Determining the viewing angle of neutron star merger jets with VLBI radio images
MMNRAS , 1–11 (2021) Preprint 15 January 2021 Compiled using MNRAS L A TEX style file v3.0
Determining the viewing angle of neutron star merger jets with VLBIradio images
Joseph John Fernández, ★ Shiho Kobayashi and Gavin P. Lamb Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK School of Physics and Astronomy, University of Leicester, University Road, Leicester, LE1 7RH, UK
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Very long base interferometry (VLBI) radio images recently proved to be essential in breaking the degeneracy in the ejecta modelfor the neutron star merger event GW170817. We discuss the properties of synthetic radio images of merger jet afterglow by usingsemi-analytic models of laterally spreading or non-spreading jets. The image centroid initially moves away from the explosionpoint in the sky with an apparent superlumianal velocity. After reaching a maximum displacement its motion is reversed. Thisbehavior is in line with that found in full hydrodynamics simulations. Since the evolution of the centroid shift and jet imagesize are significantly different in the two jet models, observations of these characteristics for very bright events might be ableto confirm or constrain the lateral expansion law of merger jets. We explicitly demonstrate how 𝜃 obs is obtained by the centroidshift of radio images or its apparent velocity provided the ratio of the jet core size 𝜃 𝑐 and the viewing angle 𝜃 obs is determinedby afterglow light curves. We show that a simple method based on a point-source approximation provides reasonable angularestimates (10 −
20% errors at most). By taking a sample of structured Gaussian jet results, we find that the model with 𝜃 obs ∼ . Key words:
Transients: gamma-ray bursts - Transients: neutron star mergers - Physical data and processes: gravitational waves- methods: numerical.
Shortly after the detection of gravitational waves (GW) from thebinary neutron star merger, GW170817, its electromagnetic (EM)counterpart was discovered in the S0 galaxy NGC4993 (Coulter et al.2017; Soares-Santos et al. 2017, etc). This transient was subject toan unprecedented follow-up campaign across the EM spectrum (e.g.Abbott et al. 2017). The counterpart was found to be made up ofseveral components: a prompt short gamma-ray burst (GRB) detected1 . 𝑠 after the merger, a kilonova, and a broad synchrotron afterglow,first detected 9 days post-merger at X-ray wavelengths; see Margutti& Chornock (2020) for a review of GW170817 and Metzger (2019),Burns (2020) and Nakar (2020) for reviews of EM counterparts toGW detectable compact binary mergers.In addition to the merger afterglow light curves, very long baselineinterferometry (VLBI) radio images were obtained (Mooley et al.2018; Ghirlanda et al. 2019). Mooley et al. (2018) presented radioimages at 75 and 230 days post-merger finding an image centroiddisplacement of ∼ . ± . 𝛽 app = . ± .
5; Ghirlanda et al. (2019)confirmed this result with a radio image obtained at 207 days postmerger. This, along with the steep post-peak afterglow decline (Lambet al. 2018, 2019b; Troja et al. 2018, 2019), broke the degeneracybetween a wide, quasi-isotropic ejecta and a narrow core-dominatedjet, confirming the emission was from the latter.VLBI images are also important for breaking degeneracies in ★ E-mail: [email protected] parameter estimation from light curves. Nakar & Piran (2020) showedthat afterglow light curves observed around their peak time 𝑇 𝑝 cannotconstrain the observing angle, 𝜃 obs , but only determine the ratioof the observing angle 𝜃 obs to the jet opening angle, 𝜃 𝑐 (or coresize for a core dominated structured jet). This leads to degeneracyamong the parameters { 𝜃 obs , 𝜃 𝑐 , 𝑙, 𝜖 𝐵 } , where 𝑙 ∼ ( 𝐸 / 𝑛𝑚 𝑝 𝑐 ) / is the Sedov length, 𝑛 the ambient density, 𝑚 𝑝 the proton mass and 𝜖 𝐵 is the fraction of the shock energy in the magnetic fields. Thisdegeneracy can be broken by the observation of afterglow images(the centroid shift around the peak time) or the decay index of thelate time afterglow light curve at the transition to the sub-relativisticphase.Gravitational waves from compact binary mergers provide aluminosity distance 𝐷 𝐿 which is independent of the cosmologicaldistance ladder (Schutz 1986; Holz & Hughes 2005). Therefore, welllocalized GW signals can in principle be used to estimate Hubble’sconstant 𝐻 if combined with EM redshift measurements (e.g.Mastrogiovanni et al. 2020). However the distance 𝐷 𝐿 and inclinationwith respect to the binary plane 𝜃 obs are entangled in the GW strain(Misner et al. 1973; Holz & Hughes 2005). Without accurate GWpolarization measurements to obtain 𝜃 obs , this degeneracy leads toadditional uncertainties in 𝐷 𝐿 .Observation of the long-lasting components of the counterpartsuch as an afterglow provide an avenue to obtain the 𝜃 obs and henceimprove the estimate of 𝐷 𝐿 . However, the degeneracy among jetparameters makes it difficult to determine 𝜃 obs from afterglow lightcurves.Radio images were used in Mooley et al. (2018); Ghirlanda © a r X i v : . [ a s t r o - ph . H E ] J a n J.J. Fernández, S. Kobayashi & G.P. Lamb et al. (2019) to constrain 𝜃 obs . This data was subsequently usedby Hotokezaka et al. (2019) and Wang & Giannios (2020) toconstrain the Hubble constant to 𝐻 = . + . − . km s − Mpc − and 𝐻 = . + − km s − Mpc − respectively. The semi-analytic afterglowmodels used in these works were limited to non-spreading jets.GRB and merger jets have been studied extensively in theliterature. Semi-analytic models based on non-spreading jets havebeen successful in the study of afterglow light curves. However,jets are expected to laterally expand at late times. The inclusion ofspreading effects is crucial to study the light curves and images. In thispaper the properties of synthetic images of both non-spreading andlaterally spreading jets are studied. In §2 we describe the numericalmethod used to model the images. In §3 we consider laterallyspreading/non-spreading jets to evaluate the afterglow images and in§4 the results are applied to a GW17017-like system. We explicitlyshow how the combination of afterglow light curve and imageobservations can break the degeneracy. Summary conclusions aregiven in §5. We consider a spherical shell, with energy 𝐸 , mass 𝑀 = 𝐸 / Γ 𝑐 expanding adiabatically with an initial Lorentz factor Γ into a coldand uniform circumburst medium (CBM) of particle density 𝑛 . Byconsidering conservation of the stress-energy tensor it can be shownthat the Lorentz factor Γ of the shell evolves with swept-up mass 𝑚 as (Pe’er 2012) 𝑑 Γ 𝑑𝑚 = − ˆ 𝛾 ( Γ − ) − ( ˆ 𝛾 − ) Γ 𝛽 𝑀 + 𝑚 [ 𝛾 Γ − ( ˆ 𝛾 − )( + Γ − )] , (1)where 𝛽 = √ − Γ is the velocity of the shock in units of 𝑐 andˆ 𝛾 is the adiabatic index of the fluid (see Pe’er 2012 and Lambet al. 2018 for details). Because of the relativistic beaming effect,the radiation from a jet can be described by a spherical model withan isotropic explosion energy 𝐸 . The actual energy in the jet with asolid angle Ω is given by ( Ω / 𝜋 ) 𝐸 , and as the bulk of the system iscausally disconnected from its edge the spherical model holds while Γ >> / 𝜃 𝑗 , where 𝜃 𝑗 is the half opening angle of the jet. As thejet decelerates, information about pressure gradients, transported bysound waves, can reach the edges, forcing them to spread laterally.To model this spreading effect we follow Granot & Piran (2012). Theopening angle 𝜃 𝑗 is constant for 𝑅 < 𝑅 𝑑 , where 𝑅 𝑑 = 𝑙 / Γ is thedeceleration radius. The shell has decelerated to Γ ≈ Γ / 𝑅 > 𝑅 𝑑 the lateral spreading is given by 𝑑𝜃 𝑗 𝑑𝑙𝑛 𝑅 = Γ 𝜃 𝑗 . (2)Since initially Γ 𝜃 𝑗 >> Γ ∼ / 𝜃 𝑗 the opening angle can growquickly (e.g. Granot & Piran 2012). The swept-up mass evolves as, 𝑑𝑚𝑑𝑅 = 𝜋𝑛𝑚 𝑝 (cid:20) ( − cos 𝜃 𝑗 ) 𝑅 +
13 sin 𝜃 𝑗 𝑅 𝑑𝜃 𝑗 𝑑𝑅 (cid:21) , (3)where it is assumed that the jet front is a conical section of a sphere.For the non-spreading case equation 3 gives 𝑚 = ( Ω / ) 𝑛𝑚 𝑝 𝑅 . This corresponds to the 𝑎 = Figure 1.
Evolution of the opening angle 𝜃 𝑗 in units of its initial value 𝜃 𝑗, . Photons emitted from a fluid element on the shell at a radius 𝑅 arrive at the observer at the time 𝑇 given by 𝑇 ( 𝑅, 𝛼 ) = ∫ 𝑅 𝑑𝑅𝛽𝑐 − 𝑅𝑐 cos 𝛼, (4)where 𝛼 is the angle between the emitter’s direction of motion andthe observer line-of-sight (LOS).To illustrate the typical jet evolution, we consider a jet withparameters 𝐸 = × ergs, Γ = 𝜃 𝑗, = ◦ and 𝑛 = − cm − ; throughout this section the on-axiscase is assumed ( 𝜃 obs = 𝑅 𝑑 = × cm, which correspondsto an on-axis deceleration time 𝑇 𝑑 ≈
300 s ∼ × − days. Figure1 shows the evolution of the opening angle 𝜃 𝑗 in units of its initialvalue 𝜃 𝑗, . The opening angle initially grows slowly and by thejet break, when Γ ∼ / 𝜃 𝑗 at 𝑇 ≈ . .
25. As the amount of swept-up massis larger in the spreading case, the spreading jet decelerates faster.This is illustrated in figure 2, which shows the evolution of the 𝑅 (solid curves) and Γ (dashed curves) as a function of 𝑇 . The bluelines correspond to the laterally spreading case, and the red linescorrespond to the non-spreading case. The non-spreading jet beginsto lead the spreading jet after 𝑇 ∼ 𝑇 =
75 dayshas a radius which 50% larger than the spreading jet. The initialevolution of the Lorentz factor Γ in both cases is similar. After theinitial coasting phase Γ decreases as Γ ∝ 𝑇 − / , as is expected fora relativistic blast wave. After 𝑇 ∼ . 𝑇 ≈ The shock accelerates the CBM electrons to a power-law distribution 𝑁 ( 𝛾 𝑒 ) 𝑑𝛾 𝑒 ∝ 𝛾 − 𝑝𝑒 𝑑𝛾 𝑒 , for 𝛾 𝑒 > 𝛾 𝑚 . Here 𝜖 𝑒 is the fraction of shockenergy given to the electrons, 𝛾 𝑚 = 𝜖 𝑒 ( 𝑝 − )( 𝑝 − ) − ( 𝑚 𝑝 / 𝑚 𝑒 )( Γ − ) is the minimum electron Lorentz factor and 𝑚 𝑒 are the massof the electron. The magnetic field strength behind the shock is 𝐵 = { 𝜋𝜖 𝐵 𝑛𝑚 𝑝 𝑐 ( Γ + )( Γ − )} / . MNRAS , 1–11 (2021) adio images and the jet viewing angle Figure 2.
Evolution of a top-hat jet radius 𝑅 (solid curves, left vertical axis)and Lorentz factor Γ (dashed lines, right vertical-axis),as a function of 𝑇 . Inboth cases 𝜃 obs =
0. Blue (red) lines are for the spreading (non-spreading)case.
Relativistic electrons emit synchrotron radiation as they gyratearound the magnetic field lines. The characteristic synchrotronfrequency is 𝜈 sync ( 𝛾 𝑒 ) = ( 𝑞𝐵𝛾 𝑒 / 𝜋𝑚 𝑒 𝑐 ) 𝛿 , where 𝑞 the electroncharge and 𝛿 = Γ − ( − 𝛽 cos 𝛼 ) − is the Doppler factor.The synchrotron spectrum is described by a broken power lawwith two break frequencies, 𝜈 𝑚 = 𝜈 𝑠𝑦𝑛𝑐 ( 𝛾 𝑚 ) and 𝜈 𝑐 = 𝜈 𝑠𝑦𝑛𝑐 ( 𝛾 𝑐 ) ,where 𝛾 𝑐 = ( 𝜋𝑚 𝑒 𝑐 )/( 𝜎 𝑇 Γ 𝐵 𝑇 ( 𝛼 = )) (e.g. Sari et al. 1998). Thespectrum is given by 𝐹 𝜈 𝐹 𝜈, max = ( 𝜈 / 𝜈 𝑚 ) / 𝜈 𝑚 > 𝜈, ( 𝜈 / 𝜈 𝑚 ) −( 𝑝 − )/ 𝜈 𝑐 > 𝜈 > 𝜈 𝑚 , ( 𝜈 𝑐 / 𝜈 𝑚 ) −( 𝑝 − )/ ( 𝜈 / 𝜈 𝑐 ) − 𝑝 / 𝜈 > 𝜈 𝑐 , (5)where the peak flux 𝐹 𝜈, max = 𝛿 ( 𝑁 𝑒 𝜎 𝑇 𝑚 𝑒 𝑐 𝐵 )( 𝜋𝑞𝐷 𝐿 ) − . Here 𝑁 𝑒 is the number of emitting electrons and 𝐷 𝐿 the luminositydistance to the source. We consider an axisymmetric jet, and assume that all jet and shockedambient material is confined to an infinitely thin region (this is knownas the thin-shell approximation). The jet is initially the polar regionof a sphere with half opening angle 𝜃 𝑗 .For numerical purposes the jet is divided into a single centralspherical cap and 𝑛 − 𝑘 = , , ..., 𝑛 −
1. The spherical cap has an opening angle 𝜃 𝑏, , andthe 𝑘 -th ring is bounded by two concentric cirlces on the sphere with 𝜃 𝑏,𝑘 and 𝜃 𝑏,𝑘 + given by 𝜃 𝑏,𝑘 = − (cid:18) 𝑘𝑛 sin 𝜃 𝑗 (cid:19) . (6)The spherical cap is regarded as a single region, and the 𝑘 -th ring isdivided in the azimuthal direction 𝜙 into 2 𝑘 + 𝜙 𝑘𝑙 = 𝜋𝑙 /( 𝑘 + ) , where 𝑙 = , , ... 𝑘 . Thisdivision yields a total of (cid:205) 𝑛 − 𝑘 = ( 𝑘 + ) = 𝑛 regions each subtending a solid angle of 2 𝜋 ( − cos 𝜃 𝑗 )/ 𝑛 (Beckers & Beckers 2012). Havingdivided the jet, the positions of the regions are defined by the radius 𝑅 and the angles ( 𝜃 𝑘 , 𝜙 𝑘𝑙 ) , where 𝜃 𝑘 = ( 𝜃 𝑏,𝑘 + 𝜃 𝑏,𝑘 − )/ 𝜙 𝑘𝑙 = ( 𝜙 𝑘𝑙 + 𝜙 𝑘𝑙 − )/
2. The coordinate vector r 𝑘𝑙 = ( 𝑥 𝑘𝑙 , 𝑦 𝑘𝑙 , 𝑧 𝑘𝑙 ) of thefluid element 𝑘𝑙 has components 𝑥 𝑘𝑙 = 𝑅 sin 𝜃 𝑘 cos 𝜙 𝑘𝑙 ,𝑦 𝑘𝑙 = 𝑅 sin 𝜃 𝑘 sin 𝜙 𝑘𝑙 ,𝑧 𝑘𝑙 = 𝑅 cos 𝜃 𝑘 . (7)Given a set of initial conditions, only the radial component 𝑅 willevolve for a non-spreading jet, whereas, for a laterally spreading jetboth 𝑅 and 𝜃 𝑘 will evolve as we discuss in section 3. To obtain light curves we use the procedure outlined in Lamb &Kobayashi (2017) and Lamb et al. (2018). We take a Cartesiancoordinate system in which the jet propagates in the 𝑧 -directionand the observer is in the 𝑦𝑧 -plane. The LOS forms an angle 𝜃 obs with the 𝑧 -axis. The direction along the LOS is defined by the unitvector ˆ n obs = sin 𝜃 obs ˆ y + cos 𝜃 obs ˆ z . The angle between the LOSand the direction of motion of the fluid element ˆ r 𝑘𝑙 = r 𝑘𝑙 / 𝑅 𝑘𝑙 ,for a non-spreading jet, is given by 𝛼 kl = cos − ( ˆ r 𝑘𝑙 · ˆ n obs ) . Thecontribution of this cell to the light curve at time 𝑇 is obtained byinverting equation 4 to obtain the observed radius 𝑅 𝑘𝑙 ( 𝑇 ) , whichdetermines the emission with the formalism detailed in section 2.2.For a laterally spreading jet the fluid element also has a sidewaysexpansion velocity. However, the lateral velocity is much smallerthan the radial velocity so the Doppler factor is evaluated by usingthe direction of radial motion only. The light curve is obtained byadding up the contribution of each individual fluid element, i.e. 𝐹 𝜈 ( 𝑇 ) = (cid:205) 𝑘,𝑙 𝐹 𝜈,𝑘𝑙 ( 𝑇 ) .The imaging plane is perpendicular to the LOS of the distantobserver. Two mutually perpendicular directions in this plane aregiven by the basis vectors ˆ˜ x = sin 𝜃 obs ˆ z − cos 𝜃 obs ˆ x , ˆ˜ y = ˆ x , where thetildes indicate vectors in the imaging plane, as seen by an off-axisobserver. This basis is chosen so that the principal jet moves in thepositive ˜ 𝑥 -direction in the imaging plane. Having defined the unitvectors in the imaging plane, the coordinates of the fluid elementsin the image are given by ˜ 𝑥 𝑘𝑙 = r 𝑘𝑙 · ˆ˜ x , ˜ 𝑦 𝑘𝑙 = r 𝑘𝑙 · ˆ˜ y . The specificintensity or brightness of a given fluid element is obtained from thespectral flux by considering the solid angle subtended by the emitterin the sky. Full hydrodynamics simulations show that when a merger jet needsto drill through surrounding ejecta, the emerging jet has a specificstructure in energy and Lorentz factor distributions (De Colle et al.2012; Xie et al. 2018; Gottlieb et al. 2020). These structures affectthe time evolution and shape of afterglow light curves (e.g. Granotet al. 2002; Wei & Jin 2003; Zhang & Meszaros 2002; Rossi et al.2004; Granot & Kumar 2003; Salafia et al. 2015; Lamb & Kobayashi2017; Beniamini et al. 2020; Takahashi & Ioka 2021).Here we consider a Gaussian jet model characterized by a core withsemi-opening angle 𝜃 𝑐 within which most of the energy is contained.The energy per unit solid angle, 𝜖 ( 𝜃 ) , and the initial Lorentz factor, Γ ( 𝜃 ) , distributions are, (cid:40) 𝜖 ( 𝜃 ) = 𝜖 𝑐 𝑒 − 𝜃 / 𝜁 𝜃 𝑐 Γ ( 𝜃 ) = + ( Γ 𝑐 − ) 𝑒 − 𝜃 / 𝜁 𝜃 𝑐 , (8) MNRAS000
2. The coordinate vector r 𝑘𝑙 = ( 𝑥 𝑘𝑙 , 𝑦 𝑘𝑙 , 𝑧 𝑘𝑙 ) of thefluid element 𝑘𝑙 has components 𝑥 𝑘𝑙 = 𝑅 sin 𝜃 𝑘 cos 𝜙 𝑘𝑙 ,𝑦 𝑘𝑙 = 𝑅 sin 𝜃 𝑘 sin 𝜙 𝑘𝑙 ,𝑧 𝑘𝑙 = 𝑅 cos 𝜃 𝑘 . (7)Given a set of initial conditions, only the radial component 𝑅 willevolve for a non-spreading jet, whereas, for a laterally spreading jetboth 𝑅 and 𝜃 𝑘 will evolve as we discuss in section 3. To obtain light curves we use the procedure outlined in Lamb &Kobayashi (2017) and Lamb et al. (2018). We take a Cartesiancoordinate system in which the jet propagates in the 𝑧 -directionand the observer is in the 𝑦𝑧 -plane. The LOS forms an angle 𝜃 obs with the 𝑧 -axis. The direction along the LOS is defined by the unitvector ˆ n obs = sin 𝜃 obs ˆ y + cos 𝜃 obs ˆ z . The angle between the LOSand the direction of motion of the fluid element ˆ r 𝑘𝑙 = r 𝑘𝑙 / 𝑅 𝑘𝑙 ,for a non-spreading jet, is given by 𝛼 kl = cos − ( ˆ r 𝑘𝑙 · ˆ n obs ) . Thecontribution of this cell to the light curve at time 𝑇 is obtained byinverting equation 4 to obtain the observed radius 𝑅 𝑘𝑙 ( 𝑇 ) , whichdetermines the emission with the formalism detailed in section 2.2.For a laterally spreading jet the fluid element also has a sidewaysexpansion velocity. However, the lateral velocity is much smallerthan the radial velocity so the Doppler factor is evaluated by usingthe direction of radial motion only. The light curve is obtained byadding up the contribution of each individual fluid element, i.e. 𝐹 𝜈 ( 𝑇 ) = (cid:205) 𝑘,𝑙 𝐹 𝜈,𝑘𝑙 ( 𝑇 ) .The imaging plane is perpendicular to the LOS of the distantobserver. Two mutually perpendicular directions in this plane aregiven by the basis vectors ˆ˜ x = sin 𝜃 obs ˆ z − cos 𝜃 obs ˆ x , ˆ˜ y = ˆ x , where thetildes indicate vectors in the imaging plane, as seen by an off-axisobserver. This basis is chosen so that the principal jet moves in thepositive ˜ 𝑥 -direction in the imaging plane. Having defined the unitvectors in the imaging plane, the coordinates of the fluid elementsin the image are given by ˜ 𝑥 𝑘𝑙 = r 𝑘𝑙 · ˆ˜ x , ˜ 𝑦 𝑘𝑙 = r 𝑘𝑙 · ˆ˜ y . The specificintensity or brightness of a given fluid element is obtained from thespectral flux by considering the solid angle subtended by the emitterin the sky. Full hydrodynamics simulations show that when a merger jet needsto drill through surrounding ejecta, the emerging jet has a specificstructure in energy and Lorentz factor distributions (De Colle et al.2012; Xie et al. 2018; Gottlieb et al. 2020). These structures affectthe time evolution and shape of afterglow light curves (e.g. Granotet al. 2002; Wei & Jin 2003; Zhang & Meszaros 2002; Rossi et al.2004; Granot & Kumar 2003; Salafia et al. 2015; Lamb & Kobayashi2017; Beniamini et al. 2020; Takahashi & Ioka 2021).Here we consider a Gaussian jet model characterized by a core withsemi-opening angle 𝜃 𝑐 within which most of the energy is contained.The energy per unit solid angle, 𝜖 ( 𝜃 ) , and the initial Lorentz factor, Γ ( 𝜃 ) , distributions are, (cid:40) 𝜖 ( 𝜃 ) = 𝜖 𝑐 𝑒 − 𝜃 / 𝜁 𝜃 𝑐 Γ ( 𝜃 ) = + ( Γ 𝑐 − ) 𝑒 − 𝜃 / 𝜁 𝜃 𝑐 , (8) MNRAS000 , 1–11 (2021)
J.J. Fernández, S. Kobayashi & G.P. Lamb where the values 𝜁 = 𝜁 = 𝑅 𝑑 does not depend on 𝜃 . The structure isimposed as initial conditions for the dynamics. From equation 2 it can be seen that the degree of lateral spreadingdepends on the Lorentz factor of the jet and that spreading becomessignificant when Γ ∼ / 𝜃 𝑗 . To include this effect in the dynamicsof a top-hat jet, equation 2 is simply applied for the edges and thespreading is applied to each fluid element accordingly.For structured jets the implementation is slightly more complex.The initial conditions are given as a function of 𝜃 by equations 8,and the discretization described above divides the jet into rings ofconstant 𝜖 ( 𝜃 𝑘 ) , Γ ( 𝜃 𝑘 ) . Each of these rings has slightly differentdynamical evolution, therefore, we assume each ring to be partof a top hat with initial opening angle 𝜃 𝑏,𝑘 + . Neglecting theinteraction between rings, the dynamical evolution of each ring,including the spreading effect, is approximated by using the top-hatjet model. Similar approaches have been considered in previousstudies, including Lamb et al. (2018) and Ryan et al. (2020). Recently, 2D (e.g. Granot et al. 2018; Zrake et al. 2018) and 3Dhydrodynamics models (Nakar et al. 2018) have been used to obtainsynthetic images in the context of the NS merger event GW170817.Semi-analytic models have been used extensively for light curvecalculation, both for simple top-hat jets and diverse structured jets.Lateral spreading is generally introduced in semi-analytic modelsby assuming a simplified model, such as the formalism presented inGranot & Piran (2012). Ryan et al. (2020) and Lamb et al. (2018) takesimilar approaches, modelling this feature as sound-speed expansionof the jet edges. However, the application of semi-analytic modelsto imaging in the literature is generally limited to non-spreadingjets. Semi-analytic imaging was presented in Gill & Granot (2018)for Gaussian and power law structured, non-spreading jets; see alsoGhirlanda et al. (2019). In Lu et al. (2020), a semi-analytic effective1-D formalism is presented for which lateral spreading is derived byconsidering momentum conservation and pressure gradients (in thispaper the authors show the evolution of the numerical grid points,but explicit imaging is not provided).Once the solid angle of the jet increases significantly, the jetdecelerates faster. The effect of lateral spreading is not substantialsignificant in the rising part of the light curves, however, the lightcurve, where 𝜈 < 𝜈 𝑚 , peaks earlier when lateral spreading isincluded. The decay index after the jet break, or peak for an off-axisobserver, depends on the lateral spreading.In this paper the synthetic radio images of laterally spreading jetsare obtained using the semi-analytic model described in the previoussection. The deceleration of the jet caused by mass build-up on theshock shell is governed by equation 1, the swept-up mass is givenby equation 3, and the evolution of the opening angle is given byequation 2.Radio imaging has been proposed as a tool to break the degeneracybetween radial and angular structured jets (Gill & Granot 2018)or between isotropic and jet-like ejecta (Mooley et al. 2018). Inparticular, obtaining the image centroid, defined as the surface brightness-weighted centre of the image,˜ 𝑥 𝑐 = ∫ 𝐼 𝜈 𝑑 ˜ 𝑥 (cid:48) 𝑑 ˜ 𝑦 (cid:48) ∫ ˜ 𝑥 (cid:48) 𝐼 𝜈 𝑑 ˜ 𝑥 (cid:48) 𝑑 ˜ 𝑦 (cid:48) , (9)was key for this purpose. For spherical blast waves the centroid doesnot move i.e. ˜ 𝑥 𝑐 =
0. This is also the case for jets observed exactly onaxis (when the LOS runs exactly along the jet axis). For jets observedoff-axis, at early times ˜ 𝑥 𝑐 moves in the principal jet direction. Forrelativistic jets we would observe superluminal motion of the jet onthe sky if the viewing angle is small.To illustrate the spreading effects on light curves and images,consider a Gaussian structured with jet 𝐸 𝑐 = 𝜋𝜖 𝑐 = × ergs, Γ 𝑐 = 𝑛 = − cm − , 𝜃 𝑗, = ◦ and 𝜃 𝑐 = ◦ , and themicroscopic parameters 𝑝 = . 𝜖 𝑒 = . 𝜖 𝐵 = .
01 (e.g. Lamb& Kobayashi 2017; Zrake et al. 2018). We consider a luminositydistance to the observer of 𝐷 𝐿 = . 𝜃 obs = ◦ , ◦ , ◦ and 45 ◦ . The on-axislight curves are very similar for the non-spreading (top panel) andspreading cases (bottom panel). For the off-axis cases, the light curvesobtained for the laterally spreading jet show a slightly faster rise, anearlier peak time, and a faster post-peak decay. The earlier brighteningis due to more mass being swept-up at smaller radii when compared tothe non-spreading case. The steeper decay after peak time is again dueto lateral spreading transporting energy away from the core (Lu et al.2020). The peak times for the non-spreading are 𝑇 𝑝 ≈ . , , 𝑇 𝑝 ≈ . , ,
51 and 122days (for 𝜃 obs = ◦ , ◦ , ◦ and 45 ◦ respectively).Figure 4 shows synthetic radio images of the afterglows for 𝜃 obs = ◦ from the jet axis, at times 𝑇 = ,
75 and 230 days after theinitial explosion. The surface brightness in each image is normalizedto 𝐼 𝜈, max , with the colour map covering the range between 0 . 𝐼 𝜈, max and 𝐼 𝜈, max . The red crosses indicate the position of the centroid ineach image. The counter-jet does not appear in the images as itsbrightness falls below the brightness threshold.In the non-spreading case (left column) the principal jet alwaysmoves in the positive ˜ 𝑥 -direction, leaving the explosion origin behind( ˜ 𝑥 =
0) in the imaging plane. The images resemble a sliced ellipsoid,and present a gradual decrease in brightness from front (right-mostedge) to back (left-most edge). A ring develops in the back ofthe image, the edges of which correspond to the wings of the jet.The bright leading section corresponds to the centre of the jet. Theextension along the ˜ 𝑥 -direction is around 2 − 𝑦 .In contrast, the morphology of the laterally spreading jet images ismuch closer to circular, due to the excess (reduced) growth along ˜ 𝑦 ( ˜ 𝑥 ). As before the images present a dim ring which encloses a brighterregion corresponding again to central, more energetic fluid elements.Similar morphology is found at higher inclinations 𝜃 obs = ◦ , ◦ .In addition, lateral spreading also causes the fluid elements to rotatearound the origin of the imaging plane. When the jet-opening anglegrows to the observing angle, 𝜃 𝑗 = 𝜃 obs , the apparent motion of itsoutermost components is in the negative ˜ 𝑥 - direction. This gives riseto the features described in following sections, such as the centroidmotion reversal. The spreading and non-spreading cases result in significantlydifferent evolutions for the centroid shifts. In figure 5 the evolutionof the image centroid is shown for observing angles 𝜃 obs = ◦ , ◦ and 45 ◦ . The centroid quickly moves away from the explosion pointin the sky. At very late times ( 𝑇 ∼ MNRAS , 1–11 (2021) adio images and the jet viewing angle Figure 3.
Light curves for a Gaussian jet obtained at frequency 𝜈 = 𝐺𝐻 𝑧 for a non-spreading jet case (top panel) and a case which includes lateralspreading (bottom panel). The light curves correspond to an on-axis observer(solid blue lines), and observers at 𝜃 obs = ◦ (orange dashed lines), 𝜃 obs = ◦ (green dashed-dotted lines) and 𝜃 obs = ◦ (red dotted lines). spreading jet the early evolution is similar, but the maximum centroiddisplacement is smaller and is reached much earlier, at 𝑇 ∼ , 𝜃 obs = ◦ , ◦ and 45 ◦ respectively. By thesetimes, the light curve flux has decreased by a factor of ∼ ,
45 and 7for the spreading jet compared to the peak flux. The centroid positioncontinues to move backwards and crosses ˜ 𝑥 =
0. It then reverses oncemore and asymptotically approaches ˜ 𝑥 = 𝑥 𝑐 from the principaland counter jet to the centroid are separated for 𝜃 obs = ◦ in figure6 (orange and green dashed lines respectively). The solid, blue lineshows the overall centroid evolution for comparison. Initially, thecontribution of the principal jet dominates in both jet models. In thenon-spreading case (left panels), when the principal-jet deceleratessufficiently the counter-jet contribution becomes relevant and thecentroid motion reverses. While at early times the jet core dominatesthe centroid calculation, as it decelerates and becomes less bright the 𝜃 obs 𝛽 app (coll./spread.) Inferred Δ 𝜃 (coll./spread.) % error20 ◦ . / . ◦ / ◦
6% / 19%30 ◦ . / . ◦ / ◦
15% / 22%45 ◦ . / . ◦ / ◦
21% / 24%
Table 1.
Observing angles inferred from the apparent velocity of the radioimage centroid at light curve peak time. contribution of the slower jet wings becomes more significant. Atvery late times 𝑇 ∼ 𝜃 𝑗 > 𝜃 obs and part of the jet begins to move backwards in the imagingplane. Consequently, the expansion becomes more isotropic, whichslows down the principal jet centroid displacement and eventuallyalso contributes to the reversal.In figure 7 apparent velocity of the centroid in units of 𝑐 , 𝛽 app , isshown as a function of 𝑇 . For all cases the centroid displacement isinitially superluminal, 𝛽 app ,𝑐 >
1. The apparent velocity presents amuch steeper decrease in the laterally spreading cases. The threshold 𝛽 app < 𝛽 app → Γ ( 𝜃 obs − 𝜃 c ) ∼
1, the apparent velocity of the centroid atthe peak time of the light curve can be used to estimate Δ 𝜃 = 𝜃 obs − 𝜃 𝑐 (e.g. Mooley et al. 2018; Nakar & Piran 2020). A point source movingwith a Lorentz factor Γ at an angle Δ 𝜃 with respect to the LOS hasthe maximum apparent velocity 𝛽 app = 𝛽 Γ = √ Γ − Δ 𝜃 = Γ − . If the apparent velocity is obtained around the peaktime, the angle Δ 𝜃 can be estimated as sin Δ 𝜃 = ( + 𝛽 ) − / .At the peak time of the light curve 𝑇 𝑝 , the simple estimate ofthe apparent velocity 𝛽 app = sin − Δ 𝜃 − ≈ . , . . 𝜃 obs = ◦ , ◦ and 45 ◦ respectively, where we have used the valuesof Δ 𝜃 set by the simulations. The values of the 𝛽 app obtained fromthe centroid shift and the inferred Δ 𝜃 are shown in table 1. The valuesof the numerical 𝛽 app are larger than those expected from the simpleestimate, implying that Δ 𝜃 is slightly underestimated in this method.The errors are larger in the spreading case. The centroid is a robust characteristic of jet images which can berelatively easily obtained from observations. For brighter jets itmight be possible to carry out more detailed analysis to obtain otherproperties (Zrake et al. 2018). Figure 9 shows the full width at halfmaximum (FWHM) along the ˜ 𝑦 -direction, as measured at ˜ 𝑥 = ˜ 𝑥 𝑐 .The case of the non-spreading jet presents only modest growth evenat late times as the expansion of the jet is preferably along ˜ 𝑥 . Aslateral spreading leads to significantly enhanced growth in ˜ 𝑦 , theFWHM grows much faster, roughly as the physical size of the imagein the sky. The centroid shift and the evolution of the FWHM canconfirm or give a constraint on the lateral expansion law of jets.Figure 8 shows the surface brightness distributions along the jet MNRAS000
1, the apparent velocity of the centroid atthe peak time of the light curve can be used to estimate Δ 𝜃 = 𝜃 obs − 𝜃 𝑐 (e.g. Mooley et al. 2018; Nakar & Piran 2020). A point source movingwith a Lorentz factor Γ at an angle Δ 𝜃 with respect to the LOS hasthe maximum apparent velocity 𝛽 app = 𝛽 Γ = √ Γ − Δ 𝜃 = Γ − . If the apparent velocity is obtained around the peaktime, the angle Δ 𝜃 can be estimated as sin Δ 𝜃 = ( + 𝛽 ) − / .At the peak time of the light curve 𝑇 𝑝 , the simple estimate ofthe apparent velocity 𝛽 app = sin − Δ 𝜃 − ≈ . , . . 𝜃 obs = ◦ , ◦ and 45 ◦ respectively, where we have used the valuesof Δ 𝜃 set by the simulations. The values of the 𝛽 app obtained fromthe centroid shift and the inferred Δ 𝜃 are shown in table 1. The valuesof the numerical 𝛽 app are larger than those expected from the simpleestimate, implying that Δ 𝜃 is slightly underestimated in this method.The errors are larger in the spreading case. The centroid is a robust characteristic of jet images which can berelatively easily obtained from observations. For brighter jets itmight be possible to carry out more detailed analysis to obtain otherproperties (Zrake et al. 2018). Figure 9 shows the full width at halfmaximum (FWHM) along the ˜ 𝑦 -direction, as measured at ˜ 𝑥 = ˜ 𝑥 𝑐 .The case of the non-spreading jet presents only modest growth evenat late times as the expansion of the jet is preferably along ˜ 𝑥 . Aslateral spreading leads to significantly enhanced growth in ˜ 𝑦 , theFWHM grows much faster, roughly as the physical size of the imagein the sky. The centroid shift and the evolution of the FWHM canconfirm or give a constraint on the lateral expansion law of jets.Figure 8 shows the surface brightness distributions along the jet MNRAS000 , 1–11 (2021)
J.J. Fernández, S. Kobayashi & G.P. Lamb
Figure 4.
Radio images ( 𝜈 = 𝐺𝐻 𝑧 ) at times 44, 75 and 230 days (top, middle and bottom panels respectiveley) after explosion for an observer with line ofsight at 𝜃 obs = ◦ from the jet axis and a distance of 𝐷 = . 𝐼 𝜈, max in eachframe, with a background threshold set at 0 . 𝐼 𝜈, max . The red crosses mark the position of the image centroid. The non-spreading case is shown in the leftcolumn and the spreading case in the right column.MNRAS , 1–11 (2021) adio images and the jet viewing angle Figure 5.
Evolution if the image centroid position ˜ 𝑥 𝑐 as a function of observertime 𝑇 and observing angle 𝜃 obs (blue: 𝜃 obs = ◦ , orange: 𝜃 obs = ◦ , green: 𝜃 obs = ◦ ). The solid (dashed) lines are for the non-spreading (spreading)case. axis, 𝐼 𝜈, mean ( ˜ 𝑥 ) = ∫ 𝐼 𝜈 ( ˜ 𝑥, ˜ 𝑦 ) 𝑑 ˜ 𝑦 Δ ˜ 𝑦 , (10)at times 𝑇 = ,
75 and 230 days after the explosion, for 𝜃 obs = ◦ .A brightness threshold 𝐼 𝜈 / 𝐼 𝜈, max ≥ .
01 was set in the imagesimage.The distributions in figure 8 trace the morphology of the fullimages in figure 4. For the non-spreading case (top panel), thedistributions move to the right (larger ˜ 𝑥 𝑐 ) and they become slightlywider as time passes. The small dips in the distributions correspondto the ring feature in figure 4. For the spreading case (bottom panel),the spreading of the distribution is more significant compared to thepropagation. The rear edge moves backward as time passes. The peakof the distribution is located slightly behind the front edge, beforea sharp dip due to the ring which encloses the central jet material.We find similar results for larger inclinations 𝜃 obs = ◦ , ◦ . Theseresults indicate that the lateral spreading introduces distinguishablefeatures in the brightness distributions from the non-spreading case.In particular, if the reverse motion of the rear edge is observed thiswould be a signature of the lateral expansion. For bright events, thecentroid shift and FWHM evolution measurements might confirm orgive a tight constraint on the lateral expansion law of merger jets. Afterglow light curves have served as powerful tools to understandthe jet physics. However, there is degeneracy among the systemparameters. Consequently, a given light curve can be compatiblewith a wide range of models, with varying model parameters. Aswe have discussed, the light curves peak when the jet core becomesvisible to the observer. Since the Lorentz factor of the core dependson the Sedov length 𝑙 ∝ ( 𝐸 / 𝑛 ) / , the peak time 𝑇 𝑝 ∝ 𝑙 Δ 𝜃 alonecan not determine the angle Δ 𝜃 = ( 𝜃 obs − 𝜃 𝑐 ) . The peak fluxalso does not provide constraints on the angles as it depends onadditional microphysical parameters which are poorly constrainedby observations. The width of the light curve peak only constrains Figure 6.
Position of the image centroid ˜ 𝑥 𝑐 (solid blue curve) as a function oftime 𝑇 for the non-spreading (top panel) and spreading case (bottom panel)for 𝜃 obs = ◦ . The brightness-weighted contribution of the principal jet(dashed orange curves) and counter jet (dashed green curves) are also shown. 𝜃 obs / 𝜃 𝑐 . For a fixed ratio jets with wider cores and larger viewingangles will produce a similar peak (Nakar & Piran 2020).This implies very different physical conditions can result in almostidentical light curves. To illustrate this point, five different Gaussianjets are considered, with parameters given in table 2. These modelswere obtained by fixing the ratio 𝜃 obs / 𝜃 𝑐 , and scaling 𝜃 obs , 𝜃 𝑐 , 𝑛 and 𝜖 𝐵 to keep 𝑇 𝑝 and 𝐹 𝜈 ( 𝑇 𝑝 ) constant. We have fixed the energy 𝐸 while varying the CBM density 𝑛 . The models span two ordersof magnitude of 𝑛 (or equivalently a factor of 10 / in 𝑙 ) and oneorder of magnitude for 𝜖 𝐵 . The models chosen to roughly matchradio ( 𝜈 = 𝐺𝐻𝑧 ) data for the afterglow of the neutron star mergerGW170817. The light curves are shown in figure 10 in which thelight curves are identical around the peak.The properties of radio images depend directly on the geometryand dynamics of the shock. By combining the constraints from thelight curves and images we can distinguish different models. Figure12 shows radio images obtained for models 1 − MNRAS000
Position of the image centroid ˜ 𝑥 𝑐 (solid blue curve) as a function oftime 𝑇 for the non-spreading (top panel) and spreading case (bottom panel)for 𝜃 obs = ◦ . The brightness-weighted contribution of the principal jet(dashed orange curves) and counter jet (dashed green curves) are also shown. 𝜃 obs / 𝜃 𝑐 . For a fixed ratio jets with wider cores and larger viewingangles will produce a similar peak (Nakar & Piran 2020).This implies very different physical conditions can result in almostidentical light curves. To illustrate this point, five different Gaussianjets are considered, with parameters given in table 2. These modelswere obtained by fixing the ratio 𝜃 obs / 𝜃 𝑐 , and scaling 𝜃 obs , 𝜃 𝑐 , 𝑛 and 𝜖 𝐵 to keep 𝑇 𝑝 and 𝐹 𝜈 ( 𝑇 𝑝 ) constant. We have fixed the energy 𝐸 while varying the CBM density 𝑛 . The models span two ordersof magnitude of 𝑛 (or equivalently a factor of 10 / in 𝑙 ) and oneorder of magnitude for 𝜖 𝐵 . The models chosen to roughly matchradio ( 𝜈 = 𝐺𝐻𝑧 ) data for the afterglow of the neutron star mergerGW170817. The light curves are shown in figure 10 in which thelight curves are identical around the peak.The properties of radio images depend directly on the geometryand dynamics of the shock. By combining the constraints from thelight curves and images we can distinguish different models. Figure12 shows radio images obtained for models 1 − MNRAS000 , 1–11 (2021)
J.J. Fernández, S. Kobayashi & G.P. Lamb
Figure 7.
Apparent velocity of the centroid in units of 𝑐 . Solid (dashed) linescorrespond to non-spreading (spreading) case. 𝑛 ( cm − ) 𝜖 𝐵 𝜃 𝑐 (rad) 𝜃 𝑗, (rad) 𝜃 obs (rad)Model 1 1 . × − .
04 0 .
05 0 .
16 0 . . × − .
02 0 .
06 0 .
19 0 . . × − .
008 0 .
07 0 .
23 0 . . × − .
004 0 .
08 0 .
26 0 . . × − .
003 0 .
09 0 .
30 0 . Table 2.
Simulation parameters for the jets in 4. The other jet parameters arefixed, 𝐸 = 𝜋 𝜖 𝑐 = . ergs, Γ 𝑐 = 𝜖 𝑒 = − . , 𝑝 = . 𝐷 𝐿 = . these cases, the model images also present a brightness distributionwhich decreases gradually from front to back, where it ends in abow shape. Note that the morphology is the same for all models,up to scaling of the image size. This is due to the ratios 𝜃 obs / 𝜃 𝑗 = . 𝜃 obs / 𝜃 𝑐 = . 𝑇 =
75 days and 𝑇 =
230 days, and the apparent velocity computed as 𝛽 app = Δ 𝑥 𝑐 / 𝑐 Δ 𝑇 ,are reported in table 3. Δ 𝜃 is also computed from 𝛽 app using thepoint particle approximation as sin Δ 𝜃 = ( + 𝛽 ) − / . In this casethe approximation results provide a much better estimate of the theinclination, with deviations of the order of a few percent for rathersmall 𝜃 𝑜𝑏𝑠 / 𝜃 𝑐 . The errors in the Δ 𝜃 estimates were larger for theparameter sets discussed in section 3.1 (i.e. table 3, 19 −
24% forthe laterally spreading jets), in which Δ 𝜃 was obtained by using thepoint-emitter approximation. We now consider model 3 in 2, andby changing the viewing angle 𝜃 obs (the other parameters including 𝜃 𝑐 are fixed), we reevaluate errors as a function of 𝜃 obs / 𝜃 𝑐 . Theresults are shown in 11 where the apparent velocity Δ ˜ 𝑥 𝑐 / Δ 𝑇 havebeen estimated for two time windows Δ 𝑇 = 𝑇 − 𝑇 = . 𝑇 𝑝 or 𝑇 𝑝 .We find that the errors (or the discrepancy) are larger for larger angleratios. However, merger afterglow observations (i.e. detailed light Figure 8.
Lateral ˜ 𝑦 -averaged brightness distributions for 𝑇 = ,
75 and 230days (blue, orange and green). Top panel: non-spreading case. Bottom panel:laterally spreading case. The distributions were obtained for 𝜃 obs = ◦ . curves and radio images) will be available only for bright events.The detail afterglow modeling is likely to be carried out only for lowinclination events, for which the point-emitter estimates are accurate,and the apparent velocity estimates are not sensitive to the choice ofthe time window.The observed centroid displacement obtained from VLBI imagingbetween 75 and 230 days post-merger, as reported in (Mooley et al.2018), is shown for comparison (green solid lines delimited bycrosses). They found a centroid displacement of 2 . ± . 𝛽 app = . ± .
5. Figure 12 showshow radio imaging can be used for parameter inference. While thedetails of the morphology of the images is the same for all models,the evolution of the centroid is determined by growth of the emittingregion, which in turn is related to how fast the shock can expand. Forexample, model 1 (5) involves a slowly (rapidly) expanding jet as thedensity is the lowest (highest) in the set. Visual inspection shows thatmodel 1 (5) results in excess (too little) centroid displacement to becompatible with observations. Models 2 , MNRAS , 1–11 (2021) adio images and the jet viewing angle Figure 9.
Vertical extent of the radio images, defined as the full width at halfmaximum (FWHM) in the ˜ 𝑦 -direction at ˜ 𝑥 = ˜ 𝑥 𝑐 . The solid lines are for anon-spreading jet, the dashed lines for a laterally spreading jet. Figure 10.
Synthetic light curves for GRB 17017A-like systems obtainedusing the parameters in table 2. Data points for 3 GHz VLA observations(from Makhathini et al. 2020) are shown as black circles. The spreading jetmodel has been assumed.
More detailed analysis would lead to tighter constraints on 𝜃 obs . Notethat the direct comparison of between the observed and numericalcentroid shift can provide a constraint on 𝜃 obs without using theapproximation Δ 𝜃 ∼ 𝛽 − . The determination of 𝜃 obs breaks thedegeneracy among the parameters { 𝜃 obs , 𝜃 𝑐 , 𝑛, 𝜖 𝐵 } .The two methods described above make it possible to break thedegeneracy in the light curve models using two radio images. Figure13 shows the centroid evolution for each of the five models wherethe ˜ 𝑥 𝑐 − 𝑇 curves do not overlap as long as the centroid does notmove backwards. Given a well localized explosion locus, in principlea single radio image and the corresponding centroid position canprovide enough information to distinguish the model parameters { 𝜃 obs , 𝜃 𝑐 , 𝑛, 𝜖 𝐵 } which best agree with observations. In all cases thecurves behave as broken power-laws ˜ 𝑥 𝑐 ∝ 𝑇 𝜔 , where it is found that Figure 11.
Error in the estimation of 𝜃 obs from synthetic images as a functionof 𝜃 obs / 𝜃 𝑐 for two different observing windows. Δ 𝑥 𝑐 (mas) 𝛽 app Δ 𝜃 (rad) % errorModel 1 3 . . .
22 3 . . . .
27 4 . . . .
32 4 . . . .
37 5 . . . .
41 5 . Table 3.
Observable parameters extracted from the synthetic radio imagesshown in 12. 𝜔 ≈ .
86 provides a good fit until the light curve peak time 𝑇 ∼ 𝜔 ≈ . Γ ∼ 𝑇 − / , the centroid shift canbe approximated as ˜ 𝑥 𝑐 ∼ 𝛽 app 𝑇 ∼ 𝑇 / . Since the jet wing propagateslower, the small contribution might make the scaling index smallerin the numerical model. We have studied the properties of synthetic merger jet radio images byusing semi-analytic models of non-spreading or laterally spreadingjets. We find our semi-analytic models reproduce the featuresexpected from hydrodynamics simulations (e.g Zrake et al. 2018,Granot et al. 2018).While spreading is not immediately apparent in the rising phaseof afterglow light curves, it affects the slope and peak time, and cancertainly affect images. We compare the evolution of the images in thespreading and non-spreading cases. We find that lateral spreading hasan important effect on the morphology of the images and the evolutionof the centroid even at early times. This shows the importance ofaccounting for this dynamical feature in parameter estimation.We study two methods to determine the viewing angle from radioimages: the point-emitter approach (e.g. Rees 1966; Mooley et al.2018) and direct comparison of two or more images. We explicitlydemonstrate how the image centroid together with light curves breaksthe degeneracy among the model parameters. By using a sample ofGaussian structured jets with parameters that can roughly explain the
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MNRAS000 , 1–11 (2021) J.J. Fernández, S. Kobayashi & G.P. Lamb
Figure 12.
Synthetic radio images ( 𝜈 = . 𝐺𝐻 𝑧 ) corresponding to the degenerate light curves in figure 10. The rows correspond to models 1 − 𝑇 = ,
150 and 230 days. The centroid positions are shown with red crosses, and the centroiddisplacment with respect to 𝑇 =
75 days are shown with red lines. The centroid displacement reported in (Mooley et al. 2018) is shown with green lines. Thesurface brightness in each image is normalized to 𝐼 𝜈, max , with the colour map covering the range 0 . 𝐼 𝜈, max − 𝐼 𝜈, max MNRAS , 1–11 (2021) adio images and the jet viewing angle Figure 13.
Centroids of the images for models 1 −
5. The dashed lines are forpower-law fits ˜ 𝑥 𝑐 ∝ 𝑡 . . The color scheme is the same as in figure 12. light curve of the GW170817 afterglow, we find estimate the viewingangle to be 𝜃 obs ≈ .
32 rad.The inclination 𝜃 obs was previously obtained by assuming thepoint-emitter approximation Δ 𝜃 ∼ 𝛽 − (e.g. Mooley et al. 2018).We find the simple method is accurate especially for the GW170817case , and our inclination estimate is consistent with the previousstudies (e.g. Mooley et al. 2018; Ghirlanda et al. 2019; Hotokezakaet al. 2019; Lamb et al. 2019a; Troja et al. 2019). ACKNOWLEDGEMENTS
This research was supported by STFC grants and a LJMUscholarship. GPL is supported by the STFC via grant ST/S000453/1.
DATA AVAILABILITY
The data underlying this article and additional plots will be sharedon reasonable request to the corresponding author.
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