Originally Posted by JayUtah

This just goes to show how tricky the calculations of NEOs that pass close to the Earth can be.

Semi-tricky.

Introductory orbital mechanics involves the two-body problem. That can be studied using closed-form mathematics. Anything beyond two bodies usually requires iterative mathematics. And iterative in this case is a placeholder word that means Extremely expensive computers chugging through solutions and consuming $2,000 of electricity a day to do it.

For some problems we consider the restricted three-body problem. That is, when the mass of one body (e.g., a spaceship or a booster stage) is insignificant compared to the mass of the other two (e.g., planets and moons) then the orbits of the two massive bodies are considered unperturbed by the small body. So the massive bodies' orbits are computed in closed form as a two-body problem and the small body's path is solved iteratively.

For other problems, such as plotting orbits of asteroids and Kuiper Belt objects, we have to use the full n-body method because those objects interact with each other as gravitational peers.