A Readily Implemented Atmosphere Sustainability Constraint for Terrestrial Exoplanets Orbiting Magnetically Active Stars

Planets beyond our solar system have become an object of fascination in recent decades, with nearly regular references in headlines and the popular media. Only recently have observational capabilities evolved to the point where potential terrestrial planets are detected around M-type dwarf stars. However, young M-type dwarfs are known to be magnetically active, often more than our middle-aged Sun. Superflares in them are a common occurrence (e.g., Maehara et al. 2012; Armstrong et al. 2016) that should result in fast and massive coronal mass ejections (CMEs; see, e.g., Khodachenko et al. 2007; Lammer et al. 2007; Vidotto et al. 2011, and several others). The CMEs are gigantic, eruptive expulsions of magnetized plasma and helical magnetic fields from the solar and stellar coronae at speeds that may well surpass local Alfvénic and sound speeds, severely but temporarily disrupting stellar winds and generating shocks around their bodily ejecta.

Contrary to solar flares that have been known since the 19th century (Carrington 1859), CMEs were only observed well into the space age (Howard 2006) due to their much fainter magnitude compared to the bright solar photospheric disk. Stellar CME detections are notoriously ambiguous, although recent efforts offer reasonable hope (Argiroffi et al. 2019). However, in strongly magnetized stellar coronae, CMEs are inevitable. In the case of intense stellar magnetic activity and the existence of an atmosphere that shields a planet, extreme pressure effects by CMEs owning to stellar mega-eruptions can, under certain circumstances, cause intense atmospheric depletion via ionization-triggered erosion (e.g., Zendejas et al. 2010). In the solar system, results from NASA's Mars Atmosphere and Volatile Evolution mission (MAVEN) seem to establish that the sustained eroding effect of solar interplanetary CMEs (ICMEs) may be responsible for the thin Martian atmosphere (Jakosky et al. 2015) after the planet's magnetic field weakened.

Our objective here is the introduction of a practical and reproducible (magnetic) atmosphere sustainability constraint (mASC), reflected on a positive, rational number ${ \mathcal R }$ and relying on CMEs and planetary magnetic pressure effects. Here ${ \mathcal R }$ is a dimensionless ratio that, in tandem with the considered habitability zone (HZ; e.g., Kopparapu et al. 2013), can provide an understanding of which terrestrial exoplanets warrant further screening for the existence of a possible atmosphere (${ \mathcal R }\lt 1$) or otherwise (${ \mathcal R }\gt 1$). Our mASC becomes fully constrained in the case of tidally locked exoplanets (e.g., Grießmeier et al. 2004) by means of an estimated planetary magnetic field, while if no tidal locking is assumed, the planetary magnetic field can be replaced by a known benchmark field, such as Earth's. Only magnetic pressure effects are taken into account in this initial study, but our mASC ${ \mathcal R }$ can be generalized at will. Given that pressure effects are extensive and additive, however, adding more terms (i.e., kinetic, thermal) to the CME pressure will only increase the adversity of possible atmospheric erosion effects for a studied exoplanet (e.g., Ngwira et al. 2014).

The magnetic activity of the exoplanets' host stars reflects on several observational facts, including the bolometric energy of their flares. From it, and assuming a Sun-as-a-star analog further reflected on the solar magnetic energy–helicity (EH) diagram (Tziotziou et al. 2012), we estimate the magnetic helicity of stellar CMEs and a corresponding stellar CME magnetic field based on the fundamental principle of magnetic helicity conservation in high magnetic Reynolds number plasmas (Patsourakos et al. 2016; Patsourakos & Georgoulis 2017). As explained in Appendix A, CMEs are inevitable in strongly magnetized stellar coronae, as they relieve stars from excess helicity that cannot be shed otherwise.

The near-star CME magnetic field is propagated self-similarly in the astrosphere until it reaches exoplanet orbits (Patsourakos & Georgoulis 2017). The mASC introduced in this study achieves a precise, quantitative assessment of whether the magnetic pressure of stellar ICMEs alone can be balanced by the estimated (tidally locked) or guessed (in the general case) magnetic pressure of observed terrestrial exoplanets at a magnetopause distance large enough to avert the erosion effects of a possible atmosphere.

There are two conceptual pillars of the mASC introduced here. First, it relies on observed stellar flare energies but does not perform a case-by-case stellar eruption analysis. In other words, it does not look at the particular eruption but points to the collective effect of numerous similar eruptions over ∼1 Gyr, or significant fractions thereof, of the young star's activity (see also Chen et al. 2020). In this sense, the suitable orientation required for an enhanced ICME planeto-effectiveness, namely, the ICMEs' ability to reconnect with the planetary magnetic field causing magnetic storms, is ignored; it is implicitly assumed that numerous such ICMEs will have the correct orientation. Second, our mASC ratio relies explicitly on a worst-case scenario (i.e., largest possible ICME magnetic field strength) magnetic pressure for stellar CMEs (∼${B}_{\mathrm{worst}}^{2}$) and a best-case scenario planetary magnetic field (i.e., largest possible planetary magnetic field) generated in the planet's core due to internal dynamo action (∼${B}_{\mathrm{best}}^{2}$). Then, our mASC becomes the ratio of planetary magnetic intensities relating to these pressure terms: the minimum planetary magnetic field (equal to B_worst) able to balance the worst-case CME magnetic field and the best-case planetary magnetic field B_best as per the modeled planetary characteristics, i.e.,

$$\begin{eqnarray}&&{ \mathcal R }=\displaystyle \frac{{B}_{\mathrm{worst}}}{{B}_{\mathrm{best}}}.\end{eqnarray} \tag{ 1 }$$

The two planetary magnetic field strengths are taken at a critical magnetopause distance from the studied planet in terms of atmospheric erosion effects (Sections 2.1 and 2.2). It is then understood that if ${ \mathcal R }\gt 1$, the planet's magnetosphere will be compressed beyond the critical threshold, presumably leading to atmospheric erosion (after processes such as thermal, nonthermal, or hydrodynamic escape, catastrophic erosion, etc., take place; see, for example, Melosh & Vickery 1989; Lundin et al. 2004; Barabash et al. 2007 for more details). Assuming that, statistically, the presumed ICME is not a unique occurrence, the planet may undergo this atmospheric stripping for hundreds of millions of years due to its star's magnetic activity. Magnetic helicity conservation, on the other hand, dictates that at least some (or even most or all) magnetic eruptions in the star will unleash powerful CMEs to shed the excess helicity generated in the star's magnetized atmosphere that is otherwise conserved and remains accumulated in the star's corona. In such situations, one casts doubt on the viability of an atmosphere in the otherwise terrestrial planet, even if the planet is seated well into the HZ of its astrosphere. The opposite is the case for ${ \mathcal R }\lt 1$.

2.1. The Worst-case CME-equipartition Magnetic Field

A critical magnetopause distance of r_mp = 2 R_p (i.e., two planetary radii, or one radius away from the surface of the planet; see Khodachenko et al. 2007; Lammer et al. 2007) was adopted as the minimum planetocentric distance in which atmospheric ionization and erosion can still be averted during extreme magnetospheric compression. Then, the equipartition planetary magnetic field B_eq (to be viewed as B_worst) that balances the ICME magnetic pressure at r_mp = 2 R_p can be estimated as (see Appendix B for a complete description)

$$\begin{eqnarray}&&{B}_{\mathrm{eq}}=8\,{B}_{\mathrm{ICME}},\end{eqnarray} \tag{ 2 }$$

where B_ICME is the worst-case ICME axial magnetic field at r_mp = 2R_p .

To infer B_ICME at any given astrocentric distance r_ICME, we first need to constrain the axial magnetic field B₀ of the CME at a near-star distance r₀ (see Appendix A for a derivation). Here B₀ is constrained by observational facts and, more specifically, by assuming a given flux-rope model and a corresponding magnetic helicity formula depending on the radius R and length L of the flux rope that can then be solved for B₀ (Patsourakos et al. 2016). Patsourakos & Georgoulis (2017) tested several linear force–free (LFF), nonlinear force–free (NLFF), and non-force-free flux-rope models and determined that the worst-case scenario B₀ for near-Sun CME flux ropes was obtained by the LFF Lundquist flux-rope model (values estimated by other models range between 2% and 80% of the Lundquist values) that gives a magnetic helicity of the form

$$\begin{eqnarray}&&{H}_{m}=\displaystyle \frac{4\pi {B}_{0}^{2}L}{\alpha }{\int }_{0}^{R}{J}_{1}^{2}(\alpha r){rdr},\end{eqnarray} \tag{ 3 }$$

where α is the constant force-free parameter and J₁() is the Bessel function of the first kind. Parameter α is inferred by the additional constraint α R ≃ 2.405, imposed by the first zero of the Bessel function of the zeroth kind, J₀(), in the Lundquist model. The flux-rope radius R corresponds to the CME front that is assumed to have a circular cross section with maximum area.

We use the Lundquist flux-rope model throughout this analysis, as this is a standard ICME model for the inner heliosphere. It gives the strongest near-Sun CME axial magnetic field B₀, provided that the twist is not excessive (see Patsourakos & Georgoulis 2017). Adopting the fundamental helicity conservation principle for high magnetic Reynolds number plasmas (e.g., Berger 1984) implies a fixed H_m and dictates that as the CME expands, B₀ decreases self-similarly as a function of 1/r², with r being the heliocentric distance. We maintain this quadratic scaling for distances relatively close to the Sun (e.g., up to the Alfvénic surface, where the solar wind speed matches the local Alfvén speed at ∼10 R_⊙). Beyond that surface, this analysis continues to adhere to helicity conservation but adopts a power-law radial falloff index a_B = 1.6 for the propagation of Lundquist flux-rope solar CMEs within the astrospheres (see Appendix A for a derivation of the exponent). As a result, B_ICME at a given astrocentric distance r_ICME is given by

$$\begin{eqnarray}&&{B}_{\mathrm{ICME}}={B}_{0}{\left(\displaystyle \frac{{r}_{0}}{{r}_{\mathrm{ICME}}}\right)}^{1.6},\end{eqnarray} \tag{ 4 }$$

where r₀ is the near-star distance up to which the CME axial magnetic field scales as (1/r²), and B₀ is this magnetic field at that distance.

2.2. The Best-case Planetary Magnetic Field

The "defense" line in the ICME pressure effects for any given planet is being held primarily by the planet's magnetic pressure. The planet's dipole magnetic moment ${ \mathcal M }$ gives rise to a planetary magnetic field

$$\begin{eqnarray}&&{B}_{p}=\displaystyle \frac{{ \mathcal M }}{{r}_{\mathrm{mp}}^{3}}\end{eqnarray} \tag{ 5 }$$

for the dayside magnetopause occurring at a planetocentric distance r_mp. To identify the best-case scenario, we examined several prominent models for the magnetic moment ${ \mathcal M }$ (Busse 1976; Stevenson et al. 1983; Mizutani et al. 1992; Sano 1993; see also Christensen 2010 for a review) to determine which would provide the strongest ${ \mathcal M }$, focusing particularly on the tidally locked regime. We concluded that an uppercase ${ \mathcal M }$ is provided by Stevenson et al. (1983) and a model variant of Mizutani et al. (1992), namely,

$$\begin{eqnarray}&&{ \mathcal M }={{ \mathcal M }}_{\mathrm{Stev}}\simeq A\,{\rho }_{c}^{1/2}{\omega }^{1/2}{R}_{c}^{3}\,{\sigma }^{-1/2}.\end{eqnarray} \tag{ 6 }$$

The other models provided values ranging between 18% and 62% of the Stevenson value. In Equation (6), A ≃ 3.45 × 10⁵ A·s·kg^−0.5 is the proportionality constant, ω corresponds to the planet's angular rotation, and ρ_c , R_c , and σ correspond to the planetary core's mean density, radius, and electrical conductivity, respectively (see Appendix C for more details on the calculations and assumptions made). The B_p for ${ \mathcal M }$ = ${{ \mathcal M }}_{\mathrm{Stev}}$ (hereafter B_Stev) is to be viewed as B_best.

Summarizing, our mASC ratio of Equation (1) translates to ${ \mathcal R }=({B}_{\mathrm{eq}}/{B}_{\mathrm{Stev}})$. The B_eq can be estimated in the case of known bolometric stellar flare energies, while B_Stev is fully constrained for tidally locked exoplanets.

Assuming tidal locking to fully determine B_Stev, Figure 1 provides the nominal values of ${ \mathcal R }$ for different stellar flare energies and an Earth-like exoplanet lying precisely at the inner (red) and outer (blue) HZ boundary of Kopparapu et al. (2013). This plot represents a different conception of Figure 2 that shows a number of exoplanets provided by the NASA Exoplanet Archive lying within and without both the tidally locked regime and the inner and outer HZ limits. Nevertheless, Figure 1 now includes ${ \mathcal R }$ as a function of stellar flare energies, while stellar masses are implicitly included in the astrocentric distances d_inner and d_outer. It comes as no surprise that higher flare energies, statistically resulting in more helical CMEs and stronger axial magnetic fields, require the planet to be orbiting its host star at a larger orbital distance to achieve ${ \mathcal R }\lt 1$. For flare energies higher than 10³³ erg, it appears that planets located precisely on the inner HZ boundary may be incapable of sustaining an atmosphere, while this is the case for energies higher than 10³⁴ erg for planets located on the outer HZ boundary.

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Importantly, our mASC method can also be applied to case studies of terrestrial exoplanets, provided that flares from their host stars are observed. If these planets are—or are assumed to be—tidally locked, then ${ \mathcal R }$ becomes fully constrained. In Figure 3, we examine six popular cases of terrestrial exoplanets, all within the tidal-locking zone and either within or close to the respective HZ of their host stars. These cases are Kepler-438b (K438b), Proxima Centauri b (PCb), and four Trappist-1 (Tp1) exoplanets, namely, Tp1d, Tp1e, Tp1f, and Tp1g. Actual mASC ${ \mathcal R }$ values and applicable uncertainties (see Appendix D) are provided in Table 1. Figure 3 offers a graphical representation of these results, where observed flare energies from host stars have been taken from Armstrong et al. (2016), Howard et al. (2018), and Vida et al. (2017) for K438b, PCb, and Tp1, respectively.

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Table 1. Six Case Studies of the mASC ${ \mathcal R }$ Value as Depicted in Figure 3

Exoplanet	Abridged	${ \mathcal R }$	Atmosphere	${a}_{{B}_{({ \mathcal R }=1)}}$	δ aB	Result
			Likely?			Robust?
Kepler-438b	K438b	5.46	No	2.48	0.99	No
Proxima Centauri b	PCb	42.84	No	3.48	0.96	Yes
Trappist-1d	Tp1d	95.35	No	4.95	1.40	Yes
Trappist-1e	Tp1e	77.87	No	4.24	1.15	Yes
Trappist-1f	Tp1f	70.61	No	3.80	0.99	Yes
Trappist-1g	Tp1g	48.53	No	3.43	0.90	Yes

Note. The values of ${ \mathcal R }$ corresponds to our nominal radial power-law falloff index a_B = 1.6. The index ${a}_{{B}_{({ \mathcal R }=1)}}$ corresponds to the equipartition a_B value for which ${ \mathcal R }=1$, while δ a_B corresponds to the uncertainty of ${a}_{{B}_{({ \mathcal R }=1)}}$. The last column assesses whether our result is robust as per the difference ${a}_{{B}_{({ \mathcal R }=1)}}\pm \delta {a}_{B}$ from the nominal a_B value.

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The uncertainty analysis mentioned above and described in Appendix D aims to determine under which circumstances a conclusion of ${ \mathcal R }\lt 1$ or ${ \mathcal R }\gt 1$ could be reversed due to applicable uncertainties. In particular, we define an equipartition scaling index ${a}_{{B}_{({ \mathcal R }=1})}$ for which ${ \mathcal R }=1$, along with its uncertainty δ a_B . If (i) $| {a}_{B}-{a}_{{B}_{({ \mathcal R }=1})}| \gt \delta {a}_{B}$ for the nominal a_B = 1.6 of Patsourakos & Georgoulis (2016, 2017) or (ii) ${a}_{{B}_{({ \mathcal R }=1})}$ is steeper than 2 beyond the applicable δ a_B , with 2 being the "vacuum" radial falloff index near the star and a_B < 2 reflecting an astrosphere with its own MHD environment, then our conclusion of ${ \mathcal R }\lt 1$ or ${ \mathcal R }\gt 1$ is unlikely to change. If the uncertainties are large enough to preclude a safe conclusion, then our main result on an exoplanet is likely to change because of these uncertainties. Analogous uncertainties may be sought in the case of additional pressure effects included in ${ \mathcal R }$.

Table 1 shows that for all six presumed tidally locked exoplanets, the 1:1 spin–orbit resonance results in planetary rotations that are slow enough to allow ${ \mathcal R }\gt 1$, rendering a sustainable atmosphere unlikely. In all cases, ${a}_{{B}_{({ \mathcal R }=1)}}$ is well above the "vacuum" value of 2, with PCb and the four Tp1 exoplanets showing ${a}_{{B}_{({ \mathcal R }=1)}}\gt 2$ beyond the applicable uncertainties. Hence, for five out of six potentially tidally locked exoplanets, our result for ${ \mathcal R }\gt 1$, meaning a nonsustainable atmosphere, seems robust. The result could be reversed for K438b, as the difference between ${a}_{{B}_{({ \mathcal R }=1)}}$ and 1.6 is within the applicable uncertainties, meaning that if a_B systematically lies in the range (1.6, 2.0), then the exoplanet might conceivably sustain an atmosphere.

The above underlines the potential value of the mASC ${ \mathcal R }$, even in its simplest form involving only magnetic pressure effects; lying in the HZ of their host stars does not necessarily make exoplanets capable of sustaining an atmosphere, a favorable Earth Similarity Index (ESI; Schulze-Makuch et al. 2011) notwithstanding. This would be the case for PCb and Tp1d, e, f, and g, although virtually all lie in the HZ. Conversely, K438b is slightly beyond the inner HZ that might inhibit an atmosphere and possible liquid water on its surface, but at the same time, due to its larger orbital distance, it might be relatively immune to at least plausible, as per flare observations, space weather from its host star.

This versatile and highly reproducible analysis shows that space weather cannot be left out of considerations for planetary habitability in stellar systems (Airapetian et al. 2020). It carries both value and promise; consider the European Space Agency CHaracterising ExOPlanet Satellite (CHEOPS) mission, for example (e.g., Sulis et al. 2020). The mission is designed to characterize only selected, confirmed exoplanets, ranging from super-Earth to Neptune sizes, aiming toward studies that extend into their potential atmospheres. Although the mission is already in orbit, analyses such as this could help assess future observing priorities. The same applies to optimizing exoplanet selection for biosignature analysis in the framework of the upcoming James Webb mission.

In anticipation of these exciting observations that will give the ultimate test of our method, we hereby supply a first round of tentative tests aiming toward its validation. For example, it was recently shown that LHS 3844b (Vanderspek et al. 2019; Kane et al. 2020), a rocky exoplanet orbiting an M dwarf star within its tidally locked zone, lacks an atmosphere. Kane et al. (2020) suggested that the mother star of LHS 3844b exhibited an active past, comparable to that of Proxima Centauri. By adopting, therefore, a maximum superflare energy identical to the one of Proxima Centauri, we obtained R = 251.08 (>1, atmosphere unlikely), which agrees with the observations. Also, our result seems robust because $| {a}_{B}-{a}_{{B}_{({ \mathcal R }=1})}| \gt \delta {a}_{B}$. ⁵ Our mASC could be indirectly validated as well by checking if other studies—also focusing on tidally locked, terrestrial worlds and having similar objectives—converge on similar results. This said, the results presented here are in qualitative agreement with MHD simulations of stellar winds, for example, by Garraffo et al. (2017), that find extreme magnetospheric compressions below ≈2.5 planetary radii for Tp1d–g and a magnetopause distance of [1.5, 4.5] planetary radii for PCb. Recent extensive (and computationally expensive) MHD models of stellar CMEs (e.g., Lynch et al. 2019), along with semiempirical approaches (e.g., Kay et al. 2016), have emerged, and both approaches take as input maps the stellar surface magnetic field inferred by Zeeman–Doppler imaging reconstructions (e.g., Donati & Brown 1997). While detailed, these studies may be rather impractical for bulk application to a large number of exoplanets. On the other hand, our introduced mASC could provide guidance to large-scale MHD simulations of stellar CMEs by efficiently scanning and bracketing the corresponding parameter space so that the simulations could be performed only for pertinent cases.

Concluding, we reiterate that mASC ${ \mathcal R }$ can be generalized at will with additional pressure terms, albeit mainly from the ICME side. In its most general form, ${ \mathcal R }={P}_{\mathrm{eq}}/{P}_{\mathrm{planet}}$, with worst- and best-case scenario pressure terms P_eq and P_planet, respectively. We note, in particular, the study of Moschou et al. (2019), where a (flare) energy versus (CME) kinetic energy diagram is inferred from stellar observations and modeling. Such statistics could be integrated into the EH diagram of this study to revise the P_eq term. The EH diagram used here is also an entirely solar one, so any possibilities of extending it to better reflect the magnetic activity of dwarf, planet-prolific stars are well warranted. Equally meaningful P_planet terms could be introduced to provide a far more sophisticated but still readily achieved mASC ratio ${ \mathcal R }$ for the screening of alien terrestrial worlds for potential atmospheres and, ultimately, life.

The authors would like to thank the anonymous referee for comments and suggestions that improved the manuscript. This work was inspired by and originated during the M.Sc. thesis of E.S., implemented at the University of Athens and the Research Center for Astronomy and Applied Mathematics of the Academy of Athens, Greece. We thank both institutions for their support and encouragement. The authors would also like to extend their acknowledgments to Prof. Dr. Stefaan Poedts from the Centre for mathematical Plasma-Astrophysics (CMPA), Department of Mathematics, KU Leuven, for his support and encouragement on this work.

Facilities: The properties of the exoplanets and host stars used in this study are available at the NASA Exoplanet Archive (https://exoplanetarchive.ipac.caltech.edu/).

Software: The code to implement the mASC for any terrestrial-like exoplanet lying within the tidally locked regime of its host star is available in the following repository: https://github.com/SamaraEvangelia/mASC-method.

A physically meaningful expression of magnetic helicity in the Sun and magnetically active stars is the relative helicity, related to the absence of vacuum and hence the flow of electric currents in the solar and stellar coronae, along with the topological settings of solar (and stellar) magnetic fields that are only partially observed and detected, on and above the stars' surface. By construction, the relative helicity must be quantitatively connected to the excess or free magnetic energy that is also explicitly due to the presence of electric currents (e.g., Sakurai 1981). An attempt to correlate the free magnetic energy with the relative magnetic helicity in solar active regions was made by Georgoulis & LaBonte (2007) for LFF magnetic fields and Georgoulis et al. (2012) for NLFF ones.

The NLFF (magnetic) energy–(relative) helicity correlation resulted in the EH diagram of Tziotziou et al. (2012). There, ∼160 vector magnetograms of observed solar active regions were treated in the homogeneous way of Georgoulis et al. (2012), aiming toward a scaling between the NLFF free magnetic energy E_c and relative helicity H_m . They found a robust scaling of the form (cgs units)

$$\begin{eqnarray}&&\mathrm{log}| {H}_{m}| \propto 53.4-0.0524{\left(\mathrm{log}{E}_{c}\right)}^{0.653}\exp \displaystyle \frac{97.45}{\mathrm{log}{E}_{c}}.\end{eqnarray} \tag{ A1 }$$

Here ∣H_m ∣ refers to the magnitude of the relative helicity H_m , which can be right- (+) or left- (–) handed. A similar expression to Equation (A1) provides a simpler power-law dependence between ∣H_m ∣ and E_c of the form (cgs units)

$$\begin{eqnarray}&&| {H}_{m}| \propto 1.37\times {10}^{14}{E}_{c}^{0.897}.\end{eqnarray} \tag{ A2 }$$

The above EH scaling was shown to hold for typical active-region free energies in the range E_c ∼ (10³⁰, 10³³) erg and respective relative helicity budgets ∣H_m ∣ ∼ (10⁴⁰, 10⁴⁴) Mx². The robustness of the EH diagram scaling was validated in multiple cases that involved not only active regions but also quiet-Sun regions and MHD models (Tziotziou et al. 2013).

Combining the EH scaling with the conservation principle of the relative magnetic helicity, E_c in Equations (A1) and (A2) is dissipated in every instability (solar flare, CME, etc.), while H_m is only shed away from the Sun via CMEs. If an active region was to completely relax (i.e., return to the vacuum energy state) by a single magnetic eruption, then for a given free magnetic energy E_c , it would expel helicity H_m . This is the core reasoning behind a worst-case scenario solar eruption originating from a given solar source. Typically, up to ∼10% of the total free energy and up to 30%–40% of the total magnetic helicity of the source are dissipated and ejected, respectively, in a solar eruption (see, e.g., Nindos et al. 2003; Moraitis et al. 2014).

The majority of CME ejecta are observed to be in the form of a magnetic flux rope (e.g., Vourlidas et al. 2013, 2017), with this geometry surviving the CMEs' inner heliospheric propagation all the way to the Sun–Earth libration point L1 and probably beyond (e.g., Zurbuchen & Richardson 2006; Nieves-Chinchilla et al. 2018). Based on the flux-rope CME geometry, tools such as the graduated cylindrical shell (GCS) model of Thernisien et al. (2009) and Thernisien (2011) processed observations by the STEREO/SECCHI coronagraph (Howard et al. 2008) to obtain the aspect ratio k = R/L between the radius R and along with the half-angular width w of the CME flux rope at some distance (typically, 10 R_⊙) away from the Sun. The length L corresponds to the perimeter of the flux rope, while the radius R corresponds to the CME front, which is assumed to have a circular cross section with maximum area.

The worst-case scenario B₀ for near-Sun CME flux ropes was provided by the LFF Lundquist flux-rope model, which was used throughout this analysis and gives a magnetic helicity of the form

$$\begin{eqnarray}&&{H}_{m}=\displaystyle \frac{4\pi {B}_{0}^{2}L}{\alpha }{\int }_{0}^{R}{J}_{1}^{2}(\alpha r){rdr},\end{eqnarray} \tag{ A3 }$$

where α is the constant force-free parameter and J₁() is the Bessel function of the first kind. Parameter α is inferred by the additional constraint α R ≃ 2.405, imposed by the first zero of the Bessel function of the zeroth kind, J₀(), in the Lundquist model.

As already mentioned, for helicity conservation imposing a fixed H_m , the Lundquist model requires that as the CME expands, B₀ decreases self-similarly as a function of 1/r² for distances close to the Sun. In this analysis, however, and for distances away from the Sun (see also Patsourakos & Georgoulis 2016), the self-similar expansion was not a priori assumed to be quadratic (i.e., 1/r²), but more generally, $(1/{r}^{{a}_{B}})$, with a_B being the absolute value of the power-law radial falloff index. This different index, still under helicity conservation that dictates the respective power laws for the increase (expansion) of the CME flux rope R and L, aimed to include all effects present during heliospheric propagation and the interaction of ICMEs with the ambient solar wind (see, e.g., Manchester et al. 2017, for an account of these effects). Prominent among them is the CME flattening (see, e.g., Raghav & Shaikh 2020, and references therein) that tends to distort the CME geometry due to plasma draping or flux pileup as the CME pushes through the heliospheric spiral. As a result, inner heliospheric propagation implies that B_ICME at a given heliocentric distance r_ICME is given by

$$\begin{eqnarray}&&{B}_{\mathrm{ICME}}={B}_{0}{\left(\displaystyle \frac{{r}_{0}}{{r}_{\mathrm{ICME}}}\right)}^{{a}_{B}},\end{eqnarray} \tag{ A4 }$$

where r₀ is the (near-Sun) distance up to which the CME axial magnetic field scales as (1/r²), and B₀ is this magnetic field at that distance.

This analysis adopts a_B = 1.6 in Equation (A4) for the propagation of Lundquist flux-rope stellar CMEs within their astrospheres (where a_B can vary in the range [1.34, 2.16]; see Salman et al. 2020). A Monte Carlo simulation for various stellar (k, w) pairs is adopted, while the stellar CME ∣H_m ∣ is inferred by the bolometric observed stellar flare energies via Equation (A2). A valid question is where in the near-star space are we to apply the model (k, w) that was taken at 10 R_⊙; assuming a 10 R_* astrocentric distance, where R_* is the radius of the host star, we should apply a "fudge-factor" correction to B₀ as follows:

$$\begin{eqnarray}&&{B}_{{0}_{(10{R}_{* })}}={B}_{{0}_{(10{R}_{\odot })}}{\left(\displaystyle \frac{{R}_{\odot }}{{R}_{* }}\right)}^{2}.\end{eqnarray} \tag{ A5 }$$

This factor is adopted for the cases of this study's six exoplanets and their host stars. In the general case, or where no assessment is taken for the stellar radius, one may start the astrospheric propagation at a physical astrocentric distance of 10 R_⊙, independently from the stellar radius (provided, of course, that this radius does not exceed 10 R_⊙).

Figure 4 provides the mean ${B}_{{0}_{(10{R}_{\odot })}}$ of the abovementioned Monte Carlo simulation for different flare energies E_c , along with a standard deviation around these values (cyan ranges). The inset in Figure 4 provides the respective median values. We notice that both mean and median ${B}_{{0}_{(10{R}_{\odot })}}$ can be adequately modeled by power laws of the flare energy E_c of the form

$$\begin{eqnarray}&&{B}_{{0}_{(10{R}_{\odot })}}\,(G)\,=\,f\,{E}_{c}^{0.23}\,(\mathrm{erg}),\end{eqnarray} \tag{ A6 }$$

where the proportionality constant f is 3.30 × 10⁻⁹ (median) or 5.94 × 10⁻⁹ (mean), with an uncertainty amplitude 1.29 × 10⁻⁸.

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Figure 5 provides the worst-case scenario axial magnetic field of ICME magnetic flux ropes (as virtually all CMEs/ICMEs are expected to be) as a function of solar/stellar flare energies and helio-/astrocentric distances. The ICME magnetic pressure effects on the planet will be determined from this magnetic field strength.

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Textbook physics dictates that either nominal solar/stellar winds or "stormy" ICMEs exert pressure that comprises magnetic (i.e., B²/(8π)), kinetic (i.e., ram; ρ υ²), and thermal (i.e., n k T) terms as per the local MHD environment (plasma number density, mass density, and speed, n, ρ, and υ, respectively, and magnetic field B). At the planet's dayside at few to several planetary radii, the only nonnegligible pressure term is magnetic pressure stemming from the planetary magnetosphere. This is because a possible atmosphere typically wanes at the planetary thermosphere or exosphere, already at small fractions of a planetary radius. In a seminal early work, Chapman & Ferraro (1930) considered a spherical magnetosphere interfacing with interplanetary ejecta. Adopting this working hypothesis and taking only the magnetic pressure term of the ICME determined by its characteristic field B_ICME, the existence of a magnetopause at planetocentric distance r_mp implies an equilibrium of the form (Patsourakos & Georgoulis 2017)

$$\begin{eqnarray}&&\displaystyle \frac{{B}_{\mathrm{ICME}}^{2}}{8\pi }=\displaystyle \frac{{B}_{\mathrm{mp}}^{2}}{8\pi }{\left(\displaystyle \frac{1}{{r}_{\mathrm{mp}}}\right)}^{6},\end{eqnarray} \tag{ B1 }$$

where r_mp is expressed in planetary radii and B_mp is the planetary magnetic field at the magnetopause. We now adopt a critical magnetopause distance, that is, the minimum planetocentric distance at extreme compression in which atmospheric ionization and erosion can still be averted at r_mp = 2 R_p (i.e., two planetary radii, or one radius away from the surface of the planet). This is a limit already adopted by several previous studies (Khodachenko et al. 2007; Lammer et al. 2007). Therefore, the equipartition planetary magnetic field B_eq that balances the ICME magnetic pressure at r_mp = 2R_p is

$$\begin{eqnarray}&&{B}_{\mathrm{eq}}=8\,{B}_{\mathrm{ICME}}\end{eqnarray} \tag{ B2 }$$

from Equation (B1). This equipartition field for the same range of flare energies and astrocentric distances (viewed in this case as the exoplanets' mean orbital distances in circular orbits) as in Figure 5 is provided in Figure 6. Equation (B2) and Figure 6, therefore, provide the requirement for the planetary magnetic field such that planets avoid atmospheric erosion due to a given ICME magnetic pressure.

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It would worth mentioning at this point that the toroidal component of the interplanetary magnetic field could become important in the case of fast rotators such as M stars (e.g., Petit et al. 2008; Kay et al. 2016; Villarreal D'Angelo et al. 2019). In such a case, an extra term is added in the pressure-balance equation at the substellar point. However, we note that in the case of active M dwarf stars, the notion of a Parker spiral for the associated interplanetary magnetic field may not be fully applicable, given the much more frequent CME activity than in the solar case that could keep the spiral significantly perturbed virtually at all times.

The planet's dipole magnetic moment ${ \mathcal M }$ gives rise to a planetary magnetic field

$$\begin{eqnarray}&&{B}_{p}=\displaystyle \frac{{ \mathcal M }}{{r}_{\mathrm{mp}}^{3}}\end{eqnarray} \tag{ C1 }$$

for the dayside magnetopause occurring at a planetocentric distance r_mp. The magnetic moment and resulting magnetic field in exoplanets are generally unknown, and one may need to use a known planetary field benchmark (i.e., Earth's or another) to estimate the equilibrium magnetopause (or standoff) distance between a planet and the stellar wind it encounters. For Earth, in particular, this distance is ∼10 R_⊕ for the unperturbed solar wind, where R_⊕ is Earth's radius. Patsourakos & Georgoulis (2017) concluded that the terrestrial magnetopause cannot be compressed to a value smaller than ∼5 R_⊕ by the magnetic pressure of ICMEs, an estimate that aligns with other findings of extreme magnetospheric compression (e.g., Russell et al. 2000).

While our knowledge of exoplanet magnetic fields is virtually nonexistent, for the subset of exoplanets that lie in the tidally locked zone of their host stars (Grießmeier et al. 2004; Khodachenko et al. 2007), we have an additional constraint: the 1:1 spin–orbit resonance (see Figure 2 to visually locate all such exoplanets). It implies that tidally locked exoplanets have a self-rotating speed equal to the rotational speed around their host stars in a synchronous rotation. While exceptions are possible, this study adopts the 1:1 spin–orbit resonance because it affords us an estimate of the angular self-rotation of the planet in case the orbital period around its host star is known from observations. Synchronous rotation statistically weakens the planetary magnetic dipole moment, but at the same time, it enables its first-order estimation, further enabling an estimation of the planetary magnetic field that is crucial for this analysis. In the case of planets that are not tidally locked to their mother star, or we do not employ tidal locking, the planetary magnetic field (B_best) will have to be hypothesized by assigning an ad hoc magnetic dipole moment in Equation (C1). No scaling law is employed then, but we can use a benchmark planetary field, such as Earth's, for example.

Emphasizing the terrestrial planets, we calculate the uppercase ${ \mathcal M }$ provided by the Stevenson et al. (1983) model by assuming a core conductivity σ = 5 × 10⁵ S m⁻¹ as per Stevenson (2003). The mean core density ρ_c that enters the relationship is further assumed to be equal to the mean density of the planet, i.e., ${\rho }_{c}=\rho =3{M}_{p}/(4\pi {R}_{p}^{3})$, where M_p and R_p are the planetary mass and radius, respectively. The assumption allows a density estimate based on direct or implicit observational facts relevant to the terrestrial planet under study. Moreover, an angular rotation (ω) equal to the planet's angular rotation around its host star is adopted for presumed tidally locked planets. In the case of planets that are not tidally locked, as explained initially, ad hoc planetary field benchmarks must be used.

Another obvious unknown is the planetary core radius, R_c . For Earth, we have R_c ≃ 0.55 R_⊕, while for Mercury, the fraction is significantly larger (R_c ≃ 0.85 R_☿). Regardless, an upper limit of R_c is the radius of the studied planet. In the frame of adopting the best-case scenario for the planetary magnetic pressure, we will use this upper limit, adopting R_c = R_p . As an example, the above settings and Equation (6) for Earth imply an equatorial magnetic field of ∼0.309 G, almost identical to the nominal terrestrial equatorial field of ∼0.305 G.

As already explained, we are essentially interested in the ratio ${ \mathcal R }={B}_{\mathrm{eq}}/{B}_{\mathrm{Stev}}$ of the equipartition magnetic field B_eq of a planet at magnetopause distance r_mp = 2 R_p to the expected upper-limit Stevenson magnetic field (B_Stev) for the planet. This paragraph details our approach to determine whether applicable uncertainties are capable of changing the outcome of ${ \mathcal R }\gt 1$ or ${ \mathcal R }\lt 1$ for a given exoplanet.

We start by asking what value of the radial falloff power-law index a_B is required for ${ \mathcal R }=1$. Under this condition and Equation (4), we find

$$\begin{eqnarray}&&{a}_{{B}_{({ \mathcal R }=1)}}=\displaystyle \frac{{\rm{log}}({B}_{{\rm{Stev}}}/(8\,{B}_{0}))}{{\rm{log}}({r}_{0}/{r}_{{\rm{ICME}}})}.\end{eqnarray} \tag{ D1 }$$

Evidently, if we find ${ \mathcal R }\gt 1$ for the nominal a_B = 1.6, then ${a}_{{B}_{({ \mathcal R }=1)}}\gt {a}_{B}$, meaning that the ICME magnetic field must decay more abruptly than assumed to achieve ${ \mathcal R }=1$ at 2R_p . The opposite is the case if we find ${ \mathcal R }\lt 1$ for a_B = 1.6 (i.e., ${a}_{{B}_{({ \mathcal R }=1)}}\lt {a}_{B}$).

Let us now assume an uncertainty δ B₀ of the near-star magnetic field of the CME, B₀. This relates to the uncertainty δ a_B on ${a}_{{B}_{({ \mathcal R }=1)}}$ as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\delta {a}_{B}}{{a}_{{B}_{({ \mathcal R }=1)}}}=\displaystyle \frac{1}{{\rm{ln}}(8{B}_{0}/{B}_{{\rm{Stev}}})}\displaystyle \frac{\delta {B}_{0}}{{B}_{0}}.\end{eqnarray} \tag{ D2 }$$

From Equation (A3), we can relate δ B₀ to the uncertainty in the CME helicity, δ H_m , as

$$\begin{eqnarray}&&\displaystyle \frac{\delta {H}_{m}}{| {H}_{m}| }=2\displaystyle \frac{\delta {B}_{0}}{{B}_{0}},\end{eqnarray} \tag{ D3 }$$

and by using the EH diagram of Equation (A2), we can find another expression for δ H_m , namely,

$$\begin{eqnarray}&&\displaystyle \frac{\delta {H}_{m}}{| {H}_{m}| }={\left[{\beta }^{2}{\left(\displaystyle \frac{\delta {E}_{c}}{{E}_{c}}\right)}^{2}+{\left(\mathrm{ln}{E}_{c}\right)}^{2}\delta {\beta }^{2}\right]}^{1/2}\simeq \mathrm{ln}{E}_{c}\,\delta \beta ,\end{eqnarray} \tag{ D4 }$$

for (δ E_c /E_c ) ≲ 4 and typical superflare energies E_c ≥ 10³⁰ erg. In Equation (D4), we have propagated the uncertainties for the free energy of the eruptive flare, δ E_c , and the power-law index in the EH diagram of Equation (A2), δ β ≃ 0.05. In Equation (D3), we have further assumed that the forward-modeled GCS geometrical properties R and L of the CME carry no uncertainties, as even when assuming (δ k/k) = (δ w/w) = 1, the dominant error term is still (δ B₀/B₀). As further shown in Equation (D4), given the term depending on the logarithm of E_c , we may ignore the contribution of δ E_c for typical superflare energies E_c ∈ (10³⁰, 10³⁸) erg, thus simplifying the uncertainty equation.

Combining Equations (D3) and (D4), we eliminate (δ H_m /∣H_m ∣) and solve for δ B₀ to find

$$\begin{eqnarray}&&\displaystyle \frac{\delta {B}_{0}}{{B}_{0}}\simeq \displaystyle \frac{1}{2}\,\mathrm{ln}{E}_{c}\,\delta \beta .\end{eqnarray} \tag{ D5 }$$

Then, substituting Equation (D5) to (D2), we reach the desired expression for δ a_B on ${a}_{{B}_{(R=1)}}$ as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\delta {a}_{B}}{{a}_{{B}_{({ \mathcal R }=1)}}}\simeq \displaystyle \frac{\mathrm{ln}{E}_{c}}{2\mathrm{ln}(8{B}_{0}/{B}_{\mathrm{stev}})}\delta \beta .\end{eqnarray} \tag{ D6 }$$

Equation (D6) allows us to assign an uncertainty to ${a}_{{B}_{({ \mathcal R }=1)}}$, and the relevant question is whether ${a}_{{B}_{(R=1)}}\ne 1.6$ beyond this uncertainty. In this case, our conclusion on ${ \mathcal R }$ is unlikely to change. In addition, if ${ \mathcal R }\gt 1$ and ${a}_{{B}_{({ \mathcal R }=1)}}\gt 2$, then ${a}_{{B}_{({ \mathcal R }=1)}}$ may be deemed unrealistic because the ICME magnetic field is required to fall more abruptly in the astrosphere than the "unobstructed" a_B = 2 case for the near-star CME expansion in order to achieve ${ \mathcal R }=1$. If ${a}_{{B}_{({ \mathcal R }=1)}}\gt 2$ beyond the applicable uncertainties, then our conclusion for ${ \mathcal R }\gt 1$ is considered solid, whereas in case $| {a}_{{B}_{({ \mathcal R }=1)}}-2| \leqslant \delta {a}_{B}$, our conclusion is again unlikely to change, but it is conceivable that ${ \mathcal R }\leqslant 1$ for a significantly steeper than 1.6 decrease of the ICME magnetic field in the astrosphere.

Figure 7 provides two examples of the ratio $(\delta {a}_{B}/{a}_{{B}_{({ \mathcal R }=1)}})$ of Equation (D6) for a wide range of superflare energies and Stevenson planetary fields, from zero to the B_Stev value for Earth (${B}_{{\mathrm{Stev}}_{\oplus }}\simeq 0.309$ G). Examples are based on two nominal B₀ values, one for E_c = 10³³ erg (Figure 7(a)) and another for E_c = 10³⁴ erg (Figure 7(b)). These are mean values stemming from the Monte Carlo simulations of Figure 4 for these two energies. Both are toward the upper end of the distribution for near-Sun CME magnetic fields as found in Patsourakos & Georgoulis (2016). Stronger B₀ increases the value of ${a}_{{B}_{({ \mathcal R }=1)}}$ (Equation (D1)) but correspondingly decreases its uncertainty fraction (Equation (D6)).

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As an example, for the strong terrestrial magnetic field, we find ${a}_{{B}_{({ \mathcal R }=1)}}=0.40\pm 0.63$ (Equations (D1) and (D6)) assuming E_c = 10³³ erg and a nominal r₀ = 10 R_⊙. The respective value for E_c = 10³⁴ erg is ${a}_{{B}_{({ \mathcal R }=1)}}=0.63\pm 0.64$. These are both very flat a_B values, flatter beyond the uncertainties than the nominal a_B = 1.6 that gives ${ \mathcal R }\simeq 0.026$ and ≃0.052, respectively, under the same settings.