Reconstructions of Total Solar Irradiance (TSI) over the past millennium disagree by roughly an order of magnitude on the Maunder-Minimum-to-present change, from essentially zero to a decrease of several W m−2\mathrm{W\,m^{-2}}. The pivotal unresolved assumption separating the low- and high-amplitude families is whether the quiet Sun's small-scale magnetic field — the mixed-polarity internetwork field sustained by the small-scale turbulent dynamo (SSD), together with weak magnetic features not connected to active regions — is invariant on multi-decadal to centennial timescales, or instead varies systematically between activity states as different as a grand minimum and the modern epoch. Penza et al. (2024)'s millennium TSI reconstruction explicitly flags this assumption as binding for 3D-MHD-based models. The problem is to determine, by physics-based simulation, observation, or a combination, whether and by how much quiet-Sun small-scale magnetization changes between a grand-minimum Sun and the modern Sun, and to quantify the resulting TSI modulation, given the established radiative sensitivity (a 10% change in quiet-Sun field strength produces a TSI change comparable to the solar cycle). This core is open as of August 2026.

Background

Total Solar Irradiance (TSI) is the spectrally integrated solar radiative flux measured at the mean Sun-Earth distance; it has been measured continuously by space radiometers only since 1978, and cross-calibrating those instruments has produced competing composites (e.g., PMOD, the composite maintained by the Physikalisch-Meteorologisches Observatorium Davos). On timescales longer than a few hours, irradiance variations are driven by the solar surface magnetic field: dark sunspots reduce, while bright faculae and the magnetic network enhance, the emitted flux. Models that reconstruct TSI before the space era therefore extrapolate the magnetic field back in time from proxies — sunspot records for the telescopic era, and cosmogenic isotopes (14^{14}C in tree rings, 10^{10}Be in ice cores) on centennial to millennial timescales, which record the modulation of galactic cosmic rays by the heliospheric open magnetic flux FoF_o (parameterized by the modulation potential ϕ\phi, a measure in MeV of the energy spectrum of the cosmic rays reaching Earth) rather than surface magnetism directly.

The quiet Sun — the surface outside sunspots, faculae, and network — is permeated by a ubiquitous, mixed-polarity, highly dynamic weak field on scales below the network (the internetwork field), believed to be sustained by the small-scale dynamo (SSD), a turbulent dynamo self-excited by near-surface convection and largely independent of the 11-year activity cycle. Grand minima, such as the Maunder Minimum (1645-1715) and the Spörer Minimum (15th-16th century), are decades-long episodes when sunspots and the large-scale (global) dynamo nearly vanish. Whether the SSD field persists unchanged through such episodes is unknown.

TSI reconstructions divide by how they treat the quiet Sun. SATIRE (Spectral And Total Irradiance REconstructions) semi-empirical models hold the quiet Sun fixed (the SATIRE reconstruction of Wu et al. 2018 gives an increase of 0.92 W m−20.92\ \mathrm{W\,m^{-2}} from the end of the Maunder minimum to the 1975–2005 mean, quoted as 0.55 W m−20.55\ \mathrm{W\,m^{-2}} for the SATIRE Maunder-minimum change by Penza et al. 2024; the Wang & Lean model gives 0.2 W m−20.2\ \mathrm{W\,m^{-2}}), while Shapiro et al. (2011) assume the quiet-Sun output itself changes strongly with activity, giving changes of several W m−2\mathrm{W\,m^{-2}} (Egorova et al. 2018 obtain 3.7-4.5 W m−2\mathrm{W\,m^{-2}}); Lockwood & Ball (2020) argue for essentially no change. Penza et al. (2024) reconstruct TSI from 971 to 2020 CE using FoF_o from cosmogenic 14^{14}C with a semi-empirical model, obtaining a Maunder-Minimum TSI about 2.2 W m−22.2\ \mathrm{W\,m^{-2}} below modern minima. Using 3D magnetohydrodynamic (MHD) simulations of the SSD, Yeo et al. (2020) computed the TSI of the Sun's least-active state — global dynamo dormant, only the SSD active — to be 2.0±0.7 W m−22.0 \pm 0.7\ \mathrm{W\,m^{-2}} below the 2019 level, bounding the Maunder-to-present rise; this bound is explicitly conditional on the SSD being not or only weakly coupled to the global dynamo. Rempel (2020) quantified the radiative stakes: the outgoing flux depends linearly on the quiet-Sun mean vertical field strength ⟨∣Bz∣⟩\langle |B_z| \rangle at photospheric optical depth τ=1\tau = 1 with relative slope ≈1.4×10−4 G−1\approx 1.4\times 10^{-4}\ \mathrm{G^{-1}}, so a 10% change in quiet-Sun field strength would produce a TSI change comparable to the ∼0.1%\sim 0.1\% solar-cycle variation. The invariance assumption is therefore not a harmless convention: plausible secular SSD changes would materially alter the inferred multi-centennial TSI modulation, a key natural forcing input to climate models (e.g., the solar forcing dataset defined for CMIP7, the Coupled Model Intercomparison Project Phase 7).

Settling the question observationally is hard. High-resolution spectropolarimetry of the internetwork exists only since about 2008, with the Hinode satellite, covering roughly one and a half activity cycles and no grand minimum; cosmogenic isotopes trace heliospheric open flux, not surface small-scale fields; Ca II K (the calcium K spectral line) photographic archives since 1892 trace plage (bright magnetic regions outside sunspots) and network, not the SSD internetwork. Cycle-scale evidence exists — internetwork magnetic power at sub-0.5-arcsec scales is constant over cycle 24 (Faurobert & Ricort 2021), and SSD simulations best reproduce observed internetwork polarization statistics (Ebert et al. 2024) — but no observation constrains centennial or grand-minimum behavior.

Problem Statement

Determine whether the quiet Sun's small-scale magnetic field — the mixed-polarity internetwork field sustained by the small-scale turbulent dynamo (SSD), together with weak magnetic features not connected to active regions — is invariant, or instead varies systematically, on multi-decadal to centennial timescales, in particular between a grand minimum of solar activity (such as the Maunder Minimum, 1645-1715) and epochs of normal activity such as the modern Sun.

The answer must be quantitative. It should deliver a determination or bound on the change in quiet-Sun small-scale magnetization between a grand-minimum Sun and the modern Sun — for example the change in mean vertical field strength ⟨∣Bz∣⟩\langle |B_z| \rangle at photospheric optical depth τ=1\tau = 1, or an equivalent measure of the SSD-sustained field — together with the consequent modulation of Total Solar Irradiance, computed via the established radiative sensitivity of the quiet Sun to its magnetization (outgoing flux varying linearly with ⟨∣Bz∣⟩τ=1\langle |B_z| \rangle_{\tau=1} with relative slope ≈1.4×10−4 G−1\approx 1.4\times 10^{-4}\ \mathrm{G^{-1}}, such that a 10% change in quiet-Sun field strength produces a TSI change comparable to the ∼0.1%\sim 0.1\% solar-cycle variation; Rempel 2020).

The determination must be accurate enough to decide whether the quiet-Sun contribution to the Maunder-minimum-to-present TSI change is small compared with the spread separating current reconstructions — from near-zero change (Lockwood & Ball 2020; the 0.2 W m−20.2\ \mathrm{W\,m^{-2}} of the Wang & Lean model; the 0.92 W m−20.92\ \mathrm{W\,m^{-2}} increase from the end of the Maunder minimum to the 1975–2005 mean found by SATIRE in Wu et al. 2018) through the intermediate 2.0±0.7 W m−22.0 \pm 0.7\ \mathrm{W\,m^{-2}} dimmest-state bound of Yeo et al. (2020) and the 2.2 W m−22.2\ \mathrm{W\,m^{-2}} of Penza et al. (2024), up to the several W m−2\mathrm{W\,m^{-2}} of quiet-Sun-varying models (Shapiro et al. 2011; Egorova et al. 2018) — i.e., whether the constant-quiet-Sun assumption built into the dimmest-state bound and into SATIRE-type reconstructions is justified, and if not, how the secular TSI modulation must be revised.

Admissible approaches include physics-based simulations (SSD simulations under imposed grand-minimum conditions, or coupled small-scale/global-dynamo modeling), observational constraints (spectropolarimetric or spectroscopic studies of solar-analog stars in Maunder-like activity states, or reanalysis of archival solar observations reaching further back than the modern instrumental era), or a combination that yields a defensible quantitative bound.

References

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