CHAPTER
II
Placement of the space settlement
II.1.
Choosing the space settlement’s location
The best solution for placing
the space settlement would be one of the Lagrange points (or Libration points),
which are locations in space where gravitational forces and the orbital motion
of a body balance each other. They were discovered in 1772 by French
mathematician Louis Lagrange in his gravitational studies about the 3body
problem consisting in how a third, small body would orbit around two orbiting
large bodies.
The solution found by Lagrange was
astronomically confirmed in 1906, when the Trojan asteroids orbiting the Sun
Jupiter L4 and L5 points were discovered. To find the Lagrange points you must
adopt a frame of reference that rotates with the system. The forces exerted on
a body at rest in this frame can be derived from an effective potential in much
the same way that wind speeds can be inferred from a weather map. The forces
are strongest when the contours of the effective potential are closest together
and weakest when the contours are far apart.
Of the
five Lagrange points, three are unstable and two are stable. The unstable
Lagrange points, labeled L1, L2, and L3 lie along the line connecting the two large
masses. An object at L1, L2 and L3 is metastable, like a sphere placed in top
of a hill. A little impulse is enough to make it move away. If the space
settlement would be situated in one of these points, it would frequently have
to use its small rocket firings to remain in the area, process in which part of
the fuel and stored energy will be lost. The L1 and L2 points are unstable on a
time scale of approximately 23 days, which will require the space settlement
parked at these positions to undergo regular course and attitude corrections.
Another inconvenient of the EarthSun L1 and L2 points, situated at
approximately 1000000 miles from Earth in opposite directions, would be that
both already shelter satellites. The L1 point of the EarthSun system affords
an uninterrupted view of the sun and is currently home to the Solar and
Heliospheric Observatory Satellite, also known as
That is why the L1, L2 and L3
points of both the EarthMoon and EarthSun systems are not a reliable option
for the space settlement’s placement, although the EarthMoon L1, L2 points are
the Lagrange points closest to the Moon.
The L4 and L5 points are home to
stable orbits so long as the mass ratio between the two large masses exceeds
0.0385 (M1/M2 < 0.0385). The condition is satisfied for both the EarthSun
and EarthMoon systems, and for many other pairs of bodies in the Solar system.
An object at L4 or L5 is truly stable, like a sphere in a bowl: when gently
pushed away, it orbits the Lagrange point without moving away farther and
farther, and without the frequent use of rocket propulsion. Another reason why
we should consider the L4 or L5 points as the most suitable for placing a space
The figure above shows the forces which appear in the
movement of our Space Settlement. The gravitational force between the Space
Settlement and Earth (Blue) and the gravitational force between the Space
Settlement and Moon (orange) are vector summing the resultant force (_{} the black one) being oriented to
the Earth’s center of mass. 
settlement is that the Sun’s pull
makes any object in EarthMoon L4 and L5 locations “orbit” the Lagrange points
in 89 days, which demonstrates the high stability of these areas. We must also
consider the L4 and L5 points’ proximity to the Moon which represents
a great advantage due to the satellite’s exploiting potential. Despite their
stability, the L4 and L5 points of the EarthSun system are not a considerable
solution because of the large, inefficient distance which separates them from a
constant source of raw materials.
Finding the positions of the 5 points requires
only simple math skills. L1, L2, and L3 are referred to as the collinear points
because they lie on the line connecting the centers of mass of M and m. L4 and
L5 are called the equilateral points because they lie at the tips of
equilateral triangles with M and m at the other two vertices.
Let's first define the distance
between M and m as a dimensionless value of 1. Next we will define A and B as
the distances from the center of mass to m and M, respectively.
_{}
The distance from center of mass
to the first 3 equilibrium points are:
_{}
_{}
Now we define:
_{}
x_{1} and x_{2}
being the distances from m to L_{1} and L_{2}, respectively,
and x_{3} being the distance from M to L_{3}. Substituting x's
into the L formulas produces the following:
_{}
Solutions to these equations must
be found by iteration. That means coming up with an initial guess for a value
for each x then plugging that guess into the equation to see what value of x it
produces. We take the new xvalue and plug it back into the equation and solve.
We repeat this process until our guess and answer match to the level of
accuracy we need.
Start with an xvalue of _{} for x_{1} and
x_{2}. For x_{3} start with _{}. Some values of B, x_{1}, x_{2}, and x_{3}
are shown in the Collinear Values Table.
Collinear
Values Table 

System 
B 
X_{1} 
X_{2} 
X_{3} 
R (mi) 
X_{1} x R 
X_{2} x R 
X_{3} x R 
EarthMoon 
.01214 
.1509 
.1678 
.9929 
239,000 
36,042 
40,076 
237,165 
SunEarth 
3.0359E06 
.010006 
.010073 
.999998 
93x10^{6} 
930,560 
936,810 
92,998,000 
L4 and L5 are easier to
calculate. They are exactly the same distance from M and m as M and m are from
each other, which puts the points 60 degrees ahead of and behind m,
respectively, on its orbital path.
Distances to Lagrange Points 

Earth to Moon 
384 300 Km 
Earth to L4/L5 
384 300 Km 
Earth to L3 
384 700 Km 
Earth to L1 
326 200 Km 
Moon to L1 
58 200 Km 
Moon to L2 
64 700 Km 
Moon to L4/L5 
384 300 Km 
Earth to Center of mass 
4700 Km 
Due to
the edifying facts above, we come to the conclusion that the space settlement
should be placed in the L4 or L5 points of the EarthMoon system.
The rotation axis of the space settlement must
be parallel with its revolution axis. It must not change its direction during
the revolution motion around Earth as the Space Station tends to keep its
kinetically momentum and would easily leave its orbit, losing the needed
mechanical equilibrium. Maintaining the rotating space settlement in an orbit
parallel with its spinning axis will also imply, in time, a considerable use of
fuel. This aspect is discussed in the next chapter.
2. Rotation and kinetic momentum
The
basic condition for generating pseudogravity inside the space settlement is
rotating the colony around its vertical axis (transversal axis). Therefore, the
spin motion of the settlement is essential for the development of human
activities in space. For a more realistic approach of the problem, we must
first lay down the exact rules of the settlement’s spinning motion.
Let’s
assume the colony is built in the position shown in the figure below, with
maximum exposure to the Sun, with the intention of keeping it in this position
during all its evolution on Earth’s orbit. The settlement’s orbital kinetic
momentum (L), the orbital speed (_{}) and the angular speed of the colony towards its rotation
centre _{} are also represented in
the figure.
The
stability of the movement involves the constancy of the orbital kinetic
momentum L in time. The constancy of the rotation radius and the flatness of
the trajectory derive from the first basic condition.
Now
let’s assume the colony begins to rotate with the angular speed of _{} around its symmetry
axis. The laws of mechanics for solid and rigid bodies show that in this kind
of situations a gyroscopic couple appears, characterized by the M_{G }momentum.
_{ }; I  the inertial momentum of the settlement
The
figure below shows the changes intervening along with the rotation of the
settlement around its axis. The momentum of the gyroscopic couple leads to the
rotation of the settlement around the direction which connects the center of
the colony with the center of rotation. Therefore, the settlement can not keep
its initial orientation towards the Sun. We can easily see that the total
kinetic momentum is prone towards the ecliptic of the Moon, which involves the
changing of the ecliptic plan orientation.
_{}
For
maintaining the trajectory and the total kinetic momentum of the settlement it
is necessary for the kinetic momentum of the spin to be parallel with the
orbital momentum. The momentum of the gyroscopic couple becomes:
_{}
_{}
_{} total kinetic momentum
_{} orbital kinetic momentum
_{}  spin kinetic
momentum
As a result, the settlement will orientate its axis perpendicularly on the ecliptic plan of the Moon. Its rotation plan will be 5^{o}9^{’} prone towards the ecliptic of the Earth. The solar rays will be incident on the surface of the settlement at an angle of approximately 84^{o} from the rotation axis of the colony.
WebWork:
Andrei Dan Costea, Flaviu Valentin
Barsan
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