On the opening angle of magnetised jets from neutron-star mergers: the case of GRB170817A
Antonios Nathanail, Ramandeep Gill, Oliver Porth, Christian M. Fromm, Luciano Rezzolla
DDraft version March 10, 2020
Preprint typeset using L A TEX style AASTeX6 v. 1.0
ON THE OPENING ANGLE OF MAGNETISED JETS FROM NEUTRON-STAR MERGERS: THE CASE OFGRB170817A
Antonios Nathanail , Ramandeep Gill , Oliver Porth , Christian M. Fromm , Luciano Rezzolla Institut f¨ur Theoretische Physik, Max-von-Laue-Strasse 1, 60438 Frankfurt, Germany Department of Physics, The George Washington University, Washington, DC 20052, USA Department of Natural Sciences, The Open University of Israel, 1 University Road, PO Box 808, Raanana 4353701, Israel Astronomical Institute Anton Pannekoek, Universeit van Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany School of Mathematics, Trinity College, Dublin 2, Ireland
ABSTRACTThe observations of GW170817/GRB170817A have confirmed that the coalescence of a neutron-starbinary is the progenitor of a short gamma-ray burst. In the standard picture of a short gamma-ray burst, a collimated highly relativistic outflow is launched after merger and it successfully breaksout from the surrounding ejected matter. Using initial conditions inspired from numerical-relativitybinary neutron-star merger simulations, we have performed general-relativistic hydrodynamic (HD)and magnetohydrodynamic (MHD) simulations in which the jet is launched and propagates self-consistently. The complete set of simulations suggests that: (i)
MHD jets have an intrinsic energyand velocity polar structure with a “hollow core” subtending an angle θ core ≈ ◦ − ◦ and an openingangle of θ jet > (cid:38) ◦ ; (ii) MHD jets eject significant amounts of matter and two orders of magnitudemore than HD jets; (iii) the energy stratification in MHD jets naturally yields the power-law energyscaling E ( > Γ β ) ∝ ( Γ β ) − . ; (iv) MHD jets provide fits to the afterglow data from GRB170817A thatare comparatively better than those of the HD jets and without free parameters; (v) finally, both ofthe best-fit HD/MHD models suggest an observation angle θ obs (cid:39) ◦ for GRB170817A. Keywords:
Gamma-ray bursts, Neutron stars, Magnetohydrodynamics INTRODUCTIONThe first detection of gravitational waves (GWs)from a binary neutron-star (BNS) merger, GW170817(The LIGO Scientific Collaboration & The Virgo Col-laboration 2017), was marked by a coincident detec-tion of a short gamma-ray burst (GRB), GRB170817A(Savchenko et al. 2017; Goldstein et al. 2017). This wasfollowed by observations across the electromagnetic (EM)spectrum, with the detection of the (The LIGO Scien-tific Collaboration et al. 2017) quasi-thermal kilonovaemission in UV, optical, and NIR followed by the de-layed detection of the non-thermal afterglow emission inthe X- ( t > . ; Troja et al. 2017), optical, and radio( t > . ; Hallinan et al. 2017) bands.The continuous brightening of the broadband after-glow flux, with its peculiar shallow rise ( F ν ∝ t . ) tothe peak at t pk (cid:39)
150 d post-merger (Lyman et al. 2018;Margutti et al. 2018; Mooley et al. 2018a), was inter-preted using two main models. The first one considereda “structured outflow” (e.g, Gill & Granot 2018), namely,a polar-structured jet with a narrow relativistic core sur- rounded by low-energy wings (e.g., Troja et al. 2017,2018; D’Avanzo et al. 2018; Margutti et al. 2018; Lazzatiet al. 2018). The second model considered a “cocoon”,namely, a wide-angle outflow expanding quasi-sphericallyand with radial velocity stratification (e.g., Kasliwal et al.2017; Gottlieb et al. 2018; Mooley et al. 2018a). The sub-sequent observation of apparent superluminal motion ofthe radio flux centroid (Mooley et al. 2018b), togetherwith the compact size of the radio image (i.e., (cid:46) mas)(Ghirlanda et al. 2019), strongly favored the structuredjet model as dominating the afterglow emission near andpost t pk .Numerical and semi-analytical models of hydrody-namic jets have been employed to explore the afterglowof GRB170817A and the models that best fit the after-glow data correspond to structured jets with angular sizeof the relativistic core of ∼ ◦ − ◦ (Mooley et al. 2018b;Ghirlanda et al. 2019; Troja et al. 2019).Most of the analysis for the outflow of GRB170817Ahas been done using semi-analytical models or relativis-tic hydrodynamic simulations that launch a jet far fromthe merger site, with launching radius of cm . These a r X i v : . [ a s t r o - ph . H E ] M a r model L t inj Γ init θ jet E B φ E B p E B p E B φ σ max β min ρ max a M tot M ej M ej M tot [ erg / s ] [ s ] [ deg ] [ erg ] [ erg ] [ g / cm ] [ M (cid:12) ] [ M (cid:12) ] %10 HD-tht.6 . − − − − − . . .
108 0 . . HD-tht.3 . − − − − − . . .
108 0 . . MHD-p2t.03 − − − − . . . .
065 0 .
13 1 . . .
108 0 .
039 36 . MHD-p2t.02 − − − −
10 2 . . .
065 0 .
13 2 . . .
144 0 .
053 37 . MHD-p2t.12 − − − − . . . .
036 0 .
13 1 . . .
108 0 .
036 34 . Table 1 . Properties of the various HD and MHD jets considered: luminosity of the HD jet ( L ), duration of the HDinjection ( t inj ), initial Lorentz factor of the HD jet ( Γ init ), initial opening angle of the HD jet ( θ jet ), toroidal and poloidalmagnetic energies ( E B φ , E B p ) and their ratio, maximum magnetization in the torus ( σ : = B / πρ ), minimum plasmaparameter in torus ( β : = p / p m , where p and p m are the fluid and magnetic pressures respectively), maximum densityof the torus ( ρ max ) and dimensionless spin parameter of the BH ( a : = J / M ), initial total mass ( M tot ), ejected mass( M ej ) and their ratio.hydrodynamic studies have been accompanied by muchfewer investigations making use of MHD simulations tostudy the properties of such outflows (Kathirgamarajuet al. 2018; Bromberg et al. 2018; Geng et al. 2019), andin two cases, the jets were launched self-consistently viathe accretion and rotation of the black hole (Fern´andezet al. 2018; Kathirgamaraju et al. 2019). In additionto such self-consistent evolutions, Kathirgamaraju et al.2019 were also the first to report afterglow lightcurves asderived from the MHD simulations.We here report on a series of two-dimensional (2D)general-relativistic MHD (GRMHD) simulations of jetsthat are self-consistently launched after a BNS mergerwhen the merger remnant has collapsed to a black hole(BH). In addition, we also carry out simulations ingeneral-relativistic hydrodynamics (HD) – where the jetis artificially powered via the injection of energy near theBH – and use these simulations to compare and contrastthe properties of the MHD and HD jets. MHD VS HD JETSWe employ
BHAC to solve the general-relativistic MHDequations in a Kerr background spacetime (Porth et al.2017). In order to describe the ejected matter and thetorus around the compact remnant that was producedafter a BNS merger, we follow the setup introduced inNathanail et al. (2019) and additional information onthe numerical setup are reported in the Appendix. Theproperties of the models simulated study are listed inTable 1.HD jets have been thoroughly studied in the contextof short GRBs from BNS mergers (Nagakura et al. 2014;Murguia-Berthier et al. 2014, 2016; Duffell et al. 2015,2018). The MHD jets in our simulations are launchedself-consistently over the timescale of the simulations,which ranges between ∼
40 ms (for most cases) and ∼
160 ms . Overall, the dynamics of the plasma can bebriefly described as follows: starting from a non self- gravitating torus with initial size r in = M = . and r out = . M = . and containing a magnetic field ofvarious topologies and strengths (cf., Table 1), the mag-netorotational instability (MRI) develops, driving the ac-cretion of matter and magnetic flux onto the BH. At thesame time, the magnetic pressure in the torus expels theouter layers with an efficiency that depends strongly onthe initial plasma β parameter in the torus. As the MRIsaturates and accretion reaches a steady state, the fun-nel region above the BH is cleared up and an MHD jet isformed. This accretion process can then continue untileither the torus is accreted and ejected, or when the BHhas lost much of its reducible energy by spinning down(Nathanail et al. 2016).As the MHD jet breaks out from the ejecta that, in oursetup, terminate at a radius of ,
200 km , it enters in a re-gion of low-density material where it does not encounterany matter pressure-gradient that contributed to its col-limation. As a result, the jet expands in the transver-sal direction while maintaining a high degree of collima-tion. More precisely, when the head of the jet reaches ∼ ,
500 km , the opening angles at a distance of ∼
500 km and ∼ ,
500 km are θ jet (cid:39) ◦ and θ jet (cid:39) ◦ , respectively.By the time the MHD jet reaches the outer boundaryof the computational domain at ∼ ,
000 km , the open-ing angle is still very small and θ jet (cid:39) ◦ . These valuesdepend in detail on the initial conditions of the jet andon the properties of the ambient medium (Tchekhovskoyet al. 2008), but do not vary significantly in the simula-tions we have considered.Another robust feature in all our MHD models re-ported in Table 1, is an almost hollow core subtending anangle θ core ≈ ◦ − ◦ , thus much smaller than the overallopening angle of the MHD jet, θ jet (cid:38) ◦ ; the latter is con-sistent with numerical-relativity simulations where thestarting point for the launching of such a jet is reached(Rezzolla et al. 2011; Kiuchi et al. 2014; Dionysopoulou agnetised jets from short GRBs . . . . . . . . ◦ HD-tht.3 . . . . . . . . . MHD-p2t.03 Γ x [10 cm] 0 . . . . . . . . ◦ hu t = − HD-tht.3 . . . . . . . . . MHD-p2t.03 hu t = − log ρ [g / cm ] x [10 cm] Figure 1 . Lorentz factor (left panel) and density (right panel) distribution for two representative models:
MHD-p2t.03 (left part of each plot) and
HD-tht.3 (right part of each plot). The dashed white line indicates a cone with openingangle of ◦ , highlighting the slow core of the MHD jet model. On the right panel the red lines denote the contour of hu t = − , so that matter above such line is gravitationally unbound; clearly the amount of ejected mass from the MHDjet is significantly larger than in the HD jet model.et al. 2015; Kawamura et al. 2016; Ruiz et al. 2016) . InFig. 1 we show a comparison between an MHD and a HDjet, where both jets have passed through the torus andthe ejected matter. The Lorentz factor, shown on theleft panel, clearly tends to unity in the inner core of theMHD jet. The appearance of a hollow cone in MHD jetshas been pointed out previously in the literature (Komis-sarov et al. 2007; Tchekhovskoy et al. 2008; Lyubarsky2009), but these were smaller than the one found here inour simulations inspired by BNS merger scenarios.The structure and opening angle of the jet models pre-sented in these studies depend strongly on the collimat-ing agent. In the case of long GRBs, this agent is rep-resented by the disk wind and the stellar layers that thejet has to bore. On the other hand, in the case of shortGRBs produced from BNS mergers, once the jet breaksout from the matter ejected by the merger, it encountersthe low-density interstellar medium (ISM), with numberdensities n ISM ∼ − − − cm − , so that no significantfurther collimation is expected after breakout.Duffell et al. (2018) have shown that as a HD jet drillsthrough merger ejecta, it does not deposit significantenergy, and thus has limited impact on the amount ofejected mass and the appearance of a kilonova. Thisis in stark contrast with what happens for MHD jets, While “hollow core” is a standard denomination, the core ofthe jet does actually contain matter, but with very small Lorentzfactor and energy. the magnetized torus produces winds, with velocities farbelow the relativistic jet but significant enough that alarge fraction of the initial matter distribution becomesunbound. On the right panel of Fig. 1 we show the dis-tribution of the rest-mass density at time t ∼
26 ms , afterthe MHD and the HD jets have broken out from themerger ejecta. To quantify how much matter becomesunbound, we employ the Bernoulli criterion and assumea fluid element to be unbound if it has a Bernoulli con-stant hu t ≤ − , where h is the specific enthalpy of thefluid (Rezzolla & Zanotti 2013). We then apply this crite-rion to measure the flux of unbound matter on a 2-sphereof ,
000 km and report in last two columns of Table 1 theamount of ejected mass and the fraction of the ejectedmass with respect to the initial mass of the torus. Notethat in all cases considered the ejected mass is betweena few percent of the initial mass and up to a maximumof ; furthermore, models with higher initial σ , havea larger fraction of unbound matter.The angular structure of the HD and MHD jets canbe better appreciated through the polar plots in Fig. 2,where we report the Lorentz factor and the energy,i.e., the volume integral up to the outer boundary of the total energy density, relative to the unbound material ofthree representative MHD jets and of the HD jet. TheLorentz factor (left panel of Fig. 2) is measured on a 2-sphere with radius r ∼ ,
000 km , and is integrated over atime interval of τ avg ∼ to capture both the variability ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ MHD-p2t.03MHD-p2t.02HD-tht.6 MHD-p2t.12
Lorentz factor Γ( θ ) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
46 47 48 49 50
MHD-p2t.03MHD-p2t.02HD-tht.6 MHD-p2t.12
Energy log ( E ) Figure 2 . Upper panels: Polar plots of the Lorentz factor for four representative outflows over a quadrant (left panel(a)), or within a cone of ◦ (right panel (b)); the thick lines show the time-averaged values, while the shaded regionthe - σ variance. Lower panel (c): Polar plot of the energy distribution for four representative models within a coneof ◦ . Γ β E ( > Γ β ) [ e r g ] HD-tht.6 Γ β MHD-p2t.03 ∝ (Γ β ) − . Γ β MHD-p2t.02 ∝ (Γ β ) − . Γ β MHD-p2t.12 ∝ (Γ β ) − . Γ β E ( Γ β ) [ e r g ] HD-tht.6 Γ β MHD-p2t.03 Γ β MHD-p2t.02 Γ β MHD-p2t.12
Figure 3 . Energy distributions shown as either as E = E ( > Γ β ) (top panels) or as E = E ( Γ β ) (bottom panels) for thefour representative models. The black and red solid lines represent the distribution at different times, t = and
20 ms respectively.and the steady features.Each of the four quadrants refers to one of the modelsconsidered, i.e.,
MHD-p2t.03 , MHD-p2t.02 , MHD-p2t.12 ,and
HD-tht.6 , with a thick line indicating the time-averaged values and with the shaded areas showing the - σ variance over the time interval τ avg , i.e., the variation of the Lorentz factor at each angle. The rightpanel of Fig. 2, on the other hand, shows the angulardistribution of the energy for the four models, where theenergy is integrated for every angle for unbound matterwith Γ > . ; such a cut-off is introduced to avoid the inclusion of comparatively slow material.In Fig. 3 we show instead the energy distribution abovea certain value of Γ β , i.e., E ( > Γ β ) , as a function of Γ β ,both for the HD jet and for the three representative MHDmodels. Since the energy E generically grows with Γ β ,the quantity E ( > Γ β ) helps capture the nonlinear growthas a deviation from a constant value and to determinethe cut-off at the highest energies. The energy is mea-sured after the jet has broken out from the merger ejecta,i.e., t =
10 ms , and is reported at three different timeswith a separation of in time. Note that the HD jet agnetised jets from short GRBs Γ β but to Γ β (cid:39) − , when it has a verysharp fall-of profile at moderate Lorentz factors. There-fore, in a HD jet a most of the energy is concentrated inthe fast-moving material.On the other hand, all the MHD jets are up to twoorders of magnitude more powerful and have a sub-lineargrowth of energy with Γ β ; at the same time, the cut-offis less abrupt and preceded by a clear power-law fall-offat high Lorentz factors, which can be approximated as E ( > Γ β ) ∝ ( Γ β ) − . . Hence, in the case of MHD jets, mostof the energy is at Γ β ∼ , but the energy distributionin the plasma can reach very large values. Note that acut-off of Γ (cid:39) is set to avoid to account for portionsof the flow where the accuracy of the numerical solutionis reduced because of the large Lorentz factors reached.It is worth noting that the bulk of our MHD jets ismoving relatively fast and overall faster than what ob-served in other simulations (Gottlieb et al. 2018) or an-alytical modellings (Mooley et al. 2018a; Gill & Granot2018), where most of the energy is in slow-moving mate-rial and the power-law behaviour Γ β −( − ) is seen alreadyfor Γ β (cid:39) . As a final remark, we note that since ourMHD jets are launched as a result of GRMHD accre-tion processes, their energetics cannot be steered fromthe initial conditions, but is the self-consistent result ofthe simulations. AFTERGLOW EMISSIONThe afterglow emission is expected to be dominatedby synchrotron radiation from electrons at the forwardshock propagating into the low-density ISM and that areaccelerated into a power-law energy distribution of thetype n e ( Γ e ) ∝ Γ − pe , where n e and Γ e are the number den-sity and Lorentz factors of the electrons, respectively;hereafter, we will assume p = . , which is consistentwith previous analysis for the afterglow of GRB170817A(Troja et al. 2019; Hajela et al. 2019). Following Sariet al. (1998), we model the emission that depends onthe microphysical parameters (cid:15) e and (cid:15) B , which describethe fraction of the total internal energy behind the shockgiven to electrons and to the magnetic field, respectively.The afterglow lightcurves are computed following the an-gular distributions of the Lorentz factor and of the energyprofile (cf., Fig. 2), together with the energy distributionin Γ β (cf., 3). The angular structure is binned uniformlyin angles along the θ direction, which yields the ini-tial Γ ( θ ) and isotropic-equivalent energy E iso ( θ ) of theflow (see Granot et al. 1999; Gill & Granot 2018, fordetails).As representative examples of our fits, we make use ofmodel HD-tht.6 and model
MHD-p2t.03 . For the data,on the other hand, we employ the most recent afterglowdata, i.e., t (cid:46)
743 d after merger (see, e.g., Hajela et al. − − − MHD-p2t.03 HD-tht.6 − − − t [d] − − − F ν [ m J y × ( D / M p c ) − ] Figure 4 . Broad band observations of GRB170817A withthe best-fit lightcurves of models
MHD-p2t.03 (red line;see main text for the fitting parameters) and
HD-tht.6 (dashed blue line; see main text for the fitting parame-ters).2019, for the latest observations in X-rays) consisting ofX-ray emission at keV and VLA radio observations at and GHz (Margutti et al. 2017, 2018; Alexander et al.2017, 2018; Hallinan et al. 2017; Mooley et al. 2018a,c;Dobie et al. 2018; Troja et al. 2018, 2019; Hajela et al.2019). The fit is performed with five free parameters,namely: the observer angle θ obs , the energy of the burst E , the fraction of the total energy in the electrons (cid:15) e ,the fraction of the total energy in the magnetic field (cid:15) B ,and the circum-merger density, n ISM . Note that the pa-rameter space is degenerate since the model parametersoutnumber the available constraints from the data (see,e.g., Gill et al. 2019). The best-fit parameters are thenfound using a genetic algorithm to optimize the parame-ter selection and minimize the reduced χ ν (Fromm et al.2019), while the fitting procedure is applied simultane-ously to the three different bands.The afterglow lightcurves relative to the set of param-eters providing the best fits for the two models MHD-p2t.03 and
HD-tht.6 , along with the observational data,are shown in Fig. 4 for a source at
40 Mpc , where the up-per and middle panels correspond to radio observationsat and GHz, while the lower panel to X-ray observa-tions at keV.Overall, the MHD jet model MHD-p2t.03 yields abetter fit to the data, with a reduced χ ν = . andparameters θ obs = . ◦ , E = . erg , log ( (cid:15) e ) = − . , log ( (cid:15) B ) = − . , and n ISM = − . cm (red line).It captures well the first data points in the afterglow, to-gether with the peak and the fall-off. The HD jet model HD-tht.6 , on the other hand, provides a less-good fitwith reduced chi-squared are χ ν = . and parameters θ obs = . ◦ , E = . erg , log ( (cid:15) e ) = − . , log ( (cid:15) B ) = − . , and n ISM = − . cm (dashed blue line); how-ever, it also yields a better match to the very late de-cay in the X-ray emission till days after the merger(model HD-tht.3 has χ ν = . and an HD jet with θ jet = ◦ has even larger reduced chi-squared). Interest-ingly, both of the best-fit models suggest an observationangle θ obs (cid:39) ◦ , which can then be taken as a robustfeature of the emission of GRB170817A. Our estimatesare thus consistent with those of Mooley et al. (2018b);Troja et al. (2019), and smaller than those coming fromthe semi-analytical and analytical modelings, which sug-gest instead θ obs (cid:39) ◦ (Hajela et al. 2019).It is worth noting that when all the physical parame-ters – i.e., E , (cid:15) e , (cid:15) B , and n ISM – are kept the same, theHD/MHD light curves show a marked difference. In-deed, while both lightcurves have similar power-law riseand fall-offs, the evolution of peak-times are considerablydifferent, with the HD having a monotonic dependence ofthe peak-times with the viewing angle, with peak-timesincreasing as viewing angles become larger. The MHDlightcurves, instead, do not have a minimum peak-time atthe smallest viewing angle, but for θ obs (cid:38) θ core ; the peak-time then increases steeply as the viewing angle grows.This considerable difference between the two afterglowlightcurves disappears for larger angles, that is, whenthe jets are observed off-axis. CONCLUSIONSWe have performed a number of general-relativistic HDand MHD simulations to model the launching of a jetafter a BNS merger and contrast the dynamics and ap-pearance of HD and MHD jets. Overall, we find that: (i)
MHD jets have an intrinsic energy and velocity struc-ture in the polar direction characterised by a “hollowcore” subtending an angle θ core ≈ ◦ − ◦ and an openingangle of θ jet > (cid:38) ◦ . HD jets, on the other hand, havea uniform energy and polar structure and much smalleropening angles of θ jet ∼ ◦ . (ii) MHD jets eject signifi-cant amounts of matter, amounting to (cid:46) of the totalmass of the system and about two orders of magnitudemore than HD jets. (iii)
The energy stratification inMHD jets naturally yields the power-law energy scaling E ( > Γ β ) ∝ ( Γ β ) − . often introduced in analytical mod-elling. This feature is robust and does not require specialtuning as is the case instead for HD jets. (iv) MHD jetsprovide fits to the afterglow data from GRB170817A inthree different bands ( GHz, GHz and keV) that arenot only very good but also comparatively better thanthose of the HD jets. While even better fits can be con-structed with suitably constructed HD jets, the fit ob-tained with MHD jets is robust and without free param-eters. (v) Both of the best-fit HD/MHD models suggest an observation angle θ obs (cid:39) ◦ for GRB170817A.While this is arguably the most comprehensive ex-ploration of jet launching from BNS mergers, exploreand contrasting for the first time HD and MHD jets,future work will have to include additional jet models,a closer comparison with other models proposed in theliterature, and a step towards imaging in the radio band. Acknowledgements.
Support comes in part also from“PHAROS”, COST Action CA16214 and the LOEWE-Program in HIC for FAIR. The simulations were per-formed on the SuperMUC cluster at the LRZ in Garch-ing, on the LOEWE cluster at the CSC in Frankfurt, andon the HazelHen cluster at the HLRS in Stuttgart.REFERENCES
Abramowicz M., Jaroszynski M., Sikora M., 1978, Astron.Astrophys., 63, 221Alexander K. D., et al., 2017, Astrophys. J. Letters, 848, L21Alexander K. D., et al., 2018, Astrophys. J. Letters, 863, L18Bovard L., Martin D., Guercilena F., Arcones A., Rezzolla L.,Korobkin O., 2017, Phys. Rev. D, 96, 124005Bromberg O., Tchekhovskoy A., Gottlieb O., Nakar E., Piran T.,2018, Mon. Not. R. Astron. Soc., 475, 2971D’Avanzo P., et al., 2018, A&A, 613, L1Dietrich T., Ujevic M., Tichy W., Bernuzzi S., Br¨ugmann B.,2017, Phys. Rev. D, 95, 024029Dionysopoulou K., Alic D., Rezzolla L., 2015, Phys. Rev. D, 92,084064Dobie D., et al., 2018, Astrophys. J. Letters, 858, L15Duffell P. C., Quataert E., MacFadyen A. I., 2015, Astrophys. J.,813, 64Duffell P. C., Quataert E., Kasen D., Klion H., 2018, preprint,( arXiv:1806.10616 )Fern´andez R., Tchekhovskoy A., Quataert E., Foucart F., KasenD., 2018, Mon. Not. R. Astron. Soc.,Fishbone L. G., Moncrief V., 1976, Astrophys. J., 207, 962Foucart F., O’Connor E., Roberts L., Kidder L. E., Pfeiffer H. P.,Scheel M. A., 2016, Phys. Rev. D, 94, 123016Fromm C. M., et al., 2019, Astron. Astrophys., 629, A4Fujibayashi S., Kiuchi K., Nishimura N., Sekiguchi Y., ShibataM., 2018, Astrophys. J., 860, 64Geng J.-J., Zhang B., K¨olligan A., Kuiper R., Huang Y.-F., 2019,Astrophys. J. Letters, 877, L40Ghirlanda G., et al., 2019, Science, 363, 968Gill R., Granot J., 2018, Mon. Not. R. Astron. Soc.,Gill R., Granot J., De Colle F., Urrutia G., 2019,arXiv:1902.10303, e-prints,Goldstein A., et al., 2017, Astrophys. J. Letters, 848, L14Gottlieb O., Nakar E., Piran T., 2018, Mon. Not. R. Astron. Soc.,473, 576Granot J., Piran T., Sari R., 1999, Astrophys. J., 513, 679Hajela A., et al., 2019, Astrophys. J. Lett., 886, L17Hallinan G., et al., 2017, Science, 358, 1579Kasliwal M. M., et al., 2017, Science, 358, 1559Kathirgamaraju A., Barniol Duran R., Giannios D., 2018, Mon.Not. R. Astron. Soc., 473, L121Kathirgamaraju A., Tchekhovskoy A., Giannios D., BarniolDuran R., 2019, Mon. Not. R. Astron. Soc., 484, L98Kawamura T., Giacomazzo B., Kastaun W., Ciolfi R., EndrizziA., Baiotti L., Perna R., 2016, Phys. Rev. D, 94, 064012 agnetised jets from short GRBs Kiuchi K., Kyutoku K., Sekiguchi Y., Shibata M., Wada T., 2014,Phys. Rev. D, 90, 041502Kiuchi K., Kyutoku K., Sekiguchi Y., Shibata M., 2018, Phys.Rev. D, 97, 124039Komissarov S. S., Barkov M. V., Vlahakis N., K¨onigl A., 2007,Mon. Not. R. Astron. Soc., 380, 51Lazzati D., Perna R., Morsony B. J., Lopez-Camara D., CantielloM., Ciolfi R., Giacomazzo B., Workman J. C., 2018, Phys. Rev.Lett., 120, 241103Lyman J. D., et al., 2018, Nature Astronomy,Lyubarsky Y., 2009, Astrophys. J., 698, 1570Margutti R., et al., 2017, Astrophys. J. Letters, 848, L20Margutti R., et al., 2018, Astrophys. J. Letters, 856, L18Mooley K. P., et al., 2018a, Nature, 554, 207Mooley K. P., et al., 2018b, Nature, 561, 355Mooley K. P., et al., 2018c, Astrophys. J. Lett., 868, L11Murguia-Berthier A., Montes G., Ramirez-Ruiz E., De Colle F.,Lee W. H., 2014, Astrophys. J., 788, L8Murguia-Berthier A., et al., 2016, Astrophys. J. Lett., 835, L34Nagakura H., Hotokezaka K., Sekiguchi Y., Shibata M., Ioka K.,2014, Astrophys. J., 784, L28Nathanail A., Strantzalis A., Contopoulos I., 2016, Mon. Not. R.Astron. Soc., 455, 4479Nathanail A., Porth O., Rezzolla L., 2019, Astrophys. J. Lett,870, L20Porth O., Olivares H., Mizuno Y., Younsi Z., Rezzolla L.,Moscibrodzka M., Falcke H., Kramer M., 2017, ComputationalAstrophysics and Cosmology, 4, 1 Radice D., Galeazzi F., Lippuner J., Roberts L. F., Ott C. D.,Rezzolla L., 2016, Mon. Not. R. Astron. Soc., 460, 3255Rezzolla L., Zanotti O., 2013, Relativistic Hydrodynamics.Oxford University Press, Oxford, UK,doi:10.1093/acprof:oso/9780198528906.001.0001Rezzolla L., Giacomazzo B., Baiotti L., Granot J., KouveliotouC., Aloy M. A., 2011, Astrophys. J. Letters, 732, L6Ruiz M., Lang R. N., Paschalidis V., Shapiro S. L., 2016,Astrophys. J. Lett., 824, L6Sari R., Piran T., Narayan R., 1998, Astrophys. J. Lett., 497, L17Savchenko V., et al., 2017, Astrophys. J. Letters, 848, L15Sekiguchi Y., Kiuchi K., Kyutoku K., Shibata M., Taniguchi K.,2016, Phys. Rev. D, 93, 124046Tchekhovskoy A., McKinney J. C., Narayan R., 2008, Mon. Not.R. Astron. Soc., 388, 551The LIGO Scientific Collaboration The Virgo Collaboration 2017,Phys. Rev. Lett., 119, 161101The LIGO Scientific Collaboration et al., 2017, Astrophys. J.Lett., 848, L12Troja E., et al., 2017, Nature, 551, 71Troja E., et al., 2018, Mon. Not. R. Astron. Soc., 478, L18Troja E., et al., 2019, Mon. Not. R. Astron. Soc., 489, 1919
APPENDIX A. NUMERICAL SETUP AND MHD MODELS
In this Appendix we provide details of the numerical setup of our simulations and further show results for an extensiveselection of MHD models in order to check the robustness of the results. As anticipated, we use
BHAC to solve thegeneral-relativistic MHD equations in a Kerr background spacetime (Porth et al. 2017). To mimic the post-mergerremnant in GW170817 and as initial condition for the launching of an MHD jet, we consider a non self-gravitatingtorus (Fishbone & Moncrief 1976; Abramowicz et al. 1978) around a BH of mass M = . M (cid:12) and various dimensionlessspins (see Table 1). The radial extent of the initial matter distribution is set to be ,
200 km , in order to account forthe expansion of the torus, and also for the matter expelled during merger, which has reached such a distance. Toaccommodate such a large extension of matter, the numerical domain has always a radius of ,
000 km . Since we herefocus on the production and launch of a jet, at the beginning of the simulation all matter is bound and set to have azero velocity. However, we do measure the mass that becomes unbound as a result of the jet launching and computeits contribution to the kilonova at the end of the simulation. The simulations are performed in two spatial dimensionsusing a spherical polar coordinate system. The computational domain is resolved with either × or × cells and with three refinement levels, thus yielding an effective resolution of × cells.Over the past several years, a robust picture has been drawn on the distribution of the ejected matter after themerger. More specifically, BNS merger simulations indicate that the polar region is not entirely empty of matter(Sekiguchi et al. 2016; Foucart et al. 2016; Radice et al. 2016; Bovard et al. 2017; Dietrich et al. 2017; Fujibayashiet al. 2018). To reproduce such conditions, we fill the polar region with matter, having density that is . orders ofmagnitude less than the maximum density of the torus and a radial profile that scales like r − . , with an exceptionfor model HD-tht.6 , where the matter in the polar region has order of magnitude higher density, but has the sameradial profile. In a typical BNS merger, the two stars have a mildly strong initial magnetic field, which is expectedto be amplified during merger, either via the Kelvin-Helmholtz or the magnetorotational instability, yielding a verymagnetic energy > erg , and with ratio between poloidal and the toroidal components that is ≈ . (Kiuchi et al.2018). To reproduce the enhancement in the magnetic field after the merger, we initialize our simulations with apoloidal nested-loop magnetic field structure and a toroidal component that traces the fluid pressure; by tuning thestrength of two components it is then possible to obtain the desired ratio in the corresponding magnetic energies.To explore a space of parameters that is as wide as reasonably possible, we vary the initial magnetic field, the ratioof the poloidal-to-toroidal magnetic-field energy, the spin of the BH, as well as the size and morphology of the torus model L t inj Γ init θ jet E B φ E B p E B p E B φ σ max β min ρ max a M tot M ej M ej M tot [ erg / s ] [ s ] [ deg ] [ erg ] [ erg ] [ g / cm ] [ M (cid:12) ] [ M (cid:12) ] %10 HD-tht.6 . − − − − − . . .
108 0 . . HD-tht.3 . − − − − − . . .
108 0 . . MHD-p2t.03 − − − − . . . .
065 0 .
13 1 . . .
108 0 .
039 36 . MHD-p2t.03-LB − − − − .
36 0 .
28 0 . . .
20 2 . . .
144 0 .
021 1 . MHD-p2t.02 − − − −
10 2 . . .
065 0 .
13 2 . . .
144 0 .
053 37 . MHD-p2t.02-LB − − − − . .
084 0 . .
002 3 .
25 2 . . .
144 0 .
002 1 . MHD-p2t.12 − − − − . . . .
036 0 .
13 1 . . .
108 0 .
036 34 . MHD-p2t.04 − − − − . . . .
065 0 .
13 1 . . .
108 0 .
033 31 . MHD-a.8-LB − − − − .
19 0 .
115 0 . . .
30 3 . . .
118 0 . . MHD-a.8-MB − − − − . .
05 0 . .
02 0 .
36 3 . . .
118 0 .
014 12 . MHD-a.8 − − − − . . . .
06 0 .
13 2 . . .
098 0 .
029 29 . MHD-rout-52.4 − − − − . .
195 0 . . .
10 10 0 . .
121 0 .
018 15 . MHD-600km − − − − . .
54 0 . .
016 0 .
52 2 . . .
127 0 . . MHD-900km − − − − . . . .
016 0 .
52 1 . . .
106 0 .
004 3 . Table A2:. Properties of the various HD and MHD jets considered: luminosity of the HD jet ( L ), duration of theHD injection ( t inj ), initial Lorentz factor of the HD jet ( Γ init ), initial opening angle of the HD jet ( θ jet ), toroidal andpoloidal magnetic energies ( E B φ , E B p ) and their ratio, maximum magnetization in the torus ( σ : = B / πρ ), minimumplasma parameter in torus ( β : = p / p m , where p and p m are the fluid and magnetic pressures respectively), maximumdensity of the torus ( ρ max ) and dimensionless spin parameter of the BH ( a : = J / M ), initial total mass ( M tot ), ejectedmass ( M ej ) and their ratio. For all models the initial torus parameters are r in = . , r out = . and the matterdistribution has a radial extent till r ext = ,
200 km , whereas model
MHD-rout-52.4 has r out = . , model MHD-600km has r ext =
600 km and
MHD-900km has r ext =
900 km . Note that models ending with MB and LB refer to matter with amedium and low magnetic-field strength, respectively, while all the other quantities are held the same.(which is ultimately dictated by the spin of the BH). The details of all the models used are listed in Table 2. Forillustrative purposes, we report in Fig. 5 the angular structure of eight outflows from Table 2, showing the Lorentzfactor within an angle of ≤ θ ≤ ◦ . Similar to Fig. 2, the Lorentz factor (thick line) is measured in slices of constantradius , i.e., r ∼ , and integrated over a time interval of τ avg ∼ , the shaded areas show the - σ varianceover the time interval τ avg , i.e., the variation of the Lorentz factor at each angle. From the two polar plots it isevident that the presence of a hollow core with an opening of ≈ ◦ − ◦ is robust in all of the MHD models consideredin our study. agnetised jets from short GRBs ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ MHD-p2t.03-LBMHD-rout-52.4MHD-a.8-MB MHD-p2t.02-LB
Lorentz factor Γ( θ ) ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦ MHD-p2t.04MHD-600kmMHD-a.8-MB MHD-900km
Lorentz factor Γ( θ ) Figure A5:. Polar plots of the Lorentz factor for eight outflows from Table 2 within a cone of ◦ , the thick lines showthe average values, while the shaded region the - σσ