Tidal effects in eccentric coalescing neutron star binaries

Dynamically formed compact object binaries may still be eccentric while in the LIGO/Virgo band. For a neutron star (NS) in an eccentric binary, the fundamental modes (f-modes) are excited at the pericenter, transferring energy from the orbit to oscillations in the NS. We model this system by coupling the evolution of the NS f-modes to the orbital evolution of the binary as it circularizes and moves toward coalescence. NS f-mode excitation generally speeds up the orbital decay and advances the phase of the gravitational wave signal from the system. We calculate how this effect changes the timing of pericenter passages and examine how the cumulative phase shift before merger depends on the initial eccentricity of the system. This phase shift can be much larger for highly eccentric mergers than for circular mergers and can be used to probe the NS equation of state.

Figure 1

The evolution of the mode energy, Eq. (12); orbital energy ($\mathrm{\Delta }{E}_{\mathrm{orb}}=E_{\mathrm{orb}}-E_{\mathrm{orb},0}$), Eq. (13); and total energy ($\mathrm{\Delta }{E}_{\mathrm{tot}}=E_{\mathrm{tot}}-E_{\mathrm{tot},0}=E_{\mathrm{mode}}+\mathrm{\Delta }{E}_{\mathrm{orb}}$) in units where $G=M_1=R_1=1$ for a binary with a single NS. We have used a $\mathrm{\Gamma }=2$ polytrope to model the NS with ${\omega }_{\alpha }=1.22(M_1/R_1^3{)}^{1/2}$ and $Q_{\alpha }=0.56$. The initial pericenter and eccentricity are $D_{\mathrm{p},0}=5.995R_1$ and $e_0=0.9$, corresponding to $|\mathrm{\Delta }{\widehat{P}}_{\alpha }|=1.6\times {10}^{-4}$ [see Eq. (15)]. This calculation does not include GR (i.e., $A_{5/2}=B_{5/2}=A_{\mathrm{PN}}=B_{\mathrm{PN}}=0$). The $l=2$, $m=(2,0,-2)$ f-modes are all accounted for in the integration. The peaks occur during pericenter passages. The mode energy away from pericenter undergoes small-amplitude oscillations over multiple orbits.

Figure 2

Same as Fig. 1, but with an initial pericenter distance of $D_{\mathrm{p},0}=6R_1$ and $|\mathrm{\Delta }{\widehat{P}}_{\alpha }|=1.5\times {10}^{-4}$. The mode energy (away from the peaks at pericenter passages) can reach larger values than in Fig. 1 due to a resonance between the mode frequency and the orbital frequency (${\omega }_{\alpha }\simeq 401{\mathrm{\Omega }}_{\mathrm{orb}}$).

Figure 3

Same as in Fig. 1, but with a smaller initial pericenter distance of $D_{\mathrm{p},0}=3R_1$ so that $|\mathrm{\Delta }{\widehat{P}}_{\alpha }|=82$. The mode energy grows chaotically over many orbits.

Figure 4

An example of how f-mode oscillations alter the orbital evolution of a coalescing, eccentric NS binary with $M_1=M_2=1.4M_{\odot }$ and $R_1=R_2=10\mathrm{km}$. The NSs are modeled as $\mathrm{\Gamma }=2$ polytropes. This system has initial eccentricity $e_0=0.9$ and pericenter distance $D_{\mathrm{p},0}=6R_1$, corresponding to a GW pericenter frequency of $f_{\mathrm{p},0}=575\mathrm{Hz}$. The solid blue lines in the top two panels show the binary separation and orbital phase (true anomaly) including tidal effects, while the dashed red lines show the same without tides. The blue (red) circles and triangles mark the times of the apocenter and pericenter. The quantity $\mathrm{\Delta }t(N_{\mathrm{p}})$ is defined as the difference in the timing of a pericenter passage for calculations with and without tides, where $N_{\mathrm{p}}$ is the number of pericenter passages prior to merger; i.e., the example in the top panel shows $\mathrm{\Delta }t(N_{\mathrm{p}}=1)$. The third panel shows $\mathrm{\Delta }\mathrm{\Phi }$, the difference in the orbital phase for calculations with and without dynamical tides [see Eq. (18)]. The bottom panel shows the evolution of the mode energies; the $m=2$ (prograde) mode dominates.

Figure 5

The gravitational waveform [Eq. (23)] that corresponds to the orbital evolution shown in Fig. 4. The amplitude is scaled by $(R_1/d)$, with $d$ the distance to the system. The bottom panel shows the difference in the phase of the GWs, $\mathrm{\Delta }\mathrm{\Psi }(t)$, for a calculation with dynamical tides and one without [Eq. (24)].

Figure 6

Cumulative time advance due to dynamical tides [see the top panel of Fig. 4 for the definition of $\mathrm{\Delta }t(N_{\mathrm{p}})$, where $N_{\mathrm{p}}$ is the number of pericenter passages prior to merger] as a function of the number of orbits for equal mass NS binaries with an initial pericenter distance of $D_{\mathrm{p},0}=5R_1$ in the left panel and $D_{\mathrm{p},0}=6R_1$ in the right panel.

Figure 7

Cumulative GW phase difference between a calculation with dynamical tides and without at $t_{\mathrm{merg}}$ (the time when $D=2.5R_1$) as a function of initial eccentricity $e_0$ for two different values of the initial pericenter distance (see Fig. 5). The two dashed lines show the result for circular orbits [see Eq. (26)].

Figure 8

The upper panel is the same as in Fig. 7 with the orbital phase rather than the gravitational phase. The lower panels show the final few orbits for calculations with (blue solid lines) and without (red dashed lines) dynamical tides for binaries with $D_{\mathrm{p},0}=5.0R_1$. In panel (a) $e_0=0$. The other five panels show values of $e_0$ that maximize [(b) and (c)] or minimize [(d), (e), and (f)] $\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{merg}}$. The dotted green circles indicate $D=2.5R_1$.

Figure 9

The GW phase shift $(\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{res}}{)}_n=2(\mathrm{\Delta }{\mathrm{\Phi }}_{\mathrm{res}}{)}_n$ [see Eq. (a10)] due to mode-orbit resonance (${\omega }_{\alpha }=n{\mathrm{\Omega }}_{\mathrm{orb}}$ with positive integer $n$) as a function of the orbital eccentricity at the resonance. The results are for the $l=m=2$ f-mode of a $\mathrm{\Gamma }=2$ polytropic NS model with $M_1=1.4M_{\odot }$ and $R_1=10\mathrm{km}$ in an equal mass binary. The dashed lines indicate $\delta N_{\mathrm{res}}=1$ [see Eq. (a12)]. The results for $(\mathrm{\Delta }{\mathrm{\Psi }}_{\mathrm{res}}{)}_n$ (solid lines) are valid when $\delta N_{\mathrm{res}}>1$, in the shaded region to the left of the dashed lines. The maximum displayed $e$ for each value of $n$ corresponds to the condition that the pericenter distance $D_{\mathrm{p}}$ exceeds $2.5R_1$.