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We study the metric perturbations induced by a classical spinning particlemoving along a circular orbit on a Schwarzschild background, limiting theanalysis to effects which are first order in spin. The particle is assumed tomove on the equatorial plane and has its spin aligned with the $z$-axis. Themetric perturbations are obtained by using two different approaches, i.e., byworking in two different gauges: the Regge-Wheeler gauge (using theRegge-Wheeler-Zerilli formalism) and a radiation gauge (using the Teukolskyformalism). We then compute the linear-in-spin contribution to the first-orderself-force contribution to Detweiler's redshift invariant up to the 8.5post-Newtonian order. We check that our result is the same in both gauges, asappropriate for a gauge-invariant quantity, and agrees with the currently known3.5 post-Newtonian results.

Detweiler's redshift invariant for spinning particles along circular orbits on a Schwarzschild background