1. 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

2. 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 (~10 6 K) line broading more bigger than Zeeman splitting

3. 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.

4. Longitudinal Zeeman-effect  Weak field (<10G)  High temperature (10 6 K) Magnetograph formula: Lin, Penn & Tomczyk 2000

5. Hanle – effect There is no information about magnetic field strength! FeXIII and FeXIV ions (Querfeld 1982)

6. 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)

7. Is This the Kind of Data Which Can Be Used in the Vector Tomography to Reconstruct B? 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 Example for Faraday-effect Contrary to scalar-field tomography, the integrand now depends on the direction the volume element is looked at.

8. A General Problem with Vector Tomography Irrotational component cannot be reconstructed Solenoidal component can be uniquely reconstructed measurements of Original FieldReconstruction 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:

9. 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 F DivB alone is minimized

10. Vector Field Tomography: 2D Example for Zeeman-effect Original Field Reconstruction ignoring any tomography data and minimizing F divB -term alone. Result of a reconstruction using a random 9% selection of a complete tomography data set. Result of a reconstruction using a random 48% selection of a complete tomography data set.

11. Reconstruction for Zeeman-effect Original Field Reconstruction with only F divB -term included Reconstruction with Zeeman- (Faraday-) effect included Vertical cross-section Equatorial cross-section

12. Reconstruction for Zeeman-effect Original Field Reconstruction with only F divB -term included Reconstruction with Zeeman- (Faraday-) effect included Vertical cross-section Equatorial cross-section

13. Reconstruction for Hanle-effect Original Field Reconstruction with only F divB -term included Reconstruction with Hanle-effect included Vertical cross-section Equatorial cross-section

14. 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 [%]

15. 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.