1. Secular motion of artificial lunar satellites H. Varvoglis, S. Tzirti and K. Tsiganis Unit of Mechanics and Dynamics Department of Physics University of Thessaloniki

2.  Artificial satellites are an indispensable tool for surveying celestial bodies and relaying communication signals back to Earth.  When talking about satellite orbits, most people think of ellipses.  However most of the bodies of the Solar System are not perfectly spherically symmetric, so that the orbits of their satellites are not Keplerian!  Hence, need to calculate orbits with specific properties (e.g. invariant plane of orbit, constant orientation, constant pericenter or apocenter etc.) around non-spherically symmetric bodies.  If this turns out to be difficult, at least calculate orbits with quasi-constant elements (present work). Motivation - results in a nutshell

3. SIZE and SHAPE of the orbit: α, e (T 2 ~ α 3 ) ORIENTATION of the orbit (Euler angles): I, g, h SIZE and SHAPE of the orbit: α, e (T 2 ~ α 3 ) ORIENTATION of the orbit (Euler angles): I, g, h  Satellites are used for a number of purposes (communications, navigation, research, intelligence, surveying, meteorology etc. )‏  Satellites' orbits are selected according to the specific mission of each satellite Lunar satellites are used for: surveying, data transmission & communications Elements of a Keplerian bounded orbit (ellipse)

4. We perform:  Numerical calculation of lunar satellite orbits, under the effect of TWO perturbations: - inhomogeneous gravitational field of the central body (Moon) - neighbouring third body (Earth)  We see that - low orbits are affected primarily by the inhomogeneous field - high orbits are perturbed primarily by Earth  We need to know (from theory!) where to look for: - Orbits that have at least one element constant (e.g. eccentricity, e, or argument of pericenter, g) (application: space mission design of surveying or intelligence)

5.  Expansion of the gravitational potential in spherical harmonics  Next best integrable approximation (besides the keplerian one): two fixed centers (e.g. see Marchal, 1986)  This work: spherical harmonics up to 2 nd and 3 rd degree  Earth's perturbations to the motion of a lunar artificial satellite Plan of the talk

6.  Motion of a satellite around a spherically symmetric body: Potential that of a point mass for r > R: V = –μ/r (μ = G M) Keplerian (exactly elliptic orbit) i.e. the orbit has a constant elliptical shape and orientation  Motion of a satellite around an asymmetric body: Potential: V(r, φ, λ) = −μ/r + B(r, φ, λ) Under the perturbation of the B-terms, the orbit is not anymore an ellipse. However if B(r, φ, λ) << μ, the orbit looks like an ellipse with slowly varying osculating elements

7. Spherical harmonics

8. Including only the J 2 term: the l-averaged system is 1-D -> integrable Molniya / Tundra type orbits: T = 12 h / 24 h, e ~ 0.7, I = 63 ο.43, apogee at constant φ

9. Axisymmetric mixed case (2 nd & 3 rd order):

10. Method of work: expand the 2-FC potential in spherical harmonics. The distance between the 2-FC and their masses are calculated by equating the numerical values of the J 2 and J 3 terms to those of the Earth. Masses and distances turn out to be complex, but the potential is real! The 2-FC problem is integrable. It is also “pathological”, but only for trajectories passing “between” the centers. Satellite orbits lie in the outer region, hence the approximation is useful. 2 – FIXED CENTERS APPROXIMATION of the axisymmetric case

11. BEYOND THE AXISYMMETRIC MODEL - 1. What happens to I c if we keep the term C 22 (near-far side asymmetry) in the expansion of the potential? - 2. What happens to the stability of the orbits if we keep 3 rd order terms? - 3. What is the effect of Earth, as a third body, in the evolution of orbits? For the specific case of the Moon : J 2 /C 22 = 9.1 (Earth: J 2 /C 22 = 689.91)

12. Delaunay variables (a.k.a. first order normalization!) Method of work

13. Equations of motion

14. “Critical” orbits for the J 2 + C 22 model : - The system of equations has finally 1 d.o.f. (there is no dependence on g!)‏ - There is no I, for which condition is satisfied! - But for some I we have (quasi-critical orbits)‏ - I qc depends on the angle h o (de Saedeleer & Henrard,2006) - The rotation of the Moon weakens considerably this dependence α=R Moon +1250 km, e=0.2

16. - Write the averaged Hamiltonian function in Delaunay variables - The system of equations has now 2 d.o.f. - Study the system using a Poincaré s.o.s. and indicators of chaotic behaviour (FLI)‏ - Look for stable periodic orbits Introduction of 3 rd order terms (mainly C 31 ) ‏

17. Poincaré map Collision limit WITHOUT rotation

18. WITH rotation (a = R Moon + 1250 km)‏ Poincaré map

19. System with only J 2 : I c =63 ο.43 (Molniya, critical inclination AND frozen eccentricity!)‏ System with J 2 and J 3 : (either critical inclination OR frozen eccentricity) - 2-FC approximation System with J 2 and C 22 : no more I c, but only I qc - without rotation: strong dependence on h o, Δg, ΔI ~ 35 o - with rotation: weak dependence on h o, Δg, ΔI ~ 0 o.05 3 rd order terms: - Without rotation: Important chaotic regions and collision orbits => no orbits of practical interest! - With rotation: Mainly ordered motion in regions of practical interest (in particular at low heights, where Earth's perturbation is not important) (next talk by Stella Tzirti) => Existence of orbits of practical interest (next talk by Stella Tzirti) Low e for any I High e for I ~ 63 o Summary and conclusions