16
CHAPTER 2 STABILITY OF CARBON DIOXIDE
The chapter was originally published in Icarus as Palmer and Brown (2008). Used with
permission.
2.1 Introduction
Iapetus is Saturn’s third largest satellite with a radius of 718 km, a bulk density of
1212 kg m-3, and a surface gravity of 0.24 m s-2. Iapetus has dark and bright faces
(Morrison et al. 1975). The dark face is hydrocarbon-polymer rich while the bright face
is composed mostly of water ice (Buratti et al. 2005). The juxtaposition of two very
dissimilar materials on the same planetary surface makes Iapetus an interesting body to
study.
During the early portion of the Cassini mission at Saturn, the Visual and Infrared
Mapping Spectrometer (VIMS) detected CO2 on the dark surface of Iapetus (Buratti et al.
2005). Buratti argues that the CO2 is complexed, either trapped in gaseous, liquid or solid
inclusions, bound in clathrates, or adsorbed on regolith grains. Complexed CO2 is very
stable and has a long residence time on the surface of Iapetus, but free CO2 ice is still
quite volatile at Iapetus’ temperatures, and without a detailed study, it is not clear whether
CO2 could exist for long as free ice on the surface of Iapetus. To that end, we have
constructed a numerical model of ballistic CO2 transport on Iapetus that takes into
account gravitational binding energy and includes detailed calculations of the seasonal
and diurnal temperature distribution on Iapetus.
Previous work on the stability of volatiles showed CO2 to be unstable over the age
of the solar system at Saturn’s distance from the Sun (Lebofsky 1975). Lebofsky
17
predicted a rate of loss between 10 and 50 mm year-1 at the equator and between 0.1 and 5
mm year-1 at 60° latitude (Lebofsky 1975; Watson et al. 1963). Thus, one would expect
that CO2 ice could not be stable on Iapetus.
Lebofsky’s work neglected the effect of gravity because his study focused on
comets with minimal gravity. For larger bodies, however, the effect of gravity becomes
important because most of the sublimated material is launched on a ballistic trajectory and
eventually impacts the surface. CO2 molecules quite literally hop around the surface until
they reach a cold trap, such as the winter pole, or attain escape velocity and leave the
system.
Here we consider the ballistic transport of CO2 molecules on Iapetus and their
eventual capture into semi-permanent cold traps. The questions we address are:
- What is the residence time of CO2 ice at mid-latitudes?
- How long will CO2 survive on Iapetus when considering gravity?
- What is the survival time of a theoretical CO2 polar cap?
- What would be the structure of a theoretical CO2 polar cap?
- How can CO2 be present today on Iapetus?
- How much CO2 must be produced to generate and maintain a polar cap.
2.2 Model
To calculate the stability, movement, and distribution of CO2, we create a model
that takes into account thermal and sublimation energy balance, and molecular migration
across the surface. After determining the appropriate mathematical relationships
18
describing each of these physical processes, we solve the resulting system of equations
numerically using a finite-differences approach (Brown and Matson 1987). Below we
describe the relevant physics of each of the aforementioned processes.
2.2.1 Thermal Model
The thermal model is critical because surface temperature distribution determines
the amount of CO2 that sublimates as well as the distribution of speeds of the sublimating
CO2 molecules, and thus, the fraction that escapes. As such, careful consideration must
be given to the thermal model and its parameters.
We modeled three energy transport mechanisms: black body radiation, thermal
conduction, and latent heat transport as described in Eq. 2.1, where S (λ,f) is the incident
solar luminosity, φ is latitude, λ is longitude, ε is the emissivity, s is the Stefan-Boltzmann
˙ is the mass flux
constant, T is temperature, L is the latent heat of sublimation for CO2, M
of sublimated material, k is the thermal conductivity, t is time, and z is depth. Specific
values for the parameters can be found in table 1.
˙ +k
S( ", # ) = $%T 4 + L M
!
dt
dz
(2.1)
2.2.1.1 Insolation
!
The incident solar flux is defined by Eq. 2.2, where So is the solar constant of
3.827x10 26 watts, A is the Bond albedo of the surface, and r is the heliocentric distance.
S( ", # ) =
!
!
So (1$ A)
cos(# sun )cos(# iap ) + sin(# sun )sin(# iap )cos( "sun $ "iap )
4 %r 2
(2.2)
19
The leading side of Iapetus is low-albedo, presumably coated with a carbon rich
material, while the poles and trailing side of Iapetus are composed of nearly pure, water
ice (Buratti et al., 2005) with an albedo similar to the icy Galilean moons.
We used the surface reflectance map generated from the Voyager data (Squyres et
al. 1984) to estimate a Bond albedo map. First, we converted his reflectance contours into
normal reflectance by using a linear scaling between the two endpoints he suggests, y = 3/5 * x + 1.6. From there, we assume that the normal reflectance is a close approximation
to the geometric albedo.
To adjust geometric albedo into a Bond albedo, we need the phase integral. Phase
functions for neither the bright or dark side on Iapetus are well known. We used Squyres'
suggested phase integral of 0.3 for the dark material (1984). We used the phase integral
found for Europa, 1.0 (Buratti and Veverka 1983), for the small and very bright region
found near the north pole of Iapetus that has an geometric albedo of .65. We scaled
linearly between these two end points. Since Iapetus’ south pole was not imaged by
Voyager, we used the albedo of the north pole. Finally, we fit a least-squares surface to
the contour levels to provide a smooth Bond albedo map.
The final Bond albedo map is represented in Fig. 2.1. The photometrically
derived albedo map is a good fit for the dark side, but is lower compared to radiometric
calculations (Loewenstein et al. 1980; Morrison et al. 1975). The photometrically derived
Bond albedo map gives a dark side Bond albedo of 0.04 and a bright side Bond albedo of
only 0.39. Radiometery suggests that the bright side of Iapetus has an average albedo of
0.6 (Loewenstein et al. 1980). Additionally, the Voyager IRIS estimates Bond albedo in
20
the range of 0.63 to 0.73 for the other Saturnian satellites (Buratti and Veverka 1984).
Figure 2.1: Bond Albedo Map of Iapetus
Bond albedo map used for the thermal model based upon the Voyager data (Squyres et al.
1984).
Saturn’s orbital semi-major axis is 1.4x109 km and its eccentricity is 0.056 (Lang
1992), resulting in a difference in insolation of about 20% throughout the Saturn year.
The sub-solar flux when Saturn is at perihelion is 16.78 W m-2 and is 13.41 w m-2 at
aphelion, Eq. 2.2. For simplicity, we assume Iapetus to be orbiting the Sun with Saturn’s
orbital elements. Iapetus’ distance from Saturn is 59 Saturn radii, three orders of
magnitude smaller than Saturn’s distance from the Sun, thus the approximation is valid
for our purposes.
21
The variations in sub-solar latitude during an orbit is of primary importance when
considering the effectiveness of polar cold traps. The higher the sub-solar latitude, the
more insolation the poles will receive, reducing their effectiveness as a long-term cold
trap. Incorporating the variations in latitude of the sub-solar point requires knowledge of
the inclination of Iapetus’ equator; the exact value can only be reliably predicted over a
span of a few hundred years (Sinclair 1974). Currently, Iapetus has an inclination relative
to the equator of Saturn of 15.1°, and to the Sun-Saturn orbital plane of 15.4°, and is
decreasing.
Due to the perturbations caused by the Sun, Titan, and Saturn’s oblateness,
Iapetus’ inclination is not constant relative to the Sun or Saturn, but rather is constant
relative to the Laplace plane. The Laplace plane is defined as the plane normal to a
satellite’s precessional pole. Near Saturn, the Laplace plane matches Saturn’s equator
(obliquity of 26.7°). Further from Saturn where the Sun’s relative influence is greater, the
Laplace plane approaches the Sun-Saturn orbital plane. At Iapetus, the Laplace plane is
inclined 14.9° relative to Saturn’s equator (Giorgini et al. 1996), and 11.7° relative to
Sun-Saturn orbital plane.
Iapetus maintains a near-constant inclination of 7.5° to the Laplace plane
(Giorgini et al. 1996), but the longitude of Iapetus' ascending node precesses with a
period of 3,000 years (Ward 1981). This results in the inclination relative to Saturn’s
orbital plane varying between 4.3° and 19.3° over a 3,000-year cycle. Sinclair suggested a
polynomial approximation for the inclination relative to the ecliptic, Eq. 2.3, (1974).
Time, t, is time in centuries since JD 240 9786.0, and io is 184570.
22
i = io " 0.9555t " 0.0720t 2 + 0.0054t 3
(2.3)
Unfortunately, Eq. 2.3 only accurately gives the inclination of Iapetus for several hundred
!
years. To estimate the long period variation inclination of Iapetus relative to Saturn’s
orbital plane, we set the precession period for the longitude of the ascending node to be
3,000 years. This enables us to describe the long-term temperature trends, though it will
be inaccurate for a specific day. Figure 2.2 shows the predicted value by Sinclair,
observed values, and our estimated function.
2.2.1.2 Black Body Radiation
Black body radiation is calculated using the Stephan-Boltzmann law assuming an
emissivity of 0.95, Eq 2.3. Figure 2.3 shows the effect of black body radiation, limiting
the maximum temperatures to 108.5K and 128.5K for the bright side and dark side
respectively, assuming a circular orbit.
2.2.1.3 Latent Heat Transfer
Owing to its high vapor pressure at Iapetus’ temperatures, CO2 will sublimate
rapidly. Lebofsky (1975) predicted that CO2 has a sub-solar sublimation rate between 10
and 50 mm year-1 for a slow rotating body with an albedo of 0.5.
23
Figure 2.2: Effective Inclination
Iapetus’ inclination varies over a 3,000 year cycle (Sinclair 1974). We assume a constant
precession for its longitude of ascending node over a 3,000 cycle giving a sinusoidal
variation. Sinclair’s polynomial fit shows good correlation over several 100 years.
24
Figure 2.3: Energy Balance and Temperature
The energy lost due to black body radiation and latent heat for a given temperature. Since
in equilibrium insolation must balance the heat loss, we can infer the insolation required
to achieve a certain temperature. The two horizontal lines show the sub-solar flux based
on the different albedos, the dark side of 0.04 and the bright side of 0.5. The effect of
conduction is too small to be plotted.
25
To account for both mass and latent heat transport, we use the formulation of
Estermann, sometimes known as the free vacuum sublimation equation, Eq. 2.4 (1955).
˙ , is in grams s-1 m-2, R is the ideal gas constant, T is temperature, P is
The mass flux, M
the partial pressure of the gas (CO2 for our purposes here), and m is the molar mass.
!
˙=P
M
m
2"RT
(2.4)
We use Eq. 2.5 for the saturation vapor pressure of CO2 (Lebofsky 1975) in Torr.
!
P = 10
"1275.62
+0.006833T +8.3071
T
(2.5)
The exponential dependence of sublimation rate on temperature results in an
!
effective maximum temperature for pure CO2 ice sublimating into a vacuum since most of
the absorbed insolation will go into latent heat loss rather than increasing the surface
temperature. Figure 2.4 shows the buffering of temperature by latent heat transport
during a diurnal cycle on Iapetus. The individual effect of latent heat transfer with respect
to insolation is shown in Figure 2.3. Sublimation is negligible below 70K. If the surface
exposure of pure CO2 ice is large enough, the temperature cannot rise over 95K.
2.2.1.4 Thermal Diffusion
Many models neglect thermal diffusion for airless bodies because most have a
low a thermal conductivity for their surface layers, and thus, conduction has a minimal
effect upon the diurnal surface temperatures. We include it here because small
26
Figure 2.4: Buffering Effect of CO2 on Temperature
The plot compares two models showing the buffering effect of sublimation on surface
temperature. The first model accounts for conduction and blackbody radiation only. The
second model considers conduction, blackbody radiation and sublimation. Note, the
plateau-like cooling effect of sublimation latent heat near 95 Kelvin.
changes in temperature can have large effects on the sublimation rate, especially when
considering a high energy flux.
To account for thermal energy transport we use the thermal diffusion equation, Eq.
2.6, with a radiative upper boundary, and an insulating lower boundary. Temperature is
T, t is time and k is the thermal conductivity. The thermal parameters are given in table
2.1.
27
"T " $ k "T '
= &
)
"t "z % #Cp "z (
(2.6)
Table 2.1: Thermal parameters used in the model
!
Constant
Value
Thermal Inertia, Γ
30 J m-2 K-1 s-1/2
Heat Capacity, Cp
800 W kg-1K-1
Thermal Conductivity, k
0.0024 J m-1s-1K-1
Thermal Diffusivity, D
6.61 x109 m2/s
Regolith Density, H2O
461.13 kg/m3
Solar Flux, So
3.827 x1026 watts
Bond Albedo
0.01 to .065
Emissivity, ε
0.95
Latent Heat of Fusion
26.3 kg/mol
Density of CO2
1718 kg/m3
Diurnal Period
Seasonal Period
Layer thickness, Δz
79.33 Earth days
29.46 Earth years
2.80 cm e(0.07i)
Ref
1
2
6
6
3
4
5
6
7
8
9
9
6
Number of Layers
40
References: 1 Spencer et al. 2005; 2 Klinger 1980; 3 Keihm et al. 1973; 4 Watson et al.
1963; 5 Morrison et al. 1975; 6 Brown and Matson 1987; 7 Giauque and Egan 1937; 8;
Keesom and Koehler, 1934; 9 Lang et al. 1992
We assume that the thermal properties of Iapetus’ surface are similar to those of
the Galilean moons. The thermal conductivity and thermal diffusivity can be derived
from the thermal inertia, " , using equations 2.7 and 2.8. Heat capacity is defined as Cp, r
is the density, and D is the diffusivity.
!
" = #Cpk
!
!
D=
k
"Cp
(2.7)
(2.8)
28
Spencer reported a thermal inertia for Iapetus of 30 J m-2 K-1 s-1/2 (2005), which is
similar to the 50 to 70 J m-2 K-1 s "1/ 2 he found for the Galilean moons (Spencer et al.
1999).
We assume that the heat capacity and density are independent of depth and
!
temperature as described in Table 1.
The upper boundary of the diffusion model is a periodic function based upon the
insolation absorbed as described in section 2.1.1. The lower boundary is an insulating
layer such that dT/dz = 0 at depth.
To ensure that the upper layer of the model is thin enough to respond to diurnal
heating, we set its thickness to be 1/3 the diurnal skin depth. The diurnal skin depth is the
scale length for diurnal thermal diffusion, Eq. 2.9. The skin depth is dz, D is the
diffusivity, and t is the time scale of interest.
dz =
Dt
2"
(2.9)
We set the top layer to 2.8 cm in thickness and allow the lower layers to have a
!
minor exponential increase in thickness, described by dz = e(0.07i) cm, with i being the
layer number (Matson and Brown 1989). The required number of layers is such that the
total depth over which the thermal solution is obtained is five times the seasonal skin
depth, ensuring that temperature cycles are minimal at depth (dT/dt = 0). We use 40
layers, corresponding to a depth of 6.4 meters, exceeding the required depth of 4.95
meters. We do not include thermal conductivity through the CO2 frost itself, but instead
use the parameters for a water ice regolith for the conductive layers.
To ensure that our equilibrium solution is independent of initial conditions we
29
start from a baseline temperature model that has been run for 50 orbits to allow the layers
to reach thermal equilibrium.
2.2.2 CO2 Transport
Once we have the temperature of the surface, we calculate the transport of the
sublimated CO2. When a CO2 molecule escapes its crystal lattice, we assume that it will
take a single sub-orbital ballistic hop. This is a good assumption because near Iapetus’
surface the mean free path is larger than its scale height. The calculation of the where the
ballistic trajectory lands requires only the initial speed and direction of a molecule. We
neglect the effects of solar radiation pressure upon the molecule.
Sublimation into a vacuum produces a half-Maxwell Boltzmann velocity
distribution, a distribution with no negative vertical speed component, Eq. 2.10. The
temperature of the surface is a major factor in the determination the distribution of speeds
as seen in Eq. 10. Velocity is V, f is the fraction of material that has a velocity between V
and V+ΔV, k is the Boltzmann constant, T is temperature, and m is the molecular mass.
3
# m & 2 2 )mV
f = 4"%
( V e 2kT
$ 2"kT '
2
(2.10)
All the CO2 molecules that have velocities greater than Iapetus' escape velocity
!
(591 m s-1) will escape from the system regardless of their initial direction, and we
assume that they are permanently lost from the system.
CO2 molecules that have orbital velocities (above 418 m s-1) are not considered
lost to the system since they will not have a stable orbit. Research on asteroid impact
30
ejecta shows that most material ejected at orbital velocities will not reach a stable orbit,
but will return to its parent body in a single orbit (Scheeres et al. 2002).
The second component of the ballistic flight of CO2 is its direction vector. The
angle at which the CO2 leaves the surface is called the angle of trajectory. The radial
direction the CO2 takes is called the azimuth. We assume that the CO2 leaves the surface
isotropically, analogous to the way a Lambert surface scatters light. Once both the speed
and the direction are known, we calculate the suborbital ballistic flight for the CO2
molecules, assuming Iapetus is a perfect sphere.
The distance a molecule travels, d, is found using the orbit’s semi-major axis, a,
eccentricity, e, true anomaly, f, and flight angle, φ, as shown by equations 2.11, 2.12, 2.13
and 2.14. The mass of Iapetus is M, G is the gravitational constant, r is the molecule’s
current orbital distance (set to the radius of the moon), and v is the molecule’s velocity.
a=
!
!
1
2 v2
"
r MG
sin 2 (# )r(2a " r)
e = 1"
a2
f = cos"1
a " ae 2 " r
re
d=(2π-2f)r
(2.11)
(2.12)
(2.13)
(2.14)
!
Once we have the initial velocity vector of the CO2, we calculate where it will
land. We do this for all variations of speed, direction and source latitude to build a
31
template of distribution. By assuming an isotropic distribution for direction and
Maxwell-Boltzmann distribution for the speed, the computation is simplified since the
surficial distribution of the mass for a given temperature will be identical from one time
step to another except for a scaling factor. As such, we do not need to track each
molecule, but can apply the template in bulk without losing accuracy. The validity of our
numerically derived templates was confirmed with a Monte Carlo simulation.
A cross section of a sample template is shown in Fig. 2.5. One can see that most
of the mass travels only a short distance; half the mass falls within 150 km. Nevertheless,
the entire moon receives some material, with a minor concentration at the antipode.
We use a time step of 1.6 Earth days in the transport model, which is the largest
stable time step that ensures the sub-solar latitude does not change substantially before an
average CO2 molecule randomly walks to a cold trap.
The fraction of sublimated CO2 that reaches escape velocity is strongly dependent
on temperature (Eq. 2.10). Table 2.2 shows what fraction of the CO2 will reach escape
velocity when it sublimates. The remaining CO2 will make a single ballistic suborbital
flight and will stick where it lands. We found that there is no appreciable warming of a
polar cap when CO2 condenses because the amount of latent heat transported by the CO2
is small compared to the radiative heat flux.
32
Figure 2.5: Distance of CO2 for a Single Time Step
The ballistic distance CO2 travels when sublimated. It assumes a Maxwell-Boltzmann
velocity distribution and an isotropic distribution for the angle of trajectory. The
distribution shows a single ballistic flight.
33
Table 2.2: Fraction of CO2 reaching escape velocity as a function of temperature.
Temperature Fraction
40 K
5.3 x 10-10
50 K
4.8 x 10-8
60 K
9.6 x 10-7
70 K
8.0 x 10-6
80 K
3.9 x 10-5
90 K
1.3 x 10-4
100 K
3.5 x 10-4
110 K
7.9 x 10-4
120 K
1.5 x 10-3
130 K
2.7 x 10-3
140 K
4.3 x 10-3
2.3 Results
2.3.1 Ablation Rate
The residence time for CO2 on the illuminated regions of Iapetus is very short.
The sublimation rate can be as high as 0.02 g m-2 s-1, such that a thick sheet of CO2 ice
cannot survive for long outside of the polar regions. When CO2 sublimates, most of it
will land near its source (see Fig. 2.5) whereupon during the next hop, some of the
material will return to the source region. As a result, the amount of CO2 removed from a
region per unit time is somewhat lower than the sublimation rate, which we will call the
ablation rate. The ablation rate is always lower than the sublimation rate.
Figure 2.6 shows how fast CO2 can ablate from the surface of Iapetus during a
single Iapetus day or 79.3 Earth days. The dark side ablation rate in the equatorial regions
can be as high as 13 mm diurnal cycle-1, showing that free CO2 ice cannot remain in the
dark region for more even a fraction of a single diurnal cycle on Iapetus.
34
Figure 2.6: Ablation Rate
The ablation rate of CO2 for a single diurnal cycle of Iapetus (29.46 Earth days). This
uses Iapetus' current effective obliquity of 15.4° and Bond albedos of 0.04 and 0.5. We
use a distance of 9.24 AU, the distance Iapetus was from the Sun during the 10 September
2007 fly-by.
35
To characterize the long-term-average ablation rate, we ran several models,
determining the ablation rate every 15° in latitude on both dark and bright terrains (0.04
and 0.5 albedo respectively), assuming a sheet of CO2 ice 12 km x 12 km wide and thick
enough not to be removed during the run of the model. The eccentricity of Saturn's orbit
results in differences in the peak insolation between the north and south poles; thus, we
include both poles in our calculation. Figure 2.7 shows the rate of CO2 ice ablation
considering both albedo and effective obliquity. The bright regions have a much lower
ablation rate, approximately 1/3 less than the dark regions. At mid-latitudes, the effect
due to the effective obliquity on the ablation rate is small; however, the ablation rate is
greatly affected by effective obliquity near the poles.
2.3.2 Polar Caps
As one might expect, CO2 will quickly move from the equator and accumulate at
the winter pole, resulting in a polar cap. Once the CO2 falls into a polar cold trap, it will
be sequestered until that pole begins its summer. As the polar solar flux increases, the
edge of the polar cap will ablate and recede. During the initial stages of the polar
summer, approximately 40% of the CO2 liberated in its ablation zone will land on the
opposite pole while the remaining 60% will land higher on the source polar cap. This will
increase the thickness of the polar cap while its latitudinal extent decreases, such that a
thin but wide polar cap will increase in thickness by an order of magnitude just before the
highest latitudes start sublimating (Fig. 2.8).
36
Figure 2.7: Long Term Ablation Rates
The ablation rate of CO2 from a sheet of CO2 ice at different inclinations, latitudes and
Bond albedos. The ablation rates are the average of an entire orbit around the Sun, giving
the long term effects. The bright terrain is set at a constant Bond albedo of 0.5 while the
dark terrain is 0.04.
37
Figure 2.8: Thickness Evolution of a Seasonal Polar Cap
The time evolved thickness and latitudinal extent of a seasonal polar cap. A seasonal
polar cap will being as a thin layer of CO2 ice when it is emplaced during the pole's
winter season. During the summer season, as the solar flux increases, the edge of the
polar cap will ablate with 40% of the ablated material random walking to the opposite
pole; however, 60% will land at higher latitudes where there still is a cold trap. The result
is a steady thickening polar cap, while its latitudinal extent retreats.
38
Polar caps fall into two categories: permanent and seasonal. If a polar cap is not
completely removed during the summer, it is a permanent polar cap; but if it fully
sublimates and migrates to the other pole, it is a seasonal polar cap. We can establish the
minimum amount of CO2 needed to make a permanent polar cap by tracking how much
CO2 a small polar cap would transfer to the other pole in a single season. We created a
model with a polar cap that was only six kilometers in radius on the north pole and
tracked how much CO2 ended on the south pole over the course of a single orbit. We find
that if there is more than 3x107 kg of CO2 on the north pole, then it will not be fully
removed over a single seasonal summer, and is thus permanent. Alternatively, if there is
less than 3x107 kg of CO2 on the north pole, all of the CO2 will sublimate and be
transferred to the opposite pole, and is thus a seasonal polar cap.
Additionally, using the same logic and knowing the latitudinal extent of a polar
cap, we can predict the minimum amount of CO2 that must be present. To that end we ran
a series of models, altering the latitudinal extent of the polar cap and tracking how much
CO2 is transferred between poles, (see Table 2.3). Each model provides us with the
minimum amount of CO2 that must be present if a polar cap of a given size exists. If the
latitudinal extent of a polar cap is known, Table 2.3 then shows how much CO2 must be
present. There will be more CO2 in the system, however, because our analysis only
considers the permanent portion of the polar cap, and not the CO2 that makes up the
seasonal part of the polar cap.
39
Table 2.3: Sublimation and movement rates for different sized polar caps.
Extent of
Polar Cap
(° Latitude)
Total
Sublimation
Kg solar orbit-1
Net
Movement
Kg solar orbit-1
Minimum Effective Obliquity 4.3
+89.5 to +90 1.0 x 100
-1
1
+88.5 to +90 1.3 x 10
-1
+87.5 to +90 2.0 x 102
-1
3
+86.5 to +90 2.0 x 10
1.7 x 103
+85.5 to +90 1.5 x 104
1.0 x 104
5
+84.5 to +90 1.1 x 10
6.1 x 104
Current Effective Obliquity 15.4
+89.5 to +90 3.2 x 107
3.0 x 107
8
+88.5 to +90 2.0 x 10
1.7 x 108
+87.5 to +90 7.0 x 108
5.6 x 108
9
+86.5 to +90 1.9 x 10
1.3 x 109
+85.5 to +90 4.3 x 109
2.7 x 109
9
+84.5 to +90 9.4 x 10
5.3 x 109
Maximum Effective Obliquity 19.3
+89.5 to +90 2.9 x 108
2.7 x 108
9
+88.5 to +90 1.9 x 10
1.6 x 109
+87.5 to +90 6.1 x 109
4.7 x 109
9
+86.5 to +90 1.5 x 10
1.0 x 1010
+85.5 to +90 3.0 x 1010
1.9 x 1010
10
+84.5 to +90 5.7 x 10
3.3 x 1010
1
- Transport is less than a monolayer.
Percent
in
Transit
85%
67%
55%
94%
85%
80%
68%
62%
56%
93%
84%
77%
67%
63%
58%
The aforementioned results are useful to us if we are able to detect a polar cap of a
specific latitudinal extent. If we know the size of the permanent cap, we can estimate the
minimum amount of CO2 that it must contain. Any less CO2 and the polar cap would not
be permanent.
The amount of CO2 that can be transported in a season is also dependent on
Iapetus, effective obliquity. During periods when Iapetus has a low obliquity, little CO2
is needed to form a permanent polar cap having an ablation rate less than a single
40
monolayer every solar orbit from latitude +90°. However, 1,500 years later, Iapetus will
be at its maximum inclination, and the same region would ablate ~0.13 mm of CO2 during
a solar orbit. Table 2.3 lists the threshold levels for latitude, but also the minimum,
maximum and current effective obliquity.
The morphology, behavior, and structure of a permanent polar cap will be defined
by the mass of CO2 in the cap and the viscosity of the CO2. During a single orbital cycle,
the accumulation of CO2 will generally be even over the entire polar cap; however, the
rate of sublimation will be higher at lower latitudes. Over time, this results in a
permanent polar cap that thickens more near the pole than at lower latitudes. The
thickness of a polar cap will grow until the basal sheer stress exceeds the yield strength of
the CO2 ice and it begins to flow. The flow of ice will be away from the pole and into a
region with more insolation. The lowest latitude of the polar cap will be determined by
the equilibrium point between viscous flow and ablation (Brown and Kirk 1994). This is
valid for high-mass polar caps only. Low-mass polar caps will not flow viscously; thus,
their shape and structure will be governed solely by the distribution of insolation. Due to
the low temperatures and gravitational force on Iapetus, we do not expect there to be any
glacial flow.
An additional consideration is the difference in insolation between Saturn's
aphelion and perihelion, which is approximately 20%. The lower flux incident upon the
north pole during its summer allows a north polar cap to persist while a southern polar cap
cannot. We find that the south polar cap loses about twice the CO2 than the north polar
cap during each season. Table 2.3 provides the sublimation and ablated amounts for a
41
north polar cap only since the north pole sets the lower limit.
While permanent polar caps require a large global reservoir of CO2, seasonal polar
caps will exist if there is any free CO2 present. The structure of a seasonal polar cap is
different than a permanent polar cap; due to the high volatility of CO2, it will only
accumulate on the unlit winter pole. Carbon dioxide transported from the summer pole
will be deposited just past the seasonal terminator of the winter pole. This will result in a
polar cap that extends from the pole to the latitude where the peak diurnal energy flux is
less than 1/2 watt m-2 (basically unlit).
2.3.3 CO2 Escape Rates
In a closed system, CO2 frost can migrate between the two poles for eternity;
however, this is implausible because there are many loss mechanisms for CO2. One
process that destroys CO2 on Iapetus is photodissociation by the Sun's UV radiation. We
calculate the photochemical time scale using Eq. 2.15, where there absorption cross
section for CO2 is denoted as σ defined by (Chan et al. 1993; Lewis and Carver 1983),
while the ultraviolet flux is Φ (Woods et al. 1998). Summing for all wavelengths
between 6nm and 200nm, we calculate a photochemical timescale of 1.7x107 s.
"=
1
1
=
J %#$
(2.15)
Alternatively, we calculate the average time it takes for a molecule to random
!
walk to the opposite pole. We use a sublimation temperature of 90K and a launch angle
of 45°, giving a Vrms = 225 m s-1 and an average time of flight of 1.7x103 s. Since it
takes approximately 350 hops to random walk between the poles, we calculate the
42
average time a CO2 molecule is in motion is 6.0x105 s, more than an order of magnitude
less than the photochemical time scale.
Carbon dioxide frost can also be sequestered in the regolith, being vaporized by
micro-meteorite impact, and adsorbed onto the surface of regolith grains, or bonding with
water forming clathrates; however, for this study these effects are neglected.
Our primary interest is how quickly CO2 can be lost from the system due to the
high velocity tail of the Maxwell-Boltzmann distribution that exceeds the escape velocity.
Our model tracks the amount of CO2 lost in this manner and we have calculated rates for
three scenarios: 1) a moon totally covered in CO2, 2) permanent polar caps, and 3)
seasonal polar caps (Fig. 2.9).
When the entire moon is covered in CO2, a large amount of CO2 will escape the
system every solar orbit (2x1012 kg solar orbit-1). The subsolar latitude, and thus the
effective obliquity, makes no difference in the amount of CO2 that reaches escape velocity
as long as the entire surface is covered in CO2.
The next regime for escape is when there are large permanent caps. We consider
polar caps stretching from the polar regions almost to the equator. As the polar caps
shrink, the amount of CO2 that sublimates gets correspondingly smaller. In general, this
results in a reduced amount of CO2 escaping from the system. The major factor affecting
the CO2 loss rate is the latitudinal extent of any polar caps. Figure 2.9 shows the loss rate
of CO2 as a function of the size of the polar cap.
43
Figure 2.9: Long Term Loss Rate of CO2 from Iapetus' Surface
The escape rate of CO2 from the surface of Iapetus showing the different regimes of CO2
escape. The maximum escape rate is found when the polar ice sheets do not cover the 30°
adjoining the equator. This is because the equatorial regions will be hotter than when
they are fully buffered by CO2 ice, allowing for a higher surface temperature and a larger
percentage of CO2 being in the high temperature tail of the Maxwell-Boltzmann
distribution.
44
Surprisingly, the maximum escape rate for CO2 (3.4x1012 kg solar orbit-1) does not
occur when the entire moon is covered in CO2, but rather when large portions of the
equatorial region are free of CO2 (about 30 degrees either side of the equator). The
reason for the increased escape rate is the effect of the higher temperature of the
effectively bare surface. This increases the percentage of CO2 molecules that are in the
high velocity tail of the Maxwell-Boltzmann distribution. When the entire moon is
covered with CO2, the temperature is buffered to a maximum of 96K, but when the
equatorial regions are free of CO2 ice, the surface temperature can reach 130K. When
only a small amount of CO2 lands there, the temperature suppression is small, allowing
the CO2 to be thermalized close to 130K, rather than 96K. A larger percentage of CO2
will thus reach escape velocity, which can result in more CO2 escaping from the system
(see Table 2.2).
The last regime we consider is when there is only enough CO2 to make a seasonal
polar cap. We believe that this is the most likely case for Iapetus since there are no
published detections of a CO2 polar cap.
The amount of CO2 that escapes depends primarily on the total amount of CO2 on
the surface, and to a lesser extent, the latitudinal extent of the polar cap. For a seasonal
polar cap, all the CO2 in the system will sublimate and move between the poles, and the
movement between the source and the sink can be seen as a random walk. During this
transit across the face of the moon, a typical molecule of CO2 will hop 350 times; many of
these hops will be at temperatures much higher than those at the pole from which it came.
45
The cumulative effect is that 6% of the mobilized CO2 will reach escape velocity while
moving from the summer pole to the winter pole. Since, to first order, the loss rate is an
exponential, we calculate a characteristic time scale for 1/2 the CO2 to be lost from the
system to be 5.8 solar orbits, or 170 years.
In general, approximately 12% of the CO2 moving between the poles escapes
during each solar orbit; however, we find that this is not a constant. Near the limit
between seasonal and permanent polar caps, the escape rate is 12% per solar orbit;
however, as the total CO2 inventory decreases, the escape fraction increases (Fig. 2.10).
This is most likely due to the increasing surface area that is not covered in CO2 allowing
for more hops to be made at unbuffered (higher) temperatures.
2.3.4 CO2 Resupply
Our previous models predict the evolution of a fixed inventory of CO2. Next, we
consider how this picture changes if the CO2 is resupplied, such as photochemically
generated or from an active vent. We consider a source region at 30° latitude and vary the
production rate of CO2. The actual position of the source region has very little effect on
the ultimate distribution of CO2 ice, since CO2 quickly migrates to the winter pole.
We find that when starting with no CO2, a seasonal polar cap will form and grow,
increasing its size until its escape rate matches the source rate. If the production or
liberation rate of CO2 is greater than the loss rate, a permanent polar cap will form. We
see this behavior for flux rates higher than 4x106 kg orbit-1, where a small region of the
north polar cap that has a Bond albedo of 0.65 will become a permanent polar cap.
46
Figure 2.10: Net CO2 Loss Rate
The percent of CO2 lost per orbit as a function of the total amount of CO2 in the system.
This graph is based upon an inclination of 15.4° and seasonal polar cap. Typically, 12%
of the CO2 escapes per solar orbit; however, as the cap gets smaller, the loss ratio
increases and the characteristic time scale for half of the CO2 to escape from the system
decreases.
47
We can use the information from Figure 2.9 to determine the size of polar caps
that can result from a given resupply rate. Since Figure 2.9 shows how much CO2 will be
lost by a polar cap of a given size, we note that with a given a specific escape rate, a polar
cap will grow in thickness and extend toward lower latitudes until it reaches steady state,
the latitudinal extent depicted in Fig. 2.9.
2.4 Discussion
One can see that the long term stability of CO2 is problematic. The first issue to
consider whether a CO2 polar cap could be primordial. A strong upper limit to the time
polar caps could exist on Iapetus can be estimated by assuming that the entire moon’s
primordial inventory of CO2 was emplaced in a single surficial ice sheet. We assume that
Iapetus' primordial inventory of CO2 can be extrapolated from the concentration of CO2
found in the plumes of Enceladus of 3% (Hansen et al. 2006). Taking the mass of Iapetus
to be 1.88x1021 kg and assuming that 3% of Iapetus' bulk mass is CO2, we get 5.6x1019
kg, or a 5-km-thick sheet of CO2 ice on the moon.
To estimate the residence time for this maximum surficial inventory of CO2, we
consider extrapolating the loss rate from an Iapetus that is both fully and partially covered
in CO2. The loss rate for Iapetus is ~1012 kg solar orbit-1 (Fig. 2.6) until the ice has
receded to +60° latitude. Using this loss rate, the entire budget of CO2 can be lost from
Iapetus in only 1.6 G.a. While it is likely that all of such CO2 would be removed over the
age of the solar system, we cannot rule out that some CO2 would remain today.
This estimate is much shorter than what would actually happen because it only
48
considers the CO2 at the poles. In actuality, a large portion of the CO2 from lower
latitudes would accumulate in the polar region making the polar cap much thicker than the
initial five kilometers. Thus, while unlikely, if the total inventory of CO2 within Iapetus
were deposited on the surface when Iapetus was formed, large polar caps would persist to
this day.
One possible endogenic source of CO2 could be an outgassing fissure similar to
the vents on Enceladus (Hansen et al. 2006). Enceladus is outgassing a large amount of
volatiles that is the source of Saturn's E ring. Unlike Enceladus, however, any vent on
Iapetus is likely to be small with a low outgassing rate for several reasons. First, the
surface of Iapetus is heavily cratered with an estimated age of 4.4 G.a. over its entire
surface (Ip 2006; Morrison 1982). This would exclude a massive outflow because that
would require substantial resurfacing. Second, neither a tenuous atmosphere nor a ring
has been observed. Thus, Iapetus cannot be effusing large amounts of gas.
Another possible exogenic source for CO2 would be a cometary impact. A
hypothetical comet can deposit a maximum CO2 of 1.7x1012 kg on Iapetus, assuming a
diameter of six kilometers, a density of 500 kg/m3, a CO2 concentration of 3%, and none
of the CO2 lost during impact. We ran a model where we deposited CO2 at the equator
and found that approximately 100 years is required for the CO2 to move to the poles
where it will form permanent polar caps about 4 degrees in latitudinal extent. These caps
will survive on the order of 75,000 years before becoming seasonal polar caps. Finally,
once the polar caps become seasonal, they lose ~ 12% of their mass each solar orbit,
exhausting the CO2 in an additional 5,000 years.
49
Finally, free CO2 could be photochemically generated from the dark material
itself. While the bright regions of Iapetus are clearly H2O dominated, the dark region has
not been fully characterized (Buratti et al. 2005). It has been speculated that the dark
surface may be:
1) A carbonaceous layer (Smith et al. 1982),
2) CH4 • x H2O and NH3 • H2O embedded in H2O (Squyres et al. 1983),
3) Composed of a nitrogen-rich tholin, amorphous carbon, and a small amount of
H2O ice (Buratti et al. 2005).
Experiments on Iapetus-like ices have shown that CO2 can be generated via both
solar and ion irradiation (Allamandola et al. 1988; Ehrenfreund et al. 1997; Gerakines et
al. 1996; Hudson and Moore 2001; Loeffler et al. 2005; Mennella et al. 2004, 2006;
Moore and Hudson 1998; Sandford et al. 1990; Strazzulla and Palumbo 1998). Specific
laboratory experiments with ultraviolet photolysis of ices composed of H2O, CH3OH,
NH3, and CO produced H2CO, CO2, CO, CH4, and HCO (Allamandola et al. 1988). A
more recent experiment with mixtures of H2O:NH3:CH4 found that irradiation with 30keV
He+ and 60 keV Ar++ results in the formation of C2H4, CO and CO2 (Strazzulla and
Palumbo 1998).
Buratti et al. (2005) notes that CO2 generated by solar UV radiation will be near
the surface and will have a short residence time. She infers that the CO2 must all be
complexed due to the short residence times, but we have shown that the gravity binds the
CO2. Our model predicts that 94% of photolytically-generated CO2 would move to
Iapetus’ poles and later be lost at a rate of 12% per solar orbit. Since the escape
50
percentage is 12% (λ=0.12 solar orbit-1), we can use the escape rate (in kg solar orbit-1), to
estimate the amount of CO2 that must be on the surface using Eq. 2.16, regardless if it is
photolytically generated or primordial. The total number of molecules moving in the
system is denoted as No, and the escape rate is dN/dt. Thus, there must be approximately
~8 times as much CO2 moving on the surface as is generated as well as lost.
"N
= N o#
"t
(2.16)
The spectra of the three suggested materials do not contain the 4.26-µm
!
asymmetric stretch absorption feature of CO2 ice. Thus, there must be some CO2 as part
of the dark material, most likely complexed due to the volatility of CO2 ice. We assume
that this CO2 is being actively produced rather than being primordial. If only complexed
CO2 is generated, then its production rate could be small to match the observed CO2
signature. It is more likely, however, that the production of CO2 would generate mostly
unbound CO2, which would allow for a higher production rate to match observed values.
The unbound CO2 would be free to ballistically move to a polar cold trap, where it might
be detected.
2.5 Conclusion
In this paper we explored mechanisms for the migration of CO2 on the surface of
Iapetus via suborbital, ballistic flight after sublimation. Any CO2 at equatorial and midlatitudes is unstable and quickly migrates to the poles via a random walk process. Once
the CO2 reaches the poles, lower energy flux there sequesters it, where the stability of ice
in this polar cold trap depends on the effective obliquity of Iapetus. Currently, the effect
51
of obliquity on polar insolation is strong enough to move 3x107 kg of CO2 from a small
polar cap of only six kilometers in radius. During the transit between poles, ~ 6% of the
CO2 will reach escape velocity and be lost to space, which results in ~ 12% lost every
solar orbit (29.46 years).
The detection of the 4.26-µm absorption feature requires that some CO2 be present
in Iapetus’ dark material, most likely being complexed. It is unlikely that this CO2 is
primordial. Since the raw materials for photolytic production of CO2 are present on
Iapetus, its production should be ongoing. We assume that most of this new CO2 would
be in the form of free CO2, which would end up forming a polar cap. Additionally, due to
the large escape rate of CO2 from the surface, any free CO2 ice found on Iapetus implies
active production, such as photochemical generation, liberation during an impact, or by an
active vent.