Department of Physics

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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

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Outline

• Assumptions
• Radiation Transfer Equation
• ------Diffusion Approximation
• Improved Feautrier Method
• Temperature Correction
• Results
• Future work

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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

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Spectrum
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P(t) ?(t) T(t)
Improved Feautrier Method
Flux const
Spectrum
Radiation transfer equation
Diffusion Approximation
Flux ?const
Unsold Lucy process
Temperature correction
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The structure of neutron star atmosphere

• Gray atmosphere (Trail temperature profile)
• Equation of state
• Oppenheimer-Volkoff

The Rosseland mean depth
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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)
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Parameters In this Case

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

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Flux const
Spectrum
Diffusion Approximation
Flux ?const
Unsold Lucy process
Temperature correction
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Radiation Transfer Equation
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Diffusion Approximation
tgtgt1 ,
(1) Integrate all solid angle and divide by 4p
(2) Times µ ,then integrate all solid angle and
divide by 4p
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Diffusion Approximation
We assume the form of the specific intensity is
always the same in all optical depth
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Radiation Transfer Equation
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Radiation Transfer Equation
(1) Integrate all solid angle and divide by 4p
(1)
(2) Times µ ,then integrate all solid angle and
divide by 4p
(2)
Note J? ?I ? dO/4p H? ?I ?µdO/4p
K? ?I ? µ2dO/4p
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Radiation Transfer Equation
From (2) ,
And according to D.A.
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Radiation Transfer Equation
substitute into (1) ,
where
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RTE---Boundary Conditions
I(t1,-µ,)0
t1,t2,t3, . . . . . . . . . . . . .
. . . . . . . . . . . . . .,tD
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RTE---Boundary Conditions

• Outer boundary

at t0
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RTE---Boundary Condition

• Inner boundary

at t8
? dO
BC1
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RTE---Boundary Condition
?µdO
at t8
BC2
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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
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Improved Feautrier Method
Initial conditions
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Improved Feautrier Method

• Put the inner boundary condition into the
  relation , we can get the uu (t)
• ? F F (t)
• Choose the delta-logtau0.01
• from tau10-7 1000
• Choose the delta-lognu0.1
• from freq.1015 1019

Note first, we put BC1 in the relation
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Spectrum
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Unsold-Lucy Process
? dO
?µdO
Note J? ?I ? dO/4p H? ?I ?µdO/4p
K? ?I ? µ2dO/4p
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Unsold-Lucy Process
define B ?B? d? , J ?J? d?, H ?H? d?, K ?K?
d?
define Planck mean ?p ??ff B? d? /B
intensity mean ?J ? ?ff J? d?/J
flux mean ?H ?(?ff?sc )H? d?/H
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Unsold-Lucy Process
Eddington approximation J(t)3K(t) and
J(0)2H(0)
Use Eddington approximation and combine above two
equation
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Spectrum
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Results
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Effective temperature 106 K
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Te106 K
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Te106 K
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Te106 K
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Te106 K
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Te106 K
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Te106 K frequency1017 Hz
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Te106 K
Spectrum
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BC1 vs BC2
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BC1 vs BC2
Te106 K
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BC1 vs BC2
Te106 K
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• 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

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Diffusion Approximationvs Non-Diffusion
Approximation
This part had been calculated by Soccer
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Te106 K
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Te106 K
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Te106 K frequency1016 Hz
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Te106 K frequency1017 Hz
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Te106 K frequency1018 Hz
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Te106 K
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Te5105 K
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Te5105 K
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Te5105 K
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Te5106 K
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Te5106 K
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Te5106 K
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• The results with higher effective temperature are
  more closed to Non-DA solutions than with lower
  effective temperature
• When ? 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

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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

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To Be Continued.
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Te106 K intensity of gray temperature
profile
?1017 Hz
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Te106 K Total flux of gray temperature
profile