Figure 1

Self-force correction to the binding energy as a function of the dimensionless parameter $y$. Increasing $y$ corresponds to decreasing orbital radius. The yellow circles show our numerical results for Eq. (13). The blue curve shows the mechanical binding energy (14) derived from the first law of binary mechanics, plotted using high-accuracy numerical data from Ref. [62]. The inset shows the absolute difference between the two (blue circles) along with a conservative estimate of our absolute numerical error (orange triangles). We note that the values for $E_{\mathrm{SF}}$ and ${\widehat{\mathcal{E}}}_0-1\approx -(y/2)$ are comparable, so the contribution of $E_{\mathrm{SF}}$ to the total binding energy is suppressed by a factor of $q$ relative to the leading term.