Multi-messenger Extended Emission from the Compact Remnant in GW170817

The observation of GW170817 (Abbott et al. 2017a, 2017b) was the first observation of a low-mass compact binary coalescence seen in a long-duration ascending gravitational-wave (GW) chirp. The accompanying GRB 170817A, identified by the Fermi-Gamma-Ray Burst Monitor (GBM) and the INTErnational Gamma-Ray Astrophysics Laboratory (INTEGRAL; Connaughton 2017; Goldstein et al. 2017; Kasliwal et al. 2017; Savchenko et al. 2017; Pozanenko et al. 2018), represents the merger of either a neutron star (NS) with another neutron star (NS–NS) or with a companion black hole (BH; NS–BH) with a chirp mass of about one solar mass. The potentially broad implications of the former has received considerable attention because of what it reveals about the origin of heavy elements (D'Avanzo et al. 2017; Kasen et al. 2017; Pian et al. 2017; Smartt et al. 2017) as well as entirely novel measurements of the Hubble constant (Freedman 2017; Guidorzi et al. 2017).

Regarding chirp mass, the nature of GW170817 is inconclusive in the absence of observing final coalescence at high GW frequencies (Coughlin & Dietrich 2019). For NS–NS coalescence, numerical simulations (e.g., Baiotti & Rezzolla 2017) show that gravitational radiation effectively satisfies the canonical model signal of binary coalescence in a run-up to about 1 kHz, beyond which the amplitude levels off and ultimately decays as the two stars merge into a single object at a maximal frequency ∼3 kHz. In contrast, NS–BH mergers include tidal break-up (Lattimer & Schramm 1976). In a brief epoch of hyper-accretion, the BH would be near-extremal with a remnant of NS debris to form a torus outside its innermost stable circular orbit (ISCO). This process is marked by gravitational radiation switching off early on at a frequency 500–1500 Hz (Vallisneri 2000; Etienne et al. 2009; Faber 2009; Ferrari et al. 2010) and possibly quasi-normal mode oscillations at yet higher frequencies (e.g., Yang et al. 2018).

Here, we report on the energy output ${ \mathcal E }$ in gravitational radiation post-merger that appears as a descending chirp of extended emission (EE) marking spin-down of a compact remnant to binary coalescence at a Gaussian equivalent level of confidence of 4.2σ (van Putten & Della Valle 2019). We give a robust estimate of ${ \mathcal E }$ using response curves determined by signal injection experiments in data of the LIGO detectors at Hanford (H1) and Livingston (L1). ${ \mathcal E }$ introduces a new calorimetric constraint that may break the degeneracy of an NS or BH central engine.

${ \mathcal E }$ reported here points to the core collapse of the merged NS produced in GW170817, inheriting its angular momentum J while greatly increasing the associated spin-energy E_J through collapse into a Kerr BH (Kerr 1963).

After our injection experiments were initiated, we learned of an independent analysis of energy considerations by single-template injections, pointing qualitatively to similar energies without, however, identifying the origin of our EE (Oliver et al. 2019).

We set out to determine response curves of our search pipeline by signal injections into LIGO data (Vallisneri et al. 2015; Figures 1–2), including whitening, butterfly filtering, and image analysis of merged (H1,L1) spectrograms (Appendix). Whitening occurs by normalizing the Fourier spectrum over an intermediate bandwidth of 2 Hz, showing GW170817 more clearly than without whitening (van Putten & Della Valle 2019).

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We recall that GW170817 is observed as an ascending chirp signifying the merger of two compact stars with a time of coalescence ${t}_{c}=1842.43\,{\rm{s}}$ followed by GRB 170817A across a gap of 1.7 s. In our injection experiments with LIGO data, we include a model DNS with the same chirp mass ${{ \mathcal M }}_{c}\,=1.188{M}_{\odot }$ as GW170817 (Figure 2) at time of coalescence of about 1818 s, producing two side-by-side ascending chirps (Figure 3). A double NS system (DNS) is described by binary masses M₁, M₂, $\mu ={M}_{1}{M}_{2}/M$, $M={M}_{1}+{M}_{2}$, at orbital separation a and orbital frequency ${\rm{\Omega }}\simeq c\sqrt{{R}_{g}/{a}^{3}}$ $(a\gg {R}_{g}$), where ${R}_{g}={GM}/{c}^{2}$ is the gravitational radius of the system, given the velocity of light c and Newton's constant G. This merger chirp has a quadrupole GW frequency ${f}_{\mathrm{GW}}={\pi }^{-1}{\rm{\Omega }}$

$$\begin{eqnarray}&&{f}_{\mathrm{GW}}(t)=A{\left({t}_{c}-t\right)}^{-\tfrac{3}{8}}\,\,(t\lt {t}_{c}),\end{eqnarray} \tag{ 1 }$$

$[A]={{\rm{s}}}^{-5/8}\,\mathrm{Hz}$, with strain $h(t)=(4\mu /D){\left(M{\rm{\Omega }}\right)}^{\tfrac{2}{3}}$, $h(t)\simeq 1.7\,\times {10}^{-22}\left(M/3{M}_{\odot }\right){\left(D/40{\rm{Mpc}}\right)}^{-1}{\left({f}_{\mathrm{GW}}/250{\rm{Hz}}\right)}^{\tfrac{2}{3}}$ and ${L}_{\mathrm{GW}}\,=(32/5){\left({{ \mathcal M }}_{c}{\rm{\Omega }}\right)}^{10/3}{L}_{0}$, where ${L}_{0}={c}^{5}/G\simeq 200,000{M}_{\odot }{c}^{2}\,{{\rm{s}}}^{-1}$ (e.g., Ferrari et al. 2010). For GW170817, $A\simeq 138\,{{\rm{s}}}^{-5/8}\,\mathrm{Hz}$. Up to 260 Hz in both H1 and L1, L_GW reaches $1.35\,\times {10}^{50}\,{\text{erg s}}^{-1}\simeq 7.5\times {10}^{-5}{M}_{\odot }{c}^{2}\,{{\rm{s}}}^{-1}$, i.e., $4\times {10}^{-10}{L}_{0}$. While small compared to ${10}^{-5}{L}_{0}$ of GW150914 at similar frequency, GW170817 produced the largest strain observed by its proximity of $D\simeq 40\,\mathrm{Mpc}$. It emitted ${E}_{0}=0.43 \% {M}_{\odot }{c}^{2}$ over 200–300 Hz with $h=(1.4\mbox{--}1.8)\times {10}^{-22}$ over ${\rm{\Delta }}t\simeq 0.25\,{\rm{s}}$ across $74\,{\rm{km}}\lt r\lt 97\,{\rm{km}}$ assuming ${M}_{1}={M}_{2}$.

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A merged (H1,L1) spectrogram shows EE post-merger below 700 Hz in the form of an exponential feature

$$\begin{eqnarray}&&{f}_{\mathrm{GW}}(t)=({f}_{s}-{f}_{0}){e}^{-(t-{t}_{s})/{\tau }_{s}}+{f}_{0}\,\,(t\gt {t}_{s})\end{eqnarray} \tag{ 2 }$$

with the observed ${\tau }_{s}=3.01\pm 0.2\,{\rm{s}}$, ${t}_{s}=1843.1\,{\rm{s}}$, ${f}_{s}\,=650\,\mathrm{Hz}$, and ${f}_{0}=98\,\mathrm{Hz}$. For illustrative purposes, we note that the isotropic equivalent strain $h={L}_{\mathrm{GW}}^{1/2}/({\rm{\Omega }}D)$ ($c=G=1$ in geometrical units) for the chirp mass of a small quadrupole mass moment $\zeta =\delta m/M$ gives $h(t)\,\simeq 2.7\,\times \,{10}^{-23}\left(\zeta /3 \% \right){\left(D/40{\rm{Mpc}}\right)}^{-1}{\left({f}_{\mathrm{GW}}/650{\rm{Hz}}\right)}^{2/3}$, ${L}_{\mathrm{GW}}\simeq 2\,\times {10}^{52}{\left(\zeta /3 \% \right)}^{2}{\left(10M/r\right)}^{5}\,{\text{erg s}}^{-1}\simeq 1 \% {M}_{\odot }{c}^{2}/{{\rm{s}}}^{-1}$.

Next we use phase coherent injections with frequency evolution (2) (see Appendix). No change in the results listed below are found after including phase incoherence by a Poisson distribution of random phase jumps over intermediate timescales, down to the duration $\tau =0.5\,{\rm{s}}$ of our butterfly templates. $\tau =0.5$ appears intrinsic, as the EE feature tends to fade out as τ approaches 1 s.

The total energy output ${ \mathcal E }={\int }_{0}^{T}{L}_{\mathrm{GW}}{dt}$ is computed numerically as sums ${ \mathcal E }={E}_{0}{K}^{-1}\sum {\nu }_{i}^{2}{h}_{i}^{2}$ (samples at t_i, $i=1,2,\,..,\,n$) covering a post-merger interval of duration T, where $K=\sum {\nu }_{0,j}^{2}{h}_{0,j}^{2}$ (samples at t_j, $j=1,2,\,\cdots ,\,m$) is a reference sum with energy E₀ over a duration T₀, and ${\nu }_{i}={f}_{\mathrm{GW}}({t}_{i})$ denotes GW frequency. E₀ is conform quadrupole emission in the same orientation as the progenitor binary by the conservation of orbital-to-spin angular momentum in transition to its remnant. Blind to any model in particular, we consider injections with power-law strain $h\propto {f}^{\alpha }$ with T = 7 s.

Figure 3 shows the outcome of a signal injection alongside GW170817EE after a calibration C_h = 0.7 for observed-to-true strain due to non-ideal H1 and L1 detector orientations relative to GW170817. Extended to multiple injections, the results show that there is no interference with the merger signal or with one another.

Figure 4 shows our estimated response curves $\chi ({ \mathcal E })$ for $h\propto {f}^{\alpha }$ ($0.1\leqslant \alpha \leqslant 1.0$). By $\hat{\chi }\simeq 7.2$ of the EE to GW170817, we infer

$$\begin{eqnarray}&&{ \mathcal E }\simeq (3.5\pm 1) \% {M}_{\odot }{c}^{2}.\end{eqnarray} \tag{ 3 }$$

For the descending chirp at hand (2), ${ \mathcal E }$ mostly derives early on at high f_GW with ${L}_{\mathrm{GW}}\lesssim 1 \% {M}_{\odot }{c}^{2}{{\rm{s}}}^{-1}$ (Figure 2).

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${ \mathcal E }$ in (3) is a significant amount of energy, exceeding the merger output observed up to about 300 Hz, emitted as a descending chirp over a secular timescale of seconds with ${f}_{\mathrm{GW}}\lt 700\,\mathrm{Hz}$ far below the characteristic frequency $c/{R}_{S}\simeq 30\,\mathrm{kHz}$ of the Schwarzschild radius ${R}_{S}=2{R}_{g}$. Important energies also appear in GRB 170817A and mass ejecta (e.g., Mooley et al. 2018a, 2018b). Of these, ${ \mathcal E }$ and f_GW will serve as primary observational constraints on the remnant, i.e., E_J of a rapidly spinning merged NS or rotating BH.

While a long-lived NS might be luminous in gravitational radiation through a baryon-loaded magnetosphere (Appendix), its spin frequency ${f}_{s}=(1/2){f}_{\mathrm{GW}}$ inferred from our EE is less than one-fifth the break-up spin frequency of about 2 kHz. This modest initial spin limits E_J to below $0.5 \% {M}_{\odot }{c}^{2}$, and probably somewhat less based on more stringent limits (e.g., Haensel et al. 2009; Oliver et al. 2019).

However, E_J greatly increases due to the core collapse of the merged NS in the immediate aftermath of GW170817, here at time of core collapse about or prior to ${t}_{s}=0.67(\pm 0.03)$ s post-merger (Figure 3), where the 30 ms refers to our time-step in t_s. By the Kerr (1963) metric,

$$\begin{eqnarray}&&{E}_{J}=2{{Mc}}^{2}{\sin }^{2}\left(\lambda /4\right)\lesssim 1{M}_{\odot }{c}^{2}\left(M/3{M}_{\odot }\right)\end{eqnarray} \tag{ 4 }$$

in terms of $a/M=\sin \lambda$, $J=a\sin \lambda$. This potentially enormous energy reservoir amply accounts for ${ \mathcal E }$ even at modest efficiency η, provided that a mechanism is in place to tap and convert E_J into gravitational radiation. Moderate frequencies ${f}_{\mathrm{GW}}\lt 700\,\mathrm{Hz}$ can be realized in catalytic conversion into quadrupole emission by a non-axisymmetric disk or torus that is sufficiently wide or geometrically thick. Exhausting E_J, a descending chirp results from the expansion of the ISCO during BH spin-down.

To add some concreteness, we estimate η in the spin-down of an initially rapidly rotating BH, losing J to matter in Alfvén waves through an inner torus magnetosphere (van Putten 1999, 2001). By heating, a non-axisymmetric thick torus is expected to generate frequencies correlated to but below those of a thin torus about the ISCO (Coward et al. 2002). In geometrical units, an extended torus produces emission from an orbital radius $r\equiv {{zR}}_{g}$ at twice the local orbital frequency, i.e., ${f}_{\mathrm{GW}}\simeq c{\pi }^{-1}\sqrt{{R}_{g}/{r}^{3}}$. Asymptotic scaling relations for large radii (modest η; van Putten & Levinson 2003) show ${L}_{\mathrm{GW}}\sim {10}^{52}\,{\text{erg s}}^{-1}$ for a non-axisymmetric torus with mass ratio $\sigma ={M}_{T}/M\simeq 0.1$. Accompanying minor output can be found in MeV neutrinos and ${E}_{w}\simeq {\eta }^{2}{E}_{J}$ in magnetic winds (van Putten & Levinson 2002a, 2002b, 2003)—most of E_J is dissipated unseen in the event horizon, increasing in area via the Bekenstein–Hawking entropy (van Putten 2015). The observed $150\,\mathrm{Hz}\lt {f}_{\mathrm{GW}}\lt 700\,\mathrm{Hz}$ indicates an effective radius of a quadrupole mass moment that is initially about three times the ISCO radius (Figure 5), suggesting a relatively thick torus. f_GW decreases with z because of the expansion of the ISCO during BH spin-down.

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Via the numerical integration of this spin-down process, the catalytic conversion of E_J gives (Figure 5)

$$\begin{eqnarray}&&{ \mathcal E }\simeq \left\langle \eta \right\rangle {E}_{J}\simeq 3.6-4.3 \% {M}_{\odot }{c}^{2}\end{eqnarray} \tag{ 5 }$$

for canonical values of initial a/M, depending somewhat on the start frequency ${f}_{s}=600\mbox{--}700\,\mathrm{Hz}$, which is consistent with (3) inferred from $\chi ({ \mathcal E })$.

The model estimate (5) uses effective values of disk mass m and K throughout. This does not readily predict $h({f}_{\mathrm{GW}})$ or the observed exponential feature (2), as m and K will be time dependent and vary with z. Effective mean values is used only for our present focus on total energy output.

While the nature of GW170817 in the chirp up to 300 Hz is somewhat inconclusive (Coughlin & Dietrich 2019), ${ \mathcal E }$ provides a novel calorimetric constraint on its remnant. ${ \mathcal E }$ in (3) challenges a hyper-massive NS (Oliver et al. 2019) yet is naturally accommodated by (5) in core collapse to a Kerr BH.

In converting E_J, ${ \mathcal E }$ is accompanied by MeV neutrinos and magnetic winds (van Putten & Levinson 2003), which is consistent with evidence for BH spin-down in normalized light curves of long GRBs (van Putten 2012).

Starting with the merged NS from a DNS, a time of collapse about or prior to ${t}_{s}\simeq 0.67(\pm 0.03)$ s (Figure 3) appears consistent—perhaps in mild tension—with the recently estimated time of collapse ${0.98}_{-0.26}^{+0.31}$ s based on jet propagation times and the mass of blue ejecta (Gill et al. 2019).

Sustained by Alfvén waves outward over an inner torus magnetosphere, a torus developing a dynamo with magnetic field $B=O\left({10}^{16}\right){\rm{G}}$ limited by dynamical stability over the lifetime of BH spin (van Putten & Levinson 2003) gives a characteristic timescale for the lifetime of rapid spin of the BH, and hence of the BH-torus system

$$\begin{eqnarray}&&{T}_{s}\simeq 1.5\,{\rm{s}}{\left(\displaystyle \frac{\sigma }{0.1}\right)}^{-1}{\left(\displaystyle \frac{z}{6}\right)}^{4}\left(\displaystyle \frac{M}{3{M}_{\odot }}\right),\end{eqnarray} \tag{ 6 }$$

which is consistent with the duration T₉₀ (90% of gamma-ray counts over background) of GRB 170817A. Over this secular timescale, the BH gently relaxes toward a nearly Schwarzschild BH as the ISCO expands. The relatively baryon-poor environment of the BH is ideally suited for it to also launch an ultra-relativistic baryon-poor jet within a baryon-rich disk or torus wind with (van Putten & Levinson 2003)

$$\begin{eqnarray}&&{E}_{j}\simeq \displaystyle \frac{{E}_{J}}{4{z}^{4}}\simeq 5\times {10}^{50}\,{\rm{erg}},{E}_{w}\simeq {\eta }^{2}{E}_{J}\simeq 4\times {10}^{51}\,{\rm{erg}},\end{eqnarray} \tag{ 7 }$$

consistent with ${E}_{j}\sim {10}^{49\mbox{--}50}\,\mathrm{erg}$ and ${E}_{k}=(1/2){M}_{\mathrm{ej}}{v}^{2}\,\simeq 4.5\times {10}^{51}$ in the relativistic ejecta of GRB 170817A and ${M}_{\mathrm{ej}}\simeq 5 \% {M}_{\odot }$ of mass ejecta at mildly relativistic velocities $v\simeq 0.3c$ (Mooley et al. 2018a, 2018b). Emission terminates abruptly as the remnant torus collapses onto the BH when ${{\rm{\Omega }}}_{H}\simeq {{\rm{\Omega }}}_{T}$ (${f}_{\mathrm{GW}}\simeq {10}^{2}\,\mathrm{Hz}$).

${ \mathcal E }\simeq (3.5\pm 1) \% {M}_{\odot }{c}^{2}$ in EE, measured by $\hat{\chi }({ \mathcal E })$ through signal injections (Figure 4), gives a powerful calorimetric constraint on the central engine. This outcome points to a Kerr BH formed in the core collapse of the merged NS in the immediate aftermath of GW170817.

At ${f}_{\mathrm{GW}}\lt 700\,\mathrm{Hz}$, our ${ \mathcal E }$ is consistent with post-merger bounds of LIGO (Abbott et al. 2017c, between the dashed lines ${E}_{\mathrm{gw}}=0.01\mbox{--}0.1{M}_{\odot }{c}^{2}$ in Figure 1) and Oliver et al. (2019). With the E_J of a Kerr BH, the concerns of Oliver et al. (2019) on the detectability of EE are unfounded. Accurate time-integration of the complex scaling ${L}_{\mathrm{GW}}\propto {\left({f}_{\mathrm{GW}}[h/{C}_{h}]\right)}^{2}$ highlights the need for measurement by signal injection, for which a one-frequency estimate of h_H1 alone (van Putten & Della Valle 2019) now appears inadequate.

Core collapse greatly enhances E_J in J inherited from the merged NS up to about $1{M}_{\odot }{c}^{2}$ in a $\sim 3{M}_{\odot }$ BH. It amply accommodates ${ \mathcal E }$ even at modest efficiencies in conversion to ${ \mathcal E }$ over durations of seconds (Figure 5). Accompanying minor emissions (7) in mass ejecta from the torus and ultra-high-energy emission from the BH agree quantitatively with observational constraints on the associated kilonova and GRB 170817A. GW170817 is too distant, however, to probe any MeV neutrino emission (Bays et al. 2012) in its MeV torus (van Putten & Levinson 2003).

Conceivably, EE does not completely exhaust E_J, permitting low-luminosity latent emission including minor output in baryon-loaded disk winds and low-luminosity jets. While outside the scope of this Letter, this might be an alternative to the same from a long-lived NS remnant that accounts for ATo2017gfo (Ai et al. 2018; Li et al. 2018; Yu et al. 2018; Piro et al. 2019).

At improved sensitivity, LIGO-Virgo O3 observations may significantly improve our ability to identify the nature of binary mergers involving an NS—including the tidal break-up in an NS–BH merger—and their remnants that might also be found in core-collapse supernovae and, possibly, the accretion-induced collapse of white dwarfs.

The authors thank the reviewer for a detailed reading and constructive comments. The first author gratefully thanks ACP, Aspen, CO, GWPop 2019 (PHY-1607611), and AEI, Hannover, where our signal injections were initiated in discussions with M. Alessandra Papa and B. Allen. We also thank A.V. Mukhanov and J. Kanner for constructive comments. We acknowledge use of the data set 10.7935/K5B8566F of the LIGO Laboratory and LIGO Scientific Collaboration, funded by the U.S. NSF, and support from NRF Korea (2015R1D1A1A01059793, 2016R1A5A1013277, 2018044640) and MEXT, JSPS Leading-edge Research Infrastructure Program, JSPS Grant-in-Aid for Specially Promoted Research 26000005, MEXT Grant-in-Aid for Scientific Research on Innovative Areas 24103005, JSPS Core-to-Core Program, Advanced Research Networks, and ICRR.

WInjection.m, whitening and signal injection (Figure 2), doi:10.5281/zenodo.2613112.

EEE.m, estimated energy and efficiency of EE (Figure 5), doi:10.5281/zenodo.2613105.

Our broadband extended GW emission (BEGE) pipeline aims for unmodeled ascending and descending chirps with a choice of an intermediate timescale of phase coherence $0\lt \tau \lesssim 1\,{\rm{s}}$, expected from extreme transient events exhausting E_J of their central engine in seconds.

• 1.  
  Butterfly filtering is matched filtering against a bank of time-symmetric chirp-like templates of intermediate duration τ, densely covering a domain in $(f(t),\left|{df}(t)/{dt}\right|\geqslant \delta$) for some choice of $\delta \gt 0$. Single-detector spectrograms are extracted as scatter plots of correlations $\rho (t,{f}_{c})$ between data segments (here, of 32 s duration) and time-symmetric chirp-like templates with central frequency f_c.
• 2.  
  To reduce noise in deep searches ($\kappa =2$), spectrograms are merged by frequency coincidences ($\left|{f}_{c,{\rm{H}}1}-{f}_{c,{\rm{H}}2}\right|\lt {\rm{\Delta }}f$) to conform causality: ${\rm{\Delta }}f$ is about $\left|{df}(t)/{dt}\right|\delta t$, where $\delta t=10\,\mathrm{ms}$ is the (maximal) signal propagation time between H1 and L1. We obtain satisfactory results with ${\rm{\Delta }}f=10\,\mathrm{Hz}$ (Figure 3).
• 3.  
  Candidate features (Figure 3) are evaluated by counting "hits:" χ($\rho \gt \kappa \sigma$) by H1&L1 over strips about a given family of curves—normalized to $\hat{\chi }$. For EE feature to GW170817, we use (2), giving $\hat{\chi }({t}_{s},{f}_{s},{f}_{0},{\tau }_{s})$. The strip is of finite width (${\rm{\Delta }}f=10\,\mathrm{Hz}$, ${\rm{\Delta }}t=0.1\,{\rm{s}}$), discretized with ${\rm{\Delta }}{t}_{s}=0.030\,{\rm{s}}$ and, for background statistics, over N = 16 steps in each parameter gathered from 1956 s of clean LIGO data in a scan over a total of 256M parameters (${N}^{3}=4096$, ${\rm{\Delta }}{t}_{s}=30\,\mathrm{ms}$; van Putten & Della Valle 2019).

The merged NS produced by GW170817 may briefly emit GWs through a magnetosphere with field B, baryon-loaded with M_b by dynamical mass ejecta and MeV-neutrino winds (e.g., Perego et al. 2014), by a quadrupole moment μ along its magnetic spin-axis misaligned with J (Kalapotharakos et al. 2012), extending out to l of its light cylinder. At Alfvén velocity ${c}_{A}=B/\sqrt{4\pi \rho }$, $B={B}_{16}{10}^{16}\,{\rm{G}}$ with matter density ρ, μ greatly exceeds that of B in a vacuum (Hacyan 2017). In geometrical units ($c=G=1$), the polar flux axis radiates like a rod with (Wald 1984) ${L}_{\mathrm{GW}}={(32/45){\mu }^{2}{{\rm{\Omega }}}^{6}\simeq (32/45)(m{\rm{\Omega }})}^{2}$, with $\mu \,={{ml}}^{2}$. A star of mass M, radius R, and Newtonian binding energy $W={M}^{2}/(2R)\simeq 0.15$ generally satisfies $M\gg W\gg {E}_{J}\,\gg {E}_{\mathrm{turb}}\gg {E}_{B}$ for turbulent motions E_turb and ${E}_{B}=(1/6){B}^{2}{R}^{3}$. Hence, ${E}_{B}=\left({E}_{\mathrm{rot}}/W\right)\left({E}_{\mathrm{turb}}/{E}_{\mathrm{rot}}\right)\left({E}_{B}/{E}_{\mathrm{turb}}\right)W\simeq {10}^{-4}M$ for fiducial ratios of 0.1 for each factor with corresponding $B\,\simeq 4\times {10}^{16}\,{\rm{G}}$. M_b enhances $m\simeq {f}_{B}{E}_{B}$ by $2{\beta }_{A}^{-2}$, ${\beta }_{A}={c}_{A}/c$, where ${f}_{B}\simeq 0.5$ for a dipole field. Accordingly, ${L}_{\mathrm{GW}}\simeq (32/45){\left(m{\rm{\Omega }}{\beta }_{A}^{-2}\right)}^{2}\simeq 2\times {10}^{52}{\left({B}_{16}/({\beta }_{A}/0.1)\right)}^{4}{\left({f}_{s}/350{\rm{Hz}}\right)}^{2}\,{\text{erg s}}^{-1}$ at rapid spin when l is a few times R. Such a burst would be short by canonical bounds on E_J of an NS.

E_J increases dramatically in the continuing core collapse into a BH. A numerical estimate of ${ \mathcal E }$ derived from the catalytic conversion of ${E}_{J}=2M{\sin }^{2}(\lambda /2)\lesssim 0.29M$ at $a/M=\sin \lambda$ (non-extremal) at modest efficiency at orbital angular velocity ${{\rm{\Omega }}}_{T}=\pi {f}_{\mathrm{GW}}$ relative to ${{\rm{\Omega }}}_{H}=\tan (\lambda /2)/(2M)$ of the BH. The estimated initial frequency of $\sim 744\,\mathrm{Hz}$ at time of coalescence t_c inferred from ${t}_{s}=0.67\,{\rm{s}}$ is below the orbital frequency at which the stars approach the ISCO of the system mass, which is about 1100 Hz at $r\simeq 16\,\mathrm{km}$. At this point, an equal-mass DNS has $a/M=0.72\lt 1$, which is consistent with numerical simulations (e.g., Baiotti & Rezzolla 2017); this allows collapse to a $\sim 3{M}_{\odot }$ Kerr BH with ${E}_{J}\simeq 24 \% {M}_{\odot }{c}^{2}$. For a torus radius K times the ISCO radius, Figure 5 shows the result of integration of the equations describing spin-down with a ${ \mathcal E }\simeq 3.6\mbox{--}4.3 \% {M}_{\odot }$ at aforementioned canonical initial values a/M, which is subject to the observed GW frequency $600\,{\rm{Hz}}\lt {f}_{\mathrm{GW}}\,\lt 700\,\mathrm{Hz}$ at ${t}_{s}=0.67\,{\rm{s}}$ post-merger. This is consistent with (3).