Gravitational waves from compact binaries in post-Newtonian accurate hyperbolic orbits

We derive from first principles a third post-Newtonian (3PN) accurate Keplerian-type parametric solution to describe PN accurate dynamics of nonspinning compact binaries in hyperbolic orbits. Orbital elements and functions of the parametric solution are obtained in terms of the conserved orbital energy and angular momentum in both Arnowitt-Deser-Misner-type and modified harmonic coordinates. Elegant checks are provided that include a modified analytic continuation prescription to obtain our independent hyperbolic parametric solution from its eccentric version. A prescription to model gravitational wave polarization states for hyperbolic compact binaries experiencing 3.5PN accurate orbital motion is presented that employs our 3PN accurate parametric solution.

Figure 1

Scaled $H_{+}{|}_Q(l)$ and $H_{\times }{|}_Q(l)$ plots for nonspinning compact binaries with total mass $m=20M_{\odot }$ and mass ratio $q=1$. We let the eccentricity $e_t$ take three values 1.5, 1.3 and 1.2, while choosing an impact parameter $b\sim 30Gm/c^2$ and inclination angle $\theta =\frac{\pi }{4}$. We observe the expected linear memory effect in the cross polarization state.

Figure 2

Trajectories and the associated scaled $H_{\times }{|}_Q(l)$ for hyperbolic compact binaries, with a choice of two different impact parameters $b$, eccentricity $e_t=1.1$, total mass $m=20M_{\odot }$, mass ratio $q=1$, and inclination angle $\theta =\frac{\pi }{4}$. For the trajectories, we adopt the geometric unit system. Newtonian and 3.5PN accurate hyperbolic orbits are denoted by black and red lines, respectively. The orbital trajectory of the relativistic system is clearly different, especially for hyperbolic passages with smaller $b$ values, which is attributed to the advance of periastron. Relativistic effects also change the nature of the waveforms, as evident from the associated $h_{\times }{|}_Q(l)$ plots.