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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

- Assumptions
- Radiation Transfer Equation
------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

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)

- 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

Spontaneous emission

Absorption

Induced emission

Scattering

n

τ>>1 ,

(1) Integrate all solid angle and divide by 4π

(2) Times μ ,then integrate all solid angle and divide by 4π

n

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

(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

From (2) ,

And according to D.A.

substitute into (1) ,

where

I(τ1,-μ,)=0

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

- Outer boundary

at τ=0

- Inner boundary

∫ dΩ

at τ=∞

[BC1]

∫μdΩ

at τ=∞

[BC2]

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

Initial conditions

- 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

∫ dΩ

∫μdΩ

Note:

Jν= ∫I ν dΩ/4π

Hν= ∫I νμdΩ/4π

Kν= ∫I νμ2dΩ/4π

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

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

Effective temperature = 106 K

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

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

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

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

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

- 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

Te=106 K intensity of gray temperature profile

ν=1017 Hz

Te=106 K Total flux of gray temperature profile