Formation of the central
engine of Long GRBs
Miguel A. Aloy
ERC Fellow
Departamento de Astronomía y Astrofísica
collaborators
N. DeBrye, C. F. Cuesta-Martínez, M. Obergaulinger, P. CerdáDurán, P. Mimica, J. A. Font

(Cerdá-Durán+’13, arXiv:1310.8290)
Supernovae and GRBs in Kyoto

Kyoto, 12 – 11 - 2013

LGRB Progenitors: Collapsars

replenishment

e roleOn thes 10%rogenm iguse ofm dearbon
erizesgravudingd notn a reonallyanger

Caveats:– Rapidly spinning stars produce low rotating coresdue to magnetic torques (Spruit’02, Heger+’05)
Solutions:accretion

MacFadyen & Woosley (1999)

OOSLEY & HEGER

Woosley (1993):– Collapse of a massive (M* ~ 30M, WR) rotatingstar that does not form a successful SN but
collapses to a BH (MBH ~ 3M) surrounded by athick accretion disk. The hydrogen envelope islost by stellar winds, interaction with acompanion, etc.

Vol. 637– Low metallicity + strong rotation ⇒ chemicallyhomogeneous evolution ⇒ cores retain high spin
(Yoon& Langer’05, Woosley & Heger’06, Yoon+‘06)

Z=0.01 Z⊙◉☉ ⇒ Z=0.07 Z⊙◉☉(Vink & de Koter 2005)– Interacting binaries
Outcomes:– LGRB?
– SNe / Unnovae?
– BH or proto-magnetar?

Woosley & Heger (2006)

2

LGRB Progenitors: Collapsars
If the progenitor forms a collapsar:– The viscous accretion onto the BH ⇒ strong heating ⇒ thermal νν-annihilating
preferentially around the axis ⇒ formation of a relativistic jet (Γ>10)?.
– Numerical models: ultrarelativistic outflow can form if luminosity > Lth ~1049 erg– Numerical simulations:core-collapse, rapid
rotation, computing GW andother aspects of theproblem:

3

Shibata'00,'03,Dimmelmeier'02,'07,'08,Fryer'04,
Cerdá-Durán'05,'07,
Dessart'08, Kiuchi'09,Kotake'09,'11,
Scheideger'10,
Sekiguchi'11,
O'Connor'10,'11,Ott'11,'13...

Aloy et al. (2000)

Formation of the central engine.

the code– General relativity:★ XCFC approximation (Isenberg 2008, Wilson et al 1996, CorderoCarrión et al 2009)★ spectral methods (LORENE library)– Godunov-type schemes for hydrodynamics.
– Spherical polar coordinates:★ ∆r = 200 m (innermost 20 km)★ logarithmic grid for r>20 km -> ∆r ~ 800 m at 100 km● General relativity:★ outer boundary 30000 km- XCFC
approximation★ ∆θ = 1.4ºCoCoNuT code

- (Isenberg 2008, Wilson et al 1996, Cordero-Carrión et al 2009)–Axisymmetry(2D) +(LORENE
equatorial symmetry- spectral
methodslibrary)● –EOS: Lattimer &schemes
Swesty’91 +Timmes& Arnett’99 (table by O'connor & Ott
Godunov-typeforhydrodynamics.
2010, LS220
in thiswork)● Sphericalpolar
coordinates:– GW:(good approx.
- Δrquadrupole= 200 mformula(innermost20 km)in PNS: Reisswig et al 2010)
4– Neutrino
leakage scheme
Bryekmet al->in Δrprep)- logarithmic
grid for(Der>20~ 800 m at 100 km- outer boundary 30000 kmoNeutrino
treatment:
Formationof theLeakage
central engine.

Ruffert et al 1996, Rosswod & Liebendörfer 2003, O'connor & Ott 2010

neutrino treatment: leakage

Ruffert et al 1996, Rosswod & Liebendörfer 2003, O'connor & Ott 2010
The real thing
Leakage simplified model
Leakage simplified scheme
Life is actually harder...

Freestreaming

Free streaming
Neutrinosphere(for meanenergy neutrino)

Diffusion

Diffusion

Neutrinosphere5(for mean neutrino energy)

The main focus of our models is not an
accurate determination of whether aparticular star develops an explosion due toneutrino heating, but specifically an
exploration of the consequences of a fSN, inwhich neutrino heating does not stop themass accretion and thereby prevent thecollapse of the inner core to a BH.
Thus, very high accuracy in the neutrino
physics is only of secondary relevancehere and we can employ simple, fast
approximations for the neutrino physics.

Janka et al 2006

Formation of the central engine.

neutrino treatment: leakage– Energy averaged: Fermi distribution•

inside ⇒ thermal + beta eq. ⇒  = eq, T= Tfluid•

outside ⇒ neutrinos scape⇒  = 0, T= T-sphere

Matching– Neutrinosphere =  threshold ⇒ ray-by-ray in radial direction– Neutrinosphere-opacity loop (computationally expensive):
opacitychemical potential

optical depth
neutrinospherelocation

6NOTE: loop can be avoided by fixing beta-equilibrium everywhere (e.g. Sekiguchi'11)

Formation of the central engine.

neutrino treatment: leakage– Diffusion region: rates based on optical depth
Includes a free parameter

Qdi↵⌫i /1;di↵t⌫ idi↵tdi↵
/k⌧⌫i
⌫i

k=0.5 ⟹ Ruffert, Janka & Schaffer’96
Rosswog & Liebendörfer’03
Cerdá-Durán et al.’13

k=1.0 ⟹ O’Connor & Ott (2011)– Effective rates: Harmonic mean of diffusion and free streaming rates

7

di↵
Q
⌫i
E⌫di↵
=i
R⌫di↵
i

11
1= di↵ + freee↵E⌫ iE⌫ iE⌫ ifreeQ⌫iE⌫free=iR⌫freei

1
1
1= di↵ + freee↵
R⌫ i
R⌫ i
R⌫ ie↵ e↵Qe↵=R⌫i⌫ i E⌫ iFormation of the central engine.

neutrino treatment: leakage– Three neutrino species: e, ̄e and X– Neutrino emission:
e- + p → n + e ; e+ + n → p + ̄e• -processes:• thermal pair annihilation: e- + e+ → i + ̄i → i + ̄i• plasmon decay:– Neutrinos diffusion:n + e → e- + p ; p + ̄e → e+ + n• absorption:
i + N → i + N ; N ∈	 {p, n, A}• scattering:– Inelastic scattering: cannot be implemented in a leakage scheme(relevant before bounce). Alternatives:• Simple deleptonization scheme (Liebendörfer 2005)• Own deleptonization tables: 1D Simulations, multi-energy, hyperbolic 28

momentum eqs. for transport (Obergaulinger & Janka 2013; Obergaulinger et al. 2013)(see talk by O. Just on Friday)

Comparison with Liebendörfer et al 2005 (G15 model)

9Model: 35OC of Woosley & Heger 2005−1∇ / (Wolf-Rayet star, Ωc∼2 rad s ), EOS: LS220

HD grid: [100 (unif.) + 1100 (log)] x 64LORENE: 17 domains x ( 17 x 9 )specific entropy
Wind lost:7 M⊙◉☉
Fe core: 2.02 M⊙◉☉
Evolve: 22 M⊙◉☉

10Typical conditions after bounce inside the shock:t - tb ~ 0.5 s, equatorial profile
convectively stable

0.5

10.0

0.0

innercore

0.20.1

hot bubble

PNS

00.3

2040
r [km]

0.060

Ye

0.4
accretion shock

15.0

5.0

11

electron-neutrinosphere

20.0
s=3

entropy per baryon [kb]25.0Time evolution of the baryonic mass
MBH0 ~ 2.7 M⊙

3.00.45 MO. s

2.00.0~100 ms
tdwell << tBH ~ 1.6 sṀbarshock dominated by thedeleptonization cooling
matter outside the shock.

1.012-1

iron core accreted

Mbar [MO. ]Ṁbarshock dominatedby the OC initialdensity profile

0220 ms— inside shock— inside PNS— inside inner core

500

1000
t - tbounce [ms]1500Formation of the central engine.

limitations of our treatment– The thermal structure and the rotation profile of the PNS evolves
from the tb to tBH

MBH0 and tBH depend on the evolution of the PNS, including thecooling by neutrinos diffusing out of the PNS and the angularmomentum redistribution.

• The presence of strong B-fields, due to the MRI, or non-axisymmetricinstabilities will probably enhance the transport of angular momentum,
decreasing the MBH0 and tBH.

• Non-magnetized axisymmetric simulations provide an upper limit tothe tBH.

• Lower limit estimate for tBH: time at which 2.41 M⊙ have accreted throughthe shock, tBHmin – tb ~ 820 ms.

• Using a stiffer EoS would allow for larger maximum masses and hence
longer collapse times.

13Gravitationalwaves
Gravitational waves- waveformbounce

14BH formation

Gravitational wave spectrum-1910

D=100 kpc
— fiducial model— slow rotation-20

D=15 Mpc- - fiducial model- - slow rotation

10-21

Adv. VIRGO

hchar

10

KAGRA
-22

10

Adv. LIGO
-23

10

ET
-24

10
15

10

1001000
f [Hz]

D hx [cm]Spectrogram analysis(a) bounce

accretion onto PNS

BH formation

Complex spectrogram, whose analysis is done by identifying frequencies
andC(i.e.,
C withC theC help
C ofA
D regions where this frequencies areB produced0 (b)other spectrograms of, e.g., density, shock position, etc.)-5050

4-20(b)

C

C
C

3C-21

2

f[kHz]

CA
1D
B

0
16-22

C

C-23

0200

400

600
800
1000
t - tbounce [ms]

1200

1400

1600

Buoyancy frequency●Local linear stability analysis (non-rotating, non-relativistic)

N 2=(N 2 >0●)∇ρ ∇ Pρ − Γ P ⋅g
1

Brunt-Väisälä frequency

Convectively stable (Ledoux criterion)

Caveats:● Rotating star: Solberg-Høiland criteria (work in progress...)● General relativity (Müller et al 2013)17

Spectrogram analysis.

Buoyancy frequency
Postshock/PNS convection excites g-modes at the lower boundariesof the unstable regions.

fg,PNS ⇠p2
Nturn2⇡

1 GMPNS⇠22⇡ RPNSs(1)mnkB T✓

GMPNS1+ 22c RPNS

◆ 4

(Murphy et al 2009, Müller et al 2013)

Outer stable layer:

 = 4/3
kBT = 15 MeV

frequency• ~ 100 Hz after bounce• monotonically increasing
frequency to a few kHz• contraction+ν-coolingtime after bounce

18

Avoided crossing of modes
During the rise: quadrupolar velocity patterns.
During the drop: quasi-radial velocity pattern. Since fqr ➞ 0 ⟹ unstable mode ⟹ BHformation (Chandrasekhar’64)
Change in behaviour of this feature likely due to an avoided crossing of two modes:
1 GMICi. g-mode (inner convectively stable layer) fg,c ⇠22⇡ RIC✓r

1ss⇢c,BHfqr ⇠ 3.3 log⇢cii. qr-mode with decreasing frequency

✓

◆1/2

GMIC
1+ 22c RIC(Gondek et al. 1997)19

frequency

maximum mass (Gourgoulhon et al. 1995;
Galeazzi et al. 2013).
• Perturbation analysis: radial- and fmodes(Gondek et al. 1997; Kokkotas & Ruoff 2001)
and crustal-modes(Gondek & Zdunik 1999).

kHzkHz

Avoided crossings have been observed:• Numerically: in NSs around its◆ 4∆s/s ∼ 2
 = 2.6

time after bounce

GWShock radius at equator

Signature of SASI on the gravitational waves• Observed from the neutrino-sphere to the shock location• Sound waves confined in a cavity• Multiple overtones20

D hx [cm]Spectrogram analysis
D: f-mode excited at bounce. Highly damped by sound waves in the hot bubbles accretion(a) bounce
onto PNSBH formation✓◆350
MPNS 10 km
kHzf⇠
0.78+1.635&CKokkotas’98)
C (Andersson
C C CfAD
B0 (b)
1.4 M
RPNS-504(b)quasi-radi
al m

1021C

ode-20-212unce)ob@citedxe(PNSeedcoamfsurf
tasdeg-mo
IC B
dlocs ing-modef[kHz]

3CS
th ASIesu moC
rfa tioce ns
Cof hitth tine gPN
CSDA-22CC-23

0200

400

600
800
1000
t - tbounce [ms]

1200

1400

1600

Detectability• How many massive, fast-rotating stars, with Z<Z⊙◉☉ produce fSNe?
• Active matters of debate:•••

fraction of massive stars producing BHs / NSs?
channels for BH formation
observational signatures? (e.g., Lovegrove & Woosley’13; Piro’13).

• Hard to estimate the rate, Rmfr (e.g., Langer’12):••••

They can be a sizable fraction of the fSNe.
Rate of fSNe ~10% of CCSNe (Woosley+’12)
1D-pistons (no rotation / binary effects) ⟹ ≲ 25% (Zhang+’08; Ugliano+’12).
SN rate problem: SN rate predicted from the star formation rate is higher thanthe SN rate by observations.

•In local Universe (≲ 10 Mpc), the rate of dim CCSNe is high (~30% − 50%)
⟹ a fraction of them are CC events producing BHs (Horiuchi+’11).

••

Paucity of observed high mass RSGs in 16.5M⊙ ≲ M ≲ 25M⊙ can be explained
if they are fSNe. fSNe rate ≲ 20% of CCSN ⟹ ~ 0.2 y−1 (Kochanek’13).

10% − 50% of massive MS stars are fast rotators (≲ 200 km s−1; Mink+’13).

➡ Fast spinning, moderate-Z, massive stars happening in nearby galaxies, mightbring detectable GW signals for the Einstein Telescope at rates of ≲ 0.1y−1.

22Conclusions•The PNS phase in the collapsar scenario is optimal for GW
emission:large amplitude : visible with ET in the Virgo cluster
long duration : ~ seconds
quasi-periodic signal
possible EM signal: long GRB, SN••23

It may provide information about the conditions in the PNS
size of PNS
contraction/accretion time-scalecooling time-scalerotation
SASI
Detectability: prospects for ~0.1 yr-1 with Einstein Telescope.