The dynamics of the elliptic Hill problem : Periodic orbits and stability

George VoyatzisSection of Astrophysics, Astronomy and MechanicsUniversity of Thessaloniki, Greece

Collaborators :I. GoliasH. Varvoglis

8th Alexander von Humboldt Colloquium for Celestial MechanicsResonances in n-body systemsBad Hofgastein, Salzburg, Austria20.03.-26.03.2011Hotel Winkler

Introduction

Hill – models

•Circular planar (Henon 1969,1970)

•Circular spatial (Henon 1973)

•Elliptic Planar (Ichtiaroglou 1980)

•Elliptic Spatial (-)

•General planar (Henon & Petit 1986)

m0

m1

m0 1m m m� �

1

1 0

0, 0 (trivial problem, 2BP)m

mm m

�� � ��

Hill’s transformation (1886) : � non-trivial, parameter free model

Applications :

• Dynamics of satellite orbits (Shen and Tremaine, 2008)

• Binary asteroids (Chauvineau and Mignard, 1990)

• Spacecraft dynamics (Villac, 2003)

1/31,X X XX X �� ��� ��

� � � ��

Contents

• Equations of the E.H. model

• Dynamical aspects of the C.H. model

• Periodic orbits, continuation and stability in the E.H. model

• Families of periodic orbits

• Stability maps – the effect of planetary eccentricity

The 3BP basic model

21 1

12 4 3 3 31 1 1 1 2

2

12 4 3 3 31

2 3 31

1

1

1 1 2

21 (1 ) 0

( ) (1 )2 2

(1 )2 2

x x x xQ Q Qx y x x y

x x x r r

Q Q Q y yy x y x x

x

x Q

x r r

x

x

x

� � �

� �

�� �� � � �

� � �� � � � �

�� � � � �

�� � �

��

�

� �

�

• The elliptic restricted problem (m=0)

• We consider as initial condition

1 10

21

210

(0) , (0) 1

(0)

x x

xQ x

�� � �

� �

�

• The general planar problem in a rotating frame

x

y

(x1,0)

(x,y)

O

r1

r2

P0

P1

P3 2

1 1 1

2

2

(1

2

)

2x y x y

y x

x x x

y

mA

x

B

C

� � �

� �

�

�

� �

� �

� �

� �

� � � �

� � �

�

� � ���� �

� � ��� �

�

�

��

11

0 1

, , are functions of , , ,m

x x ym

A B Cm

� � ��

21( (1 ) / )xQ P�� � �� � ��

(Hadjidemetriou 1975,Christides, 1978)

The Elliptic Hill model

• Hill’s transformation 1/3 1/31 , ( )0x x y� � � � �� �

2 4 210 10 10

12 3 4 3 31 1 1 1

2 4 210 10

41

1012 3 4 3 3

1

01 3 2

1

1 1 1

1

(1)

2 12 2 (2 )

1 1

10

2 2 (2 )

x x xx a

x x x x

x x xx b

x x

xx

x x

x x

�

�

� � � � � �� �

� �

� � � � � � � �� �

� � �

� �

�� � �

��� �

��

(Ichtiaroglou, 1980)

21 10x x� ��(1) is the equation of Keplerian motion with

1/3310 3

(1 ) 11, , 2

(1 ) (1( 1 1)

)e e

e x ae

eTe

�� �

� � �� ��

���

�

P1

•E.H. time dependent Hamiltonian

2 2 2 2 3 2 210 1 1

2 2 2 210 1 10 1

1 1( ) ( ) ( / 2)

2

,

H p p x x p p x

p x x p x x

�

� �

� �

� � � � � � �

� � � �� �

x

y

x10OP0

P1

�

P1 at periapsis for t=0 if e>0apoapsis e<0

The circular case

By setting e=0 we get.1

10 11 ( ) 1, ( 21, )eq

x x t a T �� � �� �

1/3

1/3

2 32

� �

�

�

� � �

� �

�� �

��� •Jacobi Integral

•Equilibriums

2 2 223HC

�� � � �� �

1/3 4/31,2 ( 3 ,0), 3L

HL C�� �

5 LHC C� �

f g

Retrogradeorbits

Progradeorbits

Poincare section ( , ), 0, 0 � � �� �

The circular caseLHC C�

4.25 LHC C� �

f gg’ g’

The circular case: Families of p.o.

Henon, 1969, 2003, 2005

RH

RH

Continuation of p.o. in the Elliptic Problem

•Periodicity conditions of symmetric periodic orbits

10 10 0 0

1 10 0 0

10 0 0

1

2

0 0 0

1 / 3 ( ), 0, 0, 0

( / 2; , , ) 0,

( / 2; , , ) 0,

( / 2

)

; , ,

(

) 0

x e O e x

x T x

T x

T x

� � � � � �

�

�

�

��

��

� �

/ 2

3sin 0( ) ( )

t T

tJ

t t�

�� �

10 0 0

10 0 0

,x

x

� � �

� � �� � � �

� � �

� � �

� � � � � �� � � �� � � �� � � �� � � �

� � �

� �

(Ichtiaroglou, 1980)

• Bifurcation condition 2 ,0J T k k� ���� ��

• Multiple Bifurcation condition ( / : resonance)2C C kk

T T T �� ��

� � �

A periodic orbit of the Circular problem with period 2k� is continued in the Elliptic problem with the same period.

A periodic orbit of the Circular problem with period 2(k/�)� is continued in the Elliptic problem with period T=2k�.

0�

• FamiliesMono-parametric continuation by varying x10 or, equivalently the eccentricity e.

/keF �

Stability of p.o. in the Elliptic Problem

1

2

3

45

6

7(0,-2)

(4,6)(-4,6)

a1

a2

•Monodromy matrix of Variational Eqs

1 1 2 2eigenvalues (1,1), ( ,1/ ), ( ,1/ )( )T � � � �� �

Stability indices :

Broucke’s indices :

1 1 1 2 2 21/ , 1/k k� � � �� � � �

1 1 2 2 1 2( ), 2a k k a k k� � � � �

•A periodic orbit is stable when 21 2 1,24 8 0, 2 2a a k� � � � � �

(Ichtiaroglou 1981, Ichtiaroglou and Voyatzis, 1990)

Families ae1/1, g’e

1/1, f e1/2 : unstable

• A family of periodic orbits, which bifurcates from an unstable periodic orbit of the circular problem,is expected to be unstable too (due to continuity reasons one pair of eigenvalues should lie on the real axis)

Bifurcations

/

2

, 1, 2,3 11e

C

k

T k T

F k�

� �

�

� �

� �

Families f 1/��

• Stable orbits exists only for multiplicity ��6 (�2BP)

-0.8 -0.4 0 0.4 0.8-0.8

-0.4

0

0.4

0.8

f1/9, e=0.5

-2 -1 0 1 2-2

-1

0

1

2

f1/2, e=0.3

Hg+

Hg-

f

RH

Families f 2/�

+e -e

g3f

• Families fe2/� start as stable for the

apoapsis case (except case 2/3)

RH

Families f 3/�

++

+

+

1( ) ,FLI t

t���

( ) : deviation vectort��

RH

Families gk/�,, g’k/�

Families ge

Families g’e

g

g’+

g’-

•Families ge1/� : unstable

•Families ge2/�

(ap) : stable

Hénon Stability maps

0 0 0 0, 0, 0, ( , )Hf C � � �� � Circular case e=0

Hénon maps in the E.H. e=0.048, case of Jupiter

mixing of phases?•An orbit is longterm stable if it is stable in both configurations (apoapsis and periapsis)

Hénon maps in the E.H.e=0.048 (Jupiter)

(Shen and Tremaine, 2008)

mixing

Hénon maps in the E.H.

E=0.056 (Saturn)

Hénon maps in the E.H.

E=0.0097 (Neptune)

Hénon maps in the E.H.

e=0.3 (an extrasolar case!)

CH

0

periapsis

apoapsis

mixing

Stability maps(150x150, tmax=30000tu)

RH

RH

g3 g3 fe2/5

of retrograde orbits (along family f )

Stability maps

Map along family g’

Bifurcationpoint

Map along family g

of prograde orbits

Conclusions

• The elliptic Hill model is an extension of the Circular one and is derived without any additional assumptions

• A large number of families of periodic orbits can be computed by considering the periodic orbits of the circular model with multiplicity �>1.

• Most of periodic orbits are unstable but stable segments are also found.

• The elliptic model estimates more efficiently the stability regions than the Circular model. Stability regions shrinks as the planetary eccentricity increases.