The Moscow State University of Geodesy and Cartography (MIIGAiK)

1. Uchaev Dm.V.1, Oberst J.1, 2, 3,Malinnikov V.A.1, Willner K.3, UchaevD.V.1, Prutov I.S.1 The Second Moscow Solar System Symposium: Moons and Planets THE PHOBOS GRAVITATIONAL FIELD MODELED ON THE BASIS OF ITS TOPOGRAPHY • The Moscow State University of Geodesy and Cartography (MIIGAiK) • Department for Geodesy and Geoinformation Science, Technical University Berlin • German Aerospace Center, Institute of Planetary Research 10-14 October 2011. Space Research Institute, Moscow

2. Outline • Introduction • The shape model of Phobos • The gravity field of Phobos • Conclusion and outlook • In future

3. Introduction Shape model Gravity field Conclusion In future Introduction Developing a model for Phobos’ gravity field is crucial for: • study of inner structure of Phobos; • investigating dynamic environment at the surface; • tracking and navigating of spacecraft near or landing on Phobos. Ways to model the gravity field of a small body: • Harmonic Expansion; • Cube Elements; • Polyhedron • Wavelets.

4. Introduction Shape model Gravity field Conclusion In future Spatial distribution of Phobos control points Fig. 1. Spatial distribution of Phobos control points of the using network (red circles) in comparison with the control point network K.Willner (green circles) (K.Willner, 2008)

5. Introduction Shape model Gravity field Conclusion In future Tikhonov regularization

6. Introduction Shape model Gravity field Conclusion In future Shape model of Phobos c a b d f e Fig. 1. Models of Phobos' leading and trailing sides, respectively, and mean residuals of the models to the GCPs after fitting expansion models as modeled by Willner (a, b, c ) (Willner, K. et all.,2010) and our group (d, e, f) with the degree and order 18 spherical harmonic functions. 794 control points were used to compute the coefficients of our expansion model.

7. Introduction Shape model Gravity field Conclusion In future Coefficients of the Spherical Expansion Models

8. Introduction Shape model Gravity field Conclusion In future Coefficients of the Spherical Expansion Models

9. Introduction Shape model Gravity field Conclusion In future Coefficients of the Spherical Expansion Models a Fig. 1. Hausdorff measure between shape model of Phobos as modeled by Willner and our group: a – leading side, b – trailing side (Red color is zero) b

10. Introduction Shape model Gravity field Conclusion In future Model for the gravity field

11. Introduction Shape model Gravity field Conclusion In future Gravitational potential on a 14km sphere Analytical result a b Fig. 1. Gravitational potential on a 14 km sphere outside Phobos obtained by analytical method: a – Shi Xian (2010), b – our result. The contours represent the topography.

12. Introduction Shape model Gravity field Conclusion In future Conclusion It is shown that the harmonic coefficients determined by developed the shape model can readily be used to calculate the gravitational potential on a sphere outside Phobos. The Tikhonov regularization method can be used with success for determination of coefficients in an expansion over spherical functions

13. Introduction Shape model Gravity field Conclusion In future In future • To simulate tidal and centrifugal potentials of Phobos • To develop a new representation of the gravity potential fields based on wavelet-multifractal approach • To create a network of control points on Phobos with higher accuracy

14. Thanks for your attention!