重力崩壊型超新星のboltzmann-hydro code - nao

16
Wakana Iwakami 2017.11.29 CfCAユーザーズミーティング 重力崩壊型超新星のBoltzmann-Hydro Code によるニュートリノ輻射流体計算

Upload: others

Post on 08-Dec-2021

9 views

Category:

Documents


0 download

TRANSCRIPT

Page 1: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Wakana Iwakami

2017.11.29CfCAユーザーズミーティング

重力崩壊型超新星のBoltzmann-Hydro Codeによるニュートリノ輻射流体計算

Page 2: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

H

He

C+O

O+Ne+Mg

Si

Fe

Core-Collapse SupernovaH

HeC+O

O+Ne+MgSiFe

Falling Matter

FeShock Wave Generation

Nuclear Density

PNS

Shock Wave

e

ex

Neutrino Heating Mechanism

Massive Star

Page 3: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

er

Momentum Space

θν

Φνp(2) = p sinθν cosφν

p(3) = p sinθν sinφν

p(1) = p cosθν p = εν

eφ er

Momentum Space

exey

ez

Φ  

t

ν

Coordinate Space µν = cosθν

Boltzmann EquationNeutrino Radiation

∂f∂t+ cosθν

∂f∂r+sinθν cosθν

r∂f∂θ

+sinθν sinφνrsinθ

∂f∂φ

−sinθνr

∂f∂θν

−cosθsinθ

sinθν sinφνr

∂f∂φν

Neutrino distribution function

f (t, r,θ ,φ;εν ,µν ,φν )

pµ pµ = −mν2= 0

=δ fδ t

⎝⎜

⎠⎟collision

Boltzmann Equation in the spherical coordinateMassless particles

Page 4: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Gµ ≡ Gsµ

s∑

Collision Term

Neutrino Energy Density (µ = 0)

Neutrino Number Density Γ ≡ Γνe−Γ

ν e

Radiation Pressure (µ = 1, 2, 3)

s : species (s = νe, νe, ν x )

Electron Capture

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

Positron Capture

Electron Capture on nuclei

Emission/Absorption Scattering

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

Neutrino-Nucleon scattering

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

Neutrino-Nuclei scattering

Neutrino-Electron scattering

ν + e ν + e [esc]

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

The Astrophysical Journal Supplement Series, 199:17 (32pp), 2012 March Sumiyoshi & Yamada

with the relation Rscat(Ω′; Ω) = Rscat(Ω; Ω′). The collision termfor the pair process is expressed by[

1c

δf

δt

]

pair= −

∫dε′ε′2

(2π )3

∫dΩ′Rpair-anni(ε, Ω; ε′, Ω′)

× f (ε, Ω)f (ε′, Ω′) +∫

dε′ε′2

(2π )3

∫dΩ′Rpair-emis(ε, Ω; ε′, Ω′)

× [1 − f (ε, Ω)][1 − f (ε′, Ω′)], (11)

where f (ε′, Ω′) denotes the distribution of anti-neutrinos. Fromthe detailed balance, the following relation holds:

Rpair-anni(ε, Ω; ε′, Ω′) = Rpair-emis(ε, Ω; ε′, Ω′)eβ(ε+ε′). (12)

We linearize the collision term, Equation (11), by assumingthat the distribution for anti-neutrinos is given by that in theprevious time step or the equilibrium distribution. This is a goodapproximation since the pair process is dominant only in high-temperature regions, where neutrinos are in thermal equilibrium.We adopt the approach with the distribution in the previous timestep in all of the numerical calculations with pair processes inthe current study. We utilize further the angle average of thedistribution when we take the isotropic emission rate as we willstate. We have also tested that the approach with the equilibriumdistribution determined by the local temperature and chemicalpotential works equally well.

As for the reaction rates, we take mainly from the conven-tional set by Bruenn (1985) with some extensions (Sumiyoshiet al. 2005). We implement the neutrino reactions,

e− + p ←→ νe + n [ecp], (13)

e+ + n ←→ νe + p [aecp], (14)

e− + A ←→ νe + A′ [eca], (15)

for the absorption/emission,

ν + N ←→ ν + N [nsc], (16)

ν + A ←→ ν + A [csc], (17)

for the isoenergetic scattering. We do not take into accountthe neutrino–electron scattering. It is well known that theinfluence of this reaction is minor although it contributes to thethermalization (Burrows et al. 2006a). As for the pair process,we take the electron–positron process and the nucleon–nucleonbremsstrahlung as follows:

e− + e+ ←→ νi + νi [pap], (18)

N + N ←→ N + N + νi + νi [nbr]. (19)

For these pair processes, we take the isotropic emission rateas an approximation, which avoids complexity but describesthe essential roles. We remark that the set of the reaction ratesadopted in the current study is the minimum, which describessufficiently the major role of neutrino reactions in the supernovamechanism. Further implementation of other neutrino reactionsand more sophisticated description of reaction rates in themodern version (Buras et al. 2006; Burrows et al. 2006b) willbe done once we have enough computing resources.

3.3. Equation of State

We utilize the physical EOS of dense matter to evaluatethe rates of neutrino reactions. It is necessary to have thecomposition of dense matter and the related thermodynamicalquantities such as the chemical potentials and the effective massof nucleon. We implement the subroutine for the evaluationof quantities from the data table of EOS as used in the othersimulations of core-collapse supernovae (Sumiyoshi et al. 2005,2007). We adopt the table of the Shen EOS (Shen et al. 1998a,1998b, 2011) in the current study. Other sets of EOSs can beused by simply replacing the data table.

3.4. Numerical Scheme

We describe the numerical scheme employed in the numericalcode for the neutrino transfer in 3D. The method of thediscretization is based on the approach by Mezzacappa &Bruenn (1993) and Castor (2004). We also refer the referencesby Swesty & Myra (2009) and Stone et al. (1992) for the othermethods of discretization of neutrino transfer and radiationtransfer.

We define the neutrino distributions at the cell centers andevaluate the advection at the cell interfaces and the collisionterms at the cell centers. We describe the neutrino distributionsin the space coordinate with radial Nr-, polar Nθ -, and azimuthalNφ-grid points and in the neutrino momentum space with energyNε-grid points and angle Nθν

- and Nφν-grid points. We explain

the detailed relations to define the numerical grid in AppendixA.2.

We discretize the Boltzmann equation, Equation (5), for theneutrino distribution, f n

i , in a finite-differenced form on the gridpoints. Here we assign the integer indices n and n + 1 for thetime steps and i for the grid position. We adopt the implicitdifferencing in time to ensure the numerical stability for stiffequations and to have long time steps for supernova simulations.We solve the equation for f n+1

i by evaluating the advection andcollision terms at the time step n + 1 in the following form:

1c

f n+1i − f n

i

∆t+

[µν

r2

∂r(r2f )

]n+1

+

[√1 − µ2

ν cos φν

r sin θ

∂θ(sin θf )

]n+1

+

(√1 − µ2

ν sin φν

r sin θ

∂f

∂φ

)n+1

+

1r

∂µν

[(1 − µ2

ν

)f

]n+1

+

[

−√

1 − µ2ν

r

cos θ

sin θ

∂φν

(sin φνf )

]n+1

=[

1c

δf

δt

]n+1

collision,

(20)

where we schematically express the advection terms for the cellcontaining f n+1

i . We evaluate the advection at the cell interfaceby the upwind and central differencing for free-streaming anddiffusive limits, respectively. The two differencing methods aresmoothly connected by a weighting factor in the intermediateregime between the free-streaming and diffusive limits. We de-scribe the numerical scheme for the evaluation of the advectionterms in Appendix A.3. We express the collision terms by thesummation of the integrand using the neutrino distributions atthe cell centers.

6

Electron-positron pair process

Nucleon-nucleon bremsstrahlung

Pair Process

Gsµ ≡ ps

µ δ fδτ

"

#$

%

&'

collision (s)

d3p∫

Γs ≡δ fδτ

#

$%

&

'(

collision (s)

d3p∫

δ fδτ

!

"#

$

%&

collision (s)

=δ fδτ

'

()*

+,emis-abs (s)

+δ fδτ

'

()*

+,scat (s)

+δ fδτ

'

()*

+,pair (s)

Page 5: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Euler Equations

Continuity Equation:

Equations of Motion:

Internal Energy Equation:

Gravitational Potential:

Evolution Equation of Electron Fraction:

EOS of Nuclear Matter (Latimer and Swesty K=220MeV, Shen, Furusawa)

Hydrodynamics

∂ρ∂t+∂∂x j

ρv j( ) = 0

Δψ = 4πGρ

∂∂t

ρYemA

"

#$

%

&'+

∂∂x j

ρYemA

v j"

#$

%

&'= −Γ

∂∂t

ρvi( )+ ∂∂x j

ρvivj +Pδi

j( ) = −ρ ∂ψ∂x j

−Gi

∂∂t

12ρv2 + e

"

#$

%

&'+

∂∂x j

12ρv2 + e+P

"

#$

%

&'v j

(

)*

+

,-= −ρv

j ∂ψ∂x j

−G0

ρ : density, v: velocity, P: pressure, e: internal energy, ψ: the gravitational potential,

G0 : neutrino radiation energy, Gi : neutrino radiation pressure, mA: the atomic mass unit, Γ : neutrino number density

G : the gravitational constant (=6.67×10−8[cm3g−1s2 ]), Ye: electron fraction,

Page 6: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

MEMBER Hiroki Nagakura, Wakana Iwakami, Hirotada Okawa, Shun Furusawa Akira Harada, Hideo Matsufuru, Kosuke Sumiyoshi, Shoichi Yamada

Boltzmann-Hydro Code

Page 7: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Two Energy Grids Approach

Collision term is calculated in the rest frame using Lagrangian Remapped Grid (LRG)

Doppler factor

※ the unit of c = G = 1

(Nagakura et al. 2014)

Advection term is calculated in the laboratory frame using Laboratory Fixed Grid (LFG)

∂f∂t+µνr2

∂∂r

r2 f( )+ 1−µν2 cosφν

rsinθ∂∂θ

f sinθ( )+1−µν

2 sinφνrsinθ

∂f∂φ

+1r∂∂µν

f (1−µµ2 )( )−

1−µµ2

rcosθsinθ

∂∂φν

f sinφν( ) = D lb δ fδ!t#

$%

&

'(collision

restframe

Boltzmann Equation

δ fδτ

!

"#

$

%&restframe

collision

δ fδ t!

"#

$

%&

laboratory

collision

=dλdt

δ fδλ

!

"#

$

%&collision

=dλdt

dτdλ

δ fδτ

!

"#

$

%&restframe

collision

=ε lbνε rfν

δ fδτ

!

"#

$

%&restframe

collision

= D lb δ fδτ

!

"#

$

%&restframe

collision

Page 8: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Advantage of Two Energy Grid Approach

δ fδ t

⎝⎜

⎠⎟

laboratory

collision

= Dlb δ fδτ

⎝⎜

⎠⎟collision

restframe

δ fδ t

⎝⎜

⎠⎟

laboratory

collision

≈δ fδτ

⎝⎜

⎠⎟collision

restframe

0th order of v/c

All orders of v/c

CfCA: 144 MPI × 4 OpenMP = 456 core (without electron scattering)Nr × Nth × Ne × Nthν × Nphν = 288× 128 × 20 × 10 × 6

Page 9: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Anisotropic Radial Flux

@CfCA

High νe number density region is formed by the convection in the PNS

High neutrino energy flux is emitted from the region.

Energy Flux

Page 10: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Our Objective

Using the Boltzmann-Hydro code,

we would like to verify the approximative method for calculating neutrino heating and cooling

we would like to know the detailed characteristics of the neutrino dynamics during supernova explosion

1)

2)

Page 11: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

arX

iv:1

702.

0175

2v1

[ast

ro-p

h.H

E] 6

Feb

201

7

Six-Dimensional Simulations of Core-Collapse Supernovaewith Full Boltzmann Neutrino Transport

Hiroki Nagakura1, Wakana Iwakami2,3, Shun Furusawa4, Hirotada Okawa2,3, Akira Harada5,Kohsuke Sumiyoshi6, Shoichi Yamada3,7, Hideo Matsufuru8 and Akira Imakura9

1TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17,California Institute of Technology, Pasadena, CA 91125, USA2Yukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8502, Japan

3Advanced Research Institute for Science & Engineering,Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

4 Frankfurt Institute for Advanced Studies, J.W.Goethe University, D-60438 Frankfurt am Main, Germany5 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

6Numazu College of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan7Department of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

8High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 308-0801, Japan and9University of Tsukuba, 1-1-1, Tennodai Tsukuba, Ibaraki 305-8577, Japan

This is the first-ever report of much awaited core-collase supernova simulations with solving theBoltzmann equations for neutrino transport, which actually amounts to a 6-dimensional (1 in time,2 in space and 3 in momentum space) problem even under axial symmetry. This is also the firststudy to find a sign of successful explosion in a computation at this level of elaboration in neutrinotransport. We also investigate the neutrino distributions in momentum space, which would notbe accessible to other approximate methods employed so far. The off-diagonal component of theEddington tensor for neutrinos derived in this letter is different from what are prescribed by handin frequently-used approximations. The results will be useful to test and possibly improve theseapproximations casually used but not validated in the literature.

PACS numbers:

Introduction.— The theoretical study of the explo-sion mechanism of core-collapse supernovae (CCSNe) hasheavily relied on numerical simulations. This is mainlybecause CCSNe are rare: it occurs a few times in a cen-tury per galaxy on average [1–7] and, in fact, SN1987Ais the only one close enough to extract some useful in-formation on what happened deep inside the massivestar from, among other things, the detection of neutri-nos [8, 9]. Since the CCSNe are intrinsically multi-scale,multi-physics and multi-dimensional (multi-D) phenom-ena, their mechanism can be addressed only with detailednumerical computations. Although the recent progress isremarkable indeed, even the most sophisticated multi-Dsimulations of CCSNe done so far employed approxima-tions one way or another in their numerical treatmentof neutrino transport [10–26]. That may be crucial totheir outcomes, since some of these calculations, whichadopted different approximations, seem to be at oddswith each other. The best way to calibrate them shouldobviously be to compare them with what is obtainedwithout appealing to those artificial approximations, i.e.,the results of first-principles simulations just like whathappened for spherically symmetric computations morethan a decade ago [27–29].

In axisymmetry, this is possible now and we achievedsuch simulations with the K supercomputer in Japan,one of the currently available best supercomputers with∼ 10PFLOPS. In this study we discretized the Boltz-mann equations both in space and momentum, which

is actually a 6-dimensional problem even under axisym-metry. Except for this almost mandatory discretizationof the basic differential equations, we do not resort toany artificial approximation or phenomenological mod-eling in the neutrino transport [32, 33]. Note that Ottet al. [10] also conducted axisymmetric simulations, dis-cretizing the Boltzmann equations, but they took in facta hybrid approach that employed the multi-group flux-limited diffusion approximation in the optically thick re-gion, i.e., theirs were not genuinely the full Boltzmann-neutrino transport. More importantly, they ignored rela-tivistic corrections completely, dropping all fluid-velocitydependent terms, and also neglected some importantneutrino-matter reactions, all of which are crucial for re-alistic modeling of CCSNe (see e.g., [30, 31]).

These days, much attention of supernova researchers isdirected to 3-dimensional simulations in space. It shouldbe remembered, however, that those computations in-evitably employ some approximations in neutrino trans-port, which have not been well validated even in axisym-metry. We believe that the results presented here are notonly a milestone for the numerical study of CCSNe, butthey are also important as realistic supernova models, al-beit in axisymmetry. They will hopefully contribute tothe resolution of the apparent contradictions among theresults of different groups, for which the approximationsthey adopted may be responsible.

Methods and Models.— We solve numerically theequations of neutrino-radiation hydrodynamics. We ap-

arX

iv:1

702.

0175

2v1

[ast

ro-p

h.H

E] 6

Feb

201

7

Six-Dimensional Simulations of Core-Collapse Supernovaewith Full Boltzmann Neutrino Transport

Hiroki Nagakura1, Wakana Iwakami2,3, Shun Furusawa4, Hirotada Okawa2,3, Akira Harada5,Kohsuke Sumiyoshi6, Shoichi Yamada3,7, Hideo Matsufuru8 and Akira Imakura9

1TAPIR, Walter Burke Institute for Theoretical Physics, Mailcode 350-17,California Institute of Technology, Pasadena, CA 91125, USA2Yukawa Institute for Theoretical Physics, Kyoto University,Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8502, Japan

3Advanced Research Institute for Science & Engineering,Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

4 Frankfurt Institute for Advanced Studies, J.W.Goethe University, D-60438 Frankfurt am Main, Germany5 Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan

6Numazu College of Technology, Ooka 3600, Numazu, Shizuoka 410-8501, Japan7Department of Science and Engineering, Waseda University, 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan

8High Energy Accelerator Research Organization, 1-1 Oho, Tsukuba, Ibaraki 308-0801, Japan and9University of Tsukuba, 1-1-1, Tennodai Tsukuba, Ibaraki 305-8577, Japan

This is the first-ever report of much awaited core-collase supernova simulations with solving theBoltzmann equations for neutrino transport, which actually amounts to a 6-dimensional (1 in time,2 in space and 3 in momentum space) problem even under axial symmetry. This is also the firststudy to find a sign of successful explosion in a computation at this level of elaboration in neutrinotransport. We also investigate the neutrino distributions in momentum space, which would notbe accessible to other approximate methods employed so far. The off-diagonal component of theEddington tensor for neutrinos derived in this letter is different from what are prescribed by handin frequently-used approximations. The results will be useful to test and possibly improve theseapproximations casually used but not validated in the literature.

PACS numbers:

Introduction.— The theoretical study of the explo-sion mechanism of core-collapse supernovae (CCSNe) hasheavily relied on numerical simulations. This is mainlybecause CCSNe are rare: it occurs a few times in a cen-tury per galaxy on average [1–7] and, in fact, SN1987Ais the only one close enough to extract some useful in-formation on what happened deep inside the massivestar from, among other things, the detection of neutri-nos [8, 9]. Since the CCSNe are intrinsically multi-scale,multi-physics and multi-dimensional (multi-D) phenom-ena, their mechanism can be addressed only with detailednumerical computations. Although the recent progress isremarkable indeed, even the most sophisticated multi-Dsimulations of CCSNe done so far employed approxima-tions one way or another in their numerical treatmentof neutrino transport [10–26]. That may be crucial totheir outcomes, since some of these calculations, whichadopted different approximations, seem to be at oddswith each other. The best way to calibrate them shouldobviously be to compare them with what is obtainedwithout appealing to those artificial approximations, i.e.,the results of first-principles simulations just like whathappened for spherically symmetric computations morethan a decade ago [27–29].

In axisymmetry, this is possible now and we achievedsuch simulations with the K supercomputer in Japan,one of the currently available best supercomputers with∼ 10PFLOPS. In this study we discretized the Boltz-mann equations both in space and momentum, which

is actually a 6-dimensional problem even under axisym-metry. Except for this almost mandatory discretizationof the basic differential equations, we do not resort toany artificial approximation or phenomenological mod-eling in the neutrino transport [32, 33]. Note that Ottet al. [10] also conducted axisymmetric simulations, dis-cretizing the Boltzmann equations, but they took in facta hybrid approach that employed the multi-group flux-limited diffusion approximation in the optically thick re-gion, i.e., theirs were not genuinely the full Boltzmann-neutrino transport. More importantly, they ignored rela-tivistic corrections completely, dropping all fluid-velocitydependent terms, and also neglected some importantneutrino-matter reactions, all of which are crucial for re-alistic modeling of CCSNe (see e.g., [30, 31]).

These days, much attention of supernova researchers isdirected to 3-dimensional simulations in space. It shouldbe remembered, however, that those computations in-evitably employ some approximations in neutrino trans-port, which have not been well validated even in axisym-metry. We believe that the results presented here are notonly a milestone for the numerical study of CCSNe, butthey are also important as realistic supernova models, al-beit in axisymmetry. They will hopefully contribute tothe resolution of the apparent contradictions among theresults of different groups, for which the approximationsthey adopted may be responsible.

Methods and Models.— We solve numerically theequations of neutrino-radiation hydrodynamics. We ap-

arXiv:1702.01752v1 [astro-ph.HE] 6 Feb 2017

Six-D

imensio

nalSim

ulatio

nsofCore-C

olla

pse

Supern

ovae

with

Full

Boltz

mann

Neutrin

oTra

nsp

ort

Hirok

iNagak

ura

1,Wakan

aIw

akami 2,3,

ShunFurusaw

a4,

Hirotad

aOkaw

a2,3,

Akira

Harad

a5,

Koh

suke

Sumiyosh

i 6,Shoich

iYam

ada3,7,

Hideo

Matsu

furu

8an

dAkira

Imak

ura

9

1TAPIR

,W

alter

Burke

Institu

teforTheoretica

lPhysics,

Mailcod

e350-17,

Califo

rnia

Institu

teofTech

nology

,Pasaden

a,CA

91125,USA

2Yukawa

Institu

teforTheoretica

lPhysics,

Kyoto

University

,Oiw

ake-ch

o,Kita

shira

kawa,Sakyo-ku

,Kyoto,606-8502,Japan

3Advanced

Resea

rchInstitu

teforScien

ce&

Engin

eering,

Wased

aUniversity

,3-4-1

Oku

bo,Shinjuku

,Tokyo

169-8555,Japan

4Frankfu

rtInstitu

teforAdvanced

Studies,

J.W

.Goeth

eUniversity

,D-60438Frankfu

rtam

Main,Germ

any

5Depa

rtmen

tofPhysics,

University

ofTokyo,7-3-1

Hongo,Bunkyo,Tokyo113-0033,Japan

6Numazu

College

ofTech

nology

,Ooka

3600,Numazu,Shizu

oka

410-8501,Japan

7Depa

rtmen

tofScien

ceandEngin

eering,

Wased

aUniversity

,3-4-1

Oku

bo,Shinjuku

,Tokyo169-8555,Japan

8High

Energy

Accelera

torResea

rchOrga

niza

tion,1-1

Oho,Tsuku

ba,Iba

raki

308-0801,Japan

and

9University

ofTsuku

ba,1-1-1,Ten

nodaiTsuku

ba,Iba

raki

305-8577,Japan

This

isthefirst-ever

report

ofmuch

awaited

core-collasesupern

ovasim

ulation

swith

solvingthe

Boltzm

anneq

uation

sfor

neu

trinotran

sport,

which

actually

amou

nts

toa6-d

imen

sional

(1in

time,

2in

space

and3in

mom

entum

space)

prob

lemeven

under

axial

symmetry.

This

isalso

thefirst

studyto

findasign

ofsuccessfu

lex

plosion

inacom

putation

atthis

levelof

elaboration

inneu

trino

transp

ort.Wealso

investigate

theneu

trinodistrib

ution

sin

mom

entum

space,

which

wou

ldnot

beaccessib

leto

other

approx

imate

meth

odsem

ployed

sofar.

Theoff

-diagon

alcom

pon

entof

the

Eddington

tensor

forneu

trinos

derived

inthis

letteris

differen

tfrom

what

areprescrib

edbyhan

din

frequen

tly-used

approx

imation

s.Theresu

ltswill

beusefu

lto

testan

dpossib

lyim

prove

these

approx

imation

scasu

allyused

butnot

validated

intheliteratu

re.

PACSnumbers:

Intro

ductio

n.—

The

theoretical

study

oftheexplo-

sionmech

anism

ofcore-collap

sesupern

ovae(C

CSNe)

has

heav

ilyrelied

onnumerical

simulation

s.This

ismain

lybecau

seCCSNeare

rare:itoccu

rsafew

times

inacen

-tury

per

galaxyon

average[1–7]

and,in

fact,SN1987A

istheon

lyon

eclose

enou

ghto

extract

someusefu

lin-

formation

onwhat

hap

pened

deep

insid

ethemassive

starfrom

,am

ongoth

erthings,

thedetection

ofneutri-

nos

[8,9].

Since

theCCSNeare

intrin

sicallymulti-scale,

multi-p

hysics

andmulti-d

imension

al(m

ulti-D

)phenom

-ena,

their

mech

anism

canbead

dressed

only

with

detailed

numerical

computation

s.Alth

ough

therecen

tprogress

isrem

arkable

indeed

,even

themost

sophisticated

multi-D

simulation

sof

CCSNedon

eso

farem

ployed

approx

ima-

tionson

eway

oran

other

intheir

numerical

treatment

ofneutrin

otran

sport

[10–26].That

may

becru

cialto

their

outcom

es,sin

cesom

eof

these

calculation

s,which

adop

teddifferen

tap

prox

imation

s,seem

tobeat

odds

with

eachoth

er.Thebest

way

tocalib

ratethem

shou

ldob

viou

slybeto

compare

them

with

what

isob

tained

with

outap

pealin

gto

those

artificial

approx

imation

s,i.e.,

theresu

ltsof

first-p

rincip

lessim

ulation

sjust

likewhat

hap

pened

forspherically

symmetric

computation

smore

than

adecad

eago

[27–29].

Inax

isymmetry,

this

ispossib

lenow

andweach

ievedsuch

simulation

swith

theK

supercom

puter

inJap

an,

oneof

thecurren

tlyavailab

lebest

supercom

puters

with

∼10P

FLOPS.In

this

studywediscretized

theBoltz-

man

nequation

sboth

inspace

and

mom

entum,which

isactu

allya6-d

imension

alprob

lemeven

under

axisy

m-

metry.

Excep

tfor

this

almost

man

datory

discretization

ofthebasic

differen

tialequation

s,wedonot

resortto

anyartifi

cialap

prox

imation

orphenom

enological

mod-

elingin

theneutrin

otran

sport

[32,33].

Note

that

Ott

etal.

[10]also

conducted

axisy

mmetric

simulation

s,dis-

cretizingtheBoltzm

annequation

s,butthey

took

infact

ahybrid

approach

that

employed

themulti-grou

pflux-

limited

diffusion

approx

imation

intheop

ticallythick

re-gion

,i.e.,

theirs

were

not

genuinely

thefullBoltzm

ann-

neutrin

otran

sport.

More

importan

tly,they

ignored

rela-tiv

isticcorrection

scom

pletely,

drop

pingall

fluid-velo

citydep

endentterm

s,an

dalso

neglected

some

importan

tneutrin

o-matter

reactions,all

ofwhich

arecru

cialfor

re-alistic

modelin

gof

CCSNe(see

e.g.,[30,

31]).

These

day

s,much

attention

ofsupern

ovaresearch

ersis

directed

to3-d

imension

alsim

ulation

sin

space.

Itshou

ldberem

embered

,how

ever,that

those

computation

sin-

evitab

lyem

ploy

someap

prox

imation

sin

neutrin

otran

s-port,

which

have

not

been

well

validated

evenin

axisy

m-

metry.

Webelieve

that

theresu

ltspresen

tedhere

arenot

only

amileston

efor

thenumerical

studyof

CCSNe,

but

they

arealso

importan

tas

realisticsupern

ovamodels,

al-beit

inax

isymmetry.

They

will

hop

efully

contrib

ute

totheresolu

tionof

theap

paren

tcon

tradiction

sam

ongthe

results

ofdifferen

tgrou

ps,for

which

theap

prox

imation

sthey

adop

tedmay

beresp

onsib

le.

Meth

ods

and

Models.—

Wesolve

numerically

the

equation

sof

neutrin

o-radiation

hydrodynam

ics.Weap

-

2

200

400

600

rad

ius

(k

m)

2DLS2DFS1DLS1DFS

(a)

0

2

4

6

8

0 50 100 150 200 250 300 0

4

8

12

L (

10

52 e

rg/s

)

Em

(M

eV

)

time after bounce (ms)

(b)

L

E

LS νe

LS νe-

LS νx

FS νe

FS νe-

FS νx

0.1

1

10

100

100 150 200 250 300 0

3

6

9

Ta

dv/T

he

at

χ

time after bounce (ms)

(c)

Tadv/Theat

χLS

FS

FIG. 1: (a) Shock radii as functions of time. The color-shadedregions show the ranges of the shock radii, red for the LSEOS and blue for the FS EOS. The solid lines are the angle-average values. For comparison, the corresponding results inspherical symmetry are displayed with dashed lines. (b) Timeevolutions of the angle-integrated luminosities (L, solid lines)and the angle-averaged mean energies (Em, dashed lines) fordifferent species of neutrinos. Both of them are measuredat r = 500km. (c) The ratio of the advection to heatingtimescales (Tadv/Theat, with solid lines) and the χ parameter(dashed lines). The dotted black line represents Tadv/Theat =1 and χ = 3 for reference.

ply the so-called discrete-ordinate method to the Boltz-mann equations for neutrino transport, taking fully intoaccount special relativistic effects. In fact, it has alreadyincorporated general relativistic capabilities as well, apart of which is utilized to track the proper motionof proto neutron star (PNS) [34]. The hydrodynamicsand self-gravity are still Newtonian: the so-called centralscheme of second-order accuracy in both space and timeis employed for the former and the Poisson equation issolved for the latter.We adopt spherical coordinates (r, θ) covering 0 ≤ r ≤

5000km and 0 ≤ θ ≤ 180 in the meridian section. Wedeploy 384(r) × 128(θ) grid points. Momentum spaceis also discretized with 20 energy mesh points covering0 ≤ ε ≤ 300MeV and 10(θ) × 6(φ) angular grid pointsover the entire solid angle. The polar and azimuthal an-gles (θ, φ) are measured from the radial direction. Threeneutrino species are distinguished: electron-type neutri-nos νe, electron-type anti-neutrinos νe and all the otherscollectively denoted by νx.We pick up a non-rotating progenitor model of 11.2

M⊙ from [35]. We employ two nuclear EOS’s: Lat-

LS - entropy

16

12

8

4

0

500 km

LS - |V|

FS - entropy

FS - |V|

3

x 109 (cm/s)

2

1

0

FIG. 2: Snapshots of entropy per baryon (upper) and fluid-speed (lower) at t = 200ms. Left and right panels are for theLS- and FS EOS, respectively.

timer & Swesty’s EOS with the incompressibility of K =220MeV [36] and Furusawa’s EOS derived from H.Shen’srelativistic mean-field EOS with the TM1 parameterset [37, 38]; the former is softer than the latter. In the fol-lowing, they are referred to as the ”LS” and ”FS” EOS’s,respectively. Neutrino-matter interactions are based onthose given by [39], but we have implemented the up-to-date electron capture rates for heavy nuclei [40–42]and incorporated the non-isoenergetic scatterings on elec-trons and positrons as well as the bremsstrahlung in nu-cleon collisions. We refer readers to [32–34] for moredetails of our code.We start the simulations in spherically symmetry and

switch them to axisymmetric computations at ∼ 1ms af-ter core bounce when a negative entropy gradient startsto develop behind the shock wave and convection is ex-pected to start. We seed perturbations of 0.1% to radialvelocities inside the radius of r = 50km. Each model isrun up to t = 300ms after bounce.Dynamics.— We find an explosion when using the LSEOS. This is the first ever successful shock revival ob-tained in a multi-D simulation at this level of elabora-tion in neutrino transport. As displayed in Fig. 1(a), theshock wave produced at core bounce expands rather grad-ually with time for the LS EOS and its maximum radiusreaches ∼ 700km at t = 300ms. Note that the explosionis globally asymmetric as shown in Fig. 2. The pend-ing explosion is apparent from the fact that the shock isstill expanding at the end of the simulation and a stan-dard diagnostic also indicates a favorable condition forexplosion: the advection timescale (Tadv = Mg/M withMg and M denoting the mass in the gain region and themass accretion rate, respectively) is much longer than the

2

200

400

600

rad

ius

(k

m)

2DLS2DFS1DLS1DFS

(a)

0

2

4

6

8

0 50 100 150 200 250 300 0

4

8

12

L (

10

52 e

rg/s

)

Em

(M

eV

)

time after bounce (ms)

(b)

L

E

LS νe

LS νe-

LS νx

FS νe

FS νe-

FS νx

0.1

1

10

100

100 150 200 250 300 0

3

6

9

Tad

v/T

heat

χ

time after bounce (ms)

(c)

Tadv/Theat

χLS

FS

FIG. 1: (a) Shock radii as functions of time. The color-shadedregions show the ranges of the shock radii, red for the LSEOS and blue for the FS EOS. The solid lines are the angle-average values. For comparison, the corresponding results inspherical symmetry are displayed with dashed lines. (b) Timeevolutions of the angle-integrated luminosities (L, solid lines)and the angle-averaged mean energies (Em, dashed lines) fordifferent species of neutrinos. Both of them are measuredat r = 500km. (c) The ratio of the advection to heatingtimescales (Tadv/Theat, with solid lines) and the χ parameter(dashed lines). The dotted black line represents Tadv/Theat =1 and χ = 3 for reference.

ply the so-called discrete-ordinate method to the Boltz-mann equations for neutrino transport, taking fully intoaccount special relativistic effects. In fact, it has alreadyincorporated general relativistic capabilities as well, apart of which is utilized to track the proper motionof proto neutron star (PNS) [34]. The hydrodynamicsand self-gravity are still Newtonian: the so-called centralscheme of second-order accuracy in both space and timeis employed for the former and the Poisson equation issolved for the latter.We adopt spherical coordinates (r, θ) covering 0 ≤ r ≤

5000km and 0 ≤ θ ≤ 180 in the meridian section. Wedeploy 384(r) × 128(θ) grid points. Momentum spaceis also discretized with 20 energy mesh points covering0 ≤ ε ≤ 300MeV and 10(θ) × 6(φ) angular grid pointsover the entire solid angle. The polar and azimuthal an-gles (θ, φ) are measured from the radial direction. Threeneutrino species are distinguished: electron-type neutri-nos νe, electron-type anti-neutrinos νe and all the otherscollectively denoted by νx.We pick up a non-rotating progenitor model of 11.2

M⊙ from [35]. We employ two nuclear EOS’s: Lat-

LS - entropy

16

12

8

4

0

500 km

LS - |V|

FS - entropy

FS - |V|

3

x 109 (cm/s)

2

1

0

FIG. 2: Snapshots of entropy per baryon (upper) and fluid-speed (lower) at t = 200ms. Left and right panels are for theLS- and FS EOS, respectively.

timer & Swesty’s EOS with the incompressibility of K =220MeV [36] and Furusawa’s EOS derived from H.Shen’srelativistic mean-field EOS with the TM1 parameterset [37, 38]; the former is softer than the latter. In the fol-lowing, they are referred to as the ”LS” and ”FS” EOS’s,respectively. Neutrino-matter interactions are based onthose given by [39], but we have implemented the up-to-date electron capture rates for heavy nuclei [40–42]and incorporated the non-isoenergetic scatterings on elec-trons and positrons as well as the bremsstrahlung in nu-cleon collisions. We refer readers to [32–34] for moredetails of our code.We start the simulations in spherically symmetry and

switch them to axisymmetric computations at ∼ 1ms af-ter core bounce when a negative entropy gradient startsto develop behind the shock wave and convection is ex-pected to start. We seed perturbations of 0.1% to radialvelocities inside the radius of r = 50km. Each model isrun up to t = 300ms after bounce.Dynamics.— We find an explosion when using the LSEOS. This is the first ever successful shock revival ob-tained in a multi-D simulation at this level of elabora-tion in neutrino transport. As displayed in Fig. 1(a), theshock wave produced at core bounce expands rather grad-ually with time for the LS EOS and its maximum radiusreaches ∼ 700km at t = 300ms. Note that the explosionis globally asymmetric as shown in Fig. 2. The pend-ing explosion is apparent from the fact that the shock isstill expanding at the end of the simulation and a stan-dard diagnostic also indicates a favorable condition forexplosion: the advection timescale (Tadv = Mg/M withMg and M denoting the mass in the gain region and themass accretion rate, respectively) is much longer than the

2

200

400

600

rad

ius

(k

m)

2DLS2DFS1DLS1DFS

(a)

0

2

4

6

8

0 50 100 150 200 250 300 0

4

8

12

L (

10

52 e

rg/s

)

Em

(M

eV

)

time after bounce (ms)

(b)

L

E

LS νe

LS νe-

LS νx

FS νe

FS νe-

FS νx

0.1

1

10

100

100 150 200 250 300 0

3

6

9

Ta

dv/T

he

at

χ

time after bounce (ms)

(c)

Tadv/Theat

χLS

FS

FIG. 1: (a) Shock radii as functions of time. The color-shadedregions show the ranges of the shock radii, red for the LSEOS and blue for the FS EOS. The solid lines are the angle-average values. For comparison, the corresponding results inspherical symmetry are displayed with dashed lines. (b) Timeevolutions of the angle-integrated luminosities (L, solid lines)and the angle-averaged mean energies (Em, dashed lines) fordifferent species of neutrinos. Both of them are measuredat r = 500km. (c) The ratio of the advection to heatingtimescales (Tadv/Theat, with solid lines) and the χ parameter(dashed lines). The dotted black line represents Tadv/Theat =1 and χ = 3 for reference.

ply the so-called discrete-ordinate method to the Boltz-mann equations for neutrino transport, taking fully intoaccount special relativistic effects. In fact, it has alreadyincorporated general relativistic capabilities as well, apart of which is utilized to track the proper motionof proto neutron star (PNS) [34]. The hydrodynamicsand self-gravity are still Newtonian: the so-called centralscheme of second-order accuracy in both space and timeis employed for the former and the Poisson equation issolved for the latter.We adopt spherical coordinates (r, θ) covering 0 ≤ r ≤

5000km and 0 ≤ θ ≤ 180 in the meridian section. Wedeploy 384(r) × 128(θ) grid points. Momentum spaceis also discretized with 20 energy mesh points covering0 ≤ ε ≤ 300MeV and 10(θ) × 6(φ) angular grid pointsover the entire solid angle. The polar and azimuthal an-gles (θ, φ) are measured from the radial direction. Threeneutrino species are distinguished: electron-type neutri-nos νe, electron-type anti-neutrinos νe and all the otherscollectively denoted by νx.We pick up a non-rotating progenitor model of 11.2

M⊙ from [35]. We employ two nuclear EOS’s: Lat-

LS - entropy

16

12

8

4

0

500 km

LS - |V|

FS - entropy

FS - |V|

3

x 109 (cm/s)

2

1

0

FIG. 2: Snapshots of entropy per baryon (upper) and fluid-speed (lower) at t = 200ms. Left and right panels are for theLS- and FS EOS, respectively.

timer & Swesty’s EOS with the incompressibility of K =220MeV [36] and Furusawa’s EOS derived from H.Shen’srelativistic mean-field EOS with the TM1 parameterset [37, 38]; the former is softer than the latter. In the fol-lowing, they are referred to as the ”LS” and ”FS” EOS’s,respectively. Neutrino-matter interactions are based onthose given by [39], but we have implemented the up-to-date electron capture rates for heavy nuclei [40–42]and incorporated the non-isoenergetic scatterings on elec-trons and positrons as well as the bremsstrahlung in nu-cleon collisions. We refer readers to [32–34] for moredetails of our code.We start the simulations in spherically symmetry and

switch them to axisymmetric computations at ∼ 1ms af-ter core bounce when a negative entropy gradient startsto develop behind the shock wave and convection is ex-pected to start. We seed perturbations of 0.1% to radialvelocities inside the radius of r = 50km. Each model isrun up to t = 300ms after bounce.Dynamics.— We find an explosion when using the LSEOS. This is the first ever successful shock revival ob-tained in a multi-D simulation at this level of elabora-tion in neutrino transport. As displayed in Fig. 1(a), theshock wave produced at core bounce expands rather grad-ually with time for the LS EOS and its maximum radiusreaches ∼ 700km at t = 300ms. Note that the explosionis globally asymmetric as shown in Fig. 2. The pend-ing explosion is apparent from the fact that the shock isstill expanding at the end of the simulation and a stan-dard diagnostic also indicates a favorable condition forexplosion: the advection timescale (Tadv = Mg/M withMg and M denoting the mass in the gain region and themass accretion rate, respectively) is much longer than the

EOS Dependence

2

200

400

600

rad

ius

(k

m)

2DLS2DFS1DLS1DFS

(a)

0

2

4

6

8

0 50 100 150 200 250 300 0

4

8

12L

(1

052 e

rg/s

)

Em

(M

eV

)

time after bounce (ms)

(b)

L

E

LS νe

LS νe-

LS νx

FS νe

FS νe-

FS νx

0.1

1

10

100

100 150 200 250 300 0

3

6

9

Tad

v/T

heat

χ

time after bounce (ms)

(c)

Tadv/Theat

χLS

FS

FIG. 1: (a) Shock radii as functions of time. The color-shadedregions show the ranges of the shock radii, red for the LSEOS and blue for the FS EOS. The solid lines are the angle-average values. For comparison, the corresponding results inspherical symmetry are displayed with dashed lines. (b) Timeevolutions of the angle-integrated luminosities (L, solid lines)and the angle-averaged mean energies (Em, dashed lines) fordifferent species of neutrinos. Both of them are measuredat r = 500km. (c) The ratio of the advection to heatingtimescales (Tadv/Theat, with solid lines) and the χ parameter(dashed lines). The dotted black line represents Tadv/Theat =1 and χ = 3 for reference.

ply the so-called discrete-ordinate method to the Boltz-mann equations for neutrino transport, taking fully intoaccount special relativistic effects. In fact, it has alreadyincorporated general relativistic capabilities as well, apart of which is utilized to track the proper motionof proto neutron star (PNS) [34]. The hydrodynamicsand self-gravity are still Newtonian: the so-called centralscheme of second-order accuracy in both space and timeis employed for the former and the Poisson equation issolved for the latter.We adopt spherical coordinates (r, θ) covering 0 ≤ r ≤

5000km and 0 ≤ θ ≤ 180 in the meridian section. Wedeploy 384(r) × 128(θ) grid points. Momentum spaceis also discretized with 20 energy mesh points covering0 ≤ ε ≤ 300MeV and 10(θ) × 6(φ) angular grid pointsover the entire solid angle. The polar and azimuthal an-gles (θ, φ) are measured from the radial direction. Threeneutrino species are distinguished: electron-type neutri-nos νe, electron-type anti-neutrinos νe and all the otherscollectively denoted by νx.We pick up a non-rotating progenitor model of 11.2

M⊙ from [35]. We employ two nuclear EOS’s: Lat-

LS - entropy

16

12

8

4

0

500 km

LS - |V|

FS - entropy

FS - |V|

3

x 109 (cm/s)

2

1

0

FIG. 2: Snapshots of entropy per baryon (upper) and fluid-speed (lower) at t = 200ms. Left and right panels are for theLS- and FS EOS, respectively.

timer & Swesty’s EOS with the incompressibility of K =220MeV [36] and Furusawa’s EOS derived from H.Shen’srelativistic mean-field EOS with the TM1 parameterset [37, 38]; the former is softer than the latter. In the fol-lowing, they are referred to as the ”LS” and ”FS” EOS’s,respectively. Neutrino-matter interactions are based onthose given by [39], but we have implemented the up-to-date electron capture rates for heavy nuclei [40–42]and incorporated the non-isoenergetic scatterings on elec-trons and positrons as well as the bremsstrahlung in nu-cleon collisions. We refer readers to [32–34] for moredetails of our code.We start the simulations in spherically symmetry and

switch them to axisymmetric computations at ∼ 1ms af-ter core bounce when a negative entropy gradient startsto develop behind the shock wave and convection is ex-pected to start. We seed perturbations of 0.1% to radialvelocities inside the radius of r = 50km. Each model isrun up to t = 300ms after bounce.Dynamics.— We find an explosion when using the LSEOS. This is the first ever successful shock revival ob-tained in a multi-D simulation at this level of elabora-tion in neutrino transport. As displayed in Fig. 1(a), theshock wave produced at core bounce expands rather grad-ually with time for the LS EOS and its maximum radiusreaches ∼ 700km at t = 300ms. Note that the explosionis globally asymmetric as shown in Fig. 2. The pend-ing explosion is apparent from the fact that the shock isstill expanding at the end of the simulation and a stan-dard diagnostic also indicates a favorable condition forexplosion: the advection timescale (Tadv = Mg/M withMg and M denoting the mass in the gain region and themass accretion rate, respectively) is much longer than the

LS: Lattimer & Swesty (K=220MeV)EOS FS: Furusawa EOS

11.2 Msol (Woosley 2002)

by Nagakura

@京

Page 12: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Rotational Effect

ボルツマン輻射流体コードを用いた 回転大質量星の重力崩壊計算 (課題番号hp160071)課題代表者:山田章一 (早稲田大学)

第四回「京」を中核とするHPCIシステム利用研究課題 成果報告会@2017/11/2 コクヨホール

3. 結果 分布関数から得られる特徴

1. 序論 超新星爆発メカニズム

概要物質の起源や恒星・宇宙の進化の鍵となる超新星爆発のメカニズム解明の手がかりを得るために、ボルツマン輻射流体コードによって自転する星の重力崩壊を計算した。その結果、ボルツマン方程式を解くことでのみ得られる特徴を明らかにした。

4. まとめ本グループで開発した多次元ボルツマン輻射流体コードを用いて、回転星の重力崩壊段階において既存の計算では求められない特徴を見出し、近似法の評価ができることを示した。

鉄コア 原始中性子星

停滞衝撃波

ニュートリノ加熱爆発!

重力 崩壊

ニュートリノ加熱メカニズム

2. 設定 ボルツマン輻射流体コードによる回転星重力崩壊計算

超新星爆発は大質量星の最期の爆発である。爆発メカニズムは未だ明らかではないが、中心の原始中性子星から放射されるニュートリノがエネルギー源になるというメカニズムが現在の最有力仮説である。この仮説を確かめる上では六次元のボルツマン方程式によるニュートリノ輻射輸送を解く必要があり、京を用いた大規模並列計算が必須である。

代表者率いるグループはかねてより、ポスト「京」研究開発枠重点課題9「宇宙の基本法則と進化の解明」を通してボルツマン輻射流体コードを開発してきており、厳密なニュートリノの扱いのもとでの超新星爆発シミュレーションを行ってきた実績がある。今回はこのコードを用い、質量が太陽の約11倍である星に自転の角速度を与え、重力崩壊計算を行った。角速度Ωのプロファイルとしては、中心からの距離rの関数として

重力崩壊中のコアは非常に密度が高いため、ニュートリノが散乱反応等で流体素片に閉じ込められる。一方でコア外部では密度が下がり、ニュートリノが抜け出せるようになる。その結果、回転星の重力崩壊においてはニュートリノは回転方向にもフラックスを持つ。これは特に実験室系でフラックスを測った時に顕著である。一方で、流体静止系でフラックスを測ると、コア内部のフラックスはほぼ0で、外側では回転方向とは逆向きのフラックスを持つ。これは、抜け出してほぼ動径方向に進むニュートリノを流体が追い越すため、流体から見るとニュートリノが回転方向とは逆向きに進んでいるように見えるからである。こうした振る舞いは、京を用いなければ計算することはできない。

を採用した。本計算では重力崩壊の開始から衝撃波形成の頃までを計算した。計算領域は半径5000 kmの球状領域であり、軸対称を仮定した上で動径方向に384分割、天頂角方向に64分割(重力崩壊中)及び128分割(衝撃波形成後)して計算を行った。運動量空間はその大きさを20分割、角度を10×6分割した。本コードはMPI+OpenMPのハイブリッド並列を採用しており、MPIは二次元の配位空間を48×16(重力崩壊中)及び48×32(衝撃波形成後)のプロセスに分割しているが、運動量空間は分割していない。また、本コードの並列化効率は87.5%、平均実行効率は12.6%である。

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

z[km]

x [km]

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Fφ[104

0cm

−2s−

1]

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

z[km]

x [km]

-1.5

-1

-0.5

0

0.5

1

1.5

Fφ[104

4cm

−2s−

1]

実験室系で測ったフラックス。回転と同じ正方向にフラックスが生じる。

流体静止系で測ったフラックス。ニュートリノを流体が追い越すことで、回転と逆の負方向にフラックスが生じる。

-150

-100

-50

0

50

100

150

-150 -100 -50 0

z[km]

x [km]

ErφBoltmzann

50 100 150

ErφM1

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

超新星爆発計算において、本課題以外のグループではニュートリノ輻射輸送を近似的に計算している。近似手法の一つにray-by-ray法と呼ばれるものがあり、これは輻射輸送を動径方向ごとに球対称化して解くものであるが、この手法では上のような結果を得ることはできない。一方、ray-by-ray法の他にM1-closure法と呼ばれる手法も使われている。輻射場の二次モーメント(エディントン・テンソル)の非対角成分はray-by-ray法では0と仮定するが、M1-closure法では輻射フラックスを用いた有限の値として与える。この手法の精度を調べるため、本課題の計算結果からM1-closure法を用いて計算したエディントン・テンソルのrφ成分を分布関数から直接計算したものと比較したところ、ニュートリノが閉じ込められていない領域で二倍程度の過剰評価をすることが判明した。特に、閉じ込め領域の境界では本来見られない振動パターンが見られ、この近似法の限界を示唆している。分布関数から計算したエディントン・テ

ンソル(左)と、M1-closure法によるものとの差(右)

ニュートリノ 閉込領域

ニュートリノ伝搬領域

ボルツマン輻射流体コードを用いた 回転大質量星の重力崩壊計算 (課題番号hp160071)課題代表者:山田章一 (早稲田大学)

第四回「京」を中核とするHPCIシステム利用研究課題 成果報告会@2017/11/2 コクヨホール

3. 結果 分布関数から得られる特徴

1. 序論 超新星爆発メカニズム

概要物質の起源や恒星・宇宙の進化の鍵となる超新星爆発のメカニズム解明の手がかりを得るために、ボルツマン輻射流体コードによって自転する星の重力崩壊を計算した。その結果、ボルツマン方程式を解くことでのみ得られる特徴を明らかにした。

4. まとめ本グループで開発した多次元ボルツマン輻射流体コードを用いて、回転星の重力崩壊段階において既存の計算では求められない特徴を見出し、近似法の評価ができることを示した。

鉄コア 原始中性子星

停滞衝撃波

ニュートリノ加熱爆発!

重力 崩壊

ニュートリノ加熱メカニズム

2. 設定 ボルツマン輻射流体コードによる回転星重力崩壊計算

超新星爆発は大質量星の最期の爆発である。爆発メカニズムは未だ明らかではないが、中心の原始中性子星から放射されるニュートリノがエネルギー源になるというメカニズムが現在の最有力仮説である。この仮説を確かめる上では六次元のボルツマン方程式によるニュートリノ輻射輸送を解く必要があり、京を用いた大規模並列計算が必須である。

代表者率いるグループはかねてより、ポスト「京」研究開発枠重点課題9「宇宙の基本法則と進化の解明」を通してボルツマン輻射流体コードを開発してきており、厳密なニュートリノの扱いのもとでの超新星爆発シミュレーションを行ってきた実績がある。今回はこのコードを用い、質量が太陽の約11倍である星に自転の角速度を与え、重力崩壊計算を行った。角速度Ωのプロファイルとしては、中心からの距離rの関数として

重力崩壊中のコアは非常に密度が高いため、ニュートリノが散乱反応等で流体素片に閉じ込められる。一方でコア外部では密度が下がり、ニュートリノが抜け出せるようになる。その結果、回転星の重力崩壊においてはニュートリノは回転方向にもフラックスを持つ。これは特に実験室系でフラックスを測った時に顕著である。一方で、流体静止系でフラックスを測ると、コア内部のフラックスはほぼ0で、外側では回転方向とは逆向きのフラックスを持つ。これは、抜け出してほぼ動径方向に進むニュートリノを流体が追い越すため、流体から見るとニュートリノが回転方向とは逆向きに進んでいるように見えるからである。こうした振る舞いは、京を用いなければ計算することはできない。

を採用した。本計算では重力崩壊の開始から衝撃波形成の頃までを計算した。計算領域は半径5000 kmの球状領域であり、軸対称を仮定した上で動径方向に384分割、天頂角方向に64分割(重力崩壊中)及び128分割(衝撃波形成後)して計算を行った。運動量空間はその大きさを20分割、角度を10×6分割した。本コードはMPI+OpenMPのハイブリッド並列を採用しており、MPIは二次元の配位空間を48×16(重力崩壊中)及び48×32(衝撃波形成後)のプロセスに分割しているが、運動量空間は分割していない。また、本コードの並列化効率は87.5%、平均実行効率は12.6%である。

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

z[km]

x [km]

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Fφ[104

0cm

−2s−

1]

-150

-100

-50

0

50

100

150

-150 -100 -50 0 50 100 150

z[km]

x [km]

-1.5

-1

-0.5

0

0.5

1

1.5

Fφ[104

4cm

−2s−

1]

実験室系で測ったフラックス。回転と同じ正方向にフラックスが生じる。

流体静止系で測ったフラックス。ニュートリノを流体が追い越すことで、回転と逆の負方向にフラックスが生じる。

-150

-100

-50

0

50

100

150

-150 -100 -50 0

z[km]

x [km]

ErφBoltmzann

50 100 150

ErφM1

-0.004

-0.003

-0.002

-0.001

0

0.001

0.002

0.003

0.004

超新星爆発計算において、本課題以外のグループではニュートリノ輻射輸送を近似的に計算している。近似手法の一つにray-by-ray法と呼ばれるものがあり、これは輻射輸送を動径方向ごとに球対称化して解くものであるが、この手法では上のような結果を得ることはできない。一方、ray-by-ray法の他にM1-closure法と呼ばれる手法も使われている。輻射場の二次モーメント(エディントン・テンソル)の非対角成分はray-by-ray法では0と仮定するが、M1-closure法では輻射フラックスを用いた有限の値として与える。この手法の精度を調べるため、本課題の計算結果からM1-closure法を用いて計算したエディントン・テンソルのrφ成分を分布関数から直接計算したものと比較したところ、ニュートリノが閉じ込められていない領域で二倍程度の過剰評価をすることが判明した。特に、閉じ込め領域の境界では本来見られない振動パターンが見られ、この近似法の限界を示唆している。分布関数から計算したエディントン・テ

ンソル(左)と、M1-closure法によるものとの差(右)

ニュートリノ 閉込領域

ニュートリノ伝搬領域

3

heating timescale (Theat = |Etot|/Qν with Etot and Qν

being the total energy and the heating rate in the gainregion, respectively), which can be seen in Fig. 1(c). Theneutrino-driven convection is dominant over the standingaccretion shock instability (SASI) in aiding the shock re-vival, which conforms to the χ parameter [43, 44] beingconsistently larger than 3 during the entire post-bouncephase (Fig. 1(c)).

Note that the explosion just presented may not be sorobust, since it is sensitive to the nuclear EOS. In fact, wedo not find shock revival when using the FS EOS, which issomewhat stiffer than the LS EOS (Fig. 1(a)). The shockwave for the FS EOS stalls at r ∼ 200km at t ∼ 100msand then starts to recede at t ∼ 250ms and shrinks backto r ∼ 100km by t ∼ 300ms. Fig. 2 compares the en-tropy and velocity distributions between the two modelsat t = 200ms. Interestingly, their post-shock morpholo-gies are quite similar to each other and only the scales aredifferent. In fact, the convection is dominant over SASIalso in the FS EOS with the χ parameter being higherthan 3 until ∼ 250ms (Fig. 1(c)). As a consequence ofthis difference in scale that is attributable to the differ-ences in the EOS, no shock revival is obtained for the FSEOS simulation, although the neutrino luminosities (L)and mean energies (Em, defined as the ratio of energydensity to number density) are almost identical betweenthe two cases (Fig. 1(b)). Note that the difference in EOSand other microphysics tends to be more remarkable inmulti-D than in spherical symmetry [26, 45]. However,this outcome may be affected by the general relativisticgravity (see [11, 46, 47, 49]), which is still lacking in thecurrent simulations and should be studied further.

ν-Distributions in Momentum Space.—Now weturn our attention to novel features of the neutrino dis-tributions in moment space. They were never accessi-ble to previous simulations, which integrated out the an-gular degrees of freedom in momentum space. We findin our calculations significant non-axisymmetry with re-spect to the radial direction in the neutrino angular dis-tributions. It is produced by lateral inhomogeneities inmatter, which are generated by hydrodynamical instabil-ities. The asymmetry hence appears inevitably in multi-D.

Fig. 3(a) shows such an example for the angular dis-tributions of νe with an energy of ε = 11.1MeV. Eachsurface displays the relative intensity of the distributionfunction at different propagation directions in the fluid-rest frame. Surfaces of different colors depict the distri-bution functions at three locations on an arbitrarily cho-sen radial ray and are normalized by their maximum val-ues. The distributions are almost isotropic at r = 23km(red surface) while they become forward peaked as ra-dius increases, a fact that is well known. What is reallynew here, as we mentioned above, is that they are non-axisymmetric with respect to the radial direction, whichis more apparent in Fig. 3(b). Note that the featureis robust, occurring irrespective of neutrino energies orspecies.

(a) er-

eφ-

eθ-

(b) er-

eφ-

eθ-

FIG. 3: (a) Angular distributions of νe in momentum spaceat tpb = 15ms for the LS EOS. Different colors correspond todifferent radial positions (red: r = 23km, green: r = 39km,blue: r = 49km) along the radial ray with the zenith angleof θ = 96. The neutrino energy is ε = 11.1MeV in thefluid-rest frame. (b) The deviations from spherical symmetryare emphasized by subtracting the minimum values and re-normalizing the resultant distribution so that the maximumvalues should be identical in all cases. The blue surface isthe same as the one in (a) while the purple surface shows thesame wired surface at the same radius of blue one, but at adifferent zenith angle, θ = 68.

The multi-angle treatment of neutrino transport inour simulations enables us to evaluate the so-calledEddington tensor (kij), which characterizes these non-axisymmetric angular distributions more quantitatively.Note that hereinafter Latin subscripts denote the spatialcomponents alone while Greek letters are used for bothspatial and temporal components. The Eddington tensoris obtained from the neutrino distribution function (f) asfollows: we first define the second angular moment Mµν

as

Mµν(ε) =1

ε

f(ε,Ωm)pµpνdΩm, (1)

where pµ is the four-momentum of neutrino and ε and Ωm

are the corresponding energy and solid angle measured inthe fluid-rest frame; then kij is given by the ratio P ij/E,where P ij and E are defined from Mµν as

P ij(ε) = γiµγ

jνM

µν(ε), (2)

E(ε) = nµnνMµν(ε), (3)

with nµ and γiµ being the unit vector orthogonal to a hy-

persurface of constant coordinate time and the projectiontensor onto this hypersurface, respectively.We pay particular attention here to one of the off-

diagonal components of the Eddington tensor, krθ. Notethat the existence of non-vanishing krθ may be an impor-tant warning sign that the ray-by-ray(-plus) approxima-

2nd angular moment:

3

heating timescale (Theat = |Etot|/Qν with Etot and Qν

being the total energy and the heating rate in the gainregion, respectively), which can be seen in Fig. 1(c). Theneutrino-driven convection is dominant over the standingaccretion shock instability (SASI) in aiding the shock re-vival, which conforms to the χ parameter [43, 44] beingconsistently larger than 3 during the entire post-bouncephase (Fig. 1(c)).

Note that the explosion just presented may not be sorobust, since it is sensitive to the nuclear EOS. In fact, wedo not find shock revival when using the FS EOS, which issomewhat stiffer than the LS EOS (Fig. 1(a)). The shockwave for the FS EOS stalls at r ∼ 200km at t ∼ 100msand then starts to recede at t ∼ 250ms and shrinks backto r ∼ 100km by t ∼ 300ms. Fig. 2 compares the en-tropy and velocity distributions between the two modelsat t = 200ms. Interestingly, their post-shock morpholo-gies are quite similar to each other and only the scales aredifferent. In fact, the convection is dominant over SASIalso in the FS EOS with the χ parameter being higherthan 3 until ∼ 250ms (Fig. 1(c)). As a consequence ofthis difference in scale that is attributable to the differ-ences in the EOS, no shock revival is obtained for the FSEOS simulation, although the neutrino luminosities (L)and mean energies (Em, defined as the ratio of energydensity to number density) are almost identical betweenthe two cases (Fig. 1(b)). Note that the difference in EOSand other microphysics tends to be more remarkable inmulti-D than in spherical symmetry [26, 45]. However,this outcome may be affected by the general relativisticgravity (see [11, 46, 47, 49]), which is still lacking in thecurrent simulations and should be studied further.

ν-Distributions in Momentum Space.—Now weturn our attention to novel features of the neutrino dis-tributions in moment space. They were never accessi-ble to previous simulations, which integrated out the an-gular degrees of freedom in momentum space. We findin our calculations significant non-axisymmetry with re-spect to the radial direction in the neutrino angular dis-tributions. It is produced by lateral inhomogeneities inmatter, which are generated by hydrodynamical instabil-ities. The asymmetry hence appears inevitably in multi-D.

Fig. 3(a) shows such an example for the angular dis-tributions of νe with an energy of ε = 11.1MeV. Eachsurface displays the relative intensity of the distributionfunction at different propagation directions in the fluid-rest frame. Surfaces of different colors depict the distri-bution functions at three locations on an arbitrarily cho-sen radial ray and are normalized by their maximum val-ues. The distributions are almost isotropic at r = 23km(red surface) while they become forward peaked as ra-dius increases, a fact that is well known. What is reallynew here, as we mentioned above, is that they are non-axisymmetric with respect to the radial direction, whichis more apparent in Fig. 3(b). Note that the featureis robust, occurring irrespective of neutrino energies orspecies.

(a) er-

eφ-

eθ-

(b) er-

eφ-

eθ-

FIG. 3: (a) Angular distributions of νe in momentum spaceat tpb = 15ms for the LS EOS. Different colors correspond todifferent radial positions (red: r = 23km, green: r = 39km,blue: r = 49km) along the radial ray with the zenith angleof θ = 96. The neutrino energy is ε = 11.1MeV in thefluid-rest frame. (b) The deviations from spherical symmetryare emphasized by subtracting the minimum values and re-normalizing the resultant distribution so that the maximumvalues should be identical in all cases. The blue surface isthe same as the one in (a) while the purple surface shows thesame wired surface at the same radius of blue one, but at adifferent zenith angle, θ = 68.

The multi-angle treatment of neutrino transport inour simulations enables us to evaluate the so-calledEddington tensor (kij), which characterizes these non-axisymmetric angular distributions more quantitatively.Note that hereinafter Latin subscripts denote the spatialcomponents alone while Greek letters are used for bothspatial and temporal components. The Eddington tensoris obtained from the neutrino distribution function (f) asfollows: we first define the second angular moment Mµν

as

Mµν(ε) =1

ε

f(ε,Ωm)pµpνdΩm, (1)

where pµ is the four-momentum of neutrino and ε and Ωm

are the corresponding energy and solid angle measured inthe fluid-rest frame; then kij is given by the ratio P ij/E,where P ij and E are defined from Mµν as

P ij(ε) = γiµγ

jνM

µν(ε), (2)

E(ε) = nµnνMµν(ε), (3)

with nµ and γiµ being the unit vector orthogonal to a hy-

persurface of constant coordinate time and the projectiontensor onto this hypersurface, respectively.We pay particular attention here to one of the off-

diagonal components of the Eddington tensor, krθ. Notethat the existence of non-vanishing krθ may be an impor-tant warning sign that the ray-by-ray(-plus) approxima-

Radiation pressure:

3

heating timescale (Theat = |Etot|/Qν with Etot and Qν

being the total energy and the heating rate in the gainregion, respectively), which can be seen in Fig. 1(c). Theneutrino-driven convection is dominant over the standingaccretion shock instability (SASI) in aiding the shock re-vival, which conforms to the χ parameter [43, 44] beingconsistently larger than 3 during the entire post-bouncephase (Fig. 1(c)).

Note that the explosion just presented may not be sorobust, since it is sensitive to the nuclear EOS. In fact, wedo not find shock revival when using the FS EOS, which issomewhat stiffer than the LS EOS (Fig. 1(a)). The shockwave for the FS EOS stalls at r ∼ 200km at t ∼ 100msand then starts to recede at t ∼ 250ms and shrinks backto r ∼ 100km by t ∼ 300ms. Fig. 2 compares the en-tropy and velocity distributions between the two modelsat t = 200ms. Interestingly, their post-shock morpholo-gies are quite similar to each other and only the scales aredifferent. In fact, the convection is dominant over SASIalso in the FS EOS with the χ parameter being higherthan 3 until ∼ 250ms (Fig. 1(c)). As a consequence ofthis difference in scale that is attributable to the differ-ences in the EOS, no shock revival is obtained for the FSEOS simulation, although the neutrino luminosities (L)and mean energies (Em, defined as the ratio of energydensity to number density) are almost identical betweenthe two cases (Fig. 1(b)). Note that the difference in EOSand other microphysics tends to be more remarkable inmulti-D than in spherical symmetry [26, 45]. However,this outcome may be affected by the general relativisticgravity (see [11, 46, 47, 49]), which is still lacking in thecurrent simulations and should be studied further.

ν-Distributions in Momentum Space.—Now weturn our attention to novel features of the neutrino dis-tributions in moment space. They were never accessi-ble to previous simulations, which integrated out the an-gular degrees of freedom in momentum space. We findin our calculations significant non-axisymmetry with re-spect to the radial direction in the neutrino angular dis-tributions. It is produced by lateral inhomogeneities inmatter, which are generated by hydrodynamical instabil-ities. The asymmetry hence appears inevitably in multi-D.

Fig. 3(a) shows such an example for the angular dis-tributions of νe with an energy of ε = 11.1MeV. Eachsurface displays the relative intensity of the distributionfunction at different propagation directions in the fluid-rest frame. Surfaces of different colors depict the distri-bution functions at three locations on an arbitrarily cho-sen radial ray and are normalized by their maximum val-ues. The distributions are almost isotropic at r = 23km(red surface) while they become forward peaked as ra-dius increases, a fact that is well known. What is reallynew here, as we mentioned above, is that they are non-axisymmetric with respect to the radial direction, whichis more apparent in Fig. 3(b). Note that the featureis robust, occurring irrespective of neutrino energies orspecies.

(a) er-

eφ-

eθ-

(b) er-

eφ-

eθ-

FIG. 3: (a) Angular distributions of νe in momentum spaceat tpb = 15ms for the LS EOS. Different colors correspond todifferent radial positions (red: r = 23km, green: r = 39km,blue: r = 49km) along the radial ray with the zenith angleof θ = 96. The neutrino energy is ε = 11.1MeV in thefluid-rest frame. (b) The deviations from spherical symmetryare emphasized by subtracting the minimum values and re-normalizing the resultant distribution so that the maximumvalues should be identical in all cases. The blue surface isthe same as the one in (a) while the purple surface shows thesame wired surface at the same radius of blue one, but at adifferent zenith angle, θ = 68.

The multi-angle treatment of neutrino transport inour simulations enables us to evaluate the so-calledEddington tensor (kij), which characterizes these non-axisymmetric angular distributions more quantitatively.Note that hereinafter Latin subscripts denote the spatialcomponents alone while Greek letters are used for bothspatial and temporal components. The Eddington tensoris obtained from the neutrino distribution function (f) asfollows: we first define the second angular moment Mµν

as

Mµν(ε) =1

ε

f(ε,Ωm)pµpνdΩm, (1)

where pµ is the four-momentum of neutrino and ε and Ωm

are the corresponding energy and solid angle measured inthe fluid-rest frame; then kij is given by the ratio P ij/E,where P ij and E are defined from Mµν as

P ij(ε) = γiµγ

jνM

µν(ε), (2)

E(ε) = nµnνMµν(ε), (3)

with nµ and γiµ being the unit vector orthogonal to a hy-

persurface of constant coordinate time and the projectiontensor onto this hypersurface, respectively.We pay particular attention here to one of the off-

diagonal components of the Eddington tensor, krθ. Notethat the existence of non-vanishing krθ may be an impor-tant warning sign that the ray-by-ray(-plus) approxima-

Radiation energy:

Eddington tensor: k ij = Pij / E

k k

4

(a)

300 km

100 km

kr θBZ

0.01

0

-0.01

kr θBZ - kr θ

M1

0

(b)kr θ

νe

0.01

0

-0.01

kr θνe -

0

FIG. 4: (a) The (rθ) component of the Eddington tensor (krθ)for νe in the northern hemisphere obtained in our simulationfor the FS EOS (left) and its deviation from the M1 prescrip-tion (right). The values of krθ are evaluated at the meanneutrino energy at each point. (b) krθ for νe (left) and νe(right) on a smaller spatial scale of 100km. The neutrino en-ergy is fixed to 8.53MeV in the fluid-rest frame. The time ist = 190ms in all cases.

tion [11–14, 16, 17, 23, 25], which neglects krθ completely,may yield inaccurate results.The left panel in Fig. 4(a) shows krθ for νe. As ex-

pected, it is almost zero inside the PNS, where mat-ter is opaque enough to make the neutrino distributionisotropic. It becomes non-zero outside the PNS, however,and increases with radius in accord with the appearanceof the non-axisymmetric structures in the neutrino angu-lar distribution (see Fig. 3). In fact, the (rθ) componentof the Eddington tensor corresponds to the mode withℓ = 2,m = 1 in the spherical harmonics expansion of thedistribution function.The right panel in Fig. 4(a) shows, on the other hand,

the evidence of a possible problem in the two-momentapproximation. This panel compares krθ obtained fromour simulation with that which is prescribed in the M1method. The Eddington tensor in the M1 prescription(kijM1) is obtained by replacing P ij in Eq. (2) with

P ijM1(ε) =

3ζ(ε)− 1

2P ijthin(ε) +

3(1− ζ(ε))

2P ijthick(ε), (4)

where ζ is referred to as the variable Eddington factor,which we set as ζ(ε) = (3+ 4F (ε)2)/(5+ 2

4− 3F (ε)2)in terms of the so-called flux factor F , which is theflux normalized with the energy density in the fluid-rest

frame. The optically thick and thin limits of P ij aredenoted by P ij

thick and P ijthin [18, 20, 22, 48]. As clearly

seen in this panel, the values of krθ are substantially dif-ferent between the two cases, which is a caution to theM1 approximation. We find that such discrepancies inkrθ are rather generic, being insensitive to the particularexpression of the Eddington factor (see [22] for variousoptions).Moreover, we find in krθ an intriguing correlation/anti-

correlation between νe and νe. The two panels ofFig. 4(b) compare krθ for νe and νe with the same en-ergy of ε = 8.5MeV. As can be seen in these panels, theyare anti-correlated with each other in the vicinity of PNS(! 50km) whereas they are positively correlated at largerradii (> 80km). The anti-correlation is particularly re-markable for low-energy neutrinos with ! 10MeV. Wefind that the sign of krθ roughly coincides with that of thelateral neutrino flux, which flows in the opposite direc-tions for νe and νe. This is due to the Fermi-degeneracyof νe at r ! 30km, which produces opposite trends inthe number densities of νe and νe. Importantly, the anti-correlation is then carried to larger radii by the outwardflux and remains non-vanishing even at r ∼ 50km, whereνe is no longer degenerate. On the other hand, at evenlarger radii, where matter is optically thin to neutrinos,krθ is correlated with the lateral velocity of matter dueto relativistic aberration. Note that this positive cor-relation at large distances is less remarkable than theanti-correlation in the vicinity of PNS (see the equato-rial region in Fig. 4(b)), since the angular distribution isno longer determined locally.

Summary and Discussion.—We have reported for thefirst time a successful shock revival in a multi-D sim-ulation with no artificial approximation except for themandatory discretization of the differential equations inthe Boltzmann-neutrino transport. Although the pro-genitor we employed in this study produced explosionsrather commonly in other approximate simulations andour results share with them the qualitative trend thatsofter EOS’s are advantageous for shock revival, it shouldbe pointed out that the shock propagation after revivalseems much less vigorous in our simulation than in others(see e.g., [14]). Detailed analyses are currently underwayand will be published elsewhere.

The complicated features in the neutrino distributionin momentum space, such as the lack of axisymmetry andthe non-vanishing off-diagonal component of the Edding-ton tensor, have never been obtained in other approxi-mate simulations done so far. We need to compare ourresults with those obtained in other approximate simula-tions quantitatively more in detail. They will provideus with invaluable information that is not only indis-pensable to understand the origin of the differences weobserved in the dynamics of shock revival but will alsoenable us to calibrate and possibly improve the prescrip-tions, which should be set by hand in the approximatetransport schemes. This is indeed important practically,since our method is very costly in terms of required nu-

M1 closure approximation:

4

(a)

300 km

100 km

kr θBZ

0.01

0

-0.01

kr θBZ - kr θ

M1

0

(b)kr θ

νe

0.01

0

-0.01

kr θνe -

0

FIG. 4: (a) The (rθ) component of the Eddington tensor (krθ)for νe in the northern hemisphere obtained in our simulationfor the FS EOS (left) and its deviation from the M1 prescrip-tion (right). The values of krθ are evaluated at the meanneutrino energy at each point. (b) krθ for νe (left) and νe(right) on a smaller spatial scale of 100km. The neutrino en-ergy is fixed to 8.53MeV in the fluid-rest frame. The time ist = 190ms in all cases.

tion [11–14, 16, 17, 23, 25], which neglects krθ completely,may yield inaccurate results.The left panel in Fig. 4(a) shows krθ for νe. As ex-

pected, it is almost zero inside the PNS, where mat-ter is opaque enough to make the neutrino distributionisotropic. It becomes non-zero outside the PNS, however,and increases with radius in accord with the appearanceof the non-axisymmetric structures in the neutrino angu-lar distribution (see Fig. 3). In fact, the (rθ) componentof the Eddington tensor corresponds to the mode withℓ = 2,m = 1 in the spherical harmonics expansion of thedistribution function.The right panel in Fig. 4(a) shows, on the other hand,

the evidence of a possible problem in the two-momentapproximation. This panel compares krθ obtained fromour simulation with that which is prescribed in the M1method. The Eddington tensor in the M1 prescription(kijM1) is obtained by replacing P ij in Eq. (2) with

P ijM1(ε) =

3ζ(ε)− 1

2P ijthin(ε) +

3(1− ζ(ε))

2P ijthick(ε), (4)

where ζ is referred to as the variable Eddington factor,which we set as ζ(ε) = (3+ 4F (ε)2)/(5+ 2

4− 3F (ε)2)in terms of the so-called flux factor F , which is theflux normalized with the energy density in the fluid-rest

frame. The optically thick and thin limits of P ij aredenoted by P ij

thick and P ijthin [18, 20, 22, 48]. As clearly

seen in this panel, the values of krθ are substantially dif-ferent between the two cases, which is a caution to theM1 approximation. We find that such discrepancies inkrθ are rather generic, being insensitive to the particularexpression of the Eddington factor (see [22] for variousoptions).Moreover, we find in krθ an intriguing correlation/anti-

correlation between νe and νe. The two panels ofFig. 4(b) compare krθ for νe and νe with the same en-ergy of ε = 8.5MeV. As can be seen in these panels, theyare anti-correlated with each other in the vicinity of PNS(! 50km) whereas they are positively correlated at largerradii (> 80km). The anti-correlation is particularly re-markable for low-energy neutrinos with ! 10MeV. Wefind that the sign of krθ roughly coincides with that of thelateral neutrino flux, which flows in the opposite direc-tions for νe and νe. This is due to the Fermi-degeneracyof νe at r ! 30km, which produces opposite trends inthe number densities of νe and νe. Importantly, the anti-correlation is then carried to larger radii by the outwardflux and remains non-vanishing even at r ∼ 50km, whereνe is no longer degenerate. On the other hand, at evenlarger radii, where matter is optically thin to neutrinos,krθ is correlated with the lateral velocity of matter dueto relativistic aberration. Note that this positive cor-relation at large distances is less remarkable than theanti-correlation in the vicinity of PNS (see the equato-rial region in Fig. 4(b)), since the angular distribution isno longer determined locally.

Summary and Discussion.—We have reported for thefirst time a successful shock revival in a multi-D sim-ulation with no artificial approximation except for themandatory discretization of the differential equations inthe Boltzmann-neutrino transport. Although the pro-genitor we employed in this study produced explosionsrather commonly in other approximate simulations andour results share with them the qualitative trend thatsofter EOS’s are advantageous for shock revival, it shouldbe pointed out that the shock propagation after revivalseems much less vigorous in our simulation than in others(see e.g., [14]). Detailed analyses are currently underwayand will be published elsewhere.

The complicated features in the neutrino distributionin momentum space, such as the lack of axisymmetry andthe non-vanishing off-diagonal component of the Edding-ton tensor, have never been obtained in other approxi-mate simulations done so far. We need to compare ourresults with those obtained in other approximate simula-tions quantitatively more in detail. They will provideus with invaluable information that is not only indis-pensable to understand the origin of the differences weobserved in the dynamics of shock revival but will alsoenable us to calibrate and possibly improve the prescrip-tions, which should be set by hand in the approximatetransport schemes. This is indeed important practically,since our method is very costly in terms of required nu-F(ε): Flux factor

(HPCI成果報告会2017ポスターより)

Eddington Factor:

by Harada

@京

Page 13: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

High Resolution Calculations (Momentum Space) @ CfCA

Chose the specific part from the whole region

Data calculated in K computer is imposed to the inner and outer boundary.

1)

2)

3)

4) Investigate how different results are between low and high resolution models.

Data calculated in K computer are used for the initial condition.

5) Detailed Analysis of the results given by very high resolution models in the intermediate region between optically thick and thin regions.

Page 14: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

FX:192(MPI) × 16(OpenMP) = 3072 coreNr × Nth × Ne × Nthν × Nphν = 120 × 128 × 20 × 10 × 6

Test Calculation for 11.2Msol Progenitor @ FX10

8km~40km 40km~200kmCalculation start from 8ms (prompt convection grow)

Boundary condition is properly imposed to both inner and outer boundaries.

Page 15: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

FX:192(MPI) × 16(OpenMP) = 3072 coreNr × Nth × Ne × Nthν × Nphν = 120 × 128 × 20 × 20 × 6

Test Calculation for 11.2Msol Progenitor @ FX10

Calculation start from 8ms (prompt convection grow)40km~200km8km~40km

Page 16: 重力崩壊型超新星のBoltzmann-Hydro Code - NAO

Summery

we would like to investigate the characteristics of the neutrino dynamics in the proto-neutron star,

Using the Boltzmann-Hydro code,

we would like to verify the approximative method for calculating neutrino heating and cooling, for example, examining the correlation between neutrino fluxes and hydrodynamical quantities.

2)

1)

we would like to do the high resolution study in the small excised region, using data obtained from the results of global simulations with K-Computer.

3)

accompanying prompt convection and rotation.