Introduction
Analytical approximation
Numerical simulation
Summary
Long time dynamical evolution of highly
elliptical satellites orbits
Eduard Kuznetsov
Polina Zakharova
Astronomical Observatory
Ural Federal University
Journees 2014
22–24 September 2014, Saint-Petersburg
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Introduction
Analytical approximation
Numerical simulation
Summary
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
Astronomical Observatory
of the Ural Federal University
Orbital evolution of HEO objects is studied by
both a positional observation method (SBG telescope)
and theoretical methods (this work)
analytical
numerical
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Introduction
Analytical approximation
Numerical simulation
Summary
Motivation
Long-term dynamical evolution near HEO
Safety of active satellites
Secular perturbations of semi-major axes
Atmospheric drag
The Poynting–Robertson effect
Long-term evolution of eccentricities and inclinations due
to the Lidov–Kozai resonance
Passage through high-order resonance zones
Formation of stochastic trajectories
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Introduction
Analytical approximation
Numerical simulation
Summary
Methods
Analytical
Resonant semi-major axis values
Critical arguments
Numerical
Positions and sizes of high-order resonance zones
Estimation of semi-major axes secular perturbations
Estimation of integrated autocorrelation function
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Introduction
Analytical approximation
Numerical simulation
Summary
Critical arguments and their frequencies
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
Critical arguments and their frequencies
Critical arguments (Allan 1967)
Φ1 = p(M + Ω + g) − qωt = ν1 t
Φ2 = p(M + g) + q(Ω − ωt) = ν2 t
Φ3 = pM + q(g + Ω − ωt) = ν3 t
Frequencies of the critical arguments
ν1 = p(nM + nΩ + ng ) − qω
ν2 = p(nM + ng ) + q(nΩ − ω)
ν3 = pnM + q(ng + nΩ − ω)
M, Ω, g are angular elements, nM , nΩ , ng are mean motions,
ω is the angular velocity of the Earth
p, q are integers
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Introduction
Analytical approximation
Numerical simulation
Summary
p:q resonances
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
p:q resonances
Types of resonances
n-resonance
ν1 ≈ 0
p:q resonance between the satellite’s mean motion nM
and the Earth’s angular velocity ω
i-resonance
ν2 ≈ 0
The position of the ascending node of the orbit
repeats periodically in a rotating coordinate system
e-resonance
ν3 ≈ 0
The position of the line of apsides of the orbit
repeats periodically in a rotating coordinate system
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Introduction
Analytical approximation
Numerical simulation
Summary
p:q resonances
17 high-order resonance relations p:q
Resonant semi-major axis values
a , k m
3 5
2 7 0 0 0
3 7
3 9
4 1
4 3
4 5
4 7
e = 0.65 and
i = 63.4◦
4 9
2 6 8 0 0
2 6 6 0 0
16 6 |p| 6 25
33 6 |q| 6 49
2
2 6 4 0 0
2 6 2 0 0
3 5
3 3
2 6 0 0 0
0
1
2
1 5
1 6
1 7
1 8
1 9
2 0
2 1
2 2
2 3
Order of resonance:
49 6 |p| + |q| 6 74
4 9
4 7
4 5
4 3
4 1
3 9
3 7
2 4
2 5
2 6
p
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Introduction
Analytical approximation
Numerical simulation
Summary
Numerical model
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
Numerical model
Numerical Model of Artificial Earth Satellites Motion
(Bordovitsyna et al. 2007)
Software developer
Research Institute of Applied Mathematics and Mechanics
of Tomsk State University
Integrator
Everhart’s method of the 19th order
Interval
24 years
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Introduction
Analytical approximation
Numerical simulation
Summary
Numerical model
The model of perturbing forces
(Kuznetsov and Kudryavtsev 2009)
the Earth’s gravitational field (EGM96, harmonics up to the
27th order and degree inclusive)
the attraction of the Moon and the Sun
the tides in the Earth’s body
the direct radiation pressure, taking into account the
shadow of the Earth (the reflection coefficient k = 1.44)
the Poynting–Robertson effect
the atmospheric drag
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Introduction
Analytical approximation
Numerical simulation
Summary
Numerical model
Initial conditions
High-elliptical orbits
a0 are consistent with resonant conditions arisen from the
analytical approximation
e0 = 0.65
Critical inclination i0 = 63.4◦
g0 = 270◦
Ω0 = 0◦ , 90◦ , 180◦ , and 270◦
Ω0 coincide with initial values of solar angle
ϕ0 = Ω0 + g0 = 270◦ , 0◦ , 90◦ , and 180◦
AMR = 0.02, 0.2, and 2 m2 /kg
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
22:45 resonance region
a , k m
3 5
2 7 0 0 0
3 7
3 9
4 1
4 3
4 5
4 7
4 9
2 6 8 0 0
2 6 6 0 0
2 2 :4 5
2
2 6 4 0 0
2 6 2 0 0
3 5
3 3
2 6 0 0 0
0
1
2
1 5
1 6
1 7
1 8
1 9
2 0
2 1
2 2
4 9
4 7
4 5
4 3
4 1
3 9
3 7
2 3
2 4
2 5
2 6
p
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the semi-major axis a
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 0◦ , AMR is 0.02 m2 /kg
a , k m
2 6 1 9 0
2 6 1 6 0
2 6 1 3 0
2 6 1 0 0
2 6 0 7 0
0
4
8
1 2
t, y e a rs
1 6
2 0
2 4
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the eccentricity e and argument of the
pericenter g near the 22:45 resonance region
a0 = 26162 km, AMR is 0.02 m2 /kg
e
0 .7 2
ϕ0 = 0◦
ϕ0 = 90◦
ϕ0 = 180◦
ϕ0 = 270◦
0 .6 8
0 .6 4
0 .6 0
2 6 0
2 6 4
2 6 8
2 7 2
2 7 6
2 8 0
g , d e g
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the inclination i
near the 22:45 resonance region
a0 = 26162 km, AMR is 0.02 m2 /kg
i, d e g
6 4 .0
6 3 .6
ϕ0 = 0◦
ϕ0 = 90◦
ϕ0 = 180◦
ϕ0 = 270◦
6 3 .2
6 2 .8
6 2 .4
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the critical argument Φ1
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 90◦ , AMR is 0.02 m2 /kg
Φ1 , d e g
3 6 0
3 0 0
2 4 0
1 8 0
1 2 0
6 0
0
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the critical argument Φ2
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 0◦ , AMR is 0.02 m2 /kg
Φ2 , d e g
3 6 0
3 0 0
2 4 0
1 8 0
1 2 0
6 0
0
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the critical argument Φ3
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 0◦ , AMR is 0.02 m2 /kg
Φ3 , d e g
3 6 0
3 0 0
2 4 0
1 8 0
1 2 0
6 0
0
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the semi-major axis a
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 0◦ , AMR is 2 m2 /kg
a , k m
2 6 2 2 0
2 6 1 6 0
2 6 1 0 0
2 6 0 4 0
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Dynamical evolution in region near the high-order resonance
Evolution of the critical argument Φ1
near the 22:45 resonance region
a0 = 26162 km, ϕ0 = 0◦ , AMR is 2 m2 /kg
Φ1 , d e g
3 6 0
3 0 0
2 4 0
1 8 0
1 2 0
6 0
0
0
4
8
1 2
1 6
2 0
2 4
t, y e a rs
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
Outline
1
Introduction
2
Analytical approximation
Critical arguments and their frequencies
p:q resonances
3
Numerical simulation
Numerical model
Dynamical evolution in region near the high-order
resonance
4
Summary
Results
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
Formation of the stochastic trajectories
The influences of the Poynting–Robertson effect
Secular decrease in the semi-major axis, which, for a
spherically symmetrical satellite with AMR = 2 m2 /kg
near the 22:45 resonance region, equals approximately
0.5 km/year
The effect weakens slightly, in resonance regions
Objects pass through the regions of high-order resonances
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
The integrated autocorrelation function A
A→1
constant time series
A → 0.5
time series representing a uniformly sampled sine wave
A tends to a finite value not far from 0.5
other periodic and quasi-periodic time series
A → 0 with a speed proportional to the inverse of the
exponential decay time
chaotic orbits
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
The integrated autocorrelation function A for the
semi-major axis a near the 22:45 resonance region
a0 = 26162 km, AMR is 0.02 m2 /kg
A
1 .2
1 .1
1 .0
ϕ0 = 0◦
ϕ0 = 90◦
ϕ0 = 180◦
ϕ0 = 270◦
0 .1 6
0 .1 2
0 .0 8
0 .0 4
0 .0 0
0
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
k
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
Conclusion
The Poynting–Robertson effect
and secular perturbations of semi-major axis
lead to the formation of weak stochastic trajectories in
HEO region.
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Introduction
Analytical approximation
Numerical simulation
Summary
Results
Thank you
for your attention!
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