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

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 )

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

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

Light curves with different temperature distribution (II)

Light curves with different temperature distribution (III)

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

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

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

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

Observational properties of RX J

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/28.8x }={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}

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

Sum of computed RX J pulsed fraction contours with values of 6% M / R=0.01 M / R=0.1 M / R=0.2 M / R=0.3

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

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

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

RX J M/R= {0.10, 0.12, 0.15, 0.2, 0.3}

RX J M/R= {0.06, 0.11, 0.15, 0.2, 0.3}

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.

Thank you!

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

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