indication of insensitivity of planetary weathering behavior and habitable zone to surface land...

7/28/2019 Indication of Insensitivity of Planetary Weathering Behavior and Habitable Zone to Surface Land Fraction

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arXiv:1208.1

760v1

[astro-ph.EP]8Aug2012

Draft version August 10, 2012Preprint typeset using LATEX style emulateapj v. 5/2/11

INDICATION OF INSENSITIVITY OF PLANETARY WEATHERING BEHAVIOR AND HABITABLE ZONETO SURFACE LAND FRACTION

Dorian S. Abbot1, Nicolas B. Cowan2, and Fred J. Ciesla1

Draft version August 10, 2012

ABSTRACT

It is likely that unambiguous habitable zone terrestrial planets of unknown water content will soon bediscovered. Water content helps determine surface land fraction, which influences planetary weatheringbehavior. This is important because the silicate weathering feedback determines the width of thehabitable zone in space and time. Here a low-order model of weathering and climate, useful forgaining qualitative understanding, is developed to examine climate evolution for planets of variousland-ocean fractions. It is pointed out that, if seafloor weathering does not depend directly on surfacetemperature, there can be no weathering-climate feedback on a waterworld. This would dramaticallynarrow the habitable zone of a waterworld. Results from our model indicate that weathering behaviordoes not depend strongly on land fraction for partially ocean-covered planets. This is powerful becauseit suggests that previous habitable zone theory is robust to changes in land fraction, as long as thereis some land. Finally, a mechanism is proposed for a waterworld to prevent complete water lossduring a moist greenhouse through rapid weathering of exposed continents. This process is named

a waterworld self-arrest, and it implies that waterworlds can go through a moist greenhouse stageand end up as planets like Earth with partial ocean coverage. This work stresses the importance ofsurface and geologic effects, in addition to the usual incident stellar flux, for habitability.Subject headings: astrobiology - planets and satellites: general

1. INTRODUCTION

The habitable zone is traditionally defined as the re-gion around a star where liquid water can exist at thesurface of a planet (Kasting et al. 1993). Since climatesystems include both positive and negative feedbacks, aplanets surface temperature is non-trivially related toincident stellar flux. The inner edge of the habitablezone is defined by the moist greenhouse, which oc-

curs if a planet becomes hot enough (surface tempera-ture of340 K) that large amounts of water can be lostthrough disassociation by photolysis in the stratosphereand hydrogen escape to space. The outer edge occurswhen CO2 reaches a high enough pressure that it canno longer provide warming, either because of increasedRayleigh scattering or because it condenses at the sur-face, which results in permanent global glaciation. Theselimits do not necessarily represent hard limits on lifeof all types. For example, life survived glaciations thatmay have been global (Snowball Earth) on Earth 600700 million years ago (Kirschvink 1992; Hoffman et al.1998). Furthermore, if greenhouse gases other than CO2are considered, e.g., hydrogen (Pierrehumbert & Gaidos2011; Wordsworth 2012), the outer limit of the habit-

able zone could be pushed beyond the CO2 condensationlimit. Alternative limits on habitability have also beenproposed. For example, a reduction in the partial pres-sure of atmospheric CO2 below 10

5 bar may prevent C4photosynthesis, which could curb the complex biosphere

[email protected] Department of the Geophysical Sciences, University of

Chicago, 5734 South Ellis Avenue, Chicago, IL 60637, USA; [email protected]

2 Center for Interdisciplinary Exploration and Research in As-trophysics (CIERA) and Department of Physics & Astronomy,Northwestern University, 2131 Tech Drive, Evanston, IL 60208,USA

(Caldeira & Kasting 1992; Bloh et al. 2005), although itwould certainly not limit most types of life.

Studies of planetary habitability have become in-creasingly pertinent as capabilities for the discoveryand characterization of terrestrial or possibly terres-trial exoplanets in or near the habitable zone have im-proved. For example, planets GJ581d (Udry et al. 2007;Mayor et al. 2009; Vogt et al. 2010; Wordsworth et al.2011), HD85512b (Pepe et al. 2011; Kaltenegger et al.

2011), Kepler-22b (Borucki et al. 2011), and GJ 677c(Anglada-Escude et al. 2012) have recently been discov-ered and determined to be potentially in the habitablezone. It is likely that the number of good candidates forhabitable terrestrial exoplanets will increase in the nearfuture as the Kepler mission, ground-based transit sur-veys, radial velocity surveys, and gravitational microlens-ing studies continue to detect new planetary systems.

The Faint Young Sun Problem discussed in theEarth Science literature is analogous to the habitablezone concept. A planet orbiting a main-sequence star re-ceives ever increasing incident flux (insolation) as thestar ages. This should tend to hurry a planet through thehabitable zone unless the planetary albedo or thermal op-

tical thickness (greenhouse effect) adjust as the systemages. As pointed out by Sagan & Mullen (1972), geologi-cal evidence for liquid water early in Earths history sug-gests that just such an adjustment must have occurredon Earth. Although there is some debate about the de-tails (Kasting 2010), it is fairly widely accepted that thesilicate-weathering feedback (Walker et al. 1981) playedan important role in maintaining clement conditionsthrough Earths history (Feulner 2012). Through thisnegative feedback the temperature-dependent weather-ing of silicate rocks on continents, which represents themain removal process for CO2 from the atmosphere, is re-
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2 Abbot, Cowan, & Ciesla

duced when the temperature decreases, creating a buffer-ing on changes in temperature since CO2 is a strong in-frared absorber. The silicate weathering feedback alsogreatly expands the habitable zone annulus around a starin space (Kasting et al. 1993), since moving a planet fur-ther from the star decreases the insolation it receives justas moving backward in time does. If insolation bound-aries are used to demarcate the habitable zone, the con-

cept can be used to consider habitability as a function ofposition relative to the star at a particular time, or as afunction of time for a planet at a constant distance froma star.

In addition to continental silicate weathering, weath-ering can also occur in hydrothermal systems in thebasaltic oceanic crust at the seafloor. Seafloor weather-ing is poorly constrained on Earth; however, it is thoughtto be weaker than continental weathering and to de-pend mainly on ocean chemistry, pH, and circulationof seawater through basaltic crust, rather than directlyon surface climate (Caldeira 1995; Sleep & Zahnle 2001;Le Hir et al. 2008). If this is true, CO2 would be lessefficiently removed from the atmosphere of a planet with

a lower land fraction, leading to higher CO 2 levels anda warmer climate. Furthermore, a planet with a lowerland fraction could have a weaker buffering to changesin insolation than a planet with a higher land fraction,which would cause it to have a smaller habitable zone.

While the weathering behavior of a planet could de-pend on surface land fraction, the water complement ofplanets in the habitable zone should vary substantially.The reason for this is that the habitable zone is in gen-eral located closer to the star than the snow line, thelocation within a protoplanetary disk outside of whichwater ice would be present and available to be incorpo-rated into solids. In the solar system, for example, thecurrent habitable zone ranges from approximately 0.8 to1.7 AU (Kasting et al. 1993) if one neglects the uncertain

effects of CO2 clouds. If CO2 clouds are assumed to pro-duce a strong warming (Forget & Pierrehumbert 1997),the outer limit of the habitable zone could be extendedto 2.4 AU (Mischna et al. 2000; Selsis et al. 2007;Kaltenegger & Sasselov 2011), although recent three-dimensional simulations including atmospheric CO2 con-densation suggest that the warming effect may in factbe fairly modest (Wordsworth et al. 2011; Forget et al.2012; Wordsworth et al. 2012). The snow line is thoughtto have been around 2.5 AU (Morbidelli et al. 2000);beyond this distance, evidence for water in the form ofice or hydrated minerals is seen in asteroids and plan-ets. The general picture for water delivery to Earth hasbeen that the orbits of bodies from beyond the snow line

were excited to high eccentricities through gravitationalscattering by massive planetary embryos and a youngJupiter. These high eccentricities would have put water-bearing planetesimals and embryos on Earth-crossing or-bits. A fraction of such bodies would have been ac-creted by Earth, stochastically delivering volatiles to theyoung planet (Morbidelli et al. 2000; OBrien et al. 2006;Raymond et al. 2009).

Efforts have been made to extend this analysis to plan-etary systems around other stars. The same dynamicaleffects are expected to occur, with the amount of waterdelivered to a potentially habitable planet being stronglydependent on the presence or orbits of giant plan-

ets (OBrien et al. 2006; Raymond 2006; Raymond et al.2009). As low mass stars are expected to have lowermass disks and therefore fewer massive bodies to pro-duce gravitational scattering, low mass stars may bemore likely to have volatile-poor habitable-zone planets(Raymond et al. 2007). Stars of mass 1 M or higher,on the other hand, could easily have habitable-zone plan-ets of similar or greater fractional mass of water than

Earth (Raymond et al. 2007).As described above, current thinking mostly focuses

on hydrated asteroids as the main source of water fora habitable planet; however, there are other dynamicalmechanisms that could allow habitable planets to accretesignificant amounts of water and volatiles. Cometary-bodies would deliver a larger amount of water for a givenimpactor mass than asteroids. Should such bodies getscattered over the course of planet formation, this mayallow larger amounts of water to be accreted than oth-erwise predicted. Furthermore, Kuchner (2003) arguedthat planets which formed outside the snow line couldmigrate inwards due to gravitational torques from a pro-toplanetary disk or scattering with other planets. Such

bodies would naturally accrete large fractions of waterice during formation beyond the snow line, which wouldthen melt and sublimate in a warmer orbit, providing ahigh volatile content to a planet close to its star.

Habitable zone planets could therefore have a wide va-riety of water mass fractions, which would lead to varyingland fractions. Furthermore, even for a constant massfraction of water, scaling relations dictate that land frac-tion depends on planetary size. At the same time, thesurface land fraction could exert strong control on theplanetary carbon cycle, which strongly influences plane-tary habitability. This warrants a general considerationof weathering behavior on planets of varying land frac-tion.

The main objective of this paper is to investigate the

effect of land fraction on the carbon cycle and weatheringbehavior of a terrestrial planet in the habitable zone. Wewill outline and use a simple analytical model for weath-ering and global climate that necessarily makes graveapproximations to the real physical processes. For ex-ample, we will use existing parameterizations of seafloorweathering, while acknowledging that observational andexperimental constraints on such parameterizations areminimal. We will only use the model, however, to makestatements that do not depend strongly on uncertain as-pects of the parameterizations. This model should beused to understand intuitively the qualitative behavior ofthe system rather than to make quantitative estimates.A major strength of the model is that it is easy to derive

and understand, yet should capture the most significantphysical processes. This type of modeling is appropriatein the study of exoplanets, for which limited data thatwould be relevant for a geochemical model exist.

We will consider an Earth-like planet with silicaterocks, a large reservoir of carbon in carbonate rocks, andat least some surface ocean. We will use equilibrium re-lations for weathering, which is reasonable for the slowchanges in insolation that a main-sequence star experi-ences. The climate model we use is a linearization ofa zero dimensional model. Although this is a severe ap-proximation, it allows the analytical progress that we feelis useful for obtaining insight into the problem. We will


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Land Fraction and Habitability 3

consider planetary surface land fractions ranging frompartial ocean coverage to complete waterworlds. In thiscontext a planet would be a waterworld if the highestland were covered by even 1 m of water, although a planetwith more water than this would qualify as a waterworldas well. However we will assume the geophysical contextis a planet with a substantial rock mantle and only up toroughly 10 times Earths water mass fraction (0.02%-

0.1%; Hirschmann & Kohlstedt 2012) rather than poten-tial waterworlds that might be O(10%) or more water bymass (Fu et al. 2010) and would therefore have vastly dif-ferent volatile cycles. Although partially ocean-coveredplanets have recently been referred to as aquaplanets(Abe et al. 2011), we will not adopt this terminology be-cause aquaplanet has a long history of being used todenote a completely ocean-covered planet in the climateand atmospheric dynamics communities.

We will find that the weathering behavior is fairly in-sensitive to land fraction when there is partial oceancoverage. For example, we will find that weatheringfeedbacks function similarly, yielding a habitable zoneof similar width, if a planet has a land fraction of 0.3

(like modern Earth) or 0.01 (equivalent to the combinedsize of Greenland and Mexico). In contrast, we will findthat the weathering behavior of a waterworld is drasti-cally different from a planet with partial ocean coverage.If seafloor weathering depends mainly on ocean acidity,rather than planetary surface temperature, no weath-ering feedback operates on a waterworld and it shouldhave a narrow habitable zone and progress through itquickly as its star ages. Finally, we will argue that itis possible for a waterworld to stop a moist greenhousein progress when continent is exposed by drawing downthe CO2 through massive weathering, which would leavethe planet in a clement state with partial ocean cover-age. We will refer to this possibility as a waterworldself-arrest.

This paper complements in two ways the recent workofAbe et al. (2011), who found that a nearly dry planetshould have a wider habitable zone than a planet withsome water. First, the calculations made by Abe et al.(2011) focused on climate modeling rather than weath-ering, although they included a qualitative description offactors that would influence weathering on a dry planet.Second, we consider a variety of land fractions up to thelimiting case of a waterworld.

The outline of this paper is as follows. We describeour model in Section 2, use it in Section 3, and performa sensitivity analysis in Section 4. We discuss the possi-bility of a waterworld self-arrest in Section 5. We outlineobservational implications and prospects for confirma-

tion or falsification of our work in Section 6. We discussour results further in Section 7, including considering thelimitations of our various assumptions, and conclude inSection 8.

2. MODEL DESCRIPTION

Here we will briefly review silicate weathering and thecarbon cycle before developing our model. The reader in-terested in more detail should consult Berner (2004) andChapter 8 ofPierrehumbert (2010). The carbon cycle ona terrestrial planet can be described as a long-term bal-ance between volcanic outgassing of CO2 and the burialof carbonates, mediated by chemical weathering of sili-

cates. Silicate minerals are weathered through reactionssuch as

CaSiO3(s) + CO2(g) CaCO3(s) + SiO2(s), (1)

where we have used CaSiO3 as an example silicate min-eral, but MgSiO3, FeSiO3, and much more complex min-erals can also participate in similar weathering reactions.

The reaction product CaCO3 is an example carbonateand silicon dioxide (SiO2) is often called silica. Thesereactions occur in aqueous solution and lead to a netdecrease in atmospheric CO2 if the resulting CaCO3 iseventually buried in ocean sediment, ultimately to besubducted into the mantle. CaCO3 is generally producedin the ocean by biological precipitation, but if there wereno biology weathering fluxes into the ocean would driveup carbonate saturation until abiotic CaCO3 precipita-tion were possible. Carbon in the mantle can be releasedfrom carbonate minerals if it reaches high enough tem-peratures, leading to volcanic outgassing and closing thecarbon cycle. The mantle reservoir of carbon is largeenough that CO2 outgassing does not depend on theamount of carbon subducted into the mantle.

Weathering reactions can occur either on continentsor at the seafloor and the weathering rate is the to-tal amount of CO2 per year that is converted into car-bonate by reactions like Equation (1) and subducted.The weathering rate is often measured in units of kilo-grams carbon per year (kg C yr1). Below we will workwith the dimensionless weathering rate, which is nor-malized such that it equals unity at the modern weath-ering rate. Because weathering reactions are aqueousand because rain increases erosion, allowing more rockto react, the continental weathering rate should increasewith increasing precipitation. Experiments suggest thatweathering reactions should increase with increasing CO2concentration, but that this dependence may be signif-

icantly reduced when land plants are present (Berner2004; Pierrehumbert 2010). Finally, weathering ratesare generally assumed to have an exponential depen-dence on temperature. This is based on temperature fits(Marshall et al. 1988) to cation, e.g., Ca2+ and Mg2+,data in river runoff (Meybeck 1979), and appears to beultimately due to an Arrhenius law for rock dissolution(Berner 1994).

These dependencies of continental weathering can becombined into the following approximation for Wl, thecontinental silicate weathering rate,

Wl

Wl0=

P

P0

a

0

beTT0Tu , (2)

where P is the precipitation rate, is the partial pres-sure of CO2, T is the surface temperature (see Table 1for a list of important model variables), and Tu = 10

C,a = 0.65, and b = 0.5 are constants. The subscript 0represents the current value and a tilde represents a di-mensionful quantity. The precipitation dependence ofcontinental weathering, a, is determined experimentallyby measuring the Ca2+ and Mg2+ solute concentrationin rivers and regressing against annual runoff. Berner(1994) and Pierrehumbert (2010) use a = 0.65 basedon a study in Kenya that found a = 0.6 (Dunne 1978)and a study in the United States that found a = 0.7


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TABLE 1List of the Most Important Model Variables

Variables DefinitionT Global mean surface temperature Partial pressure CO2 in the atmosphereS Stellar flux at planetary distance

(solar constant for Earth) Planetary land fraction

TS Climate sensitivity to external forcingW Weathering rate Outgassing rate

(Peters 1984). We will use a = 0.65 as our standardvalue, but we find in Section 4 that our main resultsare robust for a between 0 and 2. The CO2-dependenceof continental weathering, b, could be as low as 0 ifland plants eliminate the effect of CO2 on weather-ing (Pierrehumbert 2010), and the maximum value itshould take is 1 (Berner 1994). Both (Berner 1994) andPierrehumbert (2010) adopt the intermediate value of0.5, as we will here. Our main results are only sensi-tive to b if it becomes very small (Section 4). Variousestimates of the e-folding temperature, Tu, exist in the

literature, including Tu = 10.0 K (Pierrehumbert 2010),Tu = 11.1 K (Berner 1994), Tu = 21.7 K (Marshall et al.1988), and Tu = 17.7 K (Walker et al. 1981). We adoptTu = 10.0 K and show that our main conclusions arevalid for Tu = 5 25 K in Section 4.

Equation (2) can be nondimensionalized to take thefollowing form

Wl = PabeT, (3)

where Wl =WlWl0

, P = PP0

, = 0

, and T = TT0Tu

.

We parameterize the precipitation as a function of tem-perature as

P

P0= 1 + p(T T0), (4)

where p represents the fractional increase in precipi-tation per degree of warming. Climate change simula-tions in a variety of global climate models suggest thatp = 0.02 0.03, corresponding to a 2%-3% increase inglobal mean precipitation per Kelvin increase in globalmean temperature, is an appropriate value when T T0 (Schneider et al. 2010). Simulations in an idealizedglobal climate model, however, suggest that precipitationasymptotically approaches an energetically determinedconstant limit at T 300 310 K as the atmosphericoptical thickness is increased (OGorman & Schneider2008). The energetic constraint on precipitation wouldbe expected to change, however, if the insolation changes,which is the situation we will be considering below. Forsimplicity, we will take p = 0.025 in what follows.Because of the exponential dependence of continentalweathering on T in Equation (3), our results are onlyminimally affected by p, such that our results are notsubstantially altered even if we eliminate the tempera-ture dependence of precipitation by setting p = 0 (Sec-tion 4). Defining p = pTu = 0.25 we can nondimen-sionalize Equation (4) to find

P = 1 + pT. (5)

In addition to continental silicate weathering, reactionssimilar to Equation (1), which lead to the net consump-

tion of CO2, can occur at the seafloor. We will referto this low-temperature alteration of basaltic oceancrust as seafloor weathering. We do not considerhigh-temperature basaltic alteration here because itdoes not lead to net CO2 consumption. Seafloor weath-ering should depend on many factors, including oceanpH, the chemical composition of the ocean, the temper-ature at which the reactions occur, circulation of sea-

water through basaltic crust, and the seafloor spreadingrate. We will consider a constant seafloor spreading rateand assume a chemical composition similar to the mod-ern ocean. Based on laboratory measurements, Caldeira(1995) parameterized the seafloor weathering rate (Wo)as

Wo

Wo0=

[H+]m +

[H+]m0 + , (6)

where [H+] is the hydrogen ion concentration and m is aconstant. has a value of 103 and units of Mm, whereM is molar concentration, in order to satisfy dimensionalconstraints in Equation (6). This parameterization as-sumes that there is enough weatherable material at the

seafloor, and sufficient circulation of seawater through it,to accommodate arbitrary increases in ocean acidity. Ifthe weatherable material produced at ocean spreadingregions is completely used up before subduction occurs,Wo will be limited in a way not included in Equation (6).We will return to this assumption in Section 7.

Ocean chemistry leads to a functional relationship be-tween , the atmospheric CO2 concentration, and [H

+]of the ocean. We will assume that the ocean chem-istry can be described as a carbonate ion system cou-pled to an equilibrium relation for CaCO3. Althoughthis assumption neglects other sources of ocean alkalin-ity, e.g., Mg2+, Na+, and K+, it has been useful for mod-eling the carbon cycle in deep time Earth problems (e.g.,

Higgins & Schrag 2003). Furthermore, we implicitly as-sume that Ca2+ can be supplied to the ocean by sub-marine rocks in a waterworld or a planet with very lowland fraction. This system is described by the equilib-rium relations for the dissolution of CO2 and CaCO3,the disassociation of H2CO3, HCO

3 , and H2O, and al-kalinity (charge) balance (see chapter 4 of Drever 1988,for details). Using the equilibrium constants given byDrever (1988), we find that for an atmospheric CO2 con-centration between 104 and 102 bar, the dominant al-kalinity balance is 2[Ca2+] = [HC O3 ] in this system.

Since in the carbonate ion system [Ca2+] [H+]

[HCO3 ]

and [HCO3 ]

[H+] , this dominant balance leads to

[H+

] = c(T)2

3 with c(T) given by a combination of equi-librium constants. c(T) varies by only O(10%) between0C and 90C, and we therefore neglect this temperaturedependence below. This leads to the nondimensional ap-proximation [H+] =

23 , which allows Equation (6) to be

written in nondimensional form

Wo =

n +

1 +

, (7)

where Wo =WoWo0

, n = 23 m, and =

[H+]m0. Using

the Caldeira (1995) values (m = 12 and R = 103 M

12 )


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and assuming a reference ocean pH (pH=-log[H+]) of 8.2yields n = 13 and =12.6. Note that Caldeira (1995)discussed the possibility that an exponent to [H+] otherthan 12 may be appropriate and we will discuss otherscalings in Sections 4 and 7. We note that we have ne-glected any dependence of seafloor weathering on surfacetemperature, which is equivalent (e.g., Sleep & Zahnle

2001) to assuming the temperature at which seafloorweathering occurs (20-40C) is set by heat flux fromEarths interior, rather than by the temperature of sea-water. Since this assumption eliminates the potential fora weathering-climate feedback due to seafloor weather-ing (Section 3), it represents a conservative assumptionwhen considering the effect of changing the land frac-tion on weathering. We discuss the effect of allowing aseafloor weathering dependence on surface temperaturein Section 7.

Combining Eqs. (3), (5), and (7), and assuming thatcontinental weathering scales linearly with land fraction() and seafloor weathering scales linearly with oceanfraction (1-), the total weathering rate can be written

W = 0 0

(1 + pT)abeT + (1 0) 1 1 0

n

+ 1 +

,

(8)where 0 is the modern Earth land fraction and 0 =

Wl0Wl0+Wo0

is the fraction of total weathering accomplished

on continents for modern Earth conditions ( = 0,T = 0, and = 1). 0 will not generally be the ac-tual fraction of weathering accomplished on continentsfor other climatic conditions. The magnitude of theseafloor weathering rate on modern Earth is constrainedby drilling cores in the ocean floor, measuring the amountof basaltic rock that has been weathered, and scaling tothe entire ocean. Because relatively few core holes havebeen drilled and analyzed, and because weathering rates

differ significantly among them, the seafloor weatheringrate on Earth is only roughly known (Caldeira 1995).Caldeira (1995) assumes 0 = 0.67, Le Hir et al. (2008)assume 0 = 0.78, and we will take 0 = 0.75 as our stan-dard value, although we show that our results are robustas long as 0 0.51 in Section 4. Our assumption thatcontinental and seafloor weathering scale linearly withland and ocean fraction, respectively, cannot be rigor-ously justified and should be viewed as a first step. It ispossible, for example, that most seafloor weathering oc-curs near spreading centers so that total seafloor weath-ering is nearly independent of ocean fraction. If true thiswould not affect our main results significantly since theydo not depend on the details of the seafloor weathering

parameterization (Section 4), but would affect an eval-uation of the effect of seafloor weathering on the outeredge of the habitable zone, where strong seafloor weath-ering could keep CO2 below the limit imposed by CO2condensation or Rayleigh scattering (Section 7).

Global mean energy balance for a planet can be written

S

4(1 ) = (T , ), (9)

where S is the dimensionful insolation (S0=1365 W m2

for modern Earth), is the planetary Bond albedo, orreflectivity (0.3 for modern Earth), 14 is a geometric

factor representing the ratio of the area of a planet inter-cepting stellar radiation to the total planetary area, and(T , ) is the infrared radiation emitted to space (out-going longwave radiation). Below we will take = 0 tobe a constant for simplicity. We discuss this assumptionfurther in Section 7.

After nondimensionalizing ( = S0

, S = SS0

), the emit-ted thermal radiation, , can be Taylor expanded aroundcurrent Earth conditions (S = 1, T = 0, and = 1) tofind

(T, ) 1

4(1 ) + 1T 2 log , (10)

where 1 =T|T=0,=1,S=1 and 2 =

log()

|T=0,=1,S=1. Note that for a planet with

no atmosphere, T4, as one would expect for ablackbody; the current parameterization accounts forthe effects of the atmosphere, but is only a linearizationof more complete radiative transfer. Because the CO2absorption bands are saturated, the outgoing longwaveradiation is roughly linear in log (Pierrehumbert 2010),so we expand in log . We calculate 1 and 2 using

published polynomial fits to calculated using a singlecolumn radiative-convective model (Pierrehumbert2010). We find 1=0.015 using 1 = 1

TuS0

and calcu-

lating 1 as the average slope of between T = 275 Cand T = 325 K with CO2=1000 ppm and a relativehumidity of 50%. We find 2=0.007 using 2 =

2S0

and calculating 2 as the negative of the average slopeof with respect to log between CO2=280 ppm andCO2=2.8 105 ppm with a surface temperature ofT=280 K and a relative humidity of 50%. We showthat our main conclusions are unaltered if 1 and 2 arechanged by a factor or two in either direction in Section4. The linearization of (T, ) leads to the following

climate model (nondimensional form of Equation (9))

1

4(S 1)(1 ) = 1T 2 log . (11)

Taking to be the ratio of the CO2 outgassing rateto that on modern Earth, CO2 equilibrium in the at-mosphere requires = W, where W is given by Equa-tion (8). Such an equilibrium is set up on a million yeartimescale, so that kinetic (non-equilibrium) effects canbe neglected when considering variations in the weather-ing rate in response to changes in stellar luminosity. Wemake order-of-magnitude estimates on a kinetic processin Section 5 and we discuss the possibility that weather-ing cannot increase enough to establish = W in Section

7. = W, along with Equation (11), represents a cou-pled climate-weathering system that can be solved fortemperature and CO2 as a function of insolation (T(S)and (S)). For simplicity we will take = 1 in whatfollows, which is consistent with assuming a constantseafloor spreading rate and plate tectonics.

3. EFFECT OF LAND FRACTION ON WEATHERING

3.1. Simple Limits

Throughout this paper we will be interested in thestability that the carbon cycle can provide to the cli-mate system. We will consider specifically the change


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in surface temperature caused by a change in insolation(TS

), which we will call the climate sensitivity to ex-

ternal forcing. The lower TS

is, the more stable theclimate is, the wider the habitable zone will be, andthe longer a planet will stay in the habitable zone asits star evolves. We do not consider the many effectsthat could become important near habitable zone lim-its, such as water clouds on the inner edge (Selsis et al.2007; Kaltenegger & Sasselov 2011) and CO2 cloudson the outer edge (Forget & Pierrehumbert 1997;Selsis et al. 2007; Wordsworth et al. 2011; Forget et al.2012; Wordsworth et al. 2012), but T

Swill give us a

sense of how wide the habitable zone is in the broadrange between where these effects could be important.

We can calculate TS

by differentiating Equation (11)with respect to S

T

S=

1

41+

2

1

S. (12)

The change in CO2 as the insolation changes (S

) de-fines the weathering feedback strength. In the traditional

picture of the silicate weathering feedback (Walker et al.1981), the CO2 decreases as the insolation increases

(S

< 0), stabilizing the climate. If S

= 0 there is noweathering feedback, and the temperature increases at aconstant rate,

TS

0

= 141 , as the insolation increases.

To calculate S

, we need to consider weathering. Wecan generalize the weathering relations we described inSection 2 by rewriting Equation (8) as

W = f()Wl (, T) + g()W

o (, T), (13)

where f() = , g() = 1 , Wl (, T) 00

(1 +

pT)abeT, and Wo (, T)

1010

n+1+

in our stan-

dard model. The arguments in this section only requirethat f( = 1) = g( = 0) = 1 and f( = 0) = g( =1) = 0.

We will assume that the outgassing rate (), andtherefore the total weathering rate ( = W), does notdepend directly on insolation (S). This is a reason-able assumption for surface temperatures in the hab-itable zone, although the chemical composition of out-gassed material does vary at much higher temperatures(Schaefer & Fegley 2010). Using this assumption, we candifferentiate Equation (13) with respect to S to find

0 = f()

Wl

S+

WlT

T

S

+g()

W

o S + W

oT TS

. (14)

Equation (14) can be solved for S

, which can be sub-

stituted into Equation (12) to solve for TS

.Before proceeding we consider two simple limits. First

we consider a waterworld ( = 0 f = 0, g = 1) inwhich seafloor weathering does not depend on planetarysurface temperature (Wo (, T) W

o ()), as implied by

most seafloor weathering parameterizations (Section 2).

In this case all that is left of Equation ( 14) isWo

S

= 0.

If we assume that Wo has a dependence so that it can

adjust to match outgassing, we are left with S

= 0. Wetherefore see that the assumption that seafloor weather-ing does not depend on planetary surface temperatureimplies that there can be no weathering feedback on awaterworld.

T

time [Gyr]

Moist Greenhouse

Snowball

0 1 2 3 4 5 6 7 8

5

0

5

10

250

280

310

340

370

Temp.

[K]

log10

()

S

No Photosynthesis?

0.8 0.9 1 1.1 1.2 1.3 1.4

4

2

0

2

4

107

105

103

101

101

CO2

[bar]

Fig. 1. Temperature and CO2 concentration as a function ofinsolation with modern Earth volcanic outgassing ( = 1) and var-ious land fractions: = 0 (dashed), = 0.01 (light gray solid),= 0.1 (dark gray solid), and = 0.99 (black solid). Both dimen-sionless (left) and dimensionful (right) values are given on axes.For reference, the time relative to zero age main sequence for aplanet located 1 AU from a Sun-like star is plotted following Gough(1981). Gray shaded regions of the plot represent conditions thatmay be considered uninhabitable by some definition. The habit-

able zone upper temperature limit is the moist greenhouse, whichwe take to occur at 340 K (T = 5.3) (Kasting et al. 1993), and thelower temperature limit is global glaciation, which we take to occurat 265 K (T = 2.2) (Voigt et al. 2011), although it could occur10-20 K colder than this (Abbot et al. 2011; Yang et al. 2012). Thelower limit placed on CO2 for habitability in the figure, 105 bar,(Caldeira & Kasting 1992; Bloh et al. 2005) should be understoodonly as a potential limit on complex photosynthetic life, and noton other types of life.

As a second limit, we allow the land fraction ( ) to benon-zero but make seafloor weathering (Wo ) a constantindependent of temperature and CO2. While this limitdoes not accurately describe our parameterizations inSection 2, we do expect seafloor weathering to be weakerthan continental weathering and to respond weakly to

changes in CO2 (Caldeira 1995), which makes this limituseful for gaining insight into the full model. Further-more, seafloor weathering may be limited by the avail-ability of basaltic crust for reaction, which would leadto constant seafloor weathering (Sections 2 and 7). If seafloor weathering is set constant, then Equation (14)reduces to

0 = f()

Wl

S+

WlT

T

S

. (15)

Equation (15) implies that the change in CO2 as inso-

lation changes ( S

) is independent of land fraction ()


7/15


so that, by Equation (12), the climate sensitivity to ex-ternal forcing (T

S) must also be independent of land

fraction. This important result will lend insight to thefull model, in which we relax the assumption of constantseafloor weathering, where we find that T

Sis insensitive

to land fraction (), even for very small land fraction.Note that as long as f() is monotonically increasing

and all other parameters are held fixed, the continentalweathering rate decreases as the land fraction decreases(Equation (13)). This would lead to increases in temper-ature and CO2 (T and ) as land fraction () decreases,but they adjust precisely to keep T

Sconstant in the limit

of constant seafloor weathering.

3.2. Using the Full Model

We now consider the full model described in Section 2rather than just the simple limits considered in Section3.1. This means, for example, that seafloor weatheringis given by Equation (7) rather than held constant. Inthe full model there is a striking difference between theweathering behavior of a waterworld and a planet with

even a little land (Figure 1). Varying the land fraction() from 0.99 to 0.01 causes only a moderate warmingand has a very small effect on the climate sensitivity toexternal forcing (T

S). The functioning of the silicate

weathering feedback in the partially ocean-covered plan-ets can be seen from the decrease in CO 2 () with insola-tion (S; Figure 1). Decreasing from 0.99 to 0.01 wouldtherefore shift the habitable zone outward in insolation(S) slightly, but would not significantly alter its width.

Setting the land fraction to zero ( = 0), however,causes a huge shift in planetary weathering behavior.With = 0, the CO2 builds up to a high enough valuethat seafloor weathering can accommodate outgassing,and stays at this high value regardless of insolation (S),

so there is no weathering-climate feedback. This leadsto a large increase in surface temperature, CO2, and theclimate sensitivity to external forcing ( T

S) for the water-

world relative to the partially ocean-covered planets, sothe waterworld crosses the moist greenhouse thresholdmuch sooner (Figure 1). Therefore a planet with evena small amount of continent should enjoy a much widerhabitable zone, and longer tenure in it, than a water-world.

None of the curves with > 0 in Figure 1 cross themoist greenhouse threshold for S < 1.4 due to the sim-plifications in radiative transfer we have made, which isunrealistic when compared to more detailed calculations(Kasting et al. 1993); however, it is clear that a planetwith a lower land fraction will tend to be warmer andexperience a moist greenhouse at a lower insolation (S).On the other hand, a lower land fraction () leads to ahigher CO2 (), so a planet with a lower land fraction ()will cross the CO2 () threshold for C4 photosynthesis(Bloh et al. 2005) at higher insolation (S).

We can solve for climate sensitivity to external forcing(TS

) explicitly as follows

T

S=

1

41

b + (T , , )

21

(1 +ap

1+pT) + b + (T , , )

,

(16)

climate morestable

b=0.1

102

10

1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

b=0.5

S0.6 1 1.4

102

101

Fig. 2. Ratio of the climate sensitivity to external forcing ( TS

)

to the value it takes if there is no weathering feedback (TS

0

=

141

), as a function of the insolation (S) and the land fraction

() with b=0.1 (upper panel) and b=0.5 (lower panel). b is theexponent of atmospheric CO2 in the continental weathering rateparameterization (Equation (3)). Our standard value of b is 0.5,and we consider a lower value because the effect of atmosphericCO2 concentration on continental weathering may be reduced ifland plants are present (Pierrehumbert 2010).

where

(T , , ) =

(1 0)00(1 0)

n(1 )

nbeT

(1 + )(1 + pT)a

. (17)

Note that the factor multiplying 141 = TS

0in Equa-

tion (16) is always less than or equal to one, which re-

duces the climate sensitivity to external forcing ( TS) be-low

TS

0

and is a statement of the silicate weather-ing feedback. Also, if 0 = 1 (no seafloor weathering), = 0 a n d T

Shas no dependence on the land frac-

tion (), as we expected from Section 3.1. Additionally,lim,00 = and lim

TS

=TS

0

, which re-covers the waterworld limit.

We plot the climate sensitivity to external forcing ( TS

,Equation (16)) as a function of insolation (S) and landfraction () for different values of b in Figure 2. It isuseful to consider a lower value of b because the effect


8/15


of CO2 () on continental weathering may be greatly re-duced if land plants are present (Pierrehumbert 2010),as we will discuss further in Section 4. The climate sen-sitivity to external forcing ( T

S) increases for small land

fraction (), but insolation (S) has a much stronger effecton T

Sthan land fraction () does. The climate sensitiv-

ity to external forcing ( TS

) is highest at low insolation

(S) and land fraction () and decreases as S and in-crease. lim1 = 0 and 0 as S becomes largebecause T becomes large (Equation (17)). This allowsthe approximation that if either the planet is more than10% land covered and/or the insolation is as large asor greater than the modern Earth value, T

Sapproaches

the limit

T

S=

T

S

0

b

21

(1 +ap

1+pT) + b

T

S

0

b

21

+ b

. (18)

Therefore, despite the many parameters in our model(Section 2), the climate sensitivity to external forcing(TS

, Equation (16)) can be reduced to a function of onlya few relevant variables over most of the land fraction-insolation plane (Equation (18)). In particular, if we ne-glect the (usually small) term

ap1+pT

, the only weathering

parameter present in Equation (18) is b, which controlsthe strength of the scaling of land weathering with CO2(other remaining variables are from the climate model).The weaker land weathering scales with CO2 (lower b),the less T can change while maintaining a balance be-tween outgassing and weathering, and the lower T

Sis.

In the context of this paper, what is important aboutEquation (18) is that for a broad range of insolation,

the climate sensitivity to external forcing (T

S) is nearthe limit given by Equation (18) and is therefore nearlyindependent of land fraction (). This explains the in-sensitivity of the climate sensitivity to external forcing(TS

) to land fraction in this model.

4. SENSITIVITY TO MODEL PARAMETERS

In this Section we will evaluate the robustness of our re-sult from Section 3 that the weathering behavior, specif-ically the climate sensitivity to external forcing (T

S),

of a planet is relatively insensitive to surface land frac-tion (). We will compare T

Sfor a small land frac-

tion ( = 0.01) with that for an Earth-like land frac-tion ( = 0.3) while we vary model parameters. For

simplicity we consider reference insolation, outgassing,and weathering values (S = = W = 1) and define =

TS

|T=0,=1,S=1,=0.01

TS

|T=0,=1,S=1,=0.3

as a measure of the change in TS

for a large change inland fraction.

We first consider the effect on our results if current es-timates of the fraction of total weathering accomplishedon continents on modern Earth (0) were incorrect. Alower 0 would increase

TS

(Equation (16)), and this ef-fect is exaggerated when the land fraction () is smaller,increasing (Figure 3). For 0 less than approximately0.5, starts to become relatively large. Given current

estimates of 0.67 (Caldeira 1995) and 0.78 (Le Hir et al.2008) for 0, seafloor weathering on modern Earth wouldhave to be 2-3 times larger than it is thought to be toaffect our main conclusions.

Next we consider changes in the seafloor weatheringscheme. Ionic concentration in seawater may either in-crease or decrease n (Caldeira 1995), the exponent of inthe seafloor weathering parameterization (Equation (7)).

The constant m in Equation (6), however, is typicallytaken to be less than 1 (Caldeira 1995), which wouldcorrespond to n = 23 . Such a change would lead to amoderate increase in (Figure 3), although a larger in-crease in n would significantly increase . It is less clearwhether a vastly different value of could be appropriate,but only an extremely large decrease in would signif-icantly increase (Figure 3). The large effect that 0and n would have on if appropriate values were out-side current ranges highlights the importance of more ex-perimental and field work to better understand seafloorweathering.

The exponent of precipitation in Equation (3), a, andthe constant defining the increase in precipitation with

temperature (Equation (5)), p, have a small effect onTS

(Figure 3). This is because the exponential temperaturedependence of the Arrhenius component of the conti-nental weathering parameterization (Equation (3)) dom-inates changes in continental weathering due to temper-ature. Moreover, changes in a and p affect

TS

similarlyat low and high land fraction (), leading to only verysmall changes in when these parameters are changed(Figure 3).

We next consider changes in b, the exponent of CO2 inthe continental weathering parameterization. If seafloorweathering is constant , setting b = 0 yields a perfectthermostat (T

S= 0), since weathering is a function of

T alone in this case (Equation (8)). As b is decreased

in the full model, TS decreases, with larger changes forlarger land fraction (), since the model behaves morelike the constant seafloor weathering case in this limit(Figure 3). For very small b, becomes somewhat large,although this effect is greatly diminished for insolationlarger than the reference value S > 1 (Figure 2), whenthe temperature becomes large enough to make small(Section 3).

Finally, we consider the terms defining the expansionof infrared emission to space (, outgoing longwave ra-diation) as a function of temperature (1 =

T

) and the

logarithm of CO2 (2 =

log() ). We only kept the first

order terms in these expansions, and we made specific as-sumptions to calculate them (Section 2), so it is worthconsidering whether variations in 1 and 2 affect our re-sults. Decreasing 1 has the simplistic effect of increasingthe climate sensitivity to external forcing with no weath-ering feedback

TS

0

= 141 . We focus on weatheringfeedbacks in Figure 3 by normalizing the climate sensitiv-ity to external forcing by

TS

0

. 1 and 2 enter the re-lation for the weathering feedback in Equation (16) (thepart multiplying

TS

0

on the right hand side) throughthe ratio 2

1, which is a measure of the climate sensitiv-

ity to changes in CO2. Increasing the climate sensitivity


9/15


0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

0

T

S

/

(T

S

)0

0.2 0.4 0.6 0.8n

10 20 30

0 1 2a

0 0.2 0.4 0.60

0.2

0.4

0.6

0.8

1

p

T

S

/

(T

S

)0

0 1 2b

10 20 3010

3

1

5 10 1510

3

2

Fig. 3. Ratio of the climate sensitivity to external forcing (TS

) to the value it takes if there is no weathering feedback (TS

0

= 141

)

at both low land fraction (= 0.01, thick black solid line) and moderate land fraction ( = 0.3, thick gray solid line). TS

is displayed as

a function of0, the fraction of total weathering accomplished on continents on Earth today; n, the exponent of atmospheric CO2 () inthe seafloor weathering parameterization (Equation (7)); , the constant in the seafloor weathering parameterization; a, the exponent ofprecipitation in the continental weathering parameterization (Equation (3)); p, the constant defining the increase in precipitation withtemperature (Equation (5)); b, the exponent of atmospheric CO2 () in the continental weathering parameterization (Equation (3)); 1,

the first term in the expansion of the outgoing longwave radiation () with temperature ( T

; Equation (10)); and 2, the first term in

the expansion of outgoing longwave radiation with the logarithm of CO2 (

log(), Equation (10)). Circles and the thin black line signify

standard values of parameters.

to changes in CO2 (21

) increases the warming effect ofa given increase in CO2, increasing the weathering feed-back strength, and decreasing the climate sensitivity to

external forcing (Figure 3). This effect shows little de-pendence on land fraction, so large changes in 1 and 2do not alter our main results (Figure 3).

We note that Tu, the exponential temperature scalefor the direct temperature dependence in Equation (2),comes into the nondimensional relations through p =

pTu and 1 = 1TuS0

. Since reasonable variations in pand 1 do not alter our main conclusions, we can con-clude that reasonable changes in Tu would not. Indeed,we find that the climate sensitivity to external forcingdepends minimally on land fraction () when we vary Tubetween 5 and 25 K (not shown).

5. WATERWORLD SELF-ARREST

Due to the lack of a climate-weathering feedback ina waterworld, we have argued that a waterworld shouldexperience a moist greenhouse at much lower insolation(S) and have a smaller habitable zone than a planet witheven a very small land fraction. We will now discuss thepossibility that a waterworld could stop a moist green-house by engaging silicate weathering when enough oceanis lost that continent is exposed, engaging weatheringand reducing temperature (T), and thereby preventingcomplete water loss. We will refer to this possibility asa waterworld self-arrest (see Figure 4 for a schematicdiagram of the idea). The waterworld self-arrest is sim-

ilar to the idea, discussed by Abe et al. (2011), that aplanet with partial ocean coverage could experience amoist greenhouse and become a habitable land planet.

If a waterworld experiences a moist greenhouse, therewill be some amount of time before any continent is ex-posed and continental weathering is possible. We callthis timescale 0 (Figure 5), and we will not use it in thepresent argument. We call the time after the first expo-sure of continent until all remaining water is lost throughhydrogen escape to space m. To determine whether awaterworld self-arrest is possible, we must compare mto the timescale for CO2 to be reduced by continentalsilicate weathering to a level such that moist greenhouseconditions cease to exist, which we call w. If w < m,then a waterworld self-arrest is possible.

If we take the hydrogen atom mixing ratio in thestratosphere to be unity as a limiting case, the diffusion-limited rate of loss of hydrogen atoms to space is 21013

H atoms cm2 s1 (Kasting et al. 1993). Earth wouldbe essentially a waterworld if the ocean were a few timesmore deep, which would correspond to 1029 H atomscm2, so that m is of order 100 Myr. Since we seek onlyan order of magnitude here, we do not consider the effectof different topography on m, but note that larger plan-ets will tend to have less topography, which would reducem somewhat. We note also that energetic limitation onhydrogen escape could increase m (Watson et al. 1981;Abe et al. 2011).

w is much more difficult to estimate than m. Theclosest known analog to a moist greenhouse in post-


10/15


time [Gyr]

T

Moist Greenhouse

Snowball

WaterworldSelfArrest

0 1 2 3 4 5 6 7 8

4

2

0

2

4

6

0.8 0.9 1 1.1 1.2 1.3 1.4

250

280

310

340

Temp

.[K]

S

Fig. 4. Schematic diagram showing how the waterworld fromFigure 1 could experience a waterworld self-arrest as the insolation(S) increases and end up as an Earth-like planet with partial oceancoverage ( = 0.3 in this case) in the habitable zone. Axes are asin Figure 1.

0: time to expose

continent

m: time to complete

water loss after

continent exposed

water loss

Waterworld Self-Arrest Timescales

water level

initially

submergedcontinent

Fig. 5. Schematic diagram of the timescales involved in a wa-terworld self-arrest. 0 is the time the planet exists in a moistgreenhouse state before submerged continent is exposed. m is thetime to complete planetary water loss after the first exposure ofcontinent. In order to determine whether a waterworld self-arrestis possible, m must be compared with w, the timescale for CO2to be sufficiently reduced by silicate weathering that moist green-house conditions cease to exist.

Hadean Earth is likely the aftermath of the SnowballEarth events, which occurred 600700 million years ago.

The dominant explanation for Snowball Earth events(Kirschvink 1992; Hoffman et al. 1998) is that some cool-ing factor forced Earth into a global glaciation, which isan alternative stable state of climate because of the highreflectivity of snow and ice. Because it would be very coldduring a Snowball and because any precipitation wouldfall as snow, rather than rain, the continental weather-ing rate would be near zero. As we expect volcanic CO2outgassing to continue unabated, the atmospheric CO2would increase until it reached up to a thousand timesits present value ( = O(103)), at which point it wouldbe warm enough to melt the ice and end the Snowball.In the immediate aftermath of a Snowball Earth event

the planetary albedo would be very low and the CO 2concentration would be very high, which would lead tosurface temperatures in the range of a moist greenhouse.Extremely high weathering would then reduce the at-mospheric CO2 until Earths surface temperatures werecomparable to modern values (Higgins & Schrag 2003).If post-Snowball Earth was truly in a moist greenhousestate, complete water loss must not have occurred, which

would imply that w < m and a waterworld self-arrestis possible. Even if post-Snowball Earth was not a truemoist greenhouse, the drawdown of CO2 must have beenimmense, and geochronological evidence indicates thatthis took less than 4 Myr (Condon et al. 2005). Thisfurther motivates the idea that w could be small com-pared to m.

The CO2 required to end a Snowball Earth, however,may have been 1-2 orders of magnitude lower than thatquoted above (Abbot & Pierrehumbert 2010), in whichcase the Snowball aftermath would be a weaker analogfor a moist greenhouse and further consideration is re-quired. Geochemical modeling under conditions similarto a moist greenhouse yield weathering rates ofO(1013)

kg C yr1

(Higgins & Schrag 2003) and O(1012

) kg Cyr1 (Le Hir et al. 2009), although Mills et al. (2011)suggest that ion transport constraints could limit theweathering rate to 5 1011 kg C yr1 (Mills et al.2011). The waterworld CO2 is 1.65 10

4 times thepresent value (Figure 1) in our model with standard pa-rameters, which corresponds to O(1019 kg C). This yieldsan estimate ofw = O(1-10 Myr), which further suggeststhat w < m.

Although we have not performed a full analysis of thekinetic (non-equilibrium) effects, the order-of-magnitudeanalysis we have done indicates that a habitable zone wa-terworld could stop a moist greenhouse through weath-ering and become a habitable partially ocean-coveredplanet. We note that this process would not occur ifthe initial water complement of the planet is so largethat continent is not exposed even after billions of yearsin the moist greenhouse state (0 > O(10 Gyr)). Addi-tionally, we have neglected the possibility of water ex-change between the mantle and the surface, which ispoorly constrained on Earth. If, for some reason, waterwere continuously released from the planetary interiorto replace water lost through hydrogen escape, then theconclusions reached here would be invalid. This wouldrequire significantly higher mantle-surface fluxes than onpresent Earth, which are constrained by cerium mea-surements at mid-ocean ridges to be small enough torequire 10 Gyr to replace Earths surface water reser-voir (Hirschmann & Kohlstedt 2012). Finally, we note

that since we have considered a moist greenhouse causedby high atmospheric optical thickness rather than highinsolation, the waterworld would not be likely to suffera full runaway greenhouse during the process describedhere (Nakajima et al. 1992).

6. DISCUSSION OF OBSERVABILITY

The theory developed here suggests that if an Earth-size planet is detected near the habitable zone it shouldbe considered a good candidate to have a wide habitablezone and a long-term stable climate as long as it has someexposed land. The first observational point to consider,


11/15


therefore, is whether the surface land fraction could bemeasured on an Earth-like planet.

Using measurements of the rotational variations ofdisk-integrated reflected visible light from a future mis-sion such as TPF-C it will be possible to determine thesurface land fraction of an Earth-like planet orbiting aSun-like star. Cowan et al. (2009) used observations ofEarth from the Deep Impact spacecraft to demonstrate

that this is possible with multi-band high-cadence pho-tometry. The multi-band observations are critical be-cause the albedo of land and ocean have very differ-ent wavelength dependencies, while the integration timesmust be sufficiently short to resolve the planets rota-tional variability. Clouds tend to obscure surface featuresand make such a determination more difficult (Ford et al.2001; Kaltenegger et al. 2007; Palle et al. 2008), and re-gions of a planet that are consistently cloudy are im-pervious to this form of remote sensing (Cowan et al.2011). Kawahara & Fujii (2010, 2011) have extendedthis method to include seasonal changes in illumina-tion to generate rough two-dimensional maps of other-wise unresolved planets. Provided that a habitable zone

planet is not much cloudier than Earth, its land fractionshould therefore be observationally accessible with suffi-cient signal-to-noise ratio, and the present study can beused to help assess the probability of its habitable zonewidth and long-term habitability.

The presence of exposed land could be detected onan exoplanet via an infrared emission spectrum, pro-vided the overlying atmosphere is not too much of ahindrance (Hu et al. 2012). At thermal wavelengths,the land fraction of a planet could in principle be es-timated based on its thermal inertia, but in idealizedtests Gaidos & Williams (2004) found that the obliquity-inertia degeneracy impedes quantitative estimates of aplanets heat capacity. It is therefore unlikely that athermal mission, on its own, could constrain the land

fraction of an exoplanet.Direct confirmation or falsification of the theory de-

veloped here will be more difficult, requiring not onlyan estimate of land fraction, but also atmospheric CO2abundance and/or surface temperature. The differencebetween the temperature and CO2 of a waterworld and apartially ocean-covered planet could be quite large, par-ticularly near the inner (hotter) edge of the habitablezone (Figure 1). This entails measuring, in exoplan-ets, discrepancies in carbon dioxide abundance of roughlyfour orders of magnitude, and surface temperatures dif-ferences of 70 K.

Detecting carbon dioxide in an exoplanet atmo-sphere is relatively easy: the James Webb Space Tele-scope

will determine whether CO2 is present in theatmospheres of transiting super-Earths orbiting M-stars (Deming et al. 2009; Kaltenegger & Traub 2009;Belu et al. 2011; Rauer et al. 2011). The challenge is inestimating the amount of CO2 in an atmosphere. In or-der to measure the carbon dioxide abundance of an atmo-sphere, the atmospheric mass must be estimated. Thiscould be achieved for M-Dwarf habitable zone planetswith transit spectroscopy covering the Rayleigh scatter-ing slope (Benneke & Seager 2012). The unknown radiusof directly-imaged (longer-period) exoplanets may hindersuch estimates with a TPF-C type mission. At thermalwavelengths, the planetary radius will be measurable and

the pressure broadening of absorption features may offeran estimate of surface pressure and hence atmosphericmass (e.g., Des Marais et al. 2002; Meadows & Seager2010).

Using a thermal mission like TPF-I, it may be possibleto determine the surface temperature of an Earth ana-log via its brightness temperature in molecular opacitywindows, at least for a cloud-free atmosphere with few

IR absorbers (Des Marais et al. 2002; Kaltenegger et al.2010; Selsis et al. 2011). While low-lying clouds do notsignificantly degrade such estimates, high cloud coveragecan lead to surface temperature estimates at least 35 Ktoo low (Kaltenegger et al. 2007; Kitzmann et al. 2011).Continuum opacity from water vapor will also lead tolower brightness temperatures, even in opacity windows.This could lead to under-estimating the surface temper-ature of planets with high vapor content, precisely themoist greenhouse planets expected to have high surfacetemperatures. Moreover, Venus dramatically demon-strates that brightness temperatures, even in supposedwindows, are not necessarily a good probe of the surfacetemperature.

Given the similar temperature and CO2 among par-tially ocean-covered planets of varying land fraction (Fig-ure 1), it is unlikely that any mission in the foreseeablefuture would be able to distinguish between them. Ad-ditionally, factors that we have neglected here introduceuncertainties that make the temperature and CO2 abun-dance among partially ocean-covered planets effectivelyindistinguishable.

To summarize, measurements from a mission like TPF-C would provide a surface land fraction estimate for plan-ets not much cloudier than Earth. TPF-C could pro-vide an order-of-magnitude atmospheric CO2 estimateif the radius were known, which would be possible if(1) the planet were transiting, (2) its radius were esti-mated from TPF-I observations, or (3) its radius were

estimated based on a mass measurement. TPF-I couldprovide a surface temperature estimate that would likelybe good to O(10 K) for an Earth analog, but possiblymuch worse for a cloudier and/or more humid atmo-sphere. It would therefore be possible to directly testour predictions for differences in weathering behavior ofwaterworlds and partially ocean-covered planets if datafrom missions like both TPF-C and TPF-I were available,and sufficient numbers of nearby habitable zone planetsexist. Prospects are limited, but testability is still possi-ble in special cases (such as a transiting planet), if onlyone type of data is available.

A final observational point is that in Section 5 we cal-culated that a waterworld self-arrest, through which a

fully ocean planet would be transformed into a partialocean planet, is possible. Waterworlds, however, wouldstill exist and their detection would not falsify the water-world self-arrest idea. Waterworlds could have orders ofmagnitude more water than Earth (e.g., Raymond et al.2007; Fu et al. 2010), and could therefore exist in a moistgreenhouse state for many billions of years. Additionally,a waterworld in the outer regions of the habitable zonewould not be in a moist greenhouse state (Figure 4).

7. GENERAL DISCUSSION

We note again that we make numerous approximationsin our model, so that it is useful for qualitative under-


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standing rather than quantitative predictions. Our lin-earized climate model, for example, is a gross simplifi-cation of the real climate system, which is probably re-sponsible for the delayed onset of the moist greenhousein our model as the insolation is increased (Figure 1) rel-ative to more detailed calculations (Kasting et al. 1993).Our approach, however, is appropriate for a generalstudy of potential terrestrial exoplanets, whose detailed

characteristics are not known, and our main conclusionsshould be robust to more detailed modeling. For exam-ple, we point out that no weathering feedback is possi-ble (T

S=TS

0

) for a waterworld as long as seafloorweathering is independent of surface temperature (T),regardless of the specific form of the seafloor weatheringparameterization.

Nonetheless, our parameterization of seafloor weather-ing is probably the most uncertain aspect of this work.For example, we have neglected the potential effect ofa different ocean chemical composition on an exoplanet,which could lead to different base weathering rates ora different scaling of seafloor weathering with CO 2 ().Furthermore, in order to derive a scaling relationship be-

tween [H+

] and , we have approximated ocean chem-istry as a carbonate ion system with changes in alkalinitydetermined by changes in Ca2+. Real oceans have otherions present, which can alter the charge balance. Theseapproximations, however, have been useful for model-ing carbon cycling in deep time Earth problems (e.g.,Higgins & Schrag 2003) and our knowledge of typical ionconcentrations in terrestrial exoplanets is unconstrained.Finally, we have assumed that surface temperature hasa negligible effect on seafloor weathering rate, which wewill now consider in more detail.

As there is a temperature dependence of the reac-tions relevant for seafloor weathering (Brady & Gislason1997), we have effectively assumed that surface temper-ature does not strongly affect the temperature at whichseafloor weathering occurs. If a dependence of seafloorweathering on surface temperature is allowed, a climate-seafloor weathering feedback is possible, which shouldtend to reduce the effect of changing land fraction. Interms of the effect of land fraction on weathering behav-ior, we have therefore considered a conservative case inassuming no surface temperature dependence of seafloorweathering, yet we still found that land fraction has onlya small effect on weathering behavior for partially ocean-covered planets.

A strong temperature dependence of seafloor weather-ing would affect waterworld behavior at the inner edge ofthe habitable zone. We can see this by multiplying theseafloor weathering parameterization (Equation (7)) by

an exponential temperature dependence of the form eT,where = dTu

Tw, where d is the change in seafloor weather-

ing reaction temperature per unit change in surface tem-perature, and Tw 15K is the exponential scale of in-crease in seafloor weathering reaction rate with tempera-ture (Brady & Gislason 1997). The largest value couldtake would be if current reactions occur near the bot-tom water temperature (0C) and each unit increasein surface temperature creates a unit increase in bottomwater temperature (d=1, = 23 ). Our standard assump-tion is that seafloor weathering reactions occur at a tem-perature of20-40C regardless of surface temperature,

which is equivalent to d==0. Using = 23 increases theinsolation required to cause a waterworld moist green-house from S1.2 to S2.1 if=0 and prevents a water-world moist greenhouse from ever happening if =12.6.A strong temperature dependence of seafloor weatheringwould therefore make it much harder for a waterworldto experience a moist greenhouse, although we do notconsider the limiting case considered in this paragraph

likely.As noted in Section 2, we assume that seafloor spread-

ing produces a sufficient quantity of weatherable materialand a sufficient amount of water can circulate through itthat the seafloor weathering rate can increase arbitrar-ily as the ocean pH decreases. Since we assume seafloorweathering accounts for 25% of weathering on modernEarth (0 = 0.75), if the CO2 outgassing rate takes itsmodern value ( = 1), then the seafloor weathering ratemust be able to increase by a factor of four if it alone is tobalance outgassing, as would be required to establish aCO2 steady state on a waterworld. If there is not enoughweatherable material for this to be possible, for exampledue to weak circulation of seawater through weatherable

material, then the waterworld CO2 will not be limitedto the value calculated in Section 3 (Figure 1), but willinstead steadily increase due to an unbalanced sourceand sink. This would push a waterworld more quicklyto a moist greenhouse, and potentially a waterworld self-arrest, than one would otherwise expect. A related issueis that if the land fraction is very small, there may bephysical limits on continental weathering not included inEquation (3). This would lead to higher CO2 concen-trations and temperatures for small land fractions thanthose found in Section 3. The smallest land fraction weexplicitly considered, however, was = 0.01, which cor-responds to the combined area of Greenland and Mexicoon Earth. Given continuous resurfacing it is not clearthat physical limitations on weathering would be in play

on such a large land area.We have focused on the effect on land fraction on

weathering behavior relevant for the broad interior of thehabitable zone and the evolution of a planet through thehabitable zone over time. The effect of land fraction onthe outer (cold) edge of the habitable zone is of interestas well. Based on increased Rayleigh scattering or CO2condensation at the surface, the outer edge of the hab-itable zone occurs when CO2510 bar (Kasting et al.1993). Le Hir et al. (2008), however, found that seafloorweathering should limit the maximum CO2 concentra-tion on Earth to a few tenths of a bar. As pointed outby Pierrehumbert (2011), if this is the case, then theCO2 condensation limit could not be reached and seafloor

weathering would shrink the habitable zone. Using ourseafloor weathering parameterization, we can derive anupper limit on the atmospheric CO2 concentration with

Earths land fraction ( = 0): =

(1+)(10)

1n

. Us-

ing our standard parameters, we find = 7.3 104, or20 bar. depends strongly, however, on model param-eters (Figure 6), particularly n. = 103, roughly thevalue Le Hir et al. (2008) found, would be produced byour model by either a change in n from 0.33 to 0.54 ora change in from 12.6 to 2.0. Therefore, in contrastto our main results (Section 4), the weathering behavior


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we neglected effects of changing planetary size. Super-Earths would have higher gravity and therefore reducedorography. This will reduce the amount of time it takesfrom first exposure of land to complete water loss (m),which will make a waterworld self-arrest less likely. Onthe other hand, the higher gravity of a super-Earth willalso make gas escape from the atmosphere to space moredifficult, which would reduce m. Full consideration of

a waterworld self-arrest on a super-Earth is beyond thescope of this paper.

8. CONCLUSIONS

We have shown that using standard weathering pa-rameterizations, the weathering behavior of a partiallyocean-covered Earth-like planet does not depend stronglyon land fraction, as long as the land fraction is greaterthan 0.01. Consequently, planets with some continentand some ocean should have a habitable zone of simi-lar width. This is a powerful result because it indicatesthat previous habitable zone theory developed assumingEarth-like land fraction and weathering behavior shouldbe broadly applicable.

We have also pointed out that, as long as seafloorweathering does not depend directly on surface tempera-ture, a climate-weathering feedback cannot operate on awaterworld. This is a significant result because it would

imply that waterworlds have a much narrower habitablezone than a planet with even a few small continents.We find, however, that weathering could operate quicklyenough that a waterworld could self-arrest while un-dergoing a moist greenhouse and the planet would beleft with partial ocean coverage and a clement climate.If this result holds up to more detailed kinetic weatheringmodeling, it would be profound, because it implies that

waterworlds that form in the habitable zone have a path-way to evolve into a planet with partial ocean coveragethat is more resistant to changes in stellar luminosity.

9. ACKNOWLEDGEMENTS

We thank Jens Teiser for suggesting that we considerwhether weathering could draw down CO2 fast enoughto stop a moist greenhouse and David Archer and RayPierrehumbert for fruitful discussions early in the devel-opment of this project. We acknowledge input from Fran-cis MacDonald on Snowball geochronology, John Higginson weathering parameterizations, and Bob Wordsworthon the effect of CO2 clouds on the habitable zone. JacobBean, Albert Colman, Itay Halevy, Guillaume Le Hir,

Edwin Kite, Daniel Koll, Arieh Konigl, and an anony-mous reviewer provided helpful comments on early draftsof this paper. We thank Bjorn Benneke for sharing anearly draft of his manuscript.

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