The Astrophysical Journal, 775:96 (8pp), 2013 October 1

doi:10.1088/0004-637X/775/2/96


C 2013. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

THE STRUCTURE OF SURFACE H2 O LAYERS OF ICE-COVERED PLANETS WITH HIGH-PRESSURE ICE
S. Ueta and T. Sasaki
Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku,
Tokyo 152-8551, Japan; ueta@geo.titech.ac.jp, takanori@geo.titech.ac.jp
Received 2012 November 12; accepted 2013 July 30; published 2013 September 11

ABSTRACT
Many extrasolar (bound) terrestrial planets and free-floating (unbound) planets have been discovered. Whilethe existence of bound and unbound terrestrial planets with liquid water is an important question, of particularimportance is the question of these planets’ habitability. Even for a globally ice-covered planet, geothermal heatfrom the planetary interior may melt the interior ice, creating an internal ocean covered by an ice shell. In this paper,
we discuss the conditions that terrestrial planets must satisfy for such an internal ocean to exist on the timescaleof planetary evolution. The question is addressed in terms of planetary mass, distance from a central star, waterabundance, and abundance of radiogenic heat sources. In addition, we investigate the structure of the surface H2 O
layers of ice-covered planets by considering the effects of ice under high pressure (high-pressure ice). As a fiducialcase, a 1 M⊕ planet at 1 AU from its central star and with 0.6–25 times the H2 O mass of the Earth could have aninternal ocean. We find that high-pressure ice layers may appear between the internal ocean and the rock portion
on a planet with an H2 O mass over 25 times that of the Earth. The planetary mass and abundance of surface waterstrongly restrict the conditions under which an extrasolar terrestrial planet may have an internal ocean with nohigh-pressure ice under the ocean. Such high-pressure ice layers underlying the internal ocean are likely to affectthe habitability of the planet.
Key words: astrobiology – planets and satellites: composition – planets and satellites: general – planets andsatellites: interiors – planets and satellites: surfaces

only a few kilometers of ice would form at the surface of theocean (Hoffman & Schrag 2002) and life could exist in the liquidwater under the surface ice shell (Hoffman et al. 1998; Hoffman& Schrag 2002; Gaidos et al. 1999).
Another possibility is presented by planets that float in space
without being gravitationally bound to a star (free-floating
planets), as have been found thanks to recent advances inobservational techniques (Sumi et al. 2011). Although suchplanets receive no energy from a central star, even a free-floating
Earth-sized planet with considerable geothermal heat could haveliquid water under an ice-covered surface.
Considering geothermal heat from the planetary interior,Tajika (2008) discusses the theoretical restrictions for icecovered extrasolar terrestrial planets that, on the timescale ofplanetary evolution, have an internal ocean. Tajika (2008) showsthat an internal ocean can exist if the water abundance andplanetary mass are comparable to those of Earth. A planet witha mass less than 0.4 M⊕ cannot maintain an internal ocean.
For a planet with mass 4 M⊕ , liquid water would be stable
either on the planetary surface or under the ice, regardless of
the luminosity of the central star and the planetary orbit. These
are important conclusions and have important implications forhabitable planets.
In this paper, we extend the analysis of Tajika (2008) and vary
the parameter values such as the abundance of radiogenic heatsources and the H2 O abundance on the surface. Although Tajika(2008) assumed that the mass ratio of H2 O on the planetary surface is the same as that on Earth (0.023% by weight), the origin
of water on the Earth is not apparent (Genda & Ikoma 2008)
so it is possible that extrasolar terrestrial planets have someorder of the H2 O abundance of Earth. We investigate this possibility by varying the H2 O abundance in our simulations, andalso by checking whether ice appears under H2 O layers underhigh-pressure conditions (see Section 2.2). Therefore, in this

1. INTRODUCTION
Since the first extrasolar planet was discovered in 1995
(Mayor & Queloz 1995), more than 800 exoplanets have been
detected as of 2013 March, owing to improvements in bothobservational instruments and analysis methods. Although most
known exoplanets are gas giants, estimates based on boththeory and observations indicate that terrestrial planets are also
common (Howard et al. 2010). Supporting these estimates isthe fact that Earth-like planets have indeed been discovered.
Moreover, space telescopes (e.g., Kepler) have now released
observational data about many terrestrial planet candidates.
Whether terrestrial planets with liquid water exist is an importantquestion to consider because it lays the groundwork for theconsideration of habitability.
The orbital range around a star for which liquid water canexist on a planetary surface is called the habitable zone (HZ;
Hart 1979; Kasting et al. 1993). The inner edge of the HZ isdetermined by the runaway greenhouse limit (Kasting 1988;
Nakajima et al. 1992) and the outer edge is estimated from theeffects of CO2 clouds (Kasting et al. 1993; Mischna et al. 2000).
The region between these edges is generally called the HZ for
terrestrial planets with plentiful liquid water on their surfaces(ocean planets). Planets with plentiful water on the surface but
outside the outer edge of the HZ would be globally covered
with ice and no liquid water would exist on the surface. These
planets are called “snowball planets” (Tajika 2008). Moreover,an ocean planet could be ice-covered even within the HZ because
multiple climate modes are possible, including ice-free, partially
ice-covered, and globally ice-covered states (Budyko 1969;
Sellers 1969; Tajika 2008). Although such planets would be
globally ice-covered, liquid water could exist beneath thesurface ice shell if sufficient geothermal heat flows up fromthe planetary interior to melt the interior ice. In this scenario,
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Ueta & Sasaki

We assume that the surface ice is hexagonal ice (ice Ih). Between
0.5 K and 273 K, the thermal conductivity of ice Ih is known(Klinger 1975; Varrot et al. 1978). For temperatures greater than
∼25 K, the conductivity is given by Klinger (1980) aski =567[Wm−1 ]
.
T

(4)

To estimate Tib , we assume that the melting line of H2 O is astraight line connecting (0 bar, 273 K) to (2072 bar, 251 K) inthe linear pressure–temperature phase diagram. The temperature
Tib can be estimated using
Tib = 273 −Figure 1. Schematic of the model used in this study.

= 273 −

work, we consider the effect of high-pressure ice under an internal ocean and discuss its implications for habitability (seeSection 4.2). With these considerations, we discuss the conditions required for bound and unbound terrestrial planets to havean internal ocean on the timescale of planetary evolution (owing to the geothermal heat flux from the planetary interior). Our
discussion further considers various planetary masses, distancesfrom the central star, water abundances, and the abundances ofradiogenic heat sources. Finally, we investigate the structure ofsurface H2 O layers of ice-covered planets by taking into accountthe effects of high-pressure ice.

(1 − A) L+ q = εσ Ts4 ,4d 2 4π D02

To calculate the mass–radius relationships for planets with
masses in the range 0.1 M⊕ –10 M⊕ , we adjust the planetaryparameters. We assume
M
M⊕

0.27
,



−1
2
t
L(t) = 1 +
1−
L ,
5
t

(1)

as per Valencia et al. (2006), where R is the planetary radiusand M is the planetary mass. The subscript ⊕ denotes values for
Earth. The mantle thickness, core size, amount of H2 O, averagedensity, and other planetary properties are scaled according tothis equation.
The planetary surfaces are assumed to consist of frozen H2 O
and to have no continental crust. We define the planetary radius
as R = dw + l, where dw is the H2 O thickness and l is the
mantle-core radius (see Figure 1). The mass of H2 O on theplanetary surface is given byMsw =4πρw [(dw + l)3 − l 3 ],3Tib − Ts
,
q

(6)

(7)

where t = 4.7 × 109 yr and L = 3.827 × 1026 W.
From these models, we can obtain the H2 O thickness dw andthe ice thickness dh. The condition for terrestrial planets having
an internal ocean is
dw > dh.
(8)
To estimate the geothermal heat flux q through planetaryevolution, we investigate the thermal evolution of terrestrial
planets using a parameterized convection model (Tajika &Matsui 1992; McGovern & Schubert 1989; Franck & Bounama
1995; von Bloh et al. 2007; see the Appendix for details). We
assume E, which is the initial heat generation per unit time andvolume, to be 0.1E0 –10E0 , where the constant E0 is the initialheat generation estimated from the present heat flux of the Earth(see the Appendix for details).

(2)

where ρw is the density of H2 O. We vary Msw from 0.1 Msw0 to
100 Msw0 , where Msw0 = 0.00023M, with the coefficient being
the H2 O abundance of Earth (0.023% by weight).
Assuming that a heat flux q is transferred from the planetaryinterior through the surface ice shell by thermal conduction, theice thickness dh can be calculated as
dh = ki(5)

where A is the planetary albedo, d is the distance from thecentral star in AU, L is the luminosity of the central star,
D0 = 1.5 × 1011 m is a distance of 1 AU in meters, ε is theemissivity of the planet, and σ = 5.67 × 10−8 Wm−2 K−4
is the Stefan–Boltzmann constant. We assume A = 0.62 andε = 1.0 (i.e., the planetary atmosphere contains no greenhousegases, which yields an upper estimate on the ice thickness). The
increase in luminosity due to the evolution of the central star asa main sequence star (Gough 1981) is considered using

2.1. Numerical Model



22[K]
dhρw g × 10−5 ,2072[bar]where pib (bar) is the pressure at the bottom of the ice and g is
the gravitational acceleration on Earth.
Considering energy balance on the planetary surface, theplanetary surface temperature Ts is

2. METHOD

R
=
R⊕

22[K]
pib
2072[bar]

2.2. High-pressure Ice
Ice undergoes a phase transition at high pressure (Figure 2).
Unlike ice Ih, the other phases are more dense than liquid
H2 O. We call the denser ice “high-pressure ice.” Because Tajika(2008) assumes that the amount of H2 O on the planetary surface
is the same as that on the Earth’s surface Msw0 (= 0.00023M),
the only possible conditions on the planetary surface are thoselabeled 1, 2, and 3 in Figure 3. However, because we consider(3)

where ki is the thermal conductivity of ice, Tib is the temperature
at the bottom of the ice, and Ts is the temperature at the surface.
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The Astrophysical Journal, 775:96 (8pp), 2013 October 1

Ueta & Sasaki

(a)

(b)

Figure 2. Phase diagram of H2 O (after Kamb 1973).

herein that H2 O mass may range from 0.1 Msw0 to 100 Msw0 ,the H2 O–rock boundary could move to higher pressure, so weshould account for the effect of high-pressure ice (Figure 3(a)).
Therefore, types 4, 5, and 6 of Figure 3(b) are added as possible
surface conditions. Type 2 and type 5 planets both have aninternal ocean, but high-pressure ice exists in type 5 planets
between the internal ocean and the underlying rock.
We approximate the melting curve by straight lines connecting the triple points in the linear pressure–temperature phase
diagram. We also assume that the amount of heat flux from theplanetary interior that is transferred through the internal ocean
by thermal convection is the same as that transferred throughthe surface ice. Here, we presume that the temperature gradient in the liquid water part of the phase diagram is isothermal(Figures 3(a) and (c)), although a gradient in a deeper internal
ocean than what is considered in this study should be carefully
discussed. The condition for high-pressure ice to exist under theinternal ocean is
Ph < Pb ,(9)(c)

Figure 3. (a) Schematic phase diagram of H2 O (gray lines), temperature gradient(black lines), and H2 O–rock boundaries (dashed lines). (b) Types of planets thathave H2 O on their surfaces. (c) Schematic of models used to treat high-pressure
ice.

where Pb is the pressure at the H2 O–rock boundary and Ph
is the pressure in the phase diagram where the temperature
gradient and the high-pressure melting line cross (Figure 3(c)).
As a representative value, we assume that high-pressure ice hasa density of 1.2 g cm−3 . Because the characteristic featuresof high-pressure ice are poorly understood, we simplified themodel and, in particular, the thermal conductivities of thehigh-pressure ice (see below). When the thermal conductivity ofhigh-pressure ice is relatively high and the temperature gradientin the high-pressure ice part of the phase diagram is shallower
than the gradient of the melting lines, the layer of high-pressure
ice continues to the H2 O–rock boundary ((i) in Figure 3(c)).
However, when the thermal conductivity is comparatively low
and the temperature gradient is steeper than that of the melting
lines, the temperature gradient joins the melting line and goesalong with the melting lines to the H2 O–rock boundary (dashedarrows (ii) and (iii) in Figure 3(c)). Although little is known
about the conditions on the dashed arrows, we assume thatlayer to be high-pressure ice in this study. Therefore, fromthe point where the temperature–pressure line crosses into thehigh-pressure ice part, the high-pressure layer continues to the
H2 O–rock boundary.
3. RESULTS
Figures 4(a) and (b) show the surface conditions for planetswith masses from 0.1 M⊕ to 10 M⊕ 4.6 billion yr after planetary
formation, with varying H2 O masses on their surfaces or initialradiogenic heat sources. All of the planets are located 1 AU
from thier central star. We assumed E/E0 = 1 for Figure 4(a)and Msw /Msw0 = 1 for Figure 4(b). Because larger planets
have larger geothermal heat fluxes and thicker H2 O layers,these objects could have an internal ocean with a smaller H2 O
mass on the planetary surface (Figure 4(a)) and a weaker initialradiogenic heat source (Figure 4(b)). However, larger planets
also have larger gravitational accelerations. Thus, on those3

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Ueta & Sasaki

(a)

(b)

(c)

(d)

Figure 4. Surface conditions for a planet at 1 AU around a central star (L = L0 ; the present luminosity of our Sun) and temperature profiles of surface H2 O layers fortwo cases. (a) The x axis is the surface H2 O mass and the y axis is planetary mass normalized by the Earth’s mass, assuming E/E0 = 1. The two dots in this panel
indicate the conditions corresponding to the temperature profiles shown in panels (c) and (d). (b) The x axis is initial radiogenic heat, and the y axis is the planetary
mass normalized by the Earth’s mass, assuming Msw /Msw0 = 1. (c) Temperature profile at 1 M⊕ and 5 Msw0 . (d) Temperature profile at 1 M⊕ and 30 Msw0 .

planets, high-pressure ice tends to appear under the internal
ocean with a smaller H2 O mass on the surface (Figure 4(a)).
For example, if a planet of mass 1 M⊕ has an H2 O mass of
0.6 M⊕ –25 Msw0 , it could have an internal ocean. However,
if a planet has an H2 O mass > 25 Msw0 , high-pressure ice
should exist under the ocean (Figure 4(a)). Note, however,that an internal ocean can exist on a planet having a massof 1 M⊕ if the initial radiogenic heat source exceeds 0.4E0(Figure 4(b)). Figures 4(c) and (d) show the temperature profilesof surface H2 O layers. Figure 4(c) shows the conditions of aplanet parameterized by 1 M⊕ and 5 Msw0 , whereas Figure 4(d)shows the conditions of a planet parameterized by 1 M⊕ and
30 Msw0 . Given the conditions of Figure 4(c), the surface H2 O
layers consist of a conductive-ice Ih layer and a convective liquid
layer (i.e., an internal ocean). When the planet has a higher H2 O
mass, high-pressure ice could appear under the internal ocean(Figure 4(d)). Here, the planetary surface temperature seems
very low, so we assume that the planet is covered by ice that has
a higher albedo (∼0.6) than ocean/land (∼0.3).
Figures 5(a) and (b) show the surface conditions for freefloating planets (L = 0) with masses from 0.1 M⊕ to 10 M⊕ 4.6
billion yr after planetary formation. The incident flux from thecentral star affects the surface temperature, thereby affecting
the condition on the surface. Therefore, the conditions, and inparticular those shown in Figure 5(a), are different from thoseshown in Figures 4(a) and (b). The results of Figure 5(a) show
that, regardless of the amount of H2 O a 1 M⊕ planet has, aninternal ocean cannot exist under the ice shell. An internal ocean
could exist on free-floating planets under certain conditions,but the planetary size and water abundance strongly constrain
these conditions (see Figure 5(a)). For instance, if a free-floating
planet has an initial radiogenic heat source greater than 7E0 , it
can have an internal ocean (Figure 5(b)). Figures 5(c) and (d)show the temperature gradients of surface H2 O layers for freefloating planets. The parameters are set to the same values as
those in Figures 4(c) and (b), except that L = 0. The temperatureon the planetary surface, approximately 35 K, is calculated by
Equation (6) under the assumption that L = 0 and the ice Ihlayer is thicker than that for the conditions of Figures 4(c) and(d). Given the conditions of Figure 5(c), the surface H2 O layers
consist only of a conductive ice Ih layer. However, for higher
H2 O mass, high-pressure ice could appear under the ice Ih layer(Figure 5(d)).
Figure 6 shows the surface conditions for planets with masses
from 0.1 M⊕ to 10 M⊕ at varying distances from a central star,
4.6 billion yr after planetary formation. The runaway greenhouse
limit (Kasting 1988; Nakajima et al. 1992) indicating the inneredge of the HZ is not considered. The effect of the incident
flux from the central star on the surface conditions is estimatedin each graph of Figure 6 as a function of distance from thecentral star. We find that the existence of an internal ocean onplanets far from a central star depends on the planetary massand surface H2 O mass. For the conditions of Figure 6(b) on
which a planet has two times the H2 O mass (i.e., 2 Msw /Msw0 ),for example, a 2 M⊕ planet can have an internal ocean underwhich there is no high-pressure ice only out to approximately
7 AU from the central star. For a planet with five times the H2 O
mass (i.e., 5 Msw /Msw0 ; Figure 6(c)), an internal ocean withno underlying high-pressure ice can exist out to approximately
30 AU. However, if the planet has 10 times the H2 O mass (i.e., 10
Msw /Msw0 ; Figure 6(d)), an internal ocean without underlying
high-pressure ice could exist to only approximately 5 AU. The
planetary mass and surface H2 O mass strongly constrain theconditions under which an extrasolar terrestrial planet far fromits central star can have an internal ocean with no underlying
high-pressure ice.
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(a)

(b)

(c)

(d)

Figure 5. Same as Figure 4, but for a free-floating planet (L = 0).

(a)

(b)

(c)

(d)

Figure 6. Conditions for a planet orbiting central star (E/E0 = 1) for (a) Msw /Msw0 = 1, (b) Msw /Msw0 = 2, (c) Msw /Msw0 = 5, and (d) Msw /Msw0 = 10. The x
axes represent the distance from the central star and the y axes represent the planetary mass normalized by Earth’s mass.

4. DISCUSSION(4) H2 O abundance on the surface, both of which were held constant in Tajika (2008). We used Equation (1) to determine themass–radius relationships because we expected more accurate
results, although this change makes little quantitative difference in the results. Below, we discuss items (2), (3), and (4) toanalyze the models of this study.
It is known from experiments that the thermal conductivityof ice Ih depends on temperature (≈1/T ; Klinger 1980). In the

4.1. Models of Study
We expanded on the models of Tajika (2008) by (1) invoking
the mass–radius relationship (Equation (1)) to consider planetary compression by gravity, (2) considering the temperaturedependence of the thermal conductivity of ice Ih (Equation (4)),
and varying the (3) abundance of radiogenic heat sources and5The Astrophysical Journal, 775:96 (8pp), 2013 October 1

Ueta & Sasakipresent study, the surface ice shell is thicker than that consideredby Tajika (2008). For example, the surface ice shell consideredherein is about 1.1 times thicker at 1 AU and about 3.5–4.4 times
thicker on the free-floating planets (L = 0). This increased
thickness is due to the use of Equation (4) to describe the thermal
conductivity of ice Ih in contrast with Tajika (2008), wherea constant thermal conductivity (ki = 2.2) was used. Tajika(2008) treated the abundance of H2 O and heat source elements(a ratio of mass against a planetary mass) as a constant, i.e.,
Msw /Msw0 = 1 (Msw0 = 0.00023M) and E/E0 = 1, although
the amounts should vary with planetary mass, and showed thatliquid water would be stable either on the surface or beneaththe ice for a planet with a mass exceeding 4 M⊕ , regardlessof the planetary orbit and the luminosity of the central star. In
this paper, however, when we consider a planet with parameters
Msw /Msw0 = 1 and E/E0 = 1, the results shown in Figure 5
are not consistent with those of Tajika (2008). Our results
indicate that free-floating planets with masses between 0.1 M⊕
and 10 M⊕ could not have liquid water on their surfaces andalso under the ice for the planetary parameters Msw /Msw0 = 1
and E/E0 = 1. As explained, when using Equation (4) to
describe the thermal conductivity of ice Ih, the planetary surface
temperature becomes more sensitive to the thickness of thesurface ice layer, which leads to a several-times increase in
the thickness of the surface ice. These results thus differ fromthose of Tajika (2008).
In this study, we assume that the high-pressure layer continues to the H2 O–rock boundary from the point where thetemperature–pressure line crosses into the high-pressure ice part(see Section 2.2). The results of this paper could depend on thisassumption. When the thermal conductivity of high-pressure ice
is comparatively low (similar to that of ice Ih), the heat cannot
be effectively transported through the high-pressure ice, thus
the bottom of the high-pressure ice could be thermally unstable
and melt enough to form an internal ocean.
The results of the present study indicate that the highpressure layers under an internal ocean could be from 1 km
to ∼100 km thick because a super-Earth planet characterized
by a few percent H2 O by weight could have ∼100 km thickhigh-pressure ice layers. In high-pressure ice that is ∼100 km
thick, it is possible for convective ice layers to appear. To
estimate whether or not such convection layers arise, we use
the Rayleigh number Ra , which is given byRa =gαρa 3 ΔT,κηwater would change because of insufficient water on the surface
and the planet’s thermal evolution would differ from that of the
Earth. In other words, we consider only those planets that are
relatively active geothermally and have a comparable amountof water as Earth-like planets. Improving our model so that
it applies to other types of terrestrial planets is an important
problem that we leave for future work.
4.2. Habitability of an Internal Ocean
For genesis and sustenance of life, we need at least (1) liquidwater and (2) nutrient salts because these substances are required
to synthesize the body of life (Maruyama et al. 2013). Because
nutrient salts are supplied from rocks, it is necessary that liquidwater be in contact with rock to liberate the salts. A type 5planet (Figure 3(b)) is thus not likely to be habitable because theinternal ocean does not come in contract with rocks. However,
it is possible for a type 2 planet to meet this requirement.
Therefore, we presume that only type 2 planets have an internal
ocean that is possibly habitable.
Therefore, the results of this study indicate that planetary
mass and H2 O mass constitute two more conditions to add
to the previous conditions for an extrasolar planet to have aninternal ocean without high-pressure ice. In other words, these
considerations indicate that only a planet with the appropriate
planetary mass and H2 O mass can have an internal ocean that is
possibly habitable.
However, it is possible that hydrothermal activities within
the rocky crust may transfer nutrient salts to the internal
ocean through cracks in the high-pressure ice. Large terrestrial
planets such as those considered in this study are likely to haveareas of high geothermal activity along mid-oceanic ridges andsubduction zones as well as large submarine volcanos whose
tops might emerge into the internal ocean from the high-pressure
ice. Furthermore, for planets with thick, high-pressure ice layers(Section 4.1), the convective ice layer could transfer nutrient
salts from the rock to the internal ocean. In these cases,
high-pressure ice might not prevent the nutrient flux from therock floor from reaching the internal ocean.
4.3. Future WorkAs shown in Figures 4(b) and 5(b), an appropriate initialradiogenic heat source is an important factor in determining
whether or not a planet has an internal ocean. Because thevariation in the amount of initial heat generation in planets
throughout space is not known, we assume in this study that theinitial heat generation E per unit time and volume ranges from
0.1E0 to 10E0 . Thus, in order to resolve this issue, the generalamount of radiogenic heat sources for extrasolar terrestrial
planets should be estimated.
An Earth-size planet (Msw /Msw0 = 1, E/E0 = 1) orbiting at
1 AU around the Sun for 4.6 billion yr with no greenhouse gases
might be globally covered by ice (Figure 4). In contrast, the Earthis currently partially covered with ice but retains liquid water
on its surface. Therefore, it is almost certain that greenhousegases play a major role in keeping the surface warm. Even freefloating, Earth-sized planets with atmospheres rich in molecularhydrogen could have liquid water on their surfaces because
geothermal heat from their interiors would be retained due tothe greenhouse gas effect of H2 (Stevenson 1999). By applying
this model here, we can account for the greenhouse gas effectfor various values of the emissivity ε in Equation (6). In future
work, we will thus discuss how greenhouse gases modify the(10)

where α is the coefficient of thermal expansion, ρ is the density,
a is the thickness of the high-pressure ice layer, ΔT is thetemperature difference between the top and bottom part ofthe layer, κ is the thermal diffusivity, and η is the viscosity
coefficient. When the Rayleigh number Ra exceeds the criticalvalue for the onset of convection (Racrit ∼ 103 ), convective
motion spontaneously begins. If we use the parameters from
Kubo (2008), i.e., α = 10−4 K−1 , ρ = 1000 kg m−3 ,ΔT = 50 K, κ = 2 × 10−6 m2 s−1 , and η = 1015 –1018 Pa s
and the typical values from this study, g = 10 m s−2 anda = 102 km, the Rayleigh number is ∼104 –107 , which indicatesthat convective motion is possible.
In this study, we consider Msw from 0.1 Msw0 (0.0023%
by weight) to 100 Msw0 (2.3% by weight). If the planet has
significantly more H2 O on its surface, the water layer would
be very thick and our model would not apply. If the planet
has significantly less H2 O on its surface, the regassing flux of
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conditions necessary for the development of a terrestrial ocean
planet or ice-covered planet with an internal ocean.
In the present study, we assumed pure H2 O on the planetary
surface. However, even if an extrasolar terrestrial planet has
surface H2 O, the H2 O might not be pure because it might containdissolved nutrient salts such as those found in Earth’s oceans.
In this case, the melting point of the solution would differ fromthat of pure H2 O. Moreover, the phase diagram would become
more complicated and the properties of H2 O (e.g., its thermal
conductivity) could be transmuted as a result of the appearanceof phases in which H2 O ice contains nutrient salts. A very
important work would thus be to analyze multicomponent water(e.g., seawater on Earth) to see what qualitative changes such a
modified phase diagram would bring to our results.
Consider the examples Europa and Ganymede, which are
two satellites of Jupiter and are thought to have internal oceans.
Ganymede is thought to have high-pressure ice under its internal
ocean (see, e.g., Lupo 1982). Therefore, determining whether
the internal oceans of Europa and Ganymede are suitable forlife would be pertinent to the discussion of the habitability ofinternal oceans with or without high-pressure ice. In addition,a more circumstantial discussion of the habitability of internal
oceans would require considering the redox gradient within theinternal ocean (as exemplified by the discussions of Europa
Gaidos et al. 1999) and the effects of the riverine flux of nutrient
salts (Maruyama et al. 2013).

E(t) is the rate of energy production by decay of radiogenic heatsources in the mantle per unit volume.
The mantle heat flux is parameterized in terms of the Rayleighnumber Ra ask(Tm − T  ) Ra β
q=
,
(A2)
Rm − Rc
Racritwhere k is the thermal conductivity of the mantle, T  is thetemperature at the surface of the mantle, Racrit is the criticalvalue of Ra for convection onset, and β is an empirical constant.
The radiogenic heat source is parameterized as
E(t) = Ee−λt ,where λ is the decay constant of the radiogenic heat source and
E is the initial heat generation per unit time and volume. We
assume E to range between 0.1E0 –10E0 , where the constant E0
is the initial heat generation estimated from the present heat fluxof the Earth.
We obtain Ra as
Ra =Herein, we discuss the conditions that must be satisfied
for ice-covered bound and unbound terrestrial planets to havean internal ocean on the timescale of planetary evolution.
Geothermal heat flow from the planetary interior is considered as
the heat source at the origin of the internal ocean. By applying
and improving the model of Tajika (2008), we also examine
how the amount of radiogenic heat and H2 O mass affect these
conditions. Moreover, we investigate the structures of surface
H2 O layers of snowball planets by considering the effects ofhigh-pressure ice. The results indicate that planetary mass andsurface H2 O mass strongly constrain the conditions under which
an extrasolar terrestrial planet might have an internal ocean
without high-pressure ice existing under the internal ocean.

ν = νm exp(TA /Tm ),(A5)

TA = α1 + α2 x,

(A6)

Mw
,
(A7)
Mm
where νm , α1 , and α2 are constants, TA is the activation
temperature for solid-state creep, and Mm is the mass of the
mantle.
The evolution of the mantle water can be described by theregassing flux Freg and outgassing flux Fout asdM w= Freg − Foutdt
Mw
 dm fw S,= fbas ρbas dbas SRH2 O − 4  3
3πRm − Rc3(A8)
where fbas is the water content in the basalt layer, ρbas is theaverage density, dbas is the average thickness of the basalt layer
before subduction, S is the areal spreading rate, RH2 O is theregassing ratio of water, dm is the melting generation depth, andfw is the outgassing fraction of water. The regassing ratio ofwater linearly depends on the mean mantle temperature Tm via

APPENDIXPARAMETERIZED CONVECTION MODEL
By applying conservation of energy, we obtain

3

(A4)

x=

We thank the reviewer Eiichi Tajika for constructive comments. We also thank S. Ida and M. Ikoma for valuable discussions. This research was supported by a grant for the Global
COE Program “From the Earth to ‘Earths’ ” from the Ministry of
Education, Culture, Sports, Science, and Technology of Japan.
T.S. was supported by a Grant-in-Aid for Young Scientists (B),
JSPS KAKENHI grant No. 24740120.

gα(Tm − T  )(Rm − Rc )3,κν

where α is the coefficient of thermal expansion, κ is the thermal
diffusivity, and ν is the water-dependent kinematic viscosity.
The viscosity ν strongly depends on the evolution of the mass
Mw of mantle water and the mantle temperature Tm and isparameterized as

5. CONCLUSIONS

4
2
3
4π Rm
q + πρc Rm
− Rc3

(A3)

RH2 O = RT (Tm (0) − Tm ) + RH2 O,0 , 3

4= π E(t) Rm− Rc3 ,
dt
3
(A1)

 dTm

(A9)

as given by von Bloh et al. (2007), where RT is the temperaturedependence of the regassing ratio and RH2 O,0 is the initial
regassing ratio. The areal spreading rate S is

where ρ is the density of the mantle, c is the specific heat atconstant pressure, Rm and Rc are the outer and inner radii of the
mantle, respectively, Tm is the average mantle temperature, andS=7

q 2 π κA0,
4k 2 (Tm − T  )2

(A10)

The Astrophysical Journal, 775:96 (8pp), 2013 October 1

Ueta & Sasaki

Table 1
Values of the Parameter for Parameterized Convection Model
ParameterRm (1 M⊕ )Rc (1 M⊕ )ρcTm (0)k
RacritβE0λg(1 M⊕ )
α
κ
νm
α1
α2
Mm (1 M⊕ )Mw (0)(1 M⊕ )
fbas
ρbas
dbas
dm
fw
RT
RH2 O,0
A∗0

Value

Unit

Source6,271 × 103
3,471 × 103
4.2 × 106
3,000
4.2
1,100
0.3
1.46 × 10−7
0.34
9.81
3 × 10−5
10−6
2.21 × 10−7
6.4 × 104−6.1 × 106
4.06 × 1024
4.2 × 1021
0.03
2,950
5 × 10340 × 1030.194
29.8 × 10−5
0.001
3.1 × 1014mm

von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)
McGovern & Schubert (1989)
Franck & Bounama (1995)
Franck & Bounama (1995)
Franck & Bounama (1995)
von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)von Bloh et al. (2007)
McGovern & Schubert (1989)

J m−3 K−1
K
J s−1 m−1 K−1
J s−1 m−3
Gyr−1
m s−2
K−1
m2 s−1m2 s−1K
K per weight fractionkgkg
kg m−3mm
K−1m2

where A0 is the ocean basin area. The area A0 can be parameterized as


A0
R 2
=
,
(A11)
A∗0
R⊕

Hoffman, P. F., Kaufman, A. J., Halverson, G. P., & Schrag, D. P. 1998, Sci,
281, 1342
Hoffman, P. F., & Schrag, D. P. 2002, Terra Nova, 14, 129
Howard, A. W., Marcy, G. W., Johnson, J. A., et al. 2010, Sci, 330, 653
Kamb, B. 1973, in Physics and Chemistry of Ice, ed. E. Whalley, S. J. Jones, &L. W. Gold (Ottawa: Royal Society of Canada), 28
Kasting, J. F. 1988, Icar, 74, 472
Kasting, J. F., Whitmire, D. P., & Reynolds, R. T. 1993, Icar, 101, 108
Klinger, J. 1975, JGlac, 14, 517
Klinger, J. 1980, Sci, 209, 271
Kubo, T. 2008, Low Temp. Sci, 66, 123
Lupo, M. J. 1982, Icar, 52, 40
Maruyama, S., Ikoma, M., Genda, H., et al. 2013, Geoscience Frontiers, 4, 141
Mayor, M., & Queloz, D. 1995, Natur, 378, 355
McGovern, P. J., & Schubert, G. 1989, E&PSL, 96, 27
Mischna, M. A., Kasting, J. F., Pavlov, A., & Freedman, R. 2000, Icar, 145, 546
Nakajima, S., Hayashi, Y.-Y., & Abe, Y. 1992, JAtS, 49, 2256
Sellers, W. D. 1969, JApMe, 8, 392
Stevenson, D. J. 1999, Natur, 400, 32
Sumi, T., Kamiya, K., Bennett, D. P., et al. 2011, Natur, 473, 349
Tajika, E. 2008, ApJL, 680, L53
Tajika, E., & Matsui, T. 1992, E&PSL, 113, 251
Valencia, D., O’Connell, R. J., & Sasselov, D. 2006, Icar, 181, 545
Varrot, M., Rochas, G., & Klinger, J. 1978, JGlac, 21, 241
von Bloh, W., Bounama, C., Cuntz, M., & Franck, S. 2007, A&A, 476, 1365

where A∗0 is the present ocean basin area on Earth. By using
Equation (1), we obtain
A0 = A∗0



M
M⊕

0.54
.

(A12)

In Table 1, we summarize selected values for the parameters
used in the parameterized convection model.
REFERENCES
Budyko, M. I. 1969, Tell, 21, 611
Franck, S., & Bounama, C. 1995, PEPI, 92, 57
Gaidos, E. J., Nealson, K. H., & Kirschvink, J. L. 1999, Sci, 284, 1631
Genda, H., & Ikoma, M. 2008, Icar, 194, 42
Gough, D. O. 1981, SoPh, 74, 21
Hart, M. H. 1979, Icar, 37, 351

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