1. Coronal magnetic fields Thomas Wiegelmann, MPI for Solar-System Research, (Former: MPI für Aeronomie) Katlenburg-Lindau
Why are coronal magnetic fields important?
Models for magnetic field reconstruction.
Potential magnetic fields
Linear force-free fields
Non-linear force-free fields
Magnetic fields and coronal tomography
Conclusions

2. Why are coronal magnetic fields important?
Magnetic fields couples the solar interior, photosphere and atmosphere.
Magnetic field dominates in the solar corona. (Magnetic pressure >> Plasma pressure).
Knowledge of the coronal B-Field is essential to understand dynamic phenomena like coronal mass ejections and flares.

3. How to obtain coronal B-Fields
Direct measurements are extremely difficult.
Measure B on the photosphere (line of sight B or vector B) and extrapolate it into the corona.
We need assumptions regarding coronal currents: - No currents  Potential fields  Linear Force Free Fields
 Non Linear Force Free Fields

4. Coronal magnetic field models
Mathematics
Observations needed
Validity
Potential
Fields
Line of sight magnetogram
(Global) current free regions, quiet sun
Linear Force-
Free Fields
LOS magnetogram
+ observations of plasma structures
Local in active regions, low-beta plasma
Non Linear Force-Free
Vectormagnetogram
(3 times more data, ambiguities, noise)
Active regions, low beta plasma in low corona
MHS
Equilibrium
+ Tomographic Inversion of density
Helmet streamer, finite beta plasma, full solar corona

5. Global Potential Field reconstruction

6. Linear Force-Free Fields

7. Linear Force-Free Fields

8. Linear Force-Free Fields

9. Linear Force-Free Fields
EIT-image and projected magnetic field lines.
(α · L=2)
3D magnetic field lines with Kitt Peak magnetogram

10. Non-linear force-free fields
Why do we need non-linear force-free fields? - In general alpha changes in space. - Potential and linear force-free fields have no free energy to be released during an eruption.
The computation is much more difficult: - Mathematical difficulties due to non-linearity. - Vector magnetograms have ambiguities. - Transversal B-field is very noisy. - Limited field of view for current instruments. (Soon: Full disc vector magnetograph SOLIS.)

11. Non-linear force-free fields
Potential field and non-linear force free reconstruction of
a model active region regarding the same line of sight photospheric magnetic field. Our optimization code reconstructs the original analytic solution within the discretisation error.

12. Magnetic fields and coronal tomography
Coronal tomography uses line of sight integrals of the coronal density from different viewpoints.
Aim: 3D reconstruction of coronal density structure.
Density and magnetic field have to be reconstructed selfconsistently (MHS-equations + observational data).

13. Magnetic fields and coronal tomography
Use only line of sight density integrals.
Use only magnetic field data.
Use both line of sight density integrals and magnetic field as regularization operator.

14. Conclusions
Potential magnetic fields and linear force free fields are popular due to their mathematic simplicity and available data. (e.g. from MDI on SOHO, Kitt Peak)
Nonlinear force free fields are necessary to describe active regions exactly. More challenging both observational and mathematical.
A consistent 3D model of the solar corona requires tomographic inversion and magnetic reconstruction in one model.