The stability and transport of carbon dioxide on Iapetus

Icarus 195 (2008) 434–446 www.elsevier.com/locate/icarus

The stability and transport of carbon dioxide on Iapetus Eric E. Palmer ∗ , Robert H. Brown Lunar and Planetary Laboratory, 1629 E. University Blvd., The University of Arizona, Tucson, AZ 85721-0092, USA Received 5 February 2007; revised 9 November 2007 Available online 23 December 2007

Abstract Carbon dioxide has been detected associated with Iapetus’ dark material by the Cassini spacecraft. This CO2 may be primordial and/or resulting from ongoing production by photolysis of water-ice in the presence of carbonaceous material [Allamandola, L.J., Sandford, S.A., Valero, G.J., 1988. Icarus 76, 225–252]. Although any primordial CO2 would likely be complexed with the dark material and thus stable against thermal transport to Iapetus’ poles [Buratti, B.J., and 28 colleagues, 2005. Astrophys. J. 622, L149–L152], active production of CO2 would result in some fraction of the CO2 being mobile enough to allow the accumulation of CO2 at Iapetus’ poles. We develop a computer model to simulate ballistic transport of CO2 ice on Iapetus, accounting for Iapetus’ gravitational binding energy and polar cold traps. We find that the residence time of CO2 ice outside the polar regions is very short; a sheet of CO2 ice near the equator of Iapetus decreases in thickness at a rate of 50 mm year−1 . The sublimated CO2 will ballistically move around Iapetus until it reaches the polar cold traps where it can be sequestered for up to 15 years. If the total surface inventory of CO2 exceeds 3 × 107 kg, the polar ice cap will be permanent. While CO2 is moving around the surface, a small percentage will eventually reach escape velocity and be lost from the system. As such, a seasonal polar cap is lost at rate of 12% every solar orbit as the CO2 moves between the two polar cold traps. © 2008 Elsevier Inc. All rights reserved. Keywords: Iapetus; Ices

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 et al. argue 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 * Corresponding author. Fax: +1 520 621 4933.

E-mail address: [email protected] (E.E. Palmer). 0019-1035/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2007.11.020

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 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

Stability and transport of CO2

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. Model

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Table 1 Thermal parameters used in the model Constant

Value

Reference

Thermal inertia, Γ Heat capacity, Cp Thermal conductivity, k Thermal diffusivity, D Regolith density, H2 O Solar flux, S0 Bond albedo Emissivity, ε Latent heat of fusion Density of CO2 Diurnal period Seasonal period Layer thickness, z Number of layers

30 J m−2 K−1 s−1/2

Spencer et al. (2005) Klinger (1980) Brown and Matson (1987) Brown and Matson (1987) Keihm et al. (1973) Watson et al. (1963) Morrison et al. (1975) Brown and Matson (1987) Giauque and Egan (1937) Keesom and Koehler (1934) Lang (1992) Lang (1992) Brown and Matson (1987) –

800 J kg−1 K−1 0.0024 J m−1 s−1 K−1 6.61 × 109 m2 /s 461.13 kg/ m3 3.827 × 1026 W 0.01 to 0.065 0.95 26.3 kg/mol 1718 kg/m3 79.33 Earth days 29.46 Earth years 2.80 cm e(0.07i) 40

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 describing each of these physical processes, we solve the resulting system of equations numerically using a finitedifferences approach (Brown and Matson, 1987). Below we describe the relevant physics of each of the aforementioned processes. 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. (1), where S(λ, φ) is the incident solar luminosity, φ is latitude, λ is longitude, ε is the emissivity, σ is the Stefan–Boltzmann constant, T is temperature, L is the latent heat of sublimation for CO2 , M˙ is the mass flux 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. dt S(λ, φ) = εσ T 4 + LM˙ + k . dz

(1)

2.1.1. Insolation The incident solar flux is defined by Eq. (2), where S0 is the solar constant of 3.827 × 1026 W, A is the Bond albedo of the surface, and r is the heliocentric distance: S(λ, φ) =

S0 (1 − A) cos(φsun ) cos(φiap ) 4πr 2 + sin(φsun ) sin(φiap ) cos(λsun − λiap ).

(2)

The leading side of Iapetus is low-albedo, presumably coated with a carbon rich material, while the poles and trailing

Fig. 1. Bond albedo map used for the thermal model based upon the Voyager data (Squyres et al., 1984).

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 this 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 (Buratti and Mosher, 1995; Buratti et al., 2002). 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 0.65. 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. 1. The photometrically derived albedo map is a good fit for the dark side,

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Fig. 2. Iapetus’ inclination varies over a 3000 year cycle (Sinclair, 1974). We assume a constant precession for its longitude of ascending node over a 3000 cycle giving a sinusoidal variation. Sinclair’s polynomial fit shows good correlation over several 100 years.

but is lower compared to radiometric calculations (Loewenstein et al., 1980; Morrison et al., 1975). The photometrically derived Bond albedo map gives a darkside Bond albedo of 0.04 and a brightside Bond albedo of only 0.39. Radiometry suggests that the bright side of Iapetus has an average albedo of 0.6 (Loewenstein et al., 1980). Additionally, the Voyager IRIS estimates a Bond albedo in the range of 0.63 to 0.73 for the other saturnian satellites (Buratti and Veverka, 1984). Saturn’s orbital semi-major axis is 1.4 × 109 km and its eccentricity is 0.056 (Lang, 1992), resulting in a difference in insolation of about 20% throughout the Saturn year. The subsolar flux when Saturn is at perihelion is 16.78 W m−2 and is 13.41 W m−2 at aphelion, Eq. (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. 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). Sinclair reported Iapetus to have 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 3000 years (Ward, 1981). This results in the inclination relative to Saturn’s orbital plane varying between 4.3◦ and 19.3◦ over a 3000-year cycle. Sinclair (1974) suggested a polynomial approximation for the inclination relative to the ecliptic, Eq. (3). Time, t, is time in centuries since JD 240 9786.0, and i0 is 18.4570: i = i0 − 0.9555t − 0.0720t 2 + 0.0054t 3 .

(3)

Unfortunately, Eq. (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 3000 years. This enables us to describe the long-term temperature trends, though it will be inaccurate for a specific day. Fig. 2 shows the predicted value by Sinclair, observed values, and our estimated function. 2.1.2. Black body radiation Black body radiation is calculated using the Stephan– Boltzmann law assuming an emissivity of 0.95, Eq. (1). Fig. 3 shows the effect of black body radiation, limiting the maximum

Stability and transport of CO2

437

Fig. 3. 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.

temperatures to 108.5 and 128.5 K for the bright side and dark side respectively, assuming a circular orbit. 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. To account for both mass and latent heat transport, we use the formulation of Estermann (1955), sometimes known as the ˙ is in free vacuum sublimation equation. The mass flux, M, g s−1 m−2 , R is the ideal gas constant, T is temperature, P is the partial pressure of the gas (CO2 for our purposes here), and m is the molar mass:  m M˙ = P (4) . 2πRT We use Eq. (5) for the saturation vapor pressure of CO2 (Lebofsky, 1975) in Torr: P = 10

−1275.62 +0.006833T +8.3071 T

.

(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. Fig. 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 Fig. 3. Sublimation is negligible below 70 K. If the surface exposure of pure CO2 ice is large enough, the temperature cannot rise over 95 K.

2.1.4. Thermal diffusion Many models neglect thermal diffusion for airless bodies because most have a low thermal conductivity for their surface layers, and thus, conduction has a minimal effect upon the diurnal surface temperatures. We include it here because small 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. (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 1.   ∂T ∂ k ∂T (6) = . ∂t ∂z ρCp ∂z 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, Γ , using Eqs. (7) and (8). Heat capacity is defined as Cp , ρ is the density, and D is the diffusivity;  Γ = ρCp k, (7) k D= (8) . ρCp Spencer et al. (2005) reported a thermal inertia for Iapetus of 30 J m−2 K−1 s−1/2 , 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

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Fig. 4. 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 near 95 K.

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. (9). The skin depth is dz, D is the diffusivity, and t is the time scale of interest:  Dt . dz = (9) 2π 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 m, exceeding the required depth of 4.95 m, which is five times the seasonal skin depth. 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 start from a baseline temperature model that has been run for 50 orbits to allow the layers to reach thermal equilibrium. 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 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. (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 + V , k is the Boltzmann constant, T is temperature, and m is the molecular mass:  f = 4π

m 2πkT

3 2

V 2e

−mV 2 2kT

.

(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 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, θ . The radial direction the CO2 takes is called the azimuth, φ. We assume that the CO2 leaves

Stability and transport of CO2

439

Fig. 5. 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.

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, and the true anomaly, f , as shown by Eqs. (11), (12), (13) and (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 r

 e=

−

v2 MG

1−

,

sin2 (φ)r(2a − r) , a2

a − ae2 − r , re d = (2π − 2f )r. f = cos−1

(11)

(12) (13) (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 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.

Table 2 Fraction of CO2 reaching escape velocity as a function of temperature Temperature

Fraction

40 K 50 K 60 K 70 K 80 K 90 K 100 K 110 K 120 K 130 K 140 K

5.3 × 10−10 4.8 × 10−8 9.6 × 10−7 8.0 × 10−6 3.9 × 10−5 1.3 × 10−4 3.5 × 10−4 7.9 × 10−4 1.5 × 10−3 2.7 × 10−3 4.3 × 10−3

A cross section of a sample template is shown in Fig. 5. One can see that most of the mass travels only a short distance; half the mass falls within 12 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. (10). Table 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.

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Fig. 6. The ablation rate of CO2 for a single diurnal cycle of Iapetus (79.3 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.

3. Results

small; however, the ablation rate near the poles is greatly affected by effective obliquity.

3.1. Ablation rate 3.2. Polar caps 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. 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. Fig. 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 even a fraction of a single diurnal cycle on Iapetus. 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 × 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. Fig. 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

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. 8. 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 solar orbit. We find that if there is more than 3 × 107 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 3 × 107 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.

Stability and transport of CO2

441

Fig. 7. The ablation rate of CO2 from a sheet of CO2 ice at different effective obliquities, 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. Table 3 Sublimation and movement rates for different sized polar caps Extent of polar cap (◦ latitude)

Fig. 8. The time evolved thickness and latitudinal extent of a seasonal polar cap. A seasonal polar cap will begin as a thin layer of CO2 ice when it is emplaced during the pole’s winter season. During the spring 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.

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 pole (see Table 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 3 then shows how much

Total sublimation (kg solar orbit−1 )

Net movement (kg solar orbit−1 )

Percent in transition

+89.5 to +90 +88.5 to +90 +87.5 to +90 +86.5 to +90 +85.5 to +90 +84.5 to +90

Minimum effective obliquity 4.3 1.0 × 100 –a 1.3 × 101 –a 2 2.0 × 10 –a 2.0 × 103 1.7 × 103 4 1.5 × 10 1.0 × 104 1.1 × 105 6.1 × 104

– – – 85% 67% 55%

+89.5 to +90 +88.5 to +90 +87.5 to +90 +86.5 to +90 +85.5 to +90 +84.5 to +90

Current effective obliquity 15.4 3.0 × 107 3.2 × 107 8 2.0 × 10 1.7 × 108 7.0 × 108 5.6 × 108 9 1.9 × 10 1.3 × 109 4.3 × 109 2.7 × 109 9.4 × 109 5.3 × 109

94% 85% 80% 68% 62% 56%

+89.5 to +90 +88.5 to +90 +87.5 to +90 +86.5 to +90 +85.5 to +90 +84.5 to +90

Maximum effective obliquity 19.3 2.9 × 108 2.7 × 108 1.9 × 109 1.6 × 109 6.1 × 109 4.7 × 109 10 1.5 × 10 1.0 × 1010 3.0 × 1010 1.9 × 1010 10 5.7 × 10 3.3 × 1010

93% 84% 77% 67% 63% 58%

a Transport is less than a monolayer.

CO2 must be present. There will be more CO2 in the system 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.

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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 monolayer every solar orbit at latitude +90◦ . However, 1500 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 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 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 3 provides the sublimation and ablated amounts for a 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 W m−2 (basically unlit). 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. (15), where there absorption cross section for CO2 is denoted as σ defined

by Chan et al. (1993) and Lewis and Carver (1983), and the ultraviolet flux is Φ (Woods et al., 1998). Summing over all wavelengths between 6 and 200 nm, we calculate a photochemical timescale of 1.7 × 107 s. τ=

1 1 = . J σΦ

(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 90 K and a launch angle of 45◦ , giving a Vrms = 225 m s−1 and an average time of flight of 1.7 × 103 s. Since it takes approximately 350 hops to random walk between the poles, we calculate the average time a CO2 molecule is in motion as 6.0 × 105 s, more than an order of magnitude less than the photochemical time scale. Carbon dioxide frost can also be sequestered in the regolith, vaporized by micro-meteorite impact, adsorbed onto the surface of regolith grains, or bonded 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. 9. When the entire moon is covered in CO2 , a large amount of CO2 will escape the system every solar orbit (2 × 1012 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. Fig. 9 shows the loss rate of CO2 as a function of the size of the polar cap. Surprisingly, the maximum escape rate for CO2 (3.4 × 1012 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 96 K, but when the equatorial regions are free of CO2 ice, the surface temperature can reach 130 K. When only a small amount of CO2 lands there, the temperature suppression is small, allowing the CO2 to be thermalized close to 130 K, rather than 96 K. A larger percentage of CO2 will thus reach escape velocity, which can result in more CO2 escaping from the system (see Table 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

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Fig. 9. 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.

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. The cumulative effect is that about 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 10.5% per solar orbit; however, as the total CO2 inventory decreases, the escape fraction increases, Fig. 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. 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 4 × 106 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. We can use the information from Fig. 9 to determine the size of polar caps that can result from a given resupply rate. Since Fig. 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. 9. 4. Discussion One can see that the long-term stability of CO2 is problematic. The first issue to consider is 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 1.88 × 1021 kg and assuming that 3% of Iapetus’ bulk mass is CO2 , we get 5.6 × 1019 kg, or a 5-km-thick sheet of CO2 ice on the moon.

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Fig. 10. 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 a 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.

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. 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 Ga. 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 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 Ga 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 1.7 × 1012 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 5000 years. Finally, free CO2 could be photochemically generated from the dark material itself. While the bright regions of Iapetus are clearly H2 O 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 ·xH2 O and NH3 ·H2 O embedded in H2 O (Squyres and Sagan, 1983), (3) composed of a nitrogen-rich tholin, amorphous carbon, and a small amount of H2 O 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., 1988; Strazzulla and Palumbo, 1998). Specific laboratory experiments with ultraviolet photolysis of ices composed of H2 O, CH3 OH, NH3 , and CO produced H2 CO, CO2 , CO, CH4 , and HCO

Stability and transport of CO2

(Allamandola et al., 1988). A more recent experiment with mixtures of H2 O:NH3 :CH4 found that irradiation with 30 keV He+ and 60 keV Ar2+ results in the formation of C2 H4 , CO and CO2 (Strazzulla and Palumbo, 1998). Buratti et al. (2005) note that CO2 generated by solar UV radiation will be near the surface and will have a short residence time. They infer 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 percentage is 12% (λ = 0.13 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. (16), regardless if it is photolytically generated or primordial. The total number of molecules moving in the system is denoted as N0 , 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 = −N0 λ. (16) ∂t 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 would have to 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 require 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. 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 mid-latitudes 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 of obliquity on polar insolation is strong enough to move 3 × 107 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.

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