Gravitational Waves from Periodic Three-Body Systems

Three bodies moving in a periodic orbit under the influence of Newtonian gravity ought to emit gravitational waves. We have calculated the gravitational radiation quadrupolar waveforms and the corresponding luminosities for the $13+11$ recently discovered three-body periodic orbits in Newtonian gravity. These waves clearly allow one to distinguish between their sources: all $13+11$ orbits have different waveforms and their luminosities (evaluated at the same orbit energy and body mass) vary by up to 13 orders of magnitude in the mean, and up to 20 orders of magnitude for the peak values.

Figure 1

The gravitational radiation quadrupolar waveforms $h_{+,\times }\times r$ as functions of the elapsed time $t$ in units of the period $T$, for two periodic three-body orbits (in units of $Gm/c^2$; we have set $G=m=c=1$ throughout this Letter) and $r$ is the radial distance from the source to the observer. Dotted (blue) and solid (red) curves denote the $+$ and $\times$ modes, respectively. Top: Simo’s figure eight, Ref. [22]; and bottom: orbit I.B.1 Moth I. Note the symmetry of these two graphs under the (time-)reflection about the orbits’ midpoint $T/2$ during one period $T$.

Figure 2

The instantaneous (time unaveraged) luminosity $P$ of quadrupolar gravitational radiation emitted from periodic three-body orbits as a function of the elapsed time $t$ in units of the period $T$. Note the logarithmic scale for the luminosity $P$ ($y$ axis). Top left: Moore’s figure eight; second from top left: I.A.2 butterfly II; third from top left: II.B.7 dragonfly; bottom left: I.B.1 moth I; top right: $(M8{)}^7$, second from top right: I.A.3 bumblebee; second from bottom right: I.B.5 goggles; bottom right: II.B.6 butterfly IV.