CHEMODYNAMICAL DEUTERIUM FRACTIONATION IN THE EARLY SOLAR NEBULA: THE ORIGIN OF WATER ON EARTH AND IN ASTEROIDS AND COMETS

Water is one of the most abundant molecules detected in interstellar space, found throughout all stages of stellar evolution, from prestellar cores to protoplanetary disks. The ultimate role of liquid water for the origin of life on Earth is well recognized in biochemistry and astronomy (e.g., Kasting et al. 1993; Hoover 2006; Costanzo et al. 2009; Javaux & Dehant 2010; Lis et al. 2013). One of the main difficulties in tracing the origin of the Earth's water is strong evidence that it came from exogenous sources because the solar nebula was too hot in the terrestrial planet-formation zone to form water-rich planetesimals from smaller icy precursors (Boss 1998; Morbidelli et al. 2000; Min et al. 2011; Elser et al. 2012). In addition, no evidence for hydrous silicates in the inner regions of protoplanetary disks has been found (Juhász et al. 2010).

The most promising mechanism of water delivery to Earth is bombardment by carbonaceous asteroids or comets that have formed in the ice-rich outer region of the young solar system (e.g., Oró 1961; Owen et al. 1992; Morbidelli et al. 2000; Robert 2011; Izidoro et al. 2013). This idea is supported by modern sophisticated dynamical models of the early solar system, which show that complex gravitational interactions between Jupiter and Saturn had likely scattered other small bodies into the inner and far outer regions (see, e.g., Morbidelli et al. 2005; Gomes et al. 2005; Tsiganis et al. 2005; Walsh et al. 2012). More evidence of the solar system's dynamical past comes from the presence of crystalline silicates annealed at temperatures above 800 K in cometary samples collected by the Stardust mission (Wozniakiewicz et al. 2012) and revealed by infrared observations of comets (e.g., Crovisier et al. 1997; Kelley et al. 2006), suggesting that transport or highly energetic local processes occurred in the solar nebula. The importance of dynamical activity of the young solar nebula and protoplanetary disks on its chemical and isotopic evolution was also demonstrated in numerous theoretical studies (e.g., Morfill & Völk 1984; Cyr et al. 1998; Aikawa & Herbst 1999; Gail 2001; Ilgner et al. 2004; Boss 2004; Willacy et al. 2006; Tscharnuter & Gail 2007; Willacy 2007; Willacy & Woods 2009; Semenov & Wiebe 2011; Heinzeller et al. 2011; Wozniakiewicz et al. 2012).

The individual contributions from particular exogenous sources is still heavily debated. Water has likely formed on surfaces of dust grains even before the formation of the solar nebula in the protosolar molecular cloud and inherited specific isotopic composition and ortho/para ratios from that epoch (e.g., Jørgensen & van Dishoeck 2010; Cazaux et al. 2011; Taquet et al. 2013b), which may have been preserved in primitive solar system bodies such as comets. It is plausible to assume though that the water isotopic composition (and maybe even the ortho/para ratio) could have been partly or fully reset during the violent physical conditions in the inner, dynamically active solar nebula (e.g., Robert et al. 2000; Boss 2004). Other processes such as grain growth and sedimentation have been shown to have a profound effect on water abundance and deuterium fractionation (Fogel et al. 2011; Vasyunin et al. 2011; Akimkin et al. 2013). The importance of the ortho- and para-states of H₂ for deuterium fractionation is well established and also has to be taken into account (e.g., Flower et al. 2006).

The water D/H ratio remains the most essential probe for disentangling the individual contributions from the various exogenous sources. Because comets are reservoirs of pristine material in the solar nebula, a comparison of the water D/H ratios measured in comets and asteroids to that measured in protoplanetary disks is vital. These observations are challenging because most of the water is frozen out onto dust grains in disks around cool Sun-like stars (Dominik et al. 2005; Bergin et al. 2010; Hogerheijde et al. 2011). Furthermore, disks have relatively compact sizes (≲ 100–1000 AU) and are not massive (≲ 0.01 M_☉). Therefore, detailed high-resolution studies remain a challenge. Ground-based observations are further compromised by the absorption and veiling of water vapor lines in the Earth atmosphere. In order to better understand the origin of water on Earth, other solar system bodies, and possibly in other exoplanetary systems, one has to study the chemical evolution of water prior to and during the onset of planet formation.

Water evolution begins in prestellar cores, where the low temperatures cause many molecules to freeze out onto grains, and the gas remains largely void of molecules (Caselli et al. 2010). This drives a rich surface chemistry along with an efficient deuterium fractionation, resulting in high water abundances on the grains (∼10⁻⁴ relative to H) and D/H ratios (∼10⁻²). As the core contracts and begins to heat up, volatiles are released into the gas. Water has been detected in several class 0 protostellar objects (e.g., Ceccarelli et al. 1999; Kristensen et al. 2010; Visser et al. 2013; Mottram et al. 2013), but only a few detections of HDO (and D₂O) have allowed determination of D/H ratios, with values of ∼10⁻³ in the hot cores and retaining ∼10⁻² in the cold outer regions (Jørgensen & van Dishoeck 2010; Liu et al. 2011; Coutens et al. 2012; Persson et al. 2013; Taquet et al. 2013a; Coutens et al. 2013).

Several detections of water have been done in protoplanetary disks (e.g., Pontoppidan et al. 2005; Meeus et al. 2010; Bergin et al. 2010; Fedele et al. 2012), including also several using PACS/Herschel (Hogerheijde et al. 2011; Tilling et al. 2012; Fedele et al. 2012; Herczeg et al. 2012; Riviere-Marichalar et al. 2012) within the WISH program (Water in Star-Forming Regions with Herschel van Dishoeck et al. 2011). In particular, hot water vapor (≳ 500 K) has been discovered in the inner, planet-forming regions of protoplanetary disks around young solar-mass T Tauri stars, mainly by infrared spectroscopy with the Spitzer satellite (Carr & Najita 2008; Pontoppidan et al. 2008, 2010; Najita et al. 2011; Teske et al. 2011; Banzatti et al. 2012).

The first detection of the ground transition of HDO in absorption in the disk around DM Tau was reported by Ceccarelli et al. (2005). Although they have not observed H₂O, Ceccarelli et al. (2005) predicted water D/H ratios of >0.01, based on H₂O measurements in molecular clouds and protostellar envelopes. This discovery was later disputed by Guilloteau et al. (2006). Qi et al. (2008) have derived upper limits for the water column densities in the disk around TW Hya, ≲ 9.0 × 10¹⁴ cm⁻², but were not able to determine its D/H ratio.

The discovery of the ground-state rotational emission of both spin isomers of H₂O from the outer, ≳ 50–100 AU disk around TW Hya with Herschel allowed derivation of a water abundance of ≈10⁻⁷ relative to H₂ (Hogerheijde et al. 2011). The authors also derived a water ortho/para ratio of 0.77  ±  0.07, which indicates that the water formation took place on cold dust grains with temperatures of ∼10–20 K. In contrast, the water ortho/para ratios measured in comets are indicative of higher temperatures of ∼25–50 K, more typical for the inner solar nebula at ≲ 20–30 AU (Mumma et al. 1987; Woodward et al. 2007; Bockelée-Morvan et al. 2009; Shinnaka et al. 2012; Bonev et al. 2013).

Studies of carbonaceous chondrites have revealed that their phyllosilicates have D/H ratios that are very similar to that in the Earth's ocean water, ∼1.56 × 10⁻⁴, with as large overall water content as ≲ 20%, thus favoring asteroids as the main water source (e.g., Owen et al. 1992; Hoover 2006; Marty 2012; Alexander et al. 2012). Because observations of Oort-family comets have yielded water D/H ratios in their comae about two to three times higher that in the Earth's oceans (Robert et al. 2000 and references therein), and because of the low amount of volatile noble gases on Earth (e.g., Marty et al. 2013), it was concluded that comets contributed at most up to ∼10% to the water delivery to Earth. However, recent HDO observation of the short-period, Jupiter-family comet Hartley-2 with Herschel by Hartogh et al. (2011) revealed a water D/H ratio of 1.6 × 10⁻⁴, which is very close to the Earth value. Additionally, Lis et al. (2013) studied the Jupiter-family comet 45P/Honda-Mrkos-Pajdusakova but could not detect HDO with an upper water D/H ratio of 2 × 10⁻⁴. Their upper limit is still consistent with the value found by Hartogh et al. (2011) and aids in confirming the diversity of D/H ratios among different comet populations. These observations have reignited debates about the relative role of asteroids and comets in the origin of the Earth water. One has to bear in mind, however, that a major problem could be that the gases in cometary comae may have lower D/H ratios compared to the D/H ratios of the nucleus, as indicated by the experiments of Brown et al. (2012).

While we have made many advances in our understanding of the origin of Earth's oceans, we still do not fully understand the reprocessing of water during the protoplanetary disk stage, nor is there any consensus for contribution from the various exogenous sources. In this paper, we focus on the history and evolution of the early solar nebula and investigate the chemistry of frozen and gaseous water at the stage when planetesimals have not yet formed and the micron-sized dust grains were the dominant population of solids. We use an extended gas–grain chemical model that includes multiply-deuterated species, high-temperature reactions, surface reactions, and nuclear spin-states processes. This network is combined with a 1+1D steady-state α-viscosity nebula model to calculate molecular abundances and D/H ratios within the first 1 Myr of its lifetime. We consider both the laminar case with chemistry not affected by dynamics and a model with two-dimensional (2D) turbulent mixing transport. The organization of the paper is as follows. In Section 2 we describe our physical and chemical model of the solar nebula and introduce the considered deuterium processes. The results are presented in Section 3, followed by a discussion and conclusions.

2.1. Physical Model

We used a disk structure similar to Semenov & Wiebe (2011) but adopted parameters characteristic of the young Sun and early solar nebula. The physical model is based on a 1+1D steady-state α-model of a flaring protoplanetary disk described by D'Alessio et al. (1999); see Figure 1. We used the parameterization of Shakura & Sunyaev (1973) of the turbulent viscosity ν in terms of the characteristic scale height H(r), the sound speed c_s(r, z), and the dimensionless parameter α:

$$\begin{equation} \nu (r,z) = \alpha \,c_{\rm s}(r,z)\,H(r). \end{equation} \tag{ 1 }$$

From observations and detailed MHD modeling the α parameter has values of ∼0.001–0.1 (Andrews & Williams 2007; Guilloteau et al. 2011; Flock et al. 2011). A constant value of α = 0.01 was used in our simulations. Equal gas and dust temperatures were assumed.

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The central star has a mass of 1 M_☉, a radius of 1.2 R_☉, and an effective temperature of 6000 K. The nonthermal FUV radiation from the young Sun is represented by the interstellar radiation field of Draine (1978), with the unattenuated intensity at 100 AU of χ_*(100) = 10, 000χ_draine. For the X-ray luminosity of the young Sun we adopted a typical value for T Tauri stars, 10³⁰ erg s⁻¹. The model disk has an inner radius of 0.03 AU (dust sublimation front, T ≈ 1600 K), an outer radius of 800 AU, and an accretion rate of $\dot{M}= 10^{-8}\,M_\odot$ yr⁻¹. The mass of the solar nebula disk is not well known, but studies suggest it to be ≳ 0.1 M_☉ (see, e.g., Ruzmaikina et al. 1993; Desch 2007), and hence we adopt a disk model with a total mass of M = 0.11 M_☉.

We assume that the dust grains are uniform spherical particles, with a radius of a_d = 0.1 μm, made of amorphous silicates with olivine stoichiometry, with a density ρ_d = 3 g cm⁻³ and a dust-to-gas mass ratio of m_d/g = 0.01. The surface density of sites is N_s = 1.5 × 10¹⁵ sites cm⁻², and the total number of sites per each grain is S = 1.885 × 10⁶. No substantial grain growth is assumed in this early disk phase, an assumption which may be challenged in other studies.

In our simulations we assume that gas-phase species and dust grains are well mixed and coupled and transported with the same diffusion coefficient

$$\begin{equation} D_{\rm turb}(r,z) = \nu (r,z)/{\rm Sc}, \end{equation} \tag{ 2 }$$

where Sc = 1 is the Schmidt number that describes the efficiency of turbulent diffusivity (see, e.g., Shakura & Sunyaev 1973; Schräpler & Henning 2004). We treat diffusion of mantle materials similar to gas-phase molecules, without relating it to individual grain dynamics.

As boundary conditions for mixing, we assume that there is no inward or outward diffusion across boundaries of the solar nebula and that there is no flux through its central plane. All of the equations are solved on a nonuniform staggered grid consisting of 65 radial points (from 0.5 to 800 AU) and 91 vertical points.

The physical structure of the solar nebula and the chemical model with and without turbulent mixing are used to solve chemical kinetics equations (see Equation (3) in Semenov & Wiebe 2011). The equations of chemical kinetics are integrated with the diffusion terms in the Eulerian description, using a fully implicit 2D integration scheme and a sparse matrix formalism for inversion of the Jacobi matrices (Semenov et al. 2010).

2.2. Chemical Network and Model

To calculate photoreaction rates through the various disk environments, we adopt the precomputed fits of van Dishoeck et al. (2006) for a 1D plane-parallel slab, using the Draine far-UV interstellar radiation field. The self-shielding of H₂ from photodissociation is calculated by Equation (37) from Draine & Bertoldi (1996) and the shielding of CO by dust grains, H₂, and its self-shielding is calculated using the precomputed table of Lee et al. (1996). We do not take the Ly_α radiation into account.

We used the gas–grain chemical network developed by Albertsson et al. (2013), which includes an extended list of fractionation reactions for up to triply-deuterated species. In addition, the high-temperature reaction network from Harada et al. (2010, 2012) has been added. Recently, it was realized that the ortho/para ratio of H₂ and other species can lower the pace of deuterium fractionation (Flower et al. 2006; Pagani et al. 2009). The internal energy of ortho-H₂ is higher than that of para-H₂, which helps to overcome the barrier of the backward reaction of deuterium enrichment. Consequently, it results in a lower degree of deuterium fractionation in a medium having a sufficient amount of ortho-H₂ (Flower et al. 2006). Given the importance of the ortho/para ratios of H₂ and H$_3^+$ for efficiency of deuterium fractionation, particularly in the solar nebula regions with T ≳ 15–30 K, the nuclear spin-state processes and ortho-para states of H₂, H$_{2}^{+}$, and H$_{3}^{+}$ were added to our chemical network.

Reaction rates for a small number of reactions have already been measured or theoretically predicted. For this, we have added rates from several sources (Gerlich 1990; Walmsley et al. 2004; Flower et al. 2004; Pagani et al. 2009; Honvault et al. 2011), including reaction rates for the H$_{3}^{+}$ + H₂ system by Hugo et al. (2009). In order to include the nuclear spin states of any reactions including H₂, H$_{2}^{+}$, H$_{3}^{+}$, or any of their isotopologues, we employed a separation scheme similar to that described in Sipilä et al. (2013). This routine will be adopted and evaluated in Albertsson et al. (2014). Contrary to Sipilä et al. (2013), we allow reactions without H$_3^+$ or H$_2^+$ as reactants to form both ortho- and para-H₂ because we are interested in modeling chemistry in the inner warm nebula region. The size of the original network was reduced to facilitate the performance of our 2D chemodynamical model by allowing fractionation only for species with <4 hydrogen atoms, <4 carbon atoms, and species not larger than 7 atoms.

We find that the inclusion of the high-temperature reactions can increase the abundance of HDO and H₂O by up to a factor of two inside the snow line (≲ 2–3 AU). In contrast, the reduction of the original network to ∼39, 000 reactions and ∼1300 species has only insignificant effect on the calculated time-dependent abundances of the H₂O isotopologues.

We also added neutral–neutral reactions. The pre-exponential rate factors were calculated using a simple collisional theory, and the reactions barriers are adopted from Lécluse & Robert (1994). The added neutral–neutral reactions are listed below:

$$\begin{eqnarray*} &&{\rm HDO} + p\hbox{-}{\rm H_{2}} \Rightarrow {\rm H_{2}O} + {\rm HD};\quad\, 2.0 \times 10^{-10} \; {\rm cm}^{-3} \;{\rm s}^{-1}, 5170 \; {\rm K} \\ && {\rm HDO} + o\hbox{-}{\rm H_{2}} \Rightarrow {\rm H_{2}O} + {\rm HD};\quad\ 2.0 \times 10^{-10}\; {\rm cm}^{-3} \;{\rm s}^{-1}, 5000 \; {\rm K} \\ && {\rm H_{2}O} + {\rm HD} \Rightarrow {\rm HDO} + p\hbox{-}{\rm H_{2}};\quad\, 5.0 \times 10^{-11} \; {\rm cm}^{-3} \;{\rm s}^{-1}, 4840 \; {\rm K} \\ && {\rm H_{2}O + {\rm HD}} \Rightarrow {\rm HDO} + o\hbox{-}{\rm H_{2}};\quad\ 1.5 \times 10^{-10} \; {\rm cm}^{-3} \;{\rm s}^{-1}, 4840 \;{\rm K}. \end{eqnarray*}$$

However, we find that the addition of these reactions has no significant effect on our results. For initial abundances we model the chemical evolution of a cold and dark TMC-1-like prestellar core, temperature 10 K, density 10⁴ cm⁻³, and extinction A_V = 10 mag, for 1 Myr using the "low metals" initial abundance set from Lee et al. (1996, Table 11), with an initial H₂ ortho/para ratio 1:100 (as H₂ exists in para-form in cold environments). We note that the initial H₂ ortho/para ratio plays an essential role here because it affects the deuterium chemistry, and adopting the statistical 3:1 ortho/para ratio for H₂ as initial abundance for our TMC-1 model, ice HDO abundances drop by a factor of 10 and gas HDO abundances by a factor 20. Gas and ice H₂O are not significantly affected. The most abundant species in the initial abundances are listed in Table 1.

Table 1. Initial Abundances for the Models (Fractional Abundances)

Species	Abundances	Species	Abundances
p-H2	4.783 × 10−1	CH4 (ice)	5.832 × 10−6
He	9.750 × 10−2	N2	5.135 × 10−6
o-H2	2.146 × 10−2	O2 (ice)	4.619 × 10−6
H	2.681 × 10−4	O	2.661 × 10−6
H2O (ice)	7.658 × 10−5	N2 (ice)	2.464 × 10−6
CO (ice)	5.219 × 10−5	C3H2 (ice)	6.493 × 10−6
CO	1.816 × 10−5	HD	1.479 × 10−5
HNO (ice)	3.541 × 10−6	O2	9.645 × 10−6
HDO (ice)	2.887 × 10−6	NH3 (ice)	8.663 × 10−6
D	2.731 × 10−6	OH	2.295 × 10−6

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Our main set of chemical simulations consists of the two runs: (1) the laminar nebular model without transport processes, and (2) the 2D mixing model with Sc = 1. The water abundance is limited by the initial oxygen abundance in the models and changes throughout the disk. Given the rapid, ∼2–3 Myr, dynamical evolution of solids in the nebula, we modeled chemistry only within 1 Myr.

In Figure 2 we show the gas- and solid-phase water abundances between 0.8 and 30 AU for the laminar (left panel) and the 2D mixing model (middle panel). The D/H ratios from column densities are shown in the right panel. Column densities vary strongly with scale height. Both the laminar and 2D mixing show similar relative abundances of water vapor (with respect to H) going from 10⁻⁵ close to the midplane and quickly dropping to <10⁻¹² at around 0.15 scale heights in the inner regions, whereas in the outer regions relative abundances in the cold midplane are ∼10⁻¹², increasing to ∼10⁻⁶ at 0.15 scale heights and then dropping again to <10⁻¹² at ∼0.4 scale heights. Also for water ice the two models show similar relative abundances, with only smaller differences arising in the outer region ∼10 AU. In the warm inner regions relative abundances are ∼10⁻⁸–10⁻⁶ in the midplane, dropping to <10⁻¹² at ∼0.15 scale heights. Farther out in the disk, beyond the snow line at ∼2–3 AU, a thick layer of abundant ice with relative abundances of ∼10⁻⁴ stretches between 0.15 up to 0.3 scale heights, after which we see again a quick drop with relative abundances going below 10⁻¹². The calculated vertical column densities for both models are compared in the right panel.

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The same panels are shown in Figure 3 for HDO. Here layers of different relative abundances are more clear, with relative abundances of HDO vapor in the inner disk of ∼10⁻¹⁰, dropping quickly to <10⁻¹² at 0.15 scale heights. In the outer regions we have a low vapor abundance in the midplane of ∼10⁻¹⁴–10⁻¹², increasing up to 10⁻¹⁰ in the molecular layer and then at around 0.2 scale heights dropping to <10⁻¹². For HDO ice both models show relative abundances of ∼10⁻¹² in the inner region and a thick layer of abundances ∼10⁻⁸ between 0.15 to 0.3 scale heights in the outer regions. Whereas the 2D mixing model shows a smooth decline in abundances toward the inner disk, the laminar model shows larger variations, especially evident at 5–30 AU. As can be clearly seen, the snow line is not affected by slow diffusive transport and is located at ∼2–3 AU in both models, thus leaving dust grains in the Earth-formation zone barren of ices. Hence, the water delivery to Earth could occur only at a later stage, when dust grains were assembled into larger planetesimals that were able to reach 1 AU without complete loss of volatile materials.

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The resulting distribution of the D/H ratio for the total water budget (ice plus vapor) is shown in Figure 4. The water D/H ratio of the Earth's oceans and the measured value in carbonaceous meteorites ("Earth") are shown for comparison, also similar to what is observed in Jupiter-family comets. The D/H ratios for the Oort-family comets ("OFC") are also indicated.

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As can be clearly seen, the laminar model shows the Earth water D/H ratio at 0.8–2.5 AU, whereas for the 2D chemodynamical model such a low D/H value extends toward larger radii, ≲ 9 AU. Similarly, the elevated water D/H ratios representative of the Oort-family comets, ∼2.5–10 × 10⁻⁴, are achieved within ∼2–6 AU and ∼2–20 AU in the laminar and the 2D model, respectively. The reason for this behavior is a shallower radial gradient of the water D/H ratio in the 2D mixing model. Turbulent mixing slowly transports some of the water ice into warmer or irradiated regions where it desorbs and is quickly defractionated by ion-molecule and dissociative recombination processes in the gas phase. As a consequence, lower water D/H ratios can be retained farther out in the nebula, which seems to be more consistent with the delivery of water to Earth by comets. Both the laminar and the mixing nebular models have the Earth's oceans and the Jupiter-family comet water D/H ratios at radii ∼2–3 AU, where carbonaceous chondrites are believed to have formed. Thus both the laminar and dynamical nebular model can reproduce the water D/H ratios observed in carbonaceous asteroids. With regard to the water isotopic composition and the origin of the Jupiter-family and Oort-family comets, the mixing model seems to be favored over the laminar model. It allows for a larger region for their formation with the appropriate water D/H ratios, extending to the distance of ∼10–30 AU.

We have also studied how other processes may affect the results of our chemical modeling. First, given observational evidence for the presence of bigger grains in protoplanetary disks (e.g., Testi et al. 2003; van Boekel et al. 2005; Rodmann et al. 2006; Bouwman et al. 2008; Pérez et al. 2012), for reviews see Natta (2008); Henning & Meeus (2011), we have increased the uniform grain sizes to 1 and 10 μm and calculated the nebular water D/H ratios. The average 1 μm and 10 μm grain sizes imply 10 and 100 times smaller surface area (per unit volume of gas), respectively, making gas–grain interactions less prominent and allowing the high-energy radiation to penetrate deeper into the disk. The thermal desorption rates are independent of the grain size, whereas the cosmic-ray particle (CRP)-desorption rate depends on the grain size only weakly (Léger et al. 1985). The overall photodesorption rate increases because the nebula becomes more UV-transparent.

In contrast, the accretion rate onto the grains decreases when the total grain surface area per unit gas volume decreases. Consequently, in the inner warm nebular region at r ≲ 10–20 AU the amount of gaseous water increases by a factor of ∼10 for a_dust = 1 μm and ∼100 for a_dust = 10 μm, respectively, but it still constitutes only a tiny fraction of the total water budget. As a result, the water D/H ratio is not significantly affected, and the three modeled D/H radial distributions are within the uncertainties of the calculated D/H values (see Vasyunin et al. 2008; Albertsson et al. 2013). The same is found for the nebular model with the standard grains but 100 times lower UV-desorption yield of 10⁻⁵.

Next, we studied how the calculated water D/H ratios are affected if only a limited set of the nuclear spin-state processes is included in our chemical network. For that test we calculated a model in which only the para states of H₂, H$_2^+$, and H$_3^+$ isotopologues were considered, implying more efficient deuterium isotope exchange. The water D/H ratios calculated with this model coincide with the ratios calculated with our standard model with ortho/para-species (within the factor of ∼3 in intrinsic chemical uncertainties).

This result is a combination of two extreme situations. First, the initial abundances are calculated using the physical conditions of a dark, dense, 10 K cloud core. At such conditions almost the entire H₂ population exists in the lowest energy para-state, and the isotope exchange reactions involving ortho-species are still out of equilibrium. This leads to similar values of the initial water D/H ratio. Second, in the water defractionation zone in the nebula, at r ≲ 10–30 AU, temperatures are higher than 30 K. These temperatures are high enough to enable rapid gas-phase defractionation regardless of the dominant nuclear spin-state form of H₂, H$_2^+$, and H$_3^+$, mainly due to X-ray- and CRP-driven reprocessing of CO by He⁺, followed by rapid gas-phase production of H₂O. This combination of effects leads to the somewhat surprising result that a reduced network produces roughly the same abundances as the complete network.

The initial H₂ ortho/para ratio in cold dark clouds is unknown but is predicted to be low (<0.1 Pagani et al. 2009). However, later, during the warmer protostellar phase this ratio is likely modified, such that there are more ortho−H₂ and fewer para−H₂ molecules. As such the ortho/para ratio can be closer to the statistical ortho/para ratio of 3:1 in the initial phase of the protoplanetary disk. We take the statistical value 3:1 as the upper extreme of the H₂ ortho/para ratio and initiate another run of the laminar model with initial abundances resulting from a TMC-1 model with an initial 3:1 ortho/para ratio for H₂.

In Figure 5 we show the D/H ratios of solid-phase HDO between 0.8 and 30 AU for the laminar model using our previous "cold" TMC-1 initial abundance (left panel) and "warm" TMC-1 initial abundance (middle panel), and in the right panel the D/H ratio determined from column densities is shown. In the disk midplane there are small but significant differences where D/H ratios decrease from ∼10⁻² down to ∼10⁻⁴–10⁻³ around 10 AU. This causes the column density D/H ratios in the right panel to drop approximately an order of magnitude, whereas the minimum and maximum D/H ratios remain unchanged at ∼1 AU and ≳ 30 AU, respectively. Thus our conclusion that carbonaceous asteroids formed at 2–3 AU will inherit Earth-water-like D/H ratios remains valid. The fact that the H₂O D/H ratio gets lower at intermediate radii (∼10 AU) will only help us to achieve a larger zone where Hartley-2-like comets with the Earth water D/H ratio could form while still leaving a zone for the Oort comets to form at ∼10 AU.

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Finally, we discuss the effects of the underlying chemistry on the D/H ratios throughout the disk. In the midplane, temperatures are low enough to drive a rich surface chemistry, where O atoms are adsorbed on the grain surfaces and become hydrogenated, forming H₂O or reacting with deuterium atoms to form HDO or D₂O. There is a small chance for the water molecules to be released into the gas phase directly, but it is more likely that they remain on the grains and desorb as material is transported into warmer regions, i.e., the inner regions of the atmosphere, where water and its D/H ratio will be reprocessed.

The reprocessing of water occurs in the gas phase, so the relative adsorption and desorption plays an important role. These processes are heavily affected by temperature and density. In the warmer regions, the water abundances steadily increase by ≲ 30% during the first 10,000100,000 yr. As soon as the water reaches the gas phase, it is reprocessed. After that, the abundances of water ice and water vapor are at a quasi-steady-state. However, constant evaporation and reaccretion result in cycling of water between solid and gaseous phases. One important fact is that the D/H ratio never drops to the cosmic ratio, even in the warm, inner regions within a few AU of the Sun. In the inner solar nebula there are several important processes at work that set the D/H ratio. First of all, these regions experience strong X-ray- or CRP-ionization, and water is either ionized or protonated by reacting with various abundant ions, such as H$_{3}^{+}$, HCO⁺, and their isotopologues:

$$\begin{eqnarray*} && {\rm H_{2}O} + o\hbox{-}{\rm H}_{3}^{+} \Rightarrow {\rm H_{3}O}^{+} + o\hbox{-}{\rm H}_{2}; \quad\, 4.5 \times 10^{-9}\; {\rm cm}^{-3} \;{\rm s}^{-1} \\ &&\quad \beta = -0.5 \\ && {\rm H_{2}O} + p\hbox{-}{\rm H}_{3}^{+} \Rightarrow {\rm H_{3}O}^{+} + o\hbox{-}{\rm H}_{2}; \quad 2.3 \times 10^{-9}\; {\rm cm}^{-3} \;{\rm s}^{-1} \\ &&\quad \beta = -0.5 \\ && {\rm H_{2}O} + p\hbox{-}{\rm H}_{3}^{+} \Rightarrow {\rm H_{3}O}^{+} + p\hbox{-}{\rm H}_{2}; \quad 2.3 \times 10^{-9} \;{\rm cm}^{-3} \;{\rm s}^{-1} \\ &&\quad \beta = -0.5 \\ && {\rm H_{2}O} + {\rm HCO}^{+} \Rightarrow {\rm H_{3}O}^{+} + {\rm CO}; \quad\, 2.1 \times 10^{-9}\; {\rm cm}^{-3} \;{\rm s}^{-1} \\ &&\quad \beta = -0.5. \end{eqnarray*}$$

The same set of reactions applies to HDO being protonated into H₂DO⁺ (as well as heavier isotopologues). The H₃O⁺ isotopologues can then re-form water through dissociative recombination, but the D/H ratios will be reset, characteristic of the temperature in the environment at which they are re-forming:

$$\begin{eqnarray*} &&\ \ \ {\rm H_{3}O} + {\rm e}^{-} \Rightarrow {\rm H_{2}O} + {\rm H}; \quad\ 1.1 \times 10^{-7} \;{\rm cm}^{-3} \;{\rm s}^{-1} \quad\, \beta = -0.5 \\ && {\rm H_{2}DO} + {\rm e}^{-} \Rightarrow {\rm H_{2}O} + {\rm D}; \quad\ \, 7.3 \times 10^{-8} \;{\rm cm}^{-3} \;{\rm s}^{-1} \quad \beta = -0.5 \\ && {\rm H_{2}DO} + {\rm e}^{-} \Rightarrow {\rm HDO} + {\rm H}; \quad 7.3 \times 10^{-8} \;{\rm cm}^{-3} \;{\rm s}^{-1} \quad \beta = -0.5. \end{eqnarray*}$$

The reprocessing of water involves several additional reactions that compete in both lowering and increasing the D/H ratio. In the warm inner nebular region the normal H$_{3}^{+}$ fractionation pathway is not active, and instead fractionation is driven mostly via the "warm" fractionation route that includes light hydrocarbons such as CH₂D⁺ and C₂HD⁺ and is active at T ≲ 80 K (e.g., Millar et al. 1989; Roueff et al. 2005; Parise et al. 2009). Consequently, the initially high D/H ratios of light hydrocarbons, ∼10⁻³, is not fully equilibrated with the cosmic D/H ratio of ∼10⁻⁵. Neutral–neutral reactions of these hydrocarbons with atomic or molecular oxygen are able to produce normal or heavy water, e.g.,:

$$\begin{eqnarray} &&{\rm CH_{3}} + {\rm O_{2}} \Rightarrow {\rm HCO} + {\rm H_{2}O}; \quad 1.66 \times 10^{-12} \;{\rm cm}^{-3} \;{\rm s}^{-1}. \end{eqnarray} \tag{ 3 }$$

Another important channel of water formation in the gas phase is through the neutral–neutral reaction with OH or OD of H/D or ortho/para −H₂/HD:

$$\begin{eqnarray*} && {\rm OH} + {\rm H} \Rightarrow {\rm H_{2}O}; \quad\quad\quad\quad\quad\, 4.0 \times 10^{-18} \;{\rm cm}^{-3} \;{\rm s}^{-1} \\ && {\rm OH} + {\rm D} \Rightarrow {\rm HDO}; \quad\quad\quad\quad\ \ \ 4.0 \times 10^{-18} \;{\rm cm}^{-3} \;{\rm s}^{-1} \\ && {\rm OD} + {\rm H} \Rightarrow {\rm HDO}; \quad\quad\quad\quad\ \ \ 4.0 \times 10^{-18} \;{\rm cm}^{-3} \;{\rm s}^{-1} \\ && {\rm OH} + o/p\hbox{-}{\rm H}_{2} \Rightarrow {\rm H_{2}O} +{\rm H}; \quad 8.4 \times 10^{-13} \;{\rm cm}^{-3} \;{\rm s}^{-1}\quad 1040 \;{\rm K} \\ && {\rm OH} + {\rm HD} \Rightarrow {\rm HDO} + {\rm H}; \quad\quad\ 4.2 \times 10^{-13} \;{\rm cm}^{-3} \;{\rm s}^{-1}\quad 1040 \;{\rm K} \\ && {\rm OH} + {\rm HD} \Rightarrow {\rm H_{2}O} + {\rm D}; \quad\quad\ \ \, 4.2 \times 10^{-13} \;{\rm cm}^{-3} \;{\rm s}^{-1} \quad 1040 \;{\rm K}. \end{eqnarray*}$$

While OH/OD, ortho/para −H₂/HD, and H/D are formed through various processes, including the dissociation of water, the D/H ratio of OH is dominantly controlled by the equilibrium of the following forward and backward processes:

$$\begin{eqnarray} {\rm OH} + {\rm D} \Rightarrow {\rm OD} + {\rm H}; & 1.3 \times 10^{-10} \;{\rm cm}^{-3} \;{\rm s}^{-1} \nonumber \\ {\rm OD} + {\rm H} \Rightarrow {\rm OH} + {\rm H}; & 1.3 \times 10^{-10} \;{\rm cm}^{-3} \;{\rm s}^{-1} \; 810 \;{\rm K}. \end{eqnarray} \tag{ 4 }$$

At around 100 K, the equilibrium state of this reaction will lead to an enhancement by a factor of approximately 100 in OD, which will transfer into the water D/H ratio and competes with other defractionation processes, eventually leading to the enhanced D/H ratios of ∼10⁻⁴. As we move farther out in the disk the efficiency of reaction 5 gradually decreases, whereas the pace of water ice defractionation decreases due to slower cycling of water through gas and solid phases, and the water D/H ratio increases gradually until we reach the colder outer regions of the solar nebula.

Compared to Thi et al. (2010), we do not find that reaction formation of water through reactions between OH/OD and H₂/HD are the dominant neutral–neutral reactions; instead we find that OH/OD and H/D reactions are more dominant. This discrepancy likely arises due to the energy barrier of reactions with H₂ or HD, whereas reactions with H or D are exothermic. Other differences between our models may also contribute to the discrepancies. Thi et al. (2010) used a steady-state model to determine their key reactions, which are not exactly like ours. Bear in mind that time-dependent disk chemical models usually do not reach a chemical steady state even after 1 Myr of evolution. If we would allow our calculations to run over much longer time scales, we could get a set of key reactions more similar to that found by Thi et al. (2010). Furthermore, Thi et al. (2010) adopt reaction rates from Woodall et al. (2007) (H₂O) and Talukdar (1996) (HDO), which may also account for some of the discrepancy. This also results in different OH and OD column densities, where OH is approximately an order of magnitude more abundant in our model, decreasing by two orders of magnitude at ∼30 AU, whereas OD is approximately an order of magnitude more abundant for Thi et al. (2010). Finally, the inclusion of ortho-para chemistry in our network and differences in physical structures may contribute further to these differences. Most notably, the stellar parameters are significantly different, whereas Thi et al. (2010) adopts values for a less-massive (0.5 M_☉) but larger (2 R_☉) and cooler (4000 K) central protostar than our study.

We have shown the importance of 2D mixing on the water D/H gradient in the young solar nebula, which pushes the (lower) water D/H ratios from the inner disk regions toward larger heliocentric distances as water is defractionated after being transported into the inner regions. Before discussing the implications of our results, it is important to discuss the limitations of our model.

Foremost, protoplanetary disks are not static environments but are ever dynamically evolving systems. Dust grains grow into planetesimals and eventually planets, with an associated evolution of physical conditions affecting the chemistry. As we previously discussed, a moderate increase in grain sizes had no significant effect on our results (see Section 3), but this might change with even larger particles. Variations in temperature and density will likely play a significant role. Cazaux et al. (2011) predicted that significant variations in HDO/H₂O ratios can come from even small variations in dust temperatures at the time of ice formation. This would also affect initial abundances of our model because the warm-up phase following the collapse of a dark prestellar core will experience a gradual increase in both temperature and density. In addition, exact masses and structural parameters of the solar nebula remain highly uncertain.

Another large uncertainty is the postprocessing of Earth's ocean water. Such processes include biological (life and its evolution) and chemical processes. Atmospheric processes can affect the fractionation if the young Earth had a massive hydrogen atmosphere, which would experience a slow hydrodynamic escape, which will be slower for deuterium than for the lighter hydrogen isotope (Genda & Ikoma 2008). Furthermore, as was first proposed by Campins et al. (2005), processes involved in planetary accretion, such as outgassing and the evolution of hydrosphere and atmosphere, are complex and may have affected the fractionation.

Finally, the giant impact theory, which explains the formation of the moon and the similarities of its bulk composition to Earth (see, e.g., Canup 2012), requires a collision between early Earth and a large impactor. This scenario implies a heavy loss of the primordial Earth atmosphere and would have likely greatly affected the chemical composition, more specifically the D/H ratio. However, results by Saal et al. (2013) revealed a similar isotopic composition of the water in the interiors of Earth and the Moon, as well as the bulk water in carbonaceous chondrites, suggesting that this reprocessing is of minor importance.

4.1. Observations in Other Protoplanetary Disks

4.2. Previous Theoretical Studies

4.3. Solar System Bodies

In this paper the isotopic and chemical evolution of water in the early history of the solar nebula before the onset of planetesimal formation is studied. An extended gas–grain chemical model that includes multiply-deuterated species, high-temperature reactions, and nuclear spin-state processes is combined with a 1+1D steady-state α-viscosity nebula model. To calculate initial abundances, we simulated the 1 Myr of the evolution of a cold and dark TMC-1-like prestellar core, resulting in initially high D/H ratios for water and other molecules of ∼1%. Two time-dependent chemical models of the solar nebula are calculated over 1 Myr and for radii 0.8–800 AU: (1) a laminar model and (2) a model with 2D turbulent mixing transport. We find that both models are able to reproduce the Earth ocean's water D/H ratio of ≈1.5 × 10⁻⁴ at the location of the asteroid belt, ≲ 2.5–3 AU, where a transition from predominantly solid to gaseous water occurs.

The water ices there can be incorporated in growing solids, melt, and eventually produce phyllosilicates. At ≲ 2 AU, nebular temperatures are too high for the water ice to exist and the dust grains are free of water ice. Thus the planetesimals from which Earth would later form remain water-poor. We find that the radial increase of the D/H ratio in water outward is shallower in the chemodynamical nebular model. This is related to more efficient defractionation of HDO via rapid gas-phase processes because the 2D mixing model allows the water ice to be transported either inward and thermally evaporated or upward and photodesorbed. Taking the water D/H abundance uncertainties of the factor of two into account, the laminar model shows the Earth water D/H ratio at r ≲ 2.5 AU, whereas for the 2D chemodynamical model this zone is larger, r ≲ 9 AU. Similarly, the enhanced water D/H ratios representative of the Oort-family comets, ∼2.5–10 × 10⁻⁴, are achieved within ∼2–6 AU and ∼6–30 AU in the laminar and the 2D model, respectively. The characteristically slightly lower water D/H ratio that has been found for Jupiter-family comets are found farther in, and we find their possible formation location ∼1–10 AU in both models. This means that in our models we find an overlap in the possible formation location for Oort- and Jupiter-family comets. However, with regard to the water isotopic composition and the origin of the comets, the mixing model seems to be favored over the laminar model because the former allows Oort-family comets to have formed in the region of Jupiter's and Saturn's present location.

This research made use of NASA's Astrophysics Data System. The research leading to these results has received funding from the European Community's Seventh Framework Programme [FP7/2007-2013] under grant agreement No. 238258. D.S. acknowledges support by the Deutsche Forschungsgemeinschaft through SPP 1385: "The first ten million years of the solar system: A planetary materials approach" (SE 1962/1-2).