The Height Dependence of the Magnetic Field in a Sunspot Horst Balthasar Astrophysikalisches Institut Potsdam Ondřejov, December 10 2007
Partners involved: Véronique Bommier, Obs. Paris, Meudon Peter Gömöry, now AISAS, Tatranska Lomnica Bernhard Kliem Many thanks to IAC and MPS for TIP 2 and to the IAC for the SIR-code!
Reasons for these investigations The detailed height dependence of the magnetic field is not really clarified yet (0.1 G/km - 5 G/km). Study spectral lines originating in different heights of the solar atmosphere! - Is the magnetic field strength really increasing with height in the penumbra? - What are the values for the vertical gradient in the umbra? - Is the magnetic field force-free or not? - Can electric current densities serve as diagnostic tools for the penumbra?
Observations Several sunspots were observed at the VTT on Tenerife in the years 2005 and 2006 with the Tenerife Infrared Polarimeter (TIP); Spectra in all four Stokes-parameters. Here a single spot, observed 27 May 2006, is investigated. Wavelength: 1078.3-1078.7 nm (Fe I, Si I g=1.5), probing lower and mid photosphere. Parallel observations with THEMIS, MTR-mode, also full Stokes vector. Lines there: Fe I 630.15/.25, Na D1, Cr I 578.2 nm, H_alpha, covering lower photosphere to lower chromosphere.
The Telescopes The German VacuumTower Telescope VTT (left) and the French Telescope Heliographique pour l Etude du Magnetisme et des Instabilites Solaire THEMIS (right).
The sunspot AR 10886 27 May 2006 hl=7n, cmd=-6w VTT with AO! combined from spectral continuum of 3 scans
Scanning AR 10886 mosaic scanning, here original data orientation. top: image with slit intensity spectrum Stokes V, Q, U spectral lines: Fe 1078.3, (Si 1078.4, Fe 1078.5,) Si 1078.6
Inversions Inversion with SIR [Stokes Inversion based on Response functions] Ruiz Cobo & del Toro Iniesta, ApJ 398, 375 (1992) Independent runs for the spectral lines Fe 1078.3 and Si 1078.6 [nm] Find a set of start parameters applicable to all pixels and all lines! (iron line needs more freedom for temperature stratification)
Output of SIR: Temperature stratification T(tau) Total magnetic field strength B Inclination with respect to line-of-sight Magnetic azimuth (ambiguity!) Doppler velocity
Azimuth ambiguity Assume that there is an azimuth center in the spot and a radial structure of the magnetic field. Select that value which is closer to this radial structure. Polarity of the spot must be taken into account.
The magnetic vector Calculate Cartesian coordinates of B with respect to the LOS. Rotate vector by heliocentric angle of the spot. Correct for geometrical foreshortening. (Curvature is neglected.) Magnetic vector in Cartesian coordinates with respect to the solar surface. Calculate new values for the magnetic angles.
The magnetic field Total magnetic field strength derived from the two lines: Upper panel: Si 1078.6 Max: 2180 G Min(PU): 316 G Lower panel: Fe 1078.3 Max: 2308 G Min(PU): 423 G
Formation heights Formation heights estimated from interpolation using temperature maps between a quiet sun and an umbral model for which contribution functions where calculated explicitly. Upper pannel: Si 1078.6 Lower pannel: Fe 1078.3
Magnetic field strength: height gradient B(Si 1078.6) B(Fe 1078.3) -----------------------------------delta h [ G/km] Umbra: Mean = -2.60 +- 0.05 Max=-0.38 Min=-7.18 PU: Mean=-1.38 +- 0.02 Max=+0.99 Min=-3.51
Gamma Magnetic inclination for Si 1078.6 (upper panel) and Fe 1078.3 (lower panel). The field is less inclined in higher layers.
The horizontal field
Vertical Component of B Upper pannel: Si 1078.6 Max = +2171 G Min = +49 G Lower pannel: Fe 1078.3 Max= +2286 G Min = +23 G
Height dependence of Bz Bz(Si) Bz(Fe) ------------------Delta h [ G/km] Umbra: Mean=-2.15+-0.06 Max=+2.17 Min=-7.66 PU: Mean=-0.20+-0.03 Max=+1.87 Min=-3.84 Bz increases in the outer penumbra.
dbz/dz Determine horizontal differences from neighboring pixel. Si, umbra: Mean=-0.51+-0.01 Fe, umbra: Mean=-0.56+-0.01 Si, PU: Mean=-0.06+-0.01 Fe, PU: Mean=-0.08+-0.01 Values are much smaller! What s wrong?
What s wrong? Maxwell equations? - No.
What s wrong? Maxwell equations? - No. Formations heights? Possible, especially for the umbra. But, taking other investigations into account, a much more extended solar atmosphere is needed.
What s wrong? Maxwell equations? - No. Formations heights? Possible, especially for the umbra. But, taking other investigations into account, a much more extended solar atmosphere is needed. Determination of the derivatives of the magnetic field. Much higher resolution needed!
Current densities
Currents Vertical component of electric current densities. Values in ma/m^2 Upper pannel: Si 1078.6 Umbra, Mean=-11.5+-0.5 PU, Mean=-1.8+-0.3 Lower pannel: Fe 1078.3 Umbra, Mean=-10.7+-0.7 PU, Mean=-2.0+-0.3
Helicity Measure for twistedness of the magnetic field. Here:current helicity. H_z = B_z (curl B)_z In G^2/km Si 1078.6 (upper panel), Fe 1078.3 (lower panel). Negative dominance in agreement with literature, but positive values in the outer penumbra,especially in the upper left.
Is the magnetic field force-free? Lorentz-force: Integral over Maxwell-stresses. Compare with magnetic pressure force. Molodensky (1975), Low (1985), Metcalf et al. (1995), Moon et al. (2002).
Is the magnetic field force-free? Results for Fe 1078.3: (normalized) Whole area: Whole Spot: Umbra: Results for Si 1078.6: Whole area: Whole spot: Umbra: F_x -0.09-0.19-0.28 F_y -0.02-0.05-0.06 F_z -0.22-0.27-0.71-0.02-0.17-0.24-0.06-0.10-0.11-0.36-0.34-0.74 The photospheric magnetic field is NOT force-free!
Magnetic flux Can we estimate the magnetic field strength in a height of several Mm without extrapolation? Basic idea behind: H. U. Schmidt, Geoph. Astroph. Fluid Dyn. 62, 249 (1991) Question: is the penumbra deep or shallow? Measure the magnetic flux in the photosphere and the magnetic field strength at the outer edge of the penumbra. Determine a smooth surface with that B. Ellipsoid instead of hemisphere.
Magnetic flux
Magnetic flux Photospheric flux (Fe): 1.36 x 10^13 Wb B(r, Fe) = 715 G, r=6700 km Top height: 4065 km Photospheric flux (Si): 1.30 x 10^13 Wb B(r, Si) = 676 G, r=6700 km Top height: 4180 km Average gradient: 0.39 G/km
Magnetic flux Photospheric flux (Fe): 1.36 x 10^13 Wb Umbral flux(fe): 5.53 x 10^12 Wb Umbra + inner PU (Fe): 1.01 x 10^13 Wb Photospheric flux (Si): 1.30 x 10^13 Wb Umbral flux(si): 5.16 x 10^12 Wb Umbra + inner PU (Si): 8.53 x 10^12 Wb The penumbra brings more flux up than the umbra, nevertheless the outer penumbra might be shallow.
Sketch of the magnetic field
Summary Inside the spot, the magnetic field decreases with height, rather fast in deep layers and more slowly in higher ones. Outside we see magnetic canopies. The magnetic field is less inclined in higher layers. The vertical component increases with height in the outer penumbra due to the different inclination. The exact value of the magnetic gradient depends on formation heights of the spectral lines and on spatial resolution. Electric currents have a potential as diagnostic tools. The magnetic field is not force-free in the photosphere. From magnetic flux considerations I estimate 700 G in 4100 km height. The outer penumbra might be shallow.
Outlook Reduction of the THEMIS-data and combination VTT and THEMIS results. TRACE-timeseries in whitelight and the 170nm filter exist which could deliver additional information,although these series were obtained some hours before. New observations with 2D-spectropolarimeters which are available next year. Need of higher spatial resolution GREGOR
GREGOR A new solar telescope with 1.5m diameter, presently under construction on Tenerife.