A pictorial explanation of the likelihood function used to fit for the evolution of the supernova shell. We assume each cloud forms in the shell at the time of crossing. Three clouds—Cet (pink trace), LIC (salmon trace), and Leo (cyan trace)—are shown here as examples, but the full likelihood is fit over the entire ensemble of clouds. The present-day 3D positions and 3D space motions of the clouds (marked with the pink, salmon, and cyan vectors) define the backward trajectory of the clouds, subject to a deceleration term parameterized by α. Given a model for the radial and velocity evolution of the supernova shell (defined by the ambient volume density of hydrogen nuclei n₀) with the epicenter of the expansion defined by the explosion time t_exp, each cloud will first intersect the shell at a single point in time, as shown in the time snapshots in the subpanels at right. For this example, LIC crosses the shell at t ≈ −0.13 Myr (subpanel framed in salmon), Leo at t ≈ −0.43 Myr (subpanel framed in cyan), and Cet at t ≈ −0.99 Myr (subpanel framed in pink). Our likelihood function seeks to minimize the velocity difference between the clouds and the shell at their respective times of intersection, modulo a small velocity shift δv_sh, which can account for additional sources of uncertainty on the shell velocity due to the simplifications of our model.