1. Global model for neutron star surface emission -- and some application 200 3/ 12 / 18

2. Outline Assumption & goal Neutron star parameters How to build a global model Local model (I) (simple cases) Gravitation effects Changing coordinate Local model (II) (more complicated cases & strong magnetic field effects) Future work Discussion

3. Assumption & goal Neutron star is spherical symmetry Slowly rotating (Schwarzschild metric) Ignore the “dragging of inertial frame” r R is totally transparent. (Photon are emitted from the surface of an opaque sphere. ) Provide more detail result than single/two polar cap approximation does Local model → Global model Can be applied to various neutron stars to deduce their surface properties

4. Neutron star parameters For a typical neutron star: M=1.4M ⊙ R=10km T=1sec Rs=2GM/C 2 ~ 0.267R strong gravitational field!! R=3.5M M/R~0.28 θMAX = 180 ∘ R>>M M/R~0 θMAX = 90 ∘ R>3.5M M/R<0.28 90 ∘ < θMAX < 180 ∘ R~4.84M M/R~0.267 θMAX~132 ∘ θmax To observer

5. How to build a global model Flux(t): (Lightcurve) ∫I(t) cosθ’ dΩ’ Spec.:∫I ν (t) cosθ’ dΩ’ ∫I ν (t) cosθ ’ dt dΩ’ Note: cosθ= 1 Changing coordinate Z axis θ θm Magnetic Axis θb θp Surface normal

6. Local model (I) Polar cap approximationTemp. distribution due to uniform magnetic field θm θb θm = θb

7. Schaaf (1990b) for B up to 10 12 G θmθ Teff = TP if | θm -θo| ≤ 5° = 0 else Pechenick etc. ApJ 274:846 1983 Relative T v.s. θ m Relative T v.s. θ b(m)

8. Gravitational effects Lensing 1.) Self-lensing 2.) Gravitational red-shift

9. Pechenick etc. ApJ 274:846 1983 R: fixed M: changed R/M M/R 3.2 0.3125 3.40.294 4 0.25 4.84 0.267 50.2 8 0.125 e300 Relative total flux v.s ωt

10. R/M M/R 3.20.3125 3.40.294 4 0.25 4.84 0.267 50.2 8 0.125 e300 Relative total flux v.s ωt

11. Relative specific flux v.s Freq.

12. Changing coordinate R  R ’ ν  ν ’ T  T ’

14. Neutron starDistant observer Specific Flux Total Flux

15. Relative total flux v.s ωt

16. R/M M/R 40.25 e300 Relative total flux v.s ωt

17. R/M M/R 40.25 e300 ν’ ν’ ν ν  ν ’ Relative specific flux v.s Freq.

18. Local model (II) Temp. distribution due to dipole

19. Heyl etc. MNRA 324,292 2001 Best-fitting model for acos 2 θ +bsin 2 θ for 10 12 G Relative T v.s. θ b

20. Relative total flux v.s ωt Relative specific flux v.s Freq.

21. Strong Magnetic field effects Anisotropy of the surface temperature Beaming ( In magnetized electron-ion plasma, the scattering and free-free absorption opacities depend on the direction of propagation and the normal modes of EM waves) Dong Lai etc. MNRAS 327,1081 2001 core envelope atmosphere B ν cyclotron =eB/2πm e Ion cyclotron resonance occurs when The E field of the mode rotates in the same direction as the ion gyration

22. Dipole + beaming

23. Isotropic : Beaming due to B field : I ν ( T 1 ) I ν ( T 2 ) I ν ( T 3 ) I ν ( T 4 ) I ν ( T 5 ) I ν ( T 4 ) I ν ( T 3 ) I ν ( T 2 ) I ν ( T 1 ) B field T 1 = T eff I ν ( T 1 ) θb

24. .. θm Magnetic Axisθb θp θm,θp, θb, θbp Need to calculate θm, θp, θb, θbp Surface normal Harding etc. ApJ 500:862 1998 Pavlov etc. A&A 297,441 1995

25. Future work Dipole+ beaming+ limbdarkening+ line Line profile 1E1207.4-5209 Simulation Given T,A,Ro  Photon counts  Record by random  Fit Bignami etc. Nature 423:725 2003

26. Discussion Thermal surface emission INS AXPs, SGRs, Magnetars Vela Geminga PSR0656+14 : the pulsed emission has a two-component X- ray spectrum Schwarzschild metric a=JG/MC 3 ( 0 < a < 1 ) ex. For SUN: a=0.187

27. Easy to lost but cheerful after get through…