# Effects of Heliolatitudinal Anisotropy of Solar FUV/EUV Emissions on
Lyman-$\alpha$ Helioglow

M. Strumik Space Research Centre PAS (CBK PAN), Bartycka 18a, 00-716 Warsaw,
Poland  M. Bzowski Space Research Centre PAS (CBK PAN), Bartycka 18a, 00-716
Warsaw, Poland  M. A. Kubiak Space Research Centre PAS (CBK PAN), Bartycka
18a, 00-716 Warsaw, Poland

###### Abstract

We present a study of the influence of solar UV anisotropy on the heliospheric
backscatter helioglow generated by resonant scattering of solar Lyman-$\alpha$
photons on interstellar hydrogen atoms around the Sun. Simulations based on
the WawHelioGlow model suggest that the response of the helioglow pole-to-
ecliptic ratio to the anisotropy is linear, but 15% of the anisotropy (polar
darkening) generates 30–40% change in the ratio in the solar minimum and
15–20% in the solar maximum. We attribute this difference to an interplay
between the solar UV anisotropy and the latitudinal structure of the solar
wind in solar minima. The solar UV anisotropy also increases the helioglow
intensity from the downwind direction by 5–10%, due to the influence of the
anisotropy on the ionization losses and trajectories of atoms passing by the
Sun in polar regions. Consequently, mid-latitude regions (in the heliographic
and ecliptic coordinates) are least affected by the UV anisotropy.

By comparison of the simulation results with observations of the SOHO/SWAN
satellite instrument, we derive the day-by-day time evolution of the solar
Lyman-$\alpha$ anisotropy for the north and south poles over two solar cycles
from 1996 to 2022. The inferred anisotropy is $\sim$5–10% in solar minima and
$\sim$15–25% in solar maxima, the northern anisotropy being stronger than the
southern. Our study suggests that in solar minima a highly structured solar
wind is associated with relatively small solar UV anisotropy, while in solar
maxima the solar wind is more isotropic but a substantial solar UV anisotropy
appears.

††journal: ApJ††software: astropy (Astropy Collaboration et al., 2013, 2018)

##  1 Introduction

The Sun is moving through the interstellar medium with a velocity of $\sim$26
km s-1 towards the ecliptic longitude of $\sim$256$\arcdeg$ and the latitude
of $\sim$5$\arcdeg$ (Lallement et al., 2004; Witte, 2004; Bzowski et al.,
2008, 2015; Schwadron et al., 2015; Swaczyna et al., 2023; Bzowski et al.,
2023). The interaction of the solar wind and the local interstellar matter
leads to the formation of the heliosphere, bounded by the heliopause, which is
a separatrix for the heliospheric and interstellar plasma flows (Baranov,
1990; Ratkiewicz et al., 1998; Izmodenov, 2000; Pogorelov et al., 2004). The
neutral atoms penetrate this structure easily and may reach the vicinity of
the Sun. Due to the charge exchange with the solar wind protons and
photoionization processes, a cavity around the Sun is formed, where the
density of interstellar neutral hydrogen is very low in comparison with that
in the unperturbed interstellar medium (Bzowski et al., 2001).

The heliospheric backscatter Lyman-$\alpha$ glow (hereafter helioglow) is an
emission generated by the resonant scattering of solar Lyman-$\alpha$ photons
on neutral hydrogen of interstellar origin around the Sun (Bertaux & Blamont,
1971; Thomas & Krassa, 1971). The helioglow intensity depends on the
illuminating flux from the Sun and the distribution of neutral atoms in its
proximity. The density and radial velocity of the atoms play an important role
in shaping the helioglow-flux distribution in the sky (Brasken & Kyrola,
1998).

Generally, the distribution of the helioglow intensity in the sky shows a
dipole-like pattern, with the minimum and maximum in the proximity of the
hydrogen downwind and upwind directions, respectively (see, e.g., Lallement et
al., 2005). This specific pattern is related to the shape of the hydrogen
cavity around the Sun, which is organized around the upwind-downwind axis and
extended in the downwind direction. However, the position of an observer
relative to the upwind-downwind axis and dependencies of the illumination,
ionization rates, and radiation pressure on time and heliolatitude introduce
some additional modulations to the simple dipole pattern (Bzowski, 2003;
Quémerais et al., 2006; Kubiak et al., 2021a). These additional modulations
are a subject of interest in satellite missions (e.g., SOHO/SWAN, Bertaux et
al. (1995), or the upcoming IMAP/GLOWS mission, McComas et al. (2018)),
because they are a potential source of information about the solar wind
structure. The possibility of inferring the solar wind structure from the
helioglow observations has been presented in several papers (Bertaux et al.,
1995; Bzowski, 2003; Bzowski et al., 2003; Lallement et al., 2010; Koutroumpa
et al., 2019; Katushkina et al., 2019), but the obtained results are at some
tension with those obtained from analyses of interplanetary scintillations
(Sokół et al., 2013, 2015, 2020; Porowski et al., 2022, 2023), as demonstrated
by Katushkina et al. (2013).

In this paper, we focus on a relatively poorly understood question of the
influence of the anisotropy of the solar UV output on the helioglow
distribution in the sky (Cook et al., 1981; Pryor et al., 1992; Auchère et
al., 2005). The solar UV anisotropy discussed in this paper can be caused by
the presence of polar coronal holes, but also by active regions on the Sun.
The distribution of these regions on the Sun’s surface depends on both the
solar longitude and latitude. The longitudinal modulation was found to be
responsible for the so-called searchlight effect in the helioglow on the
Carrington-rotation time scale (see Bertaux et al., 2000; Quémerais & Bertaux,
2002). In this paper, we consider the UV anisotropy as averaged over the solar
longitudes (or Carrington-rotation time scale in observations), thus we assume
that it varies only with the heliolatitude.

The solar Lyman-$\alpha$ is one of the main factors responsible for the
existence of the helioglow (Kubiak et al., 2021a). The wavelength-integrated
solar Lyman-$\alpha$ flux is measured on a regular basis, but as an average
over the solar disc seen from the Earth orbit (Machol et al., 2019). This type
of measurement does not allow to estimate the heliolatitudinal structure of
the Lyman-$\alpha$ flux emitted away from the Earth’s direction, in particular
towards the solar polar regions. Indirect methods of inferring the anisotropy
include the analysis of images of the Sun in 30.4 nm and correlations between
solar emission in the 30.4 nm and Lyman-$\alpha$ wavebands (Auchère, 2005).
However, this type of correlation exhibits a relatively wide statistical
spread, which affects the accuracy of the inference. A connection between He
1086 nm images of the Sun and the anisotropy in the solar Lyman-$\alpha$ flux
was also studied as an explanation of some latitudinal-anisotropy effects in
the interplanetary Lyman-$\alpha$ (Pryor et al., 1996). Full-sky images of the
Sun in the Lyman-$\alpha$ waveband have very recently started to be provided
by the ASO-S/LST instrument (Li et al., 2019). However, to study relations
between the helioglow time evolution and solar activity in the past, we need
to know the solar Lyman-$\alpha$ anisotropy over an extended period. For this
reason, in this paper, we study an indirect method of inferring the time
evolution of the solar UV anisotropy from a comparison of the helioglow
observations and simulations.

This paper is generally intended as an extension of our previous work Strumik
et al. (2021), where a similar method of inferring the solar UV anisotropy was
discussed. In the present work, we clarify the main differences between the
effects of the solar-wind and solar-UV anisotropies on the helioglow
distribution in the sky. We discuss quantitative estimates of the effects of
the solar UV anisotropy in different regions of the sky and different phases
of the solar cycle. This allows us to identify regions in the sky that are
least affected by the anisotropy effects, which can be of interest if a model
with isotropic solar UV is compared with observations. We also investigate
temporal effects in the anisotropy-inferring method proposed by Strumik et al.
(2021), which finally leads to a more accurate estimation of the inferred
anisotropy provided in the present work.

In Section 2, we discuss briefly the WawHelioGlow model used for simulations
and its settings for the study presented in this paper. We also discuss the
results of simulations of the influence of the solar UV anisotropy on the
helioglow both for all-sky maps and for lightcurves representing specific cuts
through the sky maps. In Section 3, we compare the simulation results with
observations from the SOHO/SWAN instrument. Based on this comparison, we infer
the time evolution of the solar UV anisotropy during the past two solar cycles
(1996–2022). Our findings are summarized in Section 4.

##  2 Insights from Modeling

In this section, we use the WawHelioGlow model of the helioglow (Kubiak et
al., 2021b) to investigate the effects of the solar UV anisotropy on the
backscatter Lyman-$\alpha$ helioglow as seen by an observer located at $\sim$1
au from the Sun. We start by briefly discussing the model and its settings,
then present analyses of all-sky maps. With this, we discuss how the effects
of solar-wind and solar-UV anisotropies manifest in selected simulated
lightcurves of the helioglow for the viewing geometry similar to that planned
for the forthcoming IMAP/GLOWS experiment.

###  2.1 Model description

The WawHelioGlow simulation code calculates the intensity of the helioglow for
a selected epoch, vantage point, and viewing direction (Kubiak et al., 2021b).
The model is implemented in a modular way, where a more or less realistic
input can be used for simulations of the helioglow. The helioglow intensity is
computed from the first principles, but the model can perform data-driven
simulations if a realistic time series are provided as the model input.
Optically thin single-scattering approximation is used to compute the heliglow
intensity by integration of a source function along the lines of sight. The
source function in a given point in space is assumed to depend on the solar
irradiance and the distribution function of hydrogen atoms, mainly the density
and radial velocity.

In the WawHelioGlow model, the distribution of the hydrogen around the Sun is
computed from the (n)WTPM model (Tarnopolski & Bzowski, 2009), where the
temporal and heliolatitudinal modulation of ionization losses can be taken
into account. The effects of the charge exchange, photoionization, and
electron-impact ionization are included. The radiation pressure is computed in
a self-consistent way as the momentum change resulting from the solar
Lyman-$\alpha$ flux accounting also for the Doppler-shift effects dependent on
the radial velocity of the hydrogen atoms.

####  2.1.1 Solar UV anisotropy

In the WawHelioGlow model, the irradiances for the illuminating
(Lyman-$\alpha$) and photoionizing fluxes can be set as dependent on the
heliolatitude $\phi$ in the following way

$E=E_{\mathrm{eq}}\left(a\sin^{2}\phi+\cos^{2}\phi\right)$ (1)

(see Section 2.6.3 in Kubiak et al. (2021b) for details). The anisotropy
parameter can be defined as $a=E_{\mathrm{p}}/E_{\mathrm{eq}}$, where
$E_{\mathrm{p}}$ and $E_{\mathrm{eq}}$ are the polar and equatorial
irradiances, correspondingly. As shown by Kubiak et al. (2021b), after simple
algebraic transformations Equation (1) is equivalent to the anisotropy
definition used by Pryor et al. (1992). The anisotropy parameter for the
simulations discussed in this section is set to either $a=0.85$ or $a=1$. The
value of $a=0.85$ corresponds to 15% of solar-longitude-averaged polar
darkening of the solar UV output. One should bear in mind that the anisotropy
depends on the distance of the observer from the Sun as discussed by Auchère
et al. (2005) and Strumik et al. (2021). The WawHelioGlow setting of $a=0.85$
should be understood as the effective value for hydrogen atoms from regions
significantly contributing to the helioglow, which are located far ($\gtrsim$1
au) from the Sun. The distant-observer polar darkening of 15% corresponds to
$\sim$50% of the anisotropy close to the solar surface according to a simple
model presented by Strumik et al. (2021). In the current WawHelioGlow
implementation, the anisotropy defined by Equation (1) is constant in time and
symmetric relative to the heliographic equator, thus the same for both poles.
Results presented by Strumik et al. (2021) and further in this paper suggest
that generally different anisotropy levels are expected for the north and
south poles and the anisotropy is expected to evolve in time. These
capabilities are planned to be included in future versions of the WawHelioGlow
model, but the possibility of using these features requires prior knowledge of
a realistic time evolution of the anisotropy for both poles separately, which
could be set as the model input. Such a phenomenological model of the solar UV
anisotropy is derived in this paper.

####  2.1.2 Model input and simulation settings

In its most-realistic-input setting, the WawHelioGlow model uses the charge-
exchange rates from Porowski et al. (2022), the photoionization rates from
Sokół et al. (2020), the radiation pressure from Kowalska-Leszczynska et al.
(2020) (based on the Lyman-$\alpha$ flux from Machol et al. (2019)), and the
electron-impact ionization from Bzowski et al. (2013) (based on the OMNI2
solar wind data discussed in King & Papitashvili (2005)).

Figure 1: Time dependence of (a) ionization rates and (b) Lyman-$\alpha$
photon flux at 1 au from the Sun from 1996 to 2022. Three total ionization
rates (north – dark blue, ecliptic – green, and south – red) are compared in
panel (a) with the photoionization (black) and electron-impact ionization
(cyan) ecliptic rates. The ecliptic Lyman-$\alpha$ flux in panel (b) is based
on Machol et al. (2019).

Figure 1 illustrates the time dependence of the ionization rates and solar
Lyman-$\alpha$ photon flux at 1 au from the Sun. The total ionization rate at
the poles (red and dark blue lines) is comparable to the ecliptic rate (green)
in solar maxima of $\sim$2001 and $\sim$2014\. In solar minima ($\sim$1996,
$\sim$2010, and $\sim$2019), significant decreases in the polar rates are
observed, which suggests a substantial total-ionization-rate anisotropy. The
ionization rates affect the helioglow by changing the distribution of the
hydrogen around the Sun. The solar-cycle-related decreases of the total polar
rates seen in Figure 1(a) can be expected to change quasi-periodically the
shape of the hydrogen cavity around the Sun, moving the boundary of the cavity
towards the Sun at high heliographic latitudes near solar minima. Since the
solar Lyman-$\alpha$ irradiance changes approximately like $\propto\\!r^{-2}$
with the heliodistance $r$, such deformation of the cavity increases the
helioglow backscattered flux from polar regions.

The ecliptic photoionization (black) and electron-impact ionization (cyan)
rates shown in Figure 1(a) are small in comparison with the total rate. The
dominant contribution to the total rate is the charge-exchange reaction (see,
e.g., Sokół et al. (2020)), which depends on the density and velocity of the
solar wind. The heliolatitude-dependent solar wind speed used here is derived
from ground-based observations of interplanetary scintillations (IPS) as
presented by Porowski et al. (2022). This phenomenological model of the solar
wind parameters uses also the OMNI data from the ecliptic plane and
latitudinally-invariant solar wind energy flux for calculating the solar wind
density. The model accounts for the solar cycle variations of the solar wind,
but it is important to note that helioglow observations (e.g., from the
SOHO/SWAN instrument) are not used in any way in deriving the solar wind
structure.

The bottom panel shows the ecliptic Lyman-$\alpha$ flux based on Machol et al.
(2019), which also exhibits a solar-cycle modulation. The Lyman-$\alpha$ flux
affects the helioglow in two ways, by changing the illumination of the
hydrogen atoms and the radiation pressure. The illumination effect is the
dominant one for the helioglow intensity in comparison with the effect of
changes in the radiation pressure, which affects the trajectories of the
hydrogen atoms and thus indirectly also the helioglow.

In Figure 2 we show heliolatitudinal profiles of the total ionization rate,
photoionization rate, and Lyman-$\alpha$ photon flux for some selected dates
to illustrate in more detail how the profiles change from solar-minimum
conditions ($\sim$1997.68) to solar-maximum conditions ($\sim$2000.68).

Figure 2: Heliolatitudinal dependence of (a) total ionization rate, (b)
photoionization rate, (c) Lyman-$\alpha$ photon flux at 1 au from the Sun.
Different colors correspond to five selected dates to illustrate the evolution
of the quantities between the solar minimum and maximum. Solid lines
correspond to latitudinally-modulated quantities (referred to as 3D in the
text) and dashed lines show isotropic case (2D in the text). For the
photoionization (panel (b)) and Lyman-$\alpha$ flux (panel (c)), the
anisotropy level in Equation (1) is assumed to be $a=0.85$.

We use three types of model settings in this paper. In the first setting,
hereafter referred to as $\mathrm{SW2D/UV2D}$, the heliolatitudinal
dependencies of both the solar wind and solar UV output are neglected and the
ecliptic-equatorial parameters of the solar wind and irradiance are used for
all heliolatitudes. In particular, it means that the rates of both of the
ionization reactions (charge-exchange and photoionization) depend only on time
and radial distance from the Sun, which can be considered as a two-dimensional
model of the variability of the rates (hence the 2D label). This type of
setting corresponds to dashed lines in Figure 2. Note that in this approach,
the modeled radiation pressure depends on time and heliodistance, but one
should bear in mind also the dependence on the radial component of the
velocity of hydrogen atoms moving throughout the heliosphere.

The second setting, hereafter $\mathrm{SW3D/UV2D}$, is extended relative to
the former by including only the heliolatitudinal structure of the solar wind
and consequently the charge-exchange ionization rate. In terms of the
relations illustrated in Figure 2, it would be equivalent to a model using the
solid-line dependencies from panel (a) and dashed-line relationships from
panels (b) and (c). In the third setting, henceforth $\mathrm{SW3D/UV3D}$, the
full capabilities of the model are unleashed, and both the solar wind and
solar UV output (both illuminating Lyman-$\alpha$ and photoionizing flux) are
modeled as dependent on time, heliodistance, and heliolatitude (hence 3D
label). Solid-line relations from all panels of Figure 2 are used in the third
case.

For the simulations of $\mathrm{SW3D}$-type, the WawHelioGlow model uses a
state-of-the-art realistic time-dependent and heliolatitude-dependent solar
wind structure discussed by Porowski et al. (2022). Reducing the model to
$\mathrm{SW2D}$-type means that only OMNI-database ecliptic-equatorial
measurements are used and the solar wind is assumed to be isotropic. For the
solar UV radiation, the time dependence is taken from observations (Sokół et
al., 2020; Kowalska-Leszczynska et al., 2020; Machol et al., 2019), but the
anisotropy is handled in a simplified manner defined in Equation (1). For the
simulations of $\mathrm{UV2D}$-type, i.e., $\mathrm{SW2D/UV2D}$ and
$\mathrm{SW3D/UV2D}$, the anisotropy parameter is set to $a=1$, and for
$\mathrm{SW3D/UV3D}$ we set $a=0.85$.

In simulations presented in this paper, exactly the same parameter values for
the interstellar hydrogen inflow are used as in Kubiak et al. (2021a), which
are explicitly listed in Table 1 in Kowalska-Leszczynska et al. (2018).

###  2.2 Simulation results – comparison of all-sky maps

Figure 3 shows

Figure 3: Comparison of sky maps of the normalized intensity obtained in
WawHelioGlow simulations for three different models of the solar wind and
solar UV structure. Model name is specified by a label in the bottom left part
of each map. The difference between 2D and 3D models consists in including the
heliolatitudinal dependence (3D) in addition to the time and heliodistance
dependence (2D) for the solar wind (SW) and solar UV output (UV). Selected
dates correspond to the crosswind position of SOHO satellite (located at
$\sim$1 au from the Sun). Left column (a)-(c) shows differences for the solar
minimum conditions (Sep 6, 1997) and the right column (d)-(f) – for the solar
maximum (Sep 5, 2000). The ecliptic coordinates are used and the inverted-axis
convention for the horizontal axis (longitude) to make the maps corresponding
to the real sky. The magenta line shows the heliographic equator. The black-
line circle represents a scanning circle of the angular radius of 75° with its
center $\sim$4° off the solar direction. The arrows on the circle show the
direction of increasing spin angle and the cross shows the zero-spin position
(the northernmost point of the circle). Black vertical lines are constant-
longitude cuts through the map at the downwind (longitude of $75\fdg 4$) and
upwind (longitude of $255\fdg 4$) directions. The selected directions D1-D4
(U1-U4) close to the downwind (upwind) directions in the sky are used for
analyses presented in the paper. The white circular region was excluded from
computations due to its closeness to the Sun (black small circle).

simulated sky maps of the normalized helioglow intensity for the three models
mentioned above. The simulations were done for solar-minimum and solar-maximum
conditions for an observer located in the ecliptic plane at a crosswind point
at 1 au. The influence of the heliolatitudinal structure of the solar wind can
be apprehended by the comparison of panels (a) and (b), and the influence of
the solar UV anisotropy – by the comparison of panels (b) vs. (c) for the
solar minimum, and (e) vs. (f) for the solar maximum.

To make the comparisons easier, Figure 4 presents differences between the
three types of simulations shown in Figure 3.

Figure 4: Normalized intensity differences [see Equation (2)] between the
three models shown in Figure 3. Model-pair name is specified by a label in the
bottom left part of each map. The green lines show the zero-value isocontour
for the intensity difference. Additional elements (lines, labels, etc.) and
conventions used in the maps are described in the caption of Figure 3.

All maps in Figure 4 show a normalized intensity difference

$\Delta I=(I_{\mathrm{M1}}-I_{\mathrm{M2}})/\langle I_{\mathrm{EQ}}\rangle,$
(2)

where $I_{\mathrm{M1}}$ and $I_{\mathrm{M2}}$ are the intensities computed
from models M1 and M2, respectively. The normalization intensity is computed
as $\langle I_{\mathrm{EQ}}\rangle=(\langle I_{\mathrm{M1}}\rangle+\langle
I_{\mathrm{M2}}\rangle)/2$, where $\langle I_{\mathrm{M1}}\rangle$, $\langle
I_{\mathrm{M2}}\rangle$ are averaged values over the $\pm 10\arcdeg$ belt
around the ecliptic equator for a given pair of models, M1 and M2.

In Figure 4 (a) and (d), a difference between the models $\mathrm{SW3D/UV2D}$
(anisotropic solar wind, isotropic solar UV) and $\mathrm{SW2D/UV2D}$
(isotropic solar wind and UV) is presented. In the solar minimum (panel (a)),
the polar regions in the helioglow become brighter than the equatorial, which
illustrates the influence of the heliolatitudinal structure of the solar wind
on the helioglow. This structure typically leads to a lower charge-exchange
ionization rate in the polar regions, which changes the shape of the hydrogen
cavity, moving the boundary of the cavity towards the Sun at high heliographic
latitudes. In the solar maximum (panel (d)), the solar-wind heliolatitudinal
profile is relatively flat (compare the blue and red line in Figure 2), and
thus no substantial difference between the polar regions and the equator is
observed.

Figure 4 (b) and (e) illustrates a difference between the models
$\mathrm{SW3D/UV3D}$ (anisotropic solar wind and solar UV) and
$\mathrm{SW3D/UV2D}$ (anisotropic solar wind, isotropic solar UV), thus
related to effects introduced by the solar UV anisotropy in a model with a
realistic structure of the solar wind. A striking (but naturally expected)
feature is that the polar darkening works in the opposite direction than the
solar-wind heliolatitudinal structure, i.e., the UV anisotropy causes the
polar regions to be significantly darker, both in the solar minimum and
maximum. One can also see an increased signal from the equatorial region,
mainly from the downwind direction (longitude of $\sim$$75\fdg 4$). This is
consistent with expectations because some atoms from the downwind region
passed the Sun in high-latitude regions and were subjected to smaller
photoionization losses and radiation pressure as compared to the conditions of
isotropic solar UV output. This leads to increased hydrogen density and
helioglow intensity in the downwind direction. Consequently, the regions least
affected by the UV anisotropy (close to the green lines) are located at a
certain angular distance from the heliographic equator (magenta line) at mid
latitudes, especially in the downwind sector.

In Figure 4 (c) and (f), a difference between simulations $\mathrm{SW3D/UV3D}$
(anisotropic solar wind and solar UV) and $\mathrm{SW2D/UV2D}$ (isotropic
solar wind and solar UV) is presented. For the solar minimum (panel (c)), the
combined effects of anisotropies of the solar wind and solar UV output are
visible here, where the maximum difference between the two models occurs in
mid-latitude regions. For the solar maximum (panel (f)), the difference
between the models is similar to panel (e), because the lack of significant
solar wind anisotropy makes the UV anisotropy the main factor shaping the
distribution of the difference in the sky.

###  2.3 Simulation results – cuts through the all-sky maps

The maps shown in the previous section are convenient for a global qualitative
description of the solar-wind and solar-UV anisotropy effects on the
helioglow, but for quantitative analysis, it is useful to define some cuts
through the maps. The first type of cut is along a scanning circle shown in
all maps in Figure 4. The circle has an angular radius of 75° and its center
is shifted by $\sim$4$\arcdeg$ relative to the Sun’s position in the sky,
which represents the planned observation geometry of the GLOWS instrument
onboard the upcoming IMAP mission (McComas et al., 2018). Another type of
analyzed cut is for constant longitudes of 75$\fdg$4 (close to downwind) and
255$\fdg$4 (upwind), which are also shown in Figure 4.

Figure 5 shows a comparison of different pairs of models for the solar-minimum
conditions.

Figure 5: Comparison of scanning-circle (the first and second row) and
constant-longitude (the third row) cuts through the all-sky maps shown in
Figure 4. The geometry of the cuts is explained in the caption of Figure 4.
Helioglow-intensity differences between three types of model settings are
considered here: (a)–(c) $\mathrm{SW3D/UV2D}$ vs. $\mathrm{SW2D/UV2D}$,
(d)–(f) $\mathrm{SW3D/UV3D}$ vs. $\mathrm{SW3D/UV2D}$, and (g)–(i)
$\mathrm{SW3D/UV3D}$ vs. $\mathrm{SW2D/UV2D}$ for the solar minimum conditions
on September 6, 1997.

Differences related to the anisotropic solar wind structure
($\mathrm{SW3D/UV2D}$ vs. $\mathrm{SW2D/UV2D}$) are shown in panels (a)–(c).
In the absence of the solar wind anisotropy, a typical helioglow modulation
along the GLOWS-like scanning circle (Figure 5(a) – green line) is relatively
simple with its minimum in the downwind segment and maximum in the upwind
part. Solar wind anisotropy introduces an additional modulation (red line) for
this particular vantage point and increases the average intensity of the
helioglow. These effects are better seen in panel (b), where the normalized
difference of the intensities suggests that the brightening is from 20% in the
ecliptic region to up to 70–80% close to the poles. This is confirmed by the
latitudinal dependence of the difference shown in panel (c).

The differences caused by the solar UV anisotropy ($\mathrm{SW3D/UV3D}$ vs.
$\mathrm{SW3D/UV2D}$) are presented in Figure 5 (d)–(f). Panels (d) and (e)
show that the solar UV anisotropy on average decreases the intensity of the
helioglow and introduces a modulation anticorrelated with the effects due to
the solar wind anisotropy. As seen in panel (e), for the solar UV anisotropy
of 15% set in the model, the helioglow darkening may be up to 30% for the
polar directions. However, in the proximity of the ecliptic plane in the
downwind direction, a $\sim$10% brightening of the helioglow is observed,
which was explained in the previous section. At the upwind direction close to
the ecliptic, a very small ($\sim$1%) intensity increase is seen. Note, that
the polar darkening of 30% mentioned above can be interpreted as a change of
the pole-to-ecliptic ratio related to a change of the solar UV anisotropy by
15%.

Figure 5 (g)–(i) shows the combined effects of strong solar wind and solar UV
anisotropies ($\mathrm{SW3D/UV3D}$ vs. $\mathrm{SW2D/UV2D}$). At first sight,
the helioglow intensity (red line) shown in panels (a) and (g) is similar.
However, the extraction of modulations that are on top of the general upwind-
downwind dipole pattern is conveniently done when the pattern due to isotropic
solar wind and solar UV is subtracted. This approach has been applied in
panels (b), (e), and (h), where the combined effects of the solar wind and UV
anisotropies seem to lead to an intermediate state (panel (h)) between the two
states shown in panels (b) and (e). For a strong solar wind anisotropy, the
addition of the UV anisotropy leads to a decrease of the average intensity,
reduces the amplitude of the intensity modulation, and introduces an
additional modulation due to the anticorrelation between the solar-wind and
solar-UV anisotropy effects. In particular, the addition of the UV anisotropy
may lead to mid-latitude bumps in the normalized intensity difference as
clearly seen in panel (i).

It is well known that during solar maxima, variations of the solar wind
density and velocity with heliolatitude are rather weak (see, e.g., Porowski
et al., 2023). A case of relatively flat solar wind structure (Sep 5, 2000;
corresponding to the red line labeled “2000.68” in Figure 2) is presented in
Figure 6, where the effects of the solar UV anisotropy on the helioglow in the
presence of this weakly modulated solar wind structure are illustrated. Panels
(a)–(c) ($\mathrm{SW3D/UV2D}$ vs. $\mathrm{SW2D/UV2D}$)

Figure 6: Similar comparison as in Figure 5, but for the solar maximum
conditions on Sep 5, 2000.

show a difference of the intensity of $\sim$10%, with a very weak modulation
along the scanning circle if only the effects of the solar wind structure are
concerned. Adding a solar UV anisotropy of 15% ($\mathrm{SW3D/UV3D}$ vs.
$\mathrm{SW3D/UV2D}$) results in variations with a higher amplitude, as seen
in panels (d)–(f), but the decrease of the helioglow intensity seems to be
15–20% (panel (e)), i.e., weaker than during the solar minimum (cf. Figure
5(e)). Again, on average the solar wind anisotropy and the solar UV anisotropy
work in opposite directions. The combined effects of the solar UV anisotropy
and weakly-modulated solar wind ($\mathrm{SW3D/UV3D}$ vs.
$\mathrm{SW2D/UV2D}$) are extracted in panels (g)–(i), which suggests that the
variability of the helioglow difference (relative to $\mathrm{SW2D/UV2D}$
model) during the solar maximum is dominated by the influence of the solar UV
anisotropy. However, on average the helioglow intensity changes only slightly
(see Figure 6(h)), in contrast to the solar minimum case, where it may
increase by 20–40% (see Figure 5(h)).

###  2.4 Pole-to-ecliptic ratio vs. solar UV anisotropy

A comparison of panel (e) in Figures 5 and 6 suggests that the same level of
solar UV anisotropy influences the pole-to-ecliptic intensity ratio for the
helioglow to a different extent depending on the solar-cycle phase. This is
consistent with Figure 3 in Strumik et al. (2021), where a bunch of lines
showing the pole-to-ecliptic ratio for different anisotropy levels is squeezed
in solar maxima as compared with the solar-minima intervals. Figure 7 shows
how the

Figure 7: Dependence of the polar-to-ecliptic ratio for the helioglow on the
solar UV anisotropy. Simulated effects for times close to solar minima (1996,
2010, 2019) and close to solar maxima (2001, 2014) are illustrated here for an
observer located upwind (in our case the position of the SOHO satellite on
$\sim$June 5 of each year) at 1 au. Dashed lines show linear fits to the
simulated dependencies (circles) for each year.

ratio depends on the anisotropy for some selected cases analyzed by Strumik et
al. (2021) (upwind-observer geometry). In solar minima, 15% of solar UV
anisotropy leads to even 30-40% change in the polar-to-ecliptic ratio for the
helioglow, while in the solar maxima, the anisotropy of 15% changes the ratio
by only 15-20%. The ratio is elevated for the solar-minimum conditions (1996,
2010, 2019) relative to the solar maxima (2001, 2014), but the dependence
remains linear for both cases as shown in Figure 7. The linear dependence
suggests that this different response of the helioglow to the solar UV
anisotropy is related to a simple process, where the effects of the solar wind
structure (i.e., the average intensity increase and brigther-pole type of
modulation seen in Figure 5(b)) undergo a secondary modulation by the solar UV
anisotropy effects, which finally leads to a significant amplification of the
latter during the solar minima. During the solar maxima, a much weaker
modulation due to the solar wind structure causes a weaker amplification
effect of the solar UV anisotropy on the pole-to-ecliptic intensity ratio.

##  3 Solar FUV/EUV anisotropy retrieved from SOHO/SWAN observations

The previous section shows that the anisotropy of the solar UV output may be
an important factor affecting the distribution of the helioglow intensity in
the sky. In this section, we investigate day-by-day evolution of the solar UV
anisotropy in the Lyman-$\alpha$ wavelength based on a direct comparison of
satellite observations and simulation results. Accurate simulations of the
helioglow distribution in the sky are computationally expensive. To shorten
the computation time, we carefully selected a set of eight directions in the
sky for which WawHelioGlow simulations were performed to obtain the day-by-day
evolution of the helioglow intensity.

Table 1: Ecliptic coordinates of the directions selected for analysis Label | ecl. longitude | ecl. latitude
---|---|---
D1 | $88\fdg 5$ | $42\fdg 5$
D2 | $66\fdg 5$ | $-37\fdg 5$
D3 | $100\fdg 5$ | $84\fdg 5$
D4 | $77\fdg 5$ | $-87\fdg 5$
U1 | $273\fdg 5$ | $20\fdg 5$
U2 | $253\fdg 5$ | $30\fdg 5$
U3 | $248\fdg 5$ | $81\fdg 5$
U4 | $248\fdg 5$ | $-81\fdg 5$

Besides the vicinity of the directions to the upwind and downwind longitudes,
the main criteria for the selection included also the requirement that half of
the selected points is in the polar regions and the other half is at mid
latitudes, closer to the ecliptic equator. Moreover, the selection process
involved also the examination of the SOHO/SWAN maps for the presence of point
sources, to finally select directions possibly least affected by extra-
heliospheric contributions. The selected directions, denoted by D1–D4 (close
to the downwind longitude) and U1–U4 (close to the upwind longitude), are
shown in Figure 4, and their coordinates are listed in Table 1.

A list of dates of SOHO/SWAN observations used in this paper is restricted
with respect to all available observation days. Several criteria were used for
excluding particular days. By using these criteria, a vast majority of spikes
in the time series of the helioglow intensity evolution were removed as
illustrated in Figure 13. A more technical discussion of the data cleaning
process can be found in Appendix A.

The absolute calibration and its corrections for the sensitivity decrease are
difficult problems for satellite instruments performing measurements in the UV
range (see, e.g., Quémerais et al. (2013) or Frey et al. (2017)). As discussed
recently by Baliukin et al. (2019) and Koutroumpa et al. (2019), the SOHO/SWAN
instrument showed a significant sensitivity degradation from 2008 to 2014,
which can be expected to change its calibration over time. Due to these
calibration uncertainties, we decided to use normalized quantities for our
comparisons of the simulated and observed fluxes. Technical details of the
normalization procedure are discussed in Appendix B.

###  3.1 SOHO/SWAN vs. WawHelioGlow – mid-latitude regions

Figure 8 shows a comparison of the normalized modeled and observed

Figure 8: Comparison of WawHelioGlow simulations (red lines for the anisotropy
$a=0.85$ and green lines for $a=1$) and SOHO/SWAN observations (black lines).
Two top panels show the time evolution of the normalized intensity for the
downwind sector (directions D1 and D2), and the bottom two panels – for the
upwind sector (directions U1 and U2). Generally, all the directions are from
the mid-latitude part of the sky, where the solar UV anisotropy has little
influence on the helioglow.

intensities for the directions D1, D2, U1, and U2. The intensities are
computed as 30-day moving averages to suppress variations at the Carrington-
rotation time scale, which are usually present in the SOHO/SWAN observations
but not in the results of the WawHelioGlow model, where Carrington-period-
averaged ionization and illumination models are used. For consistency, the
averaging procedure was applied to both observations and simulations. In
Figure 8, one can see oscillations of the intensity on the one-year time
scale, which is a well-known phenomenon for the helioglow observed by
satellites operating at the Earth orbit, related to periodic changes of the
vantage point. For the downwind sector (two top panels) the effects of the
solar cycle are seen mainly as a modulation of the amplitude of the yearly
oscillations. Smaller-amplitude one-year oscillations and a solar-cycle
oscillation are clearly seen for the upwind sector (two bottom panels). Solar
cycle modulations persist even though they are partially included in the
normalization factor $I^{\mathrm{raw}}_{0}(t)$ defined in Appendix B.

The comparison of the model and observations in Figure 8 shows a general
agreement between the model and observations for the selected directions.
Small differences between the simulations with different anisotropy settings
(red and green lines) also confirm that conclusions from Sections 2.2 and 2.3
regarding the small influence of the solar UV anisotropy on mid-latitude
helioglow intensity are valid not only for the crosswind vantage point but for
all vantage points. One should bear in mind that the solar-wind
heliolatitudinal structure, and consequently the charge-exchange ionization
rates used in the simulations presented in Figure 8 are derived from
interplanetary scintillations (Tokumaru et al., 2010, 2012; Sokół et al.,
2020; Porowski et al., 2022), which is a source of information entirely
independent of the SOHO/SWAN observations. By using this observed solar-wind
heliolatitudinal structure, the WawHelioGlow model becomes a data-driven
model, but the agreement between the model and observations presented in
Figure 8 does not result from any fitting procedure, but is obtained as a
result of computations from first principles. This suggests that results of
WawHelioGlow simulations can be reconciled with SOHO/SWAN observations when
time-dependent parameters of the anisotropy of the solar UV output are
identified and implemented in the code.

###  3.2 Inferring the solar UV anisotropy from helioglow observations

As illustrated in Figure 9 for the polar directions D3, D4,

Figure 9: Comparison of the normalized intensities simulated by the
WawHelioGlow model ($a=0.85$ – red line, $a=1$ – green line) and observed by
the SOHO/SWAN instrument (black line). Four directions D3, D4, U3, U4 (from
the top) in the polar sectors are shown, where the influence of the solar UV
anisotropy is significant.

U3, and U4, the solar UV anisotropy of the magnitude of 15% generates
significant differences between the red and green lines in the normalized
intensity. The SOHO/SWAN observations (black line) are generally contained
between the two simulated curves for $a=0.85$ and $a=1$. Therefore, it is
possible to use these time series to infer the day-by-day evolution of the
solar UV anisotropy $A$ in a similar way as proposed by Strumik et al. (2021),
using the relation

$A=0.85+0.15~{}\frac{I_{\mathrm{obs}}-I_{\mathrm{a=0.85}}}{I_{\mathrm{a=1}}-I_{\mathrm{a=0.85}}},$
(3)

where $I_{\mathrm{obs}}$ is the normalized intensity observed by SOHO/SWAN
(black line in Figure 9), and $I_{\mathrm{a=0.85}}$ ($I_{\mathrm{a=1}}$) is
the modeled intensity for $a=0.85$ ($a=1$), represented by the red (green)
line in Figure 9. Equation (3) is based on an assumption of a linear
correlation between the anisotropy level and polar intensity, which was
demonstrated by Strumik et al. (2021) and is fully supported by Figure 7.

The anisotropy inferred from the time series in Figure 9 (using Equation (3))
is shown in Figure 10.

Figure 10: The solar UV anisotropy (blue and orange lines) inferred from the
time series presented in Figure 9 and Equation (3). The top panel shows the
results for the north-pole sector, and the bottom panel - for the south pole.
Results obtained by Strumik et al. (2021) are shown by black circles labeled
“SBK 2021”.

The estimates for different directions, represented by the blue and orange
lines in Figure 10, are very similar, which is consistent with the
expectations because the angular distance in the sky between the points D3 and
U3 (also between D4 and U4) is relatively small (but projection effects make
the distance apparently larger). The inferred anisotropy evolution (lines) is
similar to that presented by Strumik et al. (2021) (shown by black circles
labeled “SBK 2021”). During the solar minima, a smaller anisotropy (i.e., $A$
value closer to 1) is observed as compared to the times of solar maxima, where
$A\simeq 0.7-0.8$. The northern anisotropy is generally larger than the
southern. The anisotropies are also different during the two following solar
maxima, being larger for $\sim$2001 than for $\sim$2014.

However, a detailed inspection of Figure 10 reveals some periods when a
significant annual oscillation of the inferred anisotropy is observed (e.g.,
2018-2022 for the northern anisotropy). Such periodicity is unlikely to occur
for the solar UV anisotropy. A more likely explanation for the annual
modulation is a worse agreement between the simulations and observations for
some periods. Indeed, if we take a look at the first and third panels in
Figure 9 for the time intervals 1996-2000 and 2018-2022, we can see that the
amplitude of oscillations for the SOHO/SWAN observations (black line) is
larger than that obtained in the simulations (red and green lines). This
results in observations being closer to the anisotropic simulations (where
$a=0.85$) for the downwind position (December of each year) of the satellite,
while for other seasons the observations are generally closer to the isotropic
simulations ($a=1$).

For this reason, we decided to use a moving average over one year to eliminate
the yearly oscillations before the procedure of inferring the solar UV
anisotropy. The averaged time series of the normalized intensity are shown in
Figure 11.

Figure 11: A similar comparison of the normalized simulated and observed
intensities as in Figure 9 but for yearly-moving-average-based time series.

Except for using the averaged time series, exactly the same procedure as for
Figure 10 is applied then to infer the anisotropy $A$. The results are shown
in Figure 12,

Figure 12: The solar UV anisotropy (blue and orange lines) inferred from
Equation (3) and the yearly-moving-average-based time series presented in
Figure 11. The top panel shows the results for the north-pole sector, and the
bottom panel – for the south pole. Results obtained by Strumik et al. (2021)
are shown by the black circles labeled “SBK 2021”, and the inferred anisotropy
from Figure 10 (not-averaged case) is shown by the black line.

which shows that the annual modulations are naturally suppressed. Most of the
time the yearly-moving-average-based inferred anisotropy (blue and orange
lines) is lower than our previous estimate (black circles labeled “SBK 2021”)
based on the upwind observer position. We discuss these results in the next
section.

##  4 Discussion and Conclusions

In this paper, we studied the influence of the solar UV anisotropy on the
helioglow distribution in the sky as seen by an observer at 1 au. We show that
an anisotropy of $\sim 15$% may affect the polar-to-ecliptic ratio of the
helioglow to an extent depending on the phase of the solar cycle. During solar
minima, the ratio changes by 20–40%, while during solar maxima it is rather
15–20%. The stronger influence during solar minima seems to be related to the
solar wind anisotropy, which amplifies the effects of the solar UV anisotropy.
However, our further analysis of SOHO/SWAN observations shows that a strong
solar wind anisotropy, typical for solar minima, and a strong solar UV
anisotropy do not occur at the same time. For solar minima, the estimated
solar UV anisotropy is rather weak, $\sim$5-10%, and it may increase to
$\sim$15-25% during solar maxima.

The influence of the solar wind and solar UV anisotropies is also different in
terms of polar brightening and darkening of the helioglow. The solar wind
anisotropy increases the helioglow intensity in the polar sector and the solar
UV anisotropy works typically in the opposite direction, i.e., decreases the
polar intensity. These effects occur because the solar wind is faster and
thinner at large latitudes, as revealed by Ulysses measurements (McComas et
al., 2003, 2013) and interplanetary-scintillation analysis (Tokumaru et al.,
2010; Sokół et al., 2020). This results in lower ionization rates for hydrogen
atoms, which moves the hydrogen cavity toward the Sun and increases the
helioglow signal in polar regions. By contrast, the solar UV anisotropy is
caused by the presence of polar coronal holes during solar minima and the
clustering of the active regions at lower latitudes during solar maxima
(Auchère et al., 2005). The two causes of the UV anisotropy lead effectively
to a polar darkening of the solar UV output, and thus also the helioglow,
rather than polar brightening. From this point of view, a polar darkening of
the helioglow can be interpreted rather as a signature of a significant solar
UV anisotropy than related to the solar wind structure. Also, a latitudinally-
flat helioglow profile cannot be attributed merely to the solar wind
structure, since competition between the solar wind and the solar UV
anisotropies may lead to some cancelation effects. This suggests that
accounting for the solar UV anisotropy is important in inferring the solar
wind structure from the Lyman-$\alpha$ helioglow observations.

Based on these findings, we derived the day-by-day time evolution of the
inferred solar UV anisotropy separately for the northern and southern
hemispheres from 1996 to 2022. In comparison with our previous work Strumik et
al. (2021), this paper contains a more accurate estimation, where a better
understanding of the anisotropy-related issues allowed us to address several
difficulties. The refinements include better cleaning of observation
artifacts, using time averages (to make observations and simulations more
relevant to each other), and better normalization (based on the mid-latitude
regions in the sky as the least affected by the anisotropy). Tracking the day-
by-day time series gives us better insight into the time evolution and
possible errors in the estimation of the anisotropy, resulting from temporary
misfits between the simulations and observations. For this reason, we consider
the yearly-moving-average-based estimate presented in Figure 12 as more
reliable and accurate in comparison with that shown in Figure 10. The time
evolution of the anisotropy presented in Figure 12 is consistent with an
anisotropy obtained from synoptic maps of the Sun for 30.4 nm by Auchère et
al. (2005) for the years 1996-2003. The anisotropy evolution presented in our
paper is also qualitatively consistent with the so-called butterfly diagrams
for sunspots (see, e.g., Leussu et al. (2017)). These diagrams illustrate the
evolution of sunspots (which are typically dark regions in the visible range
but bright in the UV). According to the diagrams, the sunspots are almost not
visible during solar minima, then they appear in mid-latitude regions, their
distribution becomes wider and they migrate towards lower latitudes as the
transition to the solar maximum progresses. This time evolution seems to
correspond to the behavior of the inferred anisotropy derived in our paper.

The new estimate of the anisotropy $A$ (blue and orange lines in Figure 12)
has typically slightly smaller values (indicating a higher anisotropy) than
the former estimation by Strumik et al. (2021) (black circles) based on
observations from the upwind positions of the observer every year. Analysis of
Figure 10 suggests that for the years 1996–2011 this difference is related
mainly to spurious annual variations of the inferred anisotropy that appear
when the yearly-averaging procedure is not used. For the years 2011–2022, the
reason for the difference is presumably related to different normalizations
used in the two papers. Strumik et al. (2021) used a normalization based on
the average value over an ecliptic-equatorial belt, while in this paper we use
an improved normalization based on averages over mid-latitude regions, which
are less affected by the solar UV anisotropy as discussed in Sections 2.2 and
2.3. The most striking (but still quite small) difference between the old and
new estimate is presumably the inferred solar UV anisotropy level during solar
minima, which is now $A\simeq$0.92 in comparison with $A\simeq 1$ in the
former estimate. In this regard, the new results seem to be more consistent
with the solar disk observations by Auchère et al. (2005), where a weak but
non-negligible anisotropy is observed in the solar minimum and attributed to
the coronal holes.

This study was supported by Polish National Science Center grant
2019/35/B/ST9/01241 and by Polish Ministry for Education and Science under
contract MEiN/2021/2/DIR. We acknowledge the publicly available data with
SOHO/SWAN satellite instrument observations provided by the following webpage
http://swan.projet.latmos.ipsl.fr/data/L2/FSKMAPS. SOHO is a mission of
international cooperation between ESA and NASA.

##  Appendix A SOHO/SWAN data cleaning

The SOHO/SWAN observations for particular dates may contain contaminations
caused by solar events or various technical issues. For this reason, SOHO/SWAN
observations used in this paper are selected from all available observation
days to represent a regular evolution of the helioglow on time scales larger
than the Carrington rotation period. Several criteria were used to exclude
particular problematic dates. The first criterion excluded dates, for which
masked regions in the sky (where observations are lacking) cover more than 15%
of the entire daily map. The second criterion excluded days, for which the
average intensity (over the entire map) changed more than 30 R relative to the
previous observation day. If a multi-day gap in the observations occurred
extending over $N_{\mathrm{D}}$ days, this criterion assumed that
$N_{\mathrm{D}}\times 30$ R is the maximum allowed variation of the average
intensity between the previous and next available observation days, otherwise,
the next observation day was excluded. We also excluded those observation
days, for which MEDCPS26 field in the header of FITS files with the
observation data was larger than 20 indicating possible solar-events
contamination. By using these criteria, a majority of spikes were removed in
the time series of the helioglow intensity evolution as seen in Figure 13.

Figure 13: Comparison of averaged SOHO/SWAN observations before (top panel)
and after (bottom panel) data-cleaning procedure used in this paper. Three
types of averages are considered: all-sky (red line), northern hemisphere
(green line), and southern hemisphere (blue line) in ecliptic coordinates. The
hemispheric averages were computed after excluding the galactic equatorial
belt ($\pm 15\arcdeg$).

In this figure, the time evolutions of different types of averages are shown
to illustrate the cleaning process described in this section. The time series
in the bottom panel show solar-cycle ($\sim$11 years), one-year, and
Carrington-rotation-period ($\sim$27 days) modulations.

##  Appendix B Normalization of helioglow intensities

Sensitivity-degradation issues discussed recently by Baliukin et al. (2019)
and Koutroumpa et al. (2019) for SOHO/SWAN can be expected to change the
instrument calibration over time. These findings are confirmed by our
comparison of the observed and simulated helioglow intensities discussed
below.

Figure 14: Comparison of WawHelioGlow simulations (red lines for the
anisotropy $a=0.85$ and green lines for $a=1$) and SOHO/SWAN observations
(black lines). The left vertical-axis label refers to SOHO/SWAN observations,
while the right label – to WawHelioGlow simulations. Two top panels show the
time evolution of the raw (not-normalized) intensity for the downwind sector
(directions D1 and D2), and the bottom two panels – for the upwind sector
(directions U1 and U2).

Figure 14 shows the comparison of simulations and observations for the
directions D1, D2, U1, and U2. The intensities are computed as 30-day moving
averages to suppress variations that are usually present at the Carrington
rotation time scale. Figure 14 shows that initially (1996-2000) the SOHO/SWAN
flux (black line) is systematically higher than the simulated fluxes (red and
green line), while later it becomes systematically smaller.

Following Baliukin et al. (2019) and Koutroumpa et al. (2019), we attribute
the systematic changes to calibration uncertainties and we decide to use
normalized quantities for our comparisons of the simulated and observed
fluxes. The normalization procedure is performed for two series of
simulations, with the isotropic solar UV output ($\mathrm{SW3D/UV2D}$, where
$a=1$) and with the anisotropic UV output ($\mathrm{SW3D/UV3D}$, where
$a=0.85$), as well as for the SOHO/SWAN observations. For all of them, the
normalized intensities were computed as
$I(t)=I^{\mathrm{raw}}(t)/I^{\mathrm{raw}}_{0}(t)$, where
$I^{\mathrm{raw}}(t)$ is the raw intensity (i.e., the observed and simulated
photon flux in Figure 14) in the respective simulation or observation units.
The time-dependent normalization factor $I^{\mathrm{raw}}_{0}(t)$ is intended
to have the capability of accounting for possible changes in the instrument
calibration in time.

The factor $I^{\mathrm{raw}}_{0}(t)$ was computed in the following way. For
the time evolution of the helioglow intensity from 1996 to 2022 (shown in
Figure 14), a yearly moving average $\overline{I^{\mathrm{raw}}(t)}$ was
computed for directions D1, D2, U1, and U2. As showed in Figure 4(b) and (e),
the directions are located at mid latitudes, where the difference between the
models $\mathrm{SW3D/UV2D}$ and $\mathrm{SW3D/UV3D}$ is close to zero. In
other words, these directions are weakly affected by the solar UV anisotropy.
Therefore, this subset of points is suitable for computing the normalization
factor $I^{\mathrm{raw}}_{0}(t)$, if anisotropy-related questions are of
interest. The moving averages for the four points were then averaged to obtain
one normalization factor
$I^{\mathrm{raw}}_{0}(t)=[\overline{I^{\mathrm{raw}}_{D1}(t)}+\overline{I^{\mathrm{raw}}_{D2}(t)}+\overline{I^{\mathrm{raw}}_{U1}(t)}+\overline{I^{\mathrm{raw}}_{U2}(t)}]/4$.
This procedure was applied to both simulation sets and observations
separately. Although only four directions in the sky are used for defining the
normalization, the yearly averaging procedure provides a significant rigidity
and robustness to the normalization factor. Note also that for a given time,
the normalization factor is applied globally over the entire sky (i.e., the
same value is used for all directions D1, D2, U1, and U2).

Figure 8 shows a similar comparison like Figure 14, but for the normalized
fluxes. The systematic changes of the observed fluxes relative to the
simulated fluxes seen in Figure 14 are not present for the normalized
quantities in Figure 8. This suggests, that systematic calibration changes of
the observed intensity (on the time scales larger than one year) are naturally
addressed by the procedure of normalization described above.

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