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arxiv: 2601.16203 · v1 · submitted 2026-01-22 · 🌌 astro-ph.SR

Cyclic sunspot activity during the first millennium CE as reconstructed from radiocarbon

Ilya Usoskin , Sami K. Solanki , Natalie A. Krivova , Theodosis Chatzistergos This is my paper

Pith reviewed 2026-05-16 11:42 UTC · model grok-4.3

classification 🌌 astro-ph.SR
keywords sunspot numberssolar cyclesradiocarbongrand solar minimumcosmogenic isotopessolar activity reconstructionfirst millenniumsolar dynamo
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The pith

Radiocarbon data yield annual sunspot numbers for 1-969 CE, showing 91 cycles with a mean length of 10.6 years.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reconstructs yearly sunspot counts across the first millennium using precise radiocarbon measurements. It identifies 91 solar cycles whose lengths range from 8 to 15 years and average 10.6 years, while also locating a grand solar minimum from 650 to 730 CE and an extreme event in 774 CE. The reconstruction relies on a chain of physics models that convert measured carbon-14 into production rates, open magnetic flux, and finally sunspot numbers, with uncertainties handled through Markov Chain Monte Carlo sampling. A sympathetic reader would care because the work supplies the missing link between earlier and later solar-cycle statistics and thereby supplies concrete numbers that any model of the Sun's long-term behavior must reproduce.

Core claim

Annual sunspot numbers were reconstructed for the first millennium CE. This period includes one extreme solar event of 774 CE and one Grand solar minimum of 650-730 CE. We could identify 91 solar cycles, of which 26 were well-defined, while 24 and 41 were reasonably and poorly defined, respectively. The mean cycle length was 10.6 years, but the lengths of individual cycles vary between 8 and 15 years. The existence of empirical Waldmeier's relations remains inconclusive. No significant periodicities were found beyond the 11-year cycle.

What carries the argument

A three-step physics-based chain that first converts measured radiocarbon data into 14C production rates via a carbon-cycle box model, then derives open solar magnetic flux from a heliospheric cosmic-ray modulation model, and finally computes sunspot numbers from a model of the Sun's magnetic-field evolution, with Markov Chain Monte Carlo used to propagate uncertainties.

If this is right

  • The cycle statistics supply direct numerical constraints that solar dynamo models must satisfy over a full millennium.
  • The reconstruction fills the previously empty interval between first-millennium-BCE and second-millennium-CE records, enabling a consistent three-millennium view once a composite multi-proxy record is assembled.
  • Irradiance models can now be tested against a documented grand minimum and an extreme event within this interval.
  • The absence of periodicities other than the 11-year cycle and the inconclusive status of Waldmeier relations limit the set of allowed empirical scaling laws.
  • Cycle lengths varying between 8 and 15 years indicate that the Sun can sustain a broad range of cycle durations even without modern observational biases.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These cycle lengths and the documented grand minimum offer a benchmark for testing whether the solar dynamo operates in a single regime or switches between different modes over centuries.
  • A completed three-millennium composite could reveal whether the fraction of well-defined versus poorly defined cycles changes systematically with time.
  • The reconstruction supplies a concrete baseline against which future space-weather or climate models can quantify the effects of a grand minimum similar to the one identified here.

Load-bearing premise

The sequence of models accurately translates measured radiocarbon levels into sunspot numbers without introducing large systematic biases during the pre-telescopic era.

What would settle it

An independent reconstruction from beryllium-10 or tree-ring data that shows a substantially different number of cycles or a different length distribution over the same centuries would falsify the reconstruction.

Figures

Figures reproduced from arXiv: 2601.16203 by Ilya Usoskin, Natalie A. Krivova, Sami K. Solanki, Theodosis Chatzistergos.

Figure 1
Figure 1. Figure 1: Sequential steps of the annual solar activity recons [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Accordingly, the dataset needs to be detrended of t [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Different sources of uncertainties of the sunspot number reconstruction plotted at the 1σ level: radiocarbon measurement errors (red ∆ 14C line); geomagnetic models (blue VDM line); model uncertainties (magenta line) including both the computa￾tion of the OSF and OSF-to-SSN conversion; and the total un￾certainty (green line), which is close to the geometrical sum of the individual ones. The black dashed li… view at source ↗
Figure 6
Figure 6. Figure 6: Wavelet spectrum (Morlet basis, k=6) of the recon￾structed annual mean SN series. The black-contoured areas rep￾resent the statistically significant spectral power (p < 0.05 against the AR1 red noise). The white line bounds the cone of influence below which the results are unreliable. be imprecise for several years, while the maximum can be undefined. We found 18 such cycles. 3. The cycle is reasonably wel… view at source ↗
Figure 7
Figure 7. Figure 7: Histogram of the solar cycle lengths T (min-to-min) for all the reconstructed cycles (grey shaded bars) and for cycles cor￾responding to different values of the quality flag q as indicated in the legend. Acknowledgements. This work has received funding from the European Re￾search Council for Synergy Grant (project 101166910) and Advanced Grant (grant agreement No. 101097844 — project WINSUN). References Be… view at source ↗
read the original abstract

Context. Solar activity, dominated by the 11-year cyclic evolution, has been observed directly since 1610. Before that, indirect cosmogenic proxy data are used to reconstruct it over millennia. Recently, the precision of radiocarbon measurements has improved sufficiently to allow reconstructing solar activity over millennia. Aims. The first detailed reconstruction of solar activity, represented by annual sunspot numbers, is presented for 1-969 CE. Methods. The reconstruction of sunspot numbers from D14C was performed using a physics-based method involving several steps: using a carbon-cycle box model, the 14C production rate, corrected for the geomagnetic shielding, was computed from the measured data; The open solar magnetic flux was computed using a model of the heliospheric cosmic-ray modulation; Sunspot numbers were calculated using a model of the evolution of the Sun's magnetic field. The Markov Chain Monte Carlo approach was used to account for different sources of uncertainty. Results. Annual sunspot numbers were reconstructed for the first millennium CE. This period includes one extreme solar event of 774 CE and one Grand solar minimum of 650-730 CE. We could identify 91 solar cycles, of which 26 were well-defined, while 24 and 41 were reasonably and poorly defined, respectively. The mean cycle length was 10.6 years, but the lengths of individual cycles vary between 8 and 15 years. The existence of empirical Waldmeier's relations remains inconclusive. No significant periodicities were found beyond the 11-year cycle. Conclusions. This work fills the gap in the solar cycle statistics between the previously reconstructed first millennium BCE and the second millennium CE, providing vital constraints for the solar dynamo and irradiance models. A consistent 3-millennium-long reconstruction of sunspot numbers, based on a composite multi-proxy cosmogenic record, is pending.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reconstructs annual sunspot numbers for 1–969 CE from high-precision Δ14C data via a fixed three-step physics-based pipeline (carbon-cycle box model to obtain production rate after geomagnetic correction, heliospheric modulation model to open solar flux, and magnetic-field evolution model to sunspot number), with MCMC propagation of uncertainties. It reports 91 solar cycles (26 well-defined, 24 reasonably defined, 41 poorly defined) with mean length 10.6 yr (range 8–15 yr), identifies a grand minimum at 650–730 CE and an extreme event at 774 CE, finds no significant periodicities beyond the 11-year cycle, and leaves Waldmeier relations inconclusive.

Significance. If the model chain is shown to be unbiased outside the modern calibration interval, the reconstruction would supply the missing first-millennium segment of a multi-millennium sunspot record, furnishing direct constraints on solar-dynamo cycle statistics, grand-minimum frequency, and irradiance models. The physics-based (rather than purely empirical) approach and explicit MCMC treatment of multiple uncertainty sources are methodological strengths.

major comments (2)
  1. [Methods] Methods (three-step reconstruction procedure): no validation test is presented in which the identical pipeline is applied to post-1610 Δ14C data and the output annual sunspot numbers are compared quantitatively (e.g., via RMSE, cycle-length distribution, or amplitude correlation) to the observed Wolf or group sunspot record. Because the carbon-cycle residence times, geomagnetic dipole strength, and cosmic-ray modulation relations are calibrated on modern conditions, any systematic offset would propagate directly into the reported cycle count, classification, and 10.6 yr mean length for 1–969 CE.
  2. [Results] Results (cycle identification and classification): the quantitative criteria that distinguish the 26 well-defined, 24 reasonably defined, and 41 poorly defined cycles are not stated, nor is any sensitivity analysis shown for how changes in those thresholds affect the mean cycle length or the identification of the 650–730 CE grand minimum. This classification is load-bearing for the central claim that 91 cycles were recovered.
minor comments (2)
  1. [Abstract] Abstract: the statement that Waldmeier relations “remain inconclusive” should specify which relations were examined and the metric used to reach that conclusion.
  2. [Conclusions] Conclusions: the reference to a pending “consistent 3-millennium-long reconstruction” would benefit from explicit citations to the prior BCE and post-1000 CE segments so readers can assess continuity of the modeling framework.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed report. The comments highlight important points for strengthening the manuscript, and we will revise accordingly to address them directly.

read point-by-point responses

  1. Referee: [Methods] Methods (three-step reconstruction procedure): no validation test is presented in which the identical pipeline is applied to post-1610 Δ14C data and the output annual sunspot numbers are compared quantitatively (e.g., via RMSE, cycle-length distribution, or amplitude correlation) to the observed Wolf or group sunspot record. Because the carbon-cycle residence times, geomagnetic dipole strength, and cosmic-ray modulation relations are calibrated on modern conditions, any systematic offset would propagate directly into the reported cycle count, classification, and 10.6 yr mean length for 1–969 CE.

    Authors: We agree that an explicit end-to-end validation of the full pipeline against post-1610 data is a valuable addition. While the component models (carbon-cycle box model, heliospheric modulation, and magnetic-field evolution) have each been validated in prior literature, we will insert a new subsection in the Methods that applies the identical three-step procedure to modern Δ14C measurements and reports quantitative comparisons (RMSE, cycle-length distribution, amplitude correlation) to the observed Wolf/group sunspot record. This will directly test for systematic offsets. revision: yes

  2. Referee: [Results] Results (cycle identification and classification): the quantitative criteria that distinguish the 26 well-defined, 24 reasonably defined, and 41 poorly defined cycles are not stated, nor is any sensitivity analysis shown for how changes in those thresholds affect the mean cycle length or the identification of the 650–730 CE grand minimum. This classification is load-bearing for the central claim that 91 cycles were recovered.

    Authors: We acknowledge that the precise quantitative thresholds used to classify cycles as well-defined, reasonably defined, or poorly defined were not stated explicitly in the text. In the revision we will add a clear definition of these criteria (based on peak prominence, signal-to-noise ratio, and minimum duration) together with a sensitivity analysis that shows the effect of varying the thresholds on total cycle count, mean length, and the identification of the 650–730 CE grand minimum. revision: yes

Circularity Check

0 steps flagged

No significant circularity: physics-based model chain applied to external proxy data

full rationale

The derivation applies a fixed three-step pipeline (carbon-cycle box model inverting measured Δ14C to production rate, heliospheric modulation model to open flux, magnetic-field evolution model to sunspot numbers) with MCMC error propagation. These steps use previously published models on new input data from the first millennium CE; no parameters are fitted to the target sunspot series or cycle statistics, and the output quantities (91 cycles, 10.6 yr mean length) are not forced by construction to match any input. Self-citations to model development papers do not reduce the chain to tautology, as the models remain independently testable against modern observations and the proxy measurements are external. The reconstruction is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The reconstruction rests on three established physical models whose internal parameters are not enumerated in the abstract; these parameters function as free parameters whose values are taken from prior literature or modern calibration.

free parameters (3)
  • carbon-cycle box model parameters
    Used to invert measured Δ14C into production rate; values drawn from prior literature or modern tuning.
  • heliospheric modulation parameters
    Relate cosmic-ray intensity to open solar magnetic flux; calibrated on contemporary observations.
  • magnetic-field evolution model parameters
    Convert open flux into sunspot numbers; taken from earlier solar-cycle modeling studies.
axioms (2)
  • domain assumption The relation between open solar magnetic flux and cosmic-ray modulation remains valid over the first millennium CE
    Invoked when converting production rate to open flux in the heliospheric model step.
  • domain assumption The carbon-cycle box model accurately represents atmospheric and oceanic 14C transport for the studied period
    Used in the first inversion step from measured Δ14C.

pith-pipeline@v0.9.0 · 5654 in / 1523 out tokens · 69591 ms · 2026-05-16T11:42:33.855816+00:00 · methodology

discussion (0)

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Reference graph

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