This question is brought up due to confusion over that cool video of an asteroid cricling the Earth and not understanding the explanation of what the L1 represents. Thank you.
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3 body gravitational systems are chaotic unstable. Chaotic meaning that a small, almost insignificant change to the initial conditions will completely change the outcome of the system, and unstable meaning that almost always one of the bodies will be ejected (launched away with escape velocity by gravitational forces) resulting in a much more stable 2 body system.
Lagrange points are points that will be stable in a 3 body system where M1 >> M2 >> m. M1 being a star, M2 being a planet, and m being a satellite, and >> means much greater than.
L1, L2, and L3 were the first ones discovered because they are easy to figure out with Newtonian mechanics, that’s just where the forces of M1 and M2 on m balance out. (They originally had different names, but were renamed after L4 and L5 were discovered)
L1 is between M1 and M2, so if you place something there perfectly, the forces allow it to maintain its orbit at a slower speed than it would if M2 didn’t exist. That’s because for circular motion, a = v^2 /r, and a (acceleration) is reduced because the force from M1 is being very slightly canceled out by M2. Now this would apply at every point between M1 and M2, but it needs to be at L1 specifically so it can also have the same angular speed (have the same year length) as M2 around M1, otherwise, it would drift away from being between M1 and M2.
Since M2 >> m, we can ignore the effect m has on M2.
ω(angular velocity) = 2π/T(period) = v(linear velocity)/r
We know the periods, and therefore angular velocities of m and M2 must be the same, so we will be using ω=v/r, and remember a = v^2 /r. Combine the two, and we have a = ω^2 r. And then acceleration due to gravity is a = GM/r^2 (formula for force of gravity, but mass of the object canceled out)
For M2 we get GM1/R1^2 = ω^2 R1 and for m we get GM1/R2^2 – GM2/(R1 – R2)^2 = ω^2 R2. (R1 is M2’s distance to M1 and R2 is m’s distance to M1)
Do some math, we get GM1/R1 = ω^2 and G(M1/R2 – M2R2/(R1 – R2)^2) = ω^2, since angular velocity has to be equal, then so does angular velocity squared. GM1/R1 = G(M1/R2 – M2R2/(R1 – R2)^2)
Do some more math, M1/R1 = M1/R2 – M2R2/(R1 – R2)^2, and then M1(1/R2 – 1/R1) = M2R2/(R1-R2)^2 and then M1/M2 (R1-R2/R1R2) = R2/(R1-R2)^2
M1/M2 = R1R2^2 / (R1-R2)^3 the ratio of the masses determines the ratio of the radii
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