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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

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

Hhudson uploaded File:Icon533.png

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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

Hhudson uploaded File:533f1.png

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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: 
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].
]]

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 ==

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"]

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

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SolarNuggets

Hhudson — Wed, 05 Aug 2026 19:38:57 GMT

Added 531 and 532

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 Filament Eruptions as seen in the Sun-as-a-star H-alpha Spectrum
 {{Nugget Badge
 |title = Filament Eruptions as seen in the Sun-as-a-star H-alpha Spectrum

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Hhudson — Wed, 05 Aug 2026 19:32:59 GMT

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Hhudson — Wed, 05 Aug 2026 19:32:32 GMT

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The Source Surface Height Through the Solar Cycle: A Path to Better Solar Wind Forecasts

Hhudson — Wed, 05 Aug 2026 12:59:15 GMT

Author's fixes

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	\|number = 531		\|number = 531
	\|first_author = Sandeep KUMAR		\|first_author = Sandeep KUMAR
			\|second_author = Nandita SRIVASTAVA
	\|publish_date = July 20, 2026		\|publish_date = July 20, 2026
	\|next_nugget =		\|next_nugget =
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	The PFSS model contains only a single free parameter, the source surface		The PFSS model contains only a single free parameter, the source surface
	height, at which magnetic field lines are assumed to be exactly radial.		height, at which magnetic field lines are assumed to be exactly radial.
	A value of 2.5 <sub>R⊙</sub> for the source-surface height has		A value of 2.5 R<sub>⊙</sub> for the source-surface height has
	traditionally been adopted in most studies.		traditionally been adopted in most studies.
	Optimizing the source-surface value significantly improves model forecasts		Optimizing the source-surface value significantly improves model forecasts
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	[[File:531f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>		[[File:531f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>
	Top panel: the source-surface radial height, in black for the standard		Top panel: the source-surface radial height, in black for the standard
	fixed value of 2.5 R<sub>R⊙</sub> and in blue and gold for two		fixed value of 2.5 R<sub>⊙</sub> and in blue and gold for two
	alternative optimizations.		alternative optimizations.
	The time range covers the 100		The time range covers the 100
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	Our adjusted PFSS radii clearly improve forecasts of the solar-wind speed.		Our adjusted PFSS radii clearly improve forecasts of the solar-wind speed.
	The ~~solar wind however does not flow in~~ a ~~spherically symmetric manner,~~		The study highlights an inherent limitation of the PFSS model, which assumes a spherical source surface.
	~~either near the corona or far out~~ in the ~~heliosphere.~~		This simplifying assumption restricts further improvement in the correlation coefficient.
	~~Thus it is not surprising that even the optimizations do not yield very~~		We anticipate much greater improvements from future modeling frameworks outside that of spherical symmetry of the source surface.
	~~strong~~ correlation ~~coefficients~~.
	We anticipate much greater improvements from future modeling frameworks outside
	that of spherical symmetry.


	== References ==		== References ==

Electron-Ion equilibration in CME-driven shocks

Hhudson — Wed, 05 Aug 2026 08:19:08 GMT

Conclusion

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	[[File:532f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>		[[File:532f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>

	~~Separate~~ regions ~~are~~ highlighted in yellow, magenta, and/or cyan		A schematic map of the shock, with separate regions highlighted in yellow, magenta, and/or cyan
	based on the performance of each model based on the time of the		based on the performance of each model based on the time of the
	peaks and the intensities of the peaks. Panels showing the combined		peaks and the intensities of the peaks. Panels showing the combined
	color maps use a CMY subtractive color model to accurately reflect		color maps use a CMY subtractive color model to accurately reflect
	where the data is unable to differentiate between the models. The		where the data is unable to differentiate between the models. The
	color wheel at the top-~~left~~ serves as an approximate guide, where		color wheel at the top-right serves as an approximate guide, where
	the color changes depending on the separation from the loci for		the color changes depending on the separation from the loci for
	each model. The solid blue hexagon, for instance, represents good		each model. The solid blue hexagon, for instance, represents good
Line 88:		Line 88:

	== Conclusion ==		== Conclusion ==

			Non-equilibrium plasmas probably dominate the Universe, but they are hard to diagnose via remote-sensing
			techniques.
			A wealth of plasma physics results from T<sub>e</sub> ≠ T<sub>i</sub> and anisotropic distribution functions,
			and large-scale shock waves in the solar corona provide an excellent opportunity to study such effects.

	== References ==		== References ==

Electron-Ion equilibration in CME-driven shocks

Hhudson — Wed, 05 Aug 2026 08:11:39 GMT

No. 532 initial upload

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{{Infobox Nugget |name = Nugget
|title = Electron-Ion equilibration in CME-driven shocks
|first_author = John RAYMOND
|publish_date = August 3, 2026
|next_nugget =
|previous_nugget = {{#ask: [[Category:Nugget]] [[RHESSI Nugget Index::531]]}}
}}

== Introduction ==

Coronal mass ejections
[https://en.wikipedia.org/wiki/Coronal_mass_ejection (CMEs)]
drive shock waves into regions of very low collisionality.
The jumps in density, pressure and temperature at a shock wave in dense
gas are mediated by particle collisions, and that leads to thermal
equilibrium: equal temperatures of all particle species and a Maxwellian
velocity distribution for each species.
In a low density plasma,
on the other hand, the collision lengths are large, and the shock
jump must therefore be mediated by electromagnetic fields and plasma waves.
That can lead to non-Maxwellian velocity distributions, such as Solar
Energetic Particles (SEPs), and to differing electron and ion
temperatures.

In the solar wind, postshock electron temperatures are generally
less than proton temperatures (e.g., Ref. [1])
though there is considerable scatter.
Shocks in supernova
remnants reach higher Mach numbers, and they show a trend
of decreasing electron-to-ion temperature ratio with increasing
shock speed or Mach number (Ref. [2]). CME-driven shocks
In the solar corona are more difficult to study, but they are
observed as type II radio bursts and as faint emission in UV, EUV
and white light coronagraph spectra and images (e.g., Ref. [3]).

== How do we infer plasma temperatures? ==

In some cases, it is possible to infer the electron temperature
behind a coronal shock by comparing extreme ultraviolet images from
[https://aia.lmsal.com AIA] in different bands.
When the electrons are suddenly heated in a shock, the iron ions
are successively ionized from Fe X to Fe XII to Fe XIV to Fe XVI
(The AIA 171, 193, 211 and 335 Å bands).
If the density is known from
[https://en.wikipedia.org/wiki/Solar_radio_emission type II]
radio emission or from a global MHD model of the corona, the lags
between the appearance of the shock in the
different bands indicate the electron temperature (Refs. [4,5]).
Figure 1 shows a CME-driven shock observed
on 2010 June 13, with the 3D structure inferred from AIA and STEREO
images (Ref. [5]).
Figure 2 shows the dimensionless
parameter ΔTe at different positions along the shock front,
where ΔTe = 0 corresponds to equal electron and proton
temperatures and ΔTe = 1 corresponds to no electron heating.
The intermediate values seen in Figure 2 show that these 500-700 km/s
shocks heat the electrons about half as efficiently as the ions.

[[File:532f1.png|center|thumb|600px|caption|Figure 1: 
SDO/AIA 193 Å running difference images at three different times of
the 13 June 2010 CME-driven shock wave. The blue line shows the
outline of the geometric model - here, the data were binned by a
factor of 8 to increase the S/N when fitting the model to the data.
The software captures the irregular shape of the shock, accounting
for angle-dependent acceleration. The shock can be identified as
the bright outer ring of the structure, while the erupting prominence
driving the shock is clearly identified as the bright inner ring
structure. Bottom: Corresponding STEREO-A EUVI 195 Å difference
images, with the geometric model plotted in blue. Right: Labeled
reference of different portions of the shock structure.
]]

[[File:532f2.png|center|thumb|600px|caption|Figure 2: 

Separate regions are highlighted in yellow, magenta, and/or cyan
based on the performance of each model based on the time of the
peaks and the intensities of the peaks. Panels showing the combined
color maps use a CMY subtractive color model to accurately reflect
where the data is unable to differentiate between the models. The
color wheel at the top-left serves as an approximate guide, where
the color changes depending on the separation from the loci for
each model. The solid blue hexagon, for instance, represents good
agreement with both ΔTe = 0.5 (cyan) and
ΔTe = 0.25 (magenta) models
where ΔTe = 0 for complete equilibration and 1 for no
equilibration.
]]

== Conclusion ==

== References ==

[1] [https://ui.adsabs.harvard.edu/abs/2013SSRv..178..633G "Electron-Ion Temperature Equilibration in Collisionless Shocks: The Supernova Remnant-Solar Wind Connection"]

[2] [https://ui.adsabs.harvard.edu/abs/2023ApJ...949...50R "Electron-Ion Temperature Ratio in Astrophysical Shocks"]

[3] [http://adsabs.harvard.edu/abs/2004A%26A...413..363M "Coronal transients and metric type II radio bursts. I. Effects of geometry"]

[4] [https://ui.adsabs.harvard.edu/abs/2011ApJ...738..160M "Observations and Interpretation of a Low Coronal Shock Wave Observed in the EUV by the SDO/AIA"]

[5] [https://ui.adsabs.harvard.edu/abs/2025ApJ...989..175T "A 3D Nonequilibrium Ionization Model of a Shock Wave in the Low Corona. I. Extreme-ultraviolet Emission and Inefficient Electron Heating"]

The Source Surface Height Through the Solar Cycle: A Path to Better Solar Wind Forecasts

Hhudson — Mon, 03 Aug 2026 18:22:04 GMT

The solar cycle

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Hhudson — Mon, 03 Aug 2026 16:54:01 GMT

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Hhudson — Mon, 03 Aug 2026 16:53:35 GMT

Hhudson uploaded File:531f1.png

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The Source Surface Height Through the Solar Cycle: A Path to Better Solar Wind Forecasts

Hhudson — Mon, 03 Aug 2026 16:52:48 GMT

No. 531 initial upload

New page

{{Infobox Nugget |name = Nugget
|title = The Source Surface Height Through the Solar Cycle: A Path to Better Solar Wind Forecasts
|number = 531
|first_author = Sandeep KUMAR
|publish_date = July 20, 2026
|next_nugget =
|previous_nugget = {{#ask: [[Category:Nugget]] [[RHESSI Nugget Index::530]]}}
}}

== Introduction ==

The potential field source surface (PFSS) model serves as the basis
of many state-of-the-art space weather forecasting frameworks,
as described in our Ref. [1].
Essentially PFSS takes the photospheric magnetic field, observable
in detail by [https://en.wikipedia.org/wiki/Zeeman_effect Zeeman effect]
spectroscopy, and uses it to extrapolate into the solar corona.
This requires the assumption of just a single parameter: the radial
distance of the "source surface", a fictitious sphere separating
radial field of the solar wind, from the structured field of the photosphere.
Mathematically any field induced by external sources can be represented by a
fitting
[https://en.wikipedia.org/wiki/Potential_theory spherical harmonic]
functions at its boundaries.
This is tractable but physically it may seem terribly wrong,
since it ignores all coronal field sources (currents) within the volume itself.
But it works surprisingly well, perhaps because the currents that we
know to be present only make a minor perturbation of the basic potential
field.

== The nature of the source surface ==

The PFSS model contains only a single free parameter, the source surface
height, at which magnetic field lines are assumed to be exactly radial.
A value of 2.5 R⊙ for the source-surface height has
traditionally been adopted in most studies.
Optimizing the source-surface value significantly improves model forecasts
of the solar-wind speed as observed at the L1
[https://science.nasa.gov/solar-system/resources/faq/what-are-lagrange-points/ Lagrangian point],
just upstream of Earth in the
[https://en.wikipedia.org/wiki/Solar_wind solar wind] (Ref. [1]).
This study employed two types ("HU STD" and "HU ZPC"] of
([https://en.wikipedia.org/wiki/Global_Oscillations_Network_Group GONG)
synoptic magnetic field maps.
The fidelity of these data
was confirmed by comparing the extrapolated global
magnetic field structures with the large-scale corona observed in
the extended field of view of the PROBA2/SWAP images as shown in
Figure 1.

[[File:531f1.png|center|thumb|600px|caption|Figure 1: 
PFSS-extrapolated magnetic field lines overlaid on the
[https://www.cosmos.esa.int/web/proba-2/swap SWAP]
(top and bottom panel) image of 20 August 2017, 10:45 UT. The middle
panel shows the SWAP mosaic without PFSS extrapolation. The top
left panel shows PFSS extrapolation with HU STD map, and the bottom
left panel with HU ZPC map (on 20 August 2017, 12:14 UT). The right
column shows the zoomed-in version of the white rectangle on the
left panels.
]]

== The solar cycle ==

How must the PFSS model evolve through the
[https://www.spaceweather.gov/products/solar-cycle-progression solar cycle],
during which the solar wind obviously changes substantially?
In Ref. [3] we carried out one of the most comprehensive and
long-term studies of
SS height optimisation for solar wind prediction at L1, analyzing
synoptic magnetograms from both space-based
([http://hmi.stanford.edu SDO/HMI]) and ground-based
(GONG) observatories, across nearly three solar cycles, SC23-SC25.
This optimization used the PFSS parameters in the model of Ref. [2] to
compare with the wind-speed observations.
Our main finding is that the optimal SS height varies systematically
with the solar cycle: higher SS heights (≥2.5 R⊙) provide better
solar wind speed predictions during solar minimum, whereas lower
SS heights (<2.5 R⊙) perform better during solar maximum, suggesting
a relationship between SS height and SC phase over long timescales.
We also found that the optimized SS height depends on the choice
of magnetogram, whereas the pattern remains similar for a given
choice of magnetograms. The HMI and GONG ZPC magnetograms perform
similarly, and both provide better forecasting than GONG STD maps,
as shown in Figures 2 and 3. The results provide strong evidence
that solar-cycle-dependent optimization of the PFSS source surface
height is a practical way to improve solar wind forecasting, which
serves as the foundation for heliospheric models. ]]

[[File:531f2.png|center|thumb|600px|caption|Figure 2: 
Top panel: the source-surface radial height, in black for the standard
fixed value of 2.5 RR⊙ and in blue and gold for two
alternative optimizations.
The time range covers the 100
[https://en.wikipedia.org/wiki/Solar_rotation Carrington rotations]
from January 2018 (solar minimum) to July 2025 (high activity).
The lower panel shows Pearson correlation coefficients for predictions of
the solar-wind speed.
]]

== Conclusions ==

Our adjusted PFSS radii clearly improve forecasts of the solar-wind speed.
The solar wind however does not flow in a spherically symmetric manner,
either near the corona or far out in the heliosphere.
Thus it is not surprising that even the optimizations do not yield very
strong correlation coefficients.
We anticipate much greater improvements from future modeling frameworks outside
that of spherical symmetry.

== References ==

[1] [https://ui.adsabs.harvard.edu/abs/2025JSWSC..15...24K "On the role of source surface height and magnetograms in solar wind forecast accuracy"]

[2] [http://adsabs.harvard.edu/abs/2015SpWea..13..154R "On the role played by magnetic expansion factor in the prediction of solar wind speed"]

[3] [https://ui.adsabs.harvard.edu/abs/2026ApJ..1006...26K "Source Surface Height Optimization for Improved Solar Wind Velocity Forecasting across Solar Cycles 23, 24, and 25"]

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	synoptic magnetograms from both space-based		synoptic magnetograms from both space-based
	([http://hmi.stanford.edu SDO/HMI]) and ground-based		([http://hmi.stanford.edu SDO/HMI]) and ground-based
	(GONG) observatories, across nearly three solar cycles, SC23-SC25.		([https://gong.nso.edu GONG]) observatories, across nearly three solar cycles, SC23-SC25.
	This optimization used the PFSS parameters in the model of Ref. [2] to		This optimization used the PFSS parameters in the model of Ref. [2] to
	compare with the wind-speed observations.		compare with the wind-speed observations.
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	We also found that the optimized SS height depends on the choice		We also found that the optimized SS height depends on the choice
	of magnetogram, whereas the pattern remains similar for a given		of magnetogram, whereas the pattern remains similar for a given
	choice of magnetograms. The HMI and GONG ZPC magnetograms perform		choice of magnetograms. The HMI and GONG "ZPC" magnetograms perform
	similarly, and both provide better forecasting than GONG STD maps,		similarly, and both provide better forecasting than GONG "STD" maps,
	as shown in ~~Figures~~ 2 ~~and 3~~. The results provide strong evidence		as shown in Figure 2. The results provide strong evidence
	that solar-cycle-dependent optimization of the PFSS source surface		that solar-cycle-dependent optimization of the PFSS source surface
	height is a practical way to improve solar wind forecasting, which		height is a practical way to improve solar wind forecasting, which
	serves as the foundation for heliospheric models. ~~</i>]]~~		serves as the foundation for heliospheric models.

	[[File:531f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>		[[File:531f2.png\|center\|thumb\|600px\|caption\|Figure 2: <i>