arXiv:1401.0720v1 [astro-ph.EP] 3 Jan 2014 04 obdlie l 2012). al. al. entirely et et be Morbidelli (Raymond expec- waterworlds should 2004; generic so-called planets the water, terrestrial to in covered leads many rock that to tation relative density low lcs u hshptei osntepanteln-ems reservoir. long-term water the steam bui surface explain a Earth’s not Earth’s does of of of hypothesis content buffering this water But the blocks. of by independent early-on atmosphere, set was drosphere iuu w-a xhneo ae ewe ca and ocean between water of of con- the exchange growth and two-way the 2009), tinuous given (Harrison time remarkable over is 1974; continental This (Wise Ga remained 1999). 2.5 continen- has Eriksson least at of —freeboard— since level height constant sedimentary Gyrs sea approximately average above of 4.5 the surfaces stratigraphy tal that roughly while suggests for 2001), deposits continents al. et exposed (Wilde and 2011). Switzer & so Abbot basins 2009; their area, ocean al. while shallow et surface oceans, produces (Kite its deeper gravity have than surface to increased faster waterworlds: ought scales be planets bigger to mass likely planet’s more a are “super-Earths,” ets, A E YLN EWE CA N ATE UE-ATSNE N NEED SUPER-EARTHS : AND OCEAN BETWEEN CYCLING WATER hcg,53 ot li vne hcg,I,667 USA 60637, IL, Chicago, Avenue, Ellis South 5734 Chicago, USA 60208, IL, Universit Evanston, [email protected] Northwestern Road, Astronomy, Sheridan & 2145 Physics Scienc Planetary of & Department Earth of Department (CIERA), trophysics 3 h tcatcdlvr fwtrcmie ihits with combined water of delivery stochastic The ertlzrosidct htErhhshdboth had has Earth that indicate zircons Detrital rpittpstuigL using 2014 typeset 6, Preprint January version Draft asi&Ae(96 rpsdta h aso at’ hy- Earth’s of mass the that proposed (1986) Abe & Matsui 2 1 eateto h epyia cecs nvriyof University Sciences, Geophysical the of Department etrfrItricpiayEpoainadRsac nA in Research and Exploration Interdisciplinary for Center ics o uuemsin a etorhptei ympigthe headings: mapping Subject by pla hypothesis robust our more test planets. the can uncer value, terrestrial missions of its future source greater how greatest the discuss The inventory: water habitable. mantle are super-Earths that ntemnl.W ocueta etnclyatv ersra plan terrestrial active tectonically a that conclude wa We deep Critically a mantle. with planets. the super-Earths terrestrial so in gravity, on Motivate th surface partitioning of to water model proportional volume. two-box the a ocean to develop we long-term solutions o cycle, on water positively deep feedback and steady-state negatively stabilizing Degass depend a timescales. crust providing geological oceanic on of reservo reservoirs serpentinization interior an these and pla between wor A ocean, water other the planets. reservoir, In Earth-like surface their for continents. a w than between so exposed narrower much so-called with thermostat is planets waterworlds water, weathering for of in silicate that than covered a stable entirely lack waterworlds be because should super-Earths most xoe otnnsi t ae asfato sls than less is fraction mass water its if continents exposed ag ersra lnt r xetdt aemtdtopograp muted have to expected are planets terrestrial Large 1. EDAKO LUCK? OR FEEDBACK A T E tl mltajv 8/13/10 v. emulateapj style X lnt n aelts opsto lnt n aelts interiors satellites: and planets — composition satellites: and planets n aelts cas—paesadstlie:pyia vlto p — evolution tectonics physical satellites: and satellites: planets and — planets surfaces — satellites: oceans satellites: and 3 asv ersra plan- terrestrial Massive ioa .Cowan B. Nicolas rf eso aur ,2014 6, January version Draft tability lding ABSTRACT es, y, s- 1 oinS Abbot S. Dorian & ufc eevis h atecnan ae ohin both water contains the mantle to The comparable be to reservoirs. thought surface is but unknown, rently cleoin(ete l 05 eesr omiti a maintain to necessary phys- 2005) may and thermostat. al. weathering 1990) but et Tozer silicate (West 2011), & erosion al. (Mian et tectonics ical (Abe wider plate a the principle have could lack in planets Dry zone habitable 2012). narrower habitable al. much weathering et a (Abbot seafloor have zone weathering a should silicate and of a have thermostat existence not will the waterworlds al. feedback, zone Barring et (Kasting habitable thermostat the 1993). weathering of silicate a conception assumes traditional Crit- the temperature-dependent. ically, strongly chemi- ther- is their weathering weathering because cal silicate continents The exposed main- requires feed- 1981). mostat been weathering al. silicate have et a (Walker con- to by back time surface hypothesized geological clement of over are tained Earth Moreover, source on the and ditions heat. feedback is inertia, transport thermal water provides stabilizing helps surface vapor, water a atmospheric climate: 2012). al. planetary of et ing Abbot 1996; existence al. Holm et 1992; (Raymond the Holm Schubert & delivery & (Kasting or (McGovern water fluxes in 1989), ocean–mantle Holm luck & and (Kasting 2004) blind character surface 1992): stable geologically 1989). Schubert & (McGovern mantle h xc ae netr fErhsitro scur- is interior Earth’s of inventory water exact The h attoigo ae sakyfco nregulat- in factor key a is water of partitioning The Earth’s for explanations of classes two are There ∼ r h ate lt etnc transports tectonics Plate mantle. the ir, 0 1.1. . anyi u td sErhscurrent Earth’s is study our in tainty e yl iltn osoemr water more store to tend will cycle ter % rmtclyicesn h odds the increasing dramatically 2%, yrshr n eiesteady-state derive and hydrosphere e n fml tmdoenrde and ridges mid-ocean at melt of ing yadde cas mligthat implying oceans, deep and hy s h otnosyhbtbezone habitable continuously the ds, ae aaiyo at’ Mantle Earth’s of Capacity Water e’ ae spriind however, partitioned, is water net’s esaet nnain aty we Lastly, inundation. to are nets casadcniet fmassive of continents and oceans yrsai eflo rsueis pressure seafloor hydrostatic , eflo rsue respectively, pressure, seafloor n 2 trols hsi important is This aterworlds. lmt speitdt eless be to predicted is climate et yErhsapproximately Earth’s by d faymass any of TB WATERWORLDS BE OT a maintain can aesand lanets planets — 2 Cowan & Abbot hydrous and nominally anhydrous minerals (Hirschmann # %1"+$.*)& 2006; Hirschmann & Kohlstedt 2012). The Fe core may # *!#$)& also contain primordial water (Abe et al. 2000), which # we ignore in the present analysis. % $,$'-& Inoue et al. (2010) determined maximum water mass ,0 "0!.*)& !# fractions of 0.7% (, shallower than 410 km), "#$$,,&)$& 3.3% (, 410–520 km), 1.7% (, 520– "# 660 km), and 0.1% (perovskite, below 660 km). Com- ($)-'#& $+$)&-#& bining these estimates with the pressure-dependence of olivine water content from Hauri et al. (2006) puts Earth’s mantle water capacity at a dozen times the cur- Fig. 1.— A schematic of our hydrosphere model. Degassing of melt decreases with seafloor pressure, while hydration of ocean rent surface reservoir. Measurements of electrical con- crust increases with seafloor pressure, providing a potential stabi- ductivity (Dai & Karato 2009) suggest that Earth’s man- lizing feedback. tle contains 1–2 oceans worth of water at present-day, but other estimates have differed by a factor of a few in either critical, which occurs at nearly the pressures and temper- direction (e.g., Huang et al. 2005; Smyth & Jacobsen atures in hydrothermal systems today (Kasting & Holm 2006; Khan & Shankland 2012). 1992). If Earth’s water started in the mantle and gradually degassed to the surface, then this pressure- 1.2. The Deep dependence could explain why the oceans stabilized at their current depths. A planet starting with very deep There is a two-way flux of water between ocean and oceans (Korenaga 2008), however, would not be able mantle on Earth. Ocean crust forms at mid-ocean ridges to effectively subduct water and would remain a water- by depressurization melting of mantle, releasing volatiles world. into the ocean. Plate tectonics then drives the ocean In order to solve the problem of regassing water crust toward zones. Hydrothermal alteration, into the mantle, Holm (1996) suggested that the re- principally serpentinization, produces ocean crust that is duced efficiency of hydrothermal heat exchange beneath 5–10% water by mass. During subduction, much of the deep oceans would result in higher mantle temperatures, water is baked out of the subducting slab, producing ex- greater plate velocities and therefore efficient subduction plosive volcanism like Mount St. Helens. Some of the of hydrated crust. This mechanism may not work quan- water, however, is subducted deep into the mantle, clos- titatively, however, because hydrothermal heat transfer ing the loop (for a review of the relevant geochemistry, only accounts for 30–50% of heat flux through oceanic see Arndt 2013). crust (Stein et al. 1995). There are also qualitative prob- The current H2O degassing at mid-ocean ridges is ap- lems with this regassing argument: first of all, the same × 11 proximately 2 10 kg/yr (Hirschmann & Kohlstedt logic should apply to planets with very shallow oceans, 2012), while the regassing flux at subduction zones 12 potentially negating the supercritical circulation hypoth- is estimated to be 0.7–2.9×10 kg/yr (Jarrard 2003; esis. Moreover, hydrating the mantle lowers its viscosity, Schmidt & Poli 1998). Given the order-of-magnitude further increasing plate velocities and making this mech- agreement of these figures, and the large associated un- anism a destabilizing feedback. Lastly, although rapid certainties, Earth’s current ocean volume is typically as- subduction may aid regassing, a hotter mantle almost sumed to be in a steady state (McGovern & Schubert certainly hinders it (Bounama et al. 2001). In short, it is 1989). Moreover, realistic parameterizations of mantle 11 13 not clear that reduced heat transport through the ocean convection predict water fluxes of 10 –10 kg/yr in crust leads to net regassing of water. Earth’s geological past, implying ocean-cycling times of 8 Whatever geophysical processes govern Earth’s deep 10 yrs (McGovern & Schubert 1989). A significant flux water cycle presumably operate on other terrestrial plan- imbalance would have long ago desiccated or submerged ets with plate tectonics. The partitioning of water on the planetary surface. Earth is not only an outstanding problem in geophysics (Fyfe 1994; Langmuir & Broecker 2012), but one with 1.3. Previous Work important implications for the surface conditions and cli- A planet entirely covered in water may develop exposed mate of terrestrial exoplanets. continents by losing water to space, hydrating the crust 2. and mantle, or reshaping continents and ocean basins. HYDROSPHERE MODEL For example, an ocean-covered planet with a hot strato- As shown in Figure 1, we develop a two-box model sphere could lose water to space until continents are ex- of the deep water cycle for a terrestrial planet with posed. The resulting vigorous silicate weathering might plate tectonics: the two water reservoirs are the ocean draw down sufficient CO2 to cool the planet out of the and mantle, while the basalt and granite are important moist greenhouse state, leaving a partially water-covered for the transport of water and for setting the depth planet (Abbot et al. 2012). Detailed atmospheric simu- of ocean basins. A two-box model is appropriate for lations of such hot planets, however, indicate that the gross estimates of water partitioning over billions of years tendency of carbon dioxide to cool the stratosphere im- (McGovern & Schubert 1989; Ueta & Sasaki 2013), but pedes the loss of water (Wordsworth & Pierrehumbert may be too simple to simulate small changes in ocean 2013). volume on shorter timescales (Parai & Mukhopadhyay In an alternate hypothesis, the Rayleigh number of 2012). convecting water in hydrothermal systems at mid-ocean We assume that the total hydrosphere mass, W , is con- ridges exhibits a sharp peak when water becomes super- stant, i.e., we neglect water loss from the top of the at- Super-Earths Need Not be Waterworlds 3 mosphere. Isotopic evidence indicates that Earth’s hy- drosphere was at most 26% more massive in the TABLE 1 Model Variables & Parameters (Pope et al. 2012). Name Symbol Value 2.1. Water Fluxes a −4 planetary water mass fraction ω ω⊕ = 6.2 × 10 a The principle dependent variable in our model is the normalized gravity g˜ g˜⊕ = 1 a −4 bulk water mass fraction of the mantle, x. We assume mantle water mass fraction x x⊕ = 5.8 × 10 3 3 plate tectonics but our results should be largely insensi- density of water ρw 1.0 × 10 kg/m 3 3 density of granite ρg 2.9 × 10 kg/m tive to plate velocity: degassing and regassing are pro- 3 3 density of basalt ρb 3.0 × 10 kg/m portional to the spreading and subduction rates, respec- 3 3 density of mantle ρm 3.3 × 10 kg/m tively, which are equal in steady-state. We therefore con- 3 thickness of oceanic crust db 6 × 10 m 3 sider the change in mantle water per sea-floor overturn- melt × ′ depth of melt d 60 10 m ing, x ≡ dx/dτ, where the non-dimensional time, τ, is hydrated crust water fraction xh 0.05 related to the area of oceanic crust, Ab, spreading rate, S, subduction efficiency χ 0.23 Earth degassing efficiency fd⊕ 0.9 and the length of mid-ocean ridges, L, via τ = tLS/Ab. 3 Earth hydration depth dh⊕ 3 × 10 m The net change of mantle water per sea-floor overturn- 7 Earth seafloor pressure P⊕ 4 × 10 Pa ing is: max 3 −1 max. thickness of cont. crust dg 70 × 10 g˜ m ′ Ab ocean basin covering fraction fb 0.9 x = (w↓ − w↑), (1) mantle mass fraction fm 0.68 Mm −4 ocean mass fraction of Earth ωo 2.3 × 10 where Mm is the mantle mass, w↓ is the water content of normalized ocean basin area f˜b 1.3 b subducted crust and sediments, and w↑ is the water de- seafloor pressure dependence φ 2 b −4 gassed by the formation of ocean crust at MORs (both Earth mantle water content x⊕ 5.8 × 10 b −3 have units of kg/m2). This equation is similar to that max mantle water fraction xmax 7 × 10 a Note. — These are the variables in our model; we list here their solved by previous researchers (McGovern & Schubert b 1989; Ueta & Sasaki 2013), except that we include the nominal values for Earth. These are critical parameters whose sensitivity we test in §4.1; we list their nominal values here. first-order effects of seafloor pressure, as described be- low. The subducted water is 0.9 is the nominal degree of melt degassing on Earth today. The piecewise definition ensures that the degassed w↓ = xhρbdh(P )χ, (2) water does not exceed that present in the melt. where xh =0.05 is the mass fraction of water in the hy- 2.3. Regassing drated crust (McGovern & Schubert 1989), ρb = 3.0 × 3 3 The depth of serpentinization may also depend on the 10 kg/m is the density of basalt, dh is the depth of hydration in the crust (which depends on the seafloor pressure at the bottom of the ocean: pressure, P ), and χ = 0.23 is the fraction of volatiles P σ subducted deep into the mantle rather than outgassed dh(P ) = min dh⊕ , db , (5) P via arc and back-arc volcanism (R¨upke et al. 2004). The   ⊕   subduction efficiency, χ, is highly uncertain but we per- where σ quantifies the pressure-dependence, dh⊕ = 3 × form a sensitivity analysis below to quantify how it and 103 m is the nominal value of the hydration depth on 3 other critical parameters likely affect our results (§4.1). Earth (McGovern & Schubert 1989), and db =6 × 10 m The degassed water is is the thickness of basaltic ocean crust. Hydration w = xρ d f (P ), (3) of the lithospheric mantle below the oceanic crust is ↑ m melt d functionally identical to the hydration of oceanic crust 3 3 where ρm =3.3 × 10 kg/m is the density of the upper (R¨upke et al. 2004), but we conservatively adopt the con- 4 mantle, dmelt = 6 × 10 m is the depth of the MOR straint dh ≤ db. melting regime, and fd is the fraction of water in the It has been argued that the depth of hydration depends melt that degasses rather than remaining in the ocean on the Rayleigh number of water in hydrothermal sys- crust as it solidifies (model variables and parameters are tems (Kasting & Holm 1992). If, as they assumed, water listed in Table 1). is gradually degassed from the mantle, this amounts to a large positive σ; if water begins at the planetary surface 2.2. Degassing such a mechanism dictates a large negative σ. The fraction of water in the melt that is degassed de- 2.4. Hypsometry & Isostacy pends on hydrostatic pressure at the seafloor (Papale 1997; Kite et al. 2009). We parametrize this stabilizing We treat the planetary elevation distribution (“hyp- feedback as: sometry”) as two δ-functions: one for ocean crust and −µ another for continental crust. This is a good approxi- P mation of Earth’s strongly bimodal hypsometry (Rowley fd(P ) = min fd⊕ , 1 , (4) " P⊕ # 2013), which is presumably typical of any terrestrial   planet with plate tectonics (Stoddard & Jurdy 2012). where P is the pressure at the bottom of the ocean, P⊕ = Earth’s oceanic crust exhibits a clear age-depth rela- 4 × 107 Pa is its current value on Earth, µ> 0 quantifies tion: older crust is denser and sinks deeper into the man- the pressure-dependence of melt degassing, and fd⊕ = tle (Parsons & Sclater 1977). A larger ocean basin will 4 Cowan & Abbot therefore have a greater mean depth. The globally aver- on the ocean basin area, Ab = fbA, where A is the plan- aged ocean depth, however, should remain roughly con- etary area. We adopt fb = 0.9. In other words, 90% of stant barring secular changes in spreading rates, which the planet is covered in water and the remaining 10% is are beyond the scope of our zeroth-order treatement. exposed continent. This is sufficient dry land to maintain We assume that the modal continental height is level a silicate weathering thermostat (Abbot et al. 2012) and with the ocean surface, which is natural because of the our results would not change dramatically if we instead competing effects of erosion and deposition (Korenaga adopted fb = 0.7 (as on modern Earth) or fb = 1 (the 2008; Rowley 2013). There is a maximum thickness limiting case of a single infinitesimal island extending out that continents can achieve, however, beyond which a of the ocean). continent flows under its own weight (Rey & Houseman Given these assumptions, the maximum volume of sur- 2006). The Himalayan plateau on Earth has a crustal face water that could be accommodated by an Earth-size 18 3 thickness of 70 km and appears to be near this limit planet is 5.2 × 10 m , or a mass of 3.7Mo. (England & McKenzie 1982). We conservatively adopt a maximum continental thickness of 3. STEADY-STATE SOLUTIONS −1 We find steady state solutions to the water partitioning g dmax = 70 km , (6) on a planet by setting the upward and downward water g g  ⊕  fluxes equal to each other, where g is the surface gravity of the planet, g⊕ is that xρmdmeltfd(P )= xhρbdh(P )χ. (12) of Earth, and we have adopted the gravity-dependence of Kite et al. (2009). Using the mass-radius relation of The degassing efficiency, fd, and hydration depth, dh, Valencia et al. (2007a), a gravity of 3g⊕ corresponds to depend on seafloor pressure, P , which intimately depends a 10M⊕ super-Earth. on ocean water depth, dw: P = gρwdw. The thickness of granitic continents, dg, is related to Ocean depth can be expressed in terms of mantle water the thickness of the other layers by: content, W − xMm d = d + d + d , (7) dw = , (13) g o b m Abρw where do is the depth of the ocean basin and dm is the but it is instructive to factor out the size-dependent depth to which the continent root extends into the man- terms and define intensive quantities: tle (Figure 1). This parameterization assumes that the tops of continents are at sea level (zero freeboard). This M ω − xfm dw = , (14) is merely shorthand for a freeboard that is much smaller A fbρw than the depth of ocean basins.   Isostatic balance dictates that the pressure below each where ω = W/M is the planetary water mass fraction vertical column must be equal: and fm = Mm/M = 0.68 is the planetary mantle mass fraction. Note that the term in parentheses is propor- 2 ρwdo + ρbdb + ρmdm = ρgdg, (8) tional to gravity: M/A ∝ M/Rp ∝ g. We may therefore 3 3 3 3 write the normalized ocean depth as where the ρw =1.0×10 kg/m and ρg =2.9×10 kg/m are the density of water and granite, respectively. dw ω − xfm Combining (7) and (8) yields the following expression =˜g , (15) dw⊕ ω f˜ of isostatic balance: o b where dw⊕ = 4 km is the average depth of Earth’s do(ρm − ρw)+ db(ρm − ρb)= dg(ρm − ρg). (9) oceans,g ˜ = g/g⊕ is the normalized planetary gravity, max −4 Given the maximal crustal thickness, dg , one can ωo = Mo/M⊕ = 2.3 × 10 is the fractional mass of derive the maximum depth of water-filled ocean basins: Earth’s surface water, and f˜b = fb/fb⊕ =1.3 is the ocean basin covering fraction of the planet divided by that of max − − − max dg (ρm ρg) db(ρm ρb) Earth. do = . (10) ρm − ρw The normalized seafloor pressure is therefore max For our fiducial Earth-size parameters, d = 11.4 km. P ω − xfm o =g ˜2 . (16) Note that the thickness of oceanic crust is likely also P⊕ ω f˜ inversely proportional to g (Sleep 2012). We conserva- o b tively adopt db = 6 km regardless of planet mass, which We substitute (16) into (12) and solve for the steady- produces shallower ocean basins for massive planets. state mantle water fraction on the interval x ∈ [0,ω/fm], The densities of granite and basalt are nearly the same where the upper-limit ensures that the mantle does not (ρg ≈ ρb) and the maximum crustal thickness far exceeds contain more water than the planet as a whole. There max the thickness of oceanic crust (db ≪ dg ), so we can is also a petrological upper limit to how much water the approximate (10) as mantle can sequester, however. For Earth, that limit −3 appears to be xmax = 7 × 10 (a dozen oceans, §1.1). −1 g In cases where x > xmax, we set x = xmax. dmax ≈ 11.4 km . (11) o g Given the steady-state mantle water content, it is triv-  ⊕  ial to compute the depth of surface oceans using (15). The maximum ocean volume that can be accommo- The waterworld boundary is defined by equating surface dated while maintaining exposed continents also depends water depth with maximal ocean basin depth, (10). The Super-Earths Need Not be Waterworlds 5 solid black line in Figure 2 shows the waterworld bound- In the presence of a modest stabilizing pressure- ary for our fiducial parameters and pressure dependencies dependence (φ = 1), the steady-state mantle water frac- of µ = σ = 1. tion is 2 It is useful to compare this waterworld boundary to ωx⊕g˜ x = . (21) the null hypthesis: ignoring isostatic adjustment and ω f˜ + x f g˜2 the deep water cycle. In such a case, (15) simplifies o b ⊕ m to dw = dw⊕gω/ω˜ o, which can be combined with the In the high-gravity limit, the mantle contains the entirety approximation (11) to obtain the waterworld limit of of the planet’s water, x = ω/fm. In practice this cannot −2 ω = ωog˜ . The null-hypothesis is indicated by the solid occur for water-rich planets because of the finite water red line in Figure 2. By definition, Earth’s surface reser- capacity of the mantle (x ≤ xmax). Nonetheless, the voir (indicated by the ⊕) puts it right at the waterworld higher gravity of super-Earths biases the deep water cycle boundary under these assumptions. in favor of mantle sequestration. Water inventory and If one accounts for the ability of isostacy and erosion mantle capacity are both proportional to planetary mass, to reshape continents and keep their heads above water, producing a weak mass-dependence to the waterworld but still neglects the deep-water cycle, one obtains the boundary. dashed red line in Figure 2. 4. DISCUSSION 3.1. Analytic Approximation 4.1. Sensitivity Analysis It is possible to obtain an analytic solution to (12) As noted in §1.1, the amount of water in Earth’s man- if we ignore the piecewise nature of degassing, fd, and tle is poorly constrained. The gray broken lines in Fig- regassing, dh. This analytic approximation has intuitive ure 2 show the analytic waterworld boundary if Earth’s value so we develop it here. In this case the w↑ = w↓ can mantle water content, x⊕, is 10× greater (dashed) and be written as: 10× smaller (dotted) than our nominal value. Note that varying x⊕ in this model is mathematically equivalent to −µ σ P P varying any of the water flux parameters in (19), many of xρmdmeltfd⊕ = xhρbχdh⊕ , (17) P⊕ P⊕ which are uncertain (e.g., the subduction efficiency, χ).     Varying x⊕ by two orders of magnitude affects the wa- which can be compactly expressed as terworld boundary for a 10M⊕ super-Earth by roughly one order of magnitude; this is the dominant uncertainty x P φ in our study. = , (18) x P The maximum water capacity of the mantle, xmax is ⊕  ⊕  not well known for high-mass terrestrial planets. The where φ = µ + σ is the sum of pressure dependencies for water storage in Earth’s mantle is thought to be concen- mid-ocean ridge melt degassing and serpentinization of trated in the (410–660 km depth). If mas- oceanic crust, and Earth’s bulk mantle water content is sive planets can only sequester water in a thin transition zone, then mantle water capacity scales with planetary −1 xhχρbdh⊕ area rather than mass, or xmax ∝ g˜ . The dotted black x⊕ = . (19) ρmdmeltfd⊕ line in Figure 2 shows the minuscule effect of adopting this scaling. −4 Our fiducial parameter values yield x⊕ =5.8 × 10 , or On the other hand, the mantle of a super-Earth roughly an ocean’s worth of water in Earth’s mantle. should be primarily in the form of post-perovskite Substituting (16) into (18), we obtain (Valencia et al. 2007b), which may not have the same water capacity as Earth’s dominant mantle rock, per- φ x ω − xfm ovskite. We try setting xmax = 1 and find that the wa- = g˜2 . (20) terworld boundary is unchanged if post-perovskite can x ˜ ⊕  ωofb  hold unlimited water. The steady-state mantle water content allows us to es- Finally, the blue lines in Figure 2 show the waterworld timate the ocean depth, via (15), which we then compare boundary for pressure-dependences of φ = 3 (dashed) to the maximum ocean basin depth. The solid gray line and φ = 1 (dotted). (In both cases we use µ = σ = in Figure 2 shows the waterworld boundary for our nom- φ/2 in the numerical model.) The precise strength of inal parameter values and φ = 2. The small difference the seafloor pressure dependence of the deep water cycle between the numerical and analytic waterworld bound- affects the waterworld boundary by less than a factor of aries (solid black and gray lines, respectively) indicates two. that the piecewise definitions of fd(P ) and dh(P ) do not 4.2. critically affect our results. Plate Tectonics If φ < 0, the pressure feedback is destabilizing and Our model of the deep water cycle assumes plate there are two physical steady-state solutions: shallow tectonics, but it is currently unknown whether super- and deep oceans, respectively. The current state of a Earths are tectonically active. The increased heat flux of planet will depend on initial conditions. For φ ≥ 0, there super-Earths should produce vigorous mantle convection is a single physical root to (20) and therefore a single (Valencia et al. 2007a; van Heck & Tackley 2011), but steady-state solution. If there is no net pressure depen- the increased strength and buoyancy of crust on super- dence to the deep water cycle (φ = 0), then all planets Earths may prohibit plate tectonics (O’Neill & Lenardic have the same mantle water content as Earth, x ≡ x⊕. 2007; Kite et al. 2009). Mantle convection may even ex- 6 Cowan & Abbot

Fig. 2.— Waterworld boundary as a function of water mass fraction, ω, and normalized surface gravity,g ˜ = g/g⊕: planets in the upper-right corner are waterworlds, while those in the lower-left maintain exposed continents. The solid black line shows the waterworld boundary for our nominal parameters, including the negative feedbacks associated with seafloor pressure. The dotted black line shows the waterworld boundary if one assumes that water can only be stored in a mantle’s transition zone. The solid red line shows the waterworld boundary if one applies the g−1 dependence to Earth’s current hypsometry and presumes that all of a planet’s water resides at the surface. The dashed red line accounts for the effects of erosion and isostatic adjustment (§2.4) but not the deep-water cycle. The black symbol, ⊕, denotes Earth if one only considers its surface water reservoir; the arrow indicates Earth’s probable location if one also accounts for water present in the mantle. The gray lines indicate the analytic waterworld boundary for nominal parameters (solid), as well as Earth −3 −5 mantle water content values of x⊕ = 5.8 × 10 (dashed) and x⊕ = 5.8 × 10 (dotted). The blue lines show the waterworld boundary for seafloor pressure-dependencies of φ = 3 (dashed) and φ = 1 (dotted). A terrestrial planet with surface gravityg ˜ = 3 corresponds to a 10M⊕ super-Earth (Valencia et al. 2007b). hibit hysteresis, such that planets with identical bound- ers (differences in crustal thickness scale as g−1 for the ary conditions may or may not have plate tectonics, de- same reason as ocean basin depth). pending on initial conditions (Lenardic & Crowley 2012). Alternatively, it has been suggested that surface water 4.3. Homogeneity of the Mantle is more important than planetary mass for plate tectonics The homogeneity of volatiles in Earth’s lower man- (Mian & Tozer 1990; Korenaga 2010). In a classic case 3 of chicken-and-egg, van der Lee et al. (2008) argue that a tle is questionable. The high He abundance of ocean deep water cycle is a necessary, if not sufficient, condition islands has been attributed to a poorly-mixed lower for long-term plate tectonics. mantle (Kurz et al. 1982). By extension, this hypoth- Continental crust formation is thought to be an in- esis implies that Earth’s may hold much evitable by-product of plate tectonics in the presence of more water than what is inferred for the . Gonnermann & Mukhopadhyay (2009) argue, however, water (Rudnick 1995; Arndt 2013), so super-Earths are 3 likely to have large volumes of granitic crust. In fact, the that the He abundance of the mantle is consistent with large volume of continental crust combined with smaller homogeneous composition. maximal crustal thickness may lead to a planet entirely If the mantle is not well-mixed, then x represents the covered in continental crust. This does not greatly affect water fraction of those regions sampled by the mid-ocean our results, provided that the planet remains tectonically ridge melting and affected by subduction of oceanic crust. active and that some regions have thicker crust than oth- Indeed, the source of mid-ocean ridge basalts (MORB) appears to have maintained a constant water mass frac- Super-Earths Need Not be Waterworlds 7 tion, x, for billions of years, suggesting that subduction of We have argued, as have others (Kasting & Holm 1992; hydrated oceanic crust is depositing water in the MORB Holm 1996), that the approximately steady-state wa- source region (Hirschmann 2006). ter partitioning on Earth over geological time suggests a seafloor pressure feedback that regulates the degassing 4.4. Observational Constraints at mid-ocean ridges and/or the serpentinization and sub- sequent subduction of oceanic crust. Although ocean vol- Our model of the deep water cycle predicts that many ume may change throughout a planet’s history because super-Earths have exposed continents. It will eventu- of secular cooling, we have tackled the zeroth-order prob- ally be possible to test this hypothesis by observation- lem of steady-state solutions. ally determining the surface character of a large number Notably, seafloor pressure is proportional to a planet’s of high-mass terrestrial planets (see also discussion in surface gravity. The enhanced gravity of super-Earths Abbot et al. 2012). produces shallower ocean basins, but also leads to shal- Disk-integrated rotational multiband photometry of lower oceans. The solid black line in Figure 2 shows the Earth, essentially the changing colors of a pale blue waterworld boundary if one accounts for the pressure- dot, encode information about continents, oceans and dependence of the deep water cycle. clouds (Ford et al. 2001). Such “single-pixel” obser- The effects of isostacy, erosion, and deposition, com- vations have been used to construct coarse longitudi- bined with a pressure-dependent deep water cycle, make nal color maps of Earth and Mars (Cowan et al. 2009, super-Earths 80× less susceptible to inundation than 2011; Fujii et al. 2011; Hasinoff et al. 2011), while simu- they otherwise would be. Our model predicts that lations suggest that photometry spanning an entire plan- tectonically active 10M⊕ planets can maintain large etary orbit could be used to construct a rough 2D color exposed continents for water mass fractions less than map (Kawahara & Fujii 2010, 2011; Fujii & Kawahara 2 × 10−3. 2012). Finally, Cowan & Strait (2013) showed that disk- Exoplanets with sufficiently high water content will integrated multiband photometry of a variegated planet be water-covered regardless of the mechanism discussed can be inverted to obtain reflectance spectra of its dom- here, but such “ocean planets” may betray themselves inant surface types, even if the number and colors of the by their lower density: a planet with 10% water mass surfaces are not known a priori. fraction will exhibit a transit depth 10% greater than The bottom line is that a 5–10 m space telescope an equally-massive planet with Earth-like composition equipped with a coronagraph or starshade could produce (Sotin et al. 2007). Planets with 1% water mass frac- a coarse surface map of an Earth-analog at a distance of tion, however, are almost certainly waterworlds but may 10 pc (Cowan et al. 2009; Fujii & Kawahara 2012). Such have a bulk density indistinguishable from truly Earth- low-resolution maps would be sufficient to identify the like planets. Given that simulations of water delivery continents one expects on a tectonicaly active planet. If to habitable zone terrestrial planets predict water mass most super-Earths exhibit the bimodal surface charac- fractions of 10−5–10−2 (Raymond et al. 2004), we con- ter of Earth, it will suggest that they experience plate clude that most tectonically active planets —regardless tectonics and a deep water cycle. If, instead, large ter- of mass— will have both oceans and exposed continents, restrial planets were all determined to be waterworlds, enabling a silicate weathering thermostat. it would indicate that our hypothesis is wrong.

5. NBC acknowledges many insightful discussions with CONCLUSIONS J.P. Townsend, S.D. Jacobsen, and C.R. Bina. The It has long been suggested that super-Earths ought authors also had useful conversations with C. Andron- to be waterworlds (Stapledon 1937; Kite et al. 2009; icos, H. Gilbert, R. Jeanloz, M. Manga, D. McKen- Abbot & Switzer 2011). If one accounts for the first- zie, V.S. Meadows, D.B. Rowley, J. Rudge, S. Stein, order effects of gravity on ocean basin depth and water D.J. Stevenson, and R. Wordsworth. E.S. Kite provided inventory, then a 10M⊕ planet is not expected to have critical feedback on an early version of the manuscript. exposed continents unless it has a water mass fraction The authors thank N.H. Sleep for sharing an unpublished less than 3 × 10−5, roughly ten times drier than Earth manuscript. DSA was supported by an Alfred P. Sloan (solid red line in Figure 2). research fellowship.

REFERENCES Abbot, D. S., Cowan, N. B., & Ciesla, F. J. 2012, Astrophysical Dai, L., & Karato, S.-i. 2009, Earth and Planetary Science Journal, 756, 178 Letters, 287, 277 Abbot, D. S., & Switzer, E. R. 2011, Astrophysical Journal England, P., & McKenzie, D. 1982, Geophysical Journal of the Letters, 735, L27+ Royal Astronomical Society, 70, 295 Abe, Y., Abe-Ouchi, A., Sleep, N. H., & Zahnle, K. J. 2011, Eriksson, P. 1999, Precambrian Research, 97, 143 Astrobiology, 11, 443 Ford, E. B., Seager, S., & Turner, E. L. 2001, Nature, 412, 885 Abe, Y., Ohtani, E., Okuchi, T., Righter, K., & Drake, M. 2000, Fujii, Y., & Kawahara, H. 2012, Astrophysical Journal, 755, 101 Origin of the Earth and Moon, 1, 413 Fujii, Y., Kawahara, H., Suto, Y., Fukuda, S., Nakajima, T., Arndt, N. T. 2013, Geochemical Perspectives, 2, 405 Livengood, T. A., & Turner, E. L. 2011, Astrophysical Journal, Bounama, C., Franck, S., & von Bloh, W. 2001, Hydrology and 738, 184 Earth System Sciences, 5, 569 Fyfe, W. S. 1994, Geological Society of London Special Cowan, N. B., & Strait, T. E. 2013, Astrophysical Journal Publications, 78, 1 Letters, 765, L17 Gonnermann, H. M., & Mukhopadhyay, S. 2009, Nature, 459, 560 Cowan, N. B., et al. 2009, Astrophysical Journal, 700, 915 Harrison, T. M. 2009, Annual Review of Earth and Planetary —. 2011, Astrophysical Journal, 731, 76 Sciences, 37, 479 8 Cowan & Abbot

Hasinoff, S. W., Levin, A., Goode, P. R., & Freeman, W. T. 2011, Parsons, B., & Sclater, J. G. 1977, Journal of Geophysical in Computer Vision (ICCV), 2011 IEEE International Research, 82, 803 Conference on, IEEE, 185–192 Pope, E. C., Bird, D. K., & Rosing, M. T. 2012, Proceedings of Hauri, E. H., Gaetani, G. A., & Green, T. H. 2006, Earth and the National Academy of Sciences, 109, 4371 Planetary Science Letters, 248, 715 Raymond, S. N., Quinn, T., & Lunine, J. I. 2004, Icarus, 168, 1 Hirschmann, M., & Kohlstedt, D. 2012, Physics Today, 65, 030000 Rey, P. F., & Houseman, G. 2006, Geological Society, London, Hirschmann, M. M. 2006, Annu. Rev. Earth Planet. Sci., 34, 629 Special Publications, 253, 153 Holm, N. G. 1996, Deep Sea Research Part II: Topical Studies in Rowley, D. B. 2013, Journal of Geology, 121, 445 Oceanography, 43, 47 Rudnick, R. L. 1995, Nature, 378, 571 Huang, X., Xu, Y., & Karato, S.-i. 2005, Nature, 434, 746 R¨upke, L. H., Morgan, J. P., Hort, M., & Connolly, J. A. 2004, Inoue, T., Wada, T., Sasaki, R., & Yurimoto, H. 2010, Physics of Earth and Planetary Science Letters, 223, 17 the Earth and Planetary Interiors, 183, 245 Schmidt, M. W., & Poli, S. 1998, Earth and Planetary Science Jarrard, R. D. 2003, Geochemistry, Geophysics, Geosystems, 4 Letters, 163, 361 Kasting, J. F., & Holm, N. G. 1992, Earth and Planetary Science Sleep, N. H. 2012, International Journal of Astrobiology, 11, 257 Letters, 109, 507 Smyth, J. R., & Jacobsen, S. D. 2006, Nominally anhydrous Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icarus, minerals and Earth’s deep water cycle, Vol. 168 (American 101, 108 Geophysical Union) Kawahara, H., & Fujii, Y. 2010, Astrophysical Journal, 720, 1333 Sotin, C., Grasset, O., & Mocquet, A. 2007, Icarus, 191, 337 —. 2011, Astrophysical Journal Letters, 739, L62+ Stapledon, O. 1937, Star Maker (London: Methuen & Co., Ltd.) Khan, A., & Shankland, T. 2012, Earth and Planetary Science Stein, C. A., Stein, S., & Pelayo, A. M. 1995, Seafloor Letters, 317, 27 hydrothermal systems: physical, chemical, biological, and Kite, E. S., Manga, M., & Gaidos, E. 2009, Astrophysical geological interactions, 425 Journal, 700, 1732 Stoddard, P. R., & Jurdy, D. M. 2012, Icarus, 217, 524 Korenaga, J. 2008, Terra Nova, 20, 419 Ueta, S., & Sasaki, T. 2013, Astrophysical Journal, 775, 96 —. 2010, The Astrophysical Journal Letters, 725, L43 Valencia, D., O’Connell, R. J., & Sasselov, D. D. 2007a, Kurz, M. D., Jenkins, W. J., & Hart, S. R. 1982, Nature, 297, 43 Astrophysical Journal Letters, 670, L45 Langmuir, C. H., & Broecker, W. 2012, How to Build a Habitable Valencia, D., Sasselov, D. D., & O’Connell, R. J. 2007b, Planet (Princeton: Princeton University Press) Astrophysical Journal, 665, 1413 Lenardic, A., & Crowley, J. W. 2012, Astrophysical Journal, 755, van der Lee, S., Regenauer-Lieb, K., & Yuen, D. A. 2008, Earth 132 and Planetary Science Letters, 273, 15 Matsui, T., & Abe, Y. 1986, Nature, 319, 303 van Heck, H. J., & Tackley, P. J. 2011, Earth and Planetary McGovern, P. J., & Schubert, G. 1989, Earth and Planetary Science Letters, 310, 252 Science Letters, 96, 27 Walker, J. C. G., Hays, P. B., & Kasting, J. F. 1981, Journal of Mian, Z., & Tozer, D. 1990, Terra Nova, 2, 455 Geophysical Research, 86, 9776 Morbidelli, A., Lunine, J. I., O’Brien, D. P., Raymond, S. N., & West, A. J., Galy, A., & Bickle, M. 2005, Earth and Planetary Walsh, K. J. 2012, Annual Review of Earth and Planetary Science Letters, 235, 211 Sciences, 40, 251 Wilde, S. A., Valley, J. W., Peck, W. H., & Graham, C. M. 2001, O’Neill, C., & Lenardic, A. 2007, Geophysical Research Letters, Nature, 409, 175 34, 19204 Wise, D. U. 1974, The geology of continental margins, 45 Papale, P. 1997, Contributions to Mineralogy and Petrology, 126, Wordsworth, R. D., & Pierrehumbert, R. T. 2013, Astrophysical 237 Journal, 778, 154 Parai, R., & Mukhopadhyay, S. 2012, Earth and Planetary Science Letters, 317, 396