Earth-Moon Trajectory Performance Estimate
Trajectories for a spacecraft travelling between the Earth and the Moon are particularly difficult to
calculate. The spacecraft, Earth and Moon present a three body problem that is difficult to approximate
as two bodies 1, 2. Two techniques are generally used: 1) patched conics and 2) approximations from the
restricted three body problem. A third simple model would be to compute a trajectory that would fly-by
the moon using a Hohmann transfer to the orbit of the moon and ignoring the influence of the Moon’s
gravity entirely3,4. Of course, this would grossly over-estimate the time required to reach the Moon since
the lunar gravity will act to accelerate the fall toward the Moon shortening the transit time. Patched
conics is frequently also applied to spacecraft trajectories and is especially accurate for interplanetary
travel. As the spacecraft leaves a planet only the spacecraft and the planet are considered as a two body
problem, and the influence of the Sun is ignored. When the spacecraft crosses the sphere of influence of
the planet (with respect to the Sun) the spacecraft and the Sun are considered and the planet is ignored.
The approximation is accurate since the curvature of the trajectory, at the boundary of the sphere of
influence, is quite small making deviations due to accelerations of either the planet or the Sun over further
distances small enough to neglect. For the Earth and Moon though there is very little distance for a small
trajectory curvature compared to the sphere of influence of the Moon with respect to the Earth (about
66,300 km). Hence the approximation is likely to be inaccurate when neglecting either the influence of
the Earth or the Moon, and the only method left is the restricted problem of three bodies.
In the restricted three body problem two large mass bodies orbit in a circle. The third body is the
spacecraft with a considerably smaller mass 2, 5. The coordinate system rotates at the circular frequency
of the larger masses with its origin at the center of mass. The mass of the Earth, M e , is 81.3 times the
mass of the Moon 1,5, M m , and the mean distance between the Earth and Moon2,
the Earth and Moon the center of mass is inside the surface of the Earth.
R0 (384,400 km).
For
Figure 1 illustrates the rotating coordinate system and the relative location of the Earth and Moon. The
rotational velocity,  0 , in the inertial systems is shown. The point L1 is the first of five equilibrium
Lagrangian points, and is an unstable equilibrium point 2,5.
Figure 1: Rotating coordinate system including the unstable equilibrium point L1.
In the rotating system the equations of motion can be readily shown to be 2,5
d 2x
dy
GM  x  xe  GM m  x  xm 
  2 
,
 20
 02 x  2 e 
2
dt
dt
re  re 
rm  rm 
and
(1a)
d2y
dx
GM e  ye  GM m  ym 
2
   2   .


2



y

0
0
dt
dt 2
re2  re 
rm  rm 
(1b)
The quantities xe and xm are the x-coordinates of the center of mass of the Earth and Moon
respectively, and are given by
Mm
xe  
Me  Mm
R0,
(2a)
and
Me
R0 .
Me  Mm
xm 
(2b)
The angular velocity is given by
0 
G
is Newton’s gravitational constant,
x
and
y
G M e  M m 
.
R03
(3)
are the coordinates of the spacecraft and
t is the time.
The quantities re and rm are the distances from the center of mass of the Earth and Moon respectively,
and, of course, can be found using
re 
x  xe 2  y 2 ,
(4a)
rm
x  xm 2  y 2
.
(4b)
x, y   (R0 ,0) .
(5)
and
The Lagrangian point, L1, is located at
It can be shown for the Earth-Moon system that   0.849065 2,5.
A constant of the equations of motion,
C , can be found using Jacobi’s integral 2 and can be written as
GM e GM m
1
1
C  v 2   02 ( x 2  y 2 ) 

2
2
re
rm
,
(6)
 x, y  . Note that once the constant C is found we can use the
equation to find an appropriate speed v at any other location  x, y  .
where
v
is the velocity at the point
The Jacobi can be used to obtain more accurate estimates of required speed changes than patched conics.
We will choose the velocity that the spacecraft will go through the point L1, vL1 in the rotating system
 x, y  is xL1,0 .
vL1 is chosen as zero, in the restricted three problem, then the spacecraft
will remain at L1 as long as there are no disturbances. Smaller values of vL1 will produce smaller
where
If
values for the required mass ratios, but longer trip times. The speed in low earth orbit can be found by
setting re equal to the radius of the Earth, Re , plus the orbital altitude, he . An angle needs to be chosen
to fully specify the location in the orbit. To calculate the speed we will take the point in low earth orbit as
x, y   Re  he  xe ,0 .
(7a)
In a similar manner the speed in orbit around the Moon can be approximated by using the location
x, y   Rm  hm  xm ,0 .
(7b)
Of course, the actual angular location about the Earth or the Moon is needed to accurately determine the
speed, but since near the Earth or the Moon the term in equation (6) that will dominate is the gravitational
potential of the nearby body (i.e.,
GM e
Re  he
will dominate when in orbit about the Earth and
GM m
Rm  hm
will dominate when in orbit about the Moon). Hence the circular orbit velocity will be accurate using the
two body problem. At the Earth this will be
be
GM m
Rm  hm
GM e
Re  he
and at the Moon the circular orbit speed will
. The difference between the speeds found using equations (7a) and (7b) in equation (6)
and the circular orbit speed should be accurate.
Please note that the above assumes impulse maneuvers. This can be taken as a thrust to spacecraft mass
ratio of greater than about 0.05 times the gravitational acceleration at the surface of the Earth, g 0 .
The mass ratio,
MR , at the Earth and Moon can now be estimated using the rocket equation
MR 
where
mi
is the initial mass,
mf
 v 
mi

 exp
I g 
mf
 sp 0 
,
is the final mass, and I sp is the specific impulse.
(8)
Estimating the time to reach the Moon (or return) is more difficult. This was done by directly integrating
the equations of motion using Euler’s method with a self-adaptive time step. The time step was chosen to
be a user specified distance divided by the total velocity. Such a choice would limit how far the
spacecraft will move in any given time step to approximately the specified distance. The initial angular
location was chosen as any simulation that brought the spacecraft to within a distance Rm  hm of the
center of the Moon. This should transit times very close to the actual travel time.
Results using the above approach appear to be quite accurate. For a vL1 of 0.050 km/s the required
speed change at the Earth orbital altitude was found to be 3.071 km/s and at the Moon was 0.632 km/s.
The transit time was 2.52 days which agrees with the minimum times for Apollo lunar missions. For a
specific impulse of 425 seconds at the Earth (200 km altitude) the mass ratio is 2.09 and to enter orbit
about the Moon (a very low 60 km altitude) is 1.24 for a 300 second specific impulse. Finally to land
from orbit about the Moon would require a mass ratio of 1.75 for a 300 second specific impulse. These
appear reasonable for the specific impulse values chosen.
The above analysis was applied to a propulsion system with a specific impulse of 22,280 seconds, a thrust
of 4.9x106 N and a dry mass of 500 metric tons. The thrust-to-weight ratio 0.1 justifies the assumption of
impulse maneuvers. Table 1 below summarizes results for a variety of speeds through L1. The craft is
assumed to leave a low earth orbit at an altitude of 200 km, and enter low lunar orbit at an altitude of 60
km. The thrust-to-weight ratio is too low to take off from the Moon; hence, a separate lander will have to
be used. The propellant required to leave earth orbit and enter lunar orbit is quite small. For a rapid trip
of one-half day less than 35 metric tons of propellant is required each way.
Table 1: Transit time and propellant requirements for a dry mass of 500 metric tons
and a specific impulse of 22,800 seconds
v – km/s
Transit Time
One Way
through L1
days
Leave Earth
Orbit
Enter Lunar
Orbit
Leave Earth
Orbit
Enter Lunar
Orbit
0.05
0.1
0.2
1
2
5
10
2.52
2.52
2.50
2.10
1.57
0.81
0.43
1.014
1.014
1.014
1.014
1.015
1.019
1.032
1.003
1.003
1.003
1.004
1.006
1.018
1.040
7,000
7,000
7,000
7,000
7,250
9,500
1,500
1,500
1,500
1,750
3,200
9,000
16,000
20,000
Mass Ratio
Propellant Mass - kg
References
1. Battin, R.H., An Introduction to the Mathematics and Methods of Astrodynamics, AIAA Education
Series, American Institute of Aeronautics and Astronautics, New York, 1987.
2. Bate, R.R., D.D. Meuller and J.E. White, Fundamentals of Astrodynamics, Dover Publications, Inc.
New York, 1971.
3. Von Braun, W., F.L. Whipple, and W. Ley, Conquest of the Moon, edited by C. Ryan, Viking Press,
New York, 1953.
4. Clarke, A., The Exploration of Space, Harper & Brothers, New York, 1951.
5. Szebehely, V., Theory of Orbits: The restricted problem of three bodies, Academic Press, New York,
1967.