V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA)
description
Transcript of V.V.Sidorenko (Keldysh Institute of Applied Mathematics, Moscow, RUSSIA)
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