# ENAE 483

Principles of Space Systems Design

Fall, 2002

Question 3:

There is currently a great deal of interest at NASA in creating
a large structure assembly and repair infrastructure at the Earth-Moon
L1 point. From this location, directly between the Earth and the
Moon, low-energy transfers are possible to a variety of places
of interest, including the Earth-Sun L1 and L2 points. The radius
of the Moon's orbit around the Earth is 384,400 km. L1 is a point
in line between the Earth and the Moon at a radius of 332,600
km from Earth.

**For both ENAE 483 and ENAE 788D:**

a) Assume that L1 is just a circular equatorial orbit at the
stated radius. From an initial low earth orbit with a 500 km altitude,
calculate the delta-V required for a coplanar Hohmann transfer
to L1.

b) Your spacecraft actually starts out from the International
Space Station, which is 500 km high at an inclination of 51.2°.
Assume all plane change is done at apogee. Calculate the delta-V
for a Hohmann-type transfer from the ISS orbit to L1.

c) Find the optimum inclination change at perigee for the minimum
delta-V solution to transfer from the ISS orbit to L1. Hint: don't
get fancy and try to take deriviatives, just use Solver in Excel
or manually iterate to find the best solution.

**For ENAE 788D only:**

d) L1 is not a standard Earth orbit, as it rotates around the
center of the Earth-Moon system with the same period as the Moon
(27.322 days). The moon is also inclined 5° to the earth's
equatorial plane. Find the answer to (b) for a best-case transfer
from ISS to L1.

e) Given the Moon's orbital period in d), calculate the locations
of L2 and L3, the other Lagrange points in line with the Moon
and the Earth.