Lunar sinuous rilles have long been interpreted as features that formed as the result of surficial lava flow, though the precise mechanism responsible for channel formation (constructed versus eroded origins) is still debated. In assessing the origin of Rima Prinz, a channel interpreted to have formed by erosion, two erosion regimes, mechanical and thermal, are considered. Measurements of channel dimensions are used as inputs to analytical models to constrain the origin of Rima Prinz, including lava compositions, mechanical and thermal erosion rates, eruption durations, and lava volumes required to form the feature. Key results indicate that Rima Prinz and other large sinuous rilles could have formed as the result of thermal erosion under the weak gravity and low slope conditions characteristic of these lunar features. Further analysis indicates that lava composition has significant effects on channel formation. Model results of four considered lava compositions show that komatiite-like lava will erode a similarly composed substrate most efficiently whereas a high-Ti basalt will erode a similarly composed substrate least efficiently; ocean island basalt and low-Ti basalt erode similarly composed substrates at intermediate rates. Results indicate that Rima Prinz may have formed over 0.4–2.2 Earth years, depositing 50–250 km3 of lava over a plausible deposit area of 2450 km2. Resulting deposit thicknesses suggest that the lava that incised Rima Prinz was most likely similar in composition to a terrestrial komatiite, ocean island basalt, or lunar low-Ti basalt. Further constraints on sinuous rille formation will serve as a window into the nature of volcanic activity of the Moon's past.
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 Sinuous rilles observed on the Moon are widely accepted to represent the remains of channels formed by lava that erupted in effusive, high volume volcanic events. The exact mode of channel formation is still debated, as researchers attempt to distinguish between channels that formed due to construction of bounding levees [Spudis et al., 1988; Komatsu and Baker, 1992; Gregg and Greeley, 1993], due to mechanical erosion of the substrate [Siewert and Ferlito, 2008], due to thermal erosion of the substrate [Hulme, 1973, 1982; Head and Wilson, 1981; Wilson and Head, 1981; Williams et al., 1998, 2000; Kerr, 2009], and due to thermomechanical erosion of the substrate [Williams et al., 1998, 2001; Fagents and Greeley, 2001]. Some sinuous rilles are contiguous with distinctive source depressions, the geometry of which suggests that the lava flow forming the rille overflowed from a lava pond fed by pyroclasts falling from an optically dense fire fountain [Head and Wilson, 1980; Wilson and Head, 1980; L. Wilson and J. W. Head, Lunar sinuous rilles and their associated source depressions: The role of thermal erosion and implications for eruption conditions, submitted to Journal of Volcanology and Geothermal Research, 2011], with thermal erosion at the base of the pond as well as at the base of the flow leading to the observed morphology. In each of the proposed formation mechanisms, the composition of lava that flowed through the channel can affect the volume of lava and the amount of time required to form the observed feature. The current study simulates the formation of Rima Prinz, a lunar sinuous rille interpreted to have an eroded origin, by modeling erosion rates and expected erosion depths for four distinct lava compositions. Comparisons between expected and observed channel depths can provide constraints for the composition of the lava that formed the observed channel. These interpretations in turn can provide insight into the conditions that were present during the volcanically active era in lunar history.
 Sinuous rilles have commonly been identified as constructed or eroded features, and substantial analysis has been conducted previously in an attempt to distinguish between these two origins. Constructed channels are typically considered to form as the result of marginal cooling of a broad lava flow. As the flow cools and solidifies inward from the margins, levees form and bound the fastest moving, still molten part of the lava flow, forming a channel [i.e., Hulme, 1974]. Constructed channels, such as those found in southern Imbrium basin lava flows (Figure 1a), tend to be shallow features, forming within the initial sheet lava flow rather than incising into the substrate.
 A constructed mode of origin has also been suggested for the formation of larger sinuous rilles identified on the Moon as well as other terrestrial bodies [i.e., Spudis et al., 1988; Komatsu and Baker, 1992; Gregg and Greeley, 1993]. However, many of these larger sinuous rilles, such as Rima Prinz located east of the Aristarchus Plateau (Figure 1b), are significantly deeper features that appear to lack levees and incise into the substrate, suggesting that the development of these features is likely to have involved erosion [Hulme, 1973, 1982; Carr, 1974; Hulme and Fielder, 1977; Coombs and Hawke, 1988; Coombs et al., 1990; Pinkerton et al., 1990]. In many cases, these eroded channels, with laterally continuous, nearly parallel walls, are expected to represent open channels that developed a thin surface crust [i.e., Williams et al., 1998, 2000]. In some cases, channels might construct a structurally stable crust that remains intact once the lava recedes, or channels might form completely submerged within the substrate, forming a lava tube [i.e., Greeley, 1971]. Collapse of a lava tube roof may result in the formation of a skylight that can be observed remotely [Haruyama et al., 2009; Huang et al., 2011; Boyd et al., 2011], such as that visible in a lava tube observed west of the Marius Hills (Figure 1c). Aligned skylights might indicate the track of a subsurface lava tube. These skylights observed in association with lava tubes represent distinct features from the laterally continuous, nearly parallel walls observed in association with the open channels of eroded origin considered in this study.
 Two classes of erosion are commonly considered in the origin of eroded channels: mechanical erosion and thermal erosion. Mechanical erosion occurs when a flowing fluid removes particles that lay loosely on the ground and involves particles suspended in the flowing fluid colliding with the substrate, shearing substrate particles and incising into the substrate [i.e., Sklar and Dietrich, 1998]. While some work has been done investigating this erosion regime in the origin of lava channels [i.e., Sklar and Dietrich, 1998], this previous approach assumes that the vertical load of the lava is responsible for the erosion of the substrate rather than shear stress. However, the low viscosity expected of lunar lavas would allow for relatively high lava flow velocities, facilitating erosion by shear. The second type of erosion considered is thermal erosion, a process that involves a flowing fluid whose temperature exceeds the melting temperature of the substrate. As the hot fluid comes into contact with the substrate, the substrate is melted and assimilated into the flowing fluid, resulting in incision into the substrate. Analytical models must be used to supplement observations of channel dimensions and morphology in order to distinguish between these two erosion regimes.
 The goal of the current study is to discern the detailed origin of Rima Prinz, a lunar sinuous rille west of the Aristarchus Plateau that is interpreted to have an eroded origin. Remote observations of channel dimensions and morphology described in section 3 are used in conjunction with analytical models discussed in section 4 to calculate the erosion rates and eruption durations required in each erosion regime to form the observed channel. Effects of gravity, slope, and lava composition are explored in sections 5 and 6 to determine whether Rima Prinz formed as the result of mechanical or thermal erosion. Model results for each of four lava compositions considered are explored in order to constrain reasonable lava compositions that may have been responsible for the formation of the observed sinuous rille. This approach uses sinuous rilles as a window into the lunar interior to provide constraints for the conditions that were present during the volcanically active period of lunar history.
3. Geologic Setting
 The Aristarchus Plateau (Figure 2) has been identified as an uplifted block of lunar crust that shows compositional evidence for highlands material (excavated by the Aristarchus impact) that is superposed by Imbrium ejecta material and a regional surface veneer of dark mantle deposits [McEwen et al., 1994; Hawke et al., 1995; Chevrel et al., 2009] and is surrounded to the north and east by mare basalts of intermediate TiO2 content [Whitford-Stark and Head, 1980; Giguere et al., 2000; Hiesinger et al., 2003]. Nearly 20 volcanic vents are preserved on the plateau in the form of typically circular depressions that act as the source for sinuous rilles that extend from the plateau into the surrounding mare basalt deposits [i.e., Schubert et al., 1970; Whitford-Stark and Head, 1977]. Many additional sinuous rilles are observed in the area surrounding the plateau, making the Aristarchus Plateau – Harbinger Mountain region one of the more densely concentrated volcanic centers that was likely to have been active during the Imbrian period of lunar history [i.e., Zisk et al., 1977].
 Rima Prinz (Figure 3) is the westernmost sinuous rille in a cluster of sinuous rilles near the Harbinger Mountains located on a smaller plateau east of the Aristarchus Plateau. The source depression of Rima Prinz lies on the northern extent of ejecta associated with Prinz crater. The sinuous rille is characterized by two channels, one larger valley and a smaller, nested sinuous rille that is interpreted to have formed in a second, independent eruption after an initial eruption formed the larger valley [i.e., Strain and El-Baz, 1977; Wilson and Head, submitted manuscript, 2011]. Both channels were heavily influenced by slope during their formation, with an upper segment forming on Prinz ejecta (Figure 3b), a middle segment forming in the mare abutting the northern extent of Prinz ejecta (Figure 3c), and a lower segment forming in the mare (Figure 3d), directed down-grade to the north and terminating in Oceanus Procellarum (Figure 3e).
 Rima Prinz originates in a circular depression that is interpreted to be the site of the eruption that fed the associated lava channel [Head and Wilson, 1981; Wilson and Head, 1981]. A perspective view of Lunar Reconnaissance Orbiter Narrow Angle Camera (LROC NAC) images M104805368LE and M104805368RE (0.5 m pixel−1) overlaid on Lunar Orbiter Laser Altimeter (LOLA) data (∼120 m pixel−1) shows that the source depression for Rima Prinz has a steep, well-consolidated rim (Figure 4a). The southern depression wall (image left) is steep down to the depression floor while the northern depression wall slopes more gradually to the interpreted eruption location. This morphology is similar to that observed in the source crater of Mauna Ulu, Hawaii (Figure 4b), a feature that also has a steep, rocky rim and one wall that slopes more gradually toward the eruption location. Both source depressions are expected to have formed during a fire fountain eruption due to erosion of the surface beneath a lava lake [i.e., Head and Wilson, 1981; Wilson and Head, 1981]. Remnant deposits of this lava lake might still be visible in a long wavelength, hummocky texture on the floor of the source depression of Rima Prinz (Figure 5a), a texture that is similar to that of deposits observed in the remnant lava lake in Kilauea Iki, Hawaii (Figure 5b). These observations of similar morphologies between terrestrial fire fountain eruption features and lunar sinuous rille source depressions support the theory that lunar source depressions also formed as the result of fire fountain eruptions [i.e., Wilson and Head, 1981; Head and Wilson, 1981].
 The three channel segments of Rima Prinz connect the source depression on the ejecta of Prinz crater to the mare plains of Oceanus Procellarum to the north. Each channel segment is analyzed independently to assess channel dimensions and morphology. Dimensions of interest include 1) channel length, which is measured by averaging the lengths of the two bounding channel rims for each channel segment; 2) channel width, which is measured by averaging rim-to-rim distances along the length of each channel segment; 3) channel depth, which is measured by averaging the differences between channel rim elevation and channel floor elevation as documented by LOLA track data; 4) channel sinuosity, which is measured by averaging distances between meander extremes (i.e., meander wavelength) for each channel segment; and 5) regional slope, which is determined by measuring the average regional slope along the channel rim using LOLA gridded data. Sinuosity is measured in order to provide an estimate of the width of the lava within the currently observed valley, as the valley was unlikely to have been filled with lava [Pelletier, 2008]. Uncertainties in these measurements are generally reported as the standard deviation of measurements made along the length of each channel segment, with the exception of the uncertainty in channel length, which is reported as the variation between the rim length measurements and the length measurement of the interpreted channel thalweg. Measurements are summarized in Table 1.
Table 1. Observations and Measurements of Rima Prinz
 The upper segment of Rima Prinz (Figure 3b) is 11 km +/− 3.0 km in length, 1.8 km +/− 0.7 km in width, 230 m +/− 20 m in depth, and has a sinuosity wavelength of 1.7 km +/− 0.7 km. This channel segment formed down-gradient at a slope of 0.7° +/− 0.6°; the uncertainty is higher because the channel cut through relatively hummocky terrain consistent with ejecta deposits. The channel is characterized by steep, nearly parallel walls that lack obvious marginal levees, and the walls appear to be well-preserved, lacking evidence of substantial subsequent slumping. The nested rille remains clearly visible throughout the channel segment and typically mirrors the sinuosity of the larger outer channel.
 The middle segment of Rima Prinz (Figure 3c) is 19 km +/− 8.0 km in length, 1.0 km +/− 0.4 km in width, 210 m +/− 40 m in depth, and has a sinuosity wavelength of 2.4 km +/− 2.0 km. This channel segment formed down-gradient at a slope 0.5° +/− 0.1°, forming along the bottom of the northern extent of Prinz crater ejecta. While the northern wall of the middle channel segment remains steep and apparently well-preserved, the southern wall has been subjected to substantial slumping, possibly due to the collapse of Prinz ejecta material during a subsequent impact such as one responsible for the formation of Aristarchus crater to the southwest. This deformation of the southern wall has resulted in the concealment of the nested rille in many places along this channel segment and led to the higher uncertainty values in the reported length and depth measurements.
 The lower segment of Rima Prinz (Figure 3d) is 57 km +/− 3.0 km in length, 1.0 km +/− 0.3 km in width, 145 m +/− 30 m in depth, and has a sinuosity wavelength of 2.0 km +/− 1.0 km. This channel segment formed down-gradient at a slope of 0.4° +/− 0.1° and extends northward into Oceanus Procellarum. The lower segment of Rima Prinz is characterized by the steep, parallel, and laterally continuous walls that are observed in the upper channel segment. This portion of the channel also lacks obvious levees, consistent with a channel that formed as the result of erosion processes. The nested rille is visible for much of the length of the lower channel segment, and whereas the sinuosity of the nested channel mirrors the outer channel along most of the length of the channel segment, in some places the nested channel is significantly more sinuous. This may be a result of the duration of flow in the nested channel, with a longer duration flow forming more stable, larger meanders, or it may be the result of the low slope characteristic of this portion of the channel segment.
 Rima Prinz terminates (Figure 3e) in Oceanus Procellarum at the −1700 m contour, the same elevation at which its neighboring sinuous rille to the east, Rima “Beethoven,” terminates. Rima Prinz appears to increase in sinuosity (i.e., decrease in sinuosity wavelength) as it nears its terminus, and the walls remain nearly parallel and laterally continuous, though the channel depth has decreased substantially. Deposits are not observed beyond the currently observed channel terminus, indicating either that these deposits are too thin to be observed due to the low viscosity of the lava, or that they have been covered by subsequent mare volcanic flows. Because Rima “Beethoven” to the east terminates at the same elevation as Rima Prinz, embayment by subsequent mare volcanism is the more likely scenario, and careful inspection of the terminus of Rima Prinz indicates that mare lava may have flowed up-gradient through the channel, partially flooding the observed channel terminus for approximately 3–5 km (Figure 3e). It should be noted that the embaying lava flows may have covered the distal part of Rima Prinz, leading the length measurement for the lower channel segment to represent a minimum value of the actual length of this channel segment.
4. Theory of Lava Channel Formation by Erosion
 The morphology of Rima Prinz, specifically the lateral continuity of the channel walls and the lack of levees observed along the channel margins, supports the theory that Rima Prinz formed as the result of erosion into the substrate. The observations reported in section 3 represent the currently observed product of this erosion and can thus be used in conjunction with analytical models to distinguish between two possible erosion regimes. The analytical models considered in this study include one mechanical erosion model [Sklar and Dietrich, 1998] and two thermal erosion models [Hulme, 1973; Williams et al., 1998, 2000]. As stated earlier, mechanical erosion occurs as the result of collisions between particles in the flowing fluid and the substrate, and the rate of change in channel depth dchan as the result of mechanical erosion is given by
where Qw is the average lava flux per unit width through the channel in m2 s–1 (Qw = dlava × vlava, as calculated in equations (6) and (7)), ρ is the lava density (see Table 2 for parameter values), g is the acceleration due to gravity on the Moon, α is the ground slope, and K is a factor with units of Pa−1 that represents the erodibility of the substrate [Sklar and Dietrich, 1998; Hurwitz et al., 2010].
Table 2. Lava Compositions, Temperatures, Chemical Parameters, and Physical Parameters
 The model simulating mechanical erosion is most significantly affected by the erodibility factor K, as explained in more detail by Hurwitz et al. . Higher values of the factor K (∼10−7) represent a less consolidated substrate, like the lunar regolith, that is more susceptible to mechanical erosion at the lower slopes observed in relation to the lunar sinuous rilles. Lower values of the factor K (∼10−9) represent a more consolidated substrate, like lunar basalt, that is more susceptible to thermal erosion at lower slopes. An unconsolidated surface with an erodibility of ∼10−7 results in a modeled mechanical erosion rate that is ∼5 times higher than the mechanical erosion rate modeled for a consolidated substrate with an erodibility of ∼10−9. Slope also represents a significant parameter in the model of mechanical erosion, and analysis of how slope affects erosion rates is explored in more detail in section 4.
 As discussed in a similar study of the origin of a Martian lava channel, equation (1) can be thought of conceptually as modeling the erosion rate as a function of substrate erodibility and unit stream power, Ω, where Ω = ρ g Q sin α [Sklar and Dietrich, 1998; Hurwitz et al., 2010]. A different, more fundamental way to think about equation (1) is to separate it into different energy components: the flux term Q is a function of lava velocity and thus kinetic energy, and the term ρ g sin α represents the potential energy stored in the flowing fluid. Equation (1) therefore indicates that mechanical erosion depends on how efficiently the kinetic and potential energies stored in the flowing fluid are transferred to the substrate. Substrate erodibility is dependent on substrate composition and consolidation, and thus the thickness of the lunar regolith and the consolidation of either the ejecta associated with the upper segment of Rima Prinz or the mare basalt substrate can have a significant effect on the modeled erosion rates. In general, equation (1) predicts that a mechanically eroded channel will increase in depth faster as a higher flux of lava flows over a more poorly consolidated substrate.
 In contrast to mechanical erosion, thermal erosion occurs when the flowing fluid is hot enough to melt the substrate. The rate of change in channel depth dchan as the result of thermal erosion is a function of the energy provided by the lava flow and the energy required to melt the substrate, and is generally defined by
where T and Tmg represent the temperature of the lava and the melting temperature of the substrate, respectively, hT is the heat transfer coefficient, and Emg is the energy required to melt the substrate and is given by
where Tg is the initial temperature of the ground or substrate, cg is the specific heat of the substrate, Lg is the latent heat of fusion for the substrate, and fmg is the fraction that the substrate must be melted before being carried away by the flowing fluid [Hulme, 1973; Williams et al., 1998, 2000]. The two terms in equation (3) represent 1) the energy required to raise the temperature of the substrate to the melting temperature of the substrate and 2) the energy required to melt the substrate. The additional heat transfer coefficient term in equation (2) represents how efficiently thermal energy can be transferred from the hot flowing lava to the substrate. Two different approaches have been used to define the heat transfer coefficient: one by Hulme , given by
where k represents the thermal conductivity of the lava, μb and μg represent the bulk viscosity of the lava and the viscosity of the substrate, respectively, Re is the Reynolds number (Re = , turbulent flow that enhances erosion occurs when Re > 2000), and Pr is the Prandtl number (Pr = ). The difference between these two formulations of the heat transfer coefficient is subtle, with the formulation by Williams et al. [1998, 2000] incorporating a slightly higher weighting of the Prandtl number than the formulation by Hulme . This slightly higher weighting is used to account for the thermal boundary layer, in effect changing the efficiency at which heat is transferred across this boundary between the flowing fluid and the substrate. The models described in equations (2)–(5) indicate that thermal erosion relies most significantly on the transfer of thermal energy from the flowing lava to the substrate, but it should be noted that thermal erosion is also dependent on kinetic and potential energies, factors that are included in the model through the Re term in the heat transfer coefficient (equations (4) and (5)). Velocity, a factor in Re, is found by iteratively solving a model for moderately turbulent flows [Keszthelyi and Self, 1998], given by
where Cf is a friction factor given by
It should be noted that the full model for thermal erosion presented by Williams et al. [1998, 2000] and used in analysis of lava composition effects on erosion rate employs alternative models for velocity and friction factor. The differences are subtle and do not significantly alter the qualitative interpretations made in the current study. For a more detailed description of the full model, consult Williams et al. [1998, 2000].
 The thermal erosion models presented in equations (2)–(5) are most significantly affected by temperature, specifically in the difference between erupted temperature, assumed to be the liquidus temperature of the erupted lava, and the substrate melting temperature, assumed to be the solidus temperature of a lava of the same composition as the erupted lava. Specifically, a 100 K change in the erupted lava temperature T (i.e., the approximate difference between the erupted temperatures of a komatiite-like basalt and a low-Ti basalt) results in a change in the modeled thermal erosion rate of one order of magnitude. The temperatures used in the model, as well as the other considered parameters of thermal conductivity, specific heat, and latent heat, are all dependent on the lava composition considered, and each parameter is recalculated depending on the composition used as an input to the model. These parameter calculations are described in more detail by Williams et al. [1998, 2000].
 The models presented in equations (1)–(5) are solved initially to compare erosion rates in all three models. This is accomplished by first calculating the volume flux of the source eruption based on the geometry of the source depression and the density of the lava [Head and Wilson, 1980; Wilson and Head, 1980]. Second, the width of the lava that flowed within the observed channel is calculated based on observed sinuosity, assuming that a fully developed fluid flow meanders at a wavelength equal to 10.88 times the flow width [Pelletier, 2008]. Equations (6) and (7) are then used to solve for flow velocity by iteratively changing the depth of the lava within the observed channel. This iteration is continued until the volume flux predicted by the width, depth, and flow velocity calculations match the volume flux calculated independently. Once the lava depth and flow velocity values are constrained, Re and hT are calculated and the models in equations (1)–(5) are solved for erosion rate in each erosion regime considered.
 The three models discussed in section 4 are used to simulate the erosion rates expected for the head of Rima Prinz (at the intersection of the channel and the source depression) in order to 1) distinguish between mechanical and thermal erosion origins for the sinuous rille and to 2) compare the erosion rates predicted by the two thermal erosion models. In order to directly compare model results, the two thermal erosion models are run for only the head of the upper segment of Rima Prinz, where no contamination due to assimilation of melted substrate into the flowing lava has occurred and thus thermal and geophysical properties of the lava are assumed to be constant. These models are run for both terrestrial and lunar gravity conditions at various slopes to determine how gravity and slope affect predicted erosion rates. The most relevant model is then used to simulate the complete formation of Rima Prinz in the presence of different lavas in order to constrain the conditions present during the formation of this sinuous rille.
 Results for the mechanical and two thermal erosion models are shown in Figure 6, with results for terrestrial gravity conditions shown in Figure 6a and results for lunar gravity conditions shown in Figure 6b. These models were run assuming a lava composition similar to that of a terrestrial ocean island basalt (Table 2) because lunar lavas that erupt in high-effusion rate, high-volume eruptions are expected to originate from below the lunar crust-mantle interface, and the lavas erupt at effusion rates similar to those of terrestrial ocean island basalts [e.g., Head and Wilson, 1992; Wieczorek et al., 2001]. A lava composition similar to an ocean island basalt is considered in this study because eruption fluxes modeled for lunar eruptions and potential source depressions observed on the lunar surface are similar to those eruption fluxes and source depressions observed in conjunction with eruptions of ocean island basalts on Hawaii [e.g., Head and Wilson, 1980; Wilson and Head, 1980].
 Results for the terrestrial case indicate that mechanical erosion is more efficient than thermal erosion at slopes greater than about 0.4°. This suggests that at slopes greater than 0.4° in the terrestrial case, gravitational potential energy and thus kinetic energy are the dominant forms of energy acting during channel formation. Alternatively, at slopes less than 0.4°, potential and kinetic energies are insignificant compared to the thermal energy stored in the flowing lava, and thus thermal erosion dominates the formation process. In contrast, results for the lunar case indicate that thermal erosion dominates the channel formation process at slopes less than about 3.5°. This suggests that the lower gravity characteristic of the Moon provides insufficient potential energy and thus kinetic energy to contribute significantly to the formation of Rima Prinz (which has α < 0.7°). These results indicate that thermal erosion can dominate the formation of lunar sinuous rilles that form on consolidated substrates of basalt at low slopes even though it is not a commonly observed process in the formation of terrestrial lava channels that typically form on steeper gradients.
 Results shown in Figure 6 also indicate that predicted erosion rates are similar for both thermal erosion models considered. While these results certainly depend on the lava composition and geophysical properties assumed, predicted erosion rates are typically within ∼25% for the lava compositions considered as inputs for each model. Because the results are similar and because the full model developed by Williams et al. [1998, 2000] incorporates added complexities that allow for the analysis of affects of lava composition on channel formation, the full model developed by Williams et al. [1998, 2000] is used to simulate the detailed formation of Rima Prinz. The full model simulates the formation of a lava channel as a function of distance from the eruption, tracing how much of the substrate has been melted and assimilated into the flowing lava as well as how much olivine has crystallized in the flowing fluid, then recalculating the thermal and geophysical properties (such as Reynolds number, Prandtl number, heat transfer coefficient, thermal conductivity, and bulk viscosity) of the new lava composition [Williams et al. [1998, 2000]. In addition to heat lost to the substrate, this model also simulates the formation of a fusion crust at the top of the lava flow, a crust that can be an efficient insulator for the lava flow [Williams et al. 1998, 2000].
 The model developed by Williams et al. [1998, 2000] is used to determine how fast erosion occurs (i.e., erosion rate), and from these results and observations of depth in the head of the upper channel segment, the duration of the eruption that is required to form this uppermost portion of Rima Prinz is determined. This calculated eruption duration, which remains constant for the rest of the channel, is then used with modeled erosion rates to calculate a predicted depth for the remainder of the upper, middle, and lower channel segments, and these predicted erosion depths are compared with observed depths of Rima Prinz to confirm the merit of the model (Figure 7). In order to determine effects of lava composition on erosion rates, four compositions of lava are considered in this analysis, including compositions similar to a lunar high-Ti basalt (i.e., Apollo 17 sample 74220, [Longhi et al., 1978]), a lunar low-Ti basalt (i.e., Apollo 12 sample 12002, [Longhi et al., 1978]), a terrestrial komatiite (i.e., Kambalda, W. Australia [Lesher and Arndt, 1995]), and a terrestrial ocean island basalt (i.e., Kilauea, HI [Clague et al., 1991]; see Table 2). The two lunar lava compositions are considered because they represent lavas that have been sampled and analyzed directly from the lunar surface, though the lavas sampled from the lunar surface do not necessarily represent the lava that flowed through and formed the observed sinuous rilles. The composition of terrestrial ocean island basalt is considered because of similarities between predicted eruption fluxes and observed features associated with channels for both lunar sinuous rilles and Hawaiian eruptions of ocean island basalt lavas [e.g., Head and Wilson, 1980; Wilson and Head, 1980], and the composition of terrestrial komatiite is considered because komatiites are observed to have high liquidus and eruption temperatures, high Mg- and low Si-content, and thus a very low viscosity; lavas with a low viscosity are expected to be more efficient erosion agents [e.g., Huppert and Sparks, 1985].
 Results of the comparison between observed and predicted depths (Figure 7) indicate that the model does follow the trends in channel depth as observed using LOLA track data (Table 1) for each composition of lava considered. The gradual decreases in predicted eroded depth are a result of the decrease in erosion efficiency as lava temperature decreases and as lava contamination from assimilated substrate and thus lava viscosity increases with distance from the source vent. The sharp decreases in predicted eroded depth are artifacts due to the change in observed average channel slope incorporated into the model at the beginning of each channel segment. While in reality these slopes would vary more smoothly along the length of the channel, the predicted eroded depths adequately follow the trend in channel depths observed, supporting the earlier interpretation that thermal erosion is sufficiently simulating the formation of Rima Prinz.
 Model results also indicate that lava composition has a significant influence on the erosion rate and thus the eruption duration required to form the observed sinuous rille (Figure 8). Results are presented for scenarios in which the composition of the initial, uncontaminated lava is identical to the composition of the substrate; while the composition of the lava changes down gradient due to crystallization of olivine (which will occur as the lava temperature decreases below the liquidus temperature) and contamination by assimilated substrate, the substrate composition remains constant. The gradual decreases in predicted erosion rates mirror those in the results for predicted eroded depth and are similarly a result of the decrease in erosion efficiency as lava temperature decreases and lava viscosity increases with distance from the source vent. The steep decreases in erosion rate are artifacts of the abrupt change in slope used for each channel segment in the model, though again in reality the slope typically changes more gradually along the channel length.
 Results indicate that lava with a composition similar to that of a terrestrial komatiite will erode the substrate faster than lava similar to a lunar high-Ti basalt. Lavas with compositions similar to those of lunar low-Ti basalt and terrestrial ocean island basalt result in intermediate erosion rates. Lavas producing faster erosion rates require shorter eruption durations to incise the observed channel. Specifically, a komatiite-like lava, with erosion rates ranging from about 1.7 m d−1 at the head of the channel to 0.80 m d−1 at the channel terminus, requires about 157 Earth days (∼0.4 Earth years) to incise Rima Prinz. An ocean island basalt-like lava (OIB) has erosion rates ranging from about 0.7 m d−1 at the head of the channel to 0.4 m d−1 at the channel terminus, requiring approximately 360 Earth days (∼1 Earth year) to incise the observed channel. A low-Ti basalt has erosion rates ranging from about 0.6 m d−1 at the head of the channel to about 0.3 m d−1 at the channel terminus, requiring 435 Earth days (∼1.2 Earth years) to incise the observed channel. A high-Ti basalt has the lowest erosion rates ranging from 0.35 m d−1 at the channel head to 0.2 m d−1 at the channel terminus, and this lava requires approximately 770 Earth days (∼2.2 Earth years), to incise the observed channel. These times assume a constant eruption flux for the predicted duration of channel formation and thus represent an average time required for the formation of Rima Prinz, as eruption fluxes typically wax and wane over the course of a single eruption.
 Rima Prinz is a sinuous rille characterized by laterally continuous walls and a lack of marginal levees, and it is thus interpreted to be a lava channel that formed as the result of erosion into the lunar substrate. Analysis of analytical model results of erosion rate versus slope indicates that thermal erosion was more likely than mechanical erosion to have been the dominant process active during channel formation on the Moon (Figure 6). This dominance is due to the fact that the low gravity conditions characteristic of the Moon yield insignificant potential and thus kinetic energies in lava flowing on the lunar surface. Therefore, thermal energy contributed by hot lava flowing over a substrate is the dominant form of energy present in lava flowing over a gradually sloped lunar substrate, and thermal erosion rather than mechanical erosion was the dominant erosion regime present during lunar channel formation.
 These results are valid for lunar slopes less than about 3.5° and for a well-consolidated basalt substrate with a yield strength Y of 0.1 MPa [Schultz, 1993] and an erodibility b of 0.0005 ([Hurwitz et al., 2010] K = Y / b = 5 × 10−9 Pa−1). It should be noted that erosion rate is expected to increase in the presence of a less consolidated substrate such as the lunar regolith. Specifically for the case of lava composed similarly to that of an ocean island basalt flowing at the head of Rima Prinz (α = 0.7°; see Figure 6), mechanical erosion is expected to increase from about 1.0 m d−1 in the case of a consolidated basaltic substrate to about 5.0 m d−1 in the case of an unconsolidated regolith substrate (erodibility of 0.0025 [Hurwitz et al., 2010]). This change leads to the interpretation that mechanical erosion is more efficient than thermal erosion for slopes greater than ∼0.6°, as is the case at the head of Rima Prinz (thermal erosion rate = ∼4.0 m d−1 at α = 0.7°; see Figure 6). Therefore, mechanical erosion was likely to have been the dominant process in the initial formation of the channel when the lava is incising through regolith, but the dominant erosion regime shifted to thermal erosion once the regolith was removed and a more consolidated basaltic substrate was encountered. This initial period of more efficient mechanical erosion would be expected to decrease the duration of channel formation by about 6–10 Earth days, assuming a regolith thickness of 6–10 m [Fa and Jin, 2010; Kobayashi et al., 2010].
 Further analysis of model results indicates that lava composition has a significant effect on erosion efficiency. Specifically, komatiite lavas erode a similarly composed substrate more efficiently than lunar low-Ti basalts, ocean island basalts, and, most significantly, high-Ti basalts erode into respectively similarly composed substrates (Figure 8). The models used to simulate thermal erosion (equations (2)–(5)) suggest that the most significant parameter involved in determining erosion efficiency is the difference between the temperature of the flowing lava and the melting temperature of the substrate. In the scenarios explored in this study, the compositions of the initially erupted lava and the substrate are identical. Therefore, the lava is assumed to erupt at its corresponding liquidus temperature and the substrate is assumed to melt at its corresponding solidus temperature (Table 3). Lava with a composition similar to that of a terrestrial komatiite has the greatest difference between its corresponding liquidus and solidus temperatures (ΔT = 408°C; see Table 3), and lava with a composition similar to that of a lunar high-Ti basalt has the smallest difference between these two temperatures (ΔT = 188°C). These observations are consistent with larger differences between the liquidus and solidus temperatures resulting in a higher thermal energy contribution from the flowing lava and thus in higher erosion rates.
Erosion Rate 1 represents erosion at the head of Rima Prinz; Erosion Rate 2 represents erosion at the terminus of Rima Prinz.
Ocean Island Basalt
Low-Ti Basalt (12002)
High-Ti Basalt (74220)
 Lavas that erode more efficiently require shorter eruption durations and thus also require less lava to form the observed sinuous rille (Figure 8 and Table 3). Specifically, the fastest eroding komatiite lava requires 157 Earth days to form the channel, and, assuming a constant eruption flux of 4375 m3 s−1 [i.e., Wilson and Head, 1980; Head and Wilson, 1980] over the course of the eruption, the volume of lava that is expected to be released is 59 km3. This lava volume is similar to the volume of lava erupted in 43 flows over the course of 18 eruptions measured on Mauna Loa (25.8 km3 [Malin, 1980]) and about an order of magnitude greater than the volume of lava erupted in 44 flows over 15 eruptions measured on Kilauea (2.6 km3 [Malin, 1980]). In contrast, the slowest eroding high-Ti lunar basalt requires 766 Earth days to form the channel, and, assuming a constant eruption flux in this case of 4055 m3 s−1 over the course of the eruption, the volume of lava that is expected to be released is 268 km3. This lava volume is significantly larger than those observed in Hawaii, but this volume is still an order of magnitude less than the volume of lava that erupted in a single fissure eruption associated with the emplacement of the Columbia River flood basalts (>2,000 km3 [Hooper, 1997]). A cursory quantitative analysis of a lava composition similar to a Columbia River basalt [Murase and McBirney, 1970] indicates that this lava would be very inefficient at eroding a similarly composed substrate, supporting observations that thermal erosion does not occur in the Columbia River flood basalts [Greeley et al., 1998]. It should be noted that the lava volumes reported above represent the entire amount of lava required to form the observed sinuous rille, but that individual flows may have had smaller volumes if the channel formed over a series of events similar to those observed in Kilauea instead of as the result of a single eruptive event similar to the fissure event in the Columbia River flood basalt.
 The significant volume of lava required to form Rima Prinz must have been deposited beyond the terminus of the sinuous rille in Oceanus Procellarum. However, one of the more enigmatic characteristics of lunar sinuous rilles is the lack of observed deposits at and beyond the channel termini (Figure 3e). As suggested in section 3, these deposits may have been embayed by subsequent emplacement of mare flows that may have flowed back up the channel for 3–5 km from the currently observed channel terminus. Although the terminal deposits of Rima Prinz are not currently observed, LOLA topography data can be used to estimate the path that the lava may have taken once it left the confines of the channel (Figure 9). The main assumptions used in this analysis include, 1) the lava will always flow down-gradient, 2) on steep slopes (i.e., contours are closely spaced), lava will flow perpendicular to the contour, 3) on suddenly gradual slopes (i.e., contours are suddenly spaced far apart), lava will tend to flow along a contour until a steeper gradient is reached, 4) lava tends to flow around topographic boundaries and into basins, and 5) currently observed topography would have affected the path of the lava.
 An example of a possible lava deposit area is shown in Figure 9. The flow has a length of ∼75 km, a length that is approximately double that of the longer Kilauea flows that may have been truncated by encountering the ocean [Malin, 1980], and the flow has a surface area of approximately 2450 km2. The proposed deposit would have terminated in a basin northeast of Rima Prinz, partially but not completely filling the basin, as the basin is still observed. As previously discussed, model results indicate that a range of lava volumes (59 km3–268 km3) may have been required to form Rima Prinz, dependent on lava composition. These total lava volumes translate to average deposit depths of ∼25 m for a komatiite lava, ∼55 m for an ocean island basalt, ∼65 m for a low-Ti lunar basalt, and ∼110 m for a high-Ti lunar basalt over the proposed deposit area. Deposit depths are expected to decrease for flows deposited on steeper surfaces and increase for flows deposited on more gradually sloped surfaces or within basins. These depths indicate a minimum value for the thickness of the subsequent mare flows required to cover the deposits from Rima Prinz. The composition of the mare in the western Imbrium region has been identified as intermediate TiO2 basalts that lack evidence for extensive mixing [i.e., Zisk et al., 1977; Whitford-Stark and Head, 1980; Giguere et al., 2000; Hiesinger et al., 2003]. While the depth of the mare fill itself is dependent on the volume of lava erupted during the emplacement of these deposits, observations of irregular ‘shorelines’ along the edges of Oceanus Procellarum suggest that the mare is relatively shallow along the margins [i.e., Head, 1976]. Therefore, it might be expected that the mare lava would not be thick enough to completely embay high-Ti flows from Rima Prinz, and thus the lava that flowed through and incised Rima Prinz was most likely to have been composed similarly to lavas such as komatiites, ocean island basalts, or low-Ti lunar basalts.
 The formation of lunar sinuous rilles has long been considered an enigma, with proposed origin theories including a range of formation mechanisms: 1) origin via construction of levees that channelize flow within a cooling lava flood, 2) origin via either a constructed, structurally stable roof over a lava tube or an eroded subsurface lava tube, and 3) origin via erosion in a surface channel. Our observations of Rima Prinz, a sinuous rille located east of the Aristarchus Plateau, indicate that this channel has laterally continuous walls that lack marginal levees, supporting the theory of origin through erosion of the substrate by a surface channel. Detailed measurements of channel morphology were used as constraints in analytical models in order to determine whether mechanical or thermal erosion is the dominant process active during the formation of Rima Prinz. The most relevant model was then used to determine the detailed origin of Rima Prinz, including the lava compositions, erosion rates, and eruption durations required to produce the observed sinuous rille.
 The key interpretations of model results presented in this study indicate that the low gravity characteristic of the Moon contributes an insignificant amount of potential and thus kinetic energy to lava flowing on a surface of slopes less than ∼3.5°, allowing thermal energy and thus thermal erosion rather than mechanical erosion to dominate during the formation of a lunar sinuous rille on terrain with these slopes. Thermal erosion is expected to dominate during sinuous rille formation on the Moon even though it is not a commonly observed process in terrestrial lava channel formation, as terrestrial lava channels form under higher gravity conditions and typically on steeper gradients. Mechanical erosion remains an important process in the initial stages of lunar sinuous rille formation, however, as lava initially encounters the poorly consolidated lunar regolith that is much more susceptible to mechanical erosion than the more consolidated underlying basaltic substrate.
 Additional analysis of results from a model that includes lava composition as a model input [Williams et al., 1998, 2000] indicates that lava composition has a significant effect on erosion rates. In particular, assuming that lava erupts at its liquidus temperature and melts at its solidus temperature, and assuming that the erupted lava has the same composition as the substrate, the lava composition with the greatest difference between its solidus and liquidus temperatures will contribute the most thermal energy to melting the substrate and incising a channel. Further analysis of model results indicates that erosion efficiency increases with greater liquidus-solidus temperature differences: a lava with a composition similar to a terrestrial komatiite (ΔT = 408°C) results in the highest erosion rates (1.7 m d−1–0.8 m d−1) and a lava with a composition similar to a lunar high-Ti basalt (ΔT = 188°C) results in the lowest erosion rates (0.35 m d−1–0.2 m d−1). Lavas with compositions similar to terrestrial ocean island basalts (ΔT = 358°C) and lunar low-Ti basalts (ΔT = 290°C) result in intermediate erosion rates (0.7 m d−1–0.4 m d−1 and 0.6 m d−1–0.3 m d−1, respectively).
 As expected, greater erosion rates require less time and thus less lava to carve the observed channel. Results indicate that a komatiite-like lava would require 157 Earth days and 59 km3 of lava to form Rima Prinz, an ocean island basalt-like lava would require 362 Earth days and 137 km3 of lava, a lunar low-Ti basalt would require 435 Earth days and 157 km3 of lava, and a lunar high-Ti basalt would require 766 Earth days and 268 km3 of lava to form Rima Prinz. Although lava deposits are not currently observed at the terminus of Rima Prinz, a plausible deposit area of ∼2450 km2 is proposed based on LOLA topography data. The volumes of lavas listed above would thus result in komatiite-like lava deposit depths of ∼25 m, ocean island basalt deposit depths of 55 m, lunar low-Ti basalt deposit depths of ∼65 m, and lunar high-Ti basalt depths of ∼110 m. These deposits must have been completely embayed by subsequent mare flows of intermediate TiO2 content and relatively shallow depths in order to no longer be visible, suggesting that the most likely candidates for the composition of lava that formed Rima Prinz include komatiites, ocean island basalts, and lunar low-Ti basalts.
 Further work is needed to constrain the actual composition of the lava that formed lunar sinuous rilles. The general lack of deposits within and beyond the currently observed channel makes this a challenging task. However, the identification of possible remnant lava lake textures in the floor of the Rima Prinz source depression provide a possible starting point for further remote-sensing analyses as well as a desired destination for future ground-based studies. While there are certainly outstanding questions as to the detailed origin of some lunar sinuous rilles, it is clear from this study that thermal erosion may have played a significant role in the formation of these features and that the formation of lunar sinuous rilles represents significant events in the volcanic history of the Moon.
 We gratefully acknowledge David Williams for providing the code used extensively in the analysis presented in this paper as well as for giving a thorough and critical review of the paper. We also thank an additional anonymous reviewer for a thoughtful review. This research was supported financially by the National Aeronautics and Space Administration through grants NNX09AM54G and NNG07EK00C from the NASA Lunar Reconnaissance Orbiter project and the LRO Camera (LROC) and Lunar Orbiter Laser Altimeter (LOLA) experiments.