11institutetext: Dipartimento di Fisica e Astronomia “Ettore Majorana” –
Sezione Astrofisica, Università degli Studi di Catania, Via S. Sofia 78,
I-95123 Catania, Italy
11email<EMAIL_ADDRESS>22institutetext: INAF - Osservatorio
Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy
33institutetext: INAF - Osservatorio Astrofisico di Torino, Via Osservatorio
20, I-10025 Pino Torinese (TO), Italy 44institutetext: INAF - Osservatorio
Astrofisico di Capodimonte, Salita Moiariello 16, I-80131 Napoli, Italy
55institutetext: CNR - Istituto di Fotonica e Nanotecnologie, Via Trasea 7,
I-35131 Padova, Italy 66institutetext: Dipartimento di Fisica e Astronomia,
Università degli Studi di Firenze, Largo Enrico Fermi 2, I-50125 Firenze,
Italy

# Effects of the chromospheric Ly$\alpha$ line profile shape on the
determination of the solar wind H I outflow velocity using the Doppler dimming
technique

G. E. Capuano 11 S. Dolei 22 D. Spadaro 22 S. L. Guglielmino 1122 P. Romano 22
R. Ventura 22 V. Andretta 44 A. Bemporad 33 C. Sasso 44 R. Susino 33 V. Da
Deppo 55 F. Frassetto 55 S. M. Giordano 33 F. Landini 33 G. Nicolini 33 M.
Pancrazzi 33 M. Romoli 66 L. Zangrilli 33

###### Abstract

Context. The determination of solar wind H I outflow velocity is fundamental
to shedding light on the mechanisms of wind acceleration occurring in the
corona. Moreover, it has implications in various astrophysical contexts, such
as in the heliosphere and in cometary and planetary atmospheres.

Aims. We aim to study the effects of the chromospheric Ly$\alpha$ line profile
shape on the determination of the outflow speed of coronal H I atoms via the
Doppler dimming technique. This is of particular interest in view of the
upcoming measurements of the Metis coronagraph aboard the Solar Orbiter
mission.

Methods. The Doppler dimming technique exploits the decrease of coronal
Ly$\alpha$ radiation in regions where H I atoms flow out in the solar wind.
Starting from UV observations of the coronal Ly$\alpha$ line from the
UltraViolet Coronagraph Spectrometer (UVCS), aboard the Solar and Heliospheric
Observatory (SOHO), and simultaneous measurements of coronal electron
densities from pB coronagraphic observations, we explored the effect of the
profile of the pumping chromospheric Ly$\alpha$ line. We used measurements
from the Solar UV Measurement of Emitted Radiation (SUMER), aboard SOHO, the
Ultraviolet Spectrometer and Polarimeter (UVSP), aboard the Solar Maximum
Mission (SMM), and the Laboratoire de Physique Stellaire et Planetaire (LPSP),
aboard the Eight Orbiting Solar Observatory (OSO-8), both from representative
on-disc regions, such as coronal holes and quiet Sun and active regions, and
as a function of time during the solar activity cycle. In particular, we
considered the effect of four chromospheric line parameters: line width,
reversal depth, asymmetry, and distance of the peaks.

Results. We find that the range of variability of the four line parameters is
of about 50% for the width, 69% for the reversal depth, and 35% and 50% for
the asymmetry and distance of the peaks, respectively. We then find that the
variability of the pumping Ly$\alpha$ profile affects the estimates of the
coronal H I velocity by about 9-12%. This uncertainty is smaller than the
uncertainties due to variations of other physical quantities, such as electron
density, electron temperature, H I temperature, and integrated chromospheric
Ly$\alpha$ radiance.

Conclusions. Our work suggests that the observed variations in the
chromospheric Ly$\alpha$ line profile parameters along a cycle and in specific
regions negligibly affect the determination of the solar wind speed of H I
atoms. Due to this weak dependence, a unique shape of the Ly$\alpha$ profile
over the solar disc that is constant in time can be adopted to obtain the
values of the solar wind H I outflow velocity. Moreover, the use of an
empirical analytical chromospheric profile of the Ly$\alpha$, assumed uniform
over the solar disc and constant in time, is justifiable in order to obtain a
good estimate of the coronal wind H I outflow velocity using coronagraphic UV
images.

###### Key Words.:

Sun: chromosphere – Sun: corona – Sun: UV radiation – Sun: solar wind

##  1 Introduction

The solar wind is a tenuous plasma that continuously escapes from the Sun into
the interplanetary medium. It extends for thousands of millions of kilometers
and is mainly characterised by a fast component, with typical velocities of
about $500-800\,\mathrm{km\,s}^{-1}$ flowing from coronal holes (CHs), and a
slow component, with typical velocities of about
$300-500\,\mathrm{km\,s}^{-1}$, mainly related to the streamer belt regions
(see e.g. McComas et al., 1998, 2008).

The high coronal temperatures, of which the origin is still unclear, cause
wind expansion (e.g. Parker, 1958) and eventually the acceleration of slow and
fast solar wind. It is thought that such acceleration could be the result of,
for example, energy dissipation through the ion cyclotron resonance of high-
frequency left-hand polarised Alfvén waves (see e.g. Ofman, 2010).

Given the high coronal temperatures, the solar wind is mainly composed of
fully ionised hydrogen and helium and of free electrons. The latter component
is mainly observed through the scattering and polarisation of the white light
coming from the photosphere (Thomson scattering). Other ions of heavier minor
elements are present in smaller amounts.

The ESA/NASA Solar and Heliospheric Observatory (SOHO; Domingo et al., 1995)
spacecraft returned a wealth of data, which provided a new understanding of
the physical phenomena that contribute to the acceleration of the solar wind
in the corona. In particular, the UltraViolet Coronagraph Spectrometer (UVCS;
Kohl et al., 1995) on board SOHO allowed us to obtain crucial results, such as
information on elemental abundances and kinetic temperatures of H I and O VI
ions in different coronal structures. Antonucci et al. (2005), Susino et al.
(2008), and Abbo et al. (2010a) suggested that the main source of the slow
solar wind is the boundary region between CHs and streamers, where the wind is
dependent on the magnetic field topology. Outflows from active regions (ARs)
could even contribute to the slow wind component (Zangrilli & Poletto, 2016).
The presence of pseudo or unipolar streamers can also contribute to the slow
solar wind. Furthermore, white light observations with the Large Angle and
Spectrometric COronagraph (LASCO; Brueckner et al., 1995) aboard SOHO have
shown that outflows are connected to both large-scale ’streamer blowout’
structures, the coronal mass ejections (CMEs) closely associated with magnetic
reconnection at the current sheet above the cusp of the streamers, and to
small-scale inhomogeneities (blobs), linked to quasi-periodic emission of
plasma from cusps of helmet streamers (Sheeley et al., 1997; Wang et al.,
1998; Song et al., 2009; Viall & Vourlidas, 2015; see also the review of Abbo
et al., 2016).

Generally, observations of emission lines in the off-limb corona, such as the
H I Ly$\alpha$ (121.6 nm), which are mainly due to scattering processes of
radiation coming from the low solar atmosphere, can be used to determine the
radial component of the outflow velocity ($v_{w}$) of the specific atom or ion
that generates the scattering through the Doppler dimming technique (Hyder &
Lites, 1970; Withbroe et al., 1982; Noci et al., 1987). This technique
consists of the analysis of the intensity decrease of the coronal ultraviolet
(UV) radiation coming from flowing regions, and it relies on some assumptions
and prior knowledge of the geometrical and physical properties that are
representative of the coronal environment (such as electron density, electron
temperature, and kinetic temperature of the scattering ions) and of the source
of the UV radiation scattered (such as intensity and line profile of the
exciting chromospheric radiation). However, in those cases in which
observations come from coronagraphs designed for imaging only, these
quantities cannot be directly determined and they must be taken from the
literature. Critical works aimed at investigating how the choice of the
parameters adopted from the literature (and their uncertainties) or from
specifically constructed databases may affect the results of the Doppler-
dimming analysis have been presented very recently.

In this context, Dolei et al. (2016) analysed the H I Ly$\alpha$ line profiles
observed by UVCS instrument in a number of polar, mid-latitude, and equatorial
structures. They considered a large amount of data acquired in a time range
longer than a complete solar cycle (1996 – 2012) and determined the
temperature components for neutral hydrogen at different phases of solar
activity. They were able to derive the H I temperature radial profiles for
several heliocentric distances (from $1.3\,R_{\odot}$ to $4.5\,R_{\odot}$) and
polar angles.

Dolei et al. (2018) used coronagraphic Ly$\alpha$ observations from UVCS and
visible light (VL) images obtained with LASCO and the Mark-III Coronameter
(Mk3/MLSO; Fisher et al., 1981) to study the dependence of the derived solar
wind H I outflow velocity on the physical parameters that characterise the
scattered coronal Ly$\alpha$ emission when the Doppler dimming effect is
considered. They found that the electron temperature has the most important
impact on the determination of the H I outflow velocity, while the coronal H I
temperature has an important role at larger heliocentric distances, where
higher speeds are reached.

Dolei et al. (2019) also used UVCS and LASCO-Mk3 observations in order to
analyse the effect of chromospheric radiation inhomogeneities on the solar
wind H I outflow velocity determination with respect to the case in which a
uniform-disc brightness approximation is adopted. In particular, they created
a Carrington map of the non-uniform solar chromospheric Ly$\alpha$ radiation
using a correlation function between the H I 121.6 nm and He II 30.4 nm
intensities (Auchère, 2005), while some approximations concerning the coronal
H I temperature and a constant value of the radiance over the entire solar
disc were adopted for the uniform-disc brightness case. They found that the
uniform condition leads to an overestimated velocity of
$50-60\,\mathrm{km\,s}^{-1}$ in polar and mid-latitude regions, while
underestimated velocity values are obtained at the equatorial regions. This
difference decreases at higher altitudes.

The present work can be considered as a step forward in the investigations
carried out by Dolei et al. (2018, 2019). The aim of this paper is to study
the effects induced on the determination of the solar wind H I velocity by the
variation of the exciting chromospheric Ly$\alpha$ line profile shape, which
can occur among different regions on the solar surface, such as quiet or
active regions and CHs, or during different phases of the solar activity
cycle. For this purpose, we considered Ly$\alpha$ chromospheric observations
performed by the Solar UV Measurement of Emitted Radiation (SUMER; Wilhelm et
al., 1995) aboard the SOHO spacecraft, which include full disc observations
(Lemaire et al., 2015) and the best sets of Ly$\alpha$ solar-disc observations
currently available (Curdt et al., 2008; Tian et al., 2009a, b). We also
considered other observations concerning ARs, quiet Sun (QS) regions, and an
equatorial CH performed by the Ultraviolet Spectrometer and Polarimeter (UVSP;
Miller et al., 1981) aboard the Solar Maximum Mission (SMM; Woodgate et al.,
1980) and the LPSP instrument (Laboratoire de Physique Stellaire et
Planetaire; Artzner et al., 1977; Bonnet et al., 1978) aboard the Eight
Orbiting Solar Observatory (OSO-8). Other parameters critically affecting the
coronal Ly$\alpha$ emission, such as the electron temperature, the electron
density, and the neutral hydrogen temperature, are the same as those adopted
by Dolei et al. (2019).

In order to facilitate the comparison, bidimensional (2D) maps of outflow
solar wind H I velocity were produced by using the same numerical code as in
Dolei et al. (2018, 2019), based on the reproduction of the synthetic coronal
Ly$\alpha$ intensity along the line of sight (LOS). The numerical computations
have allowed us to change the value of the wind velocity $v_{w}$ until a match
with the observed coronal intensity is reached. Moreover, this allowed us to
estimate the solar wind velocity as a function of heliocentric distance and
latitude. In obtaining each map of solar wind H I outflow velocity, we assumed
that the chromospheric intensity and the line profile of the Ly$\alpha$
emission are unique over the entire solar disc.

Our study is of particular interest for the analysis and interpretation of
coronal Ly$\alpha$ observations that will be obtained by the Metis coronagraph
(Antonucci et al., 2020; Fineschi et al., 2020) onboard the Solar Orbiter
spacecraft (Müller et al., 2013, 2020). Indeed, Metis, which is designed for
imaging only, observes linearly polarised VL (580 nm - 640 nm) and UV
Ly$\alpha$ emission at the same time with a field of view (FOV) that ranges
from $1.6\,R_{\odot}$ to $7.5\,R_{\odot}$ (Antonucci et al., 2020), viewing
the Sun from latitudes up to about $30^{\circ}$ with respect to the equatorial
plane (Zouganelis et al., 2020).

In Sect. 2, the main mechanisms concerning the coronal emission, the analysis
technique used, and the observations are described. Sect. 3 is dedicated to
the obtained results. Finally, in Sect. 4, we discuss the results and give our
conclusions.

##  2 Data analysis

###  2.1 Mechanisms of coronal emission and the Doppler dimming phenomenon

In a static corona, the emission coming from coronal atoms/ions is due to the
excitation induced by two main mechanisms: the collision with free electrons
and the resonant scattering of chromospheric radiation. However, in the case
of the H I Ly$\alpha$ emission, the collisional term contributes only to a
small fraction of the total coronal emission (Gabriel, 1971; Raymond et al.,
1997, Landi Degli’Innocenti; private communication, quoted in Noci et al.,
1987), given the stronger dependence of the collisional component
$\mathcal{I}_{col}$ of the coronal intensity on the electron column density
$N_{e}$ (i.e. $\mathcal{I}_{col}\propto{N_{e}^{2}}$) with respect to the
resonant one $\mathcal{I}_{rad}$ (i.e. $\mathcal{I}_{rad}\propto{N_{e}}$; see
e.g. Withbroe et al., 1982) and the low-density conditions present in the
outer corona. For this reason, it can be neglected.

Under the hypothesis of low-density coronal plasma, the radiance
$\mathcal{I}_{rad}$ (in units of erg cm-2 s-1 sr-1) coming from H I atoms in a
coronal point $P$ and measured along the ${\bf{n}}$ direction, corresponding
to the line of sight, can be expressed by the following:

$\mathcal{I}_{rad}=\frac{0.833\leavevmode\nobreak\ h\leavevmode\nobreak\
B_{12}}{4\pi\lambda_{0}}\int_{-\infty}^{+\infty}{n_{e}\leavevmode\nobreak\
R_{{H\textsc{I}}}(T_{e})}\leavevmode\nobreak\ dl\\\
\times{\,\int_{\Omega}\frac{11+3({\bf{n}}\cdot{\bf{n^{\prime}}})^{2}}{12}}\leavevmode\nobreak\
{F({\bf{n^{\prime}}},v_{w},\theta)}\leavevmode\nobreak\ d\Omega,$ (1)

where $h$ is the Planck constant; $B_{12}$ is the Einstein coefficient for the
Ly$\alpha$ transition; $\lambda_{0}=121.567$ nm is the central wavelength of
the considered transition; $n_{e}$ is the electron density; $0.833$ is the
ratio between proton and electron density for a gas that is completely ionised
and contains 10$\%$ helium; $T_{e}$ is the electron temperature;
$R_{{H\textsc{I}}}(T_{e})$ is the total hydrogen ionisation fraction as a
function of $T_{e}$, meaning the ratio between the neutral hydrogen and proton
density (see e.g. Withbroe et al., 1982); $dl$ is the infinitesimal path along
the line of sight; ${\bf{n}^{\prime}}$ is the direction along which the
chromospheric radiation reaches the coronal point $P$; $\Omega$ is the solid
angle under which the point $P$ subtends the solar disc (see e.g. Fig. 1 in
Dolei et al.,
2015);$p(\Omega)=[11+3({\bf{n}}\cdot{\bf{n^{\prime}}})^{2}]/[12\leavevmode\nobreak\
(4\pi)]$ is a geometrical factor that gives the angular dependence of the
scattering process (see Beckers & Chipman, 1974); and

$F({\bf{n^{\prime}}},v_{w},\theta)=\int_{-\infty}^{+\infty}{I({\lambda^{\prime}-\lambda_{0}-\delta\lambda},{\bf{n^{\prime}}})\leavevmode\nobreak\
\Phi({\lambda^{\prime}-\lambda_{0}})\leavevmode\nobreak\ d{\lambda^{\prime}}}$
(2)

is the convolution between the chromospheric specific intensity
$I(\lambda^{\prime}-\lambda_{0}-\delta\lambda,{\bf{n^{\prime}}})$ and the
normalised coronal absorption profile $\Phi({\lambda^{\prime}-\lambda_{0}})$,
where $\lambda^{\prime}$ is the wavelength. The term

$\delta\lambda=\frac{\lambda_{0}}{c}{\bf{v}}\cdot{\bf{n^{\prime}}}=\frac{\lambda_{0}}{c}v_{w}\cos{\theta}$
(3)

is the Doppler shift seen along ${\bf{n^{\prime}}}$ by the scattering atoms
due to the velocity ${\bf{v}}$ (supposed radially symmetric) of the scattering
atoms, where $v_{w}$ is the module of ${\bf{v}}$, $\theta$ is the angle
between ${\bf{v}}$ and ${\bf{n^{\prime}}}$, and c is the speed of the
electromagnetic radiation in the vacuum (see Fig. 1 in Dolei et al., 2015 for
a clear description of the geometry of the scattering process). The specific
intensity
$I(\lambda^{\prime}-\lambda_{0}-\delta{\lambda},{\bf{n^{\prime}}})=I({\bf{n^{\prime}}})\cdot\Psi(\lambda^{\prime}-\lambda_{0}-\delta{\lambda})$
is the product between the radiance $I({\bf{n^{\prime}}})$ (in units of erg
cm-2 s-1 sr-1) and the normalised intensity profile
$\Psi(\lambda^{\prime}-\lambda_{0}-\delta{\lambda})$ (in units of Å-1) of the
chromospheric Ly$\alpha$ line. Then, $F({\bf{n^{\prime}}},v_{w},\theta)$ gives
information about the dependence on the H I atom’s velocity of the radiative
excitation efficiency by chromosperic radiation. If we assume that the
velocity distribution of the absorbing H I coronal atoms is Maxwellian, the
normalised coronal absorption profile $\Phi(\lambda^{\prime}-\lambda_{0})$ has
a width that depends on the thermal motion only (neglecting all contributions
coming from non-thermal phenomena, such as turbulence, waves, oscillations,
and so on), given by

$\Delta\lambda_{D}=\frac{\lambda_{0}}{c}\sqrt{\frac{2\leavevmode\nobreak\
k_{B}\leavevmode\nobreak\ T_{{H\textsc{I}}}}{m_{H}}},$ (4)

where $k_{B}$ is the Boltzmann constant, $T_{{H\textsc{I}}}$ is the H I
temperature, and $m_{H}$ is the mass of hydrogen atoms.

Figure 1:  Bidimensional maps of UV coronal Ly$\alpha$ intensity (top panel),
acquired by UVCS/SOHO, and VL pB emission (bottom panel), obtained from
combined observations by the LASCO/SOHO and Mk3/MLSO instruments as a function
of heliocentric distance and latitude. The maps have a FOV between
$1.5\,R_{\odot}$ and $6.0\,R_{\odot}$, and refer to observations taken on June
7, 1997.

We also define the ’Doppler factor’ $D(v_{w})$ (see Noci et al., 1987):

$D(v_{w})=\frac{\int_{\Omega}\leavevmode\nobreak\
F({\bf{n^{\prime}}},v_{w},\theta){p(\Omega)}\leavevmode\nobreak\
d\Omega}{\int_{\Omega}\leavevmode\nobreak\
F({\bf{n^{\prime}}},v_{w}=0,\theta){p(\Omega)}\leavevmode\nobreak\ d\Omega},$
(5)

which accounts for the overlapping between the coronal absorption profile and
the exciting chromospheric one as a function of $v_{w}$.

When the radiative resonantly scattered emission dominates, the outwards
expansion of the corona generates a decrease (dimming) in the Ly$\alpha$
coronal emission. This phenomenon can be explained considering the Doppler
redshift which an observer positioned in $P$ measures in the chromospheric
radiation. In fact, when the corona is in a static condition, the profile of
the chromospheric Ly$\alpha$ is centred at $\lambda_{0}$, causing the maximum
intensity of the scattered Ly$\alpha$ radiation ($D=1$). However, when the
corona is expanding, the scattered Ly$\alpha$ is dimmed because of the Doppler
shift of the chromospheric photons seen by the outflowing coronal H I atoms.
The intensity of the scattered radiation tends to vanish when the outflow
velocity becomes higher, because the chromospheric and coronal profiles no
longer overlap ($D\approx 0$). Thus, it is possible to match the synthetic
intensity $\mathcal{I}_{rad}$ and that measured by observations by numerically
tuning the $v_{w}$ value, estimating in turn the outflow velocity.

###  2.2 Observations

Figure 2:  Panel (a): Ly$\alpha$ irradiance profiles reported in Fig. 6a and
Fig. 12 of Fontenla et al. (1988) (QS and AR, respectively), as well as
profiles observed on April 17, 2009 (CH, Tian et al., 2009b), on October 28,
1996 (full disc, Lemaire et al., 2015) and in an equatorial CH at the central
meridian between November 27 and 29, 1975 (Bocchialini & Vial, 1996). Panel
(b): Parametrised Ly$\alpha$ profiles from Kowalska-Leszczynska et al. (2018)
(IKL; at maximum and at minimum) and Auchère (2005). Each profile has been
normalised to its total intensity.

We describe the observations adopted in our analysis here.

####  2.2.1 UV and pB observations

A synoptic UV map related to the observations of coronal Ly$\alpha$ intensity
and analysing data acquired on June 7, 1997 (during the minimum of solar
activity) by UVCS was obtained. This 2D map was constructed, as explained by
Bemporad (2017) and Dolei et al. (2018), starting from synoptic UVCS
observations and by performing a power-law interpolation of intensities
observed at different latitudes and altitudes. A polarised brightness pB map
in visible light was obtained by interpolating observations acquired with Mk3
between $1.1\,R_{\odot}$ and $1.5\,R_{\odot}$ and by LASCO above
$2.5\,R_{\odot}$ on the same day (see Bemporad, 2017; Dolei et al., 2018, for
more details). The maps, which are shown in Fig. 1, are those considered in
the work by Dolei et al. (2019).

It is worth mentioning that we only considered coronal observations related to
the minimum solar activity, although the analysed chromospheric observations,
as described below, concern an entire solar cycle. However, since the aim of
this work is to investigate the effects induced by the chromospheric
Ly$\alpha$ profile shape variation on the H I outflow velocity, it is not
unreasonable to adopt a simplified scenario.

####  2.2.2 Chromospheric observations of Ly$\alpha$ line profiles

In this study, we aim to explore the impact of possible variations in the
spectral profile of the pumping chromospheric radiation, such as those caused
by the solar activity cycle, on the determination of $v_{w}$. For this
purpose, we considered full disc, QS and CH observations of the chromospheric
Ly$\alpha$ line profiles carried out by SUMER/SOHO during the solar activity
cycle 23, ARs and QS profiles acquired by UVSP/SMM, and profiles referring to
an equatorial CH and a flare acquired by the LPSP instrument aboard OSO-8, in
order to take into account all the possible chromospheric regions emitting the
exciting radiation.

First, we used measurements of the Ly$\alpha$ line profile relative to the
entire solar disc. These were provided by Lemaire et al. (2015) and are
available at the Centre de Données astronomiques de Strasbourg
(CDS)111https://cdsarc.unistra.fr/viz-bin/cat/J/A+A/581/A26;
see Fig. 5 and Table A.1 of Lemaire et al. (2015) for more details.. These
data, comprising 43 profiles, do not provide information about possible
variations of the chromospheric Ly$\alpha$ profile between different regions
on the solar surface observed at the same time. Nevertheless, since they cover
an entire solar cycle, it is reasonable to expect that a good indication of
the effect of the solar activity level on the shape of the chromospheric
Ly$\alpha$ line can be achieved, at least in the coronal equatorial regions.

Table 1: Four couples of selected observed profiles representing the extreme cases of the considered profile parameters. Line width | Reversal depth
---|---
Narrowest | Broadest | Shallowest | Deepest
(AR 2340) | April 17, 2009 (CH) | (AR 2340)a | (QS, Fig. 6a in Fontenla et al., 1988)
0.58Å | 0.87Å | – | 69%
Asymmetry of the peaks | Separation of the peaks
$I\tiny{peak-blue}/I\tiny{peak-red}=1.16$ | $I\tiny{peak-blue}/I\tiny{peak-red}=0.86$ | Smallest separation | Largest separation
|  | (Equatorial CH) | April 17, 2009 (CH)
October 28, 1996 (full disc) | (see text) | 0.38Å | 0.57Å
|  $a$) No reversal
---
|  |

Ly$\alpha$ profiles observed by SUMER in different areas of the solar disc
(Curdt et al., 2008; Tian et al., 2009a, b) represent the best Ly$\alpha$
solar-disc observations currently available. We considered two profiles
observed on September 23, 2008 and on April 17, 2009, which are relevant to a
QS region at the disc centre and to a CH observed at the south pole,
respectively (for more details, see Tian et al., 2009a, b).

The UVSP profiles considered in this work were reported by Fontenla et al.
(1988) in their Figs. 3, 5, 6, and 12, where the geocoronal absorption was
removed, while the LPSP instrument on OSO-8 profiles were acquired in an
equatorial CH at the central meridian between 27 and 29 November, 1975
(Bocchialini & Vial, 1996) and in a faint flare that occurred in an AR
observed on April 15, 1978 (Lemaire et al., 1984). Analytic fits to
chromospheric Ly$\alpha$ profiles were also taken into account. In order to
examine the variation of each parameter of the Ly$\alpha$ line profile (line
width, reversal depth, asymmetry of the peaks, and separation of the peaks),
we selected four couples of profiles, each of them representing extreme cases
of the considered profile parameters. In Table 1, we report these four
couples, determined as described in the following. However, we ensured that
all the extreme cases analysed were represented by using only the five
observed profiles shown in panel (a) of Fig. 2, where each of them has been
normalised to its own total intensity.

We measured the full width at half maximum (FWHM) of each selected Ly$\alpha$
chromospheric profile once these were normalised to their total intensities.
The mean value of the two peaks’ highest intensities was adopted as the
maximum intensity of the Ly$\alpha$ line profile. Its half value was then used
to calculate the FWHM of the line. Only in the case of the considered profile
without reversal, meaning that relative to AR 2340 and reported in Fig. 12 of
Fontenla et al. (1988), did we obtain the FWHM by performing a Gaussian fit.

We find a difference between the line widths of about 50% at most over the
entire cycle, with respect to the narrowest observed Ly$\alpha$ chromospheric
profile. It is worth noting that we did not find evidence of a correlation of
the variation of the line widths with the solar cycle. Indeed, the two line
profiles that exhibit the largest difference between their FWHMs correspond to
that observed on April 17, 2009 (CH) and that reported in Fig. 12 of Fontenla
et al. (1988) (AR 2340), respectively. These profiles are shown in panel (a)
of Fig. 2 and in Fig. 3 (red dashed line and blue solid line, respectively).
They are considered extreme cases with regard to the line profile width.

The shallowest and the deepest profiles in terms of reversal depth are those
described in Fig. 12 (AR 2340, showing a negligible reversal) and Fig. 6a (QS)
of Fontenla et al. (1988), respectively. The depth was calculated by
considering the minimum central intensity value of the profile and dividing
this by the mean value between the maximum peak intensities. The reversal
depth variation of the deepest profile is equal to about 69% with respect to
the shallowest one. These profiles, representing the extreme cases related to
the profile reversal depth, are displayed in panel (a) of Fig. 2 (blue solid
line and magenta dash-dot-dot-dot line, respectively).

The peak-to-peak asymmetry between the peaks of each profile was measured by
considering the ratio $I\tiny{peak-blue}/I\tiny{peak-red}$. The profile with
the largest difference between the maximum peak intensities is that observed
on October 28, 1996 (full disc), which exhibits the highest blue asymmetry
($I\tiny{peak-blue}/I\tiny{peak-red}=1.16$). This profile is displayed in
panel (a) of Fig. 2 and in panel (c) of Fig. 4 (green dash-dotted line).

Moreover, to study the effects induced by the asymmetry of the peaks on the H
I outflow velocity, we also took into account the synthetic and reddest
asymmetric profile ($I\tiny{peak-blue}/I\tiny{peak-red}=0.86$), which was
obtained by taking the opposite of the bluest asymmetric profile
(characterised by the ratio $I\tiny{peak-blue}/I\tiny{peak-red}=1.16$), which
is the one observed on October 28, 1996 (full disc). This profile is shown in
panel (c) of Fig. 4 (red solid line).

The line profiles with the largest and smallest peak separations are those
observed on April 17, 2009 (polar CH) and in the equatorial CH at the central
meridian on 27-29 November, 1975, respectively. The separation variation is
equal to about 50% with respect to the smallest separation value. The profiles
are shown in panel (a) of Fig. 5 (red dotted line and green solid line,
respectively).

We also compared two different profiles relative to ARs. The first one,
reported in Fig. 5a of Fontenla et al. (1988) (AR 2363), shows a remarkable
reversal, while the second one, shown in Fig. 12 of Fontenla et al. (1988) (AR
2340), is characterised by a negligible reversal. In addition, we analysed the
characteristic Ly$\alpha$ profile of a faint flare observed in an AR, which is
reported in Fig. 1 of Lemaire et al. (1984), comparing it to the profile of AR
2340.

Furthermore, we considered the analytic expression for the Ly$\alpha$
irradiance profile proposed by Auchère (2005) and also used by Dolei et al.
(2018, 2019), together with two cases of the parametrised profile reported in
Kowalska-Leszczynska et al. (2018) (hereafter referred to as IKL). The first
one consists of a sum of three Gaussian components, which was introduced to
reproduce the mean Ly$\alpha$ observed profile at solar minimum. The other two
profiles were constructed by taking into account the sum of a k-function, a
straight line that simulates the background, and a reversal Gaussian function
reproducing the line reversal. Such profiles were calculated for both the
maximum and minimum irradiance values reported in Table A.1 of Lemaire et al.
(2015). These profiles are presented in panel (b) of Fig. 2 (Auchère: blue
solid line; IKL at minimum: green dashed line; IKL at maximum: red dash-dotted
line).

In order to calibrate the radiance of the spectral data from SUMER, we
considered the absolute values of line radiance
$I\,(\,\mathrm{erg\,cm^{-2}\,s^{-1}\,sr^{-1}}$) provided by the SOLar STellar
Irradiance Comparison Experiment (SOLSTICE; Rottman & Woods, 1994) aboard the
Upper Atmosphere Research Satellite (UARS; Reber, 1990), as explained, for
example, in Lemaire et al. (1998, 2002, 2015). The solid angle subtended by
the solar disc at the distance of SOLSTICE ($\sim 1$ AU) was taken into
account. The considered profiles were all normalised to the same value of
absolute line radiance observed at the minimum, such as that acquired on May
22, 1997, which corresponds to the SUMER observation closest in time to the
coronal Ly$\alpha$ map reconstructed from UVCS observations reported in this
paper. This value is $I=I_{min}=8.59\times
10^{4}\,\mathrm{erg\,cm^{-2}\,s^{-1}\,sr^{-1}}$ (see Table 1 of Lemaire et
al., 2015).

###  2.3 Parameters for the synthesis of the Ly$\alpha$ scattered intensity

In order to estimate $v_{w}$, we synthesised the intensity of the coronal
Ly$\alpha$ to be compared with UVCS observations using the code described in
Dolei et al. (2018, 2019). For the computation of the synthetic coronal
emission, we need information about the physical quantities on which the
Ly$\alpha$ intensity depends, such as $n_{e}$, $T_{e},$ and
$T_{{H\textsc{I,}}}$ under some assumptions and approximations.

The electron density $n_{e}$ is obtained through the inversion method
developed by Van De Hulst (1950) that describes the polarised brightness (due
to the Thomson scattering) as dependent only on the electron density, under
the hypothesis of cylindrical symmetry (see, e.g., Hayes et al., 2001; Dolei
et al., 2015). We used the profile of $n_{e}$ derived from the LASCO pB map,
as described in Dolei et al. (2018).

For the electron temperature $T_{e},$ we used the values derived by Dolei et
al. (2019). They obtained $T_{e}$ as a function of the heliocentric distance
and latitude, taking into account the dependence found by Gibson et al. (1999)
for equatorial regions and by Vasquez et al. (2003) for high latitude regions,
under the hypothesis of super-radial expansion of the solar wind in polar CHs
during the solar minimum. Interpolated values for mid latitudes have been
used.

The H I temperature comes from the database created by Dolei et al. (2016). We
considered a $T_{{H\textsc{I}}}$ map for a solar minimum epoch, following
Dolei et al. (2019). In order to fit the neutral hydrogen temperature profile
and to create a 2D map as a function of latitude and heliocentric distance,
the functional form given by Vasquez et al. (2003) was used, as in Dolei et
al. (2016). It is worth mentioning that the values of $T_{{H\textsc{I}}}$ are
likely overestimated, because the hydrogen line broadening includes
contributions due to non-thermal mechanisms.

For the sake of simplicity, we adopted the approximation $\theta=0^{\circ}$,
that is, ${\bf{v}}\parallel{\mathbf{n}^{\prime}}$ in all the computations
leading to the results described in Sect. 3. However, the consequence of
assuming values of $\theta\neq 0$ on the inferred values of the H I outflow
velocities has been evaluated as a separate step and is also described in
Sect. 3.

##  3 Results

In order to verify the effects of the chromospheric Ly$\alpha$ profile shape
on the determination of the solar wind H I velocity, we first studied the
behaviour of the Doppler factor dependent on the profile width. The results
are summarised in Fig. 3. The first three panels show the overlapping between
the narrowest and broadest chromospheric profiles and the coronal one that has
been calculated by assuming a H I temperature in the corona equal to
$1.5\times 10^{6}$ K. We took into account the following velocity values:
$v_{w}=0\,\mathrm{km\,s}^{-1}$ (panel (a)), $v_{w}=150\,\mathrm{km\,s}^{-1}$
(panel (b)), and $v_{w}=300\,\mathrm{km\,s}^{-1}$ (panel (c)). In addition,
for the case of the narrowest profile, we also display the grey shaded area
that is proportional to the integrand value of the function
$F({\bf{n^{\prime}}},v_{w},\theta)$ (see Eq. 2). This quantity is closely
related to the Doppler factor as it gives the overlapping between the coronal
absorption profile ($\Phi$) and the chromospheric pumping profile ($\Psi$).

In Fig. 3, panel (d), the corresponding values of $D(v_{w})$ are properly
labelled on the curves, where the Doppler factor as a function of $v_{w}$
(varying from $v_{w}=0\,\mathrm{km\,s}^{-1}$ to
$v_{w}=500\,\mathrm{km\,s}^{-1}$) is reported. We note that the curves in the
bottom panel show a small difference between them, indicating that the Doppler
dimming does not have a strong dependence on the variation of the line width
of the selected chromospheric profiles.

We studied the Doppler factor $D(v_{w})$ as a function of the profile reversal
depth. The results are reported in Fig. 4. Panel (a) of Fig. 4 shows the
overlapping of the coronal absorption profile with the shallowest and the
deepest chromospheric profiles. The same analysis was carried out for the
bluest and reddest asymmetric profiles (panel (c) of Fig. 4) and the profiles
with the smallest and the largest separation between the peaks (panel (a) of
Fig. 5). The curves in panel (b) and (d) of Fig. 4 and panel (b) of Fig. 5 do
not show any significant differences between them. Therefore, the Doppler
factor does not have a strong dependence on reversal depth, asymmetry, and
separation of the peaks with regard to all the selected chromospheric profiles
taken into account as extreme cases.

We performed a similar analysis by considering the three profiles reported in
Fig. 2 (panel (b)). From Fig. 6 (panels (a) and (b)), we infer that the
different parametrisations of these chromospheric profiles do not generate
significant differences in the Doppler factor values, even if they are larger
than in the observed profiles.

A further analysis has been carried out for AR profiles. As shown in Fig. 7,
when we compare the values of $D(v_{w})$ for the AR 2363 (AR showing line
reversal) and AR 2340 (AR with a negligible reversal), we find very small
differences. Conversely, comparing the values relevant to the AR 2340 and the
flare profiles, we observe significant differences.

Furthermore, we performed a comparison considering different values of the
angle between the flow and the line-of-sight directions, $\theta$, using the
Auchère profile. The results are shown in panel (a) of Fig. 8, where we can
note that assuming different angles $\theta$ leads to the inference of
remarkable variations of the H I outflow velocity values of about 14%.

To better understand these results, in Fig. 9 we plot the absolute value of
the differences between the values of solar wind H I velocity corresponding to
the same value of Doppler factor relative to the different chromospheric
observed and parametrised profiles. It is possible to note that in the former
case (observed profiles) the velocity differences are within a range of about
$22\,\mathrm{km\,s}^{-1}$ when we take into account the extreme cases. In the
latter (parametrised profiles), the differences are within
$30\,\mathrm{km\,s}^{-1}$, except for values of $D(v_{w})$ close to zero,
where the indetermination in the outflow velocity is much higher. However, in
both cases, up to values of $D(v_{w})\approx 0.2$, the relative velocity
differences are below about 9% and 12%, respectively. A similar result was
obtained by Dolei et al. (2015), who found that an uncertainty within the
range of H I temperature measured by UVCS can return an error up to 10% on the
determination of the theoretical coronal Ly$\alpha$ intensity, with an
influence on the estimate of the H I outflow velocity up to $\pm
10-20\,\mathrm{km\,s}^{-1}$. When we consider the cases reported in Fig. 7, AR
profiles with and without reversal and a profile observed during a flare, we
can see that for the left panels of Fig. 7 the velocity differences are below
$10\,\mathrm{km\,s}^{-1}$, while for the right panels of Fig. 7 these values
reach about $100\,\mathrm{km\,s}^{-1}$ (see also Fig. 9, panel (c)).
Therefore, we find relative values of about 3% and 21%, respectively. In Fig.
8, panel (b), concerning the Doppler factor curves relative to different
values of $\theta$, a similar result is shown. In this case, the absolute
values of the differences between the velocities corresponding to the same
value of Doppler factor are below $70\,\mathrm{km\,s}^{-1}$, with relative
values up to about 14%.

Figure 3:  Panels (a), (b), and (c): Normalised chromospheric profiles
reported in Fig. 12 of Fontenla et al. (1988) (AR 2340, blue solid line) and
observed on April 17, 2009 (red dashed line; Tian et al., 2009b), overlapped
with a normalised Gaussian coronal absorption profile (black dotted line)
computed by setting the coronal H I temperature to $1.5\times 10^{6}$ K,
considering $\theta=0^{\circ}$. Panels (a), (b), and (c) correspond to outflow
velocities equal to $v_{w}=0\,\mathrm{km\,s}^{-1}$, $v_{w}$
=$\leavevmode\nobreak\ 150\,\mathrm{km\,s}^{-1}$ , and
$v_{w}=300\,\mathrm{km\,s}^{-1}$, respectively. In the case of the exciting
narrowest profile, we also display the grey shaded area that is proportional
to the product of the coronal absorption profile ($\Phi$) and the
chromospheric pumping profile ($\Psi$). Panel (d): Doppler factor as a
function of $v_{w}$ calculated considering each chromospheric profile (AR
2340: blue solid line; polar CH: red dashed line). Triangles, squares and
asterisks indicate the Doppler dimming values corresponding to H I outflow
velocities equal to $v_{w}=0\,\mathrm{km\,s}^{-1}$, $v_{w}$
=$\leavevmode\nobreak\ 150\,\mathrm{km\,s}^{-1}$ , and
$v_{w}=300\,\mathrm{km\,s}^{-1}$, respectively.

Similarly to Dolei et al. (2016, 2018, 2019), we created 2D maps of the solar
wind H I outflow velocity assuming the chromospheric Ly$\alpha$ line shape
$\Psi(\lambda)$ as the sole variable parameter, while all the other inputs
were kept fixed. The maps have been created with a final FOV ranging from
$1.5\,R_{\odot}$ to $3.95\,R_{\odot}$.

Figure 4:  Panel (a): Normalised shallowest (blue solid line) and deepest
(magenta dash-dot-dot-dot line) profiles described in Fig. 12 and Fig. 6a of
Fontenla et al. (1988), respectively, overlapped with a normalised Gaussian
coronal profile (black dotted line) computed by setting the coronal H I
temperature at $1.5\times 10^{6}$ K, considering $\theta=0^{\circ}$. Panel
(b): Doppler factor as a function of $v_{w}$ calculated considering each
chromospheric profile. Triangles, squares, and asterisks: See Fig. 3. Panels
(c) and (d): Same as in panels (a) and (b), but concerning the bluest
asymmetric profile ($I\tiny{peak-blue}/I\tiny{peak-red}=1.16$) observed on
October 28, 1996 (green dash-dotted line; Lemaire et al., 2015) and the
reddest asymmetric profile ($I\tiny{peak-blue}/I\tiny{peak-red}=0.86$; red
solid line), respectively.

Figure 5:  Panel (a): Normalised CH profiles with smallest (green solid line)
and greatest (red dashed dotted line) peak separations observed in an
equatorial CH at the central meridian on 27-29 November, 1975 (Bocchialini &
Vial, 1996), and on April 17, 2009 (polar CH; Tian et al., 2009b),
respectively, overlapped with a normalised Gaussian coronal profile (black
dotted line) computed by setting the coronal H I temperature at $1.5\times
10^{6}$ K, considering $\theta=0^{\circ}$. Panel (b): Doppler factor as a
function of $v_{w}$ calculated considering each chromospheric profile.
Triangles, squares, and asterisks: See Fig. 3.

Figure 6:  Panel (a): Normalised parametrised chromospheric profiles reported
in Fig. 2; Auchère (2005) (blue solid line) and Kowalska-Leszczynska et al.
(2018) (green dashed line at minimum and red dot-dashed line at maximum),
overlapped with a normalised Gaussian coronal profile (black dotted line)
computed by setting the coronal H I temperature at $1.5\times 10^{6}$ K,
considering $\theta=0^{\circ}$. Panel (b): Doppler factor relative to the
profiles shown in panel (a). Triangles, squares, and asterisks: See Fig. 3.

Figure 7:  Panel (a): Normalised AR profiles with and without reversal
relevant to AR 2363 (magenta dash-dot-dot-dot line) and AR 2340 (blue solid
line) (see Fontenla et al., 1988), respectively, overlapped with a normalised
Gaussian coronal profile (black dotted line) computed by setting the coronal H
I temperature at $1.5\times 10^{6}$ K, considering $\theta=0^{\circ}$. Panel
(b): Doppler factor as a function of $v_{w}$ calculated considering each
chromospheric profile. Triangles, squares, and asterisks: See Fig. 3. Panels
(c) and (d): Same as in panels (a) and (b), but concerning the AR 2340 profile
(blue solid line; Fontenla et al., 1988) and the normalised flare profile
(green dash-dotted line; Lemaire et al., 1984), respectively.

Figure 8:  Panel (a): Doppler factor as a function of $v_{w}$ calculated
considering the normalised Auchère (2005) profile when $\theta=0^{\circ}$ and
$\theta=30^{\circ}$. Triangles, squares, and asterisks: See Fig. 3. Panel (b):
Absolute values of the outflow velocity differences determined considering the
normalised Auchère (2005) profile when $\theta=0^{\circ}$ and
$\theta=30^{\circ}$, as a function of the Doppler factor. The top axis scale
shows the velocity values corresponding to the Doppler factor values when the
Auchère (2005) profile is taken into account and $\theta=0^{\circ}$.

Figure 9:  Panel (a): Absolute values of the differences between the outflow
velocity values determined with the different observed chromospheric profiles
shown in Fig. 2 as a function of the Doppler factor. The abscissa axis is
complemented with the velocity values corresponding to the Doppler factor in
the case of the Auchère (2005) profile at the top of the panel. Panel (b):
Same as in panel (a), but with the parametrised chromospheric profiles shown
in Fig. 2 and in panel (a) of Fig. 6. Panel (c): Same as in panel (a), but
taking into account the profiles shown in panels (a) and (c) of Fig. 7.

Figure 10:  Differences between solar wind H I outflow velocity maps obtained
considering the narrowest profile (AR 2340; Fontenla et al., 1988) and the
broadest one (polar CH - April 17, 2009; Tian et al., 2009b) (top left panel),
the narrowest and the Auchère (2005) analytic profiles (top right panel), the
broadest and the Auchère (2005) analytic profiles (bottom left panel), and the
narrowest profile with that observed on September 23, 2008 (QS; Tian et al.,
2009b) (bottom right panel).

For a better visualisation of the variations induced by the use of different
chromospheric line profile shapes, in Fig. 10 we report the results obtained
in terms of the differences between the outflow velocity maps inferred from
the various chromospheric profiles adopted: narrowest and broadest profiles
(AR 2340 and SUMER April 17, 2009 CH, respectively; top left panel), narrowest
and Auchère (2005) profiles (top right panel), broadest and Auchère (2005)
profiles (bottom left panel), and narrowest and QS (SUMER, September 23, 2008)
profiles (bottom right panel). These difference maps exhibit RMS values equal
to $18\,\mathrm{km\,s}^{-1}$, $19\,\mathrm{km\,s}^{-1}$,
$20\,\mathrm{km\,s}^{-1}$, and $11\,\mathrm{km\,s}^{-1}$, respectively. As can
be seen in Fig. 10, the differences are remarkable in the left panels. In
particular, in the bottom left panel the differences are larger between about
$1.5\,R_{\odot}$ and $3.5\,R_{\odot,}$ only in the equatorial region.

##  4 Discussion and conclusions

In this paper, we investigate the influence of the chromospheric Ly$\alpha$
line profile on the determination of the solar wind H I outflow velocity
through the Doppler dimming technique (Hyder & Lites, 1970; Noci et al., 1987;
Withbroe et al., 1982).

In this analysis, we used spectroscopic measurements by SUMER (Lemaire et al.,
2015; Curdt et al., 2008; Tian et al., 2009a, b), which include the data sets
with the highest resolution of Ly$\alpha$ solar-disc observations presently
available. We also selected some profiles of ARs and QS acquired by UVSP and
described in Fontenla et al. (1988), where the correction concerning the
geocoronal absorption was applied, along with a profile reported in Lemaire et
al. (1984) characterising a solar flare, and another one acquired in an
equatorial CH (Bocchialini & Vial, 1996); both were obtained by the LPSP
instrument on OSO-8. For the sake of completeness, it is worth recalling that
the Chromospheric Lyman Alpha Spectropolarimeter (CLASP) observed the Sun in H
I Ly$\alpha$ during a suborbital rocket flight in 2015 (Schmit et al., 2017).
These observations measured the Ly$\alpha$ profile in a quiet-Sun target near
the limb with a slit 400″ long and 1$\aas@@fstack{\prime\prime}$44 wide.
Overall, these observed profiles were in agreement with the early measurements
of Gouttebroze et al. (1978), but they were affected by geocoronal absorption
and water vapor contamination. Furthermore, SUMER measurements were recently
used to derive a reference QS Ly$\alpha$ profile that would be representative
of the Ly$\alpha$ radiation from the solar disc during a minimum solar
activity (Gunár et al., 2020).

We analysed the behaviour of the Doppler factor with respect to the variation
of four parameters on which the pumping Ly$\alpha$ profile depends: line
width, reversal depth, asymmetry of the peaks, and separation of the peaks.
Moreover, taking into account the same input data adopted by Dolei et al.
(2019), that is, $T_{e}$, $T_{{H\textsc{I}}}$, $n_{e}$, and the Ly$\alpha$
coronal intensity, and using a constant value for the radiance of the exciting
chromospheric Ly$\alpha$ radiation, we computed outflow velocity maps, where
the sole variable is the profile of the exciting chromospheric Ly$\alpha$
radiation.

It is worth mentioning that just recently Cranmer (2020) obtained radial
profiles of electron and proton temperatures very different to those adopted
in the present work. Since our main goal here is to investigate the effects of
the chromospheric Ly$\alpha$ profile shape on the determination of the outflow
velocity, we feel confident that the evaluation of Cranmer’s temperature
profiles can be deferred to a future analysis without questioning the results
obtained in the present work.

Considering chromospheric Ly$\alpha$ normalised profiles observed by SUMER,
UVSP, and the LPSP instrument on OSO-8, we measured their FWHM, reversal
depth, asymmetry of the peaks, and separation of the peaks. We found a maximum
variation of about 50%, 69%, 35%, and 50% between the maximum and minimum
values of each parameter on which the Ly$\alpha$ profile depends,
respectively, referring each percentage to the minimum value of the
corresponding parameter. Using these observations, we found that the Doppler
factor $D(v_{w})$, which is a measure of the overlapping between the coronal
and chromospheric profiles as a function of the H I outflow velocity, does not
strongly depend on the observed parameters that characterise the chromospheric
Ly$\alpha$ profiles. In fact, except the case in which we consider a flare,
the relative velocity differences determined using the different observed
profiles, as a function of the Doppler factor, are below about 9%, a value
that is comparable with the uncertainty in the velocity values determined by
the Doppler dimming technique.

In order to further illustrate this effect, we also considered other analytic
profiles, which were proposed by Auchère (2005) and Kowalska-Leszczynska et
al. (2018). Also in these cases, we note negligible effects on the Doppler
factor evaluation, obtaining relative absolute values of the velocity
differences within a range of about 12% in correspondence of the same values
of the Doppler factor. Conversely, non-negligible differences appear in the
Doppler factor when we take into account a flare profile. In fact, in this
case the resulting velocity variation is significantly larger, increasing up
to about $100\,\mathrm{km\,s}^{-1}$ and returning a relative velocity
difference values of about 21% (see Fig. 9, panel (c)).

We also show the effect of considering different values of the angle between
the flow and the line-of-sight directions, $\theta$. We find that if
$\theta>0^{\circ}$, the Doppler factor curve has a smoother trend than for
$\theta=0^{\circ}$. Figure 8 shows the case of $\theta=30^{\circ}$. Therefore,
when we consider a fixed value of the Doppler factor, the corresponding
velocity value is greater when $\theta>0^{\circ}$. However, the velocity
differences remain below about $70\,\mathrm{km\,s}^{-1}$ in this case, with a
relative value equal to $\approx 14\%$. It is worth mentioning that such a
value for $\theta$ can be obtained only for very close distances from the Sun,
of about $2R_{\odot}$, so its contribution in the computation of the H I
outflow velocity is limited; this remains more valid for higher values of
$\theta$.

We used the narrowest and the broadest profiles, the one observed on September
23, 2008 (QS), and the analytic profile deduced by Auchère (2005), to derive
the 2D maps of H I outflow velocity. Taking into account the velocity
difference maps shown in Fig. 10, we found that RMS values of the difference
of the solar wind H I speed are equal to $18\,\mathrm{km\,s}^{-1}$ (narrowest
- broadest), $19\,\mathrm{km\,s}^{-1}$ (narrowest - Auchère),
$20\,\mathrm{km\,s}^{-1}$ (broadest - Auchère), and $11\,\mathrm{km\,s}^{-1}$
(narrowest - QS). These values can be considered as possible uncertainties in
the estimate of the outflow velocity with regard to the dependence on the
chromospheric profile shape. However, they are significantly smaller than
those found by Dolei et al. (2018), which are related to other parameters. In
fact, assuming a maximum uncertainty on the other physical quantities of $\pm
30$%, they estimated that the resulting uncertainties on the derived velocity
were of $67\,\mathrm{km\,s}^{-1}$, $67\,\mathrm{km\,s}^{-1}$,
$81\,\mathrm{km\,s}^{-1}$, and $45\,\mathrm{km\,s}^{-1}$ for the impact of
electron density, total chromospheric intensity, electron temperature, and H I
temperature, respectively.

Therefore, our results indicate that even the largest variations actually
observed in the parameters on which the chromospheric Ly$\alpha$ profile
depends, related to the solar magnetic activity and to different disc regions,
return small differences in the H I outflow velocity estimate, excluding the
flare case, where non-negligible effects are present. This reveals how little
effect the shape of the exciting chromospheric Ly$\alpha$ profile has on the
determination of the solar wind H I outflow velocity with respect to the other
parameters characterising the scattered coronal line. As a consequence, a
unique shape of the Ly$\alpha$ chromospheric profile can be adopted all over
the solar disc; moreover, analytical chromospheric profiles can be used, such
as those proposed by Auchère (2005) and Kowalska-Leszczynska et al. (2018),
without significantly affecting the solar wind H I velocity computation beyond
the uncertainties characterising the Doppler dimming technique.

Data coming from the Metis coronagraph (Antonucci et al., 2020; Fineschi et
al., 2020), aboard the Solar Orbiter spacecraft (Müller et al., 2020), will
return more detailed and accurate information able to update this and previous
studies, thanks to simultaneous UV and polarised VL observations with high
spatial and temporal resolution.

###### Acknowledgements.

The authors acknowledge the support of the Italian Space Agency (ASI) to this
work through contracts ASI/INAF No. I/013/12/0 and No. 2018-30-HH.0. The
authors thanks the anonymous referee for useful comments and suggestions,
which led to a sounder version of the manuscript.

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