1. A Global Model of Neutron Star Surface Emission (considering GR lensing effect) A Global Model of Neutron Star Surface Emission (considering GR lensing effect) 2003/5/22 & 29

2. Outline Outline Motive Motive Parameter & Assumptions Parameter & Assumptions Goal & Application Goal & Application Calculus (show by hand) Calculus (show by hand) Current result Current result Conclusion Conclusion

3. Motive By GR: By GR: Matter and energy tell space (and space-time) how to curve. Space tells matter how to move. how to curve. Space tells matter how to move. Curved space bend the light An index of the strength of gravitational field: An index of the strength of gravitational field: ( 引力半徑, Schwarzschild 半徑 ) Rs=2GM/C 2 ( 引力半徑, Schwarzschild 半徑 )Rs=2GM/C 2 If Rs<<R weak Rs<~R strong Rs<~R strong

4. So, taking the gravitational lensing effect into account is needed …… Lensing

5. Set G=C=1, n* : R/2M ~ 10 M=0 (flat space-time) M=0 (flat space-time) θ MAX = 90 ∘ θ MAX = 90 ∘ R/2M ~1.76 R/2M ~1.76 θ MAX = 180 ∘ R/2M ~1.6 R/2M ~1.6 θ MAX = 227 ∘ R/2M ~1.5 R/2M ~1.5 θ MAX ~ infinity (photon sphere) (photon sphere) Gravitation Misner,Thorne,Wheeler 1972

6. For n*: M=1.4M ⊙, R=10km R= 2M M/R~0.5Rs R= 3M M/R~0.33 photonsphere R=3.5M M/R~0.28 θMAX = 180∘ R>3.5M M/R<0.28 90∘< θMAX < 180∘ R~4.84M M/R~0.267 θMAX~132 ∘ R>>M M/R~0 θMAX = 90∘

7. Parameter & Assumptions(1) Parameter: Parameter: Input parameter: Input parameter: R, M, θ 0, r 0, T(θm ), I(T, δ) R, M, θ 0, r 0, T(θm ), I(T, δ) θm Magnetic Axis To Observer (as Z Axis) θ ’, φ ’ θ0θ0θ0θ0 θmax θ δ φ

8. Parameter & Assumptions(2) Assumptions(1): Assumptions(1):  Neutron star is spherical symmetry  Slowly rotating ()  Slowly rotating (Schwarzschild metric)Schwarzschild metric  Ignore the “ dragging of inertial frame ”  r R is totally transparent. (Photon are emitted from the surface of an opaque sphere. )

9. Parameter & Assumptions(3) Assumptions(2): Assumptions(2):  R/2M>1.5  r 0 → ∞  No “ beaming ” ( isotropic, independent of δ )  Strong LTE ( I=B(T) )

10. Goal & Application(1) Given T(θm ), θ 0, r 0, we can calculate: Given T(θm ), θ 0, r 0, we can calculate:  Spec  (Average) Flux  Light curve with θ 0 (t) (consider spin axis, β, γ) (consider spin axis, β, γ) Provide more detail result than single/two polar cap approximation does Provide more detail result than single/two polar cap approximation does As an ex.,the result can be applicable to thermal emission arising from accretion As an ex.,the result can be applicable to thermal emission arising from accretion Magnetic Axis Rotation Axis θ0θ0θ0θ0 β γ

11. Goal & Application(2) More, consider in detail the beaming of ration (Local model) require: More, consider in detail the beaming of ration (Local model) require:  Temperature stratification of n* atm.  The sol. of the resulting radiation-transfer problem Local model → Global model Local model → Global model Can be applied to various neutron stars to deduce their surface properties Can be applied to various neutron stars to deduce their surface properties

12. Calculus θ’θ’φ’φ’θ’θ’φ’φ’θφ TI Simply assume I=B(T)=σT 4 /π Flux= ∫ I dΩ ’ θm Given

13. Current result Code has (almost) constructed for no spin case Code has (almost) constructed for no spin case Checking if the code is right Checking if the code is right Going on: Going on: Consider spin case light curve Consider spin case light curve Consider B ν spec. Consider B ν spec. Loading....

14. Conclusion

15. Things are not necessarily easy and … Computer is important and … Computer is important Keep going ……

18. 相對論天體物理的基本概念 方勵之 1999

19. For T~1 Sec, R~10 km a~83.8 r/a~ (GM/C^2) / (MR^2*ω /MC) ~GM/CR^2*ω ~6.7*10 -8 *1.4*2*10 33 /3*10 10 *10 12 *2PI ~10 3 Kerr metric is not needed ??!