Recently, it is yet difficult to detect the existence of Martian solid inner core merely based on Mars seismic InSight data. To deal with this problem, our study intends to use the mean density and mean moment of inertia factor to constrain the size and density of Martian solid inner core. Using Mars high-degree gravity field models JGMRO120f and GMM3-120, and considering the recent precession rate, we obtained the mean density and mean moment of inertia factor, which are treated as observed values. Referring to the 4-layers internal structure model of Mars, and considering the 4 parameters including crustal density, mantle density, density of outer core, size and density of inner core, we calculated the modeled values of the Martian mean density and the mean moment of inertia factor. From the minimum residuals between observed and modeled values of mean density as well as that of mean moment of inertia factor, it is found that the two gravity fields models have the same result of distribution of free parameters. As to the optimized values of the free parameters, the two gravity field models even have the same results. Furthermore, the optimized crustal density, mantel density and density of outer core approach other studies, indicating the dependence of our results. Finally, our result demonstrates that Mars likely has a solid inner core with a size close to 840 km, and the density of inner core is near to 6950 kg·m^-3. Our result implies that Mars has an inner core not fully composed of pure iron, which is consistent with the recent study that Mars requires a substantial complement of light elements in Martian core. However, it is further needed to constrain the size and composition of Martian inner core due to the non-uniqueness of inversion results. With the improvement of processing technology on the InSight data, it can be further constrained for the size and composition of Martian inner core.

This study focuses on the size of composition of lunar core. To deal with this issue, we considered the lunar mean density and mean moment of inertia factor in our inversion. We used the degree-2 coefficients of lunar gravity field model GL990D and the lunar physical liberation parameters to compute mean moment of inertia factor, which is treated as an observed value. We also computed the observed value of the mean density according the total mass of the Moon. Based on the interior structure with various layers, we deduced the modeled expressions for the lunar mean density and mean moment of inertia factor. Summing squares of the difference between the observed and modeled values as an inversion criterion, we estimated the multi-parameters based on the simulated annealing algorithm. By considering the lunar interior structure with three layers, the estimated size of the lunar core is around 470 km, and the density of the core gets close to 5486 kg·m^-3. The computed size and density of the lunar core are close to other studies, validating our algorithm. We then considered the scenarios that the lunar core differentiated with a solid inner and a liquid outer core. The good-inversed outer core is close to 385 km, while the inner core approaches to 350 km. By using the good-inversed sizes as fixed parameters, it is found that the inner core is quite denser to 7879 kg<teshuzifu>×m^-³ than the outer core, which is estimated around 4618 kg<teshuzifu>×m^-³. Our result indicates that the outer core is composed by ferrous sulfide (FeS), while the inner core is composed by ferrous or ferro-nickel at 3.56 billion years ago when the lunar core dynamo ended.