Modeling the remnant mass, spin, and recoil from unequal-mass, precessing black-hole binaries: The intermediate mass ratio regime

We revisit the modeling of the properties of the remnant black hole resulting from the merger of a black-hole binary as a function of the parameters of the binary. We provide a set of empirical formulas for the final mass, spin, and recoil velocity of the final black hole as a function of the mass ratio and individual spins of the progenitor. In order to determine the fitting coefficients for these formulas, we perform a set of 128 new numerical evolutions of precessing, unequal-mass black-hole binaries, and fit to the resulting remnant mass, spin, and recoil. In order to reduce the complexity of the analysis, we chose configurations that have one of the black holes spinning, with dimensionless spin $\alpha =0.8$, at different angles with respect to the orbital angular momentum, and the other nonspinning. In addition to evolving families of binaries with different spin-inclination angles, we also evolved binaries with mass ratios as small as $q=M_1/M_2=1/6$. We use the resulting empirical formulas to predict the probabilities of black hole mergers leading to a given recoil velocity, total radiated gravitational energy, and final black hole spin.

Figure 1

Finding the orbital plane near merger. The upper plot shows the orbital separation $r(t)$ versus time. The inset shows $r(t)$ near merger and $\ddot{r}(t)$ (rescaled by 200 for clarity). The points ${\overrightarrow{r}}_{+}$, ${\overrightarrow{r}}_0$, and ${\overrightarrow{r}}_{-}$ correspond to the times where $\ddot{r}$ is maximized, zero, and minimized, respectively (denoted with arrows here). The plot below shows the trajectory, the points ${\overrightarrow{r}}_{+},{\overrightarrow{r}}_0,{\overrightarrow{r}}_{-}$ (large red dots) and the “merger” plane.

Figure 2

The NQ configuration. Here one BH is spinning (typically the larger one) and one is nonspinning. Numerical evolutions preserve the NQ configurations approximately.

Figure 3

The K configuration. $S_{1z}=-S_{2z}$, while $S_{1x}=S_{2x}$ and $S_{1y}=S_{2y}$, initially.

Figure 4

The RMS error in the prediction of the recoil for the K45 configurations as a function of $\varpi$ (the angle between the unit vector ${\widehat{n}}_0$ and the $x$ axis of the rotated basis) for several choices of $\zeta$. Note that $\varpi$ is measured in radians.

Figure 5

Plots of the fitted $V_1$ versus inclination angle $\theta$ and $q$ for the NQ configurations. Each data point represents the maximum of $V_1$ over a family of azimuthal configurations with the same inclination angle and mass ratio. The last plot shows an extrapolation to $q=1/10$.

Figure 6

The relative errors (residuals) in the fit of $V_{\varphi }$ versus $q$. Note that there are multiple data points for each $q$.

Figure 7

The distribution of recoils for the NQ configurations. The green boxes indicate the recoil angle as measured with respect to the initial orbital plane while the red cylinders indicate the recoil angle is measured with respect to the orbital plane at merger. Here $\iota$ is the inclination angle of the recoil with respect to the orbital plane in units of degrees and the recoil is measured in units of $\mathrm{km}{\mathrm{s}}^{-1}$.

Figure 8

Plots of the fitted $E_c$ versus inclination angle $\theta$ and $q$ for the NQ configurations. Each data point represents the value of $E_c$ for a family of azimuthal configurations with the same inclination angle and mass ratio. Note $\delta M\approx E_c$, and that a prime denotes that ${\overrightarrow{S}}_0$ was used in the fits, rather than $\overrightarrow{S}$.

Figure 9

The relative errors (residuals) in the fit of $E_c$ as a function of the binary’s parameters. Note that there are multiple data points for each $q$, and that a prime denotes that ${\overrightarrow{S}}_0$ was used in the fits, rather than $\overrightarrow{S}$.

Figure 10

Plots of the fitted $A_c$ versus inclination angle $\theta$ and $q$ for the NQ configurations. Each data point represents the value of $A_c$ for a family of azimuthal configurations with the same inclination angle and mass ratio. Note that $\alpha \approx \sqrt{A_c}$ and that a prime denotes that ${\overrightarrow{S}}_0$ was used in the fits, rather than $\overrightarrow{S}$.

Figure 11

The relative errors (residuals) in the fit of $A_c$ as a function of the binary’s parameters. Note that there are multiple data points for each $q$ and a prime denotes that ${\overrightarrow{S}}_0$ was used in the fits, rather than $\overrightarrow{S}$.

Figure 12

The integrated probability $\mathrm{\Pi }(v)$ for the remnant to recoil at speed $v$ or higher (in $\mathrm{km}{\mathrm{s}}^{-1}$). The blue (solid) curves are for cold accretion models, the red (dot-dashed) curves are for hot accretion models, and gray (dotted) curves are for dry mergers. Within a given color/line style (blue, red, gray), the dark shade indicates that the spins were aligned azimuthally, the light shade indicates that the spins were antialigned azimuthally, and the intermediate shade indicates random azimuthal alignment. The top plot shows the probabilities when modeling the recoil with $V_{459}$ and the lower panel shows the probabilities when modeling the recoil with $V_{4^{\prime }59}$.

Figure 13

The probability (nonintegrated) for a mass loss of $\delta \mathcal{M}$. The blue (solid) curves are for cold accretion models, the red (dot-dashed) curves are for hot accretion models, and gray (dotted) curves are for dry mergers. Within a given color/line style (blue, red, gray), the dark shade indicates that the spins were aligned azimuthally, the light shade indicates that the spins were antialigned azimuthally, and the intermediate shade indicates random azimuthal alignment. The upper panel displays probabilities when the radiated energy is modeled in terms of the spin variable $\tilde{S}$ and the lower panel when the variable is chosen to be ${\tilde{S}}_0$.

Figure 14

The probability (nonintegrated) for a remnant spin $\alpha$. The blue (solid) curves are for cold accretion models, the red (dot-dashed) curves are for hot accretion models, and gray (dotted) curves are for dry mergers. Within a given color/line style (blue, red, gray), the dark shade indicates that the spins were aligned azimuthally, the light shade indicates that the spins were antialigned azimuthally, and the intermediate shade indicates random azimuthal alignment. The upper panel displays probabilities when the final remnant spin magnitude is modeled in terms of the total spin variable $\tilde{S}$ and the lower panel when the variable is chosen to be ${\tilde{S}}_0$.

Figure 15

The integrated probability $\mathrm{\Pi }(v)$ for the remnant to recoil at speed $v$ or higher (in $\mathrm{km}{\mathrm{s}}^{-1}$) as predicted using $V_{p^{\prime }59}$ (black curves), the older cross kick formula (red curves), and the original hangup kick formula (green curves). The plot on the left shows the results for hot accretion, the plot in the center shows the results for cold accretion, and the plot on the right shows the results for dry mergers. In each plot, solid curves are for random azimuthal alignment, dashed curves are for azimuthal alignment, and dot-dashed curves are for antiazimuthal alignment.