Intrinsic properties of the engine and jet that powered the short gamma-ray burst associated with GW170817
DD RAFT VERSION J UNE
11, 2020
Preprint typeset using L A TEX style emulateapj v. 12/16/11
INTRINSIC PROPERTIES OF THE ENGINE AND JET THAT POWERED THE SHORT GAMMA-RAY BURSTASSOCIATED WITH GW170817 D AVIDE L AZZATI , R ICCARDO C IOLFI , , R OSALBA P ERNA , Department of Physics, Oregon State University, 301 Weniger Hall, Corvallis, OR 97331, USA INAF, Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy INFN, Sezione di Padova, Via Francesco Marzolo 8, I-35131 Padova, Italy Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794-3800, USA and Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
Draft version June 11, 2020
ABSTRACTGRB 170817A was a subluminous short gamma-ray burst detected about 1 .
74 s after the gravitational wavesignal GW170817 from a binary neutron star (BNS) merger. It is now understood as an off-axis event poweredby the cocoon of a relativistic jet pointing 15 to 30 degrees away from the direction of observation. The cocoonwas energized by the interaction of the incipient jet with the non-relativistic baryon wind from the mergerremnant, resulting in a structured outflow with a narrow core and broad wings. In this paper, we couple theobservational constraints on the structured outflow with a model for the jet-wind interaction to constrain the intrinsic properties with which the jet was launched by the central engine, including its time delay from themerger event. Using wind prescriptions inspired by magnetized BNS merger simulations, we find that the jetwas launched within about 0 . .
74 s observed delay was dominated by thefireball propagation up to the photospheric radius. We also constrain, for the first time for any gamma-ray burst,the jet opening angle at injection and set a lower limit to its asymptotic Lorentz factor. These findings suggestan initially Poynting-flux dominated jet, launched via electromagnetic processes. If the jet was powered by anaccreting black hole, they also provide a significant constraint on the survival time of the metastable neutronstar remnant. INTRODUCTION
The discovery of the gravitational wave source GW170817(Abbott et al. 2017b) marked the first detection of gravita-tional waves (GWs) from a binary neutron star (BNS) merger.The observation of the same source in the electromagneticspectrum, from the almost simultaneous γ -rays (Abbott et al.2017a; Goldstein et al. 2017; Savchenko et al. 2017) to thelater X-ray and UV, optical, IR, and radio signals (Abbott et al.2017c), allowed, among other astrophysical implications, tofirmly establish the connection between short gamma-raybursts (SGRBs) and BNS mergers (e.g., Abbott et al. 2017a;Goldstein et al. 2017; Savchenko et al. 2017; Troja et al. 2017;Hallinan et al. 2017; Kasliwal et al. 2017; Lazzati et al. 2018;Ghirlanda et al. 2019; Mooley et al. 2018).The early UV, optical, and IR radiation, detected withinabout a day from the GW/ γ -ray detection, were shown to beconsistent, both spectrally and temporally, with the expecta-tions of a kilonova (e.g., Arcavi et al. 2017; Soares-Santoset al. 2017; Pian et al. 2017), i.e. a transient powered bythe radioactive decay of heavy r-process elements synthesizedwithin the matter ejected during and after merger. The laterX-ray (Troja et al. 2017) and radio emission (Hallinan et al.2017), first detected (cid:38)
10 days after the trigger, followed asingle power-law spectrum over more than eight orders ofmagnitude in energy (Lyman et al. 2018). This suggested anorigin in a blastwave, and the spectral-temporal characteris-tics of the observed radiation were used to constrain the prop-erties of the emission region. An isotropic fireball, as well asa top-hat jet (i.e. a jet with sharp edges) were ruled out earlyon (Kasliwal et al. 2017). However, it was only with VLAobservations that the presence of a relativistic collimated jet– suggested by early modeling (Lazzati et al. 2018; Ioka &Nakamura 2018) and by the steep radio decay (Lamb et al. 2018, 2019)– was confirmed beyond doubt (Ghirlanda et al.2019; Mooley et al. 2018), hence establishing the consistencywith a standard, cosmological SGRB observed off-axis.The production of jets by astrophysical sources, which isan essential ingredient for both long and short GRBs, is anarea of much interest in astrophysics. In order to understandthe mechanisms by which jets are produced and launched, thefirst step is the characterization of their intrinsic properties,i.e. the jets properties as released by their central engines, be-fore any interaction with the surrounding material. However,what we observe are the properties of the outflow when it be-comes transparent to radiation, molded by the environment inwhich it has propagated. In the case of long GRBs this envi-ronment is the envelope of a massive star (MacFadyen et al.2001), while in the case of SGRBs it is the material expelledin a compact binary merger (e.g., Rosswog et al. 1999; Fer-nández & Metzger 2013; Ciolfi et al. 2017; Radice et al. 2018;Ciolfi et al. 2019).A model able to compute the SGRB outflow properties re-sulting from the jet interaction with the surrounding materialwas recently developed by Lazzati & Perna (2019), employ-ing a semi-analytical method calibrated via numerical simula-tions (see also Salafia et al. 2019). Such model takes as inputthe properties of the surrounding material (most importantlyits mass and velocity), those of the jet (namely its asymptoticLorentz factor, injection angle, and time delay between themerger and the jet launching), and the viewing angle, i.e. theangle between the jet axis and the line of sight. Here, weapply this model to constrain the injection parameters of thejet from GW170817. For the properties of the surroundingmaterial, we refer to the results of general relativistic mag-netohydrodynamics (GRMHD) simulations of BNS mergersperformed by Ciolfi et al. (2017). We also consider a more a r X i v : . [ a s t r o - ph . H E ] J un general parametric description as an alternative. For the jetintrinsic properties, we explore a conservative range for allthe relevant parameters.Constraints for the time interval between merger and jetlaunching have been discussed before in the literature, withsomewhat controversial results. Studies based on the needto eject enough material to support a kilonova (Gill et al.2019) and structure in the jet (Granot et al. 2017) favor a longmerger-jet delay of the order of one second. Such delay, how-ever, requires a coincidence with the propagation time of thejet to yield a total observed delay of ∼ .
74 s. This, andthe fact that the pulse duration of GRB170817A coincideswith the total observed delay favors instead a much shortermerger-jet delay (Lin et al. 2018; Zhang et al. 2018; Zhang2019). Short time delays have also been suggested by popu-lation synthesis calculations of short GRBs (Belczynski et al.2006; Beniamini et al. 2020a).Our paper is organized as follows: Section 2 describes theemployed methods, based on the model developed by Lazzati& Perna (2019), as well as the range of values allowed for theinput parameters (for the jet and the surrounding material) andthe observational constraints from GW170817/GRB 170817Athat we enforce. The results of our study are presented inSection 3. Then, we summarize and discuss our results inSection 4. METHODS
Our reference scenario is a BNS merger forming a(meta)stable massive NS remnant which might eventually col-lapse to a black hole (BH). We assume that a SGRB jet islaunched at a time ∆ t m − j after merger, either by the massiveNS or right after BH formation (see, e.g., Ciolfi 2018). Inboth cases, a nearly isotropic baryon-loaded wind from theNS remnant continuously pollutes the surrounding environ-ment for a time ∆ t m − j before the jet is launched. Our modeldescribes the propagation of the incipient jet across such anenvironment and the resulting properties and structure of thefinal escaping outflow. Throughout the manuscript, we willrefer to the incipient collimated outflow from the central en-gine with high-entropy (and eventually high Lorentz factor)as “jet”, to the wide-angle non-relativistic matter released bythe massive NS remnant prior to jet launching as “wind”, andto the ultimate structured outflow at large distances resultingfrom the jet-wind interaction as “outflow”.The analysis that we present is based on the jet-wind in-teraction model developed by Lazzati & Perna (2019). Byimposing energy conservation and pressure balance at thejet, cocoon and wind interfaces (Begelman & Cioffi 1989;Matzner 2003; Lazzati & Begelman 2005; Morsony et al.2007; Bromberg et al. 2011), they were able to develop a setof semi-analytic equations to compute the properties of theoutflow for any given jet and wind setup. The underlying as-sumptions are the following: (i) the jet has initially a top-hatstructure, with uniform properties within a half-opening angle θ j ; (ii) the engine turns on at time ∆ t m − j after merger, releas-ing a constant luminosity L j for a time T eng and then turningoff; (iii) the jet is characterized by a constant dimensionlessentropy η , which corresponds to the maximum asymptotic We note that we are not considering dynamical (tidal and shock-driven)ejecta from the merger process itself as a potential obstacle for the jet prop-agation as these are mostly expelled at high latitude (i.e. away from the jetaxis). Moreover, this matter is ejected only within ∼
10 ms from the mergertime and at larger speed, thus already far away by the time the jet is launched. o )10 E i s o ( a r b i t r a r y un i t s ) core cocoon G19 best fit ( j = 3.4 o )Best exponential profile ( j = 1.95 o )Exponential profile ( j = 3.4 o , fixed) Figure 1.
Comparison between the best fit outflow structure from Ghirlandaet al. (2019, G19 in the legend) and the exponential profile used in this work.It is found that a scaling factor of 1.75 between the core opening angles isnecessary for a good match of the angular the profiles. Also shown with adashed green line is the exponential profile for a core angle equal to the G19best fit value. Vertical dashed orange lines show the values of θ core and θ cocoon for the exponential outflow profile . Lorentz factor that the jet material would attain if the acceler-ation were complete and dissipationless.For the wind, we consider two different prescriptions. Inthe first, we model the wind following the results of GRMHDsimulations of BNS mergers by Ciolfi et al. (2017). In partic-ular, we refer to the outcome of their simulations for two pos-sible equations of state (EOS), APR4 (Akmal et al. 1998) andH4 (Glendenning & Moszkowski 1991), and for two valuesof the mass ratio, q = 1 and q = 0 . labelled as q10 and q09,respectively. For these different cases, we impose an isotropicwind with constant mass flow rate matching the value givenin Fig. 23 of Ciolfi et al. (2017) and constant velocity equal tothe reported escape velocity, namely v w =0 . c , 0 . c , 0 . c ,and 0 . c for the APR4q09, APR4q10, H4q09, and H4q10models, respectively. In our second prescription, the windis instead parametrized and we consider constant mass flowrate and velocity spanning a wide range of values, namely0 . ≤ ˙ m w / ( M (cid:12) s − ) ≤ . ≤ v w / c ≤ .
25. In allcases, the wind starts at the time of merger and persists atleast until the engine turns off.Our analysis proceeds as follows. A random set of param-eter values is first generated for the system. These are the jetentropy η , total emitted energy E j , half-opening angle θ j , du-ration of the engine activity T eng , delay time of the jet launch-ing ∆ t m − j , and viewing angle with respect to the jet axis θ l . o . s . (see Table 1). For the parametrized wind case, the list in-cludes also the mass flow rate ˙ m w and the wind velocity v w .All these parameters are randomly drawn from flat prior dis-tributions within a range that is either theoretically reasonableor constrained by observations. We assumed the followingpriors for the injection properties: • The jet is launched with an asymptotic Lorentz factor10 ≤ η = L j / ˙ mc ≤ The BNS total mass in the simulations is fixed and differs by only ≈ Table 1
Physical quantities and ranges of prior distributions for the input parameters.Symbol Range Units Explanation E j × − × erg Total jet energy T eng . − . L j derived erg/s Jet luminosity (constant over the engine activity) η − − Asymptotic Lorentz factor of the jet θ j −
45 degrees Initial jet half-opening angle at injection ∆ t m − j − .
75 s Time delay between merger and jet launching time θ l . o . s . −
45 degrees Viewing angle with respect to the jet axis
Table 2
Observational constraints on derived physical quantitiesSymbol Range Units Description E iso , l . o . s . × − × erg The outflow isotropic-equivalent energy along the line of sight Γ l . o . s . . − − The Lorentz factor of the outflow along the line of sight θ core . − ∆ t obs . − .
75 s The observed delay between merger time and prompt gamma-ray pulse • The jet total energy is limited to 5 × ≤ E j / erg ≤ × . These values are conservative compared tothe observational constraints (e.g., Fong et al. 2015). • The initial half-opening angle of the jet is limited to1 ◦ ≤ θ j ≤ ◦ . In this case we strove to consider a rangeas large as possible. The lower limit of 1 degree is setto avoid a divergence at 0, while the upper limit of 45 ◦ is conservatively larger than any successful jet that hasbeen numerically studied (Murguia-Berthier et al. 2014,2017; Nagakura et al. 2014; Lazzati et al. 2017b; Nakaret al. 2018; Xie et al. 2018; Hamidani et al. 2020; Lyu-tikov 2020). • The delay time between the BNS merger and the jetlaunching is limited to 0 ≤ ∆ t m − j ≤ .
75 s. The upperlimit in this case is set by the observed time delay (Ab-bott et al. 2017a). • The viewing angle is limited to 1 ◦ ≤ θ l . o . s . ≤ ◦ . Asfor the injection angle, the lower limit is set to avoid adivergence at 0, while the upper limit is larger than theone obtained from both gravitational waves and elec-tromagnetic observations (Abbott et al. 2017a; Mooleyet al. 2018; Ghirlanda et al. 2019). • The duration of the engine activity is limited to 0 . ≤ T eng ≤ The procedureis repeated for over 100 million random samples. The result-ing outflow properties are then checked against further obser-vational constraints and only consistent models are retained.The additional constraints that we enforce are the following(see also Table 2): The jet is launched from a nozzle at r = 10 cm with an initial Lorentzfactor of Γ = 1. • The isotropic equivalent energy of the outflow in thedirection of the line of sight has to be within the range3 × ≤ E iso , l . o . s . / erg ≤ × . The lower limit isset by assuming an efficiency of 10% for the promptgamma-ray emission (Abbott et al. 2017a). The upperlimit is obtained by analyzing various best fit modelsfrom the literature (Alexander et al. 2018; D’Avanzoet al. 2018; Lazzati et al. 2018; Nakar & Piran 2018;Mooley et al. 2018; Wu & MacFadyen 2018; Ghirlandaet al. 2019; Hotokezaka et al. 2019). • The half-opening angle of the core of the outflow (or fi-nal escaping jet) is limited to 1 . ◦ ≤ θ core ≤ ◦ . Thisconstraint comes exclusively from the modeling ofproper motion and spatial extent of the radio counter-part (Mooley et al. 2018; Ghirlanda et al. 2019). Notethat both Mooley et al. (2018) and Ghirlanda et al.(2019) use power-law outflow models, while here weuse a double exponential profile. To compensate forsuch difference, we re-scaled by a factor of 1.75 theopening angle values suggested by their analyses. Asshown in Fig. 1, this compensation provides a rathergood match between our angular profiles and theirs. • The observed time delay between the merger (or thepeak of the GW signal) and the gamma-ray detection isconstrained to be 1 . ≤ ∆ t obs ≤ .
75 s and is given bythe sum of three terms (Zhang 2019): ∆ t obs = ∆ t m − j + R bo c − β jh β jh + R ph , l . o . s . − R bo c − β l . o . s . β l . o . s . , (1)where R bo is the radius at which the jet breaks out ofthe wind, β jh is the speed of the head of the jet insidethe wind in units of c , R ph , l . o . s . is the photospheric ra-dius of the outflow, and β l . o . s . is its velocity in units of c , both measured along the line of sight of the obser-vation. Figure 2 shows the location of the various radiithroughout the evolution of the merger and subsequentoutflow. Here we have considered a fairly wide inter-val, down to 1.5 s, to take into account the fact that thebeginning of the gamma-ray emission may have beenmisidentified if initially below the the background. Figure 2.
Cartoon of the various phases of the merger/outflow phenomenology, indicating the relevant radii. Numerical values are order-of-magnitude estimates,the actual values changing for each simulation. • The initial Lorentz factor of the material moving alongthe line of sight is within the interval 1 ≤ Γ l . o . s . ≤ Calculation of the photospheric radius
A critical piece of information for constraining the observedtime delay is the calculation of the location of the photo-sphere (see Eq. 1). Calculations of the photospheric radiusin gamma-ray burst outflows have been commonly performedeither in the approximation of a thin shell or of an infinitewind (e.g., Mészáros & Rees 2000; Daigne & Mochkovitch2002). A large Lorentz factor for which (1 − β ) (cid:39) / Γ hasalso been assumed. In the case of off-axis outflows, all ap-proximations should be relaxed, since relatively slow outflowsin thick – but not infinite – shells are relevant. In addition, ithas been customary to assume a neutron free fireball in pastGRB literature, for which Y e ≡ n p n p + n n = 1. Here, n p and n n arethe proton and neutron densities, respectively, and we gener-alize the equations for the photospheric radius to the case of an outflow with Y e ≤
1. We assume our fiducial electron frac-tion to be Y e = 0 . Y e = 1.Let us consider a photon that is at the back of the outflow.If its location corresponds to the photospheric radius, then thephoton has probability 1/2 of undergoing a scattering beforeleaving the flow at the front. We can therefore write a condi-tion on the opacity such that τ = 23 = (cid:90) R ph + ∆ R ph n e σ T (cid:0) − β cos( θ γ e ) (cid:1) dr , (2)where R ph + ∆ is the outer radius of the outflow at the time atwhich the photon leaves the outflow, n e is the fireball’s elec-tron number density in the observer frame, and θ γ e is the an-gle between the photon’s and the outflow’s velocity vectors.Assuming θ γ e ∼ / Γ , we have23 = L iso , l . o . s . Y e σ T (1 − β )4 π m p c η (cid:90) R ph + cT eng1 − β R ph drr , (3)where σ T is the Thomson cross section, and we have used n e = L iso , l . o . s . Y e π r m p c η . (4) T eng (s)10 R p h ( c m ) E iso , l . o . s . =10 erg; = 8 Wind approximationShell approximationEquation 6 Figure 3.
Comparison between the solution for the photospheric radius givenin Eq. (6) and the approximations for an infinite wind and a thin shell.
We have also assumed that the fireball is fully accelerated bythe time it reaches the photospheric radius, which is reason-able for a low Lorentz factor outflow. Here we have used thesubscript l . o . s . to remind the reader that the calculated photo-spheric radius is for material moving along the line of sight tothe observer.Performing a trivial integration we obtain23 = L iso , l . o . s . Y e σ T (1 − β )4 π m p c η (cid:32) R ph − R ph + cT eng − β (cid:33) , (5)which is solved to yield R ph = (cid:114)(cid:16) cT eng − β (cid:17) + π L iso , l . o . s . Y e σ T T eng m p c η − cT eng − β . (6)Note that the latter equation is valid for any shell thickness,and that it has the correct asymptotic behavior for a high-Lorentz factor wind case, for which (1 − β ) = 1 / η :lim T eng →∞ ; η →∞ R ph = 316 π L iso , l . o . s . Y e σ T η m p c . (7)In the opposite extreme of a thin fireball, we obtain lim T eng → η →∞ R ph = (cid:115) E iso , l . o . s . Y e σ T πη m p c . (8)Figure 3 shows, for an outflow with properties similar to thoserevealed by GRB 170817A along the line of sight, how theresult of Eq. (6) depends on the engine duration T eng . Alsoshown are the two limiting cases of wind and shell approx-imations, correctly recovered. In all the calculations of thispaper, we use the more general Eq. (6). RESULTS
The results of the analysis are best shown through cornerplots, where each of the model parameters is plotted versus In this case we note that the result differs by a factor η from the equationthat was previously derived, e.g., in Lazzati et al. (2017a) since that derivationdid not consider the expansion of the outer edge of the fireball while thephoton crosses it. the other ones. In the corner plot figures, the colored panelsshow the density map of models that satisfy the observationalconstraints, while the solid lines mark the areas of 1 σ , 2 σ , and3 σ statistical significance level. Histograms on the diagonalshow the posterior probability distribution for each parame-ter marginalized over the others. Finally, histograms in theupper right part of the figures show the posterior probabilitydistribution for the observational quantities of interest.In Figure 4, we report the outcome for the wind prop-erties inspired by the GRMHD simulations of Ciolfi et al.(2017). Here, we are combining together the four differ-ent cases APR4q09, APR4q10, H4q09, and H4q10, and weshow the outcome of the simulations for our baseline casewith Y e = 0 .
5. We found that some of the parameters are wellconstrained. To begin with, the viewing angle, which wasnot directly constrained in our procedure, is constrained to θ l . o . s . = 23 . + . − . degrees (all quoted uncertainties are at the 1 σ statistical significance level, unless stated otherwise), a valuethat is in good agreement with the estimates based on high-resolution radio imaging (Mooley et al. 2018; Ghirlanda et al.2019).Parameters for which we cannot obtain direct limits fromobservations and which are also well constrained are θ j , η ,and ∆ t m − j . The injection half-opening angle, never measuredfor long or short GRBs, is found to be θ j = 17 . + . − . degrees.Additionally, we obtained a lower limit for the dimensionlessjet entropy (i.e. the maximum attainable Lorentz factor) as η >
240 at the 3 σ level. Finally, we found that the delay timebetween the merger and the injection of the jet is bound to berather small: ∆ t m − j < .
36 s. These values are also reported inTable 3, which further shows how such constraints change byconsidering different electron fractions ( Y e = 1 . .
2) andstricter constraints on Γ l . o . s . and/or the total wind mass m w .For the remaining parameters, our results favor jet energiesat the lower edge of the simulated values ( E j ∼ × erg),engine activity duration T eng ∼ Γ l . o . s . (cid:38)
6, isotropic equivalent outflow energy along theline of sight E iso , l . o . s . ∼ × erg, and a total mass of thewind in the range m wind ∼ − − − M (cid:12) . We note that thefinding on the outflow energy is in general agreement withprevious constraints from the afterglow modeling.To check whether our results are sensitive to the differentEOS and/or mass ratios under consideration, we show in Fig-ure 5 the two panels ∆ t m − j vs. η and θ l . o . s . vs. θ j , correspondingto the most constrained parameters from Figure 4, now sepa-rating the four cases. We find that the method is not able todistinguish among the four, with only a marginal differencein the θ l . o . s . vs. θ j panel for the H4q10 case (rightmost lowerpanel). This degeneracy reflects the fact that the mass flowrates and velocities are rather similar despite the different q and EOS.In Figure 6, we select the same two panels from Figure 4,but in this case we show how the result changes by imposingonly one of the four observational constraints at a time. Thelower limit on η is always reproduced independently fromwhich constraint is imposed, while for the other parametersthe outcome is significantly affected by the specific choice.Interestingly, all constraints are consistent with each other atthe 1 σ level, since the 1 σ contours have a non-null intersec-tion.We now turn to consider the results obtained with aparametrized wind, i.e. allowing for any value of the massflow rate and wind velocity within the plausible ranges Figure 4.
Corner diagram for the six parameters of the model adopting the wind prescriptions inspired by GRMHD simulations by Ciolfi et al. (2017). Theresults for the four different combinations of EOS and mass ratios are merged together. Solid contour lines show the 1 σ , 2 σ , and 3 σ confidence regions. Inaddition, probability distributions for the four derived parameters and for the breakout time t bo are shown as histograms in the upper right corner. Note that theseshow the ratio of the posterior over the prior distributions. . ≤ ˙ m w / ( M (cid:12) s − ) ≤ . ≤ v w / c ≤ .
25. The out-come, shown in Figure 7, is qualitatively similar to the previ-ous case (cf. Fig. 4), with the viewing angle and the initial jethalf-opening angle well constrained, a lower limit on the jetdimensionless entropy, and an upper limit on the time intervalbetween merger and jet launching.At a quantitative level, however, some differences emerge.The viewing and jet angles are constrained to different val-ues, namely θ l . o . s . = 30 . + . − . degrees and θ j = 10 . + . − . degrees,which remain nonetheless consistent within the 1 σ range. Theconstraints on the dimensionless entropy and on the time de-lay are less stringent: η >
150 and ∆ t m − j < . DISCUSSION AND CONCLUSIONS
In this work, we have studied the key properties of theSGRB jet that was launched by the remnant of the BNSmerger event GW170817 (Abbott et al. 2017b) and that even-tually powered the gamma-ray signal GRB 170817A (Abbottet al. 2017a; Goldstein et al. 2017; Savchenko et al. 2017). Weemployed the semi-analytic model for the jet-wind interactiondeveloped by Lazzati & Perna (2019) to obtain the proper-ties of the escaping outflow depending on (i) the propertiesof the jet at the initial injection from the central engine and(ii) the properties of the massive baryon-loaded wind expelledbeforehand by the NS remnant and acting as an obstacle forthe propagation of the jet itself. By exploring the plausibleparameter ranges with over 100 million random samples andthen selecting only cases with an outcome consistent with four
Figure 5.
Correlation plots of the two best constrained parameter pairs. The top row shows the ∆ t m − j − η plane, while the bottom row shows the θ l . o . s . − θ j plane.For both parameter pairs, four panels are shown (left to right), corresponding to the four different EOS-mass ratio combinations considered. Solid contour linesshow the 1 σ , 2 σ , and 3 σ confidence regions. Figure 6.
Correlation plots of the two best constrained parameter pairs. The top row shows the ∆ t m − j − η plane, while the bottom row shows the θ l . o . s . − θ j plane.For both parameter pairs, four panels are shown (left to right) with the results obtained by imposing only one observational constrain at a time (see text). Solidcontour lines show the 1 σ , 2 σ , and 3 σ confidence regions. main observational constraints (see Section 2), we were ableto obtain posterior distributions for the entire parameter set,and hence indications on their most favourable values.For the wind properties, we assumed an isotropic flow ex-pelled from the time of merger to the time of jet launchingwith constant mass flow rate and velocity. In our first analy-sis, the values of the latter were chosen in accordance to theresults of GRMHD BNS merger simulations by Ciolfi et al.(2017), referring to BNSs with two different EOS and twodifferent mass ratios. Then, we considered a more general parametrized wind and explored a wide range of mass flowrates and velocities.For the analysis inspired by GRMHD simulations, wefound an initial half-opening angle of the jet of θ j = 17 . + . − . degrees (at 1 σ level) and a robust 3 σ lower limit on the di-mensionless entropy η = L / ˙ mc > Figure 7.
Analogous to Figure 4 but for the case of a parametrized wind. In this case, the model has two additional parameters, i.e. the mass flow rate and thevelocity of the wind. netically driven acceleration mechanisms (Mészáros & Rees1997; Drenkhahn & Spruit 2002; Metzger et al. 2011).In addition, we obtained an upper limit on the time delaybetween the merger and the jet launching: ∆ t m − j < .
36 s atthe 3 σ level. This limit would imply that most of the ob-served delay ( ≈ .
74 s) is due to the outflow breaking outof the wind and its subsequent propagation until the photo-spheric radius is reached (along the line of sight), in agree-ment with the idea that the similarity between the gamma-raypulse duration and the total observed delay is not a simple co-incidence (Zhang et al. 2018; Lin et al. 2018). This is alsoin agreement with populations studies on short GRBs (Beni- Here we are assuming a fiducial electron fraction of Y e = 0 . ∆ t m − j . Even allowing for a quite ex-treme Y e = 0 .
2, however, the upper limit remains rather small ∆ t m − j (cid:46) .
51 s(i.e. about a factor √ . amini et al. 2020a). Such a result is likely influenced by thefairly large prompt emission energetics and, at the same time,by the fact that the Lorentz factor of the emerging outflowalong the line of sight could not be too large to account forthe late onset of the afterglow emission (Troja et al. 2017;Hallinan et al. 2017). These two features, when taken to-gether, imply that the fireball carried a significant number ofbaryons, therefore pushing the photosphere to relatively largeradii. We note that the change from a jet released with high η value to an outflow with a significant rest-mass componentrequires baryon loading during the interaction of the jet withthe wind material. Since the photosphere location is of suchimportance for estimating the propagation delay, we have de-rived in this paper a formula for the photospheric radius thatrelaxes the two commonly used approximations of either aninfinite wind or a thin shell (see Equation 6).The above upper limit ∆ t m − j (cid:46) . Table 3
Results for the four most constrained parameters: the merger-jet delay, the asymptotic Lorentz factor of the jet, the viewing angle, and the injection angle.Quoted uncertainties are at the 1 σ level, while upper and lower limits are 3 σ . We highlight in bold the results for our baseline model for both thesimulation-inspired wind and the parametric wind cases.Model ∆ t m − j (s) η θ l . o . s ( o ) θ j ( o ) Simulations; baseline (Y e = . Γ l . o . s . ≤
10; m w unconstrained) < . >
240 23 . + . − . . + . − . Simulations; Γ l . o . s . ≤ < . >
240 24 + . − . . + . − . Simulations; m w ≥ − < . >
390 23 . + . − . . + . − . Simulations; Γ l . o . s . ≤ m w ≥ − < . >
250 24 . + . − . . + . − . Simulations; Y e = 1 . < . >
260 22 . + . − . . + . − . Simulations; Y e = 0 . < . >
170 25 . + . − . . + . − . Parametric; baseline (Y e = . Γ l . o . s . ≤
10; m w unconstrained) < . >
150 30 . + . − . . + . − . Parametric; Γ l . o . s . ≤ < . >
180 34 . + . − . . + . − . Parametric; m w ≥ − < . >
420 27 . + . − . . + . − . Parametric; Γ l . o . s . ≤ m w ≥ − < . >
800 30 . + . − . . + . − . Parametric; Y e = 1 . < . >
170 32 . + . − . . + . − . Parametric; Y e = 0 . < . >
130 30 . + . − . . + . − . tant implications. In particular, under the assumption thatthe central engine launching the jet was a newly-formed BH,as currently favoured by GRMHD BNS merger simulations(Ruiz et al. 2016; Ciolfi 2020a; see Ciolfi 2020b for a recentreview), this constraint would imply a NS remnant lifetime (cid:46) . Γ l . o . s . (cid:38)
6, is at thehigher end of (but still consistent with) the range of availableestimates, for which Γ l . o . s . should not be larger than ≈ σ range) withthe latest radio observations (Mooley et al. 2018; Ghirlandaet al. 2019). Finally, the favoured range for the total massin the wind is m wind ∼ − − − M (cid:12) . We note that this isonly marginally consistent with a scenario in which (i) the jetwas launched after the collapse to a BH (Ciolfi 2020a) and(ii) the wind from the NS remnant is what mainly poweredthe early “blue” component of the associated kilonova (as as-sumed, e.g., in Gill et al. 2019); indeed, such a scenario wouldrequire a mass as high as ∼ − M (cid:12) for the unbound portionof the wind material (e.g., Villar et al. 2017).For completeness, we also checked how the constraintschange by imposing Γ l . o . s . ≤ m wind ≥ − M (cid:12) (to better accomodate the hypothesisof the blue kilonova being powered by the NS remnant windand the jet being launched after the collapse to a BH). The ad-ditional condition on Γ l . o . s . has the interesting effect of furtherreducing the upper limit on ∆ t m − j by a factor around 2, whilethe other results are poorly affected. The additional condi-tion on m wind does not show a significant effect on ∆ t m − j , butmakes the lower limit on η more stringent (although this effectdisappears when both the additional conditions are applied).The analysis based on a parametrized wind confirmed theabove overall picture, although with some quantitative differ-ences. Not surprisingly, we found that the derived constraintsare relaxed once we allow for a broader range of mass flowrates and wind velocities, especially if we consider a very low electron fraction. The merger-jet time delay, in particular, isconstrained to ∆ t m − j < . σ ), which is less restrictive.We also note that in this case small wind velocities (lowerthan 0 . c ) appear to be favoured, as well as total wind massesno larger than few × − M (cid:12) . Finally, this analysis favoursa viewing angle of θ l . o . s . = 30 . + . − . degrees that is somewhatlarger than what estimated from high resolution radio imag-ing (Mooley et al. 2018; Ghirlanda et al. 2019), causing somestrain with the observations. In this case, the additional condi-tions on Γ l . o . s . and m wind lower significantly the upper limit on ∆ t m − j , substantially enlarge the lower limit on η , and also in-crease θ j up to values similar to the simulation-inspired windcase.As a general note of caution, we remark that in this workwe assumed constant mass flow rates and velocities for thebaryon-loaded wind produced by the NS remnant. This sim-plifying assumption may have relevant effects on the outcomeof our analysis. Relaxing this assumption and employingtime-evolving wind properties (possibly motivated by BNSmerger simulation results) will be the subject of future inves-tigation.While our approach can be further refined, the present studyshows its potential. In particular, the possibility of inferringthe intrinsic jet properties at the time the jet itself is launchedby the central engine can provide a valuable input for the in-vestigation of jet launching mechanisms via numerical simu-lations. We also stress that here we applied the model to thecase of GW170817/GRB 170817A, but our method is generaland can be readily applied to any other SGRB observed in thefuture.DL acknowledges support from NASA grants80NSSC18K1729 (Fermi) and NNX17AK42G (ATP),Chandra grant TM9-20002X, and NSF grant AST-1907955.RP acknowledges support by NSF award AST-1616157.REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017a, Astrophys. J. Lett.,848, L13—. 2017b, Phys. Rev. Lett., 119, 161101—. 2017c, Astrophys. J. Lett., 848, L12Akmal, A., Pandharipande, V. R., & Ravenhall, D. G. 1998, Phys. Rev. C,58, 18040