periodic orbits in the problem of secular motion of an

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)

8

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)

10

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.