tokens: values of according toEq. ( 8 ). We chose six values for the scaling size of the core, 0.1, 0.2, 0.4, 0.6, 0.8, 0.9. All lines cross at point A which is the bulk density of a homogeneous Phobos. For core densities lower than the bulk density (left of point A shown in Fig. 3 ), the mantle is denser than the core, and vice versa, for core densities higher than the bulk density (right of point A), the mantle is less dense than the core. Figures 4 a and 4 b show the dependence of the mantle density on the size of the core for selected core densities. In Fig. 4 , only physical values (from 500 kg m −3 to 5000 kg m −3 ) for the densities of the mantle and the core are listed. For a given scaling factor of the core λ and the density of the core ρ c , 46 layering cases are considered in this paper. We removed the cases exceeding the allowable density range. Fig. 3 Range of densities for the Phobos two-layer model. The x -axis is the assumed density of the core, the y-axis is the resulting density of the mantle according to Eq. ( 8 ). Plots in Figs. 5 and 6 show that at the 3 σ level, the modeled and observed C 20 agree well when the mantle is denser than the core (see also Tables 3 and 4 ). To verify the effect of the core shape on the gravity field of the two-layer Phobos model, we used a triaxial ellipsoid core shape (homotheticto the best-fit triaxial ellipsoid of Phobos ( Willner et al. 2014 ), and recalculated the second-degree coefficients in a setting identical to the one used for Figs. 5 and 6 . The gravity potential of a homogeneous ellipsoid canbe computed using Ivory’s formula ( Garmier et al. 2002 ). Compared to the minimum absolute value of from Yang et al. (2019) , when the mantle is denser than the core, the cases listed in Table 3 (homothetic surface shapes) and Table 5 (homothetic ellipsoids) coincide. A test simulation for a core with a scaling factor of 0.6 and a density of 1400 kg m −3 produced a very small deviation of 0.097%, therefore, the deviation is not sensitive to the core shape, if the interface core-mantle is deeply buried. Only one test simulation is compatible with the data when the core is denser than the mantle. In this case, the core had a scaling factor of 0.1 and a density of 2200 kg m −3 , and the corresponding deviation was 0.002%.These results are summarized in Tables 5 and 6 . The disrupted and reaccreted body (DAR-F1) model (a rubble-pile internal structure), with a denser outer layer considered by Le Maistre et al. (2019) , is close to our two-layer model for a core with a scaling factor of 0.9 (the interface depth is about 1 km). Fig. 4 Mantle (a) and core (b) densities as a function of the core scaling factor of Phobos’ two-layer models. ( ρ c is the density of the core, λ is the scalingfactor of the core, and ρ m is the density of the mantle, units: kg m −3 ). All small bodies (radius smaller than 20 km) with known second order and degree gravity models ( Miller et al. 2000 ; Zuber et al. 2000 ;
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