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Relativistic Stars with Magnetic Fields

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.

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Relativistic Stars with Magnetic Fields

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  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 Magnetars 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 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. 27 1998 ⇒ Gamma-rays affected the ionosphere Inan et al. (99)

  5. 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 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 0limit A linear equation for flux function Y TOV equation

  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 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

  16. j q t r t r q j

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

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

  19. 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

  20. Meridional flow

  21. Ellipticity <0 Prolate Oblate

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

  23. 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

  24. 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

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