Mostly, I’m interested in L4 and L5, but the more I try to explain it to myself the more I realize I have no idea how any of them work except maybe L1- just an equilibrium of the two forces?
In: 5
The basic principle of the Lagrange points is that the net force of gravity on an object determines the orbital speed of the object. The harder gravity pulls, the faster the orbit must be, and the closer two objects are to each other, the stronger the pull of gravity.
The secondary principle of Lagrange points is that the points are “stationary” relative to the orbital body (Earth), which means that they have to complete an orbit at the same time as the orbital body does.
Since the circumference of the orbit (and therefore the distance needed to complete an orbit) increases with diameter, that means that objects farther from the Sun than the Earth must move faster, while objects closer to the Sun than Earth must move slower.
If you put these principles together, you get a paradox; an object farther from the Sun than Earth must move faster than Earth to maintain its “stationary” position, but since it’s farther from the Sun, it moves more slowly. The reverse is also true; an object close to the Sun most orbit more slowly, but gravity will cause it to orbit faster.
The Lagrange points arise because both the Sun and Earth exert gravitational pull. These can add and subtract from each other to change the orbital speed, which then changes how it moves relative to the Earth, if it moves at all (and if it doesn’t, then it’s a Lagrange point).
L1: Between Sun and Earth: The Sun dominates the gravitational pull, but Earth’s gravity negates some of the Sun’s gravity depending on how close the satellite is to the Sun or the Earth. This causes the orbital speed to decrease. At a specific point, the orbital speed matches what’s needed to be stationary relative to Earth: L1.
L2: Beyond Sun and Earth: The Sun and Earth are both pulling the satellite inwards, increasing the gravitational pull. This increases the orbital speed. Since it’s closer to Earth, Earth’s gravity varies more with distance than the Sun’s gravity, so an object that’s too far away will feel very little of Earth’s gravity but about the same Solar gravity. If you pick the right point, Earth and Solar gravity combine to get the exact orbital speed necessary to get the satellite to be stationary relative to Earth: L2.
L3: Opposite Sun and Earth: At this distance, Earth’s pull is not very significant relative to the Sun, so the satellite is acting a lot like Earth. However, Earth is still pulling a bit on it, so the satellite is a bit closer to the Sun than Earth is.
L4/L5: 60 Degrees Ahead of or Behind Earth, on Earth’s Orbit: Because the Sun and Earth are pulling in different directions (but also not directly opposite directions like L1), the forces are not as simple to describe. Satellites at both of these points are being pulled slightly inwards by the Earth, as Earth is not directly in front of or behind the satellites. However, the forward/backwards pull of the Earth is counteracted by the slight angle at which the Sun is pulling on the satellites; L4, ahead of Earth, is pulled backwards by the Earth and slightly forwards by the Sun, while L5, behind Earth, is pulled forwards by Earth and slightly backwards by the Sun.
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