On the reconstruction of the coronal magnetic field from coronal Hanle / Zeeman observations

Transcription

1 On the econstction of the coonal magnetic field fom coonal Hanle / Zeeman obsevations M. Kama & B. Inheste Max-Planck Institte fo Sola System Reseach GERMANY Linda 005

2 Coonal Magnetic Field Magnetic field contains the dominant enegy pe nit volme in the sola coona State-of-the-at detemination of the coonal magnetic field: Extapolation of photospheic magnetic soces MHD simlations. Disadvantage: These methods ae vey ill-posed, small eos in measements case big ncetainty in the coona. Difficlties of diect measements at optical wavelengths: Coonal plasma is extemely hot (~10 6 K) Magnetic fields in the qiet-sn coona ae weak (~10G) line boadening bigge than Zeeman splitting Altenative: Diect Zeeman-effect measements in IR ange (Lin et al. 004) Can this data be sed fo econstction of the coonal magnetic field?

3 Measements of magnetic field effects in the coona ae difficlt bt possible Faaday - effect Rotation of polaization plane of polaized light coming fom adio-soces and passing thogh the coona Resonance scatteing (Hanle effect) Degee and oientation of linea polaization of light scatteed by coonal FeXIII and FeXIV ions. Longitdinal Zeeman - effect Line splitting of cicla polaized infaed light scatteed by coonal FeXIII ions.

4 Coonal emission Ion B Effects fo detemination the coonal magnetic field: Hanle effect contains infomation only abot oientation of the B npolaized incident light 1 Zeeman effect gives the pojection of the B on the LOS Both ae integated along the line-of-sight (LOS) Excitation mechanisms: 1) by anisotopic npolaized adiation fom photosphee ) themal excitation

5 Longitdinal Zeeman-effect in the Coona High tempeate (10 6 K) Themal boadening Zeeman splitting Δλ Δλ D B ~ λ ~ λ Infaed wavelength ae moe favoable: λ=10747 Å of Fe XIII Weak field (~10 G) Magnetogaph fomla (Casini & Jdge 1999): dε I ( ω, Ω) e εv ( ω, Ω) = k g B cosθ, dω m ε and ε g I k = 1+ b σ V 1+ a σ is effective Lande facto, 3 / ( 3cos θ 1) ae emission coefficients fo the Stokes I σ is alignment facto, σ ( Θ, R, T, N), whee Θ = fo the10747 A line of the Fe XIII σ < 0.7 e and R, B V θ = LOS, B components,

6 Longitdinal Zeeman-effect in the Coona: Obsevations Lin et al. 004

7 Zeeman-effect: Magnetogaph fomla V dε I ( ω, Ω) e εv ( ω, Ω) = g B cosθ, dω m = LOS ε dl V = g e m e LOS e dε I ( ω, Ω) B cosθdl dω θ = LOS, B poplation density of the exited level hω ε I ( ω, Ω) = N ex. level Φ( ω) 4π if σ << 1, ε ( ω, Ω) ε ( ω) I I Voigt-fnction [ 1+ bσ (3cos θ 1) ] (no diectional dependence). σ is alignment facto, σ(θ, R, T, N e ) (Casini & Jdge 1999) di( ω) dω = LOS dε I ( ω) dl dω dε ( ω) I dω can be fond by scala field tomogaphy fom obsevations we do not need to know distibtion of the ion density and tempeate ove coona

8 Hanle effect Resonance scatteing fo lines, with lifetime >> Lamo peiod Ion B incident light Polaized intensity map of the FeXIII line emission (Habbal S.R. et al, ApJ 558, 001)

9 ( ) ( ) ( ) T T T, ),, ( ), ( ), ( ), ( ε ε ε ε ε ε ε ε ν ν ν ν = = = Ω + Ω Ω = Ω V U Q I V U Q I l E S E S K S Hanle effect (FeXIII & FeXIV) ( ) [ ] ),,, ( ), ( ), ( Re ), ( A J qq M qq M M l l l l J J J J i v B T M M i T q M M J J q M M J J A J N h l l l l l ν ρ π ν ν ε α α α α α Φ Ω + = Ω Spheical Tenso (incldes viewing geomety and oientation of magnetic field vecto) Density Matix element Voigt-fnction Ω ν ν ε d i ), ( does not depend on the stength of the magnetic field fo FeXIII and FeXIV B ~ 10 G A = 1/τ = 14 s -1 fo Fe XIII A = 1/τ = 60 s -1 fo Fe XIV

10 Hanle effect χ θ B R Θ α FeXIII and FeXIV ions (Qefeld 198) A LOS θ ε I 4Σ + Δ ε Q 3hνA Δ = ε 8 U π Δ εv hν is α Θ and the LOS to the obseve; is the angle between the local adis polaization pojected on the POS; Σ and Δ photon enegy; ae popotional to the ( 3cos θ 1)( 3cos Θ 1) ( 3cos Θ 1) sin θ cos α ( 3cos Θ sin θ sin α is the Einstein coeff. fo spontaneos emission; is the angle between the magnetic field diection is the angle between local adis and magnetic field diection (depends on the popeties of incident light, T, N); 0 and the obseved Zeeman sblevel poplations V ( Θ) = 3cos Θ 1 is the van Vleck facto Thee is no infomation abot magnetic field stength!

11 Hanle effect (FeXIII & FeXIV) Oientation of the polaization planes Dipole Field Qadopole Field Electon density N e ~ -5 (Newkik et al. 1970) The length of the lines is popotional to degee of polaization. Red lines ae magnetic field lines in the POS. Oientation of the polaization plane in espect to the magnetic field lines depends on the van Vlek facto (3cos Θ -1), whee Θ is angle between B and

12 Scala Field Tomogaphy

13 Is This the Kind of Data Which Can Be Used in the Vecto Tomogaphy to Reconstct B? Zeeman data fo example D ( t, θ ) = f ( ) B dl f ( ) can be fond by scala - field tomogaphy y Assmption: alignment facto σ 0. B B t θ B x Contay to scala-field tomogaphy, the integand now depends on the view diection. i-ray Fo 3-D case we have 3 times moe vaiables to be fond than fo scala field bt the same nmbe of eqations => Reconstction is ndedetemined

14 A Geneal Poblem with Vecto Tomogaphy Depending on the obsevation only the divegence-fee o soce-fee field component can be econstcted. Fo example, fo Zeeman-effect data: B = Φ + Ψ measements of B ds Φ Iotational component cannot be econstcted Ψ Solenoidal component can be niqely econstcted Oiginal Field Reconstction

15 Vecto Field Tomogaphy: Reglaization We need additional infomation abot field: Magnetic field is divegence-fee: B = 0 F = Nmbe of Rays ( sim obs ) D + i Di μ i= 1 Coona B dv Shold be minimized Nice popeties of this eglaization: makes the se of photospheic B obsevation as bonday condition epodces standad potential B if div-tem alone is minimized

16 Vecto Field Tomogaphy: Reglaization Poblem is badly conditioned, e.g. nmbe of nknown vaiables exceeds the nmbe of eqations Random noise in the data In eslt, thee is possible no niqe econstction. Poblem is ill-conditioned. log F DivB L-cve less filteing The optimal vale of μ F tomo sim obs ( D D ) Nmbe of Rays = i i + μ i= 1 Coona = F + μ F DivB B dv Tikhonov eglaization maximises the smoothness of the soltion. = moe filteing The optimal μ is whee the L-cve has its stongest positive cvate log F Faaday

17 Model Field Configation fo the Tests Dipole + cent loop EIT Obsevation

18 Vecto Field Tomogaphy: D Example fo Zeeman-effect Reconstction ignoing any tomogaphy data and minimizing F divb -tem alone. Oiginal Field Reslt of a econstction sing a andom 9% selection of a complete tomogaphy data set. Reslt of a econstction sing a andom 48% selection of a complete tomogaphy data set.

19 Reconstction fo Zeeman-effect Vetical coss-section Oiginal Field Eqatoial coss-section Reconstction with only F divb -tem inclded Reconstction with Zeeman- (Faaday-) effect inclded

20 Reconstction fo Hanle-effect Vetical coss-section Oiginal Field Eqatoial coss-section Reconstction with only F divb -tem inclded Reconstction with Hanle-effect inclded

21 Reconstction fo Zeeman-, Hanle-effect Zeeman-effect (solid bas) Hanle-effect (solid bas) Dashed bas - potential field econstction Angle between oiginal vecto and econstcted one [ ] Diffeence angle [ o ] Diffeence angle [ o ] Eos in absolte vale [%]

22 Reconstction fo Zeeman-, Hanle-effect Zeeman-effect (solid bas) Hanle-effect (solid bas) 4000 Dashed bas - potential field econstction Angle between oiginal vecto and econstcted one [ ] Qantity Diffeence Angle@ D Qantity Eos in absolte vale [%] Diffeence Angle@ D Qantity HÈB ecè-èb oigè,@%d oigèlèb Qantity HÈB ecè-èb oigè,@%d oigèlèb

23 Conclsion Coonal Hanle and(o) Zeeman data + constaint B=0 allows to econstct the non-potential component of the coonal magnetic field The tomogaphic invesions based on the Hanle effect and longitdinal Zeeman effect, have diffeent pecision fo the diffeent vecto components of the field, depending on the configation of the econstcting field. Paticlaly, fo the case of obsevation of a votex-like field sitated in the plane pependicla to the otation axis, the votex is hadly seen in the econstction based on the Hanle effect, while the econstction based on the Zeeman effect gives saticfactoy eslt fo this field. The invesion based on the Hanle effect gives moe pecise eslt fo the meidional component of the magnetic field than an invesion based on the Zeeman effect.

24 Conclsion Coonal Hanle and(o) Zeeman data + B = 0 allows to econstct the non-potential component of the coonal magnetic field bt...

25 Otlook Diffeent and moe ealistic coonal magnetic field configations, e.g., the field above active egions o moe ealistic steame-type field stctes shold be stdied With the code we have developed, we can stdy with test calclations systematically how mch noise is toleable to achieve a cetain pecision of the soltion. The inflence of data gaps on the invesion eslt. Obsevations of the Faaday otation of the linealy polaized adio signals taveling thogh the coona give infomation vey simila to the longitdinal Zeeman effect. It wold be inteesting to stdy how sefl these spase measements ae fo the econstction of the coonal field. In the code sed in this thesis we neglected the alignment facto σ. A finite alignment facto will modify the nmeical expessions fo the invesion of the longitdinal Zeeman-effect data by abot 30 % o less. Fo a qantitative application of o code to eal data, a calclation of the alignment facto shold be inclded. The invesion pocede pesented hee cold be looked at as a fist step towads a systematic line-of-sight invesion of all fo Stokes components which wold then yield not only the magnetic field bt also the coonal density (mainly fom the Stokes I component)

26 Otlook Inflence of data gaps. Stdy systematically how mch noise is toleable to achieve a cetain pecision of the soltion. Apply method to eal data! Altenative data which can similaly be econstcted: Faaday-otation effect M O wok is jst a fist step. Final goal: a systematic line-of-sight invesion of all fo Stokes components.