Energy Dissipation in the Upper Atmospheres of Trappist-1 Planets
DDraft version March 15, 2018
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Energy Dissipation in the Upper Atmospheres of Trappist-1 Planets
Ofer Cohen,
1, 2
Alex Glocer, Cecilia Garraffo, Jeremy J. Drake, and Jared M. Bell Lowell Center for Space Science and Technology, University of Massachusetts Lowell600 Suffolk St., Lowell, MA 01854, USA Harvard-Smithsonian Center for Astrophysics,60 Garden St., Cambridge, MA 02138, USA NASA/Goddard Space Flight Center, Greenbelt, Maryland, USA Harvard-Smithsonian Center for Astrophysics,60 Garden St., Cambridge, Massachusetts, USA National Institute of Aerospace, 100 Exploration Way, Hampton, VA 23666, USA
ABSTRACTWe present a method to quantify the upper-limit of the energy transmitted from the intense stellarwind to the upper atmospheres of three of the Trappist-1 planets (e, f, and g). We use a formalismthat treats the system as two electromagnetic regions, where the efficiency of the energy transmissionbetween one region (the stellar wind at the planetary orbits) to the other (the planetary ionospheres)depends on the relation between the conductances and impedances of the two regions. Since theenergy flux of the stellar wind is very high at these planetary orbits, we find that for the case of hightransmission efficiency (when the conductances and impedances are close in magnitude), the energydissipation in the upper planetary atmospheres is also very large. On average, the Ohmic energy canreach 0 . − W/m , about 1% of the stellar irradiance and 5-15 times the EUV irradiance. Here,using constant values for the ionospheric conductance, we demonstrate that the stellar wind energycould potentially drive large atmospheric heating in terrestrial planets, as well as in hot jupiters.More detailed calculations are needed to assess the ionospheric conductance and to determine moreaccurately the amount of heating the stellar wind can drive in close-orbit planets. Keywords: planets and satellites: atmospheres — magnetic fields — plasmas INTRODUCTIONThe recent discovery of seven Earth-size terrestrialplanets in the Trappist-1 system (Gillon et al. 2017) hasstimulated the possibility of detecting habitable planetsin nearby systems. Indeed, three of the seven Trappist-1 planets - Trappist-1e, Trappist-1f, and Trappist-1g -are in the Habitable Zone (HZ) defined as a boundeddistance from the host star at which the planetary equi-librium temperature allows water to exist in liquid formon the planetary surface. A growing number of stud-ies have been published on the Trappist-1 system in theshort time since its discovery. These include studies ofthe formation and evolution of the planetary system andits planets (e.g., Barr et al. 2017; Burgasser & Mama-jek 2017; Luger et al. 2017; Ormel et al. 2017; Quarleset al. 2017; Tamayo et al. 2017), the atmospheres ofthe Trappist-1 planets (e.g., Alberti et al. 2017; Bour- ofer [email protected] rier et al. 2017; Tilley et al. 2017; Wolf 2017), and thechance for life to exist on the Trappist-1 planets (Lingam& Loeb 2017).While detections of terrestrial planets in the HZ ofTrappist-1 are exciting, a major potential problem fortheir habitability is the fact that the HZ around faintM-dwarf stars is extremely close to the host star. Itmay be located at a distance of less than 0.1 AU andessentially inside the stellar corona. Indeed, the or-bital distances of Trappist-1 e, f, and g are 0.028 AU,0.037 AU, and 0.045 AU, respectively. In such close or-bits, the conditions of the stellar environment are muchmore extreme than those experienced by a planet lo-cated like the Earth, much further from the host star.These include increased stellar energetic radiation, en-hanced density of the stellar wind (resulting in enhanceddynamic pressure), and enhanced magnitude of the stel-lar wind magnetic field (resulting in enhanced magneticpressure). These extreme conditions may lead to evap-oration and stripping of the planetary atmosphere until a r X i v : . [ a s t r o - ph . E P ] M a r Cohen et al. they completely lost. Thus, the chance of habitabilitycould be greatly reduced.Few generic studies have been performed to esti-mate the atmospheric loss from close-orbit planets(e.g., Cohen et al. 2015; Airapetian et al. 2017; Donget al. 2017a). Recent studies have estimated the spaceenvironment conditions and the atmospheric loss onTrappist-1 (Roettenbacher & Kane 2017; Garraffo et al.2017) and the recently detected Proxima Centauri b(Garraffo et al. 2016; Dong et al. 2017b; Garcia-Sageet al. 2017). All these studies have pointed to a veryhigh mass loss rates from these close-in planets, sug-gesting their atmospheres may be completely eliminatedover their lifetimes. We stress that these estimates didnot attempt to demonstrate how the atmospheres canbe formed, which is itself another theoretical challenge.One key aspect in estimating the ability of a close-in planet to sustain its atmosphere is to quantify theenergy input from the stellar radiation and the stellarwind at the location of the planet. Detailed observa-tions and estimation of the stellar EUV and bolometricluminosity at the orbits of the Trappist-1 planets haverecently been obtained by Wheatley et al. (2017). Inthe study presented here, we quantify the total energyinput carried by the stellar wind in the vicinity of thethree potentially habitable Trappist-1 planets, and es-timate the amount of energy that is delivered to theirupper atmospheres, assuming atmospheres do exist. Weuse the radiation energies obtained by Wheatley et al.(2017) as reference for the stellar wind energy.It is known from our own solar system, that the so-lar wind energy is dissipated in upper planetary atmo-spheres in the form of Joule Heating or Ohmic dissi-pation in the ionosphere. The ionosphere is the layerof the upper atmosphere at which photoionization cre-ates a peak in the electron density so that conductivitybecomes finite. The ionosphere allows field-aligned cur-rents, that flow from the magnetosphere (in the case ofmagnetized planets) or the induced magnetosphere (inthe case of non-magnetized planets), to close throughit, while dissipating the energy due to its resistive na-ture (see e.g., Kivelson & Russell 1995; Gombosi 2004,for a complete description of the process). The dissi-pating energy depends on the solar wind driver, whichdrives the field-aligned currents, and the conductivityin the ionosphere, which depends on the atmosphericconditions, composition, and ionization. In general, thelarge-scale, ambient ionospheric conductivity is domi-nated by the so-called Pedersen conductivity, which isthe component of the conductivity tensor responsiblefor the electric field that is perpendicular to the ambi-ent magnetic field. In the case of the Earth’s ionosphere, this is the electric field that is perpendicular to both thesolar wind velocity and the solar wind magnetic field,driving a current that flows across the region where theEarth’s magnetic field is open to the solar wind. Thisregion of open field lines is called the polar cap and theelectric field across it is associated with a Cross PolarCap Potential (CPCP).There is some evidence that during a strong solar winddriver, the CPCP is saturated (see Kivelson & Ridley2008, for longer description with reference therein). Inparticular, the saturation might occur when the solarwind Alfv´enic Mach number is very low, even below one.In that case, the interaction of the moving body (magne-tized or non-magnetized) with the sub-Alfv´enic flow gen-erates the topology of
Alfv´en Wings - two standing lobesexpanding an angle that depends on the velocity of thebody and the Alfv´en speed (Drell et al. 1965; Neubauer1980, 1998). The energy transfer from the stellar windto the ionosphere during such conditions can be treatedin an idealized wave transmission manner. Kivelson &Ridley (2008) (KR08 hereafter) have treated the iono-sphere as a spherical conductor with finite conductivity,and the incoming solar wind as an electromagnetic wave.They showed that the energy transmitted from the solarwind to the ionosphere can be estimated as the trans-mitted energy of the incoming electromagnetic wave.Here, we adopt the formalism of KR08 to estimatethe energy input from the extreme stellar wind of theTrappist-1 planets onto the atmospheres of the e, f, andg planets. We describe the formalism in Section 2 andpresent the results in Section 3. We discuss the conse-quences of the results for the atmospheres of Trappist-1in Section 4 and conclude our findings in Section 5. WAVE TRANSMISSION FORMALISM OF THESTELLAR WIND ENERGY INPUTWe now review the method which is described in KR08in the context of exoplanets. The stellar wind at thevicinity of a planet has a velocity v sw and a magneticfield B sw . Therefore, the motional electric field, E sw can be obtain as: E sw = − v sw × B sw , (1)and the magnitude of the electric field is given by: | E sw | = | v sw | · | B sw | . (2)The local Alfv´en speed of the stellar wind is given by: v A = B sw √ µ ρ sw , (3) nergy Deposition at the Upper Atmospheres of the Trappist-1 Planets A = ( µ v A ) − [ Siemens ] , (4)where we can also define an associated Alfv´enicimpedance which is the inverse of the Alfv´enic con-ductivity: Σ − A = µ v A [ ohm ] . (5)The local Pedersen conductivity at certain altitude, σ P , is a function of the local electron density, N e , thecharge, e , the ion and electron masses, m i and m e , re-spectively, the ion and electron stress collision frequen-cies, ν i and ν e , respectively, and the ion and electronplasma frequencies, Ω i and Ω e , respectively (Kivelson &Russell 1995): σ p = e N e (cid:20) ν i m i ( ν i + Ω i ) + ν e m e ( ν e + Ω e ) (cid:21) [ S/m ] , (6)The height-integrated Pedersen conductance, Σ P isthe column height integral of σ P , where we can intro-duce an associated Pedersen impedance simply definedas Σ − P . Typical Earth values for the height integratedconductance are 1-10 (e.g., Kivelson & Ridley 2008).Here we test our calculations against assumed valuesof Σ p = 0 . , , , , , and
100 [ S ].Assuming the stellar wind electric field can be con-sidered as an electromagnetic wave, we can use theimpedances defined above to calculate the reflection andtransmission of the incoming electric field wave, | E i | .The reflected electric field is given by: | E r | = | E i | (cid:0) Σ − P − Σ − A (cid:1)(cid:0) Σ − P + Σ − A (cid:1) , (7)Note that when Σ − P is smaller than Σ − A , the reflectedwave has an opposite sign to that of the incoming wave.Thus, the transmitted electric field is given by: | E t | = | E r | + | E i | = 2 | E sw | Σ − P (cid:0) Σ − P + Σ − A (cid:1) (8)where the transmission of the electric field is expectedto be most significant where the Alfv´enic and Pedersenconductances are close in magnitude.Once the transmitted electric field is obtained, we canestimate the energy flux that is dissipated in the plane-tary ionosphere via Ohmic dissipation, Q t , as: Q t = j Σ P = Σ P | E t | Σ P = Σ P | E t | [ W/m ] . (9) Note that since we use the height integrated conduc-tance, the units of Q t are not of J · E but of J · E mul-tiplied by a length scale, which gives energy flux. RESULTSFigure 1 shows the stellar wind parameters along theorbits of Trappist-1 e, f, and g. The parameters were ob-tained from the MHD wind simulation presented in Gar-raffo et al. (2017) (case with an average field of 600G).The stellar wind parameters are more extreme than typ-ical solar wind conditions near the Earth, with windspeeds of 1.5-2 times that of the solar wind, magneticfield 100-1000 times larger than the solar wind magneticfield, and 100-1000 times more dense wind compared tothe solar wind near the Earth. It can be seen that dur-ing most of the orbit, the three planets reside in a low-Alfv´enic Mach number (less than 2), where Trappist-1eexperiences a plasma environment with M A ≈ V A , the Alfv´enic conductivity, Σ A , and the Alfv´enicimpedance, Σ − A , as a function of the orbital phase ofTrappist-1 e, f, and g. The conductance is below 1 formost of the orbit but reaches values of 7-10 during thestreamer crossings. The impedance values are around1 most of the orbit but become about 10 times smallerduring the streamer crossings.Figure 2 shows the transmitted energy deposited intothe planetary atmospheres of Trappist-1 e and g as afunction of orbital phase. The results for Trappist-1f liein between these two cases and therefore are not shownhere. We normalize the energy flux to three values: i)the energy flux of the stellar wind; ii) the stellar irradi-ance; and iii) the stellar EUV irradiance. We estimatethe stellar wind energy flux, in W/m , as the sum ofthe dynamic ( ρ sw v sw ) and magnetic ( B sw / π ) pressuresmultiplied by the wind velocity: F sw = (cid:0) ρ sw v sw + B sw / π (cid:1) · v sw (10)The results show that the transmitted energy is sig-nificant for values of Σ P <
10 during the orbital phaseswhere the planet resides in a low Alfv´enic Mach number.Higher values of the Pedersen conductance, or orbitalphases at which the Alfv´enic Mach number is higherthan 2, seem insufficient to enable deposition of a signif-icant amount of heating in the upper atmosphere fromthe intense stellar wind due to the reflection of mostof the Alfv´enic energy input. As expected, the energytransmission is most efficient when the values of the stel-lar wind and ionospheric conductances are close to each
Cohen et al. other. In these cases, about 10-50% of the stellar windinput energy is transmitted with Σ P = 0 . , , and P = 1 and
10 for the three planets.Note that these avreraged values do not exactly followEq. 9 DISCUSSIONWe quantify the energy deposition from the ex-treme stellar wind of Trappist-1 into its planets’ at-mospheres. We use the formalism from KR08 to relatethe impedances associated with the stellar wind andthe planetary ionosphere to the energy deposition. In away, this formalism provides an efficiency of the energytransfer from the stellar wind driver to the conductinglayer (the ionosphere) in a generalized electromagneticenergy transmission manner. The efficiency depends onthe relationship between the ability of the two mediumsto allow or suppress electric currents from flowing inthem (i.e., the conductances and impedances).It is important to note that here we compare the en-ergy inputs to the planet in terms of energy flux, andthat in the KV08 formalism, the stellar wind energy fluxis assumed to be transmitted in the area covered by theregion where planetary field lines are open to the stellarwind (the polar caps). This area depends on the plan-etary field strength which is unknown. Garraffo et al.(2017) suggested that in the Trappist-1 planets, all plan-etary field lines are open to the stellar wind so that thepolar cap covers the whole planet. Therefore, it is pos-sible that the energy transmission covers a significantarea of the planet. Additionally, we assume here thatthe angle between the stellar wind magnetic field andthe stellar wind velocity is 90 ◦ . At the Earth, the anglebetween the two is determined by the Parker Spiral andit is about 45 ◦ . In the case of the Trappist-1 planets, thetwo are more or less radial, but an angle of 5-30 ◦ betweenthe two vectors could appear due to the fast planetaryorbital motion (about 100 km s − ). A larger angle isalso possible due to the fact that the planets may residein the sub-Alfv´enic regime where the wind and magneticfield might not be fully coupled. Taking these two pointsinto account, we offer here an upper-limit for the stellarwind available energy and its transmission to the upperatmosphere, where even 10% of this energy is still veryhigh. We find that for the cases where the two impedancesare close in magnitude, the efficiency of the energy trans-mission from the wind to the ionosphere is high. Sincethe stellar wind energy flux is very large, the dissipatedenergy flux is also very large - 0.5-1% of the total stel-lar irradiance and 5-15 times higher than the EUV ir-radiance. We conclude that the upper atmospheres ofclose-orbit planets, such as the Trappist-1 planets, couldsuffer from an intense Ohmic heating sourced in the in-tense stellar wind input using constant values for theionospheric conductance. However, it is not trivial toestimate how Σ P changes with the intense EUV radi-ation in close-in planets. On one hand, the increasedionization should push the ionosphere down to regionswhere the electron density is higher. On the other hand,this will increase the collision frequencies. Therefore, amore detailed calculation of the integrated ionosphericconductance is needed using Ionosphere-Thermospheremodels (e.g., Ridley 2007; Deng et al. 2011; Bell et al.2014).Our estimates are also relevant to the problem of hot-jupiter inflation, which requires additional heating toexplain the observed inflation in this planet population.It has been suggested that Ohmic dissipation can bedriven by the strong zonal winds in tidally-locked plan-ets and the planetary magnetic field (e.g., Batygin &Stevenson 2010; Rauscher & Menou 2013). Rogers &Showman (2014); Rogers & Komacek (2014) have shownusing a full MHD model that such Ohmic dissipation ispossible, but it fragments and cannot support the nec-essary heating. Koskinen et al. (2014) have shown thatsufficient dissipation can only occur in the upper partsof the atmosphere, above the 10 mbar level. Our workhere demonstrates the potential of Ohmic dissipation inthe ionosphere, driven by the intense stellar wind, toprovide additional heating. Such a mechanism requiresinvestigation beyond the scope of this paper. CONCLUSIONSWe adopt a method to quantify the energy trans-fer and efficiency from the solar wind to the Earth’sionosphere to three of the close-orbit planets orbitingTrappist-1 in order to estimate the order of magni-tude of the Ohmic heating in the planets’ ionospheres.The method relates the conductances and impedancesof the stellar wind and the ionosphere to calculate theamount of energy transmitted into the ionosphere inthe form of Ohmic dissipation. We use an assumedset of ionospheric conductances and find that for valuesthat are less than 10 S , the dissipated energy can reach0 . − W/m , which can drive large continuous heatingin the upper atmospheres of exoplanets, and potentially nergy Deposition at the Upper Atmospheres of the Trappist-1 Planets Airapetian, V. S., Glocer, A., Khazanov, G. V., et al. 2017,ApJL, 836, L3, doi: 10.3847/2041-8213/836/1/L3Alberti, T., Carbone, V., Lepreti, F., & Vecchio, A. 2017,ApJ, 844, 19, doi: 10.3847/1538-4357/aa78a2Barr, A. C., Dobos, V., & Kiss, L. L. 2017, ArXiv e-prints.https://arxiv.org/abs/1712.05641Batygin, K., & Stevenson, D. J. 2010, ApJL, 714, L238,doi: 10.1088/2041-8205/714/2/L238Bell, J. M., Hunter Waite, J., Westlake, J. H., et al. 2014,Journal of Geophysical Research (Space Physics), 119,4957, doi: 10.1002/2014JA019781Bourrier, V., de Wit, J., Bolmont, E., et al. 2017, AJ, 154,121, doi: 10.3847/1538-3881/aa859cBurgasser, A. J., & Mamajek, E. E. 2017, ApJ, 845, 110,doi: 10.3847/1538-4357/aa7feaCohen, O., Ma, Y., Drake, J. J., et al. 2015, ApJ, 806, 41,doi: 10.1088/0004-637X/806/1/41Deng, Y., Fuller-Rowell, T. J., Akmaev, R. A., & Ridley,A. J. 2011, Journal of Geophysical Research (SpacePhysics), 116, A05313, doi: 10.1029/2010JA016019Dong, C., Huang, Z., Lingam, M., et al. 2017a, ApJL, 847,L4, doi: 10.3847/2041-8213/aa8a60Dong, C., Lingam, M., Ma, Y., & Cohen, O. 2017b, ApJL,837, L26, doi: 10.3847/2041-8213/aa6438Drell, S. D., Foley, H. M., & Ruderman, M. A. 1965,J. Geophys. Res., 70, 3131, doi: 10.1029/JZ070i013p03131Garcia-Sage, K., Glocer, A., Drake, J. J., Gronoff, G., &Cohen, O. 2017, ApJL, 844, L13,doi: 10.3847/2041-8213/aa7ecaGarraffo, C., Drake, J. J., & Cohen, O. 2016, ApJL, 833,L4, doi: 10.3847/2041-8205/833/1/L4Garraffo, C., Drake, J. J., Cohen, O., Alvarado-G´omez,J. D., & Moschou, S. P. 2017, ApJL, 843, L33,doi: 10.3847/2041-8213/aa79edGillon, M., Triaud, A. H. M. J., Demory, B.-O., et al. 2017,Nature, 542, 456, doi: 10.1038/nature21360Gombosi, T. I. 2004, Physics of the Space Environment, 357Kivelson, M. G., & Ridley, A. J. 2008, Journal ofGeophysical Research (Space Physics), 113, A05214,doi: 10.1029/2007JA012302 Kivelson, M. G., & Russell, C. T. 1995, Introduction toSpace Physics, 586Koskinen, T. T., Yelle, R. V., Lavvas, P., & Y-K. Cho, J.2014, ApJ, 796, 16, doi: 10.1088/0004-637X/796/1/16Lingam, M., & Loeb, A. 2017, Proceedings of the NationalAcademy of Science, 114, 6689,doi: 10.1073/pnas.1703517114Luger, R., Sestovic, M., Kruse, E., et al. 2017, NatureAstronomy, 1, 0129, doi: 10.1038/s41550-017-0129Neubauer, F. M. 1980, J. Geophys. Res., 85, 1171,doi: 10.1029/JA085iA03p01171—. 1998, J. Geophys. Res., 103, 19843,doi: 10.1029/97JE03370Ormel, C. W., Liu, B., & Schoonenberg, D. 2017, A&A,604, A1, doi: 10.1051/0004-6361/201730826Quarles, B., Quintana, E. V., Lopez, E., Schlieder, J. E., &Barclay, T. 2017, ApJL, 842, L5,doi: 10.3847/2041-8213/aa74bfRauscher, E., & Menou, K. 2013, ApJ, 764, 103,doi: 10.1088/0004-637X/764/1/103Ridley, A. J. 2007, Geophys. Res. Lett., 34, L05101,doi: 10.1029/2006GL028444Roettenbacher, R. M., & Kane, S. R. 2017, ApJ, 851, 77,doi: 10.3847/1538-4357/aa991eRogers, T. M., & Komacek, T. D. 2014, ApJ, 794, 132,doi: 10.1088/0004-637X/794/2/132Rogers, T. M., & Showman, A. P. 2014, ApJL, 782, L4,doi: 10.1088/2041-8205/782/1/L4Tamayo, D., Rein, H., Petrovich, C., & Murray, N. 2017,ApJL, 840, L19, doi: 10.3847/2041-8213/aa70eaTilley, M. A., Segura, A., Meadows, V. S., Hawley, S., &Davenport, J. 2017, ArXiv e-prints.https://arxiv.org/abs/1711.08484Wheatley, P. J., Louden, T., Bourrier, V., Ehrenreich, D.,& Gillon, M. 2017, MNRAS, 465, L74,doi: 10.1093/mnrasl/slw192Wolf, E. T. 2017, ApJL, 839, L1,doi: 10.3847/2041-8213/aa693a
Cohen et al.
Table 1.
Stellar Fluxes and Average Orbital Values for the Trappist-1 Planets
PlanetName Semi-majorAxis [ AU ] StellarCon-stant [ W m − ] EUVFlux [ W m − ] B sw [ nT ] v A [ km/s ] E sw [ V /m ] WindEnergyFlux[
W/m ] Q t (Σ P =1) [ W/m ] Q t (Σ P =10) [ W/m ]Trappist-1e 0.028 867 0.30 2641 842 3.15 50 8.71 3.16Trappist-1f 0.037 496 0.17 1480 636 1.85 30 4.00 1.75Trappist-1g 0.045 335 0.12 989 527 1.26 20 2.24 1.13 Figure 1.
Left: stellar wind speed (top), magnetic field (second), number density (third), and Alfv´enic Mach number (bottom)as a function of orbital phase of Trappist-1 e, f, and g (taken from Garraffo et al. 2017). Right: Alfv´enic velocity (top),conductance (middle), and impedance (bottom) along the orbits of the three planets.
Figure 2.