The document discusses numerical relativity and simulations of core-collapse supernovae. It provides background on extreme astrophysical phenomena involving strong gravity and relativistic dynamics that require general relativity. It describes the 3+1 formalism used to evolve Einstein's equations numerically and challenges such as instabilities and gauge choices. Core-collapse supernovae involve gravitational collapse, core bounce, and reviving the stalled shock, tapping into the gravitational potential energy released. Fully modeling these explosions requires solving Einstein's equations coupled to hydrodynamics and neutrino transport.
10. Numerical Relativity vs. Newtonian Simulations
C. D. Ott @ Tarusa 2016 10
G”â«
= R”⫠1
2
Rg”â«
Einstein Tensor
Ricci Tensor Ricci Scalar R = g”â«R”â«
”⫠=
1
2
g âą
(gâ«âą,” + gâąÂ”,â« g”â«,âą)
R”⫠= â”
”â«,â”
â”
”â”,â« + â”
”⫠â”
â”
” â«â”
, â
@
@x” R”⫠= g”â”gâ« Râ”
(connection coefficients; Christoffel symbols)
-> Einstein equations: 2nd derivatives of the metric in space and time
-> similar to (inhomogeneous) wave equation:
@2
@t2
U c2 @2
@x2
U = T
-> Gravitational waves!
-> Einstein equations can be written in
hyperbolic form! (time-evolution equations)
16. Practical Numerical Relativity
C. D. Ott @ Tarusa 2016 16
Have not yet specified gauge conditions: Anything goes?
âą GR dynamics will twist, squeeze, stretch
coordinates.
âą GR can develop coordinate singularities
and physical singularities.
âą For numerically stable evolution, must
avoid singularities and control
coordinate distortion.
NASA
âą ADM form of the Einstein equations is unstable in 2D/3D!
-> well-posednessproblems (-> see literature; e.g., Kidder+2001).
-> small errors in constraints get amplified exponentially over time!
âą Spherical symmetry (1D):
-> no radiative degrees of freedom, fully constrained evolution.
-> ADM with simple gauge choices: no problem.
17. C. D. Ott @ Tarusa 2016 17
Practical Numerical RelativityKey issues:
âą Initial conditions must satisfy Einstein equations.
âą No unique way to formulate evolution equations.
âą Gauge freedom â how choose gauge conditions?
âą Need combination of evolution equations + gauges that yield
to numerically stable simulations.
BSSN Formulation
Generalized Harmonic Formulation
Nakamura+87, Shibata & Nakamura 95, Baumgarte & Shapiro 99
Friedrich 85, Pretorius 05, Lindblom+ 06
âą Conformal-traceless reformulation of Arnowitt-Deser-Misner 59, York 79.
âą Additional evolution equations, conditionally strongly hyperbolic.
âą Sensitive to gauge choice; good gauges known.
âą Most widely used evolution system today.
âą Choice of coordinates so that evolution equations
wave-equation like. Symmetric hyperbolic.
âą Sensitive to gauge choices, horizon boundary conditions.
âą Used primarily by Caltech/Cornell SXS code SpEC.
18. Numerical Implementation
C. D. Ott @ Tarusa 2016 18
âą Most common:
high-order finite difference approximation
(typically 4th-order in space & time).
âą Powerful alternatives:
Spectral methods, Discontinuous Galerkin Finite Elements.
d
dt
L(q) = RHS
âą Common approach: Method of Lines
Treat problem as semi-discrete; discretize in space, then treat as
ODE, integrate in time via Runge-Kutta(or similar).
Provides for high-order coupling with additional physics
(hydrodynamics/MHD, radiation).
19. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 19
Mösta+14
Löffler+12
âą Collection of open-source software components for the
simulation and analysis of general-relativistic
astrophysical systems.
âą Supported by US National Science Foundation.
~110 users, 60 groups; ~10 active maintainers.
http://einsteintoolkit.org
20. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 20
Mösta+14
Löffler+12
âą Collection of open-source software components for the
simulation and analysis of general-relativistic
astrophysical systems.
âą Supported by US National Science Foundation.
~110 users, 60 groups; ~10 active maintainers.
http://einsteintoolkit.org
âą Cactus (framework), Carpet (adaptive mesh refinement)
âą GRHydro â GRMHD solver
âą McLachlan â BSSN/Z4c spacetime solvers.
(code auto-generated based on Mathematica script, GPU-enabled)
âą Initial data solvers / readers.
âą Analysis tools (wave extraction, horizon finders, etc.)
âą Visualization via VisIt (http://visit.llnl.gov)
Available Components:
21. The Einstein Toolkit
C. D. Ott @ Tarusa 2016 21
Mösta+14
Löffler+12
âą Regular releases of stable code versions.
Most recent: âBraheâ release, June 2016
âą Support via mailing list and weekly open conference calls.
âą Working examples for BH mergers, NS mergers, isolated
NSs, rotating, magnetized core collapse (see also arXiv:1305.5299).
Simulate a binary black hole
merger on your laptop!
27. 1010 1011 1012 1013 1014
1.0
1.5
2.0
2.5
3.0
Density (g/cm3)
AdiabaticIndexG
s = 1.2 kB/baryon
Ye = 0.3
P â Kâą
âStiffeningâ of the Nuclear EOS
27
âCore Bounceâ
C. D. Ott @ Tarusa 2016
Schematic
nuclear force
potential
=
d ln P
d ln âą
ârepulsive coreâ
32. Core-Collapse Supernova Energetics
C. D. Ott @ Tarusa 2016 32
âą Collapse to a neutron star: âŒ3 x 1053 erg = 300 [B]ethe
gravitational energy (â0.15 MSunc2).
-> Any explosion mechanism must tap this reservoir.
âą âŒ1051 erg = 1 B kinetic and internal energy of the ejecta.
(Extreme cases: 10B; âhypernovaâ)
âą âŒ 90 - 99% of the energy is radiated in neutrinos on O(10)s
-> Strong evidence from SN 1987A neutrino observations.
âą If spinning with few ms spin period, proto-NS has
âŒ1052 erg = 10 B in spin energy.
35. Neutrino Mechanism: Heating
C. D. Ott @ Tarusa 2016
Ott+ â08
35
ÂŻâ«e + p ! n + e+
â«e + n ! p + e
Cooling:
Heating via
charged-current
absorption:
Bethe & Wilson â85; also see: Janka â01, Janka+ â07
30 km 60 km 120 km 240 km
Q+
â« /
â§
1
Fâ«
Lâ«r 2
hâ2
â«i
Neutrino radiation field:
, T9
45. Resolution Comparison
C. D. Ott @ Tarusa 2016 45
(Radice+16)
dΞ,dÏ= 1.8°
dr = 3.8 km
dΞ,dÏ= 0.9°
dr = 1.9 km
dΞ,dÏ= 0.45°
dr = 0.9 km
dΞ,dÏ= 0.3°
dr = 0.64 km
âą semi-global simulations
of neutrino-driven
turbulence.
(typical resolution of
3D rad-hydro sims)
47. Summary of 2D & 3D Neutrino-Driven CCSNe
C. D. Ott @ Tarusa 2016 47els s27fheat1.00 (left column), s27fheat1.05 (center column), and s27fheat1
âą More efficient neutrino heating,
turbulent ram pressure.
âą 2D simulations explode but canât be
trusted (2D turbulence is wrong).
Explosions too weak?
But see Bruenn+16.
âą 3D simulations:
Much must be improved:
(1) Resolution
(2) Treatment of neutrino transport
(3) Treatment of gravity in
many codes
Ott+13
48. Magnetorotational Explosions
48C. D. Ott @ Tarusa 2016
he iron-core
Connor & Ott
ChiefïŹ 2006;
how an anti-
2007). The
s for rate and
s in massive
hi et al. 2005
ally symmet-
s code GR1D
d through a
equation â
rotation rel-
account for
etry nor any
e equation of
racterized by
er referred to
1 10 100 1000 10000
Radius [105
cm]
0.001
0.01
0.1
1
10
100
1000
10000
âŠ(r)[rads-1
]
12TJ
16SN
16OG
16TI
35OC
preSN
bounce
Figure 1. Angular velocity âŠ(r) versus radius r at both the pre-SN stage
(dashed lines) and at core bounce (solid lines) for selected models of Woosley
& Heger (2006). The inner homologously collapsing core maintains its initial
uniform rotation throughout collapse.
âą Core: x 1000 spin-up
âą Differential rotation -> reservoir of free energy.
âą Spin energy tapped by magnetorotational instability (MRI)?
Dessart, OâConnor, Ott â12
49. Magnetorotational Mechanism
49C. D. Ott @ Tarusa 2016
[LeBlanc & Wilson â70, Bisnovatyi-Kogan â70 & â74, Meier+76,
Ardeljan+â05, Moiseenko+â06, Burrows+â07, Bisnovatyi-Kogan+â08,
Takiwaki & Kotake â11, Winteler+ 12, Mösta+14,15]
Rapid Rotation + B-field amplification to > 1015 G
(need magnetorotationalinstability [MRI])
2D: Energetic âbipolarâ explosions.
Results in ms-period âproto-magnetar.â
-> connection to GRBs, SuperluminousSNe?
Burrows+â07
Problem: Need high core spin;
only in very few progenitor stars?
MHD stresses lead to outflows.
50. A Note on Magnetic Field Amplification
C. D. Ott @ Tarusa 2016 50
âą Precollapse magnetic field in the core?
Best observational information:
White Dwarf B-fields, max ~108 â 109 G
âą Amplification processes:
(1) flux compression
(2) linear winding (poloidal->toroidal)
(3) magnetorotational instability + dynamo
Useful estimates in
Shibata+06 &
Burrows+07
Example calculation for flux compression:
â BR2
= const. ! B /
1
R2
BPNS = BIC
â
R2
IC
R2
PNS
â
RIC â 1500 km RPNS â 30 km
BPNS â 2500BIC â 2.5 â„ 1011
1012
G
-> Flux compression alone cannot produce 1015 G magnetic field!
Winding gives another x 10. Need the MRI.
54. What is happening here?
C. D. Ott @ Tarusa 2016 54
Mösta+14, ApJL
âą B-field near proto-NS: Btor >> Bz
âą Unstable to MHD screw-pinch kink instability.
âą Similar to situation in Tokamak fusion reactors!
Braithwaite+ â06
Sherwood
Richers
Philipp Mösta
Credit: Moser & Bellan, CaltechSarff+13
56. Summary
C. D. Ott @ Tarusa 2016 56
âą Core-Collapse Supernovae are fundamentally 3D:
turbulence, magnetic fields
âą Neutrino & Magnetorotational Mechanism:
Possible solutions to the Supernova Problem.
â Neutrino mechanism may be too weak
(missing neutrino physics?).
â Magnetorotationalmechanism needs fast core
rotation, but stellar evolution predicts slow rotation.
âą 3D simulations have made great progress,
but no final answers yet.
Much work ahead!
wikipedia.org/wiki/Magnetar
58. Can this work at all?
C. D. Ott @ Tarusa 2016 58
âą Simulations of the magnetorotationalmechanism assume:
MRI works + large-scale field created by dynamo.
âą So far impossible to resolve
fastest-growing MRI mode in
global 3D simulations.
âą Unstable regions (roughly):
âą In this simulation:
fastest growing mode
λ ~ 1 km.
dark blue: most MRI unstable
Mösta+15, Nature
d ln âŠ
dr
< 0
60. 0 5 10
1014
1015
1016
t tmap [ms]
Bf[G]
Maximum in
equatorial layer
500 m
200 m
Bf = 4.0 · 1014 · e(t tmap)/t
, t = 0.5 ms
100 m
50 m
100 m
50 m
60
Local Magnetic Field Saturation
âą Initial exponential
growth resolved with
100m/50m
simulations.
âą Saturated turbulent
state within 5 ms.
C. D. Ott @ Tarusa 2016
Mösta+15, Nature
64. âą Rapidly spinning, magnetized proto-NS.
âą Global simulation in quadrant symmetry:
70 km x 70 km x 140 km box
âą Resolutions:500 m/200 m/100 m/50 m
âą hot nuclear eq. of state, neutrinos, fixed gravity, GRMHD.
âą Simulations on 130,000 CPU cores on NSF Blue Waters,
simulate for 10-20 ms.
64
Simulation Setup
âą Does the MRI efficiently build up dynamically relevant field?
âą Saturation field strength? Global field structure?
Key questions:
C. D. Ott @ Tarusa 2016
Mösta+15, Nature
67. C. D. Ott @ Tarusa 2016
67
Kolmogorov Turbulence
log E(k)
/ k 5/3
inertial
range
dissipation
range
(large spatial scale) (small spatial scale)
(Fourier-space wave number)
log k
large eddies -----------------------> small eddies
Rij = vi vj
68. C. D. Ott @ Tarusa 2016
68
Turbulent Cascade: 2D vs. 3D
and large, high-entropy bubbles emerge that push the shock outward. The explosi
convection in our simulations is very similar to that of Ott et al. (2013).
100
101
102
`
1023
1024
1025
1026
E`
r = 125 km, tpb = 150 ms
` 1
` 5/3
` 3
s15 0.95 2D
s15 1.00 2D
s15 1.00 3D
s15 1.05 3D
1
Couch & OâConnor 14
see also: Dolence+13, Hanke+12,13, Abdikamalov+â15, Radice+15ab
2D
3D
âą 2D: wrong; turbulent cascade unphysical.
âą 3D: physical; more power at small scales, less
on large scales -> harder to explode!
69. C. D. Ott @ Tarusa 2016
69
Kolmogorov Turbulence
log E(k)
/ k 5/3
inertial
range
dissipation
range
(large spatial scale) (small spatial scale)
(Fourier-space wave number)
log k
large eddies -----------------------> small eddies
Rij = vi vj
Sensitivity to
kinetic energy flux!
-> sensitivity to resolution
70. C. D. Ott @ Tarusa 2016 70
Turbulent Kinetic Energy Spectrum
(Radice+16)
âcompensatedâ spectrum
Core-collapse supernova turbulence obeys Kolmogorov scaling!
But: Global simulations at necessary resolutioncurrently impossible!
Way forward? -> Subgrid modeling of neutrino-driven turbulence?