Atmospheric escape from the TRAPPIST-1 planets andimplications for habitability

Chuanfei Dong∗1, Meng Jin2, Manasvi Lingam3, Vladimir S. Airapetian4, YingjuanMa5, and Bart van der Holst6

1Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA1Princeton Center for Heliophysics, Princeton Plasma Physics Laboratory, Princeton

University, Princeton, NJ 08544, USA2Lockheed Martin Solar and Astrophysics Lab, Palo Alto, CA 94304, USA

3Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138, USA3John A. Paulson School of Engineering and Applied Sciences, Harvard University,

Cambridge, MA 02138, USA4NASA Goddard Space Flight Center, Greenbelt, MD, USA

5Institute of Geophysics and Planetary Physics, University of California, Los Angeles,CA 90095

6Center for Space Environment Modeling, University of Michigan, Ann Arbor, MI48109, USA

Abstract

The presence of an atmosphere over sufficiently long timescales is widely perceived as oneof the most prominent criteria associated with planetary surface habitability. We addressthe crucial question as to whether the seven Earth-sized planets transiting the recentlydiscovered ultracool dwarf star TRAPPIST-1 are capable of retaining their atmospheres. Tothis effect, we carry out numerical simulations to characterize the stellar wind of TRAPPIST-1 and the atmospheric ion escape rates for all the seven planets. We also estimate the escaperates analytically and demonstrate that they are in good agreement with the numericalresults. We conclude that the outer planets of the TRAPPIST-1 system are capable ofretaining their atmospheres over billion-year timescales. The consequences arising from ourresults are also explored in the context of abiogenesis, biodiversity, and searches for futureexoplanets. In light of the many unknowns and assumptions involved, we recommend thatthese conclusions must be interpreted with due caution.

With the number of detected exoplanets now exceeding 3600 (Winn and Fabrycky, 2015),exoplanetary research has witnessed many remarkable advances recently. One of the most im-portant areas in this field is the hunt for Earth-sized terrestrial planets residing in the habitablezone (HZ) of their host stars - the HZ represents the region within which a planet can supportliquid water on its surface (Kopparapu et al., 2013); a probabilistic version of the HZ, encom-passing a wide range of planetary and stellar parameters, has also been formulated (Zsom, 2015).The importance of this endeavour stems from the fact that such planets can be subjected tofurther scrutiny to potentially resolve the question as to whether they may actually harbour life(Cockell et al., 2016).

Most of the recent attention has focused on exoplanets in the HZ of M-dwarfs, i.e. low-massstars that are much longer lived than the Sun, for the following reasons. Firstly, M-dwarfs are∗To whom correspondence should be addressed. E-mail: dcfy@princeton.edu

1

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the most common type of stars within the Milky Way (Chabrier, 2003), implying that ∼ 1010

Earth-sized planets in the HZ of M-dwarfs may exist in our Galaxy (Dressing and Charbonneau,2015). Second, owing to the HZ being much closer to such stars, it is much easier to detectexoplanets and characterize their atmospheres, if they do exist (Shields et al., 2016). Lastly, thisfield has witnessed two remarkable advances within the last year: the discovery of Proxima b(Anglada-Escudé et al., 2016) and the seven Earth-sized planets transiting the ultracool dwarfTRAPPIST-1 (Gillon et al., 2016, 2017). The significance of the former stems from the fact thatit orbits the star closest to the Solar system, and the latter is important because there exist asmany as three planets in the HZ with the possibility of life being seeded by panspermia (Lingamand Loeb, 2017a).

In light of these discoveries, the question of whether terrestrial exoplanets in the HZ ofM-dwarfs are habitable is an important one (Shields et al., 2016). Amongst the many criteriaidentified for a planet to be habitable, the existence of an atmosphere has been posited as beingcrucial for surficial life-as-we-know-it (Lammer et al., 2009; Cockell et al., 2016). It is thereforeevident that the study of atmospheric losses from exoplanets constitutes a crucial line of enquiry.Empirical and theoretical evidence from our own Solar system suggests that the erosion of theatmosphere by the stellar wind plays a crucial role, especially for Earth-sized planets where suchlosses constitute the dominant mechanism (Lammer, 2013; Brain et al., 2016) and the same couldalso be true for exoplanets around M-dwarfs (Dong et al., 2017b; Zahnle and Catling, 2017).Recent studies of atmospheric ion escape rates from Proxima b (and other M-dwarf exoplanets)also appear to indicate that the resulting ion losses are significant because of the extreme spaceweather conditions involved (Garraffo et al., 2016), potentially resulting in the atmosphere beingdepleted over a span ranging from tens to hundreds of millions of years (Dong et al., 2017b,a;Airapetian et al., 2017; Garcia-Sage et al., 2017).

Hence, in this paper we shall focus primarily on the atmospheric ion escape rates of the sevenTRAPPIST-1 planets by adapting a sophisticated multi-species magnetohydrodynamics (MHD)model which self-consistently includes ionospheric chemistry and physics, and electromagneticforces. In this work, we do not tackle the wide range of hydrodynamic escape mechanics thathave been explored for terrestrial planets (Owen and Alvarez, 2016; Zahnle and Catling, 2017)on account of the above reasons.

The stellar wind of TRAPPIST-1

In order to commence our analysis of stellar wind-induced atmospheric loss, the stellar windparameters of TRAPPIST-1 are required. Since the space weather conditions at the TRAPPIST-1 planets are presently unknown from observations, we must rely upon simulating the stellarwind of TRAPPIST-1. The latter is implemented by means of the Alfvén Wave Solar Model(AWSoM), a data-driven global MHD model that was originally developed for simulating thesolar corona and solar wind (Sokolov et al., 2013; van der Holst et al., 2014). The AWSoMhas been proven to be successful in reproducing high-fidelity solar corona conditions (van derHolst et al., 2014; Oran et al., 2013), and can readily be adapted to self-consistently modelstellar wind profiles for a wide range of stars (Vidotto et al., 2013; Cohen et al., 2014; Garraffoet al., 2016). To adapt the AWSoM for modelling the TRAPPIST-1 stellar wind, we utilize therotational period, radius, and mass of the star based on the latest estimates (Luger et al., 2017)and a mean magnetic field typical of similar late M-dwarfs (Morin et al., 2010). Further detailsconcerning our approach can be found in Appendix A.

The steady state stellar wind solution is illustrated in Fig. 1. When compared to the normalsolar wind solution (Jin et al., 2012), the stellar wind of TRAPPIST-1 is much faster (∼ 3 times)when evaluated at the same stellar distance. The critical surface is defined as the region wherevs = vf , with vs and vf representing the stellar wind and fast magnetosonic speeds respectively.The surface occurs at distances of ∼ 30 R∗ and ∼ 20 R∗ for the fast and slow stellar wind;the latter duo originate at different regions on the star and have different speeds and densities(van der Holst et al., 2014). Hence, this leads to a very unique feature of the TRAPPIST-

2

Figure 1: The steady state stellar wind of TRAPPIST-1. (a) The 3D stellar wind configurationwith selected magnetic field lines. The background contour shows stellar wind speed at theequatorial plane (z=0). The blue isosurface represents the critical surface beyond which thestellar wind becomes super-magnetosonic. The black solid lines represent the orbits of sevenplanets TRAPPIST-1b to TRAPPIST-1h. (b) The equatorial plane z=0 showing the stellarwind dynamic pressure normalized by the solar wind dynamic pressure at 1 AU. The dashed lineshows the critical surface location. (c) The equatorial plane z=0 showing the stellar wind densitynormalized by the solar wind density at 1 AU. (d)-(f) A zoom-in view of the equatorial plane atz=0 near TRAPPIST-1 depicting the stellar wind velocity, normalized dynamic pressure, anddensity, respectively. Note that the color bar for (e) is the same as (b), and that of (f) is identicalto (c).

3

1 system, namely that part of the orbit of TRAPPIST-1b (the closest planet) lies within thecritical surface, while all the other planets are embedded in the super-magnetosonic stellar wind.

This striking scenario does not exist within our own solar system primarily because of theproximity of TRAPPIST-1b to its host star (in conjunction with a strongly magnetized stellarwind). As TRAPPIST-1b orbits within the critical surface, the planet could magnetically inter-act with its host star directly. In turn, the star-planet interaction could perhaps (i) regulate therotational rate (Pont, 2009), (ii) modify the properties of a local dynamo (Cuntz et al., 2000),and (iii) even give rise to a dynamo mechanism (Cébron and Hollerbach, 2014). In this context,we observe that variations in the magnetic field occur during the stellar cycle caused by the dy-namo process. Thus, the distance of the critical surface is expected to also vary concomitantly,implying that TRAPPIST-1b could be subject to frequent transitions between sub-magnetosonicand super-magnetosonic stellar wind conditions along the lines of Proxima b (Garraffo et al.,2016).

Another distinguishing feature of the stellar wind from TRAPPIST-1 is its higher density.When combined with the higher wind speed, all of the planets are subjected to a much largerdynamic pressure compared to that experienced by the Earth. At the orbit of TRAPPIST-1b, the dynamic wind pressure is about 103-104 times greater than the solar wind dynamicpressure at Earth. Even when we consider the furthermost planet, TRAPPIST-1h, the dynamicpressure is about 100-300 times larger than the near-Earth environment. The existence of suchan extreme wind pressure has already been documented for Proxima b (Garraffo et al., 2016), andits effects on the evolution of the planet’s magnetosphere have also been thoroughly investigated(Airapetian et al., 2017; Dong et al., 2017b). The ramifications of these extreme space weatherconditions on the atmospheric ion escape rates of the TRAPPIST-1 planets are explored in thesubsequent section.

We note that the mass-loss rate from TRAPPIST-1 is ∼ 2.6 × 1011 g/sec, which is about10% of the solar mass-loss rate. Although the density and velocity of the stellar wind are higherfor TRAPPIST-1, the smaller size of the host star is responsible for yielding a value lower thanthat of the active young Sun (Airapetian and Usmanov, 2016). The mass-loss rate

(∼ 0.1M�

)obtained for TRAPPIST-1 is broadly consistent with the upper bound of ∼ 0.2M� for theslightly larger star, Proxima Centauri (Wood et al., 2001).

Lastly, the stellar wind parameters provided in this paper are useful in determining the radioauroral emission from the TRAPPIST-1 planets, which can be used to constrain their magneticfields (Grießmeier, 2015). In Appendix B, we show that the radio emission could potentially peakat O(0.1) MHz, and result in a radio flux density of O(0.1) mJy; the latter could be enhancedby 2-3 orders of magnitude during a Coronal Mass Ejection (CME) event.

Ion escape rates for the TRAPPIST-1 planets

To simulate the ion escape rates for the seven planets of the TRAPPIST-1 system, we employ thesophisticated 3-D Block Adaptive Tree Solar-wind Roe Up-wind Scheme (BATS-R-US) multi-species MHD (MS-MHD) model that has been extensively tested and validated in the Solarsystem for Venus and Mars (Ma et al., 2004; Tóth et al., 2012; Ma et al., 2013; Jakosky et al.,2015), and was recently employed to study the atmospheric losses from Proxima b (Dong et al.,2017b). The reader is referred to these papers and the Appendix for further details concerningthe details of the numerical implementation, the model equations, and the physical and chemicalprocesses encoded within the model. Note that the neutral atmosphere is the source of theproduced ions through, e.g., photoionization and charge exchange, and that only a small fractionof them will escape into space. Hence, the atmosphere will, in addition to being eroded, alsoundergo changes in the chemical composition (Dong et al., 2017b).

4

The input parameters of the model

We shall, for the most part, concern ourselves with describing and motivating our choice of thedifferent input parameters required for the BATS-R-US model. Before proceeding further, wewish to caution the readers that many of the relevant planetary and stellar wind parametersof the TRAPPIST-1 system are unknown or poorly constrained. Hence, it is important torecognize that, on account of the many uncertainties involved, the ensuing escape rates may notnecessarily be representative of the TRAPPIST-1 system.

The BATS-R-US model relies upon an atmospheric composition akin to Venus and Mars,implying that the TRAPPIST-1 planets are also assumed to possess a similar composition.There are several factors that must be noted in this context. Firstly, as seen from our Solarsystem, the ion escape rates for Venus, Mars and Earth are similar despite their compositions,sizes and magnetic field strengths being wildly dissimilar (Lammer, 2013; Brain et al., 2016),thereby indicating that the ion escape rates may be relatively sensitive to stellar wind parameterscompared to planetary properties1; this is also partly borne out by the atmospheric ion escaperate calculations for Proxima b (Airapetian et al., 2017; Dong et al., 2017b). In addition, ithas been shown recently that the ion escape rates are only weakly dependent on the surfacepressure (Dong et al., 2017b). We observe that the inner planets of the TRAPPIST-1 systemcould have experienced significant losses of H2 and water over fast timescales (Bolmont et al.,2017; Bourrier et al., 2017), leaving behind other atmospheric components. Lastly, a Venus-likeatmosphere for the inner planets cannot be ruled out empirically, as noted in (de Wit et al.,2016).2

The next two input parameters to be specified are the surface pressure and the scale heightfor each of the planets. The former remains unknown at this stage, and we work with the fiducialvalue of 1 atm at this stage. We anticipate that the surface pressure does not significantly alterthe escape rates, at least for extreme stellar wind conditions, as demonstrated in (Dong et al.,2017b). The scale height Hx is defined as

Hx =kT

mgx, (1)

where gx is the acceleration due to gravity for planet X. The latter quantity can be easilycomputed for all the planets since their masses and radii are known (Gillon et al., 2017; Lugeret al., 2017).

The stellar wind parameters are obtained from the model described in the previous section.We must also prescribe the extreme ultraviolet (EUV) fluxes received at each of these planets,since the EUV flux plays an important role in regulating the extent of photoionization and theresultant stellar heating. This is accomplished by utilizing the values for the TRAPPIST-1planets computed in (Bolmont et al., 2017; Wheatley et al., 2017; Bourrier et al., 2017).

The planetary magnetic field is a potentially important factor in regulating the ion escaperates. We consider the scenario where the planets are unmagnetized because this case yields anupper limit on the allowed escape rates (Dong et al., 2017b). Hence, if a planet was characterizedby “low” escape rates in the unmagnetized limit, it would also typically possess low escape ratesin the presence of a magnetic field. It must also be borne in mind that the planets orbitingTRAPPIST-1 are likely to be tidally locked, and it has been argued that such planets are likelyto possess weak magnetic fields (Khodachenko et al., 2007). If a planet is weakly magnetized,it is likely that the total ion escape rate will be comparable to (but slightly lower than) theunmagnetized case (Dong et al., 2017b).

1The difference in stellar wind parameters at Venus, Earth and Mars is only up to a O(1) factor.2In broader terms, gaining a thorough understanding of Venus-type exoplanets is highly relevant, because it

allows us to compare and contrast their properties against exo-Earths (Angelo et al., 2017).

5

O+ O+2 CO+

2 TotalMaximum total pressure

Trappist-1b 5.56×1027 2.09×1026 1.52×1026 5.92×1027Trappist-1c 1.54×1027 1.38×1026 1.32×1026 1.81×1027Trappist-1d 1.29×1027 3.80×1025 1.14×1025 1.34×1027Trappist-1e 7.01×1026 2.83×1025 1.10×1025 7.40×1026Trappist-1f 5.23×1026 3.37×1025 1.19×1025 5.68×1026Trappist-1g 2.17×1026 2.71×1025 1.32×1025 2.58×1026Trappist-1h 1.06×1026 1.65×1025 6.98×1024 1.29×1026

Minimum total pressureTrappist-1b 9.33×1026 4.99×1025 2.92×1025 1.01×1027Trappist-1c 4.23×1026 9.22×1025 2.76×1025 5.42×1026Trappist-1d 2.81×1026 3.07×1025 1.04×1025 3.23×1026Trappist-1e 2.20×1026 4.19×1025 1.25×1025 2.74×1026Trappist-1f 1.88×1026 4.30×1025 1.10×1025 2.42×1026Trappist-1g 9.33×1025 5.85×1025 1.38×1025 1.66×1026Trappist-1h 4.52×1025 2.69×1025 4.39×1024 7.66×1025

Table 1: Ion escape rates in sec−1.

0.01 0.02 0.03 0.04 0.05 0.06a

0.0

0.2

0.4

0.6

0.8

1.0

Nx/N

b

Numerical predictions

Analytical predictions

Figure 2: The plot of the normalized escape rate as a function of the semi-major axis for thecase with maximum total pressure. The seven distinct points represent the seven planets of theTRAPPIST-1 system.

6

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|U| in km/s (T1g-Max)

-8 -6 -4 -2 0 2X

-8 -6 -4 -2 0 2X

-8 -6 -4 -2 0 2X

Figure 3: The logarithmic scale contour plots of the O+ ion density (first row) with magneticfield lines (in white) and stellar wind speed (second row) with stellar wind velocity vectors (inwhite) in the meridional plane. The first two columns correspond to TRAPPIST-1b (minimumand maximum Ptot) while the last two represent TRAPPIST-1g (minimum and maximum Ptot).The X- and Z-coordinates have been normalized in terms of the corresponding planetary radius.

Results from the model

For each planet, we consider two limiting cases. The first corresponds to the scenario withmaximum dynamic (and total) pressure over one orbit of the planet. The second correspondsto the case with minimum total pressure, but with the maximum magnetic pressure. Thecorresponding stellar wind parameters have been provided in the Appendix, and the escaperates in Table 1.

For all seven planets, the case with maximum total pressure yields a total atmospheric ionescape rate that is a few times higher than the corresponding case with minimum total pressure.The innermost trio of planets (‘b’, ‘c’ and ‘d’) have escape rates higher than 1027 sec−1, whilethe last four have rates lower than this value when the case with maximum total pressure isconsidered. In comparison, the escape rates for Mars, Venus and Earth are ∼ 1024 − 1025

sec−1 (Lammer, 2013), while that of Proxima b is ∼ 2 × 1027 (Dong et al., 2017b). In oursubsequent analysis, we shall focus on this case (maximum total pressure) since it leads ustowards determining the upper bounds on the escape rates.

Using the mixing-length formalism of (Canto and Raga, 1991) in conjunction with the defi-nitions of the stellar and planetary mass-loss rates, it can be shown that

Nx ∝(Rx

a

)2

M?, (2)

where Nx is the atmospheric escape rate arising from stellar wind stripping, M? is the stel-lar mass-loss rate, Rx and a are the radius and semi-major axis of the planet X respectively(Zendejas et al., 2010; Lingam and Loeb, 2017b). We have normalized the escape rates forthe TRAPPIST-1 planets in terms of the escape rate for TRAPPIST-1b, and compared thenumerical and analytical predictions in Fig. 2.

An inspection of Fig. 2 reveals that the analytical formula is in good agreement with thenumerical simulations, especially for the last four planets (‘e’ to ‘h’) which are regarded as being

7

potentially capable of retaining atmospheres over Gyr timescales. Even for the inner planets, wefind that the analytical results fall within the numerical values by a factor of . 2. Hence, Eq.2 may facilitate a quick estimation of the atmospheric escape rates from unmagnetized planets.However, for both the simulations and the analytic results, it is important to note that theatmospheric escape rates were higher in the past, although quantitative estimates are difficultsince the time-dependent stellar mass-loss rates are poorly constrained.

Fig. 3 depicts how the oxygen ions escape from two planets of the TRAPPIST-1 system(‘b’ and ‘g’) along with the interplanetary magnetic field lines. From the color contours, itis evident that the ion loss rate at TRAPPIST-1b is larger than TRAPPIST-1g. The solarwind magnetosonic Mach number of TRAPPIST-1b for the case with minimum total pressure(Ptot) equals 0.94, therefore there is no shock formed in front of the planet (because of its sub-magnetosonic nature). This is particularly noteworthy since this condition is not prevalent inour Solar system for any of the eight planets. On the other hand, such a condition has beenobserved for moons located inside the planetary global magnetosphere (e.g., Ganymede andTitan). In contrast to the above case, the bow shock is present in all other cases and planets ofthe TRAPPIST-1 system.

Based on the background colors of the second row, minimum Ptot corresponds to the faststellar wind and maximum Ptot signifies the slow stellar wind, which is fully consistent with theclassical solar/stellar wind picture. The high Ptot manifested in the ‘slow’ stellar wind is mostlya consequence of the high stellar wind density. The second row of Fig. 3 also indicates thatthe stellar wind accelerates away from the star; the stellar wind velocity at TRAPPIST-1g isobviously larger than that of TRAPPIST-1b. These conclusions can also be verified through aninspection of Table 3 in the Appendix.

From Fig. 3, we observe that the plasma boundaries are compressed because of the fact thatthe total pressure is much higher at these planets compared to that experienced by Venus. Theescape of ions is primarily driven by the pressure gradient and J ×B forces in the momentumequation (Ma et al., 2013). Lastly, we refer the reader to Fig. 4 in the Appendix that depicts theionospheric profiles of TRAPPIST-1g for the cases with minimum and maximum total pressure.

Discussion and Conclusions

We have arrived at an upper bound on the ion escape rates by considering the scenario wherethe planets are unmagnetized and subject to the maximum total (or dynamic) pressure. It wasconcluded that the innermost planet TRAPPIST-1b has an upper bound of 5.92 × 1027 sec−1

while the corresponding value for the outermost planet TRAPPIST-1h is 1.29×1026 sec−1. As theplanets are approximately Earth-sized and assumed to have a surface pressure of 1 atm, we canestimate the timescales over which these planets can retain their atmospheres. The values rangefrom O

(108)years for TRAPPIST-1b to O

(1010

)years for TRAPPIST-1h. Moreover, we also

see from Fig. 2 that the overall escape rate declines monotonically as one moves outwards, fromTRAPPIST-1b to TRAPPIST-1h. Hence, taken collectively, this may suggest that TRAPPIST-1h ought to be most “stable” planet amongst them, when viewed purely from the perspective ofatmospheric ion loss. Along the same lines, it seems likely that TRAPPIST-1g will representthe best chance for a planet in the HZ of this planetary system to support a stable atmosphereover long periods.3

At this stage, we must reiterate the caveats discussed earlier. Firstly, most of the planetaryand stellar parameters are partly or wholly unknown since the appropriate observations are notcurrently existent. The presence of (i) a more massive atmosphere, (ii) a different atmosphericcomposition, and (iii) the planet’s magnetic field, are likely to alter the extent of atmospheric lossto some degree. Nonetheless, it can be surmised that the unmagnetized cases (with maximumtotal pressure) considered herein do yield robust upper bounds on the atmospheric ion escape

33D climate simulations appear to suggest that TRAPPIST-1f and TRAPPIST-1g may not be amenable tosurficial life as they enter a snowball state (Wolf, 2017), but the effects of tidal heating, which are expected tobe considerable (Luger et al., 2017), were not included in the model.

8

rates. Apart from atmospheric loss, it is possible that outgassing processes could very wellreplenish the atmosphere (Kasting and Catling, 2003). Hence, resolving the existence of anatmosphere over Gyr timescales necessitates an in-depth understanding of the interplay betweensource and loss mechanisms. Lastly, it is important to note that stellar properties evolve overtime, implying that the escape rates are also likely to change accordingly. In the case of M-dwarfs such as TRAPPIST-1, the pre-main-sequence phase is particularly long and intense, andexpected to have an adverse impact on atmospheric losses (Luger and Barnes, 2015; Shieldset al., 2016). The ensuing effect of XUV on hydrodynamic and ion escape rates during thisphase has not been investigated in the paper.

Bearing these limitations in mind, let us now turn to a discussion of the implications. Wehave argued that TRAPPIST-1h and TRAPPIST-1g represent the most promising candidatesin terms of retaining atmospheres over Gyr timescales. Instead, if the atmosphere were to bedepleted over O

(108)years, this could prove to be problematic for the origin of life (abiogenesis)

on the planet although it must be acknowledged that the actual timescale for abiogenesis onEarth and other planets remains unknown (Spiegel and Turner, 2012). Abiogenesis has beenargued to be accompanied by an increase in biological (e.g. genomic) complexity over time(Adami et al., 2000; Purvis and Hector, 2000) although this growth is not uniform, and may becontingent on environmental fluctuations. Hence, ceteris paribus, a planet capable of sustaininga stable atmosphere over long time periods (along with retaining a stable climate) might havea greater chance of hosting complex surficial organisms. The outer planets of the TRAPPIST-1system may therefore lead to more diverse biospheres eventually.

We have also shown that the ionospheric profiles for the TRAPPIST-1 planets in the HZ arenot sensitive to the stellar wind conditions at altitudes . 200 km (see Fig. 4 of the Appendix).This is an important result in light of the considerable variability and intensity of the stellarwind, since it suggests that the lower regions (such as the planetary surface) may remain mostlyunaffected from under normal space weather conditions.4

Let us turn our attention to Table 1 for the seven Earth-sized exoplanets of the TRAPPIST-1system. It is seen that the ion escape rate reduces as one moves outwards. Hence, for similarmulti-planetary systems around low-mass stars, it may be more prudent to focus on the outwardplanet(s) in the HZ for detecting atmospheres since their escape rates could be lower. Similarly,when confronted with two planets with similar values of Rx and a (see Eq. 2), we propose thatsearches should focus on stars with lower mass-loss rates and stellar magnetic activity. Furtherdetails concerning the implications and generalizations of Eq. 2 will be reported in a forthcomingpublication. Lastly, our results and implications are also broadly applicable to future planetarysystems detected around M- and K-dwarfs endowed with similar features (Lingam and Loeb,2017a).

To summarize, we have studied the atmospheric ion escape rates from the seven planets ofthe TRAPPIST-1 system by assuming a Venus-like composition. This was done by utilizingnumerical models to compute the properties of the stellar wind and the escape rates, and thelatter were shown to match the analytical predictions. We demonstrated that the outer planetsof the TRAPPIST-1 system (most notable ‘h’ and ‘g’) are capable of retaining their atmospheresover Gyr timescales. However, as many factors remain unresolved at this stage, future missionssuch as the James Webb Space Telescope (JWST) will play a crucial role in constraining the at-mospheres of the TRAPPIST-1 planets (Barstow and Irwin, 2016). In particular, a recent studyconcluded that spectral features for six of the seven TRAPPIST-1 planets could be detected with< 20 transits with 5 σ accuracy (Morley et al., 2017). Such observations would help constraintheoretical predictions, pave the way towards looking for biosignatures, and empirically estimatethe putative habitability of these planets.

4Stellar flares may have either a deleterious or beneficial effect on prebiotic chemistry that is dependent on acomplex and interconnected set of factors (Dartnell, 2011; Ranjan et al., 2017).

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Acknowledgements

The authors thank A. Burrows, J.-F. Donati, S. Kane, A. Knoll, A. Loeb and J. Winn forthe helpful discussions and comments. The support provided by the NASA Living With aStar Jack Eddy Postdoctoral Fellowship Program, administered by the University Corporationfor Atmospheric Research is acknowledged. M.J. is supported by NASA’s SDO/AIA contract(NNG04EA00C) to LMSAL. Resources for this work were provided by the NASA High-EndComputing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division atAmes Research Center. The Space Weather Modeling Framework that contains the BATS-R-UScode used in this study is publicly available from http://csem.engin.umich.edu/tools/swmf.For distribution of the model results used in this study, please contact the corresponding author.

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A Modelling the stellar wind of TRAPPIST-1

In this Section, we describe the methodology involved in modelling the stellar wind of TRAPPIST-1 in more detail.

A.1 Description of the Alfvén Wave Solar Model

In this study, the TRAPPIST-1 stellar wind is simulated by means of the Alfvén Wave SolarModel (AWSoM; van der Holst et al., 2014), a data-driven global MHD model that had beeninitially developed for simulating the solar atmosphere and solar wind. The inner boundarycondition of the magnetic field can be specified by different magnetic maps from available ob-servations. The initial conditions for the stellar wind plasma are determined through the Parkersolution, while the initial magnetic field is based on the Potential Field Source Surface (PFSS)model with the Finite Difference Iterative Potential Solver (FDIPS) described in Tóth et al.(2011).

Alfvén waves are driven at the inner boundary with a Poynting flux that scales with thesurface magnetic field. The stellar wind is heated by Alfvén wave dissipation and accelerated bythermal and Alfvén wave pressure. Electron heat conduction (that includes both collisional andcollisionless contributions) and radiative cooling are also included in the model. In the AWSoM,the electron and proton temperatures are treated separately, while the electrons and protonsare assumed to have the same bulk velocity. However, heat conduction is applied only to theelectrons, owing to their much higher thermal velocity. The system of governing equations issolved numerically using the Block Adaptive Tree Solar Wind Roe-type Upwind Scheme (BATS-R-US) code within the Space Weather Modeling Framework (Tóth et al., 2012).

By using a physically consistent treatment of wave reflection, dissipation, and heat partition-ing between the electrons and protons, the AWSoM has successfully reproduced solar coronalconditions to a high degree of precision.

A.2 Application of the AWSoM to TRAPPIST-1

To adapt the AWSoM for the TRAPPIST-1, we modify the rotational mass, radius, and periodof the star in accordance with the latest observational data (Luger et al., 2017). Hence, wespecify M? = 0.08M�, R? = 0.11R�, and P? = 3.3 days.

Due to the lack of direct surface magnetic field observations of TRAPPIST-1, we utilize asolar magnetogram under the solar minimum condition (GONG magnetogram of CarringtonRotation 2077; Jin et al., 2012) and scale the mean and radial magnetic field strength basedon the magnetic field observations of similar late M-dwarfs (Morin et al., 2010). Based on thefact that the X-ray luminosity of TRAPPIST-1 is similar to that of the quiet Sun (Bourrieret al., 2017), we modify the Poynting flux parameter of the model such that the same amountof Poynting flux is generated as in the solar case. The simulation domain is extended to 250 R?

to ensure that the orbits of all seven planets in the TRAPPIST-1 system are duly encompassed.

B Deducing the magnetic fields of the TRAPPIST-1 planets

Since the past 40 years, it has been well known that radio emission from exoplanets can be usedto detect them, and thereby determine their magnetic fields (Yantis et al., 1977). The basic ideabehind detecting the planetary magnetic field (Bp) is that the cyclotron maser instability drivesthe emission of radio waves at a frequency approximately equal to the gyrofrequency (Zarka,1998). The maximum emission frequency ω is given by

ω ≈ ωc =eB

mec, (3)

14

where me and e are the electron mass and charge respectively. Hence, we propose that theradio auroral emission from the TRAPPIST-1 planets may be detectable by ground-based ob-servatories when the frequency is above ∼ 10 MHz; the Earth’s ionosphere reflects radio wavesbelow this value thereby requiring space- or lunar-based telescopes (Grießmeier, 2015). As anillustrative example, let us suppose that Bp ≈ 0.1 G for the TRAPPIST-1 planets, which leadsus to f = ω/2π ≈ 0.3 MHz. This fiducial value is motivated by planetary dynamo scaling laws(Christensen, 2010), as well as the lower limit of the hypothesized magnetic field for Proxima b(Burkhart and Loeb, 2017).

Next, it is necessary to obtain a heuristic estimate of the radio flux density Φ at Earth toknow whether the emission would be detectable. It can be estimated via

Φ =Pradio

4π∆fD2, (4)

where bandwidth ∆f ≈ f/2, D is the distance to Earth and Pradio represents the planetaryradio power (Zarka, 2007). Thus, a knowledge of Pradio and ω would suffice to determine Φ.Although there exist several ways of computing Pradio (Grießmeier, 2015), it has been noted thatthe planetary radio power is dominated by the dissipation of the magnetic power carried by thestellar wind (Vidotto and Donati, 2017):

Pradio ∼ 2× 10−3(πB2

swr2mVsw

4π

), (5)

where Bsw is the interplanetary magnetic field (IMF) and Vsw is the stellar wind velocity, andboth of them have been listed in Table 3. Here, rm denotes the planetary magnetospheric radiusand is given by

rm = Rp

(B2

p

8πPsw

)1/6

, (6)

where Psw ≈ ρswv2sw is the dynamic pressure and ρsw is the stellar wind density provided in

Table 3. Hence, it is possible to combine Eqs. (4), (5) and (6) to arrive at an estimate of Φ.We find that the radio flux density, for the above choice of values, is O

(10−4

)Jy. However, we

wish to note that Φ can be boosted by a factor of ∼ 102 − 103 during a strong Coronal MassEjection (CME) event, which would imply that it can attain a value of ∼ 10 − 100 mJy. Thisfollows from the fact that the stellar wind parameters are enhanced during a CME event, andthe corresponding radio power is also increased accordingly (Grießmeier et al., 2007). We notethat a similar approach has also been employed in the context of Proxima b, where it was shownthat the radio flux density attained a peak value of ∼ 1− 10 Jy during a large Carrington-typeCME event.

Thus, to summarize, future space-based (or lunar) low-frequency observations may be ableto constrain the planetary magnetic fields of the TRAPPIST-1 planets by measuring the radioflux density and extrapolating backwards to determine the value of Bp.

C Atmospheric ion escape rates for the TRAPPIST-1 planets

In this Section, we briefly describe the workings of the code and provide additional resultspertaining to the atmospheric ion escape rates.

C.1 Physical model and computational methodology

The 3-D Block Adaptive Tree Solar-wind Roe Upwind Scheme (BATS-R-US) multi-species MHD(MS-MHD) model was initially developed in the context of our Solar system i.e. for studyingMars (Ma et al., 2004) and Venus (Ma et al., 2013). We rely upon the code developed forVenus, and the neutral atmospheric profiles are based on the solar maximum conditions. TheMS-MHD comprises of a separate continuity equation for each ion species, in conjunction withone momentum equation and one energy equation for the four ion fluids H+, O+, O+

2 , CO+2 (Ma

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et al., 2004, 2013). Unlike most global (Earth) magnetosphere models that commence from 2-3Earth radii, the Mars/Venus MS-MHD model contains a self-consistent ionosphere, and thusthe lower boundary extends down to an altitude of 100 km above the planetary surface. TheMS-MHD model, which serves as the basis of our paper, accounts for a diverse array of chemicalprocesses, such as charge exchange, photoionization and electron recombination.

The various chemical reactions and their corresponding rate coefficients have been delineatedin Table 2, and the reader may also consult Dong et al. (2017b) for further details. The densitiesof O+, O+

2 , CO+2 at the lower boundary satisfy the photochemical equilibrium condition as

described in Dong et al. (2017b). The model also assumes that the plasma temperature (sum ofion and electron temperatures) is approximately double that of the neutral temperature at thelower boundary because of the high collision frequency.

The grid is also taken to be non-uniform and spherical in nature to accurately capturethe multi-scale physics operating in different regions. Hence, the radial resolution ranges fromaround half the scale height at the lower boundary to several thousands of kilometers at theouter boundary. The horizontal resolution is chosen to be 3.0◦ (in both longitude and latitude),while the simulation domain ranges from −45 Rx ≤ X ≤ 15 Rx and −30 Rx ≤ Y, Z ≤ 30 Rx,where Rx denotes the radius of planet X. The code is run in the Planet-Star-Orbital (PSO)coordinate system, where the x-axis is directed from the planet towards TRAPPIST-1, the z-axisis perpendicular to the planet’s orbital plane, and the y-axis completes the right-hand system.

Chemical Reaction Rate CoefficientPrimary Photolysis in s−1

CO2 + hν → CO+2 + e− see table caption

O + hν → O+ + e− see table captionIon-neutral Chemistry in cm3 s−1

CO+2 + O → O+

2 + CO 1.64× 10−10

CO+2 + O → O+ + CO2 9.60× 10−11

O+ + CO2 → O+2 + CO 1.1× 10−9 (800/Ti)0.39

H+ + O → O+ + H 5.08× 10−10

Electron Recombination Chemistry in cm3s−1

O+2 + e− → O + O 7.38× 10−8 (1200/Te)0.56

CO+2 + e− → CO + O 3.10× 10−7 (300/Te)0.5

Table 2: Chemical reactions and associated rates adapted from Dong et al. (2017b). The pho-toionization frequencies are rescaled to account for the EUV fluxes received at each of theTRAPPIST-1 planets based on the estimates provided in Bolmont et al. (2017) and Bourrieret al. (2017).

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The multi-species MHD equations are summarized as follows:

∂ρs∂t

+∇ · (ρsu) = Ss − Ls (7)

∂(ρu)

∂t+∇ ·

(ρuu + pI +

B2

2µ0I− 1

µ0BB

)= ρG−

∑s=ions

ρs∑

n=neutrals

νsnu−∑

s=ions

Lsu(8)

∂

∂t

(ρu2

2+

p

γ − 1+B2

2µ0

)+∇ ·

[(ρu2

2+

γp

γ − 1+B2

µ0

)u− (B · u)B

µ0+ B× ∇×B

µ20σ0

](9)

=∑

s=ions

∑n=neutrals

ρsνsnms +mn

[3k (Tn − Ts)−msu

2]

+ ρu ·G

+k

γ − 1

∑s=ions

(SsTn − LsTs)ms

− 1

2

∑s=ions

Lsu2 +k

γ − 1

SeTn0 − LeTeme

∂B

∂t+∇ · (uB−Bu) = −∇×

(1

µ0σ0∇×B

)(10)

where Eqs. (7), (8) and (9) represent the conservation of mass (of each species), momentum andenergy respectively, whilst Eq. (10) is the magnetic induction equation. The total mass densityρ =

∑s=ions ρs and νsn is the elastic ion-neutral collision frequency (Ma et al., 2004, 2013). The

ratio of the specific heats γ is taken to be 5/3. The subscripts s, n and e indicate ion species s,neutral species n and electron e. The other symbols have their usual definitions (Ma et al., 2004,2013). In order to account for photoionization, we calculate the optical depth of the neutralatmosphere by applying the Chapman functions based on the numerical evaluation given bySmith and Smith (1972). The photoelectron gains an excess energy through the photoionizationprocess, as indicated by the presence of Tn0 in Eq. (9). Therefore, we include the stellarheating via photoionization as those hydrodynamic models (e.g., Tripathi et al., 2015). In theabove set of equations, note that the source (S) and loss (L) terms of species s associated withphotoionization (νph,s′), charge exchange (kis′) and recombination (αR,s) are shown as follows:

Ss = msns′

(νph,s′ +

∑i=ions

kis′ni

)(11)

Ls = msns

(αR,sne +

∑n′=neutrals

ksn′nn′

)(12)

Se = me

∑s′=neutrals

νph,s′ns′ (13)

Le = mene∑

s=ions

αR,sns (14)

In Eq. (10), σ0 is the electrical conductivity. In the model, the electrical conductivity in theplanetary ionosphere is calculated using:

σ0 =nee

2

me(νei + νen), (15)

where me is the electron mass, νei and νen are the electron-ion and electron-neutral collisionfrequencies, respectively. The collision frequency νei and νen are given by Ma et al. (2013)

νei = 54.5ni

T3/2e

, (16)

νen = 3.68× 10−8(1 + 4.1× 10−11|4500− Te|2.93

)[CO2] (17)

+8.9× 10−11(1 + 5.7× 10−4Te)T1/2e [O]

17

Nsw(cm-3) Tsw(K) Vsw(km/s) IMF(nT)Maximum total pressure

Trappist-1b 6.59×104 2.01×106 (−470, 80,−1) (381, 81,−147)Trappist-1c 2.99×104 1.68×106 (−527, 68, 0) (210, 42,−111)Trappist-1d 1.20×104 1.59×106 (−566, 56, 6) (−129, 15,−50)Trappist-1e 5.79×103 1.26×106 (−604, 50, 3) (−149, 13,−42)Trappist-1f 2.99×103 1.02×106 (−624, 44, 3) (−98, 7,−34)Trappist-1g 1.95×103 8.92×105 (−637, 40, 2) (−69, 6,−28)Trappist-1h 9.52×102 7.17×105 (−657, 37, 2) (−44, 2,−25)

Minimum total pressureTrappist-1b 4.28×103 2.40×106 (−803, 80, 10) (−2206, 96,−64)Trappist-1c 2.09×103 2.35×106 (−871, 68,−3) (−1192, 63,−64)Trappist-1d 1.06×103 2.00×106 (−923, 56,−5) (−641, 41,−59)Trappist-1e 5.68×102 1.72×106 (−972, 50, 3) (−379, 32,−38)Trappist-1f 2.93×102 1.44×106 (−1017, 44, 19) (−218, 23,−25)Trappist-1g 2.05×102 1.30×106 (−1032, 40, 2) (−159, 20,−19)Trappist-1h 1.09×102 1.07×106 (−1049, 37,−16) (−89, 14,−11)

Table 3: Stellar wind parameters for the TRAPPIST-1 planets

The above set of equations is solved using an upwind finite-volume scheme based on an approxi-mate Riemann solver, to ensure the appropriate conservation of plasma variables. To determinethe steady-state solution, the simulation starts with a 2-stage local-time stepping scheme thatenables different grid cells to select different advance time step thus accelerating convergence tothe steady-state to save computational resources. Because of the stiffness of the source terms, apoint implicit scheme is used for handling them.

C.2 Auxiliary results

In the main paper, the atmospheric ion escape rates were provided for two cases: (i) maximumdynamic and total pressure, and (ii) minimum total pressure, but maximum magnetic pressure.The corresponding stellar wind parameters for these two cases have been delineated in Table 3.

In Table 3, note that Nsw, Tsw and Vsw denote the number density, temperature and velocityof the stellar wind respectively at the locations of the seven planets, whereas IMF denotes theInterplanetary Magnetic Field. The case with minimum total pressure corresponds to the fastsolar wind, which features a higher velocity albeit with a lower density, whereas the converse istrue for the maximum total pressure that can be associated with the slow solar wind.

In Fig. 4, the ionospheric profiles for TRAPPIST-1g have been provided for the cases withminimum and maximum total pressure. An inspection of Table 3 reveals that the cases withminimum and maximum total pressure have very different stellar wind parameters, for e.g.density, velocity and interplanetary magnetic field. Despite the considerable variability in thestellar wind parameters, it is evident that the ionospheric profiles remain mostly unaffected.This would appear to indicate that the lower regions are effectively immune to the effects of thestellar wind, which lends some credibility to the fact that surface biological processes (if present)may not be significantly affected.

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Figure 4: The ionospheric profiles along the substellar line for TRAPPIST-1g for the cases of(i) maximum and (ii) minimum total pressure over its orbit.

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