Numerical modeling of core-collapse supernovaewith progress in nuclear physics and supercomputing	

Crab nebula	

K. Sumiyoshi	

(Numazu College of Technology)	

hubblesite.org	

Talk mainly on: 	• 3D calculations of neutrino-transfer	With H. Nagakura, S. Yamada,	

H. Matsufuru, A. Imakura, T. Sakurai	

Bridge @Nara, 2012. 12. 16.	

1�Bridging nuclear, astrophysics by supercomputers	

• Quark, Nuclear Physics	

– EOS, ν-reactions	• Massive Star Models	– Mass, Metallicities	

A01, A02	

A03	

• Simulations of supernovae	

• Supercomputing	- ν-radiation hydrodynamics	

A04	

A03	Explosion mechanism	 Neutrino Astronomy	
Super-Kamiokande	ν	SN1987A	KEK	YITP	ν	2�

Research products in 2008-2012	

16 papers	

• Equation of state (EOS) tables for supernovae	

– Shen EOS tables + Hyperons, Quarks	

– Mixture of nuclei, neutrino reactions	• Neutrino signals as probe of EOS	Phys. Rev x 3, ApJ x 2	JPG x 1 	→ Togashi, Furusawa	
Phys. Rev x 2, ApJ x 4	

PLB x 1 	– Neutrino bursts from proto-NS and BH	

→ Nakazato	– Dependence on massive stars, EOS	• 1D GR neutrino-radiation hydrodynamics	

• 3D neutrino-transfer for supernovae	

– Solve Boltzmann eq. in 6D	

– a new method for matrix solver	

ApJS x 1, PTEP x 1	JSIAM x 1 	

→ Imakura	
3�Neutrino-transfer is important�Neutrino heating for explosion�

4�Role of neutrinos in supernova dynamics	

Massive star	Fe coreCollapse	

ν

neutrinos produced	

e-capture	ν-trapping	mean free path↓	νν ν1000 kmνSupernova neutrinos	νCore Bounce	ν-emission	

Explosion	ν

proto-	NSνν ν

Shockwave

10 km

Neutron star	5�Problem: shockwave stalls on the way�
How to revive the shock propagation and trigger explosion?	
1053 erg	ν	Fe core	ν	Collapse	ν	ν	Explosion?	

ν	ν	 ν	Shockwave	ν	ν	 ν	Core-bounce	ν	1051 erg	ν	

• Multi-dimensional effects 	

• Microphysics (EOS, ν-reactions)	

• Neutrino heating mechanism	~1% of neutrinos is used for explosion	Sumiyoshi et al. ApJ (2005) 	

No explosion in 1D	(spherical calc.)	

6�

Neutrino heating mechanism for revival of shock	
Fe coresurface	Heating by neutrino absorption	 νe + n → e- + p, 	

νe + p → e+ + n

Transfer of energy from ν	Shock	⎛ ΔM ⎞⎛ Δt ⎞
E ν −heat ~ 2.2 ×10 ⎜
⎟erg
⎟⎜
⎝ 0.1M solar ⎠⎝ 0.1s ⎠
51~200km	

heating	p	ν	ν	Proto-NS	

Janka A&A (1996)	

ν	Shockposition	

Delayedexplosion�

€ n	

100ms after bounce	

Depends on neutrino energy/flux	

targets	νheating	time [s]	Bethe & Wilson ApJ (1985)	

7�ν-heating occurs in the intermediate region	

between diffusion& free-streaming	Need to solve neutrino-transfer	

for angle distribution� f (Eν ,θν ) freeex. Diffusion approximation
is not enough	outside	R	streaming	 ν	€ν-heating rate	

shockwave	

Janka A&A (1996)	

⎞MeV ⎛ Lν Eν2Q ≈ 110
X i ⎟
⎜ 2s⋅ N ⎝ R7 < µ > ⎠iν

heating region	θν	average energy, flux: Eν, Lν	€flux factor:	 < µ >=< cosθν >= 0 ~ 1

ν-sphere	even 10~20 % change of ν-heating 	

may affect the outcome: explosion	€diffusion	ν	ν	center	8�

Numerical simulation of supernovae	

Combination of hydrodynamics and neutrino transfer	• Hydrodynamics	

ν-heating, cooling, pressure	• Equation of state (EOS)	

(ρ, T, Ye)	pressure	

composition	

ρ, T, Ye profiles	• ν-transfer(Boltzmann eq.)	

distribution: f(t, r, Eν, θν)	Astrophysics	

• ν-reaction rates	(ρ, T, µi, Xi)	absorb	emit	scatter	

Nuclear Physics	

9�

3D Neutrino-transfer is a challenge�
Boltzmann equation in 6D�

10�

Calculation of ν-radiation transfer	• Neutrino propagation from supernova core to outside		with neutrino reactions in dense matter 	ν	Free-streaming�Surface�- Including non-equilibrium situation	

- We cannot use Fermi-distributions	

neutrino energy distribution during collpase�

Shock wave�ν	ρc=1012	

Neutrinoheating�proto-NS�ν	ρc=1014 g/cm3	& bounce	f(Eν)	

ν	ρc=1010-11 g/cm3	

10MeV	Diffusion�
←High T/ρ�Low T/ρ →	100MeV	 ν energy	

11�

To solve neutrino transfer in 3D	

• Work in 6D: 3D space + 3D momentum	z	θν	ν	θ	fν (r,θ, φ; εν ,θν , φν ; t)φν	r	

– Neutrino energy (εν), angle (θν, φν)	y	• Time evolution of 6D-distribution	

€1 ∂fν  1 ⎛ δfν ⎞+ n ⋅ ∇fν = ⎜ ⎟c ∂tc ⎝ δt ⎠ collisionx	φ	Boltzmann eq.	

– Left: Neutrino number change	

– Right: Change by neutrino reactions	€ • Energy, angle-dependent reactions	ν	– Compositions by EOS tables	12�

Status of neutrino-transfer	• 1D: first principle calculations	• 2D, 3D: approximate treatment	

Ray-by-ray method	Marek-Janka, ApJ 694 (2009)	

– Diffusion (with flux limiter)		

Only good in central part	– Ray-by-ray (radial transport)		

Neglect lateral transport 	

• Need full 3D calculations 	

1D-transport independently		- examine the approximations, to the grand challenge	• New code to solve 3D neutrino-transfer	

• For the first time in the world	

Sumiyoshi & Yamada, ApJS (2012)	

See Ott et al. ApJ(2008) for 2D Sn-method	13�€Boltzmann eq. in spherical coordinate�Sumiyoshi & Yamada, ApJS (2012)	

1− µν2 cos φν ∂1− µν2 sin φν ∂fν
1 ∂fν µν ∂ 2+ 2 (r f ν ) +(sin θf ν ) +c ∂t r ∂rr sin θ∂θr sin θ∂φ2
1−µ1 ∂
1 ⎛ δfν ⎞2
ν cos θ ∂+[(1− µν ) f ν ] +(sin φν fν ) = ⎜ ⎟r ∂µνr sin θ∂φνc ⎝ δt ⎠ collision– Discrete in conservative form (Sn method)	

– Implicit method in time	Energy, angle dependent	€• stability, time step, equilibrium	– Collision term for ν-reactions	• absorption, emission, scattering	Eν,µν	ν	E´ν,µν´	µν = cosθν

1 ⎛ δfν ⎞
1= j emission (1− f ν ) −f ν + Cinelastic [ ∫ f ν ( E νʹ′ , µνʹ′ )dE νʹ′ ]⎜ ⎟c ⎝ δt ⎠ collision
λabsoption€14

Neutrino reactions in collision term�Basic sets for supernova simulations	Bruenn (1985) +Shen	• Emission & absorption:�� e- + p ↔ νe + n ��e- + A ↔ νe + A´�� e+ + n ↔ νe + p	• Scattering:�� νi + N ↔ νi + N ��νi + A ↔ νi + A	�	• Pair-process:��e- + e+ ↔ νi + νi ��
� 3 species:	��νe, νe, νµ 	

�N + N ↔ N + N + νi +νi ��Notes: iso-energetic scattering, linearize pair process	

Lorentz transformation being implemented 	
15�Supercomputing aspects: matrix solver�• Linear equation	

Af ν = d• Neutrino distribution	

– Nspace=nr x nθ x nφ 	– Nν=nε x nθν x nφν 	

Nvector~106 x 103 	

200

Imakura et al. SIAM (2012)	

• Iterative method	

pre-conditioner by Imakura	larger time step is possible	~factor 10 faster	

number of iterations€

Nvector	

~109	

150

100

500-710-6-5

1010Δt [sec]

-4

1016�

Validation of 3D-neutrino transfer code�
diffusion	

• by analytic solutions	– Gauss packet, free space	– Formal solution along path	– Approach to equilibrium	

equlibrium	• by spherical solutions	– Fixed backgroud	• collapse, bounce stages	

– density, flux, moments	– ν-reactions, m. f. path	

utilizing 1D ν-radiation hydrodynamics (GR) 	

by Sumiyoshi et al. ApJ (2005)	

Sumiyoshi & Yamada ApJ (2012)	

17�

Applications to 2D/3D supernovae�demonstration of 3D code�

18�

Axially symmetric supernova core	• Fix the ρ, T, Ye profile & Solve 3D ν-transfer	ρ[g/cm3]	

T [MeV]	

Z	S [kB]	

Ye	Sumiyoshi, Yamada ApJ (2012)	

ref. 1D, 15Msun, tpb=100ms	19�

Axially symmetric supernova core�• From small densities, the time evolution until the stationary state�νe	density	νe	

Radial	flux	(r)	Sumiyoshi, Yamada ApJ (2012)	

νe	density	νe	Polar	flux	(θ)	t=0~10ms	

20�

Advantage of 3D neutrino-transfer�• describes polar flux	Fr	– beyond ray-by-ray	

– beyond diffusion�

Fθ	Fθ	Sumiyoshi, Yamada ApJ (2012)	

Fr	21�

2D supernova core�• Check of 		

ray-by-ray	

3D Boltzmann	-	νe density distribution�
Ray-by-ray

too strong 	

angle dependence��deviation	 n − nrbr
3D
δ=n 3D€

From Takiwaki (2011)	

11.2Msun, 2D	
100ms	

Preliminary	22�Study of 3D neutrino-transfer in supernova core	
3D profile by Takiwaki (2011)	

3D neutrino distributions, flux, spectrum	

beyond approximations to explore new effects	ν-heating/cooling rates in 3D	

Red: heating, Blue: cooling	νe	density iso-surface	Preliminary	23�Coupling with hydrodynamics� Nagakura	
working on the code for 2D/3D neutrino-radiation hydrodynamics	

• 2D hydro code is ready�• 1D collapse test�24�Computational load and future�• We have MPI parallel code + Hitachi tuning	Matsufuru, Hashimoto	3• Matrix solver costs ~M N	machine�

space�

neutrino�

memory�operations�Current�

256 x 32 x 64�

8 x 12 x 14�

2TB�

6 Tera-flop�K-computer� 512 x 64 x 128�

12 x 24 x 20�

200TB�

2 Peta-flop�

512 x 128 x 256�

24 x 24 x 24�

3PB�

80 Peta-flop�

Exascale�

Kotake et al. PTEP (2012)	

• working on KEK, planned on K-computer	• needs Exa-scale for full 3D supernovae	

25�Summary�• Neutrino-transfer in 3 dimensions is possible	

– Code to solve Boltzmann eq. in 6D	

– Collaboration with computational scientists (A04)	

• Matrix solver, Parallel coding, Tuning	• Applying to 2D/3D supernova cores	

– Unique feature: Non-radial fluxes	

– Examine the approximations used so far	

• Ready to test ν-transfer + hydro. code	– toward the grand challenge of 3D supernovae	

Support from: 	

- HPCI Strategic Program Field 5	

- Grant-in-Aid for Scientific Research (20105004, 20105005, 22540296, 24244036)	

- Supercomputing resources at KEK, YITP, UT, RCNP, K-computer	

26�