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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
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OutlineOutline
AssumptionsAssumptions Radiation Transfer EquationRadiation Transfer Equation ------Diffusion Approximation------Diffusion Approximation Improved Feautrier MethodImproved Feautrier Method Temperature CorrectionTemperature Correction ResultsResults Future workFuture work
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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
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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
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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
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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
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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
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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
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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
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Absorption Spontaneous emission
Induced emission Scattering
I
dldz
n
Radiation Transfer EquationRadiation Transfer Equation
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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
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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
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Radiation Transfer EquationRadiation Transfer Equation
* * '( )
4ff sc ff sc
R R R R
dI dI B pI
d
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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
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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) ,
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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
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RTE---Boundary ConditionsRTE---Boundary Conditions
I(τ1,-μ,)=0
τ1,τ2,τ3, . . . . . . . . . . . . . . . . . . . . . . . . . . .,τD
BI B
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RTE---Boundary ConditionsRTE---Boundary Conditions
Outer boundaryOuter boundary
0I
at τ=0
2
cF u
1 3
2R
duu
d
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RTE---Boundary ConditionRTE---Boundary Condition
Inner boundaryInner boundary
3
4
cI u F
c
BI B
Bu u at τ=∞
[BC1]
∫ dΩ
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RTE---Boundary ConditionRTE---Boundary Condition
B
R R
du du
d d
∫μdΩ
BB
R R
du duu u
d d
[BC2]
at τ=∞
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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
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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
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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
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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
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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
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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
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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
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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
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ResultsResults
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Effective temperature = 106 K
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5.670*1019 ±1%
Te=106 K
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Te=106 K
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Te=106 K
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Te=106 K
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Te=106 K
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Te=106 K frequency=1017 Hz
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Spectrum Te=106 K
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BC1BC1 vs vs BC2BC2
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BC1 vs BC2Te=106 K
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BC1 vs BC2Te=106 K
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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
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Diffusion ApproximationDiffusion Approximation
vsvs
Non-Diffusion ApproximationNon-Diffusion ApproximationThis part had been calculated by Soccer
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Te=106 K
1.2137*106 K
1.0014*106 K4.2627*105 K
3.7723*105 K
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Te=106 K
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Te=106 K frequency=1016 Hz
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Te=106 K frequency=1017 Hz
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Te=106 K frequency=1018 Hz
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Te=106 K
6.3096*1016 Hz 7.9433*1016 Hz
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Te=5*105 K
5.04614*105 K2.5192*105 K
1.9003*105 K
6.0934*105 K
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Te=5*105 K
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Te=5*105 K
3.1623*1016 Hz 5.0119*1016 Hz
3.9811*1016 Hz
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Te=5*106 K
2.1211*106 K
1.9084*106 K
4.7285*106 K
5.7082*106 K
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Te=5*106 K
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Te=5*106 K
3.9811*1017 Hz
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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
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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
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To Be Continued…….To Be Continued…….
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Te=106 K intensity of gray temperature profile
ν=1017 Hz
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Te=106 K Total flux of gray temperature profile