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. Coronal Magnetic Field.

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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

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


Coronal magnetic field

Coronal Magnetic Field

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


Measurements of magnetic field effects in the corona are difficult but possible

Measurements of magnetic field effects in the corona are difficult but possible

  • 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.


Longitudinal zeeman effect

Longitudinal Zeeman-effect

  • Weak field (<10G)

  • High temperature (106 K)

Magnetograph formula:

Lin, Penn & Tomczyk 2000


Hanle effect

Hanle – effect

  • 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)


Is this the kind of data which can be used in the vector tomography to reconstruct b

Is This the Kind of Data Which Can Be Used in the Vector Tomography to Reconstruct B?

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


A general problem with vector tomography

A General Problem with Vector Tomography

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


Vector field tomography regularization

Vector Field Tomography: Regularization

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


Vector field tomography 2d example for zeeman effect

Vector Field Tomography:2D Example for Zeeman-effect

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.


Reconstruction for zeeman effect

Vertical

cross-section

Original Field

Equatorial

cross-section

Reconstruction with

only FdivB-term included

Reconstruction with

Zeeman- (Faraday-) effect included

Reconstruction for Zeeman-effect


Reconstruction for hanle effect

Reconstruction for Hanle-effect

Vertical

cross-section

Original Field

Equatorial

cross-section

Reconstruction with

only FdivB-term included

Reconstruction with

Hanle-effect included


Reconstruction for zeeman hanle effect

Reconstruction for Zeeman-, Hanle-effect

Zeeman-effect (solid bars)

Hanle-effect (solid bars)

Dashed bars - potential field reconstruction

Angle between original vector and reconstructed one [°]

Errors in absolute value [%]


Conclusion

Conclusion

  • 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.


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