Three-Body Problem

• No analytical solution• Numerical solutions can be chaotic• Usual simplification

- restricted: the third body has negligible mass

- circular: the primaries are in circular orbits• A suitable reference frame rotates about the center of

mass with the angular velocity of the primaries• Equation of motion in the restricted problem

23

2

213

1

12

2

dd

2d

ddd

rr

Mr

r

MGr

trv

t

rtv

BR

RB

Circular Restricted Three-Body Problem

• The angular velocity is assumed to be constant

(dots indicate time-derivatives in the rotating frame)• The gravitational and centrifugal accelerations can be

written as the gradient of the potential function

• The equation of motion of the third body becomes

22

2

2

1

1

2

1r

r

M

r

MGU

rrrr

Mr

r

MGr

223

2

213

1

1

rUr 2

Lagrangian Libration Points

• The circular restricted three-body problem has five stationary solutions ( )

• Three are collinear with the primaries and are unstable(stable in the plane normal to the line of the primaries)

• Two are located 60 degrees ahead of and behind the smaller mass in its orbit about the larger mass For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable Perturbations may change this scenario

• A spacecraft can orbit around the Lagrangian points with modest fuel expenditure, as the sum of the external actions is close to zero

0rr

Lagrangian Libration Points

Use of Lagrangian Points

• Quasi-periodic not-perfectly-stable orbits in the plane perpendicular to the line of the primaries have been used around Sun-Earth L1 and L2 (halo or Lissajous orbits)

• The Sun–Earth L1 is ideal for observation of the Sun

(SOHO, ACE, Genesis, etc.)

• The Sun–Earth L2 offers an exceptionally favorable environment for a space-based observatory

(WMAP, James Webb Space Telescope, etc.

• The L3 point seems to be useless

• L4 and L5 points collect asteroids and dust clouds

Jacobi’s Integral

• Dot product of the equation of motion with gives

• The potential is not an explicit function of time and

• After substitution one obtains the Jacobi’s integral

• The specific energy of the third body in its motion relative to the rotating frame has the constant value

r

UrrrUrrrUrrr 22

t

U

t

z

z

U

t

y

y

U

t

x

x

UUr

d

d

d

d

d

d

d

d

CUvUrrUrrr 20 2

Non-dimensional equations

• The equations are made non-dimensional by means of

for masses, distances, and velocities, respectively• The non-dimensional masses of the primaries are

• The non-dimensional potential is written as

2

1 22

21

yx

rrU

2222

2221 1 zyxrzyxr

21

1

21

2 1MM

MMM

M

21

212121 RR

MMGRRMM

Surfaces of Zero Velocity

• The potential U is a function only of the position • The surfaces corresponding to a constant value C’=-2U

are called surfaces of zero velocity • The total energy of the spacecraft is -C/2• The spacecraft can only access the region with C’>C• In the Earth-Moon system a body orbiting the Earth with

- C>C1: is confined to orbit the Earth

- C1>C>C2: can reach the Moon

- C2>C>C3: can escape the system from the Moon side

- C3>C>C4: can escape from both sides

- C4>C: can go everywhere

12