Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Amplification of magnetic fields in non-rotatingcore collapse
Martin Obergaulinger, Thomas Janka, Oliver Just, Miguel Ángel AloyCAMAP, Departament d´Astronomia i Astrofı́sica, Universitat de València
Max-Planck-Institut für Astrophysik, Garching bei München

Supernovae and Gamma-Ray Burst in Kyoto, 2013-10-28 – 2013-11-01

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Magnetic fields in core collapse

Imagnetic fields need to be strong to have aneffect on SNe

But: stellar evolution theory predicts rather
weak fields in the pre-collapse core
→ efficient amplification requiredIcompression
linear winding by differential rotation
I hydromagnetic instabilities: convection,
magnetorotational instability (MRI), SASIII

Meier et al., 1976

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Magnetic fields in core collapse

energymechanism

role of ~bSASI
accretion flow
advective-acousticcycle

convection
thermalbuoyant transport
of energy/species

passive; turbulent
dynamopassive; turbulent
dynamoEndeve et al., 2008

MRI
diff. rotation
magnetic transportof angular momentum
instability driver;
turbulent dynamo

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

The neutrino distribution
The ν field is equivalently described by its
I

distribution function, f (~p,~x, t), the probability tofind a neutrino with a momentum ~p (i.e., energy = |~p|c) at position ~x, time tI

radiative intensity, I(~n, ,~x, t), the energy carriedby all ν of energy  in direction ~n = ~p/|~p|through a unit surface dA at position ~x, time t

properties
3II

I is a relativistic invariantf = (hc)c3
in local equilibrium with matter, f is given by the Fermi-Dirac
1
distribution, fFD = exp −µν +1
kB T

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

The neutrino distribution
The ν field is equivalently described by its
I

distribution function, f (~p,~x, t), the probability tofind a neutrino with a momentum ~p (i.e., energy
 = |~p|c) at position ~x, time tIradiative intensity, I(~n, ,~x, t), the energy carriedby all ν of energy  in direction ~n = ~p/|~p|through a unit surface dA at position ~x, time t

Boltzmann transport equation (mixed frame)~∂t I + ~n · ∇I=

η0 () − χ0 ()I+~n · ~v (2η0 () − ∂ η0 + [χ0 () + ∂ χ0 ()]I)

with advection, emission, absorption, and Doppler shift, aberration.

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Radiation moments
R
Expand the intensity in angular moments, M i1 i2 ...im = d~nni1 ni2 ...nim I.
R d~n  3I 0th moment: E =f , the energy density
4π hc

R
3d~n

I 1st moment: F i =ni f , the momentum density
4π hc

R
 3 i jd~n
I 2nd moment: Pij =n n f , the pressure tensor
4π hc
Ietc. ad inf. (with no straightforward physical interpretation)

Moment equations
equation for the mth moment contains the (m + 1)th moment as a flux∂t M i1 ...im + ∇j M i1 ...im j +velocity terms = Smi1 ...im⇒ infinite series of equations ≡ transport equation

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Truncated moment systemevolve only the first 2 moments
and obtain the higher (2nd and3rd ) moments by a local
algebraic closure as a functionof the lower moments

Simulations of core collapse

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Truncated moment systemevolve only the first 2 moments
and obtain the higher (2nd and3rd ) moments by a local
algebraic closure as a functionof the lower moments

0th moment
assumption for the neutrino momentum
I free streaming: ~F = cE~n; fails in
optically thick regionsc ~I diffusion: ~F = − 3κ∇E; unphysical invacuumI

flux-limited diffusion: ensure physical(not necessarily correct) vacuum limit⇒ parabolic equation

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Truncated moment system

1st moment3χ−1 i jij
set Pij = 1−χ2 δ +2 n n with a variable

evolve only the first 2 moments
and obtain the higher (2nd and3rd ) moments by a local
algebraic closure as a functionof the lower moments~F|
Eddington factor χ(E, |cE).
Ilocal approximation ⇒ simpler andless expensive than Boltzmann solvers(but less accurate)Igenuinely multidimensionalI

physical consistency: χ → {1/3, 1}
for |~F|/(cE) → {0, 1}

I

hyperbolic for suitable choice of χMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Neutrino interactions

Reactions with matter
In + νep+ + e−

Ip+ + ν̄en + e−

I(A, Z) + νeIn/p + νX

I(A, Z) + νX

Ie + νX

I−e +eI

N+N(A, Z + 1) + e−
n/p + νX
(A, Z) + νX

e + νX+

νX + ν̄X
N + N + νX + ν̄XI

implementation following Rampp& Janka (2002)

I

2d simulations below neglect thereactions in grey

→ limited scope of our models

Magnetic fields in core collapse

Two-moment neutrino transport

Tests

I

homogeneous radiating sphereI

differentially expanding
atmosphereI

neutrino transport in a
post-bounce coreI

spherical core collapse

Simulations of core collapse

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Tests

I

homogeneous radiating sphereI

differentially expanding
atmosphereI

neutrino transport in a
post-bounce coreI

spherical core collapse

Simulations of core collapse

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Tests

I

homogeneous radiating sphereI

differentially expanding
atmosphereI

neutrino transport in a
post-bounce coreI

spherical core collapse

Simulations of core collapse

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

SummaryTestss [ kB / baryon ]homogeneous radiating sphereI

differentially expanding
atmosphere412120100

20080150

100200neutrino transport in a
post-bounce core

60405000I

8250r [km]Ir [km]

0

300

50

100150t [ms]

200

250

300

0246810t [ms]500

20Aenus
VERTEX
AGILE−BOLTZTRAN

400

spherical core collapse\bar{\nu}_e300< ε > [MeV]

L [1051 erg/s]

I1520010\nu_e5νe100

total\bar{ν}e00050100150t [ms]200250300

050100150t [ms]200250300

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

SummaryTests

I

homogeneous radiating sphereI

differentially expanding
atmosphereI

neutrino transport in a
post-bounce coreI

spherical core collapse

Conclusion
in most regimes close to results ofBoltzmann codes, but some limitations
existMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Open issues
We want to study
Where do the magnetic fields of young neutron stars come from?
I Under what conditions does the field amplification during a supernova
explosion lead to dynamically important fields?
II

How do the magnetic fields react back onto the flow?

But we are not aiming to answer
I

Does a particular core produce a robust ν-driven SN explosion and, if
so, how large is the explosion energy?

IHow does the dynamics depend on the details of the equation of state
and the neutrino physics?

I

What are the consequences for nucleosynthesis?

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Open issues
We want to study
Where do the magnetic fields of young neutron stars come from?
I Under what conditions does the field amplification during a supernova
explosion lead to dynamically important fields?
II

How do the magnetic fields react back onto the flow?

But we are not aiming to answer
I

Does a particular core produce a robust ν-driven SN explosion and, ifso, how large is the explosion energy?

IHow does the dynamics depend on the details of the equation of state
and the neutrino physics?

I

What are the consequences for nucleosynthesis?

SummaryMagnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Collapse of magnetised coresI

log |b| [cgs]progenitor star: 15 M , solar
metallicity

9

no rotation → restricts
possible field amplification
mechanismsI

poloidal (off-centre dipole)magnetic fields of fieldstrength ≤ 1012 GI

axisymmetric simulationsIspectral transport (16 energybins) of νe and ν̄e10.50

S [kb / baryon]
1202.505100005000z [km]I0−5000−10000−10000−5000

0x [km]

5000

10000

t = 0.00

Field structure and entropy at t = 0

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

SummaryOverview
δYe
−0.15

δYe

S

00.15

0612

−0.10

0.10

0816

−0.15

200

200

100

100

100

−100

−200

−1000x100200

vmax = 4.76×10+9214 ms

0.15

0918

0

−200
−200

t =S [kb / baryon]

0−100−200−200114 ms0yy0z [km]200−100t =δ Ye

S0−1000x

100

200

−200

−1000x [km]vmax = 5.47×10+9

without magnetic field:−→ PNS convection, SASI and convection in the hot-bubble region

100200Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Overview
log |b| [cgs]1012.50S
150log |b| [cgs]81610S
150log |b| [cgs]

81610

200

200

100

100

100

−100−200−1000x100200

vmax = 4.92×10+90.4000

S [kb / baryon]
1509180−200−200t =12.50−100−200−2000.3000

0yy0z [km]200−100t =12.50−1000x100

200−200−1000x [km]100200vmax = 5.21×10+9

amplification of a weak field by compression, stretching and folding of fieldlines, and unstable Alfvén waves

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Overview
log |b| [cgs]1113.25

S
15.500log |b| [cgs]7141113.25

S
15.500log |b| [cgs]9181113S [kb / baryon]
1501122

400
200

200

100100z [km]

0yy

200

0

−100

−100

−200

−200

0

−200

−200

−1000x

100200vmax = 6.00×10+9−200−1000x

100

200

−400
−400

−2000x [km]200

vmax = 6.00×10+9

feedback of a strong field: field resists bending, slows down motion acrossfield lines→ modifies the growth of SASI, convection→ development of very persistent large-scale patterns of upflows anddownflows, stronger shock expansion

400Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

SummaryField amplificationlog bRMS [erg]IImost pronounced amplification
occurs during collapse bycompression
for weak fields: significant
increase in the hot-bubble onlong time scales, caused bystretching and folding of field
lines

1413

12I

I

stronger initial fields experience
less amplification, indicating
dynamic feedback
max. fields are magnetar-like

0200

400t [ms]

600

800

time evolution of the (rescaled)
r.m.s. field strength: entire volume
and hot bubble

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Dynamic feedbacklog |b| [cgs]1013

S [kb / baryon]
16−10010300

200

feedback on the flow limit theamplification of stronger seedfields

strong fields lead to much larger
unstable modes and persistent
downflows→ strong shock expansion
I100z [km]

I

0

−100

−200

−300
−300

−200

−1000x [km]100200300

2d structure of a model with strong initial field

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

Summary

Dynamic feedback
0100200t [ms]30040050025feedback on the flow limit theamplification of stronger seed
fields

16.67

8.33
0101513.67
log |b|strong fields lead to much larger
unstable modes and persistent
downflows→ strong shock expansion
I10010r [km]Ientropyr [km]

1000

100

12.33

1000

110

100

200

300

400

500

t [ms]

angularly averaged profiles of thestrongest magnetised model as a function of time.

Magnetic fields in core collapse

Two-moment neutrino transport

Simulations of core collapse

SummaryI

the moment system with local closure is a good compromise between
accuracy and effortIwe study the interplay of neutrino transport, hydrodynamics andmagnetic fields, finding efficient amplification- compression- stretching by hydro instabilities
and dynamic feedbackInext steps: switch on pair processes; rotating models

Summary