1. model spectra of neutron star surface thermal emission soccer 2005.1.6

2. Outline: Assumption and process Feautrier method Improved Feautrier method Unsold-Lucy Process Others …….

3. Assumptions Plane-parallel atmosphere( local model) Radiative equilibrium( energy transported solely by radiation ) Equation of state is ideal gas( ionized H) Hydrostatics No magnetic field The physical properties independent of time

4. 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

5. Feautrier method AbsorptionSpontaneous emission Induced emission Scattering ň Below we Omit frequency index ν

6. Feautrier method Inner boundary use diffusion approximation ….. Boundary condition: I(τ 1,-μ,)=0I(τ D,μ,)=B(τ D )+ μ∂B(τ D )/∂τ τ 1,τ 2,τ 3,...........................,τ D

7. Feautrier method τ>>1, I~B+μ ∂ B/ ∂ τ

9. Feautrier method

10. Ps: The index d mean depth, index i mean direction.

11. Feautrier method Using BD1: We have relation between P d-1 and P d Using BD2: We have P D then P D-1,P D-2 ………P 1

12. Feautrier method Using BD2: We can get R 1,R 2,…..,R D-1,R D. I(τ,μ)=P(τ,μ)+R(τ,μ) I(τ,-μ)=P(τ,μ)-R(τ,μ) Then we get intensity and flux. F (τ,μ)= ∫ I(τ,μ) μ dΩ

13. Feautrier method How to determine dellogtau, dellogfre, number of directions.

14. dellogtau=0.1, dellogtau=0.01 ---  6.4% dellogtau=0.01, dellogtau=0.001 ---  0.063%

15. Choose dellogfre=0.1 in my calaulation.

16. Choose number of directions = 9*2=18.

17. n θ φ I

18. Improved Feautrier method a=3.00000000004, b=3.00000000003 theoretical value numerical value error a+b 6.00000000007000000000 6.00000000007000000579 0.0000000000% a-b 0.00000000001000000000 0.00000000001000000083 0.0000082740% a*b 9.00000000021000000000 9.00000000021000090555 0.0000000000% a/b 1.00000000003333333333 1.00000000003333333361 0.0000000003% Error=(the-num)/the *100% The subtraction has larger error than others operations!!

19. Improved Feautrier method Feautrier: -A d P d-1 +B d P d -C d P d+1 =S d D d =(B d -A d D d-1 ) -1 C d Ψ d =(B d -A d D d-1 ) -1 (S d +A d Ψ d-1 ) The subtraction would induce more numerical round off error when B~AD. Rybicki & Hummer,A&A,245,1991 Improved Feautrier H d =-A d +B d -C d, D d =(1+F d ) -1 F d =C d-1 [H d +A d (1+F d-1 ) -1 ]F d-1 Ψ d =(1+F d ) -1 (C d ) -1 (S d +A d Ψ d-1 )

20. Compare Feautrier method and improved Feautrier method 16 effective digits : 6 effective digits : 6 effective digits :Feautrier improved Feautrier Feautrier Improved Feautrier method

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

26. Unsold-Lucy Process ∫ dΩ ∫μdΩ Ps: define J ν = ∫I ν dΩ/4π H ν = ∫I ν μdΩ/4π K ν = ∫I ν μ 2 dΩ/4π

27. Unsold-Lucy Process define B= ∫B ν dν, J= ∫J ν dν, H= ∫H ν dν, K= ∫K ν dν define plank mean κ p = ∫ κ ff * B ν dν /B intensity mean κ J = ∫ κ ff * J ν dν/J flux mean κ H = ∫( κ ff * + κ sc )H ν dν/H

28. Unsold-Lucy Process Use Eddington approximation and combine above two equation Eddington approximation: J(τ)~3K(τ), J(0)~2H(0)

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

30. Gray atmosphere: T 4 =3/4*Te 4 (2/3+τ)

31. Totalflux before temperature correction

32. 20 times Temperature correction 5.67e19±1%

33. Totalflux after temperature correction

34. 20 times Temperature correction

35. Temperature profile after 20 times temperature correction

38. frequency=1e17

40. Theta=0

41. Theta=pi

42. spectrum

43. Value of J/K

44. Then …. Check and test my code and formula is correct. Add magnetic field, line, ……