Boundary conditions for the formation of the Moon

Abstract

Recent measurements of the chemical and isotopic composition of lunar samples indicate that the Moon's bulk composition shows great similarities with the composition of the silicate Earth. Moon formation models that attempt to explain these similarities make a wide variety of assumptions about the properties of the Earth prior to the formation of the Moon (the proto-Earth), and about the necessity and properties of an impactor colliding with the proto-Earth. This paper investigates the effects of the proto-Earth's mass, oblateness and internal core-mantle differentiation on its moment of inertia. The ratio of angular momentum and moment of inertia determines the stability of the proto-Earth and the binding energy, i.e. the energy needed to make the transition from an initial state in which the system is a rotating single body with a certain angular momentum to a final state with two bodies (Earth and Moon) with the same total angular momentum, redistributed between Earth and Moon. For the initial state two scenarios are being investigated: a homogeneous (undifferentiated) proto-Earth and a proto-Earth differentiated in a central metallic and an outer silicate shell; for both scenarios a range of oblateness values is investigated. Calculations indicate that a differentiated proto-Earth would become unstable at an angular momentum L that exceeds the total angular momentum of the present-day Earth–Moon system (L 0) by factors of 2.5–2.9, with the precise maximum dependent on the proto-Earth's oblateness. Further limitations are imposed by the Roche limit and the logical condition that the separated Earth–Moon system should be formed outside the proto-Earth. This further limits the L values of the Earth–Moon system to a maximum of about L/L 0 = 1.5, at a minimum oblateness (a/c ratio) of 1.2. These calculations provide boundary conditions for the main classes of Moon-forming models. Our results show that at the high values of L used in recent giant impact models (1.8 <L/L 0 <3.1), the proposed proto-Earths are unstable before (Cuk & Stewart, 2012) or immediately after (Canup, 2012) the impact, even at a high oblateness (the most favourable condition for stability). We conclude that the recent attempts to improve the classic giant impact hypothesis by studying systems with very high values of L are not supported by the boundary condition calculations in this work. In contrast, this work indicates that the nuclear explosion model for Moon formation (De Meijer et al., 2013) fulfills the boundary conditions and requires approximately one order of magnitude less energy than originally estimated. Hence in our view the nuclear explosion model is presently the model that best explains the formation of the Moon from predominantly terrestrial silicate material.