Gravitational wave signals from a chaotic system: A point mass with a disk

We study gravitational waves from a particle moving around a system of a point mass with a disk in Newtonian gravitational theory. A particle motion in this system can be chaotic when the gravitational contribution from a surface density of a disk is comparable with that from a point mass. In such an orbit, we sometimes find that there appears a phase in which particle motion becomes nearly regular (so-called “stagnant motion” or stickiness) for a finite time interval between more strongly chaotic phases. To study how these different chaotic behaviors affect observation of gravitational waves, we investigate a correlation of the particle motion and the waves. We find that such a difference in chaotic motions reflects on the wave forms and energy spectra. The character of the waves in the stagnant motion is quite different from that either in a regular motion or in a more strongly chaotic motion. This suggests that we may make a distinction between different chaotic behaviors of the orbit via the gravitational waves.

Figure 1

Poincaré maps of orbits of a particle with $H=-0.2$ and $L=1$ across the plane $z=0$ in a system of a point mass with a disk. We set the thickness of the disk $z_0=0.5$, and its surface density (a) $\alpha =0.01$, (b) $\alpha =0.1$, or (c) $\alpha =10.0$. All figures are superpositions of trajectories starting from different initial conditions. In (a) and (c), all trajectories form regular tori. In (b), some trajectories from certain initial conditions still seem to form tori, but others do not. In fact, one initial condition, $(\varpi ,v^{\varpi },z,v^z)=(1.2,0,0,0.76)$ gives an almost two-dimensional map on which the orbital points are widely scattered, which means that the particle motion is chaotic [we call it orbit (b)]. The outermost trajectories in (a) and (c) are called orbit (a) and (c), whose initial conditions are $(\varpi ,v^{\varpi },z,v^z)=(1.2,0,0,0.76)$ and $(2.5,0,0,0.49)$, respectively.

Figure 2

The time evolutions of local Lyapunov exponents for orbits (a), (b), and (c) in Fig. 1. The Lyapunov exponents for orbits (a) and (c) settle down to very small values, but that for orbit (b) is large and changing in time. It decreases to a very small value in the time interval of $t/M=(1.6\sim 3.8)\times {10}^5$.

Figure 3

The particle motion in the $r$ direction in terms of time and its Poincaré map for the time interval between $t/M={10}^5$ and $t=4\times {10}^5$ for orbit (b). The Poincaré map shows there exist many small tori around the origin.

Figure 4

The particle motion of orbit (b) in the $r$ direction for a longer time interval than that in Fig. 3. There exists new stagnant motion as the same one in Fig. 3.

Figure 5

The power spectrum of the motion in the $r$ direction of orbit (b-2), which is part of a nearly regular motion in orbit (b) [$t/M=(1.6\sim 3.8)\times {10}^5$]. We find a power-law spectrum, e.g., a $1/f$ fluctuation. This reflects the existence of small tori in the phase space, and the particle moves almost regularly there [21].

Figure 6

The gravitational waveforms evaluated by the quadrupole formula. Top, middle, and bottom figures correspond to those for orbits (a), (b), and (c), respectively. The left and right rows give the “$+$” and “$\times$” polarization modes, respectively.

Figure 7

The energy spectra of the gravitational waves shown in Fig. 6. Orbit (b-2) gives the spectrum of the waves for the “stagnant motion,” i.e., when the particle motion of orbit (b) becomes near regular for $t/M=(1.6\sim 3.8)\times {10}^5$. (a) and (c) show many sharp peaks at certain characteristic frequencies. This is because of the regular motion. The spectrum in (b), which looks like white noise for $fM\le {10}^{-2}$, is clearly different from those in (a) and (c), but the spectrum in (b-2) does not look like white noise. It looks similar to the cases (a) and (c). However, the peaks are not sharp but rather broadened by appearing so many other spikes. Note that the typical frequency of the orbits is in the range of $fM={10}^{-2}\sim {10}^{-1}$ (see Fig. 5).

Figure 8

Magnification of energy spectra of orbits (a), (b-1), (b-2), and (c).