Shock within a shock: revisiting the radio flares of NS merger ejecta and GRB-supernovae
MMNRAS , 1–13 (2015) Preprint 29 April 2020 Compiled using MNRAS L A TEX style file v3.0
Shock within a shock: revisiting the radio flares of NS merger ejectaand GRB-supernovae
Ben Margalit (cid:63) † and Tsvi Piran Astronomy Department and Theoretical Astrophysics Center, University of California, Berkeley, Berkeley, CA 94720, USA Racah Institute of Physics, Edmund J. Safra Campus, Hebrew University of Jerusalem, Jerusalem 91904, Israel
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Fast ejecta expelled in binary neutron star (NS) mergers or energetic supernovae (SNe)should produce late-time synchrotron radio emission as the ejecta shocks into the sur-rounding ambient medium. Models for such radio flares typically assume the ejecta ex-pands into an unperturbed interstellar medium (ISM). However, it is also well-known thatbinary NS mergers and broad-lined Ic SNe can harbor relativistic jetted outflows. In thiswork, we show that such jets shock the ambient ISM ahead of the ejecta, thus evacuatingthe medium into which the ejecta subsequently collides. Using an idealized spherically-symmetric model, we illustrate that this inhibits the ejecta radio flare at early times t < t col ≈
12 yr ( E j / erg ) / ( n / − ) − / ( v ej / . c ) − / where E j is the jet energy, n the ISM density, and v ej the ejecta velocity. We also show that this can produce a sharplypeaked enhancement in the light-curve at t = t col . This has implications for radio observationsof GW170817 and future binary NS mergers, gamma-ray burst (GRB) SNe, decade-long radiotransients such as FIRST J1419, and possibly other events where a relativistic outflow precedesa slower-moving ejecta. Future numerical work will extend these analytic estimates and treatthe multi-dimensional nature of the problem. Key words: gamma-ray bursts – neutron star mergers – transients: supernovae – shock waves– radiation mechanisms: non-thermal – radio continuum: transients
Shocks are ubiquitous phenomena in astrophysical settings and arewell-studied sites of magnetic field amplification and non-thermalparticle acceleration (Bell 1978; Blandford & Eichler 1987). Conse-quently, shocks are capable of producing bright synchrotron emis-sion detectable across the electromagnetic spectrum. Models ofsynchrotron shock emission have been extremely successful in ex-plaining observations of radio supernovae (SNe; Chevalier 1982,1998), gamma-ray burst (GRB) afterglows (Paczynski & Rhoads1993; Mészáros & Rees 1997; Rhoads 1997; Sari et al. 1998; Rhoads1999) and other events associated with fast outflows and/or occur-ring in dense environments (see e.g. Fig. 5 of Margutti et al. 2018a).Binary neutron star (BNS) mergers can expel ∼ − − − M (cid:12) of material at a fraction of the speed-of-light (Lattimer & Schramm1974, 1976; Rosswog et al. 1999; Bauswein et al. 2013; Radiceet al. 2016; Siegel & Metzger 2017). Radioactive decay of freshly-synthesized r -process elements in this ejected material can sub-sequently power an optical/near-infrared kilonova (also knownas macronova; Li & Paczyński 1998; Kulkarni 2005; Metzger (cid:63) NASA Einstein Fellow † E-mail: [email protected] et al. 2010; Barnes & Kasen 2013; Tanaka & Hotokezaka 2013),such as has been recently observed in conjunction with the firstgravitational-wave (GW) detected BNS merger, GW170817 (Coul-ter et al. 2017; Cowperthwaite et al. 2017; Kasliwal et al. 2017;Tanvir et al. 2017; Abbott et al. 2017).Similarly to shocks in radio SNe, the expanding kilonova ejectahas been predicted to produce a ∼ decade-long synchrotron radioflare as the ejecta shocks the ambient interstellar medium (ISM;Nakar & Piran 2011). The work of Nakar & Piran (2011) has beenextended by various authors to account for more realistic kilonova-ejecta distributions (e.g. Piran et al. 2013; Margalit & Piran 2015;Hotokezaka & Piran 2015) with qualitatively similar results, andhas been applied to GW170817 in attempt to predict it’s radio-flare signature and observationally constrain the ejecta parameters(e.g. Alexander et al. 2017; Kathirgamaraju et al. 2019; Hajela et al.2019). It has also been used to place limits on ejecta radio flares thatwould be expected to accompany short GRBs (Metzger & Bower2014; Horesh et al. 2016; Fong et al. 2016; Klose et al. 2019).Here we point out a shortcoming in these models that has beenpreviously overlooked. Ejecta radio-flare models in the literaturetypically assume that the kilonova ejecta is expanding into a coldunperturbed (constant-density, static) ISM, however — BNS mergerevents like GW170817 produce an ultra-relativistic collimated jet © 2015 The Authors a r X i v : . [ a s t r o - ph . H E ] A p r B. Margalit & T. Piran (Mooley et al. 2018b) that precedes the kilonova ejecta. This jet runsinto and shocks the surrounding ISM before the ejecta, thereforechanging the medium with which the ejecta subsequently collides(Fig. 1). In the following we explore the effect that this “preshaping”of the ambient medium ahead of the ejecta has on the predictedejecta radio-flare.The situation described above for BNS mergers is also rele-vant in other astrophysical settings as well. A completely analogoussituation occurs for long GRBs, which are known to be accompa-nied by energetic broad-lined Ic SNe (Galama et al. 1998; Bloomet al. 1999; Hjorth et al. 2003; Woosley & Bloom 2006). The late-time radio emission from interaction of these energetic SNe ejectawith the surrounding ISM has been studied by Barniol Duran &Giannios (2015) and subsequently extended and applied to obser-vations (Kathirgamaraju et al. 2016; Peters et al. 2019). Such studieshowever similarly have yet to account for the interaction betweenthe GRB-jet and SN-ejecta, and have instead treated these as inde-pendent components propagating into an unperturbed ISM. In §4.4we apply our results to this problem and show that this interactioncan affect the predicted light-curves and implied source-parameterconstraints.This paper is structured as follows: we begin by reviewing thestandard results for jet afterglows and ejecta radio-flares producedwhen these run independent of one another into an unperturbedconstant-density ISM (§2). In §3 we extend these results and cal-culate the affect that interaction between these two componentshas on the resulting light-curve. We then discuss implications ofour results in application to various astrophysical settings (§4), andconclude with a discussion of caveats, observational implications,and directions for future work (§5). We focus on an ISM cirum-stellar medium in the main text, but generalize our results to a windenvironment in Appendix B.
We begin by discussing the relevant timescales and standard re-sults for jet afterglow and ejecta radio flares expanding into a cold,constant-density ISM. Much of this has been previously discussedby Barniol Duran & Giannios (2015) and references therein. In thefollowing, we repeat and summarize the main results for complete-ness.
From dimensional arguments alone, a characteristic timescale fora relativistic jet of total energy E j propagating into an ISM ofnumber density n is τ j = (cid:32) E j nm p c (cid:33) / ≈ .
20 yr E / , n − / . (1)Above and in the following we adopt the notation Q x ≡ Q / x forany quantity Q in appropriate cgs units.A collimated relativistic jet of half-opening angle θ j andLorentz factor Γ (cid:29) θ − will initially propagate radially without ap-preciable lateral expansion since there is no causal contact betweendifferent regions of the jet. In typical astrophysical scenarios the jet Note that we consider here the total energy of the jet and not the isotropicequivalent energy as commonly used in the afterglow literature. is double-sided, i.e. there will be two antipodally-symmetric jets (seeFig. 1), and we assume here that both have identical half-opening an-gles θ j and energy E j / E j is the total energy of thebipolar outflow). While the jet is ultra-relativistic and collimated it’sevolution is well described by the Blandford-McKee solution withthe isotropic-equivalent energy E j , iso = E j /( − cos θ j ) ≈ E j / θ (Blandford & McKee 1976).This breaks down once the jet Lorentz factor drops to Γ ∼ θ − ,at which point different regions in the jet come into causal contactand azimuthal spreading can commence. The time at which thisoccurs is known as the jet-break time, which, measured in the labframe is (Sari et al. 1999) t jb = (cid:32) E j , iso π nm p c θ − (cid:33) / = (cid:18) π (cid:19) / τ j . (2)In the observer frame this is measured to occur earlier by a factor1 / Γ ( t jb ) ∼ θ (cid:28) Following the jet-break time (eq. 2) the jet can start spreadingazimuthally. The detailed dynamics of this jet-spreading phase havebeen investigated by many works (Rhoads 1999; Granot et al. 2001;Ayal & Piran 2001; Zhang & MacFadyen 2009; Wygoda et al. 2011;De Colle et al. 2012; Granot & Piran 2012; Duffell & Laskar 2018).Initial analytic models suggested a fast exponential phase where thejet sphericizes at t ∼ t jb (Rhoads 1999), however later numericalsimulations and semi-analytic models indicate that jet spreading isdelayed and only begins in earnest once the (still-collimated) jetdecelerates to non-relativistic velocities at t = t NR (De Colle et al.2012; Granot & Piran 2012). This time is defined by the Sedov-Taylor timescale of the isotropic-equivalent one-sided jet energy E j , iso . Thus, the initially-collimated jet becomes quasi-spherical and approaches the spherical Sedov-Taylor solution for an explosionof energy E j after time t = t sph , where t sph (cid:38) t NR ∼ (cid:32) E j , iso π nm p c (cid:33) / = (cid:18) π (cid:19) / θ − / τ j (3) ≈ .
73 yr E / , n − / θ − / , − . Note that t sph can exceed t NR by a non-negligible factor ∼ − t (cid:29) t sph , t NR the “jet” is non-relativistic and sphericaland is therefore well-described by the Sedov-Taylor solution. Wecontinue to term this component the ‘jet’ even though at such timesthe outflow has lost it’s initially jet-like features (strong collimationand ultra-relativistic velocities). This is to distinguish it from thematter-dominated ejecta, which is also sub-relativistic and quasi-spherical. The jet-ISM forward shock position at times t (cid:29) t sph , t NR are thus R j = ξ (cid:18) E j nm p (cid:19) / t / ; v j = R j ( t ) t , (4)where the numerical constant ξ (cid:39) .
17 is determined by the Sedov-Taylor solution for an adiabatic index of g = / Here and in the following derived expressions we do not include explicitdependence on cosmological redshift ∝ ( + z ) − . Early work by Ayal & Piran (2001) showed that the outflow will becomefully spherical on a time scale of few thousand years (see also Ramirez-Ruiz& MacFadyen 2010). However, quasi-sphericity is in fact reached muchearlier, on the timescale that we estimate here (Zhang & MacFadyen 2009).MNRAS , 1–13 (2015) jecta Radio Flares Associated with Jets ISMshocked-ISM shocked-ISMejecta ejecta ejectajet jet ISM ISM 𝑡 !" ≤ 𝑡 ≤ 𝑡 $%& 𝑡 $%& ≤ 𝑡𝑡 ≤ 𝑡 !" !" 𝑣 !" 𝑣 " 𝑣 !" 𝑡 𝑡 Figure 1.
Cartoon illustration of the two components considered in this work — an initially ( t < t sph ; eq. 3) collimated ultra-relativistic jet (blue) andsub-relativistic spherical ejecta (orange) that expand simultaneously into an ambient ISM (green). The jet decelerates and spreads azimuthally as it sweeps-upan increasing ISM mass, until, at t (cid:38) t sph , t NR , the jet has become sub-relativistic and quasi-spherical. At this point, the jet forward-shock expanding at v = v j is well described by the Sedov-Taylor solution (eq. 4), and the ejecta which begins catching up with the jet forward shock propagates within the cavity ofjet-shocked ISM. At time t = t col the ejecta overtakes the jet-ISM forward shock (eq. 9) and subsequently expands into an unperturbed ISM. phase till radiative losses at the shock front become energeticallyimportant. This does not occur on timescales of interest for this prob-lem, and therefore we do not discuss subsequent evolution phasesof the jet-ISM shock.The bulk ejecta of SNe or BNS mergers are launched quasi-spherically at sub-relativistic velocities β ej = v ej / c <
1. Althoughdifferent launching mechanisms likely contribute to different an-gular regions of BNS merger kilonova ejecta (e.g. equatorial tidalejecta vs. polar disk winds and shock outflows; see Shibata & Ho-tokezaka 2019 for a recent review), the global structure of the kilo-nova ejecta remains quasi-spherical, especially in cases where bothpolar and equatorial outflows are present (if the remnant does notpromptly collapse to a black hole; see e.g. Margalit & Metzger2019). Deposition of thermal energy through radioactive decay of r -process material in the ejecta acts to further drive the ejecta towardsspherical-symmetry (Grossman et al. 2014). Although the oblate na-ture of this ejecta (e.g. Radice et al. 2018) can impact some detailsof the afterglow emission (Margalit & Piran 2015), the overall light-curve is reasonably approximated by spherically-symmetric mod-els. In the following we therefore assume a spherically-symmetricejecta.A further complication arises from the velocity-structure ofSNe or kilonova ejecta. Even for spherical ejecta, the outflow isexpected to have a distribution of velocities with a small, but po-tentially important, amount of material moving at velocities muchgreater than the “bulk” velocity of the ejecta (e.g. Chevalier &Soker 1989; Matzner & McKee 1999; Nakar & Piran 2011). In thefollowing we neglect these details and treat the simplified case ofa so-called ‘single-velocity shell’ (see Margalit & Piran 2015) —approximating the ejecta as an outflow with kinetic energy E ej ex-panding with a single bulk velocity v ej . This captures the qualitativefeatures of the problem and is also accurate at late times ( (cid:29) t dec ;eq. 5), however underestimates the ejecta radio-flare luminosity atearly times. In Appendix A we extend some of the results of thefollowing sections to an ejecta with a velocity structure, still withinthe framework of a non-relativistic spherical problem. For BNS mergers, the outflow velocity profile is intricately related to its non-spherical structure, a problem we leave for investigation in futurenumerical work.A massive ejecta expanding at velocity v ej into a cold constant-density ISM will initially coast at constant velocity v ej ( R ej = v ej t ).This will change and the ejecta become affected by the ambientmedium once it has swept up an ISM mass comparable to its own( ∼ E ej / v ). The characteristic time for this to occur is t dec = (cid:169)(cid:173)(cid:171) E ej π nm p v (cid:170)(cid:174)(cid:172) / ≈
27 yr E / , n − / β − / , − (5)known as the deceleration, or Sedov-Taylor, timescale. The latterstems from the fact that, at t (cid:29) t dec the ejecta asymptotes to theSedov-Taylor solution (i.e. the initial mass of the ejecta is negligiblecompared to the swept-up ISM mass, and therefore the shock isdescribable as a point-explosion of energy E ej ). Nakar & Piran(2011) showed that at typical ∼ GHz radio frequencies, the radioflare of sub-relativistic ejecta peak at t = t dec . Having discussed in the previous section the dynamics of outflowspropagating into a cold constant-density ISM, we are now in aposition to outline relevant results regarding the radio synchrotronemission produced by such interaction (see Nakar & Piran 2011,and references therein for further details).Synchrotron emission from non-relativistic shocks was firstdiscussed in the radio-SN literature (e.g. Chevalier 1982, 1998).Here we follow the formalism of Sironi & Giannios (2013) that wasdeveloped in the context of GRB afterglows. One pertinent point isthe fact that, at times of interest, both jet and ejecta are marginallywithin the ‘deep-Newtonian regime’ discussed by Sironi & Giannios(2013). When the velocity of the forward-shock propagating intothe ISM is sub-relativistic the bulk of electrons are not acceleratedto relativistic velocities and this has to be taken into account in
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MNRAS000 , 1–13 (2015)
B. Margalit & T. Piran t jb , obs = t jb θ
15 hr t θ = t jb , obs ( θ obs /θ j ) /
47 d t NR t sph .
72 yr t DN .
76 yr t eq . t col
12 yr t dec
27 yrlog( t ) -3 -2 -1 l og ( L ν )( a r b i t r a r y un i t s ) t − p − / t − p t − p t − p − / t − p +1) / t − p +1) / t ejectajet (off-axis)(on-axis) . θ . θ − / θ / . θ − / † / e . β − / . β − / . β − / ε − / ( × τ j ) Figure 2.
Schematic jet-afterglow (blue) and ejecta radio-flare (yellow) light-curves assuming these are independent of one-another (i.e. neglecting interactionbetween the two outflow components and instead assuming that both propagate into a cold constant-density ISM). The regime of focus in this present workis t eq < t < t col (shaded region), where we predict deviations from the standard light-curve illustrated above; a strong suppression at first and a brightpeak towards the end (see Fig. 3 and §3). The top horizontal axis presents the hierarchy of timescales measured in the observer frame with respect to τ j ≈ . E / , n − / (eq. 1), and as a function of the dimensionless parameters of the problem: the ratio of jet to ejecta energy ε ≡ E j / E ej , the ejectavelocity β ej = v ej / c , shock acceleration microphysical parameters ¯ (cid:15) e ≡ (cid:15) e ( p − )/( p − ) , the initial jet opening angle θ j , and the angle between the observerand jet axis θ obs . These timescales are described in the main text. The time values labeled at the very bottom are calculated for a fiducial set of parameters( E j = erg , n = − , θ j = . , θ obs = . , ¯ (cid:15) e = . , β ej = . , E ej = erg) and can change dramatically for different parameters. Temporal scalingsfor different portions of the light-curve are also listed (Sari et al. 1999; Frail et al. 2000; Nakar & Piran 2011; Sironi & Giannios 2013). The dashed portionof the blue curve shows the light-curve for an on-axis observer, while the solid is for an off-axis observer. The light-curve in the latter case peaks once the jetLorentz factor decelerates to Γ ∼ θ − , which occurs at time t θ = t jb , obs ( θ obs / θ j ) / in the observer frame. the shock accelerated synchroton emission estimates. This regimecommences once the shock velocity drops below v (cid:46) v DN = (cid:18) m e m p (cid:19) / ¯ (cid:15) − / e c ≈ . c ¯ (cid:15) − / e , − , (6)where ¯ (cid:15) e ≡ (cid:15) e ( p − )/( p − ) , 2 < p < (cid:15) e isthe fractional shock power diverted to this non-thermal electronpopulation. Synchrotron afterglow emission at times t < t DN (when v > v DN ) peaks at the frequency of emitting electrons with Lorentzfactor γ ∼ γ m (cid:29)
1, while in the deep-Newtonian regime ( t > t DN )emission is dominated by electrons with γ ∼
2. This causes theafterglow light-curve to decay less steeply with time following thedeep-Newtonian transition (Sironi & Giannios 2013).At frequencies ν > ν m of electrons with Lorentz factors γ m ,the optically-thin synchrotron luminosity is L ν ∝ γ ( dN / d γ ) B ∝ γ p − m B ( p + )/ N where B ∝ n / v is the post-shock amplified mag-netic field, and N ∝ nR the total number of radiating electrons.At early times ( t < t DN ) γ m ∝ v , while in the deep-Newtonianregime γ m , eff ∼ γ > γ m is reduced by a factor ζ ∝ v , so that N → ζ N (Sironi& Giannios 2013). The synchrotron luminosity thus scales as L ν ∝ n p + R (cid:40) v p − ; t < t DN v p + ; t > t DN . (7)At times t < t dec the ejecta coasts at a constant velocity and thereforeits luminosity scales as L ν ∝ t , whereas in the Sedov-Taylor phase— that is at t > t sph , t NR for the jet and t > t dec for the ejecta —the radio synchrotron luminosity of these outflows scales as L ν ∝ t − ( p + )/ in the deep-Newtonian regime and L ν ∝ t − ( p − )/ otherwise (Frail et al. 2000; Nakar & Piran 2011; Sironi & Giannios2013).From eq. (7) we can estimate the time t eq at which, neglect-ing any interaction between jet and ejecta , the two outflows’ radioemission would equal one another. Since by this assumption, bothejecta and jet propagate into the same cold constant-density ISM,the factor depending on n in eq. (7) is identical for both. At times t sph < t < t dec however, the dynamics of these components differ.Using eq. (4) along with R ej = v ej t we find t eq = ξ / β − / τ j (cid:16) (cid:17) ( p − ) ( p + ) ; t eq < t DN (cid:16) (cid:17) ( p + ) ( p + ) ; t eq > t DN (8) ≈ E / , n − / β − / , − , where the fiducial value in the last line is estimated for 2 ≤ p ≤ .
7. This result is consistent within factors of order unity with thepreviously derived result by Barniol Duran & Giannios (2015). Figure 2 shows the schematic radio synchrotron light-curvesboth for a jet afterglow (blue) and an ejecta radio flare (yellow),illustrating the hierarchy of timescales and temporal scalings de-scribed in this section. Note that in this figure, as in the abovediscussion, we have assumed that the observing frequency ν satis-fies ν ssa , ν m < ν < ν c where ν ssa is the synchrotron self-absorptionfrequency and ν c the cooling frequency. This regime is quite generic Note that the expression for t eq found by these authors does not explicitlydepend on p . This dependence formally arises due to the differing velocity ofjet and ejecta at fixed radius (by a factor of 2 /
5; eq. 4), though quantitativelythe result depends only weakly on the value of p .MNRAS , 1–13 (2015) jecta Radio Flares Associated with Jets in the radio band and at times of relevance (see Nakar & Piran 2011,for further details), however this may not necessarily apply in theearly GRB afterglow phase in which case the light-curve temporalscaling at these times ( (cid:28) t NR ) may differ from the quoted values inthis schematic figure (see Sari et al. 1998, for a full discussion ofthe various scenarios). In the previous section we recapitulated results for systems thathave either a jet or an ejecta expanding into a cold constant densityISM. We now turn to address systems in which both jet- and ejecta-like outflows are expected. In such systems the ejecta radio-flarewill differ from the standard results summarized in the previoussection. This is because, at early times, the ejecta does not expandinto a cold constant-density ISM. Rather, it propagates into a hot‘bubble’ of material that has been pre-shocked by the preceding jet-ISM forward shock (Fig. 1). This is relevant as long as the jet-ISMforward shock in fact precedes the ejecta (which occurs because thejet initially propagates much faster than the non-relativistic ejecta).Assuming that the ejecta, expanding within the “cavity” created bythe jet forward shock, is unaffected by the dilute medium in thiscavity, it will coast at constant velocity v ej while R ej < R j . Thisassumption is reasonable so long as E ej > E j , which is the case forastrophysical sources of interest. In this limit, we can easily find thetime t col at which the ejecta will overtake the jet forward-shock andcollide with the swept-up shell of jet-shocked ISM. From eq. (4) wefind that v ej t col = R j ( t col ) at t col = ξ / β − / τ j ≈
12 yr E / , n − / β − / , − . (9)Note that, conveniently, t eq (eq. 8) scales with t col , and for any2 ≤ p ≤ t eq / t col ∼ .
4. The collision time is alsosimply related to the decceleration time of the ejecta (eq. 5) as t col ≈ . ( E j / E ej ) / t dec .At times t > t col the ejecta has overtaken the jet forward shockand indeed runs into a cold unperturbed ISM. It is also energeticallydominant at these times and so the radio flare signature at t > t col reverts to the standard picture (Barniol Duran & Giannios 2015).At times t < t col however, this description is no longer valid sincethe jet shocks the ISM ahead of the ejecta.This pre-shocking of the ISM by the jet affects the early-timeejecta radio flare by: (i) changing the density distribution into whichthe ejecta expands — sweeping-up the ISM into a ∼ thin shell at R j and creating a dilute cavity with rising density profile interior toit; (ii) giving the shocked-ISM an outwards bulk velocity so thatthe relative velocity between the ejecta and swept-up material isreduced; and (iii) heating the post-shock gas to high temperatures.All three can act to dramatically inhibit the early afterglow signalof the ejecta at times t (cid:28) t col . However, as we show later in thissection — a sharp peak in the light-curve rising above the naiveunperturbed-ISM predictions is expected when the ejecta passesthrough the thin shell of swept-up ISM at t = t col .In the limit where the jet forward-shock sweeps the surround-ing ISM into an infinitesimally thin shell, the ejecta does not pro-duce any snychrotron emission whatsoever before colliding withthis swept-up shell. This is qualitatively consistent with what wefind in our numerical results, which indicate that the ejecta radio-flare light-curve rises sharply only close to t col . We can estimatethis from the Sedov-Taylor solution.The density profile implied by the Sedov-Taylor solution farinterior to the forward shock, r (cid:28) R j , is approximately n ( r (cid:28) R j ) ∝ -3 -2 -1 L ν ( a r b i t r a r y un i t s ) t eq t col jet ejectaejectaw/o jet J J -1 t/t col M a c hnu m b e r M < MM s Figure 3.
Top panel : synchrotron light-curve resulting from an expandingsingle-velocity shell (ejecta; yellow) with constant velocity running into anambient medium shaped by a preceding Sedov-Taylor blast-wave (jet; blue).Both are assumed to be in the deep-Newtonian regime with p = .
15. Thedashed black curve shows the naive expectation if the ejecta were to run intoan unperturbed constant-density ISM instead. The total (combined) light-curve is modified from this naive scenario at times t eq < t ≤ t col . Bottompanel : Mach number of the ejecta forward-shock as a function of time. TheMach number is modestly low because the upstream medium is pre-shockedto high temperatures by the jet. At times t < . t col the Mach numberis M < r /( g − ) t − / ( g − ) , where g is the gas adiabatic index. In the sameregime, the post-shock gas velocity is simply v ( r (cid:28) R j ) ∝ r / t . Thus,the ejecta synchrotron radio light-curve produced by interactionwith this post-shock medium would rise sharply with time, L ν, ej ( t < t col ) ∝ n (cid:0) R ej ( t ) , t (cid:1) p + R ej ( t ) (cid:2) v ej − v (cid:0) R ej ( t ) , t (cid:1)(cid:3) a ∝ t (cid:28) t col t ( p + ) (10)where the exponent a depends on whether the shock velocity is inthe deep-Newtonian regime or not (eq. 7), however the result doesnot depend on the value of a because the shock velocity (term insquare brackets) does not vary with time. In the final line we havetaken g = / t − t at t (cid:28) t col so that emission is strongly inhibited at thesetimes. The top panel of Figure 3 shows the resulting light-curvecalculated for an ejecta coasting at constant velocity v ej (appropriatein the regime E ej (cid:29) E j ) within an ambient medium described bythe Sedov-Taylor profile (yellow curve). The dashed black curveshows for comparison the afterglow light-curve predicted for thesame ejecta expanding into an unperturbed constant-density ISM,as assumed by previous models.As discussed above and illustrated in Fig. 3, the ejecta light-curve rises steeply near t (cid:46) t col and is severely inhibited at earlier We show later in this section that this emission is in fact completelyquenched at t (cid:28) t col because the ejecta propagates sub-sonically at thesetimes.MNRAS000
15. Thedashed black curve shows the naive expectation if the ejecta were to run intoan unperturbed constant-density ISM instead. The total (combined) light-curve is modified from this naive scenario at times t eq < t ≤ t col . Bottompanel : Mach number of the ejecta forward-shock as a function of time. TheMach number is modestly low because the upstream medium is pre-shockedto high temperatures by the jet. At times t < . t col the Mach numberis M < r /( g − ) t − / ( g − ) , where g is the gas adiabatic index. In the sameregime, the post-shock gas velocity is simply v ( r (cid:28) R j ) ∝ r / t . Thus,the ejecta synchrotron radio light-curve produced by interactionwith this post-shock medium would rise sharply with time, L ν, ej ( t < t col ) ∝ n (cid:0) R ej ( t ) , t (cid:1) p + R ej ( t ) (cid:2) v ej − v (cid:0) R ej ( t ) , t (cid:1)(cid:3) a ∝ t (cid:28) t col t ( p + ) (10)where the exponent a depends on whether the shock velocity is inthe deep-Newtonian regime or not (eq. 7), however the result doesnot depend on the value of a because the shock velocity (term insquare brackets) does not vary with time. In the final line we havetaken g = / t − t at t (cid:28) t col so that emission is strongly inhibited at thesetimes. The top panel of Figure 3 shows the resulting light-curvecalculated for an ejecta coasting at constant velocity v ej (appropriatein the regime E ej (cid:29) E j ) within an ambient medium described bythe Sedov-Taylor profile (yellow curve). The dashed black curveshows for comparison the afterglow light-curve predicted for thesame ejecta expanding into an unperturbed constant-density ISM,as assumed by previous models.As discussed above and illustrated in Fig. 3, the ejecta light-curve rises steeply near t (cid:46) t col and is severely inhibited at earlier We show later in this section that this emission is in fact completelyquenched at t (cid:28) t col because the ejecta propagates sub-sonically at thesetimes.MNRAS000 , 1–13 (2015) B. Margalit & T. Piran times in comparison to unshocked ISM models. However, immedi-ately before t = t col this light-curve is in fact enhanced comparedto such models, and compared to the light-curve shortly after t col (which converge at t > t col when the ejecta forward-shock hasovertaken the jet and therefore shocks an unperturbed ISM). Thisenhancement is due to the larger density of the thin-shell of swept-up ISM immediately behind the jet forward shock (cid:39) n , however,is also partly compensated by the gas bulk velocity behind the jetforward-shock which implies v ej − v ( R j ) = v ej /
10. From eq. (7;see also 10) we find that these two effects induce a jump in thelight-curve immediately following t = t col by a factor of J ≡ L ν, ej ( t − col ) L ν, ej ( t + col ) = p + (cid:16) (cid:17) p − ; t col < t DN (cid:16) (cid:17) p + ; t col > t DN . (11)Numerically this results in J ∼ p close to p ≈
2. Theabove assumes identical microphysical parameters ( (cid:15) B , (cid:15) e , p ) forthe two shocks which, however, need not be the case. If (cid:15) B and/or (cid:15) e are higher for the ejecta–hot-ISM shock than for the ejecta–unperturbed-ISM shock at t > t col then the jump in luminosity atthis transition point would be even larger.At the same time ( t = t col ) the jet synchrotron afterglow lu-minosity is one to two orders of magnitude weaker than that of theejecta. This is easy to estimate since at t = t col the jet and ejectaforward shocks are at the same radius (by definition; eq. 9), howevertheir velocities differ by a factor v j / v ej = / J ≡ L ν, ej ( t + col ) L ν, j ( t col ) = (cid:16) (cid:17) − p − ; t col < t DN (cid:16) (cid:17) − p + ; t col > t DN . (12)For p = J ≈
25 for both regimes (deep-Newtonianand otherwise). For larger values of p this ratio can be significantlylarger ( J ≈ ,
240 for p = c s ( R j ) = (cid:112) g ( g − ) R j / ( g + ) t . Gas immediately behind thejet-ISM shock front has a velocity v ( R j ) = R j / ( g + ) t . The Machnumber of the ejecta with respect to the thin shell of swept-up ISMimmediately preceding the jet forward-shock is therefore M( R j ) = v ej − v ( R j ) c s ( R j ) = ( g + ) (cid:112) g ( g − ) ≈ . , (13)and decreases at r < R j . In the last equality we have assumed g = / . Numerically, we find that
M ≤ t ≤ . t col (seeFig. 3). Thus, at early times a shock does not form and non-thermalsynchrotron emission would not be produced by the ejecta.We note that in the literature a different Mach number M s is often defined, as the ratio of upstream bulk velocity to upstreamsound speed in the rest frame of the shock-front (instead of as definedabove for M — in the downstream rest frame). The two are simplyrelated using the shock compression ratio r (M s ) = ( g + )/( g − + /M ) via the implicit relation M s = M r /( r − ) . In the bottompanel of Fig. 3 we also plot M s (solid yellow). Under this definition,the Mach number at t = t col is M s ( R j ) ≈ . (cid:28) than the shock-thickness ( ∼ ion gyro-radius) then such diffusivereflection cannot occur. For sufficiently strong shocks the thermalpool of shock-heated electrons extends to γ (cid:29)
1, and these electronscan participate in diffusive shock acceleration. For weak shockshowever, this is not the case, leading to the so-called ‘injectionproblem’, namely whether and how electrons are pre-acceleratedto γ (cid:29) M s (cid:38) . t (cid:46) . t col when M s (cid:46) . M s < .
3) are bright non-thermal syn-chrotron sources, implying that electron acceleration in such settingscan occur in Nature. Still, there are indications that this emissionis in tension with predictions of diffusive shock acceleration andmay therefore require an alternative model (e.g. Vazza & Brüggen2014; Botteon et al. 2020). One possibility is the acceleration ofrelic γ (cid:29) γ (cid:29) (cid:46) .
3) Mach numbers it seems plausible that near t = t col when M s ≈
4, synchrotron emission from the ejecta forward shock wouldbe produced as estimated in our above analysis. Our qualitativeresults therefore remain unchanged — at times t (cid:28) t col synchrotronemission by the ejecta is significantly inhibited (or even completelyshut-off) compared to the expected emission if the ejecta were toexpand in an unperturbed ISM, while immediately before t = t col the synchrotron light-curve should rise steeply and reach a localmaximum that is a factor ∼ t = t col (eq. 11).Finally, we note that diffusive-shock acceleration predicts thatthe power-law index p of accelerated electrons depend on the shockMach number. In the test-particle limit, and for non-relativisticshocks — diffusive-shock acceleration predicts (e.g. Blandford &Eichler 1987) that p = (M + )/(M − ) . Here we have implic-itly assumed an adiabatic index g = /
3. In the strong shock limit(
M → ∞ ) this expression reduces to the familiar result p =
2, andthe particle spectrum steepens with decreasing Mach number (seeSteinberg & Metzger 2020 for a related discussion in the context ofclassical novae). On both observational and theoretical grounds, thenaive p = p , and radio SNe exhibit a diversity in spectral indiceswith p > p ≈
3, whereas TypeII radio SNe are better fit with lower values of p ; Weiler et al. 2002).Nevertheless, we might expect that the qualitative trend of increas-ing p with lower M remain valid in the full non-linear regime. Inour scenario, steeper electron spectra at t < t col would act to further MNRAS , 1–13 (2015) jecta Radio Flares Associated with Jets inhibit the synchrotron emission at observed frequencies ν (cid:29) ν m .The monotonic decrease in p ( t ) towards t = t col (as M increases)would cause the light-curve at a fixed band to increase even moresharply as a function of time than our constant- p estimate shownin Fig. 3. A rapidly softening spectrum would be another interest-ing and unique feature that we therefore posit could accompany thelocally-peaking ejecta radio-flare at t = t col . In the following we apply our results to astrophysical sources anddiscuss their observational implications.
The interaction of the slow kilonova ejecta with an ISM that is pre-shocked by the successful jet seems both natural and unavoidablefor events like GW170817. In the previous sections we have shownthat this has the effect of quenching the ejecta afterglow signatureat early times when the ejecta is still expanding within the cavitygenerated by the jet-ISM forward shock.For GW170817, the jet energy and ambient ISM density areconstrained by radio and X-ray observations of the jet afterglow(Margutti et al. 2017; Troja et al. 2017; Alexander et al. 2017;Haggard et al. 2017; Alexander et al. 2018; Margutti et al. 2018a;Dobie et al. 2018; D’Avanzo et al. 2018; Troja et al. 2018; Mooleyet al. 2018a,b,c; Granot et al. 2018; Fong et al. 2019; Ghirlanda et al.2019; Troja et al. 2019; Hajela et al. 2019). The two parameters aredegenerate with one another to a large extent and have significantuncertainties from the modelling, however their ratio is somewhatbetter constrained. For example, Mooley et al. (2018b) find E j / n ∼ erg cm . The kilonova ejecta bulk-velocity is inferred from theoptical–near-infrared kilonova observations to be v ej ≈ . − . c (e.g. Villar et al. 2017). From eq. (9) it follows that the collisiontime between the ejecta and jet forward-shock should be t col ≈
80 yr (cid:16) v ej . c (cid:17) − / (cid:18) E j / n erg cm (cid:19) / . (14)The expectation is therefore that the X-ray and radio afterglowsignal of GW170817 will continue to decline until ∼
80 yr postmerger (with large uncertainties), after which a sharp rise in theafterglow luminosity, by a factor of J × J ∼
90 (eqs.11, 12), isexpected as the merger ejecta overtakes the jet forward shock. In theidealized framework discussed above, measurement of t col wouldprovide a strong constraint on v ej (eq. 14).Using the results of Sironi & Giannios (2013) (c.f. their eq. 12)with the best fit parameters for the GW170817 jet afterglow fromMooley et al. (2018b), we find that the jet component’s emission is F GHz ν, j ( t =
80 yr ) ≈ . µ Jy around t col . This is well below the de-tection threshold of the most sensitive current-day radio telescopes,however the ejecta radio flare is expected to reach a much higherpeak luminosity of ∼ µ Jy at t = t col (see eq. 15). This would beeasily detectable with next-generation radio facilities such as SKAor ngVLA which would be online well before ∼ t = t col for GW170817). High resolution numerical simulations (e.g. Kiuchi et al. 2017) reveal afast tail moving at higher velocity (up to ∼ . c ). However, the amount ofmatter and energy in this fast tail is negligible and irrelevant for our late timeconsiderations. The kilonova ejecta radio flare will peak on its Sedov-Taylordeceleration timescale, which is of order ∼
100 yrs for the low ISMdensity inferred around GW170817 (eq. 5). On shorter timescalesmore relevant for near-future detection, the rising ejecta signa-ture would naively be expected to pop up above the declining jet-afterglow after t > t eq ≈ . t col ≈
35 yr ( v ej / . c ) − / (eqs. 8,14).Instead, we have argued in this work that the ejecta signature wouldbe inhibited on these timescales and thus we would not expect theejecta radio flare to show up so early. This is particularly relevantfor efforts at constraining the kilonova ejecta properties using non-detections of a rising ejecta radio flare (e.g. Kathirgamaraju et al.2019; Hajela et al. 2019). A lacking detection of a rise in the radiolight-curve before t col ∼
80 yr is in fact expected based on our cur-rent (albeit idealized) analysis, and therefore does not necessarilyconstrain the ejecta parameters.An important caveat is our assumption of spherical symmetry.For GW170817, the time at which the jet becomes non-relativisticcan be estimated from eq. (3) to be t NR ≈
30 yr, where we havetaken θ j = .
04 as a fiducial value based on the best-fitting model ofMooley et al. (2018b) . The jet completes its azimuthal expansionaround the same time to within a factor of a few, t sph (cid:38) t NR . Thiscan become comparable to t eq and so care is needed in interpretingthe results. Secondly, we have discussed idealized single-velocitycomponent ejecta whereas realistic kilonova ejecta should have bothradial and azimuthal density stratification (e.g Radice et al. 2018;Gottlieb et al. 2018). Furthermore, constraining the synchrotronemission from a fast tail of this ejecta would require extendingour results to trans-relativistic regimes (e.g. Hotokezaka & Piran2015; Hotokezaka et al. 2018; Kathirgamaraju et al. 2019). Thisis straightforward for a spherically symmetric model, however theearly timescales on which emission from such trans-relativistic ma-terial would be relevant are almost certainly < t sph so that the spher-ical assumption for relativistic ejecta does not make much sense.For these reasons, we plan to extend our current analysis in futurework investigating the multi-dimensional nature of this problem. As discussed in the previous subsection, the ejecta radio flare andlate-time jet afterglow emission from GW170817 is expected to berelatively weak and evolve over long timescales. This is a directconsequence of the low ISM density at the location of the merger, n < − cm − (Hajela et al. 2019). If a future BNS merger occursin a denser environment, then the ejecta radio flare would be brighter( L ν ∝ n ( + p )/ ; Sironi & Giannios 2013) and evolve on shortertimescales ( t col ∝ n − / ). As an illustrative example, if an eventidentical to GW170817 but at a distance of 120 Mpc (characteristicof the LIGO O3 horizon distance; Abbott et al. 2018) were to occurinstead in an environment with n = − , we would predict t col ≈ ( v ej / . c ) − / and a (local) peak flux in the deep-Newtonianregime, F ν, ej ( t col ) ≈ . n p + E j , (cid:16) v ej . c (cid:17) p + (15) × ¯ (cid:15) e , − (cid:15) p + B , − ν − ( p + ) GHz (cid:18) d
120 Mpc (cid:19) − . This is easily observable, and shows that BNS mergers detectedin the near future may be even more promising sources thanGW170817. The prefactor can differ by a factor of a few for different values of p .MNRAS000
120 Mpc (cid:19) − . This is easily observable, and shows that BNS mergers detectedin the near future may be even more promising sources thanGW170817. The prefactor can differ by a factor of a few for different values of p .MNRAS000 , 1–13 (2015) B. Margalit & T. Piran
In recent years, late-time radio follow-up of nearby short GRBs(SGRBs) has been conducted by several authors (Metzger & Bower2014; Horesh et al. 2016; Fong et al. 2016; Klose et al. 2019). Theseobservations have all resulted (so far) in non-detections, that havebeen used to place constraints on combinations of the mass andenergy of a possible associated kilonova ejecta, the ambient ISMdensity, and microphysical parameters. These observations are mo-tivated by a class of theoretical models which assert that (possiblya subset of-) SGRBs are produced by a highly-magnetized rapidly-rotating NS (the ‘magnetar model’; Metzger et al. 2011; Rowlinsonet al. 2013). In this scenario it is expected that the magnetar’s rota-tional energy will, at least in part, be deposited into its surroundingsand increase the kilonova ejecta’s energy by an order of magnitudeor more. This energy boost would enhance the ejecta radio-flare sig-nificantly, and thus current upper-limits on such emission manageto stringently constrain this scenario.Our present work could imply that the expected kilonova radio-flare may be inhibited due to the jet pre-shocking of the ISM, whichmight naively allow for higher E ej to still be consistent with the radionon-detections and limit the constraints on the magnetar model.However, there is only a narrow parameter space where this mightapply, given that the large E ej correspond to a highly-acceleratedejecta with β ej ∼
1. This makes the relevant timescales ∼ t col very short (eq. 9), so that current observations may already be past t > t col when the ejecta radio-flare light-curve is no more affected bythe jet. Furthermore, our assumption of spherical symmetry, whichis already marginal for typical BNS merger parameters, certainlybreaks down for such fast kilonova ejecta velocities (see Fig. 4 anddiscussion in §5). In such systems the jet is viewed ∼ on-axis giventhat these were detected as classical SGRBs. In this sense, along theline of sight, the jet already obscures the ejecta and there is no needto wait till t (cid:38) t sph for this to occur. However, regions of the ejectaat other angles will expand relatively unobstructed into the ISM.Since emission from these ejecta components is not beamed (unless,perhaps, for extreme values of E ej / M ej ) and because the “obscuringjet” is optically-thin to synchrotron self-absorption, emission fromsuch ejecta would contribute to the light-curve even at times t < t sph , t col .For these reasons, we expect that constraints placed on highly-energetic kilonova ejecta remain valid and are not severely impactedby the jet-ejecta coupling discussed in this work. In contrast, wecaution that constraining ejecta properties for un-boosted (typical)kilonovae ejecta likely does need to account for such effects. MostSGRBs are too distant for current radio upper-limits to significantlyconstrain such “standard” kilonovae ejecta at present, however fu-ture radio facilities may be able to do so in the future. The scenario described in this paper is also relevant to long GRBs(LGRBs), which are known to be accompanied by very energeticbroad-lined Ic SNe (Galama et al. 1998; Bloom et al. 1999; Hjorthet al. 2003; Woosley & Bloom 2006). Motivated by this scenario,Barniol Duran & Giannios (2015) examined the radio re-brighteningof LGRB afterglows from interaction of the accompanying SN withthe ambient ISM. Applying their model to late-time radio data ofLGRBs available at the time and using the non-detection of suchre-brightening for any of their sources, Barniol Duran & Gian-nios (2015) constrained the ambient density for GRB 030329 andpredicted that it’s SN radio signature should become detectable by ∼ < t col compared to the pre-dictions of previous models, the inferred constraints from radionon-detections may be overly stringent.For GRB 030329, inferred parameters of the jet indicate E j / n ∼ ( . − ) × erg cm (Pihlström et al. 2007; Mesleret al. 2012; see also Mesler & Pihlström 2013). Observations ofthe accompanying SN 2003dh constrained the ejecta velocity to v ej ≈ ( ± . ) × km s − (Mazzali et al. 2007). From eq. (9)we therefore find that t col ≈ −
140 yr, much later than the latestepoch of observation of this source (Peters et al. 2019). We thereforeconclude that radio re-brightening by the SN ejecta should not haveoccurred yet for GRB 030329, and that a non-detection of such re-brightening does not currently constrain the source properties. Notealso that this statement is independent of the assumed microphysicalparameters (cid:15) e , (cid:15) B and does not depend on the poorly constrainedambient density, but on the slightly better constrained E j / n (this isin contrast to the standard picture of non-interacting jet and SN).Another relevant point is the fact that, on theoretical grounds,LGRB jets and their associated SNe ejecta are thought to propagateinto an ambient stellar-wind environment rather than a constant-density ISM. The situation is complicated because there is no strongobservational evidence for a wind-like ρ ∝ r − density profilein LGRB afterglow light-curves. Nonetheless, in Appendix B weextend the results of the preceding sections to a wind circum-stellardensity profile for completeness.Finally we note that two important related caveats arise in thecontext of LGRBs and also for superluminous-SNe, discussed inthe next section, if the latter harbor relativistic jets. In both casesthe jet deposits significant energy into a cocoon (Nakar & Piran2017). This generates a highly anisotropic high-velocity ejecta thatmight complicate the above picture. In fact some evidence for suchan outflow with velocities up to 0 . c has been observed in severalluminous SNe (Piran et al. 2019). Clearly addressing the evolutionof such a system requires a detailed numerical simulation. We have focused on the BNS merger or LGRB-SN scenario as amain motivation for this work, however the ideas discussed abovecan potentially be applicable to other settings as well. For example,superluminous-SNe (SLSNe; Quimby et al. 2011; Gal-Yam 2012)are extremely energetic SNe whose optical light-curve is thought tobe powered by interaction with circum-stellar material (Chevalier& Irwin 2011; Ginzburg & Balberg 2012) or by a central engine(Kasen & Bildsten 2010; Woosley 2010; Dexter & Kasen 2013).Motivated by connections between SLSNe and broad-lined SNe Icthat are accompanied by LGRBs (e.g Metzger et al. 2015), Margalitet al. (2018) proposed a mechanism by which collimated relativisticoutflows may be launched in conjunction with SLSNe. Despite sig-nificant observational follow-up in X-ray and radio bands (Coppe-jans et al. 2018; Margutti et al. 2018b; Law et al. 2019), only asingle SLSN, PTF10hgi, has been observed as a source of late-timenon-thermal emission (Eftekhari et al. 2019; Law et al. 2019). Thecurrent radio data is sparse, but one possible interpretation is the
MNRAS , 1–13 (2015) jecta Radio Flares Associated with Jets signal being an off-axis afterglow from a jet associated with thisSLSN. If indeed a subset of SLSNe are accompanied by jets, thenthe analysis above would be also relevant for the late-time radioflares of such SN ejecta.These ideas may also be relevant to the interpretation ofdecade-long radio transients. Recently Law et al. (2018) identifiedFIRST J141918.9+394036, a long-duration declining radio tran-sient that is consistent with an off-axis ‘orphan’ LGRB. Accordingto the modeling presented in Law et al. (2018), an initially off-axisGRB with total energy E j = erg and a surrounding ISM densityof n =
10 cm − can fit the data for an initial explosion epoch around ∼ R j = . ± . t (cid:46) t col rather than t < t eq ), and this allows us to place constraints onthe properties of a putative SN ejecta associated with J1419. Thesource has an initial detection epoch in 1993, which provides alower-limit on the source age t min . At the time of the EVN observa-tion, this corresponds to t min ≈
25 yr. Demanding that t min < t col ,or equivalently that v j t min < R j we find that v ej < ,
600 km s − .This is somewhat larger than typical broad-line Ic SNe velocities ∼ ,
000 km s − (Modjaz et al. 2016), and therefore currently con-sistent with the scenario.Given it’s nearby distance (87 Mpc) and age, J1419 would bean ideal target for detecting the SN radio-flare re-brightening in thefuture. In particular, using the EVN measurement of R j at time t EVN along with eqs. (4,9), we predict a re-brightening around (cid:46) t col ≈ + − yr ( t EVN /
25 yr ) − / β − / , − . This could be as early as ∼ t EVN ≥ t min ≈
25 yr). Therising ejecta-flare signature would be easily detectable with evenlow-sensitivity radio facilities (we roughly estimate the 1 GHz fluxat this time to be ∼
20 mJy; eq. 15), and we therefore encouragecontinuous follow-up of J1419 to constrain this scenario.
In this paper we have argued that predictions for radio-flares pro-duced by kilonovae or GRB-SNe ejecta must be modified at early(but not too early) times t sph < t (cid:46) t col (eq. 9) because the mediuminto which such ejecta expand is pre-shaped by the jet that accompa-nies such events. In particular, we have shown that the radio-flare ofsuch ejecta would be: significantly inhibited at t (cid:28) t col ; revert backto standard predictions at t (cid:38) t col ; and possibly experience a sharplocal peak and enhancement at t (cid:46) t col (Fig. 3). In our present anal-ysis we have, for simplicity, assumed spherical symmetry, howeverthere are clear caveats to this approach.The spherically-symmetric assumption is relevant at times t (cid:38) t sph after the initially-collimated jet decelerates and azimuthally An alternative interpretation of the signal is plerionic emission fromthe central engine, and its possible relation to fast radio burst progenitors(Metzger et al. 2017; Margalit & Metzger 2018; Eftekhari et al. 2019; Lawet al. 2019) v ej (1, 000km s ) j ( r a d ) f s p h = f s p h = f s p h = f s p h = sphericity valid( t sph < t col ) SGRBsGW170817LGRBs ej j ( d e g r ee s ) Figure 4.
Plane of initial jet opening angle θ j and ejecta velocity v ej show-ing characteristic values for LGRBs (blue), SGRBs (red), and GW170817(yellow). The spherically-symmetric approach to the problem is valid to theleft of a dashed line (eq. 16), depending on the somewhat uncertain value of f sph ≡ t sph / t NR (cid:38)
1. Narrowly collimated jets and/or fast ejecta do not obeythese requirements, motivating future multi-dimensional numerical workextending our current analysis. The light-grey dotted curve shows the anal-ogous condition for a wind-like ambient density profile that may be relevantfor LGRBs, and taking f sph =
10 (see Appendix B). expands into a quasi-spherical configuration (eq. 3). This assump-tion is valid at the characteristic timescale t col if t sph < t col . Thisdepends only on the initial jet opening angle θ j and the ejecta ve-locity v ej , θ j (cid:38) .
049 rad (cid:18) f sph (cid:19) / β / , − , (16)where f sph ∼ −
10 (De Colle et al. 2012; Granot & Piran 2012; Duf-fell & Laskar 2018) is a numerical factor relating the non-relativisticand spherical timescales of the jet, t sph = f sph t NR (eq. 3). Equa-tion (16) is a strict lower limit imposing the minimal requirement —that spherical symmetry reasonably describe the system at t = t col .Requiring that the spherical assumption be valid at earlier times t < t col would imply a larger minimal opening angle by a factor ( t / t col ) − / . Importantly, imposing this condition at time t = t eq (eq. 8), when the light-curve we predict first deviates from earliermodels (Fig. 3), demands θ j (cid:38) .
16 rad ( f sph / ) / β / , − (withsome dependence on p that has been omitted here for brevity).Figure 4 shows the parameter-space of initial jet opening-angleand ejecta velocity for LGRBs and their associated SNe (Fong et al.2015; Modjaz et al. 2016), cosmological SGRBs and their associ-ated kilonovae ejecta (Fong et al. 2015), and for GW170817 (Mooleyet al. 2018b; Villar et al. 2017). These estimates have considerableuncertainty, and in particular there are suggestions of fast-tail ejectafor both SNe (Piran et al. 2019) and kilonovae ejecta (Kiuchi et al.2017) illustrated as dashed extensions of the errorbars. Dashed greycurves show the minimal jet opening-angle such that spherical sym-metry is applicable at t = t col (eq. 16) for different values of f sph .For fast ejecta and/or very narrowly-collimated jets this conditionis not satisfied and multi-dimensional numerical tools must be usedto investigate the joint jet-afterglow ejecta-radio-flare light-curves.This will be studied in upcoming future work that will extend ourcurrent analytic treatment. In particular, the spherical condition isnot satisfied for GW170817 canonical parameters unless f sph (cid:46) MNRAS000
16 rad ( f sph / ) / β / , − (withsome dependence on p that has been omitted here for brevity).Figure 4 shows the parameter-space of initial jet opening-angleand ejecta velocity for LGRBs and their associated SNe (Fong et al.2015; Modjaz et al. 2016), cosmological SGRBs and their associ-ated kilonovae ejecta (Fong et al. 2015), and for GW170817 (Mooleyet al. 2018b; Villar et al. 2017). These estimates have considerableuncertainty, and in particular there are suggestions of fast-tail ejectafor both SNe (Piran et al. 2019) and kilonovae ejecta (Kiuchi et al.2017) illustrated as dashed extensions of the errorbars. Dashed greycurves show the minimal jet opening-angle such that spherical sym-metry is applicable at t = t col (eq. 16) for different values of f sph .For fast ejecta and/or very narrowly-collimated jets this conditionis not satisfied and multi-dimensional numerical tools must be usedto investigate the joint jet-afterglow ejecta-radio-flare light-curves.This will be studied in upcoming future work that will extend ourcurrent analytic treatment. In particular, the spherical condition isnot satisfied for GW170817 canonical parameters unless f sph (cid:46) MNRAS000 , 1–13 (2015) B. Margalit & T. Piran discussed in the present work may be more readily applicable. Qual-itatively, for systems where the jet has not yet fully sphericized weexpect that the true ejecta radio flare be inhibited at early times bya factor ∼ Ω / π , where Ω is the solid-angle subtended by the jet atthe time of interest. Thus the true result in such scenarios shouldlie in between our current estimates and those neglecting jet-ejectainteraction.Multi-dimensional numerical work into this problem will alsoallow investigation of more realistic ejecta, characterized by a ve-locity profile rather than a single ‘bulk’ velocity (see Appendix A).We also posit that additional features in the radio light-curve may beproduced when portions of the azimuthally-expanding jet forward-shock collide with the ejecta and/or the opposing counter-jet. Suchfeatures may provide additional diagnostics of the system prop-erties but can only be probed using multi-dimensional numericalsimulations.The efficiency of particle acceleration in our scenario is anotherinteresting question that we encourage future work to examine ingreater detail. The shock formed by the ejecta colliding with the hotpre-shocked medium is characterized by a low Mach number (eq. 13;Fig. 3). The presumably-enhanced magnetic turbulence in this pre-shocked medium may also play a role in changing the characteristicsand efficiency of particle acceleration in this ‘dual-shock’ scenario(as can the presence of ‘fossil’ high-energy electrons accelerated bythe preceding jet-ISM forward shock). An analogous scenario hasbeen discussed in the Solar-physics community where one model forGround Level Enhancement events (events in which the flux of SolarEnergetic Particles is greatly enhanced) posits that shocks betweenconsecutive colliding coronal mass ejections can more efficientlyaccelerate particles (Li et al. 2012; Zhao & Li 2014; Wang et al.2019). In our context, the efficiency of e − -acceleration and effective (cid:15) e , (cid:15) B , and p , at such shocks would influence the prominence anddetectability of the predicted peak in the light-curve at t = t col (see Fig. 3 and eq. 11). If conditions in such ‘dual-shocks’ areindeed conducive to particularly effective particle acceleration, thismay also have interesting implications for the astrophysical sites ofcosmic-ray acceleration.Our prediction of a sharply-rising and abruptly-declining localpeak in the light-curve when the ejecta catches-up with the thin-shellof ISM swept-up by the jet (at t = t col ) provides motivation for ob-serving strategies that differ from standard logarithmically-spacedintervals. This feature in the light-curve might be missed if the ob-servational cadence is too low. Note however that some of the sharpfeatures in the light curve illustrated in Fig. 3 should be smoothed outgiven light travel-time effects which limit ∆ t (cid:38) R / c ≈ β ej . Because β ej ∼ . The timing and prominence of this peakcan potentially shed light on important properties such as the ejectavelocity (eq. 9). If the local peak at t = t col can be compared to thesubsequent global peak of the ejecta radio-flare at t = t dec , one caninfer the ratio of jet and ejecta energies, E j / E ej ≈ ( . t col / t dec ) (eqs. 9,5). A strength of this method is the fact that it does notdepend on the ambient ISM density or on uncertain microphysicalparameters. For GW170817 the inferred ambient density is partic-ularly low, so that this peak is expected to be faint and occur onlymany decades from now (eq. 14). However, future BNS mergers A velocity profile and/or some level of asphericity in the ejecta wouldlikely also contribute to smoothing out these sharp features. may occur in regions with larger ambient densities, improving theseprospects (eq. 15).Finally, we note that Coughlin (2019) recently derived rela-tivistic corrections to the self-similar Blandford & McKee (1976)solution relevant for shocks with velocities ∼ several × . c . In ourcurrent analysis we used the standard Blandford-McKee solutionfor the jet forward shock at times t (cid:38) t sph , however in the context ofBNS mergers the jet forward-shock velocity at times of relevance( ∼ t col ) can be well within the regime affected by such relativisticcorrections. Qualitatively, these corrections: reduce the post-shockdensity at r (cid:28) R j and enhance it near r = R j ; reduce the post-shockfluid velocity; reduce the post-shock pressure (/temperature). Thus,interestingly, all three effects would contribute towards strengthen-ing the peak luminosity when the ejecta runs into the thin shell ofISM swept up by the jet (eq. 15; this would be due to an increase in J , c.f. eq. 11). ACKNOWLEDGEMENTS
BM thanks Eliot Quataert, Brian Metzger, Lorenzo Sironi, AaronTran, Stephen Ro, Adithan Kathirgamaraju and Paz Beniamini forhelpful conversations and comments. This work was conceived ininteractions that were funded by the Gordon and Betty Moore Foun-dation through Grant GBMF5076. This research was supported inpart by NASA through the NASA Hubble Fellowship grant
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APPENDIX A: EJECTA VELOCITY DISTRIBUTION
In the following we extend our discussion from an idealized ‘single-velocity shell’ ejecta, to one characterized by a velocity distribution.This can be conveniently expressed by the cumulative kinetic energyof ejecta with velocity greater than v , E ej (≥ v ) . For ease and analytictractability we focus on a power-law distribution E ej (≥ v ) = E ej (cid:18) vv ej (cid:19) − α (A1)for v > v ej and E ej (≥ v ) = E ej otherwise. We neglect relativisticeffects and assume the velocities of relevance are at most trans-relativistic.The dynamics of the forward shock between the ejecta anda cold constant density external medium of number density n are MNRAS000
In the following we extend our discussion from an idealized ‘single-velocity shell’ ejecta, to one characterized by a velocity distribution.This can be conveniently expressed by the cumulative kinetic energyof ejecta with velocity greater than v , E ej (≥ v ) . For ease and analytictractability we focus on a power-law distribution E ej (≥ v ) = E ej (cid:18) vv ej (cid:19) − α (A1)for v > v ej and E ej (≥ v ) = E ej otherwise. We neglect relativisticeffects and assume the velocities of relevance are at most trans-relativistic.The dynamics of the forward shock between the ejecta anda cold constant density external medium of number density n are MNRAS000 , 1–13 (2015) B. Margalit & T. Piran governed by the differential equation (Piran et al. 2013)4 π nm p R ( t ) v ( t ) ∼ E ej [≥ v ( t )] , (A2)which, for a distribution given by equation (A1) results in R ∝ t α + α + ; v = α + α + R ( t ) t (A3)for t < t dec . The trivial single-velocity shell result that R ∝ t for t < t dec is recovered in the limit α → ∞ .In contrast to the single-velocity shell scenario there is no well-defined “collision” time between the ejecta and jet forward shock.This is because fast-moving components of the ejecta will catch-up with the jet earlier than slower moving ejecta. What thereforeis the relevant timescale equivalent to t col ? This timescale is theone which demarcates the transition from forward-shock dynamicsdominated by the jet versus one dominated by the ejecta. This occursroughly once the blast-wave running into the external ISM becomesenergetically dominated by the ejecta instead of the jet, i.e. onceejecta with initial velocity ¯ v col , defined such that E ej (≥ ¯ v col ) ≡ E j ,has caught up to the jet-forward shock. From equation (A1) thisyields¯ v col = v ej (cid:18) E j E ej (cid:19) − / α , (A4)and making the simplifying assumption that the ejecta expandshomologously within the jet-cavity (i.e. neglecting deceleration ofejecta components still contained within R j ) — the correspondingtimescale is¯ t col = t col ( ¯ v col ) = t col ( v ej ) (cid:18) E j E ej (cid:19) / α = . (cid:18) E j E ej (cid:19) ( α + )/ α t dec . (A5)In the above, t col ( v ) is the standard single-velocity shell collisiontime (eq. 9) for the ‘bulk’ of the ejecta travelling at v = v ej .At times t (cid:28) ¯ t col the forward shock with the external ISMfollows the Sedov-Taylor solution with E = E j (eqs. 4), while attimes t (cid:29) ¯ t col the ejecta energy sets the forward-shock dynamicsthat will follow equation (A3) at t < t dec and equation (4) with E = E ej at t > t dec .Following §2, the synchrotron luminosity of ejecta collid-ing into an unperturbed constant-density ISM evolve as (see alsoKathirgamaraju et al. 2016) L ν, ej ( t < t dec ) ∝ t ( α − p + ) ( α + ) ; else t ( α − p − ) ( α + ) ; DN regime . (A6)With these scalings, the time ¯ t eq at which the declining jet after-glow and rising ejecta radio-flare signals would equal one another(neglecting jet–ejecta interaction) is given by¯ t eq = ¯ t col (cid:104) ( α + ) ( α + ) (cid:105) ( α + )( p − ) α ( p + ) ; else (cid:104) ( α + ) ( α + ) (cid:105) ( α + )( p + ) α ( p + ) ; DN regime . (A7)The term in brackets represents the ratio of jet and ejecta forward-shock velocities at time ¯ t col as determined by equations (4,A3). Onceagain, it can be verified that this result reduces to the single-velocityshell case (eq. 8) for α → ∞ , as expected. APPENDIX B: WIND AMBIENT-DENSITY
In the main text we discussed results for the case where the ambientmedium surrounding the progenitor is a constant density ISM. Herewe extend these results to the case of a density profile that decreasesas r − , relevant to systems where the progenitor may have launchedstrong stellar winds. The ambient density in this scenario can bewritten as ρ = Ar − = × g cm − A (cid:63) r − , (B1)where A = (cid:219) M w / π v w is normalized to A = × A (cid:63) g cm − appropriate for a characteristic mass-loss rate (cid:219) M w = − M (cid:12) yr − and wind velocity v w = ,
000 km s − .The Sedov-Taylor solution for the jet forward-shock in a windambient density dictates R j = (cid:18) . E j A (cid:19) / t / ; v j = R j ( t ) t . (B2)Assuming the ejecta expands within the jet-shocked ISM uninhib-ited (coasting at constant velocity) then the above implies that thecollision time at which the ejecta catches-up with the jet is t col = . (cid:18) EA (cid:19) v − ≈
49 yr A − (cid:63) E j , β − , − . (B3)Furthermore, the Sedov-Taylor solution interior to the the jetforward-shock ( r < R j ) in the wind density profile is simply ρ ∝ ( r / R j ) ρ ( R j ) and v ∝ r / t . The density at the shock front is ρ ( R j ) ∝ R − ∝ t − / . Using these results along with eq. (10), wefind that the optically-thin synchrotron light-curve produced by theejecta expanding within the jet-shocked medium scales as L ν, ej ( t < t col ) ∝ t − p (B4)prior to t col . At later times the light-curve converges to the L ν, ej ∝ t −( p − )/ behavior for a constant velocity ejecta in an r − densityprofile. Note that contrary to the constant-density ambient mediumcase, the light-curve is declining as a function of time, even priorto t = t dec (although synchrotron self-absorption would cause thelight-curve to rise at very early times).We can also calculate the jump in the light-curve immediatelybefore and after t = t col (the equivalent of eq. 11). For a winddensity profile and adiabatic index g = / t col and immediately afterwards) is 4, and the gas velocity atthe jet-wind forward shock is v ( R j ) = r / t . Thus, the jump factorcan be calculated from eq. (10) to be J = (cid:40) − ( p − ) ; t col < t DN t col > t DN . (B5)Interestingly, for the fiducial scenario where t col > t DN (and alsofor p = t > t DN regime) we find that J = t = t col is given as J = (cid:16) (cid:17) − p − ; t col < t DN (cid:16) (cid:17) − p + ; t col > t DN , (B6)analogous to eq. (12) for the constant density ISM case.A major caveat to the above estimates is the extremely lowMach number in the wind scenario. Similar to eq. (13) we find that MNRAS , 1–13 (2015) jecta Radio Flares Associated with Jets L ( a r b i t r a r y un i t s ) t col ejectaw/o jetejecta jet J t / t col M a c hnu m b e r s Figure B1.
Same as Fig. 3 but for a wind external density profile ( ρ ∝ r − ).This is calculated assuming p = .
15 and that both the jet and ejecta arewithin the deep-Newtonian regime. The light-curve temporal evolution canbe fully-solved analytically in this scenario, see text for further details. for a wind ambient density the Mach number of the ejecta withrespect to the jet-shocked wind (and an adiabatic index g = /
3) is M = /√ ≈ .
34. Furthermore, the Sedov-Taylor interior solutionin this regime implies c s ∝ r / t and therefore M = const . as afunction of time. The corresponding sonic Mach number M s ≈ . M s (cid:46) . t < t col , butagain we caution that further work is needed to clearly address thisissue. In particular, and as discussed in §3, the presence of relichigh-energy electrons accelerated at the jet-wind forward shockmay contribute to alleviating particle-acceleration inefficiencies.The sphericization timescale for the jet can be similarly gen-eralized from a constant-density to a wind ambient medium. Theanalog of eq. (3) is t sph (cid:38) t NR ∼ (cid:18) E j , iso π Ac (cid:19) ≈ .
37 yr A − (cid:63) E j , θ − , − . (B7)From the above equations we find that the assumption of sphericalsymmetry is valid at t = t col (analogous to eq. 16) if θ j (cid:38) .
015 rad (cid:18) f sph (cid:19) / β / , − . (B8)We normalized f sph in the above to a slightly lower value than ineq. (16), roughly consistent with numerical simulations, but evenfor somewhat higher values f sph ∼
10 the assumption of sphericalsymmetry is more easily satisfied in a wind environment (see Fig. 4).The decceleration timescale of the ejecta is t dec = (cid:169)(cid:173)(cid:171) E ej π A v (cid:170)(cid:174)(cid:172) ≈
190 yr A − (cid:63) E ej , β − , − . (B9)At later times the synchrotron light-curve evolves temporally as L ν, ej ( t > t dec ) ∝ (cid:40) t − p − ; t col < t DN t − p − ; t col > t DN . (B10) Therefore, in the deep-Newtonian regime there is no break in thelight-curve at t dec .The estimates above apply as long as the jet and ejecta prop-agate within the wind-zone, so that ρ ∝ r − . In a realistic setting,this profile is expected to change at distances larger than the windtermination shock, R w ≈ A / (cid:63) n − / v / , ( t / Myr ) / (Weaveret al. 1977). This should be compared to the radii of relevance inour current work ∼ R col = v ej t col ≈ . A − (cid:63) E j , β − , − . As longas R col < R w the assumption of an ρ ∝ r − wind environment isself-consistent. Otherwise, the dynamics and resulting light-curvewill be altered (e.g. van Marle et al. 2006). This paper has been typeset from a TEX/L A TEX file prepared by the author.MNRAS000