Issue 
A&A
Volume 577, May 2015



Article Number  A83  
Number of page(s)  18  
Section  Planets and planetary systems  
DOI  https://doi.org/10.1051/00046361/201424915  
Published online  07 May 2015 
Online material
Appendix A: Prior of mantle Sicontent
The prior bounds on Si_{⊚} depend on the prior bounds of Fe/Si_{⊚} and Mg/Si_{⊚}, since the mass fractions of all oxides in the NCFMAS chemical system must sum to one. The mass fraction of Si can be expressed as where μ are respective molar weights and the mass fraction of minor oxides are combined in χ_{Na2O,CaO,Al2O3}. Transforming from a uniform prior in Fe/Si_{⊚} and Mg/Si_{⊚} to the variable Si_{⊚} we find the prior of Si_{⊚} being This implies a nonuniform prior for Si_{⊚}. Outside the given interval in Table 1 the prior distribution of Si_{⊚} is essentially zero.
Appendix B: Temperature profile
We assume a fixed mantle temperature profile that is based on the Earth model. A variable temperature profile for the mantle only introduces negligible variations in the density profile (< 1%). This is demonstrated in Fig. B.1. Surface temperature is fixed at 1000 K and although temperature varies between 2100 K to 3500 K at the core mantle boundary, resultant changes in density are relatively small (Fig. B.1).
Fig. B.1
Sampled profiles of density profiles for Earth. A variable mantle temperature (gray profiles) only introduces little additional variation in density compared to profiles with fixed mantle temperature (green colors). 

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Fig. C.1
Sampled 1D marginal posterior distributions for Earth using the FMS system: prior (red), posterior (blue) and independent estimates (green) of model parameters: a) mantle Sicontent Si_{⊚}; b) mantle Fe/Si_{⊚}; c) mantle Mg/Si_{⊚}; and d) core radius R_{c}. Independent estimates are listed in Table 2. 

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Appendix C: Silicate model chemical system
To illustrate the influence of the chosen model chemical system, we solve the inverse problem for the case of the Earth as in Sect. 3.2, but using the simpler FMS system. Sampled model parameter PDFs are shown in Fig. C.1 and relative to NCFMAS (Fig. 4), we observe that with the FMS system mantle composition and core radius are less well predicted, although posterior model parameter variability is, as expected, smaller (fewer chemical components). In addition, future use of stellar determinations of Ca, Na, and Al abundances support the use of the NCFMAS system.
Appendix D: Shannon entropy measure
The Shannon entropy measure of a discrete variable X of possible states { x_{1},x_{2},..,x_{N} }, with N being the number of bins, can be written as where the x_{i} states represent the distribution histogram bins in which the probability of occurrence is given by The Shannon entropy H has a maximum H_{max} = log_{2}(N) when all N states are equally likely (e.g., a uniform distribution) and a minimum H_{min} = 0 when all realizations x_{i} fall into a single histogram bin. Clearly, H is binsize dependent. Here, the bin size used is defined such that the prior range of each model parameter is divided in ≈20 different bins.
© ESO, 2015
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