Non-Axisymmetric Dynamics and Magnetoacoustic Flux in Core Collapse

J. Craig Wheeler, Shizuka AkiyamaDepartment of Astronomy. University of Texas

Mitchell Symposium, April 13, 2006

Outline

I. Aspherical Supernovae: all core collapse supernovae are strongly aspherical, frequently approximately axisymmetric. The core-collapse explosion machine is aspherical.

II. Core collapse: jet-induced supernovae can provide the requisite asymmetry.Magneto-Rotational Instability gives inevitable production of large toroidal magnetic fields; no without B.

III. Crucial role of magnetic fields, non-axisymmetric instabilities, and of the de-leptonization phase. Supernovae make a loud, magnetoacoustic, noise.

IV. Conclusions

I. BACKGROUND

We know some supernovae leave behind pulsars - rotating,magnetic neutron stars. Are the rotation and magnetic fieldimportant for the supernova explosion?

A Crab-like field of 1012 Gauss and a Crab-like rotationof 33 ms are dynamically unimportant.

BUT

The initial field and rotation from a pulsar astronomer’spoint of view are the final field and rotation from a supernova dynamicists point of view.

What were the field and rotation during collapse and werethey dynamically important?

SN 1987ASINSKirshner, et al.

Jet

Counter jetCompact object

Crab33 ms pulsaraxis/torus structureL ~ 5x1037 erg s-1

Proper motion(Caraveo & Mignani 1999)

Kepler SN 1604

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NASA Animation: Jet Erupting Through Star

Jet-induced Core Collapse Supernovae

3D hydrodynamical calculation of jet-induced supernova (Khokhlov et al. 1999). Sufficiently strong jets can explodethe supernova (without neutrinos, in principle) and impartappropriately large asymmetries.

jet “nickel”prolate

torus, O, Ca,oblate

Axis/torusStructure

Up/Down Asymmetry=> “kick”

II. Aspherical Core Collapse

All core collapse events are aspherical.

Jets work in principle!

Role for rotation/magnetic fields.

Magneto-rotational instability - MRI (Akiyama et al. 2003).

Ultimate problem is 3-D with rotation, magnetic fields and neutrino transport - we’ve known it all along, but polarization demands it.

Khokhlov et al. 1999

FasterRotation

SlowerRotation

Direction of Angular Momentum Transport

Stretching Amplifies

B-field

S. Akiyama

Unstable if angular velocity decreases outward

Magnetorotational Instability

Field Amplification by the MRI (2D)

Balbus & Hawley 1998Balbus & Hawley 1998

Stream flow becomes turbulent

BZ <B> = 0

Criterion for instability to the MRI is a negative gradient in angular velocity, as opposed to a negative gradient in angular momentum for dynamical instability.

Specifically: N2 + ∂ 2/∂ ln r < 0 N = Brunt-Väisälä fequency (convective stability stabilizes).

Saturation field given approximately by: vAlfvén ~ r ; B2 ~ 4 r2 2

For formal fastest growing mode (Balbus & Hawley (1998):

For sub-Keplerian post-collapse rotation:

Find fields ~ 1015 - 1016 Gauss in a few tens of milliseconds

Characteristic (Blandford-Payne) MHD luminosity

LMHD = B2R3 /2 ~ 3x1052 erg s-1 B162 RNS,6

3 (PNS/10 msec)-1 ~ 1051 - 1052 erg/s

Erot = 1/2 INS NS2 ~ 1.6x1050 erg MNS RNS,6

2 (PNS/10 msec)-2

Initial Fe Core Solid Body Rotation = 0.2 s-1

~1015 GaussStable Unstable

Large toroidal fields within 10s of milliseconds after bounce (Akiyama, et al. 2003)

IMPLICATIONS

Core collapse generically creates differential rotation even if the initial iron core is in solid body rotation.

The MRI is unavoidable in the collapse ambience (supernova or GRB, neutron star or black hole).

Cannot have rotating core collapse in the absence of magnetic fields.No without B.

The magnetic field generated by the MRI must be included in any self-consistent collapse calculation - but hard, numerical resolution.

Relevant magnetohydrodynamics - large magnetic fields generated internally, primarily toroidal, not the product of twisting of external field lines

Open Issues

Magnetic effects in rotating progenitor star Dynamos, field strength Effect on equation of state Effect on neutrino transport Effect on structure, evolution of proto-neutron star Effect on jet formation Relevance to GRB, “hypernovae”

III. Non-Monotonic and Non-Axisymmetric Behavior

(Akiyama & Wheeler 2005, 2006)

Non-Monotonic Response of Proto-Neutron Star to Initial Iron Core Rotation Rate.

For modest rotation, the PNS will rotate faster as the iron core does.

Above a critical initial rotation rate, centrifugal support will lead to a less compact PNS, bounce at sub-nuclear density, and slower rotation.

Peak PNS rotation (~ 4000 rad s-1) and MRI-induced magnetic field (~ 1017 Gauss) for initial iron-core rotation rate of ~ 4 rad s-1.

Log

B

Black Hole - Neutron Star - Black Hole??

Magnetar?

(toroidal, not dipole)

Unstable to Variety of Non-Axisymmetric Modes

Rotation as “Response Filter” at Fixed Mass Iron Core(Akiyama & Wheeler 2005)

How Does All This Work for EC Collapse?

Non-Axisymmetric Instabilities(Akiyama & Wheeler 2006)

Any rotation with ratio of rotational energy to binding energyT/|W| > 0.01 will be subject to non-axisymmetric instability (Andersson 1998; Owen et al. 1998; Ott et al. 2005).

Thresholds, growth rate, saturation, depend on degree of differential rotation (Tohline & Hachisu 1990; Rampp et al. 1998; Centrella et al. 2001; Imamura & Durisen 2004; Ou et al. 2004; Shibata & Sekiguchi 2005) and will be affected by magnetic fields (Rezzolla, Lamb & Shapiro 2000, 2001a,b).

Dynamic instability to bar-like mode for T/|W| as low as ~ 0.2.Criterion T/|W| > 0.27 is neither necessary nor sufficient for dynamical instability (Shibata & Sekiguchi 2005).

Criterion for secular bar-mode instability remains at T/|W| ~ 0.14

Inside the Supernova

Most work on non-axisymmetric instabilities considers the neutron star to be in complete isolation and ignores the magnetic field; both are potentially important.

MRI will grow magnetic field faster than all but dynamical bar-modes.

Non-axisymmetric instabilities will usually occur in a magnetized medium.

Density distribution near region of peak shear and magnetic field (boundary of homologous core) is approximately independent of whether supernova has been successful or not in the interval 100 ms to 1 s after bounce.

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Ott, Burrows, et al. 2005

Rotating, 3D, post collapse behavior

Unstable m = 1 single spiral arm mode

Outward transport of angular momentum, acoustic flux

No standing shock, external environment

Magnetoacoustic Luminosity, Damping

Instabilities will perturb the magnetic field and generate fast magnetosonic waves - NOISE, damp rotation.

Lmhd ~ 6x1051 erg s-1 R64 12 cs,9 3

2 (/)2/3 f

Dissipation time:

mhd ~ 30 ms M33 R6-2 12

-1 cs,9-1 (/)-2/3 f-1

Bar-like mode: / ~ 1, filling factor f ~ 1

Other modes, one-arm spiral, r-modes, / ~ 10-2 f ~ ??

In general, mhd << de-leptonization ~ 1 s

De-Leptonization Phase

Proto-neutron star will radiate binding energy in neutrinos, contract, spin up.

de-leptonization ~ 1 s; less than time for blast wave to propagate out of C/O, He core.

De-leptonization does not occur in the vacuum of space, but takes place within the matter-filled environment in the center of the supernova, whether it is in the process of exploding or not.

The tendency for the de-leptonizing PNS to spin up will render it broadly susceptible to non-axisymmetric instabilities, the production of magnetoacoustic luminosity, and dissipation of rotation.

Evolution depends on strength of that dissipation; mhd << de-leptonization ~ 1 s.

De-Leptonization Phase

For constant angular momentum, J:

T/|W| ~ (J2/2I)/(GM2/R) ~ J2/2GM3R

T/|W| R-1

T R-2

For T/|W| = constant:

T R-1

J R-1/2

Akiyama and Wheeler (2006)

Non-axisymmetric modes and magnetoacoustic dissipation may trigger secular bar mode, stronger dissipation, evolution along locus T/W ~ 0.14, then final spin down.

Proto-neutron starDe-leptonized neutron star

J ~ constT/W ~ const

Low Rotation

Acoustic Instability (Burrows et al. 2006)

Spherical Accretion Shock Instability SASI

(Blondin et al., Foglizzo, Mezzacappa talk)

Hints that sufficient rotation damps this instability.

Modest Rotation 0.01 < T/W < 0.14

Non-axisymmetric instabilities NAXI

One-arm spirals m = 1

Resonant cavity with energy exchange through co-rotation point?

Minimum of vortensity ( x v)/, or ( x v)/ S2/ function of entropy

Rossby vortices within the (proto-)neutron star (Lovelace et al. 1999, Li et al. 2000, 2001, Ou & Tohline 2006).

Many numerical simulations use polytropic equation of state, suppress dependence of real neutron stars on entropy distribution.

Faster rotation, co-rotation point move out to between surface of neutron star and standing shock (Foglizzo), “cavity” becomes surface of (proto-)neutron star and standing shock.

High Rotation T/W > 0.14

Secular/dynamical bar modes

IV. Conclusions

The Proto-Neutron Star will be a strong, magnetoacoustic wave-generating engine!!

Supernovae make a loud noise!

Major implications for supernova physics

Major implications for pulsar fields, spins

Major implications for gravity-wave emission

JETS???

Film - scientific premise related to high energy density lasers.

Available at:

www.thekroneexperiment.com

Conclusions

All core collapse explosions are significantly polarized, asymmetric. Dynamics, radiative processes (photons, neutrinos) are asymmetric. Account of asymmetry must be made in analysis.

Core collapse is an intrinsically shearing environment. Subject to MRI. Rotation and strong magnetic fields are intrinsic to the process. True for either neutron stars or black holes, SN or GRB.

The proto-neutron star will respond non-monotonically to the iron core rotation rate.

Non-axisymmetric rotational instabilities will generate significant MHD luminosity, supplement and shape the SN, perhaps create a GRB.

Spectropolarimetry

Systematic differences between Type Ia thermonuclear explosions and core collapse supernovae (Wang et al. 1996).

All core collapse supernovae show significant polarization, ~ 1%,requires distortion axis ratios of ~ 2 to 1 in free expansion.

Core collapse polarization tends to be larger at later times when see deeper in and larger when outer hydrogen envelope is less when see deeper in, both imply it is the machinery, the core collapse mechanism itself that is strongly asymmetric (Wang et al. 1996, 2001; Leonard et al. 2001). The explosion is often (but not always) substantially bi-polar (Wang et al. 2001, 2003).

Type Ia show low continuum polarization, but dramatic line polarization before maximum, decreasing to zero after maximum (Wang et al. 2003, 2005).

Non-axisymmetric, De-leptonizing, MHD, Neutron Star Model for Long GRBs

(Wheeler & Akiyama 2006)

Implication of evolution along locus T/|W| ~ constant ~ 0.14.

T ~ (Rpns/Rns -1)Tpns ~ 4 Tpns ~ 0.6 |Wpns| ~ 6x1052 erg

Lmhd ~ 1052 erg s-1 M33R6 12 cs,9 (/)2/3

Surplus of energy, power: majority may go into heating PNS, slowing contraction; only fraction (~1%) needs to go into propagating MHD waves, jets, etc. to make GRB.

Burrows et al. 2006 Unipolar, Sonic-driven explosion (astro-ph/0510687)

Important new perspective, but need 3D for non-axisymmetric instabilities, magnetic fields for MRI, MHD, magnetoacoustic phenomena.

0.10

0.20

T/|W

|

R (10 km)

FORBIDDEN REGION

L = const

T ~ R-2

T ~ R-1

L ~ R-1/2

1 5

0.14

0.27

0.30

Minimum locus to form bar

0.10

0.20

T/|W

|

R (10 km)

T ~ R-1

L ~ R-1/2

1 5

0.14

0.27

0.30Non-axisymmetric, De-leptonizing, MHD, Neutron Star Model for Long GRBs (Wheeler & Akiyama 2006)

Any PNS born “fast” will spin down to T/|W| ~ 0.14, barely stable to secular bar mode.

Contraction will tend to spin up, trigger bar mode, spin down. Result is contraction along locus of secular bar instability, T/|W| ~ 0.14.

Final spin-down “tail”

I. Systematic Spectropolarimetry: New Tool, New Insights

Cannot “see” shape of distant supernovaSpectropolarimetry yields wavelength-dependent information on the shape of the photosphere and line-forming regions

I E2, polarization is a “quasivector,” 0o = 180o (not 360o)

Measure Stokes Vectors:

I = I0 + I90; + Q = I0 - I90; - U = I45 - I-45; -

P = (Q2/I2 + U2/I2)1/2 = (q2 + u2) 1/2 ; = 1/2 tan-1(u/q)

P = Q = U = 0: intensity the same in orthogonal directions,photosphere is circularly symmetric, supernova is sphericallysymmetric (or special viewing angle)

P, Q, U ≠ 0: intensity different in orthogonal directions,photosphere is not circularly symmetric, supernova is asymmetric

Spectropolarimetry

Systematic differences between Type Ia thermonuclear explosions and core collapse supernovae (Wang et al. 1996).

All core collapse supernovae show significant polarization, ~ 1%,requires distortion axis ratios of ~ 2 to 1 in free expansion.

Core collapse polarization tends to be larger at later times when see deeper in and larger when outer hydrogen envelope is less when see deeper in, both imply it is the machinery, the core collapse mechanism itself that is strongly asymmetric (Wang et al. 1996, 2001; Leonard et al. 2001). The explosion is often (but not always) substantially bi-polar (Wang et al. 2001, 2003).

The explosion is often (but not always) substantially axisymmetric (Wang et al. 2001, 2003):

Classic Type II Plateau SN 1999em

Sometimes there are substantial departures (Wang et al. 2003):

Type Ic “hypernova” SN 2002ap, continuum, oxygen, and calcium all showed different orientations.

Evidence for bi-polar nature: Type IIP 1999em

New techniquesto determineinterstellar polarization andnature of dust(Wang et al.(2001), and to analyzepolarization in terms of principleaxes in Q,U plane(Wang et al. 2003a)

Single dominantaxis in Q,U plane

Experimental Setup:

Stellar Hohlrum

Collapse central core of iron generates a natural cavity surrounded by quasi-spherical container composed of layers of silicon, oxygen, and carbon.

Collapse generates 1053 erg of energy, primarily in the form of neutrinos, perhaps 1% of which is deposited in the cavity along with kinetic energy, magnetoacoustic flux, Poynting .flux.

Diagnostic technique: tomography by time-dependent spectropolarimetry.

Interesting Characteristic:

Natural explanation for time scale of long GRB: de-leptonization ~ 10 s

Also time for shock to hit surface of Type Ib/c supernova progenitor

Final period after contraction with T/|W| ~ 0.14 and I ~ 2/5 MR2:

~ (5 T/|W| G M Rns-3)1/2

P ~ 9x10-4 s M33-1/2 R6

3/2

Potential test to differentiate from black hole model?