Saas Fee 39

Magnetic Fields in the Atmospheres of the Sun and Stars

Sami K. Solanki Max Planck Institute for Solar System Research Before starting...

Concentrate on observations: only few equations Will use cgs units

Many people have contributed tremendously with material, advice etc. Without their help this lecture would never have been possible. Only some are named here:

Svetlana Berdyugina, Juan-Manuel Borrero, Paul Charbonneau, Stefan Dreizler, Mark Giampapa, Andreas Lagg, John Landstreet, Theresa Luftinger, Coralie Neiner, Hardi Peter, Ansgar Reiners, Manfred Schüssler, Greg Wade ThankThank you!you! Table of contents

0. Introduction „ Basics of polarimetry and the measurement of solar magnetic fields „ The Sun's large-scale magnetic structure „ Sunspot magnetic fields „ Faculae and network magnetic fields „ Magnetic fields in the upper solar atmosphere „ Manifestations of the magnetic field in the Sun's atmosphere „ Techniques for stellar magnetic field measurements „ Magnetic fields in the Herzsprung-Russell diagram „ Activity in stellar envelopes caused by the magnetic field IntroductionIntroduction The Sun in White Light

Gas at 5800 K The Dynamic Sun

Prominence

Sunspot The Violent Sun

Flare

Solar wind and coronal mass ejections TheThe sourcesource ofof thethe SunSun’’ss activityactivity isis thethe magneticmagnetic fieldfield ÎÎ InIn orderorder toto understandunderstand thethe dynamicsdynamics andand activityactivity ofof thethe SunSun wewe needneed toto knowknow andand understandunderstand thethe magneticmagnetic fieldfield

Wiegelmann 2004 Stellar magnetic fields

40,000 20,000 10,000 7500 5500 4500 3000 (K) 106 Magnetic fields WR are found on ? G AGB 10-3-10 G 4 O-B stars throughout 10 2 10 G the HR-diagram <30% Ae-Be Post-MS 102 (103)G )

☼ 2 1-10%? RGB 10 1-103 G Often they BpAp 103-104G produce activity T Tau MS 5% 103 on the star or 1 Solar 100%? 3 Luminosity (L 1-10 G influence its ?% WD red −2 6 9 evolution (e.g. of 10 10 -10 G: 10% dwarfs NS 1-106 G: ?% 3 10-10 G Pre-MS stellar rotation) 9 15 10 -10 G 5-40% 100%? 10−4 O B A F G K M Berdyugina 2008 Spectral class BasicsBasics ofof polarimetrypolarimetry andand thethe measurementmeasurement ofof solarsolar magneticmagnetic fieldsfields Methods of solar magnetic field measurement

„ DirectDirect methods:methods: „ Zeeman effect Î polarized radiation „ Hanle effect Î polarized radiation „ Gyroresonance and Bremsstrahlung Î polarized radiation (in radio range) „ IndirectIndirect methods:methods: ProxiesProxies „ Bright or dark features in photosphere (sunspots, G- band bright points) „ Ca II H and K plage „ Fibrils seen in chromospheric lines, e.g. Hα „ Coronal loops seen in EUV or X-radiation Atom in magnetic field

„ Consider the Hamiltonian of an atom in a magnetic field (Gaussian cgs units; atom in L-S coupling) ⎛ e e2 ⎞ H h 2 ξ rrV )()( SL ⎜ SLB )2( ++⋅−+⋅++∇−= Br θ )sin( 2 ⎟ ⎜ 2 ⎟ 2m ⎝ 2mc 8mc ⎠

„ First 3 terms are kinetic energy, electronic potential, spin-orbit coupling with ξ = 22 drdVrcmr )/)(2/(1)( „ Last two terms are magnetic energy terms derived from magnetic vector potential

„ For fields up to B~10 MG (1 kT), magnetic terms are small compared to Coulomb potential. Fine structure and field treated by perturbation theory Following J. Landstreet Magnetic field regimes

⎛ e e2 ⎞ H h 2 ξ rrV )()( SL ⎜ SLB )2( ++⋅−+⋅++∇−= Br θ )sin( 2 ⎟ ⎜ 2 ⎟ 2m ⎝ 2mc 8mc ⎠

„ Perturbation theory regimes: „ Quadratic magnetic term << linear term << spin-orbit term: (linear) Zeeman effect „ Quadratic magnetic term << spin-orbit term << linear term: Paschen-Back effect un r S „ Spin-orbit term << linear termFo << quadratic magnetic term: quadratic Zeeman effect

„ Schiff 1955, Quantum Mechanics, Chapts. 23 & 39 Following J. Landstreet (Linear) Zeeman effect

„ In weak-field (Zeeman) limit, atomic energy level is only slightly perturbed by (e/2mc)B⋅( L+2 S)

„ In L-S coupling (light atoms), J and MJ are good quantum numbers. Magnetic moment of atom is aligned with J. Energy shift of level is proportional to B⋅J, so there are 2J+1 different magnetic sublevels

Ei=Ei 0 + gi (eℏ/2mc) B MJ = EEi 0++ μμ0 ggi MMJ BB where

gi=1+[ J ( J+1)+S (S+1)−L( L+1)]/ [2 J ( J+1) ] is the (dimensionless) Landé factor (L-S coupling) Zeeman splitting of atomic levels & lines

„ Transitions between B = 0 B > 0 M Zeeman split upper J and lower atomic levels J = 1 +1 lead to spectral lines 0 that are split in -1 wavelength

„ Transitions are allowed Energy between levels with ∆J = 0, ±1 & J = 0 0 ∆MJ = 0 (π), ±1 (σb, σr) (for the most common I types of transitions: electric dipole radiation) λλ Splitting patterns of lines G. Mathys „ DependingDepending onon gg ofof thethe upperupper andand lowerlower levels,levels, thethe spectralspectral lineline showsshows differentdifferent splittingsplitting patternspatterns „ Positive:Positive: ππ components:components: ΔΔMMJ == 00 „ Negative:Negative: σσ components:components: ΔΔMMJ == ±±11 „ TopTop left:left: normalnormal ZeemanZeeman effecteffect (rare)(rare) „ Rest:Rest: anomalousanomalous ZeemanZeeman effecteffect (usual)(usual) Zeeman effect observed

„ First measurement of a cosmic magnetic field, in a sunspot, was carried out 1908 by G.E. Hale

„ On Sun: Zeeman effect changes spectral shape of a spectral line (subtle in most lines outside sunspots)

„ Zeeman effect also introduces a unique polarisation signature

Î Measurement of polarization is central to measuring solar magnetic fields Polarized radiation

„ Polarized radiation is described by the 4 Stokes parameters: I, Q, U and V o o o o „ I = total intensity = Ilin(0 ) + Ilin(90 ) = Ilin(45 ) + Ilin(135 ) = Icirc(right) + Icirc(left) o o „ Q = Ilin(0 ) - Ilin(90 ) o o „ U = Ilin(45 ) - Ilin(135 )

„ V = Icirc(right) - Icirc(left) „ Note: Stokes parameters are sums and differences of intensities, i.e. they are directly measurable Polarization and Zeeman effect Zeeman effect: information content

„ Line splitting I/Ic „ Stokes I ⇒ B „ Line broadening „ Stokes I : no info on B Q/I „ Polarization

„ Stokes V ⇒ „ Stokes Q, U, V ⇒ B U/I „ Atomic diagnostics (hot gas) „ Zeeman effect (except V/I some Ap stars & WDs) „ Molecular diagnostics (cool) „ Zeeman & Paschen Back (ZIMPOL, J. Stenflo) Effect of changing field strength

Formula for Zeeman splitting (for B in G, λ in Å): -13 2 ΔλH = 4.67 10 geff B λ [Å] 1 1 )( +++= JJJJggggg +−+ ))1()1()(( eff 2 ul 4 llul uu

geff is the effective Landé factor of line 2 For large geff Bλ : ΔλH = Δλ betw. σ-component peaks

B=1600 G V

I B=200 G Zeeman splitting ~ λ2 FeFeFe III 1564.81564.8630.2 nm nmnm II VV

QQ UU Dependence on B, γ, and φ

2 2 „ II ~~ κκσ(1+(1+coscos γγ)/4)/4 ++ κκπ sinsin γγ/2/2 „ QQ ~~ BB2 sinsin2γγ coscos 22φφ „ UU ~~ BB2 sinsin2γγ sinsin 22φφ „ VV ~~ BB coscos γγ

„ VV:: longitudinallongitudinal componentcomponent ofof BB J.M. Borrero „ Q,Q, UU:: transversetransverse componentcomponent ofof BB „ AboveAbove formulaeformulae forfor Q,U,VQ,U,V referrefer toto relativelyrelatively weakweak fieldsfields (e.g.(e.g. BB andand BB2 dependencedependence ofof field)field)

„ ZeemanZeeman splittingsplitting etc.etc. isis hiddenhidden inin κκσ andand κκπ.. ForFor Q,Q, U,U, VV thesethese dependencesdependences havehave notnot beenbeen givengiven forfor simplicity.simplicity. Dependence on B, γ, and φ

2 2 „ II ~~ κκσ(1+(1+coscos γγ)/4)/4 ++ κκπ sinsin γγ/2/2 J.M. Borrero „ QQ ~~ BB2 sinsin2γγ coscos 22φφ „ UU ~~ BB2 sinsin2γγ sinsin 22φφ „ VV ~~ BB coscos γγ

„ Q, U: transverse component of B „ V: longitudinal component of B „ Formulae for Q,U,V refer to weak fields

„ κσ and κπ (splitting etc.) not given for Q,U,V for simplicity „ Magnetograph: Magnetograms Instrument to make maps of (net circular) polarization in wing of Zeeman sensitive line

„ Useful when star can be resolved, e.g. Sun

„ Image: Example of magnetogram obtained by MDI

„ Conversion of polarization into magnetic field requires a careful calibration. positive negative polarity polarity What does a magnetogram show?

„ PlottedPlotted atat left:left: „ Top: Stokes I, Q and V along a spectrograph slit „ Middle: Sample Stokes Q profile „ Bottom: Sample Stokes V profile „ Red bars: example of a spectral range used to make a magnetogram. Often only Stokes V is used (simplest to measure), gives longitudinal component of B. SynopticSynoptic chartscharts

Synoptic maps approximate the radial magnetic flux observed near the central meridian over a period of 27.27 days (= 1 Carrington rotation) Dependence on B, γ, and φ

2 2 „ II ~~ κκσ(1+(1+coscos γγ)/4)/4 ++ κκπ sinsin γγ/2/2 J.M. Borrero „ QQ ~~ BB2 sinsin2γγ coscos 22φφ „ UU ~~ BB2 sinsin2γγ sinsin 22φφ „ VV ~~ BB coscos γγ

„ Q, U: transverse component of B „ V: longitudinal component of B „ Formulae for Q,U,V refer to weak fields

„ κσ and κπ (splitting etc.) not given for Q,U,V for simplicity Measured Magnetic Field at Sun’s Surface

MonthMonth longlong sequencesequence ofof magnetogramsmagnetograms (approx.(approx. oneone solarsolar rotation)rotation) MDI/SOHOMDI/SOHO MayMay 19981998 Cancellation of magnetic polarity

SpatialSpatial resolutionresolution elementelement

Unresolved magnetic features with field strength B and filling factor = / AAf Stokes ∑ i tot V i

== positivepositive polaritypolarity magneticmagnetic fieldfield

== negativenegative polaritypolarity magneticmagnetic fieldfield Stokes V signal cancellation Stokes V signal only samples the net magnetic flux. Extreme case: negativenegative polaritypolarity positivepositive polaritypolarity magneticmagnetic fluxflux = magneticmagnetic fluxflux

++ == Scattering polarisation at Sun’s limb

„ IfIf collisionscollisions areare rare,rare, Linearly polarized lightlight isis scatteredscattered scattered photon

„ IlluminationIllumination ofof atomsatoms isis anisotropicanisotropic duedue to:to:

„ Limb darkening (dT/dz < 0, where T = temp.)

„ atom high in atmosph.

„ ScatteringScattering ++ anisotropyanisotropy ÎÎ linearlinear polarisationpolarisation parallelparallel toto limblimb „ HanleHanle effect:effect: ModificationModification Hanle effect ofof scatteringscattering polarisationpolarisation byby magneticmagnetic field.field. 22 effects:effects: „ DepolarisationDepolarisation „ depends on field orientation „ depends on B (it is complete

if ΔλH >> natural line width, i.e. for B > 0.1-100 G) „ also present for unresolved mixed polarity fields „ RotationRotation ofof polarisationpolarisation planeplane „ depends on B, γ, χ Signature of Hanle „ only if field is spatially effect for spatially resolved resolved field Hanle diagnostics: simple examples

depolarisation no depolarisation no rotation no rotation

depolarisation + rotation Example of Hanle rotation & depolarisation

„ MoreMore complexcomplex toto describedescribe HanleHanle thanthan ZeemanZeeman effecteffect „ HanleHanle parameters:parameters:

„ Depolarization factor p/pmax where p is polarization degree for B≠0, pmax is p for B=0 „ Angle of rotation β, with tan 2β = U/Q (β=0 for B=0) „ AtmosphericAtmospheric parametersparameters

„ Field strength parameter Ω, with Ω=2guωL/γN ~ B, where γN is natural damping constant, ωL is Larmor frequency, gu is Landé factor of upper level, 22 0e ωμ e γ = ωL = B N 6π ecm 2me „ Field azimuth χ, with χ=0 for B || LOS Hanle effect example (contd.)

„ HanleHanle depolarisationdepolarisation inin generalgeneral changeschanges

betweenbetween 0.2B0.2B0 andand 5B5B0

2mγ N B0 = egu

„ ExpressionExpression forfor BB0 isis equivalentequivalent toto sayingsaying thatthat Illustration for horizontal field forfor BB==BB0 wewe havehave ωωL== γγN seen exactly at limb, scattering radiation coming Stenflo 1994 exactly from below