# The energetic particle environment of a GJ 436 b-like planet

D. Rodgers-Lee1,2, P. B. Rimmer3, A. A. Vidotto4, A. J. Louca4, A. M. Taylor5,
A. L. Mesquita4, Y. Miguel6,4, O. Venot7, Ch. Helling8,9, P. Barth8,10,11,12
and E. Lacy1,13
1 Dublin Institute for Advanced Studies, School of Cosmic Physics, 31
Fitzwilliam Place, Dublin 2, D02 XF86, Ireland
2 Trinity College Dublin, School of Physics, University of Dublin, College
Green, Dublin 2, D02 PN40, Ireland
3 Cavendish Laboratory, University of Cambridge, JJ Thomson Ave, Cambridge,
CB3 0HE, United Kingdom
4 Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The
Netherlands
5 Deutsches Elektronen-Synchrotron, Zeuthen, Germany
6 SRON, Netherlands Institute for Space Research, Niels Bohrweg 4, 2333 CA
Leiden, the Netherlands
7 Université Paris Cité and Univ Paris Est Creteil, CNRS, LISA, F-75013 Paris,
France
8 Space Research Institute, Austrian Academy of Sciences, Schmiedlstrasse 6,
A-8042 Graz, Austria
9 Institute for Theoretical Physics and Computational Physics, Graz University
of Technology, Petersgasse 16 8010 Graz
10 St Andrews Centre for Exoplanet Science, University of St Andrews, North
Haugh, St Andrews, KY16 9SS, UK
11 SUPA, School of Physics and Astronomy, University of St Andrews, North
Haugh, St Andrews, KY16 9SS, UK
12 School of Earth & Environmental Sciences, University of St Andrews, Bute
Building, Queen’s Terrace, St Andrews, KY16 9TS, UK
13 School of Physics, University College Dublin, Belfield, Dublin 4, Ireland
E-mail<EMAIL_ADDRESS>
(Accepted xxxx xxxxxx xx. Received xxxx xxxxxx xx; in original form xxxx xxx
xx)

###### Abstract

A key first step to constrain the impact of energetic particles in exoplanet
atmospheres is to detect the chemical signature of ionisation due to stellar
energetic particles and Galactic cosmic rays. We focus on GJ 436, a well-
studied M dwarf with a warm Neptune-like exoplanet. We demonstrate how the
maximum stellar energetic particle momentum can be estimated from the stellar
X-ray luminosity. We model energetic particle transport through the atmosphere
of a hypothetical exoplanet at orbital distances between $a=0.01-0.2\,$au from
GJ 436, including GJ 436 b’s orbital distance (0.028 au). For these distances
we find that, at top-of-atmosphere, stellar energetic particles ionise
molecular hydrogen at a rate of $\zeta_{\rm StEP,H_{2}}\sim 4\times
10^{-10}-2\times 10^{-13}\,\mathrm{s^{-1}}$. In comparison, Galactic cosmic
rays alone lead to $\zeta_{\rm GCR,H_{2}}\sim 2\times
10^{-20}-10^{-18}\,\mathrm{s^{-1}}$. At 10au we find that ionisation due to
Galactic cosmic rays equals that of stellar energetic particles: $\zeta_{\rm
GCR,H_{2}}=\zeta_{\rm StEP,H_{2}}\sim 7\times 10^{-18}\,\rm{s^{-1}}$ for the
top-of-atmosphere ionisation rate. At GJ 436 b’s orbital distance, the maximum
ion-pair production rate due to stellar energetic particles occurs at pressure
$P\sim 10^{-3}\,$bar while Galactic cosmic rays dominate for $P>10^{2}\,$bar.
These high pressures are similar to what is expected for a post-impact early
Earth atmosphere. The results presented here will be used to quantify the
chemical signatures of energetic particles in warm Neptune-like atmospheres.

###### keywords:

planetary systems – planets and satellites: atmospheres – cosmic rays –
methods: numerical – stars: low-mass – stars: winds, outflows

††pagerange: The energetic particle environment of a GJ 436 b-like
planet–9††pubyear: xxxx

##  1 Introduction

Spectroscopic observations with JWST (Gardner et al., 2006; Rigby et al.,
2022) and future dedicated exoplanet missions, such as Ariel (Tinetti et al.,
2021), will characterise the composition of exoplanet atmospheres. Chemical
models have predicted that energetic particles lead to specific chemical
signatures in exoplanet atmospheres, such as $\rm{H_{3}O^{+}}$ for gas giants
(Helling & Rimmer, 2019; Barth et al., 2021). Bourgalais et al. (2020)
produced synthetic JWST and Ariel transmission spectra of a sub-Neptune and
showed that absorption due to $\rm{H_{3}O^{+}}$ should be observable. The
detection of such a chemical signature would be a key step to constrain the
energetic particle fluxes impacting on planets outside the solar system. Here,
we focus on modelling the fluxes of two types of energetic particles: stellar
energetic particles from the host star (also known as stellar cosmic rays) and
Galactic cosmic rays from the interstellar medium (ISM). In the future, it may
be possible to disentangle the effect of stellar energetic particles and
Galactic cosmic rays by detecting chemical signatures of energetic particles
for exoplanets at different orbital distances (as suggested in Rodgers-Lee et
al., 2020, for instance).

Galactic cosmic rays interact with, and their fluxes are suppressed by, the
magnetised winds of low-mass stars (e.g. Potgieter (2013) for the Sun and
Rodgers-Lee et al. (2021b) for solar-like stars). Previous studies have
focused on calculating the Galactic cosmic ray fluxes in the habitable zone
(i.e. the orbital distances where liquid water can exist, Kasting et al.,
1993; Pierrehumbert & Gaidos, 2011; Abe et al., 2011; Kopparapu et al., 2013;
Zsom et al., 2013) and at the orbital distance of known exoplanets for a
number of well-studied solar-type (Rodgers-Lee et al., 2021b) and M dwarf
stars (Herbst et al., 2020; Mesquita et al., 2021, 2022a, 2022b).

Here, our aim is to combine the energetic particle transport through the
stellar system with the subsequent transport through the exoplanet atmosphere.
We relate the maximum energy of the stellar energetic particles accelerated by
stellar flares to stellar magnetic field strength, following Rodgers-Lee et
al. (2021a), and show how this can be related to stellar X-ray luminosity.
Previous studies have investigated the distribution of stellar energetic
particles in the Trappist-1 and AU Mic systems (Fraschetti et al., 2019;
Fraschetti et al., 2022). In the latter case, following a perturbation of the
interplanetary medium by a coronal mass ejection (CME). Separately, the
propagation, and chemical effect, of energetic particles through a number of
Earth-like (e.g. Grießmeier et al., 2015; Tabataba-Vakili et al., 2016; Herbst
et al., 2019b; Scheucher et al., 2020) and hot Jupiter/brown dwarf (Helling &
Rimmer, 2019; Barth et al., 2021) atmospheres has been studied.

Here, we focus on the environment of the warm Neptune exoplanet GJ 436 b. GJ
436 b is scheduled for JWST observations and is also an Ariel target (Edwards
& Tinetti, 2022). We calculate the ionisation rate, ion-pair production rate
and skin-depth equivalent dose rate due to energetic particles at different
heights in the exoplanet atmosphere. In addition to studying GJ 436 b, we also
investigate the effect of artificially changing the exoplanet’s orbital
distance on these quantities. The energetic particle ionisation rate is an
important input for studying disequilibrium chemistry in exoplanet
atmospheres, with chemical kinetic models such as ARGO (Rimmer & Helling,
2016, 2019). Here, for simplicity, we do not account for the effect of
planetary magnetic fields. However, planetary magnetic fields will reduce the
energetic particle fluxes penetrating the exoplanet atmosphere (Herbst et al.,
2013, 2019b; Grießmeier et al., 2015, 2016).

This study has been separated into two companion papers. This paper (Paper I)
focuses on characterising the energetic particle environment of a GJ 436
b-like planet. The paper is structured as follows: Section 2 introduces the
models that are used for the energetic particle transport and in Section 3 our
results are presented. The discussion and conclusions are given in Sections 4
and 5. Paper II (Rimmer et al. 2023) will focus on the chemical consequences
of the energetic particles for this system.

##  2 Energetic particle spectra

We consider the transport of both Galactic cosmic rays from the ISM and
stellar energetic particles, described in Sections 2.1 and 2.2. In this
section we also describe our method to construct the stellar energetic
particle spectra. Fig. 1 shows a schematic of the system indicating the two
sources of energetic particles. The exoplanet atmosphere profiles that we use
are described in Section 2.3. Using these, we then model the energetic
particle propagation through the exoplanet atmosphere, which is described in
Section 2.4.

Figure 1: Schematic of the GJ 436 system (not to scale) and its surrounding
environment embedded in the ISM. The curved blue and green arrows represent
the advection and diffusion of Galactic cosmic rays (labelled as ‘GCRs’) and
stellar energetic particles (labelled as ‘StEPs’), respectively. The straight
blue and green arrows represent their ballistic transport through the
exoplanet atmosphere which does not account for the effect of a planetary
magnetic field. The annotations indicate where the stellar wind model and the
various energetic particle transport models are described or were first
presented. It is important to note, as indicated in the figure, that the
stellar energetic particle and Galactic cosmic ray fluxes are assumed to be
isotropic at the top of the atmosphere. This assumption is discussed further
in Section 2.4.

###  2.1 Galactic cosmic ray fluxes

The Galactic cosmic ray fluxes throughout the GJ 436 system were previously
calculated (Mesquita et al., 2021) using a diffusion-advection cosmic ray
transport model (described in Rodgers-Lee et al., 2020). The stellar wind
velocity and magnetic field profile are important quantities to model the
cosmic ray transport through the stellar system. The Galactic cosmic ray
fluxes that we adopt here correspond to ‘Case A’ from Mesquita et al. (2021),
using the stellar wind properties from Mesquita & Vidotto (2020). Fig. 2 shows
the differential intensity, $j$ (i.e. the number of energetic particles per
unit area, steradian, time and kinetic energy), as a function of energetic
particle kinetic energy for the Galactic cosmic rays. The black line
represents the Galactic cosmic ray fluxes at the orbital distance of GJ 436 b
($\sim 0.03\,$au). The coloured lines represent $j$ at the different orbital
distances between $a=0.01-0.2\,$au. For the temperature pressure profiles in
Section 2.3, we focus on a range of orbital distances between $a=0.01-0.2\,$au
in increments of 0.01 au. We select the Galactic cosmic ray fluxes from the
cosmic ray transport model closest to these orbital distances. However, the
cosmic ray transport uses a logarithmic grid and due to the model resolution
some of the lines overlap. This occurs at larger orbital distances, for
$a>0.11$ au in all but one case, where the fluxes are already very similar
(see Fig. 2). Thus, the overlap will not affect our results significantly. The
Galactic cosmic ray fluxes vary significantly for $\lesssim 10\,$GeV energy
cosmic rays depending on the orbital distance considered. The fluxes for
$\lesssim 10\,$GeV energy cosmic rays are also much smaller than the values
observed at Earth (see Fig.5 from Mesquita et al., 2021) at all of the orbital
distances considered here.

Case A corresponds to a stellar wind model which is more magnetically-
dominated than the more thermally-dominated wind (‘Case B’ in Mesquita et al.,
2021). Fig. 2 from Mesquita et al. (2021) shows the stellar wind velocity and
magnetic field profiles for Case A and B. For Case A, the surface stellar
magnetic field adopted was $B_{\star}=4\,$G, the stellar wind terminal
velocity, $v_{\infty}$, was found to be 1250 ${\rm km\,s^{-1}}$ and the
stellar mass loss rate was $\dot{M_{\star}}=1.2\times 10^{-15}M_{\odot}\,{\rm
yr^{-1}}$. The astrospheric size ($R_{\mathrm{ast}}$), which determines how
far the Galactic cosmic rays must travel through the stellar wind, was
calculated using $v_{\infty}$ and $\dot{M_{\star}}$ to be 33 au (see Eq. 11 in
Mesquita et al., 2021).

Another important quantity for the transport model is the assumed Galactic
cosmic ray spectrum outside of the astrosphere. A model fit to the local
interstellar spectrum (LIS) for Galactic cosmic rays (Vos & Potgieter, 2015)
from Voyager 1 data (Stone et al., 2013) was used as the boundary condition
for these simulations. Finally, the assumed turbulence properties for the
stellar wind are important by dictating the diffusion of the Galactic cosmic
rays from the ISM through the astrosphere. For more details see Mesquita et
al. (2021).

###  2.2 Stellar energetic particle fluxes

Unlike the LIS for Galactic cosmic rays, less is known about stellar energetic
particle spectra. It is generally assumed that stars more active than the Sun
will produce higher fluxes of stellar energetic particles (e.g. Feigelson et
al., 2002). The higher flux is expected due to the increased flare energies
and flaring rate of active stars (e.g. from Kepler and TESS observations,
Maehara et al., 2015; Günther et al., 2020).

It is also very likely for active stars that stellar energetic particles will
be accelerated to higher energies than for the Sun. The Sun itself has been
inferred to accelerate particles to GeV energies from $\gamma-$ray
observations (Ajello et al., 2014; Ackermann et al., 2014). Considering the
maximum energy that stellar energetic particles are accelerated to by their
host star is important because the energy loss rate for energetic particles is
energy dependent. The energy loss rate for GeV energy particles is much lower
than for $\sim$MeV energy particles (see Fig. 5.6 in Longair, 2011). Thus, GeV
energy particles are far more penetrating than $\sim$MeV energy particles and
can, for instance, lead to showers of secondary particles that can reach the
surface of a rocky exoplanet (Atri, 2017).

Following Rodgers-Lee et al. (2021a), there are three quantities that we use
to obtain a stellar energetic particle spectrum. These are (i) the power law
index for the spectrum, (ii) the total energy available to produce stellar
energetic particles and (iii) the maximum stellar energetic particle momentum
($p_{\rm max}$) before the spectrum decays exponentially. The power law index
is set to be $\alpha=2$ (representative of diffusion shock acceleration,
Krymskii, 1977; Bell, 1978; Blandford & Ostriker, 1978, and compatible with
acceleration due to magnetic reconnection) such that $dN/dp\propto
p^{-\alpha}e^{-p/p_{\rm max}}$, where $N$ and $p$ are the number and momentum
of the particles, respectively. The total luminosity injected in stellar
energetic particles ($L_{\mathrm{CR}}$) that we adopt is $0.1P_{\mathrm{SW}}$,
where $P_{\mathrm{SW}}=\dot{M_{\star}}v_{\infty}^{2}/2$ is the stellar wind
kinetic power. This is in line with efficiency estimates from supernova
remnants (Vink et al., 2010). We calculate $L_{\mathrm{CR}}=6\times
10^{25}{\rm erg\,s^{-1}}$ for GJ 436 with $\dot{M_{\star}}=1.2\times
10^{-15}M_{\odot}\,{\rm yr^{-1}}$ and $v_{\infty}=1250\,{\rm km\,s^{-1}}$.

####  2.2.1 Maximum stellar energetic particle momentum

In Rodgers-Lee et al. (2021a), the Hillas criterion (Hillas, 1984) was used to
estimate the maximum stellar energetic particle momentum that the Sun would
have accelerated particles to in the past. The Hillas criterion posits that
the maximum energy achieved by an accelerator is limited by the accelerator
size, i.e. if the accelerated particle’s Larmor radius is larger than the
accelerator region it will escape. Using the Hillas criterion, the maximum
momentum, $p_{\mathrm{max}}$, of accelerated particles from a star can be
expressed as

$\frac{p_{\mathrm{max}}}{p_{\mathrm{max,\odot}}}=\left(\frac{\beta_{\star}B_{\star}R_{\star}}{\beta_{\odot}B_{\odot}R_{\odot}}\right)$
(1)

Figure 2: The (a) Galactic cosmic ray and (b) stellar energetic particle
differential intensities, $j$, are plotted as a function of the energetic
particle kinetic energy at different orbital distances. The colour of the
lines represents the orbital distance. For instance, for the left hand plot,
the lowest blue line represents the fluxes at $a=0.01\,$au and the highest
green line at $a=0.2\,$au. Additionally, the solid black line represents the
fluxes at the orbital distance of GJ 436 b. The Galactic cosmic ray fluxes are
from Mesquita et al. (2021).

where $R_{\odot}$ and $R_{\star}$ are the solar and stellar radii,
respectively. $B_{\odot}$ and $B_{\star}$ are the surface solar and stellar
magnetic field strengths, respectively. $B_{\odot}=1.3\,$G and
$p_{\mathrm{max,\odot}}=0.2\,$GeV$/c$. The parameters $\beta_{\odot}$ and
$\beta_{\star}$ are the solar and stellar shock velocity in units of the speed
of light, respectively. Here, we assume $\beta_{\star}\sim\beta_{\odot}$ as a
first approximation. We consider flare accelerated particles injected at $\sim
1.4R_{\star}$ rather than particles accelerated by CMEs (see Rodgers-Lee et
al., 2021a, for a discussion of this). Eq. 1 is the same as Eq. 7 from
Rodgers-Lee et al. (2021a) for the Sun. The only difference is that
$R_{\star}$ is also included in Eq. 1.

The Zeeman Doppler Imaging (ZDI) technique has provided average large-scale
stellar magnetic field strengths for hundreds of low-mass stars (e.g. Donati
et al., 2006; Morin, 2012; Moutou et al., 2017). This technique has
traditionally focused on active stars which generally have strong magnetic
fields, best suited for the observations. However, transiting exoplanets are
typically detected around less active stars because increased stellar
activity, in the form of stellar spots, tends to obscure exoplanet signatures.
Thus, the overlap between these samples is currently extremely limited and
stellar magnetic field strengths for less active stars often must be
estimated. For instance, Vidotto et al. (2014) presented the relation between
stellar X-ray luminosity ($L_{\rm X}$) and the large-scale magnetic flux
($\Phi$) derived from a large sample of stars with ZDI maps, as

$L_{\rm X}\approx 10^{-13.7}\Phi^{1.80\pm 0.20}\mathrm{erg\,s^{-1}}$ (2)

where $\Phi=4\pi R_{\star}^{2}\langle|B|\rangle$ (Mx) and $\langle|B|\rangle$
is the average large-scale magnetic field strength derived using the ZDI
technique. Using Eq. 2, we can now re-write Eq. 1 in terms of $L_{\rm X}$ as

$\displaystyle\frac{p_{\mathrm{max}}}{p_{\mathrm{max,\odot}}}$
$\displaystyle=$
$\displaystyle\left(\frac{B_{\star}R_{\star}^{2}}{B_{\odot}R_{\odot}^{2}}\right)\left(\frac{R_{\odot}}{R_{\star}}\right)=\left(\frac{\Phi}{\Phi_{\odot}}\right)\left(\frac{R_{\odot}}{R_{\star}}\right)$
(3) $\displaystyle=$ $\displaystyle\left(\frac{L_{\rm X,\star}}{L_{\rm
X,\odot}}\right)^{\gamma}\left(\frac{R_{\odot}}{R_{\star}}\right).$

where $\gamma=0.56^{-0.06}_{+0.125}$. Thus, in the absence of a magnetic field
strength measurement the maximum stellar energetic particle momentum can be
estimated using the stellar X-ray luminosity. Here however, we have used
$B_{\star}=4$ G for GJ 436 to be consistent with Mesquita et al. (2021)111As
mentioned in Mesquita et al. (2021), $B_{\star}=4$ G was chosen because the
stellar wind model from Mesquita & Vidotto (2020) with this $B_{\star}$ value
predicts an X-ray luminosity that is consistent with what is observed for GJ
436.. This results in $p_{\rm max}=0.3$GeV$/c$ using Eq. 1. In comparison,
using Eq. 3, we find $p_{\rm max}=0.1-0.7{\rm GeV}/c$ for $L_{\rm
X,\odot}=2.7\times 10^{26}-4.7\times 10^{27}\,{\rm erg\,s^{-1}}$ (spanning the
range of values from solar minimum to maximum, Peres et al., 2000) with
$\gamma=0.56$ and $L_{\rm X,\star}=5.7\times 10^{26}{\rm erg\,s^{-1}}$ for GJ
436 (Mesquita & Vidotto, 2020).

We construct the stellar energetic particle spectrum close to the surface of
the star using the values motivated above for $L_{\mathrm{CR}}$, $\alpha$ and
$p_{\rm max}$. This spectrum is used in the energetic particle transport
equation (Eq. 1 from Rodgers-Lee et al., 2021a) to calculate the resulting
differential intensity, $j$, as a function of orbital distance. Fig. 2 shows
$j$ as a function of energetic particle kinetic energy for stellar energetic
particles at different orbital distances. The black line represents the
energetic particle fluxes at the orbital distance of GJ 436 b ($\sim
0.03\,$au). The coloured lines represent $j$ at different orbital distances
between $a=0.01-0.2\,$au in increments of 0.01 au. The stellar energetic
particle fluxes vary significantly for $\lesssim 10\,$GeV energy cosmic rays
depending on the orbital distance considered. The stellar energetic particle
fluxes are much higher than the Galactic cosmic ray fluxes at MeV energies
(see Fig. 2). For $\gtrsim$GeV energies the Galactic cosmic rays begin to
dominate due to the exponential cut-off assumed for the stellar energetic
particle spectrum; this is similar to what Barth et al. (2021) found in their
Fig. 4. For a more active/younger star than GJ 436 it would be reasonable to
assume a higher cut-off energy, as motivated by Eq. 1 which would result in
stellar energetic particles dominating up to higher energies (as discussed in
Rodgers-Lee et al., 2021a, in the context of a young Sun-like star).

Other methods for estimating stellar energetic particle spectra use
relationships between solar far-UV (Youngblood et al., 2017), X-ray flare
energy (Herbst et al., 2019a) or starspot size (Herbst et al., 2021) and
$>10\,$MeV proton flux. This is similar to our scaling of the overall spectra
with stellar wind kinetic energy. The stellar energetic particle spectrum is
normalised using these proton fluxes once a spectral shape has been selected.
The shape of the spectrum is often based on a solar energetic particle event
(e.g. from Mewaldt et al., 2005), as adopted by Rab et al. (2017) and Barth et
al. (2021) for instance, or the strongest ground level enhancement event
measured on Earth (e.g. Herbst et al., 2019b; Scheucher et al., 2020). This is
the main difference between our model and those based on solar energetic
particle events. The spectral shape obtained from our model depends on stellar
magnetic field strength which varies from star to star.

As mentioned above, we consider flare accelerated particles rather than those
from CMEs as stellar CME properties remain elusive (e.g. Veronig et al., 2021,
for a recent detection of coronal dimming of X-ray emission due to a CME and
references therein). However, Hu et al. (2022) recently modelled the expected
stellar energetic particle fluxes from a CME associated with a young solar-
like star and found higher maximum particle energies for higher CME speeds. On
the other hand, previous stellar wind simulations including CMEs (Alvarado-
Gómez et al., 2018) have suggested that strong solar-like CMEs may be
suppressed by a large-scale dipole stellar magnetic field of 75 G which would
affect the corresponding stellar energetic particle fluxes. Fraschetti et al.
(2022) showed for the AU Mic system how the stellar energetic particle flux
distribution reaching the planetary orbits is strongly altered by the passage
of a CME. To complicate matters further, solar CME-CME interactions account
for more than 25% of the major geomagnetic storms observed (Zhang et al.,
2007). Using models, Koehn et al. (2022) suggest that successive CMEs can lead
to extreme conditions at Earth. Thus, it would be expected that stellar CME-
CME interactions will also affect the associated stellar energetic particle
fluxes.

###  2.3 Exoplanet atmosphere model

The exoplanet atmosphere density ($n(z)$) in units of cm-3, is important for
the propagation of the energetic particles through the exoplanet atmosphere.
The 1D temperature pressure profiles for a Neptune-like planet in the GJ 436
system are found using the radiative transfer code HELIOS (Malik et al., 2017,
2019). We assume an adiabatic atmosphere for pressures, $P>1$ bar222We have
not included any internal temperature in the HELIOS models. Thus, taking into
account that the radiative gradient is proportional to the planet luminosity,
no adiabat will occur for our HELIOS models. Setting an internal temperature
in HELIOS could not be done for the temperature pressure profiles considered
due to non-convergence effects. We instead include an adiabat for $P>1$ bar.
This is a reasonable approximation given the uncertainties in internal
temperatures and opacities for exoplanet atmospheres.. The density is then
calculated using the ideal gas law for the entire atmosphere. The profiles,
shown in Fig. 3, are calculated for a Neptune-like planet at different orbital
distances, $a$, ranging from 0.01-0.2 au in steps of 0.01 au which includes GJ
436 b’s observed orbital distance (i.e. 0.028 au).

The required inputs for HELIOS are the planetary parameters (i.e. mass, radius
and orbital distance), the opacities and chemical abundances of included
species and the stellar spectrum. For the opacities we make use of the line
lists from the DACE database (Grimm & Heng, 2015; Grimm et al., 2021) and we
only take the most relevant species into account (i.e. CH4, CN, CO, CO2, H2O,
H2S, NH3, NaH, PH3, SiO, TiO, and VO) considering solar elemental abundance
(Asplund et al., 2009) in chemical equilibrium (as described in Louca et al.,
2022, Fig. 3). The stellar XUV spectrum is taken from the MUSCLES survey
(France et al., 2016; Youngblood et al., 2016; Loyd et al., 2016; Youngblood
et al., 2017). For higher wavelengths PHOENIX models were used and stitched to
the XUV spectrum. The stellar and planetary parameters for the GJ 436 system
are given in Table LABEL:table:stellar_parameters. For the HELIOS models where
the planet had an orbital distance (period) $<$0.069 au ($<$10 days), the
model assumed a tidally locked planet, with no heat-redistribution. Planets
with orbital distances $\geq 0.069$ au were not assumed to be tidally locked
anymore and global heat-redistribution was included. The final temperature
pressure profiles are averaged over the irradiated hemisphere.

Table 1: Stellar and planetary parameters for the GJ 436 system.

Identifier | $M$ | $R$ | $T_{\mathrm{eff}}$ | $a$ | $P_{\rm rot}$ | Ref.
---|---|---|---|---|---|---
|  |  | (K) | (au) | (days) |
GJ 436 | $0.452M_{\odot}$ | $0.437R_{\odot}$ | 3479 | - | - | 1
GJ 436 b | $0.08M_{\mathrm{J}}$ | $0.372R_{\mathrm{J}}$ | 879 | 0.028 | 2.64 | 2

(1) Knutson et al. (2011); (2) Butler et al. (2004). The effective temperature
of GJ 436 b, as shown in this table, is derived using radiative transfer
calculations with HELIOS.

Figure 3: Temperature profiles of a hydrogen-dominated warm Neptune atmosphere
for orbital distances between 0.01 au (rightmost blue line) and 0.2 au
(leftmost green line) in the GJ 436 system. The black line represents the
temperature profile at the orbital distance of GJ 436 b. The profiles were
determined using HELIOS when pressure, $P<1\,$bar, and assuming an adiabat
when $P>1\,$bar, indicated by the grey dotted line.

###  2.4 Energetic particle transport in exoplanet atmospheres

The energetic particle propagation through the exoplanet atmosphere is
performed using a Monte Carlo model (from Rimmer et al., 2012; Rimmer &
Helling, 2013). Here, we briefly outline the method which is described in
detail in Rimmer et al. (2012); Rimmer & Helling (2013). The model is based on
the continuous slowing down approximation where the energetic particles travel
through a column density, $n(z)dz$, of the exoplanet atmosphere. Here, we take
$5\times 10^{5}$ particles to ensure that the fluctuations per energy bin are
negligible. In order to calculate the energetic particle energy losses for
each height element, $dz$, in the exoplanet atmosphere, each energetic
particle is assigned two values: an individual kinetic energy, $E$, such that
the energetic particle energies are normalised to match the top-of-atmosphere
fluxes shown in Fig. 2 and a random number, $N$, of uniform distribution
between 0 and 1. The random number is compared with the energetic particle
‘optical depth’, $\sigma^{\rm ion}_{\rm p,X}n(z)dz$, for the cell $dz$ where
$\sigma^{\rm ion}_{\rm p,X}(E)$ is the ionisation cross section. The subscript
‘$\rm X$’ refers to the ionisation cross sections for $\mathrm{H_{2}}$ and He
from Rudd et al. (1985) and Padovani et al. (2009). If $N<\sigma^{\rm
ion}_{\rm p,X}n(z)dz$, the energetic particle collides and loses energy. This
requires that $\sigma_{\rm max,X}n(z)dz<1$, where $\sigma_{\rm max,X}$ is the
maximum cross section value for $\mathrm{H_{2}}$ and He. Following Rimmer et
al. (2012), based on Cravens & Dalgarno (1978), the average energy loss per
collision, $\overline{W}$, is given by $\overline{W}=7.92E^{0.082}+4.76$,
where $\overline{W}$ and $E$ have units of eV. For this model, we assume that
the atmosphere is composed of 80% H2 and 20% He. We have adopted the same
expression for $\overline{W}$ for H2 and He. This step is repeated for each
height element and new random numbers are assigned.

The transport for stellar energetic particles and Galactic cosmic rays is
treated separately. This is because if the fluxes are combined, the energetic
particle flux is dominated by the stellar energetic particles and the high-
energy Galactic cosmic rays are poorly sampled by the Monte Carlo method.

In our model we assume that the energetic particle fluxes are isotropically
distributed at the top of the atmosphere, as shown in Fig. 1. We also do not
account for the effect of a planetary magnetic field. First, a planetary
magnetic field would act to deflect low-energy energetic particles towards the
magnetic poles. Thus, the energetic particle fluxes would no longer be
isotropically distributed at the top of the atmosphere and only particles with
Larmor radii comparable to or larger than the planetary radius would penetrate
the magnetosphere. Second, while diffusion tends to isotropise energetic
particle fluxes, such as for Galactic cosmic rays, the stellar energetic
particles are more influenced by advection processes at such small orbital
distances (see Fig. 4 from Mesquita et al., 2021). Thus, for close-in
exoplanets the stellar energetic particle fluxes may be higher on the side of
the exoplanet facing the star which we do not account for in our model.

Figure 4: The $\mathrm{H_{2}}$ ionisation rates are shown as a function of
pressure in an exoplanet atmosphere at different orbital distances in the GJ
436 system for (a) Galactic cosmic rays and (b) stellar energetic particles.
The grey dotted line denotes $P=1\,$bar. It is important to note that the
$x$-axis ranges are different between the two plots which is indicated by the
grey dashed lines between the panels.

The quantities, presented in Section 3, relevant for chemical modelling of
exoplanet atmospheres are the ionisation rate, $\zeta\,$($\rm s^{-1}$), and
the ion-pair production rate, $Q=n\zeta\,(\rm cm^{-3}\,s^{-1})$. Following
Padovani et al. (2009), the ionisation rate of molecular hydrogen, $\zeta_{\rm
H_{2}}$, by protons is calculated from the energetic particle differential
intensities, $j(E)$, as:

$\zeta_{\rm H_{2}}=4\pi\int^{E_{\rm max}}_{I(\rm
H_{2})}j(E)[1+\phi(E)]\sigma^{\rm ion}_{\rm p,H_{2}}(E)\,dE$ (4)

where $I(\mathrm{H}_{2})=15.603\,$eV is the ionisation potential of $\rm
H_{2}$ and $E_{\rm max}$ is the maximum energy of the energetic particle
spectrum. The parameter, $\phi(E)$, is a correction factor that accounts for
additional ionisation of $\rm H_{2}$ by secondary electrons given by:

$\phi(E)=\frac{1}{\sigma^{\rm ion}_{\rm p,H_{2}}(E)}\int^{E_{\rm
e,max}}_{I(\rm H_{2})}\underbrace{\frac{1}{\sigma^{\rm ion}_{\rm
p,H_{2}}(E)}\frac{d\sigma^{\rm ion}_{\rm p,H_{2}}(E)}{dE_{\rm e}}}_{P_{\rm
e}(E,E_{\rm e})}\sigma^{\rm ion}_{\rm e}(E_{\rm e})\,dE_{\rm e},$ (5)

where $E_{\rm e}$ (eV) is the secondary electron energy, $\sigma^{\rm
ion}_{\rm e}(E_{\rm e})$ (cm2) is the ionisation cross section of ${\rm
H_{2}}$ by electrons and $P_{\rm e}(E,E_{\rm e})$ is the probability density
that a secondary electron with energy $E_{\rm e}$ is produced as a result of
an initial ionising proton of energy $E$. This probability density is
represented in terms of the total ionisation cross-section ($\sigma^{\rm
ion}_{\rm p,H_{2}}(E)$, cm2), and differential cross-section ($d\sigma^{\rm
ion}_{\rm p,H_{2}}(E)/dE_{\rm e}$, cm2 MeV-1). From Cravens et al. (1975),

$\frac{d\sigma^{\rm ion}_{\rm p,H_{2}}(E)}{dE_{\rm
e}}=\frac{A(E)}{1+\left(\frac{E_{\rm e}}{E_{0}}\right)^{2.1}},$ (6)

where $E_{0}=8.3\,$eV. Given that, by definition:

$\displaystyle\int\limits_{0}^{\infty}P_{\rm e}(E,E_{\rm e})\,dE_{\rm e}=1,$

$A(E)$ can be expressed as $A(E)=C\sigma^{\rm ion}_{\rm p,H_{2}}(E)$ where the
value of $C$ is determined from numerical integration.

In addition to being important for chemical modelling, energetic particles
will impact life on other planets leading to increased mutation rates (for
instance see discussion in Scalo et al., 2007; Dartnell, 2011). The equivalent
dose rate, $\dot{D}$ (Sv s-1), which is the energy absorbed per unit time and
mass, provides a measure of how damaging energetic particle fluxes are for
life-forms. We estimate the skin-depth equivalent dose rate from primary
protons as:

$\dot{D}\sim 2\pi W\int^{E_{\rm max}}_{E_{\rm min}}\frac{E\,j(E)}{R(E)}\,dE$
(7)

where $W=2$ is the radiation weighting factor for protons and $R(E)\,({\rm
g\,cm^{-2}})$ is the proton range in
water333https://physics.nist.gov/PhysRefData/Star/Text/PSTAR.html.. The
quantity $E\,j(E)$ in Eq. 7 represents the particle number flux in an energy
bin, $dE$. We have assumed that the energetic particles only impact from
above. In Section 3.3, we present $\dot{D}$ in units of ${\rm mSv\,day^{-1}}$.

##  3 Results

Here we present our results for the energetic particle propagation in the
atmosphere of a GJ 436 b-like planet at different orbital distances. Using the
energetic particle fluxes (shown in Fig. 2) and the atmospheric profiles (Fig.
3), we calculate the ionisation rate (Section 3.1), the ion-pair production
rate (Section 3.2) and the skin-depth equivalent dose rate (Section 3.3).

Figure 5: The ion-pair production rate, $Q$, as a function of pressure in an
exoplanet atmosphere for different orbital distances in the GJ 436 system for
(a) Galactic cosmic rays and (b) stellar energetic particles. The grey dotted
line denotes $P=1\,$bar. The linestyles are the same as for Fig. 4.

###  3.1 Ionisation rates

Fig. 4 shows $\zeta_{\rm H_{2}}$ from Galactic cosmic rays (Fig. 4(a)) and
stellar energetic particles (Fig. 4(b)) as a function of pressure for
different orbital distances ($a=0.01-0.2$ au) with different coloured lines.
The black line represents $\zeta_{\rm H_{2}}$ at the orbital distance of GJ
436 b. The grey dotted line denotes $P=1$ bar. At the top of the atmosphere,
$\zeta_{\rm H_{2}}$ from Galactic cosmic rays varies by approximately two
orders of magnitude ranging from $\sim 1.7\times 10^{-20}-10^{-18}\,{\rm
s^{-1}}$ between $a=0.01-0.2\,$au, shown in Fig. 4(a). In comparison, the LIS
for Galactic cosmic ray protons results in a higher value of $\zeta_{\rm
H_{2}}\sim 1.6\times 10^{-17}\,{\rm s^{-1}}$ (estimated here as $2\zeta_{\rm
H}$ calculated in Cummings et al., 2016, from Voyager data). For the orbital
distances considered in Fig. 4(a), the top-of-atmosphere $\zeta_{\rm H_{2}}$
from Galactic cosmic rays is largest at the largest orbital distance of 0.2
au. This is because the Galactic cosmic ray fluxes are highest at the LIS
values in the ISM which are then suppressed as they travel inwards through the
stellar wind. The Galactic cosmic rays diffuse into the stellar system from
the ISM. At the same time, the Galactic cosmic ray fluxes are suppressed by
advection processes due to the expanding magnetised stellar wind. In Fig.
4(a), $\zeta_{\rm H_{2}}$ starts to decrease rapidly between $P\sim
10^{-1}-10^{1}$ bar, depending on the orbital distance considered. This
illustrates that the Galactic cosmic rays travel essentially unimpeded through
the exoplanet atmosphere in regions where the density is relatively low
($P\leq 10^{-1}\,$bar).

For stellar energetic particles (Fig. 4(b)), $\zeta_{\rm H_{2}}$ begins to
decrease rapidly at $P\sim 10^{-3}$ bar. The top-of-atmosphere $\zeta_{\rm
H_{2}}$ values vary by approximately three orders of magnitude ranging from
$\sim 2\times 10^{-13}-4\times 10^{-10}\,{\rm s^{-1}}$. In contrast to
$\zeta_{\rm H_{2}}$ from Galactic cosmic rays, the highest top-of-atmosphere
$\zeta_{\rm H_{2}}$ values for the stellar energetic particles are found at
the smallest orbital distances. This is because the stellar energetic
particles originate from a $\sim$point source close to the stellar surface,
causing the fluxes to decrease with increasing orbital distance (Fig. 2) as
they are advected by and diffuse through the stellar wind in our model. As
mentioned above, the reverse is true for Galactic cosmic rays which have the
lowest fluxes closest to the star (Fig. 2) as they originate from the ISM. The
stellar energetic particle values for $\zeta_{\rm H_{2}}$ decrease much more
rapidly with increasing pressure than the Galactic cosmic ray values (compare
the black lines from Figs. 4(a) and 4(b), for instance). This is because the
stellar energetic particles are predominantly at $\sim$MeV energies which have
a higher energy loss rate than $\gtrsim$GeV energy cosmic rays. Thus, in the
upper atmosphere stellar energetic particles dominate while in deeper parts
($P\gtrsim 100\,$bar) Galactic cosmic rays are more important. Transmission
spectroscopy with JWST and Ariel will likely probe regions between $P\sim
10^{-4}-10^{-1}$ bar for $\rm{H_{2}}$-rich atmospheres (Welbanks &
Madhusudhan, 2019).

Thus, for the atmospheres and orbital distances ($a=0.01-0.2\,$au) that we
consider here for the GJ 436 system, transmission spectroscopy with JWST and
Ariel would probe regions where stellar energetic particles are more likely to
impact the chemistry than Galactic cosmic rays (also see Section 3.2 where the
ion-pair production rates are discussed). While ionisation and dissociation by
energetic particles, including photons, are likely the main sources of
chemical disequilibrium in the upper atmosphere, vertical mixing and chemical
quenching are likely the dominant sources of disequilibrium chemistry deeper
in the atmosphere (Tsai et al., 2018; Ohno & Fortney, 2022). This trend will
be dependent on the specific nature of the temperature profile. Cooler
atmospheres, especially those of rocky planets, may have other sources of
disequilibrium (e.g. degassing, deposition, condensation, escape, Noack et
al., 2014; Tosi et al., 2017; Ortenzi et al., 2020; Zahnle et al., 2020). Very
hot atmospheres may never experience chemical quenching (Kitzmann et al.,
2018).

The total $\zeta_{\rm H_{2}}$ from energetic particles as a function of
pressure can be calculated by summing the individual $\zeta_{\rm H_{2}}$
values shown in Fig. 4 for Galactic cosmic rays and stellar energetic
particles444Alternatively, the top-of-atmosphere energetic particle fluxes
could have been summed. However, as described in Section 2.4, we treat the
stellar energetic particle and Galactic cosmic ray transport separately..
These ionisation rates can be used to study the energetic particle-induced
chemistry in an exoplanet atmosphere, similar to Barth et al. (2021). Barth et
al. (2021) studied the hot Jupiter, HD 189733 b, orbiting its K dwarf host
star at 0.031 au, similar to GJ 436 b’s orbital distance of 0.028 au. As
discussed in Section 2.2, the fact that the adopted stellar energetic particle
spectra are different, in combination with the different temperature-pressure
profiles, may result in different chemical tracers for energetic particles.
The main difference in the stellar energetic particle spectra is that our
spectra has more $\sim$GeV energy particles at 0.03 au, which penetrate
further into an exoplanet atmosphere.

As shown in Fig. 4, the top-of-atmosphere values $\zeta_{\rm H_{2}}$ values
from stellar energetic particles are all larger than those due to Galactic
cosmic rays for $a=0.01-0.2\,$au. However, at 10 au we find that the value of
$\zeta_{\rm H_{2}}=7\times 10^{-18}\,\mathrm{s^{-1}}$ due to stellar energetic
particles is the same as that obtained from Galactic cosmic rays. The Galactic
cosmic ray fluxes, and the resulting ionisation rates, are highest at large
orbital distances where they have travelled through relatively little of the
stellar wind. Galactic cosmic rays from the ISM lose energy as they diffuse
through the stellar wind. At the same time, stellar energetic particle fluxes
decrease with distance from the star.

Thus, for an exoplanet at an orbital distance $>10$ au in the GJ 436 system it
is more likely that any observable chemical signatures of energetic particles
from the upper atmosphere will be due to Galactic cosmic rays rather than
stellar energetic particles. Additionally, as mentioned above, the (higher
energy) Galactic cosmic rays lose their energy less rapidly with increasing
atmospheric pressure/density in comparison to the (lower energy) stellar
energetic particles. This means, for an exoplanet at an orbital distance $>10$
au in the GJ 436 system, that Galactic cosmic rays will likely be the dominant
source of ionisation due to energetic particles at all heights in the
atmosphere. This should be true for many systems, as discussed in Rodgers-Lee
et al. (2020), though the exact orbital distance where this occurs will be
system-dependent due to different stellar wind conditions. For instance, Fig.3
of Mesquita et al. (2021) shows the effect of assuming two different stellar
wind models on the Galactic cosmic ray spectra at various orbital distances
for the GJ 436 system. The difference between these stellar wind models is
that for one model the wind is more magnetically-dominated (Case A adopted
here), whereas the other is more thermally-dominated (Case B). Thus, the value
of $\zeta_{\rm H_{2}}$ from Galactic cosmic rays at 10 au would decrease
somewhat if we had adopted the thermally-dominated stellar wind, largely due
to the larger astrosphere resulting from this stellar wind model. For other
systems which are dominated more by advective processes (see Rodgers-Lee et
al., 2021b, for instance), the difference in Galactic cosmic ray $\zeta_{\rm
H_{2}}$ values at large distances assuming different astrosphere sizes would
be more significant.

###  3.2 Ion-pair production rates

The ion-pair production rate, $Q=n\zeta$, convolves the number density of the
exoplanet atmosphere with the ionisation rate. It gives an indication of where
in the exoplanet atmosphere the energetic particles will be important for
chemistry by creating the most ions. Thus, high $Q$ values correspond to
regions where the energetic particles should be most important. Fig. 5 and 5
show the ion-pair production rate, $Q$, as a function of pressure from
Galactic cosmic rays and stellar energetic particles, respectively for
different orbital distances. The maximum Galactic cosmic ray $Q$ value occurs
at $P\sim 10-10^{2}\,$bar (rightmost green line in Fig. 5). For stellar
energetic particles instead, the maximum $Q$ value occurs much higher in the
atmosphere between $P\sim 10^{-3}-10^{-2}\,$bar, with a much greater value
(rightmost blue line). Here, we have ignored the diffusion of ions.

For comparison, Fig. 6 shows the ion-pair production rate for Galactic cosmic
rays ($Q^{\rm GCR}$, green dashed line) and stellar energetic particles
($Q^{\rm StEP}$, blue dot-dashed line) at the orbital distance of GJ 436 b as
a function of pressure. The thin solid black line represents the combination
of both. The blue shaded region illustrates where $Q^{\rm StEP}>Q^{\rm GCR}$
and the green shaded region shows where $Q^{\rm GCR}>Q^{\rm StEP}$ which
occurs at $\sim$1 bar for GJ 436 b. The fact that Galactic cosmic rays are
more important than stellar energetic particles at $10-10^{3}$ bar is
interesting because it has been suggested that a $\sim 10^{2}\,$bar atmosphere
is expected for a post-impact early Earth atmosphere (Zahnle et al., 2020;
Itcovitz et al., 2022). This suggests that Galactic cosmic rays could have
been the dominant ionisation source deep in the atmosphere at this time.

Herbst et al. (2019b) and Scheucher et al. (2020) calculate the ion-pair
production rate from Galactic cosmic rays and solar/stellar energetic
particles for an Earth-like atmosphere and for Prox Cen b, respectively. Fig.
6 of Herbst et al. (2019b) shows broadly the same behaviour as our results:
$Q^{\rm GCR}$ peaks at a lower value than $Q^{\rm StEP}$, but occurs at a
higher pressure/lower altitude. The main difference is that for our model
atmosphere for GJ 436 b, Galactic cosmic rays dominate for a broad range of
high pressures. The fact that our maximum $Q^{\rm StEP}$ value peaks at $\sim
10^{-3}\,$bar reflects that our assumed stellar energetic particle spectrum
decays exponentially at a relatively low cosmic ray kinetic energy. For a more
active star, we would expect the maximum $Q^{\rm StEP}$ value to occur at a
higher pressure.

In terms of the range of values for $Q^{\rm StEP}$ and $Q^{\rm GCR}$, our peak
$Q^{\rm GCR}$ value is lower than that found in Herbst et al. (2019b) which is
reasonable since our top-of-atmosphere Galactic cosmic ray spectrum (Fig. 2)
is reduced in comparison to values observed at Earth. Our peak $Q^{\rm StEP}$
value lies between the maximum values found in Herbst et al. (2019b) and
Scheucher et al. (2020) which is again consistent since our stellar energetic
particle fluxes at $\sim 10-100\,$MeV energies lie between those assumed for
the Earth-like atmosphere and for Prox Cen b. Our stellar energetic particle
fluxes are higher for GJ 436 b mainly because it orbits much closer to its
star than the Earth orbits the Sun, while they are lower than for Prox Cen b.
At the same time, it is complex to compare values since $Q$ depends not only
on the incident energetic particle fluxes but also on the density profile of
the planetary atmosphere. Additionally, Herbst et al. (2019b) include the
effect of an Earth-like planetary magnetic field which introduces a low-energy
particle cut-off, i.e. energetic particles below a certain energy (510 MeV in
their case for latitudes of around 60∘) cannot penetrate the planet’s
magnetosphere and do not contribute to $Q$. In this context, this indicates
that our results represent the largest possible values for $Q$ and that
exoplanetary magnetic fields would generally result in a decrease of the
values obtained for $Q$. It is likely that the $Q^{\rm StEP}$ values would be
more affected than $Q^{\rm GCR}$. This is because the top-of-atmosphere
stellar energetic particle spectra are dominated by low-energy particles,
whereas the peaks of the Galactic cosmic ray spectra occur closer to GeV
energies (see Fig. 2).

Figure 6: The ion-pair production rate, $Q$, as a function of pressure at the
orbital distance of GJ 436b for Galactic cosmic rays (labelled as ‘GCRs’,
green dashed line) and stellar energetic particles (labelled as ‘StEPs’, blue
dot-dashed line). The solid black line represents the total $Q$ value. The
grey dotted line denotes $P=1\,$bar.

###  3.3 Skin-depth equivalent dose rates

While warm Neptune exoplanets are not expected to be habitable we can
calculate the skin-depth equivalent dose rate for these planets. In the
future, the same procedure can be applied to stars like GJ 436 that could have
terrestrial planets in their habitable zones. Fig. 7 shows the skin-depth
equivalent dose rate from energetic particles as a function of pressure for
Galactic cosmic rays (green dashed line) and stellar energetic particles (blue
dot dashed line) at $a=0.2\,$au, which is the inner edge of the habitable zone
for the GJ 436 system. The thin black line represents the combination of both.
The grey star and magenta pentagon represent the average equivalent dose rate
at Earth’s surface due to Galactic cosmic rays ($9\times 10^{-4}$mSv$\,\rm
day^{-1}$, including however secondary electrons, muons and $\gamma-$rays) and
background radiation sources ($7\times 10^{-3}$mSv$\,\rm day^{-1}$). Here we
have used the temperature pressure profile at 0.2 au, described in Section
2.3, assuming the same planetary mass and radius as before.

For Galactic cosmic rays, the equivalent dose rate is approximately constant
for $P<1\,$bar. Again in comparison to Herbst et al. (2019b), our maximum
equivalent dose rates are lower (higher) for Galactic cosmic rays (stellar
energetic particles). It is important to note that Herbst et al. (2019b)
consider a terrestrial planet atmosphere whereas we are still considering a
hydrogen dominated atmosphere which affects the energetic particle transport
in the exoplanet atmosphere. In turn, this will impact the dose rates that are
calculated.

For these equivalent dose rates, and for chemical modelling that include our
energetic particle ionisation rates, it is particularly important to consider
how constant these rates would be in time. We discuss this further in Section
4.1. While indirect detections of exoplanetary magnetic fields remain
challenging, Fig. 9 of Grießmeier et al. (2016) shows that the biological dose
rate decreased significantly with increasing magnetic moments for two
different atmospheric column depths. This indicates that our results represent
an upper limit in the case that the exoplanet possesses a magnetic field.

Figure 7: The skin-depth equivalent dose rate ($\dot{D}$) is plotted as a
function of pressure for a warm Neptune exoplanet atmosphere at an orbital
distance of 0.2 au, the inner edge of the habitable zone for the GJ 436
system. The green dashed line represents $\dot{D}$ for Galactic cosmic rays
(labelled as ‘GCRs’) and the blue dot-dashed line represents $\dot{D}$ for
stellar energetic particles (labelled as ‘StEPs’).The solid black line
represents the total skin-depth equivalent dose rate. The grey star and the
magenta pentagon represent the equivalent dose rate at Earth’s surface due to
Galactic cosmic rays and background radiation sources, respectively. Figure 8:
Flare rates are plotted as a function of the maximum momentum of the stellar
energetic particles accelerated. The solid blue line represents a fit to the
solar flare frequency (from Maehara et al., 2015). The solid black line is 10
times the solar flare rate and the grey shaded region is $\pm 0.5\,$dex of
this value. The solid green line is a flare rate from Günther et al. (2020)
for an M dwarf with a rotation period of $\sim 27\,$days. The grey dashed
lines are flare rates for M dwarfs from Günther et al. (2020) without a
measured rotation period with a spectral type similar to GJ 436.

##  4 Discussion

###  4.1 How continuous are stellar energetic particle fluxes?

Stellar energetic particle fluxes may vary on timescales of days, depending on
stellar activity. Galactic cosmic ray fluxes are more likely to be
approximately constant on these timescales. Instead, they vary with the solar
cycle (e.g. Fig.3 from Vos & Potgieter, 2015), for instance, and will also
vary on stellar evolutionary timescales (Rodgers-Lee et al., 2020). Similar to
Rodgers-Lee et al. (2021a), we plot the flare rate as a function of the
maximum momentum of the stellar energetic particles in Fig. 8. This is based
on the relation between flare frequency and flare energy ($dN/dE_{\rm
flare}\propto E_{\rm flare}^{b}$, where the power law index $b$ varies from
star to star). Flare energies are related to the stellar energetic particle
maximum momenta via $p_{\rm max}\sim E_{\rm flare}^{1/2}$, normalised for
$\sim$GeV energetic particle energies at $E_{\rm flare}\sim 10^{33}\,$erg and
assuming that $E_{\rm flare}\sim B_{\star}^{2}$ (see Rodgers-Lee et al.,
2021a, for more details). The underlying relation between flare rate and
energy is plotted in Fig. 9.

The relation between flare rate and energy, and therefore the relation between
flare rate and maximum stellar energetic particle momentum is unknown for GJ
436. Thus, in Fig. 8, we plot the flare rate versus maximum stellar energetic
particle momentum for the Sun and a number of M dwarfs. In Fig. 8, the solid
blue line is a fit to the solar flare frequency versus flare energy for a
broad range of flare energies using solar observations and Kepler observations
of superflares from solar-type stars (Maehara et al., 2015). The solid black
line is 10 times the solar flare frequency fit (i.e. $10\times E\,dN/dE_{\rm
flare}$, where $dN/dE_{\rm flare}\propto E_{\rm flare}^{-1.8}$) from Maehara
et al. (2015), which is what would be expected for a solar-type star with a
rotation period between $5-10$ days. The grey shaded region is $\pm 0.5\,$dex
of this value. The solid green line reflects a fit of the flare rate versus
energy for an M dwarf with a rotation period of $\sim$27 days from Günther et
al. (2020) using TESS data. This M dwarf is selected because it is relatively
similar to GJ 436 in terms of stellar effective temperature (3850 K) and has a
relatively long rotation period. The grey dashed lines are a number of flare
rate fits for M dwarfs with similar spectral types to GJ 436 but with unknown
rotation periods (also from Günther et al., 2020).

Using 10 times the solar flare rate, Fig. 8 shows that the value of $p_{\rm
max}$=0.3 GeV$/c$ (kinetic energy of 50 MeV) assumed for GJ 436 corresponds to
a flare rate of 0.04 day-1, i.e. a flare capable of accelerating particles to
this momentum occurs approximately every 25 days with $E_{\rm flare}=3\times
10^{31}\,$erg (see Fig. 9). Considering the grey shaded region for the flare
rate corresponds to a range of a flare every 7 or 72 days.

While Günther et al. (2020) suggest that the flare rates given by the dashed
grey lines are likely to be associated with stars with $P_{\rm rot}<10$ days
(in comparison to GJ 436 with $P_{\rm rot}\sim 44\,$days), it is interesting
to note that the flare rate given by the solid green line is for an M dwarf
with $P_{\rm rot}\sim 27\,$days. This rotation period is similar to the Sun’s
rotation period but with a flare rate 2 orders of magnitude larger than the
Sun’s. If GJ 436 had such an elevated flare rate, flares capable of
accelerating particles to 50 MeV energies would occur approximately once per
day. Fig. 8 (and more broadly Fig. 11 from Günther et al., 2020) also shows
that the power law fit for the flare rate varies from star to star. Thus,
determining stellar flare rates is important to understand how continuous
stellar energetic particle fluxes of a certain energy are. Comparing flare
rates with chemical recombination rates in the future would shed light on
whether the chemistry in exoplanet atmospheres could reset between flares with
associated stellar energetic particles.

###  4.2 Connection to habitability

Understanding exoplanet habitability, and detecting the signature of life as
we know it from an exoplanet, is a key goal of modern astronomy. Energetic
particles are one of the many factors thought to affect exoplanet habitability
(e.g. Meadows et al., 2018). For instance, energetic particles can drive the
formation of prebiotic molecules (Airapetian et al., 2016; Barth et al., 2021)
that are important for the origin of life. On the other hand, high energetic
particle fluxes may be detrimental to existing life by contributing to ozone
depletion (Segura et al., 2010) which blocks harmful UV radiation reaching the
surface of an Earth-like planet. Studying exoplanets that are not expected to
be habitable (such as GJ 436 b), but that will be observed in greater numbers,
can provide important information about the conditions present in other
stellar systems.

While hydrogen dominated gas giants are not generally thought to be conducive
to life, Seager et al. (2021) have suggested how life might survive in the
cloud decks of these planets, and Seager et al. (2020) have found unique
biosignature gas compositions for microbial life as we know it in H2-rich
environments. Prebiotically relevant molecules, such as the amino acid
glycine, may also form more easily on charged cloud particles (Stark et al.,
2014). Globus & Blandford (2020) suggested muons (which are on-average spin-
polarised) that reach Earth’s surface could be responsible for the chirality
of the DNA helix by interacting with biological molecules. In this context, it
will be of interest in the future to calculate the muon fluxes associated with
our (proton) energetic particle fluxes for a terrestrial planet. While genetic
mutation rates are known to increase with increasing radiation dose (Muller,
1927), it remains unclear what the ultimate effect of high radiation doses due
to energetic particles would be on the development of life on another planet.
Lifeforms that use radiation energy for metabolism (Matusiak, 2016) may
dominate instead.

In our model, we have assumed that the top-of-atmosphere stellar energetic
particle fluxes are isotropic. In reality, stellar energetic particles will
travel along spiral flux tubes which would result in anisotropic top-of-
atmosphere fluxes. This type of transport can be modelled with the focused
transport equation, for instance (Roelof, 1969). For tidally locked planets,
if the stellar energetic particle flux is not isotropic, one side of the
atmosphere, facing toward the star, will likely receive more stellar energetic
particles than the side facing away from the star. This angular dependence
could affect the energetic particle environment and could introduce
heterogeneous chemistry in the upper atmosphere, depending on chemical
timescales and efficiency of atmospheric circulation. This could be explored
by 3D models, and is outside the scope of this paper. An anisotropic energetic
particle flux could also have implications for habitability. Again,
exoplanetary magnetic fields will complicate the picture further. For
instance, Herbst et al. (2019b) considered the impact of an Earth-like
magnetic field and indicated that $<510$ MeV energy particles were not able to
penetrate an Earth-like magnetosphere at latitudes around 60∘.

The atmospheres of the planets considered here are roughly analogous to the
most extreme transient post-impact atmospheres predicted for the early Earth
(Zahnle et al., 2020; Itcovitz et al., 2022). We can consider the energetic
particle flux at different pressures as analogous particle radiation
environments for a post-impact early Earth. If Earth post-impact had a 100 bar
${\rm H_{2}}$ atmosphere, as proposed by Zahnle et al. (2020), then Galactic
cosmic rays determine the surface irradiation environment, and the irradiation
dose is much lower than if Earth post-impact had a relatively more tenuous
atmosphere, at $<10$ bar. Then the irradiation is much more severe and due to
stellar energetic particles. This is relevant for habitability around small
cool stars, such as GJ 436, where the 0.2 au case is at the inner edge of the
liquid water habitable zone of the system. This is also relevant for prebiotic
chemistry driven by energetic particle-driven chemistry in aqueous solutions,
either on the surface or in aerosols (Kobayashi et al., 2001; Lingam et al.,
2018).

##  5 Conclusions

In this paper, we consider the effect of the stellar magnetic environment and
planetary atmosphere on the energy dependence and intensity of energetic
particles as applied to the well-studied M dwarf system, GJ 436. GJ 436 b is a
warm Neptune planet orbiting at 0.028 au from its host star (scheduled for
JWST observations and an Ariel target). We have coupled a stellar wind model
with an energetic particle transport model for the large-scale stellar system.
We have then propagated the top-of-atmosphere energetic particle fluxes
through the atmosphere of a warm Neptune-like exoplanet at various orbital
distances from GJ 436, using temperature pressure profiles from the radiative
transport code HELIOS.

We consider two sources of energetic particles: Galactic cosmic rays and
flare-accelerated stellar energetic particles. We show how the maximum
momentum that the stellar energetic particles are accelerated to can be
related to the stellar X-ray luminosity. Thus, our input stellar energetic
particle spectrum reflects in some sense the activity of the star. To our
knowledge, no other model relates the cut-off energy of the stellar energetic
particle spectrum to a stellar property. The high energy cut-off is very
important because the highest energy particles (i.e. GeV energies) are those
that can reach the surface of an Earth-like planet, for instance. Our
calculations have assumed that the stellar energetic particle fluxes are
constant in time up to our calculated maximum stellar energetic particle
momentum. These would be associated with a $\sim 10^{31}$ erg flare which
would occur once every $\sim$25 days assuming 10 times the solar flare rate
for GJ 436.

We calculate top-of-atmosphere energetic particle fluxes as a function of
star-planet separation. We also model the transport of energetic particles
through a hydrogen-dominated gas giant atmosphere at orbital distances of
$a=0.01-0.2\,$au. The peak top-of-atmosphere ionisation rate for Galactic
cosmic rays is $\sim 10^{6}$ times less than the peak ionisation rate for
stellar energetic particles at orbital distances between $a=0.01-0.2\,$au.
However, the stellar energetic particles do not penetrate as deep into the
atmospheres. This is because for GJ 436, the stellar energetic particle
spectrum that we adopt has a lower cutoff energy than for the Galactic cosmic
rays.

At the orbital distance of GJ 436 b, we find that the ion-pair production
rate, $Q$ (cm-3 s-1), peaks at $\sim 100\,$bar for Galactic cosmic rays and
$10^{-3}\,$bar for stellar energetic particles. We also estimate the skin-
depth equivalent dose rate from the primary energetic particles as a function
of height in the hydrogen-dominated atmospheres. Given the close-in orbital
distance of the inner edge of GJ 436’s habitable zone (0.2 au), the top-of-
atmosphere equivalent dose rates that we calculate are substantial
($10^{4}\,$mSv day-1). However, an important future step would be to study the
effect of a planetary magnetic field on our results.

The stellar energetic particle fluxes decrease with distance from the star and
we assume that they are isotropic at the top of the exoplanet atmosphere. At
small orbital distances, the fluxes drop close to $1/r^{2}$ as advection
processes dominate. At the same time, the magnetised, expanding stellar wind
suppresses Galactic cosmic ray fluxes, and so the greater the distance from
the star, the more intense the Galactic cosmic ray spectrum. A future step
would be to study the effect of anistropic stellar energetic particle fluxes
as they travel along magnetic flux tubes.

We find that, at a distance of 10 au, the top-of-atmosphere molecular hydrogen
ionisation rates from stellar energetic particles and Galactic cosmic rays are
equal. Top-of-atmosphere ionisation by energetic particles is dominated by
Galactic cosmic rays for $a>10\,$au. Ionisation by energetic particles at the
top of the atmosphere is dominated by stellar energetic particles for
$a<10\,$au. We anticipate that most of the ion chemistry will happen at
$P<1\mu$bar, and so the ion chemistry will be determined by stellar energetic
particles and UV photons for planets within 10 au based on the top-of-
atmosphere ionisation rates that we found. Clouds might also form in GJ 436
b’s atmosphere because of the low atmospheric temperatures. Their formation
and its effects on the atmospheric structure and the chemistry are, however,
neglected in this paper. The chemical consequences of these trends and
specific predictions for ion-neutral chemistry will be discussed in more
detail in Paper II (Rimmer et al., 2023).

## Acknowledgements

DRL would like to acknowledge that this publication has emanated from research
conducted with the financial support of Science Foundation Ireland under Grant
number 21/PATH-S/9339. DRL, ALM and AAV acknowledge funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 817540, ASTROFLOW). DRL wishes to
acknowledge the Irish Centre for High-End Computing (ICHEC) for the provision
of computational facilities and support. Research reported in this publication
was supported by the Royal Irish Academy. ChH acknowledges funding from the
European Union H2020-MSCA-ITN-2019 under Grant Agreement no. 860470
(CHAMELEON). OV acknowledges funding from the ANR project ‘EXACT’
(ANR-21-CE49-0008-01), from the Centre National d’Études Spatiales (CNES), and
from the CNRS/INSU Programme National de Planétologie (PNP). PB acknowledges a
St Leonard’s Interdisciplinary Doctoral Scholarship from the University of St
Andrews. We would like to thank the referee for helpful comments which
improved the manuscript.

## Data Availability

The output data underlying this article will be available via zenodo.org upon
publication.

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##  Appendix A Flare rates

Section 4.1 discusses how continuous stellar energetic particle fluxes can be
considered for GJ 436. Fig. 8 relates the flare rate to a maximum stellar
energetic particle momentum. This is based on the underlying relation between
flare rate and flare energy which is plotted in Fig. 9. The solar flare rate
is a fit to solar and stellar super-flare data from Maehara et al. (2015) with
a flare frequency relation of $dN/dE_{\rm flare}\propto E_{\rm flare}^{-1.8}$.
The grey shaded region surrounding the solid black line, representing 10 times
the solar flare rate, is $\pm 0.5\,$dex. While the flare rate for GJ 436 is
unknown, given the maximum stellar energetic particle momentum that we assume,
we indicated the frequency of flares accelerating to this momentum in Section
4.1.

Figure 9: Observationally inferred flare rates ($E_{\rm flare}dN/dE_{\rm
flare}$) are plotted as a function of flare energy for the Sun, 10 times the
solar flare rate and a number of M dwarf flare rates from Günther et al.
(2020). The linestyles and the grey shaded region are the same as for Fig. 8.