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

Outline  Assumptions  Radiation Transfer Equation Diffusion Approximation Diffusion Approximation  Improved Feautrier Method  Temperature Correction  Results  Future work

 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 Assumptions

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

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

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

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

Parameters In this Case  First,we consider the effective temperature is 10 6 K and effective gravity is cm/s 2

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

Absorption Spontaneous emission Induced emission Scattering n Radiation Transfer Equation

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

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

Radiation Transfer Equation

(1) Integrate all solid angle and divide by 4π (2) Times μ,then integrate all solid angle and divide by 4π Note: J ν = ∫I ν dΩ/4π H ν = ∫I ν μdΩ/4π K ν = ∫I ν μ 2 dΩ/4π (1) (2)

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

Radiation Transfer Equation substitute into (1), where

RTE---Boundary Conditions I(τ 1,-μ,)=0 τ 1,τ 2,τ 3, ,τ D

RTE---Boundary Conditions  Outer boundary at τ=0

RTE---Boundary Condition  Inner boundary at τ= ∞ [BC1] ∫ dΩ

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

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

Improved Feautrier Method Initial conditions

Improved Feautrier Method  Put the inner boundary condition into the relation, we can get the u=u ()  Put the inner boundary condition into the relation, we can get the u=u (τ)  F = F ()  F = F (τ)  Choose the delta-logtau=0.01 from tau=10 -7 ~ 1000 from tau=10 -7 ~ 1000  Choose the delta-lognu=0.1 from freq.=10 15 ~ from freq.=10 15 ~ Note : first, we put BC1 in the relation

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

Unsold-Lucy Process ∫ dΩ ∫μdΩ Note: J ν = ∫I ν dΩ/4π H ν = ∫I ν μdΩ/4π K ν = ∫I ν μ 2 dΩ/4π

Unsold-Lucy Process define B= ∫B ν dν, J= ∫J ν dν, H= ∫H ν dν, K= ∫K ν dν define Planck 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

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

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

Results

Effective temperature = 10 6 K

5.670*10 19 ±1% Te=10 6 K

Te=10 6 K frequency=10 17 Hz

Spectrum Te=10 6 K

BC1 vs BC2

Te=10 6 K

BC1 vs BC2Te=10 6 K

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

Diffusion Approximation vs Non-Diffusion Approximation This part had been calculated by Soccer

Te=10 6 K *10 6 K *10 6 K *10 5 K *10 5 K

Te=10 6 K

Te=10 6 K frequency=10 16 Hz

Te=10 6 K frequency=10 17 Hz

Te=10 6 K frequency=10 18 Hz

Te=10 6 K *10 16 Hz7.9433*10 16 Hz

Te=5*10 5 K *10 5 K2.5192*10 5 K *10 5 K *10 5 K

Te=5*10 5 K

3.1623*10 16 Hz5.0119*10 16 Hz *10 16 Hz

Te=5*10 6 K *10 6 K *10 6 K *10 6 K *10 6 K

Te=5*10 6 K

3.9811*10 17 Hz

 The results with higher effective temperature are more closed to Non- DA solutions than with lower effective temperature  When θ is large, the difference between two methods is large  The computing time for this method is faster than another  The results comparing with Non-DA are not good enough

Future Work  Including magnetic field effects in R.T.E, and solve the eq. by diffusion approximation  Compare with Non-D.A. results  Another subject: One and two-photon process calculation One and two-photon process calculation

To Be Continued…….

Te=10 6 K intensity of gray temperature profile ν=10 17 Hz

Te=10 6 K Total flux of gray temperature profile