While Jack Swigert, played by Kevin Bacon in Hollywood’s 1995 rendition of Apollo 13, may sound
cheerful, lack of radio transmission on the far side of the moon greatly limited the amount of science that the
Apollo missions could provide. An important component needed for advanced or long term lunar missions is a
suitable communication system.
Foundation
Missions to the moon are subject to dynamics known as the threebody problem, namely: the Earth,
the Moon, and a spacecraft. The gravity of the Earth and Moon interact together to form a
complex potential field. A contour plot of the gravity potential in the EarthMoon system is shown
below.
Figure 1: EarthMoon 3body System (click for cite)
Five points in the system have zero net gravity (L_{1} through L_{5}). Known as Lagrange or libration points, a
spacecraft placed at one of these points can theoretically remain there indefinitely, “balancing” between the
two bodies (13). Many unique orbits exist in the threebody problem, some which are formed around libration
points.
Fundamental Concepts
A simple way to analyze a 3body system is to restrict the motion of the two largest bodies (primary and
secondary). The following assumptions simplify the problem (11):
 The mass of the third body (the spacecraft) is negligible in comparison to the primary and
secondary
 Both primaries are subjected to Keplerian laws for twobody motion
 The primaries rotate in curcular orbits about the center of mass of the entire system (the
barycenter)
The collection of mathematics that results from these constraints is known as the Circular
Restricted Threebody Problem (CRTBP). It is convenient to represent the three body problem in a
synodic, or rotating, reference frame with the barycenter as the origin. The geometry is shown
below:
Figure 2: CRTBP Synodic Frame (3)
The problem is simplified further by making it dimensionless. The distance between the primaries and the
gravitational parameter are both set equal to 1, and the orbital period is set equal to 2π. These values are
normalized by the threebody parameter μ, which is defined as:
μ =   (1) 
where
m_{1} and
m_{1} are the masses of the primaries and
m_{1} > m_{2}.
From here we can formulate the 3body equations of motion. Given Newton’s law of gravitation (assuming
the primaries are point masses), the total force exerted on the spacecraft by the primaries can be expressed
as:
_{net} = m_{3} _{s∕c} = −G − G   (2) 
After nondimensionalizing the problem by
μ, the inertial acceleration of the spacecraft can be written
as:
or in scalar form:
ẍ  = 2ẏ + x − (1 − μ) − μ  (4)

ý  = −2ẋ + y − (1 − μ) − μ  (5)

ẍ  = −(1 − μ) − μ  (6) 
where
r_{1} and
r_{2} are the distance from the third body to the primary and secondary:
r_{1}  =  (7)

r_{2}  =  (8) 
Assuming the spacecraft’s motion is only in the xy plane, the libration points can easily be solved for by
setting the accelerations and velocity equal to zero:
0  = x − (1 − μ) − μ  (9)

0  = y − (1 − μ) − μ  (10)

  
Solving these equation results in 5 solutions, (
L_{1} through
L_{5})(11).
Advanced Concepts
Additional analysis can be done to help motivate the extension presented in this project. The equation’s of
motion (Equations 46) can be expressed in vector form:
g =   (12) 
and integrated analytically with respect to
x,
y, and
z to produce the psuedopotential, Ω:
Ω = (x^{2} + y^{2}) + +   (13) 
It can be seen the the psuedopotential is only a function of the position of the particle and
μ. Using this
potential we can simplify the equations of motion:
g = =   (14) 
where Ω
_{x}, Ω
_{y}, and Ω
_{z} are the first partial derivatives of Ω defined in Equation 13.
Using the formulations above, we can derive the state transition matrix, Φ(t,t_{0}), for the CRTBP. The
state transition matrix maps deviations in the state vector from one time to another. These state deviations
are defined from the spacecraft state.
X  = ^{T }  (15) 
whose derivative is:
Ẋ  = ^{T }  (16) 
Let δx to be some small deviation from the nominal trajectory X. The nonlinear equations of motion can
be approximated by using a Taylor series expansion:
Ẋ + δẋ = F(X) + δx + ...   (17) 
Ignoring higher order terms, the following differential equation is found.
δẋ = δx   (18) 
whose solution is of the form:
δx = Φ(t,t_{0})δx_{0}   (19) 
Thus, the state transition matrix, Φ(
t,t_{0}), is made up of the partial derivatives of the state:
Φ(t,t_{0}) =   (20) 
The state transition matrix can be propagated using the matrix of
partial derivatives
A(
t):
A(t)  =  (21)

(t,t_{0})  = A(t)Φ(t,t_{0})  (22) 
For the CRTBP, the
A matrix is defined as:
A(t)  =  (23) 
where
G_{r} and
G_{v} are the partial derivative matrices of
g from Equation 14 with respect to position and
velocity respectively:
G_{r}  = =  (24)

G_{v}  = =  (25) 
where Ω is defined in Equation 13 (11).