The Variance of Solar X-ray Flux

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1st Author:	Hugh HUDSON
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Published:	August 17, 2026
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Introduction

The time series of solar soft X-ray fluxes has been measured almost continously since the 1970s, famously by the GOES satellite series. We analyze the timeseries from its 1-8 Å photometers from the modern GOES-R data, covering the years 2017-2025.

The basic tool used in the analysis is the power spectrum, as inspired by Ref. [1] and a great deal of staring at the timeseries. These show a somewhat subtle property of the data: the RMS fluctuation, normalized to the median, tends to be smaller for lower flux levels. This can be seen real-time (currently) at the FAI ("flare anticipation index") link, which shows one-hour intervals. Note that the variance (and RMS) of this time series is undefined for long intervals because of the flat power-law occurrence distribution function of flares (Ref. [2]), the dominant source of solar X-ray fluctuations.

Correlation of variance and mean

It turns out that these quantities have a tight correlation, if assessed over finite time intervals. Figure 1 shows this correlation for 8 years of recent GOES data, using one-day intervals.

Figure 1: Variance vs. mean for one-day intervals over 8 years of GOES soft X-ray (1-8 Å) timeseries, using standard one-minute sampling. The diagonal lines show direct proportionality, as would be expected from Poisson statistics. The X-axis (the means) covers fluxes equivalent to GOES flare classes AO.1 to X1.

The figure shows clear and systematic deviations from Poisson statistics, and the correlations fit a power law over the entire dynamic range of a solar cycle:

var(S) = mean(S)α

where S is the flux and var(S) is its variance. For the 8-year sample, with no data selection at all, we find α = 3.06 ±. 0.05. There is no obvious dependence on the solar cycle (Figure 2).

Figure 1: Annual values of the slope parameter α.

The fact that the same power law fits the quiet Sun as well as the flaring Sun suggests that the flare mechanism is all that one needs to explain. This result helps to exclude the nanoflare idea for coronal heating (see Ref. [3]).

Taylor's law

This kind of relationship is known as Taylor's law, and I am grateful to ChatGPT for pointing me to Ref. [2]. It is empirical and describes many natural populations (this Taylor studied ecology), but also many physical systems and even some distributions of a purely mathematical nature. The index α follows the physical origin of the distribution: α > 1 implies correlations or bunching of events, α = 1 implies random occurrence, and α < 1 implies the suppression of occurrence by an event. Interestingly the lowest GOES flux levels (2018 in Figure 1) show a piling-up at the bottom of the scale with this property, probably produced by the necessarily finite digital levels at the background detection limit.

Conclusion

This Nugget is based on Ref. [3], and there is just a question about this: do stellar X-ray time series have the same exponent?

References

[1] "Solar flares, microflares, nanoflares, and coronal heating"

[2] "Aggregation, Variance and the Mean"

[3] Solar Physics, in review 2026