Head-on infall of two compact objects: Third post-Newtonian energy flux

Head-on infall of two compact objects with arbitrary mass ratio is investigated using the multipolar post-Minkowskian approximation method. At the third post-Newtonian order the energy flux, in addition to the instantaneous contributions, also includes hereditary contributions consisting of the gravitational-wave tails, tails-of-tails, and the tail-squared terms. The results are given both for infall from infinity and also for infall from a finite distance. These analytical expressions should be useful for the comparison with the high accuracy numerical relativity results within the limit in which post-Newtonian approximations are valid.

Figure 1

Energy flux in ADM coordinates, in units of ${\nu }^2$ (where $\nu$ is the symmetric mass ratio of the binary), as a function of the parameter $\gamma =Gm/c^{2}z$ for the head-on situation, for four different initial separations characterized by the parameter ${\gamma }_i=Gm/c^2{z}_i$: ${\gamma }_i=0.05$, ${\gamma }_i=0.02$, ${\gamma }_i=0.01$ and ${\gamma }_i=0.0$ (infinite initial separation limit) which correspond to the situations when the initial separation $z_i$ between the two objects is $20Gm/c^2$, $50Gm/c^2$, $100Gm/c^2$ and $\infty$, respectively. Curves in the top panels correspond to the value of $\nu =0$ (test-body limit) while those in the bottom panels correspond to the value of $\nu =0.25$ (equal-mass case). Left, middle, and right panels in both top and bottom panels correspond to the 2PN, 2.5PN, and 3PN accurate expressions for energy flux. The values given on the y-axis have been scaled by the factor $c^5/G=3.63\times {10}^{52}\mathrm{joules}\mathrm{\text{−}}{sec}^{-1}$. The labels on the x-axis and alternative x-axis corresponds to inverse of the parameter $\gamma$ (which is the separation between the objects under radial infall at any instant in units of $Gm/c^2$) and the values of the PN parameter $\gamma$, respectively.

Figure 2

Similar to Fig. 1 but for total energy radiated in units of ${\nu }^2{c}^{2}m$, during the head-on infall of two compact objects from a initial separation $z_i$ in ADM coordinates (related to the parameter ${\gamma }_i$) to a final separation of $z_f$ (corresponding parameter ${\gamma }_f=Gm/c^2{z}_f$).