NEUTRINO-DRIVEN WINDS IN THE AFTERMATH OF A NEUTRON STAR MERGER: NUCLEOSYNTHESIS AND ELECTROMAGNETIC TRANSIENTS

Our nucleosynthesis study is based on the first three-dimensional, Newtonian simulation of the neutrino-driven wind that emerges during the aftermath of a binary neutron star merger (Perego et al. 2014). The merger remnant is characterized by a long-lived MNS, surrounded by a quasi-Keplerian accretion disk with an inital mass of M_disk ≃ 0.19 M_⊙. For the present analysis, we repeat and extend the previous simulation to longer times, and we follow a substantially larger number of particles.

We use the parallel grid code FISH (Käppeli et al. 2011) to solve the hydrodynamical equations on a uniform Cartesian grid. Nuclear matter description is provided by the TM1 nuclear equation of state, supplemented with electron–positron and photon contributions (Timmes & Swesty 2000; Hempel et al. 2012). As an initial condition we use a late matter configuration from a three-dimensional high-resolution simulation of two non-spinning 1.4 M_⊙ neutron stars (e.g., Price & Rosswog 2006). Neutrino-matter interactions are taken into account using the multiflavor Advanced Spectral Leakage (ASL) scheme (see A. Perego et al. 2015, in preparation). It models the neutrino emission effectively by smoothly interpolating between diffusive and free-streaming rates, separately for different neutrino energies, and it has been carefully gauged at full transport calculations. Neutrino absorption is also included in optically thin conditions, based on the calculation of the neutrino densities outside the neutrino last scattering surfaces. The list of the neutrino reactions implemented inside the ASL scheme can be found in Table 1 of Perego et al. (2014). The present simulation corresponds to an extension of the one presented in Perego et al. (2014). The simulation starts at 25 ms after the beginning of the merger, including 10 ms of relaxation of the smoothed particle hydrodynamics final conditions on the grid, and follows the post-merger evolution for 190 ms. The temporal evolution of the neutrino luminosities and mean energies during the whole simulation follows the same trends reported in Figure 10 of Perego et al. (2014) for the first 90 ms. The relevant mean energies stay approximately constant: ∼11 MeV for the electron neutrinos and ∼15.5 MeV for the electron antineutrinos. The total neutrino luminosities, integrated over the whole solid angle, decrease slowly with time. At the end of the simulation (t_sim ≈ 190 ms) the electron (anti)neutrino luminosity due to cooling processes only is equal to ∼2.2 × 10⁵² erg s⁻¹ (∼2.9 × 10⁵² erg s⁻¹), while the inclusion of the neutrino absorption in optically thin conditions decreases it to ∼1.5 × 10⁵² erg s⁻¹ (∼2.4 × 10⁵² erg s⁻¹). Due to the larger optical depth along the equatorial plane, the neutrino fluxes measured far from the MNS along the polar direction are roughly three times larger than the ones in the equatorial plane (cf. Figure 12 of Perego et al. 2014).

The inclusion of the energy deposition provided by the neutrino absorption on nucleons inside the disk drives a baryonic wind on a timescale of tens of milliseconds (see Figure 1). Matter expanding inside the wind becomes unbound at a distance ≲600 km from the center. The resulting ejecta are confined within a polar angle of 60°, measured from the rotational axis of the disk. Due to the strong neutrino irradiation, the initially highly neutron-rich matter inside the disk changes its electron fraction in the wind. The dominant ν_e-absorption of neutrons raises Y_e toward equilibrium values (see e.g., Qian & Woosley 1996). The evolution toward this equilibrium value may be affected by general-relativistic treatment of the merger phase. GR simulations of a binary neutron star merger including neutrinos (Neilsen et al. 2014) and following the onset of the neutrino-driven wind (Sekiguchi et al. 2015) indicate that the luminosities can be larger than in Newtonian simulations, due to the higher temperature inside the MNS and the disk. However, the ratio between the electron neutrino and antineutrino luminosities is similar in both cases, as well as the values of the mean energies. Therefore, the equilibrium electron fraction is expected to be almost the same, while the evolution toward this Y_e should be faster in the GR simulations. This difference does not come directly from the Newtonian treatment of the disk and of the wind, but from the initial profiles resulting from the merger process.

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At the beginning of our simulation we place 10⁵ tracers inside our computational domain. The wind tracers are ejected from a region with initial density of $\rho \lesssim {10}^{10}\;{\rm{g}}\;{\mathrm{cm}}^{-3},$ as found in backtracking procedures of preliminary tests following Perego et al. (2014). Thus, we locate our tracers initially inside the thick accretion disk where the density is between 2 × 10⁶ g cm⁻³ and 2 × 10¹⁰ g cm⁻³ (see Figure 2). The number of tracers assigned to each grid cell is proportional to its mass content and the actual location of each particle within a cell is randomly assigned. Hence, the resulting tracer distribution tracks the matter density distribution inside the disk. The mass of the region of the disk where the tracers are placed is 0.0542 M_⊙. Therefore, we assign to each of them an inherent mass of m_i = m = 5.420 × 10⁻⁷ M_⊙. Every tracer records the local properties of matter (density, internal energy, electron fraction, and velocity components) at its present location by linearly interpolating the corresponding values on the computational grid. The total specific energy, e_tot, is computed as the algebraic sum of the kinetic, thermal, and gravitational specific energy. We recall here that the thermal energy is obtained from the relativistic internal energy, reduced by the rest mass energy (both obtained from the nuclear equation of state). All tracers are passively advected by the fluid and their location is evolved in time inside the grid by solving the equation $d{\boldsymbol{x}}/{\rm{d}}t={\boldsymbol{v}}$ with a second-order-accurate Euler integration scheme. A tracer particle is considered unbound if e_tot > 0 and if its radial component of the velocity is positive, steadily from a certain time until the end of the simulation. For each ejected tracer, neutrino fluxes and mean energies as a function of time come from the axisymmetric output provided by the hydrodynamic simulations (see Section 2.1 and Perego et al. 2014).

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For the post-processing of the tracers, a complete nucleosynthesis network (Winteler 2012; Winteler et al. 2012) is employed. Over 5800 nuclei between the valley of stability and the neutron drip line are considered, i.e., isotopes from H to Rg. The reaction rates are taken from the compilation of Rauscher & Thielemann (2000) for the Finite Range Droplet Model (FRDM; Möller et al. 1995). Weak interaction rates including neutrino absorption on nucleons are taken into account (Möller et al. 2003; Fröhlich et al. 2006). Furthermore, neutron capture for nuclei with Z ≳ 80 and neutron-induced fission rates are given by Panov et al. (2010) as well as β-delayed fission probabilities from Panov et al. (2005).

We perform nucleosynthesis calculations for over 17,000 ejected tracers from the hydrodynamical simulation (see Section 2.2). Our computations start when the temperature drops below T = 10 GK. Then the initial composition is determined by nuclear statistical equilibrium (NSE) and is dominated by alpha particles, neutrons, and protons. NSE is assumed to hold for T ≳ 8 GK. Between 10 GK > T > 8 GK, the network evolves the weak reactions since they are not in equilibrium and change the Y_e accordingly. As soon as the temperature undershoots the NSE threshold, the full network provides the abundances. The longest trajectories were simulated until t_sim ∼ 200 ms, therefore we extrapolate them following the prescription outlined in Korobkin et al. (2012).

Moreover, the energy generation by the r-process is calculated and its impact on the entropy is included (Freiburghaus et al. 1999b). As is common practice, we neglect the feedback of nuclear heating on the density evolution. In the case of the dynamic ejecta, it has been shown that this approximation is very good for nucleosynthesis calculations, although the long-term hydrodynamical evolution of the ejecta is heavily influenced⁵ (Rosswog et al. 2014). The heating mainly originates from β-decays and we assume that the energy is roughly equally distributed between thermalizing electrons and photons, and escaping neutrinos and photons (Metzger et al. 2010).

The composition and amount of matter ejected in the neutrino-driven wind depends on the temporal evolution of the disk and the fate of the compact central object. Here we consider a long-lived MNS. In the following, we present abundances at various times after the merger and for different latitudes to understand the angular distribution and temporal evolution of ejected matter. The potential consequences for the mixing with the dynamic ejecta and thus for the light curve are discussed in Sections 3.2 and 3.3, respectively. In the following, t_sim, refers to the time since merger. When abundances are shown at t_sim, we consider all tracers having become unbound until this time. Therefore, at later times early ejecta are also included.

An idea of the amount of matter ejected and its potential composition can be gained by exploring the dependence of the ejected mass on the electron fraction and time. This evolution is shown in Figure 3. Within the first 50 ms only a marginal amount of mass with electron fraction Y_e ≲ 0.3 gets unbound. Until t_sim = 100 ms approximately 2 × 10⁻³ M_⊙ are ejected with a central value of Y_e ≈ 0.3. After almost twice that time, i.e., t_sim = 190 ms, the mass of unbound material reaches 9 × 10⁻³ M_⊙. These late-time ejecta have relatively high electron fractions (with a mean value between 0.3 and 0.35) since neutrinos have more time to interact with nucleons. Furthermore, the growth rate of unbound mass by the late ejection of tracers remains approximately constant from 110 ms on. The substantial mass accretion rate inside the accretion disk, amounting to a few tenths of solar masses per second, provides quasi-steady-state conditions with neutrinos as the primary cooling agent. As a consequence, the neutrino luminosities decrease only slowly over the expansion timescale (cf. Figure 10 in Perego et al. 2014) and the interaction rates get close to equilibrium.

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In the following, abundances are shown using either of these two normalizations for a given nuclear mass A:

• 1.  
  Total mass:

  $$\begin{eqnarray}&&{X}_{{\rm{tot}}}(A)=\displaystyle \sum _{i=1}^{N}{X}_{i}(A)\cdot {m}_{i},\end{eqnarray} \tag{ 1 }$$

  with N, X_i, and m_i being the number of tracers considered, the mass fractions, and the mass of one tracer i, respectively. This quantity has mass units and thus allows us to compare different ejection channels to each other, as we will discuss in Section 3.2.
• 2.  
  Integrated abundances:

  $$\begin{eqnarray}&&{Y}_{{\rm{int}}}(A)=\displaystyle \frac{{\displaystyle \sum }_{i=1}^{N}{Y}_{i}(A)\cdot {m}_{i}}{{\displaystyle \sum }_{i=1}^{N}{m}_{i}}.\end{eqnarray} \tag{ 2 }$$

  Normalizing the abundances via Equation (2) allows us to explore the average nucleosynthesis yields of a certain subclass of tracers, for instance tracers ejected into a particular solid angle. Additionally, the impact of single trajectories on the overall nucleosynthesis is directly evident from the comparison with the average abundance curve.

Figure 4 shows total ejected masses for simulation times t_sim = 90, 140, and 190 ms. The heavy nuclei beyond the second r-process peak (A ∼ 130) are produced by early, very neutron-rich ejecta, as indicated by the overlap of the three curves at different simulation times. Later on, no heavy r-process elements (A ≳ 130) are produced any more. There is a threshold Y_e ∼ 0.25 below which heavy r-process elements (i.e., beyond the second peak at A = 130) can be synthesized (see also Kasen et al. 2015). As time passes, a substantial amount of tracers contributing to nuclei with A ≲ 120 becomes unbound. This evolution is in fact a direct consequence of the trend of Y_e presented in Figure 3.

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In addition, the nucleosynthesis has an angular dependency. We divide the neutrino-driven wind into four regions above and below the disk by setting cuts on the polar angle and investigate the angular dependence of the ejecta. Each of the cuts has a width of Δθ = 15°, hence the whole neutrino-driven wind is captured within 0° ≤ θ ≤ 60° (as a convention we define θ = 0° at the poles). These four regions or bins are indicated in Figure 2 with white dashed lines and are labeled with the number of the bin. The properties of the tracers ejected until t_sim = 190 ms in each of the angular bins are shown in Figures 5 and 6, when passing a sphere with a radius of 750 km. We select the nucleosynthesis-relevant quantities: electron fraction Y_e, entropy per baryon s, and radial velocity v_r. The major part of the cumulative mass is approximately equally distributed to the two angular bins close to the accretion disk. On the contrary, the two bins in the polar region contain 15%−20% of the total cumulative mass. While the entropy of the unbound tracers ranges from 10 k_B/baryon to 30 k_B/baryon (Figure 5), it is still very low and has therefore little impact on the nucleosynthesis. The radial velocity provides a measure for the dynamical timescale and is constrained to 0.04c–0.08c (Figure 6). In the neutrino-driven wind the nucleosynthesis is most sensitive to the electron fraction distribution, which varies for every angular region. As a general trend, the average electron fraction decreases as a function of the angle and reaches values down to 0.3 for the two zones closest to the disk. Both of these bins also contain extreme cases with very neutron-rich conditions of Y_e < 0.2.

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Figure 7 shows the resulting integrated abundances for tracers ejected at t_sim = 140 ms. All yields are presented in comparison to the solar abundances (dots, Lodders 2003) for one angular bin in every panel. The thin gray lines represent individual tracers, while averages are indicated by a solid thick line in every panel. The patterns in Figure 7 reveal significant differences in abundances for distinct latitudes. We find that the first r-process peak (A ∼ 80) forms for each angular region, whereas the second abundance peak (A ∼ 130) is only reached in bin 3 and bin 4. In particular, the angular zone closest to the disk, i.e., bin 4, successfully attains elemental abundances close to the solar system values up to the second abundance peak. When moving to lower polar angles (bin 1), less heavy elements are synthesized.

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To gain further insights into the dependence on time, it is instructive to compare the trends of the angular bins at various times. Total masses as a function of mass number A (Equation (1)) for unbound tracers at t_sim = 90, 140, and 190 ms are presented in Figure 8. At early times t_sim ≲ 90 ms, elements up to A ∼ 120 are produced predominantly by the ejecta in bin 3, while only bin 4 synthesizes nuclei within the vicinity of the third abundance peak (A ∼ 195). The yields resulting from both bin 3 and bin 4 clearly dominate the abundances, as is expected from the distribution of the mass and the electron fraction for the tracers among the four bins (cf. Figures 5 and 6). In contrast, bin 1 and bin 2 give marginal contributions. With time, increasing yields for lighter heavy nuclei (A ≲ 120; cf. Figure 4) are present in all considered angular regions.

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Let us take a closer look at the relevant physical quantities characterizing the nucleosynthesis. Figure 9 shows the evolution of individual electron fractions for tracers unbound at 140 ms (gray, thin lines), along with the corresponding average curves (colored, thick lines). Note that the elapsed time of a tracer is defined with respect to the moment when it gets out of NSE in the network calculations. The individual tracers provide information about the spread in the initial Y_e (i.e., at T = 10 GK), also visible in Figures 5 and 6. The slight increase of the electron fraction shortly after the beginning of the calculations is due to neutrino absorption on nucleons, whereas the steep rise at around t ≃ 10⁻¹ s is caused by β-decays.

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The electron fraction in the neutrino-driven wind depends on the competition between the expansion and the weak equilibrium timescales, on the ratio between the ν_e and ${\bar{\nu }}_{{\rm{e}}}$ luminosities, as well as on the neutrino mean energies (e.g., Qian & Woosley 1996). As we mentioned in Section 2.1, the larger luminosities predicted by GR simulations can lead to a faster evolution toward the equilibrium Y_e. Therefore, the very small amount of heavy elements (A > 130) that are produced in the initial phase of our Newtonian simulation may not be present in a GR simulation. In addition to the GR effects, an accurate calculation of Y_e would require the usage of a detailed neutrino transport scheme, which is presently computationally prohibitive in long-time three-dimensional simulations, without global symmetries. Assuming uncertainties of the order of 20% for the neutrino luminosities and of 10% for the mean energies,⁶ we estimate a potential uncertainty of ∼15% on the values of the equilibrium electron fraction. Given the broad range of Y_e obtained in the ejecta (Figures 5, 6, and 9), we consider this uncertainty as being important to be mentioned, but not crucial for our analysis.

In order to examine the production of heavy elements, we present the evolution of neutron density n_n, the average mass number $\langle A\rangle$ and the average proton number $\langle Z\rangle$ in Figure 10 for different angular regions. At the beginning, the neutron density is larger than 10³⁰ cm⁻³ and the composition is dominated by alpha particles and neutrons. The onset of the r-process nucleosynthesis is triggered when the temperature decreases to ∼3 GK (marked by the triangles in the top panel of Figure 10; see, e.g., the dependence of r-process efficiency as a function of entropy, electron fraction, and expansion timescale; Hoffman et al. 1997; Freiburghaus et al. 1999a). For t ≃ (1.5–3.0) × 10⁻¹ s, most of the neutrons are consumed as indicated by the rapid neutron density drop. After this, matter beta decays to stability and this leads to an increase of Y_e (see Figure 9). Then, the mass number stagnates, since no more heavy elements are formed (see middle panel of Figure 10). No fission takes place, resulting in the monotonic evolution of mass and proton number. The higher neutron density in bin 4 favors the build-up of heavier nuclei compared to the other regions. With decreasing polar angle the neutrons are consumed earlier, as their initial density is smaller and the expansion velocity of the corresponding tracers is larger.

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The neutrino-driven wind is only one of three nucleosynthesis-relevant ejecta in neutron star mergers. In order to have the complete nucleosynthesis picture, one would need to follow the evolution of the dynamic and disk (neutrino- and viscous-driven) ejecta, and investigate how they mix. However, this is currently too complicated to be studied in a single simulation due to the different timescales and physics involved. The dynamic ejecta expand very fast from the beginning. While neutrinos may be initially important for the Y_e, their impact is insignificant when the disk becomes transparent (Fernández & Metzger 2013; Metzger & Fernández 2014; Just et al. 2015a). Nevertheless, mixing of the different types of ejecta can occur once the unbound matter becomes considerably decelerated by the ambient medium. Where this happens depends on the merger location with respect to the galaxy. Simple estimates indicate that the timescale for being noticeably decelerated is by far greater than years (see, for example, the discussion in Section 2.1 of Rosswog et al. 2014).

In the first study of the mixing between the disk and dynamic ejecta, Fernández et al. (2015) showed that there is no significant mixing between these ejecta. This study focused on neutron star black hole mergers, where neutrino-driven ejecta are less important, as discussed also by Fernández & Metzger (2013) compared to Metzger & Fernández (2014). However, their conclusion may also apply to our results that are based on a long-lived MNS. From the nucleosynthesis perspective, Just et al. (2015a) also investigated the mixing of disk and dynamic ejecta. They modeled the long-time evolution of a black hole surrounded by a disk where matter is neutrino- and viscous-driven ejected. In a post-processing step, the nucleosynthesis of such simulations was added to the dynamic ejecta from a merger simulation (Bauswein et al. 2013).

We combine the nucleosynthesis of the neutrino-driven wind presented in Section 3.1 with the dynamic ejecta computed by Korobkin et al. (2012). Although the ejecta masses show a substantial spread depending on the parameters of the merging binary system (Rosswog 2013), the resulting abundance patterns have been found to be practically identical (Korobkin et al. 2012). In contrast to previous studies (i.e., Fernández & Metzger 2013; Just et al. 2015a) with a black hole as the central object, we assume a long-lived MNS. This has strong consequences on the amount and properties (i.e., Y_e) of the matter ejected by neutrinos as discussed by Perego et al. (2014), Metzger & Fernández (2014), and Kasen et al. (2015).

The comparison between dynamic and wind ejecta is given in Figure 11 showing the total ejected masses (Equation (1)) as a function of A for both contributions. Note that the approach here differs from the one in the previous sections, in which we treated the different times t_sim to be merely snapshots of the neutrino-driven wind. Now, we assume that the MNS collapses after t_sim, terminating the ejection of further material in the wind. Three different simulations times are considered, t_sim = 90, 140, and 190 ms. This comparison indicates that the wind ejecta complement the dynamic ejecta by producing elements below the second peak. We note that if the dynamic ejecta could produce the full r-process, as suggested by recent GR simulations (Wanajo et al. 2014; Sekiguchi et al. 2015) and by parametric studies (Goriely et al. 2015), the combination of the dynamic and wind nucleosynthesis would again allow the production of lighter heavy elements (A < 130) in addition to the heavy r-process. In the latter case, the wind ejecta would increase only the relative contribution of the lighter heavy elements. Moreover, the amount of wind ejecta becomes comparable to the dynamic ejecta if the MNS survives long enough (t_sim ≳ 140 ms in our model). These ejecta amounts will be further enhanced by viscous ejecta that can produce r-process elements from the first to the third peak (Just et al. 2015a). If the three ejecta completely mix, the wind contribution may still lead to variations in the abundances below A = 130 (i.e., Z < 50). Such variations are also expected from observations of the oldest stars (see Sneden et al. 2008 for a review). The size of the observed variations could help to constrain different contributions of the three ejecta types to the r-process abundances.

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The oldest observed stars were formed in an interstellar medium polluted by only one or few nucleosynthesis events. Therefore, their atmospheres present a unique fingerprint for the r-process. Observations indicate that there are at least two types of abundance patterns among the oldest r-process stars (see, e.g., Qian & Wasserburg 2007; Sneden et al. 2008): (1) stars with high enhancement of heavy elements (Z > 50) present a robust pattern for those and some variations for the lighter heavy elements (Z < 50); and (2) stars with low enhancement of heavy r-process. Two stars are typically identified with these trends: (1) CS 22829-052 (Sneden et al. 2003) and (2) HD 122563 (Wallerstein et al. 1963; Honda et al. 2004, 2006). Recently, Hansen et al. (2014) have shown that this second kind of pattern can be explained by the superposition of two components: an H-component producing the heavy r-process elements (and maybe also lighter ones) and an L-component contributing only below Z = 50. In neutron star mergers, the dynamic and viscous ejecta can account for the H-component while the wind ejecta would be the L-component. Matter with a perfect mixing of the three ejecta will lead to a pattern like in CS 22829-052. This is shown in the upper panel of Figure 12, where the combination of wind (L-component) and dynamic (H-component) ejecta reproduce the observed abundances. The differences around the third peak are due to nuclear physics input as shown in, e.g., Eichler et al. (2015). If the mixing is not perfect and the wind ejecta combines only with a small amount of dynamic and/or viscous ejecta, then one can explain the trend observed in HD 122563 (see also Just et al. 2015a). This is shown in the bottom panel of Figure 12 where the dynamic ejecta has been reduced by a factor of 50. In this case, one would expect to have more variability in the observations depending on the amount of the H-component that is mixed. Indeed observations show variability for Z > 50 in this kind of star (Roederer et al. 2010; Hansen et al. 2014).

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On the timescale of star formation, it is likely that all the ejecta have efficiently mixed, even if the wind expands perpendicular to the disk and the other two are more homogeneously distributed or stay in the disk plane. Do we expect stars to be polluted by a neutron star merger preferentially only receiving axial ejecta rather than disk ejecta? Or do Honda-like abundances (L-component) come from different sources (e.g., core-collapse supernovae with slightly neutron-rich conditions)? More investigation of the long-term mixing is necessary to understand the contribution of neutron star mergers to the different patterns found in the oldest observed stars. However, the dynamic and wind ejecta will not mix before they are substantially decelerated by the dilute ambient medium. For example, assuming a particle density of ∼1 cm⁻³ for the ambient medium, the estimated deceleration timescale is of the order of several years. The dynamic ejecta expand earlier and with high initial velocities of ∼0.1c. The neutrino-driven wind, in contrast, gets unbound after neutrinos have deposited enough energy. Therefore, the wind matter is ejected later and eventually reaches asymptotic velocities of up to ∼0.08c. The different ejection times and velocities make it possible to observe a kilonova signal from different parts of the ejecta with different composition, as we discuss in the next section.

The nucleosynthesis network allows us to compute radioactive heating rates for the wind outflow. Figure 13 shows the heating rates for different bins, normalized to ${\dot{\epsilon }}_{0}(t)={10}^{10}{t}_{{\rm{d}}}^{-1.3}\;{\rm{erg}}\;{{\rm{g}}}^{-1}\;{{\rm{s}}}^{-1}$ with time t_d in units of days (Metzger et al. 2010; Korobkin et al. 2012). It is interesting to point out that all the normalized heating rates show considerable excess at t ∼ 4 hr. This excess is caused by the decay of radioactive nuclei in the vicinity of the first r-process abundance peak (A ∼ 80), as those possess the highest mass fractions. The properties of the isotopes that contribute most to the heating rates in the neutrino-driven wind are listed in Table 1. On average, about 40% of the decay energy is carried away only by neutrinos. An additional fraction of energy is radiated away by escaping photons, while both the electrons and the remaining part of the photons thermalize. Therefore our hypothesis of an effective beta-decay thermalization fraction of 50% is reasonable.

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Table 1.  Properties of the Dominant β-decay Nuclei at t ∼ 1 Day; Based on the Data from NuDat 2.6 Database (http://www.nndc.bnl.gov/nudat2/)

Isotope	t1/2	Qa	${\epsilon }_{e}$ b	${\epsilon }_{\nu }$ b	${\epsilon }_{\gamma }$ b	${E}_{\gamma }^{\mathrm{avg}}$ c
	(hour)	(MeV)				(MeV)
88Kr	2.83	2.92	0.12	0.21	0.67	1.34
88Rb	0.30	5.31	0.39	0.49	0.13	1.59
87Kr	1.27	3.89	0.34	0.46	0.20	0.95
83Br	2.37	0.98	0.33	0.66	0.007	0.46
81Sr	0.37	3.93	0.28	0.37	0.35	0.42
78Ge	1.47	0.96	0.24	0.47	0.29	0.28
78As	1.51	4.21	0.30	0.39	0.31	0.94
77Ge	11.2	2.70	0.23	0.36	0.41	0.47

Note.

^aTotal energy released in the decay. ^bFraction of the decay energy released in electrons, neutrinos, and γ-rays. ^cAverage photon energy produced in the decay.

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Here we extend the semi-analytic model of Grossman et al. (2014) and Perego et al. (2014) to compute light curves. We assume that the wind shuts off immediately after the collapse of the MNS and we explore collapse times of 90, 140, and 190 ms after merger. The masses as well as average properties of the ejecta in the four bins for each of the cases are given in Table 2. The higher value of κ = 10 cm² g⁻¹ for the opacity in the last angular bin is justified by inspecting the distribution of mass fractions for heavy elements X_heavy (for mass numbers A > 130) in the neutrino-driven wind. This distribution is portrayed for extrapolated positions of the unbound tracers in Figure 14 at t = 50 s after the merger. Mass fractions of heavy elements are marginal in bins 1–3, but heap up in the area of bin 4. Reflecting that sizable fractions of heavy elements are present, we adopt an opacity similar to the one of lanthanides or actinides (Kasen et al. 2013) for the last bin.

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Figure 15 illustrates our semi-analytic model for light curve calculations, combining the results of this work and of Rosswog et al. (2014). The neutrino-driven wind is schematically shown in blue, while the gray-shaded areas tag the density of the dynamic ejecta in steps of 0.5 dex. We subdivide the wind outflow into the same four bins as in Section 2.1 above. Each bin is then approximated by a conical slice of a spherically symmetric outflow with radial density distribution averaged over the bin and expanding with averaged velocity (a similar approach is employed in Margalit & Piran 2015, in the context of radio flares). All radiation is assumed to escape from the photosphere for each bin. Vector ${\boldsymbol{n}}$ is the unit normal to the photosphere, and an observer is pointed to by a unit vector ${\boldsymbol{q}}.$ The sketch also illustrates possible obscuration effects from the dynamic, very opaque ejecta, which are concentrated in a puffed-up toroid around the equatorial plane. It is apparent that when viewed from the equatorial plane, the wind emission is completely obscured, while from the pole it is possible to observe at least the first three bins.

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For a bin k with mass Δ m_k which spans the solid angle 2ΔΩ_k (a factor of two takes into account upper and lower lobes), we take a spherically symmetric outflow with mass 4πΔm_k/2ΔΩ_k and compute its isotropic bolometric luminosity L_k,iso as described in Grossman et al. (2014; see their Section 4.1).

The luminosity is generated by radioactive heating in the bulk of the outflow and the resulting photons escape from a photosphere. Spanning the same solid angle 2ΔΩ_k for each bin, the luminosity for bin k is a proportional fraction of isotropic luminosity of the spherical model:

$$\begin{eqnarray*}&&{L}_{k}=\displaystyle \frac{{\rm{\Delta }}{{\rm{\Omega }}}_{k}}{2\pi }{L}_{k,{\rm{iso}}}.\end{eqnarray*}$$

We then compute the total luminosity of the wind outflow by summing up individual bin contributions, ignoring possible radiative flux between the bins. Individual bin contributions for the combined bolometric luminosity of the neutrino-driven wind and dynamic ejecta for three cases of the MNS collapse time delay are displayed in Figure 16. For the dynamic ejecta, we used an average case (model A from Grossman et al. 2014) with mass 1.3 × 10⁻² M_⊙. Notice that the luminosity of bin 1 is dimmer than for bins 2 and 3 because it spans a smaller solid angle. The contribution from bin 4 is not only smaller, but also peaks much later (3−4 days) due to the high opacity caused by the very heavy nuclear content (Kasen et al. 2013), such that the medium becomes delayed transparent when the temperature has decreased.

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Combined bolometric luminosities of the neutrino-driven wind and dynamic ejecta are shown in Figure 17. A first peak of L_peak ∼ 4 × 10⁴⁰ erg s⁻¹ is reached at about 4 hr after the merger. This luminosity stays roughly constant for several days while rapidly shifting into the infrared band, as shown below. For higher MNS collapse times, the light curve exhibits a double peak structure, but overall the bolometric light curves on the plots exhibit no appreciable difference between different MNS collapse times.

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Our simple model also allows us to calculate approximate light curves in different bands for different orientations of the system with respect to an observer. At any given moment t, we can compute an effective temperature T_k,eff for a bin k using the formula

$$\begin{eqnarray}&&{L}_{k,{\rm{iso}}}(t)=4\pi {r}_{k,{\rm{ph}}}^{2}\cdot ({\sigma }_{{\rm{SB}}}{T}_{k,{\rm{eff}}}^{4})\end{eqnarray} \tag{ 3 }$$

where r_k,ph(t) is the photospheric radius for the spherical model of bin k, and σ_SB is the Stefan–Boltzmann constant. We make an assumption that all the flux is emitted from the photosphere with blackbody spectrum ${B}_{\nu }({T}_{k,{\rm{eff}}})$ and isotropic intensity (following Lambert's law). Then, the spectral flux in the direction ${\boldsymbol{q}}$ is given by

$$\begin{eqnarray}&&{{\mathcal{F}}}_{\nu }({\boldsymbol{q}},t)=\displaystyle \sum _{k}{B}_{\nu }({T}_{k,{\rm{eff}}}(t)){\iint }_{({{\boldsymbol{n}}}_{k}\cdot {\boldsymbol{q}})\gt 0}({\boldsymbol{q}}\cdot d{\boldsymbol{\Omega }}),\end{eqnarray} \tag{ 4 }$$

where ${{\boldsymbol{n}}}_{k}$ is unit normal to the photosphere of bin k, and integration for each bin spans the part of the surface facing the observer (stated by the condition $({{\boldsymbol{n}}}_{k}\cdot {\boldsymbol{q}})\gt 0$). This is also illustrated in Figure 15. Essentially, the integrals in this formula are simply time-independent geometric projections of the photosphere for each bin onto the viewing plane. They can be computed beforehand and used as weighting factors ${p}_{k}({\boldsymbol{q}})$ for calculating the combined light curve:

$$\begin{eqnarray}&&{{\mathcal{F}}}_{\nu }({\boldsymbol{q}},t)=\displaystyle \sum _{k}{p}_{k}({\boldsymbol{q}}){B}_{\nu }({T}_{k,{\rm{eff}}}(t)).\end{eqnarray} \tag{ 5 }$$

After integrating ${{\mathcal{F}}}_{\nu }({\boldsymbol{q}},t)$ over certain frequency ranges, we obtain broadband light curves, shown in Figure 18. The left-hand panel shows synthetic light curves for wind outflow only, while the right-hand panel displays the combined contribution from both wind and dynamic ejecta.

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Orientation effects are displayed as a range of values for each band, with the maximum magnitude achieved when the system is viewed "face-on," i.e., from the pole. As pointed out by Kasen et al. (2015), even a small amount of dynamic ejecta can completely obscure the optical emission from the winds. Therefore, when adding up contributions from different bins and dynamic ejecta, we emulate this obscuration by excluding certain bins, depending on the inclination angle θ with respect to the observer. Our choice of which bins to exclude is motivated by the geometry of the ejecta (depicted schematically in Figure 15). Specifically, we assume that bin 4 is completely obscured for all observing angles; bins 2 and 3 are obscured when θ ≥ 60°, and for the edge-on view with θ = 90° the wind outflow emission is obscured completely. For this reason, depending on system orientation, the luminosity in different bands can vary by up to an order of magnitude, from bright blue when observed from the pole, to dim infrared when observed from the side.