Periodic orbits in the problem of secular

motion of an artificial Lunar satellite

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

Unit of Dynamics

Section of Astrophysics, Astronomy & Mechanics

Department of Physics

Aristotle University of Thessaloniki

2

• Non-axisymmetric (3rd-degree) averaged problem:

(‘J2+J3+C22+C3i+S3i(+Earth pert.)’, i=1,3)

Kepler

EarthKepler perturbations rotation

( can be omitted, since =const.)

= ( )

a

+ + +

H

HH H H H

• The rotation of the primary is included in all cases• Consideration of the Earth: how it changes the phase space• Periodic orbits (POs) for the non-axisymmetric problem:

- how they emanate from the solutions of the axisymmetric problem- their distribution in e-I, g-I diagrams for LOW and HIGH lunar orbits

α=RM+100 km α=RM+1250 km

3

• Axisymmetric problem (‘J2+J3’):

⇒ Orbits of practical interest:

- no collision with the Moon

- require minor active control (constant characteristics)

⇒ Especially low polar orbits, useful for surveying the lunar surface, determining the amplitude of gravitational harmonics

4

• The (lowest degree) averaged Earth effect:

(De Saedeleer 2006)

(μ΄=GM΄)

5

• 3rd degree model without the Earth effect (s.o.s., 2 d.o.f.)

‘critical inclination’orbit

‘frozen eccentricity’orbit

Collision limit

6

α=30

00 k

mα=

1000

0 km

• The Earth effect at high lunar satellite orbits (ho=π)

7

• The Earth effect at high inclinations (I>40 deg)

0 2 4 6 8 100.094

0.096

0.098

0.100

0.102

time [Lunar Month]

eccentricity inclination [rad]-0.86

0 2 4 6 8 10

0.324

0.326

0.328

0.330

time [Lunar Month]

(H(t)/L)2

We select In. Conditions close to a PO: α=3000 km, e=0.1, I=55o.1, h=π, g=1.6 rad

⇒ e and I perform librations with the same frequency and opposite phases

⇒ (H/L)2 has constant mean value (it would be an integral of motion if Hamiltonian contained only the Earth effect)

⇒ similar to the Kozai-Lidov effect in asteroids (perturbed by the Jupiter)

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0 20 40 60 80

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

e

Ι [deg]

'3G' model '3G-R' model '3G-R-E' model

• Position of POs on s.o.s. for HIGH satellite orbits(α=RMoon+1250 km, ho=Pi)

9

• POs for LOW satellite orbits-different models

(without the Earth effect)

0 20 40 60 80

3.6

4.0

4.4

4.8

5.2

a=RMoon+100 km, ho=Pi

g [r

ad]

inclination [deg]

J2+J3 J2+J3+C22 J2+J3+C22+C3i+S3i

0 20 40 60 800.000

0.005

0.010

0.015

0.020

0.025

0.030a=RMoon+100 km, ho=Pi

ecce

ntric

ity

inclination [deg]

J2+J3 J2+J3+C22 J2+J3+C22+C3i+S3i

The initial eccentricity and argument of pericenter of POs (black) are different from these of the axisymmetric model (red)

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0 20 40 60 80

-40

-20

0

circulation

Mea

n g

- 270

[deg

]

inclination [deg]

Without the Earth effect With the Earth effect axisymmetric problem

0 20 40 60 80180

210

240

270

300

330

g o o

f PO

s [d

eg]

inclination [deg]

• POs for LOW satellite orbits, WITH and WITHOUT the Earth effect

⇒ For low I values, g performs rotations, that is why mean g value is not shown here

⇒ The Earth effect is not important for low semi-major axis values and inclinations far from the critical value

⇒ <g>-270 ~ 2-3 degrees for polar orbits

11

• POs for HIGH satellite orbits, WITH and WITHOUT the Earth effect

⇒ The Earth affects strongly the POs, for I>58o

⇒ For I<58o, go might be quite different from the axisymmetric solution, but its mean value is very close to that (large libration amplitudes)

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go=300.5o

<g>=266o

go=306o

<g>=257o

• POs for LOW lunar satellites (ho=π)

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-0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

I0=300.5, e0=0.014, g0=4.7 rad

e Cos(g)

e Si

n(g)

α =RMoon+100 km, ho=π

Green: Collision limit

Red: PO3 integrated using 3rd degree model

Black: PO3 integrated using 7th degree model

(Earth effect included)

• PO3 using two different models - low incl. orbit

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-0.08 -0.04 0.00 0.04 0.08

-0.04

0.00I0=900.18, e0=0.017, g0=4.62 rad

e Cos(g)

e Si

n(g)

• PO3 using two different models – polar orbit

α =RMoon+100 km, ho=π

Green: Collision limit

Red: PO3 integrated using 3rd degree model

Black: PO3 integrated using 7th degree model

Collision after ~5 months

(Earth effect included)

15

• Polar frozen3 orbit

α =RMoon+100 km, ho=π

Blue: Polar frozen3 orbit,

integrated using 7th degree model

Green: Collision limit

-0,09 -0,06 -0,03 0,00 0,03 0,06 0,09-0,08

-0,06

-0,04

-0,02

0,00

0,02a=RM+100 km, I0=900, e0=0.0199, g0=3π/2

e Si

n(g)

e Cos(g)

Red: frozen orbit corrected by additional zonal termsKnezevic&Milani (1998)

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• Conclusions

⇒ Axisymmetric problem: (efrozen, gfrozen=3π/2)

⇒ Non-axisymmetric terms affect the position of the POs on Poincare s.o.s., changing significantly the g value (depending on the inclination)

⇒ Eccentricity value is affected too, especially for low orbits.

⇒ The Earth effect is important even for low semi-major axis values and inclinations close to the critical value.

⇒ It becomes more dominant for larger semi-major axis values, where it affects all inclinations.

⇒ Numerical integrations with up to 7th degree terms show that our 3rd

degree model is not adequate (some POs collide with the Moon).

⇒ Analytical Perturbation theory currently under way.