Because it’s basically a two-body calculation. Also even if it were three objects, if they are all in a line, there are very easy simplifications you can do by averaging the positions and adding the mass of two of the objects.
Lagrange points are considered a restricted three body problem which is easier to compute. A restricted 3 body problem assumes one of the bodies is massless, reducing the problem to two bodies. For an Earth-Moon-artificial satellite system, the satellite is ignored and the Lagrange points computed with respect to just the Moon and Earth.
By ignoring the mass of the third body. We just calculate were the gravity of the two bodies cancle out not how the gravity of an object put in the lagrange point changes the path of either of the other two objects.
The 3bp has no general solution. There is no equation that describes the position of any three bodies based solely on their starting positions.
But if you choose some very specific bodies, it’s all good. Totally manageable. As long as you are smart about what three bodies.
In the case of lagrange points, they mandate that:
Two of the bodies are so much more massive than the third that they are, effectively, in a 2 body system and the third body just hitchhikes on that system. This works for, say, the earth/sun/spaceprobe system but it does not work for the pluto-charon system.
The third (much smaller) body is in a very specific spot with a very specific speed. Depending on your lagrange point, there may be a bit of allowable deviation, or it may be entirely unstable and any small deviation sets your satellite on a course for chaos.
There are other solutions to the 3bp that have similar requirements. The Earth-moon-sun system, for instance. You can solve it, but you can’t create a general solution – one that describes everything.
The whole solar system is a 9-body problem, but we calculated the orbits of all the planets accurately enough in like the 1800s. Technically, every asteroid in the belt and beyond Neptune are also bodies that exert forces on all the planets. Technically, everything in the universe affects everything else.
In practice, these forces are usually so small that we can ignore them entirely and still get results that are good enough for what we need to do.
You can’t “solve” the equation for three bodies, but there are other ways to get there. Lagrange points are calculated from a limited version of 3-body, where we are solving for possible orbits of a body with insignificant mass compared to the other two. Basically we don’t take that mass into account. With this simplification you can solve the equation and find stable orbits, which are the Lagrange points.
There are other ways to “solve” the full 3-body (or n-body) solution with numerical integration. You take discrete time steps and assume simplified movement between those steps. Numerical integration is a whole another bag of worms with different algorithms that are good or bad in different ways.
eli5: lets ignore all of the complexity of orbits and lagrange points for a second. lets say im asking you to determine the total weight of three objects:
a huge solid steel anvil
a bowling ball
a hair follicle
would you even bother weighing the hair follicle? thats whats going on with the lagrange points in your example: one of the bodies is basically massless, or imparts such a low influence on the system that its treated as a rounding error or outright ignored.
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Calculating the forces involved between all 3 bodies for every point in their respective orbits is damn near impossible.
Finding where they zero out isn’t that bad
Because it’s basically a two-body calculation. Also even if it were three objects, if they are all in a line, there are very easy simplifications you can do by averaging the positions and adding the mass of two of the objects.
Lagrange points are considered a restricted three body problem which is easier to compute. A restricted 3 body problem assumes one of the bodies is massless, reducing the problem to two bodies. For an Earth-Moon-artificial satellite system, the satellite is ignored and the Lagrange points computed with respect to just the Moon and Earth.
By ignoring the mass of the third body. We just calculate were the gravity of the two bodies cancle out not how the gravity of an object put in the lagrange point changes the path of either of the other two objects.
The 3bp has no general solution. There is no equation that describes the position of any three bodies based solely on their starting positions.
But if you choose some very specific bodies, it’s all good. Totally manageable. As long as you are smart about what three bodies.
In the case of lagrange points, they mandate that:
Two of the bodies are so much more massive than the third that they are, effectively, in a 2 body system and the third body just hitchhikes on that system. This works for, say, the earth/sun/spaceprobe system but it does not work for the pluto-charon system.
The third (much smaller) body is in a very specific spot with a very specific speed. Depending on your lagrange point, there may be a bit of allowable deviation, or it may be entirely unstable and any small deviation sets your satellite on a course for chaos.
There are other solutions to the 3bp that have similar requirements. The Earth-moon-sun system, for instance. You can solve it, but you can’t create a general solution – one that describes everything.
The whole solar system is a 9-body problem, but we calculated the orbits of all the planets accurately enough in like the 1800s. Technically, every asteroid in the belt and beyond Neptune are also bodies that exert forces on all the planets. Technically, everything in the universe affects everything else.
In practice, these forces are usually so small that we can ignore them entirely and still get results that are good enough for what we need to do.
You can’t “solve” the equation for three bodies, but there are other ways to get there. Lagrange points are calculated from a limited version of 3-body, where we are solving for possible orbits of a body with insignificant mass compared to the other two. Basically we don’t take that mass into account. With this simplification you can solve the equation and find stable orbits, which are the Lagrange points.
There are other ways to “solve” the full 3-body (or n-body) solution with numerical integration. You take discrete time steps and assume simplified movement between those steps. Numerical integration is a whole another bag of worms with different algorithms that are good or bad in different ways.
eli5: lets ignore all of the complexity of orbits and lagrange points for a second. lets say im asking you to determine the total weight of three objects:
would you even bother weighing the hair follicle? thats whats going on with the lagrange points in your example: one of the bodies is basically massless, or imparts such a low influence on the system that its treated as a rounding error or outright ignored.