The Collapsar Model for

Gamma-Ray Bursts

S. E. Woosley (UCSC)

Weiqun Zhang (UCSC)

Alex Heger (Univ. Chicago)

Andrew MacFadyen (Cal Tech)

Harvard CfA Meeting on GRBs, May 21, 2002

Requirements on the Central Engineand its Immediate Surroundings

(long-soft bursts)

• Provide adequate energy at high Lorentz factor ( > 200; KE ~ 5 x 1051 erg)

• Collimate the emergent beam to approximately 0.1 radians

• Make bursts in star forming regions

• In the internal shock model, provide a beam with rapidly variable Lorentz factor

• Allow for the observed diversity seen in GRB light curves

• Last approximately 20 s, but much longer in some cases

• Explain diverse events like GRB 980425

• Produce a (Type Ib/c) supernova in some cases

• Make x-ray lines

Collapsars

Type Mass/sun BH Time Scale Distance Comment

I 15-40 He prompt 20 s all z neutrino-dominated disk

II 10-40 He delayed 20 s – 1 hr all z black hole by fall back

III >130 He prompt ~20 s z>10? time dilated, redshifted *(1+z) very energetic, pair instability, low Z

A rotating massive star whose core collapses to a black hole and produces an accretion disk.

Type I is what we are usually talking about.The 40 solar mass limit comes from assuming that all stars above 100 solar masses on the main sequence are unstable (except Pop III).

Microquasar GPS 1915in our own Galaxy – time sequence

Artist’s conception of SS433 based on observations

Quasar 3C273 as seen by the Chandra x-ray Observatory

Quasar 3C 175 as seen in the radio

Collapsar Progenitors

Two requirements:

• Core collapse produces a black hole - either promptly or very shortly thereafter.

• Sufficient angular momentum exists to form a disk outside the black hole (this virtually guarantees that

the hole is a Kerr hole)

Fryer, ApJ, 522, 413, (1999)

With decreasing metallicity, the bindingenergy of the core and the size of the silicon core both increase, making black hole formation more likely atlow metallicity. Woosley, Heger, & Weaver, RMP, (2002)

Black hole formation may be unavoidable for low metallicity

Solar metallicity

Low metallicity

In the absence of mass lossand magnetic fields, there wouldbe abundant progenitors.

Unfortunately nature has both.

15 solar mass helium core born rotating rigidly at f times break up

The more difficult problem is the angular momentum. Thisis a problem shared by all current GRB models that invokemassive stars...

note models “a-d” (withB-fields) and “e” (without)

Heger, Woosley, & Spruitin prep. for ApJ

Spruit, (2001), A&A, 381, 923

*

.

Heger and Woosley (2002) using prescription for magnetictorques from Spruit (2001)

15 M Helium Star

followed here

Some implications ....

• The production of GRBs may be favored in metal- deficient regions, either at high red shift or in small galaxies (like the SMC). The metallicity- dependence of mass loss rates for RSGs is an important source of uncertainty. (Kudritzsky (2000); Vink, de Koters, & Lamers A&A, 369, 574, (2001))

• But below some metallicity Z about, 0.1, single massive stars will not make GRBs because they do not lose their hydrogen envelope.

• GRBs may therefore not track the total star formation rate, but of some special set of stars with an appreciable evolutionary correction.

Progenitor Winds

Massive Wolf-Rayet stars – during helium burning - are known to have large mass loss rates, approximately 10-5 solar masses/yr or more. This wind may be clumpy and anisotropic and its metallicity dependence is uncertain. The density dependence of matter around a single star in vacuum could be approximately 1000 (1016 cm/R)2 cm-3 composed of carbon, oxygen, and helium, BUT

During approximately the last ~100 – 1000 years of its life, the star burns carbon (mainly) and heavier fuels. The mass loss rate of the star during these stages isunknown. No WR star has ever knowingly been observed in such a state. This means that the mass distribution inside ~ 1017 - 1018 cm is unknown (100 yrs at 1000 km/s).

Given the necessary angular momentum, black hole formation is accompanied by disk formation...

The Star Collapses (log j > 16.5)

alpha = 0.1 alpha = 0.001

MacFadyen & Woosley ApJ, 524, 262, (1999)

7.6 s 7.5 s

Neutrino flux high Neutrino flux low

In the vicinity of the rotationalaxis of the black hole, by a variety of possible processes, energy is deposited.

It is good to have an energy deposition mechanism that proceeds independently of the density and gives the jet someinitial momentum along the axis

7.6 s after core collapse; high viscosity case.

a=0.5

a=0.5

a=0

Optimisticnu-deposition

Neutrino annihilation energydeposition rate (erg cm –3 s-1)

MacFadyen & Woosley (1999)

Given the rather modest energy needs of current central engines (3 x 1051 erg?)the neutrino-powered model is still quite viable and has the advantageof being calculable.

A definitive calculation of the neutrinotransport coupled to a realistic multi-dimensional hydrodynamical model is still lacking.

Fryer (1998)

The Neutrino-Powered Model (Type I Collapsar Only)

MHD Energy Extraction

22

2 52 2 -115

o

50 -1

From the rotational energy of the black hole:

B M E ~ 0.4 ~ 4 x 10 B erg s

10 M

But only need ~ 4 10 erg s !

Sr c

x

Blandford & Znajek (1977)Koide et al. (2001)etc.

The efficiencies for converting accreted matter to energy need not be large. B ~ 1014 – 1015 gaussfor a 3 solar mass black hole. Well below equipartitionin the disk.

1a

Eventually shuts off when M can no longer sustain

such a large B-field.

Jet Initiation - 1

The jet is initially collimated by the density gradient leftby the accretion.

It will not start until the ram pressure has declined below a critical value.

MacFadyen, Woosley, & Heger, ApJ, 550, 410, (2001)

High disk viscosity (7.6 s + 0.6 s)

Low disk viscosity (9.4 s + 0.6 s)

(Energy deposition of 1.8 x 1051 erg/s commenced for 0.6 s; opening angle 10 degrees) log rho = 5 - 11.5

Jet Initiation -2

The Production of 56Ni

• Needed to power the light curve of the supernova if one is to be visible. Need 0.1 to 0.5 solar masses of it.

• A bigger problem than most realize The jet doesn’t do it – too little mass Forming the black hole depletes the innermost core of heavy elements Pulsars may have a hard time too if their time scale is > 1 sec

• In the collapsar model the 56Ni is made by a wind off of the accretion disk. It’s abundance is related to how much mass accretes into the hole

The disk wind: MacFadyen & Woosley (2001)

Neglecting electron capture in the disk

The Jet-Star Interaction

Relativistic Jet Propagation Through the Star

Zhang, Woosley, & MacFadyen (2002)

Initiate a jet of specified Lorentz factor (here 50), energy flux (here 1051 erg/s),and internal energy (here internal E is about equal to kinetic energy), at a givenradius (2000 km) in a given post-collapse (7 s) phase of 15 solar mass helium coreevolved without mass loss assuming an initial rotation rate of 10% Keplerian. Thestars radius is 8 x 1010 cm. The initial opening angle of the jet is 20 degrees.

480 radial zones120 angular zones 0 to 30 degrees80 angular zones 30 to 90 degrees

15’ near axis

Zoom out by 10

Pressure in the same model

The pressure in the jet is greater than in the star throughwhich it propagates.

The jet can be divided into three regions: 1) the unshocked jet, 2) the shocked jet, and 3) the jet head.

For some time, perhaps the duration of the burst, the jet that emerges has been shocked and has most of its energyin the form of internal energy. Information regardingthe central engine is lost.

Zhang, Woosley, & MacFadyen ApJ, to be submitted

jet head at 4.0 s

Initial opening angle 20 degrees; 1051 erg/s Initial opening angle 5 degrees; 1051 erg/s

Independent of initial opening angle, the emergent beamis collimated into a narrow beam with opening less than 5 degrees (see also Aloy et al. 2000)

The previous calculation was 2D in spherical coordinates.This puts all the resolution near the origin and also spends a lot of zones following the unshocked star.

Repeat using cylindrical coordinates and study the justthe jet’s interactions with finer zoning – but keeping the same density and temperature structure as in the staralong its rotational axis. Carry 80,000 km = 10% of the star.

i

int

50 -1

150 x 800 zones; zone size 100 km

R = 2000 km

initial jet radius = 700 km (20 deg at 2000 km)

= 10

E / KE = 20

E = 5 10 erg sx

Lorentz factor Density

Density structure at 2.2 seconds; inner80,000 km (star radius is 800,000 km).

Pressure at 2.2 seconds

Lessons Learned

• Even a jet of constant power is strongly modulated by its passage through the star.

• The variations in Lorentz factor and density can be of order unity.

• An initially collimated jet stays collimated.

• There may be important implications for the light curve – especially its time structure.

The Jet Explodes the Star

Continue the spherical calculation for a long time, at least several hundred seconds. See how the star explodes,the geometry of the supernova, and what is left behind.

8

12

50 -1 47 -1

int

o

Inner radius = 2 x 10 cm

Outer radius = 10 cm

= 10 for 20 seconds then declines to 2 at 1000 sec

E =5x 10 erg s declining to 10 erg s at 1000 s

E / KE = 20 declining to 2 at 1000 sec

20o

Density and radial velocity at 80 s (big picture)

Zoom in by 5...The shock has wrapped around and most of the star is exploding.Outer layers and material along the axis moves very fast. Most ofthe rest has more typical supernova like speeds ~ 3000 – 10,000 km s-1

80 seconds

t = 80 seconds

Shown on a magnified scale, there is still a lot ofdense low velocity material near the black hole

(Zoom in *100)

at 240 seconds

The shock has wrapped around and the whole star is exploding (initial radius wasless than one tick mark here). A lot of matter in the equatorial plane has notachieved escape velocity though and will fall back. Continuing polar outflow opensa channel along the rotational axis.

radial velocity/c

By this time the star has expandedto over ten times its initial radiusthe expansion has become (very approximately) homologous.Provided outflow continues alongthe axis as assumed, an observer along the axis (i.e., one who sawa GRB) will look deeper intothe explosion and perhaps see a bluer supernova with broaderlines (e.g., SN2001ke; Garnavich et al. 2002).

Continued accretion is occurring inthe equatorial plane.

240 seconds

Caution: Effect of disk wind not included here

Spreading of jets after they exit the star

50 -1E = 5 10 erg s high internal energy

10 = 5degrees

x

10 seconds 35 secondsZhang, Woosley, and MacFadyen (2002)

The jet properties are shown 35 seconds after its initiation. Lines give properties at 6.0,7.5, and 9.0 x 1011 cm. At15 degrees in Model W1, the equivalent isotropic energyflux is about 1046 erg s -1 (solid line).

GRB

GRB 980425Hard x-ray bursts

Unusual supernova(polarization, radio source)

A Unified Model for Cosmological Transients

5o , internal shocks

30o , external shocks?

Some Conclusions:

• The light curves of (long-soft) GRBs may reflect more the interaction of the jet with the star than the time variability of the engine itself.

•The emergent jet in the collapsar model may still contain a large fraction of its energy as internal energy. Expansion after break out of material with Lorentz factor of order 10 can still give final Lorentz factors over 100.

• Much weaker bursts are expected off axis (GRB 980425?, x-ray flashes?)

• Jet powered supernovae may have significant equatorial fall back. Jet may continue with a declining power for a long time

• Circum-burst mass distribution highly uncertain

• 56Ni synthesis given by disk wind. May be related to total mass accreted.