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.
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 entire 8-year sample, with no data selection at all, we find α = 3.06 ±. 0.05.
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 nature of the distribution: α > 1 implies correlations or bunching of events, α = 1 random occurrence, and α < 1 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 finite digital levels.
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