1988), and a black hole may or may not haveformed. Again invoking our Ansatz that evenL

jet\ vM0 c2,

for vD 0.003, we have a total jet energy of 6]1050 ergs.This is about half of the total kinetic energy inferred for SN1987A. Thus, very appreciable mixing and asymmetrywould be introduced by such a jet, provided the material

that fell back had sufficient angular momentum to accumu-late in a disk outside the compact object. However, thiswould not be enough energy to make a powerful GRB asproposed by Cen (1999). The radio and X-ray afterglow thatone might expect from the impact of such a jet on thecircumstellar medium was also not observed.

Still to be discovered.It may be that, especially withcommon GRBs, we have just seen the tip of the iceberg ofa large range of high-energy phenomena powered by hyper-

accreting, stellar mass black holes. We already mentionedthe possibility of a large population of faint, soft bursts likeGRB 980425. Other possibilities include very long GRBsbelow the threshold of BATSE, orphan X-ray afterglowsfrom jet-powered Type II supernovae, GRBs from the rstexplosions of massive stars after recombination, and more.It is an exciting time.

The authors gratefully acknowledge helpful conversa-tions on the subjects of gamma-ray bursts with Chris Fryer,David Meier, Ewald and Martin Rees. We alsoMu ller,thank the anonymous referee for useful comments. Thiswork has been supported by NASA (NAG5-8128 and MIT

SC A292701), the NSF (AST 97-31569 and INT-97-31569),the Department of Energy ASCI Program (W-7405-ENG-48), and the Alexander von Humboldt-Stiftung (1065004).We are also grateful to the Max-Planck-Institut Astro-fu rphysik for its hospitality while the nal draft of this paperwas prepared.

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a.i. macfadyen, s.e. woosley and a. heger- supernovae, jets and collapsars

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