Relativistic Stars with Magnetic Fields

Relativistic Stars with Magnetic Fields

1. Motivation: Magnetar2. Newtonian GS equation3. Relativistic GS equation4. Weak field limit5. Metric perturbation6. Numerical results

Kunihito Ioka (Penn State)

Ioka(01)MN327,639Ioka&Sasaki(03)PRD67,124026Ioka&Sasaki(04)ApJ600,296

1. Motivation: Magnetar

P

1014G

Magnetars

Production rate* 10 magnetars / 104yr ~1 magnetar / 103yr1 neutron star / 102yr

Baring & Harding (98)

Super strongly magnetized NSDiscovered in 1998

Super strongly magnetized neutron star

Deformation of neutron stars

1. Precession2. GW source (e.g., GRB)3. Influence on the oscillation

Equilibrium of magnetized stars

Magnetar

(My) Background

4410 ergsE

A giant flare from a magnetar

⇒ Gamma-rays affected the ionosphere

Inan et al. (99)

on Aug. 27 1998

Spin down410/ PP

or J I J I

Field reconfiguration ?

24

16

42 45

16

1010 G

10 erg10 G

mag

grav

grav grav

EI B

I E

BE E

1/ 2 11/ 222

455 10

10 ergs 2kHz 5kpcGW

c

E f dh

Moment of inertia:

Energy:

⇒ GW ?

Ioka(01)

Time

Woods et al. (99)

Stationary axisymmetric equilibrium

So far only poloidal fieldBonazzola & Gourgoulhon (96)Bocquet et al. (95)Konno, Obata & Kojima (99)

Circular[ ]

[ ]

0

0

T

T

, : Killing vectorst

11

22

0 0

0 0tt tg g

gg

g

g

Papapetrou (66)

Carter (69)

However, toroidal field or meridional flowviolate circularity

Toroidal

StrategyGravity Matter, Magnetic field

8G T ; 0T

Axisymmetric stationary GR ideal MHD

A master equation for flux function

GS (Grad-Shafranov) eq.

Weak magnetic field limit limit

A linear equationfor flux function

Einstein equation

TOV equation 2G T O

2. Newtonian GS equationBasic equations for ideal MHD

0 (Mass conservation), , E.O.S

1 (Euler equation)

4

0 (Faraday's law)

10 (Perfect conductivity)

4 (Poisson's equation)

v p p S

v v p B B

E

E v Bc

G

Flux function Flux surfaceconst

:"magnetic flux per meridional flow"

:"angular velocity"

:"energy"

:"angular momentum"

: entropy

C

E

L

S

Conserved quantities on flux surface

quantities,

e.g., , ,r z

B v

First integral constants

GS equation Euler equation equation of 0

Second-order, nonlinear partial differential equation

Transfield equation

transform

3. Relativistic GS equationBasic equations for GR MHD

(Baryon conservation)

(T;=0)

(Maxwell equation)

(Perfect conductivity)

(1st law)

(E.O.S)

Bekenstein & Oron (78) , , , , existE L C S

, quantitiesr

GS equation ; ,0 eq. of 0AT

transform

3 0

||

1

0

ABA B

J J u TSNMC

u u E C u u L C

2nd-order nonlinear partial differential equation

Ioka & Sasaki (03)

However it is formidable to solve GS eq. directly

4. Weak magnetic field limit0 (no magnetic field)

(no meridional flow)C

Ioka & Sasaki (04)

Zeroth order

Tolman-Oppenheimer-Volkoff (TOV) equation

First order

3 0

||

1

0

ABA B

J J u TSNMC

u u E C u u L C

GS eq.

Aboid Alfven pointsSeparable with variables

We specify the conserved functions

Separation of the angular variables

Diopole (l=1) equation

Boundary conditions: confined fields

Vector harmonics

EigenvalueMaster equationfor matter and EM

Even (-1)l

Odd (-1)l+1

2

8

g g g O

G T

5. Metric perturbation

Regge-Wheeler gauge

Linearized Einstein equation

Regge & Wheeler (57)Zerilli (70)

0,2 1,3 1,3

0,2 2

2

2

l l l

l l

l

l

t r tr

Exterior solutions

* Mass shift

Angular momentum

Mass quadrupole moment

Current hexapole moment

M

J

Q

V

Vacuum

These are to be matchedwith the interior solutions

We can solve Einstein eq. explicitly

6. Numerical resultsMagnetic fields

Magnetic field lines projected on the meridional plane (=const surface in r plane)

1 1/ , 1, 0np K n

A truncated piece of a magnetic field line on a certain flux surface(=const surface) with </2 projected onto the equatorial plane

Flux surfaceStar surface

Toroidal fieldFieldline

Meridional flow

Ellipticity equatorial radius polar radius

mean radiuse

<0

Prolate

Oblate

Frame dragging

1,3 1,3

2

*

*

*

*

l l

l

I V

Wg

r

t r

t

Vl=1,3 similar to rotating stars and Kerr black holes

Reflection symmetry about equatorial plane

Il=1,3, Wl=2: parity -1

Il=1,3 ~(M*/R*)v : meridional flow origin

Wl=2~0.1(M*/R*)(B/1018)2 : magnetic field origin

only inside the star

7. SummaryWe solve relativistic stars with toroidal field and meridional flowin the weak magnetic field limitShape is prolate not oblateReflection symmetry is violated in the frame dragging NS kick ???