1. Relativistic Stars with Magnetic Fields
Kunihito Ioka (Penn State)
Motivation: Magnetar
Newtonian GS equation
Relativistic GS equation
Weak field limit
Metric perturbation
Numerical results
Ioka(01)MN327,639
Ioka&Sasaki(03)PRD67,124026
Ioka&Sasaki(04)ApJ600,296

2. 1. Motivation: Magnetar 1014G Magnetars Production rate
Super strongly magnetized NS
Discovered in 1998
1014G
Production rate
* 10 magnetars / 104yr
~1 magnetar / 103yr
* 1 neutron star / 102yr
Baring & Harding (98)

3. Magnetar Deformation of neutron stars Equilibrium of magnetized stars
Super strongly magnetized neutron star
Deformation of neutron stars
Precession
GW source (e.g., GRB)
Influence on the oscillation
Equilibrium of magnetized stars

4. (My) Background A giant flare from a magnetar
on Aug
⇒ Gamma-rays affected the ionosphere
Inan et al. (99)

5. Field reconfiguration ?
Spin down
Woods et al. (99)
Field reconfiguration ?
Ioka(01)
Time
Moment of inertia:
Energy:
⇒ GW ?

6. Stationary axisymmetric equilibrium
Toroidal
So far only poloidal field
Bonazzola & Gourgoulhon (96)
Bocquet et al. (95)
Konno, Obata & Kojima (99)
Circular
Papapetrou (66)
Carter (69)
However, toroidal field or meridional flow
violate circularity

7. Strategy Gravity Matter, Magnetic field GS (Grad-Shafranov) eq.
Axisymmetric stationary GR ideal MHD
Gravity
Matter, Magnetic field
Einstein equation
A master equation for flux function Y
GS (Grad-Shafranov) eq.
Weak magnetic field limit
Y 0 limit
TOV equation
A linear equation
for flux function Y

8. 2. Newtonian GS equation Basic equations for ideal MHD Flux function
Flux surface

9. Conserved quantities on flux surface
First integral constants
GS equation
transform
Second-order, nonlinear
partial differential equation
Transfield equation

10. 3. Relativistic GS equation
Basic equations for GR MHD
(Baryon conservation)
(Tmn;n=0)
(Maxwell equation)
(Perfect conductivity)
(1st law)
(E.O.S)

11. Bekenstein & Oron (78) GS equation
Ioka & Sasaki (03)
transform
2nd-order nonlinear partial differential equation
However it is formidable to solve GS eq. directly

12. 4. Weak magnetic field limit
Ioka & Sasaki (04)
Zeroth order
Tolman-Oppenheimer-Volkoff (TOV) equation

13. First order GS eq. We specify the conserved functions
Aboid Alfven points
Separable with variables
GS eq.

14. Separation of the angular variables
Vector harmonics
Diopole (l=1) equation
Master equation
for matter and EM
Eigenvalue
Boundary conditions: confined fields

15. 5. Metric perturbation Linearized Einstein equation
Regge-Wheeler gauge
Regge & Wheeler (57)
Zerilli (70)
Even (-1)l
Odd (-1)l+1

17. j t r q t r q j

18. Exterior solutions Vacuum These are to be matched
We can solve Einstein eq. explicitly
These are to be matched
with the interior solutions

19. 6. Numerical results Magnetic fields
Magnetic field lines projected on the meridional plane
(Y=const surface in rq plane)

21. Toroidal field Field line Flux surface Star surface
A truncated piece of a magnetic field line on a certain flux surface
(Y=const surface) with q<p/2 projected onto the equatorial plane

22. Meridional flow

23. Ellipticity
<0
Prolate
Oblate

24. Frame dragging Vl=1,3 r q j t t r q j
similar to rotating stars and Kerr black holes

25. about equatorial plane
Il=1,3
~(M*/R*)v : meridional flow origin
Wl=2
~0.1(M*/R*)(B/1018)2 : magnetic field origin
Il=1,3, Wl=2: parity -1
Reflection symmetry
about equatorial plane
only inside the star

26. 7. Summary We solve relativistic stars with
toroidal field and meridional flow
in the weak magnetic field limit
Shape is prolate not oblate
Reflection symmetry is
violated in the frame
dragging NS kick ??? n n