NEUTRON-STAR MERGER EJECTA AS OBSTACLES TO NEUTRINO-POWERED JETS OF GAMMA-RAY BURSTS

The time evolution of several quantities characterizing the $\nu \bar{\nu }$-annihilation is illustrated in Figure 1. All modeled tori traverse the two evolutionary phases denoted as NDAF and ADAF (neutrino- and advection-dominated accretion flow, respectively; see, e.g., Metzger et al. 2008; Fernández & Metzger 2013; Just et al. 2015a). In the NDAF phase the temperatures and thus neutrino production rates are high enough for viscous heating to be efficiently compensated by neutrino emission, while in the subsequent ADAF phase neutrino reactions cease and ultimately freeze out owing to low temperatures. During the NDAF phase torus mass accreted onto the BH per unit of time, ${\dot{M}}_{\mathrm{BH}}$, is converted into energy-emission rates (i.e., luminosities) of both neutrino species, ${L}_{{\nu }_{e},{\bar{\nu }}_{e}}$, with an efficiency ${\eta }_{\nu }\equiv ({L}_{{\nu }_{e}}+{L}_{{\bar{\nu }}_{e}})/({\dot{M}}_{\mathrm{BH}}{c}^{2})$ (with the speed of light c) of a few percent (Figure 1). After being emitted, a fraction of neutrinos pair-annihilate, giving rise to a local heating rate, ${Q}_{\mathrm{ann}}$ (Figures 2–4), which is highest close to the BH, since here the neutrino densities are largest, and steeply decreases with radius. The total annihilation rate, ${\dot{E}}_{\mathrm{ann}}^{\mathrm{pol}}\equiv {\int }_{V,45}{Q}_{\mathrm{ann}}{dV}$ (with the integral performed only in the two polar cones with 45° half-opening angle), is roughly proportional to ${L}_{{\nu }_{e}}{L}_{{\bar{\nu }}_{e}}$, multiplied by additional factors accounting for the geometry of the emitting region and the preference for high neutrino mean energies $\langle \varepsilon {\rangle }_{\nu }$ (see, e.g., Goodman et al. 1987; Setiawan et al. 2006). The annihilation efficiency, ${\eta }_{\mathrm{ann}}\equiv {\dot{E}}_{\mathrm{ann}}^{\mathrm{pol}}/({L}_{{\nu }_{e}}+{L}_{{\bar{\nu }}_{e}})$, is therefore approximately proportional to ${L}_{{\nu }_{e}}$ (noting that ${L}_{{\bar{\nu }}_{e}}\sim {L}_{{\nu }_{e}}$). Because ${L}_{\nu }$ decreases roughly like ${\dot{M}}_{\mathrm{BH}}$ (Figure 1), ${\eta }_{\mathrm{ann}}$ declines earlier than ${\eta }_{\nu }$ and hence the time ${T}_{\mathrm{ann}}^{90}$, at which $90\%$ of the final ${E}_{\mathrm{ann}}^{\mathrm{pol}}$ has been deposited, is significantly shorter than the duration of the NDAF phase, ${T}_{\mathrm{NDAF}}$, defined as the time when ${\eta }_{\nu }$ drops below $\simeq 1\%$ (Table 1).

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The comparison between the models is facilitated by considering (Table 1) the global versions of the aforementioned efficiencies, ${\bar{\eta }}_{\nu }\equiv {E}_{\nu }/({m}_{\mathrm{torus}}{c}^{2})$ and ${\bar{\eta }}_{\mathrm{ann}}\equiv {E}_{\mathrm{ann}}^{\mathrm{pol}}/{E}_{\nu }$ with ${E}_{\nu }\equiv \int ({L}_{{\nu }_{e}}+{L}_{{\bar{\nu }}_{e}}){dt}$, by which means the total annihilation energy ${E}_{\mathrm{ann}}^{\mathrm{pol}}={\bar{\eta }}_{\nu }\;{\bar{\eta }}_{\mathrm{ann}}\;{m}_{\mathrm{torus}}{c}^{2}$. The emission efficiency, ${\bar{\eta }}_{\nu }$, is similar for all models (up to some reduction for higher viscosity caused by enhanced neutrino trapping at very early times). The main reason for this similarity is the comparable values of the dimensionless BH spin, ${A}_{\mathrm{BH}}$, for all models, which determine the amount of specific gravitational energy that can be released by gas before being accreted. Since approximately ${\eta }_{\mathrm{ann}}\propto {L}_{\nu }\propto {\dot{M}}_{\mathrm{BH}}$ we infer that ${\bar{\eta }}_{\mathrm{ann}}$ should grow for higher values of ${m}_{\mathrm{torus}}$ and ${\alpha }_{\mathrm{vis}}$, and a lower value of ${M}_{\mathrm{BH}}$. This turns out to be broadly consistent with our results. In the NS–BH model the positive effect of the high torus mass is more than counterbalanced by the reduced neutrino mean energies, $\langle \varepsilon {\rangle }_{\nu }$, compared to the NS–NS models (Figure 1) and by the high BH mass, which leads to a longer accretion timescale.

Finally, the upper limit of energy available for any potential jet, represented by ${E}_{\mathrm{ann}}^{\mathrm{pol}}$, is given by $\sim ({10}^{-5}\mbox{--}{10}^{-4})\times {m}_{\mathrm{torus}}{c}^{2}\sim (1\mbox{--}2)\times {10}^{49}$ erg in our models. The corresponding isotropic-equivalent energies are almost independent of the polar angle (because of the approximate isotropy of the annihilation rates, see Figures 2–4) with ${E}_{\mathrm{ann}}^{\mathrm{iso}}\approx {E}_{\mathrm{ann}}^{\mathrm{pol}}/(1-\mathrm{cos}\;45^\circ )\;\sim (3\mbox{--}8)\times {10}^{49}$ erg (Table 1).

If and how much of ${E}_{\mathrm{ann}}^{\mathrm{pol}}$ can finally end up in an ultrarelativistic outflow depends crucially on the distribution of matter surrounding the torus. This is illustrated in Figures 2–4 for three representative models, of which each exhibits a qualitatively different outflow evolution.

In model SFHO_145145 (Figure 2) the amount of baryonic pollution near the symmetry axis is so high that neutrino irradiation has hardly any impact, besides slightly accelerating the already expanding dynamical ejecta and raising their electron fraction, Y_e. The high densities are too prohibitive for annihilation to form a baryon-poor funnel, which is ultimately needed to produce high Lorentz factors.

In models SFHO_1218 (Figure 3) and TM1_13520 the configuration is better suited for the development of relativistic outflows because the poles are loaded with relatively small amounts of matter. Hence, in both models funnels are successfully created, allowing jets to form and to expand outward along the axis. Similar to jets in collapsars (e.g., MacFadyen & Woosley 1999; Bromberg et al. 2011), the jets consist of thin, ultrarelativistic beams surrounded by cocoons and mildly relativistically propagating heads. Right after the jets are launched, essentially all annihilation energy dumped into the funnels is used to power the beams, immediately increasing ${\rm{\Gamma }}h$ (where Γ and h are Lorentz factor and specific enthalpy, respectively), which for adiabatic expansion represents an estimate for the terminal Lorentz factor. Therefore, the energy carried by material with ${\rm{\Gamma }}h\gt 10$ and 100, ${E}_{{\rm{\Gamma }}h\gt 10/100}$, rises. The subsequent expansion of beam material then allows thermal energy to be converted into kinetic energy, leading to a growth also of ${E}_{{\rm{\Gamma }}\gt 10/100}$, the energy of material with ${\rm{\Gamma }}\gt 10/100$ (see Figure 1 for the time evolution of ${E}_{{\rm{\Gamma }}h\gt 10/100}$, ${E}_{{\rm{\Gamma }}\gt 10/100}$). However, the beam cannot expand freely as long as the jet is enveloped by dynamical ejecta. Hence, while being fed with annihilation energy at its base, the jet continually loses energy by drilling through the envelope. This happens at the expense of beam energy, which is transported to and dissipated at the much more slowly moving head. Once dissipated, this part of the energy is advected into the cocoon and is ultimately lost as potential contribution to a relativistic outflow. That is, only the fraction of original annihilation energy that is induced into the beam shortly before and after the time of an eventual breakout from the envelope could possibly be released within a relativistic outflow. However, in the considered NS–NS merger models the jet is not energetic enough to breakout, which is why ${E}_{{\rm{\Gamma }}h\gt 10/100}$ successively decreases and finally vanishes (Figures 1, 3).

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Model TM1_1451, associated with a NS–BH merger, offers the most optimistic configuration for relativistic outflows, because the polar regions are essentially free of dynamical ejecta. However, neutrino-driven winds and the expansion of torus matter by viscous or other effects could still prohibit or impair the emergence of relativistic ejecta. Nevertheless, model TM1_1451 develops a powerful relativistic wind that contains 61% (48%) of the original annihilation energy in the form of ${E}_{{\rm{\Gamma }}h\gt 10}$ (${E}_{{\rm{\Gamma }}h\gt 100}$), as measured at t = 0.1 s (Figure 1). The radius-dependent half-opening angle of the outflow, θ, reflects the temporal change of the torus configuration (Figure 4): at early times the torus is rather compact and can barely collimate the outflow, while subsequently the outer torus layers undergo viscous and neutrino-driven expansion and form a funnel, of which the walls collimate the outflow. Hence, the outflow consists of an ultrarelativistic (${\rm{\Gamma }}\gt 100$) core surrounded by less relativistic ($10\lt {\rm{\Gamma }}\lt 100$) wings (see Table 1 for the corresponding energies and half-opening angles). Interestingly, the collimation leads to a remarkable enhancement of the isotropic-equivalent energy of the ultrarelativistic core, ${E}_{{\rm{\Gamma }}\gt 100}^{\mathrm{iso}}\approx 2\times {10}^{50}\;$erg, compared to that of annihilation, ${E}_{\mathrm{ann}}^{\mathrm{iso}}\approx 5\times {10}^{49}\;$ erg.

The distribution of Γ and the opening angles can still change somewhat after the final simulation time of $t=0.6\;{\rm{s}}$ because of a sizable fraction of the energy still residing in internal energy. However, the coarser grid at higher radii renders the late evolution more prone to numerical diffusion, which is why ${E}_{{\rm{\Gamma }}\gt 100}$ decreases at late times and we stopped the simulation.

We investigated the ability of $\nu \bar{\nu }$-annihilation to represent the dominant agent for generating sGRB viable outflows. In contrast to previous studies, which either computed the annihilation rates on a stationary background without any hydrodynamic feedback (e.g., Setiawan et al. 2006; Birkl et al. 2007; Richers et al. 2015) or which launched the outflow artificially (e.g., Aloy et al. 2005; Nagakura et al. 2014; Duffell et al. 2015), we self-consistently followed the combined neutrino-hydrodynamic evolution of the BH-torus system, the eventual launch of the jet in response to heating by $\nu \bar{\nu }$-annihilation, and the jet propagation through the envelope of dynamical ejecta produced during the binary merger. Our examined set of models covers the most relevant NS–NS and NS–BH configurations in the sense that they are both promising regarding the annihilation mechanism, but still realistic enough to be abundantly present in nature.

We find that the total energy provided by annihilation, ${E}_{\mathrm{ann}}^{\mathrm{pol}}$, amounts to a fraction of about 10⁻⁵–10⁻⁴ of the original torus rest-mass energy. Although this fraction grows with torus mass and viscosity, and for lower BH mass, we find typical values of ${E}_{\mathrm{ann}}^{\mathrm{pol}}$ not exceeding a few 10⁴⁹ erg.

In the NS–BH merger remnant, which is essentially free of polar dynamical ejecta, annihilation heating is efficiently translated into a relativistic outflow, whose ultrarelativistic (${\rm{\Gamma }}\gt 100$) core is dynamically collimated to a half-opening angle ${\theta }_{{\rm{\Gamma }}\gt 100}\approx 8^\circ$ and an isotropic-equivalent energy ${E}_{{\rm{\Gamma }}\gt 100}^{\mathrm{iso}}\approx 2\times {10}^{50}\;$erg. However, compared to the median values of observed sGRBs (${E}_{\mathrm{obs}}^{\mathrm{iso}}\approx 2\times {10}^{51}\;$erg and ${\theta }_{\mathrm{obs}}\approx 16\pm 10^\circ$, see Fong et al. 2015), our results suggest that the annihilation mechanism is energetically too inefficient to explain the majority of sGRBs, but still could account for some low-luminosity events. This conclusion is further supported by our result that the effective time of source activity obtained for our models is ${T}_{\mathrm{ann}}^{90}\lesssim 0.1\;{\rm{s}}$ and thus much shorter than most observed times T⁹⁰ of sGRBs (Fong et al. 2015). A lower viscosity or higher BH spin could possibly alleviate but not solve this issue.

Moreover, our simulations suggest that the $\nu \bar{\nu }$-annihilation energies in NS–NS mergers are too low to launch jets energetically enough to pierce through the envelope of dynamical ejecta, even in the most optimistic case of prompt BH formation and very asymmetric progenitors, which results in less baryon-polluted polar regions. If sGRBs are connected with NS–NS mergers, our results indicate that some other, probably magnetohydrodynamic processes (e.g., Dionysopoulou et al. 2015; Kiuchi et al. 2015; Paschalidis et al. 2015) are necessary at least to create and support a baryon-poor funnel through which the outflow can propagate without major dissipation. Once a funnel is established, however, $\nu \bar{\nu }$-annihilation could still play a non-negligible role in powering the outflow. Although we consider only NS–NS systems here where the MNS collapses rather early, our conclusions probably also hold for longer delay times. Then an even greater envelope mass would have to be penetrated by the jet because of the additional neutrino-driven and magnetically driven outflows from the MNS (Perego et al. 2014; Siegel et al. 2014).

We finally point out that several simplifications entered this study, e.g., the omission of accurate general relativistic effects, the use of viscosity to mimic magnetohydrodynamic turbulence, initial models that were not fully consistent with the merger remnants, and the remaining approximations of the neutrino transport.