The magnetic configuration of a mature sunspot beneath the photosphere is still not determined by direct observation. Local helioseismic inversions measure wave-speed anomalies below spots, but the thermal (temperature and density) and direct magnetic contributions to these anomalies have never been separated, so the deep structure rests on theoretical argument. Kirichek and Solov'ev (Proc. IAU 5, 351, 2010) argued from lateral total-pressure balance, combined with the helioseismically inferred subsurface overheating below about 4 Mm, that Parker-type compact flux tubes cannot remain in equilibrium deeper than about 4 Mm: the umbral flux must expand sharply there into a diffuse configuration (the shallow sunspot), and they connected this stratification to observed low-frequency sunspot eigenoscillations. Realistic magnetohydrodynamic forward modeling has quantified how severely magnetic and thermal signatures degenerate in travel-time measurements, and first seismic detections of subsurface magnetic structure have been achieved (travel-time anisotropy from subsurface horizontal magnetic fields, and compact acoustic scatterers within about 1 Mm of the photosphere attributed to Parker-type flux fragmentation), but no validated inversion separating temperature and magnetic field beneath a real sunspot, and no direct measurement of the flux-expansion depth, exists. The problem remains open: to build and validate inversions that disentangle thermal and magnetic effects and to use them, together with eigenmode comparisons, to determine the subphotospheric magnetic geometry of sunspots.

Background

Sunspots are dark, long-lived photospheric regions with kilogauss-strength magnetic fields; their dark central umbra is surrounded by a filamentary penumbra. They are the principal agents of solar activity, yet their structure below the visible surface cannot be imaged directly and must be inferred by helioseismology, the study of solar oscillations.

Local helioseismology analyzes acoustic waves in restricted surface patches. The main techniques are time-distance helioseismology, which cross-correlates Doppler signals at pairs of surface points to measure acoustic wave travel times; ring-diagram analysis, which fits the rings of oscillation power in the horizontal-wavenumber-versus-frequency plane of local patches; helioseismic holography, which regresses the observed wave field onto acoustic foci; and Fourier-Hankel decomposition. The measured travel-time or frequency perturbations are inverted for subsurface structure using sensitivity kernels, linear weighting functions usually computed in the Born approximation (a single-scattering linearization about a quiet-Sun reference model).

Time-distance inversions mapped subsurface flows beneath a sunspot (Zhao, Kosovichev and Duvall 2001), and three-dimensional inversions of data from SOHO/MDI (the Michelson Doppler Imager instrument on the Solar and Heliospheric Observatory spacecraft; Kosovichev 2006) found beneath a large spot a two-layer wave-speed anomaly: reduced wave speed in the top 2-3 Mm (1 Mm = 1 megameter = 1000 km) and enhanced wave speed below about 4 Mm, interpreted as subsurface overheating by roughly 1000 K. That interpretation, however, assumed the direct effect of the magnetic field on wave propagation to be small at depth: the temperature and magnetic contributions to the anomaly were not separated, so the deep magnetic configuration remained a matter of theoretical modeling.

Two idealized magnetic geometries dominate the theoretical debate. In Parker's cluster ('spaghetti') model, the umbral flux is divided into many separate, strongly compressed, nearly vertical magnetic flux tubes (field bundles confined by external gas pressure) separated by almost field-free plasma that can flow between the tubes; in monolithic models the spot is a single coherent flux tube. Kirichek and Solov'ev (2010), building on Soloviev and Kirichek (2006), analyzed the lateral (transverse) balance of total pressure pgas+B2/(8π)p_{\rm gas}+B^2/(8\pi) between the interior of a flux tube and its surroundings, using helioseismologically inferred pressure and field profiles. They concluded that compact Parker-type tubes can satisfy this balance only in the cool umbral column down to about 4 Mm; in the overheated layer below, the balance fails for compact tubes, so the flux must expand sharply at about 4 Mm depth into a highly expanded, diffuse structure, the shallow sunspot. They linked this depth-dependent stratification to the observed long-period (low-frequency) eigenoscillations (natural resonant oscillation modes) of sunspots reported by the solar-physics group at Pulkovo Observatory, St. Petersburg (Efremov, Parfinenko and Solov'ev 2007; Efremov et al. 2014). Because the underlying inversions could not separate thermal from magnetic effects, neither the proposed geometry, nor the roughly 4 Mm expansion depth, nor the eigenmode interpretation was tested by direct observation.

The technical obstacles are now well characterized. Sunspot magnetic fields convert incoming acoustic waves into magnetoacoustic slow and fast waves near the equipartition layer, the depth where the Alfvén speed equals the sound speed; the Wilson depression, the lowering of the visible sunspot surface by several hundred kilometers, shifts wave paths and reflection heights; and magnetic suppression of acoustic sources inside the spot (the 'showerglass' effect) distorts the measured phases. All of these effects mimic or mask thermal wave-speed perturbations, and magnetic forces enter the wave operator quadratically in the field strength BB, so inversions linearized about a field-free reference model cannot, even in principle, return a unique separation of temperature and magnetic field.

Problem Statement

Determine, by direct observation rather than model-dependent argument, the subphotospheric magnetic configuration of a mature sunspot, which is resolved only when all three of the following are addressed jointly, since they are facets of one scientific objective; a complete result on any one item alone does not resolve the problem:

  1. Construct and validate a local-helioseismic inversion that separates the thermal (temperature and density) contributions from the direct magnetic contributions to the observed wave-speed anomalies beneath a sunspot, with quantified uncertainties, replacing the current practice of inferring a single effective scalar wave speed. Validation must demonstrate recovery of known magnetic and thermal structure on realistic magnetized sunspot simulations and robustness against each of the known confounders (mode conversion at the equipartition layer, Wilson depression, showerglass phase shifts, filtering choices).

  2. Apply such a separation to observed sunspots in order to establish the subphotospheric magnetic geometry: whether the umbral flux below the cool surface column remains in compact Parker-type flux tubes or expands sharply into a diffuse configuration, and, if the shallow-sunspot picture is correct, measure the depth and sharpness of the compact-to-diffuse transition (the source predicts an expansion beginning near 4 Mm below the photosphere). This measurement should be obtained for more than one sunspot, with uncertainties.

  3. Test whether the resulting depth-dependent magnetic stratification reproduces the observed low-frequency (long-period) eigenmode spectra of sunspots, as the source claims, by quantitative comparison of model eigenfrequencies with oscillation spectra from ground-based and spaceborne instruments across different sunspots.

The source's roughly 4 Mm expansion depth and the validity of Parker's cluster model below the cool column are the specific hypotheses to be tested by this program; they must not be replaced by an effective sound-speed anomaly, a flow measurement, or a restricted model class that assumes the answer.

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