BINARY NEUTRON STAR MERGERS: A JET ENGINE FOR SHORT GAMMA-RAY BURSTS

Abstract

We perform magnetohydrodynamic simulations in full general relativity (GRMHD) of quasi-circular, equal-mass, binary neutron stars that undergo merger. The initial stars are irrotational, n = 1 polytropes and are magnetized. We explore two types of magnetic-field geometries: one where each star is endowed with a dipole magnetic field extending from the interior into the exterior, as in a pulsar, and the other where the dipole field is initially confined to the interior. In both cases the adopted magnetic fields are initially dynamically unimportant. The merger outcome is a hypermassive neutron star that undergoes delayed collapse to a black hole (spin parameter a/M_BH ~ 0.74) immersed in a magnetized accretion disk. About 4000M ~ 60(M_NS/1.625M_⊙) ms following merger, the region above the black hole poles becomes strongly magnetized, and a collimated, mildly relativistic outflow—an incipient jet—is launched. The lifetime of the accretion disk, which likely equals the lifetime of the jet, is Δ t ~ 0.1 (M_NS/1.625M_⊙) s. In contrast to black hole–neutron star mergers, we find that incipient jets are launched even when the initial magnetic field is confined to the interior of the stars.

1. INTRODUCTION

The LIGO and Virgo Collaborations recently reported the first direct detection of a gravitational-wave (GW) signal and demonstrated that it was produced by the inspiral and coalescence of a binary black hole (BHBH) system (Abbott et al. 2016). This breakthrough marks the beginning of the era of GW astrophysics. GW signals are expected to be generated not only by BHBH binaries, but also by neutron star–neutron star (NSNS) and black hole–neutron star (BHNS) binaries.

Merging NSNSs and BHNSs are not only important sources of GWs, but also the two most popular candidate progenitors of short gamma-ray bursts (sGRBs) (Eichler et al. 1989; Narayan et al. 1992; Mochkovitch et al. 1993; Berger 2014). NSNSs and BHNSs may also generate other detectable, transient electromagnetic (EM) signals prior to (Hansen & Lyutikov 2001; McWilliams & Levin 2011; Palenzuela et al. 2013; Paschalidis et al. 2013; Ponce et al. 2014) and following (Metzger & Berger 2012; Berger 2014; Metzger et al. 2015) merger. Combining GW and EM signals from these mergers could test relativistic gravity and constrain the NS equation of state (EOS). Moreover, an association of a GW event with an sGRB (the holy grail of “multimessenger astronomy”) would provide convincing evidence for the compact binary coalescence model. However, the interpretation of EM and GW signals from such mergers will rely on a theoretical understanding of these events, which requires simulations in full general relativity to treat the strong dynamical fields and high velocities arising in these scenarios. There have been multiple studies of compact binary mergers. For NSNSs, see Faber & Rasio (2012) for a review and Paschalidis et al. (2012), Gold et al. (2012), East & Pretorius (2012), Neilsen et al. (2014), Dionysopoulou et al. (2015), Sekiguchi et al. (2015), Dietrich et al. (2015), and Palenzuela et al. (2015b) for recent results. These earlier studies have advanced our knowledge of EOS effects, neutrino transport, ejecta properties, and magnetospheric phenomena. However, few studies focused on the potential for NSNSs to power sGRBs.

Recent work by Paschalidis et al. (2015b; hereafter PRS) demonstrated that mergers of magnetized BHNS systems can launch jets and be the engines that power sGRBs. The key ingredient for generating a jet was found to be the initial endowment of the NS with a dipole B-field that extends into the NS exterior as in a pulsar magnetosphere. By contrast, if the initial magnetic field is confined to the interior of the NS, no jet is observed (Etienne et al. 2012b; Kiuchi et al. 2015).

The primary motivation for this paper is to answer the question: can NSNS mergers produce jets in the same way as BHNS systems, or does this mechanism require an initial BH? Recently, it was shown that neutrino annihilation may not be strong enough to power jets (Just et al. 2016), so MHD processes must play a major role for jet formation. Previous ideal GRMHD simulations by Rezzolla et al. (2011) suggest that NSNS mergers may launch a relativistic jet, while those by Kiuchi et al. (2014), which focus on different initial configurations, show otherwise. Both of these studies have considered only scenarios where the B-field is initially confined to the interior of the two NSs.

Here, we describe the results of ideal GRMHD simulations of NSNSs in which we follow PRS and allow an initially strong, but dynamically unimportant dipole B-field to extend from the interior of the NSs into the exterior. We call this configuration the pulsar model (hereafter P). The existence of pulsars suggests that this may be the astrophysically most common case. As in PRS, we define an incipient jet as a collimated, mildly relativistic outflow which is at least partially magnetically dominated (b² (2ρ₀)>1, where b² = B²/4 π and ρ₀ is the rest-mass density). We find that the P configuration launches an incipient jet. To study the impact of the initial B-field geometry and to compare it with previous studies, we also perform simulations where the field is confined to the interior of the NSs (hereafter the I model), keeping its strength at the stellar center the same as in the P case. In contrast to BHNS systems, we find that interior-only initial B-fields also lead to jet formation in NSNSs. Throughout this work, geometrized units (G = c = 1) are adopted unless otherwise specified.

2. METHODS

We use the Illinois GRMHD code, which is built on the Cactus⁶ infrastructure and uses the Carpet⁷ code for adaptive mesh refinement. We use the AHFinderDirect thorn (Thornburg 2004) to locate apparent horizons. This code has been thoroughly tested and used in the past in different scenarios involving magnetized compact binaries (see, e.g., Etienne et al. 2008, 2012b; Liu et al. 2008; Gold et al. 2014a, 2014b). For implementation details, see Etienne et al. (2010, 2012a) and Farris et al. (2012).

In all simulations we use seven levels of refinement with two sets of nested refinement boxes (one for each NS) differing in size and resolution by factors of two. The finest box around each NS has a half-side length of ~1.3 R_NS, where R_NS is the initial NS radius. For the I model, we run simulations at two different resolutions: a “normal” resolution (model IN), in which the finest refinement level has grid spacing 0.05 M = 227(M_NS/1.625 M_⊙) m, and a “high” resolution (model IH), in which the finest level has spacing 0.03 M = 152(M_NS/1.625 M_⊙) m. For the P model, we always use the high resolution. These choices resolve the initial NS equatorial diameter by ~120 and ~180 points, respectively. In terms of grid points per NS diameter, our high resolution is close to the medium resolution used in Kiuchi et al. (2014), which covered the initial stellar diameters by ~205 points. We set the outer boundary at 245M ≈ 1088 (M_NS/1.625M_⊙) km and impose reflection symmetry across the orbital plane.

The quasi-equilibrium NSNS initial data were generated with the LORENE libraries.⁸ Specifically, we use the n = 1, irrotational case listed in Taniguchi & Gourgoulhon (2002), Table IIIM/R= 0.14 versus 0.14, row 3, for which the rest mass of each NS is 1.625 M_⊙(k/269.6 km²)^1/2, with k the polytropic constant. This same case was used in Rezzolla et al. (2011). As in PRS we evolve the initial data up to the final two orbits prior to merger (t=t_B), at which point each NS is seeded with a dynamically unimportant B-field following one of two prescriptions:

1. The P case (Figure 1, upper left), for which we use a dipole B-field corresponding to Equation (2) in Paschalidis et al. (2013). We choose the parameters I₀ and r₀ such that the magnetic-to-gas-pressure ratio at the stellar center is β⁻¹ = 0.003125. The resulting B-field strength at the NS pole measured by a normal observer is B_pole ≃ 1.75 × 10¹⁵ (1.625 M_⊙ M_NS) G. While this B-field is astrophysically large, we choose it so that following merger, the rms value of the field strength in the hypermassive neutron star (HMNS) remnant is close to the values found in recent very-high-resolution simulations (Kiuchi et al. 2015) which showed that the Kelvin–Helmholtz instability (KHI) during merger can boost the rms B-field to 10^15.5G with local values reaching even 10¹⁷ G. Our choice of the B-field strength thus provides an “existence proof” for jet launching following NSNS mergers with the finite computational resources at our disposal. To capture the evolution of the exterior B-field in this case and simultaneously mimic force-free conditions that likely characterize the exterior, we follow PRS and set a variable-density atmosphere at t = t_B such that the exterior plasma parameter β_ext = 0.01. This variable-density prescription, imposed at t = t_B only, is expected to have no impact on the outcome (cf. PRS). With our choice of β_ext, the amount of total rest mass does not increase by more than ~0.5%.

2. The I case, which also uses a dipole field but confines it to the interior. We generate the vector potential through Equations (11), (12) in Etienne et al. (2012a), choosing P_cut to be 1% of the maximum pressure, n_b = 2, and A_b such that the strength of the B-field at the stellar center coincides with that in the P case. Unlike the P case, a variable-density atmosphere is not necessary, so we use a standard constant-density atmosphere with rest-mass density 10⁻¹⁰ρ_0,max, where ρ_0,max is the initial maximum value of the rest-mass density.

Figure 1.

Snapshots of the rest-mass density, normalized to its initial maximum value ρ_0,max = 5.9 × 10¹⁴ (1.625M_⊙/M_NS)² g cm⁻³(log scale) at selected times for the P case. The arrows indicate plasma velocities, and the white lines show the B-field structure. The bottom middle and right panels highlight the system after an incipient jet is launched. Here M=1.47 × 10⁻²(M_NS/1.625M_⊙) ms = 4.43(M_NS/1.625M_⊙) km.

In both the P and I cases, the magnetic dipole moments are aligned with the orbital angular momentum. During the evolution, we impose a floor rest-mass density of 10⁻¹⁰ρ_0,max. We employ a Γ-law EOS, P = ρ₀ε, with ε the specific internal energy, and allow for shocks.

3. RESULTS

The outcomes and basic dynamics for all our cases are similar, hence we show snapshots and discuss the evolution only for the P case, unless otherwise specified. All coordinate times will refer to the P case too, unless otherwise specified. We summarize key results for all cases in Table 1.

Table 1.

Summary of Results

Case	ΓLa	Brmsb	Ṁc	Mdisk/M0d	τdiske	LEMf
P	1.25	1016.0	0.33	1.0%	0.10	1051.3
IN	1.21	1015.8	0.54	1.1%	0.07	1050.9
IH	1.15	1015.7	0.77	1.5%	0.06	1050.7

Notes.

^aMaximum fluid Lorentz factor near the end of the simulation.

^brms value of the HMNS B-field before collapse.

^cRest-mass accretion rate in units of M_⊙ s⁻¹ at t − t_BH = 2100M ≈ 31(M_NS/1.625 M_⊙) ms.

^dRatio of disk rest mass to initial total rest mass at t − t_BH = 2100M ≈ 31(M_NS/1.625 M_⊙) ms.

^eDisk lifetime M_disk/Ṁ in (M_NS/1.625M_⊙) s.

^fPoynting luminosity in erg s⁻¹, time-averaged over the last 500M ~ 7.4(M_NS/1.625M_⊙) ms of the evolution after the jet is well -developed.

All cases evolve the same until merger. The two NSs orbit each other with their B-fields frozen in. Gravitational-radiation loss causes the orbit to shrink and the NSs make contact at t = t_merger ≈ 465M ~ 3.5(M_NS/1.625M_⊙) ms when the stars are oblate due to tidal effects (Figure 1, upper middle). Then a double-core remnant forms with the two dense cores rotating about each other and gradually coalescing (Figure 1, upper right).

Just before the HMNS collapses, the rms B-field strength is ~10^15.7–10¹⁶ G (see Table 1), a little larger than the values Kiuchi et al. (2015) reported, but consistent with Zrake & MacFadyen (2013).

During the HMNS stage from t − t_merger ≈ 115M ~ 1.7(M_NS/1.625M_⊙)ms up until BH formation, there is no significant enhancement of the B-field. This result is anticipated because we chose our initial B-fields such that they are near saturation (β ~ 100–1000) following merger. We find that inside the HMNS we resolve the wavelength λ_MRI of the fastest-growing magneto-rotational-instability (MRI) mode by more than 10 points. In addition, λ_MRI fits within the HMNS, so we conclude that MRI-induced turbulence is operating during this stage. As the total rest mass of the HMNS exceeds the maximum value allowed by uniform rotation, i.e., ~2.4M_⊙ (k/269.6 km²)^1/2 for Γ = 2 (Lyford et al. 2003), it undergoes delayed collapse to a BH (Baumgarte et al. 2000; Duez et al. 2006) at t − t_merger ≈ 1215M ~ 18(M_NS/1.625 M_⊙) ms in both the P and IN cases. In the IH case, collapse takes place later at t − t_merger ≈ 2135M ~ 31.5(M_NS/1.625 M_⊙) ms. The sensitivity of the collapse time for short-lived HMNSs to the B-field is physical and has been observed previously (Giacomazzo et al. 2011). Its dependence on resolution, even in purely hydrodynamic simulations, has been noted as well (Paschalidis et al. 2015a; East et al. 2016). In all cases the BH has a mass of M_BH ≈ 2.85 M_⊙ (M_NS/1.625 M_⊙) with spin a/M ≃ 0.74 at high resolution, or a/M ≃ 0.8 at normal resolution.

Shortly after BH formation, plasma velocities above the BH poles point toward the BH (apart from some material ejected during merger), but the winding of the B-field above the BH poles is well underway (Figure 1, bottom left). By t − t_BH ≈ 2800 M ~ 41 (M_NS/1.625M_⊙) ms, where t_BH is the BH formation time, the B-field has been wound into a helical funnel (Figure 1, bottom middle and right). Unlike in the BHNS case of PRS, the B-field does not grow following BH formation: the existence of the HMNS phase instead allows the B-field to build to saturation levels prior to BH formation. We do observe a gradual growth in b²/(2ρ₀) above the BH pole from ~1 to ~100, due to the emptying of the funnel as matter accretes onto the BH. At t − t_BH ≈ 2000M ~ 30(M_NS/1.625 M_⊙) ms, the fluid velocities begin to turn around and point outward. At t − t_BH ≈ 2900M ~ 43 (M_NS/1.625 M_⊙) ms, the outflow extends to heights greater than 100M ~ 445(M_NS/1.625 M_⊙)km. At this point an incipient jet has formed (Figure 1, bottom middle and right). We observe similar evolution in the I cases, although in the IH case the incipient jet is well-developed by t − t_BH ~ 1550M ~ 23(M_NS/1.625M_⊙) ms, while in the IN case it takes t − t_BH ~2850M ~ 42(M_NS/1.625M_⊙) ms for the incipient jet to be launched (see the arrows in Figure 2).

Figure 2.

Top: rest-mass accretion rate Ṁ. The arrows indicate the time (t_jet) at which the incipient jet reaches z = 100 M. The inset shows the outgoing EM luminosity for t > t_jet computed on a coordinate sphere of radius r=115M ≈ 510(M_NS/1.625 M_⊙) km. Bottom: GW amplitude h₊ vs. retarded time for the P case. The inset focuses on the post-merger phase, showing the (l, m) = (2, 2) and (l, m) = (2, 1) (multiplied by 20) modes of the Newman–Penrose scalar Ψ₄.

In all cases, the accretion rate Ṁ begins to settle into a steady state at t − t_BH ≈ 1350M ~ 20(M_NS/1.625M_⊙) ms and slowly decays thereafter (Figure 2, top). At t − t_BH ≈ 2100M ~ 31(M_NS/1.625M_⊙) ms, the accretion rate is roughly 0.33 M_⊙ s⁻¹, at which time the disk mass is ~0.05M_⊙ (M_NS/1.625M_⊙). Thus, the disk will be accreted in Δt~M_disk/Ṁ ~ 0.1 s, implying that the engine’s fuel will be exhausted on a timescale consistent with the duration of the very short-duration sGRBs (see, e.g., Kann et al. 2011).

In the BH–accretion disk system, we again resolve λ_MRI by more than 10 grid points. For the most part, λ_MRI fits inside the disk. MRI is thus operating in these simulations, but it is saturated, with the toroidal and poloidal B-field components approximately equal in magnitude and not growing. Near the end of the simulations the B-field magnitude above the BH pole is ~10¹⁶ G.

As in PRS, we have tracked the motion of individual Lagrangian fluid tracers. We find that they follow the expected helical motion of the outflow and that the gas pouring into the funnel and comprising the outflow originates from the disk. This test confirms the physical nature of the jet.

During the inspiral and throughout BH formation, GWs are emitted with the (l, m) = (2, 2) mode being dominant. This includes quasi-periodic spindown GWs from the transient HMNS (Shibata 2005; for a review and references, see Baumgarte & Shapiro 2010). However, we find that (l, m) = (2, 1) GW modes also develop during and after merger (Figure 2, bottom). While we see no evidence for a one-arm instability (Paschalidis et al. 2015a), m = 1 modes may be generated from asymmetries in the two cores following merger. Like post-merger m = 2 modes, m = 1 modes due to asymmetries in the two cores of double-core HMNSs may be used to constrain the NS EOS.

Figure 3 displays the ratio b²/(2ρ₀) at t − t_merger ≈ 4200M ~ 68(M_NS/1.625M_⊙)ms. Magnetically dominated areas (b² (2ρ₀)>1) extend to heights ≳20M ≈ 20r_BH above the BH, where r_BH is the BH apparent horizon radius. Using the b²/(2ρ₀) ~ 10⁻² contour as a rough definition for the funnel boundary, the funnel opening half-angle is ~20°–30°, which is consistent with PRS. The maximum value of the Lorentz factor inside the funnel is Γ_L ~ 1.1–1.25 (see Table 1), implying only a mildly relativistic flow. However, steady-state and axisymmetric jet models (Vlahakis & Königl 2003) show that the maximum attainable value of Γ_L is approximately equal to b²/(2ρ₀), which reaches values ≳100 within the funnel. Hence, these incipient jets may be accelerated to Γ_L ~ 100, as is necessary to explain sGRB phenomenology. (However, our code may not be reliable at values of b²/(2ρ₀) > 100.) In the asymptotically flat region at heights greater than 100M ~ 650(M_NS/1.625 M_⊙)km, the specific energy E = −u₀ − 1 > 0 is always positive and therefore the outflowing plasma is unbound.

Figure 3.

Ratio b²/(2ρ₀) (log scale) at t − t_merger ≈ 4135M ~ 61(M_NS/1.625M_⊙) ms for the P case. The white lines indicate the B-field lines plotted in the funnel where b²/(2ρ₀) ≥ 10⁻². Magnetically dominated areas (b²/(2ρ₀) ≥ 1) extend to heights greater than 20M ≈ 20 r_BH above the BH horizon (the black sphere). Here r_BH = 2.2(M_NS/1.625 M_⊙) km.

To assess if the Blandford–Znajek (BZ) effect (Blandford & Znajek 1977) is operating in our system as in PRS, we measure the ratio of the angular frequency of the B-field, Ω_F, to the angular frequency of the BH, Ω_H. On an x − z meridional slice passing through the BH centroid and along coordinate semicircles of radii r_BH and 2r_BH, we find that Ω_F/Ω_H ~ 0.1–0.25 within an opening angle of 20° from the BH rotation axis, within which the force-free BZ solution might apply. The observed deviation of Ω_F/Ω_H from the split-monopole value ~0.5 can be due either to the gauge in which we compute Ω_F, deviations from a split-monopole B-field, or possibly from inadequate resolution. The outgoing Poynting luminosity is L_EM ~ 10^50.7–10^51.3 erg s⁻¹ (see Figure 2, Table 1), which is only a little less than the EM power expected from the BZ effect (Thorne et al. 1986): L_EM ~ 10⁵² [(a/M_BH)/0.75]² (M_BH/2.8M_⊙)²(B/10¹⁶G)² erg s⁻¹. The BZ effect is likely operating here, but higher resolution may be necessary to match the expected luminosity more closely.

4. CONCLUSIONS

We showed that an NSNS system with an initially high but dynamically weak B-field launches an incipient jet (an unbound, collimated, and mildly relativistic outflow). This occurs following the delayed collapse of the HMNS, both for initially dipole B-fields which extend from the NS interior into its exterior and initially dipole B-fields confined to the interior. This last result differentiates NSNSs from BHNSs. The accretion timescale of the remnant disk and energy output are consistent with very short sGRBs, demonstrating that NSNSs can indeed provide the central engines that power such phenomena. Our results were obtained with high initial B-fields, which match the post-merger expectations from B-field amplification due to the KHI. However, to capture the KHI-induced B-field growth, resolutions about five times higher than those adopted here are necessary. Evolving for 4000M ~ 60(M_NS/1.625 M_⊙)ms at such high resolution would take years with current resources. The simulations presented here show that if the B-fields in NSNS mergers can be amplified by the KHI to ≳10^15.5 G, then these systems are viable sGRB engines. We anticipate, but have not yet proved, that jets can be launched even if one starts with B-field strengths of order 10¹² G. We also expect that the time interval between merger and jet launch will be longer the weaker the initial B-field.

Different EOSs affect details such as the amount of disk mass, as well as the ejection of different amounts of matter (see e.g., Hotokezaka et al. 2013; Palenzuela et al. 2015a), and therefore the ram pressure of the atmosphere surrounding the remnant BH-disk system. The above may be the main reason for the non-observance of an incipient jet in the NSNS simulations reported by Kiuchi et al. (2014). We infer from their description that the merger with their adopted EOS ejects more matter in the atmosphere with longer fall-back time than in our cases. For the outflow to emerge, the B-field must overcome the ram pressure of the infalling matter. The only way for this to happen is if the ambient matter density decreases with time, which occurs because the atmosphere becomes thinner as it accretes onto the BH. Kiuchi et al. (2014) reported that at t − t_BH ~ 26 ms, there still is matter in the atmosphere that is accreting, hence the ram pressure is larger than the B-field pressure. It is likely that their calculations require longer integration times for an incipient jet to emerge.

Recently, NSNS simulations with a weaker interior-only initial B-field were carried out by Endrizzi et al. (2016). Their resolution was too low to capture MRI or the KHI, hence the B-field did not amplify appreciably and as a result, no jets were observed.

A few caveats remain. First, several quantities are sensitive to resolution, such as the HMNS lifetime and the exact EM Poynting luminosity. However, other evolution characteristics, such as HMNS formation following NSNS merger, delayed collapse, the remnant BH mass and spin, the disk mass, and the ultimate emergence of the incipient jet are almost insensitive to resolution. The above conclusions suggest that higher resolutions than those adopted here are necessary for very accurate calculations, but the essential nature of incipient jet emergence is robust. Second, there is microphysics that we do not model here, such as the effects of a realistic hot, nuclear EOS and neutrino transport. We plan to implement such processes and address all of the aforementioned issues in future studies.