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A pulsar is a neutron star left behind by a supernova, and many of them were thrown out of their birth place at hundreds of kilometres per second. Most stars form in clusters, so if we run a pulsar's motion backwards we can ask which young cluster it might have come from.

1. We take the pulsar's measured position, distance and proper motion (ATNF pulsar catalogue) with their uncertainties. Its radial velocity is not measured, so it is drawn from a broad range around the Galactic rotation.
2. We make 1,000 slightly different copies of the pulsar and 1,000 copies of each cluster (positions and motions drawn from the cluster's measurements, Hunt & Reffert 2024).
3. Every copy is moved backwards in time through a model of the Milky Way's gravity, up to the age of the cluster, and the closest approach of each pulsar copy to its cluster copy is recorded.
4. For every pair we publish how close, how long ago and at what relative speed the closest approach happened, and what share of the copies came closer than the cluster's half-mass radius (its core) or its total radius.

The animation shows this for PSR J1302-6350 and the cluster UBC 525, whose connection is known from the literature: the copies of the pulsar (red) are shown relative to the cluster, going back in time.

• Share inside the half-mass radius: the fraction of the 1,000 copies that passed closer to the cluster centre than the radius holding half of the cluster's stars. It is the main ranking number, and the one that singled out UBC 525 for the pulsar in the example.
• Share inside the total radius: the same for the cluster's whole extent. It is larger and less specific.
• Lookback time: how long ago the closest approach was. A pulsar born in a cluster must have passed through it no earlier than the cluster's age.
• Relative speed: the speed difference at the closest approach, which would be the ejection speed. Because the pulsar's radial velocity is unknown and drawn from a broad range, these speeds mostly reflect that range and should not be read as measurements.
• A high share means that many copies met the cluster. It does not prove that the pulsar was born there: clusters are numerous, and some chance alignments are expected. The published data are the raw search output, and no control against chance alignment has been applied yet.

Each pulsar has a figure like this one: the sky positions of the best candidate clusters, the ranking by share of copies inside the cluster, and when and how close each approach was.