Models, figures and gravitational moments of Jupiter’s satellite Io: Effects of the second order approximation

The effects of the equilibrium figure theory to within terms of the second order in a small parameter α on figure parameters and gravitational moments of the Galilean satellite Io have been considered. Integro-differential equations of the theory of figure to second order have been first solved numerically. Relations between the low-order coefficients of the gravitational field for satellites in hydrostatic equilibrium are generalized according to the second order theory. To show the effects of the second approximation, two three-layer trial models of Io are used. The considered models of the Io’s interiors differ by the size and density of the core, while having the same thickness and density of the crust, and the mantle density difference is only 20 kg/m³. The corrections of second order in smallness to the gravitational moments J₂ and C₂₂ decrease the third decimal digit of model gravitational moments by two units. As the effects of third and forth harmonics are determined mostly by outer layers of Io, to distinguish between model mantle density, the gravitational moments J₄, C₄₂ and C₄₄ should be determined to accuracy with three or four decimal digits. The second order corrections mostly effect the semi-axis a, and less the semi-axes b and c.