Fast Radio Bursts from Magnetars Born in Binary Neutron Star Mergers and Accretion Induced Collapse

Fast radio bursts (FRB) are millisecond duration pulses of coherent radio emission with large dispersion measures (DM) well above the contribution from the Milky Way, thus implicating an extragalactic origin (e.g., Lorimer et al. 2007; Keane et al. 2012; Thornton et al. 2013; Spitler et al. 2014; Ravi et al. 2015; Champion et al. 2016; Petroff et al. 2016; Lawrence et al. 2017; Shannon et al. 2018; see Cordes & Chatterjee 2019; Petroff et al. 2019 for recent reviews). A cosmological origin was directly confirmed for the repeating FRB 121102 (Spitler et al. 2014, 2016) thanks to its precise localization to a low-metallicity dwarf star-forming galaxy at a redshift of z = 0.193 (Chatterjee et al. 2017; Tendulkar et al. 2017). FRB 121102 is also spatially coincident with a compact (<0.7 pc) and luminous (νL_ν ∼ 10³⁹ erg s⁻¹) persistent radio synchrotron source (Marcote et al. 2017). It exhibits an enormous rotation measure, RM ∼ 10⁵ rad m⁻² (Michilli et al. 2018; see also Masui et al. 2015), which exceeds those of other known astrophysical sources, with the exception of Sgr A* and the flaring magnetar SGR J1745-2900 located in the Galactic Center (Eatough et al. 2013).

Recently, the Australian Square Kilometre Array Pathfinder localized a second event—FRB 180924—based on the detection of a single burst (Bannister et al. 2019). This FRB has not exhibited repetition so far, it is located in a more massive and quiescent galaxy than that of FRB 121102, it is not accompanied by a persistent radio source to a limit of ∼3 times lower than FRB 121102, and it exhibits a low RM of 14 rad m⁻² (Bannister et al. 2019). Shortly thereafter, Ravi et al. (2019) reported the localization of a third burst—FRB 190523—using the The Deep Synoptic Array 10 dish prototype, whose host was similarly found to be a massive quiescent galaxy. A fourth localized burst, FRB 181112, was also subsequently reported by Prochaska et al. (2019).⁶

Although dozens of models have been proposed for FRBs, most are ruled out by a repeating, cosmological source like FRB 121102, while others are ruled out by the large all-sky rate if the bulk of FRBs are non-repeating. One compelling model for repeating FRBs are bursts generated by a young flaring magnetar (Popov & Postnov 2013; Kulkarni et al. 2014; Lyubarsky 2014; Katz 2016; Lu & Kumar 2016, 2018; Beloborodov 2017; Kumar et al. 2017; Metzger et al. 2017; Nicholl et al. 2017b). This idea is supported by the atypical properties of the host galaxy of FRB 121102, which are similar to those of superluminous supernovae (SLSNe) and long gamma-ray bursts (LGRBs; Metzger et al. 2017; Nicholl et al. 2017b; Tendulkar et al. 2017), rare explosions that are powered by engines (MacFadyen & Woosley 1999; Kasen & Bildsten 2010; Nicholl et al. 2017a). In such a model, the persistent radio source associated with FRB 121102 could be understood as emission from a compact magnetized nebula surrounding the young (decades to centuries old) magnetar and embedded in the expanding supernova ejecta (Kashiyama & Murase 2017; Metzger et al. 2017; Margalit & Metzger 2018; Omand et al. 2018). The nebula is powered by nearly continual energy release from the magnetar, likely during the same sporadic flaring events responsible for the repeated FRBs (Beloborodov 2017; Metzger et al. 2019). As shown by Margalit & Metzger (2018), the radio flux of the nebula, and its large but decreasing RM, can both be explained in this model. This notion is also supported by the recent detection of a radio source coincident with the SLSN PTF10hgi about 7.5 yr post-explosion (Eftekhari et al. 2019), although no FRBs have yet been detected from this source.

However, SLSNe/LGRBs are not the only potential sites for magnetar birth. The gravitational-wave (GW) driven merger of binary neutron stars (BNS) is generally believed to give rise to a massive magnetized neutron star (NS) remnant (e.g., Price & Rosswog 2006; Bucciantini et al. 2012; Giacomazzo & Perna 2013). In most cases this remnant is well above the Tolman–Oppenheimer–Volkoff (TOV) maximum stable mass, M_TOV, and thus is only temporarily stabilized against gravitational collapse by rapid rotation; once this rotational support is removed, by a combination of internal ("viscosity") and external (e.g., magnetic-dipole losses) stresses, the magnetar collapses to a black hole. However, depending on the cosmic population of binary NSs, the uncertain value of M_TOV, and the amount of mass loss during the merger, a modest fraction of mergers could lead to indefinitely stable NSs (e.g., Dai et al. 2006; Metzger et al. 2008; Zhang 2013; Lasky et al. 2014; Gao et al. 2016; Piro et al. 2017; Margalit & Metzger 2019). Such a stable magnetar may be similar to those formed in SLSNe and/or LGRBs, but will be surrounded by a much smaller and faster-expanding ejecta shell, and will be hosted by a different galaxy population. These magnetars may also exhibit large spatial offsets from their host galaxies, as observed for short gamma-ray bursts (SGRBs; e.g., Berger 2014). If such magnetars give rise to FRBs, they will therefore be accompanied by different nebulae and host galaxy demographics (e.g., Nicholl et al. 2017b; Yamasaki et al. 2018).

Another potential formation channel for magnetars is the accretion-induced collapse (AIC) of a white dwarf. This can occur either due to accretion from a non-degenerate binary companion (e.g., Nomoto et al. 1979; Taam & van den Heuvel 1986; Canal et al. 1990; Nomoto & Kondo 1991; Tauris et al. 2013; Schwab et al. 2015; Brooks et al. 2017) or following the merger of two white dwarfs in a binary system (e.g., Yoon et al. 2007; Schwab et al. 2016). As with BNS mergers, AIC is expected to occur in a range of host galaxy types due to the delay after star formation. However, due to the small natal kicks of the NSs formed through this channel, they may not occur with as large offsets as magnetars formed via BNS mergers. AIC has been previously suggested as a possible progenitor for FRB 121102 (Kashiyama & Murase 2017; Waxman 2017), however these models differ significantly from the flaring magnetar model discussed here.

Here, we develop predictions of the BNS merger and AIC magnetar channel for FRBs, including their host galaxy demographics and spatial locations (motivated by observations of short GRBs, the BNS merger GW170817, and SNe Ia; Section 2), their rates (Section 3), lifetimes, dispersion and rotation measure, and the properties of their accompanying persistent radio sources (Section 4). Throughout the paper we compare the results to the properties of FRB 180924 and FRB 190523.

FRBs that originate from magnetars created in SLSNe and LGRBs are expected to preferentially reside in low-metallicity dwarf galaxies known to host these classes of explosions (e.g., Fruchter et al. 2006; Stanek et al. 2006; Modjaz et al. 2008; Lunnan et al. 2014; Perley et al. 2016; Schulze et al. 2018). Moreover, both LGRBs and SLSNe tend to concentrate in bright UV regions of their hosts (Fruchter et al. 2006; Lunnan et al. 2015; Blanchard et al. 2016), with a preference for small spatial offsets from their host centers (Bloom et al. 2003; Lunnan et al. 2015; Blanchard et al. 2016). We have previously shown that the host (and location) of FRB 121102 are comparable to those of SLSNe and LGRBs (Metzger et al. 2017; Nicholl et al. 2017b). The host galaxies of FRB 180924, FRB 190523, and FRB 181112, on the other hand, do not match these expectations.

In contrast, BNS mergers are expected to occur in all types of galaxies (e.g., Belczynski et al. 2006), due to the broad delay time required for GW driven mergers. Furthermore, due to the natal kicks received by NSs at birth, BNS mergers can take place in locations spatially offset from their birth sites, sometimes outside of their host galaxies (e.g., Fryer & Kalogera 1997; Bloom et al. 1999). The consistency of these predictions with the observed host galaxy demographics and spatial offset distributions of SGRBs provided strong evidence for the association of SGRBs with BNS mergers (Berger 2014), even prior to the discovery of GW170817 (Abbott et al. 2017b, 2017a). Though we have no direct probes of white dwarf AIC, their formation channels are thought to be sufficiently similar to SNe Ia that we might expect a similar host galaxy distribution.

In Figure 1 we compare the host galaxy properties and spatial offset of FRB 180924 (Bannister et al. 2019), FRB 190523 (Ravi et al. 2019), and FRB 181112 (Prochaska et al. 2019), to those of SGRBs (Fong et al. 2013; Berger 2014; De Pasquale 2019) and the BNS merger GW170817 (Blanchard et al. 2017; Fong et al. 2017; Im et al. 2017). We also make a comparison to the host galaxies of SNe Ia (Neill et al. 2009; Gupta et al. 2011; Smith et al. 2012; Uddin et al. 2017), taking this as a proxy for the AIC scenario. We find that in terms of stellar mass, star formation rate, and stellar population age, the hosts of FRB 180924, FRB 190523, and FRB 181112 are comparable to those expected for BNS mergers and AIC. The same is true for the physical and host-normalized offset of FRB 180924 relative to the distribution for SGRBs, GW170817, and SNe Ia. FRB 190523 also shows indications of being offset from its host-galaxy, however the localization uncertainty is too large to quantitatively asses this. Thus, from the point of view of their large-scale environment it is conceivable that FRB 180924, FRB 190523, and FRB 181112 represent a BNS merger or AIC origin. We stress that a more definitive statement will require a larger sample of localized FRBs (Nicholl et al. 2017b), as well as a broader comparison set of BNS mergers from GW detectors and from the long-sought class of AIC transients from future optical or radio surveys.

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The observed SGRB rate (Wanderman & Piran 2015) and an estimated beaming factor of ∼30 (Fong et al. 2015) leads to an estimate of the BNS merger rate at z ≲ 0.5 of R_BNS ∼ 300 Gpc⁻³ yr⁻¹. This is consistent with the local merger rate estimated by Advanced LIGO/Virgo from the O1 and O2 observing runs of 110 ≲ R_BNS ≲ 3840 Gpc⁻³ yr⁻¹ (The LIGO Scientific Collaboration & the Virgo Collaboration 2019), as well as the range required if BNS mergers are the main source of heavy r-process elements in the Milky Way given the r-process yield inferred from the optical/infrared counterpart of GW170817 (e.g., Cowperthwaite et al. 2017; Hotokezaka et al. 2018).

It was proposed that during the final stages of a BNS merger inspiral the interaction between the NS magnetospheres could give rise to a single (non-repeating) FRB (Hansen & Lyutikov 2001; Totani 2013; Zhang 2014; Metzger & Zivancev 2016; Wang et al. 2016). However, given the high volumetric rate of FRBs relative to BNS mergers, such "one off" bursts can account for at most only a small fraction, ≲1%, of the total FRB population (Nicholl et al. 2017a).

If, on the other hand, FRBs are produced well after the merger by a remnant magnetar, then each magnetar produced through this channel will produce many FRBs (Nicholl et al. 2017a; Cao et al. 2018; Murase et al. 2018; Yamasaki et al. 2018). However, as discussed in the next section, the magnetar remnant is initially enshrouded within the kilonova ejecta, which must become optically thin to free–free absorption before FRBs can escape to an external observer, thus requiring a minimum magnetar lifetime of weeks to months. Meta-stable (hyper-massive or supra-massive) NS remnants will generally collapse to a black hole much earlier (e.g., Shibata & Taniguchi 2006; Ravi & Lasky 2014), unless the dipole magnetic field of the remnant is extremely weak, ≲10¹³ G, to prevent spin-down, despite the much larger fields generated during the merger process. The mergers of binaries giving rise to indefinitely stable magnetar remnants, with masses at or below M_TOV, are therefore the most promising FRB sources.

The fraction of BNS mergers that lead to stable remnants depends on the mass distribution of merging NS systems and the uncertain nuclear equation of state (Lasky et al. 2014; Gao et al. 2016; Piro et al. 2017; Margalit & Metzger 2019). Margalit & Metzger (2019) show that, for NSs drawn from the Galactic BNS population, and given current constraints on M_TOV, at most ≈3% of BNS mergers can produce stable NS remnants (although see Section 6 for a discussion of possible caveats). A low volumetric rate of stable NS remnants is also consistent with the lack of discovery of the time-evolving remnants of such objects in radio transient surveys (Metzger et al. 2015), or in late-time follow-up of SGRBs (Fong et al. 2016). Thus, the birth rate of magnetars capable of producing FRBs in the BNS merger channel is ${ \mathcal R }\lesssim 0.03{{ \mathcal R }}_{\mathrm{BNS}}\sim 3\mbox{--}100\,{\mathrm{Gpc}}^{-1}\,{\mathrm{yr}}^{-1}$, comparable or less than the volumetric rate of LGRBs and SLSNe (e.g., Prajs et al. 2017), which are considered progenitors of FRBs with dwarf-galaxy hosts like FRB 121102 (Metzger et al. 2017; Nicholl et al. 2017b; Tendulkar et al. 2017; Eftekhari et al. 2019). Nicholl et al. (2017a) show that even a volumetric birth rate ${ \mathcal R }\sim 10\mbox{--}100$ Gpc⁻³ yr⁻¹ is sufficient to explain a significant portion of the observed FRB population if each magnetar is a similarly active FRB source for decades. As discussed below (Equation (1)), the predicted FRB active lifetime of massive magnetars produced in BNS mergers is consistent with this range.

As there is no direct observational evidence for AIC, its rate remains highly uncertain. Theoretical estimates of the AIC rate are in a range comparable to that of BNS mergers but also with uncertainties of over an order of magnitude (e.g., Yungelson & Livio 1998; Tauris et al. 2013; Kwiatkowski 2015). The fraction of AIC events giving rise to magnetars via flux-freezing of the primordial white dwarf magnetic field is likely smaller than the total because of the low fraction, ≲15%, of strongly magnetized white dwarfs (e.g., Liebert et al. 2003). On the other hand, if the progenitor white dwarf is rapidly spinning, then even an initially weak magnetic field could be amplified to magnetar strengths following AIC by strong and differential rotation in the millisecond proto-NS remnant (e.g., Dessart et al. 2007). Within the significant uncertainties, the magnetar birth rate via the AIC channel could therefore be comparable to that of the BNS merger channel.

4.1. Bursts and Active Lifetime

The origin of the coherent emission process responsible for FRB emission is uncertain. One of the more developed models postulates that FRBs result from synchrotron maser emission from magnetized relativistic shocks (Lyubarsky 2014; Beloborodov 2017). In this case each individual burst could be produced by the collision between an ultrarelativistic flare from the magnetar and a baryon-loaded shell released from an earlier flare (Metzger et al. 2019). The external environment surrounding the magnetar on larger (∼parsec) scales therefore does not directly impact the burst properties in this scenario. Nevertheless, variation in the average burst properties could still result from intrinsic differences in the magnetar's activity, as may be influenced by its mass, age, and magnetic field strength.

The mass distribution of Galactic BNS systems peaks above M_TOV (Margalit & Metzger 2019). If representative of the extragalactic population, this implies that only a small fraction of mergers will leave stable NS remnants, and those will have masses just below M_TOV. Such extreme NSs may possess sufficiently high central densities to activate direct Urca cooling in their cores (Page & Applegate 1992), which exceeds by orders of magnitude the cooling rate via the "modified" Urca process that is believed to dominate in the majority of (less massive) NSs. Because the rate of ambipolar diffusion of the magnetic field inside the NS depends sensitively on the core temperature (Beloborodov & Li 2016), the timescale over which magnetic energy escapes from the stellar surface (e.g., in the form of FRB-generating flares) could speed up relative to magnetars born from core-collapse SNe or AIC. Following Beloborodov & Li (2016), we estimate the magnetic activity timescale in the direct Urca (high-mass NS) and modified Urca (normal-mass NS) case as

$$\begin{eqnarray}&&{t}_{\mathrm{mag}}\sim \left\{\begin{array}{l}20\,\mathrm{yr}\,{B}_{16}^{-1}{L}_{5}^{3/2},\,\mathrm{high} \mbox{-} \mathrm{mass}\ \mathrm{NS}\\ 700\,\mathrm{yr}\,{B}_{16}^{-1.2}{L}_{5}^{1.6},\,\mathrm{normal} \mbox{-} \mathrm{mass}\ \mathrm{NS},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where we assume δB ∼ B/2 as the amplitude of magnetic field fluctuations over a length-scale L = 10⁵L₅ cm inside the magnetar core. The magnetic energy of the magnetar, ${E}_{\mathrm{mag}}\sim 3\times {10}^{49}\,{B}_{16}^{2}\,\mathrm{erg}$, therefore implies a characteristic dissipation power

$$\begin{eqnarray}&&{\dot{E}}_{\mathrm{mag}}\sim \displaystyle \frac{{E}_{\mathrm{mag}}}{{t}_{\mathrm{mag}}}\sim \left\{\begin{array}{l}5\times {10}^{40}{B}_{16}^{3}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\,\mathrm{high} \mbox{-} \mathrm{mass}\ \mathrm{NS}\\ {10}^{39}{B}_{16}^{3.2}\,\mathrm{erg}\,{{\rm{s}}}^{-1},\,\mathrm{normal} \mbox{-} \mathrm{mass}\ \mathrm{NS}.\end{array}\right.\end{eqnarray} \tag{ 2 }$$

The most massive and highly magnetized magnetars born in BNS mergers may therefore be differentiated from lower mass more weakly magnetized NSs such as those that might be created in SLSNe, LGRBs, or AIC through their vastly different power output.

One may speculate that the shorter active lifetime of magnetars from BNS mergers could lead to shorter intervals between major ion ejection events relative to less massive magnetars, and an increased mass-loss rate. In the shock-powered maser scenario, a larger average density surrounding the magnetar into which flare ejecta collides acts to increase the peak frequency of the bursts (Equation (47) of Metzger et al. 2019), possibly to values ≫1 GHz, much higher than the sensitive range of FRB telescopes. Lower, ∼GHz frequency bursts of the more readily detectable kind might occur only in relatively rare cases in which magnetar activity ceases long enough to clear a low-density cavity around the magnetar. Similar qualitative behavior (suppressed FRB emission for extended periods of time after major ion flares) was invoked to explain the long periods of inactivity observed for FRB 121102 (Metzger et al. 2019). If the magnetars from BNS mergers undergo even longer periods of apparent inactivity at ∼1 GHz, this could help explain why repeating bursts from FRB 180924 have not been detected despite significant follow-up (Bannister et al. 2019).

Unlike in BNS mergers, the NS remnants of AIC would possess very low masses and would not be expected to undergo direct Urca cooling. Nevertheless, one cannot exclude systematically different internal magnetic field strength or topology in magnetars formed from AIC versus those from core-collapse events, which could lead to qualitatively different bursting activity than the SLSNe/LGRB case resulting in different FRB properties.

4.2. Ejecta Radio Transparency, Afterglow, and Dispersion Measure

BNS mergers are accompanied by the ejection of radioactive neutron-rich material, which powers their optical/IR emission (e.g., Li & Paczyński 1998). Axisymmetric hydrodynamical simulations find that the AIC of a slowly rotating, weakly magnetized white dwarf is accompanied by only a weak explosion that ejects a small amount of mass, ≲10⁻² M_⊙, with a small fraction of radioactive ⁵⁶Ni (e.g., Dessart et al. 2006; Abdikamalov et al. 2010). However, if the white dwarf is strongly magnetized and rapidly rotating prior to collapse (the type of events most likely to give birth to millisecond magnetars), then the amount of unbound ejecta can be substantially larger, ∼0.01–0.1 M_⊙, closer to that of BNS mergers (Dessart et al. 2007; Metzger et al. 2009).

The expanding ejecta shell surrounding the nascent magnetar in the BNS merger/AIC scenarios affects their observational signatures. First, to be observable an FRB must propagate through this confining medium, which is initially optically thick to free–free absorption at ∼GHz frequencies. The free–free optical depth is set by the ejecta density, temperature, and ionization state. Here we calculate the free–free optical depth of homologously expanding BNS merger/AIC ejecta for which the temperature and ionization state are governed by photoionization from spin-down of the nascent magnetar. We use the publicly available photoionization code CLOUDY (Ferland et al. 2013) and follow the methods described in Margalit et al. (2018) for magnetars embedded in SLSNe ejecta.

A primary difference between the SLSN/LGRB and BNS merger/AIC scenarios is the lower ejecta mass and higher ejecta velocities in the latter case, which results in a shorter free–free transparency time, ${t}_{\mathrm{ff}}$. Naively, for a fixed ionization fraction one expects ${t}_{\mathrm{ff}}\propto {M}_{\mathrm{ej}}^{2/5}{v}_{\mathrm{ej}}^{-1}$, in which case t_ff will be ∼100 times shorter for the BNS merger/AIC case, i.e., months instead of ∼decade timescale for SLSNe (Margalit et al. 2018). However, even this overestimates the free–free transparency timescale by a factor of a few because it does not account for temperature effects on the ionization state. The black curve in Figure 2 shows the result of our CLOUDY calculations for the ionization state of BNS merger/AIC ejecta, calculated assuming a central ionizing source equal to the spin-down luminosity of the magnetar for assumed dipole magnetic field strength B ∼ 10¹⁴–10¹⁶ G and birth period ∼1 ms. At 1.4 GHz we find a transparency timescale of t_ff ∼ 1 week for an assumed ejecta mass M_ej = 0.1M_⊙ and velocity v_ej = 0.52c (Equation (4)); this high value of v_ej results if the ejecta is accelerated by the spin-down energy of the magnetar early in its evolution, when radiation is still trapped and goes into PdV expansion (see Equation (4) below). A more conservative upper bound on t_ff is obtained by neglecting magnetar acceleration; adopting the ejecta mass and velocity inferred for GW170817 by Villar et al. (2017), we find t_ff ∼ month at 1.4 GHz, with the detailed result depending on the ionizing radiation field (magnetic-dipole field strength).

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In the BNS merger/AIC scenario, the rapidly diluting ejecta becomes nearly fully ionized by the magnetar radiation. The contribution to the burst DM by the ejecta, which can be extracted directly from the CLOUDY simulations (yellow curve in Figure 2), is therefore easily approximated by

$$\begin{eqnarray}&&{\mathrm{DM}}_{\mathrm{ej}}\approx \displaystyle \frac{3{M}_{\mathrm{ej}}}{8\pi {m}_{{\rm{p}}}{\left({v}_{\mathrm{ej}}t\right)}^{2}}\approx 5\,\mathrm{pc}\,{\mathrm{cm}}^{-3}\,{M}_{\mathrm{ej},-1}{\beta }_{\mathrm{ej}}^{-2}{t}_{\mathrm{yr}}^{-2},\end{eqnarray} \tag{ 3 }$$

where β_ej = v_ej/c and we have assumed a fully ionized heavy ejecta composition (i.e., atomic mass A ≈ 2 Z).

In addition to generating free–free absorption and local DM at early epochs, the BNS merger/AIC ejecta will produce a late-time (∼decade long) radio afterglow from its interaction with the ambient interstellar (e.g., Nakar & Piran 2011) or circumstellar medium (e.g., Moriya 2016). This radio signature may be particularly prominent in the scenario considered here, since formation of a long-lived magnetar remnant and its subsequent spin-down are likely to inject an enormous amount of rotational energy of up to ∼10⁵³ erg into the surrounding medium (Gao et al. 2013; Metzger & Bower 2014). In most cases in which the dipole spin-down timescale is much shorter than the radiative diffusion time, this will accelerate the BNS merger/AIC ejecta to high velocities,

$$\begin{eqnarray}&&{\beta }_{\mathrm{ej}}=\sqrt{1-{\left(1+E/{M}_{\mathrm{ej}}{c}^{2}\right)}^{-2}}\approx 0.52\,{E}_{52.5}^{1/2}{M}_{\mathrm{ej},-1}^{-1/2},\end{eqnarray} \tag{ 4 }$$

and induce a significantly stronger late-time radio signature than for BNS mergers that do not form such remnants; for GW170817, the total kinetic energy of the kilonova ejecta is estimated to be significantly smaller, ≈2.5 × 10⁵¹ erg (Villar et al. 2017), consistent with the arguments that GW170817 did not form a long-lived magnetar (Margalit & Metzger 2017). Additionally, searches for enhanced late-time radio emission following SGRBs have placed limits on the fraction of BNS mergers that form magnetar remnants (Metzger & Bower 2014; Fong et al. 2016; Horesh et al. 2016), consistent with the estimates adopted in Section 3.

The kinetic energy transferred to the ejecta via magnetic-dipole spin-down may be smaller than the total rotational energy of the magnetar due to inefficiencies in coupling the magnetar wind to the BNS merger/AIC ejecta (e.g., Bucciantini et al. 2012), hence we adopt E = 3 × 10⁵² erg as a fiducial value⁷ in Equation (4). The dashed yellow lines in Figure 3 show a simplified model for the radio light curve produced by the interaction of the magnetar-accelerated ejecta with the interstellar medium (ISM), using the formulation of Nakar & Piran (2011) and assuming M_ej = 0.1M_⊙, E = 3 × 10⁵² erg, v_ej given by Equation (4), and different ISM densities. The blue lines in Figure 3 also show a model for magnetar plerion emission described in the next section. For comparison, we also show the upper limit on persistent radio emission at the location of FRB 180924 (Bannister et al. 2019), the timescale for the ejecta to become transparent to free–free absorption at 1.4 GHz, and the time after which the ejecta DM decreases to ≲100 pc cm⁻³, consistent with upper limits on the local DM contribution of FRB 180924 (Bannister et al. 2019). Although it is unclear whether a relativistic jet is launched from long-lived magnetars in BNS mergers (e.g., Murguia-Berthier et al. 2014), we also show for comparison the measured jet radio afterglow emission of GW170817 (Alexander et al. 2018). Even if radio emissions akin to GW170817 were produced in magnetar-remnant scenarios, it would not be sufficiently luminous to be detected at the distances of FRBs such as 180924.

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Interaction of the magnetar-accelerated ejecta with the ISM may or may not contribute detectable radio emission, depending sensitively on the ISM density. SGRB afterglow modeling suggests small typical densities, n ∼ 10⁻² cm⁻³, consistent with the typical large host-galaxy offsets (Fong et al. 2015). The local ISM density in the vicinity of FRB 180924 may be similarly small, in agreement with the implied low local DM contribution and the location on the outskirts of its host galaxy. We caution that the ejecta light curves in Figure 3 are highly simplified; deviations from spherical symmetry will generally delay the peak timescale (Margalit & Piran 2015), while emission earlier than the Sedov–Taylor time of the ejecta (at which the emission peaks) is extremely sensitive to the high-velocity tail of the ejecta distribution (e.g., Hotokezaka & Piran 2015).

4.3. Nebula Persistent Radio Emission and RM

The large RM and persistent radio flux of FRB 121102 are both consistent with arising from a nebula of magnetic fields and electrons injected by the magnetar flares and confined by the SN ejecta (Beloborodov 2017; Margalit & Metzger 2018). A similar idea for rotationally powered plerionic emission from AIC and FRB sources was discussed by Piro & Kulkarni (2013) and Murase et al. (2016), respectively. These characteristics depend both on intrinsic properties of the magnetar engine, such as its energy injection rate, history, and mass loading, and also on the age of the system and circum-engine environment.

While it is not clear how the intrinsic properties of the FRB activity vary for magnetars born in different astrophysical settings (although see Equations (1) and (2) and surrounding discussion), the external environment is more robustly predicted to differ markedly for magnetars formed in SLSNe/LGRBs versus those created in BNS mergers/AIC. In the former, the magnetar nebula expands within the dense SN ejecta of M_ej ∼ several M_⊙ and v_ej ∼ 10⁴ km s⁻¹, while in the BNS merger/AIC case the confining medium within which the nebula expands is the faster, more dilute BNS merger/AIC ejecta (M_ej ∼ 0.1M_⊙ and v_ej ∼ 0.5c; Equation (4)). This allows the nebula to expand at a faster rate in the BNS merger/AIC scenario and leads to a more rapid decrease in the nebula density and magnetic field, hence also in the imprinted RM and accompanying plerionic radio emission.

The velocity of the nebula inflated within a homologously expanding uniform-density ejecta can be roughly estimated as

$$\begin{eqnarray}&&{v}_{{\rm{n}}}\sim {\left(\displaystyle \frac{\int {\dot{E}}_{{\rm{n}}}{dt}{v}_{{\rm{ej}}}^{3}}{{M}_{{\rm{ej}}}}\right)}^{1/5}\approx 0.2c\,{E}_{{\rm{n}},50}^{1/5}{\beta }_{{\rm{ej}}}^{3/5}{M}_{{\rm{ej}},-1}^{-1/5},\end{eqnarray} \tag{ 5 }$$

where we have adopted fiducial values for instantaneous (present-day) nebula energy E_n and the BNS merger ejecta. For ejecta parameters appropriate to SLSNe, Equation (5) predicts nebula velocities smaller by an order of magnitude, ∼4000 km s⁻¹, consistent with v_n adopted for the models of FRB 121102 in Margalit & Metzger (2018). To account for the lower ejecta density in a more detailed manner, we self-consistently integrate the dynamical equations for the nebula's time-dependent evolution in the thin-shell approximation as energy is deposited by the central engine (e.g., Chevalier 2005). This is a direct extension of the Margalit & Metzger (2018) model, which assumed constant v_n for simplicity.

In Figure 2 we show the resulting nebula radio light curve and RM for two models. Model A (solid blue curve) is a scaled version of the model fit to the observed persistent source of FRB 121102 (Model A of Margalit & Metzger 2018), where the characteristic timescale of magnetar energy deposition has been reduced by a factor of ∼35, in line with the shorter estimated value of t_mag (Equation (1)). Additionally, as discussed above, the nebula expansion velocity within the ejecta is explicitly integrated as a function of time (as opposed to being a constant free parameter as in Margalit & Metzger 2018), and the ejecta parameters adopted are M_ej = 0.1M_⊙, v_ej = 0.52c (Equations (4), (5)). We find that the DM contributed by the nebula is always sub-dominant to that imprinted by the BNS merger/AIC ejecta, and can thus be neglected. The scaled model of FRB 121102 is consistent with both the lack of persistent radio emission and the low measured RM of FRB 180924 (Bannister et al. 2019) for a source age of ≳10 yr following a BNS merger/AIC.

We also explore a second model (Model B; dashed blue curve in Figure 3) in which the magnetar engine deposits a total energy of E_mag = 3 × 10⁴⁹ erg at a constant rate over a timescale t_mag = 20 yr. These are plausible parameters for a magnetar with an internal magnetic field strength B_⋆ = 10¹⁶ G given the active lifetime predicted by ambipolar diffusion⁸ (Equation (1)). Figures 2 and 3 show that for this model, the upper limits on RM and persistent radio flux of FRB 180924 are satisfied by even younger magnetars, ≳1 yr old.

Motivated by the possible origin of FRB 180924 in a BNS merger, we searched the BATSE, Fermi/GBM, and Swift/BAT catalogs for SGRBs consistent with the FRB position. We found no such sources from Swift/BAT, which provides the best positional accuracy of all three detectors (typically 3' radius), but has an instantaneous sky coverage of only about 15%.

In the BATSE catalog, which spans 1991 April to 2000 May, with full sky instantaneous coverage, two SGRBs (921115 and 940114) are consistent with the location of FRB 180924 to within about 1.5 times their large error radii of 12.7° and 9°, respectively. These error regions correspond to about 1.4%–2.8% of the sky, so given the sample of about 500 SGRBs in the BATSE catalog, it is not surprising that two SGRBs would overlap the location of FRB 180924 by pure coincidence.

Similarly, in the Fermi/GBM catalog, which extends from 2008 July to the present and provides an instantaneous sky coverage of about 65%, we identify five SGRBs that overlap the location of FRB 180924 (GRBs 121014, 131006, 141128, 150805, and 170125), but again with large error radii of 9°–27° (and nominal offsets of 0.2–1 times the GBM error radius); these error circles correspond to about 0.6%–5.4% of the sky. Given a sample of 420 SGRBS in the GBM catalog, it is therefore not surprising to find several events that overlap the location of FRB 180924.

Thus, we cannot conclusively link FRB 180924 to a specific SGRB, but plausible candidates exist in both the BATSE and Fermi/GBM catalogs.

Motivated by the localization of FRBs 180924 and 190523 to the outskirts of massive early-type host galaxies (Bannister et al. 2019; Ravi et al. 2019), we explore and develop a model for FRBs arising from magnetars born in BNS mergers and/or AIC (Nicholl et al. 2017a; Cao et al. 2018; Yamasaki et al. 2018). The environments of FRB 180924, FRB 190523, and FRB 181112 (Prochaska et al. 2019) are dramatically different from that of the repeating FRB 121102, whose low-metallicity star-forming dwarf host galaxy first led to the suggested association of FRBs with magnetars born in SLSNe and/or LGRBs (Metzger et al. 2017; Nicholl et al. 2017a). Here we show that the host galaxies and offsets of FRBs 180924, 190523, and 181112 are well-matched to the distributions for SGRBs and SNe Ia, which are proxies for BNS mergers and AIC events, respectively. We search archival data for a possible SGRB coincident with the location of FRB 180924 and find plausible candidates, although a definitive association cannot be made.

We demonstrate several likely differences between the BNS merger, AIC, and SLSN/LGRB magnetar birth scenarios and how these may affect observable properties of FRBs produced by such magnetars. These generally separate into intrinsic differences in the magnetar source activity, and external differences in the surrounding environment. Although the intrinsic differences are rather uncertain, we raise the possibility that stable magnetars born in BNS mergers would form the most massive NSs and may thus cool more efficiently via direct Urca cooling than standard lower mass NSs. This leads to shorter ambipolar diffusion timescales and faster extraction of the magnetar's magnetic energy (Equations (1) and (2); see Beloborodov & Li 2016). Extrinsic differences in the ambient medium are clearer: the ejecta into which the magnetar is born is more dilute in both the BNS merger and AIC scenarios in comparison to the SLSN/LGRB channel. The density of this ambient medium governs the expansion velocity of the magnetar nebula and therefore the timescale on which synchrotron emission from the deflating nebula peaks.

In Figure 4 we show the parameter space of nebular synchrotron emission as a function of intrinsic (horizontal axis) and extrinsic (vertical axis) properties. Black (red) contours show the peak time (luminosity) at 6 GHz of the persistent radio source associated with the magnetar nebula as a function of the magnetar engine power, ${\dot{E}}_{\mathrm{mag}}$, and the surrounding ejecta density, parameterized via ${M}_{\mathrm{ej}}{v}_{\mathrm{ej}}^{-3}$. These two parameters fully describe the nebula in the simple case assumed here in which the rate of particle and energy injection into the nebula are constant in time (see Appendix). Although adopted for simplicity of presentation and analytic tractability, we note that such a toy model was shown to not be able to reproduce in detail both the RM and persistent radio luminosity of FRB 121102 (Margalit & Metzger 2018).

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Figure 4 illustrates qualitative differences between magnetar nebulae in the different scenarios. The shaded regions of different colors show expectations for the different channels (SLSLN, LGRB, AIC, BNS mergers) spanning low/high internal magnetar B-field strengths, with/without direct Urca cooling (Equation (2)), and ejecta with both high and low mass/velocity. As discussed previously, due to the high ejecta velocity and low ejecta mass, the nebular radio emission from magnetars formed in BNS mergers and/or AIC peaks at early times, as does the time at which the ejecta become transparent to free–free absorption (visible as kinks in the black contours in Figure 4). This may render the persistent nebular radio emission of such sources more difficult to detect in targeted searches of known FRB positions than in the case of SLSNe/LGRBs, consistent with the non-detection of persistent emission from FRB 180924 in comparison with FRB 121102, as well as the lower RM of FRB 180924 (Figure 2).

The discovery of FRB emission from the remnants of either NS mergers or AIC would have major implications for our understanding of these still poorly understood events. Although frequently invoked, AIC has not yet been directly observed, and indeed even whether a O/Ne WD approaching the Chandrasekhar mass will undergo AIC or a thermonuclear detonation (producing a SN Ia-like transient) remains debated (e.g., Jones et al. 2019). Likewise, the fraction of BNS mergers giving rise to long-lived stable remnants is extremely sensitive to uncertain properties of the NS equation-of-state (Margalit & Metzger 2019). If a subset of FRBs were to originate from BNS merger remnants, their observed properties and environments relative to the comparatively unbiased merger population detected by LIGO/Virgo would offer unique insights into the diverse outcome of mergers and the properties of NSs. If FRBs are observed from weeks up to decades following BNS mergers detected by LIGO/Virgo, this may provide an invaluable means of differentiating supra-massive from indefinitely stable NS remnants and place new constraints on the NS equation of state (Margalit & Metzger 2019).

The relative fraction of detected FRB sources born in BNS mergers compared with SLSNe/LGRBs depends on the relative rates of BNS mergers, SLSNe, and LGRBs, and on the fraction of BNS mergers that can form stable NS remnants (Section 3). We estimate the latter as ≲3% (Margalit & Metzger 2019); however, this depends on the maximum mass constraint from multi-messenger analysis of GW170817 (M_TOV ≲ 2.17M_⊙; Margalit & Metzger 2017), and on the assumption that the cosmic population of merging NSs is similar to the Galactic BNS population. If the NS maximal mass were slightly higher (e.g., Lasky et al. 2014; Lü et al. 2015; Gao et al. 2016; Shibata et al. 2019) then the fraction of BNS mergers that produce stable magnetar remnants would increase. This is also the case if the mass distribution of BNS mergers is "bottom heavy" compared to the Galactic BNS population. Whether direct Urca cooling is activated in BNS merger remnants and its effects on FRB production are also uncertain, as is the fraction of SLSN/LGRB born magnetars, which can become active FRB sources. Thus, the relative fraction of FRB sources observed in host galaxies representative of the SLSN/LGRB population compared to those detected in environments similar to where BNS mergers/AIC occur is highly uncertain.

We caution also that at present there are only three published well-localized FRBs (FRB 121102, 180924, 190523; although see Mahony et al. 2018 and Li et al. 2019, who attempt to constrain possible host galaxies of certain non-localized FRBs), therefore population inferences are limited by small number statistics. Increasing the sample size of well-localized FRBs will be crucial for understanding the population as a whole.

An additional signature that may be associated with FRBs from BNS mergers and/or AIC is late-time (∼decade timescale) radio emission due to interaction of the dynamical merger ejecta with the ambient medium. If accelerated by the initial spin-down energy deposition of the newborn magnetar, this signal may be detectable even at large distances and is also a useful diagnostic in constraining the NS equation of state (Margalit & Metzger 2017). As one test of the hypothesis that FRBs originate from a subset of BNS mergers that produce stable magnetar remnants, and as a future means of constraining the properties of NSs, we propose late-time searches for FRB emission from the locations of SGRBs and GW-detected BNS mergers.

Following the submission of our paper, a related work appeared by Beloborodov (2019). This work also argues that stable magnetar remnants formed in BNS mergers, with potentially short core ambipolar diffusion timescales, could produce repeating FRB sources associated with more diverse host Galaxy demographics.

B.D.M. and B.M. acknowledge many conversations with Andrei Beloborodov. B.M. would like to thank the organizers and participants of FRB2019 for a highly enjoyable and productive meeting. B.M. is supported by NASA through the NASA Hubble Fellowship grant #HST-HF2-51412.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555. B.D.M. acknowledges support from the NASA Astrophysics Theory Program (grant NNX16AB30G). The Berger Time-Domain Group is supported in part by NSF grant AST-1714498 and NASA grant NNX15AE50G. E.B. thanks the Radcliffe Institute for Advanced Study for its support through the Radcliffe Fellowship. Portions of this work were completed at KITP, supported in part by the National Science Foundation under grant No. NSF PHY-1748958.

In the following we derive an approximate solution describing the nebular radio emission in the simplified case of temporally constant energy (and particle) injection rate for a magnetar embedded within a homologously expanding ejecta with a radially constant density distribution. This model is therefore described by the two parameters $\dot{E}$ and $\zeta \equiv {M}_{\mathrm{ej}}/{v}_{\mathrm{ej}}^{3}$, where ζ is a proxy for the ejecta density (${\rho }_{\mathrm{ej}}\sim \zeta {t}^{-3}$).

If radiative losses are negligible, and treating the nebula plasma as a relativistic fluid, then the nebula expands within the confining ejecta (which is assumed to be homologously expanding) with a velocity

$$\begin{eqnarray}&&{v}_{{\rm{n}}}\sim {\left(\displaystyle \frac{\dot{E}t}{\zeta }\right)}^{1/5}\end{eqnarray} \tag{ 6 }$$

and corresponding size

$$\begin{eqnarray}&&{R}_{{\rm{n}}}=\int {v}_{{\rm{n}}}\,{dt}\sim \displaystyle \frac{5}{6}{\left(\displaystyle \frac{\dot{E}}{\zeta }\right)}^{1/5}{t}^{6/5}.\end{eqnarray} \tag{ 7 }$$

Assuming a tangled nebular magnetic field, adiabatic losses imply the field strength is determined by the magnetic energy injected within the last dynamical (nebula expansion) timescale, $\sim \sigma \dot{E}t$, where σ is the ratio of magnetic to kinetic energy injected by the magnetar. The nebula magnetic field is therefore

$$\begin{eqnarray}&&{B}_{{\rm{n}}}\sim {\left(\displaystyle \frac{6\sigma \dot{E}t}{{R}_{{\rm{n}}}^{3}}\right)}^{1/2}\sim {\left(\displaystyle \frac{6}{5}\right)}^{3/2}{\left(6\sigma \right)}^{1/2}{\dot{E}}^{1/5}{\zeta }^{3/10}{t}^{-13/10}.\end{eqnarray} \tag{ 8 }$$

A.1. Electron Distribution

We assume electrons are injected into the nebula with a characteristic Lorentz factor γ_inj and at a constant rate,

$$\begin{eqnarray}&&{\dot{N}}_{\gamma }\left(\gamma \right)\sim {\chi }^{-1}\dot{E}\delta \left(\gamma -{\gamma }_{\mathrm{inj}}\right),\end{eqnarray} \tag{ 9 }$$

where ${\gamma }_{\mathrm{inj}}\approx \chi /2{m}_{e}{c}^{2}$ and χ ≳ 0.2 GeV is the mean energy per e–p ejected in each baryon flare (the minimum value is set by the escape speed from the NS surface, and for γ_inj we assume the electrons and protons enter the nebula in equipartition after passing through the wind-termination shock).

Given the electron injection rate and some specified cooling process setting $d\gamma /{dt}={\dot{\gamma }}_{\mathrm{cool}}(\gamma ,t)$, the particle distribution can be calculated from number conservation, which implies⁹

$$\begin{eqnarray}&&{N}_{\gamma }(\gamma ,t)={\int }_{\gamma }^{\infty }{\dot{N}}_{\gamma }({\gamma }_{0},{t}_{0})\left(\displaystyle \frac{\partial {t}_{0}}{\partial \gamma }\right)\,d{\gamma }_{0}.\end{eqnarray} \tag{ 10 }$$

Here the subscript zero conveys that particles of Lorentz factor γ at time t had at earlier times t₀ < t a higher Lorentz factor γ₀ > γ, and the relation between γ, t, γ₀, t₀ is determined by the cooling process.

For adiabatic cooling,

$$\begin{eqnarray}&&{\dot{\gamma }}_{\mathrm{cool}}(\gamma ,t)={\dot{\gamma }}_{\mathrm{ad}}=-\displaystyle \frac{\gamma }{t},\end{eqnarray} \tag{ 11 }$$

and the solution to the differential equation $d\gamma /{dt}=\dot{\gamma }$ is trivial,

$$\begin{eqnarray}&&{t}_{0}=t\gamma /{\gamma }_{0},\,\mathrm{adiabatic}\,\mathrm{cooling}.\end{eqnarray} \tag{ 12 }$$

The resulting distribution implied by Equation (10) is then

$$\begin{eqnarray}\begin{array}{rcl}{N}_{\gamma }(\gamma ,t) & = & \displaystyle \frac{{\dot{N}}_{{\rm{i}}}{t}_{{\rm{i}}}}{{\gamma }_{\mathrm{inj}}}{\left(\displaystyle \frac{t}{{t}_{{\rm{i}}}}\right)}^{1-\alpha }{\left(\displaystyle \frac{\gamma }{{\gamma }_{\mathrm{inj}}}\right)}^{-\alpha }\\ & = & \displaystyle \frac{\dot{E}}{\chi {\gamma }_{\mathrm{inj}}}\left\{\begin{array}{l}0,\,\gamma \gt {\gamma }_{\mathrm{inj}}\\ {\gamma }^{0}t,\,\gamma \lt {\gamma }_{\mathrm{inj}}\end{array}\right.,\,\mathrm{adiabatic}\,\mathrm{cooling}.\end{array}\end{eqnarray} \tag{ 13 }$$

In the first equality we have generalized the particle injection (Equation (9)) to a power-law time dependence, ${\dot{N}}_{\gamma }={\dot{N}}_{{\rm{i}}}{(t/{t}_{{\rm{i}}})}^{-\alpha }\delta (\gamma -{\gamma }_{\mathrm{inj}})$, and in the last line we have reverted to the constant particle injection case considered here for simplicity (α = 0).

For synchrotron cooling with a general power-law decaying magnetic field ${B}_{n}\propto {t}^{-\beta /2}$ (as in Equation (8)),

$$\begin{eqnarray}&&{\dot{\gamma }}_{\mathrm{cool}}(\gamma ,t)={\dot{\gamma }}_{\mathrm{syn}}=-{\gamma }^{2}\displaystyle \frac{{\sigma }_{T}{{cB}}_{n}^{2}/8\pi }{{m}_{e}{c}^{2}}\equiv -{\gamma }^{2}{t}^{-\beta }A,\end{eqnarray} \tag{ 14 }$$

and the solution for the Lorentz factor evolution with time implies

$$\begin{eqnarray}&&{t}_{0}={\left[\displaystyle \frac{\beta -1}{A}\left({\gamma }^{-1}-{\gamma }_{0}^{-1}\right)+{t}^{-(\beta -1)}\right]}^{-1/(\beta -1)},\end{eqnarray} \tag{ 15 }$$

synchrotron cooling.

For the case of interest in this model (Equation (8)), we have

$$\begin{eqnarray}&&\beta =13/5,\,A=\displaystyle \frac{3{\sigma }_{{\rm{T}}}}{4\pi {m}_{e}c}{\left(\displaystyle \frac{6}{5}\right)}^{3}\sigma {\dot{E}}^{2/5}{\zeta }^{3/5}.\end{eqnarray} \tag{ 16 }$$

The electron Lorentz factor distribution (Equation (10)) can then be integrated to give

$$\begin{eqnarray}&&\begin{array}{l}{N}_{\gamma }(\gamma ,t)={\dot{N}}_{{\rm{i}}}{\left(\displaystyle \frac{{t}_{0}[\gamma ,t]}{{t}_{{\rm{i}}}}\right)}^{-\alpha }\displaystyle \frac{1}{A{\gamma }^{2}}\\ \,\,\,\times \,{\left[\displaystyle \frac{\beta -1}{A}\left({\gamma }^{-1}-{\gamma }_{\mathrm{inj}}^{-1}\right)+{t}^{-(\beta -1)}\right]}^{-\beta /(\beta -1)}\\ \longrightarrow \displaystyle \frac{\dot{E}}{A\chi }\left\{\begin{array}{ll}0, & \gamma \gt {\gamma }_{\mathrm{inj}}\\ {\gamma }^{-2}{t}^{\beta }, & {\gamma }_{\mathrm{cool}}(t)\ll \gamma \ll {\gamma }_{\mathrm{inj}}\\ {\gamma }^{\tfrac{2-\beta }{\beta -1}}{t}^{0}{\left(\tfrac{A}{\beta -1}\right)}^{\tfrac{\beta }{\beta -1}}, & \gamma \ll {\gamma }_{\mathrm{cool}}(t)\end{array}\right.,\\ \qquad \mathrm{synchrotron}\,\mathrm{cooling},\end{array}\end{eqnarray} \tag{ 17 }$$

where ${\gamma }_{\mathrm{cool}}(t)\equiv {A}^{-1}(\beta -1){t}^{\beta -1}$ and in the last equality we have again reverted to the simplified case considered here where the particle injection rate is constant in time, i.e., α = 0.

A.2. Synchrotron Radiation

Optically thick synchrotron emission from the nebula is easy to calculate first, as it does not depend on the particle distribution, which is the major complexity in analytically describing the nebula. We remember that the Lorentz factor of an electron whose peak emission is at frequency ν, is

$$\begin{eqnarray}&&\gamma (\nu ,t)\approx {\left(\displaystyle \frac{2\pi {m}_{e}c\nu }{{{eB}}_{n}(t)}\right)}^{1/2}.\end{eqnarray} \tag{ 18 }$$

The optically thick synchrotron emission can be approximated as ${L}_{\nu }\approx 8{\pi }^{2}{R}_{n}^{2}{kT}(\gamma ){\nu }^{2}/{c}^{2}$ with ${kT}(\gamma )\approx \gamma {m}_{e}{c}^{2}/3$, resulting in

$$\begin{eqnarray}&&\begin{array}{l}{L}_{\nu }\left(\nu \lt {\nu }_{\mathrm{ssa}}\right)\sim \displaystyle \frac{8{\pi }^{2}}{3}{\left(\displaystyle \frac{2\pi {m}_{e}^{3}}{e}\right)}^{1/2}{\left(\displaystyle \frac{5}{6}\right)}^{11/4}\\ \times \,{\left(6\sigma \right)}^{-1/4}{\nu }^{5/2}{\dot{E}}^{3/10}{\zeta }^{-11/20}{t}^{61/20}\\ \simeq \,8.50\times {10}^{28}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}\,{\nu }_{10}^{2.5}{\sigma }_{-1}^{-0.25}{\dot{E}}_{41}^{0.3}{\zeta }_{-5}^{-0.55}{t}_{7}^{3.05}.\end{array}\end{eqnarray} \tag{ 19 }$$

Here ν₁₀ = ν/10¹⁰ Hz, σ₋₁ = σ/0.1, ${\dot{E}}_{41}=\dot{E}/{10}^{41}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$, and t₇ = t/10⁷ s denote quantities in cgs units, while ${\zeta }_{-5}\,=\zeta /\left[{10}^{-5}\left({M}_{\mathrm{ej}}/{M}_{\odot }\right){\left({v}_{\mathrm{ej}}/{10}^{4}\mathrm{km}{{\rm{s}}}^{-1}\right)}^{-3}\right]$.

The optically thin synchrotron emission depends directly on the population of emitting electrons as

$$\begin{eqnarray}&&{L}_{\nu }\approx \displaystyle \frac{3{e}^{3}}{{m}_{e}{c}^{2}}\gamma {N}_{\gamma }{B}_{n}.\end{eqnarray} \tag{ 20 }$$

Although no simple analytic solution for the electron distribution N_γ exists in the case where both synchrotron and adiabatic cooling are important, we find in practice and in comparison with numerical calculations that Equation (17) in the regime where γ ≲ γ_cool(t) reasonably approximates the electron distribution at times of interest (near peak light curve in the GHz band) for a wide range of parameters, even when adiabatic cooling is included in addition to synchrotron cooling.

The optically thin synchrotron flux in this regime is

$$\begin{eqnarray}&&\begin{array}{l}{L}_{\nu }\left(\nu \gt {\nu }_{\mathrm{ssa}}\right)\sim {\left(\displaystyle \frac{5}{8}\right)}^{13/8}{\left(\displaystyle \frac{6}{5}\right)}^{93/32}\\ \,\times \,{\left(32\pi \right)}^{-5/16}\displaystyle \frac{3{e}^{43/16}{\sigma }_{{\rm{T}}}^{5/8}}{{m}_{e}^{21/16}{c}^{37/16}}\\ \,\times \,{\left(6\sigma \right)}^{31/32}{\chi }^{-1}{\dot{E}}^{111/80}{\zeta }^{93/160}{t}^{-143/160}{\nu }^{5/16}\\ \,\simeq \,4.46\times {10}^{28}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}\,{\nu }_{10}^{0.31}\\ \,\times \,{\chi }_{0.2\mathrm{GeV}}^{-1}{\sigma }_{-1}^{0.97}{\dot{E}}_{41}^{1.39}{\zeta }_{-5}^{0.58}{t}_{7}^{-0.89},\end{array}\end{eqnarray} \tag{ 21 }$$

where ${\chi }_{0.2\mathrm{GeV}}=\chi /0.2\,\mathrm{GeV}$. Putting aside external free–free absorption by the ejecta for the moment, the nebula synchrotron luminosity is the minimum of Equations (19) and (21). The intrinsic peak of the light curve at a given frequency therefore coincides with the synchrotron self-absorption turnover, which occurs when the optically thick and optically thin synchrotron expressions are roughly equal. Equating (19) and (21), we find the time of the self-absorption turnover,

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{ssa}} & \simeq & 8.49\times {10}^{6}\,{\rm{s}}\,{\nu }_{10}^{-350/631}{\sigma }_{-1}^{195/631}\\ & & \times \,{\chi }_{0.2\mathrm{GeV}}^{-160/631}{\dot{E}}_{41}^{174/631}{\zeta }_{-5}^{181/631}.\end{array}\end{eqnarray} \tag{ 22 }$$

The corresponding peak luminosity can be obtained by plugging this timescale back into Equation (19), giving

$$\begin{eqnarray}\begin{array}{rcl}{L}_{\nu }\left({t}_{\mathrm{ssa}}\right) & \simeq & 5.16\times {10}^{28}\,\mathrm{erg}\,{{\rm{s}}}^{-1}\,{\mathrm{Hz}}^{-1}\,{\nu }_{10}^{510/631}\\ & & \times \,{\sigma }_{-1}^{437/631}{\chi }_{0.2\mathrm{GeV}}^{-488/631}{\dot{E}}_{41}^{720/631}{\zeta }_{-5}^{205/631}.\end{array}\end{eqnarray} \tag{ 23 }$$

Equations (22) and (23) approximate the peak time and luminosity of the nebular radio emission if this peak occurs after the surrounding ejecta has already become transparent to free–free absorption. Otherwise, the observed emission peaks at the time when the ejecta eventually does become free–free transparent at radio frequencies, t_ff, and the nebular synchrotron emission is in the optically thin regime (21), i.e.

$$\begin{eqnarray}&&{t}_{\mathrm{pk}}=\max \left({t}_{\mathrm{ssa}},{t}_{\mathrm{ff}}\right)\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&{L}_{\nu ,\mathrm{pk}}=\left\{\begin{array}{l}{L}_{\nu }\left({t}_{\mathrm{ssa}}\right),\,{t}_{\mathrm{pk}}={t}_{\mathrm{ssa}}\\ {L}_{\nu \gt {\nu }_{\mathrm{ssa}}}\left({t}_{\mathrm{ff}}\right)\approx {L}_{\nu }\left({t}_{\mathrm{ssa}}\right){\left(\tfrac{{t}_{\mathrm{ff}}}{{t}_{\mathrm{ssa}}}\right)}^{-143/160},\,{t}_{\mathrm{pk}}={t}_{\mathrm{ff}}.\end{array}\right.\end{eqnarray} \tag{ 25 }$$

Even for the simple model assumed above (constant energy injection rate, delta-function Lorentz factor injection of electrons) the expressions derived in Equations (21)–(23) are only approximate estimates. This is due to approximations made in solving the electron Lorentz factor distribution function, which include among other things the neglect of adiabatic cooling, and inhibition of synchrotron cooling in the synchrotron self-absorption optically thick regime. Nevertheless, we find that these estimates typically agree with our numerical simulations (which were used to construct models A and B illustrated in Figures 2 and 3) to within a factor of ∼several in the range of parameters explored for our purposes.