ICARUS
135, 537–548 (1998)
IS985959
ARTICLE NO.
Distribution and Evolution of Water Ice in the Solar Nebula:
Implications for Solar System Body Formation
Kimberly E. Cyr
LPL/Department of Planetary Sciences, University of Arizona, 1629 East University Boulevard, Tucson, Arizona 85721
E-mail: [email protected]
William D. Sears
Computer Sciences Corp., Astronomy Programs, 100A Aerospace Road, Lanham-Seabrook, Maryland 20706
and
Jonathan I. Lunine
LPL/Department of Planetary Sciences, University of Arizona, 1629 East University Boulevard, Tucson, Arizona 85721
Received April 30, 1997; revised April 17, 1998
1. INTRODUCTION
Water is important in the solar nebula both because it is
extremely abundant and because it condenses out at 5 AU,
allowing all three phases of H2O to play a role in the composition
and evolution of the Solar System. In this paper, we undertake
a thorough examination of and model the inward radial drift
of ice particles from 5 AU. We then link the drift results to
the outward diffusion of vapor, in one overall model based on
the two-dimensional diffusion equation, and numerically evolve
the global model over the lifetime of the nebula. We find that
while the inner nebula is generally depleted in water vapor,
there is a zone in which the vapor is enhanced by 20–100%,
depending on the choice of ice grain growth mechanisms and
rates. This enhancement peaks in the region from 0.1 to 2 AU
and gradually drops off out to 5 AU. Since this result is somewhat sensitive to the choice of nebular temperature profile, we
examine representative hot (early) and cool (later) conditions
during the quiescent phase of nebular evolution. Variations in
the pattern of vapor depletion and enhancement due to the
differing temperature profiles vary only slightly from that given
above. Such a pattern of vapor enhancement and depletion in
the nebula is consistent with the observed radial dependence
of water of hydration bands in asteroid spectra and the general
trend of asteroid surface darkening. This pattern of water vapor
abundance will also cause variations in the C : O ratio, shifting
the ratio more in favor of C in zones of relative depletion,
affecting local and perhaps even global nebular chemistry, the
latter through quenching and radial mixing processes.  1998
Academic Press
Key Words: solar nebula; ices; chemistry; models.
To date, the evolution and spatial distribution of water in
the solar nebula have not been the focus of many detailed
investigations even though water can play an important
and complex role. Nebular water is important for two main
reasons: it is the most abundant condensable because oxygen is cosmochemically the third most abundant element
after hydrogen and helium, and it condenses out at p5
AU, allowing ice to become a major constituent of outer
Solar System bodies. The nebular water distribution can
impact nebular structure and evolution in a variety of ways.
Sufficiently large changes in the nebular water ice grain
distribution, for instance, will affect the disk opacity, and
thus the nebular thermal structure and transport processes.
Additionally, the distribution of water ice over the nebula’s
lifetime will influence timescales of planetesimal growth
by accretion, an important consideration for modeling the
formation of the outer planets. Moreover, the late nebular
water distribution will directly impact the composition of
subsequent Solar System bodies, both icy bodies and rocky
objects containing water of hydration, as well as possibly
affecting the supply of water to terrestrial planet surfaces.
Here we examine the transport of water vapor and condensation in the nebula, in order to consider the chemical,
compositional, and dynamical implications for the distribution of water in Solar System bodies (Fig. 1).
An early consideration of the overall transport of solid
and gaseous species in the nebula was that of Morfill and
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Copyright  1998 by Academic Press
All rights of reproduction in any form reserved.
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CYR, SEARS, AND LUNINE
FIG. 1. A schematic of the overall system under consideration: the protosun surrounded by the nebular disk. The focus for the majority of this
paper will be on the ‘‘inner solar nebula,’’ i.e., the region 1–5 AU; this includes the chemically active zone and the water condensation front
(‘‘snowline’’). The two main processes affecting the distribution of nebular H2O we examine are the diffusion of water vapor out past the condensation
front, initially at 5 AU, where the vapor condenses into ice particles, and the radial drift back inward of the ice. In all subsequent discussion, the
coordinate R represents the radial distance measured outward from the central axis of the nebula.
Völk (1984). They described average conditions of transport of dust, gas, and vapor in a turbulent protosolar nebula
by deriving analytical solutions from their model in the
limit of small particle sizes. Their general conclusions implied that material is reprocessed thermally and can be
chemically fractionated extensively and that there is a significant enhancement of solid particles just outside their
sublimation zones in the nebula which could help speed
planetesimal formation in that area.
In later work, Stevenson and Lunine (1988) considered
the diffusive redistribution and condensation of water in
the nebula with the goal of facilitating the rapid formation
of Jupiter. They modeled the outward diffusion of water
vapor in the nebula by assuming a ‘‘cold finger’’ solution—
i.e., they solved the diffusion equation in the limit that the
sink of water vapor is condensation within a narrow radial
zone, located p5 AU from the nebular center. They also
assumed that the condensate decoupled from the nebular
gas rapidly and suffered little effect from gas drag so that
small ice grains would not be carried back inward of 5 AU
due to nebular drag forces, and would grow unmolested
into larger ice bodies which remain in the condensation
zone. Given these conditions, their model predicted that
the inner 5 AU of the nebula would become severely depleted in water vapor in as little as 105 years and that the
surface density of ice in the condensation zone would be
enhanced by up to a factor of 75. This would be sufficient
enhancement to trigger formation of Jupiter’s heavy element core and thus of Jupiter itself, on a reasonable
timescale.
However, subsequent reconsideration of gas drag effects
on the ice condensate by Sears (1993) suggested that Ste-
venson and Lunine (1988) had underestimated the magnitude of aerodynamic gas drag and that drag would indeed
cause larger ice bodies to drift inward significant distances,
on relatively short timescales. An updated version of the
Stevenson and Lunine examination of the radial transport
of water through the solar nebula incorporating both inward ice drift and outward vapor diffusion is thus required
and presented here. The solar nebula framework for the
water transport model is discussed in Section 2 of this
paper, the gas drag–radial drift model is discussed in Section 3, the diffusion model expanded from Stevenson and
Lunine (1988) to include both drift and diffusion processes
is discussed in Section 4, implications for Solar System
body chemistry and formation are considered in Section
5, and conclusions are summarized in Section 6.
2. THE SOLAR NEBULA
Understanding Solar System body formation requires
an understanding of the chemical and dynamical history
of the solar nebula. In particular, modeling water transport
in the solar nebula setting requires an understanding of
both global nebular evolution as well as specific nebular
processes that could affect such transport. Both the overall
and several specific processes will be described briefly here,
so that the water transport model results (Section 5) can
be interpreted within their context.
The generally accepted scenario for Solar System formation starts with the self-gravitational collapse of a rotating
interstellar cloud into a protostar surrounded by a dusty
disk. Material raining in from the cloud has too much
angular momentum to fall directly onto the protostar, and
SOLAR NEBULA VAPOR/ICE DISTRIBUTION
so instead falls onto the disk. However, the Sun contains
p99.9% of the Solar System’s mass but only 2% of the
Solar System’s angular momentum (Boss et al. 1989); this
implies that a significant redistribution of angular momentum outward and mass inward had to occur throughout
the nebular disk by the end of its lifetime. The bulk of the
mass is ultimately transported through the disk and fed
onto the protostar, while only a relatively small amount
of matter left behind in the disk forms Solar System bodies
(Weidenschilling 1977b). The nebula is ultimately dispersed when winds from the young star blow the residual
gas and dust away.
Nebular evolution is currently understood to have occurred in two major phases. The first is the collapse phase
which lasts as long as cloud material is raining down on
the disk, and the second is the less active, quiescent phase
after infall of cloud material has ceased. The collapse phase
is believed to last on the order of 106 years, which is the
time it takes to collapse a one solar mass cloud at 10 K
(Cassen 1994); this time is also consistent with observational evidence for ages of T Tauri stars believed to be
actively accreting material (Beckwith et al. 1990). Evidence
indicates that the quiescent phase lasted an order of magnitude longer: observations show that nebular disks persist
around T Tauri stars for up to 107 years (Strom et al. 1993)
and there is also meteoritical evidence suggesting that thermal processing of nebular material occurred over a 107year period (Podosek and Cassen 1994).
During the collapse phase, it is believed that the nebula
was very active, chaotic, and potentially punctuated by
a number of transient, disruptive and poorly understood
phenomena. For instance, during mass accretion onto the
disk, density inhomogeneities may have occurred, causing
the disk to become gravitationally unstable and possibly
nonaxisymmetric (Boss 1989); nebular gas and fine dust
entrained in the gas were being relatively rapidly transported, primarily radially inward and onto the protostar
(Cassen 1994); the protostar may have undergone episodic
FU Orionis-type luminosity bursts, i.e., large, long-lived
increases in protostellar magnitude (Hartmann et al. 1993,
Bell and Lin 1994); and the nebula may have periodically
generated lightning under special circumstances, e.g., a
very dusty disk, more consistent with early stages of nebula
evolution (Gibbard et al. 1997, Pilipp et al. 1998).
Conditions are believed to be more stable during the
quiescent phase: mass infall from the cloud has ceased so
little to no new material is being accreted by the disk,
nebular gas motions are very small, and most likely no
disruptive phenomena, with the possible exception of giant
planet migration, occurred. In recent work, Trilling et al.
(1998) showed that it was possible for Jupiter-sized planets
to migrate radially inward to small heliocentric distances,
#0.1 AU, because of torques arising from the nebular disk,
protostar, and planetary mass-loss. Assuming the nebula
539
still exists at that time, such giant planets could clear a
path through the nebula, sweeping up material out to 2–3
AU around them as they migrate. Thus, the timing of
planetesimal and planetary formation is important; however, it is not well constrained. Most modelers assume,
based on meteoritic evidence (Macpherson et al. 1995),
that planetesimal formation occurred slowly during the
later, quiescent phase of nebular evolution (see Wood
(1996) for an alternate view). If planets did form during
the early chaotic stage, it is also possible that they did
not survive into the quiescent stage either because they
migrated in onto the protostar or were ejected from the
nebula. Further, Liou and Malhotra (1997) modeled the
dynamical gap in the asteroid belt via the migration of
Jupiter; parameters of some of their models require the
formation of Jupiter to be complete and the migration to
initiate at nearly 107 years, the end of the nebular lifetime.
Thus it would not be inconsistent to assume late formation
of planetesimals and planets.
For the purposes of this paper, we will investigate water
and ice transport from the start of the quiescent phase,
i.e., at the very end of the collapse phase after the last
transient and chaotic event has ceased, and then throughout quiescent evolution of the disk. We will assume that
planetesimals either did not form in or did not survive the
early chaotic stage of nebular evolution and that giant
planet migration occurred, if at all, at the end of the nebular lifetime.
3. RADIAL DRIFT MODEL
3.1. Model
The radial drift model is a modified and updated version
of Sears’s (1993) numerical model which computes gas
drag on ice particles in the solar nebula, based on the
Weidenschilling (1977a) aerodynamic gas drag formalism.
Gas drag occurs because the rotational velocity of gas in
the nebula is less than the Keplerian velocity due to gas
pressure support. Small particles move with the gas, feel
the residual inward gravitational acceleration due to gas
pressure support, and thus drift inward at terminal velocity.
Large particles move with Keplerian velocity, plowing
through the gas. The resultant ‘‘headwind’’ causes draginduced energy loss and the particles spiral inward toward
the Sun. The maximum possible drift velocity is the difference between the Keplerian and gas velocities, which in
the model is 2 3 104 cm s21 at 5 AU.
Figure 2 plots radial drift velocities due to gas drag vs
particle size. It shows that particles 1–103 cm in size will
drift significantly inward, with 10- to 100-cm-sized particles
moving the fastest through several AUs over 104–5 years.
The smallest sized particle that will decouple from the gas
and drift inward can be estimated under the assumption
of a turbulent nebula. Turbulent motions will have a char-
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CYR, SEARS, AND LUNINE
FIG. 2. Radial drift velocities caused by gas drag vs particle size.
The solid line represents velocities of particles at 5 AU while the dotted
line plots particle velocities at 1 AU, showing an increase in velocity
inward assuming a constant particle size. The shape of the plots recreates
that of Weidenschilling (1977a), reflecting the various drag laws—Stokes,
Epstein—in effect for large and small particle sizes, respectively. The
plot shows that 10- to 100-cm-sized particles will be the fastest moving
and that 1- to 103-cm-sized particles will undergo significant drift over
the 107-year nebular lifetime.
acteristic velocity, vturb . A particle decouples if its terminal
velocity, vterm , is less than vturb . To estimate a typical minimum decoupling radius, set
vterm 5 vturb p D/H p n /H,
(1)
where D is the diffusion coefficient set equal to the eddy
viscosity n, and H is the scale height of the disk obtained
from the nebular model (Wood and Morfill 1988) as
3–4 3 1013 cm at 5 AU. For this order of magnitude calculation, n is estimated as p1015 cm2 s21, a typical value, after
Stevenson and Lunine (1988) whose model provides the
basis for our calculations, discussed in Section 4. This yields
a drift velocity of p30 cm s21 and thus a decoupling radius,
read from Fig. 2, of p0.3 cm.
In addition to drag physics, the drift model also incorporates the semi-analytical Cassen (1994) nebular model.
Cassen derives a thermal profile for conditions at the end
of the collapse phase, the hottest stage in the nebula, which
we take as indicative of conditions just before the start of
the quiescent phase. Using different modeling methods,
Boss (1996) numerically derives a nebular thermal profile
similar to Cassen’s. The Boss profile is somewhat cooler
in the outer regions of the nebula, though Boss did not
incorporate any viscous heating; thus his temperature profile serves more as a lower limit for nebular temperatures.
Because of this and the general agreement between the
two profiles, we use the Cassen profile as a plausible upper
limit on the temperature corresponding to the start of the
quiescent phase of nebular evolution. Cassen’s thermal
structure migrates radially inward over time; we use temperature profiles at early (hot) and late (cool) stages of
evolution in order to provide ‘‘snapshots’’ of water transport results at various times during quiescent nebula evolution, thus tracking the changes in water transport during
the nebula’s late history.
Lastly, the full drift model incorporates not only the
Weidenschilling drag formalism and the Cassen nebular
model, but also sublimation after Lichtenegger and Kömle
(1991) and Lunine et al. (1991), condensation time scales
after Stevenson and Lunine (1988), and a numerical integration routine based on Stoer and Bulirsch (1980). Ice
particles are assumed to be spherical with r p 1 g cm23,
and particles are always assumed to be at the same temperature as the gas. Sublimation is modeled simply and is
based on the vapor pressure above a solid surface at a given
temperature. The model assumes growth by condensation
preceding grain growth by coagulation where ballistic collisions of H2O molecules are the grain growth mechanism
and the ability of the resultant snowflake to transfer collisional heat to the H2 gas is the growth limiting mechanism.
The numerical drift model starts with a given sized ice
particle moving with Keplerian velocity at a given radial
distance from the Sun. The particle is then subjected to
solar gravity and gas drag, and the model tracks the particle’s subsequent orbital motion and mass changes through
sublimation and condensation. The program ends when
the particle either becomes small enough to couple to the
gas, sublimates away completely, spirals into the Sun, or
the presumed nebular lifetime of 107 years (Cassen 1994)
has elapsed.
After modifying and updating the drift model, we
tracked various sized ice particles, initially at 5 and 7 AU,
for crystalline and amorphous ice, respectively. Crystalline
ice will condense at P p 1026 bar and T p 160 K at approximately 5 AU, while amorphous ice condenses at T # 145 K,
farther out in the nebula. Therefore, crystalline ice I, the
stable phase formed by condensation at 5 AU was assumed.
Amorphous ice was also investigated in order to determine
if ices resident in the outermost nebula, preserved molecular cloud grains, might play a role in the inner nebula. The
primary focus of the investigation, however, has been on
the vapor–ice interplay for crystalline ice and r # 5 AU.
3.2. Results
Just after cloud collapse, the quiescent phase of nebular
evolution starts. To represent conditions near the beginning of the quiescent stage, we use the Cassen (1994) temperature profile in which the ice condensation front has
migrated in to 5 AU. This occurs shortly after the end
SOLAR NEBULA VAPOR/ICE DISTRIBUTION
FIG. 3. Particle size vs time for particles of a range of sizes, initially
released at 5 AU, drifting inward. The Cassen (1994) nebular model was
used at a time just after the collapse phase of the nebula had ended and
the quiescent phase had begun. At this stage peak midplane temperatures
are p1500 K out to about 1 AU, and the midplane ice condensation
front is at 5 AU. The plot indicates the length of time it takes for particles
to drift inward and sublimate away; the fastest moving particles, 100-cmsized, take only a little over 102 years, while the slowest, 104-cm, take
just over 105 years. Also indicated is that 5- to 104-cm particles will drift
inward well within the nebular lifetime of 107 years.
of collapse and presumably after all the transitory and
disruptive phenomena characteristic of the collapse phase
have ended. Under these hot quiescent nebular conditions,
5- to 1000-cm-sized ice particles will drift back into the
inner Solar System in less than 4 3 104 years (Fig. 3), well
within the nebular lifetime, with 100-cm particles being
the fastest moving, as predicted in Fig. 2. The particles
remain intact for the bulk of their journey inward, sublimating over a relatively narrow radial zone, 2.46–3.32 AU
(Fig. 4). This suggests that the gas drag mechanism transports ice into the inner nebular, causes a pulse of water
vapor over 2.5–3.3 AU, and litters the region from 3 to
5 AU with drifting ice particles.
The nebula cools overall as it ages; the increasing presence of condensed particles in the inner Solar System will
decrease the opacity, lowering the temperatures of those
regions. Reflecting these changes, the Cassen (1994) thermal profile cools, essentially by migrating radially inward
over time. By about 2 3 106 years, the ice condensation
front has migrated in to 3 AU. Results of the radial drift
model for this temperature profile are shown in Figs. 5
and 6; particles sized 5–1000 cm take longer to drift, but
still drift inward on the order of 104 years and sublimate
over a zone from 1.47 to 2.13 AU.
It can be seen that relative to a hot nebula, cooler nebula
conditions generate some significant differences. All particle lifetimes are lengthened, so ice particles starting from
541
FIG. 4. Particle size vs radial distance for a range of sizes, initially
released at 5 AU, drifting inward (right to left in the plot) due to gas
drag. The Cassen (1994) nebular model for the start of the quiescent
phase of nebular evolution was used. The plot indicates the range of
radial distance over which particles sublimate, 2.46–3.32 AU. It also
shows that larger particles drift further inward before completely sublimating and that larger particles sublimate over a broader radial zone.
the same radial distance in the nebula spend more time
drifting through the cooler nebula before sublimating.
Also, in a cool nebula drifting particles sublimate later on
their path inward; this shifts the location of the sublimation
zone further radially inward, by p1 AU over roughly the
FIG. 5. Particle size vs time for particles of a range of sizes, initially
released at 5 AU, drifting inward. The Cassen (1994) nebula temperature
model was used; at this later time during quiescent nebular evolution the
midplane ice condensation front has migrated in to 3 AU. The cooler
nebula shows the effects of sublimation, causing ice particles to last longer
(cf. Fig. 3) before sublimating to their final end state.
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CYR, SEARS, AND LUNINE
fore it reaches 5 AU and therefore would only augment
the crystalline ice population at 5 AU. The crystalline ice
drift results would remain the same. Figure 7, though, does
serve to show that varying the vapor pressure relation of
inward-drifting icy particles can modify drift model results,
primarily by changing the width of the sublimation zone.
4. DIFFUSION MODEL
4.1. Model
FIG. 6. Particle size vs radial distance for a range of sizes, initially
released at 5 AU, drifting inward due to gas drag. As in Fig. 5, the cooler,
older Cassen (1994) nebula temperature profile was used. The cooler
temperature profile causes particles to sublimate over a narrower radial
zone and about 1 AU farther in, 1.47–2.13 AU, than in the hotter, younger
nebula (Fig. 4).
first 2 3 106 years of the quiescent phase. Last, the lifetimes
of different sized particles are lengthened to different degrees. In general, the smaller particles, 5–10 cm, are not
as well buffered against hotter nebula conditions by their
mass as the larger 103–4-cm particles; thus the smallest
particles are more sensitive to temperature changes and
benefit the most from cooler conditions, traveling proportionally farther inward before significantly sublimating.
This dispersion in lifetime lengthening reduces the width
of the sublimation zone, by about 0.2 AU, in the cooler
nebula. In general, as the nebula quiescently evolves we
expect the sublimation zone to shift radially inward and
shrink.
The initial, simplistic amorphous ice results from the
drift model showed that this kind of ice, formed by condensing farther out in the nebula by direct infall from the
surrounding molecular cloud, travels farther and faster
than crystalline ice, presumably due to its different structure and different vapor pressure relation. Figure 7 shows
that in the cool nebula 5- to 103-cm-sized particles impact
the inner Solar System, sublimating over a slightly broader
radial range of 0.6–2.2 AU (cf. Fig. 5). However, further
consideration indicates that taking into account the phase
transition of amorphous to crystalline ice at T . 145 K
would eliminate this effect. The phase transition is exothermic as the ice structure transits to a more organized, lower
energy state. If the heat given off is sufficient to vaporize
the ice in whole or part, the vapor would then recondense
into crystalline ice, as the rest of the ice reorganized into
crystalline ice. Amorphous ice becomes crystalline ice be-
During the collapse phase in nebular evolution, radial
motion of nebular gas is most likely dominated by the
redistribution of mass and angular momentum as gas
streams through the nebular disk and ultimately is accreted
onto the protostar (Cassen 1994). However, during the
quiescent phase radial motion of the gas due to protostellar
accretion may have been insignificant; most of the gas
remaining in the quiescent nebula is thought to have been
dispersed back into interstellar space at the end of the
nebular lifetime rather than transported onto the protostar
(Shu et al. 1993). Thus diffusion, rather than being a small
perturbation on vapor motion as it likely was during the
collapse phase, may have been a significant mode of vapor
transport during the bulk of the nebula’s lifetime. Though
there are indications that advective motions in the midplane may have also affected vapor transport (Cuzzi et al.
FIG. 7. Particle size vs radial distance for a range of sizes of amorphous ice particles, initially released at 7 AU. The cool Cassen (1994)
nebular model was used. The plot indicates that outer nebula ice can
drift into the inner nebula regions and sublimate over a broader zone,
0.6–2.2 AU, than crystalline ice (cf. Fig. 6) by the end of the nebular
lifetime. However, when phase changes of the ice (e.g., amorphous to
crystalline) are taken into account, this effect is eliminated. The plot still
serves to show how different vapor pressure relations of icy particles
can affect drift model results, primarily by changing the width of the
sublimation zone.
SOLAR NEBULA VAPOR/ICE DISTRIBUTION
1993), see Section 6 for further discussion, for the purposes
of this parameter study we assume a simple diffusive
nebula.
The radial drift results of the previous section suggest
that water vapor is reintroduced into the inner nebula over
approximately a 1-AU-wide zone. In addition to the fact
that this vapor will in turn rediffuse outward toward 5
AU, is the possibility that, time scales permitting, ice will
continuously condense and drift inward, sublimate, and
the resultant vapor diffuse outward. Numerous repetitions
of this vapor-diffusion, ice-drift cycle are thus possible by
the end of the nebular lifetime. It is necessary, then, to
track the continuous diffusion–drift cycle throughout the
evolution of the nebula in order to determine whether
water vapor in the inner nebula survives and, if so, what
breadth and amplitude the final vapor zone has.
In order to track the diffusion–drift cycle, we expanded
the Stevenson and Lunine (1988) nebular vapor diffusion
model. The Stevenson and Lunine cold-finger model tracks
the diffusion of water vapor in the solar nebula over the
nebula’s lifetime, and solves the diffusion equation in the
limit that water vapor condenses within a narrow radial
zone at the snowline, initially p5 AU from the nebula
center. Stevenson and Lunine assumed no other sources
or sinks of vapor; in contrast, we incorporate the ice drift
results through the addition of source terms of varying
magnitude. The expanded diffusion model numerically
solves the two-dimensional diffusion equation in cylindrical coordinates with azimuthal symmetry, after Stevenson
and Lunine (1988) and Barrer (1970),
dc/ dt 2 (2D/R) dc/ dR 2 Dd 2c/ dR 2 1 S(R) 5 0,
(2)
where c is the concentration of H2O molecules, D is the
diffusion coefficient set equal to the eddy viscosity of the
turbulent nebula (which we numerically choose to be 1015
cm2 s21), R is the radial distance from the nebular center,
and S(R) is the source term. Boundary conditions are
c 5 solar abundance for R , 5 AU,
c 5 0 for R 5 5 AU,
all t,
t50
indicating that at the start of the calculation the concentration of water vapor is constant across the inner nebula
(0.1–5 AU) and is that which results from a solar composition gas, while the concentration of vapor is zero at (and
beyond) 5 AU, at all times, because vapor condenses out
as ice at and beyond that point. The explicit, forward time
centered space (FTCS) differencing method (Press et al.
1992) was used to numerically evolve and solve the system
over the nebular lifetime.
Through the source term, S(R), drift model results are
incorporated into the diffusion equation. S(R), expressed
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in units of concentration of water vapor per time, describes
the rate of the resupply of vapor from the drifting ice
particles, at various radial distances throughout the inner
nebula. Drift model runs were performed to determine a
variety of source terms (rates of resupply of vapor) at
various radial distances and times, creating in effect a table
of values for S(R) that the program reads in at appropriate
time and distance intervals. No other sources, such as outer
nebula amorphous ice contributions (other than our one
amorphous ice run) or sinks of H2O were assumed, so the
total H2O budget present as ice plus vapor always remained
constant. Thus, all water vapor no longer present in the
inner Solar System, lost due to diffusion, is assumed to
have condensed into ice either residing at 5 AU or drifting
inward. Most of the ice was assumed to be in the fastest
moving sizes, 10–100 cm, though a small amount, #5%,
was assumed to be in the largest 103–4 cm sizes, in order
to study the impact of large inward-drifting ice bodies.
Inherent in the diffusion–drift model are various timescales which ultimately control whether the diffusion or
the drift process dominates. Relevant timescales include
the nebular lifetime of 107 years as well as drift, sublimation, diffusion, and grain growth timescales. Drift times
(Figs. 3, 5) vary, but range from 102 to 103 years for the
fastest (smaller) moving particles and from 104 to 106 years
for the larger particles. Significant particle sublimation,
e.g., particle radius dropping to 60% or less of original,
occurs over # the last 4% of the particle’s lifetime for all
particle sizes considered and for all choices of nebular
temperature profile. Thus sublimation occurs effectively
instantaneously and timescales over which sublimation occurs are negligible. Diffusion timescales can be taken from
Stevenson and Lunine (1988) or Fig. 8, which reproduces
their results. Significant diffusion, e.g., total nebular vapor
content from 1 to 5 AU dropping to 60% of original, starts
at p104 years. Unlike the foregoing timescales, however,
grain growth rates are less well constrained and can vary by
orders of magnitude depending on the mechanism invoked.
For instance, the Stevenson and Lunine (1988) model of
simple condensation and coagulation yields a 1-cm particle
in 105 years, but with the caveat of the possibility of growth
to 104 cm in 104 years due to collisional processes (Nakagawa et al. 1981), which were not included in their model.
Due to the uncertainties in grain growth rates, a range of
rates, 103 –105 years, were investigated reflecting some to
no collisional growth of ice grains. In physical terms, grain
growth rates directly limit the time before ice populations
can start to drift inward and thus constrain the amount of
ice that can drift at any given time. Given that it is not
unreasonable to expect some amount of collisional processing of ice grains in the nebula, and thus a relatively
efficient grain growth rate, we would expect the short drift
time scales of the fastest particles to dominate, preserving
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CYR, SEARS, AND LUNINE
some amount of vapor in the inner nebula by the end of
nebular evolution.
4.2. Diffusion Results
Figure 8 shows the result of the diffusion–drift model
when no sources are present; i.e., no significant inward ice
particle drift occurs. This is equivalent to, and reproduces,
the Stevenson and Lunine (1988) results. Each curve represents a different moment in time, with the area under any
curve representing the amount of vapor present over 1–5
AU. Figure 8 shows the nebula inward of 5 AU cleared
of water vapor by 2 3 105 years. The diffusion process
alone will result in a drier inner nebula, where the dryness
is both relative to initial assumptions of solar abundance
concentrations and absolute, in that little to no water vapor
is present. Other results of the simple diffusion case to
note are that in the absence of inward ice drift or other
mechanisms the inner nebula is also devoid of ice particles,
that the inner nebula is uniformly depleted in water vapor
across 1–5 AU—i.e., there is no radial dependence to the
depletion, and that since in this case all vapor that diffuses
out to 5 AU condenses into ice and remains in the condensation zone, there is a large enhancement in the concentration of ice located at p5 AU.
Figure 9 shows the results of our diffusion–drift model,
at the beginning of the quiescent phase of evolution, for
the relatively efficient grain growth rate in which the fastest
sized particles, 10–100 cm, grow in p103 years. The 1to 5-AU region still experiences something of a general
depletion in water vapor relative to solar abundance concentrations. However, for this the hottest stage of quiescent
evolution there is also a significant local vapor enhancement from 0.1 to 2.5 AU of p60% over the no-sources
case (Fig. 8), which drops off gradually out to 5 AU. This
suggests that while there is still some tendency toward
overall drying of the inner nebula relative to solar values,
there is additionally a local ‘‘wet’’ zone of relative vapor
enhancement which diminishes with radial distance until
it disappears near 5 AU.
Diffusion–drift results at later, cooler times in the nebula
are similar to those in Fig. 9. At cooler times (Fig. 10),
the enhancement spreads over a smaller zone, 0.1–2 AU
before tapering off out through 5 AU, and the peak enhancements are less (p40%) than during hotter conditions
in the nebula. The differences in the hotter nebula conditions relative to the cool nebula are due primarily to the
location and to a lesser extent to the increased breadth of
the sublimation zone. Because the sublimation zone has
shifted radially outward, the enhancements are also generally shifted outward. Moreover, because vapor is resupplied closer to the condensation front and because diffusion
affects vapor closer to this front first, more of the newly
injected vapor and less of the original nebula vapor diffuses
outward. Thus, in the hot nebula case, the sublimation
zone acts as a better buffer against diffusion, preserving
more of the original nebula vapor inward of the zone.
The other end member case of very slow grain growth,
growth of 10- to 100-cm particles in p105–6 years, yields
an even more extreme distribution of water that is effectively a step function: vapor enhancement 10 times solar
abundance released over 1.5–2.0 AU, negligible amounts
of vapor elsewhere. Though it is perhaps possible that such
slow grain growth rates may have predominated throughout nebular evolution, we expect that the more efficient
grain growth rates yielding 10- to 100-cm particles in 103–4
years and which incorporate some amount of collisional
processing, are more likely to have occurred. These more
efficient grain growth rates all generate the same qualitative diffusion–drift plot behavior, with vapor enhancements of 20–100% over the no source case from 0.1–2 AU,
tapering off through 4–5 AU.
As the nebula evolves, we would expect more and more
of the original nebula vapor to be thermally cycled across
the condensation front, the outer edge of the zone of peak
vapor enhancement to shrink inward from 2.5 to less than
1.8 AU, the peak level of vapor enhancement to decrease,
and the vapor enhancement to taper off out to 5 AU over
a correspondingly larger radial zone.
FIG. 8. Diffusion model assuming no significant radial drift. This plots the simple diffusion of water vapor out to 5 AU, where it condenses
out as ice, assuming no sources of vapor, i.e., no radial drift and subsequent sublimation of ice particles. Each curve represents the water vapor
distribution at a different time during the nebula’s evolution. The plot reproduces the Stevenson and Lunine (1988) model result, and shows that
the inner nebula becomes effectively completely devoid of water vapor by the end of the nebular lifetime for this limiting case.
FIG. 9. Diffusion model with radial drift. Diffusion of water vapor out to the 5 AU condensation front is modeled as in Fig. 8, but also
incorporates the inward drift and subsequent sublimation of ice particles. A relatively efficient grain growth rate of 10- to 100-cm-sized particles
forming in 103 years was used, as was the hot Cassen (1994) temperature profile corresponding to the start of the quiescent phase of nebular
evolution. Relative to Fig. 8, the case in which drift effects were not incorporated, this plot shows that by 105 years, there is a significant enhancement
of water vapor from 1 to 2.5 AU with the vapor enhancement falling off out through 5 AU. Maximum enhancement, from 0.1 to 2.5 AU, is p60%
relative to the no-sources case.
FIG. 10. Diffusion model with radial drift, all parameters as in Fig. 9, but based on the cooler Cassen temperature profile corresponding to a
later stage of quiescent nebula evolution in which the condensation front has migrated inward to 3 AU. The qualitative results are the same as in
Fig. 9; however, the breadth of the zone of enhancement is narrower (0.1–1.8 AU) and the amount of enhancement is less; maximum enhancement
is now 40% relative to the no-sources case (Fig. 8).
SOLAR NEBULA VAPOR/ICE DISTRIBUTION
5. IMPLICATIONS
The major conclusion of Stevenson and Lunine (1988)
was that the diffusion process facilitated the rapid formation of Jupiter by concentrating a greater abundance of
ice at p5 AU. Our results indicate that relatively efficient
grain growth rates coupled with the inward drift of ice
particles will serve to deplete the feeding zone for Jupiter
by converting some of the ice to vapor and spreading out
545
the location of the condensate as the condensation front
migrates inward. This might lead to a slower formation
time for Jupiter’s heavy element core than that suggested
by Stevenson and Lunine (1988). However, if radial migration of giant planets is significant before bodies accrete
one Jupiter mass, something not yet studied, accretion of
material during migration could offset the drift-induced
losses.
In the relatively recent past there have been a series of
546
CYR, SEARS, AND LUNINE
intriguing detections of hydration features (Lebofsky et al.
1981, A’Hearn and Feldman 1992) and/or ammonia features (King et al. 1992) in asteroid spectra as well as other
circumstantial evidence that asteroid Ceres may have once
had significant amounts of free water in its interior (Fanale
and Salvail 1989). The number of asteroids with detected
hydration features is correlated with heliocentric distance
R, with fewer detections near 2 and 4 AU and more detections near 3 AU. Jones (1988) has interpreted these and
other data as suggesting that C-class asteroids may have
initially accreted from an unequilibrated mix of anhydrous
high temperature minerals, organic material, and water ice
and then were subjected to a heliocentric heating event.
The heating event would have vaporized the ice of nearer
asteroids, melted the ice of mid-range asteroids, but not
have affected the ice in asteroids farther out. Inner and
outer asteroids would have no detectable hydration features either because the ice was vaporized and blown off,
or because the ice never melted and thus did not react
with the minerals allowing detection. Mid-range asteroids
would have undergone sufficient melting such that chemical alteration of silicates would occur and be detectable.
The diffusion-drift model results provide a direct means
for causing hydration features in asteroids and an alternative to other mechanisms. The enhancement in water vapor
out to 3 AU or so would have increased the amount of
aqueous alteration of minerals incorporated into asteroidal
bodies; the drop off of vapor enhancement out to 5 AU
would have limited the amount of aqueous alteration incorporated in asteroids further out. Heating events may still
have occurred and given that ice particles drifting inward
would have been present from 3 to 5 AU and thus incorporated in asteroids, the mechanism posited by Jones is not
completely precluded by our model—both may have
been operative.
In terms of nebular chemistry, the diffusion–drift mechanisms produce an overall decrease of water vapor from 2
to 5 AU coupled with the decreasing amount of vapor with
distance over that region, due to the tail-off of local vapor
enhancement. A decrease in H2O implies a decrease in
oxygen abundance, which will shift the C : O ratio in favor
of carbon over that region. This could cause more carbonrich molecules to form, their relative abundance increasing
out to 5 AU. There is a general trend of observed darkening
of asteroids with radial distance from the Sun; Jones (1988)
and others have interpreted the darkness as being due to
increasing thicknesses of organic materials coating asteroid
surfaces. The gradual decrease in water vapor with radial
distance and thus the gradually increasing reducing nature
of the nebula could explain the asteroidal darkening. Previous diffusion models, like Stevenson and Lunine (1988),
which resulted in a flat radial vapor distribution are not
easily able to reproduce such darkening. More detailed
chemical calculations (in process) will quantify the importance of this effect.
Further, the general drying of the inner nebula by diffusive processes is potentially consistent with the oxidation
state of the enstatite chondrites (Hutson 1996), with the
caveat that other meteorite types with higher oxidation
states must be understood in the context of this model as
well. The overall problem of wide variation in oxidation
state of the environments of various meteorite types remains a daunting one, and local environments created by
parent bodies with liquid water and heat sources remains
one poorly constrained explanation for the chemistry.
Last, other implications of the model include changes
in the predicted CO/CH4 ratio produced by nebular gas
phase chemistry in the inner 1 AU of the solar nebula.
Removal of oxygen would tend to favor higher CH4 abundances relative to the predictions assuming solar composition, such as those summarized in Prinn and Fegley (1989).
The implications for the outer solar nebula, and objects
formed there, depend critically on the efficiency of radial
mixing of the nebula. Our solar nebula model, in which
tubulent diffusion is the principal mechanism for mixing
matter, would argue against significant transfer of material
between the outermost disk (beyond 10 AU) and the chemically active zone at 1 AU, based on the work of Stevenson
(1990). However, other types of advective motions could
produce more efficient mixing (Prinn 1990, Cuzzi et al.
1993), and our more reducing inner nebula could supply
some of the methane seen in outer Solar System objects
such as Triton, Pluto, and some comets. While we still
favor the source of methane as the nascent molecular cloud
(Lunine et al. 1995), a more reducing inner nebula may
be relevant in preserving some of this material against
oxidation into CO.
6. CONCLUSIONS
We have constructed a model of the evolution of water
in the inner portion of the solar nebula, i.e., inward of the
condensation front located at a midplane distance of 5 AU.
We have gone beyond the diffusional cold finger model of
Stevenson and Lunine (1988) by including the growth and
radial drift of snowballs inward of the condensation front
and their eventual sublimation. We have found an overall
drying of the inner portion of the nebula associated with
outward diffusion and trapping of water vapor, as in Stevenson and Lunine (1988) but a relative, local enhancement of the water vapor abundance appears around the
midplane region at 0.1–2 AU for a cool nebula or 0.1–3
AU for a hot nebula. This relative enhancement may have
its signature in the radial dependence of water of hydration
bands seen in asteroids. Conversely, the gradual relative
depletion of water vapor from 2 or 3 to 5 AU may have
its signature in the radial dependence of observed darken-
SOLAR NEBULA VAPOR/ICE DISTRIBUTION
ing of asteroids as well as possibly playing a role in the
oxidation states of enstatite meteorites.
A quiescent nebula with advective motions in addition
to turbulent diffusion would display a different time-dependent radial profile of water vapor than shown here. In
particular, the advective nebular models of Cuzzi et al.
(1993) produce a systematic flow of warm gas outward
across condensation fronts, such as that of water ice. Their
findings pertain to a thin midplane layer within which radial
advection occurs. The amount of material advected across
the water ice boundary over nebular lifetimes was found
to be significant relative to water budgets in the giant
planets. However, Cuzzi et al. (1993) did not evaluate the
results of the advection in terms of depletion of the water
vapor inward of the condensation zone; nor did they explicitly model the transport of condensible water including the
effects of inward drift of growing icy particles. Incorporation of such flows into a time-dependent history of nebular
water is a worthy next step, as is explicit consideration of
the nebular temperature dependence in the vertical direction. Because of water’s large overall abundance, and its
ability to condense at a key place in the nebula, tracking
its history is crucial to understanding how the solar nebula
evolved into the planetary system we witness today.
ACKNOWLEDGMENTS
This work was supported in part by NASA GSRP Grants NGT-51127
and NGT-51646, and the NASA Origins program. Special thanks go to
Andrew Rivkin for his discussion of asteroid hydration-feature spectra
and to referees J. Wood and M. Podolak for their helpful comments.
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