Department of PhysicsDepartment of PhysicsNational Tsing Hua UniversityNational Tsing Hua University

G.T. ChenG.T. Chen2005/11/32005/11/3

Model Spectra of Neutron Model Spectra of Neutron Star Surface Thermal Star Surface Thermal

EmissionEmission---Diffusion Approximation ---Diffusion Approximation

Model Spectra of Neutron Model Spectra of Neutron Star Surface Thermal Star Surface Thermal

EmissionEmission---Diffusion Approximation ---Diffusion Approximation

OutlineOutline

AssumptionsAssumptions Radiation Transfer EquationRadiation Transfer Equation ------Diffusion Approximation------Diffusion Approximation Improved Feautrier MethodImproved Feautrier Method Temperature CorrectionTemperature Correction ResultsResults Future workFuture work

Plane-parallel atmosphere( local model).Plane-parallel atmosphere( local model). Radiative equilibrium( energy transported Radiative equilibrium( energy transported

solely by radiation ) .solely by radiation ) . Hydrostatics. All physical quantities are inHydrostatics. All physical quantities are in

dependent of timedependent of time The composition of the atmosphere is fully The composition of the atmosphere is fully

ionized ideal hydrogen gas. ionized ideal hydrogen gas. No magnetic fieldNo magnetic field

AssumptionsAssumptions

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 process

Oppenheimer-VolkoffOppenheimer-Volkoff

Diffusion Approximation

SpectrumRadiation 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 atmosphere

Oppenheimer-VolkoffOppenheimer-Volkoff

Diffusion Approximation

The structure of neutron star atmosphereThe structure of neutron star atmosphere

Gray atmosphereGray atmosphere (Trail temperature profile)(Trail temperature profile)

Equation of stateEquation of state

Oppenheimer-VolkoffOppenheimer-Volkoff

4 43 2

4 3e RT T

31

2 2 2 2

*

*

4 2(1 )(1 )(1 )

R

dP Gm P z P Gm

dz z c mc zc

dPg

dz

dP g

d

The Rosseland mean depth R Rd dz

kT

mP p

2

0

0

1

1

R

Bd

T

Bd

T

*sc ff

The Rosseland mean opacity

where

If given an effective temperature( Te ) and effective gravity ( g* ) , we can get

( )

( )R

R

T T

P P

(The structure of NS atmosphere)

The structure of neutron star atmosphereThe structure of neutron star atmosphere

Parameters In this CaseParameters In this Case

First ,we consider the effective First ,we consider the effective temperature is 10temperature is 106 6 K and effective K and effective gravity is 10gravity is 101414 cm/s cm/s22

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 Method

Oppenheimer-VolkoffOppenheimer-Volkoff

Diffusion Approximation

Absorption Spontaneous emission

Induced emission Scattering

I

dldz

n

Radiation Transfer EquationRadiation Transfer Equation

Diffusion ApproximationDiffusion Approximation

4

cuJ B

4

3

3

4

BF

B F

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

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

BI B

Diffusion ApproximationDiffusion Approximation

3

4

cI u F

c

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

I

dldz

n

Radiation Transfer EquationRadiation Transfer Equation

* * '( )

4ff sc ff sc

R R R R

dI dI B pI

d

Radiation Transfer EquationRadiation Transfer Equation

* *ff ff

R R R

dHJ B

d

*ff sc

R R

dKH

d

(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

Hν= ∫I νμdΩ/4π

Kν= ∫I ν μ2dΩ/4π

(1)

(2)

4

4

cuJ

FH

Radiation Transfer EquationRadiation Transfer Equation

*R

ff sc R

dKH

d

2*

R

ff sc R

dF I d

d

3

4

cI u F

c

*

1

3 3R

ff sc R R

du duc cF

d d

And according to D.A.

From (2) ,

Radiation Transfer EquationRadiation Transfer Equation

* *1

3 ff ff B

R R R R

dudu u

d d

4Bu Bc

substitute into (1) ,

*ff sc

R

where

RTE---Boundary ConditionsRTE---Boundary Conditions

I(τ1,-μ,)=0

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

BI B

RTE---Boundary ConditionsRTE---Boundary Conditions

Outer boundaryOuter boundary

0I

at τ=0

2

cF u

1 3

2R

duu

d

RTE---Boundary ConditionRTE---Boundary Condition

Inner boundaryInner boundary

3

4

cI u F

c

BI B

Bu u at τ=∞

[BC1]

∫ dΩ

RTE---Boundary ConditionRTE---Boundary Condition

B

R R

du du

d d

∫μdΩ

BB

R R

du duu u

d d

[BC2]

at τ=∞

Improved Feautrier MethodImproved Feautrier Method

1

1i

i ii

uu

F

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

1

1

1

1

1

1

1

1

i ii i

i i

i i ii

i ii i

i

AFF H

C F

U A

AFC H

F

where

Improved Feautrier MethodImproved Feautrier Method

2

1 1

2

1 1

*

*

4ln

4ln

3

3( )

ii i i i i

ii i i i i

ffi

R i

ff Bi

R i

C

A

H

U u

1 1 2 1

1

3

20

F

Initial conditions

Improved Feautrier MethodImproved Feautrier Method

Put the inner boundary condition into the relaPut the inner boundary condition into the relation , we can get the u=u (tion , we can get the u=u (τ) )

F = F (F = F (τ)) Choose the delta-logtau=0.01Choose the delta-logtau=0.01 from tau=10from tau=10-7-7 ~ 1000 ~ 1000 Choose the delta-lognu=0.1Choose the delta-lognu=0.1 from freq.=10from freq.=101515 ~ 10 ~ 101919

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 process

Oppenheimer-VolkoffOppenheimer-Volkoff

Diffusion Approximation

Unsold-Lucy ProcessUnsold-Lucy Process

4

)( '**

dpIBI

d

dI

R

sc

R

ff

R

scff

4

)1()('

dpIdl

dlIedlBedlIdI

sc

kTh

ffkT

h

ffscff

Hd

dK

BJd

dH

R

scff

R

ff

R

ff

*

**

∫ dΩ

∫μdΩ

Note:

J ν= ∫I ν dΩ/4π

Hν= ∫I νμdΩ/4π

Kν= ∫I ν μ2dΩ/4π

Unsold-Lucy ProcessUnsold-Lucy Process

Hd

dK

BJd

dH

R

H

R

P

R

J

define B= ∫Bν dν , J= ∫J ν 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

30

3

4

0

194e

*

0

***

0

4

])0(2')'(3[

)4

,(

])0(2')'(3[

)4

10*5.69

4

TH*(

])0(2')'(3[

])0(2')'(3[

T

dHd

HdH

T

T

BT

TdBB

d

HdHdHB

d

dHHdHB

d

dHHdHB

P

R

R

H

P

J

P

R

R

H

P

J

P

R

R

H

P

J

P

R

R

H

P

J

Unsold-Lucy ProcessUnsold-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 process

Oppenheimer-VolkoffOppenheimer-Volkoff

Diffusion Approximation

ResultsResults

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

Spectrum Te=106 K

BC1BC1 vs vs BC2BC2

BC1 vs BC2Te=106 K

BC1 vs BC2Te=106 K

The results of using The results of using BC1 and BC2 BC1 and BC2 are are almost the same almost the same

BC1 has more physical meanings, so BC1 has more physical meanings, so we take the results of using BC1 to we take the results of using BC1 to compare with Non-diffusion compare with Non-diffusion approximation solutions calculated approximation solutions calculated by Soccerby Soccer

Diffusion ApproximationDiffusion Approximation

vsvs

Non-Diffusion ApproximationNon-Diffusion ApproximationThis part had been calculated by Soccer

Te=106 K

1.2137*106 K

1.0014*106 K4.2627*105 K

3.7723*105 K

Te=106 K

Te=106 K frequency=1016 Hz

Te=106 K frequency=1017 Hz

Te=106 K frequency=1018 Hz

Te=106 K

6.3096*1016 Hz 7.9433*1016 Hz

Te=5*105 K

5.04614*105 K2.5192*105 K

1.9003*105 K

6.0934*105 K

Te=5*105 K

Te=5*105 K

3.1623*1016 Hz 5.0119*1016 Hz

3.9811*1016 Hz

Te=5*106 K

2.1211*106 K

1.9084*106 K

4.7285*106 K

5.7082*106 K

Te=5*106 K

Te=5*106 K

3.9811*1017 Hz

The results with higher effective The results with higher effective temperature are more closed to Non-temperature are more closed to Non-DA solutions than with lower DA solutions than with lower effective temperatureeffective temperature

When θ is large , the difference When θ is large , the difference between two methods is largebetween two methods is large

The computing time for this method The computing time for this method is faster than anotheris faster than another

The results comparing with Non-DA The results comparing with Non-DA are not good enough are not good enough

Future WorkFuture Work

Including magnetic field effects in R.T.E, anIncluding magnetic field effects in R.T.E, and solve the eq. by diffusion approximation d solve the eq. by diffusion approximation

Compare with Non-D.A. results Compare with Non-D.A. results

Another subject: Another subject: One and two-photon process calculation One and two-photon process calculation

To Be Continued…….To Be Continued…….

Te=106 K intensity of gray temperature profile

ν=1017 Hz

Te=106 K Total flux of gray temperature profile