Role of planetary obliquity in regulating atmospheric escape: G-dwarf vs. M-dwarf Earth-like exoplanets
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Role of planetary obliquity in regulating atmospheric escape: G-dwarf vs. M-dwarf Earth-like exoplanets
Chuanfei Dong,
1, 2
Zhenguang Huang, and Manasvi Lingam Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA Princeton Center for Heliophysics, Princeton Plasma Physics Laboratory, Princeton University, Princeton, NJ 08544, USA Center for Space Environment Modeling, University of Michigan, Ann Arbor, MI 48109, USA Institute for Theory and Computation, Harvard University, Cambridge MA 02138, USA
ABSTRACTWe present a three-species (H + , O + and e − ) multi-fluid magnetohydrodynamic (MHD) model,endowed with the requisite upper atmospheric chemistry, that is capable of accurately quantifying themagnitude of oxygen ion losses from “Earth-like” exoplanets in habitable zones, whose magnetic androtational axes are roughly coincidental with one another. We apply this model to investigate the roleof planetary obliquity in regulating atmospheric losses from a magnetic perspective. For Earth-likeexoplanets orbiting solar-type stars, we demonstrate that the dependence of the total atmospheric ionloss rate on the planetary (magnetic) obliquity is relatively weak; the escape rates are found to varybetween 2 . × s − to 2 . × s − . In contrast, the obliquity can influence the atmosphericescape rate ( ∼ s − ) by more than a factor of 2 (or 200%) in the case of Earth-like exoplanetsorbiting late-type M-dwarfs. Thus, our simulations indicate that planetary obliquity may play aweak-to-moderate role insofar as the retention of an atmosphere (necessary for surface habitability) isconcerned. INTRODUCTIONOver the past decade, much attention has been di-rected toward understanding what factors contribute toexoplanetary habitability (Cockell et al. 2016). In par-ticular, it is widely accepted that orbital parametersplay a major role in governing habitability (Shields et al.2016). One of the chief orbital parameters is the obliq-uity (axial tilt). The fact that Earth’s obliquity is sub-ject to only mild fluctuations is believed to play a vitalrole in maintaining its stable climate.In consequence, numerous studies have analyzed howobliquity affects planetary climate (Spiegel et al. 2009;Ferreira et al. 2014; Rose et al. 2017; Kang 2019). Thereis broad consensus that climate is sensitive to changes(even minimal ones) in obliquity as it can transitionfrom one stable state to another (Williams & Pollard2003; Linsenmeier et al. 2015; Kilic et al. 2017; Coloseet al. 2019). A number of observational techniques basedon amplitude and frequency modulations in light curvesarising from thermal emission and scattered light havebeen proposed for inferring planetary obliquity (Gaidos& Williams 2004; Schwartz et al. 2016; Rauscher 2017;Kane & Torres 2017). Thermal phase curves of HotJupiters have already yielded constraints on planetary
Corresponding author: Chuanfei [email protected] obliquity; for example, CoRoT-2 b has an inferred obliq-uity of 45 . ◦ ± . ◦ (Adams et al. 2019).Exoplanets around M-dwarfs are typically anticipatedto have very low (or zero) obliquities due to rapid tidalenergy dissipation (e.g., Heller et al. 2011). This effectmay be particularly pronounced for the inner planetsof multi-planet systems such as the Kepler-186 system(Bolmont et al. 2014). Nevertheless, there are mech-anisms that permit the existence of high obliquity M-dwarf exoplanets. Perhaps the most famous amongthem are the “Cassini states” that involve the precessionof the planet’s spin and orbital angular momenta at thesame rate (Colombo 1966; Peale 1969; Winn & Holman2005); the Moon has an non-zero obliquity of 6 . ◦ dueto this reason. Hence, it is feasible for certain M-dwarfexoplanets to have high obliquities (Dobrovolskis 2009;Wang et al. 2016; Shan & Li 2018); see also Millholland& Laughlin (2019) and Millholland & Batygin (2019).Another factor that plays a vital role in regulatingsurficial habitability is the presence of an atmosphere.Moreover, an atmosphere also permits the detection ofbiosignature gases (e.g., molecular oxygen) via spec-troscopy (Kaltenegger 2017; Schwieterman et al. 2018;Madhusudhan 2019). Recent numerical and theoreti-cal studies indicate that both magnetized and unmag-netized planets around M-dwarfs might be particularlysusceptible to the depletion of ∼ a r X i v : . [ a s t r o - ph . E P ] J u l (Garcia-Sage et al. 2017; Dong et al. 2017b, 2018a,b;Lingam & Loeb 2018, 2019; Airapetian et al. 2019).In view of the preceding discussion, it is worthwhileasking the question: how does obliquity regulate atmo-spheric escape rates? It is, however, important to notethat the rotational and magnetic axes of Earth are sep-arated only by ∼ ◦ . Hence, it is plausible that thesetwo axes are potentially aligned for Earth-like planetsas well, viz., the angle between the two might be fairlysmall. In other words, the magnetic obliquity, i.e. theangle between the magnetic axis and orbital axis, mayapproximately coincide with the conventional planetaryobliquity. Furthermore, as the magnetic obliquity deter-mines the orientation of the planetary magnetic field, itcan influence magnetospheric properties and the resul-tant escape of atmospheric ions, which forms the subjectof this Letter.Thus, in this study, we opt to perform a paramet-ric analysis of how magnetic obliquity affects the atmo-spheric ion loss from magnetized exoplanets. We focuson two distinct examples due to their astrobiological rel-evance: an Earth-like planet around a solar-type starand an Earth-like planet around a late-type M-dwarfusing TRAPPIST-1 as a proxy. MULTI-FLUID MHD MODELA multi-fluid MHD model is utilized herein to sim-ulate the interaction between stellar winds and exo-planets and the concomitant atmospheric ion loss fromEarth-like worlds. On such worlds, note that the majorneutral and ion species in planetary upper atmosphereare atomic oxygen and O + respectively (Mendillo et al.2018), owing to which we focus on O + - the dominantescaping species - in our model. The multi-fluid MHDmodel we develop and employ herein possesses separatecontinuity, momentum and pressure equations for eachspecies (see the Appendix for detail). Despite the highercomputational expense incurred by the multi-fluid MHDapproach, they are more realistic and accurate than tra-ditional single-fluid approach (T´oth et al. 2012; Donget al. 2014, 2018c).To summarize, (1)-(14) in the Appendix permit us tosimulate the O + ions, stellar wind protons (H + ) andelectrons (e − ) with complete self-consistency; all essen-tial interactions between fluids as well as their individ-ual evolution are accounted for. More specifically, bothelastic and inelastic collisions and the associated heat-ing and cooling terms are explicitly present in our model.It is important to recognize that in addition to photo-electron heating, our model also incorporates Joule heat-ing , as seen from manipulating the second term in thesquare brackets on the RHS of (4) along the followinglines. (cid:88) t =O , O + ρ e ν et m e + m t (cid:20) m t ( u t − u e ) (cid:21) ∝ J σ e = J · E (1) where m t / ( m e + m t ) ≈ SIMULATION SET-UPAs mentioned in Sec. 1, we investigate two differentEarth-like planets by changing the host star they or-bit; this endeavor is important because stellar parame-ters influence numerous aspects of habitability (Lingam& Loeb 2019). The first involves a G-type star withthe Sun constituting the proxy. The second is based onTRAPPIST-1 because it is a well-known example of lateM-dwarfs.To investigate Earth-like planets, we make use ofEarth’s thermospheric temperature profile (Schunk &Nagy 2009) and specify a fiducial surface pressure of 1bar and magnetic dipole moment equal to the Earth’s. Itis worth noting that Earth’s thermosphere is much hot-ter ( ∼ ∼
20 K inducedby obliquity as per climate models (Kang 2019) is notlikely to affect the upper atmosphere significantly. Asnoted previously in Sec. 2, O + is the primary ion under-going atmospheric escape in our model because we areinterested in exoplanets that resemble Earth. It is fea-sible to select alternative atmospheres (e.g., resemblingVenus), along the lines of Dong et al. (2017b, 2018a),and study the effects of varying magnetic obliquity butthis is left for future publications.Our simulation domain commences at 100 km, wherethe density of O + ions is roughly in photochemical equi-librium. In our numerical model, float boundary condi-tions are employed for the velocity u and the magneticfield B . The simulation box is extended up to 200 plan-etary radii by means of a non-uniform spherical grid. Inthe lower ionosphere and thermosphere, the lowest spa-tial resolution of ∼
10 km - several times smaller thanthe thermospheric scale height - is attained to capturefine-scale variations in the upper atmosphere. The an-gular (i.e., horizontal) resolution is 3 ◦ × ◦ . The equa-tions in the Appendix are solved by means of an upwindfinite-volume scheme (T´oth et al. 2012); see Dong et al.(2017a) for further details.Since we are studying Earth-like planets around solar-type stars and late M-dwarfs, we require the appropriatestellar wind parameters to compute self-consistent ionescape rates. For the Sun, current solar wind param-eters are adopted from Schunk & Nagy (2009). Giventhat we use TRAPPIST-1 as a proxy, we use the simu-lated stellar wind parameters from Dong et al. (2018a) at Table 1.
The elastic collision rates and chemical reaction rates used in the multi-fluid MHD code. For elastic collisions, Z s , m s , n s and T s denote the charge state, mass (in amu), number density (in cm − ) and temperature (in K) for a given species.Note that m st = m s m t m s + m t and T st = m s T t + m t T s m s + m t represent the reduced mass and temperature.Elastic Collision Rate (s − ) Referenceion-ion ( ν st ) 1 . × Z s Z t √ m st m s n t T / st Schunk & Nagy (2009)ion-neutral ( ν sn ) C sn n n a Schunk & Nagy (2009)electron-ion ( ν es ) 54 . × n s Z s T / e Schunk & Nagy (2009)ion-electron ( ν se ) 1 . × √ m e m s n e Z s T / e Schunk & Nagy (2009)electron-neutral ( ν en ) 8 . × − n n (1 + 5 . × − T e ) T / e Schunk & Nagy (2009)Chemical Reaction Rate (s − )Primary Photolysis and Particle ImpactO + hν → O + + e − see table footnote b Schunk & Nagy (2009)e − + O → e − + O + + e − see reference Cravens et al. (1987)Ion-Neutral Chemistry Rate (cm s − )H + + O → H + O + . × − Schunk & Nagy (2009)Electron Recombination Chemistry Rate (cm s − )O + + e − → O 3 . × − (cid:16) T e (cid:17) . , Schunk & Nagy (2009)H + + e − → H 4 . × − (cid:16) T e (cid:17) . , Schunk & Nagy (2009) a Both ν H + O and ν O + O are for resonant ion-neutral interactions, where ν H + O = 6 . × − n O T / H + (1 − .
047 log T H + ) for T r > K ,and ν O + O = 3 . × − n O T / r (1 − .
064 log T r ) for T r > K . T r = ( T i + T n ) /
2, where T i and T n are the ion and neutraltemperatures, respectively. Atomic oxygen number density n O has units of cm − . b The photoionization rate has been appropriately rescaled based on the extreme ultraviolet (EUV) values at Earth and TRAPPIST-1g.The photoionization rate equals 4.13 × − s − for Earth, and 32.96 × − s − for TRAPPIST-1g. TRAPPIST-1g. The parameters for the two cases arepresented in Table 2. The Planet-Star-Orbital (PSO)coordinates are used herein. In this system, the X-axisis directed from the planet toward the star, the Z-axisis normal to the planet’s orbital plane, and the Y-axisis perpendicular to the X- and Z-axes. RESULTSRecall that the our primary intention is to determinehow planetary obliquity regulates the atmospheric ionloss from a magnetic perspective. The final results aredepicted in Table 2. There are several interesting fea-tures that stand out in both cases, viz., solar-type starsand late M-dwarfs.First, the atmospheric ion escape rate is O (10 s − )in the case of G-dwarf Earth-like planets, whereas theescape rate increases to O (10 s − ) for M-dwarf plan-ets due to the extreme stellar wind conditions and highenergy radiations in the close-in habitable zones (HZs).In other words, a ∼ O (10 ) yrs to be depleted for a G-type star and O (10 ) yrs for an M-dwarf based on nor-mal stellar wind conditions.Second, the variation in the atmospheric ion escaperates is virtually independent of the magnetic obliquityfor an Earth-analog around a solar-type star. We findthat the total variation is less than 10%. In contrast, when we consider an Earth-like planet around a late M-dwarf, we determine that the variation is modest (butnon-negligible); in quantitative terms, the maximum es-cape rate is more than twice (or 200%) the minimumvalue. The chief reason why the obliquity plays a weakrole in determining the escape rate for solar-type starsstems from the temperate stellar wind and radiation inHZs.As shown in Fig. 1, the magnetosphere of the G-dwarf planet is larger than that of the M-dwarf planet;therefore, regardless of magnetic obliquity’s value, theionosphere does not experience much difference. On theother hand, for an Earth-analog around TRAPPIST-1,the dual effect of the compressed magnetosphere andhigh energy radiation makes the ion sources (e.g., elec-tron impact ionization and charge exchange) more sen-sitive to the magnetic configuration.Third, we see that the maximal ion escape rate is at-tained at a magnetic obliquity of 90 ◦ whereas the min-imum occurs at 0 ◦ or 180 ◦ (Fig. 2). While the at-mospheric escape rates at 0 ◦ and 180 ◦ are nearly thesame, there is a clear distinction between 90 ◦ and 270 ◦ .The reason behind the latter behavior has to do withthe relative orientation of the interplanetary magneticfield (IMF) and the planetary magnetic field. At themagnetic obliquity of 90 ◦ , the IMF can directly connectto the dayside planetary surface due to the field polar- Figure 1.
Logarithmic scale contour plots of the O + ion density with magnetic field lines (in white) in the meridional planebased on the stellar wind conditions at current Earth (top two rows) and at TRAPPIST-1g (bottom two rows). Different plotscorrespond to various choices of the planetary magnetic obliquity. Note the same colorbar range but the different box sizebetween the top two rows and the bottom two rows. ity, whereas the IMF can only connect to the nightsidesurface at the magnetic obliquity of 270 ◦ ; see the thirdcolumn of Fig. 1. Therefore, stellar wind particles, es-pecially electrons (with relatively low energy) can betransported along the field lines and ionize the atomicoxygen in the upper atmosphere via impacts as shownin Fig. 3.Fig. 1 depicts the contour plots of the O + ion densityfor an Earth-like planet around a solar-type star (toptwo panels) and a late M-dwarf (bottom two panels) atdifferent values of the magnetic obliquity. Consistent with the results in Table 2, more O + ions escape fromthe M-dwarf planet due to the extreme stellar wind andradiations. Fig. 2 shows the corresponding atmosphericoxygen ion escape rates, where two peaks clearly occurat magnetic obliquities of 90 ◦ and 270 ◦ ; the former isslightly higher than the latter.In the 90 ◦ or 270 ◦ cases, the cusp region (i.e., theregion filled with open magnetic field lines) directly facesthe star and thus the stellar wind. Therefore, the stellarwind particles can impact the dayside upper atmosphererelatively easily and deposit their energy. Compared to Table 2.
The stellar wind input parameters are based on: (i) typical solar wind parameters at 1 AU in the current epoch(Schunk & Nagy 2009), (ii) stellar wind parameters simulated at TRAPPIST-1g (Dong et al. 2018a). For a fair comparison, weassume that both stellar wind velocities only have an x component, and stellar magnetic fields are located in the x-y plane asthe nominal case. Here, the radiation flux received at Earth refers to the solar cycle moderate conditions and the radiation fluxreceived at TRAPPIST-1g is based on estimates provided in Bolmont et al. (2017) and Bourrier et al. (2017).n sw v sw IMF Radiation Obliquity O + loss rate Normalization a (cm − ) (km/s) (nT) flux (degree) (s − ) (%)0 2.19 × × × × × × × × × × × × × × × × a The normalization refers to the oxygen ion escape rate normalized to the canonical zero obliquity case.
Figure 2.
Oxygen ion escape rate for different values of the planetary (magnetic) obliquity. Note that the scales of the verticalaxis in the two panels are different. the 270 ◦ case, the 90 ◦ case possesses a better magneticconnection with the planet (based on our model setup), thereby achieving a slightly higher escape rate primarilydue to the high electron impact ionization. Figure 3.
First row:
Logarithmic scale contour plots of theO + ion density with O + ion velocity vectors (in black) andmagnetic field lines (in white) in the meridional plane forthe M-dwarf planet with obliquities of 90 ◦ and 270 ◦ , respec-tively. The black arrows depict both the direction and themagnitude of O + ion velocities. Second to fourth rows:
Log-arithmic scale contour plots of the photoionization rate R
Phi O + ,electron impact ionization rate R Imp O + , and charge exchangerate R CX O + (with stellar wind protons) of O + ions. Last row:
The difference in total ionization rate and electron impactionization rate between 90 ◦ and 270 ◦ . In order to demonstrate this point, the oxygen ion den-sity and velocity (first row) and different ionization rates(second to fourth rows) are illustrated for an Earth-likeplanet around a late M-dwarf in Fig. 3. The ion out-flow of O + (namely, the polar wind) driven primarilyby the electron pressure gradient, ∇ p e , is rendered inthe upper panel of Fig. 3. The bottom panel of Fig. 3shows the difference of the total ionization rate between90 ◦ and 270 ◦ (left column), resembling the difference inelectron impact ionization rate (right column); in otherwords, the difference in total ionization rate is mainlyregulated by electron impact ionization.Lastly, over an extended period of time, due to thepolarity reversals of stellar and planetary magnetic fields(Glatzmaier 2013), variations in the escape rate withobliquity may get averaged out. CONCLUSIONSIn this Letter, we have described a sophisticated nu-merical code for simulating the escape of atmosphericions from Earth-like (exo)planets. We have incorpo-rated the appropriate upper atmospheric chemistry andevolve each species separately via the multi-fluid MHDequations of Sec. 2. This model was applied to studytwo different exoplanets: the first around a solar-typestar and the second orbiting a late M-dwarf, for whichTRAPPIST-1 was used as a proxy. Our goal was to de-termine how the ion escape rates vary with the planetaryobliquity from a magnetic perspective.We found that the maximum escape rates arose atobliquities of 90 ◦ or 270 ◦ (depending on field polarities),whereas the minimum rates were attained at 0 ◦ or 180 ◦ .The reason is that the cusp (comprising open field lines)faces the stellar wind at obliquities of 90 ◦ or 270 ◦ , andallows the stellar wind particles to deposit their energyin the planetary upper atmosphere. For Earth-like plan-ets around solar-type stars, it is found that the escaperate is virtually independent of the obliquity. On theother hand, for late M-dwarfs, we determined that theescape rate varies by more than a factor of ∼ ∼ O (10 ) and O (10 ) yrs, for solar-type stars and lateM-dwarfs, respectively. If we assume that the source ofatmospheric O is water from oceans, we find that themass of Earth’s oceans ( M oc ) cannot be depleted overthe main-sequence lifetime of a solar-type star. In con-trast, for a late M-dwarf we determine that M oc couldbe depleted over a timescale of O (10 ) yrs, which isshorter than the star’s lifetime.There are two conclusions to be drawn from this find-ing. When studying Earth-like planets around solar-type stars, at least insofar as the atmospheric ion escaperates are concerned, the effects of obliquity is ostensi-bly minimal. In contrast, when it comes to exoplanetsaround late M-dwarfs, obliquity might play an impor-tant role. There are, however, two different cases toconsider for M-dwarf exoplanets. In the first case, if theatmosphere is completely depleted over a timescale thatis orders of magnitude smaller than 1 Gyr, changing thisvalue by a factor of ∼ ◦ to 90 ◦ . Itis therefore instructive to carry out a thought experi-ment. Suppose that an Earth-like planet can retain anatmosphere for 4 Gyr for an obliquity of 0 ◦ and thatbiological evolution unfolds in a similar fashion as onEarth; at an obliquity of 90 ◦ , the depletion timescaleis 1 . ◦ obliquity, enough time might exist for the emergence ofcomplex multicellular organisms, whereas an obliquityof 90 ◦ may not suffice for the evolution of eukaryoticanalogs and complex multicellularity (Knoll 2015).Some caveats regarding our treatment are worth em-phasizing here. We have chosen to vary only the stellarparameters, thus leaving planetary parameters (e.g., sizeand magnetic field strength) fixed. In actuality, habit-able exoplanets are probably very diverse and the es-cape rates will change accordingly; however, it is plausi-ble that the qualitative trends described herein are stillpartly valid. Moreover, we have incorporated only O + as it constitutes the dominant ion species in the Earth’s ionosphere, but subsequent treatments should incorpo-rate additional minor species. As we utilize a multi-fluid MHD model, kinetic effects contributing to atmo-spheric ion escape are not included in our model (e.g.,see Strangeway et al. 2005). Finally, the issue of atmo-spheric depletion is difficult to address comprehensivelybecause it also necessitates knowledge of other pertinentissues including outgassing, bolide impacts, (super)flaresand associated phenomena (e.g., coronal mass ejections).To summarize, planetary magnetic obliquity does notappear to affect atmospheric ion escape rates for hab-itable planets around solar-type stars, whereas it has aweak-to-moderate influence on the escape rates for lateM-dwarf exoplanets.The authors acknowledge fruitful discussions with Yu-tong Shan, Anthony Del Genio, Michael Way, Fei Dai,Gongjie Li, and Joshua Winn. CD was supportedby NASA grant 80NSSC18K0288. ML was supportedby the Institute for Theory and Computation at Har-vard University. Resources for this work were pro-vided by the NASA High-End Computing (HEC) Pro-gram through the NASA Advanced Supercomputing(NAS) Division at Ames Research Center. The SpaceWeather Modeling Framework that contains the BATS-R-US code used in this study is publicly available fromhttp://csem.engin.umich.edu/tools/swmf. For distribu-tion of the model results used in this study, please con-tact the corresponding author.APPENDIXIn this section, we describe the multi-fluid MHD model, endowed with the electron pressure equation, that is usedto simulate the oxygen ion loss from Earth-like exoplanets.The multi-fluid MHD model comprises three fluids, of which two of them are ionic O + and H + (with subscript s ) andthe last is the electron fluid with subscript e . For the background neutral species, we employ the subscript n . In themulti-fluid MHD equations, ρ , u , p , ←→ I , k B and γ = 5 / ∂ρ s ∂t + ∇ · ( ρ s u s ) = S s − L s (1) ∂ ( ρ s u s ) ∂t + ∇ · (cid:16) ρ s u s u s + p s ←→ I (cid:17) = n s q s ( u s − u + ) × B + q s n s en e ( J × B − ∇ p e ) + ρ s G + ρ s (cid:88) t =all ν st ( u t − u s ) + S s u n − L s u s (2) This is clearly an idealization, but one we adopt to carry outthe thought experiment to fruition. ∂p s ∂t + ( u s · ∇ ) p s = − γ s p s ( ∇ · u s )+ (cid:88) t =all ρ s ν st m s + m t (cid:20) k B ( T t − T s ) + 23 m t ( u t − u s ) (cid:21) + k B S s T n − L s T s m s + 13 S s ( u n − u s ) (3) ∂p e ∂t + ( u e · ∇ ) p e = − γ e p e ( ∇ · u e )+ (cid:88) t =s , n ρ e ν et m e + m t (cid:20) k B ( T t − T e ) + 23 m t ( u t − u e ) (cid:21) − k B L e T e m e + 23 n n ( ν ph,n E excns − ν imp,n E potns ) − n e n n R inelasticen + 13 S e ( u n − u e ) (4) ∂ B ∂t = ∇ × ( u + × B − η J ) (5)where ν signifies the collision frequency between two species and u + refers to the charge-averaged velocity, u + = (cid:88) s = ions q s n s u s en e (6)while η is the magnetic diffusivity, η = 1 µ σ e = 1 µ (cid:18) σ en + 1 σ ei (cid:19) (7)where σ e is the electron conductivity and is composed of both electron-neutral ( σ en = e n e / Σ n (cid:48) ν en (cid:48) m e ) and electron-ion( σ ei = e n e / Σ s (cid:48) ν es (cid:48) m e ) collisions.The preceding set of equations consists of source ( S ) and loss ( L ) terms: S s = m s n s (cid:48) (cid:32) ν ph,s (cid:48) + ν imp,s (cid:48) + (cid:88) i =ions k is (cid:48) n i (cid:33) (8) L s = m s n s (cid:32) α R,s n e + (cid:88) t (cid:48) =neutrals k st (cid:48) n t (cid:48) (cid:33) (9) S e = m e (cid:88) s (cid:48) n s (cid:48) ( ν ph,s (cid:48) + ν imp,s (cid:48) ) (10) L e = m e n e (cid:88) s =ions α R,s n s . (11)Note that the source and loss terms consist of photoionization ( ν ph,s (cid:48) ), charge exchange ( k is (cid:48) ), electron impact ionization( ν imp,s (cid:48) ), and recombination ( α R,s ).In addition, inelastic collisions between electrons and O (neutral oxygen) are effective at cooling the former in thelower thermosphere, where collisions occur regularly. Therefore, we incorporate the cooling rate coefficient R inelasticen (in eV cm s − ) in (4) by adopting the formalism in Schunk & Nagy (2009): R inelasticen = D − { S (cid:0) − exp (cid:2) . (cid:0) T − e − T − n (cid:1)(cid:3)(cid:1) + S (1 − exp (cid:2) . (cid:0) T − e − T − n (cid:1) ] (cid:1) + S (1 − exp (cid:2) . (cid:0) T − e − T − n (cid:1) ] (cid:1) } , (12)where D = 5 + exp( − . T − n ) + 3 exp( − . T − n ), S = 1 . · − , S = 1 . · − , and S = 8 . · − T . e exp( − . T − n ).The optical depth of the neutral atmosphere is determined by employing the numerical formula in Smith & Smith(1972) to study photoionization. Photoelectrons gain excess energy E excns during the photoionization process and theylose the ionization energy of neutral oxygen, E potns , during electron impact ionization (Schunk & Nagy 2009), as seenfrom Eq. (4). The number density and velocity of electrons is easy to calculate after imposing quasineutrality andexpressing it in terms of the current (T´oth et al. 2012), n e = 1 e (cid:88) s =ions n s q s , (13) u e = u + − J en e = u + − ∇ × B µ en e (14)with the last equation following from Amp´ere’s law.REFERENCES Adams, A. D., Millholland, S., & Laughlin, G. P. 2019, AJ,arXiv:1906.07615Airapetian, V. S., Barnes, R., Cohen, O., et al. 2019, arXive-prints, arXiv:1905.05093Bolmont, E., Raymond, S. N., von Paris, P., et al. 2014,ApJ, 793, 3Bolmont, E., Selsis, F., Owen, J. E., et al. 2017, MNRAS,464, 3728Bourrier, V., Ehrenreich, D., Wheatley, P. J., et al. 2017,A&A, 599, L3Cockell, C. S., Bush, T., Bryce, C., et al. 2016,Astrobiology, 16, 89Colombo, G. 1966, AJ, 71, 891Colose, C. M., Del Genio, A. D., & Way, M. J. 2019, arXive-prints, arXiv:1905.09398Cravens, T. E., Kozyra, J. U., Nagy, A. F., Gombosi, T. I.,& Kurtz, M. 1987, J. Geophys. Res., 92, 7341Dobrovolskis, A. R. 2009, Icarus, 204, 1Dong, C., Bougher, S. W., Ma, Y., et al. 2014, GeoRL, 41,2708Dong, C., Huang, Z., Lingam, M., et al. 2017a, ApJL, 847,L4Dong, C., Jin, M., Lingam, M., et al. 2018a, PNAS, 115,260Dong, C., Lingam, M., Ma, Y., & Cohen, O. 2017b, ApJL,837, L26Dong, C., Lee, Y., Ma, Y., et al. 2018b, ApJL, 859, L14Dong, C., Bougher, S. W., Ma, Y., et al. 2018c, J. Geophys.Res., 123, 6639Ferreira, D., Marshall, J., O’Gorman, P. A., & Seager, S.2014, Icarus, 243, 236Gaidos, E., & Williams, D. M. 2004, NewA, 10, 67Garcia-Sage, K., Glocer, A., Drake, J. J., Gronoff, G., &Cohen, O. 2017, ApJL, 844, L13 Glatzmaier, G. A. 2013, Introduction to ModelingConvection in Planets and Stars: Magnetic Field, DensityStratification, Rotation (Princeton University Press)Heller, R., Leconte, J., & Barnes, R. 2011, A&A, 528, A27Huang, Z., T´oth, G., Gombosi, T. I., et al. 2016, J.Geophys. Res. A, 121, 4247Kaltenegger, L. 2017, ARA&A, 55, 433Kane, S. R., & Torres, S. M. 2017, AJ, 154, 204Kang, W. 2019, ApJL, 876, L1Kilic, C., Raible, C. C., & Stocker, T. F. 2017, ApJ, 844,147Knoll, A. H. 2015, Life on a Young Planet: The First ThreeBillion Years of Evolution on Earth, 2nd edn. (PrincetonUniversity Press)Lingam, M., & Loeb, A. 2018, IJAsB, 17, 116—. 2019, RvMP, 91, 021002Linsenmeier, M., Pascale, S., & Lucarini, V. 2015,Planet. Space Sci., 105, 43Madhusudhan, N. 2019, ARA&AMendillo, M., Withers, P., & Dalba, P. A. 2018, NatAs, 2,287Millholland, S., & Batygin, K. 2019, ApJ, 876, 119Millholland, S., & Laughlin, G. 2019, NatAs, 3, 424Peale, S. J. 1969, AJ, 74, 483Rauscher, E. 2017, ApJ, 846, 69Rose, B. E. J., Cronin, T. W., & Bitz, C. M. 2017, ApJ,846, 28Rubin, M., Combi, M. R., Daldorff, L. K. S., et al. 2014,ApJ, 781, 86Schunk, R., & Nagy, A. 2009, Ionospheres: Physics, PlasmaPhysics, and Chemistry, Cambridge Atmospheric andSpace Science Series (Cambridge Univ. Press),doi:10.1017/CBO9780511635342Schwartz, J. C., Sekowski, C., Haggard, H. M., Pall´e, E., &Cowan, N. B. 2016, MNRAS, 457, 9260