Evolution of highly eccentric binary neutron stars including tidal effects

This work is the first in a series of studies aimed at understanding the dynamics of highly eccentric binary neutron stars, and constructing an appropriate gravitational-waveform model for detection. Such binaries are possible sources for ground-based gravitational wave detectors, and are expected to form through dynamical scattering and multibody interactions in globular clusters and galactic nuclei. In contrast to black holes, oscillations of neutron stars are generically excited by tidal effects after close pericenter passage. Depending on the equation of state, this can enhance the loss of orbital energy by up to tens of percent over that radiated away by gravitational waves during an orbit. Under the same interaction mechanism, part of the orbital angular momentum is also transferred to the star. We calculate the impact of the neutron star oscillations on the orbital evolution of such systems, and compare these results to full numerical simulations. Utilizing a post-Newtonian flux description we propose a preliminary model to predict the timing of different pericenter passages. A refined version of this model (taking into account post-Newtonian corrections to the tidal coupling and the oscillations of the stars) may serve as a waveform model for such highly eccentric systems.

Figure 1

$|I_{\pm 2}|$ and $|I_0|$ [Eq. (26)] as functions of $R_0/R_{*}$. In the Newtonian limit, when $R_0=2R_{*}$ the surfaces of two undeformed stars are in marginal contact at the periastron passage.

Figure 2

The ratio between the total mode energy (angular momentum) deposited onto stars and the energy (angular momentum) carried away by GWs during the main burst, as a function of the normalized periastron frequency $f/f_c$. Here $R_{*}{c}^2/(GM)$ is taken to be 5.88. The dots shown in the top plot represent estimates from the numerical simulations listed in Table 1.

Figure 3

A comparison of the GWs from a BH-NS and NS-NS case with $R_0/M_{\text{tot}}=11.5$. The top and middle panels show the real part of the (2,2) component of ${\mathrm{\Psi }}_4$, the former emphasizing the GW burst from the close encounter, and the latter emphasizing the GW oscillations from the $f$-mode excitations in the NS(s). The bottom panel shows the post fly-by GWs ($t-r>500M_{\text{tot}}$) in the frequency domain. The peak at approximately $f_{\mathrm{GW}}\approx 0.027/M_{\text{tot}}$ associated with the $f$-mode oscillations is a factor of 2 larger in the binary NS case, as expected if the NSs are tidally perturbed by the same amount in both cases.

Figure 4

The coordinate separation versus orbital angle at large distances post-fly-by for a BH-NS and NS-NS system with the same initial binary parameters. We also show the Newtonian orbits with the best-fit parameters for these two cases.

Figure 5

Plot of PN eccentricities for a sequence of pericenter passages, indexed by $n$. The quantities $e_t$, $e_r$, and $e_{\varphi }$ are not gauge invariant and their values are given in modified harmonic coordinates. The initial pericenter separation is $R_{0,1}=3R_{*}$, and the initial $e_r$ is set to 0.9. The quantities $e_{\mathrm{t},\text{no mode}}$, $e_{\mathrm{r},\text{no mode}}$, and $e_{\varphi ,\text{no mode}}$ are obtained without including the star’s oscillations; $e_{\mathrm{t},\text{no init}}$ is computed including the star’s oscillation but zero $A_{\mathrm{init}}$; $e_{\mathrm{t},\mathrm{init}}$ is computed taking into account the evolution of the mode amplitude over time.

Figure 6

The same setup as in Fig. 5. The top plot represents the evolution of the orbital period; the middle panel represents the evolution of the pericenter frequency; the bottom panel represents the evolution of the pericenter distance.