# Particle content of very inclined air showers for radio signal modeling

Marion Guelfand,11footnotetext: Corresponding author. Simon Chiche  Kumiko
Kotera,22footnotetext: Also at Some University. Simon Prunet  Tanguy Pierog

###### Abstract

The reconstruction of very inclined air showers is a new challenge for next-
generation radio experiments such as the AugerPrime radio upgrade, BEACON, and
GRAND, which focus on the detection of ultra-high-energy particles. To tackle
this, we study the electromagnetic particle content of very inclined air
showers, which has scarcely been studied so far. Using the simulation tools
CORSIKA and CoREAS, and analytical modeling, we explore the energy range of
the particles that contribute most to the radio emission, quantify their
lateral extent, and estimate the atmospheric depth at which the radio emission
is strongest. We find that the distribution of the electromagnetic component
in very inclined air showers has characteristic features that could lead to
clear signatures in the radio signal, and hence impact the reconstruction
strategies of next-generation radio-detection experiments.

##  1 Introduction

Recently, several radio experiments (AERA, CODALEMA, TREND, LOFAR) [1, 2] have
demonstrated that the information contained in the radio signal of vertical
air showers allows one to perform an accurate reconstruction of the primary
particle parameters (nature, arrival direction, energy) [3]. The next
generation of extended radio arrays (GRAND, BEACON, AugerPrime radio upgrade)
[4, 5, 6] focuses on very inclined extensive air showers (EAS) that leave
large radio footprints on the ground. Thanks to this specific feature, one can
deploy detectors over gigantic arrays with a sparse density to provide large
detector effective area at affordable costs.

The study of the radio signal produced by these very inclined air showers has
evidenced that the emission presents drastically different characteristics
compared to vertical showers. In particular, the emission can become
incoherent, translating into a drop in the electric field strength [7]. In
order to understand the specific signatures related to this very inclined
regime, we develop phenomenological models and use the Monte-Carlo simulation
tools CORSIKA and CoREAS to study the leptonic content of these air showers,
directly linked to the radio signal. We examine the particle populations
mainly responsible of the radio emission and the particle distribution in the
shower front, taking into account shower-to-shower fluctuations.

##  2 Dynamics of particles for very inclined air showers

When ultra-high energy particles travel through the atmosphere, they interact
with air molecules and produce a cascade of secondary particles, that
propagates toward the ground at relativistic speed. The electromagnetic
component of the shower is responsible for the production of the radio signal:
a beamed coherent emission in the MHz regime. The geomagnetic effect [8] is
the major ingredient for radio emission in air: it corresponds to the
deflection of electrons and positrons in opposite directions due to the
Lorentz force of the Earth magnetic field, balanced by particle scattering
with the air molecules. Ultimately, particles are deflected in a plane
perpendicular to that of the shower propagation, creating an electric current
in the atmosphere. This current varies with time since the number of electrons
and positrons changes during the shower development and the accelerated
charges thus induce a radio signal. The emission is beamed in the forward
direction inside a narrow cone due to the relativistic speed of the particles.
The half-width of this cone is equal to the Cherenkov angle, typically
1–$2^{\circ}$ in the air [9]. For an observer located at this specific angle,
the radio signals emitted all along the shower arrive simultaneously, boosting
the signal along this Cherenkov ring and resulting in brief and high amplitude
pulses.

Very inclined air showers (with zenith angles $\theta$ larger than $\gtrsim
70^{\circ}$) present different geometries compared to vertical ones. They can
extend over several hundreds of kilometers, as opposed to the $\sim 10$ km
thickness of the troposphere for vertical showers. Moreover, they develop
higher in the atmosphere where the air density $\rho_{\rm air}$ is lower (see
Figure 1). This implies that shower particles have larger mean free paths for
collisions with the air molecules, and hence experience more strongly the
effect of the Earth magnetic field.

###  2.1 Scattering processes in air showers and deflections by the
geomagnetic field

Electrons and positrons in the air-shower undergo Bremsstrahlung radiation at
energies $\epsilon=\gamma m_{\rm e}c^{2}\gtrsim\epsilon_{\rm c}$, where
$\epsilon_{\rm c}=88\,$MeV is the critical energy below which energy losses
due to ionization become dominant — hence most particles are absorbed in the
atmosphere [10]. In the following, we focus on the high energy regime where
$\epsilon\gtrsim\epsilon_{\rm c}$ and neglect both ionization and Coulomb
scattering, which becomes a dominant process below $\epsilon_{\rm
Coulomb}=20\,\rm MeV$ [10]. The attenuation length for these processes can
thus be written [8]:

$\displaystyle l_{\rm rad}=\frac{X_{0}}{\rho_{\rm air}}\sim 4\times
10^{2}\,{\rm m}\,\left(\frac{\rho_{\rm air}}{1\,{\rm kg\,m^{-3}}}\right)^{-1}\
,$ (2.1)

where $X_{0}=36.7\,{\rm g}\,{\rm cm}^{-2}$ is the electronic radiation length.

Electrons and positrons also experience deflections in opposite directions by
the geomagnetic field $B$, with Larmor radius $r_{\rm
L}=\beta\epsilon{\rm\sin\alpha}/(ceB)\ \sim 3.3\times 10^{4}{\rm
m}\,{\rm\sin\alpha}(\epsilon/100\,{\rm MeV})(B/10\,\mu{\rm T})^{-1}$, with the
reduced velocity $\beta\sim 1$ for relativistic particles, $c$ the speed of
light, $e$ the elementary charge and $\alpha$ the so-called geomagnetic angle
between the shower arrival direction and the local geomagnetic field
direction. In the following, we take ${\rm sin\alpha}\sim 1$.

Figure 1: Atmospheric density at the maximal shower development for various
inclinations. The chosen atmosphere is the U.S. standard atmosphere
parameterized according to Linsley [11]. For a given inclination, the maximal
shower development $X_{\rm max}$ position can be at different depths due to
shower-to-shower fluctuations. Very inclined air showers develop in less dense
atmospheres.

###  2.2 Analytical estimate of the air-shower lateral extent

In this section, we derive an analytical estimate of the particle lateral
extent at the shower front, which we define in this work as the lateral
separation induced by the geomagnetic field. As explained in Section 2.1,
electrons and positrons in the air-shower undergo a circular motion depending
on the geomagnetic field $B$ whose radius is the Larmor radius $r_{\rm L}$.
The transverse acceleration of electrons yields $a_{t}(t)={{\rm
d}v_{t}(t)}/{{\rm d}t}={\beta^{2}c^{2}}/{r_{\rm L(t)}}$ which in the limit
$\beta\sim 1$ gives ${{\rm d}v_{t}(t)}/{{\rm d}t}={c^{3}eB}/{\epsilon}$.
Assuming a continuous Bremsstrahlung particle energy loss in the air of
electrons with initial energy $\epsilon_{0}$, we get
$\epsilon(t)=\epsilon_{0}e^{-t/\tau}$. The transverse particle acceleration in
the shower front at time $t$ thus reads:

$\frac{{\rm d}^{2}x_{\rm lat}}{{\rm
d}t^{2}}=\frac{c^{3}eB}{\epsilon_{0}\,\exp(-t/\tau)}\,$ (2.2)

where $x_{\rm lat}$ is the particle transverse position (orthogonal to the
shower axis) and $\tau=l_{\rm rad}/c$ the Bremsstrahlung energy loss timescale
[8]. It yields a transverse velocity:

$\displaystyle v_{t}(t)=\frac{{\rm d}x_{\rm lat}}{{\rm
d}t}=\int_{0}^{t}\frac{{\rm d}v_{t}(t^{\prime})}{{\rm d}t^{\prime}}{\rm
d}t^{\prime}$ $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{c^{3}eB}{\epsilon_{0}e^{-t^{\prime}/\tau}}{\rm
d}t^{\prime}\ $ (2.3) $\displaystyle=$ $\displaystyle\frac{c^{3}\tau
eB}{\epsilon(t)}\left(1-\frac{\epsilon(t)}{\epsilon_{0}}\right).$

where $t=0$ corresponds to the creation of the particle. Integrating over
time, we finally obtain the transverse position of particles in the shower as
a function of time [7]: $x_{\rm
lat}(t)=\tau^{2}c^{3}eB\rm\,sin\,\alpha\,(e^{t/\tau}-1-{t}/{\tau})/\epsilon_{0}$.
The shower lateral extent then reads:

$\displaystyle\begin{split}l_{\rm lat}=2x_{\rm
lat}(t=\tau)=&2\left(\frac{l_{\rm rad}}{c}\right)^{2}\frac{c^{3}eB(\rm
e^{1}-2)}{\epsilon_{0}}\ \\\ \sim&3\times 10^{1}\,{\rm m}\left(\frac{\rho_{\rm
air}}{1\,{\rm{kg\,m^{-3}}}}\right)^{-2}\left(\frac{B}{50\,\rm\mu\rm
T}\right)\left(\frac{\epsilon_{0}}{100\,\rm MeV}\right)^{-1}\ .\end{split}$
(2.4)

where the factor 2 accounts for the dynamics of both positrons and electrons.

This lateral extent is thus specifically the extent induced by the geomagnetic
field for particles at high energy regime, which are the main particles
involved in the radio signal as shown in Section 3.

As demonstrated in Section 3.4, most of the emission is produced in the region
around the shower axis where the electron/positron density is the higher, and
more specifically at $X_{\rm max}$. The amplitude of the radio signal scales
linearly with the number of electrons and positrons and its power scales
quadratically.

If the shower lateral extent at $X_{\rm max}$ is larger than the spatial
coherence length, the signal becomes incoherent [7].

##  3 Contribution of air-shower particles in radio emission

One of the first steps to link the particle content of very inclined EAS with
the specific features of the radio signal consists in identifying the
particles that are specifically involved in the radio emission. For this
purpose, we will rely on microscopic numerical simulations.

###  3.1 Microscopic simulations with CORSIKA and CoREAS

We simulate air showers with CORSIKA 7.7500 [12], a detailed Monte-Carlo code
which follows the individual trajectories of the particles. Quantum variations
on $X_{\rm max}$, between air showers with identical primaries, are
intrinsically modelled. QGSJetII is chosen as the high-energy interaction
model and URQMD 1.3.1 as the low-energy interaction model. The primary
particle is a proton with two possible energies: $10^{17}$ eV or $10^{18}$ eV,
with a zenith angle between $30^{\circ}$ and $90^{\circ}$ and two possible
azimuth angles ($0^{\circ}$, $45^{\circ}$). The elevation site (1140 m) and
magnetic field correspond both to those of a site similar to Dunhuang, China,
where the GRAND prototype is being deployed [13]. The atmosphere is
approximated with the standard Linsley model [11]. We store the information on
the particles of the EAS (position, energy) that cross the shower plane at
$X_{\rm max}$ (if no other precision) thanks to the inclined plane option in
CORSIKA. The shower plane is the plane perpendicular to the direction of
propagation of the air shower. In this plane, the axes are usually aligned
with the Lorentz force and thus with the $({\bf v}\times{\bf B})$ direction
and with the $[{\bf v}\times({\bf v}\times{\bf B})]$ to get an orthogonal
coordinate system.

For the radio emission computation, we use CoREAS V1.4 [14], the extension of
CORSIKA that allows to simulate the electromagnetic radiation produced by the
particles of the shower. The same input parameters as CORSIKA are used, and
antennas are located on the ground, on the North axis at the position where
the radio signal is maximal (i.e., on the Cherenkov cone). This position is
found by running simulations and recording the signal for a set of antennas on
the North axis and keeping only the antenna position where the signal is
maximal. The time traces obtained thanks to CoREAS are used to define the
contribution of different shower stages and particle energy regimes in the
emission of the radio signal. Only the signal from the Cherenkov cone is kept
so that the comparison of the radio signal obtained between the simulations
does not depend on the position of the antennas.

Figure 2: (Top) Electrons normalized energy spectra at different shower depths
($X$) and inclinations ($\theta).$ The primary particle is a proton with
energy $10^{18}\,\rm eV$ with $\theta=0^{\circ}$ (dashed lines) and
$\theta=80^{\circ}$ (lines). (Bottom) Ratio of the very inclined
($\theta=80^{\circ}$) to vertical ($\theta=0^{\circ}$) electrons energy
spectra.

###  3.2 Energy spectra of leptons in air showers

We study the particle energy content in air showers at fixed primary energies
and different inclinations with CORSIKA simulations. We store the energy of
electrons and positrons of the air showers when they cross the shower plane at
different stages of the shower development.

The energy spectra of electrons for two different zenith angles are presented
in Figure 2 (top). The colored lines and dashed lines indicate the total
energy of particles contained in each energy bin, at a given atmospheric depth
$X$ of the shower for zenith angles $80^{\circ}$ and $0^{\circ}$ respectively.
The depth $X$ refers to a physical location along the longitudinal axis of the
air-shower depending on the amount of crossed atmosphere. At early stages
(with lower shower development stage $X$ and so at less amount of crossed
atmosphere), the shower contains higher energy particles than at later stages.
This feature is expected as these particles lose energy during the shower
development. Furthermore, the peak energy is shifted from a few tens of MeV
for vertical air showers to a few hundreds of MeV for very inclined air
showers. Figure 2 (bottom) shows the ratio of the two energy spectra around
the peak energy range: the shower contains more particles around and above the
critical energy $\epsilon_{\rm c}$ for very inclined air showers.

###  3.3 Contribution of particle of different energies to radio emission

Figure 3: Traces of the radio signal created at different electron/positron
energy (Lorentz factor) ranges for an air shower with inclination
$\theta=70^{\circ}$ and azimuth $\phi=0^{\circ}$. The primary particle is a
proton with energy $10^{17}\,\rm eV$. The antenna is located at the Cherenkov
cone, at a distance of 1100 m of the shower core on the North axis. The lines
represent the electric fields produced by particles in specific energy ranges
as indicated.

Figure 4: (Top) Contribution of different particle energy regimes responsible
for the radio signal. Each point corresponds to the mean value of the fraction
of the integral of the electric field in a given energy range on a set of 50
simulations. The error bars represent the uncertainty $\sigma/\sqrt{N}$ on
this value, where $N$ is the number of simulations. The points and error bars
have been artificially shifted for the same zenith angle for more visibility.
(Bottom) Sum of the contribution of particle energy regimes for $\gamma>100$
and $\gamma<100$.

In the next step, we examine the contribution of particles of various energies
to the radio signal. We run CoREAS simulations for a fixed primary with
different zenith angles, and record the electromagnetic radiation emitted by
the particles of the air showers in specific energy bins —i.e., in specific
Lorentz factor bins— and for the whole range of energy. In Figure 3, we
represent the traces of the radio signal for a zenith angle of $70^{\circ}$.
Only the East-West polarisation of the electric field is represented, as it is
largely dominant in this specific configuration: for an air shower with
azimuth angle $\phi=0\,^{\circ}$ and an antenna located on the North axis, the
North-South polarization of the electric field is negligible compared to the
East-West polarization.

The radiation energy, which is defined as the energy emitted by the air shower
in the form of radio waves, comes from the radiation generated by the
electromagnetic part of the air shower. [15]. To determine this radiation
energy, one method is to interpolate and integrate the measured energy fluence
on the ground [16]. This energy fluence $f$, which is the energy deposited per
area, can be broken down into two components: $f=f_{{\bf v}\times{\bf
B}}+f_{{\bf v}\times({\bf v}\times{\bf B})}$ with:

$\displaystyle f_{{\bf v}\times{\bf B}}=\epsilon_{0}c\Delta
t\sum_{i}E^{2}_{{\bf v}\times{\bf B}}({\bf r},t_{i})$ (3.1) $\displaystyle
f_{{\bf v}\times({\bf v}\times{\bf B})}=\epsilon_{0}c\Delta
t\sum_{i}E^{2}_{{\bf v}\times({\bf v}\times{\bf B})}({\bf r},t_{i})$ (3.2)

with $\epsilon_{0}$ the vacuum permittivity, $c$ the speed of light, $N$ the
total number of electrons and positrons and $\Delta t$ the sampling interval
of the electric field ${\bf E}({\bf r},t)$ which is broken down into two
components.

According to Eq 3.1 and Eq 3.2, the energy fluence is the time-integral of the
radio pulse. We then integrate the electric field of each pulse over time and
normalize by dividing it by the integral of the total electric field, i.e., by
the sum of each integral. This enables us to compare the contribution of the
different particle populations to the radio signal for each zenith angle.

Figure 4 (top) shows the mean values and uncertainties of this fraction
obtained for a set of 50 simulations on each zenith angle to take into account
the shower-to-shower fluctuations. It appears that the electric field is
dominated by contributions from increasingly energetic particles as the
inclination increases. It is highlighted by Figure 4 (bottom) that represents
the sum on the mean values split into two energy ranges: $\gamma>100$ and
$\gamma<100$. For inclinations larger than $70^{\circ}$, the contribution from
low energy particles can be neglected.

We can therefore consider that the electric field is dominated by [50, 1500]
MeV particles, which would be consistent with the energy spectra of Figure 2,
peaking in this range. Energy losses are mainly due to Bremmstrahlung in this
energy range. For showers with a lower inclination, another energy regime is
dominant, in the range [10, 500] MeV.

###  3.4 Radio emission point in the air-shower

The contribution of different shower phases can also be estimated for air
showers with different zenith angles. Following the same procedure as before,
we generate with CoREAS the electric field emitted by particles with energy in
the range [50, 1500] MeV or [10, 500] MeV according to the zenith angle (see
Section 3.3). Figure 5 represents the time traces of the East-West
polarisation of the electric field for $\theta=85^{\circ}$. Comparing the
fraction of the integral of the pulses that arrive at an antenna located on
the Cherenkov cone as shown in Figure 6, we can see that the most
representative region of the radio emission is around the maximal development
of the shower $X_{\rm max}$ whatever the inclination. This confirms the
commonly admitted notion that the radio emission can be assumed to be produced
at $X_{\rm max}$, also for very inclined air showers.

Figure 5: Traces of the radio signal created at different stages of the shower
development by particles in the energy range 100 < $\gamma$ < 3000\. The
primary particle is a proton with energy $10^{17}\,\rm eV$, zenith angle
$\theta=85^{\circ}$ and azimuth $\phi=0^{\circ}$. The antenna is located at
the Cherenkov cone, on the North axis at a distance of 14000 m from the shower
core. Figure 6: Contribution of shower stages in radio emission. Each point
corresponds to the fraction of the integral of the electric field in a given
slant depth range. For all the simulations, $X_{\rm max}=585\,\rm g/cm^{2}$.

##  4 Particle distribution and lateral extension of very inclined air showers

In the previous sections, we found that the energy range of the bulk of the
particles is shifted towards higher energies for very inclined air showers.
The identification of particles responsible for the radio signal allow us to
study the particle content of air showers in the relevant energy ranges and to
highlight the distinct features on the particle distribution for different
inclinations. We also found that the shower development stage $X$ that
produces the radio signal does not vary with the zenith angle. The robustness
of this last parameter can be used to give a universal definition of the
lateral extent of electrons and positrons in the air shower in two energy
regimes depending on the inclination.

Figure 7: Density of electrons and positrons in the shower plane at $X_{\rm
max}$. The color (linear) contours delimit the region within which the number
of electrons (positrons) divided by their total number in the shower at
$X_{\rm max}$ within the selected energy range is as indicated. The primary
particle is a proton with energy $10^{18}$ eV and azimuth $\phi=0^{\circ}$.
(Top) $\theta=60^{\circ}$ and only particles in the energy range [10, 500] MeV
are selected. (Bottom) $\theta=87^{\circ}$ and only particles in the range
[50, 1500] MeV are selected.

Figure 8:  Electron and positron distributions projected on the $({\bf
v}\times{\bf B})$ and $({\bf v}\times[{\bf v}\times{\bf B}])$ axis at $X_{\rm
max}$ for various inclinations. The primary particle is a proton with energy
$10^{17}$ eV. For $\theta=80^{\circ}$ and $\theta=87^{\circ}$, only particles
in the energy range [50, 1500] MeV are selected whereas for
$\theta=40^{\circ}$ and $\theta=60^{\circ}$, only particles in the energy
range [10, 500] MeV are selected. Each point corresponds to the mean value of
the number of electrons or positrons on a set of 100 simulations.

###  4.1 Particle distribution of air showers for various inclination regimes

We study the lateral distribution of electrons and positrons in the shower
plane $[{\bf v}\times{\bf B},{\bf v}\times({\bf v}\times{\bf B})]$ at $X_{\rm
max}$ with CORSIKA simulations, $\bf v$ being the shower axis and $\bf B$ the
local magnetic field direction. Figure 7 represents the 2D particle density
contained inside isocontours in the shower plane for a vertical
($\theta=60^{\circ}$) and a very inclined ($\theta=87^{\circ}$) air showers in
specific energy ranges (see Section 3.3). For both zenith angles, most of the
particles are distributed at the center of the shower, where most secondary
particles are produced, but the spatial distribution look different. For the
vertical air shower, electrons and positrons are isotropically dispersed
around their barycenter. The charge separation along the $({\bf v}\times{\bf
B})$ axis, which is a deflection in opposite directions as mentioned in
Section 2 under the influence of the Lorentz Force $\bf F=\pm q{\bf
v}\times{\bf B}$, is concentrated in the first meters around the shower axis
of coordinates $(0,0)$. In this inclination regime, the particle positions are
constrained within the Moliere radius $R_{\rm M}$ given by the multiple
Coulomb scatterings from high air density [17]: there is therefore a
competition between Coulomb diffusion and charge separation. At a height of 4
km (typical height of the shower maximum for vertical air showers), the
atmospheric density is $\rho_{\rm air}\sim 0.82{\rm mg.cm^{-3}}$ that yields
to $R_{\rm M}\sim 117{\rm m}$ [8]. For the very inclined air shower, the
charge separation between electrons and positrons is stronger and coupled with
a drastic lateral extent increase in the $({\bf v}\times{\bf B})$ direction.
It is a regime where air showers propagate in less dense atmosphere, resulting
in a higher mean free path between collisions and particle deflection
dominates, giving anisotropic particle distributions.

These specific features are highlighted in Figure 8 which represents the mean
number of electrons and positrons projected along the $({\bf v}\times{\bf B})$
or $[{\bf v}\times({\bf v}\times{\bf B})]$ axis for
$\theta=40^{\circ},60^{\circ},80^{\circ},87^{\circ}$ on a set of $100$
simulations. While the lateral extent clearly drastically increases in the
$({\bf v}\times{\bf B})$ direction for very inclined air showers (there are
still 10% of electrons at $1000$ m from the shower core in the $-({\bf
v}\times{\bf B})$ direction for a very inclined air shower for only 0.02% of
electrons at the same distance for a vertical air shower), their charge
separation is also larger. Along the $[{\bf v}\times({\bf v}\times{\bf B})]$
direction, the size of the shower front does not depend on deflection induced
by the geomagnetic field: particles are thus isotropically dispersed around
the shower axis for all inclinations.

###  4.2 Energy distribution in the shower plane

The mean energy distribution of electrons and positrons in the specific energy
range responsible for the radio signal is studied in the $({\bf v}\times{\bf
B})$ direction of the shower plane. Figure 9 shows this distribution at
$X_{\rm max}$ for the same 4 zenith angles as previously:
$\theta=40^{\circ},60^{\circ},80^{\circ},87^{\circ}$ on a set of 100
simulations. As expected, we observe that the most energetic particles are
contained at the center of the shower while the less energetic ones are much
more deflected by the Earth magnetic field. It is consistent with the expected
rigidity of the particles $R=r_{\rm L}B$ which is a measure of the particle
resistance to deflection by a magnetic field. It depends on the Larmor radius
$r_{\rm L}$, which defines the circular motion of a particle in a uniform
magnetic field, and which is given in Section 2.1 by $r_{\rm L}\sim 3.3\times
10^{4}\,{\rm m}\,\sin\alpha\,(\epsilon/100\,{\rm MeV})(B/10\,\mu{\rm
T})^{-1}$. The Larmor radius scales linearly with the particle energy and thus
more energetic particles have greater rigidity than less energetic ones.

Figure 9: Electrons/positrons mean energy distribution on the $({\bf
v}\times{\bf B})$ axis at $X_{\rm max}$ for various inclinations. (Top) For
vertical showers with $\theta=40^{\circ}$ and $\theta=60^{\circ}$ where only
particles in the energy range [10, 500] MeV are selected. (Bottom) for very
inclined air showers with $\theta=80^{\circ}$ and $\theta=87^{\circ}$ where
only particles in the energy range [50, 1500] MeV are selected.

###  4.3 Lateral extent estimation

The spatial position of particles in the shower plane $[{\bf v}\times{\bf
B},{\bf v}\times({\bf v}\times{\bf B})]$ at $X_{\rm max}$ is easily accessible
with CORSIKA, as shown in Figure 7. To estimate the lateral extension of the
particles, we therefore work on the projection of the particle positions along
the $\textbf{v}\times\textbf{B}$ axis, as this deflection produces a time
varying current mainly responsible for the radio emission. Figure 10
represents the cumulative fraction of particle energies at fixed distances of
the shower core and the corresponding particle fractions for three different
inclinations. It appears that the relationship between energy fraction and
particle number at a given distance is almost a power law whatever the zenith
angle.

At the distance from the shower axis where the shower contains 90% of the
particles, more than 90% of the energy is also contained. This distance is
thus representative of the lateral extent and is chosen as the lateral extent
estimator in this work.

Figure 10:  Fraction of energy contained at fixed distance in the shower plane
at $X_{\rm max}$ and corresponding fraction of particles for 3 zenith angles..
The abscissa axis represents the fraction of particles contained along the
$\textbf{v}\times\textbf{B}$ axis for 3 zenith angles in specific energy
ranges: for $\theta=40^{\circ}$, only particles in the range [10, 500] MeV are
selected, for $\theta=82^{\circ}$ and $\theta=70^{\circ}$, the range [50,
1500] MeV is selected. For each fraction of particles and each zenith angle
corresponds a fixed distance from the shower core. For each of this fixed
distance the cumulative fraction of energy contained from the shower core is
calculated. Figure 11: Lateral extent of particles in the shower plane at
$X_{\rm max}$ for air showers with inclinations between $30^{\circ}$ and
$85^{\circ}$ on a set of 200 simulations. The air-density at $X_{\rm max}$ is
also represented. The dashed line is the analytical calculation of the lateral
extent given in Section 2.4 and the dotted line stands for the Molière radius.

###  4.4 Lateral extent as a function of inclination

The lateral extent is computed on a set of 200 simulations obtained with
CORSIKA for air showers with zenith angles varying between $30^{\circ}$ and
$85^{\circ}$ –and thus for different atmospheric densities as shown in Figure
11. According to Sections 3.2 and 3.3, the energy of the particles responsible
for the bulk of the radio emission is a function of the inclination and two
different ranges of particle energies are selected: [50, 1500] MeV for
particles above $70^{\circ}$ and [10, 500] for particles below.

We observe that the lateral extent increases with zenith angle, because
inclined air showers develop in thinner atmosphere than vertical ones. The
black dashed line corresponds to the analytical calculation of the extension
induced by the geomagnetic field, presented in Eq. (2.4) for an energy of 100
MeV. The corresponding particles are indeed representative of the leptonic
content in very inclined air showers (see Section 3.2). The blue dotted line
is the Molière radius given in Ref. [8] by $R_{\rm M}=(9.6\,\rm
g/cm^{2})/\rho_{\rm air}$ which represents the spread due to scattering.
Hence, two distinct regimes are highlighted at low and high inclinations as
mentioned in Section 4.1: whereas the lateral extent of vertical air showers
is driven by multiple scattering, very inclined air showers experience a
drastic lateral extent increase due to the Earth’s magnetic field.

##  5 Conclusion

Using CORSIKA and CoREAS simulations and taking into account shower-to-shower
fluctuations (i.e. variations on the size of the shower, on the location of
$X_{\rm max}$ for air showers with exactly the same first parameters), we
demonstrate the existence of two different particle energy regimes responsible
for the radio signal depending on the inclination: the range [10, 500] MeV for
vertical air showers and [50, 1500] MeV for very inclined ones. Thus, the
population size is larger at higher energies for very inclined air showers. We
also demonstrate that the main region of radio signal emission is around
$X_{\rm max}$ for all zenith angles.

Using these results, we study the particle content of air showers subjected to
a strong magnetic field $(B=56\,\mu T)$ for various inclinations in the shower
plane. We find a drastic lateral extent increase for very inclined air showers
in the $({\bf v}\times{\bf B})$ direction whereas this extent is constrained
for vertical ones. Based on these observations, we construct a global
estimator of the particle lateral extent at $X_{\rm max}$ in the two different
energy regimes. The distance to the shower axis where 90% of the particles are
contained can be safely used as an estimator of the lateral extent.

We also propose an analytical computation of the lateral extent for very
inclined showers based on the geomagnetic effect which is in agreement with
the estimator of the lateral extent coming from our simulations. We thus find
that the lateral extent is driven by deflections due to the Earth’s magnetic
field for very inclined air showers because they develop in thinner atmosphere
whereas it mainly depends on diffusion due to multiple Coulomb scattering for
vertical air showers. Thus, the drastic lateral extent increase induced for
very inclined air showers subjected to a strong enough magnetic field results
in a loss of coherence and in a drop of the radio signal. The transition to
incoherent radio emission for $B=56{\rm\mu T}$ and a radio wave frequency
$\nu=50{\rm MHz}$ is expected for $\rho_{\rm air}<0.27{\rm kg.m^{-3}}$
($\theta>75^{\circ}$) (more details can be found in Ref [18]). It corresponds
to air showers with a typical lateral extent $l_{\rm lat}\sim 400{\rm m}$ as
shown in Figure 11, which is of a same order of magnitude as the coherence
length of the radio emission coming from an air shower with same parameters
[18]. This specific feature will strongly impact the reconstruction strategies
of the next generation of extended radio arrays and could, for example, allow
us to discriminate between cosmic-ray air showers and neutrinos that also have
very inclined trajectories but which develop in much denser atmosphere [18].

## Acknowledgments

The authors thank V. Niess and M. Tueros for their careful reading, as well as
the GRAND-Paris team, K. de Vries, T. Huege and the GRAND team at KIT for
fruitful discussions. This work was supported by the Programme National des
Hautes Energies of CNRS/INSU with INP and IN2P3, co-funded by CEA and CNES.
Simulations were performed using the computing resources at the CCIN2P3
Computing Centre (Lyon/Villeurbanne – France), partnership between CNRS/IN2P3
and CEA/DSM/Irfu. This work is part of the NuTRIG project, supported by the
Agence Nationale de la Recherche (ANR-21-CE31-0025; France) and the Deutsche
Forschungsgemeinschaft (DFG; Projektnummer 490843803; Germany).

## References

  * [1] T. Huege, _Radio detection of cosmic ray air showers in the digital era_ , _Physics Reports_ 620 (2016) 1.
  * [2] F.G. Schröder, _Radio detection of cosmic-ray air showers and high-energy neutrinos_ , _Progress in Particle and Nuclear Physics_ 93 (2017) 1.
  * [3] A. Nelles, S. Buitink, A. Corstanje, J. Enriquez, H. Falcke, J. Hörandel et al., _The radio emission pattern of air showers as measured with LOFAR—a tool for the reconstruction of the energy and the shower maximum_ , _Journal of Cosmology and Astroparticle Physics_ 2015 (2015) 018.
  * [4] J. Álvarez-Muñiz, R.A. Batista, A.B. V., J. Bolmont, M. Bustamante, W. Carvalho et al., _The giant radio array for neutrino detection (GRAND): Science and design_ , _Science China Physics, Mechanics and Astronomy_ 63 (2019) .
  * [5] D. Southall, C. Deaconu, V. Decoene, E. Oberla, A. Zeolla, J. Alvarez-Muñiz et al., _Design and initial performance of the prototype for the BEACON instrument for detection of ultrahigh energy particles_ , _Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment_ 1048 (2023) 167889.
  * [6] A. Castellina and Pierre Auger Collaboration, _AugerPrime: the Pierre Auger Observatory Upgrade_ , in _European Physical Journal Web of Conferences_ , vol. 210 of _European Physical Journal Web of Conferences_ , p. 06002, Oct., 2019, DOI [1905.04472].
  * [7] S. Chiche et al., _New paradigm and radio signatures for very inclined air showers_ , in _ICRC Conference Proceedings_ , 2023.
  * [8] O. Scholten, K. Werner and F. Rusydi, _A macroscopic description of coherent geo-magnetic radiation from cosmic-ray air showers_ , _Astroparticle Physics_ 29 (2008) 94.
  * [9] F.G. Schröder, _Radio detection of cosmic-ray air showers and high-energy neutrinos_ , _Progress in Particle and Nuclear Physics_ 93 (2017) 1.
  * [10] T.K. Gaisser, R. Engel and E. Resconi, _Cosmic Rays and Particle Physics_ (2016).
  * [11] A.M. Hillas, _The Origin of Ultra-High-Energy Cosmic Rays_ , _Annual Review of Astron and Astrophys_ 22 (1984) 425.
  * [12] D. Heck, J. Knapp, J.N. Capdevielle, G. Schatz and T. Thouw, _CORSIKA: a Monte Carlo code to simulate extensive air showers._ (1998).
  * [13] J. Torres de Mello Neto, _The Giant Array for Neutrino Detection (GRAND)_ , in _ICRC Conference Proceedings_ , 2023.
  * [14] T. Huege, M. Ludwig and C.W. James, _Simulating radio emission from air showers with CoREAS_ , in _AIP Conference Proceedings_ , AIP, 2013, DOI.
  * [15] C. Glaser, M. Erdmann, J.R. Hörandel, T. Huege and J. Schulz, _Simulation of radiation energy release in air showers_ , _Journal of Cosmology and Astroparticle Physics_ 2016 (2016) 024.
  * [16] A. Aab, P. Abreu, M. Aglietta, E. Ahn, I.A. Samarai, I. Albuquerque et al., _Measurement of the radiation energy in the radio signal of extensive air showers as a universal estimator of cosmic-ray energy_ , _Physical Review Letters_ 116 (2016) .
  * [17] Particle Data Group collaboration, _Review of particle physics_ , _Phys. Rev. D_ 98 (2018) 030001.
  * [18] S. Chiche et al., _Loss of coherence and change in emission physics for radio emission from very inclined air showers_ , _Physical Review Letters (submitted)_ .


* Fiorentino et al. (2015) Fiorentino, G., Bono, G., Monelli, M., et al. 2015, ApJ, 798, L12, doi: 10.1088/2041-8205/798/1/L12
  * Fiorentino et al. (2017) Fiorentino, G., Monelli, M., Stetson, P. B., et al. 2017, A&A, 599, A125, doi: 10.1051/0004-6361/201629501
  * Fitzpatrick (1999) Fitzpatrick, E. L. 1999, PASP, 111, 63, doi: 10.1086/316293
  * Flaugher et al. (2015) Flaugher, B., Diehl, H. T., Honscheid, K., et al. 2015, AJ, 150, 150, doi: 10.1088/0004-6256/150/5/150
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306, doi: 10.1086/670067
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1, doi: 10.1051/0004-6361/201833051
  * Gallart et al. (2004) Gallart, C., Aparicio, A., Freedman, W. L., et al. 2004, AJ, 127, 1486, doi: 10.1086/381955
  * Gallart et al. (2019) Gallart, C., Bernard, E. J., Brook, C. B., et al. 2019, Nature Astronomy, 3, 932, doi: 10.1038/s41550-019-0829-5
  * Górski et al. (2005) Górski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759, doi: 10.1086/427976
  * Gravity Collaboration et al. (2019) Gravity Collaboration, Abuter, R., Amorim, A., et al. 2019, A&A, 625, L10, doi: 10.1051/0004-6361/201935656
  * Greco et al. (2005) Greco, C., Clementini, G., Held, E. V., et al. 2005, arXiv e-prints, astro.  https://arxiv.org/abs/astro-ph/0507244
  * Guldenschuh et al. (2005) Guldenschuh, K. A., Layden, A. C., Wan, Y., et al. 2005, PASP, 117, 721, doi: 10.1086/431178
  * Helmi (2008) Helmi, A. 2008, A&A Rev., 15, 145, doi: 10.1007/s00159-008-0009-6
  * Helmi et al. (2018) Helmi, A., Babusiaux, C., Koppelman, H. H., et al. 2018, Nature, 563, 85, doi: 10.1038/s41586-018-0625-x
  * Hernitschek et al. (2016) Hernitschek, N., Schlafly, E. F., Sesar, B., et al. 2016, ApJ, 817, 73, doi: 10.3847/0004-637X/817/1/73
  * Holl et al. (2018) Holl, B., Audard, M., Nienartowicz, K., et al. 2018, A&A, 618, A30, doi: 10.1051/0004-6361/201832892
  * Homma et al. (2018) Homma, D., Chiba, M., Okamoto, S., et al. 2018, PASJ, 70, S18, doi: 10.1093/pasj/psx050
  * Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Iorio et al. (2018) Iorio, G., Belokurov, V., Erkal, D., et al. 2018, MNRAS, 474, 2142, doi: 10.1093/mnras/stx2819
  * Ivezic et al. (2004) Ivezic, Z., Lupton, R., Schlegel, D., et al. 2004, in Astronomical Society of the Pacific Conference Series, Vol. 317, Milky Way Surveys: The Structure and Evolution of our Galaxy, ed. D. Clemens, R. Shah, & T. Brainerd, 179.  https://arxiv.org/abs/astro-ph/0309074
  * Ivezić et al. (2005) Ivezić, Ž., Vivas, A. K., Lupton, R. H., & Zinn, R. 2005, AJ, 129, 1096, doi: 10.1086/427392
  * Jerzykiewicz & Wenzel (1977) Jerzykiewicz, M., & Wenzel, W. 1977, Acta Astron., 27, 35
  * Johnston et al. (2008) Johnston, K. V., Bullock, J. S., Sharma, S., et al. 2008, ApJ, 689, 936, doi: 10.1086/592228
  * Johnston et al. (2002) Johnston, K. V., Spergel, D. N., & Haydn, C. 2002, ApJ, 570, 656, doi: 10.1086/339791
  * Jones et al. (2001) Jones, E., Oliphant, T., Peterson, P., et al. 2001, SciPy: Open source scientific tools for Python.  http://www.scipy.org/
  * Kafle et al. (2012) Kafle, P. R., Sharma, S., Lewis, G. F., & Bland -Hawthorn, J. 2012, ApJ, 761, 98, doi: 10.1088/0004-637X/761/2/98
  * Karczmarek et al. (2017) Karczmarek, P., Pietrzyński, G., Górski, M., Gieren, W., & Bersier, D. 2017, AJ, 154, 263, doi: 10.3847/1538-3881/aa9574
  * Koposov et al. (2019) Koposov, S. E., Belokurov, V., Li, T. S., et al. 2019, MNRAS, 485, 4726, doi: 10.1093/mnras/stz457
  * Kuehn et al. (2013) Kuehn, C. A., Dame, K., Smith, H. A., et al. 2013, AJ, 145, 160, doi: 10.1088/0004-6256/145/6/160
  * Lyke et al. (2020) Lyke, B. W., Higley, A. N., McLane, J. N., et al. 2020, ApJS, 250, 8, doi: 10.3847/1538-4365/aba623
  * MacLeod et al. (2012) MacLeod, C. L., Ivezić, Ž., Sesar, B., et al. 2012, ApJ, 753, 106, doi: 10.1088/0004-637X/753/2/106
  * Marconi (2012) Marconi, M. 2012, Memorie della Societa Astronomica Italiana Supplementi, 19, 138
  * Marconi et al. (2015) Marconi, M., Coppola, G., Bono, G., et al. 2015, ApJ, 808, 50, doi: 10.1088/0004-637X/808/1/50
  * Martínez-Vázquez et al. (2015) Martínez-Vázquez, C. E., Monelli, M., Bono, G., et al. 2015, MNRAS, 454, 1509, doi: 10.1093/mnras/stv2014
  * Martínez-Vázquez et al. (2016a) Martínez-Vázquez, C. E., Stetson, P. B., Monelli, M., et al. 2016a, MNRAS, 462, 4349, doi: 10.1093/mnras/stw1895
  * Martínez-Vázquez et al. (2016b) Martínez-Vázquez, C. E., Monelli, M., Gallart, C., et al. 2016b, MNRAS, 461, L41, doi: 10.1093/mnrasl/slw093
  * Martínez-Vázquez et al. (2017) Martínez-Vázquez, C. E., Monelli, M., Bernard, E. J., et al. 2017, ApJ, 850, 137, doi: 10.3847/1538-4357/aa9381
  * Martínez-Vázquez et al. (2019) Martínez-Vázquez, C. E., Vivas, A. K., Gurevich, M., et al. 2019, MNRAS, 490, 2183, doi: 10.1093/mnras/stz2609
  * Mateu et al. (2018) Mateu, C., Read, J. I., & Kawata, D. 2018, MNRAS, 474, 4112, doi: 10.1093/mnras/stx2937
  * Matthews (1975) Matthews, B. W. 1975, BBA, 405, 442, doi: 10.1016/0005-2795(75)90109-9
  * McConnachie (2012) McConnachie, A. W. 2012, AJ, 144, 4, doi: 10.1088/0004-6256/144/1/4
  * Medina et al. (2018) Medina, G. E., Muñoz, R. R., Vivas, A. K., et al. 2018, ApJ, 855, 43, doi: 10.3847/1538-4357/aaad02
  * Mo et al. (2010) Mo, H., van den Bosch, F. C., & White, S. 2010, Galaxy Formation and Evolution (Cambridge University Press)
  * Morganson et al. (2018) Morganson, E., Gruendl, R. A., Menanteau, F., et al. 2018, PASP, 130, 074501, doi: 10.1088/1538-3873/aab4ef
  * Munoz et al. (2018) Munoz, R. R., Cote, P., Santana, F. A., et al. 2018, arXiv e-prints, arXiv:1806.06891.  https://arxiv.org/abs/1806.06891
  * Nakoneczny et al. (2019) Nakoneczny, S., Bilicki, M., Solarz, A., et al. 2019, A&A, 624, A13, doi: 10.1051/0004-6361/201834794
  * Nandra et al. (1997) Nandra, K., George, I. M., Mushotzky, R. F., Turner, T. J., & Yaqoob, T. 1997, ApJ, 476, 70, doi: 10.1086/303600
  * Neilsen et al. (2016) Neilsen, Jr., E. H., Bernstein, G., Gruendl, R., & Kent, S. 2016, Limiting Magnitude, $\tau$, $t_{eff}$, and Image Quality in DES Year 1, Tech. Rep. FERMILAB-TM-2610-AE-CD, Fermilab, doi: 10.2172/1250877
  * Neilsen et al. (2019) Neilsen, Eric H., J., Annis, J. T., Diehl, H. T., et al. 2019, arXiv e-prints, arXiv:1912.06254.  https://arxiv.org/abs/1912.06254
  * Nidever et al. (2019) Nidever, D. L., Olsen, K., Choi, Y., et al. 2019, ApJ, 874, 118, doi: 10.3847/1538-4357/aafaf7
  * Oosterhoff (1939) Oosterhoff, P. T. 1939, The Observatory, 62, 104
  * Pace & Li (2019) Pace, A. B., & Li, T. S. 2019, ApJ, 875, 77, doi: 10.3847/1538-4357/ab0aee
  * Paterno (2004) Paterno, M. 2004, doi: 10.2172/15017262
  * Pedregosa et al. (2012) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2012, arXiv e-prints, arXiv:1201.0490.  https://arxiv.org/abs/1201.0490
  * Pieres et al. (2017) Pieres, A., Santiago, B. X., Drlica-Wagner, A., et al. 2017, MNRAS, 468, 1349, doi: 10.1093/mnras/stx507
  * Pieres et al. (2020) Pieres, A., Girardi, L., Balbinot, E., et al. 2020, MNRAS, 497, 1547, doi: 10.1093/mnras/staa1980
  * Pila-Díez et al. (2015) Pila-Díez, B., de Jong, J. T. A., Kuijken, K., van der Burg, R. F. J., & Hoekstra, H. 2015, A&A, 579, A38, doi: 10.1051/0004-6361/201425457
  * Plummer (1911) Plummer, H. C. 1911, MNRAS, 71, 460, doi: 10.1093/mnras/71.5.460
  * Powers (2008) Powers, D. 2008, Mach. Learn. Technol., 2, 37
  * Price-Whelan et al. (2019) Price-Whelan, A. M., Mateu, C., Iorio, G., et al. 2019, AJ, 158, 223, doi: 10.3847/1538-3881/ab4cef
  * Prudil et al. (2018) Prudil, Z., Grebel, E. K., Dékány, I., & Smolec, R. 2018, MNRAS, 480, 669, doi: 10.1093/mnras/sty1885
  * Ramos et al. (2020) Ramos, P., Mateu, C., Antoja, T., et al. 2020, arXiv e-prints, arXiv:2002.11142.  https://arxiv.org/abs/2002.11142
  * Rimoldini et al. (2019) Rimoldini, L., Holl, B., Audard, M., et al. 2019, A&A, 625, A97, doi: 10.1051/0004-6361/201834616
  * Rousseeuw & Croux (1993) Rousseeuw, P. J., & Croux, C. 1993, Journal of the American Statistical Association, 88, 1273.  http://www.jstor.org/stable/2291267
  * Rusakov et al. (2020) Rusakov, V., Monelli, M., Gallart, C., et al. 2020, arXiv e-prints, arXiv:2002.09714.  https://arxiv.org/abs/2002.09714
  * Saha (1985) Saha, A. 1985, ApJ, 289, 310, doi: 10.1086/162890
  * Saha et al. (2010) Saha, A., Olszewski, E. W., Brondel, B., et al. 2010, AJ, 140, 1719, doi: 10.1088/0004-6256/140/6/1719
  * Sanderson et al. (2017) Sanderson, R. E., Secunda, A., Johnston, K. V., & Bochanski, J. J. 2017, MNRAS, 470, 5014, doi: 10.1093/mnras/stx1614
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103, doi: 10.1088/0004-637X/737/2/103
  * Schlegel et al. (1998) Schlegel, D. J., Finkbeiner, D. P., & Davis, M. 1998, ApJ, 500, 525, doi: 10.1086/305772
  * Schneider et al. (2010) Schneider, D. P., Richards, G. T., Hall, P. B., et al. 2010, AJ, 139, 2360, doi: 10.1088/0004-6256/139/6/2360
  * Sesar et al. (2011) Sesar, B., Jurić, M., & Ivezić, Ž. 2011, ApJ, 731, 4, doi: 10.1088/0004-637X/731/1/4
  * Sesar et al. (2010) Sesar, B., Ivezić, Ž., Grammer, S. H., et al. 2010, ApJ, 708, 717, doi: 10.1088/0004-637X/708/1/717
  * Sesar et al. (2014) Sesar, B., Banholzer, S. R., Cohen, J. G., et al. 2014, ApJ, 793, 135, doi: 10.1088/0004-637X/793/2/135
  * Sesar et al. (2017) Sesar, B., Hernitschek, N., Mitrović, S., et al. 2017, AJ, 153, 204, doi: 10.3847/1538-3881/aa661b
  * Sevilla-Noarbe et al. (in prep.) Sevilla-Noarbe, I., Bechtol, K., et al. in prep., arXiv
  * Sharma et al. (2011a) Sharma, S., Bland-Hawthorn, J., Johnston, K. V., & Binney, J. 2011a, ApJ, 730, 3, doi: 10.1088/0004-637X/730/1/3
  * Sharma et al. (2011b) Sharma, S., Johnston, K. V., Majewski, S. R., Bullock, J., & Muñoz, R. R. 2011b, ApJ, 728, 106, doi: 10.1088/0004-637X/728/2/106
  * Sheffield et al. (2014) Sheffield, A. A., Johnston, K. V., Majewski, S. R., et al. 2014, ApJ, 793, 62, doi: 10.1088/0004-637X/793/1/62
  * Shipp et al. (2018) Shipp, N., Drlica-Wagner, A., Balbinot, E., et al. 2018, ApJ, 862, 114, doi: 10.3847/1538-4357/aacdab
  * Simm et al. (2015) Simm, T., Saglia, R., Salvato, M., et al. 2015, A&A, 584, A106, doi: 10.1051/0004-6361/201526859
  * Simon et al. (2020) Simon, J. D., Li, T. S., Erkal, D., et al. 2020, ApJ, 892, 137, doi: 10.3847/1538-4357/ab7ccb
  * Smith (1995) Smith, H. A. 1995, Cambridge Astrophysics Series, 27
  * Sokolovsky et al. (2017) Sokolovsky, K. V., Gavras, P., Karampelas, A., et al. 2017, MNRAS, 464, 274, doi: 10.1093/mnras/stw2262
  * Soszyński et al. (2017) Soszyński, I., Udalski, A., Szymański, M. K., et al. 2017, Acta Astron., 67, 103, doi: 10.32023/0001-5237/67.2.1
  * Soszyński et al. (2019) —. 2019, Acta Astron., 69, 87, doi: 10.32023/0001-5237/69.2.1
  * Stetson (1996) Stetson, P. B. 1996, PASP, 108, 851, doi: 10.1086/133808
  * Stetson et al. (2014) Stetson, P. B., Fiorentino, G., Bono, G., et al. 2014, PASP, 126, 616, doi: 10.1086/677352
  * Stringer et al. (2019) Stringer, K. M., Long, J. P., Macri, L. M., et al. 2019, AJ, 158, 16, doi: 10.3847/1538-3881/ab1f46
  * Tie et al. (2017) Tie, S. S., Martini, P., Mudd, D., et al. 2017, AJ, 153, 107, doi: 10.3847/1538-3881/aa5b8d
  * Tolstoy et al. (2009) Tolstoy, E., Hill, V., & Tosi, M. 2009, ARA&A, 47, 371, doi: 10.1146/annurev-astro-082708-101650
  * Torrealba et al. (2015) Torrealba, G., Catelan, M., Drake, A. J., et al. 2015, MNRAS, 446, 2251, doi: 10.1093/mnras/stu2274
  * Torrealba et al. (2019) Torrealba, G., Belokurov, V., Koposov, S. E., et al. 2019, MNRAS, 488, 2743, doi: 10.1093/mnras/stz1624
  * Udalski et al. (2001) Udalski, A., Wyrzykowski, L., Pietrzynski, G., et al. 2001, Acta Astron., 51, 221\.  https://arxiv.org/abs/astro-ph/0109446
  * Van Der Walt et al. (2011) Van Der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2
  * Vivas et al. (2020a) Vivas, A. K., Martínez-Vázquez, C., & Walker, A. R. 2020a, ApJS, 247, 35, doi: 10.3847/1538-4365/ab67c0
  * Vivas & Zinn (2006) Vivas, A. K., & Zinn, R. 2006, AJ, 132, 714, doi: 10.1086/505200
  * Vivas et al. (2001) Vivas, A. K., Zinn, R., Andrews, P., et al. 2001, ApJ, 554, L33, doi: 10.1086/320915
  * Vivas et al. (2004) Vivas, A. K., Zinn, R., Abad, C., et al. 2004, AJ, 127, 1158, doi: 10.1086/380929
  * Vivas et al. (2017) Vivas, A. K., Saha, A., Olsen, K., et al. 2017, AJ, 154, 85, doi: 10.3847/1538-3881/aa7fed
  * Vivas et al. (2020b) Vivas, A. K., Walker, A. R., Martínez-Vázquez, C. E., et al. 2020b, MNRAS, 492, 1061, doi: 10.1093/mnras/stz3393
  * Walker (1989) Walker, A. R. 1989, PASP, 101, 570, doi: 10.1086/132470
  * Wang et al. (2019) Wang, M. Y., de Boer, T., Pieres, A., et al. 2019, ApJ, 881, 118, doi: 10.3847/1538-4357/ab31a9
  * Watkins et al. (2009) Watkins, L. L., Evans, N. W., Belokurov, V., et al. 2009, MNRAS, 398, 1757, doi: 10.1111/j.1365-2966.2009.15242.x
  * Welch & Stetson (1993) Welch, D. L., & Stetson, P. B. 1993, AJ, 105, 1813, doi: 10.1086/116556
  * White & Frenk (1991) White, S. D. M., & Frenk, C. S. 1991, ApJ, 379, 52, doi: 10.1086/170483
  * Xue et al. (2015) Xue, X.-X., Rix, H.-W., Ma, Z., et al. 2015, ApJ, 809, 144, doi: 10.1088/0004-637X/809/2/144
  * Zinn et al. (2014) Zinn, R., Horowitz, B., Vivas, A. K., et al. 2014, ApJ, 781, 22, doi: 10.1088/0004-637X/781/1/22