A Parametric Approximate Solution to the Collinear Three-Body Problem

We present a family of parametric trajectories for the collinear gravitational three-body problem with equal masses. Beginning from the scalar equations of motion, we derive a quadratic-in-time ansatz for the relative distances between the bodies. The key mathematical step is the reduction of the absolute-value gravitational terms to rational functions under the assumption of constant sign, which renders the equations integrable in closed form. Integration yields expressions involving arctangent functions with eleven free parameters. Through systematic numerical optimisation over approximately 360,000 parameter combinations, we identify a configuration that approximately satisfies the equations of motion over a finite time interval to high precision: the total squared residual over twelve evaluations (three equations at four time points) is , corresponding to a per-equation root-mean-square error of approximately 0.103 acceleration units. At the optimal parameters, the centre of mass moves with exactly constant velocity, confirming that the solution describes a genuinely isolated system with no external forces. The optimal parameter values are simple rational numbers (, , , , ), suggesting the possibility of an underlying algebraic structure worthy of further investigation.