Broucke, R. A.

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FREE-FALL TRAJECTORIES FROM II_E MOON TO THE LAGRANGIAN EQUILIBRIUM
POINTS; R. A. Broucke, Jet Propulsion Laboratory, Pasadena, California 91103.
In the last two years some interest has developed in the construction of
large manned space stations. These stations would be orbiting in the Earth-
Moon system and would be constructed and assembled in space from materials
taken partly from the Earth, but mostly from the Moon. Several transportation
systems could be considered. They range from magnetic accelerators located
on the Moon to space-shuttle type vehicles traveling back and forth. Several
possible locations for the large manned station have also been proposed, and
the equilibrium points (Lagrange Points) are favorite candidates, especially
the triangular points L4 (60 degrees ahead of the Moon) and L5 (60 degrees
behind the Moon), or even the collinear points L1 (in front of the Moon) or
L2 (behind the Moon).
We have thus undertaken a systematic study of the possible free-fall
trajectories connecting the Moon with any of the equilibrium points LI, L2,
L3 and L4. We only consider here those free-fall trajectories which are of
enough practical value: the trajects should be as direct as possible, with
short transit times and with small velocity at the arrival at the Lagrangian
point. As a model, we have taken, for the present initial exploration, the
well known circular restricted three-body problem in two dimensions. The
Earth and the Moon are the only two acting bodies; they are point-masses in
circular orbits around each other. The true radius of the Moon does not
enter the problem. A Levi-Civita regularization was used for the numerical
integration of trajectories in proximity of the Moon.
We have used the symmetry properties (mirror image theorem) of the
restricted problem in order to reduce the exploratory work: to each trajec-
tory corresponds a mirror image on the other side of the Earth-Moon line
(Syzygy-axis). For instance, to each Moon-to-L 4 trajectory corresponds a
symmetric L5-to-Moon trajectory (traveled in the same time).
In the examples that are given below, we specify the trajectories by
their final condition at LI, L2, L3, or L4 rather than the initial conditions
at the Moon. l_nis can be reduced to a set of initial conditions at the Moon
by integrating backwards or more simply by using the mirror image theorem.
The parameters which are used to specify a given trajectory are the 2 coordin-
ates of the libration point together with the velocity and flight path angle
at the point. The flight path angle is measured with respect to the Earth-
Moon line and is zero in the Earth-to-Moon direction (180 degrees in the
Moon-to-Earth direction). These angles (and also the velocities) are relative
quantities measured in the rotating Earth-Moon system.
https://ntrs.nasa.gov/search.jsp?R=19910073673 2020-03-24T07:04:12+00:00Z
FREE=FALLTRAJECTORIESFROMMOON
Broucke, R. A.
159
An example of the trajectory from L5 to the Moon {or from the Moonto L4)
has been computedwith a velocity of 90m/sec, a flight path angle of 205
degrees and a transit time of iS days. This trajectory is mostly inside the
Moon's orbit and is faster than the trajectories from L4 to the Moon (or from
the Moonto L5) which are retrograde and mostly outside the Moon's circular
orbit. Twosuch trajectories were found with a velocity of 90m/sec at L4.
In the first one the transit time is 19 days and the angle at L4 is 96 degrees,
while for the second one these numbersare 206 degrees and 25 days.
Several similar trajectories have also been computed for the Lagrange
points L1 and L2. The point L] can be reached via a fairly direct path.
The hidden point L2 can be reached with a residual velocity of 230m/sec in
2.8 days. The flight-path angle at arrival at L2 is 70 degrees.
Becauseso muchwork has been done on the study and classification of
periodic orbits in the restricted three-body problem it is of interest to
relate these trajectories to someof the knownclassical families of periodic
orbits, it is found, for instance, that the above-mentioned trajectory from
the moonto L4 is very nearly the knownperiodic collision orbit of the
retrograde family around the equilibrium point L2.
Inspection of the complete trajectory shows that it would be possible
to take off from the Moonat the sub-Earth point, travel to L4, keep on
going behind the Moon[intersect the syzygy-axis at the right angle at an
apogee about 200,000 km behind the Moon)and then travel on to L5 and finally
return to the Moonand land at the departure point. Let us also mention that
all the membersof this family of periodic orbits have unstable characteristic
components. Also, that this family corresponds to class a of Stromgren's
early investigations.
The examples that we have elected to describe here seemto be someof
the best compromisesbetween fuel consumption and time of flight, but we
intend to study more in detail the possibilities of other practical trajec-
tories in the Earth-Moon system. Also, the different approach and landing
paths on the Moonshould be studied, as well as the sensitivities of the
different trajectories.


Free-fall trajectories from the Moon to the Lagrangian equilibrium points

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