Diffusion in magnetic fields G. Alecian (CNRS, Observatoire de - - PowerPoint PPT Presentation

Diffusion in magnetic fields

• G. Alecian

(CNRS, Observatoire de Paris)

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Ap magnetic

Magnetic Ap star

Also:

• Ambipolar diffusion

(Babel & Michaud, 1991)

• Effect on models

(LeBlanc, Michaud & Babel, 1994)

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Ion diffusion velocity without magnetic field

VDi ≈ Dip − Ai − Zi −1 2       mp kT g + Ai mp kT gi

rad +…

     

Microscopic diffusion velocity of an ion (Zi) :

gravity

Radiative acceleration

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The effect of horizontal magnetic fields

• n the diffusion velocity
• Correcting factor in the direction orthogonal to magnetic lines
• The corrected average diffusion velocity (approximated) for horizontal field

ti is the « collision » time ωi / 2π is the cyclotron frequency fslow,i = 1+ ωi

2ti 2

( )

−1

VD ≈ Ni fslow,iVDi

i

∑

Ni

i

∑

ωi = ZeH mic 0 < fslow,i ≤1

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Some typical results

• Silicon: Vauclair, Hardorp & Peterson (1979)

have shown the consequence of the ions trapping by the magnetic field

Fig4a of Alecian & Vauclair (1981)

Similar studies for:

• Oblique rotator model (Michaud, Megessier

& Charland, 1981)

• Ga in Ap stars (Alecian & Artru, 1987)
• Ca, Sc, Ti, Mn, Cr, Sr in 53 Camelopardalis

(Babel & Michaud, 1991)

• Al in Ap stars (Hui Bon Hoa, Alecian &

Artru, 1996)

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Oblique magnetic lines

Alecian & Vauclair, 1981 : and, an horizontal component! VH,Zi = Vi fslow,i + 1 fslow,i sin2θ       VH,Xi = fslow,iVi 2 sin2θ

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Horizontal diffusion !

• Mégessier, 1984 (Si distribution on Bp and Ap stars with dipolar field and

applying the previous formulae) However, the time scale for the appearance of significant inhomogeneities through horizontal diffusion onto the stellar surface is about 107y ! This implies that magnetic structures should be stable over a long period. Moreover, time scales are much shorter for vertical diffusion, which remains dominant in the stratification process Therefore, abundance patches are more likely formed through the Vz component (by inhomogeneous vertical diffusion according to the local field angle).

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Radiative accelerations, with and without magnetic field

g i

rad =

n i ,k ni Am pc

k,m> k

∑

e.I

( )Ω dΩdν

Ω

∫

ν∫

gi

rad =

n i,k ni Am pc σ k,m

∞

∫

φ ν,n i

( ) dν

k,m >k

∑

(magnetic)

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The CARAT code

• Polarized radiation transfer in LTE (Alecian & Stift, 2004)

– VALD data base, Kurucz ATLAS9 models, plane-parallel – Magnetic field up to 60 000 Gauss, any angle – Detailed opacities up to 5 mA of resolution – radiative accelerations for 329 ions – parallel computing

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Effet du champ sur grad

Raie du Mg II

SrII λ 4077

6.0 5.5 5.0 4.5 4.0 3.5 log gv (SrII)

• 6
• 4
• 2

2 log τ5000 0T CARAT 0T BM 1.5T CARAT 1.5T CARAT without M-O 1.5T BM

(a)

6 5 4 3 2 1 log g (SrII)

• 6
• 4
• 2

2 log τ5000

log gv (SrII):

CARAT CARAT without M-O BM

log gh (SrII):

BM CARAT without M-O (-x) CARAT (+x) CARAT (-x) β = 60°

(b)

Teff=8500K

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Blends

6.0 5.5 5.0 4.5 4.0 log grad

• 6
• 4
• 2

2 log τ5000 5.0 4.8 4.6 4.4 4.2 4.0 3.8 6 5 4 3 2 0T, with blends 4T, with blends 0T, alone 4T, alone Cr Fe Ag

Effect of Blends

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Zeeman amplification

0.4 0.3 0.2 0.1 0.0

• 0.1
• 0.2

ε

• 6
• 4
• 2

2

log τ5000 4T vs 0T, 0°

Be B C O Mg Al Si P S Cl Ca Sc Ti Cr Mn Fe Co Ni Cu Zn Ga Ge Zr Ag Cd In Sn Pr Os Ir Au Hg Bi

log(amplification) Teff = 12 000K

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Preliminary results (complete computation with CARAT) Al

Teff=12000K

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Preliminary results (2) Al

comparing fluxes

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Future developments

• Stratification at equilibrium (already in progress for zero field case, see

LeBlanc et al.) –  Self-consistent models –  NLTE

• 2D stratifications (modelling oblique rotators)
• + all potentially important processes (ambipolar diffusion,

hydrodynamics,…)

• Connection to internal structure
• Far in the future: time-dependent stratifications (LTE,…)