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