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Appendix A: Computing the gravitational potential energy of the system

The gravitational potential energy of n different subsystems with volumes D_i and densities (MacMillan 1958) is (A.1)where and the potential¹⁰ of each subsystem (MacMillan 1958) is given by (A.2)In our case we have three constituents (i,j = m,f,s), hence (A.3)The difficulty in evaluating this expression is related to the geometry of the region occupied by the fluid. To overcome this problem, we take into account (Escapa & Fukushima 2011) that (A.4)the superscript (0) referring to the configuration when R_m = R_s. In this way, from Eqs. (A.2) and (A.4) the gravitational potential of the fluid can be written in the form (A.5)The potential is that of a fluid with density ρ_f and volume given by an sphere of radius d_m centered at the mantle barycenter; the potential corresponds to a fluid with density ρ_f and volume given by an sphere of radius d_s centered at the inner core barycenter.

Accordingly, the gravitational potential energy of the system is (A.6)The subsystems involved in this equation are spheres or spherical shells with constant or spherical symmetric density functions. Their potentials can be obtained from that of a spherical shell (B) with interior radius d₁, which would be 0 for a sphere, and exterior radius d₂, whose density is a piecewise continuous function ρ of the distance from the center. This has the expression (A.7)If in this equation the density is , we recover the formulas given in Escapa & Fukushima (2011, Eq. (B3)¹¹).

The application of Eqs. (A.7) to (A.6) provides the constant k₁, arising from the terms and , and from , , , and . The first group is constant because it is the case for the potential when evaluated at interior points of the spherical shell. The second group is constant because it provides the gravitational self-energy of spheres or spherical shells referring to the same region of space.

The remaining terms in can be cast as (A.8)With the expression of arising from Eq. (A.7), it reads (A.9)A change of variable R′′ = R−R_s = R−R_m−r allows computing this integral with respect to a reference system whose origin is located at the center of sphere . In this way, we obtain (A.10)where k₂ comprises all the constant terms resulting from the integration, and Vol Vol D_s is the volume of the inner core.

Therefore, the gravitational potential energy of the system is (A.11)since the constant terms play no role for our purposes, we will omit them.

Appendix B: Expression of the amplitude ratio of the mantle to the inner core for the stratification of the rigid layers assumed in Coyette et al. (2012)

Here, we follow the same notation as Coyette et al. (2012). With this, we can write¹² for the mass of the mantle, the fluid, and the inner core (B.1)The mean density of the inner core reads as (B.2)The amplitude ratio of the mantle to the inner core (Eq. (9)) can be written as (B.3)From Eqs. (B.1) and (B.2), with ρ_f = ρ_{ic + 1}, we obtain (B.4)where [ρ]_j = ρ_j−ρ_{j + 1},j = 1,...,N−1, [ρ]_N = ρ_N.

Equation (B.4) is exactly the same formula as derived in Coyette et al. (2012, Eq. (3)).

© ESO, 2015