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All issues Volume 16 (2026) J. Space Weather Space Clim., 16 (2026) 13 Full HTML
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J. Space Weather Space Clim.
Volume 16, 2026
Topical Issue - Space Climate: Solar Extremes, Long-Term Variability, and Impacts on Earth’s System
Article Number 13
Number of page(s) 18
DOI https://doi.org/10.1051/swsc/2026012
Published online 07 May 2026

© N. Eugene, et al., Published by EDP Sciences, 2026

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The transport of charged, energetic particles in the astrospheres of exoplanet-hosting stars has been the subject of an increasing number of studies in recent years, as these particles, whether of stellar or galactic origin, may influence the habitability of these planets (see, e.g., Herbst et al. 2022). Modelling particle transport in an astrosphere other than our own heliosphere is hampered by large uncertainties as to the large- and small-scale astrospheric plasma quantities that influence the various transport mechanisms, such as diffusion and drift (see, e.g., Engelbrecht et al. 2022), that these particles are subject to. Accordingly, the role of galactic cosmic rays (GCRs) has been the subject of scientific debate: while the majority of studies show that their contribution to the exoplanetary radiation environment, for various types of stellar astrospheres, is typically negligible relative to that of stellar energetic particles (see, e.g., Sadovski et al. 2018; Mesquita et al. 2021; Rodgers-Lee et al. 2021; Mesquita et al. 2022; Rodgers-Lee et al. 2023), some studies argue that GCR intensities at some exoplanets could in fact be similar to, or even higher than, those observed at Earth (Herbst et al. 2020b; Engelbrecht et al. 2024; Light et al. 2025; Scherer et al. 2025). These differences, which can arise from differing astrospheric input parameters for the different GCR modulation models, can also arise due to the fact that most studies of GCR modulation in astrospheres employ 1D, steady-state solvers for the 3D, time- and energy-dependent Parker (1965) transport equation governing the transport of these particles. As such, they suffer from inherent limitations (Engelbrecht & Di Felice 2020) and cannot take into account inherently 3D transport processes such as anisotropic diffusion, drifts due to gradients and curvatures of 3D astrospheric magnetic fields, and drifts along possible current sheet structures (for a discussion of this, see Light et al. 2025). To date, no study has been made of the time-dependent transport of GCRs in other astrospheres. Given that activity-cyclic behaviour, encompassing cyclic temporal changes in stellar plasma parameters, has been observed in many stars to date (see Jeffers et al. 2023; Suarez Mascareno et al. 2016b), ranging from M-dwarfs (Suarez Mascareno et al. 2016b; Yadav et al. 2016; Wargelin et al. 2017; Irving et al. 2023; Ibañez Bustos et al. 2025) to G-type stars (Jeffers et al. 2022; Chahal et al. 2025). Intriguingly, some stars have been observed to undergo magnetic polarity reversals (Bellotti et al. 2025), much like those observed over the Sun’s 22-year Hale cycle, but with polarity reversal timescales that appear to decrease with increased rotation period (see Nigro 2022). It is an open question how these variations would influence the transport of GCRs in the astrospheres of these stars.

Since the earliest observations of temporal variations in sunspot numbers, the Sun has been known to display cyclical variations in activity, most notably with the approximately 11-year Schwabe cycle, characterised by, for example, changes in the observed heliospheric magnetic field (HMF) magnitude, as well as in the tilt angle between the Sun’s rotational and magnetic axes. During periods of solar maximum, these quantities are observed to display significantly higher values than during periods of low solar activity, solar minima (see Hathaway 2015, and references therein). This behaviour can be observed in the top two panels of Figure 1, showing spacecraft observations of the HMF magnitude at Earth (blue lines, sourced from OMNI data1, see King & Papitashvili 2005) and the yearly-averaged tilt angle (red lines, sourced from the Wilcox Observatory2, see Hoeksema 1995) as a function of time. The polarity of the HMF has also been observed to reverse over the approximately 22-year Hale cycle. Such changes accordingly influence the transport and hence the intensities of GCRs in the heliosphere, which, when detected by spacecraft or ground-based neutron monitors, display a temporal profile clearly anti-correlated with solar cycle-related changes in, for example, the HMF magnitude at Earth, with peak intensities during solar minimum and significantly lower intensities during solar maximum (e.g., Hedgecock 1975; Moraal 1976; Ahluwalia 2000; Agarwal & Mishra 2008; Mavromichalaki et al. 2007; Kharayat et al. 2016; Usoskin 2023). This can clearly be seen in spacecraft observations of the intensities of GCR proton proxies with a rigidity of 1.28 GV at Earth reported on by Gieseler et al. (2017), shown as a function of time in the bottom panel of Figure 1. Therefore, in the heliosphere, intensities during solar minima are approximately twice as high as those observed during s-olar maxima. Alternating solar minimum GCR temporal intensity profiles also differ, displaying peaked profiles during periods of positive magnetic polarity (A >  0) and more plateau-like profiles during periods of negative magnetic polarity (A <  0)3 (see, e.g., Forbush 1958; Shea & Smart 1981; Quenby 1984; McDonald 1998; Kóta 2013; Caballero-Lopez et al. 2019). This additional 22-year periodic behaviour is a result of the 22-year Hale magnetic polarity cycle of the Sun and is a direct consequence of GCR drift effects: during A >  0, positively charged GCRs experience drift due to gradients in and curvatures of the HMF inwards from the polar regions of the heliosphere and drift outward along the heliospheric current sheet (the neutral sheet separating regions of differing magnetic polarity, see Smith 2001b; Khabarova et al. 2021), these directions reversing during A <  0 (Isenberg & Jokipii 2023; Jokipii & Thomas 1981; Kota & Jokipii 2013; Reinecke & Potgieter 1994; Strauss et al. 2012; Mohlolo et al. 2022; Troskie et al. 2024). As such, a considerable number of heliospheric GCR modulation studies have been devoted to studying time-dependent GCR modulation, from early, steady-state models adapted so as to take into account changes in heliospheric background plasma parameters (such as the tilt and HMF magnitude), such as that employed by Kota & Jokipii (2013) to time-dependent models of increasing complexity (e.g. le Roux 1999; Manuel et al. 2011; Moloto et al. 2018; Wang et al. 2019), to models that are observation-driven, taking into account even the observed solar cycle variations in HMF turbulence parameters (see Zhao et al. 2018; Burger et al. 2022) to yield computed intensities in reasonable to good agreement with spacecraft observations (e.g. Engelbrecht & Moloto 2021; Moloto et al. 2003).

Thumbnail: Figure 1. Refer to the following caption and surrounding text. Figure 1.

Top panel: Heliospheric magnetic field magnitude at 1 au as function of time, from OMNI data. Middle panel: heliospheric tilt angles, classical model results from the Wilcox Solar Observatory. Bottom panel: 1.28 GV GCR proton proxy intensities reported by Gieseler et al. (2017).

The present study aims to study the influence of a stellar activity cycle on the transport of GCR protons and helium in the astrosphere of Proxima Centauri, an M5.5-dwarf located 1.3 pc from the Sun with a rocky exoplanet Prox Cen b4 located at 0.0485 au away from its star in the potential habitable zone with an orbital period of 11 days (Anglada-Escude et al. 2016). It should be noted that Proxima Centauri has been observed to be a strongly flaring star displaying many transient events (e.g. Shapley 1951; Fuhrmeister et al. 2011; Vida et al. 2019; Alvarado-Gomez et al. 2019), which leads to structures similar to the coronal mass ejections (Webb & Howard 2012; Moschou et al. 2019; Zic et al. 2020) and possibly global merged interaction regions such as those observed in the heliosphere during periods of higher solar activity (see, e.g., Burlaga & Ness 2000). This star is chosen as the transport of GCRs within its astrosphere has already been studied, albeit for approximately stellar minimum conditions, using a 3D GCR modulation model by Engelbrecht et al. (2024) and Light et al. (2025), and due to its relative proximity resulting in comparatively more observational information, particularly pertaining to cyclically-varying astrospheric parameters known to influence GCR transport, being available in the literature. As such, Proxima Centauri is a relatively slow-rotating star with a rotation period of 89.8 days (Klein et al. 2021); a surface magnetic field strength reported by Reiners & Basri (2008) using Zeeman broadening to be 600 G (see also the discussion by Garraffo et al. 2022), and by Klein et al. (2021) to be 200 G with a dipole component of 135 G using Zeeman-Doppler imaging (note that these different techniques sample magnetic fields on different scales); and a stellar cycle duration of approximately 7 years (Wargelin et al. 2017; Klein et al. 2021). Furthermore, close to maximum activity, a stellar tilt angle of 51° was reported by Klein et al. (2021). Beyond these observations, several studies have reported on the results of magnetohydrodynamic simulations for plasma parameters essential to the study of GCR modulation in Proxima Centauri’s astrosphere (see, e.g., Herbst et al. 2020b; Engelbrecht et al. 2024), increasingly using Zeeman-Doppler imaging maps as direct inputs (e.g. Garraffo et al. 2016; Alvarado-Gomez 2020; Kavanagh et al. 2021; Garraffo et al. 2022). These provide key inputs for modulation models. The present study will employ observational (where available) and simulation-based inputs for simple, cyclically-varying analytical models for large-scale plasma parameters such as the astrospheric magnetic field (AMF) magnitude and stellar tilt angle, as well as for small-scale turbulence parameters such as magnetic variances and correlation scales. Although no observations or simulations for these latter quantities exist, these are known to be extremely important in modelling the diffusion coefficients of GCRs (see Engelbrecht et al. 2022), and have been observed to vary cyclically in the heliosphere, assuming larger values during periods of solar maximum, and smaller during solar minimum (e.g. Zhao et al. 2018; Engelbrecht & Wolmarans 2020). These will be scaled following the assumed AMF cyclic behaviour, following the observed temporal behaviour of these quantities in the heliosphere (Burger et al. 2022). These temporal scalings will be used as inputs to the numerical modulation code employed by Light et al. (2025), which solves the fully 3D Parker (1965) GCR transport equation in a physics-first manner, employing diffusion coefficients derived from the quasilinear theory (QLT, Jokipii 1966) and the nonlinear guiding center theory (NLGC, Matthaeus et al. 2003). Although to date no observations exist for a magnetic polarity cycle for Proxima Centauri, this behaviour has been observed in other slow-rotating M-dwarf stars (e.g. Lehmann et al. 2024), and will be taken into account in this study, in order to investigate the possible influence of such a polarity cycle on the astrospheric transport of GCRs. It should, however, be noted that cyclic magnetic polarity reversals have been reported on for a variety of stars (see, e.g., Jeffers et al. 2014; Boro Saikia et al. 2016; Rosen et al. 2016; See et al. 2016; Boro Saikia et al. 2022; Bellotti et al. 2025; Alvarado-Gomez et al. 2025). Temporally-varying GCR proton differential intensity spectra so calculated at the location of Proxima Centauri b will then be compared with heliospheric observations at 1 au.

As discussed in previous studies (e.g., Herbst et al. 2019b, 2024), the planetary high-energy particle environment influences (exo)planetary atmospheres, eventually leading to changes in atmospheric chemistry and climate. This, in turn, may affect potential biosignature signals such as ozone and methane, particularly in the case of Earth-like (i.e., N2-O2 dominated Grenfell et al. 2012) atmospheres. Thus, in general, including cosmic ray studies is essential for understanding and interpreting observations from the James Webb Space Telescope (JWST, e.g., Gardner et al. 2006, 2023) as well as future transmission spectra from the Atmospheric Remote-sensing Exoplanet Large-survey (Ariel, e.g., Tinetti et al. 2022) or the Extremely Large Telescope (ELT, see Padovani & Cirasuolo 2023). Cosmic rays further can drive the formation of prebiotic molecules (see, e.g., Rimmer et al. 2014; Airapetian et al. 2016), the building blocks of life, and high-energy GCRs in particular might have indirectly influenced the helicity of DNA (Globus & Blandford 2020). However, an enhanced flux of these energetic particles within an exoplanetary atmosphere can lead to enhanced radiation exposure and, with that, can induce DNA damage (see, e.g., Kennedy 2014). Thus, the manifold effects of cosmic rays within (exo)planetary atmospheres cannot be neglected in the context of (exo)planetary habitability.

The next section is devoted to a brief description of the GCR modulation model used, as well as the temporal profiles assumed for the various plasma inputs. The following section then introduces the atmospheric interaction model, after which model results will be presented. The study closes with a section discussing said results, and future prospects.

2. Modulation model

The present study solves the 3D Parker GCR transport equation stochastically (see, e.g., Pei et al. 2010; Engelbrecht & Burger 2015b; Strauss & Effenberger 2017, for more detail on this numerical technique), given in terms of the omnidirectional GCR phase space density f, related to to the differential intensity by j T  = p 2 f where p is the particle momentum (see, e.g., Moraal 2013), and given by

∂ f ∂ t = ∇ · ( K · ∇ f ) − V sw · ∇ f + 1 3 ( ∇ · V sw ) ∂ f ∂ ln p . Mathematical equation: $$ \begin{aligned} \frac{\partial f}{\partial t} = \nabla \cdot \left( \mathbf K \cdot \nabla f \right) - \mathbf V _{sw} \cdot \nabla f + \frac{1}{3} \left( \nabla \cdot V_{sw} \right) \frac{\partial f}{\partial \ln p}. \end{aligned} $$(1)

The above equation models the influence of various processes on GCR intensities as these particles enter an astrosphere, which include adiabatic energy changes (last term on the right) and convection with a stellar wind travelling at speed V s w (middle term on the right). Diffusion and drift effects are contained in the diffusion tensor K in the first term on the right hand side, given in AMF-aligned coordinates by (Burger et al. 2008)

K ′ = [ κ ⊥ , 3 κ A 0 − κ A κ ⊥ , 2 0 0 0 κ ‖ ] Mathematical equation: $$ \begin{aligned} \mathbf K^{\prime } =\left[\begin{array}{ccc} \kappa _{\perp ,3}&\kappa _{A}&0\\ -\kappa _{A}&\kappa _{\perp ,2}&0\\ 0&0&\kappa _{\parallel }\end{array}\right] \end{aligned} $$(2)

with the subscripts on the diffusion coefficients κ denoting whether they describe diffusion parallel or perpendicular to the local AMF, related to an appropriate mean free path (MFP) by κ = v λ/3, with v the particle speed (see Shalchi 2009). In the stochastic approach, the Parker transport equation is rewritten as a set of equivalent Itō type differential equations (Zhang 1999) which can be solved by tracing a large number (in this study 10 000) of pseudoparticles per energy considered in a time-backward manner from the location of the exoplanet (in this case, Proxima Centauri b at 0.0485 au) to the simulation boundary. This is assumed to be at the location of Proxima Centauri’s termination shock, which is in this study modelled to be the latitudinally and azimuthally-dependent radial distance yielded by the MHD simulations of Proxima Centauri’s astrosphere presented by Engelbrecht et al. (2024). This distance can vary considerably, and is illustrated as function of azimuth in the ecliptic plane in Figure 2, where it extends to larger radial distances in the tail of the astrosphere (at 180°) than in the nose (at 0°). Although considerable GCR modulation is observed in the heliosheath (Stone et al. 2013), the termination shock location is chosen as the modulation boundary here, as Engelbrecht et al. (2024) demonstrated that most GCR modulation in Proxima Centauri’s astrosphere appear to happen within the termination shock. At this non-spherical modulation boundary, the differential intensity spectrum at the initial position (subscript ‘o’) can be calculated from the unmodulated boundary spectrum jB, often referred to as the local interstellar spectrum (LIS), using (Strauss et al. 2011)

Thumbnail: Figure 2. Refer to the following caption and surrounding text. Figure 2.

Radial distances at which Proxima Centauri’s termination shock is located in the ecliptic plane as function of azimuth, as yielded by the 3D MHD modelling presented by Engelbrecht et al. (2024), and employed in this study. Note that the astrospheric nose is located at 0°, and the tail at 180° azimuth.

j ( x i o , t o ) = 1 N ∑ k = 1 N j B ( x i , k e , t k e ) , Mathematical equation: $$ \begin{aligned} j(x_{i}^{o},t^{o})=\frac{1}{N}\sum _{k=1}^{N}j_{B}(x_{i,k}^{e},t_{k}^{e}), \end{aligned} $$(3)

where superscript e denotes an exit time or position. In this study, it is assumed that the LIS does not vary much over the relatively small (on galactic scales) distance between the Sun and Proxima Centauri, and accordingly employ the GCR proton expression proposed by Burger et al. (2008), given by

j B = 19.0 ( P / P 0 ) − 2.78 1.0 + ( P / P 0 ) 2.0 , Mathematical equation: $$ \begin{aligned} j_{B} = 19.0 \frac{(P/P_0)^{-2.78}}{1.0 + (P/P_0)^{2.0}}, \end{aligned} $$(4)

which is a function of rigidity P, with P 0 = 1.0 GV, and is expressed here in units of particles m−2 s−1 sr−1 MeV−1. This expression is chosen for the purposes of comparison with the results of Engelbrecht et al. (2024), and it should be noted that Light et al. (2025) employ a boundary spectrum constructed by Moloto & Engelbrecht (2020) to fit Voyager observations at the heliospheric termination shock, in order to accommodate possible modulation effects in the astrosheaths of the various astrospheres those authors consider. For helium, a fit to the GALPROP5 simulations of GCR helium in the Galaxy by Boschini et al. (2020) as suggested by Engelbrecht et al. (2024) is used, given by

j B He = 12.0 ( P / P 0 ) − 2.7 ( 2.5 + 5.0 ( P / P 0 ) − 3.3 ) 0.6 , Mathematical equation: $$ \begin{aligned} j^{He}_{B} = 12.0 \frac{(P/P_0)^{-2.7}}{(2.5 + 5.0(P/P_0)^{-3.3})^{0.6}}, \end{aligned} $$(5)

in the same units as equation (4). For more detail as to the modulation code employed here, see Engelbrecht & Burger (2015b) and Light et al. (2025). Time-dependent differential intensities are calculated using an approach similar to that of Nagashima & Morishita (1980a); Nagashima & Morishita (1980b): astrospheric plasma parameters corresponding to a particular time are used as inputs to the model, which is then run sequentially for incremental timesteps. This approach has been shown to successfully capture the main temporal features of the solar cycle dependent modulation of GCRs in the heliosphere (see, e.g., Moloto & Engelbrecht 2020; Engelbrecht & Wolmarans 2020).

Diffusion coefficients derived assuming the composite slab+2D turbulence model (Matthaeus et al. 1990; Bieber et al. 1996) are used as inputs for Eq. (2). For the parallel MFP, the QLT expression derived by Teufel & Schlickeiser (2003) for a slab turbulence power spectrum with a wavenumber-independent energy-containing range and a Kolmogorov inertial range with spectral index s, is employed,

λ ‖ ( T ) = 3 s ( s − 1 ) R ( T ) 2 k m ( T ) B 0 2 ( T ) δ B sl 2 ( T ) [ 1 4 π + 2 R − s ( T ) π ( 2 − s ) ( 4 − s ) ] , Mathematical equation: $$ \begin{aligned} \lambda _{\parallel }(T) = \frac{3s}{\left(s - 1\right)}\frac{R(T)^2}{k_m(T)}\frac{B^2_0(T)}{\delta {B^2_{sl}(T)}}\left[\frac{1}{4{\pi }} + \frac{2R^{-s}(T)}{{\pi }(2 - s)(4 - s)}\right], \end{aligned} $$(6)

where δ B s l 2 denotes the slab variance, k m the inverse of the slab correlation scale, and R = R L k m . This quantity is modelled to vary with time T in years after solar maximum as the various input parameters vary with this time. The perpendicular MFP use here is the NLGC result derived by Shalchi et al. (2004) for a 2D power spectrum with the same form as the slab spectrum assumed for λ ∥, with an inertial range spectral index given by 2ν = 5/3, and given by

λ ⊥ ( T ) = [ α d 2 3 π 2 ν − 1 ν Γ ( ν ) Γ ( ν − 1 / 2 ) λ 2 D ( T ) δ B 2 D 2 ( T ) B 0 2 ( T ) ] 2 / 3 λ ‖ 1 / 3 ( T ) , Mathematical equation: $$ \begin{aligned}&\lambda _{\perp }(T) =\nonumber \\&\left[{\alpha _d}^2{\sqrt{3{\pi }}}\frac{2\nu - 1}{\nu }\frac{\Gamma (\nu )}{\Gamma (\nu - 1/2)}{\lambda }_{2D}(T)\frac{{\delta }B^2_{2D}(T)}{B^2_0(T)}\right]^{2/3}{\lambda }_{\parallel }^{1/3}(T), \end{aligned} $$(7)

where the subscript ‘2D’ denotes a 2D turbulence quantity, and it is assumed that α d 2 = 1 / 3 Mathematical equation: $ \alpha_{d}^{2}=1/3 $ following the test-particle simulations of Matthaeus et al. (2003), although this quantity may be smaller in the heliosphere (see Els et al. 2024).

The off-diagonal coefficients κA in equation (2) describe drift effects due to a possible astrospheric current sheet, as well gradient and curvature drifts due to the geometry of the AMF. In the low-turbulence, weak scattering limit, these are given in terms of the maximal particle Larmor radius RL by (Forman et al. 1974)

κ A = v R L 3 , Mathematical equation: $$ \begin{aligned} \kappa _A = \frac{vR_L}{3}, \end{aligned} $$(8)

and can be related to the drift velocity by v d  = ∇ × (κ A e B ) (e.g. Jokipii & Thomas 1981), where e B denotes a unit vector in the direction of the AMF. The drift velocity is then modelled following the approach of Burger (2012), assuming an angular extent of the astrospheric current sheet given by (Kota & Jokipii 2013)

θ ns ( T ) = π 2 − tan − 1 [ tan α ( T ) sin ( Ω r V sw ) ] , Mathematical equation: $$ \begin{aligned} \theta _{ns}(T) = \frac{\pi }{2} - \tan ^{-1} \left[ \tan \alpha (T) ~ \sin \left( \frac{\Omega r}{V_{sw}} \right) \right], \end{aligned} $$(9)

where the stellar tilt angle α is now also a function of the time T in years after stellar maximum (see, e.g., Engelbrecht et al. 2019; Mohlolo et al. 2022, for further discussion on modelling drift effects numerically). The turbulent reduction of drift effects (see, e.g., Engelbrecht et al. 2017; van den Berg et al. 2021) are not considered here, as turbulence levels (as modelled here) are not expected to strongly influence particle drifts (see, e.g., the numerical simulations of Minnie et al. 2007; Els et al. 2024).

In order to model GCR transport coefficients self-consistently, temporal and spatial dependences for the large-scale plasma quantities they depend upon, as well as the small-scale turbulence quantities, need to be chosen. In terms of the spatial dependence of the astrospheric magnetic field, a Parker (1958) model is assumed, motivated by MHD simulation results (e.g. Herbst et al. 2022), given by

B A ( r , θ , T ) = A B 0 ( T ) ( r 0 r ) 2 ( r ̂ − tan Ψ ϕ ̂ ) , Mathematical equation: $$ \begin{aligned} B_{A}(r, \theta , T) = A B_0(T) \left( \frac{r_0}{r} \right)^2 \left( \hat{r} - \tan \Psi \hat{\phi } \right), \end{aligned} $$(10)

which is normalised to a temporally-varying value of B 0(T) at 1 au, with parameter A = ±1 governing the sign of the magnetic field. The magnetic field lines of this AMF form spirals on cones of constant latitude, which are wound with an angle given by

tan Ψ = Ω ( r − r s ) sin θ V sw Mathematical equation: $$ \begin{aligned} \tan \Psi = \frac{\Omega (r - r_s) \sin \theta }{V_{sw}} \end{aligned} $$(11)

with Ω the stellar rotation rate, and V s w the stellar wind speed. As a first approach, a constant value of 1500 km s−1 is chosen for this latter quantity, motivated by the MHD simulations of Proxima Centauri’s astrosphere presented by Herbst et al. (2020b) within the termination shock. It should be noted that Kavanagh et al. (2021) report on a non spherically-symmetric stellar wind with a latitude-dependent speed that assumes a maximum value of 1200 km s−1, and such a stellar wind speed profile will be implemented in future studies, in a manner similar to that employed in heliospheric modulation studies (for an example, see Troskie et al. 2024). It is assumed that B 0(T) increases from a stellar minimum value of 2.4 nT (Engelbrecht et al. 2024) to 4.8 nT, making the assumption that the activity cycle reaches its peak when the magnetic cycle does, following Yadav et al. (2016). This doubling also reflects a factor of approximately 2 change in Proxima Centauri’s surface magnetic field inferred from X-ray observations over its cycle and convective dynamo simulations (see Wargelin et al. 2017; Alvarado-Gomez 2020; Garraffo et al. 2022), and is similar to the solar cycle-dependent behaviour of the heliospheric magnetic field (see Fig. 1). It should be noted however, that the Parker model may not fully capture the complexity of the astrospheric magnetic field. Indeed, it does not fully describe the complexities of the heliospheric magnetic field at solar maximum (Balogh & Smith 2001; Owens & Forsyth 2013), but is here implemented as a first modelling approach. The stellar tilt angle is assumed to vary from a stellar minimum value of 5°, as opposed to the assumption of 0° made in prior studies (implying a completely flat current sheet, see equation (9)) and motivated by heliospheric observations, to a maximum value of 51°, motivated by observations of Proxima Centauri (Klein et al. 2021). The magnetic polarity cycle length for Proxima Centauri is not yet known. As a first approach, it is here assumed that the AMF polarity will change at the time of full stellar maximum, as is the case for the Sun, which implies a magnetic polarity cycle twice the length of the stellar cycle. Such a long cycle choice is motivated by the slow rotation of Proxima Centauri: from observations of other stars, Bellotti et al. (2025) find that a faster magnetic polarity cycle is often associated with a greater rotation rate, albeit for a relatively small sample of stars.

Astrospheric turbulence quantities cannot yet be directly observed, and therefore are modelled here following the approach of Engelbrecht et al. (2024), where heliospheric models are scaled up (or down) by a factor related to the relative magnitude of the AMF to HMF magnitude at 1 au, which can then be varied as function of the stellar cycle through the accompanying variations in the astrospheric magnetic field. This choice is motivated by heliospheric observations of the solar cycle variations of these quantities, which increase or decrease with increases/decreases in the heliospheric magnetic field magnitude (Burger et al. 2022). Spatially, simple power-law radial dependences are chosen for magnetic variances and correlation scales. These are chosen so as to be in agreement with heliospheric observations (see, e.g., Zank et al. 1996; Smith et al. 2001; Pine et al. 2020; Burger & McKee 2023) as well as the results of turbulence transport modelling (see, e.g., Breech et al. 2008; Oughton et al. 2011; Adhikari et al. 2021; Oughton & Engelbrecht 2021). For the total magnetic variance,

δ B T 2 ( T ) = 12.5 nT 2 × ( B A ( T ) B H ) ( r r 0 ) − 2.5 Mathematical equation: $$ \begin{aligned} \delta B^2_T (T) = 12.5\,\mathrm{nT} ^2 \times \left(\frac{B_A (T)}{B_H}\right)\left(\frac{r}{r_0}\right)^{-2.5} \end{aligned} $$(12)

where a heliospheric value of 12.5 nT2 (Smith et al. 2006) at r 0 = 1 au is modified by the ratio of the astrospheric to the solar minimum heliospheric magnetic field magnitude, chosen here to be B H  = 5 nT. As a first approach, the total variance is assumed to be composed of an 80% 2D component and a 20% slab component, after Bieber et al. (1994). The spatial dependence of the 2D correlation scale is also modelled following heliospheric observations (see Smith et al. 2001; Cuesta et al. 2022), assuming a value at 1 au that is scaled up from the heliospheric value reported by Weygand et al. (2011), such that

λ 2 D ( T ) = 0.0074 au × ( B H B A ( T ) ) ( r r 0 ) 0.5 Mathematical equation: $$ \begin{aligned} \lambda _{2D} (T) = 0.0074\,\mathrm{au} \times \left(\frac{B_H}{B_A (T)}\right)\left(\frac{r}{r_0}\right)^{0.5} \end{aligned} $$(13)

with a temporal dependence again governed by that of the AMF. The slab correlation scale is modelled after the heliospheric ratio of these quantities reported on by Weygand et al. (2011), such that

λ sl = 2.55 λ 2 D ( T ) . Mathematical equation: $$ \begin{aligned} \lambda _{sl} = 2.55 \lambda _{2D} (T). \end{aligned} $$(14)

Given that detailed observations are not yet available, the temporal dependences of the quantities discussed above are modelled using a simple cosine dependence, given by

x = a + b cos [ 2 π T P c ] Mathematical equation: $$ \begin{aligned} x=a+b\cos \left[\frac{2\pi T}{P_{c}}\right] \end{aligned} $$(15)

with P c  = 7 the stellar cycle length in years, and where a = (x max + x min)/2 and b = (x max − x min)/2, quantities that refer to the maximum and minimum values for the various plasma quantities modelled here, which are listed in Table 1. Note that the temporal profiles used here, as modelled with equation (15), do not take into account any transient structures due to flaring that may propagate outwards into the astrosphere, such as corotating interaction regions or merged interaction regions. As a first approach, however, these effects are omitted here, as it is at present unclear how such structures will evolve in Proxima Centauri’s astrosphere, which would require magnetohydrodynamic modelling (for a heliospheric example of such a study, see Wiengarten et al. 2015). The top two panels of Figure 3 illustrate the modelled stellar cycle dependences for the AMF magnitude at 1 au and the stellar tilt angle. Both drop from maximal values during stellar maximum (T = 0, 7 and 14 years) towards stellar minimum (T = 3.5 and 10.5 years), under the assumption that both will change in phase. As a first approach, the temporal profile employed here for the tilt angle α is not the same as that observed for the heliosphere, shown in Figure 1, which does not display a symmetric temporal profile in the years around to solar maximum (see, e.g., Smith 2001b). Should future observations motivate the use of such a profile, the analytical fit proposed by Burger et al. (2008) can potentially be modified and employed in subsequent studies. Magnetic variances and correlation scales (not shown) follow similar temporal profiles. The bottom panel of Figure 3 shows the parallel MFP (Eq. (6), red line), the perpendicular MFP (Eq. (7), blue line), and the drift scale (which is here, assuming the weak scattering limit, equal to the maximal Larmor radius RL found in equation (8), and shown as the green line in the figure) calculated at a rigidity of 1.28 GV as function of time after stellar maximum at 1 au. The parallel MFP assumes larger values during stellar maximum than during stellar minimum, and remains roughly two orders of magnitude larger than λ⊥ for all times shown, a factor similar to that expected in the heliosphere (see, e.g., Palmer 1982; Lang et al. 2024). The perpendicular MFP assumes larger values during stellar minima than during maxima, the opposite being true for the drift scale due to its dependence on the AMF magnitude, and remains larger than λ⊥, even during times of stellar maximum. This is surprising, as the drift scale drops below λ⊥ during solar maximum in the heliosphere, leading to GCR intensities relatively unaffected by drift effects during those periods (see, e.g., Engelbrecht & Moloto 2021).

Table 1.

Stellar minimum and maximum model inputs. See text for details.

Thumbnail: Figure 3. Refer to the following caption and surrounding text. Figure 3.

Top panel: Assumed temporal profile for Proxima Centauri’s AMF magnitude at 1 au, as function of years after full stellar maximum. Middle panel: Assumed temporal profile for Proxima Centauri’s tilt angle. Bottom panel: 1.28 GV GCR proton parallel and perpendicular MFPs (red and blue lines, respectively) as well as drift scales (green line), at 1 au. See text for details.

3. Atmospheric interaction model

Stellar cycle-induced GCR intensity changes will directly influence the atmospheres of planets orbiting stars. To study the atmospheric effects of cosmic rays, Banjac et al. (2019) developed the Atmospheric Radiation Interaction Simulator (AtRIS), a GEANT4-based model allowing to compute induced secondary particle showers, ionisation rates, and radiation exposure in diverse exoplanetary atmospheres. AtRIS has successfully been validated and applied in solar system studies (e.g. Herbst et al. 2019a, 2020a; Guo et al. 2019; Winant et al. 2023) and beyond (e.g., Scheucher et al. 2020; Herbst et al. 2019b, 2024; Scherer et al. 2025; Light et al. 2025).

One of the main indicators of the atmospheric cosmic ray impact is the atmospheric ionisation which is directly linked to atmospheric chemistry and climate (e.g., Herbst et al. 2019b, 2024). Numerically speaking, the induced atmospheric ion pair production rate Q is described as

Q ( E c , x ) = ∑ i ∫ E c ∞ j T ( E ) · Y i ( E , x ) d E . Mathematical equation: $$ \begin{aligned} Q(E_c,x) = \sum _i \int _{E_c}^{\infty } j_T(E)\cdot Y_i(E,x)\,\mathrm{d}E. \end{aligned} $$(16)

Thereby, Y i (E, x) refers to the atmospheric ionisation yield given by 2π∫dθcos(θ)sin(θ)⋅(Δ E i /Δ x)/E ion, with E ion the average atmospheric ionisation energy6 and Δ E i /Δ x the mean specific energy loss of a primary particle of type i at a specific altitude x. The variable E c refers to the cutoff energy of the primary particle, the energy needed to reach specific locations at certain altitudes. In this study, E c is set at 10 MeV reflecting low magnetic shielding (i.e., polar regions at Earth).

AtRIS further provides a pre-calculated relative ionisation efficiency ℐ R, j (E i ), the ratio between the average ionisation energy caused by a secondary particle of type j produced (e.g., p, n, e±, etc.) in a pre-selected phantom. We selected a human-mimicking phantom to study the effect on life as we know it from Earth. In particular, the International Commission on Radiation Units and Measurements (ICRU) phantom (McNair 1981) has been used. Of particular interest are the absorbed dose rates D j ¯ Mathematical equation: $ \bar{D_j} $ given by

D ¯ j ( E i , r ) = I R , j ( E i ) · E i m ph ( r ) · Mathematical equation: $$ \begin{aligned} \bar{D}{j}\left(E_i, r\right) = \mathcal{I} _{R,j}(E_i) \cdot \frac{E_i}{m_{\mathrm{ph} }(r)}\cdot \end{aligned} $$(17)

Here, m ph denotes the phantom’s mass7. Convolving the results with the primary particle spectrum and summing over all energy bins and particle types gives the absorbed dose rate profiles.

4. Results

The top panel of Figure 4 shows differential intensities of GCR protons as function of kinetic energy computed at Proxima Centauri b during times of full stellar minimum (solid lines) and full stellar maximum (dashed lines) for both A >  0 (red lines) and A <  0 (blue lines) magnetic polarity conditions. Also shown are heliospheric observations of GCR protons at 1 au, taken during solar minimum periods of positive and negative magnetic polarity, reported by McDonald et al. (1992). As observed in the heliosphere, GCR intensities during stellar minimum are larger than during stellar maximum, indicating a clear influence of stellar cycle related changes in astrospheric plasma parameters on computed GCR intensities. Stellar minimum intensities are close to those reported on by Engelbrecht et al. (2024), but differ in that the difference in intensities for A <  0 and A >  0 is somewhat larger here. This is due to the fact that a tilt angle of 5° is assumed here, while Engelbrecht et al. (2024) assume a completely flat current sheet. This is reminiscent of the relationship between GCR intensity and tilt angle observed in the heliosphere, where intensities during A >  0 decrease sharply with small increases in α directly after solar minimum, while intensities remain relatively constant for similar changes during A <  0 (see, e.g., Lockwood & Webber 2005). It should be noted that stellar minimum intensities reported on here are larger than those reported on by Light et al. (2025), due to differences in the boundary spectra. The intensities for both stellar maximum and stellar minimum in Figure 4 also remain well above heliospheric observations. This is true even during stellar maximum, and is due to the slow rotation of the star, which leads to a highly underwound AMF. which in turn leads to diffusion parallel to the AMF contributing largely towards the inward transport of GCRs (Light et al. 2025). During stellar minimum, A <  0 intensities remain above A >  0 intensities, as reported on by Engelbrecht et al. (2024), and in contrast to what is observed in the heliosphere. This can again be related to the slow stellar rotation rate, in that drifts due to gradients in, and curvatures of, the AMF, which govern the inward transport of GCR protons during A >  0, are accordingly less effective than current sheet drift, which governs the inward transport of GCRs during A <  0. This picture is somewhat reversed during stellar maximum, where, in contrast with what is observed in the heliosphere during solar maximum, drift effects still appear to play a role, and where A >  0 intensities are now slightly larger than A <  0 intensities. This is not entirely unexpected, as from the bottom panel of Figure 3 it can be seen that the drift scale remains larger than the perpendicular diffusion coefficient even over stellar maximum periods, in contrast to the heliosphere. As the astrospheric current sheet as modelled by equation (9) would be considerably wavier due to the larger tilt angle, particles have further to drift along this structure, leading to lower intensities. It should be noted that the stellar maximum spectra shown here differ from those computed by Light et al. (2025) where a tilt of 51° was assumed, with all other parameters held to stellar minimum values. From their results, it could be concluded that stellar cyclic effects would be negligible during A <  0, and considerably smaller than can be seen in Figure 4 for A >  0. These differences arise because that study did not include the effects of a larger AMF magnitude during stellar maximum and did not vary magnetic variances and correlation scale with stellar cycle either, as is done here, thereby not fully modelling the influence of stellar cycle variations on GCR transport coefficients, and highlighting the need to do so. GCR Helium intensities computed at Proxima Centauri b are shown in the middle panel of Figure 4. These behave very similarly to the GCR protons, with intensities during stellar minimum and maximum remaining well above observed heliospheric intensities at 1 au reported by Adriani et al. (2013). Charge-sign dependent modulation can also still be discerned during full stellar maximum.

Thumbnail: Figure 4. Refer to the following caption and surrounding text. Figure 4.

Top panel: GCR proton differential intensities as function of kinetic energy, calculated at 0.0485 au for A >  0 (red lines) and A <  0 (blue lines) magnetic polarity conditions. Solid lines indicate intensities calculated during full stellar minimum, dashed during full stellar maximum. Also shown are the Burger et al. (2008) LIS (green line), and heliospheric observations at Earth reported by McDonald et al. (1992). Middle panels: Same as top panel, but for GCR Helium. Green dashed line denotes the Boschini et al. (2020) LIS employed here, while stars indicate A <  0 PAMELA observations reported by Adriani et al. (2013). Bottom panel: 1.28 and 0.44 GV GCR proton intensities (green and black lines, respectively) calculated at 0.05 au as function of time after full stellar maximum.

The bottom panel of Figure 4 shows 1.28 GV (0.65 GeV, green line) and 0.44 GV (0.1 GeV, black line) GCR intensities computed as function of years after full stellar maximum at Proxima Centauri b. Given the similarity in relative behaviour between stellar minimum and maximum intensities for GCR protons and helium discussed above, only proton intensities are shown here. Red and blue shadings indicate positive and negative magnetic polarity conditions, respectively, over a time period of twice the stellar cycle length. Note that the discrete jumps/drops in intensities just past the 7-year period is indicative of a change in magnetic polarity. This, similar to Kota & Jokipii (2013) is done discretely in the model, and not propagated outwards with time. Continuously propagating such a polarity change outward into the astrosphere will lead to a more continuous transition, as has been demonstrated in heliospheric models by, e.g., Moloto & Engelbrecht (2020), but is beyond the scope of the present study. The GCR modulation model is run consecutively for incrementally increased times, so that each run, and resulting intensity, is separated by three months, as indicated by the dots on the lines. The resulting intensity profiles are remarkably different to that seen for 1.28 GV GCR protons at 1 au in the heliosphere (bottom panel of Fig. 1). For the 1.28 GV protons at Proxima Centauri b, only a small variation from minimum to maximum can be seen, and the stellar minimum peak profiles during A >  0 are only moderately different from those computed for A <  0 conditions. As can be seen in the top panel of Figure 4, the amplitude of the variation in GCR intensity from a peak during stellar minimum to a trough during stellar maximum varies as function of energy, being more marked for the 0.44 GV protons (where more modulation occurs), as well as being larger during A <  0 than during A >  0. For the 1.28 GV protons, the ratio between full stellar minimum (at 3.5 and 10.5 years here) and full maximum (at 0 and 14 years in the figure) intensities is 1.13 during A >  0, and 1.12 during A <  0, in contrast with the peak-to trough ratio of an approximately consistent value of 3 observed in the heliosphere at this rigidity, demonstrating a more modest dependence of GCR intensities on stellar cycle-varying plasma parameters, arising from the influence of the slow rotation of the parent star on GCR transport that leads to less modulation. At lower energies, however, the effect of taking into account the stellar cycle variation of parameters becomes much more significant: at 0.1 GeV n−1, the A >  0 ratio is 1.49, while during A <  0 it is 1.84. Differences in peaks during different magnetic polarity conditions are still not as striking as observed in the heliosphere: the A <  0 peak profile at 0.44 GV is only somewhat sharper than the very slightly plateaued A >  0 profile. Lastly, it should be noted that the smoothness of the GCR temporal profiles in the bottom panel of Figure 4 reflects the smoothness of the input temporal profiles of the plasma quantities employed in the model (Eq. (15)). Taking into account more complex, transient phenomena such as corotating interaction regions or global merged interaction regions, would greatly alter this behaviour. These have been shown to lead to decreases in GCR intensities in the heliosphere, as they act as transient barriers to the inward diffusion of GCRs (see, e.g., Burlaga 1984; Potgieter et al. 1993; le Roux & Fichtner 1999; Richardson et al. 2022; Burlaga et al. 2003) due to enhanced turbulence (see, e.g., Wiengarten et al. 2015; Strauss et al. 2016), particularly towards solar maximum. As such, the GCR intensities shown in Figure 4 can possibly be interpreted as representing an upper estimate.

Based on the GCR spectra discussed in Section 2, Figure 5 shows the atmospheric GCR-induced ion-pair production rate (left panel) and absorbed dose rate (middle panel) profiles modelled with AtRIS. Here, the contributions due to the primary proton (dashed lines) and helium (dotted lines) GCRs are separated and the total values (solid lines) used in what follows are displayed. The right panel further shows the helium particle contribution to the total ion pair production rates (blue line) as well as the absorbed dose rates (orange line). The results account for the A <  0 stellar minimum conditions. As can be seen, the helium contribution to the ionisation rates at altitudes above 23 km and absorbed dose rates above 11 km exceeds 20%, highlighting the necessity of including helium spectra, particularly when ionisation rates – crucial inputs to atmospheric chemistry and climate models – are derived.

Thumbnail: Figure 5. Refer to the following caption and surrounding text. Figure 5.

GCR-induced atmospheric ion-pair production rates (left panel) and absorbed dose rates (middle panel) due to primary protons (dashed lines) and helium (dotted lines); solid lines represent the sum of both. The right panel shows the helium contribution.

In addition, Figure 6 shows the stellar cycle induced changes in atmospheric ionisation (left panels) and absorbed dose rates (right panels), calculated using both the primary proton and helium GCR contributions. The upper panels display the full stellar maximum condition induced changes in years 0, 7, and 14 corresponding to stellar maximum while the lower panels refer to the full stellar minimum conditions during A >  0 conditions (i.e., at year 3.5 after maximum) and during A <  0 conditions (i.e., at year 10.5 after maximum). In all cases, the petrol bands represent the induced changes over the entire time frame after full stellar maximum. The maximum induced changes over the entire time frame (i.e., years 0 and 10.5) are up to 28% in atmospheric ionisation rates and up to 5% in absorbed dose rate values. The changes of the full stellar maximum conditions in years 0 and 7 are almost identical while showing an increase of up to 8% and 2%, respectively. The A <  0 conditions induce changes in the order of up to 12% (2%) in atmospheric ionisation (absorbed dose rates) higher than the A >  0 conditions.

Thumbnail: Figure 6. Refer to the following caption and surrounding text. Figure 6.

GCR induced ion pair production rates (left panels) and absorbed dose rates (right panels), during the full stellar maximum conditions in years 0, 7, and 14 (the first two give identical values) in the upper panels and the full stellar minimum conditions in years 3.5 (A >  0) and 10.5 (A <  0) in the lower panels. The petrol-shaded envelope shows the entirety of the variations within these 14 years.

5. Conclusions

Stellar cycle-related changes in the astrospheric magnetic field magnitude and stellar tilt angle have been demonstrated here, for the first time, using a 3D ab initio cosmic ray modulation model, to lead to stellar cycle-related changes in GCR proton and Helium intensities at the location of Proxima Centauri b. Qualitatively, these intensity variations follow the same pattern as those observed in the heliosphere: peak intensities occur during stellar minima, and minimum intensities during periods of high stellar activity. Quantitatively, the stellar cycle variations of GCR intensities in Proxima Centauri’s astrosphere are much more modest than those observed at Earth. During stellar minima the intensities computed for negative magnetic polarity conditions are larger than those computed for positive polarity conditions, again in contrast with what is usually observed during solar minima, and drift effects persist even during stellar maxima, which again contrasts with the essentially drift free GCR propagation observed in the heliosphere during solar maximum. Intriguingly, even though GCR proton and Helium temporal intensity profiles display minima during stellar maximum periods, these intensities remain significantly above solar minimum observations at Earth. This is due to a combination of factors, as reported by Engelbrecht et al. (2024) and Light et al. (2025). The slow rotation of Proxima Centauri leads to an underwound astrospheric magnetic field, which allows for enhanced particle diffusion into the astrosphere along astrospheric magnetic field lines, which is coupled with the effects of a smaller magnetic field magnitude, leading to larger particle Larmor radii and hence more efficient drift effects. The significance of the interplay of such inherently 3D phenomena highlights the need for 3D modelling of energetic particle transport in stellar astrospheres. Furthermore, the idiosyncrasies of stellar cycle-related effects on the temporal variation of GCR intensities in the astrosphere highlight the differences in these effects from what would be expected of them based on heliospheric observations. We further investigated the impact of GCR modulation-induced stellar cyclicity effects on an assumed Earth-like atmosphere for Prox Cen b as employed by Engelbrecht et al. (2024). Similar to the solar system, stronger modulation results in lower GCR-induced atmospheric ionisation and radiation exposure. However, changes exceeding 20% moderately affect the atmosphere due to variations in GCR flux at altitudes above 20 km. As exoplanetary atmospheric transmission spectra rely on data regarding atmospheric chemistry in the upper layers, this highlights the necessity for more precise information on GCR-induced background ionisation and radiation exposure in order to effectively interpret future observations from JWST, Ariel and the ELT for transiting exoplanets orbiting stars that display stellar cyclic behaviour.

As more observations of cyclic behaviour become available for a greater number of (potentially) exoplanet-hosting stars, these effects will need to be taken into account in particle transport modelling endeavours, so as to ascertain their significance in influencing the ionizing radiation environments of their (potential) exoplanets. The present study provides a first theoretical and modelling framework to do so in as self-consistent a manner as possible. However, future GCR modulation studies will also need to take into account the influence of transient, large-scale structures, such as corotating interaction regions, that can arise as a result of stellar flaring. The current model also employs parametric scalings for the spatial dependences of basic turbulence quantities used to compute GCR diffusion coefficients. This is a limitation, in that these are motivated by heliospheric observations. Future studies will include turbulence transport modelling, such as already has been done for the heliosphere (see, e.g. Oughton & Engelbrecht 2021; Adhikari et al. 2021) and for a young-Sun proxy star (Engelbrecht et al. 2026), to provide more theoretically self-consistent turbulence inputs for our GCR modulation code that are appropriate to Proxima Centauri’s astrosphere, so as to provide as realistic as possible estimations of the radiation environments of exoplanets orbiting active stars.

Acknowledgments

We acknowledge use of NASA/GSFC’s Space Physics Data Facility’s OMNIWeb service, and OMNI data. The editor thanks two anonymous reviewers for their assistance in evaluating this paper.

Funding

Part of this work was funded by the Research Council of Norway (RCN), through its Centre of Excellence funding scheme, project number 332523 (PHAB, Centre for Planetary Habitability).

Conflict of interest

The authors declare no Conflict of Interest.

Data availability statement

The data generated for this study are available on request from the corresponding author.

References

3

During A >  0 the HMF in the northern hemisphere points away from the Sun and towards it in the southern hemisphere, the opposite being true during A <  0.

4

By convention (see Hessman et al. 2010), exoplanets are named by adding a lowercase letter, starting at ’b’ and proceeding alphabetically in order of discovery, to the name of their host star.

6

For the N2-O2 dominated atmosphere assumed in this study, an average atmospheric ionisation energy of 32 eV is used (e.g., Simon Wedlund 2011).

7

With m ph = ρ · 4 3 π · r ph 3 Mathematical equation: $ m_{\mathrm{ph}}=\rho \cdot \frac{4}{3}\pi \cdot r_{\mathrm{ph}}^3 $ (see, e.g., Herbst et al. 2020a).

Cite this article as: Engelbrecht NE and Herbst K. 2026. On stellar activity cycle-related cosmic ray modulation effects in the astrosphere of Proxima Centauri. J. Space Weather Space Clim. 16, 13. https://doi.org/10.1051/swsc/2026012.

All Tables

Table 1.

Stellar minimum and maximum model inputs. See text for details.

In the text

All Figures

Thumbnail: Figure 1. Refer to the following caption and surrounding text. Figure 1.

Top panel: Heliospheric magnetic field magnitude at 1 au as function of time, from OMNI data. Middle panel: heliospheric tilt angles, classical model results from the Wilcox Solar Observatory. Bottom panel: 1.28 GV GCR proton proxy intensities reported by Gieseler et al. (2017).
In the text
Thumbnail: Figure 2. Refer to the following caption and surrounding text. Figure 2.

Radial distances at which Proxima Centauri’s termination shock is located in the ecliptic plane as function of azimuth, as yielded by the 3D MHD modelling presented by Engelbrecht et al. (2024), and employed in this study. Note that the astrospheric nose is located at 0°, and the tail at 180° azimuth.
In the text
Thumbnail: Figure 3. Refer to the following caption and surrounding text. Figure 3.

Top panel: Assumed temporal profile for Proxima Centauri’s AMF magnitude at 1 au, as function of years after full stellar maximum. Middle panel: Assumed temporal profile for Proxima Centauri’s tilt angle. Bottom panel: 1.28 GV GCR proton parallel and perpendicular MFPs (red and blue lines, respectively) as well as drift scales (green line), at 1 au. See text for details.
In the text
Thumbnail: Figure 4. Refer to the following caption and surrounding text. Figure 4.

Top panel: GCR proton differential intensities as function of kinetic energy, calculated at 0.0485 au for A >  0 (red lines) and A <  0 (blue lines) magnetic polarity conditions. Solid lines indicate intensities calculated during full stellar minimum, dashed during full stellar maximum. Also shown are the Burger et al. (2008) LIS (green line), and heliospheric observations at Earth reported by McDonald et al. (1992). Middle panels: Same as top panel, but for GCR Helium. Green dashed line denotes the Boschini et al. (2020) LIS employed here, while stars indicate A <  0 PAMELA observations reported by Adriani et al. (2013). Bottom panel: 1.28 and 0.44 GV GCR proton intensities (green and black lines, respectively) calculated at 0.05 au as function of time after full stellar maximum.
In the text
Thumbnail: Figure 5. Refer to the following caption and surrounding text. Figure 5.

GCR-induced atmospheric ion-pair production rates (left panel) and absorbed dose rates (middle panel) due to primary protons (dashed lines) and helium (dotted lines); solid lines represent the sum of both. The right panel shows the helium contribution.
In the text
Thumbnail: Figure 6. Refer to the following caption and surrounding text. Figure 6.

GCR induced ion pair production rates (left panels) and absorbed dose rates (right panels), during the full stellar maximum conditions in years 0, 7, and 14 (the first two give identical values) in the upper panels and the full stellar minimum conditions in years 3.5 (A >  0) and 10.5 (A <  0) in the lower panels. The petrol-shaded envelope shows the entirety of the variations within these 14 years.
In the text

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