The Cassini State of the Moon's Inner Core

Abstract

We present a model of the precession dynamics of the Moon that comprises a fluid outer core and a solid inner core. We show that three Cassini states associated with the inner core exist. The tilt angle of the inner core in each of these states is determined by the ratio between the free inner core nutation frequency ( ficn ) and the precession frequency Ω p = 2 ∕18.6 year −1 . All three Cassini states are possible if | ficn | > 2 ∕16.4 year −1 , but only one is possible otherwise. Assuming that the lowest energy state is favored, this transition marks a discontinuity in the tilt angle of the inner core, transiting from −33 ∘ to 17 ∘ as measured with respect to the mantle figure axis, where negative angles indicate a tilt toward the orbit normal. Possible Lunar interior density structures cover a range of ficn , from approximately half to twice as large as Ω p , so the precise tilt angle of the inner core remains unknown, though it is likely large because Ω p is within the resonant band of ficn . Adopting one specific density model, we suggest an inner core tilt of approximately −17 ∘ . Viscoelastic deformations within the inner core and melt and growth at the surface of a tilted inner core, both neglected in our model, should reduce this amplitude. If the inner core is larger than approximately 200 km, it may contribute by as much as a few thousandths of a degree on the observed mantle precession angle of 1.543 ∘ . Plain Language SummaryIt is well known that the reason we see only one face of the Moon is because the period of its revolution around Earth is equal to the period of Moon's rotation around itself. Yet the full description of the Moon's rotation is a bit more complex. The Lunar spin axis is inclined by 1.54 ∘ in space, and its direction is precessing, like a spinning top, at a period of 18.6 years. This precession is caused by the precession of the plane of the Lunar orbit at the same period, an arrangement known as a Cassini state, named after the seventeenth century Italian astronomer. Like the Earth, the Moon is suspected to have a metallic iron core, its outer region being fluid but the central part being solid. In this work, we calculate the tilt of the spin axis of this solid inner core so that it obeys a Cassini state. We show that this angle can be large, 17 ∘ with respect to the mantle for one specific model but that it is also very sensitive to the specific size and mass of the solid and fluid parts of the Lunar core which are not well known.

The Influence of a Fluid Core and a Solid Inner Core on the Cassini State of Mercury

“…Viewed in the frame attached to the mantle rotating at sidereal frequency Ω o , the Cassini plane is rotating in a retrograde direction at frequency ω Ω o (see Figure 2b), where ω , expressed in cycles per Mercury day, is equal to The factor δω = Ω p /Ω o = 4.933 × 10 −7 is the Poincaré number, expressing the ratio of the forced precession to sidereal rotation frequencies. The invariance of the Laplace plane normal as seen in the mantle frame is expressed as or equivalently, by Equation (19e) of Stys and Dumberry (2018) This expresses a formal connection between θ p and θ m which is independent of the interior structure of Mercury. Using Equation and cos( θ m ) → 1, this connection can be rewritten as and thus the relative amplitudes of θ m and θ p depend of the Poincaré number δω .…”

“…Here, we present a model of Mercury's Cassini state that comprises a fluid core and solid inner core. The model is an adaptation of a similar model developed to study the Cassini state of the Moon (Dumberry & Wieczorek, 2016; Organowski & Dumberry, 2020; Stys & Dumberry, 2018). The specific questions that motivate our study are the following.…”

Cassini states of a rigid body with a liquid core

“…In contrast, when the body possesses a liquid core, the problem has only been studied in the vicinity of low obliquities (e.g., Touma and Wisdom, 2001;Henrard, 2008;Dufey et al, 2009;Noyelles et al, 2010;Noyelles, 2012;Peale et al, 2014). Yet, recently Stys and Dumberry (2018) obtained a set of equations truncated in eccentricity but valid at all obliquity whose solutions provide the location of all the Cassini states of a three-layered body composed of a rigid mantle, a fluid outer core and a rigid inner core. This analysis applied to the Moon has been used to infer the orientation of its inner core (ibid).…”

Viscoelastic Relaxation within the Moon and the Phase Lead of Its Cassini State

“…This framework was adapted to model the Cassini state of the Moon in Dumberry and Wieczorek (2016), henceforth referred to as DW16. A further extension of the framework is presented in Stys and Dumberry (2018), henceforth referred to as SD18. We give an outline of this rotational model below, but the interested reader is referred to DW16 and SD18 for more details.…”

“…Third, the tilt angle of the inner core relative to the lunar mantle can theoretically be large (Dumberry & Wieczorek, 2016; Stys & Dumberry, 2018). In the reference frame of the lunar mantle, a tilted inner core precesses with a period of one lunar day.…”