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 Iν= Planck function

Consider Limb-darkening Iν=

(require )

The dependence of model spectra and pulse profiles on the

input parameters

ζ: the viewing angle α: the inclination angle

Light curveswith different temperature distribution (I)

Light curveswith different temperature distribution (II)

Light curveswith different temperature distribution (III)

Light curveswith and without limb-darkening

Dash: without limb-darkening

Dot: with limb-darkening

Spectrawith different magnetic field from

, Tp=2 x106K

Solid curve: Planck function (with suitable normalization) of temperature 2 x106 K

Determining the

inclination angle and

viewing angle of neutron stars from their X-ray thermal emissions

Observation:

Blackbody Best Fit Temperature(AσT4=4πd2F)

1)Observed Flux2)Pulse Fraction

Input parameters:1) Hot Spot Size2) M / R3) α and ζ

Inferred possible range of α and ζ

Consistent?

Model fluxModel Pulsed fraction

Yeah

No

Observational properties of RX J0806.4-4123

Apply T hot spot = 95.6 eVM/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}

The shapes of computed pulsed fraction contours are not change too much for fixed mass-to-radius ratio

Computed pulsed fraction contoursfor 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 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

Pulse profilese.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 J0420.0-5022M/R=0.01 0.1

0.2 0.3

{0.10, 0.12, 0.15, 0.2, 0.3}

RX J0720.4-3125

M/R=0.01 0.1

0.2 0.3

{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