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Quasi-satellite orbits in the context of coorbital dynamics. V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA) A.V.Artemyev, A.I.Neishtadt, L.M.Zelenyi (Space Research Institute, Moscow, RUSSIA). Moscow , 2012. Quasi-satellite orbits. - PowerPoint PPT Presentation
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V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA)
A.V.Artemyev, A.I.Neishtadt, L.M.Zelenyi (Space Research Institute, Moscow, RUSSIA)
Quasi-satellite orbits in the context of coorbital
dynamics
Moscow, 2012
Quasi-satellite orbits
1:1 mean motion resonance!
Resonance phase ’ librates around 0 (and ’ are the mean longitudes of the asteroid and of the planet)
(based on pictures from http://www.astro.uwo.ca/~wiegert/quasi/quasi.html)
Historical background:
•J. Jackson (1913) – the first(?) discussion of QS-orbits;
•A.Yu.Kogan (1988), M.L.Lidov, M.A.Vashkovyak (1994) – the consideration of the QS-orbits in connection with the russian space project “Phobos”
•Namouni(1999) , Namouni et. al (1999), S.Mikkola, K.Innanen (2004),… - the investigations of the secular evolution in the case of the motion in QS-orbit
Quasi-satellite orbits
Real asteroids in QS-orbits:2002VE68 – Venus QS; 2003YN107, 2004GU9, 2006FV35 – Earth QS;2001QQ199, 2004AE9 – Jupiter’s QS……………………
Secular effects: examples
Nonplanar circular restricted three-body problem “Sun-Planet-
Asteroid”
Parameters: 2, 1 coshP e i
Phase portrait of the slow motion: mathching of the trajectories on the
uncertainty curve
Scaling
A – the motion in QS-orbit is perpetual
21 hP
2 2~ i e
Time scales at the resonanceT1 - orbital motions periods
T2 - timescale of rotations/oscillations of the resonant
argument (some combination of asteroid and planet mean longitudes)
T3 - secular evolution of asteroid’s eccentricity e,
inclination i, argument of prihelion ω and ascending node longitude Ω .
T1 << T2 << T3
Strategy: double averaging of the motion equations
Nonplanar circular restricted three-body problem “Sun-Planet-
Asteroid”
Resonant approximationScale transformation
0
0
( ) / ,
, 1
L L t
L
Slow-fast system
,
,
dx W dy W
d y d x
d d W
d d
2
3
( , , , )
1 cos
h
h
W W x y P
P e i
SF-Hamiltonian2
( , , , )2 hH W x y P
d d dy dx
Slow variables
Fast variables
Symplecticstructure
-approximate integral of the problem
Regular variables
2(1 ) cos , 2(1 ) sinh hx P g y P g
2 2( , ) ( ) { 2(1 )}h hx y D P x y P
Relationship with the Keplerian elements:
2 2 2 2 22 2
1( ) 4 ( ) , cos
4 2 ( )hP
e x y x y ix y
2 22 2
2 2
2 arccos , 0
( 0)
arccos , 0
xy
x yx y
xy
x y
Averaging over the fast subsystem solutions on the level Н = ξ
,dx W dy W
d y x
( , , , )
0
1( , , ( , , , , ), ))
( , , , )
,
hT x y P
h hh
V Wx y x y P P d
T x y P
x y
Problem: what solution of the fast subsystem should be used
for averaging ?
QS-orbit or HS-orbit?
The accuracy of )(O over time intervals /1~
Asteroid 164207 (2004GU9)
Asteroid 164207 (2004GU9)
Asteroid 164207 (2004GU9 )
Asteroid 164207 (2004GU9)
Asteroid 2006FV35
Asteroid 2006FV35
Conclusions:
•Row classification of slow evolution scenarios is presented;•The criterion to distinguish between the perpertual and temporarily motion in QS-orbit is established