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

Thermal Emission Thermal Emission

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

Thermal Emission Thermal Emission

Soccer 2005.10.20

Assumptions

• Plane-parallel atmosphere( local model).

• Radiative equilibrium( energy transported solely by radiation ) .

• Hydrostatics.

• The composition of the atmosphere is fully ionized ideal hydrogen

gas.

• B~1012 gauss, T~ 106K, g*~1014cm/s2.

• All physical quantities are independent of time

The Structure of neutron star atmosphere

Radiation transfer equation

Temperature correction

Flux ≠const

Flux = constSpectrum

P(τ) ρ(τ) T(τ)

Feautrier or Improved Feautrier

Unsold Lucy process

Oppenheimer-VolkoffOppenheimer-Volkoff

The structure of neutron star atmosphere

• Gray atmosphere

• Equation of state

• Oppenheimer-Volkoff

4 4 63 210

4 3e eT T T K

kT

mP p

2

21

2 2 2 2

*

*

4 2(1 )(1 )(1 )

sc

dP Gm P z Gm

dz z c mc zc

dPg

dz

dP g

d

We adopted the Thomson depth, .scd dz

'

'

, , ,

, ' ' ' ',

( )( , ) [ ( , ) ( , )] ( , ) ( ) ( , )(1 ) ( )

2

( , ) ( , ) ( ) ( , , ) ( , ) ( )

hi i i i i kTff sc ff

h sci i jkTff

j

B ldI l k l k l k I l k l dl l k e l dl

dl k e I l k l dl i k j k I l k d l dl

d

Absorption Spontaneous emission

Induced emission Scattering

3

2

2 1

1h

kT

hB

c e

I

R

dldz

ň

Radiation transfer equation:

Electromagnetic wave in magnetized plasma

.dv q

m qE v B m vdt c

33333333333333333333333333333333333333333333333333333333

( )1 00

( )1 00

( )10 1 0

, 0.

, 0.

,| | | | .

i k r t

i k r t

i k r t

v v v e v

E E E e E

B B B e B B

3333333333333333333333333333333333333333333333333333 3333

3333333333333333333333333333333333333333333333333333333333333333333333

3333333333333333333333333333333333333333333333333333333333333333333333

1 11 12 2 2 2 2 2 2 2

( )[ ]

( ) ( )c c c

c c

E Eq qv i E

m m

33333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333333

We considered fully ionized hydrogen gas in homogenous magnetic field. The equation of motion of the gas is

Assuming cold plasma that is neglect the thermal motions of gas.

0

0

c

qB for electronsmc

qB for protonsmc

33333333333333

3333333333333333333333333333 1 i

0

1/ 2

2 2

2 2

4

0

0

0 0

1 , , 1

( ) , ( )

i i ii

ij

i i i

pc

P PJ J c M

t t

D E P

J q n v

ig

ig

v vu vg

u u

u v

3333333333333333333333333333333333333333333333333333333333333333333333

333333333333333333333333333333333333333333

3333333333333333333333333333

From the above formulas we can get the dielectric tensor for cold plasma.

If w>>wci, w>>wpi, we can neglect ion component.

Assuming neutral plasma that is J0=0 and neglecting the volume magnetic moment we have M=0.

The dielectric tensor describes the properties of the plasma in the magnetic field.

0

0

22

2

2 2 2

2

2 2 2

4 0

4 1 1

0

( ) 0

cos sin cos

0 0

sin cos 0 sin

i j i ij ij j

x

y

z

D

D DH J

c c t c t

BE

c t

B

k k k Ec

N ig N E

ig N E

N N E

33333333333333

33333333333333333333333333333333333333333333333333333333

3333333333333333333333333333

33333333333333

From Maxwell equations we can solve index of reflection( a complex number).

x

y

zB

k θ

Solving above equation we obtain N2 for X-mode and O-mode.

22

2 2 2 2 4 2 2 2

2( )[( ) ( )]

2( )( ) sin sin 4 ( ) cos

v u v v vN

v u v uv u v uv v

Plus sign for X-mode ; minus sign for O-mode.

x

y

zB

k θ

Then we can solve Ex, Ey, Ez in the coordinate that the magnetic field is parallel to z-axis.

Define

e+=(Ex+iEy)/21/2

e-=(Ex-iEy)/21/2

ez=Ez.

Here

|e+|2 +|e-|2 +|ez|2=1

The Thomson scattering opacity

2 2 21 12 2 22 2

1 1[ | | | | | | ](1 ) (1 )

j j j jsc sc ze e e

u u

* * 2 2 21 12 2 22 2

1 1[ | | | | | | ](1 ) (1 )

j j j jff ff ze g e g e g

u u

The free-free opacity

' ' '2

' ' 2 * * * 21 122 2

1 1 1( , , ) ( ) | |

1 1

i j i j i jscz z

p e

d ek i k j e e e e e e

d m m c u u i

'

'

, , ,

, ' ' ' ',

( )( , ) [ ( , ) ( , )] ( , ) ( ) ( , )(1 ) ( )

2

( , ) ( , ) ( ) ( , , ) ( , ) ( )

hi i i i i kTff sc ff

h sci i jkTff

j

B ldI l k l k l k I l k l dl l k e l dl

dl k e I l k l dl i k j k I l k d l dl

d

Absorption Spontaneous emission

Induced emission Scattering

3

2

2 1

1h

kT

hB

c e

I

R

dldz

ň

Radiation transfer equation:

'

'

* *, , ,

, ' ' ' '

[ ( , ) ( , )] ( , )( , ) ( )( , )

2

1( , , ) ( , )

i i iiff sc ffi

Rsc sc

sc j

jsc

k k kdI k BI k

d

di k j k I k d

d

cos Rdl dz cosR R *, , (1 )

hi i kTff ff e

BI

x

y

z

Θ is the angle between B and I.θB

θR

ΦR

n

Use diffusion approximation for inner boundary.

Boundary condition:

Ii(τ1,-μR)=0 Ii(τD, μR)=(B(τD)+ μR∂B(τD)/∂τ)/2

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

' '

'

'

'

* *' ' '

( , ) ( , )( , )

2

( , ) ( , )( , )

2

( )

2, , ( , , )

2

i ii

i ii

ii i

R

ii i i ij j

Rhemispherej

i i isc ff ffi i ij sc

sc sc sc

I k I kP k

I k I kR k

dPR

d

dRP S M P k

d

dBS M i k j k

d

Feautrier method

2 1( )R

hemisphere

d dPP S MP

d d

Combine above two equation and use matrix form for two modes.

We can solve P and then obtain R and intensity I immediately.

Boundary conditions:

PD=B/2

P1=R1

Unsold-Lucy Process(Mihalas , 1st edition ,1970)

tau tau

log(tau) tau

tem

pera

ture

tem

pera

ture

flux

flux

4i

i i i ij j

j

ii i

dHJ S J

d

dKH

d

∫ dΩ

∫μRdΩ

'

'

* *, , , , ' ' '1

( , , )2

i i iiff sc ff sci j

Rjsc sc sc

ddI BI i k j k I d

d d

' ' '

' '

'

* *, , , ,

*, ,

, ' ' ' '

, ' '

4 4 4

4

1( , , ) 4

4

1( , , )

4

i i i iff sc ff sci i i i

sc sc

i iff sci

sc

sc j ij j ij j

jj jsc

scij

sc

ij ij

d d dI I J

d

d di k j k I d I d J

d

d di k j k

d

'

'

* *, , , ,

*,

4

4 4 4

2 4

i i i iff sc ff sci i i i

R Rsc sc

iffi

sc

d

d d dI I H

B dS

J P

H

dHJ B

ddK

Hd

1 2 1 2 1 2

1 1 2 2

1 2

1 11 21 1 2 12 22 2

, ( ) , ( ) , ( )

( )

( )

[( 4 4 ) ( 4 4 ) ]

H

p

J

B B d J J J d H H H d K K K d

H H H d

B S S d

J J J d

0

0

4 19* *e

4

3

03

1[3 ( ') ' 2 (0)]

1[3 ( ') ' 2 (0)]

T 5.69*10(H , )

4 4

,4

1[3 ( ') ' 2 (0)]

4

JH

P P

JH

P P

JH

P P

dHB H d H

d

d HB H d H

d

H H H

T Bby B B d T

T

d HH d H

dT

T

Combine above two equation, we have

Assume:

J(τ)~3K(τ) , J(0)~2H(0)

1.The following results are in the condition of θB=0 that is surface

normal parallel to magnetic field.

2.Dellogtau=0.01, dellogfre=0.1, number of direction in hemisphere

is 25.

3.The magnetic filed=1012 Gauss, Teff=106 K, g*=1e14 cm/s2.

4.Only the radiation damping term was adopted in opacities.

*

*

( )

( )

( , , )

sc

i isc ff

i isc ff

sc

i

d dz

d dz

v k e

Further works…..

1. First, there are still some problem about the modes in index of refraction.

2. There are some gap where wave could not propagate. We should use

a reasonable way to deal with it.

3. Add other damping terms( radiation damping, Doppler damping, collision damping).

4. Introduce the opacities which include higher harmonic components.

Later powerpoints are prepared for questions………………

Why do we need two boundary conditions?

(0, ) 0...... 1 0( , )( ( , ), ( , ))...... .. :

( , ) ......0 1

IdIf I I boundary conditions

I Bd

I

τ

μ

I(0, μ)=0I(T, μ)=B

About 1E 1207-5209 In August 2002 by XMM-Newton from De Luca, Mereghetti, Caraveo, Moroni, Mignani, Bignami, 2004, ApJ 418.

supernova remnant G296.5+10.0

1E 1207.4-5209

Red represents photons in the 0.3-0.6 keV band, green and blue correspond to the 0.6-1.5 keV and 1.5-8 keV bands respectively.

P~424ms

P derivative~1.4*10-14ss-1

Figure 5: Fit of the phase-integrated data. The model (double blackbody plus line components) is described in the text. From top to bottom, the panels show data from the pn, the MOS1 and the MOS2 cameras. In each panel the data are compared to the model folded through the instrumental response (upper plot); the lower plot shows the residuals in units of sigma.

Figure 6: Residuals in units of sigma obtained by comparing the data with the best fit thermal continuum model. The presence of four absorption features at ~0.7 keV,~1.4 keV, ~2.1 keV and ~2.8 keV in the pn spectrum is evident. The three main features are also independently detected by the MOS1 and MOS2 cameras.

From pn: 0.68/0.24 : 1.36/0.18

Four absorption features have central energies colse to the ratio 1:2:3:4

The dispersion relation k-w for X-mode at θ=0 is

2 24

2c c p

c

k

ω

222

2[1 ]

( )p

c

kc

The dispersion relation k-w for O-mode at θ=0 is

2 24

2c c p

k

ω

222

2[1 ]

( )p

c

kc

The dispersion relation k-w for X-mode at θ=1.57 is

2 24

2c c p

2 2c p

2 2 2 2 222

2 2 2 2 2

( )[ ]( )

p c

c p

kc

k

ω

2 24

2c c p

The dispersion relation k-w for O-mode at θ=1.57 is

p

k

ω

222

2 2[1 ]pk

c

'

* *' ' ' '

* *

1 2 *

*

*

1( , , )

2

1

2

,

~2

(0) ( ) ~ ( )

x xsc sc

sc sc

i i iiff sc ffi jsc

Rjsc sc sc

i i iiff sc ffi i i

scisc sc sc

i isc ff

xxffsc

sc sc

xff

sc

ddI BI i k j k I d

d d

dI BI I

d

I I

dI BI

d

I I e Be I e

xsc

sc

Roughly estimate the criterion for outer boundary.

^__^

Thank you…………….