Figure 8 shows our model for heating, melting, cooling, and crystallization on Vesta, based on the results of our chemical models and existing physical models for magma oceans on Vesta and other planets/planetesimals. The early history of Vesta can be thought of in four stages:
Stage 2. Surface Processing, Destruction of the Primitive Lid, and Rapid Cooling and Equilibrium Crystallization of the Magma Ocean
The thickness of the primitive lid is an important consideration because it exerts a first-order control on the rate at which Vesta heated and cooled. Early planetesimals were heated by 26Al decay, but had a cold surface in contact with the interplanetary medium. Even when Vesta's interior had developed a convecting magma ocean, it should have retained a conductive lid.
Since the pioneering work of Ghosh and McSween (1998), thermal models of planetesimal heating have been largely based on solving the 1-D heat conduction equation (Merk et al. 2002; Hevey and Sanders 2006; Gupta and Sahijpal 2010; Elkins-Tanton et al. 2011; Moskovitz and Gaidos 2011; Neumann et al. 2012; Šrámek et al. 2012). All of these models attempt to incorporate some combination of other processes that affect the distribution of heat and rate of heat loss, such as continuous accretion (Merk et al. 2002; Gupta and Sahijpal 2010; Elkins-Tanton et al. 2011; Neumann et al. 2012; Šrámek et al. 2012), convection (Hevey and Sanders 2006; Gupta and Sahijpal 2010; Elkins-Tanton et al. 2011; Neumann et al. 2012), melt migration (Gupta and Sahijpal 2010; Moskovitz and Gaidos 2011; Neumann et al. 2012), sintering (Hevey and Sanders 2006; Gupta and Sahijpal 2010; Moskovitz and Gaidos 2011; Neumann et al. 2012), and even impacts (Šrámek et al. 2012). However, only the Gupta and Sahijpal (2010) and Elkins-Tanton et al. (2011) models deal with planetesimals that have radii larger than 120 km or smaller than 500 km (RVesta is approximately 270 km).
Understandably, no model adequately deals with heat loss from a convecting magma ocean on a Vesta-sized planetary embryo through a thin, dynamic lid that is subject to gravitational readjustment, penetration by impacts, advection of the melt to the surface, and periodic rapid radiative heat loss. However, the heat budget of growing planetesimals is strongly controlled by surface processes because all heat loss occurs at the surface. The thickness and processing of the cold lid are therefore vitally important for the thermal–chemical evolution of the asteroid.
A thin lid (<few km) would be easily broken up by impacts (e.g., Davison et al. 2012) and gravitational readjustment of the vestan surface. It would be prone to foundering and easily admixed into the magma ocean, which would expose hot magma ocean material to space. This would lead to rapid radiative cooling, as well as the formation of quench crust, which, being denser than the magma ocean, would itself founder into the interior (Walker et al. 1980), exposing more magma and forming more quench crust. This cycle, assisted by large impacts, would lead to rapid cooling of the magma ocean and very little preservation of compositional heterogeneity.
Conversely, a thicker lid (>10 km) would be porous and insulating, difficult to break up, and would inhibit the transport of magma to the surface due to its low density (Elkins-Tanton et al. 2011). The result would be slower cooling and the preservation of large-scale heterogeneity in the form of primitive chondritic material in the shallow vestan subsurface.
Understanding and quantifying the dynamics of the upper thermal boundary layer is an important next step for thermal models of asteroidal heating. However, for the immediate purposes of this study, we can take a broad look at lid behavior for Vesta. All caveats aside, the modeling results of Gupta and Sahijpal (2010) and Elkins-Tanton et al. (2011) indicate lid thicknesses that vary from approximately 1 to >25 km, depending on accretion time, accretion rate, and lid porosity. We can also take a simpler look at minimum lid thickness by using HED isotopic data to determine the timing of peak magma ocean conditions, and then deriving the lid thickness at this time from the conduction equation:
where q is heat flux at the surface, ∆T is the heat difference across the conductive lid (approximately 1500 K), κ is the thermal conductivity of the lid (approximately 1.5 W m−1 K−1; Opeil et al. 2010), and z is the lid thickness. Peak magma ocean conditions on Vesta were likely reached approximately 2–3 Ma after CAIs (Lugmair and Shukolyukov 1998; Schiller et al. 2011). Adjusting heat production estimates for an asteroid composed of CI chondrites from Hevey and Sanders (2006) to account for the revised decay energy of Al26 (Castillo-Rogez et al. 2009) and the more Al-rich composition of Vesta compared with CI chondrites (Ghosh and McSween 1998), the total heat production on Vesta during peak magma ocean conditions was approximately 2.5–8 × 1012 W. Assuming a near-steady state magma ocean and lid temperature profile, this heat production rate corresponds to a surface heat flux of approximately 3–9 W m−2 during peak magma ocean conditions, which implies a lid thickness of 250–750 m. Although this is a crude estimate, this order of magnitude is reasonable if Vesta accreted early enough to reach peak magma ocean conditions by approximately 2 Ma after CAIs, for which there is ample chemical evidence (see the Chronology of Early Processes on Vesta section).
Gupta and Sahijpal (2010) and Schiller et al. (2011) note that a major difficulty in creating HEDs in a magma ocean scenario on Vesta is cooling the magma ocean rapidly enough. However, this is not an obstacle in the case of efficient lid removal and foundering of quench crust. The ancient HED ages are therefore not a problem for the time scales of thermal models.
The Vestan subsurface does not appear to contain a significant quantity of relict chondritic crust. Diogenites and most eucrites are homogeneous in their O isotopes (Greenwood et al. 2013), and very few contain any visible chondritic material (Zolensky et al. 1996). The presence of chondritic material in some howardites (Zolensky et al. 1996; Cartwright et al. 2012) and the observation of highly variable exotic material on the surface of Vesta (Reddy et al. ) probably represent more recent impact delivery rather than a relict lid (Reddy et al. ). The absence of relict lid material suggests a lid sufficiently thin (<few km) to be efficiently destroyed and reprocessed into the vestan interior during the magma ocean phase. This is broadly consistent with estimates from existing models and simple physical considerations.
Extensive processing of the surface and exposure of the magma ocean to space would result in rapid radiative cooling and equilibrium crystallization. Although a small amount of olivine settling was probably inevitable, the low gravity, relatively high convective velocities, and rapid solidification would severely limit settling. Furthermore, there is no robust evidence to suggest that the vestan magma ocean was ever above the liquidus, and only 70–80% melting is required to satisfy both isotopic homogeneity and core-mantle equilibria. If the vestan magma ocean retained at least a 10–20% crystal fraction at all stages, settling on a Myr time scale would have been essentially impossible due to crystal–crystal interactions (Suckale et al. 2012).
Stage 3. Convective Lockup and Melt Extraction
Our best-fit models require large-scale melt extraction from the magma ocean after 60–70% equilibrium crystallization. This is what the chemistry tells us, but the physical rationale for this process requires a brief consideration of crystal-liquid mixtures. The melt is more buoyant than the crystals (by approximately 450 kg m−3 in our models), so a crystal + liquid mush can only be maintained (i.e., melt extraction can only be prevented) if convection is sufficiently vigorous to mix the mush on a more rapid time scale than that of melt extraction.
Convection slows dramatically in a crystal suspension once crystals begin to interact with each other. For crystal shapes most relevant to an olivine + orthopyroxene crystal mush (crystal aspect ratio close to 1), an initial framework is generated at a crystal fraction of 20–30% (Saar et al. 2001; Baker et al. 2002; Walsh and Saar 2008). Above this crystal fraction, the mush has a yield strength, which increases near-exponentially as a function of crystal content until solid-like viscosities are reached at high crystal fractions (Walsh and Saar 2008). Modeling and experimental studies suggest that the rheology of the mush becomes overwhelmingly controlled by the rigid crystal framework at crystal fractions of 45–70% (Costa 2005; Costa et al. 2009, and references therein), dramatically increasing the viscosity and slowing convection.
Efficient melt extraction is possible once the buoyancy-driven ascent rate of melt exceeds the convective velocity of the system. This does not require convection to cease, but will instead occur below a critical convective velocity. Righter and Drake (1997) estimated that melt extraction would occur after 80% crystallization based on Kraichnan's (1962) condition for turbulent convection. Using the method of Righter and Drake (1997) for our model, we would obtain a critical crystal fraction of 74%. However, in Kraichnan's analysis, the turbulent regime does not suddenly switch from dominating the system to shutting off, but instead becomes infinitesimally small before disappearing. As such, the majority of the vestan magma ocean would be in a viscous regime that permitted significant melt extraction before reaching this critical crystal fraction, even if a diminishing region in the center of the mush was still capable of turbulent convection. Removal of melt into shallow magma chambers would increase the crystal fraction in the mush, further promoting melt extraction and fully locking up the convective system.
After convective lockup, the quench-foundering cycle dramatically slows, thickening and stabilizing the lid, which results in slower cooling because the upper boundary of the system is now purely conductive. The convectively locked crystal framework slowly compacts as melt ascends out of it and into multiple adjacent (and possibly interacting) shallow magma chambers.