Lagrange Points [Location]

In The 22nd-Century Humanity's Territory is Divided into The Earthsphere and The Outer Sphere. The Earth Sphere is a Conceptual Region of Space which Encompasses The Earth, The Moon, and Nearly All of The Artificial Satellites Orbiting Them along The Lagrange Points. The Outer Sphere Contains all The Species and Colonies that have claimed independence or are so far from Earth's Ruling Governments that most if not all Laws and Restrictions cannot not be Strictly Enforced.

In Celestial Mechanics, A Lagrange Point is something that occurs in a System of Multiple Gravitational Pulls, like that of The Earth and The Moon. These Points occur in Any Two-Body Gravitational System. This of course means that Lagrange Points exist in The Earth-Moon System, as well as in The Sun-Earth System. These Points in The System are Gravitationally Neutral, The **** of Gravity is The Same in All Directions such that there is no Net **** Exerted in that part of Space.

There are always Five Lagrange Points in any of The Systems Described Above, and they are all Designated L1-L5. If an object were to be placed at a Lagrange Point it will remain there indefinitely.

Because of this Stability, Lagrange Points are The Perfect Places to Build Space Colonies.

Lagrange 1:

Lagrange 1 (or L1) lies within a Small Asteroid Field consisting of a Series of Five Asteroids, Labelled MO-I to V, That were moved into The Earth Sphere to provide Resources for The Construction and Maintenance of Space Colonies, These Resource Satellites are Run and Maintained by The Winter family and are Valuable Supply Points for Ships.

The Location of L1:

The L1 Point lies on The Line Defined by The Two Large Masses M1 and M2, and between them. It is The Most Intuitively understood of The Lagrangian Points: The One Where The Gravitational Attraction of M2 Partially Cancels M1 Gravitational Attraction.

• Example: An Object which Orbits The Sun more closely than The Earth would normally have a Shorter Orbital Period than The Earth, but that ignores The Effect of The Earth's own Gravitational Pull. If the object is Directly between The Earth and The Sun, then The Effect of The Earth's Gravity is to Weaken The **** pulling The Object towards The Sun, and therefore increase The Orbital Period of The Object. The closer to Earth The Object is, The Greater this Effect is. At The L1 Point, The Orbital Period of The Object becomes Exactly Equal to The Earth's Orbital Period.

L1 Highlights:

The Sun–Earth L1 is Ideal for making Observations of The Sun. Objects here are never shadowed by The Earth or The Moon. The Solar and Heliospheric Observatory (SOHO) is stationed in a Halo Orbit at L1, and The Advanced Composition Explorer (ACE) is in a Lissajous Orbit, also at The L1 Point. The Earth–Moon L1 allows Easy Access to Lunar and Earth Orbits with Minimal Change in Velocity and is ideal for a Half-Way Manned Space Station intended to help Transport Cargo and Personnel to The Moon and back.

Lagrange 2:

The L2 Point lies on The Line Defined by The Two Large Masses, Beyond The Smaller of The Two. Here, The Gravitational Forces of The Two Large Masses Balance The Centrifugal **** on The Smaller Mass.

• Example: On The Side of The Earth away from The Sun, The Orbital Period of an Object would normally be Greater than that of The Earth. The Extra Pull of The Earth's Gravity Decreases The Orbital Period of The Object, and at The L2 Point that Orbital Period becomes Equal to The Earth's.

The Sun–Earth L2 is a Good Spot for Space-Based Observatories. Because an Object around L2 will maintain The Same Orientation with respect to The Sun and Earth, Shielding and Calibration are much Simpler. The Wilkinson Microwave Anisotropy Probe is already in Orbit around The Sun–Earth L2. The Future Planck Satellite, Herschel Space Observatory, Gaia Probe, and James Webb Space Telescope have been placed at The Sun–Earth L2. The Earth–Moon L2 home to several Communications Satellites covering The Moon's Far Side.

If The Mass of The Smaller Object (M2) is Much Smaller than The Mass of The Larger Object (M1) then L1 and L2 are at approximately Equal Distances r from The Smaller Object, Equal to The Radius of The Hill Sphere, given by:

Where R is the distance between the two bodies.

This Distance can be Described as being such that The Orbital Period, Corresponding to a Circular Orbit with this Distance as Radius around M2 in The Absence of M1, is that of M2 Around M1, Divided by

Examples:

• Sun and Earth: 1,500,000 Kilometers (932,056.8 Miles) From The Earth
• Earth and Moon: 61,500 Kilometers (382,143.3 Miles) From The Moon

Lagrange 3:

Like Lagrange 1, Lagrange 3 (or L3) is situated within an Asteroid Field used as Colony-Building Resource Satellites.

The L3 Location:

The L3 Point lies on The Line Defined by The Two Large Masses, Beyond The Larger of The Two.

• Example: L3 in The Sun–Earth System exists on The Opposite Side of The Sun, a little outside The Earth's Orbit but slightly closer to The Sun than The Earth is. (This Apparent Contradiction is because The Sun is also affected by The Earth's Gravity, and so Orbits around The Two Bodies' Barycentre, Which is However, Well Inside The Body of The Sun.) At The L3 Point, The Combined Pull of The Earth and Sun again causes The Object to Orbit with The Same Period as The Earth.

L3 Highlights:

The Sun–Earth L3 Point is also where "Counter-Earth" is positioned. The Sun–Earth L3 is Highly Unstable, because The Gravitational Forces of The Other Planets outweigh that of The Earth (Venus, For Example, Comes within 0.3 AU of L3 every 20 Months).

Lagrange 5

Lagrange 5 is an Area of Gravitational Stability between a Celestial Body and its Satellites.

Overview:

The L4 and L5 Points lie at The Third Corners of The Two Equilateral Triangles in The Plane of Orbit whose Common Base is The Line between The Centers of The Two Masses, Such that The Point lies behind (L5) or ahead of (L4) The Smaller Mass with regard to its Orbit around The Larger Mass.

The Reason these Points are in Balance is that, at L4 and L5, The Distances to The Two Masses are Equal. Accordingly, The Gravitational Forces from The Two Massive Bodies are in The Same Ratio as The Masses of The Two Bodies, and so The Resultant **** acts through The Barycenter of The System; Additionally, The Geometry of The Triangle ensures that The Resultant Acceleration is to The Distance from The Barycenter in The Same Ratio as for The Two Massive Bodies. The Barycenter being both The Center of Mass and Center of Rotation of The System, This Resultant **** is exactly that required to keep a body at The Lagrange Point in Orbital Equilibrium with The Rest of The System. (Indeed, The Third Body need not have Negligible Mass; The General Triangular Configuration was Discovered by Lagrange in work on The 3-Body Problem.)

L4 and L5 are sometimes called Triangular Lagrange Points or Trojan Points. The Name Trojan Points comes from The Trojan Asteroids at The Sun–Jupiter L4 and L5 Points, Which themselves are Named after Characters from Homer's Iliad (The Legendary Siege of Troy). Asteroids at The L4 Point, Which leads Jupiter, Are Referred to as The 'Greek Camp,' While at The L5 Point they are Referred to as The 'Trojan Camp.' These Asteroids are (Largely) named after Characters from The Respective Sides of The Trojan War.

Examples:

• The Sun–Earth L4 and L5 Points lie 60° ahead of and 60° behind The Earth as it Orbits The Sun.
• The Earth–Moon L4 and L5 Points lie 60° ahead of and 60° behind The Moon as it Orbits The Earth. They may Contain Interplanetary Dust in what is called Kordylewski Clouds.
• The Sun–Jupiter L4 and L5 Points are occupied by The Trojan Asteroids.
• Neptune has Trojan Kuiper Belt Objects at its L4 and L5 Points.
• Saturn's Moon Tethys has Two Much Smaller Satellites at its L4 and L5 Points Named Telesto and Calypso, Respectively.
• Saturn's Moon Dione has Smaller Moons Helene and Polydeuces at its L4 and L5 Points, Respectively.
• The Giant Impact Hypothesis suggests that an Object Named Theia formed at L4 or L5 and crashed into The Earth after its Orbit Destabilized, Forming The Moon.

L5 Highlights:

Lagrange Point 5 is the location of the Space Colony clusters known as Side 1 and Side 6.

L5 is also The Position of The Whole PLANT Homeland, Where The First PLANT-Type Space Colonies have been built and are located.

The Eclipse Colony though incomplete is located in L5.