Editing Orbit (section)

===Approaches to many-body problems===
Rather than an exact closed form solution, orbits with many bodies can be approximated with arbitrarily high accuracy. These approximations take two forms:
:One form takes the pure elliptic motion as a basis and adds [[perturbation (astronomy)|perturbation]] terms to account for the gravitational influence of multiple bodies. This is convenient for calculating the positions of astronomical bodies. The equations of motion of the moons, planets, and other bodies are known with great accuracy, and are used to generate [[ephemeris|tables]] for [[celestial navigation]]. Still, there are [[secular phenomena]] that have to be dealt with by [[Parameterized post-Newtonian formalism|post-Newtonian]] methods.
:The [[differential equation]] form is used for scientific or mission-planning purposes. According to Newton's laws, the sum of all the forces acting on a body will equal the mass of the body times its acceleration (''F = ma''). Therefore accelerations can be expressed in terms of positions. The perturbation terms are much easier to describe in this form. Predicting subsequent positions and velocities from initial values of position and velocity corresponds to solving an [[initial value problem]]. Numerical methods calculate the positions and velocities of the objects a short time in the future, then repeat the calculation ad nauseam. However, tiny arithmetic errors from the limited accuracy of a computer's math are cumulative, which limits the accuracy of this approach.

Differential simulations with large numbers of objects perform the calculations in a hierarchical pairwise fashion between centers of mass. Using this scheme, galaxies, star clusters and other large assemblages of objects have been simulated.<ref>{{Cite journal |last1=Carleton |first1=Timothy |last2=Guo |first2=Yicheng |last3=Munshi |first3=Ferah |last4=Tremmel |first4=Michael |last5=Wright |first5=Anna |title=An excess of globular clusters in Ultra-Diffuse Galaxies formed through tidal heating |journal=Monthly Notices of the Royal Astronomical Society |year=2021 |volume=502 |pages=398–406 |doi=10.1093/mnras/stab031 |doi-access=free |arxiv=2008.11205 }}</ref>