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 d

v

 d

t B

 d 2

r

 d

t

2

R

 2  

v



R

   

r

   d  d

t B



r

  

G M

1

r

1 3 

r

1 

M

2

r

2 3 

r

2

Circular Restricted Three-Body Problem

• The angular velocity is assumed to be constant

r

   

G M

1

r

1 3

r

 1 

M

2

r

2 3

r

 2    

r

   2  

r

 (dots indicate time-derivatives in the rotating frame) • The gravitational and centrifugal accelerations can be written as the gradient of the potential function

U

 

G

 

M

1

r

1 

M

2

r

2    1  2

r

2 2 • The equation of motion of the third body becomes

r

   

U

 2  

r



Lagrangian Libration Points

• The circular restricted three-body problem has five 

r

 • 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

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 L 1 • The Sun–Earth L 1 and L 2 (halo or Lissajous orbits) is ideal for observation of the Sun (SOHO, ACE, Genesis, etc.) • The Sun–Earth L 2 offers an exceptionally favorable environment for a space-based observatory (WMAP, James Webb Space Telescope, etc.

• The L 3 point seems to be useless • L 4 and L 5 points collect asteroids and dust clouds

Jacobi’s Integral

r

 

r

   

r

  

U

 2

r

  

r

  

r

  

U

 2

r

 

r

   

r

  

U

• The potential is not an explicit function of time and

r

  

U

 

U



x

d

x

d

t

 

U



y

d

y

d

t

 

U



z

d

z

d

t

 d

U

d

t

• After substitution one obtains the Jacobi’s integral

r

 

r

  

r

  

U



r

 

r

    0 

v

2  2

U

 

C

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

Non-dimensional equations

• The equations are made non-dimensional by means of

M

1 

M

2

R

1 

R

2

G M

1

R

1 

M

2 

R

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

M

1

M

2 

M

2 1   

M

1

M

1 

M

2 • The non-dimensional potential is written as

U

  1 

r

1   

r

2 

x

2  2

y

2

r

1    

x

 2 

y

2 

z

2

r

2   1   

x

 2 

y

2 

z

2

Surfaces of Zero Velocity

• The potential

U

is a function only of the position • The surfaces corresponding to a constant value

C

’=-2

U

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

>

C

1 : is confined to orbit the Earth

C

1 >

C

>

C

2 : can reach the Moon

C

2 >

C

>

C

3 : can escape the system from the Moon side

C

3 >

C

>

C

4 : can escape from both sides

C

4 >

C

: can go everywhere

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