Gravitational waves from neutron star excitations in a binary inspiral

In the context of a binary inspiral of mixed neutron star–black hole systems, we investigate the excitation of the neutron star oscillation modes by the orbital motion. We study generic eccentric orbits and show that tidal interaction can excite the $f$-mode oscillations of the star by computing the amount of energy and angular momentum deposited into the star by the orbital motion tidal forces via closed form analytic expressions. We study the $f$-mode oscillations of cold neutron stars using recent microscopic nuclear equations of state, and we compute their imprint into the emitted gravitational waves.

Figure 1

Distribution of the rate of energy absorbed by the fundamental NS oscillation mode divided by $K$, defined in Eq. (23), as a function of the harmonic $j$ of the fundamental mode in eccentric orbits, for nine equally spaced values of eccentricity (from $e_0=0$ in blue, through $e_i=i/10$ until $e_8=0.8$ in yellow). For each value of eccentricity, the curves for $x=0.01$ and $x=0.07$ are shown, with $x$ defined in Eq. (4). For the latter value of $x$, the contribution of the $\ell =2$ mode alone is isolated in the dashed curve. For $x=0.07$, the resonant absorption peaks $\ell =2$ are visible for $e=0.5$, 0.6, 0.7, 0.8, and for $e=0.6$, 0.7, 0.8 the resonant peak of $\ell =3$ is also visible, as $\overline{j}{\omega }_0=\overline{j}{x}^{3/2}/(G_{N}M)\simeq 18.1\mathrm{kHz}(\overline{j}/34)(6.965M_{\odot }/M)(x/0.07{)}^{3/2}$ where for this plot $M_{\mathrm{BH}}=5M_{\odot }$ and we used the equation of state A (APR) of Ref. [20] and central density ${\rho }_0=1.5\times {10}^{15}\mathrm{gr}/{\mathrm{cm}}^3$; see Table 1. In this case, the $n=0$ $f$ mode has frequency ${\nu }_f^{\ell =2}=2.888\mathrm{kHz}$ [we have verified that for $x<0.07$ the NS is safe from tidal breaking, whose condition requires $\mathcal{D}\lesssim 0.3R_{\mathrm{NS}}(M_{\mathrm{NS}}/M_{\mathrm{BH}}{)}^{1/3}{x}^{1/2}(M/M_{\mathrm{BH}}{)}^{1/2}$; see 36].

Figure 2

Rate of energy absorbed ${\dot{E}}_{*}$ as a function of the eccentricity, with $M_{\mathrm{BH}}=5M_{\odot }$, the NS with equation of state A (APR) of [20] for different values of the central density ${\rho }_0=(1.5,1.2,0.99)\times {10}^{15}\mathrm{gr}/{\mathrm{cm}}^3$; lines of increasing thickness show results for increasing ${\rho }_0$. For comparison, we also plot the GW luminosity for the two values of $x$; all functions are divided by the Newtonian GW luminosity at zero eccentricity ${\dot{E}}_{\mathrm{GW}0}$ given by Eq. (24). Plots for the other equations of states described in Appendix pp3 are shown in Fig. 7 and are qualitatively similar.

Figure 3

Rate of angular momentum absorbed as a function of the eccentricity, the same parameters as in Fig. 2. Here ${\dot{L}}_{\mathrm{GW}}$ is the Newtonian angular momentum loss in GWs for small eccentricities ${\dot{L}}_{\mathrm{GW}}=\frac{32}{5}{\eta }^{2}M\frac{x^{7/2}}{(1-e^2{)}^2}(1+\frac{7}{8}{e}^2)$ and ${\dot{L}}_{\mathrm{GW}0}\equiv {\dot{L}}_{\mathrm{GW}}{|}_{e=0}$.

Figure 4

Second derivative of the quadrupole ${\ddot{Q}}_{22}$ divided by the reduced mass $\mu \equiv \eta M$: the contribution from orbital dynamics compared with the (magnified) contribution from the NS oscillation $Q_{*22}$ for an inspiral with initial conditions $x_i=0.04$, $e_i=0.4$, and $M_{\mathrm{BH}}=5M_{\odot }$ and the parameter for the NS given by equation of state B (SLy4) [21] with ${\rho }_0=2\times {10}^{15}\mathrm{gr}/{\mathrm{cm}}^3$.

Figure 5

Given the same parameters of Fig. 4, here are displayed the $f$-mode displacements $q_{0\ell \ell }$ for $\ell =2$, 3, 4 (magnified by a factor of $10^3$). Also shown are the main gravitational wave frequency $f_{\mathrm{GW}}\equiv {\omega }_0/\pi$ and the eccentricity along the inspiral dynamics considered.

Figure 6

Distribution of the energy per unit of mass absorbed by a single NS oscillation $f$ mode ${\dot{E}}_j$ divided by the quantity $K$ defined in Eq. (23) as a function of the harmonic mode $j$ of the fundamental orbital frequency in eccentric orbits for the equation of state [21] in Table 2 for ${\rho }_0=2.0\times {10}^{15}\mathrm{gr}/{\mathrm{cm}}^3$.

Figure 7

Rate of energy absorbed ${\dot{E}}_{*}$ as a function of eccentricity, with $M_{\mathrm{BH}}=5M_{\odot }$, NS with the equation of state given by Ref. [21] in Table 2. For comparison, we also plot the GW luminosity for two values of $x$, and all functions are divided by the Newtonian GW luminosity at zero eccentricity ${\dot{E}}_{\mathrm{GW}0}$ given by Eq. (24). Note that for large eccentricity $e>0.7$ absorption by the NS as computed in this approximation is not negligible compared to GW emission. For each equation of state, results for the three values of the central density reported in the corresponding tables are reported, with increasing line thickness denoting a higher central density.

Figure 8

Rate of angular momentum absorbed as a function of eccentricity, the same parameters as in Fig. 2. Here ${\dot{L}}_{\mathrm{GW}}$ is the Newtonian angular momentum loss in GWs for small eccentricities ${\dot{L}}_{\mathrm{GW}}=\frac{32}{5}{\eta }^{2}M\frac{x^{7/2}}{(1-e^2{)}^2}(1+\frac{7}{8}{e}^2)$ and ${\dot{L}}_{\mathrm{GW}0}={\dot{L}}_{\mathrm{GW}}{|}_{e=0}$, with $M\equiv M_{*}+M_{\mathrm{BH}}$, $\eta \equiv M_{*}{M}_{\mathrm{BH}}/M^2$.