Elemental Fractionation During Rapid Accretion of the Moon Triggered by a Giant Impact
Recently, Ida et al. made an N-body simulation of lunar accretion from a protolunar disk formed by a giant impact. One of their important conclusions is that the accretion time of the Moon is as short as one month. Such rapid accretion is a necessary consequence of the high surface density of a lunar mass disk accreting just beyond the Roche limit (about 3Re); the Safronov accretion time (a few days) is even shorter. The energy of accretion always exceeds the gravitational binding energy of newly arriving matter. Hence, without an energy sink, the accreting body is thermally unstable. For the Earth and other planets, radiation acts as the sink. However, in such a short accretion time, the Moon cannot radiate the accretional energy. Even radiating at a silicate cloudtop temperature of roughly 2000 K, it would take more than 100 yr to radiatively cool the Moon. The plausible alternative heat sinks are heat capacity, latent heat of vaporization, and thermal escape of the gas to space (i.e., hydrodynamic blowoff). The latter becomes plausible for the Moon because the scale height at 2000 K (about 300 km) is a significant fraction of the lunar radius. The early stages of lunar (or "lunatesimal") growth release relatively little energy and can occur simply by heating the material, especially if the accreting material is originally cold. However, the material is unlikely to be cold, because the disk itself is hot and cooling time is long, while the lunar accretion time iss very short. Therefore, the moon is likely to accrete condensed material just after it condenses. Accordingly, the newly accreted material will be on the verge of vaporization and will have very little heat capacity to spare. The immediate heat sink is the latent heat of vaporization. Most of the vapor will escape from the moon, because the thermal energy in the gas can be used to drive escape. However, vaporization is generally incomplete. the latent heat of vaporization exceeds the energy of accretion. Viewed globally, the accretional energy is about half the energy required to vaporize the entire Moon. Thus to first approximation, half of the Moon-forming material can be vaporized and lost during accretion. During this process, we would expect preferential loss of relatively volatile elements. Escape will retard the rate of accretion. To test these ideas, we computed detailed models of the thermal state of the Moon during accretion. We pay special attention to the structure of the silicate atmosphere and its loss rate by calculating the chemical species at equilibrium. We used the PHEQ program which includes 12 elements (H,O,C,Mg,Si,Fe,Ca, Al, Na,Ti, and N.) and 272 compounds (including ionic compounds). Because of the large heats of vaporization and ionization, the adiabatic atmosphere is nearly isothermal and massive escape is expected. The pressure of the atmosphere is determined by the balance between vaporization of a accreting material and escape. If the accretion time is one month, a 0.3 bar atmosphere is expected. Elemental fractionation depends strongly on the temperature of the accreting material. The initial temperature of the material can be estimated from the condition of gravitational instability in the protolunar disk. As shown by Ida et al, accretion starts when gravitational instability occurs when more than 99% of the material condenses. At this point, all of Ca, Al, Si, Mg, and Fe, and 95% of Na (probably K also), are in condensed phases. If the moon is formed from the accretion of such material, volatile elements such as Na, and K are retained by the moon only early in accretion. At later times, K and Na are lost and a fraction of the MG, Si and Fe is lost. However, refractory elements such as Ca and Al are retained and so achieve a mild degree (factor 2) of superabundance.