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Computational modelling of stellar magnetic fields from observational b oundary values
M. S. Wheatland
Scho ol of Physics Sydney Institute for Astronomy The University of Sydney Cosmic Magnetism Kiama 8 June 2010

Model for solar AR 10953 from Hinode/SOT data


Overview
Background Zeeman effect on spectral lines The Sun Modelling active regions The data ­ vector magnetograms Nonlinear force-free modelling The inconsistency problem Self-consistent nonlinear force-free modelling Other cool stars Zeeman Doppler Imaging (ZDI) Modelling Summary


Nobody can measure physical quantities of the solar atmosphere
­ Del Toro Iniesta & Ruiz Cobo (1996), Sol. Phys. 164, 169


Background: Zeeman effect on spectral lines

Classical model: dipole-oscillator atom
absorbs light near e
-

(Sakurai 1989; Jefferies et al. 1989)

oscillation frequency 0
-

Introduce field B: motion of e

parallel to field unaffected

in plane perpendicular to B the e - precesses frequency of precession is Larmor frequency B = eB /4 me motion described in terms of frequencies 0 ± B superposed CCW and CW motions at 0 + B and 0 - B

Wavelength shift (Landґ factor gL is quantum correction) e B = B 2 /c = 11.7g 0
L

B [G] 1000 G

0 [° A] 5000 ° A

2

[m° A]

(1)

small effect except for large fields


Viewed along B: observe circular motions
line replaced by two shifted circularly-polarized lines -components

Classical explanation of longitudinal Zeeman effect (Sakurai 1989)


Viewed transverse to B: observed two linear motions of e
central unshifted linearly-polarized component two shifted linearly-polarized -components

-

Classical explanation of transverse Zeeman effect (Sakurai 1989)


For oblique B interpretation requires radiative transport
for stellar case specification of an atmospheric model result is not a measurement of B but an inference
"Nobody can measure..."

1

Unno & Rachovsky analytic solution

(Unno 1956; Rachkovsky 1962; 1967)

radiative transfer with uniform B and simple atmosphere often the basis for interpreting spectro-polarimetric data simpler weak-field approximation also used (e.g. Ronan et al. 1987)

1

For more details see e.g. Landi degl'Innocenti & Landolfi (2004).


The Sun: Modelling active regions
Sunspot magnetic fields power solar activity:
solar flares ­ magnetic explosions in the atmosphere (corona) Coronal Mass Ejections (CMEs) ­ expulsions of material

Space weather: CMEs influence local conditions
storms of energetic particles (Solar Proton Events)

A flare and a sunspot: 12 Dec 2006 (Hino de/SOT)


Large active regions flare repeatedly
e.g. ARs 10484 and 10486 in Oct-Nov 2003
2

Problem: model the coronal magnetic fields of these regions

29 October 2003 AR 10484

AR 10486
ARs 10484 and 10486 produced a sequence of huge flares in October-November 2003 [MDI]

2

A good read: Stuart Clark 2007, "The Sun Kings," Princeton University Press


The Sun: The data ­ vector magnetograms
Stokes profiles I (), Q (), U (), V () measured Stokes inversion: vector magnetic field inferred3
nonlinear least-squares fitting to Unno-Rachovsky solution
(Auer et al. 1977; Skumanich et al. 1987; Skumanich & Lites 1987; Lites & Skumanich 1990)

line-of-sight and transverse field are parameters of fit transverse field subject to a 180 ambiguity

180 ambiguity must be resolved

(Metcalf 1994; Metcalf et al. 2006)

Vector magnetogram: photospheric map of B = (Bx , By , Bz ) Vertical current density Jz may be calculated at photosphere: Jz = 1 µ0 By Bx - x y at z = 0 (2)

locally planar approximation to photosphere

3

For more details see e.g. Landi degl'Innocenti & Landolfi (2004).


New generation of instruments
Hinode Solar Optical Telescope (SOT) Spectro-Polarimeter
(Tsuneta et al. 2008)

Solar Dynamics Observatory Helioseismic & Magnetic Imager
(Borrero et al. 2007)

Hinode-derived vector magnetogram for active region 10953

Active region AR 10953 on 30 April 2007. Left: Bz . Right: Jz (Wheatland & Leka in preparation; Hinode/SOT).


The Sun: Nonlinear force-free modelling
Vector magnetograms provide boundary conditions for models
coronal magnetic field reconstruction

Force-free model for coronal magnetic field: J в B = 0 and ·B=0 (3)

J = µ-1 в B is electric currrent density 0 physics: Lorentz force dominates over other forces coupled nonlinear PDEs

Writing J = B/µ0 (J is parallel to B): B· = 0 and в B = B (4)

is the force-free parameter = µ0 Jz /Bz at z = 0 defines values over vector magnetogram


Boundary conditions

(Grad & Rubin 1958)

:

Bn in boundary in boundary over region where Bn > 0 or where Bn < 0
over one polarity we label the polarities P and N respectively

Vector magnetograms give two sets of boundary conditions
values of = µ0 Jz /Bz over both P and N are available

Eqs. (4): methods of solution are iterative Current-field iteration (Grad & Rubin 1958)
at iteration k solve the linear system B
[k -1]

(e.g. Wiegelmann 2008)

·



[k ]

=0

and
[k ]

вB

[k ]

= [k ] B

[k -1]

(5)

BCs imposed on [k ] and Bz Wheatland (2007): a fast implementation


The Sun: The inconsistency problem

Force-free methods work for test cases but fail for solar data
(Schrijver et al. 2006; Metcalf et al 2008; Schrijver et al. 2008; DeRosa et al. 2009)

different methods give different solutions P and N solutions do not agree for the same method

Vector magnetogram data inconsistent with force-free model
errors in field determination field at photospheric level is not force free (Metcalf et necessary conditions for a force-free field not met
al. 1995) (Molodenskii 1969)


AR 10953 on 30 April 2007
P (blue) and N (red) solutions from vector magnetogram

Force-free solutions from K. D. Leka's vector magnetogram data for AR 10953


The Sun: Self-consistent nonlinear force-free modelling
Find the closest force-free solution to the observed data Self-consistency procedure (Wheatland & Rґ nier 2009) eg
P and N solutions constructed (current-field iteration) Bayesian probability plus solutions used to modify BCs on
taking into account relative uncertainties in boundary values

procedure iterated until the P and N solutions agree

Wheatland & Rґ egnier (2009): demonstrated on AR 10953
method shown to work but uncertainties were not available for the boundary data self-consistent solution was close to potential (current-free) result was considered a proof of concept

Problem re-visited with data including uncertainties
solution with large currents obtained
(Wheatland & Leka in preparation)


AR 10953 on 30 April 2007
New self-consistent solution(s): P (blue) and N (red)

Self-consistent nonlinear force-free solutions for AR 10953


Soft X-ray image of AR 10953 on 30 April 2007

Hinode/XRT broadband soft X-ray image (Hinode/XRT)


Other cool stars: Zeeman Doppler Imaging (ZDI)
Permits determination of surface field over cool stars Proposed by Semel (1989)4
applicable to rapidly rotating stars assumes field evolves on a time scale longer than a period

Basic technique:
combine Stokes V (, t ) profiles for many lines to improve SNR fit composite profiles to profiles for a surface field model Unno-Rachovsky solution or weak-field approximation used

Donati et al. (2006) model: B = [Br (, ), B (, ), B (, )]
components expanded in spherical harmonics fitting determines coefficients in the expansion

(6)

4

Further developmeants e.g. Brown et al. (1991); Donati & Brown (1997); Donati (2001).


Evidence for stellar global polarity switches

(Donati et al. 2008)

planet-hosting F8 star Boo successive polarity switches of field components over two years

Surface distribution of Br inferred by ZDI for Boo (Donati & Landstreet 2009)


Other cool stars: Modelling
Source surface modelling
(e.g. Jardine et al. 1999; Jardine et al. 2002)

a potential (current-free) model for global field developed for the Sun (Altschuler & Newkirk 1969; Schatten et al. 1969) mimics radial stretching of field at height due to stellar wind

Source surface model field (which satisfies

в B = 0): (7)

B(r , , ) = - = (Br , B , B )
boundary conditions: Br (R , , ) = BrZDI (, ) B (Rs , , ) = B (Rs , , ) = 0 field is purely radial at source surface Rs 3R - 5R field components may be expanded in spherical harmonics
coefficients determined by imposing boundary conditions

(8) (9)

ZDI values of B , B inconsistent with potential model
non-potential models also tried
(e.g. Hussain et al. 2002)


Summary
Stellar magnetic fields are inferred not measured
inferred surface values permit coronal field modelling

The Sun
active region modelling motivated photospheric vector magnetogram nonlinear force-free modelling has boundary data is inconsistent with self-consistency solution presented by activity/space weather data is available been developed the model

Other cool stars
inference of surface fields using Zeeman Doppler Imaging coronal field modelling e.g. source surface solutions

List of solar sites including pictures and movies:
http://sydney.edu.au/science/physics/wheat/5

5

Easier: search for Mike Wheatland on go ogle.