Gravitational waves from core collapse supernovae - LUTh

Gravitational waves from
core collapse supernovae
Ewald Müller
Max-Planck Institut für Astrophysik

Gravitational waves
(Einstein quadrupole formula)
h jk = 2G
c 4
time-dependent mass-energy quadrupole moment
in core collapse supernovae due to
- convection in proto-neutron star
- convection in neutrino heated hot bubble
- anisotropic neutrino emission
- any other non-radial instability (e.g. SASI, AAC)
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R
and due to rotation and magnetic fields
d 2 Q jk
dt 2
~ R s
R
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-20 *
R =1 km , v/c=0.1 , R=10kpc ---> h ~ 10 s
generically produced by any CCSN
* [ measuring the distance earth-sun with an accuracy of 1 nm ]
v 2
c 2

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● An
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important
technical
aspect !

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● Stability
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against pseudo
radial modes (collapse)
of self-gravitating
rotating configurations

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The effective adiabatic index

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● A
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brief history
the efforts to predict
GW signature
core collapse supernovae

Most studies aimed at studying the GW signature of core collapse
supernovae based on greatly simplified parameterized models
* spheroidal / ellipsoidal, one-zone models
(Saenz & Shapiro '81) �E/Mc 2 ~ 10 -6 ... 10 -4
* Newtonian ''realistic'' prompt explosion models, parameterized
non-equilibrium progenitor models
(Müller '82, Mönchmeyer etal '89) �E/Mc 2 ~ 10 -10 ... 10 -7
* Newtonian polytropes, parameter study of non-equilibrium
progenitor models
(Finn & Evans '90, Yamada & Sato '95, Zwerger & Müller '87,
Rampp, Müller & Ruffert '98 (3D) )

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Collapse dynamics & waveform types in rotational core collapse
(Newtonian gravity, axisymmetric models)
● Mönchmeyer et al., '91: 4 models with tabulated EOS
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● typical signal
strength &
frequency:
● h ~ 10 -20
at 10 kpc
● f ~ 500Hz -
1 kHz
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● Zwerger & Müller '97: 78 models with simple analytic EOS

* GR collapse(CFC), parameter study of equilibrium progenitor
models, simplified equation of state, no transport
(Dimmelmeier, Font & Müller '02 ; Shibata '03)
* microphysical equation of state, Newtonian gravity, some
treatment of weak interactions and crude neutrino ''transport''
(Fryer, Holz & Hughes '02, '04 ; Fryer etal '03 ;
Kotake, Yamada & Sato '03 ; Ott etal '03)
* simple equation of state, Newtonian gravity, MHD models
(Kotake etal '04 ; Yamada & Sawai '04 ; Obergaulinger etal '04)
---> �E GW < 10 -6 M sun c 2

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• GR collapse of rotating polytropes using CFC approximation
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(H. Dimmelmeier, J.A. Font & EM '02, '03)
• toroidal density stratification at bounce for an increasing rate and
degree of differential rotation

Rotational core bounce can occur at subnuclear densities!
Gravitational waves provide direct diagnostic tool
c e n tra l d e n s ity
w a ve a m p litu d e
bounce due to the stiffening of the equation
of state beyond nuclear matter density
c e n tra l d e n s ity
w a ve a m p litu d e
multiple bounce due to rotation
GRHD simulations of polytropes (Dimmelmeier, Font & Müller 2002)

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GW amplitude vs frequency for GR and Newtonian models
(Dimmelmeier, Font & Müller '02)
centrifugal bounce less likely to occur in GR
at 10 kpc

* presently most advanced microphysical study (concerning
the GW signature of core collapse supernovae)
(Müller, Rampp, Buras, Janka & Shoemaker '04)
- microphysical equation of state
- progenitor models from stellar evolutionary calculations
- detailed modelling of all relevant weak interaction processes
- multi-flavour Boltzmann neutrino transport
- effective relativistic gravitational potential
- Newtonian dynamics of axisymmetric flows

GWs from a non-rotating 11.2 M sol star
Models with detailed micro and transport physics
Müller, Rampp, Buras, Janka & Shoemaker '04

GWs from a rotating 15 M sol star
b o u n c e s ig n a l
Müller, Rampp, Buras, Janka & Shoemaker '04

GWs from simplified models of CCSNe:
- miss important physics (convection)
- initial models rotate too fast according to state-of-the-art
stellar evolution models and spin rates of young pulsars
GWs from elaborate models of CCSNe: LIGO II
- proto-neutron star convection (at least): ~ 10 kpc
- convection in non-rotating progenitors: ~ 15 kpc
both type of signals are generically produced by any CCSN
- state-of-the-art rotating progenitors: ~ 100 kpc
[ only bounce signal: ~ 5kpc ]

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● Towards
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● relativistic
core collapse simulations
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microphysics
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● neutrino
transport

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● most sophisticated simulations of GR rotational core collapse
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up to now include: (Ott, Dimmelmeier, et al., 06)
● - coupled relativistic gravity (BSSN, CFC) and GRHD
(Cactus/Carpet/Whisky & CoCoNut codes)
● - tabulated microphysical EOS (Shen et al., '98; Marek et al., '05)
● - Newtonian quadrupole formula for GW signal
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● - parameterized, approximate
treatment of deleptonization
(Liebendörfer '05: 1D!!)
● Y e ~ ρ and ρ trap
fine until shortly after bounce!

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Simple vs. microphysical EOS (Ott, Dimmelmeier, et al., 06)
● slow & (almost) uniform rotating progenitor („best“ stellar evolution)
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● simulations with microphysical EOS:
- influence of rotation on dynamics & GW signal less pronounced
● - no longer multiple centrifugal bounces & type-II GW signals
(for very rapid rotators: type-I collapse dynamics & GW signal)

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Detection prospects of GW from core collapse (to NS)
● - bounce signal of a galactic supernova detectable by current detectors
● - microphysical EOS: GW signal frequency range significantly narrower
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● low frequency
GW signals
(i.e. multiple
centrifugal
bounces)
are suppressed
in simulations
with GR &
a microphysical
EOS!
● (Ott, Dimmelmeier,
et al., 06)

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Cause of suppression: relativistic gravity?
● Newtonian study of the collapse of rotating polytropes (Zwerger & Mueller, '97)
repeated in relativistic gravity
(Dimmelmeier, Font & Muller, '02; Dimmelmeier et al., 05; Cerda et al., '05;
Shibata & Segikuchi, '05, '06)
● relativistics effects: deeper potential --> larger bounce densities, more
compact PNS
● less multiple centrifugal bounces (less type II GW signals)

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Cause of suppression: microphysical EOS?
● - neglect deleptonization and adopt adiabatic index for simplified
EOS according to microscopic EOS
● - simple EOS (piecewise polytropic + thermal part to mimic shocks)
● γ < 4/3 for ρ < ρ nuc = 2 10 14 g/cm 3
● γ = 2.5 for ρ > ρ nuc
● similar to microphysical EOS
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● - centrifugal bounce: (below ρ nuc )
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γ is always very close to 4/3
● ---> approximating microphysical EOS with simple EOS (with γ=1.32)
predicts correct collapse & GW signal type!

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Cause of suppression: deleptonization?
● - when including effects of deleptonization: no centrifugal multiple
bounce found! (neither in Newtonian nor relativistic gravity)
(Dimmelmeier et al., in prep.)
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Conclusions
Relativistic gravity important for explosion mechanism & GW signal
Its effects can be well modelled (for not too extreme models) by
means of an effective relativistic potential (during collapse to NS)
Models of (rotational) core collapse including a microphysical EOS,
some treatment of the core's deleptonization, and (approximate)
relativistic gravity show no multiple centrifugal bounce
Open questions: GR post-bounce evolution & GW signal
GR & magnetic fields (collapsars)
BH formation from core collapse

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● Additional
aspects
3D effects (angular momentum conservation during collapse)
fast rotating
axisymmetric
cores may
become triaxial
but
low density at
bounce
rotating polytropes (Rampp, Ruffert & Muller '97)

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● Additional
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aspects
asymmetries (��/� ~ 10%) in progenitor star due to oxygen &
silicon burning
4 s -1
& Arnett '98)
0.25 s -1
h = ~ 20% of amplitude of fast rotating core (Fryer, Holz & Hughes '04)
10 s -1
0 s -1

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● Relativistic
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core collapse simulations:
alternative approach

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Characteristic GR collapse of axisymmetric polytropes
Initial value problem of general relativity: Choice of foliation?
Cauchy/ADM
(boundary conditions?)
characteristic/light cone
(unambigous extraction of GW)

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Application: Spherical and non-spherical core collapse
(Siebel et al., 02) [polytropes, hybrid EOS (bounce + shock)]
Evolution of central density:
Increase by 4.5 orders of magnitude
spacetime
diagram
shock
formation &
propagation

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Gravitational waves at null infinity: Bondi news function
- gauge contributions must be considered
- shock propagation creates numerical noise
Work still in progress!
first time characteristic
2D numerical relativity
is applied to a matter
flow problem

Magnetohydrodynamic
effects in supernova

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Basic MHD processes in core collapse
(Meier, Epstein, Arnett & Schramm '76)
● infall R≈R 0 /� � E G ≈�E G0 , E rot ≈� 2 E rot0 , E mag ≈ �E mag0
(if collapse approx. homologous)
● poloidal circulation (due to centrifugal force) � �(�)~� -2
● 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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Meier, Epstein, Arnett & Schramm '76
● two parameters determine
evolution:
● � R0 = E rot0 / E G0
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● � M0 = E mag0 / E G0
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● model C (LeBlanc & Wilson '70):
● 7 M sun Fe-Ni core
● � R0 = 0.0025 ⇔ � = 0.42 rad/s
� M0 = 0.00025 ⇔ B = 9.2 10 11 Gauss
Heger etal '05: �=0.05s -1 , � R0 = 5x10 -4
observed WD fields: � M0 < 10 -10

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Other MHD processes
● - Thompson & Murray ('01) field amplification and transport by rapid
(semi-analytic) convective overturn in PNS for ~10 s ; if
turbulent = magnetic stresses � B equi ~ 10 15 G
● - Wheeler etal ('00, '02) asymmetric CCSNe & MHD jets (like in AGNs);
(semi-analytic) field amplification by differential rotation
and possibly by an �-� dynamo
● - Akiyama etal ('03) importance of MRI (exponential growth ~� -1 )
(1D HD) � strong MHD activity in PNS & behind
stalled shock; even for moderate initial
conditions: Bsat ~10 15 ...10 16 G ?
● - Sato's group
Yamada & Sawai ('04) parameter studies with polytropic & realistic
Kotake etal ('04a, '04b) EOS; neutrino leakage scheme;
Takiwaki etal ('04) field contribution to GW signal
Kotake etal ('05)
Sawai etal ('05) --> no MRI?
(2D MHD)

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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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– 2D MHD simulations: Obergaulinger, Aloy & Müller 2005
– (TVD MHD code of Pen, Arras & Wong 2003
based on the relaxing TVD scheme of Jin & Xin 1995 )
● - B-field from toroidal
current loop in equat.
plane (100≤r/km≤800)
● central field strength
10 10 – 10 13 Gauss
(realist. B < 10 9 Gauss)

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Evolution of central density and rotation rate:
non-magnetized Newtonian initial model bouncing due to EOS
(models A1B3G3-D3Mx, x∈ {10, 12, 13); Obergaulinger, Aloy & Müller '05)
central density rotation rate
● ----- 10 10 Gauss ----- 10 12 Gauss ----- 10 13 Gauss

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Evolution of central density and rotation rate:
non-magnetized Newtonian initial model bouncing due to rotation
(models A3B3G3-D3Mx, x∈ {10, 12, 13); Obergaulinger, Aloy & Müller '05)
central density rotation rate
● ----- 10 10 Gauss ----- 10 12 Gauss ----- 10 13 Gauss

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Gravitational wave quadrupole amplitude:
TOV (solid line) vs Newtonian (dashed line) models:
(A1B3G3-D3M1x; Obergaulinger, Aloy & Müller '06)
● 10 10 Gauss 10 13 Gauss

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Gravitational wave quadrupole amplitude:
TOV (solid line) vs Newtonian (dashed line) models:
(A3B3G3-D3M1x; Obergaulinger, Aloy & Müller '06)
● ---- 10 10 Gauss ----- 10 13 Gauss

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Gravitational wave spectra:
(A3B3G3-D3M1x-T; Obergaulinger, Aloy & Müller '06)
● ---- 10 10 Gauss ----- 10 13 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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Snapshot of model A3B3G5-D3M13 about 49 ms post-bounce
strong initial B field & rapid rotation --> collimated outflow
(Obergaulinger, Aloy & Müller
'05)
● gas & magnetic pressure
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● velocity & final velocity
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kinetic energy)

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● Conclusions from MHD studies up to now
● - weak (realistic!) initial fields (B < 10 11 G) do neither change
collapse dynamics nor resulting GW signal
● - strong initial fields (B ≥ 10 12 G)
---> - slow down core efficiently (even retrograde rotation occurs!)
- qualitatively different dynamical evolution & GW signal
- highly bipolar, mildly relativistic outflows
● - shape of GW signal reflects dynamical behavior of the model
(in particular the collimated outflow)
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● Still waiting exploration: - MHD models with realistic microphysics
and neutrino-driven convection
● - secular evolution ?
● - 3D effects, MHD instabilities (MRI) &
possible MHD dynamo