Stellar versus Galactic: The intensity of cosmic rays at the evolving Earth and young exoplanets around Sun-like stars

Energetic particles, such as stellar cosmic rays, produced at a heightened rate by active stars (like the young Sun) may have been important for the origin of life on Earth and other exoplanets. Here we compare, as a function of stellar rotation rate ($\Omega$), contributions from two distinct populations of energetic particles: stellar cosmic rays accelerated by impulsive flare events and Galactic cosmic rays. We use a 1.5D stellar wind model combined with a spatially 1D cosmic ray transport model. We formulate the evolution of the stellar cosmic ray spectrum as a function of stellar rotation. The maximum stellar cosmic ray energy increases with increasing rotation i.e., towards more active/younger stars. We find that stellar cosmic rays dominate over Galactic cosmic rays in the habitable zone at the pion threshold energy for all stellar ages considered ($t_*=0.6-2.9\,$Gyr). However, even at the youngest age, $t_*=0.6\,$Gyr, we estimate that $\gtrsim\,80$MeV stellar cosmic ray fluxes may still be transient in time. At $\sim1\,$Gyr when life is thought to have emerged on Earth, we demonstrate that stellar cosmic rays dominate over Galactic cosmic rays up to $\sim$4$\,$GeV energies during flare events. Our results for $t_*=0.6\,$Gyr ($\Omega = 4\Omega_\odot$) indicate that $\lesssim$GeV stellar cosmic rays are advected from the star to 1$\,$au and are impacted by adiabatic losses in this region. The properties of the inner solar wind, currently being investigated by the Parker Solar Probe and Solar Orbiter, are thus important for accurate calculations of stellar cosmic rays around young Sun-like stars.


INTRODUCTION
There is much interest in determining the conditions, such as the sources of ionisation for exoplanetary atmospheres, that were present in the early solar system when life is thought to have begun on Earth (at a stellar age of ∼1 Gyr, Mojzsis et al. 1996). This allows us to postulate what the important factors that led to life here on Earth were. These studies can then be extended to young exoplanets around solar-type stars whose atmospheres may be characterised in the near future by upcoming missions, such as the James Webb Space Telescope (JWST, Gardner et al. 2006; Barstow & Irwin 2016).
Cosmic rays represent a source of ionisation (Rimmer & Helling 2013) and heating (Roble et al. 1987;Glassgold et al. 2012) for exoplanetary atmospheres. In this paper we compare the contributions from two distinct populations of E-mail: drodgers@tcd.ie energetic particles: stellar cosmic rays accelerated by their host stars and Galactic cosmic rays. Galactic cosmic rays reach Earth after travelling through the heliosphere (see review by Potgieter 2013). These cosmic rays originate from our own Galaxy and constitute a reservoir of relativistic particles in the interstellar medium (ISM) that diffuse through the magnetised solar wind in a momentum-dependent way.
The propagation of Galactic cosmic rays through the stellar winds of younger solar-type stars has previously been studied (Svensmark 2006;Cohen et al. 2012;Rodgers-Lee et al. 2020a). The intensity of Galactic cosmic rays that reached the young Earth (∼1 Gyr old) is thought to be much reduced in comparison to present-day observed values. This is due to the increased velocity and magnetic field strength present in the stellar wind of a young (∼Gyr old) solar-type star in comparison to the present-day solar wind (assuming that the turbulence properties of the wind remain constant with stellar age). Similar to what has been estimated to occur for Galactic cosmic rays, the changing physical condi-tions of the stellar wind throughout the life of a Sun-like star will affect the propagation of stellar cosmic rays. The propagation of Galactic cosmic rays through the astrospheres 1 of a number of M dwarf stars has also recently been considered Mesquita et al. 2021).
Here, we investigate the intensity of solar, or more generally stellar, cosmic rays as they propagate through the wind of a solar-type star throughout the star's life, particularly focusing on the intensity at the orbital distance of Earth. Solar/stellar cosmic rays are also known as solar/stellar energetic particles. We focus on solar-type stars so that our results can also be interpreted in the context of the young Sun and the origin of life on Earth.
Cosmic rays are thought to be important for prebiotic chemistry and therefore may play a role in the origin of life (Dartnell 2011;Rimmer et al. 2014;Airapetian et al. 2016;Dong et al. 2019). Cosmic rays may also result in observable chemical effects in exoplanetary atmospheres by leading to the production of molecules such as NH + 4 , H + 3 and H3O + (Helling & Rimmer 2019;Barth et al. 2020). In addition to this, cosmic rays may lead to the production of fake biosignatures via chemical reactions involving NOx (Grenfell et al. 2013). Biosignatures are chemical signatures that are believed to be the chemical signatures of life, such as molecular oxygen (Meadows et al. 2018). Thus, in order to interpret upcoming observations which will focus on detecting biosignatures we must constrain the contribution of cosmic rays to fake biosignatures.
An interesting aspect that we focus on in this paper is the fact that the intensity and momentum of stellar cosmic rays accelerated by a solar-type star most likely increase for younger stars due to their stronger stellar magnetic fields, unlike the Galactic cosmic ray spectrum which is assumed to remain constant with time. The present day Sun, despite being an inactive star, has been inferred to accelerate particles to GeV energies in strong solar flares (Ackermann et al. 2014;Ajello et al. 2014;Kafexhiu et al. 2018). There is also evidence from cosmogenic nuclides to suggest that large solar energetic particle events occurred even in the last few thousand years (Miyake 2019). Thus, it is very likely that a younger Sun would accelerate particles at a higher rate, and to higher energies, due to the stronger magnetic field strengths observed for young Sun-like stars (e.g. Johns-Krull 2007;Hussain et al. 2009;Donati et al. 2014) and at a more continuous rate due to an increased frequency of stellar flares (Maehara et al. 2012(Maehara et al. , 2015. A scaled up version of a large solar energetic particle event is often assumed as representative of stellar cosmic rays around main sequence M dwarf stars or young pre-main sequence solar-type stars. The work presented in this paper builds upon this research and aims to contribute towards a clearer and broader understanding of the spectral shape and intensity of stellar cosmic rays which reach exoplanets around solar-type stars as a function of age. There has also been a significant amount of research concerning the propagation of stellar cosmic rays through the magnetospheres and atmospheres of close-in exoplanets around M dwarf stars (Segura et al. 2010; Tabataba-Vakili 1 The more general term for the heliosphere of other stellar systems et al. 2016), as well as comparisons with Galactic cosmic rays (Grießmeier et al. 2015). Atri (2020) also investigated the surface radiation dose for exoplanets resulting from stellar cosmic ray events starting at the top of an exoplanetary atmosphere considering an atmosphere with the same composition as Earth's. Our results can be used in the future as an input for these types of studies.
In this paper we compare the relative intensities of stellar and Galactic cosmic rays of different energies as a function of stellar age. This allows us to estimate the age of a solar-mass star when the intensities of stellar and Galactic cosmic rays are comparable, at a given energy. This also depends on the orbital distance being considered. Here we focus mainly on the habitable zone of a solar-mass star where the presence of liquid water may be conducive to the development of life (Kasting et al. 1993).
Previous studies have estimated the intensity of Galactic cosmic rays at ∼1 Gyr when life is thought to have begun on Earth. Stellar cosmic rays have separately been considered in the context of T-Tauri systems (Rab et al. 2017;Rodgers-Lee et al. 2017, 2020bFraschetti et al. 2018;Offner et al. 2019) and more generally in star-forming regions (see Padovani et al. 2020, for a recent review). Fraschetti et al. (2019) also investigated the impact of stellar cosmic rays for the Trappist-1 system. Scheucher et al. (2020) focused on the chemical effect of a large stellar energetic particle event on the habitability of Proxima Cen b. However, the propagation of stellar cosmic rays through stellar systems has not yet been investigated as a function of stellar rotation rate or at the potentially critical time when the Sun was ∼1 Gyr old. We also compare the relative intensities of stellar and Galactic cosmic rays for the HR2562 system (Konopacky et al. 2016) which we focused on previously in Rodgers-Lee et al. (2020a).
The paper is structured as follows: in Section 2 we describe the details of our model and the properties that we have adopted for the stellar cosmic rays. In Section 3 we present and discuss our results in relation to the young Sun and the young exoplanet, HR 2562b. Finally, we present our conclusions in Section 4.

FORMULATION
In this section we motivate our stellar cosmic ray spectrum as a function of stellar rotation rate. We also briefly describe the cosmic ray transport model and stellar wind model that we use (previously presented in Rodgers-Lee et al. 2020a).

From solar to stellar cosmic rays
The present-day Sun is the only star for which we can directly detect solar cosmic rays and determine the energy spectrum of energetic particles arriving to Earth. We use these observations of the present-day Sun to guide our estimate for the stellar cosmic ray spectrum of solar-type stars of different ages, which are representative of the Sun in the past. The shape of the energy spectrum and the overall power in stellar cosmic rays are the two quantities required to describe a stellar energetic particle spectrum (Section 2.4). Solar energetic particle (SEP) events can broadly be divided into two categories known as gradual and impulsive events (Reames 2013;Klein & Dalla 2017, for instance). Gradual SEP events are thought to be mainly driven by the acceleration of particles at the shock fronts of coronal mass ejections (CMEs) as they propagate. These events produce the largest fluences of protons at Earth. Impulsive SEP events are associated with flares close to the corona of the Sun and while they result in lower proton fluences at Earth they occur more frequently than gradual events. The terms 'gradual' and 'impulsive' refer to the associated X-ray signatures.
While gradual SEP events associated with CMEs produce the largest fluences of protons detected at Earth it is unclear what energies the CMEs would have and how frequently they occur for younger solar-type stars (Aarnio et al. 2012;Drake et al. 2013;Osten & Wolk 2015). Very large intensities of stellar cosmic rays associated with very energetic, but infrequent, CMEs may simply wipe out any existing life (Cullings et al. 2006; Atri 2020) on young exoplanets rather than helping to kick start it. On the other hand lower intensities of stellar cosmic rays, associated with impulsive flare events, at a more constant rate may be more of a catalyst for life (Atri 2016;Lingam et al. 2018;Dong et al. 2019). In the context of the potential impact of stellar energetic particles on exoplanetary atmospheres we restrict our focus here to protons. This is because only protons can be accelerated to ∼GeV energies, rather than electrons which suffer from energy losses.
Many white light (referring to broad-band continuum enhancement, rather than chromospheric line emission, for instance) flares have been detected by the Kepler mission (Koch et al. 2010). An increase in the frequency of superflares (bolometric flare energies of > 10 33 erg) with increasing stellar rotation (i.e. younger stars) has also been found (Maehara et al. 2015). Some of the most energetic white light flares are from pre-main sequence stars in the Orion complex detected in the Next Generation Transit Survey (Jackman et al. 2020). Since SEP events often have associated optical and X-ray emission, the detection of very energetic white light flares from younger stars/faster rotators is presumed to lead to a corresponding increase in X-rays. Indeed, young stars are known to be stronger X-ray sources in comparison to the Sun (Feigelson et al. 2002). Therefore, it seems likely that stars younger than the Sun will also produce more stellar energetic particles than the present day Sun (see , for a recent estimate of stellar proton fluxes derived using the empirical relation between stellar effective temperature and starspot temperature).
Somewhat surprisingly, given the number of superflares detected with Kepler, there have only been a small number of stellar CME candidate events (Argiroffi et al. 2019;Moschou et al. 2019;Vida et al. 2019;Leitzinger et al. 2020). To investigate the possibility that stellar CMEs are not as frequent as would be expected by extrapolating the solar flare-CME relation (Aarnio et al. 2012;Drake et al. 2013;Osten & Wolk 2015), Alvarado-Gómez et al. (2018) illustrated using magnetohydrodynamic simulations that a strong large-scale stellar dipolar magnetic field (associated with fast rotators) may suppress CMEs below a certain energy threshold. Another line of argument discussed in Drake et al. (2013) suggests that the solar flare-CME relationship may not hold for more active stars because the high CME rate expected for active stars (obtained by extrapolating the solar flare-CME relationship) would lead to very high stellar mass-loss rates. This has not been found for mass-loss rates inferred from astrospheric Lyα observations (Wood et al. 2002;Wood 2004;Wood et al. 2014) or from transmission spectroscopy, coupled to planetary atmospheric evaporation and stellar wind models (Vidotto & Bourrier 2017). At the same time the number of stars with estimates for their massloss rates remains small.
Thus, as a first estimate for the intensity of stellar energetic particles impinging on exoplanetary atmospheres we consider flare-accelerated protons that we inject close to the surface of the star. We do not consider stellar energetic particles accelerated by shocks associated with propagating CMEs due to the current lack of observational constraints for the occurence rate and energy of stellar CMEs as a function of stellar age. Our treatment of the impulsive stellar cosmic ray events is described in the following section.
Our investigation treats the injection of stellar cosmic rays as continuous in time during a given epoch of a star's life. Two key factors here that control the applicability of such an assumption are that young solar-type stars (i.e. fast rotators) are known to flare more frequently than the present-day Sun, and their associated flare intensity at a given frequency is more powerful (Salter et al. 2008;Maehara et al. 2012Maehara et al. , 2015. In order to focus on stellar cosmic rays injected at such a heightened rate and power, and thus can be treated as continuous in time, we restrict our results to stellar rotation rates greater than the rotation rate of the present-day Sun. We discuss this assumption in more detail in Section 3.3.

Transport equation for stellar cosmic rays
To model the propagation of stellar cosmic rays from a solartype star out through the stellar system we solve the 1D cosmic ray transport equation (derived by Parker 1965, for the modulation of Galactic cosmic rays in the solar system), assuming spherical symmetry. We use the same numerical code as presented in Rodgers-Lee et al. (2020a) which includes spatial diffusion, spatial advection and energy losses due to momentum advection of the cosmic rays. The 1D transport equation is given by where f (r, p, t) is the cosmic ray phase space density, κ(r, p, Ω) is the spatial diffusion coefficient, v(r, Ω) is the radial velocity of the stellar wind and p is the momentum of the cosmic rays (taken to be protons) 2 . Q is defined as which represents the volumetric injection of stellar cosmic rays per unit time and per unit interval in momentum, injected at a radius of rinj ∼ 1.3R .Ṅ = dN/dt is the number of particles injected per unit time. Q varies as a function of stellar rotation rate, Ω. The details of how we treat the injection rate are discussed in Section 2.4. The numerical scheme and the simulation set-up are otherwise the same as that of Rodgers-Lee et al. (2020a). We assume an isotropic diffusion coefficient, which varies spatially by scaling with the magnetic field strength of the stellar wind (and on the level of turbulence present in it) and depends on the cosmic ray momentum (see Eq. 3 of Rodgers-Lee et al. 2020a). Here, for simplicity we take the level of turbulence to be independent of stellar rotation rate using the same value as motivated in Rodgers-Lee et al. (2020a). Spatial advection and the adiabatic losses of the cosmic rays depend on the velocity and divergence of the stellar wind.
In the context of the modulation of Galactic cosmic rays spatial and momentum advection collectively result in the suppression of the local interstellar spectrum (LIS) of Galactic cosmic rays that we observe at Earth. For stellar cosmic rays the situation is slightly different due to their place of origin in the system. Stellar cosmic rays still suffer adiabatic losses as they travel through the stellar wind via the momentum advection term, but the spatial advective term now merely advects the stellar cosmic rays out through the solar system. Spatial advection only operates as a loss term if the stellar cosmic rays are advected the whole way through and out of the stellar system.

Stellar wind model
In our model the stellar wind of a Sun-like star is launched due to thermal pressure gradients and magneto-centrifugal forces in the hot corona overcoming stellar gravity (Weber & Davis 1967). The wind is heated as it expands following a polytropic equation of state. Stellar rotation is accounted for in our model leading to (a) angular momentum loss via the magnetic field frozen into the wind and (b) the development of an azimuthal component of an initially radial magnetic field. We use the same stellar wind model as in Rodgers-Lee et al. (2020a) that is presented in more detail in Carolan et al. (2019). Our 1.5D polytropic magneto-rotator stellar wind model (Weber & Davis 1967) is modelled with the Versatile Advection Code (VAC, Tóth 1996;Johnstone et al. 2015) and here we provide a brief summary of it.
By providing the stellar rotation rate, magnetic field, density and temperature at the base of the wind as input parameters for the model we are able to determine the magnetic field strength, velocity and density of the stellar wind as a function of orbital distance out to 1 au. Beyond 1 au the properties of the stellar wind are extrapolated out to the edge of the astrosphere as discussed in Section 2.3. In the stellar wind model the base, or launching point, of the wind is chosen to be 1 R for the instances in time that we investigate. Table 1 provides the stellar rotation rates/ages that we consider here. The corresponding stellar surface magnetic field strength, base density and temperature that we use are given in Table 1 of Rodgers-Lee et al. (2020a). Generally, the magnetic field strengths and velocities of the stellar wind increase with increasing stellar rotation rate.

Stellar cosmic ray spectrum
We assume a continuous injection spectrum for the stellar cosmic rays such that dṄ /dp ∝ dN/dp ∝ p −α . We adopt a power law index of α = 2 which in the limit of a strong non-relativistic shock is representative of diffusive shock acceleration (DSA, as first analytically derived by Krymskii 1977;Bell 1978;Blandford & Ostriker 1978) or compatible with acceleration due to magnetic reconnection. We relate the total injected kinetic power in stellar cosmic rays, LCR (which we define in Section 2.4.2), to dṄ /dp in the following way where m is the proton mass, c is the speed of light and T (p) = mc 2 ( 1 + (p/mc) 2 − 1) is the kinetic energy of the cosmic rays. pmax is the maximum momentum that the cosmic rays are accelerated to which is discussed further in Section 2.4.1. The logarithmically spaced momentum bins for the cosmic rays are given by p k = exp{k × ln(pM /p0)/(M − 1) + ln p0} for k = 0, ..., M with M = 60. The extent of the momentum grid that we consider ranges from p0 = 0.15 GeV/c to pM = 100 GeV/c, respectively. We have normalised the power law in Eq. 4 to a momentum of 2mc since this demarks the part of integrand which dominates the integral (for spectra in the range 2 < α < 3 of primary interest to us). To illustrate this, following Drury et al. (1989), Eq. 3 can be approximated as ∞ 0 dṄ dp T (p)dp ≈ dṄ dp 2mc which has split the integral into a non-relativistic and relativistic component given by the first and second term on the righthand side of Eq. 5, respectively. Eq. 5 implicitly assumes that p0 2mc. For α = 2, Eq. 5 can be estimated as Thus, considering pmax ∼ 3 GeV/c indicates that the first and last term contribute approximately equally in Eq. 6.
p 2 dṄ dp |2mc is chosen to normalise the integral to the required value of LCR.

Spectral break as a function of stellar rotation rate
The maximum momentum of the accelerated cosmic rays, pmax, is another important parameter that we must estimate as a function of Ω. It physically represents the maximum momentum of stellar cosmic rays that we assume the star is able to efficiently accelerate particles to. Both DSA and magnetic reconnection rely on converting magnetic energy to kinetic energy. The magnetic field strength of Sun-like stars is generally accepted to increase with increasing stellar rotation rate, or decreasing stellar age (Vidotto et al. 2014a;Folsom et al. 2018). This indicates that more magnetic energy would have been available at earlier times in the Sun's life or for other stars rotating faster/younger than the present-day Sun to produce stellar cosmic rays. Therefore we evolve pmax as a function of stellar magnetic field strength. In our model this effectively means that the maximum injected momentum evolves as a function of stellar rotation rate. We assume that We chose pmax, = 0.2 GeV/c, corresponding to a kinetic energy of Tmax = 20 MeV (Kouloumvakos et al. 2015, for instance, report impulsive stellar energetic particle events with kinetic energies 50 MeV). We also investigate the effect of assuming pmax, = 0.6 GeV/c (corresponding to Tmax = 200 MeV). The values for pmax(Ω) are given in Table 1. The maximum value that we use is 3.3GeV/c for a stellar rotation rate of 3.5Ω at t * = 600 Myr using pmax, = 0.6 GeV/c. In comparison, Padovani et al. (2015) estimated a maximum energy of ∼30 GeV for the acceleration of protons at protostellar surface shocks for t * 1 Myr. The maximum momentum of 3.3 GeV/c that we adopt corresponds to a surface average large-scale magnetic field of ∼ 8 G at 600 Myr. If we investigated younger stellar ages when it would be reasonable to adopt an average large-scale stellar magnetic field of ∼ 80 G then we would also find a maximum energy of ∼30 GeV.
Eq. 7 is motivated by the Hillas criterion (Hillas 1984) which estimates that the maximum momentum achieved at a shock can be obtained using where βs = vs/c and vs is the characteristic velocity associated to the scattering agent giving rise to acceleration (eg. shock velocity or turbulence velocity), Bs is the magnetic field strength within the source, and Rs is the size of the source region. If we assume that the size of the emitting region (a certain fraction of the Sun's surface) and the characteristic velocity do not change as a function of stellar rotation rate we simply obtain pmax ∝ Bs as adopted in Eq. 7. Indeed, high energy γ−ray observations from Fermi-LAT found that for strong solar flares the inferred proton spectrum, located close to the surface of the Sun, displays a high maximum kinetic energy break of 5 GeV (Ackermann et al. 2014; Ajello et al. 2014). Generally, the spectral break occurs at lower kinetic energies, or momenta, for less energetic but more frequent solar flares. Since the power law break in the SEP spectrum shifts to higher energies during strong solar flares this is a good indicator that pmax is likely to have occurred at higher momenta in the Sun's past when solar flares were more powerful. For instance, Atri (2017) uses a large SEP event as a representative spectrum for an M dwarf star which has a higher cut-off energy at approximately ∼GeV energies.

Total kinetic power in stellar cosmic rays
We use the kinetic power in the stellar wind, PSW = M (Ω)v∞(Ω) 2 /2, calculated from our stellar wind model to estimate LCR as a function of stellar rotation rate assuming a certain efficiency, shown in Fig. 1.Ṁ (Ω) and v∞(Ω) are the mass loss rate and terminal speed of the stellar wind, respectively. Here we assume that LCR ∼ 0.1PSW which is shown on the righthand side of Fig. 1. Such a value is typical of the equivalent efficiency factor estimated for supernova remnants (Vink et al. 2010, for instance). Without further evidence to go by, we simply adopt the same value here for young stellar flares. Adopting a different efficiency of 1% or 100%, for instance, would change the values for the differential intensity of stellar cosmic rays presented in Section 3 by a factor of 0.1 and 10, respectively. The values that we use here are broadly in line with the value of LCR ∼ 10 28 erg s −1 from Rodgers-Lee et al. (2017) which was motivated as the kinetic power of stellar cosmic rays produced by a T-Tauri star.

Comparison of solar, stellar and Galactic cosmic ray spectra
We include a schematic in Fig. 2 which shows representative values for the differential intensity of solar and Galactic cosmic rays as a function of kinetic energy. The differential intensity of cosmic rays, j, is often considered (rather than the phase space density given in Eq. 1) as a function of kinetic energy which we plot in Section 3. These quantities are related by j(T ) = dN/dT = p 2 f (p). Fig. 2 includes the estimate for our most extreme Tm a x 2 0 2 0 0 M e V LIS = 2 . 7 8 G a la c t ic C R s @ 1 a u Figure 2. This sketch illustrates typical cosmic ray spectra, both solar/stellar (impulsive and gradual events) and Galactic in origin, at various orbital distances in the solar/stellar system. The solid black line represents an approximation for the LIS of Galactic cosmic rays outside of the solar system. The green line represents a typical Galactic cosmic ray spectrum observed at Earth. The blue dashed line is a typical spectrum for a gradual SEP event (averaged over the duration of the event). The spectral slope γ at high energies is also indicated in the plot. The solid blue line is the estimate for an impulsive stellar energetic particle event spectrum that we motivate in this paper for a young solar-type star (∼ 600Myr old) which includes a spectral break at much higher energies than the typical present-day (gradual) SEP spectrum (blue dashed line). See Section 2.5 for more details.
steady-state spectrum for stellar cosmic rays injected close to the stellar surface for Ω = 4Ω or t * = 600 Myr (solid blue line). This is representative of an impulsive stellar energetic particle event. The spectral break occurs at ∼GeV energies as motivated in the previous section. The resulting spectrum at 1 au (and other orbital distances) is presented in Section 3.
A fit to the Galactic cosmic ray LIS, constrained by the Voyager 1 observations (Stone et al. 2013;Cummings et al. 2016;Stone et al. 2019) outside of the heliosphere, is denoted by the solid black line in Fig. 2 (Eq. 1 from Vos & Potgieter 2015). A fit to the modulated Galactic cosmic ray spectrum measured at Earth is given by the solid green line (using the modified force field approximation given in Eq.10 of Rodgers-Lee et al. 2020a, with φ = 0.09 GeV). We also indicate the spectral slope, γ = 2.78, at high energies on the plot. Note, this represents dN/dT ∝ T −γ rather than dN/dp ∝ p −α . The power law indices γ and α are related. At relativistic energies, γ = α since T = pc and at nonrelativistic energies α = 2γ − 1. It is important to note that the measurements at Earth change a certain amount as a function of the solar cycle. Here, however we treat the LIS and the Galactic cosmic ray spectrum at Earth as constant when making comparisons with the stellar cosmic ray spectrum as a function of stellar rotation rate 3 .
3 It is important to note that the Galactic cosmic ray LIS may have been different in the past. Supernova remnants are believed to be a major contributor to the Galactic component of the LIS (Drury 1983(Drury , 2012. The star formation rate (SFR, which can be linked to the number of supernova remnants using an initial mass function) of the Milky Way in the past therefore should influence the LIS in the past. For instance, high ionisation rates (with large uncertainties) have been inferred for galaxies at high redshifts which have higher SFRs than the present-day Milky Way (Muller et al. 2016;Indriolo et al. 2018). In these studies, the inferred ionisation rate is attributed to galactic cosmic rays. Recently, using observations of the white dwarf population in the solar neighbourhood (d < 100pc), Isern (2019) reconstructed an effective SFR for the Milky Way in the past. They found evidence of a peak in star formation ∼ 2.2 − 2.8 Gyr ago, an increase by a factor of ∼3 in comparison to the present-day SFR. Using a sample of late-type stars, Rocha-Pinto et al. (2000) also found an increase in the SFR by a factor of ∼2.5 approximately 2 − 2.5 Gyr ago. Since these results suggest that the SFR has been within a factor of ∼3 of its present-day value for the stellar ages that we focus on, we did not vary the LIS fluxes with stellar age in Rodgers-Lee et al. (2020a). Table 1. List of parameters for the simulations. The columns are, respectively: the age (t * ) of the solar-type star, its rotation rate (Ω) in terms of the present-day solar value (Ω = 2.67 × 10 −6 rad s −1 ), its rotation period (Prot), the astrospheric radius (R h ), the radial velocity (v 1au ) and the magnitude of the total magnetic field (|B 1au |) at r = 1 au.Ṁ is the mass-loss rate. L CR is the power we inject in stellar cosmic rays which we relate to the kinetic power in the stellar wind. The second and third last columns are the momentum and kinetic energy for the stellar cosmic rays at which the exponential break in the injected spectrum occurs. In order to reproduce our injected cosmic ray spectrum (Q in Eq. 1-2): first, the values of L CR and pmax given below can be used in Eq. 4 to determine ( dṄ dp )| 2mc for a given value of Ω. Then, dṄ dp , and therefore Q, can be calculated for a given value of Ω. The last column gives the kinetic energy below which the stellar cosmic ray intensities dominate over the Galactic cosmic ray intensities at 1 au. On the other hand, the differential intensities for SEPs observed at Earth cannot be treated as constant in time.
The SEP spectrum at 1 au, shown by the blue dashed line in Fig. 2, is not continuous in time for the present-day Sun. The differential intensity given by the dashed blue line represents the typical intensities of SEPs at Earth that are derived from time-averaged observations of particle fluences (such as those presented in Mewaldt et al. 2005). This spectrum is representative of a gradual SEP event. This type of SEP event lasts approximately a few days. Rab et al. (2017) estimated the stellar cosmic ray spectrum for a young pre-main sequence star (shown in their Fig. 2) representing the present-day values for a typical gradual SEP event multiplied by a factor of 10 5 (the motivation for which is given in Feigelson et al. 2002). Tabataba-Vakili et al. (2016) similarly use a typical spectrum for a gradual solar energetic particle event and scale it with 1/R 2 to 0.153 au in order to find the values for the differential intensity of stellar cosmic rays at the location of a close-in exoplanet orbiting an M dwarf star. In both of these examples the spectral shape is held constant, whereas here it is not. The propagation of stellar cosmic rays from the star/CME through the stellar system is not the focus of either of these papers. This type of treatment for estimating the spectrum of stellar cosmic ray events at 1 au, or other orbital distances, around younger stars (and later type stars) and the impact of the underlying assumptions are what we investigate in this paper. This can be used as a starting point towards deriving more realistic stellar cosmic ray spectra in the future that can be constrained by upcoming missions like JWST and Ariel (Tinetti et al. 2018).
Transmission spectroscopy using JWST will be able to detect emission features from molecules in exoplanetary atmospheres. Stellar and Galactic cosmic rays should produce the same chemical reactions. Thus, close-in exoplanets around young and/or active stars are the best candidates to detect the chemical signatures of stellar cosmic rays as they should be exposed to high stellar cosmic ray fluxes. In comparison, the Galactic cosmic ray fluxes at these orbital distances should be negligible. Helling & Rimmer (2019) and Barth et al. (2020) identify a number of "fingerprint ions" whose emission, if detected in an exoplanetary atmosphere, would be indicative of ionisation by cosmic rays. These fingerprint ions are ammonium (NH + 4 ) and oxonium (H3O + ). Barth et al. (2020) also suggest that stellar and Galactic cosmic rays contribute (along with other forms of high energy radiation, such as X-rays) to the abundance of the following key organic molecules: hydrogen cyanide (HCN), formaldehyde (CH2O) and ethylene (C2H4). Barth et al. (2020) indicate that CH2O and C2H4 may be abundant enough to possibly be detected by JWST.

Overview of the simulations
We consider 7 cosmic ray transport simulations in total for our results. Additional test case simulations are presented in Appendix A for physical set-ups with known analytic solutions verifying that our numerical method reproduces well these expected results. Six of the 7 simulations that we ran represent the result of varying the stellar rotation rate. The remaining simulation, for Ω = 3.5Ω , investigates the effect of increasing the value of the pmax which is discussed in Appendix B. The parameters for the simulations are shown in Table 1.
The values for the astrospheric radii, R h (Ω), are given in Table 1 which is the outer radial boundary. These values were derived by balancing the stellar wind ram pressure against the ram pressure of the ISM (see Section 2.3.3 of Rodgers-Lee et al. 2020a). The logarithmically spaced radial bins for i = 0, ..., N are given by ri = exp{i×ln(rN /r0)/(N − 1) + ln r0} where r0 = 1 R and rN = R h (Ω) with N = 60.

RESULTS
In this section we investigate the evolution of the stellar cosmic ray spectrum at different orbital distances as a function of stellar rotation rate. Five parameters vary with Ω for these simulations: B(r), v(r), R h , LCR and pmax. The value of R h does not play much of a role in our simulations since it is always much larger than the orbital distances that we are interested in.
After travelling through the stellar wind, stellar cosmic rays can interact with a planet's atmosphere. If a planetary magnetic field is present this will also influence the propagation of the stellar cosmic rays through the atmosphere (e.g. Grießmeier et al. 2015). Higher energy cosmic rays will be less easily deflected by an exoplanetary magnetic field. Cosmic rays with energies that are capable of reaching the surface of an exoplanet are of interest for the origin of life. For this, the pion production threshold energy of 290 MeV should be significant. Pions produce secondary particles which can trigger particle showers (as discussed in Atri 2020, for instance). Sufficiently energetic secondary particles, such as neutrons, can reach the surface of a planet which are known as ground level enhancements. Solar neutrons have been detected even on Earth with neutron monitors since the 1950s (Simpson 1951). Thus, our aim is to determine the range of stellar rotation rates for which the differential intensity of stellar cosmic rays dominates over Galactic cosmic rays at energies above the pion threshold energy.
3.1 Stellar cosmic rays as a function of stellar rotation rate (or age) Fig. 3 shows the stellar cosmic ray differential intensities as a function of kinetic energy for our simulations. In each of the panels the blue shaded region represents the values of differential intensities for stellar cosmic rays present in the habitable zone for a solar-mass star. For comparison, the green shaded region shows the differential intensities for Galactic cosmic rays in the habitable zone (from the simulations presented in Rodgers-Lee et al. 2020a). The habitable zone of a solar-mass star evolves with stellar age which we have incorporated in the shaded regions of Fig. 3. We follow the formalism of Selsis et al. (2007) with the recent Venus and early Mars criteria, using the stellar evolutionary model of Baraffe et al. (1998). At 600 Myr the young Sun was less luminous and had an effective temperature slightly smaller than its present day value. Thus, the habitable zone at 600 Myr was located closer to the Sun between r ∼ 0.64 − 1.58 au in comparison to the present day values of r ∼ 0.72−1.77 au (using the recent Venus and early Mars criteria). Given the finite resolution of our spatial grid some of the blue shaded regions in Fig. 3 are slightly smaller than the calculated habitable zone. Finally, for comparison in each of the panels the solid black line shows the LIS values (from Vos & Potgieter 2015). The vertical grey dashed line represents the pion threshold energy at 290 MeV.
Figs. 3(a)-(f) show at the pion threshold energy that stellar cosmic rays dominate over Galactic cosmic rays in the habitable zone for all values of stellar rotation rate (or age) that we consider. The energy that they dominate up to differs though as a function of stellar rotation rate (given in Table 1). For instance, at Ω = 1.3 Ω , the transition from stellar cosmic rays dominating over Galactic cosmic rays occurs at ∼ 1.3 GeV. It increases up to ∼ 13 GeV for Ω = 4 Ω . The stellar cosmic ray fluxes also increase in the habitable zone as a function of stellar rotation rate. At the same time, the Galactic cosmic ray fluxes decrease.
The red dashed line and solid lines in Fig. 3(e) are the values for the differential intensities at 20 au for Galactic and stellar cosmic rays, respectively. We previously discussed the Galactic cosmic ray differential intensities for Ω = 3.5Ω (Fig. 3(e) here) in Rodgers-Lee et al. (2020a) in the context of the HR2562 exoplanetary system. HR2562 is a young solar-like star with a warm Jupiter exoplanet orbiting at 20 au. Although Galactic cosmic rays (dashed red line) represent a source of continuous cosmic ray flux, stellar cosmic rays can dominate (solid red line) at approximately the orbital distance of the exoplanet for 5 GeV. This would happen at times of impulsive events.
The solid blue line in Fig. 2 shows the steady-state spectrum close to the star corresponding to Ω = 4Ω . By comparing with the values for the fluxes found in the habitable zone, shown in Fig. 3(f), we can determine by how many orders of magnitude the stellar cosmic ray fluxes have decreased between ∼ 1 R and ∼1 au (∼ 200 R ). The decrease is slightly greater than 4 orders of magnitude. The decrease is the combined result of diffusive and advective processes. In Appendix A, we discuss the effect of the different physical processes, shown in Fig. A1. Fig. 4 shows the timescales for the different physical processes for Ω = 4Ω . The diffusion timescales for 0.015, 0.1, 1 and 10 GeV energy cosmic rays are shown by the solid lines in Fig. 4 where t diff = r 2 /κ(r, p, Ω). The magenta dots represent an estimate for the momentum advection timescale t madv ∼ 3r/v. For r 1 au, Fig. 4 shows that the spatial and momentum advection timescales are shorter than the diffusion timescale for cosmic rays with kinetic energies GeV. These low energy cosmic rays are affected by adiabatic losses in this region and are being advected by the stellar wind, rather than propagating diffusively. Since the stellar cosmic rays are injected close to the surface of the star only the cosmic rays with kinetic energies GeV, and therefore short diffusion timescales, propagate diffusively out of this region.
We also investigated the sensitivity of our results on our choice of pmax in Appendix B. Fig. B1 shows the results of adopting a higher maximum cosmic ray momentum for Ω = 3.5Ω . We find that the location of the stellar cosmic ray spectral break is an important parameter to constrain and that it affects our results significantly, with the maximum energy at which stellar cosmic rays dominate Galactic cosmic rays being an increasing function of pmax.
3.2 Differential intensities as a function of rotation rate at 1 au Fig. 5 shows the differential intensities of the stellar cosmic rays at 1 au as a function of Ω. The differential intensities obtained at 1 au increase as a function of stellar rotation rate. The increase in the differential intensities is almost entirely due to the corresponding increase in LCR. The red shaded region indicates the values for different stellar rotation rates with the same stellar age, t * = 600 Myr. The shift in the maximum energy to higher energies with increasing stellar rotation rate can also been seen by comparing the Ω = 1.3Ω (dashed blue line) and Ω = 4Ω (dashed red line) cases. The slope of the spectrum at 10 −2 − 1 GeV energies becomes less steep with increasing stellar rotation and starts to turn over at slightly higher energies.  Figure 3. The differential intensity of stellar cosmic rays (blue shaded regions) and Galactic cosmic rays (green shaded regions) in the habitable zone as a function of kinetic energy. Each panel represents a different value for the stellar rotation rate. Ω = 2.1Ω corresponds to t * = 1 Gyr, shown in (c), when life is thought to have begun on Earth. Also shown are the differential intensities of stellar (solid red line) and Galactic cosmic rays (red dashed line) at 20 au, the orbital distance of HR 2562b, in panel (e). The black solid line is a fit to the Voyager 1 data for the LIS. The grey dashed line represents the pion threshold energy, 290 MeV. See text in Section 3.1. Fig. 3 shows that the intensities of stellar cosmic rays are greater than those of the Galactic cosmic rays at energies around the pion energy threshold for all values of stellar rotation rate that we consider. However, we must also estimate the energy up to which these stellar cosmic rays can be treated as continuous in time.

Assumption of continuous injection
In order for the stellar cosmic ray flux to be considered continuous, the rate of flare events (producing the stellar cosmic rays) must be larger than the transport rate for a given cosmic ray energy. We use 1/t diff at 1 au where it is approximately independent of radius as a reference value for the transport rate. We estimate the maximum stellar cosmic ray energy which can be taken as continuous by considering the relation between flare energy and flare frequency (dN/dE flare ∝ E −1.8 flare from Maehara et al. 2015). First, from Fig. 4 of Maehara et al. (2015) we can obtain the flare rate, by multiplying the flare frequency by the flare energy, as a function of flare energy. Fig. 2 of Maehara et al. (2015) also indicates that stars with rotation periods between 5-10 days flare approximately 10 times more frequently than slow rotators, like the Sun. Thus, as an estimate we increase the flare rate by an order of magnitude for fast rotators as a function of flare energy (Fig. 4 of Maehara et al. 2015). We determine pmax for a given flare energy by equating the flare energy with magnetic energy such that E flare ∝ B 2 (similar to . Therefore, using the Hillas criterion given in Eq. 8, pmax ∝ E 1/2 flare . In Fig. 6, we plot the flare rate (solid lines) and diffusion rates (dashed lines) as a function of momentum. The diffusive timescale for the slow rotator/∼solar case is based on the stellar wind properties presented in Rodgers-Lee et al. (2020a) for the present-day Sun, Ω = 1Ω . For the slow rotator/solar case, this plot indicates that the maximum continuously injected cosmic ray momentum is pc,max = 0.11 GeV/c (Tc,max = 5 MeV). For fast rotators, it indicates that pc,max = 0.4 GeV/c (Tmax = 80 MeV). Thus, even for our most extreme case, flare-injected stellar cosmic rays cannot be considered as continuous beyond 80 MeV in = 4 tdiff,15MeV tdiff,100MeV tdiff,GeV tdiff,10GeV tmadv tadv Figure 4. Timescales for the different physical processes for the stellar wind properties corresponding to a stellar rotation rate of Ω = 4Ω , corresponding to t * ∼ 600 Myr. The solid lines represent the diffusion timescale for cosmic rays with different energies. The magenta dotted line and the grey dashed line represent the momentum advection and advection timescales, respectively. For 10 GeV cosmic rays, t madv t diff at r 0.03 au and t madv t diff at r 0.5 au for GeV energies. This illustrates the importance of adiabatic losses for the stellar cosmic rays at small orbital distances. energy. The plot has been normalised such that ∼GeV cosmic ray energies correspond to E flare ∼ 10 33 erg. It is important to note that here we have determined quite low values of pc,max by comparing the diffusive transport rate with the flare rate. However, a comparison of the flare rate with the chemical recombination rates in exoplanetary atmospheres may result in higher values for pc,max.

DISCUSSION & CONCLUSIONS
In this paper we have investigated the differential intensity of stellar cosmic rays that reach the habitable zone of a solartype star as a function of stellar rotation rate (or age). We motivated a new spectral shape for stellar cosmic rays that evolves as a function of stellar rotation rate. In particular, the maximum injected stellar cosmic ray energy and total injected stellar cosmic ray power evolve as a function of stellar rotation rate. We consider stellar cosmic rays injected at the surface of the star, which would be associated with stellar flares whose solar counterpart are known as impulsive SEP events. The values for the total injected stellar cosmic ray power and the maximum stellar cosmic ray energy that we provide in this paper can be used to reproduce our injected stellar cosmic ray spectrum. We then used the results of a 1.5D stellar wind model for the stellar wind properties (from Rodgers-Lee et al. 2020a) in combination with a 1D cosmic ray transport model to calculate the differential intensity of stellar cosmic rays at different orbital distances.
Our main findings are that, close to the pion threshold energy, stellar cosmic rays dominate over Galactic cosmic rays at Earth's orbit for the stellar ages that we considered, t * = 0.6 − 2.9 Gyr (Ω = 1.3 − 4Ω ). Stellar cosmic rays dominate over Galactic cosmic rays up to ∼ 10 GeV energies for stellar rotation rates > 3Ω , corresponding approximately to a stellar age of 600 Myr. The differential intensities of the stellar cosmic rays increases with stellar rotation rate, almost entirely due to the increasing stellar cosmic ray luminosity. At 1 Gyr, when life is thought to have begun on Earth, we find that high fluxes of stellar cosmic rays dominate over Galactic cosmic rays up to 4 GeV energies. However, based on stellar flare rates, we estimate that the stellar cosmic ray fluxes may only be continuous in time up to MeV energies even for the fastest rotator cases that we consider. For momenta where the diffusive transport rate is larger than the flare rate, the flare injection cannot be treated as continuous. The transition point corresponds to pc,max = 0.1 and 0.4 GeV/c, or to Tc,max = 5 and 80 MeV, for the slow rotator/solar and young solar cases, respectively.
Our results overall highlight the importance of considering stellar cosmic rays in the future for characterising the atmospheres of young close-in exoplanets. They also highlight the possible importance of stellar cosmic rays for the beginning of life on the young Earth and potentially on other exoplanets.
We find for the young exoplanet HR 2562b, orbiting its host star at 20 au, that stellar cosmic rays dominate over Galactic cosmic rays up to ∼ 4 GeV energies despite the large orbital distance of the exoplanet. However, these stellar cosmic ray fluxes may not be continuous in time.
Our results presented in Fig. 3, for a stellar age of 600 Myr (Ω = 4Ω ), demonstrate that low energy stellar cosmic rays (<GeV) move advectively as they travel out through the stellar wind from the injection region to 1 au. In this region the low energy cosmic rays are also impacted by adiabatic losses. Beyond 1 au the low energy cosmic rays are influenced to a greater extent by diffusion. This finding is quite interesting because the velocity of the solar wind at close distances is currently unknown. NASA's Parker Solar Probe (Bale et al. 2019) and ESA's Solar Orbiter (Owen et al. 2020) have only recently begun to probe the solar wind at these distances. Thus, our simulation results are sensitive to parameters of the solar wind that are only now being observationally constrained. If the solar wind is faster in this region than what we have used in our models then the fluxes of stellar cosmic rays that we calculate at larger radii will be smaller.
Our results are based on a 1D cosmic ray transport model coupled with a 1.5D stellar wind model. In reality, stellar winds are not spherically symmetric. Latitudinal variations are seen in the solar wind which also depend on the solar cycle (e.g. McComas et al. 2003) and magnetic maps of other low-mass stars also show that the magnetic field structure is not azimuthally symmetric (e.g. Llama et al. 2013;Vidotto et al. 2014b). Gradients in the magnetic field can lead to particle drifts which we cannot investigate with our models. Our results are based on steady-state simulations which means that effects occurring on timescales shorter than the rotation period of the star are neglected in the cosmic ray transport model. The fact that flares may also occur at positions on the stellar surface which then do not reach Earth is not taken into account in our models. It will be of great interest in the future to use 2D or 3D cosmic ray transport models in combination with 3D stellar wind models (e.g. Kavanagh et al. 2019;Folsom et al. 2020) to study in greater detail the stellar cosmic ray fluxes reaching known exoplanets. Our results represent some type of average behaviour that could be expected: at particular times during a stellar cycle the stellar cosmic ray production rate via flares could be increased, whereas at other times during the minimum of a stellar cycle the production rate would be lower. However, due to the present lack of observational constraints for the stellar cosmic ray fluxes in other stellar systems using a simple 1D cosmic ray transport model and a 1.5D stellar wind model is justified.
Finally, it is also worth bearing in mind that the stellar cosmic rays considered here are representative of impulsive events. The stellar cosmic ray fluxes produced by CMEs are likely to be far in excess of those presented here. These fluxes would be even more transient in nature than the stellar cosmic ray fluxes presented here. In light of these findings, future modelling of stellar cosmic rays from transient flare events and gradual events appears motivated. Computing (ICHEC) for the provision of computational facilities and support. DRL would like to thank Christiane Helling for very helpful discussions which improved the paper. We thank the anonymous reviewer for their constructive comments.

APPENDIX A: TEST CASES
We present three simulations to illustrate that the code reproduces the expected analytic results for a number of simple test cases. We isolate the effect of different physical terms in Eq. 1, giving additional insight into the system. The test cases use the stellar wind parameters for the Ω = 3.5Ω simulation unless explicitly stated otherwise. For all of the test simulations the same power law is injected as described in Eq. 4 with pmax = 1.03 GeV/c and LCR = 4.16×10 28 erg s −1 .
The three test cases are simulations with: (a) a constant diffusion coefficient only, (b) the momentum-dependent diffusion coefficient derived from the magnetic field profile for Ω = 3.5Ω only and (c) the diffusion coefficient used for (b) along with the spatial and momentum advection terms. These test cases are described below in more detail. The results from these tests are shown in Fig. A1 for 1 au.
The first test case consisted of using a constant diffusion coefficient in momentum and space with −v · ∇f = ((∇ · v)/3)(∂f /∂lnp) = 0 from Eq. 1 (κ/βc = 0.07 au using B = 10 −5 G). Thus, a continuous spatial point source injection close to the origin (at ∼ 1.3 R in our case) with a p −2 profile in momentum should result in a steady-state solution with the same momentum power law of p −2 at all radii until the cosmic rays escape from the spatial outer boundary. The blue dots in Fig. A1 represent the cosmic ray intensities as a function of kinetic energy from the simulation at r ∼ 1 au. The dashed line overplot a p −2 e −p/pmax /β profile for comparison and show that our results match well the expected result.
The second case (green dots in Fig. A1) illustrates the effect of a spatially varying diffusion coefficient which also depends on momentum (κ = κ(r, p), as is used in the simulations generally and using the magnetic field profile for the Ω = 3.5Ω case). For a continuous spatial point source injection the particles now diffuse in a momentum-dependent way and the expected profile is p −3 e −p/pmax /β. The green dashed line overplots a p −3 e −p/pmax /β profile for comparison and show that our results match well the expected result.
Finally, we include all three terms in the transport equation which is shown by the solid magenta line in Fig. A1. In MeV 1 ] r = 1 au = C p 2 e p/pmax / All terms = (r, p) p 3 e p/pmax / Figure A1. The differential intensity for stellar cosmic rays as a function of kinetic energy at 1 au are shown here for a number of test cases, described in Section A. The blue dots are the values obtained using a constant diffusion coefficient, test case (a). The green dots represent the model with only spatial diffusion, test case (b). Finally, the magenta solid line includes all terms considered in our models, test case (c). Different power laws are shown by the dashed lines.
comparison to the diffusion only case, the cosmic ray fluxes are decreased at 1 au by nearly 2 orders of magnitude due to spatial and momentum advection.

APPENDIX B: INFLUENCE OF THE MAXIMUM MOMENTUM
Here, we investigate the sensitivity of our results on our choice of pmax, which is used to normalise the scaling relation in Eq. 7. We increase pmax, to 0.6 GeV/c, increasing the maximum momentum to 3.3 GeV/c for Ω = 3.5Ω . We compare the results of the simulation using this higher maximum momentum with the value adopted in the previous section. Fig. B1 shows the differential intensities obtained from these simulations. The red dashed line in Fig. B1 represents the results obtained using pmax = 3.30GeV/c. The red dots are the same as the results shown in Fig. 5 using the lower value of pmax = 1.03GeV/c. The red dash-dotted line represents the differential intensities for Galactic cosmic rays at 1 au. The effect of changing the maximum momenta is quite significant. The higher spectral break means that stellar cosmic rays would dominate over Galactic cosmic ray fluxes up to ∼33 GeV, in comparison to ∼10 GeV for the lower spectral break.
This increase in the intensities occurs because of the timescales for the different physical processes (shown in Fig. 4 for Ω = 4Ω ). By increasing the spectral break to 3.3GeV/c there are sufficient numbers of GeV energy cosmic rays that can avoid momentum losses in the innermost region of the stellar wind. ] pmax = 1.03GeV/c pmax = 3.30GeV/c GCRs@1au Figure B1. The differential intensities for stellar (with two different spectral breaks) and Galactic cosmic rays at 1 au are plotted for Ω = 3.5Ω . The dotted lines represent the same values as in Fig. 5.

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