Guido Guido BarbielliniBarbiellini (University and INFN, Trieste)Annalisa CelottiAnnalisa Celotti (SISSA, Trieste)

Francesco LongoFrancesco Longo (University and INFN, Trieste)

The The fireworks fireworks model model for for GRBGRB

OUTLINE

n Introductionn Available energyn Vacuum breakdownn The formation of the fireballn Two phases mechanismn Jet angle estimation

ABSTRACT

n The energetics of the long duration GRB phenomenum is compared withmodels of a rotating Black Hole (BH) in a strong magnetic field generatedby an accreting torus. A rough estimate of the energy extracted from arotating BH with the Blandford-Znajek mechanism is evaluated with avery simple assumption: an inelastic collision between the rotating BHand the torus. The GRB energy emission is attributed to an high magneticfield that breaks down the vacuum around the BH and gives origin to a e±fireball. Its following evolution is hypothized, in analogy with the in-flightdecay of an elementary particle, to evolve in two distinct phases. The firstone occurs close to the engine and is responsible of energizing andcollimating the shells. The second one consists of a radiation dominatedexpansion, which correspondingly accelerates the relativistic photon-particle fluid and ends at the transparency time. This mechanism predictsthat the observed Lorentz factor is determined by the product of theLorentz factor of the shell close to the engine and the Lorentz factorderived by the expansion. The angular distribution of the emitted shells isthus determined by the bulk Lorentz factor at the end of the collimationphase.

Introduction

n The connection of GRB with massive stars and BHs iswell established (e.g. Vietri et al. 2001)

n There are indications for jetted emission from GRBs(Frail et al. 2001)

n Jet energy vs angle relationship (Rossi et. al. 2002)n The opacity problem for GRB is solved with relativistic

expansion (e.g. Piran 1999)n X-ray flashes observations

Available Energy

n Blandford-Znajek mechanism for GRBn ˜

¯

ˆÁË

Ê⋅=M

ME bhBZ

54103.0¡

Blandford & Znajek (1977)Brown et al. (2000)Barbiellini & Longo (2001)

Figure from McDonald, Price and Thorne (1986)

Available Energy

n Inelastic collision between a rotating BH (10 M¡)and a massive torus(0.1 M¡) that falls down onto the BH from the last stable orbit

n Conservation of angular momentum:

n Available rotational energy:

n Available gravitational energy:

n Total available energy:

W=W+W III ttbhbh

˜̃¯

ˆÁÁË

Ê-W=˜

¯

ˆÁË

Ê -WªD 33

232 12121

MM

MI

IIE bhbh

bhbhrot bhbh

2,

23

83

332 cMMM

EMM

ME tbh

tbhrot

bh

tbhrot bh

ªª˜̃¯

ˆÁÁË

ÊWªD

2

31

3cM

RMGM

RMGM

E tbh

bht

bh

bhtgrav ª-=D

5310@D+D=D gravrot EEE erg

Vacuum Breakdown

Polar cap BH vacuum breakdown

Figure from Heyl 2001

Vacuum Breakdown

n Critical magnetic field:

n Wald mechanism:

n Electric field:

n Pair volume:

13105.4 ⋅=cB Gauss

161022 ⋅ª= BJQ C

3

bhRVc ª

15102 ⋅ªE V/cm

The formation of the fireball

n Pair density (e.g. Fermi 1966):

n Magnetic field density:

n Energy per particle:

n Energy in plasmoid:

n Number of plasmoids:

29108 ⋅=±en cm-3

25108 ⋅=BU erg cm-3

40 10

-ª acche erg4510ª= Bcplasmoid UVE erg

810Bplasmoid

Bplasmoid EE

N hh ªD

=

The formation of the fireball

n Acceleration time scale in E field:

n Particle collimation by B field:

Curvature radius:

n Randomisation time scale by Compton Scattering inradiation field with temperature T0:

s1010 19

22

acceacc

acc eEccm

t hh -ªª

s10sinsin

19-ªªl

hl

r acccoll c

t

cm)Gauss()GeV(

106BE

=r

s10 16 accrandt h-ª

K101

8T 10

412

0 ª˜̃¯

ˆÁÁË

Ê⋅=a

Bp

Two phase expansion

n Phase 1 (acceleration and collimation) ends when:

n Assuming a dependence of the B field: this happens at

n Parallel stream with

n Internal “temperature”

collrand tt =3-µ RB

cm1081 ªR

acch301 =G

1'1

ªG

Two phase expansion

n Phase 2 (adiabatic expansion) ends at the smallerof the 2 radii:ß Fireball matter dominated:

ß Fireball optically thin to pairs:

n R2 estimationn Fireball adiabatic expansion

20 McE

RR =h

41

430

0 43

˜˜¯

ˆÁÁË

Ê=

ppair TR

ERR

p

02 50RR =

0

2'

'2

1RR

=GG

Jet Angle estimation

ß Analogy with the in-flight particle decay

Figure from Landau-Lif_its (1976)

Jet Angle estimation

n Lorentz factors

n Opening angle

n Result:

Energy Angle relationship

Predicted Energy-Angle relation