The two-body problem
For two point masses, the relative motion obeys r̈ = −μr/r³, whose solutions are conic sections: circles, ellipses, parabolas and hyperbolas. Angular momentum h = r × v and specific energy ε = v²/2 − μ/r are conserved; e and a follow from them.
Perturbations
Real orbits drift. Earth's equatorial bulge (the J₂ term) makes orbital planes precess — exploited by Sun-synchronous orbits, whose planes turn about 0.986° per day to keep constant lighting. Atmospheric drag, solar radiation pressure and third-body gravity add further perturbations.
Three bodies
In the circular restricted three-body problem a small mass moves under two large ones. Five equilibrium points appear: the Lagrange points. L1 and L2 of the Sun–Earth system, about 1.5 million km from Earth, host observatories such as SOHO (L1) and Webb and Gaia (L2), which follow halo orbits around them.