A Readily Implemented Atmosphere Sustainability Constraint for Terrestrial Exoplanets Orbiting Magnetically Active Stars
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A Readily Implemented Atmosphere Sustainability Constraint for Terrestrial Exoplanets OrbitingMagnetically Active Stars
Evangelia Samara,
1, 2
Spiros Patsourakos, and Manolis K. Georgoulis Royal Observatory of Belgium, 1180 Brussels, Belgium Centre for Mathematical Plasma Astrophysics, KU Leuven, 3001 Leuven, Belgium Department of Physics, University of Ioannina, 45110 Ioannina, Greece Research Center of Astronomy and Applied Mathematics (RCAAM) of the Academy of Athens, 11527 Athens, Greece
ABSTRACTWith more than 4,300 confirmed exoplanets and counting, the next milestone in exoplanet research isto determine which of these newly found worlds could harbor life. Coronal Mass Ejections (CMEs),spawn by magnetically active, superflare-triggering dwarf stars, pose a direct threat to the habitabilityof terrestrial exoplanets as they can deprive them from their atmospheres. Here we develop a readilyimplementable atmosphere sustainability constraint for terrestrial exoplanets orbiting active dwarfs,relying on the magnetospheric compression caused by CME impacts. Our constraint focuses on asystems understanding of CMEs in our own heliosphere that, applying to a given exoplanet, requiresas key input the observed bolometric energy of flares emitted by its host star. Application of ourconstraint to six famous exoplanets, (Kepler-438b, Proxima-Centauri b, and Trappist-1d, -1e, -1f and-1g), within or in the immediate proximity of their stellar host’s habitable zones, showed that onlyfor Kepler-438b might atmospheric sustainability against stellar CMEs be likely. This seems to alignwith some recent studies that, however, may require far more demanding computational resources andobservational inputs. Our physically intuitive constraint can be readily and en masse applied, as isor generalized, to large-scale exoplanet surveys to detect planets that could be sieved for atmospheresand, perhaps, possible biosignatures at higher priority by current and future instrumentation.
Keywords:
Exoplanets — Habitable zone — Solar coronal mass ejections — Solar flares — Stellarcoronal mass ejections — Stellar flares INTRODUCTIONPlanets beyond our solar system have become an object of fascination in recent decades, with nearly regular refer-ences in headlines and the popular media. Only recently have observational capabilities evolved to the point wherepotential terrestrial planets are detected around M-type dwarf stars. However, young M-type dwarfs are known to bemagnetically active, often more than our middle-aged Sun. Superflares in them is a common occurrence (e.g., Maeharaet al. 2012; Armstrong et al. 2016) that should be resulting 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). CMEs are gigantic, eruptiveexpulsions of magnetized plasma and helical magnetic fields from the solar and stellar coronae at speeds that maywell surpass local Alfv´enic and sound speeds, severely but temporarily disrupting stellar winds and generating shocksaround their bodily ejecta.Contrary to solar flares that are known since the 19th century (Carrington 1859), CMEs were only observed wellinto 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 case of intense stellar magnetic activityand the existence of an atmosphere that shields a planet, extreme pressure effects by CMEs owning to stellar mega-
Corresponding author: Evangelia [email protected] a r X i v : . [ a s t r o - ph . E P ] F e b Samara, Patsourakos & Georgoulis 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 (MAVEN)mission seem to establish that the sustained eroding effect of solar interplanetary CMEs (ICMEs) may be responsiblefor 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 R and relying on CME and planetary magnetic pressure effects. 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 possibleatmosphere ( R < R > R can be generalized at will. Given thatpressure effects are extensive and additive, however, adding more terms (i.e., kinetic, thermal) to the CME pressurewill only increase the adversity of possible atmospheric erosion effects for a studied exoplanet (e.g., Ngwira et al. 2014). (MAGNETIC) ATMOSPHERE SUSTAINABILITY CONSTRAINT (MASC)The magnetic activity of the exoplanets’ host stars reflects on several observational facts, including the bolometricenergy of their flares. From it, and assuming a Sun-as-a-Star analogue further reflected on the solar magnetic energy –helicity (EH) diagram (Tziotziou et al. 2012), we estimate the magnetic helicity of stellar CMEs and a correspondingstellar CME magnetic field based on the fundamental principle of magnetic helicity conservation in high magneticReynolds 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 beshed 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 ofwhether 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 toavert erosion effects of a possible atmosphere.There are two conceptual pillars of the mASC introduced here: first, it relies on observed stellar flare energies butdoes not perform a case-by-case stellar eruption analysis. In other words, it does not look at the particular eruptionbut points to the collective effect of numerous similar eruptions over the ∼ ∼ B worst ) 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 best ). Then, our mASC becomes the ratio of planetarymagnetic intensities relating to these pressure terms: the minimum planetary magnetic field (equal to B worst ) ableto balance the worst-case CME magnetic field and the best-case planetary magnetic field B best as per the modeledplanetary characteristics, i.e., R = B worst B best (1)The two planetary magnetic field strengths are taken at a critical magnetopause distance from the studied planet interms of atmospheric erosion effects (Sections 2.1, 2.2). It is then understood that if R >
1, the planet’s magnetospherewill be compressed beyond the critical threshold, presumably leading to atmospheric erosion (after processes such asthermal, nonthermal or hydrodynamic escape, catastrophic erosion and others, take place; see, for example, Melosh& Vickery (1989); Lundin et al. (2004); Barabash et al. (2007) for more details). Assuming that, statistically, thepresumed ICME is not a unique occurrence, the planet may undergo this atmospheric stripping for hundreds ofmillions of years due to its star’s magnetic activity. Magnetic helicity conservation, on the other hand, dictates that atleast some (or even most, or all) magnetic eruptions in the star will unleash powerful CMEs to shed away the excesshelicity generated in the star’s magnetized atmosphere that is otherwise conserved and remains accumulated in the he mASC method for Terrestrial Exoplanets R < 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 theplanet, see Khodachenko et al. (2007); Lammer et al. (2007)) was adopted as the minimum planetocentric distancein 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) B eq = 8 B icme , (2)where B icme is the worst-case ICME axial magnetic field at r mp = 2 R p .To infer B icme at any given astrocentric distance r icme , we first need to constrain the axial magnetic field B ofthe CME at a near-star distance r (see Appendix A for a derivation). B is constrained by observational facts andmore specifically by assuming a given flux rope model and a corresponding magnetic helicity formula depending onthe radius R and length L of the flux rope, that can then be solved for B (Patsourakos et al. 2016). Patsourakos &Georgoulis (2017) tested several linear (LFF), nonlinear (NLFF) and non-force-free flux rope models and determinedthat the worst-case scenario B for near-Sun CME flux ropes was obtained by the LFF Lundquist flux-rope model(values estimated by other models range between 2-80 % of the Lundquist values) that gives a magnetic helicity of theform H m = 4 πB Lα (cid:90) R J ( αr ) rdr , (3)where α is the constant force-free parameter and J () is the Bessel function of the first kind. Parameter α is inferredby the additional constraint αR (cid:39) . J (),in the Lundquist model. The flux rope radius R corresponds to the CME front that is assumed to have a circularcross-section with maximum area.We use the Lundquist flux-rope model throughout this analysis, as this is a standard ICME model for the innerheliosphere. It gives the strongest near-Sun CME axial magnetic field B , provided that the twist is not excessive(see Patsourakos & Georgoulis (2017)). Adopting the fundamental helicity conservation principle for high magneticReynolds number plasmas (e.g., Berger 1984) implies a fixed H m and dictates that as the CME expands, B decreasesself-similarly as a function of 1 /r , with r being the heliocentric distance. We maintain this quadratic scaling fordistances relatively close to the Sun (e.g., up to the Alfv´enic surface where the solar wind speed matches the localAlfv´en speed at ∼ R (cid:12) ). Beyond that surface this analysis continues to adhere to helicity conservation but adopts apower-law radial fall-off index a B = 1 . B icme at a given astrocentric distance r icme is givenby B icme = B ( r r icme ) . (4)where r is the near-star distance up to which the CME axial magnetic field scales as (1 /r ) and B is this magneticfield 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 magneticpressure. The planet’s dipole magnetic moment M gives rise to a planetary magnetic field B p = M r mp (5)for the dayside magnetopause occurring at a planetocentric distance r mp . To identify the best-case scenario, weexamined several prominent models for the magnetic moment 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 M , Samara, Patsourakos & Georgoulis focusing particularly on the tidally locked regime. We concluded that an upper-case M is provided by Stevenson et al.(1983) and a model variant of Mizutani et al. (1992), namely M = M Stev (cid:39)
A ρ / c ω / R c σ − / . (6)The other models provided values ranging between 18-62 % of the Stevenson value. In Equation (6), A (cid:39) . × A · s · kg − . is the proportionality constant, ω corresponds to the planet’s angular rotation and ρ c , R c and σ correspondto the planetary core’s mean density, radius and electrical conductivity, respectively (see Appendix C for more detailson the calculations and assumptions made). B p for M = M Stev ( B Stev , hereafter) is to be viewed as B best .Summarizing, our mASC ratio of Equation (1) translates to R = ( B eq /B Stev ). B eq can be estimated in case ofknown bolometric stellar flare energies, while B Stev is fully constrained for tidally locked exoplanets. APPLICATION OF THE MASC METHOD
Figure 1.
The mASC ratio R = ( B eq /B Stev ) for Earth twins, but with different equatorial magnetic fields, lying on the inner(red) and outer (blue) HZ boundary inferred by Kopparapu et al. (2013) for different stellar flare energies, each represented by adifferent curve thickness. The astrocentric distance in the abscissas implicitly includes the stellar mass shown in Figure 2. Thelimit R = 1 (dotted lines) separates an apparent non-viability of an atmosphere ( R >
1) from an apparently likely atmosphere( R <
1) for both cases.
Assuming tidal locking to fully determine B Stev , Figure 1 provides the nominal values of R for different stellar flareenergies and for an Earth-like exoplanet lying precisely at the inner (red) and outer (blue) HZ boundary of Kopparapuet al. (2013). This plot represents a different conception of Figure 2 that shows a number of exoplanets provided bythe NASA Exoplanet Archive lying within and without the tidally locked regime and within and without the inner andouter HZ limits. Nevertheless, Figure 1 now includes R as a function of stellar flare energies while stellar masses areimplicitly 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 itshost star at a larger orbital distance to achieve R <
1. For flare energies higher than 10 erg it appears that planetslocated precisely on the inner HZ boundary may be incapable to sustain an atmosphere while this is the case forenergies higher than 10 erg for planets located on the outer HZ boundary.Importantly, our mASC method can also be applied to case studies of terrestrial exoplanets, provided that flaresfrom their host stars are observed. If these planets are –or are assumed to be– tidally locked, then R becomes fully he mASC method for Terrestrial Exoplanets Astrocentric distance (AU) M s t a r ( M s un ) Proxima-Centauri b Kepler-438bTrappist-1 exoplanets tidal-locking limit potentially tidal-locking limit
Figure 2.
An ensemble of 1,771 confirmed exoplanets by the NASA Exoplanet Archive plotted on a diagram of stellar massvs. astrocentric distance. We restrict the analysis to stars less or equally massive to the Sun while exoplanets shown have aconfirmed orbital semi-major axis. Inner and outer HZ limits are indicated by solid lines enclosing the pink-shaded HZ area,while the black and gray dashed lines indicate internal and potential external tidal-locking limits, respectively. The locations ofnine exoplanets, six of which are studied here due to their proximity to, or presence within, the HZ, are also highlighted. constrained. In Figure 3, we examine six popular cases of terrestrial exoplanets, all within the tidal-locking zone andeither within, or close to, the respective HZ of their host stars. These cases are Kepler-438b (K438b), Proxima-Centaurib (PCb), and four Trappist-1 (Tp1) exoplanets, namely Tp1d, Tp1e, Tp1f and Tp1g. Actual mASC R -values andapplicable uncertainties (see Appendix D) are provided in Table 1. Figure 3 offers a graphical representation of theseresults 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 Keppler-438, Proxima Centauri and Trappist-1, respectively.The uncertainty analysis mentioned above and described in Appendix D aims to determine under which circumstancescould a conclusion of R < R > equipartition scaling index a B ( R =1 ) for which R = 1, along with its uncertainty δa B . If (i) | a B − a B ( R =1 ) | > δa B forthe nominal a B = 1 . a B ( R =1 ) is steeper than 2 beyond applicable δa B , with 2 being the ’vacuum’ radial fall-off index near the star and a B < R < R > R .Table 1 shows that for all six presumed tidally locked exoplanets the 1:1 spin-orbit resonance results in planetaryrotations that are slow enough to allow R >
1, rendering a sustainable atmosphere unlikely. In all cases, a B ( R =1) iswell above the ’vacuum’ value of 2, with PCb and the four Tp-1 exoplanets showing a B ( R =1) > R >
1, meaning a non-sustainable atmosphere, seems robust. The result could be reversed for K438b as the difference between a B ( R =1) and1.6 is within applicable uncertainties, meaning that if a B systematically lies in the range (1 . , .
0) then the exoplanetmight conceivably sustain an atmosphere.The above underline the potential value of the mASC R , even in its simplest form involving only magnetic pressureeffects: 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 forPCb and Tp1d,e,f, and g although virtually all lie in the HZ. Conversely, K438b is slightly beyond the inner HZ thatmight inhibit an atmosphere and possible liquid water on its surface but, at the same time, due to its larger orbitaldistance it might be relatively immune to at least plausible, as per flare observations, space weather from its host star. Samara, Patsourakos & Georgoulis
Figure 3.
Testing the viability of a potential atmosphere from the value of the mASC R for six confirmed exoplanets, namely(clockwise from top left) K438b, PCb, Tp1e, Tp1g, Tp1f and Tp1d. Ordinates correspond to stellar flare energies and abscissasto astrocentric distances. The color scales correspond to log( R ). Each planet is represented by a point corresponding tocoordinates set by the maximum observed flare energies from its host star and confirmed orbital distances, highlighted byvertical lines. Exoplanet Abridged R Atmosphere a B ( R =1) δa B Resultlikely? robust?Keppler-438b K438b 5.46 No 2.48 0.99 NoProxima Centauri b PCb 42.84 No 3.48 0.96 YesTrappist-1d Tp1d 95.35 No 4.95 1.40 YesTrappist-1e Tp1e 77.87 No 4.24 1.15 YesTrappist-1f Tp1f 70.61 No 3.80 0.99 YesTrappist-1g Tp1g 48.53 No 3.43 0.90 Yes
Table 1.
Six case studies of the mASC R -value as depicted in Figure 3. The values of R correspond to our nominal radialpower-law fall-off index a B = 1 .
6. The index a B ( R =1) corresponds to the equipartition a B -value for which R = 1, while δa B corresponds to the uncertainty of a B ( R =1) . The last column assesses whether our result is robust as per the difference a B ( R =1) ± δa B from the nominal a B -value. 4. CONCLUSIONSThis versatile and highly reproducible analysis shows that space weather cannot be left out of considerations forplanetary habitability in stellar systems. It carries both value and promise: consider the European Space AgencyCHEOPS (Characterizing Exoplanet Satellite) mission, for example ( e.g., Sulis et al. 2020). The mission is designed he mASC method for Terrestrial Exoplanets selected , confirmed exoplanets, ranging from super-Earth to Neptune sizes, aiming toward studiesthat extend into their potential atmospheres. Although the mission is already in orbit, analyses such as this couldhelp assess future observing priorities. The same applies to optimizing exoplanet selection for biosignature analysis inthe framework of the upcoming James Webb mission.In anticipation of these exciting observations that will give the ultimate test to our method, we hereby supplya first round of tentative tests aiming towards its validation. For example, it was recently shown that LHS 3844b(Vanderspek et al. 2019; Kane et al. 2020), a rocky exoplanet orbiting a M-dwarf star within its tidally locked zone,lacks an atmosphere. Kane et al. (2020) suggest that the mother star of LHS 3844b exhibited an active past, comparableto that of Proxima Centauri. By adopting, therefore, a maximum super-flare energy identical to the one of ProximaCentauri, we obtained R = 251.08 ( >
1, atmosphere unlikely) which agrees with the observations. Also, our resultseems robust because | a B − a B ( R =1 ) | > δa B . Our mASC could be indirectly validated as well, by checking if otherstudies – focusing also on tidally locked, terrestrial worlds and having similar objectives – converge on similar results.This said, results presented here are in qualitative agreement with MHD simulations of stellar winds, for example byGarraffo et al. (2017), that find extreme magnetospheric compressions below ≈ . , .
5] planetary radii for PCb. Recent extensive (and computationally expensive) MHDmodels of stellar CMEs (e.g., Lynch et al. 2019) along with semi-empirical approaches (e.g., Kay et al. 2016) haveemerged, and both approaches take as input maps of the stellar surface magnetic field inferred by Zeeman-Dopplerimaging reconstructions (e.g., Donati & Brown 1997). While detailed, these studies may be rather impractical forbulk application to a large number of exoplanets. On the other hand, our introduced mASC could provide guidanceto large-scale MHD simulations of stellar CMEs, by efficiently scanning and bracketing the corresponding parameterspace so that the simulations could be performed only to pertinent cases.Concluding, we reiterate that mASC R can be generalized at will with additional pressure terms, albeit mainly fromthe ICME side. In its most general form, R = P eq /P 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 vs. (CME)kinetic energy diagram is inferred from stellar observations and modeling. Such statistics could be integrated into theenergy-helicity diagram of this study to revise the P eq -term. The energy-helicity diagram used here is also an entirelysolar one, so any possibilities to extend it to better reflect the magnetic activity of dwarf, planet-prolific stars are wellwarranted. Equally meaningful P planet -terms could be introduced to provide a far more sophisticated, but still readilyachieved, mASC ratio R for the screening of alien terrestrial worlds for potential atmospheres and, ultimately, life.ACKNOWLEDGMENTSThe authors would like to thank the anonymous referee for his comments and suggestions that improved themanuscript. This work was inspired by and originated during the M.Sc. Thesis of ES, implemented at the Uni-versity 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 theiracknowledgements to Prof. Dr. Stefaan Poedts from the Centre of Mathematical Plasma-Astrophysics (CMPA) KULeuven, for his support and great incitement on this work. Facilities:
The properties of the exoplanets and host-stars used in this study are available at the NASA ExoplanetArchive (https://exoplanetarchive.ipac.caltech.edu/).
Software:
The code to implement the mASC for any terrestrial-like exoplanet lying within the tidally locked regimeof its host star is available in the following repository: https://github.com/SamaraEvangelia/mASC-method For this specific exoplanet which orbits very close to its mother star, we maintain the quadratic scaling of B with astrocentric distance( 1 /r ) for distances up to 7 R ∗ and not 10 R ∗ . Beyond 7 R ∗ , we adopt the same power-law radial fall-off index a B = 1.6 for the propagationof Lundquist-flux-rope solar CMEs within the astrospheres. Samara, Patsourakos & Georgoulis
APPENDIX A. NEAR- SUN/STAR CMES AND HELIO-/ASTRO-SPHERIC PROPAGATIONA physically meaningful expression of magnetic helicity in the Sun and magnetically active stars is the relativehelicity, related to the absence of vacuum and hence the flow of electric currents in the solar and stellar coronae, alongwith the topological settings of solar (and stellar) magnetic fields that are only partially observed and detected, onand above the stars’ surface. By construction, the relative helicity must be quantitatively connected to the excess orfree magnetic energy that is also explicitly due to the presence of electric currents (e.g., Sakurai 1981). An attempt tocorrelate the free magnetic energy with the relative magnetic helicity in solar active regions was taken by Georgoulis &LaBonte (2007) for linear force-free (LFF) magnetic fields, and then by Georgoulis et al. (2012) for nonlinear force-free(NLFF) ones.The NLFF (magnetic) energy - (relative) helicity correlation resulted in the energy - helicity (EH) diagram ofTziotziou et al. (2012). There, ∼
160 vector magnetograms of observed solar active regions were treated in the ho-mogeneous way of Georgoulis et al. (2012), aiming toward a scaling between the NLFF free magnetic energy E c andrelative helicity H m . They found a robust scaling of the form (CGS units)log | H m |∝ . − . E c ) . exp 97 . E c . (A1)Here | H m | refers to the magnitude of the relative helicity H m , that can be right- (+) or left- (–) handed. A similarexpression to Equation (A1) provides a simpler, power-law dependence between | H m | and E c , of the form (CGSunits) | H m |∝ . × E . c . (A2)The above EH scaling was shown to hold for typical active-region free energies in the range E c ∼ (10 , ) erg andrespective relative helicity budgets | H m |∼ (10 , ) Mx . The robustness of the EH diagram scaling was validatedin multiple cases that involved not only active regions but also quiet-Sun regions and magnetohydrodynamic 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. Ifan active region was to completely relax (i.e., return to the vacuum energy state) by a single magnetic eruption, thenfor a given free magnetic energy E c it would expel helicity H m . This is the core reasoning behind a worst-case-scenariosolar 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 pointL1 and probably beyond (e.g., Zurbuchen & Richardson 2006; Nieves-Chinchilla et al. 2018). Based on the flux ropeCME geometry, tools such as the Graduated Cylindrical Shell (GCS) model of Thernisien et al. (2009); 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 solar radii, R (cid:12) ) away from the Sun. The length L corresponds to the perimeter of the flux rope whilethe radius R corresponds to the CME front, that is assumed to have a circular cross-section with maximum area.The worst-case scenario B for near-Sun CME flux ropes was provided by the LFF Lundquist flux-rope model whichwas used throughout this analysis and gives a magnetic helicity of the form H m = 4 πB Lα (cid:90) R J ( αr ) rdr , (A3)where α is the constant force-free parameter and J () is the Bessel function of the first kind. Parameter α is inferredby the additional constraint αR (cid:39) . J (), inthe Lundquist model. he mASC method for Terrestrial Exoplanets H m , the Lundquist model requires that as the CMEexpands, B decreases self-similarly as a function of 1 /r for distances close to the Sun. In this analysis however andfor distances away from the Sun (see also Patsourakos & Georgoulis (2016)), the self-similar expansion was not a prioriassumed to be quadratic (i.e., 1 /r ), but more generally, (1 /r a B ), with a B being the absolute value of the power-lawradial fall-off index. This different index, still under helicity conservation that dictates respective power laws for theincrease (expansion) of the CME flux rope R and L , aimed to include all effects present during heliospheric propagationand the interaction of ICMEs with the ambient solar wind (see, e.g., Manchester et al. 2017, for an account of theseeffects). Prominent among them is the CME flattening (see, e.g., Raghav & Shaikh 2020, and references therein) thattends to distort the CME geometry due to plasma draping or flux-pileup as the CME pushes through the heliosphericspiral. As a result, inner-heliospheric propagation implies that B icme at a given heliocentric distance r icme is given by B icme = B ( r r icme ) a B (A4)where r is the (near-Sun) distance up to which the CME axial magnetic field scales as (1 /r ) and B is this magneticfield at that distance.This analysis adopts a B = 1 . a B can vary in the range [1.34, 2.16], see Salman et al. (2020)). A Monte Carlo simulationfor various stellar ( k, w )-pairs is adopted, while the stellar CME | H m | is inferred by the bolometric, observed stellarflare energies via Equation (A2). A valid question is where in the near-star space are we to apply the model ( k, w ),that were taken at 10 R (cid:12) : assuming a 10 R ∗ astrocentric distance, where R ∗ is the radius of the host star, we shouldapply a ’fudge-factor’ correction to B as follows: B (10 R ∗ ) = B (10 R (cid:12) ) ( R (cid:12) R ∗ ) . (A5)This factor is adopted for the cases of this study’s six exoplanets and their host stars. In the general case, or whereno assessment is taken for the stellar radius, one may start the astrospheric propagation at a physical astrocentricdistance of 10 R (cid:12) , independently from the stellar radius (provided, of course, that this radius does not exceed 10 R (cid:12) ). Figure 4.
Mean (main plot) and median (inset) values of the near-star CME axial magnetic field B at a physical distanceof 10 R (cid:12) from the star. The modeled data are shown by blue points, while least-squares power-law best fits are shown by redcurves. Standard deviations around each B -value are represented by cyan segments. Figure 4 provides the mean B (10 R (cid:12) ) of the above-mentioned 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 respectivemedian values. We notice that both mean and median B (10 R (cid:12) ) can be adequately modeled by power laws of the flareenergy E c of the form0 Samara, Patsourakos & Georgoulis B (10 R (cid:12) ) ( G ) = f E . c ( erg ) , (A6)where the proportionality constant f is 3 . × − (median) or 5 . × − (mean), with an uncertainty amplitude1.29 × − . B. ICME EQUIPARTITION MAGNETIC FIELDFigure 5 provides the worst-case scenario axial magnetic field of ICME magnetic flux ropes (as virtually allCMEs/ICMEs are expected to be) as a function of solar/stellar flare energies and helio-/astro-centric distances. TheICME magnetic pressure effects on the planet will be determined from this magnetic field strength. r (AU) E f l a r e ( e r g ) B I C M E ( G ) Figure 5.
Worst-case scenario axial magnetic field for ICME magnetic flux ropes as a function of source flare energy andastrocentric distance, up to one astronomical unit (AU). A wide range of flare energies are provided, from flares observed in theSun (i.e., up to 10 erg) to orders of magnitude stronger superflares with energies up to 10 erg. Textbook physics dictates that either nominal solar/stellar winds or ’stormy’ ICMEs exert pressure that comprisesmagnetic (i.e., B / (8 π )), kinetic (i.e., ram; ρυ ) and thermal (i.e., n k T ) terms as per the local magnetohydrodynam-ical (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 non-negligible pressure term is magnetic pressurestemming from the planetary magnetosphere. This is because a possible atmosphere typically wanes at the planetarythermosphere 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 workinghypothesis 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) B icme π = B mp π ( 1 r mp ) , (B7)where r mp is expressed in planetary radii and B mp is the planetary magnetic field at the magnetopause. We nowadopt a critical magnetopause distance, that is, the minimum planetocentric distance at extreme compression in whichatmospheric ionization and erosion can still be averted at r mp = 2 R p (i.e., two planetary radii, or one radius awayfrom 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 magneticpressure at r mp = 2 R p is B eq = 8 B icme , (B8)from Equation (B7). This equipartition field for the same range of flare energies and astrocentric distances (viewedin this case as the exoplanets’ mean orbital distances in circular orbits) as in Figure 5 is provided in Figure 6.Equation (B8) and Figure 6, therefore, provide the requirement for the planetary magnetic field such that planetsavoid atmospheric erosion due to a given ICME magnetic pressure. he mASC method for Terrestrial Exoplanets r (AU) E f l a r e ( e r g ) - - - - - - Earth B e q ( G ) Figure 6.
Same as in Figure 5 but showing the equipartition magnetic field of the planet at the adopted atmospheric-erosioncritical threshold of two planetary radii. An indication labeled ’Earth’ in the color bar shows the uncompressed terrestrialequatorial magnetic field, for reference.C.
BEST-CASE PLANETARY MAGNETIC FIELD IN TIDALLY LOCKED REGIMESThe planet’s dipole magnetic moment M gives rise to a planetary magnetic field B p = M r mp (C9)for the dayside magnetopause occurring at a planetocentric distance r mp . The magnetic moment and resulting magneticfield in exoplanets is generally unknown and one may need to use a known planetary field benchmark (i.e., Earth’s oranother) to estimate the equilibrium magnetopause (or stand-off) distance between a planet and the stellar wind itencounters. For Earth, in particular, this distance is ∼ R ⊕ for the unperturbed solar wind, where R ⊕ is Earth’sradius. Patsourakos & Georgoulis (2017) concluded that the terrestrial magnetopause cannot be compressed to a valuesmaller than ∼ R ⊕ by the magnetic pressure of ICMEs, an estimate which aligns with other findings of extrememagnetospheric compression (e.g., Russell et al. 2000).While our knowledge of exoplanet magnetic fields is virtually non-existent, for the subset of exoplanets that lie inthe tidally-locked zone of their host stars (Grießmeier et al. 2004; Khodachenko et al. 2007) we have an additionalconstraint: the 1:1 spin-orbit resonance (see Figure 2 to visually locate all such exoplanets). It implies that tidallylocked exoplanets have a self-rotating speed equal to the rotational speed around their host stars, in a synchronousrotation. While exceptions are possible, this study adopts the 1:1 spin-orbit resonance because it affords us an estimateof 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 itsfirst-order estimation, further enabling an estimation of the planetary magnetic field that is crucial for this analysis.In case of planets that are not tidally locked to their mother star, or we do not employ tidal locking, the planetarymagnetic field (B best ) will have to be hypothesized by assigning an ad hoc magnetic dipole moment in Equation C9.No scaling law is employed then, but we can use a benchmark planetary field, such as Earth’s, for example.Emphasizing on terrestrial planets, we calculate the upper-case M provided by Stevenson et al. (1983) model byassuming a core conductivity σ = 5 × S/m as per Stevenson (2003). The mean core density ρ c that enters therelationship is further assumed equal to the mean density of the planet, i.e., ρ c = ρ = 3 M p / (4 πR p ) where M p and R p are the planetary mass and radius, respectively. The assumption allows a density estimate based on direct orimplicit observational facts relevant to the terrestrial planet under study. Moreover, an angular rotation ( ω ) equal tothe planet’s angular rotation around its host star is adopted, for presumed tidally locked planets. In case of planetsthat are not tidally locked, as explained initially, ad hoc planetary field benchmarks must be used.2 Samara, Patsourakos & Georgoulis
Another obvious unknown is the planetary core radius, R c . For Earth, we have R c (cid:39) . R ⊕ while for Mercury thefraction is significantly larger ( R c (cid:39) . R (cid:39) ). Regardless, an upper limit of R c is the radius of the studied planet. Inthe 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 ∼ . G ,almost identical to the nominal terrestrial equatorial field of ∼ . G . D. SENSITIVITY ANALYSISAs already explained, we are essentially interested in the ratio R = B eq /B 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 ofchanging the outcome of R > R < a B is required for R = 1. Under this conditionand Equation (4), we find a B ( R =1) = log( B stev / (8 B ))log( r /r icme ) . (D10)Evidently, if we find R > a B = 1 .
6, then a B ( R =1) > a B , meaning that the ICME magnetic field mustdecay more abruptly than assumed to achieve R = 1 at 2 R p . The opposite is the case if we find R < a B = 1 . a B ( R =1) < a B ).Let us now assume an uncertainty δB of the near-star magnetic field of the CME, B . This relates to the uncertainty δa B on a B ( R =1) as follows: δa B a B ( R =1) = 1 ln (8 B /B stev ) δB B . (D11)From Equation (A3) we can relate δB to the uncertainty in the CME helicity, δH m , as δH m | H m | = 2 δB B , (D12)and by using the EH diagram of Equation (A2) we can find another expression for δH m , namely δH m | H m | = [ β ( δE c E c ) + (ln E c ) δβ ] / (cid:39) lnE c δβ , (D13)for ( δE c /E c ) (cid:46) E c ≥ erg. In Equation (D13) we have propagated the uncertaintiesfor the free energy of the eruptive flare, δE c , and the power-law index in the EH diagram of Equation (A2), δβ (cid:39) . R and L of theCME carry no uncertainties, as even assuming ( δk/k ) = ( δw/w ) = 1 the dominant error term is still ( δB /B ). Asfurther shown in Equation (D13), given the term depending on the logarithm of E c we may ignore the contribution of δE c for typical superflare energies E c ∈ (10 , ) erg, thus simplifying the uncertainty equation.Combining Equations (D12) and (D13), we eliminate ( δH m / | H m | ) and solve for δB to find δB B (cid:39) lnE c δβ . (D14)Then, substituting Equation (D14) to Equation (D11) we reach the desired expression for δa B on a B ( R =1) as follows: δa B a B ( R =1) (cid:39) lnE c ln (8 B /B stev ) δβ . (D15)Equation (D15) allows us to assign an uncertainty to a B ( R =1) and the relevant question is whether a B ( R =1) (cid:54) = 1 . R is unlikely to change. In addition, if R > a B ( R =1) >
2, then a B ( R =1) may be deemed unrealistic because the ICME magnetic field is required to fall more abruptly in the astrospherethan the ’unobstructed’ a B = 2 case for the near-star CME expansion in order to achieve R = 1. If a B ( R =1) > R > | a B ( R =1) − |≤ δa B our he mASC method for Terrestrial Exoplanets Figure 7.
Normalized uncertainties ( δa B /a B ( R =1) ) of the radial power-law fall-off index a B required for an equipartition( R = 1) between the ICME and planetary magnetic pressures at planetocentric distance equal to 2 planetary radii. B -valuesshown correspond to the mean near-Sun CME axial magnetic fields (Figure 4) for E c = 10 erg (a) and E c = 10 erg (b).Diamonds in images and indications in the color bar show ( δa B /a B ( R =1) ) for Earth for the two cases. conclusion is again unlikely to change but it is conceivable that R ≤ δa B /a B ( R =1) ) of Equation (D15) for a wide range of superflare energiesand Stevenson planetary fields ranging from 0 to the B Stev -value for Earth ( B Stev ⊕ (cid:39) .
309 G). Examples are basedon two nominal B -values; one for E c = 10 erg (Figure 7a) and another for E c = 10 erg (Figure 7b). These aremean values stemming from the Monte Carlo simulations of Figure 4 for these two energies. Both are toward the upperend of the distribution for near-Sun CME magnetic fields as found in Patsourakos & Georgoulis (2016). Stronger B increase the value of a B ( R =1) (Equation (D10)) but correspondingly decrease its uncertainty fraction (Equation (D15)).As an example, for the strong terrestrial magnetic field we find a B ( R =1) = 0 . ± .
63 (Equations (D10) and (D15))assuming E c = 10 erg and a nominal r = 10 R (cid:12) . The respective value for E c = 10 erg is a B ( R =1) = 0 . ± . a B -values, flatter beyond uncertainties than the nominal a B = 1 . R (cid:39) .
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