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
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
““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
- 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))
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