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3D Inversion of the Magnetic Field from Polarimetry Data of Magnetically Sensitive Coronal IonsPowerPoint Presentation

3D Inversion of the Magnetic Field from Polarimetry Data of Magnetically Sensitive Coronal Ions

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3D Inversion of the Magnetic Field from Polarimetry Data of Magnetically Sensitive Coronal Ions

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3D Inversion of the Magnetic Field from Polarimetry Data of Magnetically Sensitive Coronal Ions

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3D Inversion of the Magnetic Field from Polarimetry Data of Magnetically Sensitive Coronal Ions

M. Kramar, B. Inhester

Max-Planck Institute for Solar System Research

GERMANY

COSPAR, July 2004, Paris

Magnetic field contains the dominant energy per unit volume in the solar corona

- State-of-the-art determination of the coronal magnetic field:
- Extrapolation of photospheric magnetic sources which are measured with
the Zeeman-effect in photospheric lines.

- MHD simulations.
Disadvantage: These methods are very ill-posed, small errors in the photospheric

magnetic field measurement cause big uncertainty in the corona.

Difficulties of direct measurements at optical wavelengths :

- Magnetic fields in the quiet-Sun corona are weak (~10G)
- Coronal plasma is extremely hot (~106 K)

line broading more

bigger than Zeeman

splitting

- Faraday - effect
Rotation of polarization plane of polarized light coming from radio-sources and passing through the corona

- Hanle - effect
Degree and orientation of linear polarization of light scattered by coronal FeXIII and FeXIV ions.

- Longitudinal Zeeman - effect
Line splitting of circular polarized infrared light scattered by coronal FeXIII ions.

- Weak field (<10G)
- High temperature (106 K)

Magnetograph formula:

Lin, Penn & Tomczyk 2000

- Resonance scattering for λ ,
- which lifetime >> Larmor period

- From measuring Stokes U,Q we
- obtain the orientation of B in the
- plane of the sky (POS).
- No magnitude of B estimation available

Polarized intensity map of the

FeXIII line emission

(Habbal S.R. et al, ApJ 558, 2001)

Example for Faraday-effect

Contrary to scalar-field tomography,

the integrand now depends on the

direction the volume element is looked at.

Data (for Faraday-effect):

For 3-D case we have 3 times more

variables to be found than for scalar

field with the same number of equations

Depending on S||,,…, divergence-free or source-free fields, or

combinations are in the null-space of the tomography operator.

For example, for Zeeman-effect data we have:

measurements of

Irrotational

component

cannot be

reconstructed

Original Field

Reconstruction

Solenoidal

component

can be uniquely

reconstructed

It is necessary to introduce additional information about field.

Magnetic field is divergencefree:

Should be

minimized

- Nice properties of this regularization:
- make the use of photospheric B observation as bounary conditions
- reproduce standard potential B if FDivB alone is minimized

Reconstruction ignoring any tomography data

and minimizing FdivB-term alone.

Result of a reconstruction using a random

9% selection of a complete tomography

data set.

Original Field

Result of a reconstruction using a random

48% selection of a complete tomography

data set.

Vertical

cross-section

Original Field

Equatorial

cross-section

Reconstruction with

only FdivB-term included

Reconstruction with

Zeeman- (Faraday-) effect included

Vertical

cross-section

Original Field

Equatorial

cross-section

Reconstruction with

only FdivB-term included

Reconstruction with

Hanle-effect included

Zeeman-effect (solid bars)

Hanle-effect (solid bars)

Dashed bars - potential field reconstruction

Angle between original vector and reconstructed one [°]

Errors in absolute value [%]

- Inversion code for tomographic reconstruction of vector field has been written
- Vector tomography on the basis of Faraday-, Hanle- and Zeeman-effect measurements can improve the reconstruction of magnetic field rather than it is possible from the surface observations alone.

Future plan

- Influence of data incompleteness on the reconstruction
- Reconstruction of the coronal magnetic field on the basis of real data from polarization measurements during solar eclipse.