1. Model Spectra of Neutron Star Surface Thermal Emission Soccer 2005.10.20

2. Assumptions Plane-parallel atmosphere( local model). Radiative equilibrium( energy transported solely by radiation ). Hydrostatics. The composition of the atmosphere is fully ionized ideal hydrogen gas. B~10 12 gauss, T~ 10 6 K, g*~10 14 cm/s 2. All physical quantities are independent of time

3. The Structure of neutron star atmosphere Radiation transfer equation Temperature correction Flux ≠const Flux = const Spectrum P(τ) ρ(τ) T(τ) Feautrier or Improved Feautrier Unsold Lucy process Oppenheimer-Volkoff

4. The structure of neutron star atmosphere Gray atmosphere Equation of state Oppenheimer-Volkoff We adopted the Thomson depth,

5. Absorption Spontaneous emission Induced emission Scattering ň Radiation transfer equation:

6. Electromagnetic wave in magnetized plasma We considered fully ionized hydrogen gas in homogenous magnetic field. The equation of motion of the gas is Assuming cold plasma that is neglect the thermal motions of gas.

7. From the above formulas we can get the dielectric tensor for cold plasma. If w>>w ci, w>>w pi, we can neglect ion component. Assuming neutral plasma that is J 0 =0 and neglecting the volume magnetic moment we have M=0. The dielectric tensor describes the properties of the plasma in the magnetic field.

8. From Maxwell equations we can solve index of reflection( a complex number). x y z B kθ

9. Solving above equation we obtain N 2 for X-mode and O-mode. Plus sign for X-mode ; minus sign for O-mode.

10. x y z B kθ Then we can solve E x, E y, E z in the coordinate that the magnetic field is parallel to z-axis. Define e + =(E x +iE y )/2 1/2 e - =(E x -iE y )/2 1/2 e z =E z. Here |e + | 2 +|e - | 2 +|e z | 2 =1

11. The Thomson scattering opacity The free-free opacity

12. Absorption Spontaneous emission Induced emissionScattering ň Radiation transfer equation:

13. B I x y z Θ is the angle between B and I. θBθB θRθR ΦRΦR n

14. Use diffusion approximation for inner boundary. Boundary condition: I i (τ 1,-μ R )=0I i (τ D, μ R )=(B(τ D )+ μ R ∂B(τ D )/∂τ)/2 τ 1,τ 2,τ 3,...........................,τ D

15. Feautrier method

16. Combine above two equation and use matrix form for two modes. We can solve P and then obtain R and intensity I immediately. Boundary conditions: P D =B/2 P 1 =R 1

17. Unsold-Lucy Process (Mihalas, 1st edition,1970) tau log(tau)tau temperature flux

18. ∫ dΩ ∫μ R dΩ

21. Combine above two equation, we have Assume: J(τ)~3K(τ), J(0)~2H(0)

22. 1.The following results are in the condition of θ B =0 that is surface normal parallel to magnetic field. 2.Dellogtau=0.01, dellogfre=0.1, number of direction in hemisphere is 25. 3.The magnetic filed=10 12 Gauss, T eff =10 6 K, g*=1e 14 cm/s 2. 4.Only the radiation damping term was adopted in opacities.

60. Further works….. 1. First, there are still some problem about the modes in index of refraction. 2. There are some gap where wave could not propagate. We should use a reasonable way to deal with it. 3. Add other damping terms( radiation damping, Doppler damping, collision damping). 4. Introduce the opacities which include higher harmonic components.

61. Later powerpoints are prepared for questions………………

62. Why do we need two boundary conditions? I τ μ I(0, μ)=0 I(T, μ)=B

63. About 1E 1207-5209 In August 2002 by XMM-Newton from De Luca, Mereghetti, Caraveo, Moroni, Mignani, Bignami, 2004, ApJ 418. supernova remnant G296.5+10.0 1E 1207.4-5209 Red represents photons in the 0.3-0.6 keV band, green and blue correspond to the 0.6-1.5 keV and 1.5-8 keV bands respectively. P~424ms P derivative~1.4*10 -14 ss -1

64. Figure 5: Fit of the phase-integrated data. The model (double blackbody plus line components) is described in the text. From top to bottom, the panels show data from the pn, the MOS1 and the MOS2 cameras. In each panel the data are compared to the model folded through the instrumental response (upper plot); the lower plot shows the residuals in units of sigma.

65. Figure 6: Residuals in units of sigma obtained by comparing the data with the best fit thermal continuum model. The presence of four absorption features at ~0.7 keV,~1.4 keV, ~2.1 keV and ~2.8 keV in the pn spectrum is evident. The three main features are also independently detected by the MOS1 and MOS2 cameras. From pn: 0.68/0.24 : 1.36/0.18 Four absorption features have central energies colse to the ratio 1:2:3:4

66. The dispersion relation k-w for X-mode at θ=0 is k ω

67. The dispersion relation k-w for O-mode at θ=0 is k ω

68. The dispersion relation k-w for X-mode at θ=1.57 is k ω

69. The dispersion relation k-w for O-mode at θ=1.57 is k ω

70. Roughly estimate the criterion for outer boundary.

71. ^__^ Thank you…………….