circular restricted three body model

8/10/2019 Circular Restricted Three Body Model

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Circular Restricted Three Body Model

Space Systems Lectures

Pierpaolo Pergola

Chiara Finocchietti

[email protected]

Pisa, November 2012 [email protected]


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IntroductionThe general problem of Three Bodies

Reference Frame

Jacobi & HillLibration Points

Equilibrium Region

Periodic Orbits

Manifolds

Manifolds Structures

Poincar Sections

Prescribed Itineraries

Contents


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Law of Universal Gravitation ( inverse square distance force law ):

in which:

Neglecting any other action, the motion of nbodies under theirmutual gravity fields is ruled by the law of universal gravitation:

respect to an inertial reference system

nbody Problem

Fg

! "!

j!i="G

mim

j

rji

3 r!

ji

d 2r

!

i

dt2 =

!G

mj

rji

3 (r

!

i

!r

!

j)j=1j"i

n

#Y

Z

X r2

rn

r 1

r i

M1

M2

Mn

Mi

FgnF

g 2

Fg1

G = 6.6726!10"11 m

3

kg # s2

!

rji =!

ri "!

rj

i =1,...,n


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n=2 Analytic solutions exist (Classical Keplerian motion)

n=3 Analytic solutions exist onlyin configurations of the 3bodies in which the distances are instantaneously constant(straight line, equilateral triangle and eightshape curve)

n>3 No analytic solutions

Generally, the problem of n!

3 bodies is solvedby numerically integrating the differential equations of motion and

introducing some simplifications

(e.g. hypothesis on the mass or motion of the bodies)

nbody Problem


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n=3 and no other forces

The General Three Bodies Problem Three particles with arbitrary masses attract each other according

to the universal law of gravitation; they are free to move in space and

are initially moving in any given manner; find their motion

1 simplification: tostudy the motion of a smallmass that does notaffect the motion of two more massive attractors (primaries).

Two massive bodies move in space under the influence of their

mutual gravitational attraction; describe the motion of a third very

less massive body attracted by the previous two but not

influencing their motion.

General and Restricted Threebody Problem

The Restricted Three Body Problem


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2 simplification: toconsider the two primaries revolving in the same

plane around the common center of mass on circularorbits (withthe same angular velocity).

Circular Restricted Threebody Problem

The Circular Restricted Three Body

Problem CR3BP)

Two bodies revolve around their center of

mass in circular orbits under the influence oftheir mutual gravitational attraction; describe

the motion of a third body attracted by the

previous two but not influencing their

motion.


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The approximately circular motion of the planets around the Sun

and the small masses of the asteroids and satellites compared toplanetary masses suggested the formulation of the CR3BP.

Brief History:

Formulated by Euler in 1767: dynamics and 3equilibria;Few years later Lagrange (1772) determined other 2equilibria;

In 1836 Jacobi discovered the integral of the motion in thesynodic (rotating) reference system;

Poincar (1899) investigated about stability and qualitativecharacteristics.

Circular Restricted Threebody Problem


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Assumptions:

Only three bodies and no other forces 3-Body Problem

m3


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General problem setting:

Equations of motion can be written by using the force functionFthat gives on the right-hand side the forces acting on a body due to

the presence of the other two masses:

The General Problem of Three Bodies

mi

!""ri =

!F!!

ri

i =1,2,3

with F = k2 m1m2

|

!

r21|

+

m2m

3

|

!

r32|

+

m3m

1

|

!

r13|

"

#

$$%

&

''

Gaussian gravitational constant:

k = GMsun

= 0.01720209895A3/2

S1/2 D

A: astronomical unit

D: mean solar day

S: Sun mass

!

r1 = (q

1,q

2,q

3)

!

r2 = (q

4,q

5,q

6)

!

r3 = (q

7,q

8,q

9)

!

"#

$#

%

!

r21

=

!

r1&

!

r2

!

r32

=

!

r2&

!

r3

!

r13

=

!

r3&

!

r1

!

"#

$#


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In the general nbody problem the set of first order differential

equations depend on the number nof bodies 6nEqs.

The general problem of three bodies is described by:

3second order Eqs. for ri

9second order Eqs. for qi

18first order Eqs. for qi

The CR3BP requires 6 first order differential Eqs. (4 for the

PCR3BP), since only the m3motion is studied.


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!ri!m

i

!"ri

i=1

3

" =!

c

mi

!"ri!#""ridt ="

i=1

3

# $F

$!ri

! !"ridt"

i=1

3

#

m1

!"r1

2+m

2

!"r2

2+m

3

!"r3

2= 2F%C

mi

!"ri =

!

a

i=1

3

!

mi

!

ri =

!

at +

!

b

i=1

3

!

"

#

$

$

%

$$

6 first integrals

3 first integrals

Without external forces thecenter of mass moves of

rectilinear uniform motion.

Without external forces theangular momentum of thesystem is conserved.

The total energy of thesystem is conserved.

The 18th-order system is reducible to a 8th-order system by means

of the first integrals of the motion.

First integrals of the nbody problem are 10,regardless from the

number of bodies:

1 first integral


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The generic unconstrained system of n bodies requires 6nfirst

integrals for the integrability. 10are given.

If n=2, the order of the system is reduce to 2 by the 10 classical firstintegrals and the solutions are computed by quadrature imposing

the initial conditions ( ):Integrating :

Substituting in and integrating:


1

2 (!r

2+ r

2 !!2

)!

r

= "= const

r2 !! = h = const !(t) =

h

r2 (t)

dt

t0

t

! +!0

r2 !! = h

t! t0=

1

2

1

!+

r!h

2

r2

r0

r

" dr

!0 ,r0


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Burns, Painlev and Poincar proved that do not exist other first

integrals independent in involution on these 10. All the others aredependent.

The nbody problem with n

3can not be integrated via first

integrals and quadrature.10 first integrals are given.

The 6 first integrals related to the motion of the center of mass areused to place the center of mass in the origin of an inertial

reference system.The remaining 4 first integrals are the only available to integrate thesystem.

Numerical integrations are always possible (today up to n=107).



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The existence of solution is assured by the Cauchy theorem, locally

in the neighborhood of a regular point.

Semianalytic and computeraided methods allow to identifyspecial solutions also for very long (but limited) time intervals.

Such direct solutions can be implemented only for given initialconditions, but they do notrepresent general solutions.

A general solution would consist of a set of 3ncouples (one for

each position and velocity coordinate) of linearly independentfunctions satisfying the system for a generic time instant and a set of6nconstants given by the initial conditions.



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From the general to the restricted three body problem:

Additional integration constants (the primaries motion) areintroduced.

Assuming m3=0:

Decouples the equations for the primaries motion.

Causes the m3equation to be useless (0=0) the trueassumption is to consider m

3

very small, such that it does not

affect the motion of

m

1

and

m

2

, but

!0.


Further reduction of the system from 8th to 6th order


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In an inertial frame, it is required a balancing between gravitational

and centrifugal forces to have the primaries in circular motionaround their center of mass:

The Circular Restricted Problem of Three Bodies

k 2m

1m

2

l 2

= m2an

2= m

1bn

2

k 2m

1= an

2l 2

k 2m

2 = bn

2l 2

k 2(m

1

+m2

) = n2l 3

n = mG

a3

Third Kepler law

=Angular velocity or mean motion


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Considering a force function the equations ruling m3can be written

as:

with:

Due to the time dependence of the m

1

and m

2

coordinates, the

time appears explicitlyin the equations of motion:

Limits of an Inertial Frame

d 2X

dt*2

=!F

!X

d 2Y

dt*2

=!F

!Y

d

2

Zdt

*2 = !F!Z

"

#

$$

%

$

$

X1 = bcos(nt*) X

2 = !acos(nt*)

Y1 = bsin(nt*)

Z1 = 0

Y2 = !asin(nt*)

Z2 = 0

"

d

2

Xdt*2 = #

F(X

,Y,t*

)#X

d 2Y

dt*2

=

#F(X ,Y,t*)#Y

d 2Z

dt*2 =

#F(X ,Y,t*)

#Z

$

%

&&&

'

&&

&

F = k2(m

1 R

1+m

2 R

2)

R1= (X ! X

1)2 + (Y !Y

1)2

R2 = (X !X2 )2 + (Y

!Y2)2

Negative Potential


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The primaries appear at rest in a synodic frame rotating along with

them in such a frame they have zero velocity.

The synodictoinertial coordinate transformation is given by asimple time dependent rotation around the z axis:

Rotating Frame

X

Y

Z

!

"

##

#

$

%

&&

&

=At*'

x

y

z

!

"

##

#

$

%

&&

&

!X

!Y

!

Z

!

"

##

#

$

%

&&

&

= At

*'

!x( y

!y+ x

!z

!

"

####

$

%

&&&&

where At

* =

cos(nt* ) (sin(nt* ) 0

sin(nt* ) cos(nt

* ) 0

0 0 1

!

"

###

#

$

%

&&&

&

Synodic reference with:

the origin in the center of mass ofthe primaries (G);

(x,y) defining the plane of motionof the primaries;

xaxis connecting the primaries;

m2


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In order to have general equations, units are scaled choosing:

m1+ m2as the mass unit (MU).the mean distance, l , between m1and m2 as distance unit(DU).

the time unit (TU) such that the period of m1and m2aroundtheir center of mass is 2!.

From these it follows that:

G=1;

"=1, "is the angular velocity of the system and of primaries.

To obtain dimensional values:

Nondimensionalization

d[Km]= ld[DU] v Km

s

!

"

#$

%

&=V v DU

TU

!

"

#$

%

& t[s]=T

2!

'

(

)*

+

,t[TU]

Mass Unit 1 MU=m1+m

2

Distance Unit 1 DU= l

Time Unit 1 TU= T/2"


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Let us assume .

A restricted three body model can be identified by means of a

single parameter, themass parameter:

Due to the balancing between gravitational and centrifugal forcesthere is a strong relation between masses and distances:

The Mass Parameter

m1> m

2!m

3

m2 =

m

1

=1!

1= m

1 (m

1+ m

2) = b l

2= m

2 (m

1+ m

2) = a l

! 1+

2=1

Two body problem:=0 Sun(Earth+Moon):=3. 036 x 106

EarthMoon:=1. 215 x 102 SunJupiter:=9. 537 x 104

m2 in(1!,0,0)

m1in(!,0,0)

2

1 2

10,2

m

m m

! "= # $ %+


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Kinetic energy:

Gravitational potential energy (independent from the referenceframe):

where r1and r2are the distancesof m3from m1and m2 :

Note: the

gravitationalpotential of apoint mass is:

Equations of Motion: Lagrangian Approach

T =1

2( !X 2 + !Y

2+ !Z

2 ) =1

2[(!x! y)

2+ ( !y+ x)

2+ !z

2]

r1

2= (x+)2 + y2 + z2

r22

= (x!

1+)2

+ y2

+ z2 !

#

r1 r

2

M1

M2

P

1!

U =!

(1!)

r1

!

r2!

1

2(1

!

)

Conventional constant

not affecting eqs. of

motion)

Gravitational potential

due to m

2


due to m

1

V =Gm

r


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TheLagrangian of the system is just an expression of its total

energy and in the rotating frame it reads:

The EulerLagrange equations rule the evolution of L, thus

provide the Eqs. of motion (qi

are the generalized coordinates):

These Eqs. have to be specialized to the Lagrangian underconsideration where the generalized coordinates in the rotatingframe are 6: .


L(x,y,z, !x, !y,!z) =T !U =1

2[(!x! y)

2+ ( !y+ x)

2+ z

2]+1!

r1

+

r2

+

1

2(1!)

d

dt

!L

!!qi

"!L

!qi

= 0

x, y,z,!x, !y,!z


25/100

The Eqs. take the form:

These can be rewritten as a set of time independent 3scalar Eqs. ofthe second order or 6scalar Eqs. of the first one :


q1=x! d

dt

"L"v

x

#"L"x

= 0! ddt

(!x# y) = !y+x#Ux

q2 = y!

d

dt

"L

"vy

#"L

"y= 0!

d

dt( !y#x) = #(!x# y)#U

y

q3=z!

d

dt

"L

"vz

#"L

"z= 0!

d

dt(!z) = #U

z

!!x!2 !y ="x

!!y+2!x ="y

!!z ="z

!x =vx

!vx= 2v

y+!

x

!y =vy

!vy="2v

x+!

y

!z =vz

!vz=!

z

Equations of motion in Lagrangian formulation


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Equations of motion are expressed by means of derivatives of anEffective Potential. It is composed by a centrifugal term and agravitational one:

Extension of the gravitational potential to the rotating frame.

Thetotal energy Ein the synodic frame is:


!(x,y,z) =1

2(x2 +y

2 )"U(x,y,z)=1

2(x2 +y

2 )+1"

r1

+

r2

+

1

2(1")

Centrifugal potential


E =Trel !" =

1

2(!x2 + !y2 + !z2)!"(x, y,z)


27/100

The Legendre Transformation provides the passage from the

Lagrangian to the Hamiltonian formulation:

Once H is known, the canonical equations are directly applicable:

For the problem under consideration these reduce to:

Equations of Motion: Hamiltonian Approach

pi =

!L

!!qi

; H(qi,p

i) = p

i !q

i! "L(qi, !qi)

!qi =

!H

!pi

!pi = !

"H

"qi

Conjugate

Momenta

q1 =x p1 = px =!L

!!x=

!x" y

q2= y p

2= p

y =

!L

! !y= !y+x

q3=z p

3= p

z =

!L

!!

z

= !z


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29/100

The Lagrangian and Hamiltonian formulations are equivalent andgive both an autonomous system:

Conjugate momenta have been introduced instead of velocities.

The phase space size is unaffected, just some variables are replaced

the system of motion is a 6 dimensional dynamical system

in either or .

Equations of Motion: Hamiltonian Approach

!x = px+ y !p

x= p

y!x+"

x

!y = py! x !p

y=!p

x! y+"

y

!z = pz

!pz="

z

!x =vx

!vx= 2v

y+!

x

!y =vy

!vy=!2v

x+"

y

!z =vz

!vz=!

z

(x, y,z, !x, !y,!z) (x, y,z, px, p

y, p

z)


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The Eqs. of motion reveal the symmetries of the CR3BP:

W.r.t. the(x,y) plane:

[x(t),y(t),z(t)] is solution [x(t),y(t),z(t)] is solution

W.r.t. the (x,z) plane for inverting times:

[x(t),y(t),z(t)] is solution [x(t),y(t),z(t)] is solution

W.r.t. to the mass parameter changing #1 :[x(t),y(t),z(t)] is solution for [x(t),y(t),z(t)] is solution for 1.

Equations of Motion: Symmetries


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32/100

Let us compute the following:

This means that Cis conserved along the motion in rotating frame:

Is Cdifferent from H(or E)?

The smaller the energy, the larger the Jacobi Constant.

Equations of Motion: First Integral

ddt

(!x2+ !y

2+ !z

22 ) = 2(!x!!x+ !y!!y+ !z!!z) =

= 2[!x(2 !y+!x)+ !y("2!x+!

y)+ !z!

z]= 2

d

dt(!)#

d

dt("2T

rel + 2!) = 0

E =Trel !" =

1

2(!x2 + !y

2+ !z

2 )!"(x, y,z)

C(x,y,z, !x, !y,!z) = !(!x2 + !y2 + !z2 )+ 2"2C E=!

C(x,y,z,!x, !y,!z) := !(!x2 + !y2 + !z2 )+ 2"#d

dtC = 0

Jacobi constant


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Cis known as the Jacobi Constant. Furthermore, there is not anyother conserved quantity.

Therefore this single and unique first integral reduces thedimension of the phase space only by one the CR3BP can notbe integrated by quadrature, neither in its planar version.

The restricted problem requires 6 first integrals (it studies themotion of a single body).

The synodic reference frame definition already exploits the 6

integrals related to the center of mass motion.The 3 first integrals of the angular momentum are useless as theyrules the motion of the primaries.

The Jacobi constant reduces the degree of freedom by one, but still

there are 5 (3D) or 3 (2D) missed first integrals.

Equations of Motion: First Integral


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Effective Potential

Both around m1andm2there is a gravity

well:

A further well is at

infinity, as centrifugalforces dominate:

r1!0

lim "#( )=

r2!0

lim "#( )= "$

r!"

lim #$( ) = #"

!"(x,y) (planar case,z = 0)


35/100

Effective Potential

The gravity well due to m2is directly related to .Decreasing m2, this welltends to zero.

For a Keplerian model thesize of the m1well (the onlyone) is related to the

energy:

EarthMoon system

2a

! =

"


36/100

Energy Surfaces are 5D manifolds in the 6D phase space (or 3D

in the 4D one for the 2D case) where the energy remainsconstant, e:

Setting the kinetic energy equal to zero we define the boundariesof the admissible position space for the motion :

The Hill Regionsare regions of the phase space where the kineticenergy is positive; i.e. they are defined by:

Hill Regions

M(,e) ={(x,y,z,!x, !y,!z) |E(x,y,z, !x, !y,!z) = e}

!" #e

M(,e) ={(x,y,z) |E(x,y,z) = e}

M(,e) ={(x,y,z) |!"(x,y,z) = e}


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Theadmissible and forbidden regions for the motion can be seen

by sections of the Effective Potential plot for different energy values.

The Hill Regions are bounded by T=0.

This condition identifies the Zero Velocity Curves( ).

Hill Regions

!"(x,y) (planar case,z = 0)

0 | | 0T v= ! =

l


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e

Indeed, the body can move only where its kinetic energy is nonnegative (T!0):

Zero Velocity Curves

Zero velocity curves depend

on e.e

is defined by the initial

conditions.

These curves can be closed

around m1and m2or openedamong the realms.

These curves can not be

crossed during the motion.

E =T !"=1

2(!x

2+ !y

2+ !z

2 )!"(x,y,z) = e

T # 0

$

%&

'&

( e+"# 0(!") e

Hill R i


39/100

For a given , there exist five possible configurations of the Hill

Regions depending on the value of E(i.e. on the initial conditions):

For even higher energies (E>E4=E

5) the whole space is allowed.

Ei(i=1,..,5) is the energy value of a particle at rest at the equilibriumpoint Li.

Once the energy of the body is know, the regions allowed for themotions are known.

Hill Regions

Allowed Reagion

Forbidden Reagionm1Realm m2 Realm

Exterior Realm

L i P i


40/100

Lagrangian Points

Points with horizontal

tangent.

Points where the right handside of Eqs. of motion

vanish.

Points whit zero accelerationin the rotating frame.

Points where a massremains forever whenplaced with zero velocity.

Equilibrium points.

Lagrangian or Libration Points are:

Points: Li(i=1,,5)

Energies: Ei(i=1,,5)

L i P i


41/100

Three libration points (L1,L2,L3) are collinear with the twoprimaries Collinear Points.

Two libration points (L4, L5) define an equilateral triangle with

primaries Triangular Points.

Lagrangian Points

L i P i


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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.52.8

3

3.2

3.4

3.6

3.8

4

4.2

4.4

CLi()

CLi

L1

L2

L3

L4

L5

The critical values of the energy in the equilibrium points are:

Since the energy is measured in a rotating frame, we cannotdetermine the stability proprieties of the equilibrium points by

their energy.

! !"!# !"$ !"$# !"% !"%# !"& !" !"' !"'# !"#%"%

%"$

%

$"(

$")

$"*

$"+

$"#

$"'

ELi()

ELi

L1

L2

L3

L4

L5

Lagrangian Points

C1>C

2 >C

3>C

4 =C

5= 3

E1< E

2< E

3< E

4 = E

5=!1.5

L i P i


43/100

Non-dimensional synodic coordinates of the Lagrangian points:

Triangular points:

Collinear points?

The Effective Potential for a collinear point (ye=ze=0) is:

The zeros of the derivatives of this function are the collinear

points.

Lagrangian Points

!(xe,0,0) =

1

2x

e

2+

1"

| xe

+ |+

| xe

"1+ |

L4 =

1

2!,

3

2,0

"

#$$

%

&'' L5 =

1

2!,!

3

2,0

"

#$$

%

&''

C lli P i C di


44/100

Analyzing the Effective Potential for a collinear point:

Collinear Points Coordinates

!(xe

,0,0) = 12x

e

2+ 1"

| xe+ |

+

| xe

"1+ |

x!"

x!#

x!1#

$

%&

'&

((xe

,0,0)!+"A stationarypoint has to be

located in:

( , )

( ,1 )

(1 , )

!" !#$

! !%

$ ! "&

Figure: The solid line is theintersection of with the plane

y=0. At the xlocation of m1andm2the function plunges to -#.Maxima correspond to unstablecollinear points.

!"

C lli P i t C di t


45/100

Setting to zero the first derivative of results in a 5 order

algebraic equation:

In which $%

is the distance of Li

from the nearest primary:


!(xe,0,0)

d

dx!(x

e,0,0) = 0

!1

5! (3!)!

1

4+ (3! 2)!

1

3!!

1

2+ 2!

1

!= 0

!2

5+ (3!)!

2

4+ (3! 2)!

2

3!!

2

2! 2!

2!= 0

!3

5+ (2+)!

3

4+ (1+ 2)!

3

3! (1!)!

3! (1!) = 0

L1

L2

L3

!i =

L1m

2 if L

1(i =1)

L2m

2 if L

2(i = 2)

L3m

1 if L

3(i = 3)

!

"

##

$

##

C lli P i t C di t


46/100

The 5 order algebraic equations have to be solved numerically

(e.g. Newton method).Also series expansions are available.

Both the numerical approach and the series approximation start

from:o

Series expansions read:


rh =

3

!

"#

$

%&

1

3

! = 7

12( )

Hill Radius (i.e. Radius of the sphere

surrounding m2in position space

where the m2 gravity field dominates

over that of m1)

L1

L3

L2

!1= r

h(1!

1

3rh!

1

9rh

2+ ...) !

2= r

h(1+

1

3rh!

1

9rh

2+ ...)

!3=1!"(1+

23

84"

2+

23

84"

3+ ...)

C lli E ilib i


47/100

Location of the collinear

libration points versus

Collinear Equilibria

System X coord L

1

[DU]

X coord L

2

[DU]

Sun(Earth+Moon)

Numeric: 0.9899Analytic: 0.9899


EarthMoon Numeric: 0.8367Analytic: 0.8373


#

$ #

% #

&

'$#

&

'%#

EarthMoon andSun(Earth+Moon) values:

Li i ti f th ti


48/100

To investigate about the stability of a libration point and the motion

near itwe linearize the CR3BP equations of motion around thatpoint. Three linearization methods are available:

Linearization of the Eqs. of motion using Taylor series around an

equilibrium point Classical Approach

Expansion to the second order of the Hamiltonian around anequilibrium point Hamiltonian Approach

Expansion to the second order of the Lagrangian around anequilibrium pointt Lagrange Approach

Linearization of the motion

Li i ti Cl i A h


49/100

The Taylor series of a generic function around anequilibrium point is:

Linearizing (to the first order) it becomes:

After the linearization of the CR3BP equations around a librationpoint (Li), the stability can be investigated by introducing a

perturbation :

Linearization: Classic Approach

!

g( !

x) =!

g( !

x0)+

!

g(n) (

!

x0)

n!n=0

!

" !

x# !

x0( )

n

!

g( !

x) =!

g( !

x0)+ !

!

g(

!

x)!!

x !x=

!

x0

!

x" !

x0( )

!

x0

!

g( !

x)

x =x0+x

y = y0+ y where

z =

z0+

z

x ,y,z( )t

Li = (x

0,y

0,0)t

m3L

! "!!!

i = (x,y,z)

t


50/100

Linearization: Hamilton Approach


51/100

It is a more practical method for the motion around a collinear

libration points.Let us focus on one of the three collinear libration points

x

e

)

in the

planarcase:

To linearize the canonical Eqs. quadratic terms of H are required

(derived once w.r.t. the state variables):

H is Taylor expanded around a collinear point (xe,0,0,xe)

Origin is moved into (xe,0,0, xe).

Second order H:


2 2 2 2( , , , ) ( , , , )/x y x y

i i

HH x y p p x y p pp q

!" # "! !

! !

Hl =

1

2[(p

x + y)2 + (p

y!x)2 !ax2 +by2]

with a = 2c+1 e b= c!1 c = |

xe

!

1+

|

!3+

(1!

) |x

e

+

|

!3



52/100

From Hl, the linearEqs. of motion (planar case) in the rotating

frame centered in the collinear point read:

Since His just another expression of E, also the energy assumes asecond order form:

Surfaces characterized by El=0are energy surfaces passingthrough Li.

For 3D case the linear Eq. of motion along the z-axis is:


2 2 2 21

( )2l x yE v v ax by= + ! +

!x =!H

l

!px

= px + y !p

x = "

!Hl

!x= p

y"x+ax

!y =!H

l

!py

= py"x !p

y = "

!Hl

!y

= "px"y"by

!!z = !cz

Linearization: Lagrange Approach


53/100

In the Lagrangian approach the state of a generic collinearpoint

reads (planarcase): (xe,0,0,0).

The inverse Legendre transformation performs the HamiltoniantoLagrangian passage.

Thus, the linear Eqs. of motion (planar case) in the Lagrangeformulation read (w.r.t. a rotating frame centered in the collinearpoint):

For 3D case the linear Eq. of motion along the z-axis is:

Linearization: Lagrange Approach

!x =vx !vx = 2vy +ax

!y =vy

!vy = !2v

x!by

!!z = !cz


54/100

Collinear Equilibria Stability Investigation


55/100

Investigating about the sign of the second derivative of the

same stability considerations can be drawn:

L

1

, L

2

, L

3

are unstable equilibria.

Collinear Equilibria Stability Investigation

!"(xe,0,0)

d 2(!")

dx2

x=xe

= !1! 1!

| xe+ |3

!

| xe

!1+ |3

!"(xe,0,0) = !

1

2x

e

2!

1!

| xe

+ |!

| xe

!1+ |

The sign is always negative(concave)

!"(xe,0,0)

Equilateral Equilibria Stability Investigation


56/100

Specializing the linear Eqs. of motion to :

Four of the eigenvalues (the other two ) of the system read:

The points are stable if alleigenvalues have zerorealpart

, thus:

L

4

, L

5

are stable equilibria for


57/100

The gravity field of m1and m2shows 5points where forces actingon

m

3

vanish.

Due to their fast instability L

1

and L

2

are the most interesting for

space mission design.

Libration Points: Stability Summary

approaching

eaving

Configuration in the Position Space


58/100

Configuration in the Position Space

The most interesting caseis when E2


59/100

The aim is to investigate about the shapeof the solution

considering the linearized Eqs. near collinearequilibria. Let usfocus on theplanarcase.

Let us recall the linear Eqs. in the Lagrangian version:

The 4 eigenvalues (roots ofp) are ( ):

Geometry of Solution Near Equilibria

!x!y

!vx

!vy

!

"

#####

$

%

&&&&&

=

0 0 1 0

0 0 0 1

a 0 0 2

0 'b '2 0

(

)

****

+

,

----

.

" #$$$ %$$$

/

xy

vx

vy

!

"

#####

$

%

&&&&&

0 p(!)= !4 + (2 ' c)!2 + (1' c' 2c2 )'ab

" #$ %$

! = !2! p(!) = 0

!1=

c!2+ 9c2 !8c

2> 0"

"1= !

1

"2 = ! !

1

#$%

&%!

2 =

c!2! 9c2 !8c

2< 0"

"3= i !

2

"4= !i !

2

#$%

&%

Characteristic polynomial



60/100

Let us define:

The eigenvalues take the form:

From eigenvectorsassume the general form:

Thus:


! = "1and !

xy= !

2

! and i!xy

with!> 0 and !xy > 0

!"#

$#

Au = !u

u = (k1,k

2,k

3,k

4)

k3= !k

1 "k

1+ 2k

4= !k

3

k4= !k

2 !bk

2!2k

3= !k

4

"

k1!0 otherwise ki=0.

Thus, we choose k1=1.



61/100

The relative eigenvectors are:

We can write the complex eigenvectors trough two real vectors

as: .


with! =2 !"

"2+b

> 0 and # = "$

xy

2+a

2 !$xy

< 0

!

!

u1 = (1,!!,",!"!)

!

u2 = (1,!,!",!"!)

!

w1= (1,

!i#,i$xy,$xy#)

!

w2 = (1,i#,!i$

xy,$

xy#)

i!xy

!

w1=

!

!1! i

!

!2,

!

w2=

!

!1+ i

!

!2

!

!1,!

!2



62/100

Let us now consider the eigenvectors as axes of the reference

frame:The new coordinates are: .

In a reference frame where eigenvectors are the basis, the

constant coefficient matrix is diagonal with the eigenvalues in the

relative position along the diagonal:

For the imaginary eigenvectors the differential equations of therelative coordinates are imaginary: .


(!,",w1,w

2)

!!

!"

!w1

!w2

!

"

#####

$

%

&&&&&

=

# 0 0 0

0 '# 0 0

0 0 i$xy

0

0 0 0 'i$xy

(

)

*****

+

,

-----

.

!

"

w1

w2

!

"

#####

$

%

&&&&&

!w1 = i!

xyw

1, !w

2 = !i!

xyw

2



63/100

To obtain all real equations we search for a real block diagonal

coefficient matrix (Jordan form).

Thus, we apply a coordinate change defining as reference axisinstead of :

The phase space remains with coordinates: .

!Re : !!1 ="

xy!

2

!Im : !!2 = ""

xy!1

!w1 = i!xyw1

!w2= !i!

xyw

2

"#$

%$&

!!1! i!!2 = !w1 = i!xyw1 = i!xy("1 ! i"2) =!xy"2 + i!xy"1

!!1+ i!!

2= !w

2= !i!

xyw

2= !i!

xy("

1+ i"

2) =!

xy"

2! i!

xy"

1


!4

!

w1, !

w2

(!,",#1,#

2)

!

!1,!

!2



64/100

The Eqs. of motion in the planar case take the very easy form:

The solutions of Eqs. read:

are the initial conditions.

and divergent components.

(

periodic components.


!(t) = !0e

"t

#(t) =#0e!"t

$(t) = $1(t)+ i$

2(t) = $

0e!i%

xyt

"

#$$

%

$

$

!! = "! !#1 =$xy#2

!%= !"% !#2 = !$

xy#1

"#$

%$

!0,"

0e !

0

The coordinate

transformation decouples

the coordinates. Only the

complex conjugate pair

shows cross dependence.



65/100

In such a system also the energy takes the very easy form:

Since this system is a low order (linear) approximation of the true

motion additional first integrals are available besides the energy.

Besides the energy, other two constant functions (the elements ofthe linear energy equation) along solutions are:


El = !"#+$xy

2(!

1

2+!

2

2)

2 2 2

1 2| |

cost

cost

!"

# # #

=$%

= + =

Flow in the Equilibrium Region


66/100

The Equilibrium Region, R, is the region in the phase space

sufficiently close to the equilibria where the linear approximationholds.

This region defines boundaries on the coordinates/eigenvalues:

Considering that and assuming :

R is the product between a 4Dsphere and an interval.


El =! = cost and |"!#|" c !,c > 0

lE !=

!

4

("2 +#2)+$

xy

2

(%1

2+%

2

2 ) = &+!

4

("!#)2 !"#"

#

$

%|"!#|& c

'

(

)

*)

2 2( ) ( )4 4

! !!"# " # " # = + $ $

4D-sphere

equation

Constant for a value

of on the

interval I=[-c,c]

!!"( )Is a constant for a

value of on

the interval I= [-c, c]

!!"( )


67/100



68/100

Let us project the solution flow onto the ((1,(2) plane:

The projections of the solutions are circumferencesof different radius onto the ((1,(2) plane Centre.

In the 3D case, the projection onto the plane of the eigenvectorsrelative to the motion along the z-axis are analogues, so the 3D

solution of the linearized motion in the equilibrium region is:


Projection in the Position Space


69/100

In the position space the Equilibrium Region is the region

connecting two realms.In the planar case, the general solution of the linear system near acollinear point in rotating coordinates reads (verifiable bysubstitution):

where *1and *2are real and +=+1+i+2 complex.

The xcomponent of solution is:


[x(t),y(t),!x(t), !y(t)]=!1e"tu

1+!

2e!!tu

2+ 2Re(!ei"xytw

1)

x(t) =!1e"t +!

2e!!t + 2(!

1cos("

xyt)!!

2sin("

xyt))

1

1

1

0 ( )

0 ( )

0 ( )

for t if x t

if x t bounded

if x t

!

!

!

" +# $ < $ " %#

= $

> $ " +#

2

2

2

0 ( )

0 ( )

0 ( )

for t if x t

if x t bounded

if x t

!

!

!

" #$ % < % " #$

= %

> % " +$



70/100

Combinations of *1and *2return the same types of orbits

highlighted in the eigenvectorreferred axes:

1=2=0periodic orbits

12=0asymptotic orbits

to the periodic one

120nontransit orbits

The asymptotic orbits computed for each point of the periodic

orbit project in to 2 strips S1and S2in the x-y plane.


Periodic Orbits


71/100

In the Equilibrium Region there exist periodic solutions around

massless points.

Recalling the linearized system of 3D motion near a collinear pointw.r.t the rotating frame centered in the point:

and its 6 eigenvalues:

The general solution of the linear system reads:

In which: can be calculated from the initialconditions and are the angular phases.

b

!!x! 2!y! (2c+1)x = 0

!!y+2!x+ (c!1) y = 0

!!z+cz = 0

"

#$

%$

x(t) =!1e

!t+!

2e!!t

+Axcos(!

xyt +!)

y(t) =!k1!1e

!t

+ k2!2e!!t

!k2Axsin(!xyt +!)

z(t) = Azcos(!

zt +!)

!1,!

2,A

xand A

z

! and"

k1=

2c+1!!2

z

2!z

k2=

2c+1+!2xy

2!xy

! i!xyand i!

z

Periodic Orbits


72/100

Toanalyze only the periodic motion, let us assume:

Thus, the linearized system becomes:

The different types of periodic orbits near a collinear point are:

Always familiesof periodic orbits in Hamiltonian systems. Specific

orbit parameterized by means of an energyrelated parameter.

x(t) =Axcos(!

xyt +")

y(t) = !k2A

xsin(!

xyt +")

z(t) = Azcos(!

zt +#)

!1 =!

2 = 0

!xy!!

z

In the linear motionit is always:

Ax ! 0"A

z = 0

Az ! 0"A

x = 0

Az ! 0"A

x ! 0"!

xy =!

z

Az ! 0"Ax ! 0"!xy !!z

Planar Lyapunov orbits

Vertical Lyapunov orbits

Lissajous orbits quasi-periodic)

Halo orbits

Adding non-linear terms

Periodic orbits: Halo Orbits


73/100

Parameterized by means of the outofplane amplitude (Az).

Halo orbits generate when the amplitudeAz is sufficiently large forallowing non-linear terms to generate equal eigenfrequencies!

xy=!

z

Periodic Orbits: Lissajous Orbits


74/100

Not exactly periodic orbits.

j

The actual orbits used in realspace missions.

Stationkeeping

maneuvers aremandatory.

Two amplitudes as free

parameters; more than oneorbit with the same energy.


75/100

Periodic Orbits: Kinds


76/100

The Poincar Section approach allows for the fast identification of

all objects.

Periodic Orbits: Computation


77/100

With the aim at computing the non-linear periodic orbits near a

collinear point, either analytic and numerical approximationmethods are necessary.

p

Periodic orbitcomputation procedure

Analytic approximation(e.g. Richardsons method)

Numerical methodto correct the analytic approximation

(e.g. Differential correction)

Periodic Orbits: Analytic Approximation


78/100

Richardsons Method Halo orbits):

It is chosen a rotating reference frame centered in the collinearpoint;

The Lagrangian of the CR3BP is rewritten by exploiting therecursive relations of the Legendre polynomials;

Appling the Euler-Lagrange equations to the rewritten Lagrangianthe equations of motion can be computed;

Non-linear periodic analytic solutions approximated to the third

order are computed by using frequency corrections to keep the

solution bounded;

The analytic solution so computed isa function of the amplitudesA

x

and A

z

of the linearized solutions of motion.

y pp

Periodic Lyapunov Orbits: Analytic Approximation


79/100

SettingAz=0 the Richardson method can be apply to compute

Planar Lyapunov orbits and settingAx=0 to compute VerticalLyapunov orbits.

Planar Lyapunov orbits

Vertical Lyapunov orbits

Periodic Orbits: Analytic Approximation


80/100

The Richardson approximation is too poor to produce actual

periodic orbits when used as initial conditions for numericallyintegrating the nonlinear Eqs. of motion.

Planar Lyapunov orbit

Vertical Lyapunov orbit

Periodic Orbits: Numerical Approximation


81/100

Differential Correction:

The numerical differential correction procedure modifies theanalytic approximation in the actual force model to obtainperiodicity.

Starting from the Richardson approximation, periodic orbits are

constructed by using the state transition matrix.

The numerical scheme employs a correction of the initialconditionsby means of a first order expansion of the final state

until periodicity conditions are obtained.Due to the symmetries of the problem, an orbit two timesperpendicularly intersecting the x,z) synodic plane is periodic (Royand Ovenden Theorem).


82/100

Periodic Orbits: Numerical Approximation


83/100

Differential Correction continue):

= Analytic approximation

= Differential Correction

Az=80.000 km

Planar Lyapunov orbit

Vertical Lyapunov orbit

Manifolds


84/100

Associated to each collinear equilibria and periodic orbit near them

exist a set of lines and tubes respectively, which naturally tendingasymptotically toward (stable) or moving from (unstable) them:

Stable Manifold toward the collinear

point/periodic solution for ;

Unstable Manifold toward the collinearpoint/periodic solution for .

t!+"

t!"#

Onedimensional Manifolds


85/100

Manifolds associated to the collinear libration points:

The unstable dynamics (saddlepart) of the collinear points give riseto a stable(real negative eigenvalue of the linear system ) and anunstable(real positive eigenvalue ) manifold.

The stable eigenvalue shows an exponentially fast convergence

toward the libration point.The unstable eigenvalue shows an exponentially fast divergencefrom the libration point.

Qualitative results obtainable by a linear approximation of the full

motion can be extended to the nonlinear case

[Moser].

Manifold can be computed by propagating backward/forward intime a small perturbation along the stable/unstable eigenvectors(u

1

/u2

) of the linear system (manifold globalization).!

x0

s/u=

!

x0d

!

u1/2

!

!!

Onedimensional Manifolds


86/100

Manifolds associated to the first and second collinear libration point

of the Sun-Jupiter CR3BP system:

Manifolds

associated to L

1

Manifolds

associated to L

2

termas orbit


87/100

Twodimensional Manifolds


88/100

X [DU]

Unstable

Stable

The linear approximation of manifolds associated with a point x0

on a periodic orbit is given by the stable/unstable eigenvectorsof the monodromy matrix.

(vs/v

u)

Applying the Mosertheorem, manifold tubes

can be computed bypropagating backward/forward in time a smallperturbation (d) along the

stable/unstableeigenvectors of themonodromy matrix(manifold globalization).

Twodimensional Manifolds


89/100

The stable/unstable eigenvector is

the one associated with the realeigenvalue of modulus smaller/

larger than 1.

x0

s= x

0dv

s! stable

x0

u=x

0dv

u!unstable

Manifolds structures: Homoclinic Orbits


90/100

Orbits asymptotic to the sameunstable periodic orbit (libration

point) both for and for .

Orbits living on the stable and unstable manifold of the sameperiodic orbit.

t!+" t!"#

Manifolds structures: Homoclinic Orbits


91/100

Mc Gehee (1969) demonstrated that the surface of allowed motionare broken upfurther into regions bounded by invariant tori.These invariant tori project in two annuli A

1

and

A

2

) in theposition space that can not be crossed during the motion.

Every orbit leaving the vicinity of one unstable periodic orbits ofL1/L2evolves inside the annuli T1/T2in the counterclockwise/

clockwise direction before returning to that vicinity.

Manifolds structures: Heteroclinic Orbits


92/100

Orbits asymptotic to

two

different

unstable equilibrium points, ortwo periodic orbits with same energy associated to differentcollinear points, both for and for .

Orbits living on the stable manifold of one orbit and on theunstable manifold of the other.

t!+" t!"#

Homoclinic-Heteroclinic Chain


93/100

The combination of two homoclinic orbits, in the interior andexterior realm, with an heteroclinc orbit of the same energy andassociated to the same periodic orbits generates a so calledhomoclinic-heteroclinic chain of orbits.

It connects the interior, exterior and m2realm.

Poincar Sections


94/100

The idea is to exploit a lowdimensional map preserving all theinformation of the original system.

The complete state of the system can be reconstructed by using

flowtransversal planes to store the coordinate not fixed by the

plane location. Using the energy integral we obtain the 4

th

coordinate to complete the state.

.

.rbits

.. .Poincar Map

1

st

cut

2

nd

cut

Poincar Sections


95/100

Poincar Sections of stable/unstable periodic orbit manifolds are

areas.Images of manifolds inherit their characteristics.

Points within these areas have the same manifold dynamics.


96/100

Poincar Sections: Prescribed Itineraries


97/100

Prescribed Itinerary: [X,J,S,J,X]

red=unstable manifold; green=stable manifold

Poincar Sections: Prescribed Itineraries


98/100

Twisting:

The closer is an orbit to the manifold and the more deformed willbe its image when it leaves the equilibrium region.With infinitesimal changes in the initial conditions e.g. a small

velocity change in a fixed point), we can dramatically modify the

destination of the orbit.

Interplanetary Superhighways


99/100

The Interplanetary Superhighways are the network of trajectories

generated by the collection of invariant manifolds of all of theunstable periodic orbits within the Solar System.

The Solar System is modeled as a series of coupled CR3BPs.

References


100/100

V. Szebehely, Theory of Orbits, The Restricted Problem of Three Bodies, AcademicPress inc., New York, 1967

E.A. Belbruno and J.K. Miller, SunPerturbed EarthtoMoon Transfers with BallisticCapture, Journal of Guidance, Control and Dynamics, Vol. 16, No. 4, pp. 770775,1993

M. Kim, Periodic orbits for future spacebased deep space observations ,

Diplomarbeit, sterreich, 2001Shane David Ross, Cylindrical Manifolds and Tube Dynamics in the Restricted ThreeBody Problem, Thesis, California Institute of Technology Pasadena, California, 2004

Conley, C., Low Energy Transit Orbits in the Restricted ThreeBody Problem, SIAMJournal Appl. Math., No. 16, pp. 732746, 1968

D.L. Richardson, Analytic Construction of Periodic Orbits about the Collinear Points,Celestial Mechanics, Vol. 22, No. 3, pp. 241253, 1980

BernelliZazzera F., Topputo F. and Massari M., Assessment of Mission DesignIncluding Utilization of Libration Points and Weak Stability Boundaries, Final Report,

circular restricted three body model

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