1. Sounding of the interior structure of Galilean satellite Io using the parameters of the theory of figure and gravitational field in the second approximation V.N.Zharkov and T.V.Gudkova Schmidt Institute of Physics of the Earth B.Gruzinskaya, 10 123995 Moscow - RUSSIA (e-mail: zharkov@ifz.ru)

2. All Galilean satellites are in synchronous rotation; their orbits are nearly circular and lie in the equatorial plane of Jupiter. Io is the large satellite closest to Jupiter. Therefore, the influence of Jupiter’s tidal potential on the equilibrium figure and gravitational field of Io is appreciably stronger than it is on the remaining large satellites. For the theory of Io’s figure to be consistent with currently available observational data, it must include effects of the second order in smallness. In the first approximation, the ratio of the moments J 2 and C 22 : J 2 =10/3 C 22. The parameters of the gravitational field for Io determined in the Galileo space mission have shown that relation holds with a high accuracy. For Io, J 2 and C 22 are given to the fourth decimal place. This main relation that is used to judge whether Io has an equilibrium figure was derived in the first approximation. Therefore, the following question arises: With what accuracy is this theoretical ratio valid? To answer this question, we must construct a theory in the next (second) approximation by including the terms of order  2 (  is the small parameter of theory of figure, defined as  =3π/(Gρ 0 τ 2 ), where ρ 0 and τ are the average density and rotation period of Io, respectively. This is the main goal of our analysis.

3. 1. Introduction Figure theory To zeroth approximation the equilibrium figure of a body has the form of a sphere with the mean radius s1. In the first approximation the sphere transforms to a triaxial ellipsoid (normal figure) with the equatorial semiaxes a, b and the polar semiaxis c. In the second approximation equilibrium figure is deviated from a triaxial ellipsoid.

4. www.ifz.ru 4 The second basic problem

5. www.ifz.ru 5 The first and the second approximations

6. www.ifz.ru 6 Results

7. ALGEBRAIC RELATIONS – THE DUALISM IN THE EQUILIBRIUM FIGURE THEORY The figure parameters specify the shape of the figure of an equilibrium satellite. The gravitational moments specify the external gravitational field of an equilibrium satellite. (A) or inversly: (B) If the satellite is in hydrostatic equilibrium, then the coefficients in the expansion of its external gravitational field in terms of spherical functions can be determined by measuring its figure parameters. The reverse is also true. Relations (А) and (B) express the dualism in the figure theory.

8. Models The Galilean satellites of Jupiter are in a state close to hydrostatic equilibrium, then data on their figures and gravitational fields allow us to impose a constraint on the density distribution in the interiors of these bodies and, thereby, to make progress in modeling their internal structure. To show the effects of the second approximation, two three-layer trial models of Io are used [2]. The considered models of the Io’s interiors differ by the size and density of the core Io1 (the core density  c =5150 kg/m 3 : Fe-FeS eutectic) and Io3 (  c =4600 kg/m 3 : FeS with an admixture of nickel), while having the same thickness and density of the crust, and the mantle density difference is only 20 kg/m 3. The models have a thin 50-km crust with the density  crust=2700 kg/m 3. An important parameter of the silicate reservoir is the magnesium number Mg#, the ratio of the number of magnesium atoms to the sum of the numbers of magnesium and iron atoms (occasionally, this ratio multiplied by 100 is used). Theoretically, such an important parameter as the mantle silicate iron content (Fe#=1-Mg#) in the model Io3 is equal to 0.265. This value is a factor of two and half higher than the value for mantle silicates in the Earth. The high iron content in the Io3 model leads to high mantle density. Therefore, one can assume, that if the accuracy of the gravity field determination for Io is significantly improved, the effects of the second approximation will put restrictions on the value of average mantle density of Io.

9. . n The method for solving the equations of the figure theory is described in detail in [1]. A computer code was developed to calculate figures functions s 4 (s), s 42 (s) and s 44 (s) for the Io models. Tables 1 and 2 give results of numerical solutions of figure equations for the Io1 and Io3 models [2] at s=s 1 (for visual perception tables list the normalized figure parameters and gravitational moments. As seen from Table 1, including the second order terms in J 2 and C 22 decreases the values by two units in the third decimal digit. To make clear insight into the problem, we plot normalized figure functions s 2 (s), s 22 (s) (Fig. 1a) and s 4 (s),s 42 (s) (Fig. 1b). Figure functions s 2 (s),s 22 (s) and s 31 (s),s 33 (s) are proportional to the Love functions h 2 (s) and h 3 (s)(Fig.2), respectively. The outer regions of Io’s interiors influence more the Love number h 3 (s), then h 2 (s). The same fact is seen, when comparing the figure functions s 2 (s), s 22 (s) (Fig. 1a) and figure functions s 4 (s), s 42 (s), s 44 (s) (Fig. 1b). In that way, functions of the second approximations 4 (s), s 42 (s), s 44 (s) sound the density distribution of the external zones to a greater extent than functions of the first approximation s 2 (s), s 22 (s). Table 1: Normalized figure parameters s n and gravitational coefficients J n and C nm (the first order) and the second order corrections for the model Io1. Table 2: Normalized figure parameters s n and gravitational coefficients J n and C nm (the second order) for the models Io1 and Io3 Fig. 1: The distribution of the parameters of the equilibrium figure of Io s2(s), s22 (s) (a) and s4(s), s42(s), s44(s) (b) along the planetary radius. Thfunctions s2(s), s22 (s) and s4(s), s42(s), s44(s) are not normalized.e.

10. Fig. 2. Love numbers h 2 and h 3 along the planetary radius.

11. The effects of the second approximation on the figure parameters and gravitational moments of the satellite Io have been considered. It turns out that the account of the second order values decrease gravitational moments J 2 and C 22 by 2 units in the third decimal digit. To calculate the figure parameters s 4, s 42 and s 44 and consequently gravitational moments J 4, C 42 and C 44, three integro- differential equations [3] for trial model density distributions were first solved numerically and, as a result, all second order corrections were obtained. We have estimated the contribution from the effects of the second approximation to the lengths of the semiaxes a, b, and c for the equilibrium figure of Io. Including the corrections of second order changes the a, b, and c semiaxes by 55, 9, and -4 m, respectively. Conclusion

12. 12 Thank you! 10, Bolshaya Gruzinskaya str. Moscow 123995, Russia zharkov@ifz.ru