Lunar Halo Orbits
Introduction

CAPCOM-GOLD: “Aquarius, this is Houston. Expect loss of signal in approximately ten seconds.”
JACK SWIGERT: “So long Earth. Catch you on the flip side.” (1)

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 three-body 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 Earth-Moon system is shown below.

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Figure 1: Earth-Moon 3-body System (click for cite)

Five points in the system have zero net gravity (L1 through L5). 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 three-body problem, some which are formed around libration points.


Fundamental Concepts

A simple way to analyze a 3-body system is to restrict the motion of the two largest bodies (primary and secondary). The following assumptions simplify the problem (11):

  1. The mass of the third body (the spacecraft) is negligible in comparison to the primary and secondary
  2. Both primaries are subjected to Keplerian laws for two-body motion
  3. 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 Three-body 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:

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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 three-body parameter μ, which is defined as:

μ = ---m2----
m1 + m2 (1)

where m1 and m1 are the masses of the primaries and m1 > m2.


From here we can formulate the 3-body 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:

⃗Fnet = m3š⃗r s∕c = Gm3m1--
  r31⃗r1 Gm3m2--
 r32r⃗2 (2)

After non-dimensionalizing the problem by μ, the inertial acceleration of the spacecraft can be written as:

š
⃗r s∕c = G1 −-μ-
 r3
  1⃗r1 Gμ-
r3
 2⃗r2 (3)

or in scalar form:
= 2 + x (1 μ)x + μ
--r3--
   1 μx − 1 + μ
----r3----
     2 (4)
ý = 2 + y (1 μ)y
--
r31 μ y
--
r32 (5)
= (1 μ)z-
r31 μ-z
r32 (6)

where r1 and r2 are the distance from the third body to the primary and secondary:

r1 = ∘ -------2----2----2
  (x + μ) +  y + z (7)
r2 = ∘ ----------------------
  (x − 1 + μ)2 + y2 + z2 (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)x + μ
--r3--
   1 μx − 1 + μ
----r3----
     2 (9)
0 = y (1 μ)y
--
r31 μ y
--
r32 (10)

Solving these equation results in 5 solutions, (L1 through L5)(11).

Advanced Concepts

Additional analysis can be done to help motivate the extension presented in this project. The equation’s of motion (Equations 4-6) can be expressed in vector form:

g = ⌊  ⌋
  šx
⌈ šy⌉

  šz (12)

and integrated analytically with respect to x, y, and z to produce the psuedo-potential, Ω:

Ω = 1
2-(x2 + y2) + 1 − μ
--r---
   1 + μ
r-
 2 (13)

It can be seen the the psuedo-potential is only a function of the position of the particle and μ. Using this potential we can simplify the equations of motion:

g = ⌊                 x+μ-    x−1+μ⌋
  2˙y + x − (1 − μ) r31 − μ   r32
|⌈  − 2x˙+ y − (1 − μ) y3-− μ y3 |⌉
                 z-   r1-z   r2
        − (1 − μ)r31 − μr32 = ⌊          ⌋
  2y˙+ Ωx
⌈− 2˙x + Ωy ⌉
     Ω
      z (14)

where Ωx, Ωy, and Ωz are the first partial derivatives of Ω defined in Equation 13.

Using the formulations above, we can derive the state transition matrix, Φ(t,t0), 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 = [                 ]
 x  y  z  x˙ y˙ z˙T (15)

whose derivative is:
= [                 ]
 ˙x  y˙ z˙ xš  šy  šzT (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:

+ δ = F(X) +   --
∂-F(X-)
  ∂Xδx + ... (17)

Ignoring higher order terms, the following differential equation is found.

δ =  --
∂F-(X-)
  ∂Xδx (18)

whose solution is of the form:

δx = Φ(t,t0)δx0 (19)

Thus, the state transition matrix, Φ(t,t0), is made up of the partial derivatives of the state:

Φ(t,t0) =  ∂X (t)
-------
∂X  (t0) (20)

The state transition matrix can be propagated using the matrix of partial derivatives A(t):

A(t) = ∂-X˙(t)
∂X  (t) (21)
Φ˙(t,t0) = A(t)Φ(t,t0) (22)

For the CRTBP, the A matrix is defined as:

A(t) = [        ]
  0    I
 Gr   Gv (23)

where Gr and Gv are the partial derivative matrices of g from Equation 14 with respect to position and velocity respectively:

Gr =   --
∂-g
∂r-- = ⌊ Ω    Ω     Ω  ⌋
⌈   xx    xy   xz⌉
  Ωyx  Ωyy   Ωyz
  Ωzx  Ωzy   Ωzz (24)
Gv =   --
∂-g
∂v-- = ⌊  0   2  0⌋
⌈          ⌉
  − 2  0  0
   0   0  0 (25)

where Ω is defined in Equation 13 (11).