Journal of Geophysical Research: Planets

Viscoelastic evolution of lunar multiring basins

Authors


Abstract

[1] We investigate the evolution of multiring basins on the Moon. The orbital geophysical and geochemical data collected by Lunar Prospector allow three distinct groups of basins to be identified. Group 1 basins show well-preserved topography, a bull's-eye gravity signature, and little mare basalt fill. Group 2 basins contain large amounts of mare basalt and show mare-dominated gravity and topography signatures. Group 3 basins are characterized by low-amplitude topographic relief and weak gravity anomalies. While the crustal structures of group 1 and 2 basins appear to date from the collapse of the transient cavity, group 3 basins are highly degraded and show very little crustal thinning. Using a viscoelastic model, we explore the evolution of the lunar multiring basins, with the goal of explaining the degraded condition of group 3 basins. Our results show that while viscous relaxation is a plausible explanation for the state of group 3 farside basins, requiring heat flux > 25–40 mW m−2, the thin nearside crust renders subsolidus ductile flow in the crust unlikely. The existence of many degraded basins on the nearside points to a weakened, and possibly partially molten, lower crust. That South Pole–Aitken appears not to have relaxed suggests that similar conditions did not exist there and that the nearside was hotter than the farside early on. This is likely a consequence of the nearside-farside crustal thickness dichotomy, which would have caused the farside to reach full crystallization first, leaving the nearside with the remaining liquid and an excess of heat-producing elements.

1. Introduction

[2] Multiring impact basins are the principal topographic features on the Moon and the most common source of large-scale gravity and compositional anomalies [e.g., Spudis et al., 1994; Zuber et al., 1994]. It was discovered early on that several are associated with positive gravity anomalies; these became known as the “mascon” basins [Muller and Sjogren, 1968]. Because many of the lunar basins appear to date from the earliest lunar history, they provide important clues for understanding the early evolution of the Moon. In particular, the mechanical state of the basins may yield important constraints on the rheological structure of the Moon at the times of formation of the basins and thus on the thermal evolution of the Moon.

[3] It is now generally accepted that the lunar crust and upper mantle crystallized from a global magmasphere [e.g., Shearer and Papike, 1999; Warren, 1985]. One-dimensional models of lunar evolution and isotopic constraints suggest that most of the magmasphere solidified within 100–200 Myr and the residual liquid up to several hundred million years later [Solomon and Longhi, 1977; Nyquist and Shih, 1992]. However, recent global mapping of the Moon by the Clementine and Lunar Prospector spacecraft has shown that lunar evolution was more complex than these 1-D models suggest. Global maps of gravity, topography and elemental abundances have established the existence of a profound asymmetry in crustal thickness, volcanism, and abundance of radioactive elements [e.g., Joliff et al., 2000].

[4] The asymmetry shows that different regions of the Moon must have undergone very different thermal and mechanical evolutions. In particular, the lunar crust can be divided into three regions: the Procellarum KREEP Terrane (PKT), a region resurfaced by mare basalt and rich in incompatible elements; the Feldspathic Highlands Terrane (FHT), a region characterized by thick anorthositic crust and low iron abundance; and the South Pole–Aitken Terrane (SPAT), a mafic region created by a giant impact [Joliff et al., 2000]. Thermal models of the Procellarum KREEP Terrane (PKT), which appears to contain a large fraction of the Moon's inventory of heat-producing elements, show that it may have remained anomalously hot (and perhaps partially molten) for billions of years following the crystallization of the magma ocean [Wieczorek and Phillips, 2000].

[5] Few lunar basins have been accurately dated, and the ages of the oldest basins are almost completely unconstrained. Most authors agree on an age of ∼3.85 Ga for Imbrium [e.g., Nyquist and Shih, 1992; Hartmann et al., 2000]. Serenitatis and Crisium are inferred to have formed ∼3.89 Ga, with Orientale forming ∼3.82 Ga. The age of Nectaris basin was initially believed to be ∼3.92 Ga [e.g., Nyquist and Shih, 1992], but reinterpretation of the data in the light of Lunar Prospector data suggests that an age of 4.1 Ga is at least equally valid [Korotev et al., 2002; Warren, 2003]. Eleven or twelve other basins appear to date from the period between these two [Wilhelms, 1987; Stöffler and Ryder, 2001], the Nectarian age. No accurate dates exist for pre-Nectarian basin-forming impacts, although Wilhelms [1987] speculated that South Pole–Aitken and Procellarum basins formed between 4.1–4.2 Ga and other ancient basins after 4.1 Ga.

[6] Under the assumption that the young Nectaris age is correct, the clustering of up to 14 basins in a ∼100 Myr range gave rise to the hypothesis that this was a period of intense bombardment known as the “terminal cataclysm” or late heavy bombardment (LHB). Indeed, recent dynamical models of early solar system evolution have shown that such an event may have been the result of catastrophic destabilization of the asteroid belt, triggered by the migration of the orbits of giant planets early on [e.g., Gomes et al., 2005]. Even if Nectaris did form at 4.1 Ga, the evidence still supports the LHB hypothesis, but the number of basins that formed during the LHB would be uncertain.

[7] High-resolution gravity data from the Lunar Prospector mission has allowed decent modeling of the crustal structure of several nearside lunar basins. Wieczorek and Phillips [1999], taking into account estimates of mare fill, reconstructed the crustal structure and excavation cavities of 11 basins. Their models showed that the ratio of excavation depth (dex) to diameter is consistent with proportional scaling [Croft, 1980] for all but the three largest basins. The authors also showed that many of the mascon basins were out of isostatic equilibrium prior to mare flooding. Concurring with Neumann et al. [1996], they proposed that the mantle rebounded to a superisostatic position while the target was fluidized and froze in place once strength returned.

[8] However, the reconstructed dex of Imbrium, Serenitatis, and South Pole–Aitken (SPA) were found to be anomalously shallow. SPA appears to be in isostatic equilibrium at the present time, while Imbrium and Serenitatis may have been close to isostasy prior to mare filling [Wieczorek and Phillips, 1999]. Because of the unique nature of the PKT, it has been suggested that certain basins in that vicinity (i.e., Imbrium and Serenitatis) may have undergone viscous relaxation [Wieczorek and Phillips, 1999]. On the other hand, these three basins are the largest on the Moon and proportional scaling arguments predict that the impacts must have excavated the entire crust.

[9] Many other ancient impact basins are highly degraded and have very low topographic relief. As a result, these were not modeled by Wieczorek and Phillips [1999]. The characteristics of these basins are consistent with either (1) relaxation of topographic relief by ductile flow on a geologic timescale [e.g., Solomon et al., 1982] or (2) enhanced crater collapse or very rapid relaxation immediately following impact, as has been suggested for craters on icy bodies [Schenk, 2002; Turtle and Ivanov, 2002]. Both scenarios require that the basins formed early in lunar history, when the Moon was hot and the lower crust more ductile.

[10] In this paper, we investigate the evolution of multiring impact basins on the early Moon. First, we identify the main characteristics of lunar basins using orbital geophysical data and classify them according to their inferred mechanical evolution. Using a semianalytic model, we explore the viscoelastic evolution of these basins under conditions appropriate for the early Moon. We determine the conditions under which viscous flow can occur and whether it is a plausible explanation for the appearance of the oldest basins. Finally, we discuss the implications of the results for the thermal evolution of the Moon.

2. Lunar Basins

[11] In an attempt to impose qualitative constraints on the evolution of the major lunar basins, we here classify them on the basis of their gravity, topography, and geology. These characteristics will later be used in conjunction with the viscoelastic relaxation model to shed light on the early thermal evolution of the Moon. Table 1 shows the basins considered in this study, and they are identified on a global map in Figure 1a. The relative ages are assigned numbers following Halekas et al. [2003], on the basis of the classification of Wilhelms [1987]. In this system, the letters I, N, and P denote the Imbrian, Nectarian, and pre-Nectarian ages, respectively, and the numbers increase with absolute age (from 1 to 14). The minimum crustal thickness (Dmin) is determined from an updated (M. Wieczorek, personal communication) crustal thickness inversion of Wieczorek and Phillips [1998], which employs a thinner crustal thickness constraint (31 km) [Lognonné et al., 2003; Chenet et al., 2006] at the Apollo 12/14 sites. A global map of the spherical harmonic crustal thickness model (expanded up to l = 65) is shown in Figure 1c.

Figure 1.

Global maps of (a) topography, (b) free-air gravity, and (c) crustal thickness. The location, name, and group assignment of each basin are indicated in Figure 1a. See text for details.

Table 1. Properties of the Major Lunar Basins
BasinCentera,bRegioncDiameter, kmReliefdDmin,e kmAgefGroup
  • a

    Neumann et al. [1996].

  • b

    Spudis [1993].

  • c

    The basins are assigned to the following regions: farside, nearside, limb, and PKT.

  • d

    Topographic relief is characterized as “deep” (≥2 km) or “shallow” (<2 km).

  • e

    Dmin is the minimum crustal thickness in the central part of the basin.

  • f

    Relative ages are assigned following Halekas et al. [2003]: I, Imbrian; N, Nectarian; and P, pre-Nectarian. The numbers increase with age.

  • g

    Wilhelms [1987].

South Pole-Aitken56°S, 180°ESPA2500adeep6P-141
Imbrium36°N, 19°WPKT1160ashallow13I-32
Crisium17°N, 58.5°Enear1060a (740b)deep3N-42
Fecunditatis4°S, 52°Enear990ashallow21P-133
Orientale19.5°S, 94°Wlimb930adeep8I-11
Australe51.5°S, 94.5°Elimb880ashallow35P-133
Nectaris16°S, 34°Enear860adeep6N-62
Smythii2.5°S, 86.5°Elimb840a (740b)deep7P-112
Humorum24°S, 39.5°WPKT820 (425b)deep9N-42
Tranquilitatis7°N, 40°Enear800 (700b)shallow35P-133
Serenitatis26°N, 19°Enear740 (920b)shallow9N-42
Mutus-Vlacq52°S, 21°Enear700bshallow29P-133
NW Procellarum40°N, 62.5°WPKT700ashallow25P3
Tsiolkovsky-Stark15°S, 128°Efar700ashallow50P-143
Nubium21°S, 15°WPKT690ashallow24P-133
Mendel-Rydberg50°S, 94°Wlimb630a (420b)deep20N-61
Lomonosov-Fleming19°N, 105°Efar620gshallow33P-133
Humboldtianum58°N, 84°Elimb600a (650b)deep8N-41
Freundlich-Sharonov18.5°N, 175°Efar600adeep21P-81
Hertzsprung2.5°N, 128°Wfar570adeep49N-41
Coulomb-Sarton52°N, 123°Wfar530a (440b)deep30P-111
Balmer15°S, 70°Enear500bshallow28P(≥11)3
Keeler-Heaviside10°S, 162°Efar500bshallow45P-123
Korolev4.5°S, 157°Wfar440adeep58N-61
Moscoviense26°N, 147°Efar440a (420b)deep17N-61
Grimaldi5.5°S, 68.5°WPKT-adj430a (440b)deep27P-72

[12] Figure 1a shows a map of the gridded lunar topography data used in this study, obtained by Clementine LIDAR [Smith et al., 1997]. Free-air gravity data (Figure 1b) are taken from the JGL100K1 spherical harmonic model (up to l = 100) derived from line-of-sight tracking of Lunar Prospector and previous spacecraft [Konopliv et al., 2001]. Because of the lack of direct tracking of the spacecraft on the farside, the gravity data are poorly resolved for that hemisphere, producing uncertainties as great as 200 mGal in the central farside [Konopliv et al., 2001]. Nevertheless, the improved quality of the post-Prospector models has permitted several new farside mascons to be identified. The data sets were obtained from the Planetary Data System (http://pds-geosciences.wustl.edu/).

[13] We divide the major lunar basins into three classes based on gravity, topography, and geology. Figure 2 shows their relationship on a plot of free-air gravity versus depth. The depths are measured from rim crest to floor [from Williams and Zuber, 1998] and the free-air amplitude is measured relative to the regional value. The three groupings are indicated on the plot and we propose that they reflect three distinct evolutionary paths. Groups 1 and 2 form a continuum between end-members of unfilled and completely filled basins, and thus the boundary is somewhat arbitrary. Between them, they include all of the mascon basins in the size range considered here. Despite the large uncertainty in the gravity anomalies of farside basins, Konopliv et al. [2001] have identified the group 1 farside basins on the plot as likely mascons, though poorly resolved ones. The grouping itself should not be significantly affected by the large uncertainty, as it relies on several criteria. Graphs of topography and free-air gravity are shown for typical examples of basins from each group in Figures 3, 4, and 5.

Figure 2.

Plot of free-air anomaly amplitude versus depth. Group 1, 2, and 3 basins are represented by diamonds, circles, and triangles, respectively. Open symbols are farside basins, and filled symbols are nearside basins.

Figure 3.

Azimuthally averaged topography and free-air anomaly for three group 1 basins: (a) Orientale, (b) Mendel-Rydberg, and (c) Freundlich-Sharonov.

Figure 4.

Group 2 basins, labeled as in Figure 3: (a) Crisium, (b) Nectaris, and (c) Humorum.

Figure 5.

Group 3 basins, labeled as in Figure 3: (a) Keeler-Heaviside, (b) Lomonosov-Fleming, and (c) Balmer.

[14] Group 1 basins, including Orientale, Mendel-Rydberg, and Freundlich-Sharanov, are the best preserved. They are characterized by great depth, intermediate-amplitude free-air anomaly (<200 mGal), and little or no mare basalt fill. Their free-air gravity signature (shown in Figure 3) consists of a large positive anomaly in the center, surrounded by a negative gravity ring in the outer part of the basin and a positive ring beyond that. The negative free-air rings appear to stem from subsurface density contrasts that are not well correlated with surface topography. The crustal structure, as estimated from models, is characterized by a superisostatic Moho uplift in the center, surrounded by a ring of thickened crust [Neumann et al., 1996; Wieczorek and Phillips, 1999]. Similarities between the crustal structure of these basins and the tectonic structure of the terrestrial Chicxulub basin [see Morgan et al., 2000], along with basin formation models [e.g., Collins et al., 2002], suggest that the crustal structures of group 1 basins have been preserved since the collapse of their transient cavities. This is consistent with the findings of Neumann et al. [1996] and Wieczorek and Phillips [1999].

[15] Many of the large nearside basins, including Crisium, Nectaris, and Humorum, have been flooded with mare basalt. Group 2 encompasses those basins with thick mare basalt fill, whose gravity and topographic expressions appear to be dominated by the fill. The basins show flat, shallow floors and high-amplitude positive free-air anomalies in the center, as illustrated in Figures 2 and 4. Although their current gravity and topography are very different from group 1 basins, crustal thickness models show that the group 2 basins (with the exception of Imbrium and Serenitatis) have similar crustal structures to group 1 basins [Neumann et al., 1996; Wieczorek and Phillips, 1999]. Those in or near the PKT appear to have been closer to isostasy prior to mare filling than those that formed outside [Wieczorek and Phillips, 1999].

[16] Group 3 basins show the least topographic relief and low-amplitude free-air gravity anomalies. The typical example of this class is Tranquilitatis, a pre-Nectarian multiring basin whose topographic signature has been almost completely erased [Wilhelms and McCauley, 1971; Spudis, 1993]. Group 3 basins are listed in Table 1 and a few are shown in Figure 5. Of these, Australe, Fecunditatis, Balmer, Keeler-Heaviside (K-H), Tsiolkovsky-Stark (T-S) and Mutus-Vlacq (M-V) have been identified as true multiring basins [Pike and Spudis, 1987; Wilhelms, 1987; Spudis, 1993; Spudis et al., 1994]. Lomonosov-Fleming (L-F) and Nubium are believed to be also, although multiple rings have not been identified [Spudis, 1993]. The existence of a basin in NW Procellarum was proposed by McEwen et al. [1994] and has since been confirmed by Clementine topographic data [Zuber et al., 1994] and Lunar Prospector gamma ray data [Gasnault et al., 2002].

[17] South Pole–Aitken (SPA) basin does not fall easily into any of these categories. It is characterized by great depth, little mare fill, and gravity anomalies consistent with isostatic compensation [Zuber et al., 1994]. However, similarities between SPA and the young Imbrium and Serenitatis basins, the only others >1000 km in diameter, lead us to believe that postimpact isostasy (or near isostasy) may be a feature of basins in this size range. The crustal thickness models of Wieczorek and Phillips [1999] suggest that (1) prior to mare filling, Imbrium and Serenitatis were, like SPA, near isostasy, and (2) all three giant basins are characterized by anomalously shallow reconstructed excavation cavities for their size. The improved crustal thickness model used in this paper also shows that (3) these basins are underlain by very thin crust. We interpret these three observations to indicate that all three basins excavated the entire crust (as would be suggested by proportional scaling arguments) and subsequently reached a near-isostatic state. If we can attribute SPA's state of compensation to its size, it appears to fit best in group 1, because of its well-preserved topography and dearth of mare fill.

3. Model

[18] We model the relaxation of lunar impact basins with the goal of learning the conditions under which viscous flow would have been an important process in basin evolution, particularly for group 3 basins. We introduce a semianalytic, self-gravitating viscoelastic relaxation model based on the purely viscous models of Grimm and Solomon [1988], Ricard et al. [1984], and Richards and Hager [1984], as well as the viscoelastic model of Zhong and Zuber [2000]. The governing equations are [e.g., Zhong and Zuber, 2000]

equation image
equation image
equation image

where ν is the velocity, σ is the stress tensor, ρ is the density, Δρ is the density anomaly, ϕ is the gravitational potential due to the density anomaly, G is the gravitational constant, and δir is the Kronecker delta. For a viscoelastic body, or Maxwell solid, stress and strain (ɛ) are related by

equation image

where η is the viscosity, μ is the shear modulus, and P is the pressure. As shown by Zhong and Zuber [2000], performing a Laplace transform on equations (1)(4) reduces it to a viscous problem in the Laplace domain. Equations (1)(3) remain unchanged, and equation (4) becomes

equation image

with p = P/(1 + M), ηs = η/(1 + M), τM = η/μ is the Maxwell time, and s is the Laplace variable.

[19] In order to approximate temperature-dependent viscosity, the domain is divided into several constant viscosity layers. We use a semianalytic method to solve the equations in the Laplace domain (see Appendix A for details), as opposed to the analytic propagator matrix method used by Zhong and Zuber [2000]. Once the Laplace domain solution is obtained, an inverse Laplace transform produces the solution, of the form

equation image

where hi is the topography at interface i (taken to be s or m for surface or Moho topography, respectively), M is the number of relaxation modes, Rsj is the residue and τj the relaxation time of the jth mode, and t is time. One mode is introduced for each density interface, and two for each viscosity interface. Results from our model are in good agreement with that of Zhong and Zuber [2000]. Following these authors and Zhong [2002], we also benchmarked the model against the lithospheric flexure model of Willemann and Turcotte [1981], using high- and low-viscosity layers to represent the lithosphere and asthenosphere, respectively.

[20] The stress produced by topography at a density interface is applied as a discontinuity at that interface. This formulation assumes that the topography at the boundaries of a given layer is small with respect to the thickness of the layer, an assumption which is clearly violated for large impact basins. The use of linearized boundary conditions here introduces three distinct sources of error: (1) the stresses generated away from the interfaces are neglected, (2) the reduced thickness of the crustal channel in the center of the basins is neglected, and (3) the low temperature at the base of the thinned crust is ignored.

[21] The first of these is significant in any investigation where stress is produced by finite (as opposed to infinitesimal) amplitude topography. Nunes et al. [2004] compared the results of an analytic model with linearized boundary conditions to a finite element model in their investigation of relaxation of crustal plateaus on Venus. They found that while linearization introduces a small error in the relaxation time, the overall flow field is almost identical. They observed that the two models produce identical results if the viscosity in the analytic model is reduced by ∼30% with respect to the numerical model.

[22] The second and third occur because impact basins are zones of crustal thinning. The large diameter of these basins with respect to the thickness of the crust ensures that any flow in the crust will be in the channel flow regime [e.g., Zhong, 1997; McKenzie et al., 2000; Nunes et al., 2004]. As a result, the timescale of flow will be sensitive to the thickness of the crust. Similarly, the limiting viscosity of flow (i.e., that at the base of the crust) will be substantially higher where the crust is thin.

[23] Clearly, these sources of inaccuracy cannot be eliminated completely without using a fully numerical approach, but the effects can be mitigated. We consider the third source to be the principal obstacle, as the error introduced by the first is small relative to the inherent uncertainties in the viscosity and the second is also likely small because of the area of the thinned region relative to the overall scale of the flow. In addition, these two effects have opposite sign and will offset to some degree. The thin crust and uplifted mantle beneath the floors of the basins, on the other hand, will prevent flow altogether if sufficiently cold. Accordingly, we set the viscosity of the region extending from the depth of maximum Moho uplift to the base of the background crust equal to that at the top of this interval.

[24] The effective viscosity of each layer is calculated using the following relationships:

equation image
equation image

where equation image is the strain rate, A is the creep parameter, σ is the average driving stress, n is the stress exponent, Q is the activation energy, R is the gas constant, and T is the temperature. The 3 in the denominator on the right side of equation (8) is necessary (instead of 2) because the rheological constants were determined in triaxial experiments [e.g., Ranalli, 1995; Nunes et al., 2004; Albert et al., 2000].

[25] For the purpose of calculating the viscosity, we assume σ = 30 MPa, a typical value for the pre-mare stress state of mascon basins [Wieczorek and Phillips, 1999]. Where the driving stress exceeds the yield stress [Byerlee, 1978], we use Byerlee's rule to determine a maximum “effective viscosity” [e.g., Nunes et al., 2004], as shown in Figure 6c. The rheology of rocks is strongly dependent on plagioclase content [Mackwell et al., 1998], so we use dry, high-plagioclase (∼70 vol%) Columbia diabase for the crust. Dry olivine is used for the mantle rheology [Karato and Wu, 1993]. The relevant parameters are shown in Table 2.

Figure 6.

Plots of (a) temperature, (b) strength envelope (for equation image = 10−15 and 10−17 s−1, indicated by black and gray lines, respectively), and (c) viscosity (for σ = 30 MPa) with depth at 4 Ga for Tb = 1350 K and various crustal thicknesses (indicated by thin horizontal lines). Crustal thicknesses of 30, 45, and 60 km correspond to heat fluxes of 41, 33, and 29 mW m−2, respectively, as indicated on each plot. A viscosity discontinuity occurs at ∼15 km because the driving stress exceeds the yield strength above that level.

Table 2. Rheological Parameters
 A, Pa−n s−1nQ, kJ mol−1
Columbia diabasea1.2 × 10−264.7485
Dry olivineb2.4 × 10−163.5540

[26] We approximate the thermal structure of the Moon using simple three-parameter models in order to minimize the number of free parameters. First, we assume a “surface” temperature of 250 K, which is the temperature beneath the lunar regolith (at ∼100 m depth) [Langseth et al., 1976]. Following Warren and Rasmussen [1987], we assume that there is a megaregolith extending to a depth of ∼2 km, with a porosity of ∼17%, a thermal conductivity (kMR) of 0.2 W m−1 K−1, and a density (ρMR) of 2300 kg m−3. The underlying bedrock (crust) has kBR = 1.5 W m−1 K−1 and the mantle km = 3 W m−1 K−1 [Warren et al., 1991]. We parameterize the models using the temperature at the base of the crust (Tb), average crustal thickness (D), and the radiogenic heat production in the crust (HC). The concentrations of radioactive elements adopted for the feldspathic crust, the lunar mantle (HM), and incompatible-rich LKFM rocks (HLKFM) are shown in Table 3; these values are extrapolated to 4 Ga for the model runs. LKFM is the likely composition of the crust ejected by the Imbrium impact [Korotev, 2000]. HM is not varied, as its value does not have a significant effect on the results. It is assumed that the rate of heat production in the megaregolith is equal to that of the regular feldspathic crust. The steady state temperature in the megaregolith can be expressed as [Turcotte and Schubert, 1982]

equation image

where z is the depth, zMR is the base of the megaregolith, q0 is the heat flux at the surface, and ρMR is the density of the megaregolith. Below, the temperature is calculated as

equation image
equation image

where qb = q0ρMRHczMRρcHc (DzMR) is the heat flux at the base of the crust and

equation image

[27] Table 4 lists the parameters used in the following calculations, while Figure 6 shows temperature profiles, strength envelopes and viscosity structures corresponding to the final stage of the magma ocean (Tb ∼ 1300–1350 K) [Shirley and Wasson, 1981; Hess and Parmentier, 1995], demonstrating the presence of a low-viscosity lower crust following the crystallization of the magma ocean. This temperature represents the maximum lower crustal temperature for which it is reasonable to assume that the crust is solid. As the rheological models on which we rely are only valid for solid materials, we cannot model relaxation under supersolidus conditions. The minimum and maximum viscosities are chosen so that the corresponding Maxwell times are much less and much greater, respectively, than geologic timescales [Zhong and Zuber, 2000]. For simplicity, we use six-layer models (12 modes) in all runs, which should produce reliable results [Vermeersen et al., 1996]. A core radius of 350 km is used [e.g., Williams et al., 2001; Khan et al., 2004].

Table 3. Radioactive Element Abundances of Different Zones
 Feldspathic Crust,a ppmLKFM,b ppmMantle,c
U0.142.246.8 ppb
Th0.538.225 ppb
K480380017 ppm
Table 4. Model Parameters
ParameterVariableValueUnits
Shear modulusμ5 × 1010Pa
Megaregolith densityρMR2300kg m−3
Crustal densityρc2800kg m−3
Mantle densityρm3350kg m−3
Megaregolith thermal conductivitykMR0.2W m−1 K−1
Crustal thermal conductivitykc1.5W m−1 K−1
Mantle thermal conductivitykm3W m−1 K−1
Surface temperatureTs250K
Gravitational accelerationg1.62m s−2
Maximum viscosityηmax1029Pa s
Minimum viscosityηmin1019Pa s
Planetary radiusR1740km
Core radiusRc350km

4. Results

4.1. Uniform Compensation

[28] In order to better understand the physical processes involved, we first explore the response of a viscoelastic Moon to initially uniformly compensated (i.e., wavelength-independent compensation) topography. We define the degree of compensation:

equation image

[29] Our ultimate goal is to test the hypothesis that the group 3 basins were produced by viscoelastic relaxation of group 1 basins. Though group 1 basins show strong variations in compensation, both spatially and spectrally, their primary characteristic is the superisostatic central Moho uplift. Accordingly, we assume an initial value of C = 2, comparable to that in the center of the well-preserved Mendel-Ryberg and Orientale basins. Of the 12 viscoelastic modes, three distinct types can be discerned: lithospheric modes, where surface and Moho deflect together to compensate the load; ductile modes, where surface and Moho relief are both reduced; and upper crustal modes, where only topographic relief relaxes. The lithospheric modes have short relaxation times τ ∼103 years and represent flexural adjustment of the lithosphere to the load. Ductile modes represent relaxation of surface and Moho relief on a timescale that is strongly dependent on the viscosity structure of the crust and upper mantle. Upper crustal modes have the longest relaxation times and represent the relaxation of the topographic component that is supported by the elastic strength of the upper crust.

[30] Ductile relaxation of Moho and surface topography is generally dominated by a single mode (or cluster of modes), with a relaxation time τD. Though the total number of modes is dependent on the number of layers used, the clustering of the ductile modes is not. Figure 7a shows the wavelength dependence of τD for three thermal profiles. The three sets of (Tb, D) chosen delineate the boundary between relaxation and preservation of surface topography. As a result, the relaxation times are comparable for the three cases. The monotonic decrease of τD as harmonic degree increases and the length scale of flow decreases is a consequence of the large ratio of wavelength to crustal thickness, demonstrating that channel flow is indeed the dominant style of flow. Once the wavelength drops below a few times the crustal thickness, τD begins to increase again [Zhong, 1997; McKenzie et al., 2000; Nunes et al., 2004]. The kink in the D = 30 km curve at l = 16–17 is essentially an artifact resulting from the fact that τD represents what is, at shorter wavelengths, a cluster of modes.

Figure 7.

(a) Relaxation time of the first ductile mode for three thermal profiles. (b) Fraction of topography with relaxation time >109 years under the same conditions. The remaining topography is normalized with respect to the initial value.

[31] Because Moho topography declines monotonically to near-zero relief (unlike surface topography), the amplitude of Moho relief of a basin should be a good indicator of the degree of relaxation that the basin has undergone. None of the group 3 basins have a substantial Moho uplift, and they are therefore consistent with complete relaxation. Of the group 1 basins, Hertzsprung, C-S, and Korolev appear to have substantially lower Moho relief than other group 1 basins of comparable size, as indicated by the low mascon gravity anomaly. However, as they are located in the central farside—where uncertainty in the gravity data may be up to ∼200 mGal [Konopliv et al., 2001]—no conclusions can be reached regarding the relaxation of these basins until better observations of farside gravity are available.

[32] A fraction of the topography is supported by the elastic lithosphere and relaxes on a timescale controlled by the Maxwell time of the upper crust. The fraction retained increases with increasing lithospheric thickness and can be used to constrain the thermal gradient during relaxation. Figure 7b shows the fraction of topography with relaxation time greater than 1 Gyr (which does not relax during geologic time) as a function of harmonic degree. Supported topography decreases with decreasing harmonic degree until membrane support begins to dominate, producing a minimum at l ∼ 6. The sharp upturn in the D = 60 km curve at l < 3 occurs because τD > 1 Gyr at such long wavelengths; it is not primarily due to elastic support. From the above results, one would expect relaxed basins to have negligible Moho topography and low-amplitude surface topography dominated by short wavelengths (with a small contribution from the longest wavelengths).

[33] Figure 8 shows the evolution of degree of compensation and surface topography over time under conditions sufficient to allow relaxation of Orientale-sized (and smaller) basins on a timescale of ∼108 year, with background crustal thicknesses of 30, 45, and 60 km. On short timescales, the lithospheric response produces rapid subsidence because of the overcompensated topography, particularly at long wavelengths. This response is identical to the flexure of an elastic shell of appropriate thickness. At longer timescales, ductile relaxation of surface and Moho topography becomes important. The short-wavelength relief at both interfaces relaxes relatively quickly, because τD is low, reducing the degree of compensation at this scale to zero (8a and 8c). The longer wavelengths relax more slowly, as expected from Figure 7a. Increasing the crustal thickness facilitates ductile flow, thereby decreasing the minimum basal temperature required for relaxation. While Moho relief relaxes “completely” (that is, until the driving stress becomes too low to sustain the flow), the surface topography relaxes until it reaches a level which can be supported by the upper crust (see Figure 7b).

Figure 8.

From an initial degree of compensation of 2, shown are: (a, c, and e) compensation at various times for D = 30, 45, and 60 km, respectively, and Tb = 1400, 1300, and 1200, respectively; (g) compensation for H = 0.5HLKFM (see text for details), D = 20 km, and Tb = 1200 K; and (b, d, f, and h) corresponding topography at various times (normalized to the initial topography). The numbers denote time elapsed in Myr. In Figure 8g, 50+ indicates that compensation does not change after 50 Myr.

[34] Figure 8 also demonstrates the decrease in the amplitude of subsidence (due to overcompensated topography) and increase in the remaining topography as lithospheric thickness increases (heat flux decreases). This is most obvious by comparing Figures 8b and 8f. In the latter case, the thick lithosphere obstructs surface deflection, both initially and in response to decaying Moho topography, while the surface moves more freely in the former. This underscores the important point that lateral flow occurs only in the ductile lower crust; the surface topography responds to the extent permitted by the lithosphere.

[35] We investigate the effect of increased crustal radiogenic heat production by increasing HC to 0.5 HLKFM. Though much of the PKT may have abundances of heat-producing elements approaching that of LKFM, we use half of that value to be conservative. The increase in radiogenic heat production results in an increased thermal gradient in the megaregolith and a decreased gradient in the crust below, thus increasing the thickness of the ductile channel and decreasing the lithospheric thickness (for a given Tb). Figures 8g and 8h show the evolution of compensation and topography for D = 20 km, Tb = 1200 K and H = 0.5HLKFM. In this case, the lithosphere is almost nonexistent and relaxation proceeds isostatically, leaving no topography.

4.2. Basin Relaxation

[36] Because of its size, the outer parts of the Moon must have cooled rapidly following the crystallization of the global magma ocean. This is consistent with the well-preserved state of late pre-Nectarian and younger basins, as well as estimates of the lithospheric thickness [e.g., Solomon and Head, 1980; Arkani-Hamed, 1998]. As several of these basins have been shown by radiometric dating to have formed after 4 Ga, successful basin relaxation would have to take place on a timescale of <500 Myr. Using this criterion, we explore the evolution of multiring basins on a hot Moon, assuming that group 3 basins initially resembled group 1 basins of comparable size. We use M-R, Orientale, and SPA to represent the initial condition. We expand the surface and Moho topography of each basin in spherical harmonics, the latter derived from the updated crustal thickness model (shown in Figure 1c). Figure 9 shows the viscosity layers used when D = 45 km, Tb = 1350 K.

Figure 9.

Crustal structure of Mendel-Rydberg basin; gray lines indicate the boundaries of the viscosity layers for D = 45 km and Tb = 1350 K (q0 = 35 mW m−2).

[37] Table 5 shows the heat flux—calculated from equation (12)—required to produce timely basin relaxation as a function of basin size and D. Because of the uncertainty in the activation energy, we investigate the effect of ± 20% change in Q (for the crust). We find that the effect on Tb is about ± 100–150 K (± 5 mW m−2 for q0). Figure 10 shows the evolution over time of a basin initially similar to M-R, for D = 45 km and Tb = 1350 K (q0 = 35 mW m−2). Relaxation takes place over ∼100–500 Myr, leaving only short-wavelength topography with amplitude ∼0.5–0.75 km.

Figure 10.

Evolution of (a) surface topography, (b) Moho relief, and (c) free-air anomaly of Mendel-Rydberg over time for D = 45 km and Tb = 1350 K. The current topography and free-air gravity of Lomonosov-Fleming and the topography of Keeler-Heaviside are shown for comparison. The free-air gravity of the latter basin is not shown because of the low resolution of the gravity data over the central farside.

Table 5. Heat Flux Required to Allow Relaxation as Function of Basin Size and Da
 M-ROriSPA
  • a

    Units are in mW m−2.

  • b

    Subsolidus relaxation does not occur; the heat flux listed is that which results when the base of the crust is at the magma ocean solidus.

D = 60 km253035
D = 45 km3540b40b
D = 30 km50b50b50b

[38] Figure 10 also compares the model topography and free-air gravity with those of two group 3 basins, K-H and L-F, at the present day. The topography of K-H shows marked similarities in amplitude and wavelength to the 500 Myr (completely relaxed) model curve, while L-F shows more subdued surface topography. K-H is located in the central farside, while L-F is on the eastern limb of the Moon. As a result, the free-air anomaly of K-H is not shown. The gravity signature of L-F is essentially identical to the model curves, though lower in amplitude. The gravity and topography signatures of these two basins are similar to those expected of relaxed basins. In addition, the subdued topography and free-air anomaly of L-F suggest a higher thermal gradient during its relaxation than K-H, which is consistent with its older age, as indicated in Table 1.

[39] As seen in Table 5, viscous relaxation of nearside basins, where D < 30 km, cannot occur if the temperature of the lower crust is below the magma ocean solidus. This is because impact-induced crustal thinning obstructs flow in the lower crust. With excavation depths of 20–30 km [e.g., Wieczorek and Phillips, 1999], subsolidus lower crustal flow is unachievable if the preimpact crustal thickness is ≤30 km. As a result, relaxation of nearside basins by subsolidus flow would have been very unlikely.

[40] A possible exception to this is the PKT region, where extremely high concentrations of radioactive elements may have allowed low viscosities to exist at shallow depth. As illustrated by Figures 8g and 8h, subsolidus relaxation may be possible under such conditions, if a sufficient thickness of unexcavated crust remains. We have not modeled the relaxation of a basin under PKT conditions, as lateral temperature variations and partial melt would likely be important factors, but we cannot rule out the possibility. It is clear that nearside group 3 basins outside of the PKT have been modified in such a way as to remove almost all surface and Moho topography. This suggests that another mechanism must be responsible for the degradation of these basins.

[41] In contrast, SPA, the oldest and largest basin on the Moon, has not noticeably relaxed. The extremely thin crust beneath SPA (min ∼6 km) would have almost completely prevented subsolidus ductile flow. Even using the crustal model of Khan and Mosegaard [2002], with a thickness of 38 km at the Apollo 12/14 sites, the crust near the center of SPA would be less than 15 km. The great size of the basin also requires very long wavelength flow to occur in order to relax. Figure 11 shows the evolution of the basin for D = 45 km and Tb = 1500 K (ignoring the possibility of supersolidus behavior). Even with such a high heat flux, the top of the Moho uplift remains cold and unable to flow, thus preventing relaxation.

Figure 11.

Evolution of the topography of SPA over time for D = 45 km and Tb = 1500 K.

5. Discussion

[42] The lunar multiring basins show three main types of gravity/topography signatures: (1) well-preserved group 1 basins have high-amplitude topography and most are associated with a “bull's eye” mascon gravity anomaly; (2) mare-filled group 2 basins have shallow, flat floors and a positive, high-amplitude, mare-dominated gravity anomaly; (3) old, degraded group 3 basins show little topographic relief and low-amplitude free-air gravity. The crustal structures of group 1 and 2 basins are consistent with formation during the collapse of the transient cavities, while those of group 3 basins have been almost completely erased.

[43] The evolutionary paths followed by the basins in this study appear to be related to the regions in which they lie. The mare-filled (group 2) basins are restricted to the nearside, while few unmodified (group 1) basins are found there, because of the ubiquity of mare volcanism on the nearside. Highly degraded (group 3) basins are found on the farside and nearside, but only on the farside have any retained substantial topographic relief. This is consistent with the compositionally distinct terranes revealed by Lunar Prospector [e.g., Joliff et al., 2000].

[44] Using a viscoelastic model, we have investigated quantitatively the hypothesis that viscous relaxation of multiring basins has occurred on the Moon, producing the group 3 basins. The results of our modeling show that relaxation of basins similar to group 1 by ductile flow in the lower crust is a plausible explanation for the state of group 3 basins on the farside, where the crust is thick. The heat flux required to produce relaxation increases with decreasing crustal thickness and increasing basin diameter. Farside basins can relax for q0 > 25–40 mW m−2 (depending on preimpact crustal thickness and basin size), while the amplitude of topography associated with Keeler-Heaviside suggests that q0 was not far above 35 mW m−2 when this basin formed. The existence of relaxed basins on the farside requires that the lower crust remained ductile for a few hundred million years following the crystallization of the magma ocean. This result supports the contention that the Moon was initially hot [e.g., Pritchard and Stevenson, 2000] and suggests higher crustal temperatures and heat flow than previous models [e.g., Zhong and Zuber, 2000].

[45] The thinner nearside crust (D ∼ 30 km), on the other hand, does not allow relaxation of large basins by solid-state creep. This result is essentially model-independent, as the viscosity of the crust and uplifted mantle beneath the center of the basins is too high (>1023 Pa s) to permit flow on a reasonable timescale. Consequently, the degraded state of group 3 nearside basins requires that the crust was weakened at the time of their formation. The most likely mechanism is the presence of partial melt in the lower crust, perhaps as a remnant of the global magma ocean, which might also have facilitated enhanced structural collapse during formation or very rapid relaxation.

[46] Several authors have investigated the response of the Moon to the loads produced by the mascon basins. Solomon and Head [1979] determined that the concentric rilles around Serenitatis are graben caused by the superposition of a global extensional stress field with flexure of the elastic lithosphere beneath the mare load. They concluded that the period of global extension must have ended prior to the cessation of rille formation (3.6 ± 0.2 Gyr ago [Lucchita and Watkins, 1978]). Assuming that the loading was due only to mare basalt fill, Solomon and Head [1980] determined lithospheric thicknesses beneath the mascon basins at the time of rille formation to be ≤25 km for Grimaldi, 40–50 km for Serenitatis, Orientale and Humorum, 50–75 km for Imbrium, and >75 km for Nectaris, Smythii, and Crisium. Arkani-Hamed [1998] found the gravity and topography relationships of the mascon basins to be consistent with lithospheric thicknesses of 20 km (Orientale), 30 km (Smythii and Humorum), 35 km (Crisium), and 50 km (Imbrium, Serenitatis, and Nectaris). Using a numeric viscoelastic model, Freed et al. [2001], estimated that the lithosphere beneath Serenitatis was ∼25 km thick at the time of rille formation.

[47] Using the lower bounds on the heat flux estimated above, we can construct lower bounds on the lithospheric thickness on the nearside and farside at time of formation of group 3 basins following the method of McNutt [1984]. Using a strain rate of 10−15 s−1—which is consistent with our model results for flexural adjustment—we estimate that the lithospheric thickness was <30 km when the farside group 3 basins formed and <15 km when those on the nearside formed. Unsurprisingly, our thickness estimate for the nearside is unambiguously less than all estimates of the lithosphere beneath the mascon basins listed above (all are >25 km). That on the farside, on the other hand, is comparable to the lower end of the estimates. As the farside group 3 basin are unambiguously older than the nearside mare basins and the same age as the nearside group 3 basins, this suggests that the farside cooled faster than the nearside.

[48] Increased heat production in the crust has the effect of increasing the surface heat flux for a given Tb, and thereby increasing the temperature beneath the megaregolith and thinning the lithosphere. The thickness of the ductile channel at the base of the crust also increases, allowing ductile flow at a lower Tb. In cases where heat production is very high, relaxation can occur for Tb as low as 1200 K. Under this circumstance, the lithosphere is almost nonexistent, and the basin relaxes isostatically. The PKT appears to be extremely enriched in heat-producing elements, but also has the thinnest crust on the Moon. While subsolidus relaxation would have been unlikely, the PKT is believed to have remained partially molten longer than other regions [e.g., Wieczorek and Phillips, 2000]. This is likely responsible for the condition of Nubium and NW Procellarum basins.

[49] Wieczorek and Phillips [1999] speculated that the anomalously thick crust beneath Imbrium and Serenitatis, both of which lie within or overlap the PKT, might be the result of viscous relaxation. The thermal models of Wieczorek and Phillips [2000] predict that temperatures ∼2000 K prevailed at the base of the crust at the time of the Imbrium impact, which would certainly be sufficient to allow basin relaxation. However, both models adopted the then-prevailing seismic estimate [e.g., Goins et al., 1981] of a thick, 60-km crust, which now appears unlikely [Khan and Mosegaard, 2002; Lognonné et al., 2003]. The updated crustal thickness model used in this paper shows that the crust beneath the two basins is ≤15 km thick. Not only would such thin crust mechanically restrict ductile flow, but it limits the amount of crustal thickening which can have been accomplished by ductile flow (or any other process). It therefore appears unlikely that either Imbrium or Serenitatis have significantly relaxed.

[50] Where did even this thin crust come from if the impacts excavated the entire crust? The answer may lie in the prodigious volume of melt produced by giant impacts. While most of the melted crust would be expelled, over half of the total melt volume would be retained within the crater [Cintala and Grieve, 1998]. Geological evidence [Morrison, 1998; Therriault et al., 2002; Zieg and Marsh, 2005] and numerical models [Ivanov, 2006] suggest that the thick melt sheets generated by large impacts differentiate, producing “new” crust. Because of the anomalously high subsurface temperatures in the PKT, Imbrium and Serenitatis likely produced an unusually large volume of melt, in addition to an unknown thickness of volcanics. It is likely therefore that the present-day crust beneath Imbrium, Serenitatis, and SPA is not primarily a remnant of the preimpact crust, but a product of the impacts and subsequent volcanism.

[51] Though it is probably the oldest basin on the Moon, SPA does not appear to have relaxed. Subsolidus relaxation would certainly be precluded by its great size and thin crust, but it is interesting that SPA did not relax by a similar mechanism to the nearside basins. Was the crust beneath (and surrounding) SPA not weak enough? If so, then one must conclude that the weakening mechanism which allowed many group 3 basins to relax did not operate globally, but preferentially on the nearside. Assuming that it was a thermally activated process, this provides further evidence of a cooler farside.

[52] The crustal model used in this paper employs a thickness of ∼30 km at the Apollo 12/14 sites and the crustal and mantle densities listed in Table 4, resulting in an average thickness of ∼40 km, which is favored by the most recent seismic and gravity results [Lognonné et al., 2003; Chenet et al., 2006]. However, considerable uncertainty exists with respect to the average thickness of the lunar crust. Wieczorek et al. [2006] suggest that the average thickness can be constrained to be in the range 33–66 km.

[53] It has been shown in previous studies that modeling of viscous relaxation of crustal thickness variations can provide an upper bound on the average crustal thickness [Zuber et al., 2000; Nimmo and Stevenson, 2001]. SPA is the oldest and deepest basin on the Moon and therefore provides the best constraint on the crustal thickness. Our modeling shows that an increase of ∼15 km in the average crustal thickness would be sufficient to allow substantial relaxation of SPA, and would therefore be inconsistent with its current state of preservation. As a result, we consider a crust ≥15 km thicker than the model used here to be unlikely unless the crust-mantle density contrast is significantly less than current estimates, which would allow for greater lateral variations.

[54] Parmentier et al. [2002] have shown that the concentration of mare basalts on the nearside and the great depth of melting required by high-Ti glasses can be explained by long-wavelength instability of an ilmenite-rich cumulate layer, followed by melting ∼400–500 Myr later. However, this model does not address the crustal thickness asymmetry, nor does it explain the extremely high concentrations of radioactive elements in rocks from the PKT (e.g., KREEP, LKFM). We propose that the thermal, volcanic, and radioactive element abundance asymmetries are, in fact, consequences of the crustal thickness asymmetry.

[55] Because of the thick crust there, the farside would have reached full solidification first, leaving the nearside with the remaining liquid. As a result, the nearside would inherit the bulk of the heat-producing elements from the magma ocean, providing a significant heat source to the nearside and leaving the farside with very little heat production, allowing it to cool more rapidly [Shearer et al., 2006]. The layer of ilmenite-rich cumulates, believed to have crystallized from the last 5–10% of the magma ocean [Snyder et al., 1992; Parmentier et al., 2002], would be thick on the nearside and thin to nonexistent on the farside. The sinking and later remelting of these cumulates would subsequently have occurred almost exclusively on the nearside, thus producing the observed distribution of mare basalt without requiring degree 1 convection. Further cooling would have concentrated the dregs of the magmasphere in the PKT, where they now appear to reside.

[56] When did these basins form? Wilhelms [1987] has suggested that the crust had solidified sufficiently to record impact events between 4.4 and 4.2 Ga. Thermal evolution models and isotopic systematics predict that the magma ocean had mostly crystallized by ∼4.4 Ga, while the residual liquid likely took an additional 200–300 Myr (and possible longer) to crystallize [Solomon and Longhi, 1977; Nyquist and Shih, 1992]. Studies of short-lived radionuclides suggest that the early stage cumulates had begun crystallizing from the magma ocean by ∼4.5 Ga, while the late-stage ilmenite-rich cumulates may have crystallized ∼100 Myr later [Kleine et al., 2005; Borg and Wadhwa, 2006]. As the group 3 nearside basins likely formed prior to the concentration of the final dregs (i.e., ur-KREEP) in the PKT, it appears unlikely that any formed later than 4.2 Ga. Indeed, the isotopic results suggest that the majority may have formed substantially earlier.

6. Conclusions

[57] The results of this study impose significant constraints on the thermal evolution of the Moon. We have shown that subsolidus relaxation is a plausible explanation for the degraded state of group 3 farside basins. The topography and free-air gravity of these basins is consistent with the relaxation of well-preserved group 1 basins of comparable size. Relaxation of farside basins would likely have required heat flows of 25–40 mW m−2 (depending on the preimpact crustal thickness). The lithospheric thicknesses implied by these heat flow estimates—combined with previous estimates of lithospheric thicknesses beneath the younger nearside mare basins—suggest that the farside cooled faster than the nearside. However, subsolidus relaxation of large (diameter > 400 km) nearside basins cannot occur, because of the thin crust (≤30 km). This conclusion is not strongly dependent on the assumed rheology, as the nearside crust is thin enough to have been almost entirely excavated by the impacts, hence impeding lower crustal channel flow. The degraded state of nearside group 3 basins suggests that crust must have been weakened by some other mechanism, most probably as a result of a partially molten lower crust.

[58] The revised crustal thickness model used in this paper shows that the crust beneath SPA is very thin, suggesting that the basin has probably not relaxed. This implies that the same weakening mechanism that operated on the nearside was not available when it formed, and supports the idea that the nearside remained hot longer than the farside. Serenitatis and Imbrium also appear to have undergone little or no relaxation, although this is less surprising, as they formed much later.

[59] It is well known that a large fraction of the Moon's budget of heat-producing elements resides on the nearside, concentrated in the PKT [Joliff et al., 2000]. One explanation for the results in this paper is that the late-stage magma ocean crystallized primarily beneath the nearside crust, thus providing the nearside with the bulk of the heat-producing elements in the magmasphere. The degraded nearside basins must then have formed before the dregs became concentrated in the PKT. Thermal evolution models and isotopic analyses of lunar basalts suggest that this occurred prior to 4.2 Ga.

Appendix A

[60] For axisymmetric flow, a similarity solution is obtained by expanding the variables in zonal spherical harmonics [e.g., Ricard et al., 1984]:

equation image

where r is the radius, equation image is the colatitude, η0 = η(rs) is the viscosity at the surface, ρ0 = ρc is the crustal density. I define V = (u1l, u2l, u3l, u4l, u5l, u6l)T and ν = ln(r/R0). This leads to a system of first-order linear differential equations [e.g., Ricard et al., 1984; Richards and Hager, 1984]:

equation image
equation image

where L = l(l + 1). We then nondimensionalize the equations using R0 as the reference length, ρ0 as the reference density, η0 as the reference viscosity, and t0 = equation image as the reference time,

[61] We use linearized boundary conditions where the stress due to topography is applied at the surface [e.g., Grimm and Solomon, 1988; Nunes et al., 2004] and crust-mantle boundary topography produces a discontinuity in σ and equation image at the Moho. Shear stress is assumed to be zero at the surface and a free-slip condition is applied at the core-mantle boundary (CMB).

equation image
equation image
equation image
equation image
equation image

where hi is the topography at interface i, the subscripts s, m, and c refer to the surface, Moho, and CMB, respectively, Δρm = equation image is the density discontinuity across the Moho, and γ = equation image. The condition satisfied by u5 at the CMB is

equation image

where.

[62] To satisfy the boundary conditions, we use a linear combination of four independent solution vectors, following Grimm and Solomon [1988]. Three are specified at the CMB and integrated to the surface using a fourth-order Runge-Kutta method:

equation image
equation image
equation image

The fourth solution takes into account the mass discontinuity at the Moho and is integrated from there to the surface:

equation image

[63] Using Vij to denote the jth element of Vi, evaluated at the surface, (A3) and (A4) are rewritten as

equation image
equation image

Thus

equation image

[64] The coefficients are then expressed:

equation image

e, f, and g are uniquely determined from (A4), (A5), and (A8):

equation image

and

equation image

where det V = V13V24V14V23. The motion of the two interfaces is calculated from vr:

equation image
equation image

[65] After performing a Laplace transform, (A17) and (A18) become

equation image

where hs0 and hm0 are the initial relief at the surface and Moho, respectively, and

equation image

[66] The Laplace domain solution is then

equation image
equation image

[67] These solutions contain several singularities, and the inverse Laplace transform is performed by evaluating the residue at each singularity. In this case, they are simple poles, so the residue is

equation image

where N(s) is the numerator, B(s) is the denominator, Sj is the jth singularity and Rsj is the corresponding residue. The solution at interface i is then

equation image

Acknowledgments

[68] We would like to thank Daniel Nunes and Jeffrey Hanna for their help at several stages of this process and Randy Korotev for helpful discussions. We would also like to thank Mark Wieczorek, Shijie Zhong, and an anonymous reviewer for their thorough reviews. This research was supported by a McDonnell Center for the Space Sciences fellowship and by grant NAG5-13243 from the NASA Planetary Geology and Geophysics Program.

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