The smallest structures of the solar photosphere - the dark intergranular lanes (DIL) between granules, micropores, and facular knots - are described by Solov'ev et al. (2019, Astrophys. Space Sci. 364, 222) with a stationary magnetic-diffusion model in which converging convective flows balance the ohmic spreading of magnetic flux. The model predicts characteristic scales of 20-30 km for lane thickness and for the diameter of the thinnest magnetic flux tubes, and 200-400 km for micropores, but leaves four gaps open: the gas-kinetic conductivity sigma inside lanes and micropores is an unconstrained guess that enters all scale estimates under a square root; the predicted minimal scales lay below the resolution of every then-existing telescope, so direct verification was impossible; exact stationary solutions for strongly non-axisymmetric azimuthal modes of the micropore perimeter were not obtained; and the facular-knot model's predicted pressure, density, and temperature maps were never quantitatively tested against high-resolution spectropolarimetry. DKIST has now resolved 20-50 km photospheric striae at about 0.03 arcsec - bringing the observational frontier to the predicted scale range without yet measuring lane widths, flux-tube diameters, or lane internal structure. All four strands remain open; the observational part has become marginally feasible for the first time.

Background

High-resolution observations of the solar photosphere reveal a granulation pattern: bright granules, the tops of convective upflows roughly 700-1400 km across (1 arcsec corresponds to about 725 km on the Sun) living 10-20 minutes, separated by a continuous network of dark intergranular lanes (DIL), where plasma cools and sinks. Convection sweeps magnetic field into these lanes, concentrating it into kilogauss magnetic flux tubes - vertical bundles of magnetic field with strengths of order a kilogauss (about 0.1 tesla) - which appear as small bright points in the photosphere and, on larger scales, as faculae (bright magnetic regions outside sunspots). Micropores are small (a few hundred km), dark, penumbra-free magnetic concentrations, and a facular knot is a compact facular assembly whose dark central core is essentially a micropore.

Solov'ev et al. (2019) proposed that the DIL and micropores are maintained in a stationary magnetic-diffusion regime. In magnetohydrodynamics (the fluid description of conducting plasma), the induction equation with ohmic diffusion spreads magnetic structures at a rate set by the magnetic diffusivity η=c2/(4πσ)\eta = c^2/(4\pi\sigma), where σ\sigma is the gas-kinetic (electrical) conductivity of the plasma. The authors posit that converging horizontal and vertical plasma flows into the lane or micropore continuously compensate this ohmic spreading, so that for a structure of inverse spatial scale kk and dynamic time γ−1\gamma^{-1} the stationarity condition ηk2≃γ\eta k^2 \simeq \gamma holds. All characteristic scales then follow: taking a granule dynamic time of order 10310^3 s and conductivities interpolated between the hot photosphere (σ∼1012 s−1\sigma \sim 10^{12}\,\mathrm{s^{-1}}) and the cool sunspot umbra (σ∼1010 s−1\sigma \sim 10^{10}\,\mathrm{s^{-1}}), the model yields a DIL thickness of 20-30 km, a thinnest flux-tube diameter of about 25 km, and micropore diameters of 200-400 km with lifetimes near one day. Because σ\sigma enters under a square root, every predicted scale inherits the (moderate) uncertainty of the lane and micropore conductivity - which has never been observationally constrained; independently, an analysis of quiet-Sun magnetic scaling laws (Stenflo 2012) places the peak of the size distribution of flux tubes invisible in magnetograms (maps of the solar magnetic field) at 10-100 km, in exactly this range.

Two circumstances made these predictions untestable in 2019. First, adaptive-optics telescopes (ground-based telescopes with real-time wavefront correction) of the 0.7-1.6 m class then available - the Swedish 1-m Solar Telescope, the Goode Solar Telescope, GREGOR, the Vacuum Tower Telescope - are diffraction-limited to about 0.1-0.18 arcsec (70-130 km), and the practical resolution limit for the smallest magnetic elements was 40-70 km; structures of 20-30 km were entirely out of reach. Second, the theory itself is incomplete for strongly non-axisymmetric azimuthal modes of the micropore perimeter (real micropore boundaries are highly irregular): in the authors' words the exact stationary solutions are 'almost impossible to obtain', so the treatment rests on the approximations mi2−ni2=0m_i^2 - n_i^2 = 0 and 4ηaik2≃γi4\eta a_i k^2 \simeq \gamma_i, whose validity across the parameter space was not delineated. In parallel, the same group's analytic facular-knot model (Solov'ev & Kirichek 2019) predicts full pressure, density, and temperature maps, including a central temperature dip that deepens with field strength - a trend supported so far only by statistics at the resolution of NASA's Solar Dynamics Observatory (SDO) satellite (Riehokainen et al. 2019), never by a quantitative comparison with simultaneous high-resolution spectropolarimetry (measurements of the four Stokes parameters that encode the vector magnetic field and atmospheric thermodynamics).

The 4-m Daniel K. Inouye Solar Telescope (DKIST), with first light in 2021-2022, reaches a diffraction limit near 0.022-0.03 arcsec in blue wavelengths (16-22 km), placing the predicted minimal scales at or just below the observational frontier for the first time. That makes the completion of this program - constraining σ\sigma, directly resolving the predicted structures, completing the non-axisymmetric theory, and testing the facular-knot model - a well-posed and newly feasible open problem.

Problem Statement

Complete the quantitative verification of the stationary magnetic-diffusion picture of the finest photospheric magnetic structures, as left open by Solov'ev et al. (2019), in four jointly supporting parts: (1) Constrain the gas-kinetic conductivity σ\sigma (equivalently the magnetic diffusivity η\eta) inside dark intergranular lanes and micropores, where it is currently only a guessed intermediate value between photospheric and umbral conditions, and propagate that constraint into the predicted structure widths and lifetimes, whose estimates depend on σ\sigma under a square root (at fixed dynamic time the predicted width scales as σ−1/2\sigma^{-1/2}). (2) Directly observe the predicted minimal scales - lane widths of 20-30 km, thinnest flux-tube diameters near 25 km - and the internal structure of the dark intergranular lanes, now that DKIST-class telescopes reach effective resolutions of order 0.03 arcsec (about 22 km) but no such measurements have been published; the model's micropore diameters of 200-400 km with lifetimes near one day are already consistent with published observations. (3) Obtain the stationary solutions for fully general, strongly non-axisymmetric azimuthal modes of the micropore perimeter, or rigorously delineate the parameter ranges in which the source's approximations (mi2−ni2=0m_i^2 - n_i^2 = 0, 4ηaik2≃γi4\eta a_i k^2 \simeq \gamma_i) are valid. (4) Quantitatively confront the facular-knot model's predicted pressure, density, and temperature maps - including the field-strength dependence of its central darkening - with high-resolution observations over a sample of facular knots. Together, the four parts establish, or refute, the stationary diffusion-advection balance as the mechanism setting the minimal scale of photospheric magnetic structure.

References

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