Model spectra of neutron star surface thermal emission diffusion approximation
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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

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

Department of Physics

National Tsing Hua University

G.T. Chen

2005/11/3


Outline
Outline

  • Assumptions

  • Radiation Transfer Equation

    ------Diffusion Approximation

  • Improved Feautrier Method

  • Temperature Correction

  • Results

  • Future work


Assumptions
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


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


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


The structure of neutron star atmosphere

  • Gray atmosphere(Trail temperature profile)

  • Equation of state

  • Oppenheimer-Volkoff

The Rosseland mean depth


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)


Parameters in this case
Parameters In this Case

  • First ,we consider the effective temperature is 106 K and effective gravity is 1014 cm/s2


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


Radiation transfer equation

Spontaneous emission

Absorption

Induced emission

Scattering

n

Radiation Transfer Equation


Diffusion approximation
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 approximation1

n

Diffusion Approximation

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



Radiation transfer equation2
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


Radiation transfer equation3
Radiation Transfer Equation

From (2) ,

And according to D.A.


Radiation transfer equation4
Radiation Transfer Equation

substitute into (1) ,

where


Rte boundary conditions
RTE---Boundary Conditions

I(τ1,-μ,)=0

τ1,τ2,τ3, . . . . . . . . . . . . . . . . . . . . . . . . . . .,τD


Rte boundary conditions1
RTE---Boundary Conditions

  • Outer boundary

at τ=0


Rte boundary condition
RTE---Boundary Condition

  • Inner boundary

∫ dΩ

at τ=∞

[BC1]


Rte boundary condition1
RTE---Boundary Condition

∫μdΩ

at τ=∞

[BC2]


Improved feautrier method
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 method1
Improved Feautrier Method

Initial conditions


Improved feautrier method2
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


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


Unsold lucy process
Unsold-Lucy Process

∫ dΩ

∫μdΩ

Note:

Jν= ∫I ν dΩ/4π

Hν= ∫I νμdΩ/4π

Kν= ∫I νμ2dΩ/4π


Unsold lucy process1
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


Unsold lucy process2
Unsold-Lucy Process

Eddington approximation: J(τ)~3K(τ)

and J(0)~2H(0)

Use Eddington approximation and combine above two equation


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




5.670*1019 ±1%

Te=106 K


Te=106 K


Te=106 K


Te=106 K


Te=106 K


Te=106 K frequency=1017 Hz


Te=106 K

Spectrum


Bc1 vs bc2
BC1 vs BC2


Te=106 K

BC1 vs BC2


Te=106 K

BC1 vs BC2


  • 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


Diffusion approximation vs non diffusion approximation
Diffusion ApproximationvsNon-Diffusion Approximation

This part had been calculated by Soccer


1.2137*106 K

1.0014*106 K

4.2627*105 K

3.7723*105 K

Te=106 K


Te=106 K


Te=106 K frequency=1016 Hz


Te=106 K frequency=1017 Hz


Te=106 K frequency=1018 Hz


6.3096*1016 Hz

7.9433*1016 Hz

Te=106 K


6.0934*105 K

2.5192*105 K

5.04614*105 K

1.9003*105 K

Te=5*105 K



3.9811*1016 Hz

3.1623*1016 Hz

5.0119*1016 Hz

Te=5*105 K


5.7082*106 K

2.1211*106 K

4.7285*106 K

1.9084*106 K

Te=5*106 K



3.9811*1017 Hz

Te=5*106 K


  • 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
Future Work closed to Non-DA solutions than with lower effective temperature

  • 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


To be continued
To Be Continued……. closed to Non-DA solutions than with lower effective temperature


Te=10 closed to Non-DA solutions than with lower effective temperature6 K intensity of gray temperature profile

ν=1017 Hz


Te=10 closed to Non-DA solutions than with lower effective temperature6 K Total flux of gray temperature profile


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