Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.gao.spb.ru/english/as/j2014/presentations/kuznetsov.pdf Äàòà èçìåíåíèÿ: Sun Sep 28 20:12:05 2014 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 05:55:04 2016 Êîäèðîâêà: Ïîèñêîâûå ñëîâà: òóìàííîñòü àíäðîìåäû

Introduction

Analytical approximation

Numerical simulation

Summary

Long time dynamical evolution of highly elliptical satellites orbits
Eduard Kuznetsov Polina Zakharova

Astronomical Obser vator y Ural Federal University

Journees 2014 22­24 September 2014, Saint-Petersburg

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Introduction

Analytical approximation

Numerical simulation

Summary

Outline
1 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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3

4


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 per turbations of semi-major axes
Atmospheric drag The Poynting­Rober tson effect

Long-term evolution of eccentricities and inclinations due to the Lidov­Kozai 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 per turbations Estimation of integrated autocorrelation function

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Introduction

Analytical approximation

Numerical simulation

Summary

Critical arguments and their frequencies

Outline
1 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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3

4


Introduction

Analytical approximation

Numerical simulation

Summary

Critical arguments and their frequencies

Critical arguments (Allan 1967) 1 = p(M + + g ) - q t = 1 t 2 = p(M + g ) + q ( - t ) = 2 t 3 = pM + q (g + - t ) = 3 t Frequencies of the critical arguments 1 = p(nM + n + ng ) - q 2 = p(nM + ng ) + q (n - ) 3 = pnM + q (ng + n - ) M , , g are angular elements, nM , n , ng are mean motions, is the angular velocity of the Ear th p, q are integers

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Introduction p:q resonances

Analytical approximation

Numerical simulation

Summary

Outline
1 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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Introduction p:q resonances

Analytical approximation

Numerical simulation

Summary

Types of resonances
n-resonance 1 0 p:q resonance between the satellite's mean motion nM and the Ear th's angular velocity i -resonance 2 0 The position of the ascending node of the orbit repeats periodically in a rotating coordinate system e-resonance 3 0 The position of the line of apsides of the orbit repeats periodically in a rotating coordinate system

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Introduction p:q resonances

Analytical approximation

Numerical simulation

Summary

17 high-order resonance relations p:q
Resonant semi-major axis values
27000 26800 26600

a,km

35

37

39

41

43

45

47

49

e = 0.65 and i = 63.4 16 33 |p| |q | 25 49

2

26400 26200 26000 0 1 2 15 16

33
17

35
18

37
19

39
20

41
21

43
22

45

47

49

Order of resonance: 49 |p| + |q | 74
25 26
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p

23

24


Introduction Numerical model

Analytical approximation

Numerical simulation

Summary

Outline
1 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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3

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Introduction Numerical model

Analytical approximation

Numerical simulation

Summary

Numerical Model of Ar tificial Ear th Satellites Motion (Bordovitsyna et al. 2007)

Software developer Research Institute of Applied Mathematics and Mechanics of Tomsk State University Integrator Everhar t's method of the 19th order Interval 24 years
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Introduction Numerical model

Analytical approximation

Numerical simulation

Summary

The model of per turbing forces (Kuznetsov and Kudryavtsev 2009)

the Ear th'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 Ear th's body the direct radiation pressure, taking into account the shadow of the Ear th (the reflection coefficient k = 1.44) the Poynting­Rober tson effect the atmospheric drag
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Introduction Numerical model

Analytical approximation

Numerical simulation

Summary

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 0 = 0 , 90 , 180 , and 270 0 coincide with initial values of solar angle 0 = 0 + 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 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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3

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Introduction

Analytical approximation

Numerical simulation

Summary

Dynamical evolution in region near the high-order resonance

22:45 resonance region
a,km

27000 26800 26600

35

37

39

41

43

45

47

49

2

2 2 :4 5

26400 26200 26000 0 1 2 15 16

33
17

35
18

37
19

39
20

41
21

43
22

45

47

49

p

23

24

25

26
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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, 0 = 0 , AMR is 0.02 m2 /kg
26190 26160 26130 26100 26070 0 4 8

a,km

t, y e a rs

12

16

20

24

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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
0 .7 2

e

0 .6 8

0 .6 4

0 = 0 0 = 90 0 = 180 0 = 270

0 .6 0

260

264

268

272

276

g,deg

280
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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
6 4 .0 6 3 .6 6 3 .2 6 2 .8 6 2 .4 0 4 8 12 16 20

i, d e g

0 = 0 0 = 90 0 = 180 0 = 270

t, y e a rs

24

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Introduction

Analytical approximation

Numerical simulation

Summary

Dynamical evolution in region near the high-order resonance

Evolution of the critical argument 1 near the 22:45 resonance region
a0 = 26162 km, 0 = 90 , AMR is 0.02 m2 /kg
360 300 240 180 120 60 0 0 4 8 12 16 20 24

1 , d e g

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 2 near the 22:45 resonance region
a0 = 26162 km, 0 = 0 , AMR is 0.02 m2 /kg
360 300 240 180 120 60 0 0 4 8 12 16 20 24

2 , d e g

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 3 near the 22:45 resonance region
a0 = 26162 km, 0 = 0 , AMR is 0.02 m2 /kg
3 , d e g
360 300 240 180 120 60 0 0 4 8 12 16 20 24

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, 0 = 0 , AMR is 2 m2 /kg
26220

a,km

26160

26100

26040 0

4

8

12

16

20

t, y e a rs

24

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Introduction

Analytical approximation

Numerical simulation

Summary

Dynamical evolution in region near the high-order resonance

Evolution of the critical argument 1 near the 22:45 resonance region
a0 = 26162 km, 0 = 0 , AMR is 2 m2 /kg
1 , d e g
360 300 240 180 120 60 0 0 4 8 12 16 20 24

t, y e a rs

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Introduction Results

Analytical approximation

Numerical simulation

Summary

Outline
1 2

Introduction Analytical approximation Critical arguments and their frequencies p:q resonances Numerical simulation Numerical model Dynamical evolution in region near the high-order resonance Summar y Results
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3

4


Introduction Results

Analytical approximation

Numerical simulation

Summary

Formation of the stochastic trajectories

The influences of the Poynting­Rober tson 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 Results

Analytical approximation

Numerical simulation

Summary

The integrated autocorrelation function A
A1 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 propor tional to the inverse of the exponential decay time chaotic orbits
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Introduction Results

Analytical approximation

Numerical simulation

Summary

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
1 .2 1 .1 1 .0 0 .1 6 0 .1 2 0 .0 8 0 .0 4 0 .0 0 0 100 200 300 400 500 600

A

0 = 0 0 = 90 0 = 180 0 = 270

k

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Introduction Results

Analytical approximation

Numerical simulation

Summary

Conclusion

The Poynting­Rober tson effect and secular per turbations of semi-major axis lead to the formation of weak stochastic trajectories in HEO region.

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Introduction Results

Analytical approximation

Numerical simulation

Summary

Thank you for your attention!

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