3. The Restricted Three-Body Problem If two of the bodies in the problem move in circular, coplanar orbits about their common centre of mass and the mass of the third body is too small to affect the motion of the other two bodies, the problem of the motion of the third body is called the circular, restricted, three -body problem.

3-2. Equation of Motion x y   r2 P r1 r (2, 2, 2) (, , ) (3.1) We consider the motion of a small particle of negligible mass moving under the two masses m1 and m2. We assume that two masses have circular orbit around their common mass center. Then the two masses keep the constant distance and have the same angular velocity. Consider the geometry in the right-hand figure. Let the unit of mass be chosen such that =G(m1+ m2)=1 If we assume that then in this system of units the two masses are The unit of length is chosen such that the constant separation of the two masses is unity. It then follows that the common mean motion, n, of the two masses is also unity. (, , ) (1, 1, 1) (2, 2, 2) r r1 r2 m2 m1 x y   m1 m2 P (3.1) (3.2) nt

The equation of motion of the particles are Let the coordinates of the particle in the inertial or sidereal system (, , ). The equation of motion of the particles are (3.3) (3.4) (, , ) (3.5) (2, 2, 2) where (3.6) (1, 1, 1) (3.7) We assume that the circular orbits of two masses, which imply the distance to these two is kept constant. In this condition, we can consider that the motion of the particle in a rotating reference frame in which the locations of the two masses are also fixed.

Consider a new, rotating coordinate system that has the same origin as the , ,  system but which is rotating at a uniform rate n in the positive direction. The direction of the x-axis is chosen such that the two masses always lie along it with coordinates (x1,y1,z1)=(-2,0,0) and (x2,y2,z2)=(+1,0,0). Hence from Eq. (3.6~7) we have ------(3.6~7) (3.8) (3.9) y where (x,y,z) are the coordinates of the particle with respect to the rotating, or synodic system. (x,y,z) r1 r2 m1 m2 x O (-, 0, 0) (+, 0, 0)

If we now differentiate each component in Eq. (3.10) twice we get The coordinates (x,y,z) are related to the coordinates in the sidereal system by the following rotation (3.10) sidereal coordinate 항성의 synodic coordinate 삭망 If we now differentiate each component in Eq. (3.10) twice we get (3.11) Centrifugal acceleration Corioli’s acceleration -----(skip)----- (3.12)

+ + Using Eq. (3.12), (3.10), Eq. (3.3-5) become  cosnt  (-sinnt) (3.3) (3.4) (3.5)  cosnt  (-sinnt) +  sinnt  cosnt + (3.13) (3.14) (3.15) (3.16) (3.17) (3.18)

We conventionally define the negative potential, and then (3.16) (3.17) (3.18) These accelerations can also be written as the gradient of a scalar function U: (3.19) (3.20) Centrifugal potential (3.21) Gravitational potential (3.22) Corioli’s term: It depend on the velocity of the particle in the rotating reference frame. We conventionally define the negative potential, and then (3.23-25)