Lagrange Points of the Earth-Moon System

In Death by Black Hole by Neil Degrasse Tyson, I just read about spots called Lagrange points in which the gravitational and centrifugal forces between two bodies balance and thus basically cancel each other out. As calculated by Joseph-Louis Lagrange, the first spot (termed L1) of the Earth-Moon system falls between the Earth and the Moon, slightly closer to Earth than the point at which the Earth and Moon's gravitational forces (excluding centrifugal) balance precisely. In other words, closer to the Moon than the Earth. At this point, any shift sideways will cause the object to lose the balance and thus plummet to either the Earth or the Moon. L2 lies on the Earth-Moon system and exists on the far side of the moon while L3 exists on the side of the Earth opposite the Moon. At these two points, objects can orbit Earth with the same monthly orbit of the Moon. The balances of these two point are less precarious because it takes only a little fuel to reenter the orbit. L4 and L5, however are much more useful. Lying on either the far left or the far right side of the Moon, each point forms an equilateral triangle with the centers of the Earth and Moon. That being said, despite drifting in any direction an object can remain in orbit. Raw materials there would be under very little stress from gravity and the orbits are very stable making L4 and L5 ideal spots for space control centers.These points are occupied by satellites such as the Hubble Space Telescope and the International Space Station. Lagrangian points exist for all two-body planets. Tyson suggests that since "interplanetary trajectories that begin at Lagrangian points require very little fuel to reach other Lagrangian points or even other planets", these points can eventually be used as fueling stations to allow for more extensive space exploration.

The Paradoxicality

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