Post-Newtonian diagnosis of quasiequilibrium configurations of neutron star–neutron star and neutron star–black hole binaries

We use a post-Newtonian diagnostic tool to examine numerically generated quasiequilibrium initial data sets for nonspinning double neutron star and neutron star–black hole binary systems. The post-Newtonian equations include the effects of tidal interactions, parametrized by the compactness of the neutron stars and by suitable values of “apsidal” constants, which measure the degree of distortion of stars subjected to tidal forces. We find that the post-Newtonian diagnostic agrees well with the double neutron star initial data, typically to better than half a percent except where tidal distortions are becoming extreme. We show that the differences could be interpreted as representing small residual eccentricity in the initial orbits. In comparing the diagnostic with preliminary numerical data on neutron star–black hole binaries, we find less agreement.

Figure 1

Best-fit eccentricity estimated from the energy (thick lines) and the angular momentum (thin lines) for irrotational, equal-mass neutron star–neutron star binaries. Top: we fix $\Gamma =2.00$ and consider different values of the compactness. Bottom: we fix $M/R=0.14$ and consider different values of $\Gamma$. Results on the left use the Newtonian apsidal constants, results on the right use the “relativistically corrected” apsidal constants of Eq. (a6).

Figure 2

Binding energy and angular momentum vs $m\Omega$ for equal masses, $\Gamma =2.0$, $(M/R{)}_1=(M/R{)}_2=0.12$. Top: circles are the numerical data points, solid line is the PN diagnostic for circular orbits using the Newtonian apsidal constant values (Table ) for $\Gamma =2$, dashed line has no tidal contributions; dot-dashed line uses the uniform density values for the apsidal constants; dotted line uses the “relativistically corrected” apsidal constants of Eq. (a6). Bottom: Numerical data (circles), point mass (dashed line) and uniform density (dot-dashed line) plotted relative to $\Gamma =2$ PN diagnostic as the baseline. Triangles denote numerical data plotted relative to $\Gamma =2$ circular-orbit PN diagnostic using the relativistically corrected apsidal constants.

Figure 3

Same as Fig. 2, for $\Gamma =2.0$, $(M/R{)}_1=(M/R{)}_2=0.18$.

Figure 4

Same as the bottom panels of Fig. 2, for various values of $\Gamma$ and $M/R$, for equal-mass binaries. Baseline is the PN circular-orbit diagnostic with Newtonian values of the apsidal constants.

Figure 5

Same as Fig. 4, but for binaries with different masses.

Figure 6

Comparison of PN diagnostic eccentricities or errors with truncation errors in the PN approximation and with virial errors in the numerical simulations.

Figure 7

Binding energy and angular momentum vs $m\Omega$ for a black hole–neutron star binary with mass ratio 5. Plotted are data from Taniguchi et al. (circles) and from Grandclément (triangles), along with the corresponding PN diagnostic curve for circular orbits.

Figure 8

Best-fit eccentricity estimated from the energy (thick lines) and the angular momentum (thin lines) for irrotational, black hole–neutron star binaries. Left: data from Taniguchi et al. [19], for a compactness factor of 0.145, and four mass ratios. Right: data from Grandclément, for a single mass ratio of $5:1$, and a range of compactness factors. The dotted curves in each figure correspond to a mass ratio 5 and similar compactness factors; they also correspond to the data plotted in Fig. 7.