The Effect of Multiple Heat Sources on Exomoon Habitable Zones

Home , Tidal heating

arXiv:1703.02447v1 [astro-ph.EP] 7 Mar 2017 aial Wlim ta.19;Hle 04 amre al. et Lammer 2014; Heller current 1997; potentially 2014 with be 2014). al. et Heller would (Williams detectable they If 2009; habitable and be 2015) al. 2015a,b). et Angerhausen (Kipping could and Pudritz Hippke and technology moons Heller near-future these 2014; or po al. exist, formation it et (Heller a they today, from around view of stars, exist Sun-like of as could at exomoons discovered planets Mars-mass super-Jovian been that shown has been exomoon has no Although Introduction 1. oe ohv ena bu . Udrn h nlsae fthe reser of water stages large final these the Hence, during 1981). 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As Endre for Csatkai Centre Research Institute, Geophysical and Geodetic ne,H12 okl hg ilsú 51,Bdps,Hung Budapest, 15-17, út Miklós Thege Cen Konkoly Research H–1121 Institute, ences, Astronomical Miklós Thege Konkoly ff cso ia etn nteptnilo as oEarth-size to Mars- of potential the on heating tidal of ects & Astrophysics srbooy–mtos ueia lnt n satellites and planets – numerical methods: – astrobiology .Dbs e-mail: Dobos, V. : / cetd1Jn 2015 June 1 Accepted aucitn.tidal no. manuscript eaDobos Vera [email protected] ff rn tla itne nseicstar-planet-satellite specific in distances stellar erent + 1 , illum_aa_corr2 2 eéHeller René , ffi in nor cient n efidta h temsa e ‘thermostat the that find We on. rubtosfo h star. the from erturbations voirs olsi oe.I diint ia etn,w osdrs consider we heating, tidal to addition In model. coelastic e .0 n .1i h oiateeg orebyn – AU 3–5 beyond source energy dominant the is 0.01 and 0.001 een a aial oe hr tla aito speaet W prevalent. is radiation stellar where zone, habitable lar ABSTRACT ee,tee the wever, ff y 84,4IyLn,Pyo al rneo,N,USA NJ, Princeton, Hall, Peyton Lane, Ivy 4 08544, ty, atrzto feoon osbei h erftr.Here future. near the in possible exomoons of racterization rnm n at cecs ugra cdm fSciences, of Academy Hungarian Sciences, Earth and tronomy int raudn ufc aiasbyn h oa ytm Sever system. solar the beyond habitats surface abundant er Lei-e ,307Gtign Germany Göttingen, 37077 3, -Liebig-Weg p- in l- ; t mcldt ult fucmn pc isosadground-b and missions space upcoming of quality data omical r o srnm n at cecs ugra cdm fS of Academy Hungarian Sciences, Earth and Astronomy for tre vre(P) nvriyo oy,2788,515Kashiwa 5-1-5 227-8583, Tokyo, of University (WPI), iverse btterhs tr no erteselrhbtbezones, habitable stellar the near or in stars host their rbit aeltst otlqi ufc ae,adw opr the compare we and water, surface liquid host to satellites d 3 ary atrwudb airt eet(fte xs) u hi o their but exist), they (if detect to easier be would latter e aeilo h on( moon the t of a material with ( parameterization eq factor through from habitabil- dissipation e.g. calculated surface models, usually longterm tide is to exomoons librium of key heating be Tidal would ity. exomoon large a host their from AU several at stars. 2016) Marley He and moo 2014; Sengupta of Albrecht transits heat- 2016; and the (Heller tidal planets observe giant to extreme luminous or around with 2013) Turner exomoons and image (Peters ing to possible be ever, separati typically orbital star, small relatively the implies transi This transits conventional star. multiple their wi after the exomoons certainty detect with statistical only significant can achieved surveys transit be because method, cannot stars their exomoons large of surfaces the separations. on stellar life wide for available be could a edsrbdb cntn-ielg CL oe,wher perturber model, (CTL) planetary ”constant-time-lag“ a its by described and be can equilibrium moon Alternatively, mo 2013). the the Makarov of and connecting bulge (Efroimsky tidal line the between the phases and tidal the co calle of a lag is assume stant they models because equilibrium models tidal (CPL) the “constant-phase-lag” of family This stants. n di .Turner L. Edwin and , eea lnt n aelts interiors satellites: and planets – general : ff tti ieadsac rmishs tr ia etn in heating tidal star, host its from distance a wide this At from AU several at orbiting moons of observations However, cso osbemo topeeaengetd emap We neglected. are atmosphere moon possible a of ects < ff Q c’o h iceatcmdli significant is model viscoelastic the of ect’ n ihauiomrgdt fterocky the of rigidity uniform a with and ) U(zb ta.20) tmgt how- might, It 2006). al. et (Szabó AU 1 µ 4 ,bt fwihaecniee con- considered are which of both ), ofiuain,addtriethose determine and configurations, , 5 ril ubr ae1o 8 of 1 page number, Article elrradiation, tellar lofidthat find also e eexplore we lsearches al

H–9400 c S 2017 ESO thas it noha, nfotof front in rbital from ased tidal n from ons ci- es the d tides idal ller it e ui- on ns n- th at t A&A proofs: manuscript no. tidal+illum_aa_corr2 is assumed that the tidal bulge of the moon lags the line between using a polynomial fit to the data given by Fortney et al. (2007, the two centers of mass by a constant time. Table 4, line 17). The albedos of both the planet and the moon In reality, however, all these parameters depend on tempera- (in the optical and also in the infrared) were set to 0.3, that is, to ture. Another problem with equilibrium tide models is that the Earth-like values. values of the parameters (e.g. of Q) are difficult to calculate We estimate the runaway-greenhouse limit of the globally (Remus et al. 2012) or constrain observationally even for solar averaged energy flux (FRG), which defines the circumplanetary system bodies: Q can vary several orders of magnitude for dif- habitable edge interior to which a moon becomes uninhabit- ferent bodies: from ≈ 10 for rocky planets to > 1000 for giant able, using a semi-analytic model of Pierrehumbert (2010) as planets (see e.g. Goldreich and Soter 1966). described in Heller and Barnes (2013). In this model, the outgo- For these reasons, Dobos and Turner (2015) applied a vis- ing radiation on top of a water-rich atmosphere is calculated us- coelastic model previously developed by Henning et al. (2009) ing an approximation for the wavelength-dependent absorption to predict the tidal heating in hypothetical exomoons. Viscoelas- spectrum of water. The runaway greenhouse limit then depends tic models are more realistic than tidal equilibrium models, as exclusively on the moon’s surface gravity (i.e. its mass and ra- they consider the temperature feedback between the tidal heat- dius) and on the fact that there is enough water to saturate the ing and the viscosity of the material. Due to the phase transi- atmosphere with steam. Our test moons are assumed to be rocky tion at the melting of rocky material, viscoelastic models result bodies with masses between the mass of Mars (0.1 Earth masses, in more moderate temperatures than do tidal equilibrium mod- M⊕)and 1 M⊕. Our test host planets are gas giants around either els, and so they predict a wider circumplanetary habitable zone a sun-like star or an M dwarf star. (Forgan and Dobos 2016). We apply the model of Kopparapuet al. (2014) to calcu- In this work, we compare the effects of two kinds of tidal late the borders of the circumstellar HZ, which are different for heating models, a CTL model and a viscoelastic model, on the the 0.1 and the 1 Earth-mass moons. The corresponding energy habitable edge for moons within the circumstellar HZ. We also fluxes are then used to define and evaluate the habitability of our calculate the contribution of viscoelastic model to the total en- test moons, both inside and beyond the stellar HZ. ergy flux of hypothetical exomoons with an emphasis on moons far beyond the stellar habitable zone. 2.2. Tidal heating models Tidal heat flux is calculated using two different models: a vis- 2. Method coelastic one with temperature-dependent tidal Q and a CTL 2.1. Energy flux at the moon’s top of the atmosphere one, the latter of which converges to a CPL model for small eccentricities as the time-lag of the principal tide becomes 1/Q We neglect any greenhouse or cloud feedbacks by a possible (Heller et al. 2011). For the CTL model we use the same frame- moon atmosphere and rather focus on the global energy budget. work and parameterization as in the work of Heller and Barnes The following energy sources are included in our model: stellar (2013), which goes back to Leconte et al. (2010) and Hut (1981). irradiation, planetary reflectance, thermal radiation of the planet, In particular, we use the following parameters: k2 = 0.3 (as in and tidal heating. The globally averaged energyflux on the moon Henning et al. 2009; Heller and Barnes 2013), and tidal time lag, is estimated as per (Heller et al. 2014, Eq. 4) τ = 638s, which was measured for the Earth (Lambeck 1977; Neron de Surgy and Laskar 1997). In this model, the tidal heating function is linear in both − 2 τ ∼ 1/(nQ) and k2. Hence, changes in these parameters usu- glob L∗ 1 αs,opt xs πRpαp Lp 1 − αs,IR Fs = + + +hs +Ws , ally do not have dramatic effects on the tidal heating. In contrast,  4 f 2a2  2 2 4πa2 1 − e2 s s 4πas fs 1 − es changes in the radius or mass of the tidally distorted object, or p p   q   p in the orbital eccentricity or semi-major axis result in significant (1) changes of the tidal heating rates. For the viscoelastic tidal heating calculations, we use the where L and L are the stellar and planetary luminosities, re- ∗ p model described by Dobos and Turner (2015), which was origi- spectively, α is the Bond albedo of the planet, α and α p s,opt s,IR nally developed by Henning et al. (2009). Tidal heat flux is cal- are the optical and infrared albedos of the moon, a and a are p s culated from the semi-major axes of the planet’s orbit around the star and of 1 the moon’s orbit around the planet , ep and es are the eccentric- 5 5 2 ities of the planet’s and the satellite’s orbit, Rp is the radius of 21 R n e h = − Im(k ) s s s , (2) the planet, xs is the fraction of the satellite’s orbit that is not s,visc 2 2 G spent in the shadow of the host planet, and fs describes the ef- ficiency of the flux distribution on the surface of the satellite. where Im(k2) is the complex Love number, which describes the The first and second terms of this equation account for illumi- structure and rheology in the satellite (Segatz et al. 1988). In the Maxwell model, the value of Im(k ) is given by (Henning et al. nation effects from the star and the planet, whereas hs indicates 2 2009) the tidal heat flux through the satellite’s surface. Ws denotes ar- bitrary additional energy sources such as residual internal heat from the moon’s accretion or heat from radiogenic decays, but 57ηω we use W = 0. We also neglect eclipses, thus x = 1, and we −Im(k2) = , (3) s s 19µ 2 η2ω2 assume that the satellite is not tidally locked to its host planet, + + 4ρgRm 1 1 2 hence f = 4. The planetary radius is calculated from the mass,  2ρgRm ! µ  s     1 We assume that the moon’s mass is negligible compared to the plan- where η is the viscosity, ω is the orbital frequency, and µ etary mass and that the barycenter of the planet-moon system is in the is the shear modulus of the satellite. The temperature depen- center of the planet. dency of the viscosity and the shear modulus is described by

Article number, page 2 of 8 Vera Dobos et al.: The Eﬀect of Multiple Heat Sources on Exomoon Habitable Zones

Fischer and Spohn (1990) and Moore (2003); and the adapted 2.3. Simulation setup values and equations are listed by Dobos and Turner (2015). Since only rocky bodies like the Earth are considered as satel- We consider various reference systems of a star, a planet, anda lites in this work, the solidus and liquidus temperatures, at which moon. the material of the rocky body starts melting and becomes com- pletely liquid, were chosen to be 1600 K and 2000 K, respec- 2.3.1. Comparison of equilibrium and viscoelastic tides tively. We assume that disaggregation occurs at 50% melt fraction, which implies a breakdown temperature of 1800 K. First, we examine the different effects of tidal heating in either The viscoelastic tidal heating model also describes the con- a CTL or in a viscoelastic tidal model on the location of the vective cooling of the body. The iterative method described by circumplanetary habitable edge (see Section 3.1). We consider a Henning et al. (2009) was used for our calculations of the con- star with a sun-like radius (R⋆ = R⊙) andeffective temperature vective heat loss: (Teff = 5778K), a 5 Jupiter-mass gas giant at 1AU orbital distance with an eccentricity of 0.1, and a 0.5 M⊕ moon. The orbital period and eccentricity of the moon are varied between 1 Tmantle − Tsurf and 20 days and between 0.01 and 1, respectively. q = k , (4) BL therm δ(T) 2.3.2. Viscoelastic tides beyond the stellar habitable zone where ktherm is the thermal conductivity (∼ 2W/mK), Tmantle and Tsurf are the temperatures in the mantle and on the surface, re- Second, we map the circumplanetary habitable zone over a wide spectively, and δ(T) is the thickness of the conductive layer. We range of circumstellar orbits (see Section 3.2) using four differ- use δ(T) = 30km as a first approximation, and then for the iter- ent star-planet-moon configurations. The star is either sun-like ation (see Section 2.3.1) or an M3 class main sequence star (M∗ = 0.36 M⊙, R∗ = 0.39 R⊙, Teff = 3250K and L∗ = 0.0152 L⊙, Kaltenegger and Traub 2009, Table 1). The planet is a Jupiter- d Ra −1/4 δ(T) = (5) mass gas giant, and the satellite’s mass is either 0.1 M⊕ or 1 M⊕ 2a2 Rac ! (i.e. a Mars or Earth analogue). The density of the moon is that of the Earth. The stellar luminosity and the temperature values is used, where d is the mantle thickness (∼ 3000 km), a2 is the are used for our calculations of the HZ boundaries, and the stel- flow geometry constant (∼ 1), Rac is the critical Rayleigh num- lar radius and temperature values are required for the incident ber (∼ 1100) and Ra is the Rayleigh number which can be ex- stellar flux calculation. pressed as

3. Results 4 α g ρ d qBL Ra = . (6) 3.1. The habitable edge η(T) κ ktherm We calculated the total energy flux (using Eq.1) at the top of −4 with α (∼ 10 ) as the thermal expansivity, κ = ktherm/(ρ Cp) as the moon’s atmosphere as a function of distance to the planet in the thermal diffusivity, and Cp = 1260 J/(kgK). The iteration both the CTL (Fig. 1) and the viscoelastic (Fig. 2) frameworks. of the convective heat flux ends when the difference of the last In both Figs. 1 and 2, colours show the amount of the total flux two values becomes smaller than 10−10W/m2. Once the stable received by the moon, and the white contours at 288W/m2 indi- equilibrium temperature is found, we compute the tidal heat flux. cate the habitable edge defined by the runaway-greenhouselimit This viscoelastic model was already used together with a cli- (Heller and Barnes 2013). Interestingly, in the CTL model the mate model by Forgan and Dobos (2016) with the aim of deter- habitable edge is located closer in to the planet than in the vis- mining the location and width of the circumplanetary habitable coelastic model. zone for exomoons. The 1D latitudinal climate model included Black contour curves show the tidal heat flux alone at eclipses, the carbonate-silicate cycle and the ice-albedo feedback 288, 100 and 2W/m2, the latter is the global mean heat flow of the moon (in addition to tidal heating, stellar and planetary ra- from tides on Io as measured with the Galileo spacecraft diation). The ice-albedo positive loop in the climate model along (Spenceret al. 2000). This curve in Fig. 2 has a different slope with eclipses result in a relatively close-in outer limit for circum- than the one labeled ‘tidal heat at 100W/m2’. The different planetary habitability, if the orbit of the satellite is not inclined. slope is caused by the changing equations in the viscoelastic The climate model, however, can only be used for bodies of sim- model. The tidal flux (and also the convective cooling flux) is ilar sizes to the Earth. In this work we investigate the effect of described by different formulae below or above certain temper- the viscoelastic model on smaller exomoons,as well, and for this atures (namely the solidus, the breakdown and the liquidus tem- reason, instead of a climate model, we apply an orbit-averaged peratures). In other words, if the tidal heating is low, different illumination model. Beside solar-like host stars, we made calcu- equations will be used, than in the case of higher tidal forces. lations also for M dwarfs. Figs. 1 and 2 show substantial differences. The viscoelastic Both the CPL and the viscoelastic models are valid only for model predicts moderate (≤ 2 W/m2) heating beyond 5.3 days small orbital eccentricities, that is, for e.0.1. For larger eccen- of an orbital period for e = 0.01, while the CTL model needs the tricities, the instantaneous tidal heating in the deformed body moontobeascloseas3.5 days to generate the same tidal heating can differ strongly from the orbit-averaged tidal heating rate and with the same orbital eccentricity. In other words, the viscoelas- the frequency spectrum of the decomposed tidal potential could tic model predicts significant tidal heating in wider orbits. At involve a wide range of frequecies (Greenberg 2009). Both as- close orbits, however, the CTL model products extremely high pects go beyond the approximations involved in the models, and fluxes. With a 1.6 day orbital period (similar to Io, see arrows at hence our results for e&0.1 need to be taken with a grain of salt. the top), our test moon with e = 0.01 would generate a tidal heat

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M M M M 2 2 s = 0.5 ⊕ , p = 5 Jup (CTL model) (tidal heat ≤ 100W/m ), while above 100W/m the viscoelas- 1.00 [W/m2] tic model produces lower heating rates, where the CTL model IGC runs away. As a consequence, in this specific example of a 0.5 1600 Earth-mass moon around a 5 Jupiter-mass planet, the habitable edge (red contour curve in the figures) is located at a larger dis- 1400 tance from the planet than for the CTL model. It is caused by the 2 1200 stronger tidal heating rate below 100W/m . Stellar and planetary illuminations can also have important 1000 0.10 2 2 effects. They can break the feedback loop between tidal heat-

2 800 ing and convective cooling: in an extreme case the system will 600 not even ﬁnd a stable convective heat transport rate. However,

2 = Runaway Greenhouse Limit the most relevant cases of tidally heated exomoons will involve

moon’s orbital eccentricity 400 tidal heattidal at 288heat W/mat 100 W/m systems orbiting so far from the star that illumination heating is 200 irrelevant and around planets so old that planetary illumination 288 W/m tidal heat at 2 W/m 0.01 0 is also small. In any cases, such systems will be the ones that can 1 10 be most easily understood, if they exist. orbital period [days]

Fig. 1. Energy flux at the top of the atmosphere of a 0.5 Earth-mass 3.2. Circumplanetary tidal habitable zone moon orbiting a 5 Jupiter-mass planet at 1 AU distance from a Sun-like star; tidal heating is calculated with a CTL model. The planetary or- In addition to the inner habitable edge, we also want to locate the bit has 0.1 eccentricity. The stellar radiation, the planetary reflectance, outer boundary of the circumplanetary habitable zone for dif- thermal radiation of the planet and tidal heating were considered as en- ferent satellite sizes and stellar classes. Figs. 3 and 4 show the ergy sources. White contour curve indicates the runaway greenhouse circumplanetary habitable zones for our Jupiter-like test planet limit considering all energy sources, while the black curves show the with a moon around either a sun-like or an M dwarf star, respec- tidal heating flux alone. The arrows at the top of the figure indicate the tively. The left and right panels of the figures show the cases of orbital periods of Io (I), Ganymede (G) and Callisto (C). Note that the 0.1 and 1 Earth-mass satellites, respectively. The eccentricity of CTL model is valid only for small orbital eccentricities. Heating con- the moon’sorbitis 0.001in the top panels and 0.01 in the bottom tours in the upper half of the panel (above 0.1 along the ordinate) could panels. be off by orders of magnitude in real cases. Grey horizontal lines in each panel indicate zero tidal heat

M M M M ﬂux. Above these lines, only stellar radiation is relevant since

the reflected and thermal radiation from the planet are also negligibly small. Green areas illustrate habitable surface conditions with tidal heating rates below 100Wm−2, whereas diagonally striped orange areas visualize exomoon habitability with tidal heating rates above 100Wm−2 (color coding adopted from Heller and Armstrong 2014). As expected, for smaller satellite masses the HZ is located closer to the planet. At zero tidal flux, small plateaus are present which are consequences of the viscoelastic tidal heating model. At the horizontal line tidal forces are turned on, hence there is a change in the position of the green area. At the plateau the tidal heat flux elevates from zero to about 10W/m2 or more. The sudden change is caused by the fact that the tidal heating model gives result only if the tidal heat flux and the convective cooling flux has stable equilibrium. It means that if tidal forces are weak, thenthe convectivecoolingwill be weak too, or not present in the body at all, hence there is no equilibrium. In other words, tidal Fig. 2. Energy flux at the top of the atmosphere of a 0.5 Earth-mass heating is insufficient to drive convection in the body. There will moon orbiting a 5 Jupiter-mass planet at 1 AU distance from a Sun- still be some heating, but a different heat transport mechanism like star; tidal heating is calculated with a viscoelastic model. The same (probably conduction) would be in play. A more accurate model colours, contours and signatures were applied as in Fig. 1. Note that the viscoelastic model is valid only for small orbital eccentricities. Heating would consider all heat transport mechanisms at all temperatures contours in the upper half of the panel (above 0.1 along the ordinate) and would probably not exhibit such discontinuities. could be off by orders of magnitude in real cases. In the inspected cases the two dominating energy sources are the stellar radiation and the tidal heating. Since these two effects are physically independent, one would not expect them flux that is sufficient to trigger a runaway greenhouse effect. In to be of comparable importance except in a small fraction of this case, stellar illumination is not even required to make such a cases. In general, however, one effect will dominate the other. moon uninhabitable. In contrast, with the viscoelastic model the To explore this interplay between stellar illumination and tidal moon would need to have a 1 day orbital period to trigger the heating in more detail, we calculated the distance from the star same effect. at which tidal heating (for a given hypothetical moon and or- A comparison of these two plots shows the ‘thermostat ef- bit) equals stellar heating. These critical distances are indicated fect’ of the viscoelastic model described by Dobos and Turner with blue dashed curves in Figs. 3 and 4. We find that beyond (2015). The CTL model yields lower tidal heating rates than 3-5AU around a G2 star and beyond 0.4-0.6AU around an M3 the viscoelastic model in the weak-to-moderate heating regime star, tidal heating for moons with eccentricities between 0.001

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e =0.001, Mm =0.1MEarth e =0.001, Mm =1.0MEarth ]

J 20 20 R

10 10 9 9 8 8 2 stellar radiation dominates 7 7 Ftidal = 0 W/m 6 stellar radiation dominates 6 F = 0 W/m2

] 5 tidal 5 tidal heating dominates J

R 4 tidal heating dominates 4 F >100 W/m2 3 3 tidal F >100 W/m2 tidal stellar HZ stellar HZ 2 2

semi-major axis of the moon [ themoon of axis semi-major 0.9 1 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.9 1 2.0 3.0 4.0 5.0 6.0 7.0 8.0

e =0.01, Mm =0.1MEarth e =0.01, Mm =1.0MEarth 20 20 F = 0 W/m2 tidal stellar radiation dominates stellar radiation dominates F = 0 W/m2 10 tidal 10 tidal heating dominates 9 9 8 tidal heating dominates 8 7 7 semi-major axis of the moon [ themoon of axis semi-major 6 6 2 Ftidal >100 W/m 5 2 5 Ftidal >100 W/m 4 4

3 3 stellar HZ stellar HZ

2 2 0.9 1 2.0 3.0 4.0 5.0 6.0 7.0 8.0 0.9 1 2.0 3.0 4.0 5.0 6.0 7.0 8.0 semi-major axis of the planetsemi-major [AU] axis of the planet [AU]

Fig. 3. Tidal heating habitable zone for 0.1 and 1 Earth-mass exomoons (left and right panels, respectively) around a Jupiter-mass planet hosted by a G2 star as functions of the semi-major axes of the planet and the moon. The red curve indicates the runaway greenhouse limit (habitable edge). The green and the diagonally striped orange areas cover the habitable region, where the striped orange colour indicates that the tidal heating flux is larger than 100 W/m2. The orbital eccentricity of the moon is 0.001 in the top panels, and 0.01 in the bottom panels. Vertical dashed lines indicate the inner and outer boundaries of the circumstellar habitable zone. At the dashed blue contour curve the tidal heating flux equals to the stellar radiation flux, hence it separates the tidal heating dominated and the stellar radiation dominated regime. At the grey horizontal line the tidal flux is zero. and 0.01 will be the dominant source of energy rather than stel- line is shown as a shaded region in grey and labeled as ‘Hill lar radiation (see the overlapping of the grey and dashed blue unstable’. As a consequenceof the closeness of this Hill unstable curves). This is the case for both satellite masses investigated. limit, the circumplanetary volume for habitable satellites ismuch For smaller stellar distances, the dominant heat source depends smaller in systems of M dwarfs than in systems with sun-like on the circumplanetary distance of the moon. stars, in particular if the moons have substantial eccentricities. Based on the fact that Io is a globalvolcano world, one might Note how the circumplanetary space between the red (runaway take the 2 W/m2 limit as a conservative limit for Earth-like sur- greenhouse) curve and the Hill unstable region becomes smaller face habitability. But other worlds might still be habitable even as the moon’s eccentricity or its mass increases. in a state of extreme tidal heating near 100W/m2. In fact, the surface habitability of rocky, water-rich planets or moons has not been studied in this regime of extreme tidal heating to our 4. Discussion 2 knowledge. Nevertheless, although 100 W/m seems a lot of in- From the results shown in Section 3, the following general find- ternal heating compared to the internal heat flux on Earth, which ings can be concluded. 2 is less than 0.1 W/m (Zahnle et al. 2007), it could still allow the The circumplanetary habitable edge calculated with the vis- presence of liquid surface water as long as the total energyflux is coelastic model (Fig. 2) is located at a larger distance than in the below the runaway greenhouse limit. The striped orange colour CTL model (Fig. 1). If the outer boundary of the circumplane- indicates that the body might become volcanic or tectonically tary habitable zone is defined by Hill stability, than it means that active in this regime. the viscoelastic model reduces the habitable environment. How- It is supposed that the gravitational pull of an M dwarf can ever, Forgan and Yotov (2014) showed that the moon can enter force an exomoon whose orbit is within the stellar HZ into a into a snowball phase, if the ice-albedo feedback and eclipses very eccentric circumplanetary orbit (Heller 2012). The resulting are also taken into account. It means that the outer boundary will tidal heating might ultimately prevent such moons from being be significantly closer to the planet, than in the case when it is habitable. Since the planetary Hill sphere in the stellar HZ of defined by Hill stability. Since the viscoelastic model resulted in M dwarfs is not so large, any moon needs to be in a tight orbit. higher tidal fluxes below 100W/m2, than the CTL model, we Prograde (regular) moons are only stable out to about 0.5 Hill expect that the outer boundary defined by the snowball state will radius (Domingoset al. 2006). In Fig. 4, the area beyond this be farther from the planet. Altogether, the circumplanetary HZ is

Article number, page 5 of 8 A&A proofs: manuscript no. tidal+illum_aa_corr2

e =0.001, Mm =0.1MEarth e =0.001, Mm =1.0MEarth ]

J 20 20 R

Hill unstable Hill unstable 10 10 9 9 8 8 2 stellar radiation dominates 7 7 Ftidal = 0 W/m 6 6 F = 0 W/m2 stellar radiation dominates

] 5 tidal 5

J tidal heating dominates

R 4 tidal heating dominates 4 2 Ftidal >100 W/m 3 2 3 Ftidal >100 W/m stellar HZ stellar HZ 2 2

semi-major axis of the moon [ themoon of axis semi-major 0.1 1.0 0.1 1.0

e =0.01, Mm =0.1MEarth e =0.01, Mm =1.0MEarth 20 20

2 Ftidal = 0 W/m

Hill unstable Hill unstable stellar radiation dominates F = 0 W/m2 stellar radiation dominates 10 tidal 10 9 9 8 tidal heating dominates 8 tidal heating dominates 7 7 semi-major axis of the moon [ themoon of axis semi-major 6 6 2 Ftidal >100 W/m 5 2 5 Ftidal >100 W/m 4 4

3 3

stellar HZ stellar HZ 2 2 0.1 1.0 0.1 1.0 semi-major axis of the planetsemi-major [AU] axis of the planet [AU]

Fig. 4. Tidal heating habitable zone for 0.1 and 1 Earth-mass exomoons (left and right panels, respectively) around a Jupiter-mass planet hosted by an M3 dwarf as functions of the semi-major axes of the planet and the moon. The orbital eccentricity of the moon is 0.001 in the top panels, and 0.01 in the bottom panels. The same colours and contours were applied as in Fig. 3. The grey area in the upper left corner covers those cases where the moon’s orbit is considered unstable. not likely to be thinner in the viscoelastic case, but it is located than the circumplanetary disk, as in the case of the Moon. In in a larger distance from the planet. such cases, larger mass ratios are reasonable, too. Tidal heating in close-in orbits is more moderate with the We have considered a large range of orbital eccentrici- viscoelastic model and does not generate extremely high rates as ties for the moon with values up to 1. Beyond the fact that predicted by the CTL model. our models are physically plausible only for moderate val- Moons that are primarily heated by tides, that is, moons be- ues (≤ 0.1), tidal circularization will act to decrease eccen- yond the stellar habitable zone, have a wider circumplanetary tricities to zero on time scales that are typically much shorter range of orbits for habitability with the viscoelastic model than than 1 Gyr (Porter and Grundy 2011; Heller and Barnes 2013). with the CTL model. This is in agreement with the findings Hence, even moderate eccentricities can only be expected in of Forgan and Dobos (2016). In Heller and Armstrong (2014, real exomoon systems if the star has a significant effect on Fig. 2), it was shown that the circumplanetaryhabitable zone cal- the moon’s orbit (Heller 2012), if other planets can act as or- culated with the CTL model thins out dramatically as the planet- bital perturbers (Gong et al. 2013; Payne et al. 2013; Hong et al. moon binary is virtually shifted away from their common host 2015), if other massive moons are present around the same star. This is due to the very strong dependence of tidal heating in planet, or if the planet-moon system has migrated through or- the CTL model on the moon’s semi-major axis. bital resonances with the circumstellar orbit (Namouni 2010; In the calculations we considered 0.1, 0.5 and 1 Earth-mass Spalding et al. 2016). moons. Lammer et al. (2014) found that about 0.25 Earth masses Hot spots (hot surface areas generated by geothermical heat, (or 2.5 Mars masses) are required for a moon to hold an atmo- that could generate volcanoes) might be important on both ob- sphere during the first 100 Myrs of high stellar XUV activity, servational and astrobiological grounds. Regarding observations, assuming a moon in the habitable zone around a Sun-like star. hot spots produce variability in the brightness and spectral en- Considering this constraint, a 0.1 Earth-mass moon may not be ergy distribution of thermal emission from the moon. As Io il- habitable in the stellar HZ because of atmospheric loss, how- lustrates, hot spots can potentially be big and prominently hot ever, at larger stellar distances the stellar wind and strong stellar (see Spencer et al. 2000, Fig. 4). This could let one determine activity do not present such severe danger to habitability. the moon’s spin period and may indicate a geologically ‘active’ From a formation point of view, smaller moons are more body. Hot spots also allow liquid water to locally exist and per- probable to form around super-Jovian planets than Earth-mass sist over long periods of time even if the mean surface temper- satellites, according to Canup and Ward (2006), who gave an up- ature is far below the freezing point. Enceladus may provide an per limit of 10−4 to the mass ratio of the satellite system and the example for such phenomenon in the Solar System, where tidal planet. However, moons can also originate from collisions rather heating maintains a liquid (and probably global) ocean below

Article number, page 6 of 8 Vera Dobos et al.: The Effect of Multiple Heat Sources on Exomoon Habitable Zones the ice cover, and contributes to the eruption of plumes at the axes of the moon defines whether stellar radiation or tidal heat- southern region of the satellite (Thomas et al. 2016). ing dominates. On the other hand, hot spots can be effective in conducting From an observational point of view, M dwarfs are better the internal heat. The calculated fluxes in this paper are average candidates to detect habitable exomoons, since both the stellar surface fluxes, but in reality hot spots are probableto form. These and the circumplanetary tidal HZs and closer to the star, mean- areas are much higher in temperature, meaning that other areas ing that the orbital period of the planet is shorter, hence more on the surface must be somewhat colder than the average. As a transits can be observed. However, in the stellar HZ, the possible result, the temperature of the surface (excluding the vicinity of habitable orbits for exomoons are constrained by Hill stability. the hot spots) can be lower than that is calculated. At larger stellar distances the circumplanetary tidal HZ and the Two end-member models exist for spatial distribution of tidal Hill unstableregiondo not overlap,so there is no such constraint. heat dissipation. In model A the dissipation mostly occurs in the deep-mantleof the body,and in model B it occursin the astheno- Acknowledgements sphere, which is a thin layer in the upper mantle (Segatz et al. 1988). In model A the surface heat flux is higher at the polar re- We thank Duncan Forgan for a very helpful referee report. VD gions, while in model B the flux is mostly distributed between thanks László L. Kiss for the helpful discussion. This work was −45 ◦ and +45 ◦ latitudes. From volcano measurements of Io it supported in part by the German space agency (Deutsches Zen- seems that model B is more consistent with the location of hot trum für Luft- und Raumfahrt) under PLATO grant 50OO1501. spots (Hamilton et al. 2011, 2013; Rathbun and Spencer 2015). This research has been supported in part by the World Pre- The global distribution of volcanoes on the surface is random mier International Research Center Initiative, MEXT, Japan. VD (Poisson distribution), but closer to the equator they are more has been supported by the Hungarian OTKA Grants K104607, widely spaced (uniform distribution, Hamilton et al. 2013). K119993, and the Hungarian National Research, Development On Io the total power output of volcanoes and paterae is and Innovation Office (NKFIH) grant K-115709. about 5 · 1013 W (Veeder et al. 2012a,b). 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