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The Variance of Solar X-ray Flux

Hhudson — Mon, 24 Aug 2026 16:01:20 GMT

Correlation of variance and mean

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	var(S) = mean(S)<sup>α</sup>		var(S) = mean(S)<sup>α</sup>

	where S is the flux and var(S) is its variance. For the ~~entire~~ 8-year		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.		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).		There is no obvious dependence on the solar cycle (Figure 2).

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Hhudson — Sat, 22 Aug 2026 15:52:41 GMT

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Hhudson — Sat, 22 Aug 2026 15:52:23 GMT

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Hhudson — Sat, 22 Aug 2026 15:51:58 GMT

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The Variance of Solar X-ray Flux

Hhudson — Sat, 22 Aug 2026 07:36:47 GMT

Taylor's law

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	expected from		expected from
	[https://en.wikipedia.org/wiki/Poisson_distribution Poisson statistics].		[https://en.wikipedia.org/wiki/Poisson_distribution Poisson statistics].
			The X-axis (the means) covers fluxes equivalent to GOES flare classes
			AO.1 to X1.
	</i>]]		</i>]]

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	Annual values of the slope parameter α.		Annual values of the slope parameter α.
	</i>]]		</i>]]

			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 ==		== Taylor's law ==
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	ecology), but also many physical systems and even some distributions of		ecology), but also many physical systems and even some distributions of
	a purely mathematical nature.		a purely mathematical nature.
	The index α follows the physical ~~nature~~ of the distribution:		The index α follows the physical origin of the distribution:
	α > 1 implies correlations or bunching of events, α = 1 random		α > 1 implies correlations or bunching of events, α = 1 implies random
	occurrence, and α < 1 the suppression of occurrence by an event.		occurrence, and α < 1 implies the suppression of occurrence by an event.
	Interestingly the lowest GOES flux levels (2018 in Figure 1) show a		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		piling-up at the bottom of the scale with this property, probably produced
	by finite digital levels.		by the necessarily finite digital levels at the background detection limit.

	== Conclusion ==		== Conclusion ==

SolarNuggets

Hhudson — Fri, 21 Aug 2026 19:59:42 GMT

Added No. 533

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	Welcome to the [[SolarNuggets]] collection, which extends the series of [[RHESSI]] Nuggets. The following is a time-ordered list of the latest Nuggets added to the HelioWiki. An [[:Category:Nugget\|alphabetical list of the SolarNuggets]] is also available as well as [[:Category:RHESSI Nugget List\|yearly lists]]. One can search on author, topic, IAU flare identifier, etc.). We welcome volunteer authors - please see our page of [[Help:For_Authors\| help for authors]] or just send an email to the Curator at (hugh.hudson@glasgow.ac.uk).		Welcome to the [[SolarNuggets]] collection, which extends the series of [[RHESSI]] Nuggets. The following is a time-ordered list of the latest Nuggets added to the HelioWiki. An [[:Category:Nugget\|alphabetical list of the SolarNuggets]] is also available as well as [[:Category:RHESSI Nugget List\|yearly lists]]. One can search on author, topic, IAU flare identifier, etc.). We welcome volunteer authors - please see our page of [[Help:For_Authors\| help for authors]] or just send an email to the Curator at (hugh.hudson@glasgow.ac.uk).

			{{Nugget Badge
	~~Filament Eruptions as seen in the Sun~~-as-~~a-star H-alpha Spectrum~~		\|title = The Variance of Solar X-ray Flux
			\|number = 533
			\|first_author = Hugh HUDSON
			\|publish_date = 17 August 2026
			\|description = Taylor's law describes solar X-ray variability all the way
			\|image=Icon533.png}}

	{{Nugget Badge		{{Nugget Badge
	\|title = Electron-Ion equilibration in CME-driven shocks		\|title = Electron-Ion equilibration in CME-driven shocks

File:Icon533.png

Hhudson — Fri, 21 Aug 2026 19:58:53 GMT

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The Variance of Solar X-ray Flux

Hhudson — Fri, 21 Aug 2026 19:46:14 GMT

Correlation of variance and mean

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	There is no obvious dependence on the solar cycle (Figure 2).		There is no obvious dependence on the solar cycle (Figure 2).

	[[File:533f2.png\|center\|thumb\|~~600px~~\|caption\|Figure 1: <i>		[[File:533f2.png\|center\|thumb\|400px\|caption\|Figure 1: <i>
	Annual values of the slope parameter α.		Annual values of the slope parameter α.
	</i>]]		</i>]]

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Hhudson — Fri, 21 Aug 2026 19:45:51 GMT

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The Variance of Solar X-ray Flux

Hhudson — Fri, 21 Aug 2026 19:45:25 GMT

Correlation of variance and mean

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	the modern GOES-R data, covering the years 2017-2025.		the modern GOES-R data, covering the years 2017-2025.

	The basic tool used in the analysis is the power spectrum,		The basic tool used in the analysis is the
			[https://mathworld.wolfram.com/PowerSpectrum.html power spectrum],
	as inspired by Ref. [1] and a great deal of staring at the timeseries.		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,		These show a somewhat subtle property of the data: the RMS fluctuation,
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	for 8 years of recent GOES data, using one-day intervals.		for 8 years of recent GOES data, using one-day intervals.

	[[File:~~533f1~~.png\|center\|thumb\|~~800px~~\|caption\|Figure 1: <i>		[[File:533_f1.png\|center\|thumb\|600px\|caption\|Figure 1: <i>
	Variance vs. mean for one-day intervals over 8 years of GOES		Variance vs. mean for one-day intervals over 8 years of GOES
	soft X-ray (1-8 Å) timeseries, using standard one-minute		soft X-ray (1-8 Å) timeseries, using standard one-minute
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	where S is the flux and var(S) is its variance. For the entire 8-year		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.		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).

			[[File:533f2.png\|center\|thumb\|600px\|caption\|Figure 1: <i>
			Annual values of the slope parameter α.
			</i>]]

	== Taylor's law ==		== Taylor's law ==
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	== Conclusion ==		== Conclusion ==

	~~Just~~ a question: do stellar X-ray time series have the same exponent?		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 ==		== References ==
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	[2] [https://ui.adsabs.harvard.edu/abs/1961Natur.189..732T "Aggregation, Variance and the Mean"]		[2] [https://ui.adsabs.harvard.edu/abs/1961Natur.189..732T "Aggregation, Variance and the Mean"]

			[3] Solar Physics, in review 2026

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Hhudson — Fri, 21 Aug 2026 19:34:36 GMT

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The Variance of Solar X-ray Flux

Hhudson — Fri, 21 Aug 2026 19:31:28 GMT

No. 533 initial upload

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{{Infobox Nugget |name = Nugget
|title = Variance of solar X-ray flux
|first_author = Hugh HUDSON
|publish_date = August 17, 2026
|next_nugget =
|previous_nugget = {{#ask: [[Category:Nugget]] [[RHESSI Nugget Index::532]]}}
}}

== Introduction ==

The time series of solar soft X-ray fluxes has been measured almost
continously since the 1970s, famously by the
[https://www.spaceweather.gov/products/goes-x-ray-flux 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
[https://www.astro.gla.ac.uk/users/hsh3f/scratch/fai.html 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
[https://en.wikipedia.org/wiki/Solar_flare 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.

[[File:533f1.png|center|thumb|800px|caption|Figure 1: <i>
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
[https://en.wikipedia.org/wiki/Poisson_distribution Poisson statistics].
</i>]]

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)<sup>α</sup>

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 ==

Just a question: do stellar X-ray time series have the same exponent?

== References ==

[1] [http://adsabs.harvard.edu/abs/1991SoPh..133..357H "Solar flares, microflares, nanoflares, and coronal heating"]

[2] [https://ui.adsabs.harvard.edu/abs/1961Natur.189..732T "Aggregation, Variance and the Mean"]

File:533 f1.png

Hhudson — Fri, 21 Aug 2026 19:28:41 GMT

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