Wayne L. Slattery

The origin of the Moon and the single-impact hypothesis III

In previous papers in this series the smoothed particle hydrodynamics method (SPH) has been used to explore the conditions in which a major planetary collision may have been responsible for the formation of the Moon. In Paper II (W. Benz, W.L. Slattery, and A.G.W. Cameron 1987, Icarus 71, 30-45) it was found that the optimum conditions were obtained when the mass ratio of the impactor to the protoearth was 0.136. In the present paper we investigate the importance of the equation of state by running this optimum case several times and varying the equation of state and other related parameters. The two equations of state compared are the Tillotson (used in the previous papers) and the CHART D/CSQ ANEOS. Because of differences in these equations of state, including the fact that different types of rocks were used in association with each, it was not possible to prepare initial planetary models that were comparable in every respect, so several different simulations were necessary in which different planetary parameters were matched between the equations of state. We also used a new version of the SPH code. The results reaffirmed the previous principal conclusions: the collisions produced a disk of rocky material in orbit, with most of the material derived from the impacting object. These results indicate that the equation of state is not a critical factor in determining the amount of material thrown into orbit. This confirms the conclusions of Paper II that gravitational torques, and not pressure gradients, inject the orbiting mass. However, the way this mass is distributed in orbit is affected by the equation of state and the choice of rock material, the Tillotson equation for granite giving slightly larger mean orbital radius for the particles left in orbit than the ANEOS dunite for the same impact parameter. We also find, compared to Paper II, that in all subsequent cases the new SPH code leads to a slightly less extended prelunar accretion disk. We think this is due to the new shape adopted for the kernel. A few additional calculations were made to test the effects of increasing the impact parameter on the calculations, other parameters remaining unchanged. The motivation for this was that solar tides will have reduced the Earth-Moon angular momentum somewhat over the course of time. An increment of 6% in the angular momentum of the collision increases the amount of iron-free material in orbit and its mean orbital radius, but more than that leaves increasing amounts of iron in orbit (the iron has a small mean orbital radius). The debris from the destroyed impacting object tends to form a straight rotating bar which is very effective in transferring angular momentum. If the material near the end of the bar extends well beyond the Roche lobe, it may become unstable against gravitational clumping.

Collisional stripping of Mercury's mantle

Abstract We investigated the conditions under which a giant collision between a hypothetical proto-Mercury and a planet one-sixth its mass would result in the loss of most of the silicate mantle of the planet, leaving behind an iron-rich planet and thus explaining the anomalously high density of Mercury. We carried out a series of numerical simulations using our three-dimensional smoothed particle hydrocode, varying the impact parameter and the relative velocity between the planet and impactor. We demonstrate that the details of the equation of state do not play an important role. We show that a head-on collision at 20 km/sec and an off-axis (impact parameter equal to half the radius of proto-Mercury) collision at 35 km/sec are about equivalent as far as damage to proto-Mercury is concerned. Both collisions leave behind a remnant that has the required characteristics of the present Mercury. Whether this scenario is actually successful depends on the size of the condensates in the ejected cloud of debris. Preliminary estimates show that most of the ejected mass is probably removed from Mercury-crossing orbits. If this turns out to be true, a giant collision is a plausible explanation for the strange density of Mercury.

Giant impacts on a primitive Uranus

Abstract Using smooth particle hydrodynamics, we have carried out a series of simulations of collisions between a model of a primative Uranus and impactors with masses ranging from 1.0 to 3.0 M⊕. These impactors were assumed to be differentiated and composed of iron, rock, and ices in solar proportions. The model Uranus had a mass equal to present day Uranus minus the mass of the impactor and was also assumed to be differentiated and composed of iron, rock, and ices in solar proportions with an additional 2 M⊕ mixed into the ices. We also assumed that the planet was not rotating prior to the collisions. All simulations reported here have been carried out assuming a relative velocity at infinity between the planets of 5 km/sec. For each impactor, a series of collisions were simulated varying the total angular momentum in the system (effectively varying the impact parameter since the velocity at infinity was kept constant). There was a range of angular momenta for each of the cases, except for the 1.0 M⊕ case, in which Uranus was set into rotation with a period less than the present one of 17.24 hr. Most of the runs left ices in orbit, derived from the Uranus atmosphere or the impactor (or both), and a subset of these runs also left some rock or iron from the impactor in orbit. From these simulations we conclude that there is a fairly large range of giant impacts that could have produced the present period and inclination of the spin axis to the plane of the ecliptic, and there is a subset of these that could have deposited suitable material in orbit from which the regular satellites of Uranus could have been assembled.

Numerical simulation of strongly coupled binary ionic plasmas

Abstract New lengthy Monte Carlo simulations of the energy equation of state of binary ionic mixture fluids in a uniform background show that deviations from the linear mixing rule are small, positive, and nearly constant as a function of Λ. Deviations from linear mixing for the Helmholtz free energy are positive and behave an 1n Λ. Quantitative results are obtained from the correction to the thermonuclear reaction rate.

Harmonic Lattice theory of coulomb solids and comparison with Monte Carlo simulations

The potential energy of the OCP in the fluid phase is known to great accuracy from Monte Carlo (MC) data obtained from very long simulations. For the crystalline phase, bcc or fcc, the MC simulations are equally accurate but can also be represented nearly analytically in the Harmonic Lattice (HL) approximation. We give here recent results for the anharmonic effects in the classical regime T ≫ Tp (where Tp is the on plasma temperature) both exact and obtained in the HL approximation. We also give results for the harmonic potential energy in a quantum OCP crystal and discuss the importance of quantum effects for the fluid-lattice phase transition.

Pair distribution of ions in Coulomb crystals

The pair distribution function g(r) of ions in body-centered-cubic and face-centered-cubic Coulomb crystals is calculated using the harmonic-lattice (HL) approximation in a wide temperature range, from the high-temperature classical regime (T⪢ℏωp,ωp is the ion plasma frequency) to the low-temperature quantum regime (T⪡ℏωp). The radial pair distribution function g(r) is calculated by averaging g(r) over orientations of r. In the classical limit, g(r) is also obtained from extensive Monte Carlo (MC) simulations. MC and HL results are shown to be in good agreement. With decreasing temperature T, the correlation peaks of g(r) and g(r) become narrower and finally freeze at T⪡ℏωp being solely determined by zero-point ion vibrations.

Numerical Simulation of Coulombic Freezing

The fluid to crystalline solid first order phase transition of the classical one component plasma (OCP) has been studied by Monte Carlo simulation in three dimensions for temperatures below the thermodynamic freezing temperature (Γ = Z2e2/akT = 180, a = Wigner-Zeitz radius). With N = 686 we found freezing from a metastable supercooled fluid into microcrystals for values of Γ ranging from 250 to 700. In one case, Γ = 500, the system froze into a perfect bcc lattice from a random start. With more particles the system froze into two or more crystals one bcc and one fcc and smaller examples of hcp. The lattice planes were examined and various kinds of crystal defects could be observed. We developed a program for determining the local environment of each particle as bcc, fcc, hcp, or fluid in order to identify microcrystals in the system at any stage of the freezing process. Generally freezing proceeds rapidly when any single miciutrystal attains a sufficient size or roughly 60 to 70 particles. The observations of freezing seem to agree with classical nucleation theory. No separate glass phase (non-crystalline) was seen.

Pair distribution of ions in Coulomb lattice

The pair distribution function g(r) ≡ g(x, y, z) and the radial pair distribution function g(r) of ions in body-centred-cubic and face-centred-cubic Coulomb crystals are calculated within the harmonic-lattice (HL) approximation in a wide temperature range, from the high-temperature classical limit (T wp, wp being the ion plasma frequency) to the low-temperature quantum limit (T wp). In the classical limit, g(r) is also calculated by the Monte Carlo (MC) method. MC and HL results are demonstrated to be in good agreement. With decreasing T, the correlation peaks of g(r) and g(r) become narrower. At T wp they become temperature independent (determined by zero-point ion vibrations).

PROPERTIES OF STRONGLY COUPLED MULTI-IONIC PLASMAS

A reexamination of the OCP fluid equation of state using new Monte Carlo results gives the EOS as — (7frac9o10;) Γ + b Γ 3 with s ~ ⅓ rather than the ¼ value obtained previously. The EOS for the bcc crystalline phase indicates that first order anharmonic corrections are present. A recalculation of the freezing transition leaves the bcc transition unchanged at about Γ = 178, but the transition to fcc is lower, about 192. For ionic mixtures the linear mixture rule has near universal validity for fluid and solid mixtures. An extreme charge asymmetry test case using HNC equations shows near perfect agreement with the linear mixing rule.

EQUATION OF STATE FOR BINARY IONIC PLASMAS, FLUID AND SOLID PHASES

This paper gives a summary of the most recent Monte Carlo simulation data for the internal energy, U/NkT, for the the One Component Plasma (OCP) and Binary Ionic Mixtures (HIM) for both fluid and solid phases. The Monte Carlo results are compared with Molecular Dynamics results of nearly the same accuracy. The OCP fluid and solid data are used to give the most accurate available equation of state for both the fluid and the crystalline (bcc) phases of the OCP. For the BIM mixture with 5% charge 2. and 95% charge 1. the deviations from linear mixing are given to the best available accuracy. These results make it possible to obtain the classical contribution to the screening enhancement of thermonuclear reaction rates for white dwarf stars. Current results on deviations from linear mixing are also given for random ionic mixtures for which the two different charges are placed randomly on bcc lattice sites. It is found that the deviation from linear mixing in the crystalline solid is roughly an order of magnitude larger than deviations from linear mixing for BIM fluids. The data supports the conclusion that the phase diagram for a BIM mixture is spindle shaped for carbon and oxygen mixtures found in white dwarf stars.