Cannonball model diagnosis of the short gamma ray burst 170817A
The rich and complex data obtained from multi-wavelength observations of SHB170817A, the short hard gamma ray burst (SHB) associated with GW170817 –the first neutron stars merger (NSM) event detected in gravitational waves (GWs)– are analyzed in the framework of the cannonball model of SHBs. In this model a highly relativistic jet is launched by the nascent neutron star (or black hole) into a surrounding glory (light fireball) which was present already before the merger. The SHB was produced by inverse Compton scattering of glory photons by the jet, which was viewed far off-axis. The fading fireball, which produced the initial UVOIR afterglow, was powered by a neutron star remnant. It was overtaken by a late time X-ray, UVOIR and radio afterglow produced by synchrotron radiation from the decelerating jet in the interstellar medium of the host galaxy. If the radio afterglow of SHB170817A was indeed produced by the jet, it should display a superluminal motion relative to the SHB location, still detectable in VLA and VLBI radio observations.
Physics Department, Technion, Haifa 32000, Israel \affiliationPhysics Department, Technion, Haifa 32000, Israel \affiliationTheory Division, CERN, Geneva, Switzerland; IFT, Universidad Autónoma, Madrid, Spain
Gamma ray bursts (GRBs) were discovered fifty years ago by the American spy satellites Vela (Klebesadel, Strong & Olson 1973). For decades their origin and production mechanism remained mysterious. Thirty years ago, Goodman, Dar and Nussinov (1987) suggested that GRBs may be produced in extragalactic neutron star mergers (NSMs) by an fireball (Goodman 1986) formed by neutrino-antineutrino annihilation around the nascent compact object –a massive neutron star or a black hole. But shortly after the launch of the Compton Gamma-Ray Burst Observatory (CGRO), it became clear that such neutrino-annihilation fireballs are not powerful enough to produce observable GRBs at the very large cosmological distances indicated by the CGRO observations (Meegan et al. 1992).
Consequently, Meszaros and Rees (1992) suggested that the fireball produced in compact mergers may be collimated into a conical fireball by funneling through surrounding matter. Shaviv and Dar (1995) argued that GRBs are produced by inverse Compton scattering (ICS) of the fireball light by narrowly collimated jets of highly relativistic plasmoids (cannonballs, CBs) of ordinary matter, launched in mergers of compact stars due to the emission of gravitational waves (GWs), in phase transition of neutron stars to quark stars in compact binaries following mass accretion, or in stripped-envelope core-collapse supernova explosions (Dar et al. 1992). Li and Paczynski (1998), countered that all GRBs are produced by the thermal radiation emitted from fireballs formed by the radioactive decay of r-process elements synthesized from the tidally-disrupted neutron-star surface material in compact binaries undergoing a merger by GW emission.
It has also been observed long ago that GRBs may be roughly classified into two distinct species, long-duration soft gamma-ray bursts (GRBs) that usually last more than 2 seconds and short hard bursts (SHBs) typically lasting less than 2 seconds (Norris et al. 1984; Kouveliotou et al. 1993), and that a large fraction of GRBs are produced in broad line stripped envelope supernova (SN) explosions of type Ic akin to SN1998bw (see Galama et al. 1998 for the first observed GRB-SN association; Dado et al. 2002, Zeh et al. 2004 and references therein for early photometric evidence; Stanek et al. 2003 and Hjorth et al. 2003 for the first spectroscopic evidence, and Della Valle et al. 2016 for a recent review. See also Dado, Dar & De Rújula 2003 for the prediction of the discovery date and properties of the SN associated with GRB030329).
Based on indirect evidence, it was widely believed that SHBs were associated with NSMs (see, e.g., Fong and Berger 2013, Berger 2014). In particular, a faint infrared emission from SHB130603B was claimed to be the first observational evidence for a kilonova produced by a NSM (e.g. Berger et al. 2013, Tanvir et al. 2013). The first indisputable SHB/NSM association (SHB170817A/GW170817) was observed only recently (von Kienlin et al. 2017, Abbott et al. 2017a,b,c,d, Goldstein et al. 2017). Two days before these ground breaking observations, Dado and Dar predicted (2017) that because of the relatively small detection horizon of NSMs by Ligo-Virgo, the smoking guns of their NSM events, beside GWs will be only far off-axis SHBs or orphan afterglows.
The underlying process in the cannonball (CB) model of GRBs and SHBs is the ejection of highly-relativistic narrowly-collimated jets of CBs in stripped-envelope SNeIc and NSMs, respectively (e.g. Dar & De Rujula 2004, Dado, Dar & De Rújula 2002, 2009a, Dado, Dar & De Rújula 2009b). The photon-generating mechanism is the ICS of the glory’s light (pre-supernova light scattered by the earlier remnants of pre-SN ejections) in SNeIc. In NSMs the ICS takes place on a glory of radius cm, whose origin is not yet established, which was present already before the merger.
In this paper we show that ICS of photons in such a fireball surrounding the merging neutron stars, by a highly relativistic jet launched in the NSM and viewed far off-axis, can explain the prompt emission in SHB170817A. Its optical afterglow can be understood as the emitted radiation from the decaying fireball powered by its compact neutron star remnant —as opposed to a black-hole remnant— which is taken over at late time by synchrotron emission from the decelerating jet in the local ISM of the host galaxy.
In our CB model analysis, the absence of an extended emission in SHB170817A (Troja et al. 2017) following the prompt -ray emission is explained by the merger site not being in a densely illuminated region (Levan et al. 2017), unlike the globular clusters in which a considerable fraction of SHBs occur. The observed late-time X-ray afterglow, faint relative to that of ordinary SHBs (Troja et al. 2017), is synchrotron radiation from the far off-axis decelerating jet. So is the observed radio afterglow. If the radio afterglow is emitted by the jet, it should display a superluminal motion relative to the SHB’s location (Dar & De Rújula 2000a, Dado, Dar & De Rújula 2016) hopefully still detectable by VLA and VLBI radio observations.
2 Prompt emission
To explain the prompt emission of SHBs, the CB model requires a “glory” of target photons, Compton up-scattered by the CBs electrons to -ray energies. The observed duration of the SHB pulses implies a glory of typical size cm. The natural candidate is the plerion (Weiler & Panagia 1978) of the merging neutron stars, illuminated by one or both of them being pulsars.
In the CB model SHBs and GRBs share many properties, since they are produced by the same mechanism: ICS of ambient light by a narrow jet of CBs with large Lorentz factors . Their most probable viewing angles relative to the approaching-jet direction are and the polarization of their radiation is predicted to be linear and large: (Shaviv & Dar 1995; Dar & De Rújula 2004 and references therein).
The CB model entails very simple correlations between the main observables of SHBs and GRBs (Dar & De Rújula 2000a). For a burst at redshift and Doppler factor , for instance, the peak energy of their time-integrated energy spectrum satisfies , while their isotropic-equivalent total gamma-ray energy is . Hence, ordinary GRBs and SHBs, mostly viewed from an angle , obey
while the far off-axis () ones satisfy
Updated results on the correlation of Eq.(1), later empirically discovered by Amati et al. (2002), are shown in Figure 1. They satisfy well the CB model predicted correlation for ordinary GRBs.
In Figure 2 we plot the observations for the correlation Eq.(2) for far off-axis GRBs (Dar & De Rújula 2000a), which is also well satisfied.
In Figure 3 we plot results for the entire population of SHBs with known , and , along with the predictions of Eqs.(1),(2). Ordinary (near axis) SHBs satisfy Eq.(1). The prediction of Eq.(2) for low luminosity (far off-axis) SHBs cannot be tested because of the incompleteness of the current data (unknown and/or ) on the few low luminosity SHBs.
The reported peak energy of SHB170817A, keV (von Kienlin et al. 2017) is similar to that of the first-detected far off-axis GRB, 980425. This SHB’s is well above that expected from Eq.(1). But it is what is expected from Eq.(2), if the relatively small is due to being viewed far off-axis. The same conclusion can also be drawn from its X-ray afterglow (Troja et al. 2017), which is simply impossible to fit in the CB model with , as we shall see in detail in Section 6. In many respects, SHB170817A-NSM170817 is similar to GRB980425-SN1998bw, both of which were the first of their kind.
3 Pulse shape
The -ray data on this SHB (Goldstein et al. 2017) are insufficient to perform a detailed pulse-shape study. Thus we present a simplified derivation of its expected shape as a function of time, integrated in energy upwards from a given value, .
At the earliest times, , after the beginning of a CB’s pulse, the SHB’s number flux increases as , due to the increase of the effective cross section of the CB as its radius increases, , and it is crossing a region of radius wherein the ambient photons it scatters are approximately isotropic, i.e. the glory in Figure 4 (Dar & De Rújula 2004).
Let be the distance from a CB to the central engine at observer’s time , amplified from by time aberration. At distances the black-body number density of ambient photons intercepted by the CB decreases with distance as , and for the the values relevant here, it is in its Wien tail, so that:
In the CB’s rest frame the longitudinal momentum of the photons it intercepts is red-shifted by a factor , relative to its value in the engine’s rest frame. Thus, in that frame, the photon’s parallel momenta are negligible compared to their transverse momenta, unchanged by the Lorentz boost.
Let , as in Figure 4, be the transverse distance of the emission point relative to the CB’s direction of motion of a photon intercepted by the CB at . Its transverse momentum is and its energy in the CB’s rest frame is . The observer measures an energy and a photon number flux:
Integrate from upwards to obtain:
As a simple parametrization interpolating the small and large behaviors of the pulse’s shape, we write:
A fit of this expression to the observed pulse shape (Goldstein et al. 2017) returns very precise values of the fitted , but an extremely large range of allowed values of . The theoretical curve shown in Figure 5 is for a best fit s and a chosen s, the value of at which , with the SHB’s parameters estimated in Section 2. The fit has a .
4 No extended emission
A considerable fraction of SHBs show an extended emission (EE) after the prompt SHB (Villasenor et al. 2005; Norris & Bonnell 2006 Fong & E. Berger 2013; Berger 2017 for a review). Such SHBs may take place in rich star clusters or globular clusters (GCs) (Dado, Dar & De Rújula 2009b), where the ratio of binary neutron stars to ordinary stars is much higher than in the regular interstellar medium of galaxies. ICS of ambient light in GCs by the highly relativistic jets, which produce the SHBs, can explain the origin of their extended emission (Dado, Dar & De Rújula 2009b; Dado & Dar 2017). SHB170817A did not take place in a GC or a bright location in its host galaxy NGC4993 (Levan et al. 2017) and, indeed, as expected, no extended emission following its prompt emission was observed.
5 The fireball afterglow
It has been shown that the bolometric light curves of ordinary (Dado & Dar 2013) and superluminous (Dado & Dar 2015) SNe of Type Ia can be successfully described by a “master formula” involving five parameters. The underlying physics is complex, but the fitted values of the parameters turn out to be, in the cases that were studied, very close to the values expected and determined by simple physics considerations. To make this note self-contained we give in this section a brief description of the derivation of the master formula, required for our discussion of SHB170817A, and based on energy conservation in the fireball powered by an energy source and loosing energy by expansion and radiation.
Let be the time after the beginning of the formation of a fireball. As long as it is highly opaque to optical photons and rays, its thermal energy density is dominated by black body radiation, , at a temperature that we assume for simplicity to be spacially uniform. The fireball’s total radiation energy is , with the fireball’s volume. For a constant velocity of expansion and the resulting energy loss is simply , since and for black-body radiation.
Photon emission constitutes a second mechanism of energy loss by a fireball, corresponding to a bolometric luminosity , where the photon’s mean diffusion time is . The photon’s mean free path, , is dominated by their Thomson scattering on free electrons and positrons, , with their number density. For a fireball expanding at a constant velocity, , whose total number of free electrons and positrons is , , with . For a Type Ia SN, can be estimated to be days (Dado & Dar 2015).
Neutrino-antineutrino annihilation and the decay of radioactive isotopes synthesized in the merger ejecta make a positive contribution to the energy balance within a fireball, at a rate . Gathering all three contributions to , the time variation of , we conclude that energy conservation implies:
during the photospheric phase.
The solution of Eq. (7) is
Consequently, the bolometric luminosity, , is given by the simple analytic expression
This simple master formula, which was first derived by Dado & Dar (2013), provides an excellent description of the bolometric light curve of Supernovae Type Ia and of superluminal supernovae (Dado & Dar 2015). For a short energy deposition time by neutrino annihilation and r-processes, the late-time () behavior of Eq.(9) is
Such a bolometric light-curve is expected if the compact remnant of the NSM170817 is a stellar black hole. A best fit of Eq.(10) to the bolometric light curve of SHB170817A reported by Smartt et al. (2017), Evans et al. (2017) and Pian et al. (2017), shown in Figure 6, yields a rather unsatisfactory for d and erg/s.
For a fireball that at late time () is mainly powered by a pulsar with a period (Dar & Dado 2017 and references therein)
where . The time dependence of is slow relative to that of the rest of the integrand in Eq.(9). As a consequence it is a good approximation to factor it out of the integral, to obtain:
A best fit of Eq.(12) to the bolometric light curve of SHB170817A reported by Smartt et al. (2017), Evans et al. (2017), and Pian et al. (2017), shown in Figure 7, yields erg/s, d, and d, with an entirely satisfactory . For these parameters, the approximation of Eq.(12) differs from the “exact” result of substituting Eq.(11) into Eq.(9) by 15% at the peak luminosity and % at d.
The CB model best fits to the measured bolometric light curves of the UVOIR afterglow of SHB170817A, as shown in Figures 6,7 suggest that the compact remnant of NSM170817 is probably a pulsar rather than a stellar mass black hole.
On the other hand, best fits to the bolometric light curve reported by Cowperthwaite et al. (2017) and shown in Figures 8,9 are much less conclusive: for d and erg/s for a black hole remnant, while for a neutron star remnant, for erg/s, d, and d.
6 Rising Afterglow Of Far Off-Axis CB
The circumburst medium in the path of a CB moving with a Lorentz factor is completely ionized by the CB’s radiation. The ions of the medium that the CB sweeps in generate within it turbulent magnetic fields. The electrons that enter the CB with a Lorentz factor in its rest frame are Fermi accelerated there, and cool by emission of synchrotron radiation, an isotropic afterglow in the CB’s rest frame. As for the rest of the CB’s radiations, the emitted photons are beamed into a narrow cone along the CB’s direction of motion, their arrival times are aberrated, and their energies boosted by the Doppler factor and redshifted the cosmic expansion.
The observed spectral energy density of the unabsorbed synchrotron afterglow has the form (e.g., Eq. (28) in Dado, Dar & De Rújula 2009a)
where is the spectral index of the emitted radiation at frequency .
The swept-in ionized material decelerates the CB’s motion. Energy-momentum conservation for such a plastic collision between a CB of baryon number , radius , and an initial Lorentz factor yields the deceleration law (e.g. Eq. (3) in Dado and Dar 2012)
where is the slow-down time scale.
The frequency and time dependence of the afterglow implied by Eqs. (13),(14) depend only on three parameters: the product , the slow-down time-scale , and the spectral index . For , Eq. (14) yields . Hence, for far off-axis GRBs and SHBs,
which is valid as long as . Consequently, Eq.(13) yields rising light curves, for the X-ray, UVOIR and radio afterglows.
In Figure 10 we compare the 3 GHz light curve of the radio afterglow of SHB170817A first discovered by Hallinan et al. (2017) and followed up with the Karl G. Jansky Very Large Array (VLA), the Australia Telescope Compact Array (ATCA) and the upgraded Giant Metrewave Radio Telescope (uGMRT) and summarized by Mooley et al.(2017), and the CB model prediction. This CB model shape follows from Eqs.(13) with the measured radio spectral index , and Eq. (14) for . The value was obtained from the measured , and days, corresponding to a break time of days, from a best fit to the data.
In Figure 11 we compare the X-ray light curve measured with the Chandra X-ray observatory (CXO) and that predicted by Eqs.(13), and (14) using the same parameters as in Figure 10. Either Eq.(13) with Eq.(14), or Eq.(15) describe well the rising phase of the keV X-ray light curve measured with CXO (Troja et al. 2017a,b; Margutti et al. 2017a,b; Haggard et al. 2017).
Further optical and FIR observations of the counterpart of GW170817 with the Hubble Space Telescope, which took place on 6 Dec 2017 (Levan et al. 2017) recovered the source in optical filters, but did not detect it in the infrared, where the background from the galaxy is higher. The measured magnitudes of the source in the optical bands are broadly consistent with the extrapolation from the 93 day radio epoch (Mooley et al. 2017) to the near contemporaneous observations with CXO (Troja et al. 2017b; Margutti et al. 2017; and Haggard et al. 2017).
7 Superluminal motion
A very specific prediction of the CB model concerns the apparently superluminal motion in the plane of the sky, relative to the engine that produced them (Dar & De Rújula 2000b,; Dado, Dar & De Rújula 2016 and references therein), of the CBs moving towards the observer at a small but not vanishing angle .
In the CB model the peak value of a GRB or XRF pulse is . For a typical eV and –as in the case of SHB170817A– and , the Doppler factor simplifies to , so that . Equating this result to the observed keV, we estimate . The corresponding apparent velocity of a CB relative to the engine that emitted it is (e.g., Dar and De Rújula 2000b, Dado, Dar & De Rújula 2016 and references therein). For cases such as SHB170817A:
with as in Eq.(14). For days as estimated in Section 6, for d, and for d.
The angular displacement from the location of the neutron star merger to the CB’s later position is:
with Mpc the angular distance to SHB170817A in the standard cosmology. The function is plotted in Figure 12.
In a VLA or VLBI observation of SHB170817A, the angular Fresnel scale is of order as, considerably smaller that the angular size of a CB. This may lead one to expect that a CB’s image would scintillate (Taylor et al. 2004). But typical integration times of these observations are 100 minutes. At an early time of observation (), the image of the CB of SHB170817A would move by an angle as, and as at day . This shifting position while the data are accumulated would obliterate the scintillations (DDD 2016).
The temporal and spatial coincidence of NSM170817 and SHB170817A has shown that at least a fraction, if not most SHBs, are produced in NSMs.
SHB170817A and its afterglow at redshift ( Mpc) appeared to be very different from all other SHBs with known redshift, including SHB130603B at redshift ( Mpc) Mpc, where a faint infrared emission was claimed to be the first observational evidence for a kilonova produced by a NSM (e.g., Berger et al. 2013, Tanvir et al. 2013). But the observations of the low luminosity SHB170817A and its afterglow produced by NSM170817 can be naturally explained, as we have shown, by the cannonball model of GRBs, if SHB170817A was beamed along a direction far off from its line of sight and took place in the low density and low luminosity halo of its host galaxy NGC 4993, rather than in a dense stellar region.
In fact, based on the cannonball model of SHBs, we predicted (Dado & Dar 2017) that Ligo-Virgo NSM detections will be accompanied only by low-luminosity (far off-axis) SHBs (or by orphan SHB afterglows). This is due to the much smaller detection horizon of Ligo-Virgo compared to the mean distance of SHBs estimated from SHBs with known redshift.
Our detailed analysis indicates that the compact remnant of NSM170817 probably was a neutron star and not a black hole, in agreement with the evidence from the afterglow of all other SHBs with known redshift and well sampled X-ray afterglow (Dado & Dar 2017).
The existence of a glory surrounding the merger site already a day before the NSM merger is required by the CB model analysis of SHB170817A. Its origin is not clear. The most likely explanation is a plerion surrounding the binary neutron stars and powered by the emission of radiation and high energy particles by one or both of them. A fastly expanding kilonova powered by the radioactive decay of r-processed elements which exists already a day before the NSM when the merging neutron stars are separated by more than cm does not seem plausible. Observations of more NSM-SHB events and theoretical studies will be required in order to identify the origin of the UVOIR fireball.
The large superluminal speeds and angular displacements we have discussed could perhaps be observed by VLA and VLBI follow-up measurements of the late-time location of the radio afterglow of the jet, relative to the location of SHB170817A. Such observations would be most decisive if a non-vanishing angular displacement could be measured at least two different times. It goes without saying that the precise values of the predicted are quite uncertain, the crucial point being the trend shown in Figure 12.
Acknowledgment: ADR acknowledges that this project has received funding/support from the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 690575.
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