# WawHelioGlow: a model of the heliospheric backscatter glow. I. Model
definition

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

###### Abstract

The helioglow is a fluorescence of interstellar atoms inside the heliosphere,
where they are excited by the solar EUV emission. So far, the helioglow of
interstellar H and He have been detected. The helioglow features a
characteristic distribution in the sky, which can be used to derive both the
properties of interstellar neutral gas and those of the solar wind. This
requires a simulation model capable of catching with a sufficient realism the
essential coupling relations between the solar factors and interstellar. The
solar factors include the solar wind flux and its variation with time and
heliolatitude, as well as the heliolatitude and time variation of the solar
EUV output. The ISN gas inside the heliosphere features a complex distribution
function, which varies with time and location. The paper presents the first
version of a WawHelioGlow simulation model for the helioglow flux using an
optically thin, single scattering approximation. The helioglow computations
are based on a sophisticated kinetic treatment of the distribution functions
of interstellar H and He provided by the (n)WTPM model. The model takes into
account heliolatitudinal and spectral variations of the solar EUV output from
observations. We present a formulation of the model and the treatment of the
solar spectral flux. The accompanying Paper II illustrates details of the line
of sight evolution of the elements of the model and a brief comparison of
results of the WawHelioGlow code with selected sky maps of the hydrogen
helioglow, obtained by the SWAN instrument onboard the SOHO mission.

##  1 Introduction

The heliospheric backscatter glow is a fluorescence of the interstellar
neutral (ISN) gas inside the heliosphere, resonantly excited by the solar EUV
radiation. It is an observationally discovered phenomenon (Bertaux & Blamont,
1971; Thomas & Krassa, 1971) that revealed the existence of the heliosphere
(Blum & Fahr, 1970) around the Sun that moves through a warm, magnetized,
partially ionized cloud of interstellar matter.

A helioglow photon is created when a neutral atom is excited by a photon of an
appropriate frequency, and subsequently de-excited with emission of a photon
with a slightly different frequency in a random direction. Details of this
process have been extensively discussed by Brasken & Kyrola (1998). At a given
location in space, the atoms forming the local population of ISN gas are
excited with a rate that depends on one hand on the magnitude of the
illuminating spectral flux, and on the other hand on the distribution of
radial speeds of the atoms, which are responsible for the Doppler-tuning of
individual atoms to the spectral flux. All atoms from the given population
that have identical radial velocities have identical probabilities of being
excited. Depending on the species, either all atoms are deep within the
Doppler width of the illuminating solar line (as is the case for H) and are
subject to excitation, or the illuminating line is so narrow that only a
portion of the atoms from the given population are within its span, and the
rest does not contribute to the formation of the helioglow (as may happen at
certain locations for He).

In the atom de-excitation following immediately the excitation, photons are
emitted at random directions, given by the so-called scattering function,
which depends on the species. Thus, the gas in the given location becomes a
volume source of resonant radiation. An observer at a selected vantage point
in space measures the intensity of the helioglow (regardless of its wavelength
within a certain spectral band), i.e., counts the photons that reach its
vantage point from a line extending towards a selected direction from the
observer to infinity. Conversely, in a given location in space only the
photons that are directed towards the observer will contribute to the
helioglow signal that this observer registers.

The problem of calculating the intensity of the heliospheric backscatter glow
of ISN gas has been dealt with by many authors, including Weller & Meier
(1974); Meier (1977); Keller & Thomas (1979); Keller et al. (1981); Quémerais
& Bertaux (1993); Quémerais (2000, 2006); Scherer & Fahr (1996); Fayock et al.
(2013). All these models derive from a series of seminal papers by Hummer
(Hummer, 1962, 1964, 1968, 1969a, 1969b) and assume that the temperature of
the ISN gas inside the heliosphere is isotropic. However, the anisotropy of
the distribution function of ISN gas within a few au from the Sun is strong
and can be adequately represented neither by an isotropic nor anisotropic
Maxwell-Boltzmann function (see, e.g., Fahr, 1979; Kubiak et al., 2019; Sokół
et al., 2015b, 2019) even when one neglects the charge exchange reaction
between the charged and neutral interstellar components in the outer
heliosheath.

In this paper, we derive a model of the intensity of the backscatter resonance
glow of ISN H and He taking into account the full distribution function
returned by the hot model of ISN gas (Fahr, 1978, 1979), calculated using the
nWTPM model of ISN gas (Tarnopolski & Bzowski, 2009; Sokół et al., 2015b)
where a fully kinetic approach is applied without any analytic approximation
for the velocity distribution function inside the heliosphere. We present a
first version of the WawHelioGlow model, which uses the optically thin
approximation and ignores absorption of the solar illuminating radiation by
the ISN gas, as well as effects related to finite optical thickness of the ISN
gas. These effects will be included in future versions of the model. In the
present version, we implement effects of the time- and heliolatitude-dependent
ionization of ISN gas and radiation pressure (which is also dependent on the
atom radial velocity), as well as the currently most reliable estimates for
the evolution of the illuminating solar spectral lines of the Lyman-$\alpha$
and He I 58.4 nm radiation.

The model is devised as a tool in analysis of future observations from the
GLOWS experiment on the NASA Interstellar Mapping and Acceleration Probe
mission (IMAP McComas et al., 2018). The objective of the paper is to present
the baseline version of the model. A qualitative comparison of the model
results with selected helioglow observations from the Solar Wind ANisotropy
experiment onboard the SOlar and Heliospheric Observatory (SWAN/SOHO; Bertaux
et al., 1995) is presented in the accompanying Paper II 111M.A. Kubiak,
WawHelioGlow: a model of the heliospheric backscatter glow. II. The helioglow
buildup and potential significance of the anisotropy in the solar EUV output .
An in-depth analysis of these observations is postponed to a future work.

In Section 2 we present the definition of the helioglow intensity for a
selected line of sight, differentiating between the cases of the ISN hydrogen
and helium glows. In particular, we discuss the models of the solar
illumination of ISN H and He by the relevant portions of the solar EUV
spectrum. We start with a presentation of the geometry of the line of sight
and proceed to discuss the definition of the source function of the helioglow,
the scattering phase function, the distribution function of the ISN gas and
its partial density, the excitation function of the gas, and the illumination
function. We close this long section with presentation of some numerical
aspects of the integration of the helioglow intensity, and close the paper
with a summary and an outlook leading to the accompanying Paper II. In
Appendix A we present some textbook derivations of the photon absorption cross
sections and the radiation pressure for ISN H atoms, In Appendix B, we
summarize available measurements of spectral features of the solar He I 58.4
nm line, to better justify the choices of the parameter values of the model of
this line used in the paper.

##  2 Model definition

Table 1:  The most important quantities defined in the model, with their respective equations Defined quantity | Equation No | Description
---|---|---
$I_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{r}}_{\text{obs}},\hat{\boldsymbol{{l}}})$ |  1, 2.7, 34 | helioglow intensity
$J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})$ | 2 | helioglow source function
$\hat{\boldsymbol{{l}}}$ |  3, Fig.1 | look (i.e., line of sight) direction
$\boldsymbol{{r}}_{\text{LOS}}\left(\boldsymbol{{l}}\right)$ |  4, Fig.1 | heliocentric radius-vector of a point belonging to the line of sight (LOS)
$\beta$ |  5, 6, Fig.1 | scattering phase angle
$S(\boldsymbol{{l}},v_{r})$ | 7 | spectral source function of the helioglow
$\psi$ |  8, 9 | scattering phase functions
$\eta(\boldsymbol{{l}},v_{r})$ | 10 | partial density of ISN gas, dependent on $v_{r}$
$\sigma(\nu)$ | 11 | cross section for absorption of a photon by a H, He atom
$E(\tilde{S}_{\sun},\boldsymbol{{l}},\nu),E(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})$ |  13 , 14, 16 | excitation function of ISN gas
$\tilde{S}_{\sun}(v_{r},t)$ |  15, 17, 27 | illuminating solar spectral flux
$\mu$, $\langle\mu\rangle$ |  18, 33 | radiation pressure factors: instantaneous and average
$s(v_{r})$ |  18, 20, 24 | model of the solar spectral line profile, unnormalized
$p$ |  19, 26 | normalization factor for the line profile $s$
$v_{D}$ |  21, 22, 23, 25 | Doppler speed
$I_{\sun}(\boldsymbol{{r}}_{\text{LOS}},t)$ |  28, 29, 30, 31 | total solar illuminating flux at $\boldsymbol{{r}}_{\text{LOS}}$
$I_{\sun,\text{E}}(\phi,t)$ | 31 | total solar illuminating flux at the reference distance $r_{\text{E}}$ and heliolatitude $\phi$
$I_{\lambda\phi}(\phi)$ | 29 | solar flux at heliolatitude $\phi$ relative to that at $\phi=0$

The objective of the WawHelioGlow model is to calculate a wavelength-
integrated flux density (i.e., the total intensity per steradian) of the
heliospheric backscatter glow (the helioglow), observed by an observer at a
location given by radius-vector $\boldsymbol{{r}}_{\text{obs}}$, looking into
a direction specified by the spherical coordinates
$\lambda_{\text{LOS}},\phi_{\text{LOS}}$. The gas is illuminated by a solar
emission line: Lyman-$\alpha$ for H and the 58.4 nm line for He. The spectral
intensity of the solar illumination at a given moment $t$ is given by the
illuminating solar spectral flux $\tilde{S}_{\sun}(\nu)$ (Equation 17), which
varies with time, possibly heliolatitude, and depends on photon frequency
$\nu$. It is assumed that photons from the Sun propagate instantaneously and
are scattered only once, which greatly simplifies the radiation transfer
equation (Hummer, 1969b). The helioglow intensity
$I_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{r}}_{\text{obs}},\hat{\boldsymbol{{l}}})$
is given by a line-of-sight (LOS) integral of the helioglow source function
$J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})$:

$I_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{r}}_{\text{obs}},\hat{\boldsymbol{{l}}})=\int\limits_{0}^{l}J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})\,dl^{\prime},$
(1)

where $\boldsymbol{{l}}=l^{\prime}\,\hat{\boldsymbol{{l}}}$. A precise
definition of the geometry of the line of sight is given in Section 2.1. The
partial helioglow intensity $I_{\text{glow}}(l)$ is defined so that the upper
boundary of the integration is finite and equal to $l$. For
$l\rightarrow\infty$, the partial helioglow intensity becomes the full
intensity. The helioglow source function $J_{\text{glow}}$ is defined as an
integral over atom velocity of the helioglow spectral source function
$S(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})$:

$J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})=\int\limits_{-\infty}^{\infty}S(\tilde{S}_{\sun}(v_{r}),\boldsymbol{{l}},v_{r})d{v_{r}},$
(2)

where we have converted $\tilde{S}_{\sun}$ from the dependence on photon
frequency $\nu$ to that on the radial velocity of atoms $v_{r}$ due to the
Doppler effect (Equation 12). The source function $S$ (Equation 7) is a
product of the excitation function $E$ of individual atom (Equation 13) for a
given radial velocity $v_{r}$, the number of atoms suitable for excitation at
this radial speed $v_{r}$, given by the partial density $\eta$ (Equation 10),
and the probability of emission of the re-emitted photons towards the
observer, given by the scattering phase function $\psi$ (Equations 8, 9). The
atom excitation function $E$ depends on the solar spectral flux
$\tilde{S}_{\sun}$ at a given location in space (Equations 15, 17, 27) and on
the cross section $\sigma$ for photon absorption by an atom of a given species
(Equation 11).

In the following, we define and discuss all relevant factors, starting from
the definition of the geometry of the line of sight and proceeding to the
aforementioned factors composing the spectral source function. For readers
convenience, we summarize in Table 1 the most important quantities and the
equations where they are defined. The geometry of observations assumed in the
model is presented in Figure 1.

###  2.1 Line of sight $\boldsymbol{{l}}$ and other geometrical definitions

Figure 1: The geometry of the line of sight defined in Equations 3 and 4, with
three example points $\boldsymbol{{r}}_{{}_{\text{LOS}},{1-3}}$ along this
line. The observer is in the location in space defined by the radius-vector
$\boldsymbol{{r}}_{\text{obs}}$, anchored at the Sun. Along the line of sight,
the three scattering phase angles $\beta_{1-3}$ (Equations 5, 6) at the three
selected points are presented, along with the corresponding lengths along the
line of sight $l_{{}_{\text{LOS}},{1-3}}$ (Equation 3).

The geometry of the ine of sight is presented in Figure 1. The observer
position is defined by the vector $\boldsymbol{{r}}_{\text{obs}}$ in a Sun-
centered frame. The line of sight in space is defined by its origin (i.e.,
$\boldsymbol{{r}}_{\text{obs}}$) and the look direction
$\hat{\boldsymbol{{l}}}$. The unit vector $\hat{\boldsymbol{{l}}}$ is defined
by the spherical coordinates $\lambda_{\text{LOS}},\phi_{\text{LOS}}$ of the
direction of observation as follows:

$\boldsymbol{{\hat{l}}}\left(\lambda_{\text{LOS}},\phi_{\text{LOS}}\right)=\left(\begin{array}[]{c}\cos\lambda_{\text{LOS}}\,\cos\phi_{\text{LOS}}\\\
\sin\lambda_{\text{LOS}}\,\cos\phi_{\text{LOS}}\\\
\sin\phi_{\text{LOS}}\end{array}\right),$ (3)

and the points in the 3D space belonging to the line of sight are given by the
relation:

$\boldsymbol{{r}}_{\text{LOS}}\left(\boldsymbol{{l}}\right)=\boldsymbol{{r}}_{\text{obs}}\boldsymbol{{l}}=\boldsymbol{{r}}_{\text{obs}}l\,\hat{\boldsymbol{{l}}},\text{
for }l\in(0,...,\infty).$ (4)

Note that a given point belonging to a line of sight can be alternatively and
fully equivalently identified either by $\boldsymbol{{r}}_{\text{LOS}}$ or
$(\boldsymbol{{r}}_{\text{obs}},\boldsymbol{{l}})$.

For any point $\boldsymbol{{r}}_{\text{LOS}}$, the scattering phase angle
$\beta$, i.e., the angle at which the radiation incoming at
$\boldsymbol{{r}}_{\text{LOS}}$ from the Sun is scattered towards the observer
at $\boldsymbol{{r}}_{\text{obs}}$ is defined as the angle between
$\boldsymbol{{r}}_{\text{LOS}}$ and $-\boldsymbol{{\hat{l}}}$:

$\beta\equiv\angle(\boldsymbol{{r}}_{\text{LOS}},-\boldsymbol{{\hat{l}}});$
(5)

Cosine of this angle is given by the projection of the unit vector
$\hat{\boldsymbol{{r}}}_{\text{LOS}}$ on the direction towards the observer:

$\cos\beta=-\hat{\boldsymbol{{l}}}\cdot\hat{\boldsymbol{{r}}}_{\text{LOS}}.$
(6)

###  2.2 Definition of the spectral source function $S$

The spectral source function of the helioglow $S$ is a product of the
scattering phase function, partial density of the ISN gas, and the excitation
function of this gas. Both the partial density and the excitation function
depend on the illuminating solar spectral flux $\tilde{S}_{\sun}$. The
dependence of $E$ on $\tilde{S}_{\sun}$ is direct and that of the partial
density is indirect. The solar spectral flux excites the atoms (the direct
dependence), which emit the helioglow, and affects the partial density because
it is responsible for the radiation pressure effect, which modifies the
trajectories of the atoms in space and thus the distribution function of ISN
atoms along the line of sight (the indirect dependence).

The helioglow spectral source function
$S(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})$ at a location $\boldsymbol{{l}}$
within the line of sight for an ISN atom traveling at a velocity
$\boldsymbol{{v}}=(v_{r},v_{T_{1}},v_{T_{2}})$ is defined as follows:

$S(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})=\psi(\beta)\,\eta(\boldsymbol{{l}},v_{r})\,E(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r}).$
(7)

Here,
$v_{r}=\boldsymbol{{v}}\cdot\hat{\boldsymbol{{r}}}_{\text{LOS}}(\boldsymbol{{l}})$
is the radial speed of an individual atom from the ISN gas at
$\boldsymbol{{l}}$. The quantities $v_{T_{1}},v_{T_{2}}$ are the two
components of the Cartesian vector perpendicular to the solar direction. In
this formulation, the reference system can be arbitrarily selected provided
that the $v_{r}$ axis is heliocentric. Here, we integrate over $v_{T1},v_{T2}$
to form the partial density $\eta$ of ISN gas. For definition of $\eta$, see
Equation 10.

$E(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})$ is the excitation function for
individual atoms, and $\psi(\beta)$ is the phase function of the scattering
between the direction of incidence given by
$\hat{\boldsymbol{{r}}}_{\text{LOS}}$ and the direction of re-emission, given
by $-\hat{\boldsymbol{{l}}}$ (Equations 5, 6). Thus, the product $\eta\,E$
gives the number of excited atoms per unit time and unit volume, and $\psi$
determines the fraction of photons resulting from de-excitation that are
directed towards the detector. A precise definition of the function $E$ is
presented in Section 2.5. The partial density $\eta$ is presented in Section
2.4, and the scattering phase function $\psi$ in Section 2.3.

###  2.3 The phase function $\psi$

Figure 2: Phase function for scattering the solar radiation by ISN H and He
atoms, adopted after Meier (1977) and defined in Equations 8 and 9.

The phase function $\psi(\beta)$ determines the probability that if a photon
arrives from the direction $\hat{\boldsymbol{{r}}}$, it will be re-emitted
towards the direction defined by $-\hat{\boldsymbol{{l}}}$. The angle between
these directions is denoted $\beta$, see Section 2.1 and Fig.1. The formulae
for the phase functions for H and He, adopted after Meier (1977), is defined
in the following equations:

$\psi_{\text{H}}(\beta)=\frac{1}{12}\left(11+3\cos^{2}\beta\right)\,\,\text{for
H},$ (8)
$\psi_{\text{He}}(\beta)=\frac{3}{4}\left(1+\cos^{2}\beta\right)\,\,\text{for
He}$ (9)

and presented in Figure 2.

###  2.4 The distribution function of the ISN gas and the partial density
$\eta$

The term $\eta(\boldsymbol{{l}},v_{r})$ in Equation 7 is the partial density
of ISN gas at $\boldsymbol{{l}}$, defined as follows:

$\eta(\boldsymbol{{l}},v_{r})=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f_{\text{ISN}}(\boldsymbol{{l}},v_{r},v_{T_{1}},v_{T_{2}})\,dv_{T_{1}}\,dv_{T_{2}}.$
(10)

In this definition, $f_{\text{ISN}}(\boldsymbol{{l}},\boldsymbol{{v}})$ is the
distribution function of ISN gas at the location $\boldsymbol{{l}}$ for an
atom with a velocity $\boldsymbol{{v}}=(v_{r},v_{T_{1}},v_{T_{2}})$. The
magnitude of $f_{\text{ISN}}(\boldsymbol{{l}},\boldsymbol{{v}})$ depends on
the survival probability of ISN atoms against ionization losses and on the
radiation pressure, calculated for this time and location. Radiation pressure
is conveniently expressed by a coefficient of compensation of solar gravity by
the radiation pressure force $\mu$. In the heliosphere, only H is affected by
this effect; for He, we adopt $\mu\equiv 0$. The survival probability is
determined by a function $\beta_{I}(\phi,t)$ that calculates the instantaneous
ionization rate of a given species for a given heliolatitude $\phi$ and time
$t$. The functions $\beta_{I}$ and $\mu$ are used in the calculation of the
distribution function $f_{\text{ISN}}$, which involves tracking the atom
trajectories in time and space between $\boldsymbol{{r}}_{\text{LOS}}$ and the
entrance to the heliosphere (Ruciński & Bzowski, 1995; Ruciński et al., 2003;
Bzowski et al., 2002). Therefore, formally $f_{\text{ISN}}$ is a function of
these quantities as well as the velocity and $\boldsymbol{{r}}_{\text{LOS}}$:
$f_{\text{ISN}}\equiv
f_{\text{ISN}}(\boldsymbol{{r}}_{\text{LOS}},\boldsymbol{{v}}_{\text{LOS}},\beta_{I},\mu)$,
but further in the paper we will omit them for brevity.

The distribution function is marginalized over the two components
$v_{T_{1}},v_{T_{2}}$ perpendicular to $v_{r}$ and to each other; integrating
$\eta(\boldsymbol{{l}},v_{r})$ over $v_{r}$ yields $n(\boldsymbol{{l}})$ – the
local density of ISN gas.

Basically, the distribution function can be evaluated using any reasonable
approximation, including that of the hot model of the ISN gas (e.g. Fahr,
1978; Wu & Judge, 1979) or in a simplified form that of the Maxwell-Boltzmann
function with the density, velocity, and temperature obtained from moments of
the function derived from the hot model. In the latter case, some of the
integrals in the derivation of the glow intensity can be performed
analytically.

In this paper, the distribution function of ISN gas at $\boldsymbol{{l}}$ is
calculated fully kinetically, using the numeric version of the Warsaw Test
Particle Model (nWTPM) of the ISN gas. For H, it is computed using a version
of this model in which radiation pressure is taken into account in addition to
solar gravity (Tarnopolski & Bzowski, 2009). The model of radiation pressure
is based precisely on the same profile of the solar Lyman-$\alpha$ radiation
as that used for the illuminating function (Section 2.6.4). For He, the
density is computed using the version presented by Sokół et al. (2015a), where
the motion of He atoms is governed solely by solar gravity.

The WTPM method of calculation of the local distribution function of ISN gas
was originally developed by Ruciński & Bzowski (1995). The local distribution
function is calculated using the hot-model paradigm (Fahr, 1978, 1979; Wu &
Judge, 1979), but the atom trajectories are tracked numerically with the
effects of time-, latitude- and radial velocity-dependent radiation pressure
taken into account (Tarnopolski & Bzowski, 2009). The survival probabilities
of the atoms against ionization processes, which also vary with the solar
distance, heliolatitude, and time (Bzowski et al., 2013b, a), are calculated
by numerical integration of the ionization probability along the trajectories
(Bzowski et al., 2002). In this paper, for illustration purposes, we adopt a
model of these factors by Sokół et al. (2020), but any reasonable ionization
model can be used instead. In particular, what counts is the calculation of
the ionization losses of ISN atoms, i.e., the total rate of their ionization,
without the need to calculate the rates of the constituent ionization
processes separately. This feature enables determining the spatial structure
of the ionization rates of ISN gas based on observations of the helioglow, as
it has been successfully performed in the past (see, e.g. Bertaux et al.,
1999; Bzowski, 2003; Bzowski et al., 2003; Katushkina et al., 2019; Koutroumpa
et al., 2019; Kyrölä et al., 1998; Lallement et al., 2010; Summanen et al.,
1993, 1997; Summanen, 2000).

Since the helioglow model is optically thin, single scattering, it is possible
to account for multiple populations of ISN gas by calculating their local
distribution functions and the helioglow contributions separately and co-add
the results. For H, we take the primary and secondary populations of ISN gas
with the inflow parameters obtained by Bzowski et al. (2008) from analysis of
pickup ion observations on Ulysses, with subsequent adjustment of their
parameters by Kowalska-Leszczynska et al. (2018a). This latter adjustment was
to account for a larger temperature of the primary population, obtained by
Bzowski et al. (2015) from observations of ISN He on IBEX, and for a
distortion of the heliosphere from axial symmetry by the interstellar magnetic
field, which results in an angular shift of the apparent inflow direction of
the secondary population from that of the primary (Kubiak et al., 2016).

Specifically, we adopted for the primary population of hydrogen the J2000
longitude 255.745°, latitude 5.169°, velocity 25.784 km s-1, temperature 7443
K, density 0.031 cm-3. For the secondary population, we adopted the longitude
251.57°, the latitude 11.95°, the velocity 18 km s-1, the temperature
$16\,300$ K, and the density 0.054189 cm-3. For He, we adopted one population,
with the parameters identical to those for the primary H population with the
density equal to 0.015 cm-3 (Gloeckler et al., 2004; Möbius et al., 2004); the
secondary population was neglected.

In the optically thin approximation, the absolute intensity of the helioglow
is linearly proportional to the absolute density of the ISN gas in front of
the heliosphere. This approximation will not exactly hold when multiple
scattering effects are taken into account.

It is also possible to use an alternative method to calculate the distribution
function of ISN He, defined by Bzowski et al. (2017) (see also Bzowski et al.,
2019). In this method, the primary and the secondary populations are not
clearly separated. The local distribution function is obtained within the hot-
model paradigm extended by solutions of the production and loss equations of
ISN He atoms along individual atom trajectories from the unperturbed
interstellar medium through the perturbed interstellar plasma outside the
heliopause down to a given location at $\boldsymbol{{r}}_{\text{LOS}}$. This
approach, currently available for ISN He, may be adopted in a future version
of the model.

The distribution function of ISN gas
$f_{\text{ISN}}(\boldsymbol{{l}},\boldsymbol{{v}})$ bears imprints from the
ionization processes and radiation pressure, which vary with time. This
variation adds to the time variation of the spectral source function $S$
resulting from variations in the solar spectral illumination flux
$\tilde{S}_{\sun}$, but while the time scale of the latter is on the order of
the solar rotation period, the time scale of the former is more typically from
a month to a year, depending on the distance from the Sun.

###  2.5 The excitation function $E$

The excitation function $E(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})$ is the
rate (a frequency) of exciting an atom traveling with a radial speed $v_{r}$
from energy level 1 to level 2 given the illuminating spectral flux
$\tilde{S}_{\sun}(v_{r})$. The atom can be excited by photons from a narrow
spectral range around the central frequency $\nu_{0}$, corresponding to the
energy difference $h\,\nu_{0}$ between the two energy levels ($h$ being the
Planck constant). The cross section for absorption of a photon in the rest
frame of the atom is given by the following formula (see Equation A.2, derived
in Appendix A.2):

$\sigma_{\nu}=\frac{\pi\,e^{2}}{m_{e}\,c}\,f_{\text{osc}}.$ (11)

When an atom is traveling with a velocity
$\boldsymbol{{v}}=(v_{r},v_{T_{1}},v_{T_{2}})$ relative to the Sun, then in
the atom rest frame the Sun is moving with respect to the atom with the
velocity $-(v_{r},v_{T_{1}},v_{T_{2}})$ and photons emitted by the Sun are
Doppler-shifted by a frequency corresponding to $-v_{r}$, based on the
relation

$v_{r}=\left(\frac{\nu}{\nu_{0}}-1\right)c=\left(\frac{\lambda_{0}}{\lambda}-1\right)\,c.$
(12)

The excitation function in the atom rest frame is defined as

$E(\tilde{S}_{\sun}(\nu_{0},r_{\text{LOS}}),\boldsymbol{{l}},\nu)=\int\limits_{0}^{\infty}\tilde{S}_{\sun}(\nu,r_{\text{LOS}})\sigma(\nu_{0},\nu)\,d\nu,$
(13)

where $\tilde{S}_{\sun}(\nu,r_{\text{LOS}})$ is a Doppler-shifted illuminating
solar spectral flux for the frequency $\nu$ at the atom location in the
Lyman-$\alpha$ or 58.4 nm line. Inserting Equation 11 and integrating over
$\nu$ one obtains the excitation function in the atom rest frame, which can be
immediately transferred to the solar inertial frame, yielding

$E(\tilde{S}_{\sun}(\nu_{0},r_{\text{LOS}}),\boldsymbol{{l}},\nu_{0})=\frac{\pi\,e^{2}}{m_{e}\,c}\,f_{\text{osc}}\,\tilde{S}_{\sun}(\nu_{0},r_{\text{LOS}}),$
(14)

Converting now the excitation function so that it depends on radial velocity
instead of the frequency (Equation 12) and assuming there is no absorption of
the solar spectral flux between the solar surface and
$\boldsymbol{{r}}_{\text{LOS}}$

$\tilde{S}_{\sun}(v_{r},r_{\text{LOS}})=\tilde{S}_{\sun}(v_{r},r_{\text{E}})\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2}$
(15)

we obtain

$E(\tilde{S}_{\sun}(v_{r}),\boldsymbol{{l}},v_{r})=\frac{\pi\,e^{2}}{m_{e}\,c}\,f_{\text{osc}}\,\tilde{S}_{\sun}(v_{r},r_{\text{E}})\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2},$
(16)

where $r_{\text{E}}$ is the solar distance for which the illuminating solar
spectral flux $\tilde{S}_{\sun}(v_{r})$ is known (typically $r_{\text{E}}=1$
au). As it is implied by this equation, it is assumed that there is no
modification of the illuminating solar spectral flux with the distance from
the Sun other than scaling by the square of solar distance. This assumption
may be removed in the future.

###  2.6 The illuminating solar spectral flux $\tilde{S}_{\sun}$

The factors relevant for the helioglow production are the spectral shape of
the solar EUV emission in spectral bands characteristic for the helioglow of
ISN H and He, and the total intensity of radiation within these wavebands in
the geometric locations along the line of sight. It is assumed that the
illuminating solar spectral flux $\tilde{S}_{\sun}$ factorizes as follows:

$\tilde{S}_{\sun}(v_{r})=p\,s(I_{\sun,\text{E}},v_{r}).$ (17)

The spectral shape of the function $\tilde{S}_{\sun}$ is defined by a profile
function $s(I_{\sun,\text{E}},v_{r})$, which may be unnormalized;
normalization is obtained by multiplication by a scaling factor $p$. The
factor $p$ may or may not be a function of the solar flux $I_{\sun,\text{E}}$,
depending on the definition of $s$. Thus, the profile function $s$ may vary
along the line of sight $\boldsymbol{{l}}$ because $\boldsymbol{{l}}$
traverses various heliolatitudes and $I_{\sun,E}$ may depend on heliolatitude
(Equation 31). It is assumed that time variation in the solar illuminating
flux propagates instantaneously to all points within the LOS.

The elements of the solar illuminating flux function $I_{\sun,\text{E}}$ and
its components are discussed in Sections 2.6.3 and 2.6.4, after presentation
of the functions $s$ and the factors $p$ for H and He (Sections 2.6.1 and
2.6.2).

####  2.6.1 The spectral profile for hydrogen $s_{\text{H}}$ and the scaling
factor $p_{\text{H}}$

The central wavelength of the Lyman-$\alpha$ line is equal to
$\lambda_{\text{H}}=121.56701$ nm, and the oscillator strength is
$f_{\text{osc},\text{H}}=0.41641$ (Wiese & Fuhr, 2009). The solar
Lyman-$\alpha$ line is relatively wide compared with the Doppler range
corresponding to the velocity spread of interstellar neutral hydrogen in the
heliosphere. The line can be approximately described by a kappa profile
sitting on top of a sloped background, with a chromospheric Gaussian-like
self-reversal, which results in a characteristic two-horn profile (Figure 3,
left panel). The shape of the profile (e.g., the horn-to-center ratio) evolves
with the evolution of the solar activity, as evidenced by the different shapes
for the two selected epochs shown in the aforementioned figure. The spectral
irradiance is expressed as a function of Doppler speed $v_{r}$.

The shape of the solar Lyman-$\alpha$ profile and its evolution during the
solar activity cycle are known relatively well owing to a series of spectral
observations performed by SUMER/SOHO in the years 1996–2009 (Lemaire et al.,
2015). Based on these observations, a model of the evolution of solar resonant
radiation pressure for H atoms during the cycle of solar activity was
developed by Kowalska-Leszczynska et al. (2018b, a), which is parameterized by
the composite Lyman-$\alpha$ flux $I_{\text{tot,H}}(t)$ (Woods et al., 1996,
2000). Here, we adopt the most recent version of this model by Kowalska-
Leszczynska et al. (2020), which uses the most recent version of
$I_{\text{tot,H}}(t)$, compiled by Machol et al. (2019). Specifically, we use
Equation 14 from Kowalska-Leszczynska et al. (2018b) with the parameters from
Table 1 in Kowalska-Leszczynska et al. (2020) and obtain the radiation
pressure factor $\mu(I_{\sun,\text{E}},v_{r})$ (Equation 18), i.e., the factor
of compensation of the solar gravity force for a selected Doppler velocity.
$I_{\sun,\text{E}}(t,\phi)$ is obtained from Equation 31. In our paper, we
apply this result directly in the calculation of the distribution function of
ISN H, and after an appropriate rescaling, in the calculation of the solar
illumination.

To this latter end, renormalization of the system defined by Kowalska-
Leszczynska et al. (2018b) must satisfy that in Equation 18 we insert
$I_{\sun}=I_{\text{tot,H}}$ and integrate
$s_{\text{H}}(r_{\text{E}},I_{\text{tot,H}}(t),v_{r})$ over $v_{r}$ we obtain
$I_{\text{tot,H}}(t)$. The integration interval is defined so that the radial
speed boundaries correspond to the wavelength range $(121,122)$ nm, because
this is the interval used when measuring the solar composite Lyman-$\alpha$
flux (Woods et al., 2005). Therefore, the adopted model of the solar
Lyman-$\alpha$ line does not require any further normalization other than
rescaling to the units of ph cm-2 s-1 nm-1.

$\displaystyle
s_{\text{H}}(r_{\text{E}},I_{\sun},v_{r})=\mu(I_{\sun,\text{E}}(t,\phi),v_{r})=\mu(I_{\lambda\phi}I_{\text{tot,H}},v_{r})$
(18)

and the normalization factor $p_{\text{H}}$ is

$\displaystyle p_{\text{H}}$ $\displaystyle=$
$\displaystyle\left[f_{\text{osc},\text{H}}\left(\frac{\pi\,e^{2}}{m_{e}c^{2}}\,h\,\lambda_{\text{H}}\right)\frac{r_{\text{E}}^{2}}{G\,M_{\sun}m_{\text{H}}}\right]^{-1}=3.34467\times
10^{12}\text{ ph s}^{-1}\,\text{cm}^{-2}\,\text{nm}^{-1}.$ (19)

For derivation, see Appendix A.4. To obtain the illuminating solar spectral
flux, the quantities $s_{\text{H}}$ and $p_{\text{H}}$ thus defined must be
inserted into Equation 17.

The left panel of Figure 3 presents profiles of $s_{\text{H}}$ for the solar
minimum and maximum conditions. The two profiles are linearly scaled so that
they are equal to 1 for $v_{r}=0$. This illustrates the variation of the shape
of the solar line profile with $I_{\text{tot}}$, which is more complex than a
simple linear scaling. In particular, the horn/center ratios vary during the
cycle of solar activity, thus the illuminating solar spectral flux
$\tilde{S}_{\sun}$ varies during the solar cycle both in the absolute
magnitude and in the relation of the flux at one wavelength to another.

####  2.6.2 The spectral profile for helium $s_{\text{He}}$ and the scaling
factor $p_{\text{He}}$

The central wavelength of the He I 58.4 nm line is equal to
$\lambda_{\text{He}}=58.43339$ nm, and the oscillator strength
$f_{\text{osc},\text{He}}=0.27625$ (Wiese & Fuhr, 2009). The solar He I 58.4
nm line is so narrow that available measurements of its profile are few and
far between (see Appendix B). It is expected that the profile is Gaussian-like
with a flat spectral background or kappa-like (e.g., Jeffrey et al., 2017). In
our model, the Full Width at Half maximum (FWHM) of the profile was adopted as
$\text{FWHM}_{\lambda}=0.0118$ nm, independent of the phase of the solar cycle
(McMullin et al., 2004), which corresponds to FWHM${}_{v}=60.54$ km s-1. A
rationale behind this choice and a discussion on available measurements of
this line is provided in Appendix B.

We define a normalized Gaussian line profile with a background as follows:

$s_{\text{He}}(v_{r},v_{\text{D}},s_{\text{bkg}})=\frac{1}{1+s_{\text{bkg}}}\left(\exp\left[-\left(\frac{v_{r}}{v_{\text{D}}}\right)^{2}\right]+s_{\text{bkg}}\right).$
(20)

The profile is parametrized by a Doppler speed, a Doppler width, and a
background level. The relation between the line width expressed by the central
wavelength $\lambda$, line width $\Delta\lambda$, and Doppler speed
$v_{\text{D}}$ is

$v_{\text{D}}=(\Delta\lambda/\lambda)c,$ (21)

where $c$ is the speed of light.

It is expected that the $s_{\text{bkg}}$ is small relatively to the spectral
irradiance at the line center. In the definition in Equation 20, the spectral
irradiance at the central wavelength is normalized to 1. The variation in the
intensity of the solar line is accounted for so that the profile maintains its
shape and is scaled linearly with $I_{\text{tot,He}}(t)$.

For $s_{\text{bkg}}=0$, the relation between the full width at half maximum
(FWHMv) and $v_{\text{D}}$ is given by the formula:

$v_{\text{D}}=\frac{\text{FWHM}_{v}}{2\sqrt{\ln 2}}.$ (22)

For $0<s_{\text{bkg}}<1$, this relation becomes

$v_{\text{D}}=\frac{\text{FWHM}_{v}}{2\sqrt{\ln\frac{2}{1-s_{\text{bkg}}}}}.$
(23)

Alternatively, one can define the profile by a kappa function:

$s_{\text{He}}(v_{r},v_{\text{D}},\kappa)=\left[1+\frac{1}{\kappa}\left(\frac{v_{r}}{v_{\text{D}}}\right)^{2}\right]^{1-\kappa},$
(24)

where $v_{\text{D}}$ of the profile is related to FWHMv by

$v_{\text{D}}=\frac{\text{FWHM}_{v}}{2\sqrt{\kappa\left(2^{\frac{1}{\kappa-1}}-1\right)}}.$
(25)

In fact, as illustrated in the right panel of Figure 3, a Gaussian profile
with a moderate background differs from a kappa profile with identical FWHM
only in the tails even for relatively low magnitudes of the $\kappa$
parameter.

The definitions presented above include a normalization with
$s_{\text{He}}(v_{r}=0)=1$. In the calculations, we have the magnitude of the
line-integrated flux $I_{\text{tot}}$ corresponding to a certain waveband
$\lambda_{1},\lambda_{2}$; $\lambda_{2}-\lambda_{1}=1$ nm, i.e., within a
nanometer-wide wavelength interval that includes the central wavelength of the
He I 58.4 nm line. The boundaries $(\lambda_{1},\lambda_{2})$ can be expressed
by the Doppler velocities $(v_{r,1},v_{r,2})$ (Equation 12). The normalization
factor $p_{\text{He}}$ for the line profile must satisfy the following
condition:

$I_{\sun,\text{E}}=I_{\lambda\phi}(\phi)I_{\text{tot}}(t)=p_{\text{He}}\int\limits_{v_{r,1}}^{v_{r,2}}s_{\text{He}}(v_{r})\,dv_{r},$
(26)

where $s_{\text{He}}(v_{r})$ is the function defined in Equation 20 or 24 with
the relevant parameters invariable with time.

Ultimately, the illuminating solar spectral function

$\tilde{S}_{\sun,\text{He}}(v_{r},t)=p_{\text{He}}(I_{\text{tot,He}})\,s_{\text{He}}(v_{r}),$
(27)

where, unlike for H, the factor $p_{\text{He}}$ (Equation 26) depends on the
line-integrated solar flux $I_{\text{tot,He}}$, and consequently on time;
$I_{\lambda\phi}$ is given by Equation 29.

####  2.6.3 Modulation of the illuminating solar flux with heliolatitude
$I_{\lambda\phi}$

The solar EUV output is a superposition of components from the quiet Sun, the
low corona, and all kinds of active regions (Amblard et al., 2008). These
latter ones are mostly responsible for the variation during the solar activity
cycle and solar rotations. The fact that this component originates in discrete
regions on the solar disk, which tend to cluster in heliolatitudinal bands,
results in a latitudinal and longitudinal variation of the line-integrated
solar flux in space $I_{\sun}$, evolving with time.

The dimensionless factor $I_{\lambda\phi}$ is a modulating function,
representing variations of the line-integrated solar flux with the
heliolatitude and the heliolongitude. This latter effect is caused by solar
active regions, located at the surface of the rotating Sun.

The variation of $I_{\lambda\phi}$ in the longitude was pointed out in the
context of the helioglow by Bertaux et al. (2000); Pryor et al. (1992) as
responsible for the “searchlight effect”, i.e., an apparent motion of a
reflection of an active region in the sky. The Sun features a rotation-related
flickering in the Lyman-$\alpha$ line with the statistically most likely
magnitude of $\sim 7$%, as we derived from the daily time series of
$I_{\text{tot}}$ by Machol et al. (2019), restricted to a portion of these
data that originate directly from Lyman-$\alpha$ measurements, and not from
proxies. The median value of the maximum to minimum ratio within individual
Carrington rotations is equal to 9% and the mean to 11%, while magnitudes
larger than 24% are detected during 3% of days in the discussed time series.
These variations are reflected in brighter sky regions several dozens of
degrees across. The magnitude of variations in the helioglow intensity
observed along selected lines of sight depends on the viewing geometry and is
the largest for antisolar lines of sight. For lines of sight close to
perpendicular to the Sun-observer line, they are typically 5–10% percent, as
obtained from analysis of selected SWAN data by Bzowski et al. (2003).
Accounting for the searchlight effect is easiest for the viewing geometries
close to anti-solar; this is because for these geometries, the largest portion
of the line of sight is affected by the active region-related brightening. In
this case it seems sufficient to use for illumination a daily value of
$I_{\text{tot}}$ instead of the monthly-averaged one, which is used to
calculate the distribution of ISN H.

In the present version of our model, we neglect this variation and assume that
the modulation factor is only a function of the heliolatitude and does not
vary with time:
$I_{\lambda\phi}(t,\lambda_{\text{helio}},\phi_{\text{helio}})\equiv
I_{\lambda\phi}(\phi_{\text{helio}})$. In the following, we will abbreviate
$\phi_{\text{helio}}\equiv\phi$. We define this function requiring a
normalization such that

$I_{\lambda\phi}(\phi=0)=1,$ (28)

i.e., it is equal to 1 at 1 au at 0 heliolatitude, i.e., close to the latitude
where measurements of $I_{\sun}$ are taken.

The dependence of the solar EUV output on heliolatitude has not been fully
investigated (see discussion in Bzowski et al., 2013b). Studies by Pryor et
al. (1992) and Auchère (2005) suggest that the intensity of solar EUV at polar
latitudes is somewhat lower than in the ecliptic, and that this difference
changes little with the solar activity. Pryor et al. (1992) considered a
superposition of contributions from the quiet Sun, assumed to be spherically
symmetric, and another one from active regions at the solar surface, organized
by heliolatitude. Effectively, the heliolatitude dependence of
$I_{\sun,\text{E}}(\phi)$ in this approach was given by a constant spherically
symmetric portion and a portion with a certain amplitude, modulated as
$\cos^{2}\phi$, i.e., $I_{\sun,\text{E}}(\phi)=c_{1}+c_{2}\cos^{2}\phi$. These
authors connected the $c_{1}$ and $c_{2}$ parameters with the intensity for
the quiet Sun and a portion related to the latitude distribution of active
regions and their surface density. These parameters can be obtained from solar
disk observations. This approach is worth a more thorough analysis but in our
paper we use a formula with a reasonable but arbitrary parameter value because
we only want to investigate the sensitivity of the helioglow to an anisotropy
in the illumination. To that end, we tentatively adopt the following formula:

$I_{\lambda\phi}(\phi)=a\,\sin^{2}\phi+\cos^{2}\phi,$ (29)

where $a=I_{\text{pole}}/I_{\text{tot}}$ (note that from Equation 30,
$I_{\text{tot}}=I_{\text{eqtr}}$ ) is an adjustable parameter, equal to 0.85,
both for H and He. This function, however, can and most likely will be
redefined in the future, when a better insight is available. Setting $a=1$
removes the latitude dependence of the solar illumination in the model. Note
that for the Pryor et al. (1992) formula,
$I_{\text{pole}}/I_{\text{tot}}=c_{1}/(c_{1}+c_{2})$. It can be shown using
simple algebra that substituting $a=c_{1}/(c_{1}+c_{2})$ to
$I_{\sun,\text{E}}=I_{\text{tot}}\,I_{\lambda\phi}=I_{\text{tot}}(a\sin^{2}\phi+\cos^{2}\phi)$
one obtains
$I_{\sun,\text{E}}(\phi)=(I_{\text{tot}}/(c_{1}+c_{2}))(c_{1}+c_{2}\cos^{2}\phi)$.

Figure 3: Profiles of the solar Lyman-$\alpha$ (left) and He I 58.4 nm lines
(right), for illustration purposes normalized to the spectral irradiance at
their respective central wavelengths of $\sim 121.6$ nm and $\sim 58.4$ nm.
The profiles are defined as a function of the Doppler shifts, expressed in km
s-1. The left panel presents the shapes of the Lyman-$\alpha$ profile defined
by Kowalska-Leszczynska et al. (2020), for the solar minimum and maximum
conditions, the epochs 1996.43 and 2001.43, respectively. The spectral
irradiances for the line center for these epochs are equal to $2.76\times
10^{12}$ and $4.49\times 10^{12}$ cm-2 s-1 nm-1. This panel illustrates the
variation of the profile shape with the solar activity level. The right panel
presents alternative profiles for the He I 58.4 nm line, defined in Equations
20 and 23 for Gaussian profiles (G) with background equal to 0 and 0.01 and
for a kappa profile (K) with an identical FWHM, $\kappa=5$ (Equations 24 and
25). The vertical bars mark multiplicities of the adopted $v_{\mathrm{D}}$
speeds. The horizontal bar marks half of the height, to illustrate that with
the adopted definitions, the spectral flux defined by the Gaussian and kappa
profiles within the FWHM range are practically identical.

####  2.6.4 The total illuminating solar flux $I_{\sun}$

The total illuminating solar flux $I_{\sun}(t,\boldsymbol{{r}}_{\text{LOS}})$
measured at $\boldsymbol{{r}}_{\text{LOS}}$ is integrated over the solar
surface hemisphere defined by the unit vector
$\hat{\boldsymbol{{r}}}_{\text{LOS}}$. The flux $I_{\sun}$ varies
quadratically with the distance from the Sun (assuming no absorption). In the
paper, we assume that for a time $t$, the total solar flux is known at a
reference distance $r_{\text{E}}=1$ au at the solar equator ($\phi=0$) and we
denote this quantity

$I_{\text{tot}}(t)=I_{\sun}(t,r=r_{\text{E}},\phi=0).$ (30)

This quantity is measured by Earth-orbiting satellites for H and He (Woods et
al., 2005, 2012). The total illuminating flux
$I_{\sun}(t,\boldsymbol{{r}}_{\text{LOS}})$ at $\boldsymbol{{r}}_{\text{LOS}}$
in the spherical heliographic coordinates is then given by

$I_{\sun}(t,\boldsymbol{{r}}_{\text{LOS}})=I_{\sun,\text{E}}(t,\phi)\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2}=I_{\lambda\phi}(\phi)I_{\text{tot}}(t)\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2},$
(31)

that is, the variation of the solar output
$I_{\sun}(t,\boldsymbol{{r}}_{\text{LOS}})$ far away from the Sun
$(r_{\text{LOS}}\gg r_{\sun})$ is equal to the solar output
$I_{\sun,\text{E}}(t,\phi)$ measured at a heliolatitude $\phi$ and a reference
distance $r_{\text{E}}$, and scaled with the square of solar distance.
$I_{\sun,\text{E}}$ is factorized into a (heliolongitude, heliolatitude)
variation $I_{\lambda,\phi}$ and the global time dependent part
$I_{\text{tot}}(t)$ measured at the solar equator.

In reality, the solar flux is attenuated by the intervening gas. This
phenomenon was recently discussed by Kowalska-Leszczynska et al. (2018a) – see
their Figures 8 and 9. Since absorption is wavelength dependent, both the
magnitude and the spectral profile of the solar illuminating flux at
$\boldsymbol{{r}}_{\text{LOS}}$ must vary with $\boldsymbol{{r}}_{\text{LOS}}$
(differently for different heliolongitudes, heliolatitudes and solar
distances). Therefore, while the solar Lyman-$\alpha$ and 58.4 nm line
profiles at $r_{\text{E}}=1$ au can be assumed to be identical to those just
outside the solar corona (because of the negligible optical thickness of the
ISN gas inside 1 au), outside 1 au they are gradually modified. Consequently,
the illuminating solar spectral flux profile is a function of wavelength and
the location along the line of sight. The effect of absorption will be
introduced to a future version of the model, but to the function $s$
describing the solar line profile rather than to the function of the total
illuminating flux.

The time variations in the ecliptic plane have been demonstrated to be highly
correlated with the well-known proxies of the solar activity. The daily
Lyman-$\alpha$ flux measured at the Earth location is publicly available in
the form of so-called solar composite Lyman-$\alpha$ flux (Woods et al., 1996;
Woods & Rottman, 1997; Woods et al., 2005), with the most recent version given
by Machol et al. (2019). This data product is based on observations from space
from several spacecraft since 1970s, with gaps filled with the proxies (see
Woods et al., 2000; Bzowski et al., 2013b). This product is adopted as the
basis for the solar illuminating flux $I_{\text{tot,H}}(t)$ for hydrogen. We
use the original time series, available at a daily time resolution, averaged
over Carrington rotation period to average over variations in heliolongitude
and we assume that for a given time moment $t$ the Carrington period-averaged
$I_{\text{tot}}(t)$ is identical for all heliolongitudes.

The solar 58.4 nm line for helium, $I_{\text{tot,He}}(t)$, has been measured
by SDO (Woods et al., 2012) and also features correlations with the solar
proxies. We adopted $I_{\text{tot,He}}$ as the data product from SDO/EVE level
3 lines dataset version 6 provided by LASP Interactive Solar Irradiance
Datacenter (http://lasp.colorado.edu/lisird/data/sdo_eve_lines_l3/), averaged
over Carrington period identically as in the case of hydrogen.

###  2.7 Numerical integration of the source function to yield the helioglow
intensity $I_{\text{glow}}$

Combination of Equations 1, 2, 7, 10, 15, 16, 17 and 31 results in the
following integration formula:

$\displaystyle
I_{\text{glow}}(\boldsymbol{{r}}_{\text{obs}},\boldsymbol{{\hat{l}}})$
$\displaystyle=$
$\displaystyle\int\limits_{0}^{\infty}\int\limits_{-\infty}^{\infty}E(\tilde{S}_{\sun},\boldsymbol{{l}},v_{r})\,\eta(\boldsymbol{{l}},v_{r})\,\psi(\beta)\,dv_{r}\,dl=$
$\displaystyle=$
$\displaystyle\frac{\pi\,e^{2}}{m_{e}\,c}\,f_{\text{osc}}\,p\int\limits_{0}^{\infty}dl\,\psi(\beta)\,\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2}$
$\displaystyle\int\limits_{-\infty}^{\infty}dv_{T_{1}}\int\limits_{-\infty}^{\infty}dv_{T_{2}}\int\limits_{-\infty}^{\infty}dv_{r}\,s(I_{\sun,\text{E}}(t,\phi),v_{r})f_{\text{ISN}}(\boldsymbol{{l}},v_{r},v_{T_{1}},v_{T_{2}}).$

Note that the result of integration in the last row of this equation for
hydrogen is proportional to the radiation pressure averaged over the entire
population at the location, given by the radius vector $\boldsymbol{{l}}$
within the line of sight. The result of this integration is a function of this
location, and the result, of course, does not depend on the details such as
selection of the coordinate system. Therefore, it is not needed to change the
coordinate system along the line of sight such to always keep one of the axes
heliocentric. In our WawHelioGlow code, the integration is performed in the
spherical heliographic coordinates, to facilitate the use of latitudinally
anisotropic models of ISN gas ionization and of the illuminating solar flux
$I_{\lambda\phi}$. We perform a four-dimensional numerical integration of the
helioglow intensity $I$ according to Equation 2.7, and to calculate the
integrand function
$f_{\text{ISN}}(\boldsymbol{{l}},v_{r},v_{T_{1}},v_{T_{2}})$, we perform
tracking of an individual atom trajectory by calculation of numerical solution
of the equation of motion of a H atom between $\boldsymbol{{l}}$ until this
atom reaches a predefined distance from the Sun (typically selected about 150
au).

If we wanted to calculate an average radiation pressure for the sample of H
atoms at $\boldsymbol{{l}}$, we would need to calculate

$\langle\mu(\boldsymbol{{l}})\rangle=\frac{\int\mu(I_{\sun,\text{E}},v_{r})f_{\text{ISN}}(\boldsymbol{{l}},\boldsymbol{{v}})\,d^{3}v}{\int
f_{\text{ISN}}(\boldsymbol{{l}},\boldsymbol{{v}})\,d^{3}v}.$ (33)

The term in the denominator is equal to the local density $n$, so effectively
for H, Equation 2.7 can be written down as:

$\displaystyle
I_{\text{glow}}(\boldsymbol{{r}}_{\text{obs}},\boldsymbol{{\hat{l}}})$
$\displaystyle=$
$\displaystyle\frac{\pi\,e^{2}}{m_{e}\,c}\,f_{\text{osc,H}}\,p_{\text{H}}\int\limits_{0}^{\infty}dl\,\psi(\beta)\,\left(\frac{r_{\text{E}}}{r_{\text{LOS}}}\right)^{2}\,n(\boldsymbol{{l}})\,\langle\mu(\boldsymbol{{l}})\rangle.$
(34)

In the numerical calculations, we integrate over $l$ up to a limiting distance
$L\approx 60$ au for H, and $L\approx 20$ au for He, and add a correction
term, which accounts for the integration to $\infty$:

$\displaystyle
I_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{r}}_{\text{obs}},\hat{\boldsymbol{{l}}})=\int\limits_{0}^{L}J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})\,dl+I_{\text{rest}},$
(35) $\displaystyle
I_{\text{rest}}=\int\limits_{L}^{\infty}J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{l}})\,dl.$

Because for large distances from the Sun we assume $l\approx r_{\text{LOS}}$
and using Equation (15) we obtain
$J_{\text{glow}}(\tilde{S}_{\sun},l)=J_{\text{glow}}(\tilde{S}_{\sun},L)(r_{\text{E}}/l)^{2}$.
After analytic integration of $\int 1/l^{2}\,dl$ we have

$I_{\text{rest}}=-\frac{J_{\text{glow}}(\tilde{S}_{\sun},L)\,r_{\text{E}}}{l}\Big{|}_{L}^{\infty}=\frac{J_{\text{glow}}(\tilde{S}_{\sun},\boldsymbol{{L}})\,r_{\text{E}}}{L}.$
(36)

Selection of $L\approx 60$ au and $L\approx 20$ au is based on our experience
with test runs of the simulation program for observer locations at about 1 au
from the Sun. The integration defined in Equation 2.7 is performed using an
adaptive-step numerical method that increases the numerical integration step
when it determines that the increment of the integrand function becomes small.
Consequently, at 20 au and 60 au from the Sun the integration step may be
quite large. To save the calculation time, we use a loose criterion for
finishing the numerical part of the integration, requiring that the routine
exceed the $L$ limit but need not achieve it precisely. At these distances
from the Sun the lines of sight originating at 1 au run very close to the
radial direction, and radial gradients of the density, velocity, and thermal
spread of ISN gas are very small, which makes the approximation defined in
Equation 35 very reasonable. Nevertheless, the correction term
$I_{\text{rest}}$ in most cases cannot be neglected in comparison with our
target numerical accuracy of $\approx 2$%.

##  3 Summary and outlook

In this paper, we present Warsaw Heliospheric Glow model (WawHelioGlow) – a
numerical code for simulating the helioglow intensity of ISN H and He. The
model, based on the (n)WTPM model of the ISN gas distribution in the
heliosphere, takes into account the time evolution and heliolatitudinal
dependence of the solar factors: spectral illumination, radiation pressure,
and ionization losses of the ISN gas inside the heliosphere. The evolution of
these factors in the simulations is included by using well-established models
based on relevant observations. The construction of the code facilitates
replacing these models with alternative ones.

Presented is the first version of the WawHelioGlow code, which uses an
optically thin, single scattering approximation of radiation transfer.
However, the distribution function of the ISN gas is simulated fully
numerically, without the frequently used approximation of Maxwell-Boltzmann
distribution function for the ISN gas inside the heliosphere. Including
effects of multiple scattering of the solar Lyman-$\alpha$ photons will be
implemented in a future version of the model.

Effects of multiple scattering were discussed by several authors, e.g., Ajello
et al. (1994); Hall (1992); Keller & Thomas (1979); Quémerais & Bertaux
(1993); Quémerais (2000); Quémerais et al. (2019); Scherer & Fahr (1996).
Their magnitude depends on details of approximations made in the multiple
scattering approach, in particular on the model of frequency redistribution.
Multiple-scattering codes are notorious for their high demand on computer
power. Ajello et al. (1994) and Quémerais (2000) suggested pre-calculating the
ratio of multiple-scattering and optically thin approximations and using them
as correction terms in further analyzes of observations using single-
scattering modeling. The corrections are the largest for the downwind
direction, on the order of 1.25 (see Table 5 in Quémerais, 2000). This
approach seems to work best for viewing geometries close to anti-solar.

The multiple scattering correction to an optically thin model is not an
optimum solution especially for LOS geometries much different from antisolar
because the additional excitation of the gas due to the scattered light is
basically a function of the angular distance from the upwind direction and the
distance from the Sun; when the line of sight traverses a wide range of offset
angles, then it is not straightforward to decide on the magnitude of
correction to apply even for observer locations at 1 au. Clearly, it is best
to have a multiple-scattering code. WawHelioGlow will account for multiple
scattering effects in one of its future versions.

Another effect that will be addressed in a future version of the code is the
27-day variation associated with active regions at the Sun’s surface (Pryor et
al., 1992, 1996; Bertaux et al., 2000). Accounting for this effect is easiest
in the viewing geometries close to anti-solar. The amplitude of the variation
is largest for these geometries; this is because for these geometries, the
largest portion of the line of sight is affected by the active region-related
brightening. In future versions of the code it is planned to use daily values
of the solar flux $I_{\text{tot}}$ for illumination of the gas and Carrington
rotation-averaged $I_{\text{tot}}$ values for the calculation of the ISN H
distribution.

The WawHelioGlow model includes provisions to account for a heliolatitudinal
anisotropy in the solar EUV output. This anisotropy is poorly investigated
but, based on available evidence, must be regarded as a distinct possibility.
The EUV anisotropy affects on one hand the illumination of the gas, and on the
other hand the spatial distribution of the ISN H and He populations because of
their sensitivity to the radiation pressure (only H) and photoionization. For
H, this anisotropy is a secondary anizotropization effect, in addition to the
primary effect of the solar wind anisotropy impressed in the gas by charge
exchange. For He, the EUV anisotropy is the main anizotropization factor
because He is the most sensitive to photoionization. In the code, the
anisotropy is currently implemented in a simplified way as a single-parameter
analytical formula, but it can be easily replaced with a more refined model.

The spectral illumination in the code can be defined in a variety of ways. The
code does not rely on any particular analytic definition or normalization of
the profile, the solar line profiles can be defined arbitrarily. In the
present version, for He it implements a Gaussian profile with a uniform
spectral background or, alternatively, a kappa profile. In the case of
hydrogen, the present version uses a state of the art model of the evolution
of the solar Lyman-$\alpha$ line. This model is based on spectral measurements
performed in the ecliptic plane throughout the cycle of the solar activity
(Lemaire et al., 2015). It assumes that the spectral shape depends solely on
the total Lyman-$\alpha$ flux. In connection with the model of the anisotropy,
this results in an extrapolation of the measurement-based spectral model
towards low values of the total solar flux during low solar activity at high
heliolatitudes. This feature of the WawHelioGlow model must be regarded as an
educated speculation. However, it is easy to implement a different approach
and test the results against observations.

The WawHelioGlow model is optimized towards analysis of future observations of
the helioglow by the GLOWS experiment onboard the planned NASA IMAP mission.
It is also well suited for analysis of photometric observations of the
helioglow from the SWAN experiment onboard SOHO, as well as other photometric
observations of the helioglow of both H and He, performed within a few au from
the Sun.

A presentation of the behavior of the factors making up the helioglow signal
for various observation geometries and phases of the solar cycle, for H and
He, is given in the accompanying paper. That paper also shows a qualitative
comparison of selected maps of the Lyman-$\alpha$ helioglow observed by SWAN
during a minimum and a maximum of the solar activity. To these comparisons we
used the results of the WawHelioGlow code obtained with the use of a state of
the art model of the ionization factors and solar illumination, and either
including or excluding the model of the latitudinal anisotropy of the solar
EUV output. That paper illustrates that WawHelioGlow model will potentially be
able to represent the observed helioglow intensity using the existing models
of the evolution of the solar ionization factors when latitudinal variation of
the solar EUV output is better investigated.

_Acknowledgments_. This study was supported by Polish National Science Center
grants 2019/35/B/ST9/01241, 2018/31/D/ST9/02852, and by Polish Ministry for
Education and Science under contract MEiN/2021/2/DIR.

##  Appendix A Cross section for photon absorption, the G factor, and
radiation pressure

###  A.1 Introduction

In this appendix, we present textbook derivations of some of the quantities
relevant for modeling the helioglow, the resonant absorption and radiation
pressure, and related quantities. This is a _pro memoria_ description, also
intended as a quick-reference source.

Throughout the helioglow work, we use a common base of the numerical values of
physical quantities. To facilitate maintaining this system, in Table 2 we
present a list of these physical constants and their units in the cgs system.
We will use them to derive normalization factor $p_{\text{H}}$ that allows us
to convert between radiation pressure profile in the so-called $\mu$-units and
the solar spectral flux in the physical units, as well as the $g$-factor,
informing on the frequency of atom excitation at a given distance to the Sun.

Table 2:  Physical constants in cgs units. In our numerical simulations, we use more precise values based on the NIST standards.  constant | symbol | value | unit
---|---|---|---
elementary charge | $e$ | $4.8032\times 10^{-10}$ | $\text{cm}^{3/2}\text{ g}^{1/2}\text{ s}^{-1}$
electron mass | $m_{\text{e}}$ | $9.1094\times 10^{-28}$ | g
speed of light | $c$ | $2.9979\times 10^{10}$ | $\text{cm s}^{-1}$
Lyman-$\alpha$ wavelength | $\lambda_{\text{H}}$ | $121.567$ | nm
He I wavelength | $\lambda_{\text{He}}$ | $58.433$ | nm
gravitational constant | $G$ | $6.6743\times 10^{-8}$ | $\text{cm}^{3}\text{ g}^{-1}\text{ s}^{-2}$
astronomical unit | $r_{\text{E}}$ | $1.496\times 10^{13}$ | cm
solar mass | $M_{\sun}$ | $1.989\times 10^{33}$ | g
hydrogen mass | $m_{\text{H}}$ | $1.6726\times 10^{-24}$ | g
Gaus gravity constant | $k^{2}$ | $2.959122130672713\times 10^{-4}$ | au3 day${}^{-1}\,M_{\sun}$
spectral flux | $\pi F_{\lambda}$ | time dependent | $\text{ph cm}^{-2}\text{s}^{-1}\text{ nm}^{-1}$
radiant intensity | $I_{\nu}$ | time dependent | $\text{erg s}^{-1}\text{cm}^{-2}\text{ sr}^{-1}\text{ Hz}^{-1}$
H oscillator strength | $f_{\text{osc},\text{H}}$ | $0.41641$ | dimensionless
He oscillator strength | $f_{\text{osc},\text{He}}$ | $0.27625$ | dimensionless
combination | $\frac{\pi\text{e}^{2}}{m_{\text{e}}\text{c}}$ | $0.0265402$ | $\text{cm}^{2}\text{ s}^{-1}$

###  A.2 Cross section

Neutral atoms in the heliosphere are exposed to EUV radiation from the Sun.
Sometimes, they absorb a photon and then immediately re-emit it in a different
direction and with a slightly different frequency. The process must conserve
the total momentum of the atom and the photon. As a result, due to the photon
absorption the momentum of the atom is incremented by that of the photon. In
one act of absorption, the momentum of the atom changes by:

$\Delta\text{p}=\frac{\text{h}}{\lambda}$ (A1)

and the direction of the momentum increment is anti-solar. After re-emission,
the momentum changes by a very similar amount in a random direction, with the
likelihood proportional to the phase function (Equations 8, 9). Since,
however, the momentum changes due to re-emission average to zero change
because the scattering phase function is axially symmetric about the photon
impact direction, the momentum change due to re-emission is neglected in the
further discussion.

A measure of how often photons are absorbed by the gas atoms is the cross
section for absorption. The absorption is considered in the inertial reference
frame of the atom. In the following, we derive the cross section using the
classical theory. Fundamentally, the same result can be obtained using the
quantum theory, which returns the correct species and transition-dependent
result. The classical result must be scaled to the quantum-mechanical one
using a proportionality constant referred to as the oscillator strength
$\text{f}_{\text{osc}}$ for the given atom and transition.

Considering an atom in a given quantum state as a classical damped oscillator
with radiation as an exciting force, we find that the cross section can be
expressed as:

$\displaystyle\sigma(\omega)$
$\displaystyle=\frac{8\pi\text{e}^{4}\omega^{4}}{3\text{m}_{\text{e}}^{2}c^{4}}\frac{1}{(\omega^{2}-\omega_{0}^{2})^{2}\gamma^{2}\omega^{2}},$
(A2) $\displaystyle\gamma$
$\displaystyle=\frac{2\text{e}^{2}\omega_{0}^{2}}{3\text{m}_{\text{e}}\text{c}^{3}},$
(A3)

where $\omega$ is the eigenfrequency, and $\gamma$ is the damping constant of
the considered oscillator.

Since we consider frequencies near the resonance
($\Delta\omega=\omega-\omega_{0}\ll\omega_{0}$), we can apply the following
approximation:

$\omega^{2}-\omega_{0}^{2}=(\omega-\omega_{0})(\omega+\omega_{0})\approx
2\omega_{0}(\omega-\omega_{0}).$ (A4)

Every other appearance of $\omega$ in Equation A2 can be substituted by
$\omega_{0}$:

$\sigma(\omega)=\frac{2\pi\text{e}^{2}}{\text{m}_{\text{e}}\text{c}}\frac{\frac{\gamma}{2}}{(\omega-\omega_{0})^{2}(\frac{\gamma}{2})^{2}}.$
(A5)

This equation is very convenient to interpret. The second part is known as the
Lorentzian function that has very well defined properties, the first part is a
proportionality constant.

Equation A5 can be expressed either as a function of the radiation frequency
or as a function of its wavelength:

$\displaystyle\Delta\omega$ $\displaystyle=2\pi\Delta\nu$ (A6)
$\displaystyle\Delta\omega$
$\displaystyle=\frac{2\pi\text{c}}{\lambda^{2}}\Delta\lambda.$ (A7)

With this, we can write the following formulae expressing the spectral cross
section (dependent on the frequency or alternatively the wavelength) and the
total cross sections, integrated over the frequency. In the reality, atoms
should be described using quantum mechanics rather than the classical theory.
Therefore, in Equations A8—A.2 we introduced the oscillator strength
$\text{f}_{\text{osc}}$, which is a scaling factor that makes the classical
result in agreement with the quantum-mechanical one.

$\displaystyle\sigma(\nu)$
$\displaystyle=\frac{\text{e}^{2}}{\text{m}_{e}\text{c}}\text{f}_{\text{osc}}\frac{\frac{\gamma}{4\pi}}{(\nu-\nu_{0})^{2}(\frac{\gamma}{4\pi})^{2}},$
(A8) $\displaystyle\sigma_{\nu}=\int_{0}^{\infty}d\nu\sigma(\nu)$
$\displaystyle=\int_{-\infty}^{\infty}d(\Delta\nu)\sigma(\Delta\nu)$
$\displaystyle=\frac{\text{e}^{2}}{\text{m}_{e}\text{c}}\text{f}_{\text{osc}}\int_{-\infty}^{\infty}d(\Delta\nu)\frac{\frac{\gamma}{4\pi}}{(\Delta\nu)^{2}(\frac{\gamma}{4\pi})^{2}}$
$\displaystyle=\frac{\pi\text{e}^{2}}{\text{m}_{e}\text{c}}\text{f}_{\text{osc}}$
$\displaystyle[\text{cm}^{2}\text{Hz}],$ (A9)

or the wavelength:

$\displaystyle\sigma(\lambda)$
$\displaystyle=\frac{\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\lambda_{0}^{2}}{\text{c}}\text{f}_{\text{osc}}\frac{\frac{\gamma\lambda_{\text{0}}^{2}}{4\pi\text{c}}}{(\lambda-\lambda_{\text{0}})^{2}(\frac{\gamma\lambda_{\text{0}}^{2}}{4\pi\text{c}})^{2}}$
(A10) $\displaystyle\sigma_{\lambda}=\int_{0}^{\infty}d\lambda\sigma(\lambda)$
$\displaystyle=\int_{-\infty}^{\infty}d(\Delta\lambda)\sigma(\Delta\lambda)$
$\displaystyle=\frac{\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\lambda_{0}^{2}}{c}\text{f}_{\text{osc}}\int_{-\infty}^{\infty}d(\Delta\lambda)\frac{\frac{\gamma\lambda_{\text{0}}^{2}}{4\pi\text{c}}}{(\Delta\lambda)^{2}(\frac{\gamma\lambda_{\text{0}}^{2}}{4\pi\text{c}})^{2}}$
$\displaystyle=\frac{\pi\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\lambda_{0}^{2}}{c}\text{f}_{\text{osc}}$
$\displaystyle[\text{cm}^{2}\text{nm}].$ (A11)

The oscillator strength can be calculated theoretically as well as directly
measured.

###  A.3 Factor g

The so-called g factor is a measure of how many photons are absorbed by an
atom in a unit time given an incident spectral flux. For an atom stationary
relative to the Sun, it can be computed in following way:

$\text{g}=\int_{0}^{\infty}d\lambda\pi\text{F}(\lambda_{0})\sigma(\lambda),$
(A12)

where $\pi\text{F}(\lambda_{0})$ is the solar spectral flux in the center of
the line (defined by wavelength $\lambda_{0}$) expressed in the units
$\text{ph cm}^{2}\text{s}^{-1}\text{nm}^{-1}$.

For the Lyman-$\alpha$ radiation, which is relevant for H atoms, we do have
direct measurements of the spectral flux in the center of the line (Lemaire et
al., 2005). In the case of the He I line, measurements of the profile are very
few and far between (see Appendix B) and on a longer basis, there are only
measurements of the total flux integrated over whole line available (Woods et
al., 2015). This is because the He I line is much narrower than the
Lyman-$\alpha$ line. Therefore, we will follow the approach presented in
Equation 4 of Grava et al. (2018) and calculate the flux in the center of line
assuming the Gaussian shape. Either way, the flux $\pi\text{F}(\lambda_{0})$
is a wavelength-independent value that can be taken out of the integral. Then,
using Equation A.2 and Equation A12, we can write a simple formula for the g
factor:

g
$\displaystyle=\pi\text{F}(\lambda_{0})\frac{\pi\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\lambda_{0}^{2}}{\text{c}}\text{f}_{\text{osc}}$
$\displaystyle[\text{ph atom}^{-1}\text{s}^{-1}].$ (A13)
$\displaystyle\lambda_{0}$
$\displaystyle=\begin{cases}\lambda_{\text{H}}&\text{Hydrogen}\\\
\lambda_{\text{He}}&\text{Helium}.\end{cases}$
$\displaystyle\text{f}_{\text{osc}}$
$\displaystyle=\begin{cases}\text{f}_{\text{osc,H}}&\text{Hydrogen}\\\
\text{f}_{\text{osc,He}}&\text{Helium}.\end{cases}$
$\displaystyle\pi\text{F}(\lambda_{0})$
$\displaystyle=\begin{cases}\text{I}_{\text{Lya}}(\lambda_{\text{H}})&\text{Hydrogen}\\\
\frac{\text{I}_{\text{tot,He}}\frac{\lambda_{\text{He}}}{\text{hc}}10^{3}}{1.064\text{
FWHM}_{\lambda}}&\text{Helium}\end{cases}$

where $\text{I}_{\text{Lya}}(\lambda_{\text{H}})$ is the spectral flux in the
center of the Lyman-$\alpha$ line expressed in units of $\text{ph
cm}^{2}\text{s}^{-1}\text{nm}^{-1}$, $\text{I}_{\text{tot,He}}$ is the total
flux in He I line expressed in units of W m-2, $\text{FWHM}_{\lambda}=0.0136$
nm (see Appendix B) is the Full Width at Half Maximum of the He I line
assuming its Gaussian shape, the $10^{3}$ factor is for conversion between
units. For the product $G\,M_{\sun}$ it is best to use the Gauss gravity
constant squared: $G\,M_{\sun}=k^{2}$ (see Table 2). This is because this
quantity is much more precisely known than the gravitational constant and the
solar mass separately.

For an atom with a non-zero radial velocity $v_{r}$ relative to the Sun, the
spectral flux at the line center in Equation A13 must be replaced with the
spectral flux corresponding to the Doppler-shifted wavelength given by
Equation 12.

###  A.4 Radiation pressure

Radiation pressure is relevant only for H atoms because of a much lower
spectral flux and the mass of He atoms being four-fold larger than that of H
atoms. Therefore, from this point on we will be referring only to the
quantities characteristic for this chemical element. Each scattering event
transfers momentum between the interacting photon and atom. A cumulative
effect of many scatterings can be described as a continuous force acting on
atoms up to a distance $\text{r}\sim 5000$ au from the Sun. Beyond this
distance, the mean time between scattering events is comparable to the time
needed for an atom to travel the length equivalent to its heliospheric
distance and the continuous force approximation is no longer valid (see
section 3.2.11 in Hall (1992)). The continuous force caused by the momentum
transfer, based on Equations A1, A13, can be expressed as:

$\text{P}_{\text{rad}}=\text{g}\,\Delta\text{p}=\text{I}_{\text{Lya}}(\lambda_{\text{H}})\frac{\pi\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\text{h}\lambda_{\text{H}}}{\text{c}}\text{f}_{\text{osc},\text{H}}.$
(A15)

Radiation pressure acts in the opposite direction to the solar gravitational
force
$\text{F}_{\text{g}}=-\frac{\text{GM}_{\sun}\text{m}_{\text{H}}}{\text{r}^{2}}$.
Since in the optically thin approximation the solar spectral flux drops with
the square of solar distance, then the radiation pressure force is directly
proportional to that of solar gravity at all distances. We define a
dimensionless factor $\mu$ that is very commonly used in the literature. It is
defined as a ratio between the force caused by the momentum transfer due to
scattering events and gravitational force at a given distance. When it is
equal to 1, there is no effective force acting on a moving atom. Since we have
direct measurements of solar irradiance at the distance $r=r_{\text{E}}$, we
will use it in the following considerations.

$\displaystyle\mu$
$\displaystyle=\frac{\text{P}_{\text{rad}}}{|\text{F}_{\text{g}}|}$
$\displaystyle=\text{I}_{\text{Lya}}(\lambda_{\text{H}})\frac{\pi\text{e}^{2}}{\text{m}_{e}\text{c}}\frac{\text{h}\lambda_{\text{H}}}{\text{c}}\text{f}_{\text{osc},\text{H}}\frac{\text{r}_{\text{E}}^{2}}{\text{GM}_{\sun}\text{m}_{\text{H}}}$
$\displaystyle=\frac{\text{I}_{lya}(\lambda_{\text{H}})}{\text{p}_{\text{H}}},$
(A16) $\displaystyle\text{p}_{\text{H}}$
$\displaystyle=\left[\frac{\pi\,\text{e}^{2}}{\text{m}_{\text{e}}\,c}\,\frac{\text{h}\lambda_{\text{H}}}{\text{c}}\text{f}_{\text{osc},\text{H}}\frac{\text{r}_{\text{E}}^{2}}{\text{G}\,\text{M}_{\sun}\text{m}_{\text{H}}}\right]^{-1}=3.34467\times
10^{12}$ $\displaystyle\text{ ph s}^{-1}\,\text{cm}^{-2}\,\text{nm}^{-1}.$
(A17)

When the single-scattering optically-thin approximation is not used, then
still the radiation pressure force at a given distance to the Sun can be
expressed as a fraction $\mu$ of the solar gravity force at this distance, but
it is no longer distance-independent.

##  Appendix B Available measurements of the solar He I 58.4 nm line

One of the first measurements of the 58.4 nm solar line was given by Hall &
Hinteregger (1970) who reported the radiance in the He I 58.4 nm line on 11th
March 1967, 09:45 UT equal to $0.89\cdot 10^{9}$ phot cm-2 s-1 and the helium
continuum level (measured at 50.4 nm) equal to $0.5\cdot 10^{9}$ phot cm-2
s-1. For 15th May 1967, the flux was reported larger by a factor of 1.6 at
58.43 nm.

Another measurement of this line was done by Doschek et al. (1974), who
obtained a geocorona-uncorrected FWHMλ equal to $0.014\pm 0.0015$6 nm and
reported no departures from the Gaussian shape.

Cushman et al. (1975) based his analysis of the 58.4 nm solar line on a rocket
underflight on August 30, 1973. The resolution was $\sim 0.002$ nm, and since
the measurement was taken from within the geocorona, there was a 0.003 nm wide
geocoronal absorption feature in the profile, for which the measurement was
compensated. The appearance of the resulting profile is kappa-like, even
though the authors do not discuss the differences between the assumed Doppler
(Gaussian) shape and that actually measured. In addition, there was an active
region in the field of view, which did not cover the entire solar disk. The
FWHMλ of the helium line was found to be 0.01 nm for the active region and
0.008 nm for the quiet Sun. The measured spectral irradiance at the line
center for the quiet Sun observation was found at $\sim 2\cdot 10^{10}$ phot
(s Å cm${}^{2})^{-1}$. The total intensity was reported at $1.3\cdot
10^{9}\,\text{phot }(\rm{cm}^{2}\,s)^{-1}\pm 40$%. The disk coverage was $\sim
3$%.

Radiance variations in the solar He I 58.4 nm line from 5th March 1996 until
8th August 1999 were presented by Schühle et al. (2000). The radiance seems to
increase approximately linearly in the log scale from 0.4 to 0.5 W m-2 sr-1
when measured on a portion of the disk corresponding to quiet Sun. These
authors also show relative variations of the network and cell areas, i.e., the
brightest and the weakest regions in the disk. While variations are by a
factor of 2, these measurement feature an increase from solar minimum to solar
maximum conditions similar to that for the quiet Sun.

Del Zanna & Andretta (2006) provide a series of EUV irradiance in the helium
line between $\sim 1998$ and $\sim 2006$ observed by CDS/SOHO and obtain the
absolute values between 1.5 and $2.0\cdot 10^{9}$ phot cm-2 s-1 (for
comparison, the total irradiance in the Lyman-$\alpha$ line during the minimum
of solar activity is about $3.5\cdot 10^{11}$ phot cm-2 s-1). Del Zanna &
Andretta (2015) present a continuation of the irradiance measurement from 1998
until 2015, spanning an interval $(1.0-2.0)\cdot 10^{9}$ phot cm-2 s-1.

We adopt the profile of the solar He I 58.4 nm line as a Doppler (Gaussian)
with a non-zero background, parametrized by a radial velocity of a scattering
atom relative to the Sun $v_{r}$. The central wavelength of the transition
responsible for the 58.4 nm ISN He backscatter glow is $\lambda_{\text{He I
58.4}}=58.43350$ nm (CHIANTI database), which is equivalent to the frequency
$\nu_{\text{He I 58.4}}=\nu_{0}=5.1304895\cdot 10^{15}$ Hz. The relation
between the atom rest frame frequencies and radial speeds are given by
Equation 12.

The evolution of the width of the solar He I 58.4 nm line was presented by
McMullin et al. (2004) based on SUMER/SOHO observations carried out between
1996 and 2001, i.e., from a solar minimum to a maximum. The conclusion was
that the Doppler width $v_{\mathrm{D}}$ was constant during the solar cycle
and equal to $36.5\pm 1.7$ km s-1, where the uncertainty is the standard
deviation of the measurement. The minimum and maximum 10% of the measured
widths were found to be at 27.7 and 49.5 km s-1, respectively. The relation
between the widths expressed by the wavelengths and the Doppler speeds used by
McMullin et al. (2004) is $v_{\mathrm{D}}=(\Delta\lambda/\lambda)c$, identical
to that we use (Equation 21).

McMullin et al. (2004) state that the widths $v_{\mathrm{D}}$ were fitted to a
“single Gaussian function and a linear background”, but they do not provide
the formula for the profile. Lallement et al. (2004) refers to the paper by
McMullin et al. (2004) and states that based on the data from this paper, the
widths of this line are between 92 and 155 mÅ, with the preferred value at 118
mÅ. A function defined as
$\exp\left[-\left(\frac{x}{2\sigma}\right)^{2}\right]$ takes the value of
$\frac{1}{2}$ for $x=\pm 2\sqrt{\ln 2}\,\sigma$, so $\mathrm{FWHM}=4\sqrt{\ln
2}\sigma$, which corresponds to $\mathrm{FWHM}_{v}=0.118/584\,c\cong 60.6$ km
s-1. From this relation one obtains that
$2\sigma=\mathrm{FWHM}_{v}/(2\sqrt{\ln 2})\cong 36.4$ km s-1, which we adopt
in the model.

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