Secular motionSecular motionof artificial lunar of artificial lunar satellitessatellites

H. Varvoglis, S. Tzirti and K. TsiganisH. Varvoglis, S. Tzirti and K. Tsiganis

Unit of Mechanics and DynamicsUnit of Mechanics and Dynamics

Department of PhysicsDepartment of Physics

University of ThessalonikiUniversity of Thessaloniki

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 Motivation - results in a nutshellnutshell

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)Elements of a Keplerian bounded orbit (ellipse)

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)

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

Plan of the Plan of the talktalk

Motion of a satellite aMotion of a satellite aroundround a spherically symmetric body: 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

2 1

sin( ) cos( ) sin( ) (sin )n nn

n n nm nm nmn m

R RB J P C m S m P

r r r

Motion of a satellite aMotion of a satellite aroundround an asymmetric body: 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

Spherical harmonics

Including only the J2 term: the l-averaged system is 1-D -> integrable

Molniya / Tundra type orbits: T = 12 h / 24 h, e ~ 0.7, I = 63ο.43, apogee at constant φ

22

2 oc2 22

3 (1 5cos ) 0 for =63 .43 ( , , =const.)

4(1 ): n J I R

g g I a e Ie a

J

2 2 2 2 22 3

2 2 22

32

32 2

o c o o o.

.

2 3

3 1 5cos 1 sin cos sin 1

4(1 ) 2 1 sin

3 sin 1 5cos cos

8 (1 )

= , , ,

('critical inclin0

0

+ :n J I JR R I e I g

ge a J a e I e

n Re J I I g

e a

I I a e g

g

e

J J

.

o o o

ation' orbits)

g = ± π/2, , such that 0

('frozen eccentricity' orbits)

a e g

Axisymmetric mixed case (2Axisymmetric mixed case (2ndnd & 3 & 3rdrd order):order):

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.

2 – FIXED CENTERS 2 – FIXED CENTERS APPROXIMATIONAPPROXIMATION

of the axisymmetric caseof the axisymmetric case

BEYOND THE AXISYMMETRIC MODEL

- 1. What happens to Ic if 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 3rd 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:

J2 /C22= 9.1 (Earth: J2 /C22 = 689.91)

2, = 1- , = cosL a G L e H G I

( , , ) Earthr VV Br

Keple Earr asymmetric trotation h ( , , , ) = K v r K K K K

( , , , ) { ( , , , , , ) } ( , , , , , )K v r K a e I h g l K L G H l g h

sat. Moon

2π

0

=>

1Averaging with respect to the "fast" angle d :

2

T T

l H K l

3 d.o.f. 2 d.o.f.

DelaunayDelaunay variables variables

(a.k.a. first order normalization!)

Method of Method of workwork

/ , / 0

/ , /

/ , /

l L L l

g G G g

h H H h

H H

H H

H H

Equations of Equations of motionmotion

““CriticalCritical”” orbits for the orbits for the JJ22 + + CC22 22 modelmodel ::

- 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)

- Iqc depends on the angle ho

(de Saedeleer & Henrard ,2006)

- The rotation of the Moon weakens considerably this dependence

0g

0t

g

0g

α=RMoon+1250 km, e=0.2

α = RMoon + 1250 km, e = 0.2

- 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 3Introduction of 3rdrd order terms order terms ((mainlymainly CC3131))

Poincaré map

Collision limitCollision limit

WITHOUT rotationWITHOUT rotation

WITH rotationWITH rotation ((aa == RRMoonMoon ++ 1250 km)1250 km)

Poincaré map

System with only J2: Ic=63ο.43 (Molniya, critical inclination AND frozen eccentricity!)

System with J2 and J3: (either critical inclination OR frozen eccentricity)

- 2-FC approximation

System with J2 and C22: no more Ic , but only Iqc

- without rotation: strong dependence on ho , Δg, ΔI ~ 35o

- with rotation: weak dependence on ho , Δg, ΔI ~ 0o.05

3rd 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)

=> Existence of orbits of practical interest (next talk by Stella Tzirti)(next talk by Stella Tzirti)

LowLow ee for any for any II

High High ee forfor II ~~ 6363oo

Summary and conclusionsSummary and conclusions