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 3-body 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 meta-stable, 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 Earth-Sun 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 Earth-Sun system affords an uninterrupted view of the sun and is currently home to the Solar and Heliospheric Observatory Satellite, also known as SOHO. The Microwave Anisotropy Probe (M.A.P.) is in a halo orbit around the Sun–Earth L2 position, about a million miles in the opposite direction. The L2 point will also be home of the Next Generation Space Telescope. There are, however, orbits around the L1, L2 and L3 points of the Earth-Moon system, in the planes perpendicular to the axis connecting the two major bodies, which are almost stable, requiring only small occasional corrections. The orbit around L1 would be a suitable location for the temporary parking of shuttles, crews and materials during the construction phase. It could also serve as waiting point for the space ships demanding permission to anchor the colony’s cosmodrome. As the International agreement has set aside the far side of the Moon for being the only place in our Solar system shielded from terrestrial electronic transmissions, the Lagrangian L2 is not suitable for building space colonies or temporary “work camps”.

That is why the L1, L2 and L3 points of both the Earth-Moon and Earth-Sun systems are not a reliable option for the space settlement’s placement, although the Earth-Moon 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 Earth-Sun and Earth-Moon 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

settlement is that the Sun’s pull makes any object in Earth-Moon 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 Earth-Sun 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:

x₁ and x₂ being the distances from m to L₁ and L₂, respectively, and x₃ being the distance from M to L₃. 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 x-value 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 x-value of  for x₁ and x₂. For x₃ start with . Some values of B, x₁, x₂, and x₃ are shown in the Collinear Values Table.

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.

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 Earth-Moon 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 pseudo-gravity 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^o9^’ 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.

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