magnetorotational supernovae and magnetorotational instability

Magnetorotational supernovae and magnetorotational

instability .

S.G. Moiseenko, G.S. Bisnovatyi-Kogan,

Space Research Institute,

Moscow, Russia

N.V. Ardeljan,

Moscow State University, Moscow, Russia

Magnetorotational mechanism for the supernova explosion Bisnovatyi-Kogan (1970)

First 2D calculations: LeBlanck&Wilson (1970) ->too large initial initial magnetic fields.Meier et al. 1976, Ardeljan et al.1979, Mueller & Hillebrandt 1979, Symbalisty 1984, Ardeljan et al. 2000, Wheeler et al. 2002, Yamada & Sawai 2004, Kotake et al. 2004

1D calculations of the magnetorotational mechanism with realistic magnetic field values (Ardeljan et al.1979)

The realistic values of the magnetic field are: Emag<<Egrav ( 10-8-10-12)

Small initial magnetic field -is the main difficulty for the numerical simulations.

The hydrodynamical time scale is much smaller than the magnetic field amplification time scale (if magnetorotational instability is neglected).

Explicit difference schemes can not be applied. (CFL restriction on the time-step).

Implicit schemes should be used.

jet axialE~E grav0mag0

mag

grav

E

E

Basic equations: MHD +selfgravitation, 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

Axis symmetry ( ) and equatorial symmetry (z=0) are supposed.0

Notations:

Additional condition divH=0

.,R,G

,, ,

, , ,z),(r,,

indexadiabaticconstantgasttanconsnalgravitatio

energyinternalpotentialnalgravitatiofieldmagnetic

pressurepdensityvelocitytdt

d

H

uxu

Boundary conditions

r

poloidal q

0 : rot rot 0,

0,

0 : or

0,

outer boundary: 0,

r r

z z

zz

r u u H H

u H

z

Bu

zP T H

H H

H H

(from Bio-Savara law)

Quadrupole field

Dipole field

Difference scheme Lagrangian, triangular grid with rezoning,

Complete conservation=>angular momentum conserved automatically

Grid reconstructionElementary 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

=>

Difference scheme

Method of basic operators (Samarskii) – grid analogs of basic 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)

Implicit scheme. Time step restrictions are weaker for implicit schemes.

The scheme is completely conservative:Conservation of the mass, momentum and energy.

The scheme is Lagrangian=> conservation of angular momentum.

Numerical method testing

The method was tested on the following tests:

1. Collapse of a dust cloud without pressure2. Decomposition of discontinuity,3. Spherical stationary solution with force-free magnetic field,4. MHD piston problem,

Example of the triangular grid

Presupernova Core Collapse

Equations of state take 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 radiation pressure and an ideal gas.

Neutrino losses were taken into account in the energy equations.

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

To obtain the collapse we increase the density at each point by 20% and we also impart uniform rotation on it.

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 )

RZ

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 )

Initial state, spherically symmetrical stationary state, initial angular velocity 2.519 (1/sec)

Temperature distribution

T

3/2T1.2042 sunM M

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 0.01 0.020

0.005

0.01

0.015

0.02

TIME= 4.12450792 ( 0.14246372sec )

Neutron star formation in the center and formation of the shock wave

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 )

R

Z

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

TIME= 5.29132543 ( 0.18276651sec )

Mixing

Shock wave does not produce SN explosion :(

R

Z

0 0.002 0.004 0.006 0.0080

0.002

0.004

0.006

Angular velocity357.283310.745293.484276.224258.963241.703224.442207.181189.921172.66155.399138.139120.878103.61886.356969.096351.835634.57517.3144

TIME= 4.15163360 ( 0.14340067sec )

Angular velocity (central part of the computational domain). Rotation is VERY differential.

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

Angula

rve

locity

0.1 0.2

50

100

150

200

The period of rotation of the young neutron star is about 0.001- 0.003 sec

Z

R

O u te rb o u n d ary

Initial toroidal current Jφ

,vRc

13

dRJ

H

),( zr HHJ

Biot–Savar law

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^2Maximal values of Hf=2.5 ·10(16)G

After SN explosion at the border of neutron star H=2 1014G

Temperature and velocity field

Specific angular momentum rV

Time evolution of the energies

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 evolution of the energies

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

Gravitational energy Internal energy

Time evolution of the energies

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

Neutrino losses (ergs) Neutrino luminosity (ergs/sec)

Ejected energy and mass

Ejected energy 510.6 10 ergParticle is considered “ejected” –

if its kinetic energy is greater than its potential energy

Ejected mass 0.14M

2.1

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)

Magnetorotational supernova in 1D

mag0

grav0explosion

1,

E

Et

Example: 2explosion10 0.1,t

12 6explosion1 !10 !!0 t

Ardeljan et al. 1979

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 for the different 0 2 12

0

10 10mag

grav

E

E

Magnetorotational instability mag. field grows exponentially (Dungey 1958,Velikhov 1959, Balbus & Hawley 1991,

Spruit 2002, Akiyama et al. 2003)

Dependence of the explosion time frommag0

grav0

E

E

explo 1sion 0~ log ( )t (for small )

Example:

6explosion10 ~ 6,t

12explosion10 ~ 12.t

Toy model for MRI in the magnetorotational supernovaArdeljan, Bisnovatyi-Kogan & Moiseenko MNRAS 2005, 359, 333

; r

dH dH r

dt dr

beginning of the MRI => formation of multiple poloidal differentially rotating vortexes

0r v

r

dH dH l

dt dl

( )vdl H H

dl

in general we may approximate:

Assuming for the simplicity that is a constant during the first stages of MRI, and taking as a constant we come to the following equation:

ddrr A

H 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 the initial stage of the process * :H H const,r

dH r

dr

Central part of the computational domain . Formation of the 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 )

Magnetorotational instability

Initial magnetic field – dipole-like symmetryMoiseenko, Ardeljan, Bisnovatyi-Kogan MNRAS 2006, 370, 501

R

Z

0 1 2 3 40

0.5

1

1.5

2

2.5

3

3.5

Magnetorotational explosion for the dipole-like magnetic field

Magnetorotational explosion for the dipole-like magnetic field

Magnetorotational explosion for the dipole-like magnetic field

Ejected energy and mass (dipole)

Ejected energy 510.5 10 erg Particle is considered “ejected” –

if its kinetic energy is greater than its potential energy

Ejected mass 0.14M

time, sec0 0.5 1 1.5

0

1E+50

2E+50

3E+50

4E+50

5E+50

time, sec0 0.5 1 1.5

0

1E+50

2E+50

3E+50

4E+50

5E+50

time, sec0 0.5 1

0.02

0.04

0.06

0.08

0.1

0.12

0.14

Characteristic time of the magnetic field reconnection

Petcheck mechanism – characteristic reconnection time

Our estimations show: conductivity ~ 8 1020c-1

Magnetic Reynolds number ~1015

Characteristic time of the magnetic field reconnection For the magnetorotational supernova is:(approximately 10 times larger than characteristic time of magnetorotational supernova explosion). .

Reconnection of the magnetic field does not influence significantly on the supernova explosion.

Mirror symmetry violation of the magnetic field in rotating stars

• а. Initial toroidal field

• b. Initial dipole field

• с. Generated toroidal field

• d. Resulting toroidal field

Mirror symmetry violation of the magnetic field in rotating stars

+ =

Resulting toroidal filed is larger in the upper hemisphere.

Violation of mirror symmetry of the magnetic field in magnetorotational explosion leads to:

Onesided ejections,

Rapidly moving radiopulsarss (up to 300 km/s).

In reality we have dipole + quadrupole + other multipoles…

(Lovelace et al. 1992)

Dipole ~

Quadrupole ~

3

1

r

4

1

r

The magnetorotational supernova explosion in reality will be always

asymmetrical.

Cassiopea A- supernova with jets-an example of the magnetorotational supernova

(Una Hwang, J. Martin Laming, et al. ApJL, 2004, 615, L117 )

ConclusionsConclusions

1.Magnetorotational mechanism (MRM) produces enough energy for the core collapsed supernova.

2.The MRM is weakly sensitive to the neutrino cooling mechanism.

3.MRI helps to produce the magnetorotational explosion in a short time.

3.The shape of the explosion is always asymmetrical.

4.The shape of the explosion qualitatively depends on the initial configuration (symmetry type) of the magnetic field.

Roadmap of the supernova science evolution