Analytic self-force calculations in the post-Newtonian regime: Eccentric orbits on a Schwarzschild background

We present a method for solving the first-order Einstein field equations in a post-Newtonian (PN) expansion. Our calculations generalize the work of Bini and Damour and subsequently Kavanagh et al. to consider eccentric orbits on a Schwarzschild background. We derive expressions for the retarded metric perturbation at the location of the particle for all $\ell$-modes. We find that, despite first appearances, the Regge-Wheeler gauge metric perturbation is $C^0$ at the particle for all $\ell$. As a first use of our solutions, we compute the gauge-invariant quantity $⟨U⟩$ through 4PN while simultaneously expanding in eccentricity through $e^{10}$. By anticipating the $e\to 1$ singular behavior at each PN order, we greatly improve the accuracy of our results for large $e$. We use $⟨U⟩$ to find 4PN contributions to the effective one body potential $\widehat{Q}$ through $e^{10}$ and at linear order in the mass ratio.

Figure 1

The effect of resumming the small-$e$ expansion at each PN order as seen by comparing to numerical data, all computed at $p=1000$. We see that especially as eccentricity increases, our resummation greatly improves convergence. Also, note the consistency of our convergence throughout the orbit. Our results are no less effective at periapsis than apoapsis. The dips in the residuals are from zero crossings, and not meaningful. See the discussion in the text for more details.

Figure 2

Comparison of our PN expressions with numerical data from Table II of Akcay et al. [22]. Dots are numerical values and lines are our analytic calculations. The top row is ${\log }_{10}$ of the absolute value of the full $⟨U{⟩}_{\mathrm{gsf}}$. Successive rows show residuals after subtracting each term in the PN series. The consistency of our agreement out to $e=0.4$ is only possible because of our resummation of the small-$e$ expansion. Unevenness of the final residuals for $p=100$ is likely a numerical artifact.