Twobody

The twobody problem, in a physical context often called the two-body problem, is the classical mechanics question of determining the motion of two bodies interacting through a central force, most commonly Newtonian gravity. It assumes no other forces and ignores distortions from other bodies, providing a fundamental model for planetary orbits and binary star systems.

In Newtonian gravity the motions can be reduced to the relative position r = r2 − r1 between the bodies. With m1 and m2 the masses and μ = m1 m2 / (m1 + m2) the reduced mass, the equation of motion becomes μ r'' = − G m1 m2 r / r^3, or equivalently r'' = − G (m1 + m2) r / r^3. The problem then reduces to a single particle of mass μ moving in the gravitational potential of total mass M = m1 + m2.

The orbit of the relative motion is a conic section. In polar form, r(θ) = p / (1 + e cos θ), where the semi-latus rectum p = h^2 / (G M) and h = |r × v| is the specific angular momentum. The specific orbital energy ε = v^2/2 − G M / r determines the orbit type: ε < 0 yields ellipses (including circular orbits), ε = 0 a parabola, and ε > 0 a hyperbola. The eccentricity e satisfies e^2 = 1 + 2 ε h^2 / (G M)^2.

Both bodies move about the center of mass, with positions r1 = −(m2 / M) r and r2 = (m1 / M) r. The twobody problem under Newtonian gravity thus has a closed-form solution and forms the basis for more complex celestial mechanics, perturbations, and relativistic corrections. Extensions include non-Newtonian central forces, perturbations from additional bodies, and general relativistic effects that introduce precession.