Gravitational waves from binaries on unbound orbits

A generalized true anomaly-type parametrization, convenient to describe both bound and open orbits of a two-body system in general relativity is introduced. A complete description of the time evolution of both the radial and angular equations of a binary system taking into account the first order post-Newtonian (1PN) corrections is given. The gravitational radiation field emitted by the system is computed in the 1PN approximation including higher multipole moments beyond the standard quadrupole term. The gravitational waveforms in the time domain are explicitly given up to the 1PN order for unbound orbits, but the results are also illustrated on binaries on elliptic orbits with special attention given to the effects of eccentricity.

Figure 1

Time dependence of the polarization states ($h_{+}$ depicted by solid line and $h_{\times }$ corresponds to dashed line) at the Newtonian order for the Hulse-Taylor (left) and for the J0737-3039 pulsars (right). The amplitudes are multiplied by the factor ${10}^{20}$.

Figure 2

Time dependence of the first corrections to the polarization states at 0.5PN order. Again, $h_{+}$ is depicted by the solid line and $h_{\times }$ by the dashed line. The corrections for the Hulse-Taylor pulsar are shown on the left and for the J-0737 pulsar on the right. For further comparison amplitudes are multiplied by the factor ${10}^{20}$.

Figure 3

Multipolar corrections to the polarization states at 1PN order. Again, the corrections for the Hulse-Taylor pulsar are shown on the left and for the J0737-3039 pulsar on the right. The solid line shows the $h_{+}$ polarization state and the dashed line shows $h_{\times }$. Again, the amplitudes are multiplied by the factor ${10}^{20}$.

Figure 4

1PN order corrections to the polarization states, which arise from the perturbative description of the elements of the motion. The corrections for the Hulse-Taylor pulsar are shown on the left and for the J0737-3039 pulsar on the right. The solid line shows the $h_{+}$ polarization state and the dashed line shows $h_{\times }$. To compare the different terms of these corrections, the amplitudes are multiplied by the same factor as before, namely ${10}^{20}$.

Figure 5

In these figures we plot the full 1PN corrections to the polarization states for the Hulse-Taylor pulsar (left) and for the J0737-3039 pulsar (right). The solid line represents the $h_{+}$ polarization state and the dashed line shows $h_{\times }$.

Figure 6

Time dependence of the polarization states at Newtonian order for the two open orbit sources. The solid line shows the parabolic case and the dashed line shows the waveform of the hyperbolic ($ϵ=2$) source. The $h_{+}$ polarization state is depicted on the left figure and $h_{\times }$ is shown on the left. The amplitudes are multiplied by the factor ${10}^{20}$.

Figure 7

Time dependence of the first corrections to the waveform polarization states at 0.5PN order. Again, the corrections to the $h_{+}$ polarization are shown on the left and for $h_{\times }$ on the right. The solid line shows the parabolic case and the dashed line shows $ϵ=2$ hyperbolic source. For further comparison amplitudes are multiplied by the factor ${10}^{20}$.

Figure 8

Multipolar corrections to the polarization states at 1PN order. Again, the corrections to $h_{+}$ are shown on the left and to $h_{\times }$ on the right. Parabolic and hyperbolic orbit cases are depicted by solid and dashed lines, respectively. Again, the amplitudes are multiplied by the factor ${10}^{20}$.

Figure 9

1PN order corrections to the polarization states, which arise from the perturbative description of the elements of the motion. The notation is the same as on the former figures in this subsection. To compare the different terms of these corrections, the amplitudes are multiplied by the same factor as before, namely ${10}^{20}$.

Figure 10

In these figures we plot the full 1PN corrections to the polarization states for the parabolic (soled line) and hyperbolic (dashed line) sources. The corrections to the $h_{+}$ polarization states are depicted on the left, while to $h_{\times }$ on the right.