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**Annales Geophysicae**
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- Abstract
- Introduction
- The magnetosheath plasma and a simplified theoretical approach to sheath ENAs
- Shrinking of the escape cone
- The Lyman-alpha glow contribution by H-ENAs and a look into self-consistency in H-ENA production
- The Lyman-alpha glow emission of terrestrial H-ENAs
- Conclusions
- Data availability
- Appendix A
- Competing interests
- Acknowledgements
- References

**Regular paper**
21 Mar 2018

**Regular paper** | 21 Mar 2018

Neutralized solar wind ahead of the Earth's magnetopause as contribution to non-thermal exospheric hydrogen

- Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

- Argelander-Institut für Astronomie, Universität Bonn, Auf dem Hügel 71, 53121 Bonn, Germany

**Correspondence**: Hans J. Fahr (hfahr@astro.uni-bonn.de)

**Correspondence**: Hans J. Fahr (hfahr@astro.uni-bonn.de)

Abstract

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In a most recent paper by Qin and Waldrop (2016), it had been found that the scale height of hydrogen in the upper exosphere of the Earth, especially during solar minimum conditions, appears to be surprisingly large. This indicates that during minimum conditions when exobasic temperatures should be small, large exospheric H-scale heights predominate. They thus seem to indicate the presence of a non-thermal hydrogen component in the upper exosphere. In the following parts of the paper we shall investigate what fraction of such expected hot hydrogen atoms could have their origin from protons of the shocked solar wind ahead of the magnetopause converted into energetic neutral atoms (ENAs) via charge-exchange processes with normal atmospheric, i.e., exospheric hydrogen atoms that in the first step evaporate from the exobase into the magnetosheath plasma region. We shall show that, dependent on the sunward location of the magnetopause, the density of these types of non-thermal hydrogen atoms (H-ENAs) becomes progressively comparable with the density of exobasic hydrogen with increasing altitude. At low exobasic heights, however, their contribution is negligible. At the end of this paper, we finally study the question of whether the H-ENA population could even be understood as a self-consistency phenomenon of the H-ENA population, especially during solar activity minimum conditions, i.e., H-ENAs leaving the exosphere being replaced by H-ENAs injected into the exosphere.

**Keywords. **Magnetospheric physics (plasmasphere; solar wind-magnetosphere interactions) – solar physics, astrophysics, and astronomy (energetic particles)

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How to cite.

Fahr, H. J., Nass, U., Dutta-Roy, R., and Zoennchen, J. H.: Neutralized solar wind ahead of the Earth's magnetopause as contribution to non-thermal exospheric hydrogen, Ann. Geophys., 36, 445–457, https://doi.org/10.5194/angeo-36-445-2018, 2018.

1 Introduction

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The density and temperature structure of the Earth's exosphere is a
long-standing research subject, which in the past, due to missing
observations, was mainly accessible by theoretical studies which considered
the particular motions of collision-less or collision-poor atoms in the
region above the exobase. In these nearly collision-free exospheric regions,
ballistic particles, satellite particles and escaping particles have to be
treated differently, and their different height-dependent contributions to
the total local exospheric density is a complicated matter of theoretical
calculations (see early work by Opik and Singer, 1961; Chamberlain, 1963;
Brinkmann, 1970, or later work by Chamberlain, 1978, 1979; Fahr, 1970, 1976;
Fahr and Nass, 1978; Fahr and Weidner, 1977, or Fahr and Shizgal, 1983). The
general assumption in all of these studies is that from the
collision-dominated region below the exobase, atoms are emitted upwards into
the exosphere by Maxwellian velocity distribution functions associated with
the local gas temperature at the exobase *T*=*T*_{ex}.

This, however, is a critical and unsafe assumption, because in the transition region between collision-dominated and collision-free domains, thermodynamic equilibrium conditions are perturbed, i.e., temperatures of the different exobasic gas constituents due to the weak thermodynamic coupling deviate from each other. Especially for hydrogen, due to the strongly pronounced escape flux of these light atoms (see Jeans, 1923, or e.g., later work by Fahr and Weidner, 1977), the downward branch of the velocity distribution at the branch of super escape velocities is unpopulated if external sources of H atoms are neglected. This, in fact, has the effect to reduce the effective exobasic hydrogen temperature with respect to the exobasic oxygen temperature and thus reduces the effective H-atom escape flux (see Fahr, 1976).

Nevertheless, these reasons for H-atom deviations from local exobasic thermodynamic equilibrium conditions do not explain what surprisingly and most recently has been observed by Qin and Waldrop (2016). These authors find that the upper exospheric hydrogen density distribution, especially during solar minimum conditions, is substantially higher than can be understood as a result of a terrestrial H–exosphere connected with exobasic oxygen temperatures (see Hodges, 1994). Their exciting result is inferred from radiative transfer modeling of Lyman-alpha resonance glow measurements made with the satellite TIMED/GUVI. To best fit these results with their Lyman-alpha multiple scattering code, the authors apply two independent Maxwellian distributions for two separate exobasic hydrogen populations, one with the usual exobasic temperature according to the NRLMSISE-00 model (Picone et al., 2002) and the other with a much higher temperature with no clear origin, which thus needs to be fitted from Lyman-alpha glow data. It then, as a surprise, turns out that the presence of the hotter H component is especially pronounced under solar minimum conditions, just at times when normal exobasic temperatures should be the lowest.

In the following paper, we are going to show that this interesting finding can at least at larger exospheric heights partly be ascribed to energetic hydrogen atoms (H-ENAs) impinging onto the lower exosphere and originating from beyond the Earth's magnetopause via charge-exchange processes on the solar side of the subsonic solar magnetosheath plasma region. That the Earth's magnetopause is in fact an actively emitting H-ENA site has already directly been proven by IBEX-H-ENA measurements under suitable observational conditions from regions far above the exobase (Goldstein and McComas, 2013; also see our Fig. 1 from Fuselier et al., 2010). In the following part of the paper we shall develop a simplified description of this hot exospheric H component in order to elaborate a qualitative picture of the relevance of this H-ENA component for the upper exosphere.

2 The magnetosheath plasma and a simplified theoretical approach to sheath ENAs

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Outside of the magnetopause and inside of the Earth's magneto bow shock, the shocked solar wind, the so-called “magnetosheath plasma”, is deflected from its original antisolar direction and its flow is forced to become tangential to the outer magnetopause surface. The resulting global plasma flow, in reality, is rather complicated in its geometric structure, its asymmetries and its time dependencies. With all of these complications concerning time-dependent and space-dependent specialities in densities, bulk velocities and temperatures, this global plasma system does not allow for an easy theoretical representation.

First predictions for this plasma flow were made on the basis of gasdynamic models by Spreiter and Stahara (1980). In these models, it is assumed that the bulk properties of the shocked solar wind flow can be described by a monofluid hydrodynamic approach with frozen-in magnetic fields only included by a kinematic approximation. To better understand the influence of the self-consistent magnetic fields involved, Zwan and Wolf (1976) proposed a plasma depletion model including the effect of magnetic fields on the magnetosheath flow, which, however, was later criticized by Southwood and Kivelson (1995) since it did not allow for the needed forces to drive the sheath flow. Furthermore, the effect of magnetic field reconnection in the magnetosheath was considered by Cowley and Owen (1989) and Cooling et al. (2001). Then Song et al. (1992a, b) and Fuselier et al. (2002) have compared model predictions with data from the Active Magnetospheric Particle Tracer Explorers/Ion Release Modul (AMPTE/IRM) and found that while gasdynamic models predict too high bulk velocities, magneto-hydrodynamic (MHD) models including reconnection propose just the opposite, in both cases strongly deviating from measurements. In a more recent study by Cooling (2003), based on data from the satellites GEOTAIL and WIND, it was found that remarkable asymmetries in the dawn and dusk values of plasma density and velocity are pronounced, which in addition are subject to time-dependent variations. In a recent paper by Parks et al. (2016), even more complications in the features of the plasma flow downstream of the Earth's magneto bow shock were pointed out based on data from the CLUSTER II space mission. It is shown there that dependent on the upstream solar wind data, the downstream bulk velocities sometimes remain super-Alfvénic and ion distribution functions, though most often nearly isotropic, can sometimes be very complex showing multiple beams of reflected, gyrating or shock-escaping ions. These very complicating facts are even more stressed by Longmore et al. (2005) in view of more recent CLUSTER in situ data. These latter data show that the magnetosheath plasma does not at all have an axisymmetrical shape concerning flow and density structures, in contrast significant asymmetries exist between dawn and dusk sectors and between northern and southern hemispheres. The magnetosheath flow close to the magnetopause at highest and lowest latitudes are found to be sub-Alfvénic, while at mid-latitudes and at the flanks of the magnetopause they tend to be super-Alfvénic. In addition, everything seems to be strongly subject to time-variations and does not have a clear trigger in the solar wind magnetic field direction.

This all taken together shows clearly that there does not exist an easy way towards an appropriate 3-D representation of the global structure of the magnetosheath plasma, and hence for the ongoing considerations in this paper we therefore have to make essential simplifications guided by modern MHD simulations of multi-component interactions in case of the heliospheric plasma interface (see, for example, the review by Izmodenov and Baranov, 2006).

As at least evident, neutral exospheric hydrogen atoms originating at and
ascending from the exobase, as shown in the analysis presented by Zoennchen
et al. (2013), propagate to large geocentric distances. In fact, their
density extends to beyond the magnetopause region (i.e., to beyond 8 up to 10
Earth radii, *r*_{E}, depending on solar wind conditions, e.g., see
Shue et al., 1998). In this outer region of shocked, subsonic solar wind
plasma, these atoms undergo resonant charge-exchange reactions with the
ambient protons of the shocked solar wind and, as newly appearing energetic
neutral H atoms (H-ENAs), they thereafter partly impinge on the Earth's
exobase if their inherited velocity direction is within the appropriate space
angle $\mathrm{d}\mathrm{\Omega}=\mathit{\pi}\left({r}_{\mathrm{ex}}^{\mathrm{2}}/{r}_{\mathrm{mp}}^{\mathrm{2}}\right)$. Assuming for the solar side of the bow shock a strong perpendicular MHD
shock (compression
ratio *s*≃4), then according to Rankine–Hugoniot standards the
shocked solar wind plasma should be highly subsonic there (*M*_{s}≤0.1).
This implies that the energetic magnetosheath protons (keV) in this subsolar region
are practically represented by an isotropic suprathermal Maxwellian
distribution in velocity space. Hence via resonant H–H^{+} charge exchange,
these protons colliding with low-energy exobasic H atoms effectively produce
isotropically distributed high-energy (keV) neutral atoms (H-ENAs), which
are decoupled from the plasma flow and from magnetosheath magnetic fields and
thus are partly and directly shot downwards onto the Earth's exobase.

When impinging onto the exobase, these H-ENAs are elastically colliding with
the most abundant atomic species of this region, which is the mono-atomic
oxygen atom, i.e., 16-times more massive than H atoms. In a velocity space
diagram showing the O- and the H-atom velocities before and after the elastic
collision (see Fig. 2), one can see that the center of mass velocity
of this system is close to the O-atom velocity and the H-atom velocity after
the collision is found by turning the relative velocity vector by an angle
*χ* around the “center of mass” velocity vector. For hard-sphere
collisions, modeling elastic O–H collisions reasonably well, it is known that
all turning angles *χ* are equally probable. Therefore it can be found
that with a high probability of $\mathrm{\Pi}\left(\mathrm{up}\right)\simeq {\sum}_{\mathrm{1}}^{j}(\mathrm{1}/\mathrm{2}{)}^{i}$ (i.e., for *j*≃4, cut-down because of successive
energy losses, thus with a probability of 93 %), H-ENAs impinging on the
exobase are finally, after consecutive collisions, reflected back into the
upward hemisphere through the upper exosphere back to space (see also the
collision geometry illustrated in Fig. 2).

The whole global 3-D charge-exchange scenario, in reality and in a rigorous treatment, is highly asymmetric both in configuration- and velocity-space (see Fig. 3) and it would be hard to describe it in an adequately detailed 6-D manner.

One can, however, dare to simplify and symmetrize the problem for the sake of
calculating the main features of this physical context of hot hydrogen atoms
(H-ENAs) reflected upwards from the exobase back to space in order to
investigate qualitatively their role for the exosphere. Thus, for
simplification reasons, we approximate things here only considering the H-ENA
irradiation of the solar side of the spherical exobase surface with radius
*r*_{ex}. This part is essentially irradiated by H-ENAs originating in
the subsolar sphere of the magnetosheath, as is indicated in
Fig. 4. As supported by CLUSTER measurements, the magnetosheath
plasma in the subsolar region is characterized by a strongly shocked solar
wind plasma with a subsonic signature approaching the nose of the magnetopause
with decreasing bulk velocity. Here, we consider only the solar exobase
irradiation by those H-ENAs originating in this subsolar magnetosheath region
(Fig. 4), though H-ENA fluxes from magnetosheath regions at high
latitudes or from the flanks do certainly also contribute, but as minor
contributions they are not considered here. This is because in these regions
the plasma bulk velocities are growing again and are oriented away from the
exobase, so that the proton distribution function contains less and less
velocity vectors directed towards the exobase.

For the purpose of the following estimation, the following assumptions might
appear acceptable: the density of the supersonic solar wind upstream of the
Earth's bow shock at about 1 AU is adopted with
*n*_{p1}=5 cm^{−3}. Downstream of the subsolar part of the Earth's
bow shock, assumed to act as a strong perpendicular shock with a compression
ratio of *s*≃4, the shocked solar wind then, according to a monofluidal
gasdynamic Rankine–Hugoniot approach, becomes compressed by a factor of 4 and
the downstream plasma becomes strongly subsonic with Mach numbers of the
order of *M*_{s}≤0.1 (see e.g., Serrin, 1959; Gombosi, 1998; Erkaev
et al., 2000). As a consequence of that low Mach number flow (see Fahr and
Siewert, 2015; Fahr et al., 2015), one can expect, as is also in the case of the
heliosheath plasma, a nearly incompressible magnetosheath plasma density with
${n}_{\mathrm{p}\mathrm{2}}=s\cdot {n}_{\mathrm{p}\mathrm{1}}=\mathrm{20}\phantom{\rule{0.125em}{0ex}}$cm^{−3} (indices “1” and “2”
characterize solar wind quantities upstream and downstream of the bow shock,
respectively). The shocked solar wind has an energy of the order of
$k{T}_{\mathrm{p}\mathrm{2}}\simeq (\mathrm{3}/\mathrm{8})\cdot (m{U}_{\mathrm{1}}^{\mathrm{2}}/\mathrm{2})\simeq \mathrm{1}$ keV (i.e.,
thermalized solar wind; *U*_{1}≃400 km s^{−1}; *T*_{p2}≃10^{6} K) with a velocity distribution function, which due to the very low
Mach number *M*_{s}≃0.1 can be considered as quasi-isotropic
Maxwellian with a nearly negligible bulk velocity shift, in view of the small
bulk velocity *U*_{2} associated with the high thermal proton velocities of
the order of $\sqrt{\mathrm{8}k{T}_{\mathrm{p}\mathrm{2}}/\mathit{\pi}m}$.

On the basis of that, we shall now calculate the flux of H-ENAs produced by
means of the charge-exchange reactions of the shocked solar wind protons with
cold exobasic hydrogen atoms in the subsolar region of the magnetosheath.
These latter exospheric hydrogen density distributions have been found and
published by several authors like Rairden et al. (1986), Hodges (1994),
Ostgaard et al. (2003), Bailey and Gruntman (2011) or Zoennchen et al. (2011,
2013). Here we start with the result derived from TWINS-LAD data by Zoennchen
et al. (2013) for summer solstice conditions yielding at larger radial
distances *r*≥*r*_{0} an asymptotic radial density profile of exobasic
H atoms in the form

$$\begin{array}{}\text{(1)}& {n}_{\mathrm{H}}={n}_{\mathrm{H}\mathrm{0}}\cdot {\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{2.7}},\end{array}$$

with a reference density *n*_{H0}=50 cm^{−3} at a reference height
*r*_{0}=8 *r*_{E}.

Hence along a radial line of sight cutting through the subsolar magnetosheath
(see Fig. 4), the following total production rate
Γ_{ENA}(*v*) of H-ENAs with a velocity *v* hitting the exobase
can be calculated:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\Gamma}}_{\mathrm{ENA}}\left(v\right)\phantom{\rule{0.125em}{0ex}}{\mathrm{d}}^{\mathrm{3}}\phantom{\rule{0.125em}{0ex}}v=\\ \text{(2)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\underset{{r}_{\mathrm{MP}}}{\overset{{r}_{\mathrm{BS}}}{\int}}\stackrel{{v}_{\mathrm{H}}}{\int}{f}_{\mathrm{p}}^{\mathrm{ss}}\left({v}_{\mathrm{p}}\right)\phantom{\rule{0.125em}{0ex}}{f}_{\mathrm{H}}\phantom{\rule{0.125em}{0ex}}\left({v}_{\mathrm{H}}\right)\mathit{\sigma}\left({v}_{\mathrm{rel}}\right){v}_{\mathrm{rel}}{\mathrm{d}}^{\mathrm{3}}{v}_{\mathrm{p}}\phantom{\rule{0.125em}{0ex}}{\mathrm{d}}^{\mathrm{3}}{v}_{\mathrm{H}}\phantom{\rule{0.125em}{0ex}}\mathrm{d}r,\end{array}$$

where ${f}_{\mathrm{p}}^{\mathrm{ss}}$ and *f*_{H} denote the local
distribution functions of solar wind magnetosheath protons in the subsolar
region and the exobasic H atoms, respectively; *σ* is the charge-exchange
cross section for proton–H-atom collisions;
${v}_{\mathrm{rel}}=\left|{\mathit{v}}_{\mathrm{p}}-{\mathit{v}}_{\mathrm{H}}\right|$
is the relative velocity between these collision partners;
d^{3}*v*_{p} and d^{3}*v*_{H} denote the
differential 3-D velocity space volumes of protons and H atoms, respectively;
and d*r* is the increment along the radial line of sight from the
magnetopause *r*_{MP} up to the bow shock *r*_{BS}. The above
expression Γ_{ENA}(*v*)d^{3}*v* due to the conditions
mentioned above (i.e., suprathermal protons, cold exospheric H atoms; i.e.,
*v*_{H}≪*v*_{p}) can then be sufficiently well approximated by
the following expression:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\Gamma}}_{\mathrm{ENA}}\left(v\right){\mathrm{d}}^{\mathrm{3}}v\simeq {n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot \mathit{\sigma}\left(v\right)\cdot v\\ \text{(3)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \left[{A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]\cdot \underset{{r}_{\mathrm{MP}}}{\overset{{r}_{\mathrm{BS}}}{\int}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{2}}\mathrm{\Omega}}{\mathrm{4}\mathit{\pi}}}{\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{2.7}}\mathrm{d}r,\end{array}$$

where ${\mathrm{d}}^{\mathrm{2}}\mathrm{\Omega}=\mathit{\pi}({r}_{\mathrm{ex}}/r{)}^{\mathrm{2}}$ denotes the space
angle under which the exobase appears from a space point at distance *r* and
the quantity *A*_{2} is the Maxwell normalization factor defined by
${A}_{\mathrm{2}}=(m/\mathrm{2}\mathit{\pi}k{T}_{\mathrm{p}\mathrm{2}}{)}^{\mathrm{3}/\mathrm{2}}$. Hence one obtains

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\Gamma}}_{\mathrm{ENA}}\left(v\right)\phantom{\rule{0.125em}{0ex}}{\mathrm{d}}^{\mathrm{3}}v\simeq {n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot \mathit{\sigma}\left(v\right)\cdot v\\ \text{(4)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \left[{A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]\cdot \underset{{r}_{\mathrm{MP}}}{\overset{{r}_{\mathrm{BS}}}{\int}}{\displaystyle \frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{\mathrm{4}{r}^{\mathrm{2}}}}{\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{2.7}}\mathrm{d}r,\end{array}$$

yielding

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\Gamma}}_{\mathrm{ENA}}\left(v\right)\phantom{\rule{0.125em}{0ex}}{\mathrm{d}}^{\mathrm{3}}v\phantom{\rule{0.33em}{0ex}}\simeq {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot \mathit{\sigma}\left(v\right)\cdot v\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \left[{A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]{\displaystyle \frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{{r}_{\mathrm{0}}^{\mathrm{2}}}}\\ \text{(5)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \underset{{r}_{\mathrm{MP}}}{\overset{{r}_{\mathrm{BS}}}{\int}}{\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{4.7}}\mathrm{d}r,\end{array}$$

or with ${\mathit{\psi}}_{\mathrm{0}}=\frac{\mathrm{1}}{\mathrm{2}}{n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot {r}_{\mathrm{0}}\cdot {x}_{\mathrm{ex}}^{\mathrm{2}}$ reads

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{\Gamma}}_{\mathrm{ENA}}\left(v\right){\mathrm{d}}^{\mathrm{3}}v\simeq {\mathit{\psi}}_{\mathrm{0}}\mathit{\sigma}\left(v\right)\cdot v\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \left[{A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]\underset{{x}_{\mathrm{MP}}}{\overset{{x}_{\mathrm{BS}}}{\int}}{x}^{-\mathrm{4.7}}\mathrm{d}x\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\mathit{\psi}}_{\mathrm{0}}\mathit{\sigma}\left(v\right)\cdot v\cdot \left[{A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]\\ \text{(6)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot {\displaystyle \frac{\mathrm{1}}{\mathrm{3.7}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right],\end{array}$$

where the normalized radial distance is given by $x=r/{r}_{\mathrm{0}}$, and the reader may be reminded that rough values give ${x}_{\mathrm{MP}}\simeq \mathrm{10}/\mathrm{8}$ and ${x}_{\mathrm{BS}}\simeq \mathrm{15}/\mathrm{8}$.

Let us assume now that this flux is more or less homogeneously hitting the
exobase from an extended area of the subsolar magnetosheath region, i.e.,
from an associated space angle d^{2}Ω_{in} which we
can calculate with the help of Fig. 4. The center of the H-ENA
emitting subsolar region may have a radial distance of
${r}_{\mathrm{ss}}={r}_{\mathrm{MP}}+(\mathrm{1}/\mathrm{2})({r}_{\mathrm{BS}}-{r}_{\mathrm{MP}})=(\mathrm{1}/\mathrm{2})({r}_{\mathrm{MP}}+{r}_{\mathrm{BS}})$.
Then the space angle for the H-ENA input is given by

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathrm{d}}^{\mathrm{2}}{\mathrm{\Omega}}_{\mathrm{in}}={\displaystyle \frac{\mathit{\pi}\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}}{{r}_{\mathrm{ss}}^{\mathrm{2}}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\mathit{\pi}{\displaystyle \frac{(\mathrm{1}/\mathrm{4})({r}_{\mathrm{BS}}-{r}_{\mathrm{MP}}{)}^{\mathrm{2}}}{(\mathrm{1}/\mathrm{4})({r}_{\mathrm{BS}}+{r}_{\mathrm{MP}}{)}^{\mathrm{2}}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\mathit{\pi}{\displaystyle \frac{({r}_{\mathrm{BS}}-{r}_{\mathrm{MP}}{)}^{\mathrm{2}}}{({r}_{\mathrm{BS}}+{r}_{\mathrm{MP}}{)}^{\mathrm{2}}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\mathit{\pi}{\left({\displaystyle \frac{\mathrm{5}}{\mathrm{15}}}\right)}^{\mathrm{2}}\\ \text{(7)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathit{\pi}}{\mathrm{9}}}.\end{array}$$

Since this H-ENA injection rate, as we shall show below, induces a
hemispherical symmetric, isotropic ENA outflux
Φ_{ENA}(*v*_{out}) from the exobase, each subsolar exobase
point consequently has to balance inflows and outflows by the relation

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\mathrm{2}\mathit{\pi}{\mathrm{\Phi}}_{\mathrm{ENA}}\left({v}_{\mathrm{out}}\right)\left(\mathit{\pi}{r}_{\mathrm{ex}}\right){\mathrm{d}}^{\mathrm{3}}{v}_{\mathrm{out}}\\ \text{(8)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\mathrm{\Gamma}}_{\mathrm{ENA}}\left({v}_{\mathrm{in}}\right){\mathrm{d}}^{\mathrm{3}}{v}_{\mathrm{in}}\left(\mathit{\pi}{\mathrm{\Delta}}^{\mathrm{2}}{r}_{\mathrm{ss}}\right)\cdot {\mathrm{d}}^{\mathrm{2}}{\mathrm{\Omega}}_{\mathrm{in}},\end{array}$$

where we have distinguished between incoming and associated outgoing
velocities *v*_{in} and *v*_{out}, as we shall also explain further
below. Due to the typical energies of the incoming H-ENAs (∼ 1 keV)
compared to the H-atom escape energy ${E}_{\mathrm{esc}}\simeq \mathrm{7}\cdot {\mathrm{10}}^{-\mathrm{4}}$ keV, the influence of the Earth's gravitational field on these
H-ENA trajectories can be completely neglected (i.e., straight trajectories).
This is also true for the outgoing H-ENAs as we shall also show next.

When an incoming energetic H-ENA approaches the exobase, it most probably
will undergo an elastic collision with the most abundant atomic atmospheric
species there, i.e., with the mono-atomic oxygen atoms (NB
*m*_{O}=16*m*_{H}). During such elastic collisions, the colliding
ENA experiences a deflection of its initial relative velocity by an angle
*χ* with a probability $\mathit{\delta}\left(\mathit{\chi}\right)=(\mathrm{d}\mathit{\chi}/\mathrm{d}p)\mathit{\delta}p$ around the center of mass of the colliding atoms (*p* being the
collisional impact parameter; for a sketch of the collision geometry in
velocity space see the sketch given as Fig. 12 in Fahr,
1978, or its reproduction given in Appendix A).
While the velocity of the center of mass *v*_{M} is conserved
during the collision process, the relative velocity Δ*v*_{H} of the H atom with respect to *v*_{M} is
turned around this conserved center-of-mass velocity by the angle *χ* (see
Fig. 2 in this paper and Figs. 11 and 12 in Fahr, 1978). The two
relative velocities of the O and the H atom with respect to the
center-of-mass velocity are thus given by

$$\begin{array}{}\text{(9)}& \left|\mathrm{\Delta}{\mathit{v}}_{\mathrm{H}}\right|={\displaystyle \frac{{m}_{\mathrm{O}}}{{m}_{\mathrm{H}}+{m}_{\mathrm{O}}}}{v}_{\mathrm{H}}={\displaystyle \frac{\mathrm{16}}{\mathrm{17}}}{v}_{\mathrm{H}}\end{array}$$

and

$$\begin{array}{}\text{(10)}& \left|\mathrm{\Delta}{\mathit{v}}_{\mathrm{O}}\right|={\displaystyle \frac{{m}_{\mathrm{H}}}{{m}_{\mathrm{H}}+{m}_{\mathrm{O}}}}{v}_{\mathrm{H}}={\displaystyle \frac{\mathrm{1}}{\mathrm{17}}}{v}_{\mathrm{H}},\end{array}$$

where ${v}_{\mathrm{H}}\simeq \sqrt{\mathrm{3}k{T}_{\mathrm{p}\mathrm{2}}/m}\gg {v}_{\mathrm{esc}}$ is the
mean thermal proton velocity in the subsonic solar wind inherited via charge
transfer by the H-ENA. With the above result, one can conclude that the
distribution function of the reflected H-ENAs being emitted from the exobase
to outer space is also a Maxwellian; however, with income velocities
*v*_{in} converted into reduced outcome velocities
${v}_{\mathrm{out}}=(\mathrm{16}/\mathrm{17}){v}_{\mathrm{in}}$ and with a space angle
d^{2}Ω_{out}=2*π* instead of an income space angle
${\mathrm{d}}^{\mathrm{2}}{\mathrm{\Omega}}_{\mathrm{in}}=\mathit{\pi}/\mathrm{9}$. The resulting exobasic density
*n*_{ex,ENA} is then obtained from the following ENA balance request
for the local vertical fluxes

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{n}_{\mathrm{ex},\mathrm{ENA}}{r}_{\mathrm{ex}}^{\mathrm{2}}\cdot \int {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{out}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot \underset{\mathrm{0}}{\overset{{\mathit{\vartheta}}_{\mathrm{out}}}{\int}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\vartheta}\mathrm{d}\mathit{\vartheta}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\mathit{\psi}}_{\mathrm{0}}\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}\int \mathit{\sigma}\left(v\right)\cdot {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-{\displaystyle \frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{in}}\\ \text{(11)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot {\displaystyle \frac{\mathrm{1}}{\mathrm{3.7}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right]\underset{\mathrm{0}}{\overset{{\mathit{\vartheta}}_{\mathrm{in}}}{\int}}{\mathrm{cos}}^{\mathrm{2}}\mathit{\vartheta}\mathrm{d}\mathit{\vartheta},\end{array}$$

yielding

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{n}_{\mathrm{ex},\mathrm{ENA}}={\mathit{\psi}}_{\mathrm{0}}{\mathit{\sigma}}_{\mathrm{0}}\left({v}_{\mathrm{0}}\right){\displaystyle \frac{\mathrm{1}}{\mathrm{3.7}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right]\\ \text{(12)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot {\displaystyle \frac{\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}}{{r}_{\mathrm{ex}}^{\mathrm{2}}}}{\displaystyle \frac{\int {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-\frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{in}}}{\int {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-\frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{out}}}}{\displaystyle \frac{\mathrm{1}+\mathrm{cos}{\mathit{\vartheta}}_{\mathrm{in}}}{\mathrm{1}+\mathrm{cos}{\mathit{\vartheta}}_{\mathrm{out}}}},\end{array}$$

where the quantity *ψ*_{0} has already been defined by
${\mathit{\psi}}_{\mathrm{0}}=\frac{\mathrm{1}}{\mathrm{2}}{n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot {r}_{\mathrm{0}}\cdot {x}_{\mathrm{ex}}^{\mathrm{2}}$, and the reference value
${\mathit{\sigma}}_{\mathrm{0}}\left({v}_{\mathrm{0}}\right)={\mathrm{10}}^{-\mathrm{15}}$ cm^{2} (relying on its very small variability
in the covered velocity range) has been introduced as the charge-exchange
cross section at the reference velocity ${v}_{\mathrm{0}}=\sqrt{\mathrm{3}k{T}_{\mathrm{p}\mathrm{2}}/m}$.
Furthermore, *ϑ*_{in} and *ϑ*_{out} denote the
border angle limitations of the input cone and the output cone, respectively.
Due to the hemispherical ENA emission from the exobase, one evidently has
${\mathit{\vartheta}}_{\mathrm{out}}=\mathit{\pi}/\mathrm{2}$. For the limiting input angle
*ϑ*_{in}, we obtain from the following definition

$$\begin{array}{}\text{(13)}& {\mathrm{d}}^{\mathrm{2}}{\mathrm{\Omega}}_{\mathrm{in}}={\displaystyle \frac{\mathit{\pi}}{\mathrm{9}}}=\mathrm{2}\mathit{\pi}\underset{\mathrm{0}}{\overset{{\mathit{\vartheta}}_{\mathrm{in}}}{\int}}\mathrm{sin}\mathit{\vartheta}\mathrm{d}\mathit{\vartheta}=\mathrm{2}\mathit{\pi}\left(\mathrm{1}-\mathrm{cos}{\mathit{\vartheta}}_{\mathrm{in}}\right),\end{array}$$

leading to $\mathrm{cos}{\mathit{\vartheta}}_{\mathrm{in}}=\mathrm{17}/\mathrm{18}$. That then leads to the following expression:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{n}_{\mathrm{ex},\mathrm{ENA}}\simeq \mathrm{2}{\mathit{\psi}}_{\mathrm{0}}{\mathit{\sigma}}_{\mathrm{0}}\left({v}_{\mathrm{0}}\right){\displaystyle \frac{\mathrm{1}}{\mathrm{3.7}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right]\\ \text{(14)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot {\displaystyle \frac{\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}}{{r}_{\mathrm{ex}}^{\mathrm{2}}}}{\displaystyle \frac{\int {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-\frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{in}}}{\int {\left[v\cdot {A}_{\mathrm{2}}\mathrm{exp}\left[-\frac{m{v}^{\mathrm{2}}}{\mathrm{2}k{T}_{\mathrm{p}\mathrm{2}}}\right]{v}^{\mathrm{2}}\mathrm{d}v\right]}_{\mathrm{out}}}}.\end{array}$$

Reminding ourselves that both of the above requested integrals as Maxwellian moments represent the mean thermal velocity of the associated distribution functions, one thus simply finds

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{n}_{\mathrm{ex},\mathrm{ENA}}\simeq \mathrm{2}{\mathit{\psi}}_{\mathrm{0}}{\mathit{\sigma}}_{\mathrm{0}}\left({v}_{\mathrm{0}}\right){\displaystyle \frac{\mathrm{1}}{\mathrm{3.7}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right]\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\cdot {\displaystyle \frac{\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}}{{r}_{\mathrm{ex}}^{\mathrm{2}}}}{\displaystyle \frac{{\sqrt{\mathrm{3}k{T}_{\mathrm{p}\mathrm{2}}/m}}_{\mathrm{in}}}{{\sqrt{\mathrm{3}k{T}_{\mathrm{p}\mathrm{2}}/m}}_{\mathrm{out}}}}\\ \text{(15)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\mathrm{0.6}\cdot {\mathit{\psi}}_{\mathrm{0}}{\mathit{\sigma}}_{\mathrm{0}}\left({v}_{\mathrm{0}}\right){\displaystyle \frac{\mathrm{\Delta}{r}_{\mathrm{ss}}^{\mathrm{2}}}{{r}_{\mathrm{ex}}^{\mathrm{2}}}}\left[{x}_{\mathrm{MP}}^{-\mathrm{3.7}}-{x}_{\mathrm{BS}}^{-\mathrm{3.7}}\right].\end{array}$$

Putting in numbers for *ψ*_{0} and *σ*_{0}(*v*_{0}), one obtains

$$\begin{array}{}\text{(16)}& {n}_{\mathrm{ex},\mathrm{ENA}}\simeq \mathrm{4.3}\cdot {\mathrm{10}}^{-\mathrm{3}}\phantom{\rule{0.125em}{0ex}}{\mathrm{cm}}^{-\mathrm{3}}.\end{array}$$

While the exobasic H-atom density is falling off like *r*^{−2.7}, the H-ENAs
being reflected from the exobase in view of their high velocities do not
care for the Earth's
gravitational field and hence their density falls off like *r*^{−2}. In
principle, this should lead to a cross-over point of the two H-density
profiles as indicated in Fig. 5.

Looking at the numbers given, however, this cross-over point would be far beyond the magnetopause.

3 Shrinking of the escape cone

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As a small refinement, another geometrical consideration may be carried out
for the case of the H-ENAs evaporating from the exobase and arriving at
distances *r*>*r*_{ex}, namely that the widespread velocity cone of
H-ENAs being hemispherically reflected from the exobase, with an initial cone
angle of ${\mathit{\beta}}_{\mathrm{c},\mathrm{ex}}=\mathit{\pi}/\mathrm{2}$, will systematically be reduced with
increasing height to an increasingly narrower velocity cone in velocity
space, the larger the radial distance *r* from the exobase becomes (see
Fig. 6).

This shrinking of the velocity cone of the H-ENA velocities increases the
cone-averaged upward bulk velocity. This evident increase in the effective
radial H-ENA bulk velocity 〈*v*_{ENA,z}〉
can be calculated with the following expression:

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}\u2329{v}_{\mathrm{ENA},\mathrm{z}}\u232a={v}_{\mathrm{H}}{\displaystyle \frac{\frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}{\iint}_{\mathrm{0}}^{{\mathit{\beta}}_{\mathrm{c}}}\mathrm{cos}\mathit{\beta}\mathrm{sin}\mathit{\beta}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\beta}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\varphi}}{\frac{\mathrm{1}}{\mathrm{4}\mathit{\pi}}{\iint}_{\mathrm{0}}^{{\mathit{\beta}}_{\mathrm{c}}}\mathrm{sin}\mathit{\beta}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\beta}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\varphi}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}_{\mathrm{H}}{\displaystyle \frac{\mathrm{1}-{\mathrm{cos}}^{\mathrm{2}}{\mathit{\beta}}_{\mathrm{c}}}{\mathrm{1}-\mathrm{cos}{\mathit{\beta}}_{\mathrm{c}}}}\\ \text{(17)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}_{\mathrm{H}}\left(\mathrm{1}+\mathrm{cos}{\mathit{\beta}}_{\mathrm{c}}\right),\end{array}$$

where *β*_{c} is the critical, local cone angle of the H-ENA
velocity distribution function at *r*. For H-ENAs escaping from the exobase
*r*_{ex}, it can be calculated by the expression

$$\begin{array}{}\text{(18)}& \mathrm{cos}{\mathit{\beta}}_{\mathrm{c}}={\displaystyle \frac{L}{r}}={\displaystyle \frac{\sqrt{{r}^{\mathrm{2}}-{r}_{\mathrm{ex}}^{\mathrm{2}}}}{r}}=\sqrt{\mathrm{1}-{\displaystyle \frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{{r}^{\mathrm{2}}}}},\end{array}$$

(for *L* see Fig. 6) meaning that with the shrinkage of this cone, the radial
ENA bulk velocity is growing with radial distance by

$$\begin{array}{}\text{(19)}& \u2329{v}_{\mathrm{ENA},\mathrm{z}}\u232a={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{v}_{\mathrm{H}}\left(\mathrm{1}+\sqrt{\mathrm{1}-{\displaystyle \frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{{r}^{\mathrm{2}}}}}\right).\end{array}$$

This expression for large distances asymptotically yields the result $\u2329{v}_{\mathrm{ENA},\mathrm{z}}(r\to \mathrm{\infty})\u232a={v}_{\mathrm{H}}$. This finally shows that the ENA density at radial distances *r*≫*r*_{ex}, instead by (1∕*r*^{2}), rather falls off like

$$\begin{array}{}\text{(20)}& {n}_{\mathrm{ENA}}\left(r\right)={n}_{\mathrm{ex},\mathrm{ENA}}\cdot {\displaystyle \frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{{r}^{\mathrm{2}}}}{\displaystyle \frac{\mathrm{1}}{\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{1}+\sqrt{\mathrm{1}-\frac{{r}_{\mathrm{ex}}^{\mathrm{2}}}{{r}^{\mathrm{2}}}}\right)}},\end{array}$$

which, however, is only a small correction with respect to the
(1∕*r*^{2}) drop-off used in our calculations above.

4 The Lyman-alpha glow contribution by H-ENAs and a look into self-consistency in H-ENA production

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Thinking of H-ENA densities perhaps growing over the exobasic H densities
beyond some critical radial distance *r*_{c}, one could be seduced to
ask whether or not the escaping H-ENAs in fact do partly reproduce themselves
by a second-generation charge exchange in the magnetosheath and thus
contribute to the H-ENA production in the trans-magnetopause region. This
idea is especially interesting, because it is not a priori clear what sort of
H atoms, i.e., exobasic H atoms or H-ENAs, are in fact responsible for the
Lyman-alpha signal registered at the largest exospheric heights. For the
exospheric H densities inferred from TWINS-LAD Lyman-alpha measurements, it
can not be excluded that H densities derived from such Lyman-alpha
measurements at heights around 8 *r*_{E} (see Zoennchen et al., 2013)
are partly or even fully due to H-ENAs themselves. In that case of being
fully due to H-ENAs, the presence of these H-ENAs originating from charge-exchange processes
in the magnetosheath should in fact be self-replacing,
i.e., these H-ENAs could be a phenomenon of a self-consistency. For that
purpose we are carrying out the following study of the specific contribution
of H-ENAs to the Lyman-alpha glow emission at the largest exospheric heights
with $r\ge {r}_{\mathrm{0}}\simeq \mathrm{8}{r}_{\mathrm{E}}$.

Their relative contribution to the full ENA production is the ratio ${\mathit{\xi}}_{\mathrm{ENA}}={\mathit{\beta}}_{\mathrm{ENA}}^{\ast}/{\mathit{\beta}}_{\mathrm{ENA}}$ of the two production rates due to H-ENAs and due to normal exospheric H atoms, given by

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\beta}}_{\mathrm{ENA}}^{\ast}\simeq {n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{ENA},\mathrm{0}}\cdot {\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}\cdot \underset{{r}_{\mathrm{MP}}}{\overset{\mathrm{\infty}}{\int}}{\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{2}}\phantom{\rule{0.125em}{0ex}}\mathrm{d}r\\ \text{(21)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{ENA},\mathrm{0}}\cdot {\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}\cdot {r}_{\mathrm{0}}{\left({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{1}}\end{array}$$

and

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\beta}}_{\mathrm{ENA}}\simeq {n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a\cdot \underset{{r}_{\mathrm{MP}}}{\overset{\mathrm{\infty}}{\int}}{\left({\displaystyle \frac{r}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{2.7}}\mathrm{d}r\\ \text{(22)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathrm{1}}{\mathrm{1.7}}}\cdot {n}_{\mathrm{p}\mathrm{2}}\cdot {n}_{\mathrm{H}\mathrm{0}}\cdot \u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a\cdot {r}_{\mathrm{0}}{\left({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}\right)}^{-\mathrm{1.7}},\end{array}$$

where ${\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}$ and 〈*σ**v*_{rel}〉 are average charge-exchange
frequencies between hot H-ENAs or cold exobasic H atoms, respectively, and
the subsonic hot magnetosheath protons. This then consequently means

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\xi}}_{\mathrm{ENA}}={\displaystyle \frac{{\mathit{\beta}}_{\mathrm{ENA}}^{\ast}}{{\mathit{\beta}}_{\mathrm{ENA}}}}\\ {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\frac{\mathrm{1}}{\mathrm{2}}{n}_{\mathrm{p}\mathrm{2}}{n}_{\mathrm{ENA},\mathrm{0}}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}{r}_{\mathrm{0}}{\left(\frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}\right)}^{-\mathrm{1}}}{\frac{\mathrm{1}}{\mathrm{1.7}}{n}_{\mathrm{p}\mathrm{2}}{n}_{\mathrm{H}\mathrm{0}}\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a{r}_{\mathrm{0}}{\left(\frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}\right)}^{-\mathrm{1.7}}}}\\ \text{(23)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}={\displaystyle \frac{\mathrm{1.7}}{\mathrm{2}}}({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}{)}^{\mathrm{0.7}}\cdot {\displaystyle \frac{{n}_{\mathrm{ENA},\mathrm{0}}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}}{{n}_{\mathrm{H}\mathrm{0}}\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}\end{array}$$

and thus leads to the expression

$$\begin{array}{}\text{(24)}& {\mathit{\xi}}_{\mathrm{ENA}}=\mathrm{0.85}\cdot ({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}{)}^{\mathrm{0.7}}\cdot {\displaystyle \frac{{n}_{\mathrm{ENA},\mathrm{0}}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}}{{n}_{\mathrm{H}\mathrm{0}}\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}.\end{array}$$

Now one can make use of the fact that the charge-exchange cross section at
energies of ${E}_{\mathrm{rel},\mathrm{0}}=(m/\mathrm{2})\cdot {v}_{\mathrm{rel},\mathrm{0}}^{\mathrm{2}}$ is varying
only very mildly with *E*_{rel} or
*v*_{rel} and hence one can use a Taylor series expansion of the cross
section with respect to the deviation Δ*v*_{rel} from
*v*_{rel,0} in the form developed by Fahr (2003) and with the
charge-exchange cross section given by Maher and Tinsley (1977), which then
leads to

$$\begin{array}{ll}{\displaystyle}& {\displaystyle \frac{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}^{\ast}\u232a}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}={\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{0}}\left[\mathrm{1}+\frac{\mathrm{2}B}{\sqrt{{\mathit{\sigma}}_{\mathrm{0}}}}\left(\mathrm{1}-\frac{{x}_{\mathrm{rel}}}{{X}_{\mathrm{rel}}}\right)\right]\left({v}_{\mathrm{rel},\mathrm{0}}+\mathrm{\Delta}{v}_{\mathrm{rel}}\right)}{{\mathit{\sigma}}_{\mathrm{0}}\cdot {v}_{\mathrm{rel},\mathrm{0}}}}\\ \text{(25)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\left[\mathrm{1}+{\displaystyle \frac{\mathrm{2}B}{\sqrt{{\mathit{\sigma}}_{\mathrm{0}}}}}\left(\mathrm{1}-{\displaystyle \frac{{x}_{\mathrm{rel}}}{{X}_{\mathrm{rel}}}}\right)\right]\left(\mathrm{1}+{\displaystyle \frac{\mathrm{\Delta}{v}_{\mathrm{rel},\mathrm{0}}}{{v}_{\mathrm{rel},\mathrm{0}}}}\right),\end{array}$$

where the quantities *x*_{rel} and *X*_{rel} are defined by

$$\begin{array}{}\text{(26)}& {x}_{\mathrm{rel}}={\displaystyle \frac{{v}_{\mathrm{rel},\mathrm{0}}}{\sqrt{\frac{\mathrm{2}k{T}_{p,\mathrm{2}}}{m}}}}\simeq \mathrm{1}\end{array}$$

and

$$\begin{array}{}\text{(27)}& {X}_{\mathrm{rel}}=\sqrt{{\displaystyle \frac{\mathrm{64}}{\mathrm{9}\mathit{\pi}}}\left(\mathrm{1}+{\displaystyle \frac{{T}_{\mathrm{H}}}{{T}_{\mathrm{p},\mathrm{2}}}}\right)+{M}_{\mathrm{s}}^{\mathrm{2}}},\end{array}$$

with *T*_{H}≃10^{3} K; ${T}_{p,\mathrm{2}}\simeq {\mathrm{10}}^{\mathrm{6}}$ K; and
*M*_{s}≃0.1. The above expression thus leads to

$$\begin{array}{ll}{\displaystyle \frac{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}^{\ast}\u232a}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}& {\displaystyle}=\left[\mathrm{1}+{\displaystyle \frac{\mathrm{2}B}{\sqrt{{\mathit{\sigma}}_{\mathrm{0}}}}}\left(\mathrm{1}-{\displaystyle \frac{\mathrm{1}}{\sqrt{\frac{\mathrm{64}}{\mathrm{9}\mathit{\pi}}}}}\right)\right]\left(\mathrm{1}+{\displaystyle \frac{\mathrm{\Delta}{v}_{\mathrm{rel},\mathrm{0}}}{{v}_{\mathrm{rel},\mathrm{0}}}}\right)\\ \text{(28)}& {\displaystyle}& {\displaystyle}=\left[\mathrm{1}+{\displaystyle \frac{\mathrm{2}B}{\sqrt{{\mathit{\sigma}}_{\mathrm{0}}}}}(\mathrm{1}-\mathrm{0.66})\right]\left(\mathrm{1}+{\displaystyle \frac{\mathrm{\Delta}{v}_{\mathrm{rel},\mathrm{0}}}{{v}_{\mathrm{rel},\mathrm{0}}}}\right),\end{array}$$

and hence one obtains with 2B $/\sqrt{{\mathit{\sigma}}_{\mathrm{0}}}=\left(\mathrm{2}\cdot \mathrm{6.8}/\sqrt{\mathrm{20}}\right)\cdot {\mathrm{10}}^{-\mathrm{2}}=\mathrm{0.14}$ (see Maher and Tinsley, 1977)

$$\begin{array}{ll}{\displaystyle \frac{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}^{\ast}\u232a}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}& {\displaystyle}=[\mathrm{1}+\mathrm{0.34}\cdot \mathrm{0.14}]\left(\mathrm{1}+{\displaystyle \frac{\mathrm{\Delta}{v}_{\mathrm{rel},\mathrm{0}}}{{v}_{\mathrm{rel},\mathrm{0}}}}\right)\\ \text{(29)}& {\displaystyle}& {\displaystyle}=\mathrm{1.048}\cdot \left(\mathrm{1}-\sqrt{{\displaystyle \frac{\mathrm{64}}{\mathrm{9}\mathit{\pi}}}}\right)\simeq \mathrm{1.6}.\end{array}$$

This means that the average ENA charge-exchange production per atom is higher by a factor of 1.6 for hot H-ENAs as collision partners compared to cold exobasic H atoms. Thus, we finally come to the result

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{\mathit{\xi}}_{\mathrm{ENA}}=\mathrm{0.85}\cdot {\left({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}\right)}^{\mathrm{0.7}}\cdot {\displaystyle \frac{{n}_{\mathrm{ENA},\mathrm{0}}{\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}^{\ast}}{{n}_{\mathrm{H}\mathrm{0}}\u2329\mathit{\sigma}{v}_{\mathrm{rel}}\u232a}}\\ \text{(30)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}=\mathrm{1.33}\cdot {\left({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}\right)}^{\mathrm{0.7}}\cdot {\displaystyle \frac{{n}_{\mathrm{ENA},\mathrm{0}}}{{n}_{\mathrm{H}\mathrm{0}}}}.\end{array}$$

When reminding ourselves of the density representations found in this article, namely Eq. (1) and

$$\begin{array}{ll}{\displaystyle}& {\displaystyle}{n}_{\mathrm{ENA}}={n}_{\mathrm{ex},\mathrm{ENA}}\cdot {\left({\displaystyle \frac{r}{{r}_{\mathrm{ex}}}}\right)}^{-\mathrm{2}}\\ \text{(31)}& {\displaystyle}& {\displaystyle}\phantom{\rule{2em}{0ex}}\ge \mathrm{0.048}\cdot {n}_{\mathrm{H},\mathrm{ex}}\cdot {\left({\displaystyle \frac{r}{{r}_{\mathrm{ex}}}}\right)}^{-\mathrm{2}},\end{array}$$

this will then tell us that

$$\begin{array}{}\text{(32)}& {\mathit{\xi}}_{\mathrm{ENA}}\ge \mathrm{0.048}\cdot {\left({\displaystyle \frac{{r}_{\mathrm{MP}}}{{r}_{\mathrm{0}}}}\right)}^{\mathrm{0.7}}=\mathrm{0.048}\cdot {\mathrm{\Psi}}^{\mathrm{0.7}}.\end{array}$$

So, even under extremely unlikely magnetopause locations of
$({r}_{\mathrm{MP}}/{r}_{\mathrm{0}})\simeq \mathrm{2}$, the above result would still mean that
*ξ*_{ENA}≃0.08, i.e., H-ENAs can only reproduce themselves by
less than 10 %. Hence one can conclude that the H-ENA component discussed
in this article can not be a self-consistent phenomenon, i.e., the
Lyman-alpha glow observed by TWINS-LAD (Zoennchen et al., 2013) is due to
exobasic H atoms only. To say it in other words, the self-consistency of
magnetosheath ENAs could only then be expected under conditions of a
magnetosheath plasma that is “optically thick” with respect to
charge-exchange collisions of H-ENAs with magnetosheath protons, i.e.,
*τ*_{sheath}≃1. It turns out, however, that this “optical
thickness” *τ*_{sheath} of the magnetosheath plasma with its
extension $D={r}_{\mathrm{BS}}-{r}_{\mathrm{MP}}$ has a value of only

$$\begin{array}{ll}{\displaystyle}{\mathit{\tau}}_{\mathrm{sheath}}& {\displaystyle}\simeq {n}_{\mathrm{sheath}}\u2329{\mathit{\sigma}}_{\mathrm{ex}}{v}_{\mathrm{rel}}\u232a{\displaystyle \frac{D}{{v}_{\mathrm{H}-\mathrm{ENA}}}}\\ {\displaystyle}& {\displaystyle}\simeq {n}_{\mathrm{sheath}}\u2329{\mathit{\sigma}}_{\mathrm{ex}}\u232a\left({r}_{\mathrm{BS}}-{r}_{\mathrm{MP}}\right)\\ \text{(33)}& {\displaystyle}& {\displaystyle}=\mathrm{4.5}\cdot {\mathrm{10}}^{-\mathrm{4}}.\end{array}$$

5 The Lyman-alpha glow emission of terrestrial H-ENAs

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Finally, it may be interesting to study how much the above discussed H-ENAs in the upper exosphere contribute to the terrestrial Lyman-alpha glow emission that, for example, is measured with a broad-band Lyman-alpha photometer by the satellite package TWINS-LAD (see Zoennchen et al., 2011, 2013; Bailey and Gruntman, 2013). For that purpose, one has to look at the spectral profile of the solar Lyman-alpha line emission that enters the Earth's exosphere from the solar direction. This profile of the full disk solar Lyman-alpha emission is a strongly inverted Gaussian with a strong spectral depletion near the center of the line (see observations from the Orbiting Solar Observatory, OSO, shown by Bonnet et al., 1978). As a reasonable approach to these OSO observations the following functional form of the solar line can be used (see Scherer et al., 2000):

$$\begin{array}{l}{\displaystyle \frac{I\left(\mathit{\lambda}\right)}{{I}_{\mathrm{0}}}}={I}_{\mathrm{norm}}\left\{a\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left[-{\left({\displaystyle \frac{\mathit{\lambda}-{\mathit{\lambda}}_{\mathrm{0}}}{\mathrm{\Delta}a}}\right)}^{\mathrm{2}}\right]\right.\\ \text{(34)}& {\displaystyle}\phantom{\rule{2em}{0ex}}\left.-b\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left[-{\left({\displaystyle \frac{\mathit{\lambda}-{\mathit{\lambda}}_{\mathrm{0}}}{\mathrm{\Delta}b}}\right)}^{\mathrm{2}}\right]\right\},\end{array}$$

where the following numbers are used:

$$\begin{array}{ll}{\displaystyle}a& {\displaystyle}=\mathrm{7.9589}\\ {\displaystyle}b& {\displaystyle}=\mathrm{5.6930}\\ {\displaystyle}\mathrm{\Delta}a& {\displaystyle}=\mathrm{0.351}\phantom{\rule{0.125em}{0ex}}\mathit{\AA}\\ {\displaystyle}\mathrm{\Delta}b& {\displaystyle}=\mathrm{0.2}\phantom{\rule{0.125em}{0ex}}\mathit{\AA}.\end{array}$$

The quantity *I*_{norm} normalizes the area below the spectral curve
to unity by

$$\begin{array}{}\text{(35)}& {I}_{\mathrm{norm}}={\left(\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}{\displaystyle \frac{I\left(\mathit{\lambda}\right)}{{I}_{\mathrm{0}}}}\phantom{\rule{0.125em}{0ex}}\mathrm{d}\mathit{\lambda}\right)}^{-\mathrm{1}}={\displaystyle \frac{\mathrm{1}}{\sqrt{\mathit{\pi}}\left(a\mathrm{\Delta}a-b\mathrm{\Delta}b\right)}}\end{array}$$

and *I*_{0} denotes the full-disk average of the solar Lyman-alpha flux at 1 AU.

Now we look at those H-ENAs bouncing off the dayside exobase with a
thermally broad distribution around the bulk velocity *U*_{ENA} of
roughly 300 km s^{−1} and moving off in zenith direction into the solar
direction. These H atoms are resonantly absorbing from the blueshifted wing
of the solar profile with a mean blueshift of $\mathrm{\Delta}{\mathit{\lambda}}^{z}={\mathit{\lambda}}_{\mathrm{0}}({U}_{\mathrm{ENA}}/c)\simeq $ (1216∕1000) Å = 1.2 Å
and with an equal spread around this absorption peak due to the thermally
broad H-ENA distribution. Thus, the resonant excitation of these H-ENAs would
be centered around the spectral place
${\mathit{\lambda}}_{\mathrm{ENA}}^{z}={\mathit{\lambda}}_{\mathrm{0}}+\mathrm{\Delta}\mathit{\lambda}$, where the spectral
intensity of the incoming solar line is larger than at line center. On the
other hand, when looking into the dawn–dusk regions of the geocorona, the
H-ENA Lyman-alpha resonance glow would be connected with H-ENAs moving off
the exobase into the direction perpendicular to the solar line. This is why
for these ENAs the spectral Doppler shift vanishes, as it does for the
exobasic H atoms, and these H atoms are thus excited mainly in the line
center where the spectral solar intensity is smaller by a factor of about
1∕2 compared to the spectral intensity maxima. For an illustrative view,
see Fig. 7.

At this moment it is, however, hard to make clear predictions on the anisotropy of the H-ENA radiation glow without more exact resonant scattering calculations on the basis of exact H-ENA distribution functions. It may, nevertheless, be already predictable now that the sun-dusk H-ENA glow anisotropy could come up to a factor of 2, also meaning that the density of H-ENAs concerning their glow contribution in the solar direction would also be weighted by such a factor of 2 with respect to the corresponding contribution from exobasic H atoms.

6 Conclusions

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At the end of this article, we once again want to refer back to the beginning
where we looked at the problem raised by Qin and Waldrop (2016) who had found
indications for the existence of a non-thermal exospheric hydrogen component
in their Lyman-alpha glow observations with the satellite TIMED/GUVI, a
component for which no explanation as yet is available. In this article,
we have discussed a non-thermal hydrogen component originating from charge-exchange
interactions of exobasic terrestrial hydrogen atoms with protons of
the shocked solar wind plasma ahead of the magnetopause. Though this
H component definitely represents a contribution to non-thermal hydrogen in
the terrestrial exosphere with a relative relevance increasing with height,
it has also become clear from the calculations presented in this article that
this contribution cannot explain the observations made by Qin and
Waldrop (2016). Concerning the Lyman-alpha resonance glow, the
trans-magnetopause contribution discussed in this article due to its rather
low relative density in heights below three Earth radii can only become
recognizable at large heights of about five to eight Earth radii, while Qin and
Waldrop (2016) claim to clearly identify the
influence of a non-thermal H component from their glow measurements already at heights of 3000 km. Thus,
an idea, alternative to ours presented here and bringing in suprathermal
H atoms at lower heights, might be to study whether the observationally
indicated non-thermal H component could perhaps be due to a non-thermalized
form of H atoms originating in the upper thermosphere via photo-dissociation
of H_{2}O through OH producing photo-dissociative H atoms with velocities
of about 8 km s^{−1} (e.g., see Keller, 1990), which as non-thermalized
H products represent a hydrogen component with an effective temperature of
about 4000 K.

Data availability

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Data availability.

No original data were generated for this paper, but all data sets used are given in Qin and Waldrop (2016).

Appendix A

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This figure, reproduced from the paper by Fahr (1978), is provided to
illustrate the process of an elastic collision of two particles with
different masses *m* and *M* and velocities *v*_{1} and *v*_{2}
before the collision. At the course of the collision, the magnitude of the
relative velocity ${v}_{\mathrm{rel}}=[{\mathit{v}}_{\mathrm{1}}-{\mathit{v}}_{\mathrm{2}}]$ between the
two particles is conserved, as is the center-of-mass velocity. The newly
originating velocities ${\mathit{v}}_{\mathrm{1}}^{\prime}$ and ${\mathit{v}}_{\mathrm{2}}^{\prime}$, resulting after the
collision has occurred, are connected with a turn of the relative velocity
vector around the center-of-mass velocity. Thus, the resulting velocities are
found to be placed on the two shells around the center-of-mass velocity, the
smaller for the heavier mass, the larger for the smaller mass, all in
connection with the resulting turning angle *χ*.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

The authors thank the TWINS team (PI Dave McComas) for making this work
possible. We also acknowledge the support of the German Federal Ministry of
Economics and Technology (BMWi) through the DLR grants FKZ 50 OE 1401 and
FKZ 50 OE 1701.

The topical editor,
Anna Milillo, thanks Herbert O. Funsten, Iannis Dandouras, and one anonymous
referee for help in evaluating this paper.

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Short summary

We investigate what fraction of the hot hydrogen atoms recently found from Lyman-alpha measurements in the Earth's upper exosphere could have their origin as energetic neutral atoms via charge exchange from protons of the shocked solar wind ahead of the magnetopause.
Our calculations show that this contribution, although definitely present at larger exospheric heights, cannot explain the observations at lower altitudes.

We investigate what fraction of the hot hydrogen atoms recently found from Lyman-alpha...

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