1. Pu, Hung-Yi Institute of Astronomy, National Tsing-Hua University The Effects of Photon Path Bending on the Observed Pulse Profile and Spectra of Surface Thermal Emission from Neutron Stars

2. Spectra calculation: ∫I’ ν (t) cosθ’ dΩ’ ( Include photon path bending ) Motivation: Application: Determining the inclination and viewing angles for Radio-quiet neutron stars from their X-ray thermal emission Neutron stars with X-ray thermal emission  I ν = Planck function Consider Limb-darkening  I ν = (require )

3. The dependence of model spectra and pulse profiles on the input parameters

4. ζ: the viewing angle α: the inclination angle Light curves with different temperature distribution (I)

5. Light curves with different temperature distribution (II)

6. Light curves with different temperature distribution (III)

7. Light curves with and without limb-darkening Dash: without limb-darkening Dot: with limb-darkening

8. Spectra with different magnetic field from, T p =2 x10 6 K Solid curve: Planck function (with suitable normalization) of temperature 2 x10 6 K

9. Determining the inclination angle and viewing angle of neutron stars from their X-ray thermal emissions

10. Observation: Blackbody Best Fit Temperature (AσT 4 =4πd 2 F) 1)Observed Flux 2)Pulse Fraction Input parameters: 1) Hot Spot Size 2) M / R 3) α and ζ Inferred possible range of α and ζ Consistent? Model flux Model Pulsed fraction Yeah No

11. Observational properties of RX J0806.4-4123

12. Apply T hot spot = 95.6 eV M/R=0.2 Different hot spot size: 12,13,15,16 from top to down (in unit of canonical polar cap size, ~0.25 degree) Left: computed flux {Flux/28.8x10 -13 }={0.9,1.0,1.1,1.2} Right: computed pulsed fraction {0.05,0.06,0.07,0.1,0.2,0.3,0.4,0.5}

13. The shapes of computed pulsed fraction contours are not change too much for fixed mass-to-radius ratio Computed pulsed fraction contours for different mass-to-radius ratio M / R=0.01 M / R=0.1 M / R=0.2 M / R=0.3

14. Sum of computed RX J0806.4-4123 pulsed fraction contours with values of 6% M / R=0.01 M / R=0.1 M / R=0.2 M / R=0.3

15. Pulse profiles e.x. M/R= 0.1, hot spot size equal to 22 times of the canonical polar cap sizes

16. 117 Radio pulsar geometries reported by Rankin ( 1993 ) Distribution of Radio pulsars in α-ζ plane

17. M / R=0.01 M / R=0.1 M / R=0.2 M / R=0.3

18. RX J0420.0-5022 M/R=0.010.1 0.20.3 {0.10, 0.12, 0.15, 0.2, 0.3}

19. RX J0720.4-3125 M/R=0.010.1 0.20.3 {0.06, 0.11, 0.15, 0.2, 0.3}

20. Summary Pulse fraction in general decreases as the gravity increases; Beaming effect in general increases the pulse fraction. The inferred possible geometry range is broad in the α-ζ plane and we cannot tell if the distribution for radio-quiet neutron stars and radio pulsars are different. Our computed result depends on the beaming pattern we use.

21. Thank you!

26. ζ: the viewing angle α: the inclination angle Lyne 1998 Rotation Axis Magnetic Axis Hot Spot

27. ∫I’ ν (t) cosθ’ dΩ’ Calculate (specific) Observed Flux: dΩ ’ Distant Observer