Secular motion of artificial lunar satellites

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

2. Motivation - resultsin a nutshell • 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).

3. Elements of a Keplerian bounded orbit (ellipse) 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 numberofpurposes (communications, navigation, research, intelligence, surveying, meteorology etc.)‏ • Satellites' orbits are selected according to the specificmission ofeachsatellite Lunar satellites are used for: surveying, data transmission & communications

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 bythe 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. Plan of the talk • 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 2nd and 3rd degree • Earth's perturbations to the motion of a lunar artificial satellite

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 J2 term: the l-averaged system is 1-D -> integrableMolniya / Tundra type orbits: T=12 h / 24 h, e~0.7, I=63ο.43, apogee at constant φ

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

10. 2 – FIXED CENTERS APPROXIMATION of the axisymmetric case • 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 J2 and J3 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.

11. BEYOND THE AXISYMMETRIC MODEL • 1. What happens to Icif we keep the term C22(near-far side asymmetry) in the expansion of the potential? • 2. What happens to the stability of the orbits if we keep 3rdorder terms? • 3. What is the effect of Earth, as a third body, in the evolution of orbits? For the specific case of the Moon: J2/C22= 9.1 (Earth: J2/C22= 689.91)

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

13. Equations of motion

14. “Critical”orbitsfortheJ2 + C22 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)‏ • Iqcdepends on the angle ho (de Saedeleer & Henrard ,2006) • The rotation of the Moon weakens considerably this dependence α=RMoon+1250 km, e=0.2

15. α=RMoon+1250 km, e=0.2

16. Introductionof 3rdorderterms(mainlyC31)‏ • 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

17. WITHOUT rotation Poincaré map Collisionlimit

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

19. Summaryandconclusions • SystemwithonlyJ2: Ic=63ο.43(Molniya, critical inclination AND frozen eccentricity!)‏ • System with J2 and J3: (either critical inclination OR frozen eccentricity) - 2-FC approximation • SystemwithJ2andC22: nomoreIc, butonlyIqc - withoutrotation: strongdependenceonho , Δg, ΔI~35o - withrotation: weakdependenceonho, Δg, ΔI~0o.05 • 3rdorderterms: - Withoutrotation: Importantchaoticregionsandcollisionorbits => noorbitsofpracticalinterest! - Withrotation: Mainlyorderedmotioninregionsofpracticalinterest (inparticularatlowheights, whereEarth's perturbationisnotimportant) => Existenceoforbitsofpracticalinterest(next talk by Stella Tzirti) HigheforI~63o LoweforanyI