ELECTROMAGNETIC EMISSION FROM LONG-LIVED BINARY NEUTRON STAR MERGER REMNANTS. II. LIGHT CURVES AND SPECTRA

According to our model presented in Paper I, predictions for the EM emission from a BNS merger remnant result from a self-consistent numerical evolution of a set of coupled differential equations based on several input parameters, which we list in Table 1. In this paper, we explore physically plausible ranges for these parameters (see "Range" in Table 1) and employ a set of specific values labelled as "fiducial" in Table 1 as a reference for comparison.

The majority of influential model input parameters can be determined, estimated, or constrained using numerical relativity simulations of the BNS merger and early post-merger phase. In particular, ${\dot{M}}_{\mathrm{in}}$, v_ej,in, $\bar{B}$, E_rot,NS,in, P_c, R_e, M_NS,in, and T_ej,NS,in can be directly read off from a simulation (e.g., Dessart et al. 2009; Siegel et al. 2014; Siegel & Ciolfi 2015a). Furthermore, the neutrino-cooling timescale t_ν can be estimated from the thermal energy content of the NS, given the total neutrino luminosity (e.g., Dessart et al. 2009). The timescale for removal of differential rotation, t_dr, can be estimated by the Alvén timescale, which is given in terms of the magnetic field strength $\bar{B}$, the radius R_e, and the mass M_NS,in of the NS (Shapiro 2000). In summary, the initial conditions for our model can essentially be set by an appropriate final snapshot of a BNS merger simulation.

There are further parameters, such as ${\eta }_{{B}_{{\rm{p}}}}$, I_pul, ${\eta }_{{B}_{{\rm{n}}}}$, η_TS, γ_max, and Γ_e, which cannot be constrained from simulations of the merger and early post-merger process, as they refer to properties of the pulsar and the PWN in Phase II and III of the evolution. Fortunately, ${\eta }_{{B}_{{\rm{p}}}}$ can largely be absorbed into the parameter $\bar{B}$ (cf. Section 5.1), I_pul can essentially be absorbed in E_rot,NS,in, varying η_TS results in a trivial change (just renormalizing the light curves), and the numerical results are not particularly sensitive to the remaining parameters.

Some of the parameters listed in Table 1 are not varied and, instead, set to some representative value. These are parameters that are either well constrained and do not influence the numerical evolution significantly, only change the results in a trivial way, and/or they can essentially be absorbed into other parameters. One of these parameter values merits further discussion.

The value for η_TS corresponds at an order-of-magnitude level to the efficiency of the Crab Nebula in converting pulsar wind power into particle motion, which is currently estimated to be ≳10% (Kennel & Coroniti 1984; Bühler & Blandford 2014; Olmi et al. 2015). For PWNe formed in BNS mergers according to our model, this efficiency factor could be different. However, different values for η_TS result in different offsets for the light curves, i.e., in a renormalization of the numerical results. Therefore, as long as our model is not used to fit observational data, this parameter can be kept fixed.

Our model characterizes the ejecta material by a mean opacity κ, which we assume to be constant over time, $\kappa ={\kappa }_{\mathrm{es}}\approx 0.2\;{\mathrm{cm}}^{2}\;{{\rm{g}}}^{-1}$. This value corresponds to Thomson electron scattering, which is the dominant source of opacity at optical and UV wavelengths in the case of iron-rich material (resulting from a high electron fraction). The other important source for opacity of iron-rich material at optical/UV wavelengths, bound–bound absorption, is of the same order of magnitude or less (∼0.1 cm² g⁻¹; Pinto & Eastman 2000; Kasen et al. 2013). For neutron-rich material (low electron fraction), instead, κ can become as high as 10 cm² g⁻¹ due to bound–bound opacities generated by transitions of the valence electrons in lanthanide elements (Kasen et al. 2013). However, we assume here that even if the ejecta material was neutron-rich, irradiation from the PWN is strong enough to ensure a very high degree of ionization of the lanthanide elements, such that κ ∼ κ_es. At X-ray wavelengths, which is what we are most interested in, an important contribution to the opacity can come from bound-free absorption, depending on the ionization state of the material. Here we assume again that the PWN is sufficiently luminous to ensure a very high degree of ionization of the ejecta material on the timescales of interest, such that κ ∼ κ_es. A more detailed computation of the opacity at X-ray wavelengths based on partial ionization of the ejecta material, e.g., along the lines of Metzger et al. (2014) and Metzger & Piro (2014), can, in principle, be included into our model as well. Nevertheless, we take κ as an input parameter of our model (cf. Table 1) and explore the effect of higher values of κ in Section 5.1.

For the time being, we adopt two further simplifying assumptions. First, we neglect effects of thermal Comptonization in the PWN in Phase III, i.e., we set ${\dot{n}}_{{\rm{C}}}^{{\rm{T}}}\equiv 0$ in the photon balance equation (cf. Section 4.3.1 of Paper I). This reduces the procedure of solving the coupled set of integro-differential equations for the photon and particle spectra (Equations (78) and (79) of Paper I) of the PWN to the scheme outlined in Section 5.2 of Paper I, which severely lowers the computational cost for the overall numerical evolution of our model.

Second, for the time being, we neglect effects of further acceleration in Phase III after t = t_shock,out, i.e., we set a_ej ≡ 0 in Equation (11) of Paper I. This is because we find that for typical input parameter values, the assumption of quasi-stationarity adopted for the description of the radiative processes in the PWN (see Sections 4.3.1 and 5.4 of Paper I) would otherwise not be well satisfied and we expect errors of the order of unity in this case (see also Section 4.2). We therefore postpone a discussion of results for non-zero acceleration of the ejecta shell until a time-dependent framework for the radiative physics of the PWN has been implemented in future work. Such a time-dependent formalism is also required to include a non-zero albedo of the ejecta material, which is why we neglect it here throughout the entire evolution.

Figure 1 shows the luminosity of escaping radiation from the system throughout the entire evolution. In the following, we discuss the evolution of the system focusing on the top panel, which reports the thermal luminosity L_rad (cf. Equations (34), (74), and (100) of Paper I) and non-thermal luminosity L_rad,nth (cf. Equation (104) of Paper I) as computed in the lab frame (rest frame of the NS; cf. Appendix B of Paper I). The evolution of our model can be initialized by data from a numerical relativity simulation (see Section 3.1). We associate the time of merger of the BNS with t = 0 and start the numerical evolution of our model a few to tens of milliseconds after merger, typically at t = t_min = 10⁻² s, once a roughly axisymmetric state of the NS has been reached.

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During Phase I, the NS ejects a significant amount of mass through a baryon-loaded wind (see Section 4.1.1 of Paper I) that carries thermal and EM energy (see Section 4.1.2 of Paper I). Density profiles of this wind for selected times during Phase I are reported in Figure 2. Radiative energy loss from this wind is, however, comparatively inefficient due to the very high optical depth of the ejecta material at these early times (see Figure 3). The total luminosity remains at a level of ≲10³⁹ erg s⁻¹ and rises appreciably only toward the end of Phase I when further mass ejection from the NS is heavily suppressed (see Equation (17) of Paper I) and the bulk of the material has already moved far away from the NS, such that the average density of the baryon-loaded wind and thus the optical depth strongly decreases. This steep decrease of the average density toward the end of Phase I is evident from Figure 2.

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At t = t_pul,in, once the density in the vicinity of the NS has become sufficiently small, a pulsar magnetosphere is set up and Phase II begins (cf. Section 4.2.1 of Paper I). The resulting pulsar wind inflates a PWN behind the less rapidly expanding ejecta. This PWN is highly overpressured with respect to the ejecta material and thus drives a strong shock through it, which sweeps up all the material into a layer of shock-heated ejecta material. Initially, the shock front (denoted by R_sh in Figure 4) moves essentially with the speed of light and decelerates to non-relativistic speeds when encountering higher density material shortly before reaching the outer ejecta radius R_ej at t = t_shock,out (which terminates Phase II). At any time, this shock front separates the ejecta into shocked (${R}_{{\rm{n}}}\lt r\lt {R}_{\mathrm{sh}}$) and unshocked (${R}_{\mathrm{sh}}\lt r\lt {R}_{\mathrm{ej}}$) material, where R_n denotes the outer radius of the PWN. Phase II can be much shorter than Phase I depending on parameters. Its duration is indicated by the black dashed lines in Figures 1, 3, and 4 (see the inset figures, in particular). During this phase the lab-frame luminosity is ever increasing due to a rapidly decreasing optical depth, which, in turn, is caused by the fact that the thickness of the unshocked ejecta layer decreases as the shock front is moving across the baryon-loaded wind.

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When R_sh = R_ej (cf. Figure 4), Phase III begins. A higher expansion speed than in Phase I and II (cf. Section 4.3.2 of Paper I) induces a rapid decrease of the optical depth (cf. Figure 3), which results in a steep rise of the total luminosity. The luminosity reaches its maximum at ∼10⁴⁸ erg s⁻¹ when a significant fraction of the energy reservoir of the ejecta material has been consumed. This maximum brightening of the system occurs at ∼10³ s, i.e., on the timescale of hours after the BNS merger. We note that this brightening is a very robust feature of our model and it occurs on roughly similar timescales for almost any other set of parameters (see the following sections). If the ejecta shell is further accelerated after t = t_shock,out (not considered here for reasons discussed in Section 3.2), this maximum brightening is expected to be shifted to slightly earlier times. Preliminary results indicate that in determining this timescale, acceleration is likely to dominate other effects such as employing different opacities.

After having radiated away most of its energy, the ejecta shell enters an asymptotic regime, in which the internal energy of the ejecta layer is essentially determined by the flux of inflowing energy from the PWN. The energy deposited and thermalized in the ejecta shell per time step is again radiated away during the same time step, thanks to the ever decreasing photon diffusion timescale of the ejecta shell. Hence, there is no further energy acquisition anymore. In this regime, the overall luminosity is determined by the luminosity of escaping photons from the PWN, which is again set by the spin-down luminosity L_sd of the pulsar as the PWN conserves energy (see Appendix D of Paper I and the discussion in Section 5.6 of Paper I). Therefore, the total luminosity follows the spin-down luminosity (green line in the top panel of Figure 1; see also Section 4.2.1 of Paper I) at a constant ratio set by the efficiency factor η_TS (cf. Table 1). Hence, it also shows the characteristic ∝t⁻² scaling at late times $t\gg {t}_{\mathrm{sd}}$, where t_sd is the spin-down timescale (cf. Section 4.2.1 of Paper I; indicated by the green dashed line in Figures 1 and 3).

In this asymptotic regime, the evolution equation for E_th (Equation (14) of Paper I) becomes stiff and an alternative scheme is used to further evolve the set of coupled differential equations of our model (see the discussion in Section 5.6 of Paper I). The transition time at which we switch over to this alternative formulation is called t_res, and it is indicated by a black dotted line in Figures 1 and 3.

As the optical depth approaches unity the ejecta shell becomes transparent to the radiation from the PWN. From Equations (99) and (103) of Paper I, one obtains the scaling ${\rm{\Delta }}{\tau }_{\mathrm{ej}}\propto {R}_{\mathrm{ej}}^{-2}\propto {t}^{-2}$, where the second proportionality assumes a constant expansion speed. This scaling only holds once the shell thickness Δ_ej has become much smaller than R_ej, which is not yet the case at t = t_shock,out for the fiducial run as illustrated by Figure 4. From Figure 3 it is evident, however, that this scaling is reached at times $t\gtrsim {10}^{2}\;{\rm{s}}$. Once Δτ_ej has approached unity, radiation from the PWN is not absorbed anymore by the ejecta material, such that the thermal emission from the ejecta surface rapidly decreases and, instead, non-thermal radiation from the PWN is emitted toward the observer (cf. Figure 1). This transition from predominantly thermal to non-thermal spectra is discussed in more detail in Section 4.3. The time of transition between the optically thick and thin regime is indicated by a red dashed line in Figures 1 and 3. We refer to Section 4.3.3 of Paper I for a discussion of how this transition is implemented in the model evolution equations.

As the PWN conserves energy (see Appendix D of Paper I), the total non-thermal luminosity L_rad,nth also scales as L_sd once the ejecta shell is optically thin. We note that in the fiducial setup, the PWN becomes optically thin prior to the ejecta matter (see Figure 3). Hence, radiation from the deep interior of the nebula is already being emitted toward the observer once the PWN radiation is free to escape from the system.

The middle panel of Figure 1 shows the total thermal and non-thermal luminosity (L_obs,th and L_obs,nth) as seen by a distant observer in comparison with the total lab-frame luminosity L_rad. These observer light curves are reconstructed using the method presented in Section 5.7 of Paper I and take into account the combined effects of relativistic beaming, the relativistic Doppler effect, and the time-of-flight effect for radiation emerging from the surface and/or the interior of a relativistically expanding sphere. As the expansion speed of the ejecta material (v_ej,in = 0.01c in Phase I and II, v_ej = 0.064c in Phase III; cf. Section 4.3.2 of Paper I) remains non-relativistic for the fiducial setup (as we are neglecting further acceleration for the time being, see Section 3.2), these relativistic effects induce only small corrections to the observer light curve, which closely follows the lab-frame light curve.

The delayed onset of the observer luminosity with respect to L_rad at very early times is due to the time-of-flight effect: radiation from high latitudes (small θ) of the expanding sphere reaches the observer earlier than radiation from lower latitudes (high θ, where θ = 0 defines the direction of the observer; see Section 5.7 of Paper I for more details). The offset between the time grids of the lab frame and the remote observer is calibrated in such a way that a photon emitted from the spherical surface of the expanding ejecta at θ = 0, r = R_min, with R_min being the inner spatial boundary of our model, at a time t = t_min after the BNS merger is received by the observer at a time t' = t_min (see Section 5.7 of Paper I). This time-of-flight effect is also evident during Phase II, where the peak observer luminosity appears to precede the end of Phase II by R_ej(t_shock,out)/c ≈ 0.16 s (compare the inset figures in the two upper panels of Figure 1). This is because the main contribution to the observer luminosity originates from high latitudes, while the delayed radiation from low latitudes only forms a tail that broadens the peak. Furthermore, we note that the thermal observer luminosity is slightly higher than the lab-frame luminosity in Phase III, which is due to the relativistic Doppler effect. Finally, the time-of-flight effect can be noticed again when the ejecta shell becomes optically thin. The onset of the non-thermal radiation as seen by the observer precedes the corresponding onset in the lab frame (red-dashed line in Figure 1) by R_n/c. Moreover, the observer light curve starts to deviate from the lab-frame light curve at very late times (barely visible in Figure 1). This is because the PWN is optically thin and radiation from the entire volume of the nebula can reach the observer. As radiation is being delayed while the overall luminosity is decreasing, this results in a less steep slope for the observer light curve.

In summary, the relativistic corrections to the light curve are very small and barely visible for the fiducial parameter setup. However, for higher velocities and/or in presence of further acceleration during Phase III, these relativistic corrections can significantly reshape the observer light curve, which needs to be taken into account when comparing the predictions of our model to observational data.

The total energy radiated away form the system is distributed over many orders of magnitude in frequency. Depending on the energy band of interest, very different morphologies for the observer light curves are obtained (cf. the lower panel of Figure 1). A remarkable feature evident from Figure 1 is that the typical temperature of the ejecta shell during Phase I, II, and up to the time of maximum brightening in Phase III corresponds to an energy in the X-ray band, such that almost all of the radiated energy falls into the sensitivity regime of the XRT instrument. This is a very robust feature that we also find in case of almost all other parameter settings: the radiation from the system up to its maximal brightness is predominantly thermal and of X-ray nature. Soon after this, the UV and optical emission dominates, until, finally, most of the energy is radiated away in the radio band once the ejecta shell becomes transparent to the radiation from the nebula. This radio emission is generated by synchrotron cooling of the electron-positron pairs in the PWN (see also Section 4.3). Nevertheless, appreciable X-ray luminosities of the order of $\sim {10}^{42}\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ may still be present at times as late as t ∼ 10⁵ s. For the remainder of this paper, we focus specifically on X-ray luminosities and spectra, as they appear to be most relevant in the context of early afterglows of a BNS merger event.

Our approach for modeling the radiative processes in the PWN (cf. Section 4.3.1 of Paper I) is based upon two important assumptions: quasi-stationarity and negligible influence of synchrotron self-absorption. The validity of these assumption needs to be routinely verified during the numerical integration of the model evolution equations. As discussed in Sections 5.4 and 5.5 of Paper I, this can be achieved by monitoring several timescales and the optical depth to synchrotron self-absorption, which we shall discuss here.

Figure 5 reports the evolution of the light crossing time τ_l of the nebula, the timescales for change of particle and photon injection into the nebula (τ_e and τ_ph, respectively), and the total particle cooling timescale τ_c of the nebula throughout Phase III (for definitions, see Section 5.4 of Paper I). In general we find that

$$\begin{eqnarray}&&{\tau }_{{\rm{c}}}\ll {\tau }_{\mathrm{ph}},{\tau }_{{\rm{c}}}\ll {\tau }_{{\rm{e}}},\end{eqnarray} \tag{ 2 }$$

which shows that the nebula particle distribution can adjust fast enough to changes of the exterior conditions and thus justifies the assumption of stationarity concerning the particle distribution in the nebula (see Section 5.4 of Paper I). Only at late times t ≳ 10⁷ s these conditions are not satisfied anymore for part of the particle energy spectrum. However, we are not interested in the evolution of the system at such late times. This is a general result that qualitatively holds throughout the parameter space considered in Table 1: at times of interest, the stationarity assumption regarding the particle distribution is very well satisfied. Moreover, Figure 5 also shows that typically

$$\begin{eqnarray}&&{\tau }_{{\rm{l}}}\ll {\tau }_{\mathrm{ph}},\end{eqnarray} \tag{ 3 }$$

except for a short transition phase around $t\sim {10}^{5}\;{\rm{s}}$. This is the time when the ejecta shell becomes optically thin and the timescale for change of the photon spectrum is mostly determined by the auxiliary function f_ej to model this transition (cf. Section 4.3.3 of Paper I). In general and across the parameter space we find, however, that Equation (3) is well satisfied, which justifies the stationarity assumption concerning the photon distribution inside the PWN (see Section 5.4 of Paper I). This picture can change when further acceleration of the ejecta shell during Phase III is considered. We find that in this case τ_ph can become similar to τ_l and errors of the order of unity for the photon and particle spectra are expected. This is why we refrain from discussing results for this case in the present paper. To overcome this problem, a time-dependent framework for the radiative processes in the nebula has to be developed, which we postpone to future work.

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Figure 6 monitors the validity of the second central assumption: negligible influence of synchrotron self-absorption. It shows the normalized dimensionless frequency which separates the optically thick and thin regimes of the photon spectrum to synchrotron self-absorption (for a definition, see Section 5.5 of Paper I). As we have ${x}_{{\rm{\Delta }}{\tau }_{\mathrm{syn}}=1}\lt 4\times {10}^{-4}$ except for the first few time steps in Phase III, the spectrum in the XRT band (which is what we are mostly interested in) is unlikely to be affected by effects of synchrotron self-absorption. We find that this conclusion typically also holds across the entire parameter space considered in Table 1.

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After having discussed the light curves for the fiducial parameter setup in Section 4.1, this subsection presents results for the associated spectral evolution.

Figure 7 shows the intrinsic spectrum L_PWN(ν, t) of the radiation escaping from the nebula for selected times during Phase III (cf. Equation (91) of Paper I). These spectra result from the numerical solution to the photon and particle balance equations in the PWN (cf. Section 4.3.1 of Paper I) and account for the combined effects of synchrotron losses, (inverse) Compton scattering, pair production and annihilation, Thomson scattering off thermal particles, and photon escape.

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At low frequencies, the spectra are dominated by synchrotron radiation. The spectrum peaks at a frequency around ${\nu }_{{\rm{c}},{\gamma }_{\mathrm{min}}}$ (cf. the dot–dashed lines in the three panels), where

$$\begin{eqnarray}&&{\nu }_{{\rm{c}},\gamma }=\displaystyle \frac{3}{4\pi }\displaystyle \frac{{{eB}}_{{\rm{n}}}}{{m}_{{\rm{e}}}{c}^{2}}{\gamma }^{2}\end{eqnarray} \tag{ 4 }$$

is the critical frequency above which the synchrotron spectrum of a single radiating particle of Lorentz factor γ sharply decreases, and γ_min = 1 is the minimum Lorentz factor of the particle distribution (see also Appendix C of Paper I). The synchrotron spectrum extends up to ${\nu }_{{\rm{c}},{\gamma }_{\mathrm{max}}}$ (cf., e.g., the dashed vertical lines in the bottom panel) and shows power-law decline between ${\nu }_{{\rm{c}},{\gamma }_{\mathrm{min}}}$ and ${\nu }_{{\rm{c}},{\gamma }_{\mathrm{max}}}.$ In particular, this is evident at late times (bottom panel), when the spectrum is dominated by synchrotron radiation below ${\nu }_{{\rm{c}},{\gamma }_{\mathrm{max}}}.$ This behavior reflects the underlying particle distribution N(γ) (cf. Section 4.3.1 of Paper I), which is also of power-law nature at late times.

At UV and higher energies other processes can determine the spectral shape. One prominent feature is typically caused by the strong photon field with luminosity L_rad,in of thermal radiation from the inner surface of the confining ejecta envelope (cf. Equation (101) of Paper I). A corresponding thermal "bump" becomes visible around log t ≈ 2.7 (cf. the upper panel) and persists until the ejecta material becomes optically thin around log t ≈ 5. The maxima of the thermal injection spectra at energies corresponding to $\approx 2.8\;{k}_{{\rm{B}}}{T}_{\mathrm{eff}}$ are indicated by dotted lines in the middle panel, where ${T}_{\mathrm{eff}}^{4}={T}_{\mathrm{eff},\mathrm{com}}^{4}/\zeta {\gamma }_{\mathrm{ej}}$ defines the effective temperature of the ejecta material as measured in the lab frame (see Equation (101) of Paper I). It is the full non-linear interaction of these injected photons with the particle distribution via (inverse) Compton scattering and pair creation/annihilation that essentially determines the spectral shape from UV to gamma-ray energies up to x_max = γ_max.

The non-thermal radiation escaping from the PWN is reabsorbed and thermalized by the surrounding ejecta layer as long as the confining envelope is optically thick. The spectra as seen by a remote observer are thus characterized by thermal spectra at early times (cf. Figure 8, upper panel), a transition from thermal to non-thermal spectra when the ejecta material becomes optically thin (cf. Figure 8, middle panel), and non-thermal spectra at late times (cf. Figure 8, bottom panel). We refer to Section 5.7 of Paper I for a discussion of how these observer spectra are reconstructed. The thermal part of the spectrum is plotted as dotted curves in the lower two panels and shows how quickly this contribution fades away when the optical depth of the ejecta approaches unity. We note that the effective temperature of the ejecta material at early times falls into the XRT band, which typically also holds for all other runs across the entire parameter space (see also Section 4.1). This is a remarkable feature that makes XRT an ideal instrument to observe and analyze the radiation escaping from the system at ≲10⁴–10⁵ s after the BNS merger.

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According to our model, the observable radiation at early times is purely thermal. However, due to the high degree of ionization of the material, there are numerous free electrons upon which photons can inverse Compton-scatter to form a non-thermal high-energy tail of the spectrum. Furthermore, we have employed a single effective temperature to characterize the ejecta layer. The ejecta shell will, however, be characterized by strong temperature gradients, which will give rise to a superposition of individual Planck spectra of different temperatures. Hence, the actual observable spectrum at early times might be different from the simple Planck spectrum employed here and could even appear as a power law in the XRT band. The transition from this "thermal" spectrum at early times to an intrinsically non-thermal spectrum at later times, which carries the signatures of the PWN, is a characteristic prediction of our model that is common to essentially all parameter settings. The timescale for this transition depends on the optical thickness of the ejecta material, which depends on a number of parameters, such as the initial density of the ejecta shell (i.e., t_dr, ${\dot{M}}_{\mathrm{in}}$, σ_M), its opacity κ, and the expansion velocity v_ej. Depending on these parameters, it can occur significantly earlier than for the fiducial parameter set.

Non-collapsing model runs correspond to a scenario in which the merger results in an NS that is indefinitely stable against gravitational collapse. We restrict the discussion here to those parameters that influence the model results significantly, i.e., $\bar{B}$, ${\eta }_{{B}_{{\rm{p}}}}$, t_dr, ${\dot{M}}_{\mathrm{in}}$, E_rot,NS,in, and κ. We find that the predictions of our model concerning the dynamics of the system and the observer light curves are not significantly influenced by the remaining parameters.

Figure 9 compares the prediction for the X-ray light curve in the XRT band and the fiducial parameter setup (cf. Table 1) with runs employing different values for $\bar{B}$, ${\eta }_{{B}_{{\rm{p}}}}$, t_dr, and ${\dot{M}}_{\mathrm{in}}$. The top panel shows that a higher total initial magnetic field strength in the outer layers of the newly formed NS already leads to an appreciably higher X-ray luminosity in Phase I and II. This is because the baryon-loaded wind is endowed with a much higher Poynting flux for higher values of $\bar{B}$, which is dissipated in the ejecta material (${L}_{\mathrm{EM}}\propto {\bar{B}}^{2};$ cf. Equation (32) of Paper I). At the beginning of Phase III, higher values of $\bar{B}$ lead to a significantly steeper rise of the luminosity, because the dipolar magnetic field strength at the pole of the NS, B_p, is also higher (cf. Equation (41) of Paper I). As the spin-down luminosity scales as ${L}_{\mathrm{sd}}\propto {B}_{{\rm{p}}}^{2}$ (cf. Section 4.2.1 of Paper I) and the PWN conserves energy (cf. Appendix D of Paper I), more energy is deposited in the ejecta material at earlier times. Furthermore, the maximum value for the luminosity is essentially determined by L_sd,⁵ which is why the maxima of the runs spread over two orders of magnitude. The time of this maximal brightening of the system is roughly ∼10³ s and varies only marginally among the different runs. This timescale is essentially determined by the optical depth of the ejecta shell, which does not directly depend on $\bar{B}$ (see also below). After the maximum in the luminosity, the light curve decays steeper for higher values of $\bar{B}$, which is due to the fact that the spin-down timescale scales as t_sd ∝ B_p⁻² and, hence, energy injection into the ejecta material fades away more rapidly. Moreover, it is important to note that varying either $\bar{B}$ or ${\eta }_{{B}_{{\rm{p}}}}$ by the same factor yields roughly identical results in Phase III (compare the cyan and blue and the red and purple lines, respectively). This is because with all other parameters being identical, those runs also share the same value of B_p, which determines the spin-down luminosity and therefore the energy output in Phase III. In Phase I, however, the luminosity differs, as $\bar{B}$ directly enters the source term L_EM (see above). Consequently, as far as Phase III is concerned, the parameter ${\eta }_{{B}_{{\rm{p}}}}$ can be absorbed into $\bar{B}$.

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The parameter t_dr is highly influential in determining timescales, as shown by the middle panel of Figure 9. With all other parameters fixed, it does not only set the duration of Phase I, which is directly proportional to t_dr (the timescale for the density in the surrounding of the NS to decrease is t_dr/σ_ρ; see Sections 4.1.1 and 4.2.1 of Paper I). It also impacts the time of maximum brightness, which lies between ∼10²–10⁴ s for the cases considered here. This is because with higher values of t_dr, the total ejected mass ${M}_{\mathrm{ej}}\approx {\dot{M}}_{\mathrm{in}}{t}_{\mathrm{dr}}/{\sigma }_{M}$ (cf. Equation (31) of Paper I) increases and thus the ejecta shell needs more time to expand in order to reach a comparable average density and thus a comparable optical depth. At the same time, however, the material cools down, which leads to a lower peak brightness as t_dr increases.

Smaller initial mass loss rates ${\dot{M}}_{\mathrm{in}}$ induce moderately higher luminosities in Phase I, II, and up to the global maximum of the X-ray light curve in Phase III (see bottom panel of Figure 9), thereby also shifting the time of maximum brightness to earlier times. This is because lower values of ${\dot{M}}_{\mathrm{in}}$ result in a smaller amount of ejected mass M_ej(t) up to time t in Phase I (cf. Equation (31) of Paper I) and, hence, in a lower average density and optical depth of the ejecta material. As the total amount of ejected material ${M}_{\mathrm{ej}}\approx {\dot{M}}_{\mathrm{in}}{t}_{\mathrm{dr}}/{\sigma }_{M}$ is also reduced, the ejecta shell starts off with a lower average density at t = t_shock,out and is thus characterized by a lower optical depth throughout Phase III. The red line in the lower panel of Figure 9 corresponds to a case in which both t_dr and ${\dot{M}}_{\mathrm{in}}$ differ from the fiducial value, but combine to give the same total ejected mass as in the ${\dot{M}}_{\mathrm{in}}={10}^{-3}\;{M}_{\odot }$ case (blue curve). The evolution up to the global maximum is different in the two cases, showing that the two parameters cannot be reduced to a single one (e.g., to M_ej). This is because t_dr has a much stronger effect on shifting the onset of Phase II. However, on much longer timescales this shift becomes irrelevant and the two curves essentially agree.

Different initial rotational energies of the newly born NS after merger affect the X-ray light curve only by rescaling the global maximum (see the top panel of Figure 10). This scaling can roughly be explained by the fact that ${L}_{\mathrm{sd},\mathrm{in}}\propto {E}_{\mathrm{rot},\mathrm{NS},\mathrm{in}}^{2}$ (assuming that the rotational energy of the pulsar is approximately given by E_rot,NS,in; cf. Section 4.2.1 of Paper I). This is similar to the scaling found for $\bar{B}$ and ${\eta }_{{B}_{{\rm{p}}}}$ (see above).

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Figure 10 (bottom panel) shows the effect of a much higher (by a factor of 50) mean opacity κ than in the fiducial case (cf. the discussion in Section 3.1). This increases the optical depth, which thus results in a dimmer X-ray light curve and a delayed global maximum (as it takes more time to radiate away the acquired energy). The cooling timescale of the ejecta material after the global maximum in the luminosity, however, depends mostly on the energy input, which is similar in the two runs. Therefore, both light curves fall off in a very similar way after ∼10⁴ s as the temperature of the Planck spectrum moves out of the XRT band. Furthermore, we note that concerning the behavior in Phase III, higher opacities have a very similar effect as a higher initial mass-loss rate for the baryonic wind in Phase I, as shown by the blue curve. This shows that part of the uncertainty in the opacities can effectively be absorbed into, e.g., the initial mass-loss rate.

Collapsing models correspond to the most frequent case in which the NS is supramassive (or hypermassive) at birth (cf. Section 1 of Paper I) and it is thus doomed to collapse to a BH. Everything that has been noted in Section 4 and in the previous subsection still applies (at least up to the time of collapse), with the additional complication that the NS collapses to a BH at t = t_coll, which alters the subsequent evolution of the system. After t = t_coll, the evolution equations are numerically integrated as described in Section 4.4 of Paper I. We first concentrate on the more likely case of a gravitational collapse during Phase III (Section 5.2.1), and return to the possibility of a collapse during Phase I in Section 5.2.2.

5.2.1. Collapse During Phase III

If the NS is supramassive at birth it can survive until a substantial fraction of its rotational energy has been dissipated. Its lifetime is therefore expected to be of the order of the spin-down timescale t_sd. The top panel of Figure 11 shows the evolution of the system for the fiducial parameter set, but different times of collapse. Dotted lines indicate t = t_coll, which is set in units of t_sd by the parameter f_coll (see Table 1).

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If the collapse occurs prior to the time of maximal brightening, it manifests itself only as a "kink" in the X-ray light curve (cf., e.g., the blue curve). The latter still increases up to a maximum after the time of collapse due to the fact that there is still a substantial amount of internal energy (stored in the ejecta material) yet to be radiated away. The sooner the time of collapse, the dimmer is this global maximum of the light curve, since further energy injection by L_PWN is substantially reduced at t = t_coll (see below). After the maximum the light curve gradually decreases (cf. blue and green curves) until a plateau-like phase is reached.

In contrast, if the NS collapses after the time of maximal brightening, the light curve directly decreases to a plateau-like phase. This is because the ejecta shell enters an "asymptotic" phase shortly after having reached the global maximum (cf. Section 5.6 of Paper I). In this phase, most of the acquired internal energy has already been radiated away and the present luminosity depends on the present influx of energy from the PWN. This influx is severely reduced after t = t_coll, as only gradual cooling of the PWN can still supply the ejecta shell with further energy (as described in Section 4.4 of Paper I).

It is this supply with residual energy from the PWN that creates the plateau phase characteristic of all runs across the entire range for f_coll (cf. the upper panel of Figure 11). The characteristics of the light curves after t = t_coll are, however, better illustrated by plotting the luminosity as a function of the time after collapse (cf. the bottom panel of Figures 11 and 12).

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Depending on the collapse time, the light curves show a characteristic two-plateau structure after t = t_coll (cf. the bottom panel of Figures 11 and 12). Only for early collapse times, the light curve essentially shows a single rising plateau plus a "bump" of radiation (cf., e.g., the case f_coll = 0.1) caused by the release of energy stored in the ejecta shell (see above). In this case, the light curve after t = t_coll strongly depends on the previous history as the radiation escaping from the system is still sourced by energy acquired by the ejecta material prior to the collapse. However, for runs in which the ejecta shell has already entered or is about to enter the asymptotic regime (see above; cf., e.g., the cases f_coll = 1, 2, 3) its properties strongly depend on the present state of the nebula and the total observer luminosity L_obs,tot clearly shows a two-plateau structure (see the bottom panel of Figure 12). The duration of the first plateau in these runs can be roughly estimated by the photon diffusion timescale t_diff,ej of the ejecta layer at the time of collapse (cf. Equation (122) of Paper I and the top panel of Figure 12). This timescale represents the time needed for the ejecta material to adjust to the sudden (instantaneous) change of the PWN conditions at t = t_coll.⁶ The latter process is nicely reflected by the aforementioned light curves depicted in the lower panel of Figure 12, in which case t_diff,ej at the time of collapse (indicated by dotted lines) indeed provides reliable predictions for the duration of the first plateau.

The duration of the second plateau phase in the total observer luminosity L_obs,tot (cf. Section 2) is essentially determined by the photon diffusion time t_diff,n of the PWN (cf. Section 4.4 of Paper I and the top panel of Figure 12), which sets the cooling timescale of the nebula after the collapse. As t_diff,n is monotonically decreasing after t = t_coll (cf. Equation (105) and (106) of Paper I), the diffusion time at t = t_coll provides an upper bound on the plateau duration (see the bottom panel of Figure 12). This fact has also been noted by Ciolfi & Siegel (2015b) based on a simplified analysis. As t_diff,n varies only slightly across the different collapse times considered here (cf. upper panel of Figure 12), the plateau durations are remarkably similar (∼10⁵ s; cf. the bottom panel of Figure 12). Moreover, the absolute luminosity levels L_obs,tot during the second plateau phase are also very similar. This is because the total luminosity during this phase is essentially determined by the ratio of the internal energy of the nebula E_nth (which is very similar among the different runs) and t_diff,n (cf. Section 4.4 of Paper I).

It is important to note that the ejecta material cools down and that the maxima of the thermal spectra drift out of the XRT band on the timescales considered here. Therefore, after t = t_coll the X-ray light curves L_obs,XRT differ somewhat from the total luminosity L_obs,tot (cf. the bottom panel in Figures 11 and 12). In particular, the second plateaus of the X-ray light curves are characterized by lower luminosity levels and shorter durations of the order of ∼10⁴ s. Furthermore, as the PWN becomes optically thin before t = 10⁵ s (cf. also Figure 3), a short breakout of luminous, non-thermal radiation from the interior of the PWN becomes visible in the X-ray band shortly after the second plateau (cf. Figure 11). This breakout is only short-lived as further cooling of the nebula prevents such high X-ray luminosities soon after the onset. In the case of f_coll = 0.1, the PWN has even cooled down to such an extent that this transition to the optically thin regime of the PWN does not lead to appreciable X-ray luminosities at all.

Finally, we note that in the recently proposed time-reversal scenario (Ciolfi & Siegel 2015a, 2015b), the X-ray light curves depicted in the lower panel of Figure 11 correspond to the observable X-ray afterglow of the SGRB as seen in the XRT band. We note that the two-plateau structure found here is remarkably similar to the structure of X-ray afterglows observed in a number of SGRB events (e.g., Gompertz et al. 2014). Furthermore, as the prompt SGRB emission is associated with the collapse of the NS in the time-reversal scenario, the X-ray emission predicted by our model prior to the collapse (as, e.g., depicted in the upper panel of Figure 11) corresponds to a precursor signal that can be searched for by future X-ray missions. If found, it would constitute a remarkable piece of evidence in favor of the time-reversal scenario. The essence of "time reversal" in this scenario is nicely illustrated by Figures 11 and 12: the energy radiated away from the system after t = t_coll has been extracted from the NS prior to the collapse. This energy diffuses outward on the respective diffusion timescales and produces a characteristic two-plateau structured X-ray light curve.

5.2.2. Collapse During Phase I

In this section, we return to the problem of predicting the EM transient signal associated with an early collapse of the NS during Phase I. Such an early collapse is expected to occur either if the NS is hypermassive at birth or if it is supramassive but collapses due to (magneto-) hydrodynamic instabilities. In either case we numerically integrate the corresponding evolution equations (Equations (1)–(2) of Paper I) after the collapse as described in Section 4.4 of Paper I. In this case, the time of collapse, t = t_coll, is parametrized in units of t_dr, f_coll,PI = t_coll/t_dr.

Figure 13 shows X-ray light curves of radiation in the XRT band (cf. Section 2) for the fiducial parameter setup varying the time of collapse, σ_M, and κ. These are the most influential parameters concerning this early collapse scenario. At t = t_coll mass ejection from the NS suddenly stops and the ejected material starts to form a spherical shell of thickness ${R}_{\mathrm{ej}}\mbox{--}{R}_{\mathrm{in}}$ (cf. Section 4.1.2 of Paper I) that moves outward with a constant head speed v_ej,in (see Figure 14). After t = t_coll the total luminosity L_obs,tot typically increases as a power law in time⁷ up to a maximum at a timescale ∼10³–10⁴ s after the BNS merger. For the fiducial setup (cf. Table 1), the shell thickness remains constant (since ${\sigma }_{v}=0$; cf. Section 4.1.1 of Paper I) and the peak emission frequency remains in the XRT band until late times. This is why the corresponding X-ray light curves show this power-law behavior and follow the total luminosity until shortly after the global maximum (cf. the f_coll,PI = 0.1, 1, 3 curves in Figure 13). When σ_M < 2, σ_v > 0 and the shell thickness increases ∝t. Hence, the temperature decreases more rapidly and the corresponding X-ray light curves deviate from a power-law increase soon after t = t_coll as the peak emission frequency moves out of the XRT band (cf. the σ_M = 1, 1.5 cases in Figure 13). Finally, we note that the higher opacity of κ = 10 cm² g⁻¹ leads to an overall dimmer light curve, with the power-law increase and global maximum being shifted to larger timescales.

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In this early-collapse scenario, the maximum of the light curve is always followed by a monotonic and abrupt decrease in contrast to the typical (two-)plateau morphology found in the case of a collapse during Phase III. This is due to the absence of a PWN, which could further supply the ejecta matter with energy once the energy reservoir has been depleted.

Figure 15 shows five different categories of X-ray light curve morphologies, which we briefly discuss below. In contrast to the time-reversal case (Section 6.2), these light curves are characterized by a "delayed onset": it typically takes ∼1–10 s after the SGRB to reach appreciable X-ray luminosities of the order of ≳10⁴² erg. This is mainly due to the very high optical depths of the ejecta matter at early times (cf. Section 4.1). The time of peak brightness is typically reached between ∼10² and 10⁴ s after the SGRB with luminosities of ∼10⁴⁶–10⁴⁹ erg s⁻¹.

Cat. M—These light curves show a single maximum followed by a steep decay (upper left panel of Figure 15). They correspond to thermal radiation from the ejecta shell, with a non-thermal "tail" of radiation from the nebula at late times in some cases (cf. t_dr = 0.1 and ${\dot{M}}_{\mathrm{in}}={10}^{-3}\;{M}_{\odot }$). Because of the single maximum and abrupt decay these light curves are qualitatively similar to the shapes of the observed X-ray afterglows of, e.g., GRB 080905A, GRB 090515, GRB 090607, GRB 100117A.

Cat. MM—These light curves show a global maximum due to thermal radiation from the ejecta shell, followed by a steep decay and a separated second "bump" of non-thermal radiation from the nebula (upper right panel of Figure 15). The non-thermal radiation typically decreases as a power law ∝t^−a, where a ≳ 2. This could explain some more moderate declines of X-ray luminosity at late times in some observed cases. Characteristic of these light curves is also the absence of high X-ray luminosity at intermediate times. Potential candidates for such a morphology could be, e.g., GRB 051227, GRB 061201, GRB 060313, GRB 070724A, GRB 071227, GRB 110112A, GRB 121226A, GRB 131004A. Furthermore, regarding timescales and luminosity levels, the second maximum of non-thermal radiation is consistent with the late-time X-ray rebrightening observed in some SGRB events, such as GRB 050724, GRB 080503, and the r-process powered kilonova candidate GRB 130603B (Grupe et al. 2006; Perley et al. 2009; Fong et al. 2014). Based on their models, Metzger & Piro 2014 and Gao et al. 2015 argued that such a rebrightening for GRB 080503 and GRB 130603B could be caused by radiation from a stable magnetar surrounded by a PWN (similar to our setup here) and thus called those GRBs candidate events for a "magnetar driven transient" or "magnetar-driven merger-novae." We note that, in particular, Fong et al. (2014) reported a late-time (t ≳ day) X-ray excess of $L\simeq 4\times {10}^{43}{(t/\mathrm{day})}^{\alpha }\;\mathrm{erg}\;{{\rm{s}}}^{-1}$ for GRB 130603B, where α = −1.88 ± 0.15. As noted above, some of our light curves also show a power-law behavior with an exponent close to −2. Furthermore, the timescales and luminosity levels of some of our models are consistent with such an X-ray excess at late times.

Cat. P—These light curves are characterized by a global maximum (as in the previous cases), followed by a plateau that is caused by the collapse of the NS (cf. Section 5.2.1) and a final steep decay (lower left panel of Figure 15). The radiation originates from the ejecta shell throughout the evolution and is thus thermal. The plateaus typically extend up to ∼10⁴ s and the luminosity levels are typically roughly two orders of magnitude lower than the preceding peak at ∼10²–10³ s. This morphology together with the timescales and luminosity levels seems to resemble some of the SGRBs considered by Gompertz et al. (2013, 2014). More specifically, potential candidates for such a morphology could be, e.g., GRB 051227, GRB 070714B, GRB 070724A, GRB 071227, GRB 080123, GRB 111121A.

Cat. PF—These X-ray afterglow light curves are very similar to Cat. P, except for the fact that there is a noticeable flare-like event of non-thermal radiation from the nebula after the plateau (bottom row of Figure 15, middle panel). This is due to the fact that the ejecta shell becomes optically thin during the plateau phase of the total luminosity (cf. Section 5.2.1). A potential candidate for this morphology could be, e.g., GRB 050724, which shows a flare-like event just after the plateau phase. However, as such flares are not always well separated from the plateau itself, this morphology can also be confused with Cat. P in some cases. Therefore, some of the GRB candidates for Cat. P could also apply to this category and vice versa.

Cat. GF—These are more peculiar light curves shown in the lower right panel of Figure 15. They are similar to Cat. PF, but characterized by an abrupt decrease after the global maximum and by absence of strong X-ray radiation (a "gap") at intermediate times. The final flare dominates the luminosity at intermediate times.

Figure 16 depicts examples for five different X-ray afterglow light curve morphologies in the time-reversal scenario. In contrast to the standard scenario (cf. Section 6.1) and except for a few peculiar cases, these light curves typically do not show a delayed onset. They are rather marked by very high X-ray luminosities starting from the SGRB itself. This is because at the times of collapse, the system is typically already radiating at a very high luminosity level. We note that in the time-reversal scenario, the X-ray luminosity predicted by our model prior to the collapse as discussed in Sections 6.1 and 4 and 5 represents a precursor signal that can be searched for by future X-ray missions.

Cat. SP—These are X-ray light curves characterized by a single plateau lasting up to 10⁴–10⁵ s, followed by a rather steep decay (upper left panel of Figure 16). The radiation is purely non-thermal in the cases shown (broad-band synchrotron plus inverse Compton, cf. Section 4.3), as the collapse of the NS occurs at late times when the ejecta shell is already optically thin. This type of morphology seems to resemble some of the X-ray plateaus discussed by Rowlinson et al. (2013). Potential candidates for this morphology could be, e.g., GRB 051221A, GRB 060313, GRB 061201, GRB 070809, GRB 090510, GRB 111020A, 130603B, GRB 140903A. Some of these cases require a contribution from the standard X-ray afterglow of the SGRB jet itself in combination with our prediction (see Rowlinson et al. 2013 and Section 7).

Cat. TP—Together with Cat. TPF this family of light curves represents the most typical prototype for the time-reversal scenario, as it shows the characteristic two-plateau morphology (see Section 5.2.1; middle panel of the upper row in Figure 16). The plateau durations are determined by the photon diffusion times of the ejecta shell and the PWN (see Section 5.2.1). The luminosity levels and plateau durations of typically ∼1–10³ s and ∼10³–10⁴ s for the first and the second plateau, respectively, are in very good agreement with some of the observed two-plateau structures discussed by Gompertz et al. (2014). Potential candidates for this morphology are GRB 051227, GRB 060614, GRB 070714B, GRB 070724A, GRB 071227, GRB 080123, GRB 111121A.

Cat. TPF—Same as Cat. TP, but with a late-time "flare" of non-thermal radiation from the nebula as the ejecta shell becomes optically thin (upper right panel of Figure 16). A potential candidate for this morphology is GRB 050724, which shows such a flare at late times attached to the second plateau (cf. Gompertz et al. 2014). However, as these flares are typically not well separated from the second plateau, many of such events could be confused with Cat. TP and some of the GRB candidates mentioned there could also apply to this category and vice versa.

Cat. RP—This family of light curves essentially shows a single plateau, but with a rising luminosity level (lower left panel of Figure 16). After the maximum there is a very short second "plateau" up to typically ∼10³–10⁴ s, followed by a steep decay. The radiation is thermal throughout. Candidate SGRBs for this category could be, e.g., GRB 070714A, GRB 090607, GRB 131004A.

Cat. NP—These light curves are again more peculiar examples without a (dominant/characteristic) plateau (lower right panel of Figure 16). This family is rather marked by a single maximum or "bump" of thermal or purely non-thermal radiation. A maximum at early times ≲10³ s (as in the case of a collapse during Phase I; cf. the magenta curve and Section 5.2.2) is typically characterized by thermal radiation and could resemble some of the candidate GRBs noted in Cat. M of Section 6.1. The other light curves with maxima at later times (∼10⁴–10⁵ s) are either thermal (cyan curve; collapse during Phase I) or non-thermal (remaining curves; collapse during Phase III). The absence of strong X-ray radiation at early times in these cases is either due to a high optical depth (collapse during Phase I) or a low temperature of the ejecta material (collapse during Phase III). These late-time maxima could be consistent with the afterglow rebrightening observed in some SGRBs, such as GRB 050724, GRB 080503, and the r-process powered kilonova candidate GRB 130603B (Grupe et al. 2006; Perley et al. 2009; Fong et al. 2014; see also the discussion of Cat. MM in Section 6.1). The timescales for rebrightening in the aforementioned cases of ∼10⁴–10⁵ s are in good agreement with the timescales for maximum brightness found here. This rebrightening is to be understood as an additional component to the fading standard X-ray afterglow of the SGRB jet itself, as the early X-ray emission in the aforementioned cases cannot be explained by the light curves such as those in the lower right panel of Figure 16. Such an interpretation is similar to Metzger & Piro (2014) and Gao et al. (2015), who, according to their models, noted that the observed X-ray excess at late times in some of the aforementioned cases can be consistent with a magnetar driven transient (or "merger-nova") in addition to the standard X-ray afterglow.