Core collapse supernovae:
influence of magnetic
fields
Martin Obergaulinger, Pablo Cerdá-Durán, Ewald Müller
Max-Planck-Institut für Astrophysik
Magnetohydrodynamic
effects in supernova
infall R≈R0 /  EG≈EG0, Erot≈2Erot0 , Emag≈ Emag0
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(if collapse approx. homologous)
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poloidal circulation (due to centrifugal force)  ()~-2
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linear field amplification (toroidal field) by differential rotation
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Up to 2000 only a few
numerical studies:
LeBlanc & Wilson ('70)
Bisnovatyi-Kogan etal ('76)
Müller & Hillebrandt ('79)
Ohnishi ('83)
Symbalisty ('84)
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Heger etal '05: =0.05s-1, R0= 5x10-4
Meier, Epstein, Arnett & Schramm '76 observed WD fields:  < 10-10
M0
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two parameters determine
evolution:
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R0 = Erot0 / EG0
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and
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M0 = Emag0 / EG0
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model C (LeBlanc & Wilson '70):
7 Msun Fe-Ni core
R0 = 0.0025
⇔  = 0.42 rad/s
M0 = 0.00025 ⇔ B = 9.2 1011Gauss
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additional MHD processes of importance
- Thompson & Murray ('01)
(semi-analytic)
field amplification by rapid convective
overturn in PNS for ~10 s ; if
turbulent = magnetic stresses  Bequi ~ 1015G
- Wheeler etal ('00, '02)
(semi-analytic)
asymmetric CCSNe & MHD jets (like in AGNs);
field amplification by differential rotation
and possibly by - dynamo
- Akiyama etal ('03)
(1D HD)
importance of Magneto Rotational Instability
 strong MHD activity even for moderate
initial conditions: Bsat ~1015...1016 G
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- Tokyo group (2D/3D MHD)
Yamada, Kotake, Sawai, parameter studies with polytropic & microTakiwaki, Nishimura
physical EOS; neutrino leakage scheme;
2004-2008
B-field contribution to GW signal;
nonuniform & offset B-fields
MRI in action ?
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- Moiseenko etal ('06, '08)
2D MHD - MRI found?
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- Scheidegger etal ('08)
3D MHD
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- Mezzacappa etal ('08)
compressional folding by SASI
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2D MHD simulations: Obergaulinger, Aloy & Müller 2006
(TVD MHD code of Pen, Arras & Wong 2003
based on the relaxing TVD scheme of Jin & Xin 1995 )
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Initial models:
- rigidly & diff. rotating
polytropes (as in
Zwerger & Müller '97)
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- approx. rel. gravity
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- B-field from toroidal
current loop in equat.
plane (100≤r/km≤800)
central field strength
1010 – 1013 Gauss
(realist. B < 109 Gauss)
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Evolution of central density and rotation rate
non-magnetized slowly rotating Newtonian model bounces due to EOS
Obergaulinger, Aloy & Müller '06
central density
rotation rate
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----- 1010 Gauss
----- 1012 Gauss
----- 1013 Gauss
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Snapshot (49 ms post-bounce) of a model with a strong initial
B-field & rapid differential rotation --> collimated outflow
(Obergaulinger, Aloy &
Müller 2006)
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gas & magnetic pressure
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velocity & final velocity
(total energy -->
kinetic energy)
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Gravitational wave quadrupole amplitude
TOV (solid line) vs Newtonian (dashed line) gravity for a rapidly rotating model
Obergaulinger, Aloy & Müller '06
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---- 1010 Gauss
----- 1013 Gauss
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Gravitational wave spectrum of a rapidly rotating initial model
Obergaulinger, Aloy & Müller '06
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---- 1010 Gauss
----- 1013 Gauss
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enhancement of spectral power at low frequencies in strong
field model (right) is due to collimated outflow (jet)
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MHD studies have shown
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- efficient angular momentum transport by B-fields possible
--> affects structure & rotation rate of PNS
- conversion of rotational energy into thermal energy
--> additional (isotropic) pressure contribution
- weak, but realistic initial fields (B < 1011G) do neither change collapse
dynamics nor resulting GW signal
- strong initial fields (B ≥ 1012G) slow down core efficiently, change
dynamics qualitatively, modify GW signal, and cause bipolar, mildly
relativistic outflows
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Still waiting exploration: - MHD models with realistic microphysics
and neutrino-driven convection
- secular evolution ?
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- 3D effects, MHD instabilities (MRI) &
possible MHD dynamo
A closer look to
field amplification mechanisms
in core collapse supernovae
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compressional amplification
feeds off kinetic energy of
infall
emag ~ - b2/2 div v
amplifies field energy by a
factor of 100...1000 during
collapse
works irrespective of field
strength and geometry
13. III. 2008
MRI in core-collapse supernovae
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energy source: differential rotation
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requires poloidal seed field
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13. III. 2008
winding by differential rotation
creates & amplifies toroidal field
component
linear in time; time scale set by
rotational period
MRI in core-collapse supernovae
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turbulent dynamo
energy source: turbulent
kinetic energy
turbulence excited by
different effects
genuinely 3D effect
13. III. 2008
MRI in core-collapse supernovae
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The magneto-rotational instability (I)
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main & original application: accretion disks
(Balbus & Hawley 1991ff; based on Velikhov 1959 & Chandrasekhar 1960)
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MRI-driven turbulence can explain disk viscosity?
-viscosity (Shakura-Sunyaev); see, however, King et al. 2007
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large mean Maxwell stresses transport angular momentum
outward (dependence on hydrodynamic background state?)
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energy source is differential rotation
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local linear instability with exponential growth
13. III. 2008
MRI in core-collapse supernovae
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The magneto-rotational instability (I)
Properties of the MRI
- stability criteria from analysis of wave-like (WKB)
perturbations exp[i (k*r + t)]
- accretion disks (Balbus & Hawley, 1991 ff)
- weak fields |vA| << min { cS, |v| }
- incompressible gas (Boussinesq approx.)
- angular velocity constant on cylinders
- thin Keplerian disk and polytropic EOS
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The magneto-rotational instability (II)
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MRI in SN cores quite different from MRI in accretion disks
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results from accretion disks may not apply, because
- sub-Keplerian rotation
- general rotation law (,z)
- pressure forces are important (important global gradients)
- thermal stratification (entropy gradients!)
- different microphysics (e.g., radiative heating/cooling)
- dynamic background
===> necessity for a MRI
more
general
stability analysis
in core-collapse
supernovae
13. III. 2008
(Balbus 1995, Urpin 1996)
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Properties of the MRI
---> stability criterium: ℜ = ∂2 > 0
if criterium is violated, modes with wavenumbers
are unstable!
2
growth rate of the fastest growing mode (Balbus & Hawley '92)
independent of B; at wavenumbers close to the critical one
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Properties of the MRI
general dispersion relation (Balbus 1995, Urpin 1996,
Obergaulinger, Cerdá-Durán & Müller 2008
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Properties of the MRI
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convenient quantity
for stability analysis:
general dispersion relation
Obergaulinger, Cerdá-Durán
& Müller (2008)
modes with wavenumbers
smaller than the
critical wavenumber
are unstable and grow
Two branches of unstable modes (Urpin 1996):
Alfven modes:
C< 0
Buoyant modes: C < - 4
magneto-shear regime:
-4<C< 0
mixed regime:
-8<C<-4
magneto-convective regime:
C<-8
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Physical and technical problems
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dispersion relation imposes severe resolution problem
(very difficult to resolve the MRI properly)
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relevant physical length scales
101 km ... 103 km
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global scales:
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scale heights of physical quantities:
Hydrodynamics
13. III. 2008
MRI in core-collapse supernovae
1 km
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Physical and technical problems
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dispersion relation imposes severe resolution problem
impossible to resolve properly the MRI
relevant physical length scales
101 km ... 103 km
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global scales:
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scale heights of physical quantities:
1 km
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MRI wavelength
1 cm ... 10 m
magnetic field
13. III. 2008
MRI in core-collapse supernovae
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Physical and technical problems
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dispersion relation imposes severe resolution problem
impossible to resolve properly the MRI
relevant physical length scales
101 km ... 103 km
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global scales:
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scale heights of physical quantities:
1 km
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MRI wavelength
1 cm ... 10 m
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viscous and resistive scales
13. III. 2008
MRI in core-collapse supernovae
≪ 1 cm
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Physical and technical problems
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MRI requires 3d simulations
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numerical grid can cover a factor of 102...104 in length scales
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impossible to cover all scales appropriately
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where to place the numerical grid?
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global simulations
do not resolve MRI properly, but capture global dynamics
13. III. 2008
MRI in core-collapse supernovae
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Physical and technical problems
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MRI requires 3d simulations
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numerical grid can cover a factor of 102...104 in length scales
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impossible to cover all scales appropriately
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where to place the numerical grid?
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local simulations
(more or less) well resolved, but neglect global dynamics
13. III. 2008
MRI in core-collapse supernovae
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Global simulations
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simulate a large region of the core at a fairly coarse resolution
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2d axisymmetric or 3d
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include detailed physics
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use detailed initial models
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MRI found; study its implications for dynamics
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BUT: MRI is underresolved !
=> use stronger initial fields, relying on efficient amplification
13. III. 2008
MRI in core-collapse supernovae
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Global simulations
magnetic energy
13. III. 2008
MRI in core-collapse supernovae
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Local simulations
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simulate a small box with a fine resolution
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2d, 3d Cartesian / spherical / cylindrical
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simplified physics
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representative initial conditions
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choice of boundary conditions may be crucial
periodic (shearing box), reflecting, outflow
resolve the MRI for weak initial fields
13. III. 2008
MRI in core-collapse supernovae
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Local simulations
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problems: neglect of global dynamics may distort results
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large parameter space
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background models (rotation, stratification)
initial field strength and geometry
transport coefficients
grid size
resolution
unclear dependence of the saturation state on the
numerical parameters (e.g., grid size, boundary treatment)
(Umurhan & Regev, 2008)
13. III. 2008
MRI in core-collapse supernovae
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Semi-global simulations
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local simulations of MRI in SN cores (Obergaulinger etal '2008)
with shearing disk boundary conditions allowing for the inclusion of
global gradients (Klahr & Bodenheimer 2003)
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box (edge length 1km; located 15 km from center of star) taken from
an idealized model of a post-collapse core
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simplified EOS, no neutrino transport
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variation of the initial magnetic field strength and geometry
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new high-resolution finite-volume flux-conservative MHD code
[ TVD-PLM, MP5 or WENO4, with approx. Riemann solvers (MUSTA) and CT ]
13. III. 2008
MRI in core-collapse supernovae
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Semi-global simulations
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development of channel flows in
an axisymmetric model with an
initially uniform B-field
13. III. 2008
MRI in core-collapse supernovae
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Semi-global simulations
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MRI grows rapidly (within a fraction of a millisecond)
saturation depends on the field geometry, the grid, and the
dimensionality
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2d, uniform B-field: occurrence of coherent channel flows
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2d, vanishing net B-flux: turbulence
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3d: channel flows unstable against parasitic instabilities,
turbulence develops;
channels may reorganise spontaneously
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saturation level set by rotational equipartition (~ 1015 Gauss)
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Maxwell stresses transport
angular momentum efficiently
MRI in core-collapse supernovae
13. III. 2008
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Evolution of magnetic field strength in semi-global 3d models
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(1km)3, 1003
0 = 15.5 km
b0 = 2e13 G
t = 16 ms: channel
flows grow
0 = 1900 s-1
 = -1.25
t = 27 ms: channels
are disrupted
t = 43 ms: reappearance
of coherent flow
13. III. 2008
MRI in core-collapse supernovae
volume rendering of |B| (blue to green)
iso- surface
[   z]
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Conclusions
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simulation of MRI hindered by resolution requirements
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combination of global and local simulations suggested
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semi-global simulations
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confirm efficient amplification of magnetic stresses by
the MRI, turbulence, and angular momentum transport
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preliminary analysis leaves open several issues
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detailed dependence on initial conditions
possible dependence on the numerics
effects of neglected physics?
MRI studies shouldMRIbe
extended to GRMHD
in core-collapse supernovae
13. III. 2008