Waveform degeneracy of binary systems and Lagrange three-body systems

Abstract

A particular solution to the three-body problem is the circular Lagrange three-body system, where the masses move in circular orbits such that they always constitute an equilateral triangle. Such a system has been found to emit gravitational waves with a waveform similar to that of a binary system. In this work, we study the gravitational waveform degeneracy between quasi-circular binary systems and Lagrange three-body systems up to 0.5PN order. Assuming we know the parameters of a given binary system, we determine the parameters of the Lagrange triple that produces the same waveform as that of the binary. We show that there exists a mass quadrupole degeneracy in both the plus and cross modes, characterized by two parameters. We also find that there are binary systems and linearly stable Lagrange three-body systems that can have the same mass quadrupole waveform up to the coalescence time. In such cases, the normalized overlap of the waveforms with respect to the power spectral density of the advanced LIGO design remains above 0.97 as long as the binary has nearly symmetric masses. Beyond the mass quadrupole, there is a unique degeneracy at the 0.5PN. However, the Lagrange triple that satisfies this degeneracy is unstable.

Figures (9)

• Figure 1: Schematic diagram of the Lagrange three-body system. Each mass moves in a circular orbit about the COM.
• Figure 2: Plot of the total mass $M_{\mathrm{(3B)}}$ in solar masses as a function of the normalized masses $\beta_1$ and $\beta_2$ for a chirp mass of $\mathcal{M}_{\mathrm{(3B)}} = 5 \ \mathrm{M}_{\odot}$.
• Figure 3: Contour plot of $r_{\mathrm{(3B)}}/r_{\mathrm{(2B)}}$ in equation \ref{['eq:quadDegeneracy_r_3B_both']} as a function of $M_{\mathrm{(3B)}}/\mathcal{M}_{\mathrm{(2B)}}$ and $\beta_1$. The white region is the Newtonian stability region of the Lagrange triple.
• Figure 4: Values of $\beta_1$ satisfying equation \ref{['eq:equalChirpMass']} for different values of $\mathcal{M}_{\mathrm{(2B)}}/M_{(\mathrm{3B})}$. The gray region is the Newtonian stability region of a Lagrange triple, and the black dashed line and the blue dash-dot lines are approximately the last values of $\mathcal{M}_{\mathrm{(2B)}}/M_{(\mathrm{3B})}$ when $\beta_1$ falls within the Newtonian stability region and when $\beta_1 < 0.5$, respectively
• Figure 5: Waveforms of SymmetricBinary and a linearly stable Lagrange triple ($b_0 = 3.0411 \times 10^{-11} \ \mathrm{pc}$, $m_{1,\mathrm{(3B)}} = m_{2,\mathrm{(3B)}} = 2.25 \ \mathrm{M}_{\odot}$, $m_{3,\mathrm{(3B)}} = 145.5 \ \mathrm{M}_{\odot}$, $r_{\mathrm{(3B)}} = 39744 \times 10^{6} \ \mathrm{pc}$, $\iota_{\mathrm{(3B)}} = 30^{\circ}$). In both plots, the binary coalesces first, and the matches up to the coalescence time are $M\qty(h^{+}_{\mathrm{(2B)}}, h^{+}_{\mathrm{(3B)}}) = M\qty(h^{\times}_{\mathrm{(2B)}}, h^{\times}_{\mathrm{(3B)}}) = 0.838$.