**Why Isn't the Earth Completely Covered in Water?**

Joseph A. Nuth III1, Frans J. M. Rietmeijer2 and Cassandra L. Marnocha1,3 *1Astrochemistry Laboratory, Code 691 NASA's Goddard Space Flight Center, Greenbelt 2Dept. of Earth and Planetary Sciences, MSC03-2040 University of New Mexico, Albuquerque 3University of Wisconsin at Green Bay, Green Bay USA* 

## **1. Introduction**

44 Space Science

Wik, M.; Viljanen, A.; Pirjola, R.; Pulkkinen, A.; Wintoft, P. & Lundstedt, H. (2008).

pp. 1775-1787

11 pp.

induced currents on Swedish technical systems. *Annales Geophysicae*, Vol.27, No.4,

Calculation of geomagnetically induced currents in the 400 kV power grid in southern Sweden. *Space Weather*, Vol.6, No.7, S07005, doi: 10.1029/2007SW000343,

> There is considerable discussion about the origin of Earth's water and the possibility that much of it may have been delivered by comets either within the first several hundred million years or possibly over geologic time [Drake, 2005]. Typical models for the origin of the Earth begin by assuming the accumulation of some combination of chondritic meteorites (Javoy, 1995; Ringwood, 1979; Wanke, 1981), yet it is highly likely that the asteroids that went into the terrestrial planets are no longer represented to any significant extent in the present-day asteroid population (Nuth, 2008). The argument concerning the composition of the building blocks of the Earth is typically phrased in terms of the chemical composition of the terrestrial mantle as compared to that of primitive meteorites [Righter et al., 2006] or to the isotopic composition of the Earth's oceans as compared to that of cometary water [Righter, 2007]. Both of these considerations yield important constraints on the problem. However, we will demonstrate that a completely novel examination of the problem based on models of nebular accretion, terrestrial planet formation and the evolution of primitive bodies makes using any modern meteorite type as the basis for understanding the volatile content of the Earth inappropriate. Unfortunately, while this approach yields terrestrial planets with sufficient water to easily explain the Earth's oceans, it also introduces a new problem: How do we get rid of the massive excess of water that this model predicts?

> The mechanism for the formation of the terrestrial planets has been the subject of considerable debate. Gravitational instabilities in a dusty disk (Goldreich & Ward, 1973; Youdin and Shu, 2002) may have been responsible for planetesimal formation on very rapid timescales compared to the lifetime of the nebula. On the other hand, collisional accretion of larger aggregates starting from primitive interstellar dust grains should also occur in the nebula (Blum, 1990; Blum and Wurm, 2000), and these aggregates could continue to evolve into kilometer scale planetesimals or even into proto-planetary scale objects. While it is not clear if dust and gas can be concentrated sufficiently to trigger the gravitational accretion (Cuzzi and Weidenschilling, 2006) of planetesimals or proto-planets, it is clear that some level of collisional accretion must occur in proto-planetary nebulae in order to at least make chondrule precursors and probably to make components of meteorite parent bodies, meters

Why Isn't the Earth Completely Covered in Water? 47

results to determine the composition of the population of planetesimals at 1 A.U. from

Using Weidenschilling's results together with Hayashi's model for the solar nebula (Hayashi, 1981), based on the minimum mass solar nebula, we were able to construct a simple model maintaining the nebular structure used in both authors' works. Hayashi's model provides equations for the surface density of gas, rock, and solids as a function of annular distance. The surface density for rock (r) is given as 7.1 r -1.5 g cm-2 and the surface density for solids (s) as 30 r -1.5 g cm-2. These equations also apply to the assumptions of surface density at 30 AU used in Weidenschilling's model. Assuming that the composition of the nebula does not change as we increase nebular mass, then the "surface density" phase coefficient for rock or solids in higher mass nebulae is easily calculated by increasing each by the same factor by which one wishes to increase the nebular mass. These surface density coefficients are used to calculate the mass of either rock or solids (ice plus rock) between two

where Q is the phase coefficient for a given nebular mass in units of g cm-2, [1.496x 1013]2 converts( A.U.)2 to cm2, rb is the annular distance in A.U. at which the planetesimal begins accreting material, and ra is the annular distance at which the planetesimal ends its accretion

Weidenschilling (W97) used numerical methods to calculate that a planetesimal must start at about 200 AU in order to grow to 15 kilometers by the time it drifts in to 5 - 10 A.U. Using the solid surface density coefficient (s = 84) for Weidenschilling's model, as well as his stated "start" and "end" points, we calculate that a planetesimal must travel through 2.494 x 1030 grams of material in order to accrete to 15 km. This equates to an accretion efficiency of ~1.41 parts in 1012 depending on the density assumed for the accreting planetesimal. This "accretion rate" is used throughout our calculations such that: planetesimal mass (in g) =

Note, we have assumed a density of 1.0 g/cc for all planetesimals. Determining approximate percentages of rock and ice found in a planetesimal was dependent on the starting and ending radii in reference to the snowline (which in this very simple model we have assumed to remain at 5 A.U.). Planetesimals beginning and ending accretion inside the snowline will be completely dry and 100% rock. Planetesimals that begin to accrete beyond the snowline and ending their accretion within the snowline must have the total mass (e.g., rock) from the snowline to the end point calculated, as well as the mass of solids (e.g., rock and ice) calculated from the starting point to the snowline. Again, using the surface density coefficients allows us to "separate" the mass of rock and ice accreted beyond the snowline from the pure rock

Several important points must be made concerning our use of Weidenschilling's results before we begin to discuss our own. First, Weidenschilling reported extensive results from numerical simulations that explicitly took into account a wide range of factors such as the sticking of grains to a growing body. We have lumped all of these effects into an efficiency

Total Mass (g) = 4 Q [rb1/2 – ra1/2][1.496 x 1013]2 (1)

Mass (g) = (4 Q [rb1/2 – ra1/2][1.496 x 1013]2)/1.41 x 1012 (2)

which the terrestrial planets may have formed.

given points via the following equation:

(presumably merging into a larger body).

total mass of solids and/or rock between rb and ra /1.41 x 1012, or

accreted within, and thereby calculates approximate percentages for each.

to tens of meters in diameter. Youdin and Goodman (2005) have suggested that turbulent shear instabilities might serve to concentrate mixtures of chondrules and dust to sufficient density for gravitational instabilities to directly form meteorite parent bodies. As these bodies grow from dust grains to larger sizes, they also drift inward due to gas drag and can be lost into the sun in about a century (Youdin, pers. comm.). This drift might also serve to bring ices to the terrestrial planet region from beyond the snowline, and could thus be a primary source of water that has not been accounted for in models of planet growth. The purpose of this effort is to investigate the potential for such a mechanism to deliver water to the terrestrial planets as they grow.

Weidenschilling [1997: Hereafter W97] published an excellent model for the formation of comets in a minimum mass solar nebula. In this model, because the growing icy agglomerates slowly decouple from the gas as they gain mass and become more compact, comets begin to form at nebular radii between about 100 - 200 A.U. and fully decouple from the gas at 5 to 10 A.U. having grown into planetesimals on the order of 10 - 15 km in diameter. In more massive nebulae, the feeding zone for materials incorporated into a growing planetesimal would be proportionally smaller and icy agglomerates that begin accreting at 200 A.U. might easily reach diameters of 10 - 15 km before leaving the region of the Kuiper Belt. We have tried to extend a very simplified version of this general model to delineate the feeding zones for planetesimals that might have been accreted into the early Earth.

In the sections that follow we will describe our very simple calculations of the feeding zones from which the planetesimals formed that aggregated into protoplanets at 1 A.U. We will discuss the results of these calculations in terms of their dependence on the mass of the solar nebula and demonstrate that if this accretion model is applicable to even some planetesimals, then the Earth may have accreted an enormous quantity of water, the majority of which must have escaped early in our planet's history. We will very briefly discuss reasons why meteorites in our modern collections are unlikely to represent the materials that accreted to form our planet 4.5 billion years ago, and how our modern sample is likely to differ from these more primitive materials. We will also discuss possible alternative scenarios for the accretion of the proto-Earth, and how these scenarios might change our conclusions.

#### **2. Methods**

Weidenschilling (W97) published what we consider to be one of the best models for the formation of comets in a low-mass nebula. The model is based on the simple accretion of nebular solids and the effect that increased mass has on an object in orbit about the sun in a gas filled nebula. Specifically, tiny solids are initially very closely coupled to the gas. As accretion proceeds, a growing body begins to orbit independently and decouples from the gas. However, while the local gas and dust is partially supported by gas pressure, requiring a lower velocity to maintain its position in the nebula, a growing dusty snowball soon begins to drift inward without the support of the surrounding gas. This drift brings the accreting planetesimals into contact with fresh materials across a broad feeding zone, and continues until the body has grown to a size where its orbital velocity is no longer significantly affected by gas drag. Weidenschilling's (W97) purpose was to identify the radial dimensions of the nebula that produced the comets that were scattered by the giant planets into the Oort Cloud. It is our intention in this work to extend Weidenschilling's 46 Space Science

to tens of meters in diameter. Youdin and Goodman (2005) have suggested that turbulent shear instabilities might serve to concentrate mixtures of chondrules and dust to sufficient density for gravitational instabilities to directly form meteorite parent bodies. As these bodies grow from dust grains to larger sizes, they also drift inward due to gas drag and can be lost into the sun in about a century (Youdin, pers. comm.). This drift might also serve to bring ices to the terrestrial planet region from beyond the snowline, and could thus be a primary source of water that has not been accounted for in models of planet growth. The purpose of this effort is to investigate the potential for such a mechanism to deliver water to

Weidenschilling [1997: Hereafter W97] published an excellent model for the formation of comets in a minimum mass solar nebula. In this model, because the growing icy agglomerates slowly decouple from the gas as they gain mass and become more compact, comets begin to form at nebular radii between about 100 - 200 A.U. and fully decouple from the gas at 5 to 10 A.U. having grown into planetesimals on the order of 10 - 15 km in diameter. In more massive nebulae, the feeding zone for materials incorporated into a growing planetesimal would be proportionally smaller and icy agglomerates that begin accreting at 200 A.U. might easily reach diameters of 10 - 15 km before leaving the region of the Kuiper Belt. We have tried to extend a very simplified version of this general model to delineate the feeding zones for

In the sections that follow we will describe our very simple calculations of the feeding zones from which the planetesimals formed that aggregated into protoplanets at 1 A.U. We will discuss the results of these calculations in terms of their dependence on the mass of the solar nebula and demonstrate that if this accretion model is applicable to even some planetesimals, then the Earth may have accreted an enormous quantity of water, the majority of which must have escaped early in our planet's history. We will very briefly discuss reasons why meteorites in our modern collections are unlikely to represent the materials that accreted to form our planet 4.5 billion years ago, and how our modern sample is likely to differ from these more primitive materials. We will also discuss possible alternative scenarios for the accretion of the proto-Earth, and how these scenarios might

Weidenschilling (W97) published what we consider to be one of the best models for the formation of comets in a low-mass nebula. The model is based on the simple accretion of nebular solids and the effect that increased mass has on an object in orbit about the sun in a gas filled nebula. Specifically, tiny solids are initially very closely coupled to the gas. As accretion proceeds, a growing body begins to orbit independently and decouples from the gas. However, while the local gas and dust is partially supported by gas pressure, requiring a lower velocity to maintain its position in the nebula, a growing dusty snowball soon begins to drift inward without the support of the surrounding gas. This drift brings the accreting planetesimals into contact with fresh materials across a broad feeding zone, and continues until the body has grown to a size where its orbital velocity is no longer significantly affected by gas drag. Weidenschilling's (W97) purpose was to identify the radial dimensions of the nebula that produced the comets that were scattered by the giant planets into the Oort Cloud. It is our intention in this work to extend Weidenschilling's

the terrestrial planets as they grow.

change our conclusions.

**2. Methods** 

planetesimals that might have been accreted into the early Earth.

results to determine the composition of the population of planetesimals at 1 A.U. from which the terrestrial planets may have formed.

Using Weidenschilling's results together with Hayashi's model for the solar nebula (Hayashi, 1981), based on the minimum mass solar nebula, we were able to construct a simple model maintaining the nebular structure used in both authors' works. Hayashi's model provides equations for the surface density of gas, rock, and solids as a function of annular distance. The surface density for rock (r) is given as 7.1 r -1.5 g cm-2 and the surface density for solids (s) as 30 r -1.5 g cm-2. These equations also apply to the assumptions of surface density at 30 AU used in Weidenschilling's model. Assuming that the composition of the nebula does not change as we increase nebular mass, then the "surface density" phase coefficient for rock or solids in higher mass nebulae is easily calculated by increasing each by the same factor by which one wishes to increase the nebular mass. These surface density coefficients are used to calculate the mass of either rock or solids (ice plus rock) between two given points via the following equation:

$$\text{Total Mass (g)} = 4 \text{ } \pi \text{ Q } [\text{r}\_b 1^{1/2} - \text{r}\_a 1^{1/2}] [1.496 \times 10^{13}]^2 \tag{1}$$

where Q is the phase coefficient for a given nebular mass in units of g cm-2, [1.496x 1013]2 converts( A.U.)2 to cm2, rb is the annular distance in A.U. at which the planetesimal begins accreting material, and ra is the annular distance at which the planetesimal ends its accretion (presumably merging into a larger body).

Weidenschilling (W97) used numerical methods to calculate that a planetesimal must start at about 200 AU in order to grow to 15 kilometers by the time it drifts in to 5 - 10 A.U. Using the solid surface density coefficient (s = 84) for Weidenschilling's model, as well as his stated "start" and "end" points, we calculate that a planetesimal must travel through 2.494 x 1030 grams of material in order to accrete to 15 km. This equates to an accretion efficiency of ~1.41 parts in 1012 depending on the density assumed for the accreting planetesimal. This "accretion rate" is used throughout our calculations such that: planetesimal mass (in g) = total mass of solids and/or rock between rb and ra /1.41 x 1012, or

$$\text{Mass (g)} = \left(4 \text{ } \pi \text{ Q } [\text{r}\_{\text{b}}1^{1/2} - \text{r}\_{\text{a}}1^{1/2}][1.496 \times 10^{13}]^2\right) / 1.41 \times 10^{12} \tag{2}$$

Note, we have assumed a density of 1.0 g/cc for all planetesimals. Determining approximate percentages of rock and ice found in a planetesimal was dependent on the starting and ending radii in reference to the snowline (which in this very simple model we have assumed to remain at 5 A.U.). Planetesimals beginning and ending accretion inside the snowline will be completely dry and 100% rock. Planetesimals that begin to accrete beyond the snowline and ending their accretion within the snowline must have the total mass (e.g., rock) from the snowline to the end point calculated, as well as the mass of solids (e.g., rock and ice) calculated from the starting point to the snowline. Again, using the surface density coefficients allows us to "separate" the mass of rock and ice accreted beyond the snowline from the pure rock accreted within, and thereby calculates approximate percentages for each.

Several important points must be made concerning our use of Weidenschilling's results before we begin to discuss our own. First, Weidenschilling reported extensive results from numerical simulations that explicitly took into account a wide range of factors such as the sticking of grains to a growing body. We have lumped all of these effects into an efficiency

Why Isn't the Earth Completely Covered in Water? 49

line in this work, and still only in the most massive nebulae do the smallest planetesimals at 1 A.U. contain pure rock. All planetesimals 10 km in diameter and larger contain a

1 10 20 30 40 50

10 122 8.1 5.8 5.05 4.71 4.52 15 1068 25.2 12.1 8.81 7.36 6.55 20 5594 85.2 31.2 19.11 14.17 11.56

 Percentage of Ice in the Final Planetesimal (%) 10 74 52 27 3 0 0 15 76 69 62 55 47 40 20 76 73 70 67 64 61

Diameter (km) Distance from the Proto-Sun where Aggregation Begins (A.U.)

Table 1. Radius of Feeding Zone and Percentage of Ice in the Final Planetesimal as a

As noted above we made a number of simplifying assumptions in our calculations that could affect our results. First, we assumed that the snowline remains at 5 A.U. no matter the mass of the nebula. However, since any likely accretion scenario for the Solar Nebula would have a mass considerably larger than the Hayashi Minimum Mass, the snowline for more massive nebulae would occur closer to the proto-sun. This would tend to increase the number of planetesimals containing ice and increase the ice content as a fraction of planetesimal mass for small bodies at 1 A.U. In fact Desch (2008) calculated that the snowline for a nebula 25 times the Hayashi Minimum Mass would occur at 2.5 A.U., well inside the outer asteroid belt. Under these conditions, some outer main belt asteroids would

Second, we assumed that we could extend the results of Weidenschilling's (W97) numerical calculations from the Outer Planets region to the Terrestrial Planets region and that the efficiency for particulate aggregation would remain unchanged. We find that reasonable changes in the efficiency factor used in these calculations do not change our basic results that smaller planetesimals at 1 A.U. accreted in more massive nebulae contain more rock while larger bodies in lower mass nebulae contain much more ice. Certainly the detailed results are modified with changes to this factor; however, the uncertainty in the actual mass of the Solar Nebula is more important in determining the ice content of the planetesimals than is the exact value of the accretion efficiency factor employed in the calculations. We also use the same factor for the accretion of both icy and anhydrous dust. The relative sizes of the particles and the velocities of the collisions appear to be more important than the exact composition of the accreting material, but the modest experimental results that we have to date indicate that collisions between icy particles are more likely to result in sticking than are collisions between dry rocks and pebbles. Our calculations therefore may

Function of Nebular Mass and Planetesimal Diameter

**4. Effects of our assumptions on the results** 

certainly contain significant quantities of ice in their interiors.

overestimate the efficiency of forming ice-free planetesimals.

Total Nebular Mass (Hayashi Minimum Mass Nebula)

significant fraction of ice.

Planetesimal

factor and have used this same efficiency factor to represent the accretion of both icy dust (outside the snowline) and dry rock (inside the snow line). This is obviously an oversimplification of a very complex and poorly understood process. Second, other processes may have created planetesimals in the solar nebula such as gravitational instabilities or large-scale vortices. Our study only examines the results of hierarchical accretion. Third, we assume that the snowline represents a discontinuity between a mixture of dry dust and hydrous gas inside the line and a mixture of dry gas and icy dust on the outside of this sharp divide. This obviously neglects the bodies that may have accreted to ten-meter or even kilometer scales outside this divide, yet drifted inward to some extent due to gas drag or gravitational interactions. Finally, we have assumed that the position of the snowline (at 5 A.U.) does not migrate as we increase the mass of the nebula, but instead remains fixed no matter how we change the mass of the system.

The effects of each of these simplifications will be examined below, after we have presented the results of our calculations. However, we contend that the effects of many of these factors would not tend to favor the accretion of rock over ice into planetesimals at 1 A.U. and that the uncertainties in nebular conditions, particularly in the mass of the nebula itself, are sufficiently large that our simplified calculations are an appropriate first step in examining the potential incorporation of ice and water into the progenitors of the terrestrial planets.

## **3. Results**

From Weidenschilling's work [W97] we calculated an efficiency factor for the accretion of primitive planetesimals based on the final diameter of the body, the density of nebular solids (including ice) and the total mass of the nebular disk. We first validated this efficiency factor by using it to reproduce other examples of accretion calculations shown in Weidenschilling's (W97) paper. We then used this efficiency factor to calculate the size of the initial feeding zone for planetesimals that had grown to 10, 15 and 20 km in diameter by the time they reached 1 A.U. as a function of nebular mass. In other words, assuming that 10, 15 and 20 km sized planetesimals were present at 1 A.U., were still drifting inward but were available to be incorporated into growing protoplanets, where did these bodies begin to accrete? The results of our calculations are presented in Table 1 where we show the nebular radius where accretion begins as a function of the diameter of the planetesimal at 1 A.U. and the mass of the solar nebula. In all cases we assumed that the snowline is located at exactly 5 A.U. As can be seen from Table 1, the size of the planetesimal feeding zone and the percentage of ice in the final planetesimal are strong functions of the nebular mass and the size of the planetesimal itself: higher mass planets in lower mass nebulae contain much more ice.

We based our calculations on the total nebular mass expressed in units of the Hayashi Minimum Mass Nebula [Hayashi, 1981, Hayashi et al., 1985] and for each nebular mass we calculated where aggregation must begin in order to produce the planetesimal size of interest based on the accretion efficiency discussed above. We note that Weidenschilling [W97] used a value 2.8 times the Hayashi Minimum Mass in his model of comet formation, and more recently, Desch [2008] has estimated that the primitive solar nebula must have been at least 25 times the Hayashi Minimum Mass, but with a somewhat steeper slope (e.g. less mass in the outer regions of the nebula), thus more centrally concentrating the material available for planet formation. In this scenario, the snow line may be as close as 2.8 A.U. from the protosun. We decided to adopt the more conservative 5 A.U. position for the snow 48 Space Science

factor and have used this same efficiency factor to represent the accretion of both icy dust (outside the snowline) and dry rock (inside the snow line). This is obviously an oversimplification of a very complex and poorly understood process. Second, other processes may have created planetesimals in the solar nebula such as gravitational instabilities or large-scale vortices. Our study only examines the results of hierarchical accretion. Third, we assume that the snowline represents a discontinuity between a mixture of dry dust and hydrous gas inside the line and a mixture of dry gas and icy dust on the outside of this sharp divide. This obviously neglects the bodies that may have accreted to ten-meter or even kilometer scales outside this divide, yet drifted inward to some extent due to gas drag or gravitational interactions. Finally, we have assumed that the position of the snowline (at 5 A.U.) does not migrate as we increase the mass of the nebula, but instead

The effects of each of these simplifications will be examined below, after we have presented the results of our calculations. However, we contend that the effects of many of these factors would not tend to favor the accretion of rock over ice into planetesimals at 1 A.U. and that the uncertainties in nebular conditions, particularly in the mass of the nebula itself, are sufficiently large that our simplified calculations are an appropriate first step in examining the potential incorporation of ice and water into the progenitors of the terrestrial planets.

From Weidenschilling's work [W97] we calculated an efficiency factor for the accretion of primitive planetesimals based on the final diameter of the body, the density of nebular solids (including ice) and the total mass of the nebular disk. We first validated this efficiency factor by using it to reproduce other examples of accretion calculations shown in Weidenschilling's (W97) paper. We then used this efficiency factor to calculate the size of the initial feeding zone for planetesimals that had grown to 10, 15 and 20 km in diameter by the time they reached 1 A.U. as a function of nebular mass. In other words, assuming that 10, 15 and 20 km sized planetesimals were present at 1 A.U., were still drifting inward but were available to be incorporated into growing protoplanets, where did these bodies begin to accrete? The results of our calculations are presented in Table 1 where we show the nebular radius where accretion begins as a function of the diameter of the planetesimal at 1 A.U. and the mass of the solar nebula. In all cases we assumed that the snowline is located at exactly 5 A.U. As can be seen from Table 1, the size of the planetesimal feeding zone and the percentage of ice in the final planetesimal are strong functions of the nebular mass and the size of the planetesimal itself:

We based our calculations on the total nebular mass expressed in units of the Hayashi Minimum Mass Nebula [Hayashi, 1981, Hayashi et al., 1985] and for each nebular mass we calculated where aggregation must begin in order to produce the planetesimal size of interest based on the accretion efficiency discussed above. We note that Weidenschilling [W97] used a value 2.8 times the Hayashi Minimum Mass in his model of comet formation, and more recently, Desch [2008] has estimated that the primitive solar nebula must have been at least 25 times the Hayashi Minimum Mass, but with a somewhat steeper slope (e.g. less mass in the outer regions of the nebula), thus more centrally concentrating the material available for planet formation. In this scenario, the snow line may be as close as 2.8 A.U. from the protosun. We decided to adopt the more conservative 5 A.U. position for the snow

remains fixed no matter how we change the mass of the system.

higher mass planets in lower mass nebulae contain much more ice.

**3. Results** 

line in this work, and still only in the most massive nebulae do the smallest planetesimals at 1 A.U. contain pure rock. All planetesimals 10 km in diameter and larger contain a significant fraction of ice.


Table 1. Radius of Feeding Zone and Percentage of Ice in the Final Planetesimal as a Function of Nebular Mass and Planetesimal Diameter
