A Magnetar Origin for the Kilonova Ejecta in GW170817

The gravitational wave chirp from the binary neutron star (NS) merger GW170817 (Abbott et al. 2017b) was followed within seconds by a short burst of gamma-rays (Abbott et al. 2017a; Goldstein et al. 2017; Savchenko et al. 2017). Roughly half a day later, a luminous optical counterpart was discovered in the galaxy NGC 4993 at a distance of only ≈40 Mpc (Abbott et al. 2017c; Arcavi et al. 2017; Coulter et al. 2017; Covino et al. 2017; Díaz 2017; Drout et al. 2017; Evans et al. 2017; Hu et al. 2017; Kilpatrick et al. 2017; McCully et al. 2017; Smartt et al. 2017; Soares-Santos et al. 2017; Utsumi et al. 2017; Buckley et al. 2018). The observed emission started out blue in color, with a featureless thermal spectrum that peaked at UV/optical frequencies (e.g., Evans et al. 2017; McCully et al. 2017; Nicholl et al. 2017), before rapidly evolving over the course of a few days to become dominated by emission with a spectral peak in the near-infrared (NIR; Chornock et al. 2017; Pian et al. 2017; Tanvir et al. 2017). Simultaneous optical (e.g., Nicholl et al. 2017; Shappee et al. 2017) and NIR (Chornock et al. 2017) spectra around day 2.5 appear to demonstrate the presence of distinct optical and NIR emission components (however, see Smartt et al. 2017; Waxman et al. 2017 for an alternative interpretation).

The properties of the optical/NIR emission agreed remarkably well with those predicted for "kilonova" (KN) emission powered by the radioactive decay of r-process nuclei (Li & Paczyński 1998; Metzger et al. 2010; Roberts et al. 2011; Barnes & Kasen 2013; Tanaka & Hotokezaka 2013; Grossman et al. 2014; Martin et al. 2015; Fontes et al. 2017; Wollaeger et al. 2017; Tanaka et al. 2018; also see Fernández & Metzger 2016; Metzger 2017 for reviews), a conclusion reached independently by several groups (e.g., Drout et al. 2017; Kasen et al. 2017; Kasliwal et al. 2017; Murguia-Berthier et al. 2017; Tanaka et al. 2017; Waxman et al. 2017). The early-time blue emission is well-explained by radioactive material with a relatively low opacity (a "blue" KN; Metzger et al. 2010; Roberts et al. 2011), as would be expected if the outer ejecta layers contain exclusively light r-process nuclei synthesized from matter with a relatively high electron fraction, Y_e ≳ 0.25 (Metzger & Fernández 2014). On the other hand, an upper bound of Y_e ≲ 0.3–0.35 is needed to produce a radioactive decay rate consistent with the smooth decline in the bolometric light curve (e.g., Lippuner & Roberts 2015; Rosswog et al. 2017). The late NIR is instead well-explained by radioactive material with high line opacity, consistent with the presence of at least a moderate mass fraction X_La ≳ 10⁻² of lanthanide or actinide nuclei (a so-called "red" or "purple" KN; Barnes & Kasen 2013; Kasen et al. 2013; Tanaka & Hotokezaka 2013). Motivated by the theoretical expectation of distinct ejecta components with different compositions and opacities (Metzger & Fernández 2014; Perego et al. 2014), a two-zone ("red"+"blue") KN model provides a reasonable fit to the photometric and spectroscopic data; these give best-fit values for the ejecta masses and mean velocities of the blue and red components of ${M}_{\mathrm{blue}}\approx 2\times {10}^{-2}{M}_{\odot },{v}_{\mathrm{blue}}\approx 0.25c$ and M_red ≈ 0.05M_⊙, v_red ≈ 0.1 c, respectively (e.g., Cowperthwaite et al. 2017; Drout et al. 2017; Kasen et al. 2017; Perego et al. 2017; Villar et al. 2017).

Several groups have made the case that the data is also consistent with a single ejecta component with a range of velocities and single low opacity (Smartt et al. 2017; Tanaka et al. 2017; Waxman et al. 2017). However, the thermal NIR continuum requires a higher line opacity in this wavelength range than that supplied by light r-process nuclei (with exclusively d-shell valence electron shell structure), implicating the presence of some lanthanides/actinide nuclei in the inner ejecta layers (Kasen et al. 2017). Synthesizing a small but nonzero lanthanide mass fraction (e.g., ${X}_{\mathrm{La}}\,\approx \,{10}^{-3}$) requires fine tuning of the ejecta Y_e distribution: the value of X_La increases from ≲10⁻⁴ to ≳0.1 over an extremely narrow range in Y_e centered at Y_e ≈ 0.25 (Tanaka et al. 2018). Ejecta with a smoothly varying Y_e outward to higher velocities, with distinct lanthanide-rich and lanthanide-poor regions, therefore provides a more physical explanation for the observations (though we point out how delayed mixing of the ejecta might provide a loophole to the lanthanide fine tuning argument in Section 4).

The total quantity of KN ejecta required to explain the observations ≳0.06M_⊙ exceeds that of the dynamical ejecta for any general relativistic (GR) NS merger simulation published to date; furthermore, the velocity of the red/purple KN ejecta, v_red ≈ 0.1 c, is significantly less than that predicted by simulations for the dynamical ejecta (which typically find v ≳ 0.2–0.3 c; e.g., Rosswog et al. 1999; Oechslin & Janka 2006; Bauswein et al. 2013; Hotokezaka et al. 2013; Sekiguchi et al. 2016; Bovard et al. 2017; Dietrich et al. 2017). The red KN ejecta is instead more naturally explained as being an outflow from the remnant accretion torus, which is created around the central compact object following the merger (e.g., Metzger et al. 2008a, 2009; Fernández & Metzger 2013; Metzger & Fernández 2014; Perego et al. 2014; Just et al. 2015; Martin et al. 2015; Wu et al. 2016; Lippuner et al. 2017; Siegel & Metzger 2017b). A three-dimensional GR MHD simulation of the post-merger disk evolution demonstrates that ≈40% of the initial mass of the torus will be unbound over a timescale of ∼1 s post-merger at an average velocity of v ≈ 0.1 c (e.g., Siegel & Metzger 2017a, 2017b; see also Metzger et al. 2008a).

If the merger end product is a hypermassive NS (HMNS) remnant (as was likely the case for GW170817; e.g., Bauswein et al. 2017; Granot et al. 2017; Ma et al. 2017; Margalit & Metzger 2017; Perego et al. 2017; Rezzolla et al. 2017; Ruiz et al. 2018), then the formation of a torus generically results from the outward transport of angular momentum, which removes centrifugal support and results in the collapse of the HMNS to a black hole (e.g., Shibata & Taniguchi 2006). However, creating a torus of sufficient mass ≳0.1M_⊙, as is needed to supply the M_red ≳ 0.04M_⊙ of wind ejecta inferred for GW170817, requires a fairly stiff NS equation of state. For instance, Radice et al. (2018) find that producing a torus of sufficient mass translates into a lower limit on the dimensionless NS tidal deformability parameter of $\tilde{{\rm{\Lambda }}}\gtrsim 400$, consistent with the upper limit $\tilde{{\rm{\Lambda }}}\lesssim 800$ obtained from the absence of detectable tidal losses on the inspiral gravitational waveform (Abbott et al. 2017b). This bound $\tilde{{\rm{\Lambda }}}\gtrsim 400$ translates into a rough lower limit on the radius of a 1.4M_⊙ NS of ≳12 km.

The source of the less neutron-rich ejecta responsible for the early blue KN is not so easily understood. Table 1 summarizes several possible origins, with their successes and shortcomings. While the high velocities v_blue ≈ 0.2–0.3 c and composition (Y_e ≳ 0.25) agree well with predictions for the shock-heated dynamical ejecta (Oechslin & Janka 2006; Wanajo et al. 2014; Goriely et al. 2015; Radice et al. 2016; Sekiguchi et al. 2016), the large quantity ${M}_{\mathrm{blue}}\,\approx \,2\,\times \,{10}^{-2}{M}_{\odot }$ is more challenging to explain. Bauswein et al. (2013) perform a suite of equal-mass⁴ merger simulations for a range of different equations of states, finding M_blue ≳ 0.01M_⊙ of dynamical ejecta only in cases where the NS radius is very small⁵ R_ns ≲ 11 km (Nicholl et al. 2017), near the lower bound of R_ns ≳ 10.7 km established by Bauswein et al. (2017) based on the inferred lack of a prompt collapse in GW170817. As outlined above, such a small NS radius is also in tension with that needed to create the massive torus needed to explain the red KN (Radice et al. 2018), further challenging the dynamical explanation for the blue KN ejecta.

Table 1.  Potential Sources of the Fast Blue KN Ejecta in GW170817

Ejecta Type	Quantity?	Velocity?	Electron Fraction?	References
Tidal Tail Dynamical	Maybe, if ${M}_{1}/{M}_{2}\,\lesssim \,0.7$ a	✓	Too Low	e.g., 1, 2
Shock-Heated Dynamical	Maybe, if Rns ≲ 11 kmb	✓	✓if NS long-lived	e.g., 3–5
Accretion Disk Outflow	✓if torus massive	Too Low	✓if NS long-lived	e.g., 6–9
HMNS Neutrino-Driven Wind	Too Low	Too Low	Too High?	e.g., 11, 12
Magnetized HMNS Wind	✓if NS long-lived	✓	✓	e.g., 12, 13

Notes.

^aWhere here M₁, M₂ are the individual NS masses (see Dietrich et al. 2017; Dietrich & Ujevic 2017; Gao et al. 2017). ^bHowever, a small NS radius may be in tension with the creation of a large accretion disk needed to produce the red KN ejecta (Radice et al. 2018). References: (1) Rosswog et al. (1999); (2) Hotokezaka et al. (2013); (3) Bauswein et al. (2013); (4) Sekiguchi et al. (2016); (5) Radice et al. (2016); (6) Fernández & Metzger (2013); (7) Perego et al. (2014); (8) Just et al. (2015); (9) Siegel & Metzger (2017b); (10) Dessart et al. (2009); (11) Perego et al. (2014); (12) Metzger et al. (2008b); (13) Metzger et al. (2008c).

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The high velocity tail of the KN ejecta might not be an intrinsic property, but instead the result of shock-heating of an originally slower ejecta cloud by a relativistic jet created following some delay after the merger (Bucciantini et al. 2012; Duffell et al. 2015; Bromberg et al. 2017; Gottlieb et al. 2017, 2018; Kasliwal et al. 2017; Piro & Kollmeier 2017). However, such a jet-accelerated ("cocoon") scenario still cannot account for the large quantity of high-Y_e ejecta because a relativistic jet contributes negligible baryon mass. The jet energy required to explain the kinetic energy of the blue KN ejecta ${M}_{\mathrm{blue}}{v}_{\mathrm{blue}}^{2}/2\approx {10}^{51}$ erg furthermore exceeds the beaming-corrected kinetic energies of cosmological short gamma-ray bursts (≈10⁴⁹–10⁵⁰ erg; Nakar 2007; Berger 2014) by a factor of ∼10–100.

Here we propose an alternative source for the blue KN ejecta: a magnetized neutrino-irradiated wind, which emerges from the HMNS remnant over ≈0.1–1 s prior to its collapse to a black hole. Though an HMNS remnant was previously proposed as a potential ejecta source (e.g., Evans et al. 2017), we emphasize the key role that strong magnetic fields play in endowing the wind with the high ejecta mass and velocity needed to explain the observations. The properties of such proto-magnetar winds have been extensively explored previously in the context of gamma-ray bursts from core collapse supernovae (Thompson et al. 2004; Bucciantini et al. 2006, 2007, 2008, 2009; Metzger et al. 2007, 2008c, 2011; Vlasov et al. 2014, 2017) and NS mergers (Metzger et al. 2008b; Bucciantini et al. 2012). Bringing this theory to bear on GW170817, we find that a temporarily stable magnetar remnant with a surface field strength B ≈ (1–3) × 10¹⁴ G can naturally provide the blue radioactive ejecta which lit up GW170817.

The end product of an NS–NS merger is typically an HMNS remnant,⁶ which is supported against gravitational collapse by its rapid and differential rotation (e.g., Shibata & Taniguchi 2006; Hanauske et al. 2017). Once its differential rotation has been removed (e.g., by gravitational waves or internal "viscosity" resulting from various MHD instabilities) the HMNS collapses into a black hole; however, the timescale for this process, and thus the remnant lifetime t_rem, is challenging to predict theoretically due to uncertainties in the supranuclear density equation of state and due to difficulties in numerically resolving all of the relevant physical processes over sufficiently long times (e.g., Siegel et al. 2013; Radice et al. 2018; Shibata & Kiuchi 2017; Shibata et al. 2017).

As shown below, the mass-loss rate of the HMNS remnant over its first several seconds of cooling is so high as to preclude the formation of an ultrarelativistic jet (e.g., Metzger et al. 2008b; Murguia-Berthier et al. 2016). Thus, if the gamma-rays observed in coincidence with GW170817 (Abbott et al. 2017a; Goldstein et al. 2017; Savchenko et al. 2017) are in some way associated with an off-axis—but otherwise typical—short GRB jet, then the formation of a black hole must have occurred prior to the arrival of the gamma-rays, setting an upper limit of t_rem ≲ 1.7 s on the HMNS lifetime. On the other hand, as we discuss in Section 4, the velocity of the magnetized HMNS wind could grow to become mildly relativistic over the neutrino cooling timescale of the remnant of a few seconds (Metzger et al. 2008b), in which case the gamma-rays could have been produced by the relativistic shock breakout created by internal collisions within the HMNS wind itself.

2.1. Standard (Unmagnetized) Neutrino-driven Wind

One source of post-dynamical ejecta is a neutrino-driven wind from the hot NS remnant (Dessart et al. 2009; Metzger & Fernández 2014; Perego et al. 2014; Martin et al. 2015), which is qualitatively similar to that expected from the hot proto-NS created by the core collapse of a massive star (Duncan et al. 1986; Qian & Woosley 1996). The steady-state mass-loss rate due to neutrino heating from an unmagnetized NS is approximately given by (Qian & Woosley 1996; Thompson et al. 2001)

$$\begin{eqnarray}{\dot{M}}_{\nu } & \simeq & 4\times {10}^{-4}{M}_{\odot }\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{{L}_{\nu }}{2\times {10}^{52}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{5/3}\\ & & \times {\left(\displaystyle \frac{{\epsilon }_{\nu }}{15\mathrm{MeV}}\right)}^{10/3}{\left(\displaystyle \frac{{M}_{\mathrm{ns}}}{2.4{M}_{\odot }}\right)}^{-2}{\left(\displaystyle \frac{{R}_{\mathrm{ns}}}{15\mathrm{km}}\right)}^{5/3},\end{eqnarray} \tag{ 1 }$$

where we have combined the contributions of electron neutrinos and antineutrinos to the heating into a single product of the neutrino luminosity L_ν and mean neutrino energy _ν defined by ${L}_{\nu }{\epsilon }_{\nu }^{2}\equiv {L}_{{\nu }_{e}}| {\epsilon }_{{\nu }_{e}}{| }^{2}+{L}_{{\bar{\nu }}_{e}}| {\epsilon }_{{\bar{\nu }}_{e}}{| }^{2}$. The fiducial values used above are motivated by axisymmetric hydrodynamical simulations of the post-merger remnant including neutrino transport by Dessart et al. (2009), who find L_ν ≈2 × 10⁵² erg s⁻¹ on timescales of t ≲ 30 ms post-merger, with average neutrino energies in the range _ν ≈ 13–17 MeV. In agreement with the above, Dessart et al. (2009) found $\dot{M}\,\approx \,{10}^{-3}\mbox{--}{10}^{-4}{M}_{\odot }$ s⁻¹ over the first t ≈ 20–100 ms of evolution.

Absent of magnetic fields, the wind is accelerated entirely by thermal pressure, and its asymptotic velocity is approximately given by (Thompson et al. 2001)

$$\begin{eqnarray}&&{v}_{\nu }\approx 0.12\,{\text{}}c\,{\left(\displaystyle \frac{{L}_{\nu }}{2\times {10}^{52}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{0.3}.\end{eqnarray} \tag{ 2 }$$

The asymptotic electron fraction in the wind is set by the competition between the weak interactions ${\nu }_{e}+n\to p+{e}^{-}$ and ${\bar{\nu }}_{e}+p\to n+{e}^{+};$ its equilibrium value is approximately given by (Qian et al. 1993)

$$\begin{eqnarray}&&{Y}_{e,\nu }\simeq {\left(1+\displaystyle \frac{{L}_{{\bar{\nu }}_{e}}}{{L}_{{\nu }_{e}}}\displaystyle \frac{{\epsilon }_{{\bar{\nu }}_{e}}-2{\rm{\Delta }}+1.2{{\rm{\Delta }}}^{2}/{\epsilon }_{{\bar{\nu }}_{e}}}{{\epsilon }_{{\nu }_{e}}+2{\rm{\Delta }}+1.2{{\rm{\Delta }}}^{2}/{\epsilon }_{{\nu }_{e}}}\right)}^{-1}\approx 0.4-0.6,\end{eqnarray} \tag{ 3 }$$

where ${\rm{\Delta }}\equiv ({m}_{n}-{m}_{p}){c}^{2}$ is the neutron–proton mass difference. This equilibrium is achieved in the wind because the gravitational binding energy of a nucleon on the surface of the HMNS of GM_nsm_p/R_ns ≈ 200 MeV greatly exceeds the mean neutrino energy _ν; each nucleon must therefore necessarily absorb several neutrinos in the process of escaping to infinity and the outflow is protonized as compared to the much lower value of Y_e ≲ 0.1 at the base of the wind in the degenerate surface layers of the star. In the final line, Y_e,ν is given as a range because the difference between the spectra of electron neutrinos and antineutrinos from the cooling merger remnant which enters the expression for Y_e,ν depends on the detailed transport processes in this environment and are thus theoretically uncertain (e.g., Roberts et al. 2012; Martínez-Pinedo et al. 2012 for recent work in the core collapse context).

The electron fraction of an unmagnetized PNS wind is sufficiently high Y_e ≳ 0.4−0.5 to synthesize exclusively Fe-group nuclei or light r-process nuclei with the low opacities needed to produce blue KN emission. However, the quantity ${\dot{M}}_{\nu }{t}_{\mathrm{rem}}\lesssim {10}^{-3}{M}_{\odot }$ and velocity v_ν ∼ 0.1 c of the neutrino-driven wind ejecta are too low compared to observations of GW170817. The predicted composition may also be problematic; the radioactive energy input of Y_e ≈ Y_e,ν ≳ 0.4 matter is dominated by a few discrete nuclei (Lippuner & Roberts 2015), inconsistent with the observed smooth decay of the KN bolometric light curve (Rosswog et al. 2017). Matter with lower Y_e can be unbound by neutrino heating of the surrounding accretion disk (e.g., Metzger & Fernández 2014; Perego et al. 2014; Martin et al. 2015), but the velocity of this material ≲0.1 c is also too low (Table 1).

2.2. Magnetized, Neutrino-heated Wind

A standard neutrino-heated wind cannot explain the observed properties of the blue KN, but the prospects are better if the merger remnant possesses a strong magnetic field. Due to the large orbital angular momentum of the initial binary, the remnant is necessarily rotating close to its mass-shedding limit, with a rotation period P = 2π/Ω ≈ 0.8–1 ms, where Ω is the angular rotation frequency. The remnant is also highly magnetized, due to amplification of the magnetic field on small scales to ≳10¹⁶ G by several instabilities (e.g., Kelvin–Helmholtz, magnetorotational) which tap into the free energy available in differential rotation (e.g., Price & Rosswog 2006; Siegel et al. 2013; Zrake & MacFadyen 2013; Kiuchi et al. 2015). As a part of this process, and the longer-term MHD evolution of its internal magnetic field (e.g., Braithwaite 2007), the rapidly spinning remnant could acquire a large-scale surface field, though its strength is likely to be weaker than the small-scale field.

In the presence of rapid rotation and a strong ordered magnetic field, magnetocentrifugal forces accelerate matter outward from the HMNS along the open field lines in addition to the thermal pressure from neutrino heating (Figure 1). A magnetic field thus enhances the mass-loss rate and velocity of the HMNS wind (Thompson et al. 2004; Metzger et al. 2007), in addition to reducing its electron fraction as compared to the equilibrium value obtained when the flow comes into equilibrium with the neutrinos, Y_e,ν (e.g., Metzger et al. 2008c).

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A key property quantifying the dynamical importance of the magnetic field is the wind magnetization

$$\begin{eqnarray}&&\sigma =\displaystyle \frac{{{\rm{\Phi }}}_{{\rm{M}}}^{2}{{\rm{\Omega }}}^{2}}{{\dot{M}}_{\mathrm{tot}}{c}^{3}}=\displaystyle \frac{{B}^{2}{R}_{\mathrm{ns}}^{4}{f}_{\mathrm{open}}{{\rm{\Omega }}}^{2}}{\dot{M}{c}^{3}},\end{eqnarray} \tag{ 4 }$$

where ${{\rm{\Phi }}}_{{\rm{M}}}={f}_{\mathrm{open}}{{BR}}_{\mathrm{ns}}^{2}$ is the open magnetic flux per steradian leaving the NS surface, B is the average surface magnetic field strength, f_open is the fraction of the NS surface threaded by open magnetic field lines, $\dot{M}$_tot = f_open$\dot{M}$ is the total mass-loss rate, and $\dot{M}$ is the wind mass-loss rate when f_open = 1 limit (which in general will be substantially enhanced from the purely neutrino-driven value estimated in Equation (1)). In what follows, we assume the split-monopole magnetic field structure (f_open = 1), which is a reasonable approximation if the magnetosphere is continuously "torn open" by latitudinal differential rotation (Siegel et al. 2014), neutrino heating of the atmosphere in the closed-zone region (Thompson 2003; Komissarov & Barkov 2007; Thompson & ud-Doula 2017), and by the compression of the nominally closed field zone by the ram pressure of the surrounding accretion disk (Parfrey et al. 2016). However, our results can also be applied to the case f_open ≪ 1, as would characterize a more complex magnetic field structure, provided that the ratio ${B}^{2}/\dot{M}\propto {f}_{\mathrm{open}}^{-1}$ can be scaled-up accordingly to obtain the same value of σ needed by observations.

Upon reaching the fast magnetosonic surface (outside of the light cylinder), the outflow achieves a radial four-velocity vγ ≃ cσ^1/3 (Michel 1969). Winds with σ ≫ 1 thus become ultrarelativistic, reaching a bulk Lorentz factor γ ≫ 1 in the range σ^1/3 ≲ γ ≤ σ, depending on how efficiently additional magnetic energy initially carried out by Poynting flux is converted into kinetic energy outside of the fast surface. By contrast, winds with σ < 1 attain subrelativistic speeds given by⁷

$$\begin{eqnarray}{v}_{B} & \simeq & \sqrt{3}c{\sigma }^{1/3}=\sqrt{3}{\left(\displaystyle \frac{{B}^{2}{R}_{\mathrm{ns}}^{4}{{\rm{\Omega }}}^{2}}{\dot{M}}\right)}^{1/3}\\ & \approx & 0.22{\text{}}c{\left(\displaystyle \frac{B}{{10}^{14}{\rm{G}}}\right)}^{2/3}{\left(\displaystyle \frac{\dot{M}}{0.01{M}_{\odot }{{\rm{s}}}^{-1}}\right)}^{-1/3}{\left(\displaystyle \frac{P}{1\mathrm{ms}}\right)}^{-2/3},\end{eqnarray} \tag{ 5 }$$

where in the final line we have taken R_ns = 15 km and the factor $\sqrt{3}$ accounts for the additional conversion of the wind Poynting flux (two-thirds of its flow energy near the fast surface) into bulk kinetic energy at larger radii.

Figure 2 shows the values of σ (or, equivalently, asymptotic four-velocity; top axis) and $\dot{M}$ from a suite of steady-state, one-dimensional, neutrino-heated, magnetocentrifugal wind solutions calculated by Metzger et al. (2008c) for an assumed neutrino luminosity ${L}_{\nu }\approx 1.6\,\times \,{10}^{52}$ erg s⁻¹, similar to that from the hot post-merger remnant at early times ∼0.1–1 s after the merger (Dessart et al. 2009). Solutions are shown for different values of the surface magnetic field strength B = 10¹⁴, 10¹⁵, 10¹⁶ G (denoted by different colors, from left to right) and several rotation rates Ω = 6000, 7000, 8000 rad s⁻¹ (P = 1.05, 0.90, 0.79 ms; bottom to top) in the range expected for the post-merger remnant.

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The strong magnetic field increases both the kinetic energy and mass-loss rate of the wind. $\dot{M}$ is increased by a factor of ≳100–1000 for B ∼ 10¹⁴–10¹⁶ G as compared to the unmagnetized value given in Equation (1). At fixed B, $\dot{M}$ also rapidly increases with faster rotation, while the outflow velocity slightly decreases for larger Ω because of the ${v}_{{\rm{B}}}\propto \,{\dot{M}}^{-1/3}$ dependence.

The additional magnetocentrifugal acceleration reduces the time matter spends in the neutrino-heated region and reduces the final electron fraction Y_e of the outflow compared to the neutrino-driven equilibrium value Y_e,ν (Equation (3)), as marked next to each solution in Figure 2. While a relatively weak magnetic field (10¹⁴ G) and slow rotation results in a value Y_e ≈ Y_e,ν ≃ 0.46 similar to the unmagnetized case, the more strongly magnetized and rapidly rotating cases have Y_e ≪ Y_e,ν. In these cases, the magnetic field accelerates matter away from the NS surface so rapidly that electron neutrinos have insufficient time to convert the initially neutron-rich composition at the electron-degenerate base of the wind back into one with a nearly equal number of neutrons and protons by much larger radii (where the r-process itself takes place).

Magnetocentrifugal enhancement of $\dot{M}$ is crucially tied to the conditions required for Y_e ≪ Y_e,ν, as both are related to the timescale matter has to absorb neutrinos near the base of the wind. In the limit that the magnetic field dominates the wind dynamics in the neutrino heating region, an outflow that is sufficiently neutron-rich (Y_e ≲ 0.25) to synthesize lathanide nuclei is necessarily accompanied by a minimum mass-loss rate (Metzger et al. 2008c; their Equation (20))

$$\begin{eqnarray} & & {\dot{M}}_{\mathrm{blue}}\approx 20\,\left(\displaystyle \frac{{{GM}}_{\mathrm{ns}}{m}_{p}}{{R}_{\mathrm{ns}}{\epsilon }_{\nu }}\right)\,{\dot{M}}_{\nu }\approx 0.1{M}_{\odot }\,{{\rm{s}}}^{-1}\\ & & \times {\left(\displaystyle \frac{{L}_{\nu }}{2\times {10}^{52}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{5/3}{\left(\displaystyle \frac{{\epsilon }_{\nu }}{15\mathrm{MeV}}\right)}^{7/3}{\left(\displaystyle \frac{{M}_{\mathrm{ns}}}{2.4{M}_{\odot }}\right)}^{-1}.\end{eqnarray} \tag{ 6 }$$

The factor (GM_nsm_p/R_nsν) accounts for the additional energy per nucleon supplied that must be by the magnetic acceleration, above the normal value from neutrino heating acting alone, in order to obtain Y_e ≲ Y_e,ν. The numerically calibrated pre-factor of 20 is that needed to reduce Y_e further to ≲0.25. Thus, one can place a strict upper limit on the quantity of blue KN ejecta (Y_e ≳ 0.25) of

$$\begin{eqnarray}&&{M}_{\mathrm{blue}}^{\max }\sim {\dot{M}}_{\mathrm{blue}}{t}_{\mathrm{rem}}\sim {10}^{-2}({t}_{\mathrm{rem}}/0.1\,{\rm{s}}){M}_{\odot },\end{eqnarray} \tag{ 7 }$$

where we have taken ${L}_{\nu }\approx {10}^{52}$ erg s⁻¹ as the roughly constant neutrino luminosity of the remnant on timescales of t_rem ≲ 1 s (e.g., Dessart et al. 2009).

The 1D solutions of Metzger et al. (2008c) were calculated along flow lines that emerge along the equatorial plane, near the last closed field line. Because these low-latitude lines cover a large solid angle and carry most of the total mass flux of the wind, the results are similar to the total mass-loss rate and mass-averaged value of the outflow Y_e integrated across the full open magnetosphere (Vlasov et al. 2014, 2017). Nevertheless, the wind velocity and composition do show moderate variation with polar latitude θ of the outflow launching point; for instance, Vlasov et al. (2014) found that for P ≈ 1 ms and ${L}_{\nu }\approx 2\,\times \,{10}^{52}$ erg s⁻¹, Y_e varied from ≈0.40 to ≈0.32 as θ increased from ≲0.35 to ≈0.5 (flow lines that carry ≈20% and ≈80% of the total mass loss, respectively).

2.2.1. Application to the Blue KN of GW170817

On the timescale t ≲ t_rem ≲ 1 s before the HMNS remnant collapses into a black hole, neither its rotation rate nor its neutrino luminosity will evolve substantially. The average rotation frequency Ω cannot change substantially because only a moderate loss of rotational support would trigger gravitational collapse, while the neutrino luminosity decreases by a factor of ≲2 over the first second of the Kelvin–Helmholtz cooling evolution; e.g., Pons et al. 1999). Our analytic and numerical results at fixed Ω can be therefore used to approximate what type of magnetar remnant, if any, can explain all three key properties (mass, velocity, and composition) the blue KN from GW170817 (Figure 2).

For a wind duration equal to the magnetar lifetime t_rem, producing the observed quantity of ejecta ${M}_{\mathrm{blue}}\approx 2\,\times {10}^{-2}{M}_{\odot }$ (e.g., Cowperthwaite et al. 2017; Nicholl et al. 2017; Perego et al. 2017; Villar et al. 2017) requires an average wind mass-loss rate

$$\begin{eqnarray}&&\dot{M}\approx 2\times {10}^{-2}{({t}_{\mathrm{rem}}/1{\rm{s}})}^{-1}{M}_{\odot }\,{s}^{-1}.\end{eqnarray} \tag{ 8 }$$

According to Equation (5), explaining the velocity of the ejecta v_blue ≈ 0.2–0.3 c = v_B requires a wind magnetization σ ≈0.002–0.005, which, from Equation (4), requires a surface magnetic field strength of (for Ω = 8000 rad s⁻¹, R = 15 km)

$$\begin{eqnarray}&&B\approx {(1\mbox{--}2)\times {10}^{14}({t}_{\mathrm{rem}}/1{\rm{s}})}^{-1/2}\,{\rm{G}}\end{eqnarray} \tag{ 9 }$$

or B ∼ (1–3) × 10¹⁴ G for t_rem ∼ 0.1–1 s. Extrapolating between the B = 10¹⁴ G and 10¹⁵ G solutions in Figure 2 shows that winds with mass-loss rates corresponding to t_rem ≲ 1 s indeed have electron fractions Y_e ≈ 0.25–0.4 in agreement with the range required to power the blue KN. This "concordance" region is shaded blue in Figure 2.

Following its discovery at t ≈ 11 hr, the first few days of optical emission following GW170817 is well-explained by KN emission powered by the decay of heavy r-process nuclei. This agreement is independent of the precise origin of the ejecta because, by this late time, most of the initial thermal energy released during the ejection process on timescales t₀ ≲ 1 s has been degraded by adiabatic expansion, such that the observed luminosity is dominated by the radioactive decay of isotopes with half-lives ∼t. However, different sources of blue KN ejecta will make quantitatively distinct predictions for the emission at earlier times, within the first few hours after the merger. Though not available for GW170817, such early observations may be possible for future GW-detected mergers, such as those which occur above the North American continent at night (e.g., Kasliwal & Nissanke 2014). Early UV follow-up would also be possible with a dedicated wide-field satellite, such as the proposed ULTRASAT mission (e.g., Waxman et al. 2017).

A key distinguishing feature of different ejecta sources is the timescale over which the mass loss occurs. Dynamical ejecta emerges almost immediately, within ≲1–10 ms following the merger, and freely expands thereafter without additional heating other than from radioactivity. By contrast, outflows from a magnetar or accretion disk emerge over longer timescales Δt ∼ 0.1–1 s, in which case outflow variability acting over the same characteristic timescale would create internal shocks that inject additional thermal energy on radial scales $\sim {\rm{\Delta }}t\cdot {v}_{\mathrm{blue}}\sim {10}^{9}\mbox{--}{10}^{10}$ cm, which greatly exceed⁸ those of the central engine ≲10–100 km (e.g., Beloborodov 2014).

As the HMNS cools following the merger, approximately over its Kelvin–Helmholtz diffusion timescale Δt ∼ 1 s (e.g., Pons et al. 1999), a factor of ≈2 decrease in its neutrino luminosity would reduce the wind mass-loss rate by a factor of ≈3 (Equation (1)) and increase the wind outflow speed ${v}_{{\rm{B}}}\propto {\dot{M}}^{-1/3}$ by a factor of ∼1.5 (Equation (5)); growth of the magnetic field strength from a dynamo, or emergence of magnetic flux from the interior, would also increase the wind speed ${v}_{{\rm{B}}}\propto {B}^{2/3}$. The rising wind velocity will result in the development of internal shocks, heating the ejecta as discussed above.

A crude but illustrative estimate of the time evolution of the wind velocity is shown in Figure 4, separately for a case in which the HMNS magnetic field is held constant in time, and when the field is assumed to grow exponentially on a timescale of ≈0.5 s. The wind four-velocity is calculated from Equation (5) based on the evolution of $\dot{M}\propto {L}_{\nu }^{5/3}{\epsilon }_{\nu }^{10/3}$ (Equation (1)), where L_ν(t) and _ν(t) are taken from the proto-NS cooling calculations of Pons et al. (1999; which should qualitatively capture the cooling evolution of the post-merger remnant). In the case of the temporally strengthening field, the internal shocks in the magnetar wind could grow to transrelativistic velocities γβ ≳ 1 on a timescale of a couple seconds, if the HMNS has not already collapsed by this time. Relativistic shock breakout (e.g., Nakar & Sari 2012) from this interaction might provide an alternative source for the gamma-ray emission from GW170817, which does not necessarily implicate black hole formation or the creation of an ultrarelativistic jet on this timescale. Future hydrodynamical modeling with radiative transfer is required to assess whether the internal shocks generated within the magnetar wind can explain in detail the properties (mean energy, luminosity, and duration) of the gamma-ray emission.

To illustrate the influence of radially extended heating on the early-time KN emission, Figure 3 shows several KN models that fit the bolometric luminosity ${L}_{\mathrm{bol}}\approx {10}^{42}$ erg s⁻¹ measured at t ≈ 11 hr (see the Appendix for details) yet do not overproduce the luminosity at 1.5 days (at which time the slower inner layers, which produce the red/purple KN are beginning to contribute). Solid black lines show models for ejecta for different assumptions about the time t₀ = 0.01, 0.1, 1 s after the merger at which the thermal energy of each layer of the ejecta was last comparable to its asymptotic kinetic energy (at times t ≥ t₀, the only heating is by the decay of r-process nuclei). As discussed above, this timescale can be roughly associated with the timescale Δt over which a significant fraction of the mass loss is occurring, e.g., the HMNS lifetime t_rem in the case of a magnetar wind or the viscous timescale in the case of a jet or accretion disk outflow (e.g., Fernández & Metzger 2013). Figure 3 shows that the luminosity at one hour post-merger is ∼3–10 times higher if the mass loss emerges and continues to self-interact through internal shocks over an extended timescale of t₀ ∼ 0.1–1 s (e.g., magnetar or accretion disk wind) than if the ejecta is dynamical in origin and freely expands from the merger site (t₀ ≲ 0.01 s).

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Additional thermal energy could be deposited into the expanding matter also by the passage of a relativistic jet through the blue ejecta, creating an expanding tail of high velocity shock-heated matter (e.g., Bromberg et al. 2017; Gottlieb et al. 2017, 2018; Kasliwal et al. 2017; Piro & Kollmeier 2017). However, reasonable constraints can be placed on the magnitude of this effect based on the observed kinetic energies of on-axis cosmological short duration gamma-ray bursts. The isotropic kinetic energies of short GRB jets as inferred from their nonthermal afterglows are typically ${E}_{\mathrm{iso},{\rm{K}}}\approx {10}^{50}\mbox{--}{10}^{51}$ erg (Nakar 2007; Berger 2014; Fong et al. 2015). The local (z = 0) rate of binary NS mergers as measured by Advanced LIGO after GW170817 is ${{ \mathcal R }}_{\mathrm{BNS}}={1540}_{-1220}^{+3200}$ Gpc⁻³ yr⁻¹, while the observed rate of short GRBs is ${{ \mathcal R }}_{\mathrm{SGRB}}\approx 2\mbox{--}6$ Gpc⁻³ yr⁻¹ (Wanderman & Piran 2015). This places an upper limit on the beaming fraction of the emission of f_b ≲ 0.02, which in turn implies a beaming-corrected GRB jet energy of ≲10⁴⁹ erg which is 2 orders of magnitude smaller than the measured kinetic energy of the blue KN ejecta of ∼10⁵¹ erg. Thus, unless the jet cocoon contains ≳10–100 times more energy than is typical of the ultrarelativistic GRB itself, cocoon heating contributes insignificantly to the early-time optical emission. This argument would not be valid if cosmological short GRBs do not originate from NS–NS mergers.

The early KN emission can also be enhanced due to the radioactive decay of free neutrons in the outermost layers of the ejecta (Metzger et al. 2015; see also Kulkarni 2005). Free neutrons may be present in the outermost ∼1% of the ejecta if these layers are expanding sufficiently rapidly for neutron-capture reactions to freeze-out prior to completion of the r-process (as occurs on a timescale of ∼1 s; Bauswein et al. 2013). The anomalously long half-life of a free neutron τ_n ≈ 10 minutes, as compared to a typical r-process isotope (τ ∼ 1 s), enables free neutron decay to dominate r-process heating during the first several hours of emission. Figure 3 shows that the presence of free neutrons in the outermost $\sim {10}^{-4}\mbox{--}{10}^{-3}{M}_{\odot }$ of the expanding material can mimick to some extent the effects of a long heating timescale t_rem ≳ 0.1–1 s, including a cocoon.

While the presence of luminous optical (blue) KN emission from the first NS merger was not a surprise (e.g., Metzger et al. 2010), the large ejecta mass, high velocity, and relatively neutron-poor composition are in tension with traditionally considered ejecta sources (Table 1). Instead, we have proposed that the blue KN ejecta in GW170817 originates from the neutrino-irradiated, magneto-centrifugally driven outflow from a temporarily stable rapidly rotating HMNS remnant (Figure 1). Figure 2 shows that such a millisecond magnetar wind can simultaneously explain the large ejecta mass (due to magnetocentrifugal enhancement of the neutrino-driven mass-loss), high velocity (magnetocentrifugal enhancement of the wind acceleration), and electron fraction (from moderate, but not excessive, neutrino irradiation near the wind launching point), given only two free parameters: the surface magnetic field strength B ∼ (1–3) × 10¹⁴ G and HMNS lifetime t_rem∼ 0.1–1 s.

The HMNS generally collapses to a black hole once its centrifugal support from internal differential rotation is removed by the outward transport of angular momentum (e.g., Kaplan et al. 2014). While this process has been modeled as local "effective viscosity" (e.g., Radice et al. 2018; Shibata et al. 2017), angular momentum redistribution could in principle be dominated by torques from large-scale magnetic fields, even if they are weaker in magnitude than the smaller-scale turbulent ones. In this case, solid-body rotation will be established roughly on the radial Alfvén crossing timescale of the remnant⁹ given its large-scale magnetic field B,

$$\begin{eqnarray}&&{t}_{{\rm{A}}}=\displaystyle \frac{{R}_{\mathrm{ns}}}{{v}_{{\rm{A}}}}\approx 1.0{\left(\displaystyle \frac{B}{{10}^{14}{\rm{G}}}\right)}^{-1}{\rm{s}},\end{eqnarray} \tag{ 10 }$$

where ${v}_{{\rm{A}}}=B/\sqrt{4\pi \rho }$ is the Alfvén velocity and $\rho \approx 3{M}_{\mathrm{ns}}/(4\pi {R}_{\mathrm{ns}}^{3})$ is the mean density of the NS. Interestingly, the value of t_A for values of B ∼ (1–3) × 10¹⁴ G we find are needed to explain whether the blue KN is similar to the HMNS collapse time t_rem ≈ 0.1–1 s inferred independently from the KN emission (and also possibly the arrival delay of the gamma-ray burst).

The hypermassive magnetar scenario makes predictions for how the blue KN emission will correlate with the GW-inferred properties of the NS–NS binary progenitor and, potentially, the timescale of the gamma-ray emission. In general, more massive binaries will create less-stable HMNS remnants that collapse to black holes earlier (smaller t_rem), which (for fixed B) would reduce the quantity of the blue KN ejecta from the magnetar wind (M_blue ∝ t_rem). On the other hand, if t_A (Equation (10)) also plays a role in the collapse time, then a merger remnant of total mass otherwise similar to that in GW170817, but with a stronger magnetic field B, will produce both a higher KN ejecta velocity ${v}_{{\rm{B}}}\,\propto \,{B}^{2/3}$ and a shorter lifetime ${t}_{\mathrm{rem}}\,\propto \,{B}^{-1}\,\propto \,{v}_{{\rm{B}}}^{-3/2};$ the latter may also correlate with the timescale of the observed gamma-ray burst, assuming it is powered by an ultrarelativistic jet that is generated soon after black hole formation.

Although disk outflows provide a satisfactory explanation for the slower, red KN ejecta (e.g., Siegel & Metzger 2017b), it is important to address whether the latter can also be explained by a magnetar wind with time-evolving properties. If the magnetic field of the HMNS remnant were to grow in time, then Figure 2 shows that Y_e will decrease and $\dot{M}$ will increase (for fixed Ω), although the decaying neutrino luminosity as the remnant cools will counter the latter. If Y_e were to decrease in time, this would be consistent with the observed late emergence of red/purple KN ejecta (Y_e ≲ 0.25) following the earlier blue KN ejecta (Y_e ≳ 0.25). However, an increasing magnetic field would also cause the wind velocity to grow in time (Figure 4), which, as discussed in Section 3, would give rise to powerful internal shocks. This interaction would cause a redistribution of the radial velocity stratification and might lead to mixing of the high- and low-Y_e material (see below). We conclude that a time-evolving magnetar wind could, in principle, supply both the blue and red KN ejecta, though detailed numerical modeling of the magnetized outflow, including neutrino transport from the remnant, will be needed to confirm this idea quantitatively.

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We have discussed ways to observationally disentangle different sources for the ejecta based on the first few hours of the KN (Figure 3). In particular, the KN luminosity will be substantially higher at these early times if the ejecta is reheated above its launching point by internal shocks. The latter is generally expected for any long-lived ejecta source, e.g., a magnetar of lifetime t_rem ≳ 0.1–1 s (e.g., Metzger et al. 2008b) or an accretion disk of lifetime ∼1 s (e.g., Fernández & Metzger 2013), and is not a unique prediction of a (failed or successful) ultrarelativistic jet (Gottlieb et al. 2018). This will unfortunately make the source of early ejecta heating (jet, magnetar wind, and disk outflow) challenging to identify. Early-time blue emission from shocks is also degenerate with heating from free neutron decay in the outer ejecta layers (Metzger et al. 2015). It is therefore important to explore any additional signatures of late-time ejecta heating or shocks.

Several groups have argued that the entire KN emission from GW170817 can be explained by a single homogeneous ejecta with a low lanthanide mass fraction, e.g., ${X}_{\mathrm{La}}\approx {10}^{-3}$ (e.g., Smartt et al. 2017; Waxman et al. 2017; Tanaka et al. 2018). However, as already discussed, endowing the ejecta with a low but nonzero lanthanide abundance appears to require fine tuning of the Y_e distribution to be tightly peaked around Y_e ≈ 0.25 (see also Tanaka et al. 2018).

One loophole to creating a homogeneous low-X_La composition would be to mix separate ejecta components with vastly different lanthanide abundances, e.g., combine 1% of the mass with Y_e ≲ 0.25 and X_La ≈ 0.1 with a much larger fraction ≳99% that was lanthanide-free (Y_e ≳ 0.25). Importantly, however, such mixing would need to occur after the r-process is complete, i.e., after free neutrons are captured into nuclei on a timescale of t_capture ∼ 1 s after ejection. While such late-time mixing would not be consistent with a dynamical origin for the ejecta, it could result naturally if the engine lifetime was indeed t_rem ≳ 0.1–1 s ∼ t_capture, as predicted for a long-lived magnetar. As already discussed, the moderate rise in the magnetar outflow velocity, even over its short lifetime, will indeed produce internal shocks within the ejecta (Metzger et al. 2008b) that would also enable mixing. Moderate variations in Y_e are also expected with polar latitude in the wind (Vlasov et al. 2014) as well as in time, both secularly as the HMNS cools and on shorter timescales due to periodic magnetic reconnection events instigated by opening of closed magnetic field lines by differential rotation and neutrino heating (Thompson 2003).

B.D.M. is supported by NASA through the ATP program, grant numbers NNX16AB30G and NNX17AK43G. E.Q. was supported in part by a Simons Investigator Award from the Simons Foundation and the Gordon and Betty Moore Foundation through Grant GBMF5076. We thank Niccolo Bucciantini for previous collaborations and many useful discussions on this topic.

Multi-velocity semi-analytic one-dimensional kilonova models have been considered in several previous works (e.g., Piran et al. 2013; Metzger 2017; Waxman et al. 2017). Following the merger and any subsequent shock heating, e.g., by the GRB jet, the radial velocity structure of the ejecta ultimately approaches homologous expansion (e.g., Rosswog et al. 2014). We approximate the distribution of mass with velocity greater than a value v as a power law of index β,

$$\begin{eqnarray}&&{M}_{v}=M{(v/{v}_{0})}^{-\beta },v\gtrsim {v}_{0},\end{eqnarray} \tag{ 11 }$$

where M is the total mass and v₀ is the minimum (∼average) velocity. The thermal energy of each mass shell evolves as a function of time t after the merger according to

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{v}}{{dt}}=-\displaystyle \frac{{E}_{v}}{t}-\displaystyle \frac{{E}_{v}}{{t}_{d,v}+{t}_{{lc},v}}+{\dot{Q}}_{v},\end{eqnarray} \tag{ 12 }$$

where the first term accounts for PdV losses, the second term is the radiative luminosity, where

$$\begin{eqnarray}&&{t}_{d}=\displaystyle \frac{3{M}_{v}\kappa }{4\pi {cvt}}\end{eqnarray} \tag{ 13 }$$

is the photon diffusion time through layer v and t_lc,v = (v/c)t is the light crossing time. We have modeled the opacity produced by r-process lines as a gray opacity of value κ. Tanaka et al. (2018) show that bolometric luminosities are reasonably well-produced for κ = 0.5 cm² g⁻¹ for light r-process nuclei (lanthanide-free, Y_e ≳ 0.25) and κ = 10 cm² g⁻¹ (full lanthanide-fraction, Y_e ≲ 0.25).

The final term in Equation (12) is the radioactive heating rate, which for r-process nuclei is reasonably approximated by (Metzger et al. 2010; Roberts et al. 2011)

$$\begin{eqnarray}&&{\dot{Q}}_{v}={\epsilon }_{\mathrm{th},v}{M}_{v}{\dot{Q}}_{1,v}{(t/1\mathrm{day})}^{-1.3},\end{eqnarray} \tag{ 14 }$$

where the normalization depends on both the thermalization efficiency _th,v of the decay products and the overall normalization ${\dot{Q}}_{1,v}$ of the specific radioactive energy loss rate at 1 day. The thermalization efficiency varies with the density of the mass shell, with a high value at early times ≈0.5–0.7 due to the efficient exchange of beta decay and gamma-ray products (Metzger et al. 2010; Barnes et al. 2016; Hotokezaka et al. 2016; Waxman et al. 2017). The value of ${\dot{Q}}_{1,v}$ depends on the nucleosynthesis products and varies with Y_e from ≈1.5 × 10¹⁰ erg g⁻¹ s⁻¹ at low Y_e ≲ 0.3 to a value ≈(1–4) × 10¹⁰ erg g⁻¹ s⁻¹ at higher Y_e. In what follows, we take ${\dot{Q}}_{1,v}=2\times {10}^{10}$ erg g⁻¹ s⁻¹.

In some cases, we also include radioactive heating from free neutrons (decay timescale τ_n ≈ 900 s), which exist in the outermost layers (${M}_{{\rm{n}}}\approx {10}^{-4}\mbox{--}{10}^{-3}{M}_{\odot }$) of the ejecta and produce a radioactive heating rate (in lieu of Equation (14)) of

$$\begin{eqnarray}{\dot{Q}}_{{\rm{v}}} & = & 3.2\times {10}^{14}(1-2{Y}_{e})\mathrm{erg}\,{{\rm{g}}}^{-1}\,{{\rm{s}}}^{-1}\\ & & \times \exp (-t/{\tau }_{{\rm{n}}}),{M}_{{\rm{v}}}\lesssim {M}_{{\rm{n}}},\end{eqnarray} \tag{ 15 }$$

where 1–2Y_e is the neutron fraction; because the outer layers of the ejecta typically have Y_e ≪ 1, this factor is usually ≈1.

Boundary conditions on the thermal energy are derived by fixing E_v = v²/2 for each shell until times t = t₀, after which time the energy is evolved according to Equation (12).