Core-Collapse Supernovae: Magnetorotational Explosions and Jet

Formation

G.S.Bisnovatyi-Kogan,

JINR and Space Research Institute, Moscow

Dubna, August 21, 2006

.

1. Presupernovae 2. Development of SN theory 3. Magnetorotational mechanism of explosion 4. 2-D MHD: Numerical method. 5. Core collapse and formation of rapidly rotating neutron

star. 6. Magnetorotational supernova explosion 7. Magnetorotational instability 8. Jet formation in magnetorotational explosion 9. Mirror symmetry breaking: Rapidly moving pulsars.

Content

Supernova is one of the most powerful explosion in the Universe, energy (radiation and kinetic) about 10^51 egr

End of the evolution of massive stars, with initial mass more than

8 Solar mass.

Tracks in HR diagram of a representative selection of stars from the main sequence till the

end of the evolution.

Iben (1985)

Explosion mechanism.

1. Thermonuclear explosion of C-O degenerate core (SN Ia)

2. Core collapse and formation of a neutron star, gravitational energy release 6 10 erg, carried away by

neutrino (SN II, SN Ib,c)

53

W.Baade and F.Zwicky, Phys.Rev., 1934, 45, 138 (Jan. 15)

The Hydrodynamic Behavior of Supernovae Explosions

Astrophysical Journal, vol. 143, p.626 (1966)

S.Colgate, R.White

The behavior of a massive star during its final catastrophic stages of evolution has been investigated theoretically, with particular emphasis on the effect of electron-type neutrino interactions. The methods of numerical hydrodynamics, with coupled energy transfer in the diffusion approximation, were used. In this respect, this investigation differs from the work of Colgate and White (1964) in which a "neutrino deposition" approximation procedure was used. Gravitational collapse initiated by electron capture and by thermal disintegration of nuclei in the stellar center is examined, and the subsequent behavior does not depend sensitively upon which process causes the collapse. As the density and temperature of the collapsing stellar core increase, the material becomes opaque to electron-type neutrinos and energy is transfered by these neutrinos to regions of the star less tightly bound by gravity. Ejection of the outer layers can result. This phenomenon has been identified with supernovae. Uncertainty concerning the equation of state of a hot, dense nucleon gas causes uncertainty in the temperature of the collapsing matter. This affects the rate of energy transfer by electron-type neutrinos and the rate of energy lost to the star by muon-type neutrinos. The effects of general relativity do not appear to become important in the core until after the ejection of the outer layers.

Gravitational collapse and weak interactions D. Arnett

Canadian Journal of Physics, 44, Vol. 44, p. 2553-2594 (1966)

Dynamics of supernova explosion

Nauchnye Informatsii, Vol. 13, p.3-91 (1969)

Magnetorotational explosion (MRE): transformation of the rotational energy of the neutron star into explosion energy by

means of the magnetic field in core collapse SN

Transformation of the neutrino energy into kinetic one - ???

In a simple 1-D model neurino deposition cannot give enough energy to matter (heating) for SN explosion

Neutrino convection leads to emission of higher energy neutrino, may transfer more energy into heating

Results are still controversial

Most of supernova explosions and ejections are not spherically symmetrical. A lot of stars are rotating and have magnetic fields.Often we can see one-side ejections.

Magnetorotational mechanism: transforms rotational energy of the star to the explosion energy.

In the case of the differential rotation the rotational energy can be transformed to the explosion energy by magnetic fields.

.

The Explosion of a Rotating Star As a Supernova Mechanism.

Soviet Astronomy, Vol. 14, p.652 (1971)

G.S.Bisnovatyi-Kogan

The magnetohydrodynamic rotational model of supernova explosion

Astrophysics and Space Science, vol. 41, June 1976, p. 287-320

Calculations of supernova explosion are made using the one-dimensional nonstationary equations of magnetic hydrodynamics for the case of cylindrical symmetry. The energy source is supposed to be the rotational energy of the system (the neutron star in the center and the surrounding envelope). The magnetic field plays the role of a mechanism of the transfer of rotational momentum. The calculations show that the envelope splits up during the dynamical evolution of the system, the main part of the envelope joins the neutron star and becomes uniformly rotating with it, and the outer part of the envelope expands with large velocity, carrying out a considerable part of rotational energy and rotational momentum. These results correspond qualitatively with the observational picture of supernova explosions.

alpha=0.01, t=8.5

1-D calculations of magnetorotational

explosion

.

tвзрыва~1

Small is difficult for numerical calculations with EXPLICIT numerical schemes

because of the Courant restriction on the time step, “hard” system of equations: determines a “hardness”.

In 2-D numerical IMPLICIT schemes have been used.

The main results of 1-D calculations:

Magneto-rotational explosion (MRE) has an efficiency about 10% of rotational energy.

For the neutron star mass the ejected mass 0.1М,Explosion energy 1051 ergEjected mass and explosion energy depend very weekly on the parameter Explosion time strongly depends on .

Explosion time =

A Numerical Example of the Collapse of a Rotating Magnetized Star

Astrophysical Journal, vol. 161, p.541 (1970)

J.LeBlanc, J.Wilson

Jets from collapse of rotating magnetized star: density and

magnetic flux

LeBlanc and Wilson (1970)

Astrophys. J. 161, 541.

First 2-D calculations.

Difference scheme (Ardeljan, Chernigovskii, Kosmachevskii, Moiseenko)

Lagrangian, on triangular grid

The scheme is based on the method of basic operators - grid analogs of the main differential operators:

GRAD(scalar) (differential) ~ GRAD(scalar) (grid analog)

DIV(vector) (differential) ~ DIV(vector) (grid analog)

CURL(vector) (differential) ~ CURL(vector) (grid analog)

GRAD(vector) (differential) ~ GRAD(vector) (grid analog)

DIV(tensor) (differential) ~ DIV(tensor) (grid analog)

The scheme is implicit. It is developed and its stability and convergence are investigated

by the group of N.V.Ardeljan (Moscow State University)

The scheme is fully conservative:conservation of the mass, momentum and total energy,

correct calculation of the transitions between different types of energies.

Difference scheme: Lagrangian, triangular grid with grid reconstruction(completely conservative=>angular momentum conserves automatically)

Grid reconstruction

Elementary reconstruction: BD connection is introducedinstead of AC connection. The total number of the knotsand the cells in the grid is not changed.

Addition a knot at the middle of the connection:the knot E is added to the existing knots ABCDon the middle of the BD connection, 2 connectionsAE and EC appear and the total number of cells is increased by 2 cells.

=>

Removal a knot: the knot E is removed from the grid and the total number of the cells is decreased by 2 cells

Grid reconstruction (example)

Presupernova Core Collapse

Equations of state takes into account degeneracy of electrons and neutrons, relativity for the electrons, nuclear transitions and nuclear

interactions. Temperature effects were taken into account approximately by the addition of the pressure of radiation and of an

ideal gas.

Neutrino losses and iron dissociation were taken into account in the energy equations.

A cool iron white dwarf was considered at the stability border with a mass equal to the Chandrasekhar limit.

To obtain the collapse we increase the density at each point by 20% and switch on a uniform rotation.

Ardeljan et. al., 2004, Astrophysics, 47, 47

sunMM 0042.1

Fe -iron dissociation energy

F(,T) - neutrino losses

Gas dynamic equations with a self-gravitation, realistic equation of state, account of neutrino losses and iron dissociation

3)(),(

4

0

TTPTPP

.6,2,10

),1()(

)1()419.8(lg)(

0

13/1

13/5

1)1(

00 kforaP

forcbPP

kkbk kc

k

4

0

3( , ) ( ) ( , ),

2 Fe

TT T T

0

20

0 .)(

)( dxx

xP

Equations of state (approximation of tables)

115/2359 ])/101.7(1/[)(103.1),( cgergTTTf

,20,31.664

,207),20(024.5131.664

,7,1

)(

T

TT

T

T

.910 TT

Neutrino losses:URCA processes, pair annihilation, photo production of neutrino, plasma neutrino

URCA:

( , ) ( , )F T f T e Approximation of tables from Ivanova, Imshennik, Nadyozhin,1969

, 0,

1, 0,

( , ) ,p

b Fe FeFe

m Fe Fe

E T TT

A T T

Fe –dis-sociation

R

Z

0 0.5 10

0.25

0.5

0.75

1

TIME= 0.00001000 ( 0.00000035sec )

R

Z

0 0.5 10

0.25

0.5

0.75

1 Density18 5.0704417 4.7324416 4.3944315 4.0564314 3.7184213 3.3804212 3.0424111 2.7044110 2.36649 2.02848 1.690397 1.352396 1.014385 0.6763794 0.3383743 0.04602092 0.005446171 0.000797174

TIME= 0.00001000 ( 0.00000035sec )

T

3/2T

sunMM 0042.1

Initial State

Spherically Symmetric configuration, Uniform rotation with angular velocity 2.519 (1/сек). Temperature distribution:

Density contours+ 20% Grid

R

Z

0 0.25 0.5 0.750

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Density27186222007820713319418718124116829515535014240411651210356690620.764729.251783.425891.912946.1607.295307.362150.5420.341389

TIME= 4.12450792 ( 0.14246372sec )

R

Z

0 0.005 0.01 0.015 0.020

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02 Density27186222007820713319418718124116829515535014240411651210356690620.764729.251783.425891.912946.1607.295307.362150.5420.341389

TIME= 4.12450792 ( 0.14246372sec )

Maximal compression state

R

Z

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

TIME= 5.29132543 ( 0.18276651sec )

R

Z

0 0.2 0.4 0.6 0.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

TIME= 5.29132543 ( 0.18276651sec )

Mixing

Shock wave does not produce SN explosion :

Distribution of the angular velocity

R

Z

0 0.005 0.01 0.015 0.020

0.0025

0.005

0.0075

0.01

0.0125

0.015

0.0175

0.02 Density27186222007820713319418718124116829515535014240411651210356690620.764729.251783.425891.912946.1607.295307.362150.5420.341389

TIME= 4.12450792 ( 0.14246372sec )

R

Angularvelocity

0.1 0.2

50

100

150

200

The period of rotation at the center of the young neutron star is about 0.001 sec

2-D magnetorotational supernova

Equations: MHD + self-gravitation, infinite conductivity:

, div 0,

1grad div( ) grad

div ( , ) 0, ( , ), ( , ),

G ,

.

dx d

dt dtdu

pdt

dp F T p P T T

dt

d

dt

u u

H HH H

u

HH u

Axial symmetry ( ) , equatorial symmetry (z=0).0

Additional condition: divH=0

N.V.Ardeljan, G.S.Bisnovatyi-Kogan, S.G.Moiseenko MNRAS, 2005, 359, 333.

Boundary conditions

qpoloidal

r

,0 :boundaryouter

,0

or

,0

:0

,0rotrot:0

HH

HH

HTPz

Bu

Hu

z

HHuur

zz

zz

rr

Quadrupole-like field

Dipole-like field

Rotational axis:

Equatorial plane

Z

R

O u te rb o u n d ary

Initial toroidal current Jφ

,vRc

13

d

RJH

),( zr HHJ

Bio-Savara law

(free boundary)

Initial magnetic field –quadrupole-like symmetry

R

Z

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5TIME= 0.00000000 ( 0.00000000sec )

R

Z

0 0.01 0.020

0.005

0.01

0.015

0.02

TIME= 0.00000779 ( 0.00000027sec )

Toroidal magnetic field amplification.

pink – maximum_1 of Hf^2 blue – maximum_2 of Hf^2

Maximal values of Hf=2.5 10(16)G

The magnetic field at the surface of the neutron star after the

explosion is

H=4 1012 Gs

Temperature and velocity field Specific angular momentumrV

Time dependences

time,sec0.1 0.2 0.3 0.4 0.5

0

5E+50

1E+51

1.5E+51

2E+51

2.5E+51

3E+51

3.5E+51

4E+51Ekinpol

Erot

Emagpol

Emagtor

time,sec0 0.1 0.2 0.3 0.4

-6.4E+52

-6.3E+52

-6.2E+52

-6.1E+52

-6E+52

-5.9E+52

-5.8E+52

-5.7E+52

-5.6E+52

-5.5E+52

-5.4E+52

Egrav

time,sec0 0.1 0.2 0.3 0.4 0.5

3.6E+52

3.7E+52

3.8E+52

3.9E+52

4E+52

4.1E+52

4.2E+52

Eint

time,sec0 0.1 0.2 0.3 0.4

1E+52

1.1E+52

1.2E+52

Neutrinolosses

time,sec0 0.1 0.2 0.3 0.4

5E+51

6E+51

7E+51

8E+51

9E+51 Neutrinoluminosity

Gravitational energy Internal energy

Neutrino losses Neutrino luminosity

Rotational energyMagnetic poloidal energyMagnetic toroidal energy Kinetic poloidal energy

Ejected energy Ejected mass 0.14M

time,sec0 0.1 0.2 0.3 0.4

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

Ejected mass/Masssun

time,sec0.1 0.2 0.3 0.4

1E+50

2E+50

3E+50

4E+50

5E+50

6E+50

Ejected energy (ergs)

0.6 10 эрг

51

51

Particle is considered “ejected” if its kinetic energy is greater than its potential energy

time,sec0 0.5 1 1.5

0

5E+50

1E+51

1.5E+51

2E+51

2.5E+51

3E+51

3.5E+51

4E+51

4.5E+51

Erot, (ergs)

time,sec0 0.5 1 1.5

0

1E+50

2E+50

3E+50

4E+50

5E+50Emagtor, (ergs)

time,sec0 0.5 1

0

1E+50

2E+50

3E+50

4E+50

5E+50

6E+50

Ekinpol, (ergs)

time,sec0 0.25 0.5 0.75

0

1E+50

2E+50

3E+50

4E+50

5E+50

6E+50

Emagpol, (ergs)

Magnetorotational explosion at different

Magnetorotational instability exponential growth of magnetic fields.Different types of MRI:

Dungey 1958,Velikhov 1959, Balbus & Hawley 1991, Spruit 2002, Akiyama et al. 2003

2 1210 10

Dependence of the explosion time on mag0

grav0

E

E

(for small )

6explosion10 ~ 6,t

12explosion10 ~ 12.t

2explosion10 10,t

12 6explosion1 10 (!0 )t

1-D calculattions:Explosion time

2-D calculattions:Explosion time

1взрываt

log( )взрываt

astro-ph/0410234

astro-ph/0410330

Inner region: development of magnetorotational instability (MRI)

R

Z

0.01 0.015 0.020.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

TIME= 34.83616590 ( 1.20326837sec )TIME= 34.83616590 ( 1.20326837sec )TIME= 34.83616590 ( 1.20326837sec )TIME= 34.83616590 ( 1.20326837sec )

R

Z

0.01 0.015 0.020.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

TIME= 35.08302173 ( 1.21179496sec )TIME= 35.08302173 ( 1.21179496sec )TIME= 35.08302173 ( 1.21179496sec )TIME= 35.08302173 ( 1.21179496sec )

R

Z

0.01 0.015 0.020.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

TIME= 35.26651529 ( 1.21813298sec )TIME= 35.26651529 ( 1.21813298sec )TIME= 35.26651529 ( 1.21813298sec )TIME= 35.26651529 ( 1.21813298sec )

R

Z

0.01 0.015 0.020.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

0.011

0.012

0.013

0.014

TIME= 35.38772425 ( 1.22231963sec )TIME= 35.38772425 ( 1.22231963sec )TIME= 35.38772425 ( 1.22231963sec )TIME= 35.38772425 ( 1.22231963sec )

Toroidal (color) and poloidal (arrows) magnetic fields(quadrupole)

Toy model of the MRI development: expomemtial growth of the magnetic fields

; r

dH dH r

dt dr

MRI leads to formation of multiple poloidal differentially rotating vortexes. Angular velocity of vortexes is growing (linearly) with a growth of H.

0r v

r

dH dH l

dt dl

( )vdl H H

dl

2

02( )

r

dH H AH H H

dt

0

0

( )0

3 2 1 2( )0

0 1 .

r

r

A H t tr

A H t trr r

H H H

H

e

H HA

e

at initial stages * :H H const,

dr A

dr

Microquasar

GRS 1915+105

Jet ejection

MERLIN 5GHz

Fender (1999)

X-ray image of

microquasar

XTE J1550-564(center)

with two jets.

0.3-7 keV

11 March 2002

Chandra

Kaaret et al. (2002)

X ray image of

microquasar

XTE J1550-564 (left)

and western jet.

Upper panel is from

11 March 2002.

Lower panel is from

19 June 2002.

Chandra, 0.3-7 keV

Kaaret et al. (2002)

X-ray binary with jets

(sketch)

Fender (2001)

Dipole-like initial magnetic field

R

Z

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

Jet formation in MREMoiseenko et al. Astro-ph/0603789

Jet formation in MRE: velocity field evolution

Jet formation in MRE:

entropy evolution

Jet formation in MRE: (dipole magnetic field)

Energy of explosion0.6·1051эрг Ejected mass 0.14M

Toroidal (color) and poloidal (arrows) magnetic fields (dipole)

Toroidal magnetic field (color) and poloidal velocity field(dipole)

Why time of MRE depends logarithmically on alpha in presence of MRI

Exitation of NS oscillations during MRE with magnetic dipole

Violation of mirror symmetry of magnetic field(Bisnovatyi-Kogan, Moiseenko, 1992)

1. Initial toroidal field

2. Initial quadrupole field

3. Generated toroidal field

4. Resulted toroidal field

Kick velocity, along the rotational axis, due to the asymmetry of the magnetic field ~ up to 300km/sec

Bisnovatyi-Kogan, Moiseenko (1992) Astron. Zh., 69, 563

Kick velocity, along the rotational axis, ~ up to 1000km/sec

Bisnovatyi-Kogan, 1993, Astron. Ap. Transactions 3, 287

In reality we have dipole + quadrupole + other multipoles… Lovelace et al. 1992

The magnetorotational explosion will be always asymmetrical dueto development of MRI.

Interaction of the neutrino with asymmetric magnetic field

S. Johnston et al. astro/ph 0510260 Evidence for alignment of the rotation and velocity vectorsin pulsars

We present strong observational evidence for a relationship between the direction of a pulsar's motion and itsrotation axis. We show carefully calibrated polarization data for 25pulsars, 20 of which display linearly polarized emission from the pulselongitude at closest approach to the magnetic pole… we conclude that the velocity vector and the rotation axis arealigned at birth.

W.H.T. Vlemmings et al. astro-ph/0509025

Pulsar Astrometry at the Microarcsecond Level Determination of pulsar parallaxes and proper motionsaddresses fundamental astrophysical questions. We have recentlyfinished a VLBI astrometry project to determine the proper motions andparallaxes of 27 pulsars, thereby doubling the total number of pulsarparallaxes. Here we summarise our astrometric technique and presentthe discovery of a pulsar moving in excess of 1000 kms, PSR B1508+55.

CP violation in week procecces in regular magnetic field: does not work, because MRI leads to formation of highly chaotic field.

ConclusionsConclusions

1. In the magnetorotational explosion (MTE) the efficiency of transformation of rotational energy into the energy of explosion is 10%. This is enough for producing core – collapse SN from rapidly rotating magnetized neutron star.

2. Development of magneto-rotational instability strongly accelerate MRE, at lower values of the initial magnetic fields.

3. The new born neutron star has inside a large (about 10^14 Gauss) chaotic magnetic field.

4. Jet formation is possible for dipole-like initial topology of the field: possible relation to cosmic gamma-ray bursts; equatorial ejection happens at prevailing of the quadrupole-like component.

5. MRI leads to violation of mirror symmetry, asymmetry in MRE explosion,

and in the neutrino flux, producing kick.

..

.