By inverting full Stokes spectra of the He I 10830 Å triplet and the photospheric Si I 10827 Å / Ca I 10833 Å lines of a single sunspot observed on two days, Joshi, Lagg, Hirzberger & Solanki (2017) found that the umbral field strength is reduced by a factor of 1.30–1.65 in the upper chromosphere, that the field decreases upward at 0.5–0.9 G/km in the umbra and more slowly in the penumbra, that the chromospheric penumbral field is more vertical than the photospheric one, that a magnetic canopy extends beyond the visible spot boundary, and that the differential twist between the two layers increases toward the outer penumbra. All of these quantities are conditional on an assumed geometrical height separation (1000 km, alternatively 1250 km) between the He I and photospheric diagnostics, and on a one-spot, two-day sample. The open problem is to place this photosphere-to-chromosphere characterization on secure ground: (i) determine the actual formation height of the He I 10830 Å triplet in sunspot umbra and penumbra, and its spot-to-spot variation, so that measured field differences become true vertical gradients; and (ii) establish, from a sample of sunspots spanning size, evolutionary stage, and viewing angle, which of the reported properties are common, their statistical distributions, and the effect of unresolved mixed-polarity flux in the deep photosphere. No multi-spot survey and no in-spot He I height calibration are known in the published literature; the associated sunspot gradient discrepancy is explicitly documented as unresolved by Balthasar (2018).

Background

A sunspot is a long-lived, strongly magnetized region on the solar surface appearing dark because suppressed convection and magnetic pressure evacuate and cool the gas. Its dark central umbra is surrounded by a filamentary penumbra of nearly horizontal fields; bright lanes called light bridges sometimes cross the umbra. Because a sunspot's magnetic field controls its stability, decay, and eruptive activity, its three-dimensional structure — how the field strength BB, inclination, and azimuth change with geometrical height — is a central observable of solar physics.

The magnetic field vector is measured by spectropolarimetry: the four Stokes parameters (II, QQ, UU, VV) of a magnetically sensitive spectral line encode its strength and direction through the Zeeman effect (the splitting and polarization of the line in a magnetic field), and are converted to physical parameters by fitting model atmospheres to the observed profiles, a procedure called an inversion. In the near-infrared window around 10830 Å, the Si I 10827 Å and Ca I 10833 Å lines sample two layers of the photosphere, while the He I 10830 Å triplet — a set of three transitions of neutral helium — samples the upper chromosphere, roughly one to a few megameters above the photosphere. The helium lines are special: neutral helium in the upper chromosphere is ionized by extreme-ultraviolet radiation from the overlying corona and then recombines, so the line's opacity and its geometrical formation height depend on the local coronal illumination and thermal structure rather than on a fixed optical-depth surface. In the kilogauss fields of sunspots the linear polarization of the triplet is dominated by the transverse Zeeman effect, which makes it a comparatively direct vector-field diagnostic, but the Stokes QQ and UU signals are intrinsically weak and therefore noisy.

Joshi, Lagg, Hirzberger & Solanki (2017) inverted simultaneous He I, Si I, and Ca I Stokes spectra of the leading sunspot of active region NOAA 11124 (observed 14–16 November 2010 with the TIP-2 polarimeter at the Vacuum Tower Telescope) and compared the two layers pixel by pixel. They found the umbral field weaker in the upper chromosphere by a factor of 1.30–1.65; a vertical gradient ΔB/Δd\Delta B/\Delta d of 0.5–0.9 G/km in the umbra decreasing to 0–0.7 G/km in the penumbra; a chromospheric field 5–20° more vertical than the photospheric one in the penumbra; a magnetic canopy — field from the expanding spot overlying weaker, more horizontal field outside the visible boundary, with the chromospheric field up to ~300 G stronger than the photospheric field there; a cusp-like field above a light bridge; and a differential twist (azimuth difference ϕCh′−ϕPh′\phi_{\rm Ch}'-\phi_{\rm Ph}' between chromosphere and photosphere) that is small on average but grows toward the outer penumbral boundary. Two systematic limitations qualify all of these numbers. First, converting the field difference between the two diagnostics into a gradient requires their geometrical height separation: the authors adopted 1000 km from the phase-spectra seismology of Centeno et al. (2006) but note that the alternative 1250 km of Joshi et al. (2016) lowers the umbral gradient to 0.4–0.7 G/km, and the separation may differ between umbra and penumbra and from spot to spot. Second, unresolved fine-scale opposite-polarity flux in the deep photosphere biases the photospheric reference field, while the weak He I QQ/UU limits the reliability of small-scale chromospheric structure. The companion study of the same group (Joshi et al. 2017, A&A 599, A35) showed that even within the photosphere, gradient determinations are method-sensitive.

This local problem sits inside a documented, unresolved discrepancy: Balthasar (2018) reviewed all height-dependence determinations of sunspot magnetic fields and found that photospheric multi-line profile slopes give gradients of −2 to −3 G/km, while divergence-free reconstructions and chromospheric-line determinations give shallow gradients near −0.5 G/km, with no accepted reconciliation. Related open aspects are the origin of sunspot field twist and its radial variation, and how chromospheric mass flows couple to the field geometry: the Evershed flow is the well-known outward flow along photospheric penumbral filaments, and the inverse Evershed flow is its chromospheric counterpart — an inflow of material along inclined canopy field lines that descends near the spot boundary; chromospheric plumes are persistent, dense, downflowing structures observed in the sunspot atmosphere.

Problem Statement

Determine the structure of the sunspot magnetic field between the photosphere and the upper chromosphere as a calibrated, statistically established property of sunspots in general, comprising two linked requirements: (1) Height calibration — determine the geometrical formation height of the He I 10830 Å triplet relative to the photospheric Si I 10827 Å and Ca I 10833 Å diagnostics inside sunspots, separately for umbra and penumbra and including its variation from sunspot to sunspot, so that observed photosphere-chromosphere field-strength differences can be converted into genuine vertical gradients ΔB/Δd\Delta B/\Delta d; the present determinations (1000 km from umbral phase-spectra seismology, i.e. the height separation inferred from the phase difference between oscillations observed in the photospheric and He I lines, and 1250 km from the offset between the photospheric and chromospheric magnetic neutral lines — the loci where the line-of-sight field reverses sign — in a penumbra) are the values adopted in the source paper rather than in-spot measurements: the source describes 1000 km as a conservative estimate and 1250 km as a conditional upper limit, and the actual separation, being set by the local coronal EUV illumination, may be region-dependent and may lie outside this range. (2) Statistical survey — from spectropolarimetric observations of a sample of sunspots covering a range of sizes, evolutionary stages, and viewing angles, establish which of the properties reported for one spot by Joshi et al. (2017) are common to sunspots and determine their statistical distributions: the umbral chromospheric-to-photospheric field-strength reduction factor, the radial dependence of the vertical field gradient, the degree to which the chromospheric field is more vertical than the photospheric field, the radial extent of the magnetic canopy beyond the visible boundary, the prevalence of cusp-shaped fields above light bridges, and the radial dependence of the differential twist. The survey must also determine the effect of unresolved mixed-polarity flux in the deep photosphere. The problem is considered answered when a qualified expert can judge, from published observations and analysis, the commonality and full range of the upper-chromospheric magnetic-field properties of sunspots on a calibrated height scale. The origin of the radially increasing differential twist toward the outer penumbra and the relation of the inverse Evershed flow and chromospheric plumes to the three-dimensional magnetic topology are related open aspects discussed in the Background, but they are not part of the completion criterion of this problem.

References

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