Modelling stellar atmospheres with full Zeeman treatment

Modelling stellar atmosphereswith full Zeeman treatment

Katharina M. Bischof, Martin J. Stift

M. J. Stift’s Supercomputing GroupFWF project P16003

Institute f. AstronomyVienna, Austria

CP#AP workshop 12th of September 2007

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MotivationThe CAMAS Code

Findings

Contents

1 Motivation

2 The CAMAS Code

3 Findings

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Findings

Contents

1 Motivation

2 The CAMAS Code

3 Findings

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Findings

Contents

1 Motivation

2 The CAMAS Code

3 Findings

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FindingsOutline of the problem

Outline of the problem

Bagnulo et al., 2001, A&A 369, 889

Properties of magnetic CP stars:upper main sequence starsTeff ranging from 8.000 to 15.000 Kspectrum and photometric variabilitypeculiar and stratified abundancesmagnetic fields,|B| ranging from ∼ 100 G to 35 kG,up to 100 kG at magnetic poles

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Bagnulo et al., 2001, A&A 369, 889


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Bagnulo et al., 2001, A&A 369, 889


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Bagnulo et al., 2001, A&A 369, 889


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requirements specification for the atmospheric model codetemperature range (of interest): 8.000 to 15.000 Kpeculiar and stratified abundancesarbitrary inclination of the field in plane parallel modelsfull Zeeman treatment, polarised Feautrier solverhydrostatic equilibrium with magnetic pressureVALD line data, CoCoSsimplifications: plane parallel, Kurucz continuum routines,no dynamic phenomena considered, no microturbulence.

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Findings

Software engineeringPolarised radiation transferTemperature correction

The CAMAS Code

Program features:ATLAS12 continua for comparabilitywith standard modelsconsistent with spectral synthesis(COSSAM) and radiative diffusion(CARAT) codewritten in Ada95thread parallelmodularised, object oriented

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Findings


The CAMAS Code


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Findings


The CAMAS Code


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Findings


The CAMAS Code


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Findings


Polarised radiation transfer equation

ddz I = −K I + K (S,0,0,0)†

Stokes vectorI = (I,Q,U,V )†

absorption matrixK = κc 1 + κo Φ

line absorption matrix

Φ =

φI φQ φU φVφQ φI φ

′

V −φ′UφU −φ′V φI φ

′

QφV φ

′

U −φ′Q φI

γ

χ

~B

~x ′ ~y ′

~z ′,~I

Show details

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Findings


Zeeman Feautrier solver

Feautrier equation can be generalised to the magneticcase in the presence of blends (see Alecian and Stift, 2004, A&A

416, 703)

d~Jdτ5000

= X~H and d~Hdτ5000

= X(~J − ~S) with X := Kκ5000µ

ddτ5000

(X−1 d~Jdτ5000

) = X(~J − ~S) (inner points)

dX−1

dτ5000

d~Jdτ5000

+ X−1 d2~Jdτ2

5000= X(~J − ~S) (boundary condition)

N equations, N unknowns

B1~J1 − C1

~J2 = ~L1

− An ~Jn−1 + Bn ~Jn − Cn ~Jn+1 = ~Ln

− AN~JN−1 + BN

~JN = ~LN

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Findings


Temperature correction

Dreizler’s Lucy Unsold scheme adapted for polarised radiationtransport equations:(see Dreizler, 2003, ASPC 288, 69)

momenta of the polarised radiation transport equationdifferences (∆X = XEquilibrium − XModel) of flux, intensity andradiation pressuretwo flux criteria

local balance of emitted versus absorbed energyd(HI )zdτ5000

= 0 (constant flux)cannot be used in deep layers where the atmospherebecomes diffusive and S ∼ J ∼ local Planck function.nonlocal condition of constant flux:∞∫0

∮HIdΩdν − σ

4πT 4eff = 0 (desired value of the flux)

inefficient in regions with small opacitiesShow momenta

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Findings


Combining the flux criteria

∆T =π

4σT 3„d1

„SR∞

0 [Kν~Jν ]IdνR∞0 [Kν ]I,ISνdν

− S«

| z local energy conservation

+ d2SR∞

0 [Kν~Jν ]IdνJIR∞

0 [Kν ]I,ISνdνf0∆HI(0)

fg| z surface flux

+ d3SR∞

0 [Kν~Jν ]IdνJIR∞

0 [Kν ]I,ISνdν1fZ τ

0

R∞0 [Kν ~Hν ]Idν

HIκ5000∆HIdτ5000| z

global energy conservation

«

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Findings

Comparison with Carpenter’s resultsComparisons with LLMODELS

Summary

Results and comparisons

Two effectsenhanced line blanketingmagnetic pressure

Comparison with the results ofCarpenter (enhanced line blanketing and magneticpressure see Carpenter, 1985, ApJ 289, 660, Carpenter, 1983, PhD Thesis )LLMODELS (enhanced line blanketing, see Kochukhov et al., 2005,

A&A 433, 671, Khan and Shulyak, 2006a, A&A 448, 1153, Khan and Shulyak,

2006b, A&A 454, 933)

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Findings


Summary

Comparison with Carpenter’s results

CAMAS results Carpenter, 1985, ApJ 289, 660

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Findings


Summary

Comparisons with LLMODELS

differences between the field less and the “isotropic” model

CAMAS results Kochukhov et al., 2005, A&A 433, 671

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Findings


Summary

Comparisons with LLMODELS

differences between the field less and the “isotropic” model

CAMAS results Kochukhov et al., 2005, A&A 433, 671

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Findings


Summary

Depth scales

τross is affected by

(magnetic) line

blanketing, τ5000 is

not directly affected

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Findings


Summary

Depth scales

τross is affected by (magnetic) line blanketing, τ5000 is not directly affected

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Findings


Summary

Comparison with LLMODELS

differences between the anisotropic and the “isotropic” modelCAMAS results

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Findings


Summary


differences between the anisotropic and the “isotropic” modelKhan and Shulyak, 2006b, A&A 454, 933

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Findings


Summary


differences between the anisotropic and the “isotropic” model

CAMAS results Khan and Shulyak, 2006b, A&A 454, 933

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Findings


Summary

Summary

The Zeeman effect enhances the opacity. This affects theoptical depth and the temperature structure.CAMAS essentially confirms the results of LLMODELS. Theisotropic models are largely identical, however theanisotropic models with strong magnetic fields exhibitnotable differences that need to be clarified.The atmospheres computed with CAMAS will be used(among others) for the modelling of diffusion processes.

Acknowledgements: this work was supported by the Austrian Science Funds(FWF project P16003 of M.J.Stift). KMB wants to thank Bob Kurucz for theuseful discussions on the ATLAS code, Christian Stutz and the VALD team forthe line data and Holger Pikall for tree weeks of CPU-time.

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AppendixReferences

Bibliography I

Alecian, G. and Stift, M. J.: 2004, A&A 416, 703Bagnulo, S., Wade, G. A., Donati, J.-F., Landstreet, J. D.,

Leone, F., Monin, D. N., and Stift, M. J.: 2001, A&A 369, 889Carpenter, K. G.: 1983, Ph.D. thesis, AA(Ohio State Univ.,

Columbus.)Carpenter, K. G.: 1985, ApJ 289, 660Dreizler, S.: 2003, in I. Hubeny, D. Mihalas, and K. Werner

(eds.), ASP Conf. Ser. 288: Stellar Atmosphere Modeling,Vol. 288 of ASP Conference Series, p. 69, Review oftemperature correction schemes, Damping

Khan, S. A. and Shulyak, D. V.: 2006a, A&A 448, 1153Khan, S. A. and Shulyak, D. V.: 2006b, A&A 454, 933Kochukhov, O., Khan, S., and Shulyak, D.: 2005, A&A 433, 671

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AppendixReferences


differences between the anisotropic and the “isotropic” model

CAMAS resultsKhan and Shulyak, 2006b, A&A 454, 933

Return

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AppendixReferences

first depth point

d~Jdτ5000

|1 = X1(~J1 − (1− e−τ1X1)~S1)

~J1 = ~J2−∆τ(2,1)d~Jdτ |1−

(∆τ(2,1))2

2d2~Jdτ2 |1− . . . (“forward” Taylor series)

A1 = 0

B1 = 1 + ∆τ(2,1)X1 −(∆τ(2,1))2

2X1

dX−1

dτ5000|1X1 +

(∆τ(2,1))2

2X2

1

C1 = 1

~L1 = (∆τ(2,1) −(∆τ(2,1))2

2X1

dX−1

dτ5000|1)X1(1− e−τ1X1 )~S1 +

(∆τ(2,1))2

2X2

1~S1

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AppendixReferences

inner depth points

ddτ

X−1 d~Jdτ

∣∣∣∣∣n

=X−1

n+ 12

~Jn+1−~Jn∆τ(n+1,n)

− X−1n− 1

2

~Jn−~Jn−1∆τ(n,n−1)

∆τ(n+1,n)+∆τ(n,n−1)

2

An =(Xn−1 + Xn)−1

∆τ(n,n−1)∆τ(n+1,n−1)

Bn =(Xn−1 − Xn)−1

∆τ(n,n−1)∆τ(n+1,n−1)+

(X−1n+1 + Xn)−1

∆τ(n+1,n)∆τ(n+1,n−1)+ Xn

Cn =(Xn+1 + Xn)−1

∆τ(n+1,n)∆τ(n+1,n−1)

~Ln = Xn~Sn

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AppendixReferences

Flowchart of CAMAS

read input, initialiseenter radiative equilibrium loop

check convectionrecompute equidistant τ5000 and interpolate model data

← EXIT if convergence criteria are metpretabulate continua if necessaryreselect lines if necessaryenter integration loop (> 90% of computation time)

independent tasks use a magnetic Feautrier solver for allwavelength points and add the results

calculate temperature correctionscalculate and apply pressure correctionssearch for noise in the flux distribution and checkconvergenceapply temperature corrections if not converged yetapply Ng acceleration (optional)

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AppendixReferences

Feautrier solver

solve radiation transport equation as boundary valueproblem

dI±

dτ5000= I± − S

define flux like and intensity like quantitiescombine equations for outward and inward raysdiscretize and add boundary conditions

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AppendixReferences

Feautrier solver

solve radiation transport equation as boundary valueproblemdefine flux like and intensity like quantities

Hn =I+n − I−n

2, Jn =

I+n + I−n

2

combine equations for outward and inward raysdiscretize and add boundary conditions

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AppendixReferences

Feautrier solver

solve radiation transport equation as boundary valueproblem

dI±

dτ5000= I± − S

define flux like and intensity like quantitiescombine equations for outward and inward rays

dJdτ5000

= H ,dH

dτ5000= J − S︸︷︷︸

d2Jdτ2

5000= J − S

discretize and add boundary conditions

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AppendixReferences

Feautrier solver

solve radiation transport equation as boundary valueproblemdefine flux like and intensity like quantitiescombine equations for outward and inward raysdiscretize and add boundary conditions

I−(τ = 0) = 0, no incident radiation on the surface

dJdτ5000

|τ=0 = J(τ = 0)

HN = I+N − JN , diffusive at innermost depth-point

dJdτ5000

|N = I+N − JN with I+

N = SN +dS

dτ5000|N

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AppendixReferences

Angle dependent opacity Return

line absorption terms

φI = 14 (2φp sin 2γ + (φr + φb)(1 + cos2 γ))

φQ = 14 (2φp − (φr + φb)) sin2 γ cos 2χ

φU = 14 (2φp − (φr + φb)) sin2 γ sin 2χ

φV = 12 (φr − φb) cos γ

φ p, b, r ... line absorption profiles

Faraday terms

φ′

Q = 14 (2φ

′p − (φ

′r + φ

′

b)) sin2 γ cos 2χ

φ′

U = 14 (2φ

′p − (φ

′r + φ

′

b)) sin2 γ sin 2χ

φ′

V = 12 (φ

′r − φ

′

b) cos γ

φ′

p, b, r ... anomalous dispersion profiles

θβ

ψ

~B~I

~x ~y

~z

γ

χ

~B

~x ′ ~y ′

~z ′,~I

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AppendixReferences

Combining the flux criteria

0th moment: d∆HIdτ5000

=R∞

0 [Kν∆~Jν ]Idνκ5000

−R∞

0 [Kν ]I,I∆Sνdνκ5000

dHeq

dτ5000= 0, 0th moment for dHmod

dτ5000R∞0 [Kν~Xν,eq]I dν

Xeq∼

R∞0 [Kν~Xν,mod]I dν

Xmodel

∆S∆T = 4σT 3

π

4σT 3

π ∆T =S

R∞0 [Kν~Jν ]IdνR∞

0 [Kν ]I,ISνdν − S + SR∞0 [Kν ]I,ISνdν

R∞0 [Kν~Jν ]Idν

JI∆JI

1st moment: d∆KIdτ5000

=R∞

0 [Kν∆~Hν ]Idνκ5000

integrationvariable Eddington factors fτ = Kτ

Jτand g = H0

J0

f ∆JI︸︷︷︸∆KI

= f0g ∆HI(0) +

∫ τ0

R∞0 [Kν∆~Hν ]Idν

κ5000dτ5000

Return

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Modelling stellar atmospheres with full Zeeman treatment