Download - Relativistic Stars with Magnetic Fields
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 ???