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Model Spectra of Neutron Star Surface Thermal Emission ---Diffusion Approximation

Model Spectra of Neutron Star Surface Thermal Emission ---Diffusion Approximation. Department of Physics National Tsing Hua University G.T. Chen 2005/11/3. Outline. Assumptions Radiation Transfer Equation ------Diffusion Approximation Improved Feautrier Method

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Model Spectra of Neutron Star Surface Thermal Emission ---Diffusion Approximation

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  1. Model Spectra of Neutron Star Surface Thermal Emission---Diffusion Approximation Department of Physics National Tsing Hua University G.T. Chen 2005/11/3

  2. Outline • Assumptions • Radiation Transfer Equation ------Diffusion Approximation • Improved Feautrier Method • Temperature Correction • Results • Future work

  3. Assumptions • Plane-parallel atmosphere( local model). • Radiative equilibrium( energy transported solely by radiation ) . • Hydrostatics. All physical quantities are independent of time • The composition of the atmosphere is fully ionized ideal hydrogen gas. • No magnetic field

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

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

  6. The structure of neutron star atmosphere • Gray atmosphere(Trail temperature profile) • Equation of state • Oppenheimer-Volkoff The Rosseland mean depth

  7. The structure of neutron star atmosphere The Rosseland mean opacity where If given an effective temperature( Te ) and effective gravity ( g* ) , we can get (The structure of NS atmosphere)

  8. Parameters In this Case • First ,we consider the effective temperature is 106 K and effective gravity is 1014 cm/s2

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

  10. Spontaneous emission Absorption Induced emission Scattering n Radiation Transfer Equation

  11. Diffusion Approximation τ>>1 , (1) Integrate all solid angle and divide by 4π (2) Times μ ,then integrate all solid angle and divide by 4π

  12. n Diffusion Approximation We assume the form of the specific intensity is always the same in all optical depth

  13. Radiation Transfer Equation

  14. Radiation Transfer Equation (1) Integrate all solid angle and divide by 4π (1) (2) Times μ ,then integrate all solid angle and divide by 4π (2) Note: Jν= ∫I ν dΩ/4π Hν= ∫I νμdΩ/4π Kν= ∫I νμ2dΩ/4π

  15. Radiation Transfer Equation From (2) , And according to D.A.

  16. Radiation Transfer Equation substitute into (1) , where

  17. RTE---Boundary Conditions I(τ1,-μ,)=0 τ1,τ2,τ3, . . . . . . . . . . . . . . . . . . . . . . . . . . .,τD

  18. RTE---Boundary Conditions • Outer boundary at τ=0

  19. RTE---Boundary Condition • Inner boundary ∫ dΩ at τ=∞ [BC1]

  20. RTE---Boundary Condition ∫μdΩ at τ=∞ [BC2]

  21. Improved Feautrier Method To solve the RTE of u , we use the outer boundary condition ,and define some discrete parameters, then we get the recurrence relation of u where

  22. Improved Feautrier Method Initial conditions

  23. Improved Feautrier Method • Put the inner boundary condition into the relation , we can get the u=u (τ)  F = F (τ) • Choose the delta-logtau=0.01 from tau=10-7 ~ 1000 • Choose the delta-lognu=0.1 from freq.=1015 ~ 1019 Note : first, we put BC1 in the relation

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

  25. Unsold-Lucy Process ∫ dΩ ∫μdΩ Note: Jν= ∫I ν dΩ/4π Hν= ∫I νμdΩ/4π Kν= ∫I νμ2dΩ/4π

  26. Unsold-Lucy Process define B= ∫Bν dν , J= ∫Jν dν, H= ∫Hν dν, K= ∫Kν dν define Planck mean κp= ∫κff* Bν dν /B intensity mean κJ= ∫ κff* Jν dν/J flux mean κH= ∫(κff*+κsc )Hν dν/H

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

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

  29. Results

  30. Effective temperature = 106 K

  31. 5.670*1019 ±1% Te=106 K

  32. Te=106 K

  33. Te=106 K

  34. Te=106 K

  35. Te=106 K

  36. Te=106 K frequency=1017 Hz

  37. Te=106 K Spectrum

  38. BC1 vs BC2

  39. Te=106 K BC1 vs BC2

  40. Te=106 K BC1 vs BC2

  41. The results of using BC1 and BC2 are almost the same • BC1 has more physical meanings, so we take the results of using BC1 to compare with Non-diffusion approximation solutions calculated by Soccer

  42. Diffusion ApproximationvsNon-Diffusion Approximation This part had been calculated by Soccer

  43. 1.2137*106 K 1.0014*106 K 4.2627*105 K 3.7723*105 K Te=106 K

  44. Te=106 K

  45. Te=106 K frequency=1016 Hz

  46. Te=106 K frequency=1017 Hz

  47. Te=106 K frequency=1018 Hz

  48. 6.3096*1016 Hz 7.9433*1016 Hz Te=106 K

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