Magnetic field
extrapolations
Thomas Wiegelmann
Andreas Lagg
Sami K. Solanki
Max Planck Institute for Solar System ResearchMagnetic loops from new diagnosticof magnetic vector at coronal base
Magnetic loops
deduced frommeasurementsof He I 10830 Å
Stokes profiles
in an emerging
flux region.
Left projection:
Field strength
Right projection:
Vertical velocity
Solanki et al.
2003, Nature

Testing Magnetic Extrapolations
using B measurements
 Non-linear force-free fields reproduce the loops

reconstructed from observations signif. better than the linear
force-free ones which are much better than potential fields.
 Loops harbour strong currents while still emerging.

Observed

Potential

linear force freeWiegelmann et al. 2005

non-linear ff

What if coronal B-field is not known?
 Extrapolate from photospheric magnetic field(if linear force-free field, then with arbitrary value of α) Compute magnetic field lines and their projection

onto an EUV-image.

 Use EUV-image to compute intensity and intensity

gradient along the projected field lines.

 Check how well field lines and plasma features

agree (See next slide for a quantitative measure, C)

 For a linear force-free model repeat procedure with different

value α until global minimum in C is reachedCheck how well projected fieldlines and plasma features agreel∫0
C= l 2l ( ∫ ρ ds )0|∇ρ | ds

(Wiegelmannet al. Sol.
Phys. 2005)

where l is the projected loop length, ρ the intensity,∇ρ is the gradient along the field line.
The smaller the C, the better the reconstruction.
This measure gives higher weights to brightand long loops. (Used/Optimized for SOHO/EIT)Example: Active Region AR 7953
Linear force-free model, different values of force-free parameter α.

EIT-image & projected magnetic field lines for a potential
field (α = 0) (poor agreement)

Linear force-free fieldwith α = +0.01 [Mm-1](even worse agreement)Linear force-free field with
α = –0.01 [Mm-1](better agreement)

3D-magnetic field lines,
linear force-free α= – 0.01
[Mm-1]Example of use of 3D extrapolated magnetic field

downupSUMER Dopplergram in NeVIII(λ 77 nm) and a 2-D-projectionof some field lines.

Marsch et al. A&A 2004

Mass flux density inferred from
Dopplershift and intensity fromSUMER observations.

Extrapolations: some problems
 How much does noise in magnetograms influence

extrapols?

 White noise has all freq. Î affects all layers of extrapols
 Problem mainly for non-linear ff extrapols, since noise intransverse field > 10x larger than in longitudinal field 180o ambiguity, how to get round it?
 Only affects non-linear ff extrapolations.
 Major problem: information simply isn’t there. All methods

to resolve problem make assumptions about field.

 ÎUse comparison with loops to solve problem?

 Can thermal energy of gas really be neglected?
 Not a problem for HMI+AIA (lower corona)?

Extrapolations: some problems
 How much does noise in magnetograms influence

extrapols?

 White noise has all freq. Î affects all layers of extrapols
 Problem mainly for non-linear ff extrapols, since noise intransverse field > 10x larger than in longitudinal field 180o ambiguity, how to get round it?
 Major problem: information simply isn’t there. All methods

to resolve problem make assumptions about field.

 ÎUse comparison with loops to solve problem?

 Can thermal energy of gas really be neglected?
 Mainly a concern in outer corona (streamers)

Ambiguity

180 degreeambiguity
The method initiallychooses the ambiguity
to minimize the angle
with a corresponding
potential magnetic field.
In subsequent steps
flips in the ambiguity ofadjacent points areremoved. The method
is limited to strong fieldregions, where thesignal for thetransversal field iswell above the noiselevel.

Solar-B and extrapolations
 Solar-B: ideal mission!

photospheric vector maps Å Æ coronal structuresÆ benefit for extrapolation techniques extrapolations fill the observational gap of Solar-Bin chromosphere additional information from extrapolations (eg.
currents)
 helps in resolving 180° ambiguity
Æ benefit for Solar-B data interpretation

Æ Solar B and extrapolations are complementary