  We continue a previous work on the comparison between the post-Newtonian (PN)
approximation and the gravitational self-force (SF) analysis of circular orbits
in a Schwarzschild background. We show that the numerical SF data contain
physical information corresponding to extremely high PN approximations. We find
that knowing analytically determined appropriate PN parameters helps
tremendously in allowing the numerical data to be used to obtain higher order
PN coefficients. Using standard PN theory we compute analytically the leading
4PN and the next-to-leading 5PN logarithmic terms in the conservative part of
the dynamics of a compact binary system. The numerical perturbative SF results
support well the analytic PN calculations through first order in the mass
ratio, and are used to accurately measure the 4PN and 5PN non-logarithmic
coefficients in a particular gauge invariant observable. Furthermore we are
able to give estimates of higher order contributions up to the 7PN level. We
also confirm with high precision the value of the 3PN coefficient. This
interplay between PN and SF efforts is important for the synthesis of template
waveforms of extreme mass ratio inspirals to be analysed by the space-based
gravitational wave instrument LISA. Our work will also have an impact on
efforts that combine numerical results in a quantitative analytical framework
so as to generate complete inspiral waveforms for the ground-based detection of
gravitational waves by instruments such as LIGO and Virgo.
