Intermittent mildly magnetized jets as the source of GRBs
MMNRAS , 1–10 (2021) Preprint 2 February 2021 Compiled using MNRAS L A TEX style file v3.0
Intermittent mildly magnetized jets as the source of GRBs
Ore Gottlieb (cid:63) , Omer Bromberg, Amir Levinson, Ehud Nakar
School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel
ABSTRACT
Gamma-ray bursts (GRBs) are powered by relativistic jets that exhibit intermittency over a broad range of timescales - from ∼ ms to seconds. Previous numerical studies have shown that hydrodynamic (i.e., unmagnetized) jets that are expelled froma variable engine are subject to strong mixing of jet and cocoon material, which strongly inhibits the GRB emission. In thispaper we conduct 3D RMHD simulations of mildly magnetized jets with power modulation over durations of 0.1 s and 1 s,and a steady magnetic field at injection. We find that when the jet magnetization at the launching site is σ ∼ .
1, the initialmagnetization is amplified by shocks formed in the flow to the point where it strongly suppresses baryon loading. We estimatethat a significant contamination can be avoided if the magnetic energy at injection constitutes at least a few percent of the jetenergy. The variability timescales of the jet after it breaks out of the star are then governed by the injection cycles rather than bythe mixing process, suggesting that in practice jet injection should fluctuate on timescales as short as ∼
10 ms in order to accountfor the observed light curves. Better stability is found for jets with shorter modulations. We conclude that for sufficiently hotjets, the Lorentz factor near the photosphere can be high enough to allow efficient photospheric emission. Our results imply thatjets with 10 − < σ < ∼
10 ms duty cycle are plausible sources of long GRBs.
Key words: gamma-ray bursts — MHD — instabilities — methods: numerical
Despite years of extensive research, the underlying mechanism ofthe prompt gamma-ray emission is still one of the most fundamentalmysteries of Gamma-ray bursts (GRBs). The prompt emission is as-sociated with several notable characteristics of GRBs, such as highradiation efficiency (e.g. Granot et al. 2006; Ioka et al. 2006; Zhanget al. 2007; Beniamini et al. 2016) and rapid temporal variabil-ity (e.g. Ramirez-Ruiz & Fenimore 2000; Nakar & Piran 2002a,b;MacLachlan et al. 2012; Bhat 2013). The latter could emerge from avariable engine (Levinson & Eichler 1993; Sari & Piran 1997; Mac-Fadyen & Woosley 1998; Fenimore et al. 1999; Aloy et al. 2000), orfrom the interplay between the jet and a high pressure cocoon whichit inflates (e.g. Aloy et al. 2002; Matzner 2002; Morsony et al. 2007;Gottlieb et al. 2019). The jet-cocoon interplay induces the variabil-ity by virtue of forming local hydrodynamic instabilities along thejet-cocoon interface (JCI; Gottlieb et al. 2021) which render the jetstructure irregular . The non-linear nature of the jet physics and theturbulent behavior of the cocoon imply that a complete study of thejet evolution to the emission zone can only be obtained through 3Dsimulations.A numerical and analytic work by Gottlieb et al. (2019) showedthat continuously powered hydrodynamic (i.e., unmagnetized) jetsyield highly efficient photospheric emission for any reasonable set ofjet parameters. Their 3D simulations also showed that all hydrody-namic jets are subject to local hydrodynamic instabilities that growalong the collimation shock at the jet base (see also Meliani & Kep- (cid:63) [email protected] pens 2010; Matsumoto & Masada 2013a,b, 2019; Matsumoto et al.2017; Toma et al. 2017; Gourgouliatos & Komissarov 2018). Thegrowth of the instabilities leads to efficient mixing of jet and cocoonmaterial along the JCI. The mixing is reflected in the light curveas variations in the radiation efficiency along the jet. Therefore, ifthe mixing is not too intense such that the terminal Lorentz factor is (cid:38) continuously launched jets exhibit both high efficiency and rapidvariability. Gottlieb et al. (2019) argued that an additional advan-tage of this model is that pair production in the downstream of thecollimation shock serves as a thermostat, which leads to a spectralpeak of the photospheric emission that is consistent with observa-tions. This model however has two shortcomings. First, the degreeof mixing changes with time, indicating that some temporal evolu-tion is expected in the light curve, which has not been observed.Second, and more importantly, if there is no additional dissipationprocesses between the collimation shock and the photosphere, thenthe spectrum of the emission is expected to have an exponential cut-off above the spectral peak. This is inconsistent with the observedprompt emission broken power-law spectrum. A potential solutionto this problem is an additional dissipation process that acts near thephotosphere, such as internal shocks. Gottlieb et al. (2019) foundthat the variable mixing leads to internal shocks, however it is un-clear whether those are strong enough to reshape the emerging spec-trum.The naive expectation is that a variable engine will lead to strongefficient internal shocks. It is also reasonable to expect that the cen-tral engine operates intermittently as its dynamical time is on orderof ms, much shorter than the burst duration. Thus, in a following © 2021 The Authors a r X i v : . [ a s t r o - ph . H E ] J a n O. Gottlieb et al. study Gottlieb et al. (2020b) examined the effect of engine variabil-ity on the propagation of hydrodynamic jets and on the resultingemission. They carried out numerical simulations of intermittentlylaunched hydrodynamic jets where the jet power has high and lowpower episodes. Interestingly, they found that such jets are subjectto very heavy baryonic entertainment from the cocoon, much heav-ier than the one seen in continuous jets. The heavy mixing emergesduring the low power episodes as the high pressure of the cocoon“squeezes” the region between the last high power episode and thenext one, filling it with heavy cocoon material. As a result, thenext powerful jet episode encounters the dense cocoon material andthis interaction leads to the heavy mixing. Ultimately, the jet ma-terial launched during high-power episodes dissipates all its energyon pushing the dense cocoon material that stands in its way suchthat both components are heavily mixed during the process. Subse-quently the loaded jet reaches its photosphere with a Lorentz factorthat is too low to generate the prompt GRB emission. The conclu-sion of these two studies is that continuous hydrodynamic jets canpotentially be the sources of GRBs, although it is unclear whetherthey can generate the observed spectrum, while intermittent hydro-dynamic jets cannot generate GRBs.A different, and arguably more realistic, picture of GRB jets is thatthey are at least weakly magnetized, as the jet launching is likely tobe driven by magnetic fields (Blandford & Znajek 1977; Komissarov2001). As magnetic fields are known for suppressing the growth oflocal hydrodynamic instabilities (e.g. Millas et al. 2017; Matsumoto& Masada 2019), magnetized jets would naturally yield differentstructures than those of hydrodynamic jets. Gottlieb et al. (2020a)performed 3D simulations of continuously launched weakly mag-netized jets and found that the growth of instabilities is inhibited injets with a magnetization of σ (cid:38) − (see also Matsumoto et al.2020). The suppression of the instabilities renders the jet more sta-ble, thereby keeping its terminal Lorentz factor high to power an ef-ficient photospheric emission, while avoiding the temporal evolutionthat results from the change in the mixing. However, such jets do notfeature the temporal variability in their light curve, owing to the lowbaryon loading. To conclude, continuously injected magnetized jetsgenerate efficient radiation however lack high variability, while in-termittent hydrodynamic jets naturally provide persistent variabilitybut lack the high radiative efficiency.Intermittent jets with subdominant magnetic fields may resolvethis issue by stabilizing the jets against the mixing, thereby allowingboth efficient and highly variable emission. However, this is not nec-essarily the case as the dominant mixing process in continuous andintermittent jets is different. In continuous jets the mixing is inducedby the growth of the instabilities along the JCI, and can be sup-pressed by sub-dominant magnetic fields. In intermittent hydrody-namic jets, most of the mixing originates in the interaction betweenthe jet material at the front of the high-power episodes and the highpressure cocoon material that squeezes the low-power jet. This inter-action is similar to the one that occurs at the head of the jet, wheremixing is taking place in shock. This type of mixing is not neces-sarily affected by sub-dominant fields. Therefore, in order to testthe effect of sub-dominant field on intermittent jets a full numericalanalysis is required. In this work we perform 3D relativistic mag-netohydrodynamic (RMHD) simulations of variable mildly magne-tized (10 − (cid:46) σ (cid:46)
1) jets propagating in a dense medium, and studythe evolution of such systems for the first time. We find that a sub-stantial magnetic component enables intermittent jets to avoid heavymixing while propagating inside the surrounding medium, and thusis likely to be essential in order to explain the GRB prompt emis-sion. The outline of this paper is as follows. In §2 we set up the numerical framework and present the models that we consider. In§3 and §4 we examine the stability and the magnetization of the jet,respectively. In §5 we discuss the expected post-breakout structure.We summarize and discuss the implications of our results in §6.
We perform a set of high resolution (see discussion and convergencetests in Appendix A) 3D simulations with
PLUTO v4.2 (Mignoneet al. 2007), using the RMHD module and a relativistic ideal gasequation of state. For the integration we employ a third order Runge-Kutta time stepping, piece-wise parabolic reconstruction and anHLL Riemann solver. To impose ∇ · B = PLUTO ’sconstrained transport scheme.Our setup is based on models of continuously injected magneticjets, LM − and LM − from Gottlieb et al. (2020a), denoted hereas models W ∞ and S ∞ , respectively. In these models the jet is carry-ing a toroidal magnetic field and is injected with an initial Lorentzfactor Γ = × cm at an altitudeof z = × cm into a non-rotating star with a mass M (cid:63) = (cid:12) ,a radius R (cid:63) = cm and a density profile: ρ (cid:63) ( r ) = π × gcm r − (cid:32) R (cid:63) − rR (cid:63) (cid:33) . (1)We define the toroidal magnetic field profile at the injection pointfollowing Mignone et al. (2009, 2013); Gottlieb et al. (2020a): b φ = (cid:113) π h ρ j σ c (cid:40) r / r j , r < r j , r j , r (cid:16) − ( r − r j , / ) ( r j , − r j , / ) (cid:17) r j , < r < r j , (cid:41) , (2)where ρ j is the jet’s mass density and σ is the peak magnetizationat half of the nozzle radius. In models W ∞ and S ∞ the values of σ are 0.01 and 0.1, respectively. The jet is injected hot with a specificenthalpy h ≡ + p t / ρ j c + b φ / πρ j c = ( + σ ) , where p t is the thermal pressure. It expands conically to an opening angle of θ ≈ . / Γ = .
14 (Mizuta & Ioka 2013; Harrison et al. 2018),before it is collimated by the cocoon. The maximal terminal proper-velocity (if no mixing occurs) is thus η ≡ (cid:113) Γ h − ≈ ( + σ ) .To each model W ∞ and S ∞ we apply step-function modulationsof 0.1 s and 1 s in the luminosity, such that it jumps between 20%and 100% of a total (two-sided) luminosity, L = erg s − . Westress that the central engine variability is likely to be on shortertimescales, but these are not feasible for 3D RMHD simulationsover the range of length scales used in this work, given our avail-able computational resources. To avoid strong currents on the lowerboundary that result from similar jumps in the tangential magneticfield, we keep the injected magnetic field profile constant throughoutthe duration of the simulation, and vary only the injected ρ j ∝ L . Itimplies that σ ∝ b φ h − ρ − j ∝ L − in the low power episodes, σ , h ,is 5 times higher than in the high power episodes, σ , l . We compareour results with jet models from our previous studies: (i) two con-tinuously injected magnetic jets models, W ∞ and S ∞ ( LM − and LM − in Gottlieb et al. 2020a); (ii) two intermittent hydrodynamicmodels H . and H (models E and G in Gottlieb et al. 2020b). Thereference models maintain the same engine variability and hydro-dynamic or magnetohydrodynamic (with σ = σ , l ) parameters asthose in our models. The rest of the parameters in our simulationsare listed in Table 1. MNRAS000
14 (Mizuta & Ioka 2013; Harrison et al. 2018),before it is collimated by the cocoon. The maximal terminal proper-velocity (if no mixing occurs) is thus η ≡ (cid:113) Γ h − ≈ ( + σ ) .To each model W ∞ and S ∞ we apply step-function modulationsof 0.1 s and 1 s in the luminosity, such that it jumps between 20%and 100% of a total (two-sided) luminosity, L = erg s − . Westress that the central engine variability is likely to be on shortertimescales, but these are not feasible for 3D RMHD simulationsover the range of length scales used in this work, given our avail-able computational resources. To avoid strong currents on the lowerboundary that result from similar jumps in the tangential magneticfield, we keep the injected magnetic field profile constant throughoutthe duration of the simulation, and vary only the injected ρ j ∝ L . Itimplies that σ ∝ b φ h − ρ − j ∝ L − in the low power episodes, σ , h ,is 5 times higher than in the high power episodes, σ , l . We compareour results with jet models from our previous studies: (i) two con-tinuously injected magnetic jets models, W ∞ and S ∞ ( LM − and LM − in Gottlieb et al. 2020a); (ii) two intermittent hydrodynamicmodels H . and H (models E and G in Gottlieb et al. 2020b). Thereference models maintain the same engine variability and hydro-dynamic or magnetohydrodynamic (with σ = σ , l ) parameters asthose in our models. The rest of the parameters in our simulationsare listed in Table 1. MNRAS000 , 1–10 (2021)
RBs from intermittent mildly magnetized jets Model σ , l σ , h T [s] t b [s] t f [s] H . H W ∞ − − ∞ W . − × − W − × − S ∞ − − ∞ S . − × − S − × − Table 1.
The models’ parameters. σ , l and σ , h are the initial values of σ during the high and low power episodes, respectively. T is the time cycle, t b is the breakout time of the forward shock from the star, and t f is the time atwhich the simulation ends. The numerical grid used in the simulations includes three patcheson the ˆ x and ˆ y axes and one on the ˆ z -axis. The inner patch on theˆ x and ˆ y axes is uniform with 640 cells in the inner | . × | cm.The outer patches are logarithmic with 80 cells in each directionfrom | . × | cm up to | . × | cm. On the ˆ z -direction we use2000 uniform cells from z to 2 R (cid:63) = × cm. In total we have800 × × = . × cells. We provide convergence testsin Appendix A. Continuous jet-cocoon interaction throughout the jet propagation inthe star leads to mixing of jet and cocoon material along the JCI.The mixing reduces the terminal proper-velocity of the outflow to η s ≡ (cid:112) Γ s h s − < η such that under intense mixing conditions,the jet becomes radiatively inefficient. Magnetic fields can stabilizethe JCI and considerably reduce the mixing. Recent 3D RMHD sim-ulations of continuously injected jets by Gottlieb et al. (2020a) haveshown that the presence of a toroidal magnetic field can suppress thegrowth of the local hydrodynamic instabilities that emerge on theJCI. The magnetization degree required for jet stabilization dependson various jet parameters such as the jet power and initial openingangle. Low power or wider jets need stronger fields for stabilizationwhereas high power and narrower jets are more stable and thus therequired magnetization is smaller. Since in our models of modulatedjets we vary the jet power and its magnetization such that L ∝ σ − ,the stabilization of the jet interface may also vary between low andhigh power modes. We take the maximal luminosity (minimal σ )to be that of the stable continuously injected jets in Gottlieb et al.(2020a), so that the σ values are at least as high as those which sta-bilize continuous jets. It is therefore expected that if the jet stabilitydepends solely on the magnetic field’s ability to suppress the growthof local hydrodynamic instabilities, our intermittent magnetized jetswill also remain stable. If the mixing originates from the pulsationnature of the jet rather than the instabilities along the JCI, like in thecase of intermittent hydrodynamic jets (Gottlieb et al. 2020b), thenmagnetic fields might have only a minor effect on the jet composi-tion.At early times intermittent magnetic jets share similarities withboth continuously launched magnetic jets and intermittent hydro-dynamic jets. On one hand, they are supported by a toroidal mag-netic field which stabilizes the jet and keeps η s ≈ η , similar tocontinuous jets (Gottlieb et al. 2020a). On the other hand, the mod-ulations in the jet power destabilize the jet, since the low power The subscript s reflects the value of the quantity after mixing. episodes cannot support the jet against the confining cocoon, whichcollapses inwards towards the jet spine. All the energy of the pre-vious high power episode has been used to accelerate heavy mate-rial with η s (cid:28) η at the jet head, and the jet has to be rebuilt inthe next powerful episode. When the front of the jet-cocoon systemfinally crosses a substantial part of the star, a structure of multiplemini-jets emerges, as was found in hydrodynamic jets (Gottlieb et al.2020b). However, unlike hydrodynamic jets, here the jet magnetiza-tion which is kept at a level of σ ≥ − , has a stabilizing effect onboth high power and low power episodes, and is able to prevent fromthe weaker jet episodes to completely mix with cocoon material.Figure 1 shows iso-contours of log ( η s ) in intermittent jet models,shortly after the jet broke out from the star. Strong colors: red, or-ange and yellow mark higher η s whereas blue-gray colors portraylower η s values (see figure caption for accurate values). While alljets contain rather mixed material at their fronts, due to completedissipation of the early modulations, the stability of the jet spine atlower parts differs between models. It is prominent that jets withstrong fields (models S . and S on the right) are more stable thanthe rest of the models. The stability of the weaker magnetized jets(models W . and W in the middle) seems to be similar to that ofthe non-magnetized jets (models H . and H on the left). It is alsoshown that the magnetized jets exhibit some helical motion neartheir heads. In models S . and S the jet head deviates from the sym-metry axis by ∼ ◦ . This behavior can be attributed to two factors:(i) kink instability as these jets contain regions with σ ∼ σ (cid:38) . p m dictates thejet stability and dynamics. It depicts meridian cuts of log ( η s ) (left)and the reciprocal of the plasma beta: log ( β − ) ≡ log ( p m / p t ) (right)in models S . (top) and S (bottom). One can see that jet elementswith ultra-relativistic η s (cid:38)
100 have a substantial p m component( β (cid:46) p m is subdominant, baryon entrainment from the cocoon penetratesinto the jet and increases the mixing (white double arrows). In lower σ models p m is dynamically subdominant throughout the jet, allow-ing strong mixing at all times. While the magnetic pressure in thejet suppresses baryon entrainment into the jet, the magnetic pressurein the JCI dictates the jet dynamics on large-scales. It can be seenthat the magnetic pressure support is asymmetric between the twosides of the jet. As a result the jet tilts towards the side where β ishigher, with larger asymmetry ( S . ) leads to a larger deviation fromthe axis. The magnetic pressure in the JCI may also contribute to thejet head helical motion together with the expected kink instabilities.We stress that the value of β upon injection is not held constantbetween low and high power episodes. This comes as a result ofholding the magnetic pressure fixed throughout the injection andvarying the gas pressure as p t , ∝ ρ j ∝ L in order to keep h con-stant at all times. It then follows that the jet is launched with β ∝ L ,so that high power episodes are less supported by the magnetic pres-sure upon injection. We find however that during the jet propagationin the star, β changes and becomes lower in the high power episodesthan that in the low power ones. The reason is twofold: (i) Highpower episodes sustain stronger collimation shocks which amplify MNRAS , 1–10 (2021)
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H0.1 W0.1 S0.1H1 W1 S1
Figure 1.
3D isocontours of log ( η s ) shortly after jet breakout from the star. Top: 0.1 s modulations H . (left), W . (middle) and S . (right). Bottom: 1 smodulations: H (left), W (middle) and S (right). The values of log ( η s ) are 0.5 (gray), 0.8 (blue), 1.1 (green), 1.4 (yellow), 1.7 (orange) and 2 (red).MNRAS000
3D isocontours of log ( η s ) shortly after jet breakout from the star. Top: 0.1 s modulations H . (left), W . (middle) and S . (right). Bottom: 1 smodulations: H (left), W (middle) and S (right). The values of log ( η s ) are 0.5 (gray), 0.8 (blue), 1.1 (green), 1.4 (yellow), 1.7 (orange) and 2 (red).MNRAS000 , 1–10 (2021) RBs from intermittent mildly magnetized jets Figure 2.
2D maps of ˆ x − ˆ z planes in models S . (top) and S (bottom)upon jet breakout from the star. Shown are log ( η s ) (left) and magnetic tothermal pressure ratio log ( β − ) (right). The white arrows demonstrate therelationship between low η s regions and larger β . the magnetization in the jet (see §4), thereby decreasing the value of β ; (ii) High power jets are more stable (Gottlieb et al. 2021) suchthat the high power episodes keep their σ better than low powerepisodes in which β increases faster due to the mixing.Finally, when the modulations are longer ( S ), there are larger re-gions in the jet with low p m , allowing substantial baryon entrainmentfrom the cocoon into the jet. In model S . only small parts along thejet have subdominant p m , and thus the mixing remains lower whenthe modulations are shorter. This is in contrast to intermittent hydro-dynamic jets, in which longer modulations increase the jet stability(Gottlieb et al. 2020b). This behavior is due to the fact that hydrody-namic jets do not have any stabilization effect between high power episodes. Thus, small mini-jets dissipate their energy fast whereaslarger ones can keep their structure for longer times. This lookoutmight be important as the time cycles in nature are more likely tobe on dynamical timescales of an order of ∼ ms, so that magneticstabilization effect should be even more prominent. The intermittent jets are launched with toroidal magnetic fields intounmagnetized medium. As they propagate, they form cocoons ofshocked magnetized jet and unmagnetized medium material, mixedtogether due to turbulence that grow in the hot plasma. The turbu-lence also generate poloidal fields, which remain subdominant in allregions where σ is high.Figure 3a shows the average of σ weighted by the energy E (ex-cluding rest-mass): < σ > = E − (cid:82) σ dE , as a function of the ter-minal proper-velocity η s , taken at the last snapshot of each simu-lation. Regions in the plot that are associated with the jet, the co-coon or the JCI are highlighted with different background colorsaccording to their η s . All models exhibit the same behavior, inde-pendent of the modulation time, featuring high σ in unmixed jetmaterial (yellow background) and low σ in mixed cocoon material(blue background), in agreement with §3. The magnetization in thecocoon ( η s (cid:46)
3) roughly scales as < σ > ∼ η s , demonstrating thatshocked jet material (high η s ) has substantially higher < σ > thanthe shocked stellar material (low η s ). The dependency of < σ > on η s becomes weaker at the JCI and the jet, until it peaks with a value ∼ < σ > ≡ ( σ , l + σ , h ) / η .Figures 3b,c depict the normalized energy distribution per loga-rithmic scale of σ at the last snapshot of each simulation. Solid linesmark the total distribution in the system whereas dashed lines con-sider only material at the inner part of the star, z < R (cid:63) . We findthat qualitatively the total energy (solid lines) is distributed equallyin logarithmic scales of σ up to < σ > , the average of the injectedmagnetization peaks. The only exception is model S ∞ which exhibitsa bump at ∼ σ , indicating that the jet remains stable and the mix-ing is low. More subtle differences are notable between models S . , W . and W ∞ , with continuously injected ( W ∞ ) and strongly mag-netized ( S . ) jets exhibit less mixing and lower cocoon/jet energyratio.The left side of the distribution ( σ (cid:46) − ) marks the magne-tization in the cocoon. Magnetized cocoon material flows into thecocoon from the jet head with initial magnetization of σ ∼ < σ > .Over time this material undergoes mixing with shocked medium ma-terial and its σ decreases. Close to the jet base the magnetization islower than σ ∼ − and therefore the dashed lines and the solidlines coincide at this region. When the jet is intermittent, furthermixing takes place as different jet episodes interact with each other,rendering the cocoon more energetic compared to the jet, as can beseen in Figure 3c.The right side of the distribution depicts the magnetization in thejet. The strong collimation shocks at the base of the jet amplify themagnetic field after the shock up to σ ∼ .
1. In models W . , W and W ∞ the amplification of the magnetic field is reflected by theextension of the energy distribution up to σ (cid:38) . > σ , h , with allelements with σ (cid:38) . z < R (cid:63) where the collimationshock resides, so that dashed and solid lines coincide. In models S . , S and S ∞ the amplified field is comparable with σ and thus isnot as prominent as in the lower σ jets. In model S ∞ the dashed andsolid lines do not coincide at σ ∼ σ since the mixing is minimalsuch that the magnetization remains high at all radii. When the cen- MNRAS , 1–10 (2021)
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W W W S S S -4 -2 -3 -2 -1 (b) W W S S -4 -2 -3 -2 -1 (c) W W
S S
Figure 3.
Panel (a) depicts the total energy (excluding rest-mass) weightedaverage of σ at each terminal proper-velocity value η s . Background colorsmark the regions of the cocoon (light blue), JCI (light red) and jet (light yel-low), as found by Gottlieb et al. (2021). Panels (b) and (c) depict energydistributions as functions of σ . Solid lines delineate the total distributionwhereas dashed lines represent the distribution at z < R (cid:63) . We present a com-parison of different time cycles (b) and between continuous and intermittentjets (c). tral engine is variable, the amplification of the magnetic field alsooccurs in the internal shocks that are induced by the modulations,and are shown as longer high σ tails of the distributions of modu-lated jets. The observed radiation is released at large radii, long after the jetbreaks out of the star. Thus, in order to estimate the resulting emis-sion one needs to follow the post-breakout jet evolution up to thepoint where the flow becomes homologous. However, tracking the post-breakout evolution of magnetized jets requires keeping veryhigh resolution grid outside of the star. With our computational re-sources we can follow the evolution only to the point where thejet head reaches ∼ R (cid:63) . In some simulations even our highest res-olution turns out to be insufficient, and the numerical integrationcrashes soon after the jet breakout, limiting our ability to analyzethe post-breakout behavior any further. Hence, in order to estimatethe effect of the jet structure of intermittent magnetized jet on theobserved emission, we compare the outflow structure at this lim-ited range with the most stable intermittent hydrodynamic models H . and H . These simulations of hydrodynamic jets reach largerradii than magnetized jets. Thus we can use the comparison betweenmagnetized and unmagnetized jets at early time together with com-parison to the distribution of hydrodynamic jets at late times to learnabout the expected late time evolution of the magnetized jets.Figure 4 depicts the energy distribution of material that brokeout of the star as a function of η s , when the jet head is at differ-ent radii, r h . The figure compares the distributions of various hy-drodynamic and magnetized intermittent jets. The intermittent hy-drodynamic models contain predominant mildly-relativistic materialat early times. The temporal evolution of the energy distribution ofmodel H . shows two trends: (i) The distribution converges to aprominent peak, owing to internal shocks between the fast elementswith the mildly-relativistic quasi-spherical cocoon in the front; (ii)The peak is shifted to higher velocities (blue to yellow curves), dueto acceleration of the outflow by freshly less mixed elements thatexit the star. However, even after ∼
30s of engine activity (corre-sponds to the line of r h = R (cid:63) ), the peak is at η s ∼
10, implying thatthis model cannot generate a GRB. In model H those trends arenot observed due to a longer time cycle of the modulations whichleads to longer times between internal shocks. The late time analy-sis conducted by Gottlieb et al. (2019) showed that a prominent peakof η s ≈
30 is obtained when r h is at a few dozen stellar radii (stillbelow the photosphere), implying that jets with longer modulationscannot produce an efficient emission either.By contrast to the hydrodynamic models, η s of intermittent mag-netized jets recovers over time and converges to a rather flat energydistribution up to η . This comes as a result of a reduction in themixing of matter that breaks out from the star at later times, aswas also found in the post-breakout evolution of the continuouslyinjected magnetized jets W ∞ and S ∞ (Gottlieb et al. 2020a). There-fore, while e.g. the distributions of H and W . are rather similarwhen the jet breaks out (as also seen in Figure 1), the magnetizedjet evolution shows an extension to higher η s whereas the hydro-dynamic jet converges to a peak at low η s . Furthermore, model W exhibits a flat distribution of the outflow all the way to η alreadywhen r h = . R (cid:63) . As we discussed in §3, the jets with the strongermagnetic fields (models S . and S ) are the most stable ones. Thisis also demonstrated here with model S . being more stable thanmodel W . . However, the higher magnetization also causes morenumerical noise, limiting our ability to analyze model S far fromthe star, as its simulation crashes soon after the jet breakout. In sum-mary, Figure 4 implies that the resulting photospheric emission fromintermittent hydrodynamic jets will be inefficient, owing to low ter-minal η s , whereas magnetized jets become more stable over time,such that most of their energy lies in ultra-relativistic velocities topower an efficient photospheric emission. MNRAS000
30 is obtained when r h is at a few dozen stellar radii (stillbelow the photosphere), implying that jets with longer modulationscannot produce an efficient emission either.By contrast to the hydrodynamic models, η s of intermittent mag-netized jets recovers over time and converges to a rather flat energydistribution up to η . This comes as a result of a reduction in themixing of matter that breaks out from the star at later times, aswas also found in the post-breakout evolution of the continuouslyinjected magnetized jets W ∞ and S ∞ (Gottlieb et al. 2020a). There-fore, while e.g. the distributions of H and W . are rather similarwhen the jet breaks out (as also seen in Figure 1), the magnetizedjet evolution shows an extension to higher η s whereas the hydro-dynamic jet converges to a peak at low η s . Furthermore, model W exhibits a flat distribution of the outflow all the way to η alreadywhen r h = . R (cid:63) . As we discussed in §3, the jets with the strongermagnetic fields (models S . and S ) are the most stable ones. Thisis also demonstrated here with model S . being more stable thanmodel W . . However, the higher magnetization also causes morenumerical noise, limiting our ability to analyze model S far fromthe star, as its simulation crashes soon after the jet breakout. In sum-mary, Figure 4 implies that the resulting photospheric emission fromintermittent hydrodynamic jets will be inefficient, owing to low ter-minal η s , whereas magnetized jets become more stable over time,such that most of their energy lies in ultra-relativistic velocities topower an efficient photospheric emission. MNRAS000 , 1–10 (2021)
RBs from intermittent mildly magnetized jets -1 H -1 H -1 W -1 W -1 S -1 S Figure 4.
Energy distribution of matter outside the star as a function of η s . Shown are intermittent hydrodynamic models (top) and magnetized jets (center andbottom), with modulations of 0.1s (left) and 1s (right). Each model is shown at different times, manifested by the jet head location r h . Previous numerical studies have shown that hydrodynamic jets aresubject to hydrodynamic instabilities that grow on the JCI bound-ary, induce mixing between the jet and the cocoon and give rise torapid variability (Gottlieb et al. 2019). When magnetic fields are in-troduced in continuous flows, the growth of the instabilities is sup-pressed and thus these jets cannot account for the observed high vari-ability of GRB light curves (Gottlieb et al. 2020a). Continuous jet in-jection over the crossing time of the stellar envelope seems unlikelygiven the dynamical time (ms) of the putative engine. Recent sim-ulations by Gottlieb et al. (2020b) indicate that hydrodynamic jetswith modulated power are subject to another type of mixing. In thosejets the mixing emerges at the fronts of the high-power jet episodes as they slam into the heavy cocoon material that squeezes the low-power jet to fill up the regions between the high-power episodes.This mixing takes place via shocks in a manner that is similar to themixing that arises at the jet head, as if each high-power episode de-velops its own head. We denote this type of mixing as “head-like”.This type of mixing gives rise to heavy loading that inhibits emis-sion nearly completely, such that intermittent hydrodynamic jets arealso inconsistent with GRB observations. Nonetheless, under the hy-pothesis that jets are produced by magnetic extraction of the BH (ormagnetar) spin energy, the jet is likely to be magnetized well insidethe star, and the question remains as to how this might affect themixing of modulated jets.In this paper we report on high resolution 3D simulations of
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O. Gottlieb et al. the propagation of modulated, mildly magnetized (0 . ≤ σ ≤ . σ (cid:38) . .Due to numerical limitations we were unable to simulate the jetevolution outside of the star beyond ∼ ∼ R (cid:63) and show that over time theenergy distribution in log ( η s ) space converges to a rather flat profile,similar to the profiles of continuous hydrodynamic jets and flatterthan profiles of intermittent hydrodynamic jets. Following Gottliebet al. (2019) we calculate the average η s value on the jet axis inmodels W . and W above the collimation shock in the last snapshotof the simulation. We find average values of < η s > ( W . ) ≈ < η s > ( W ) ≈ W . with shorter modulations has a typical higher η s value, which indicates a higher stability. The measured < η s > values are larger by about an order of magnitude from the valuesmeasured in intermittent hydrodynamic jets (Gottlieb et al. 2020a).This implies that intermittent unmagnetized jets are less stable andhave higher baryon contamination than magnetized jets. A similaranalysis of higher σ jets cannot be preformed due to their deviationfrom the jet axis. The source of the shift, which is caused by animbalance in the magnetic pressure at the cocoon is still unclear.However, since the jets in models S . and S seem to be more stablethan those in models W . and W , based on our analysis in §3 and§5, we expect their < η s > to be larger than 200 as well.The high η s values and flat energy distributions in log ( η s ) space,seen in intermittent mildly magnetized and continuous hydrody-namic jets, imply that the jets should share some similarities in theirobservable characteristics. Gottlieb et al. (2019) found that in theirsimulated steady hydrodynamic jets the photospheric radius and ra-diative efficiency were r ph ≈ cm and ε (cid:38) .
5, respectively.Since in our simulations σ <
1, the specific enthalpy at the pho-tosphere is likely to be dominated by the radiation thermal pressure,and thus intermittent magnetized jets are expected to show similarlyhigh radiative efficiencies at their photosphere.An important difference between steady hydrodynamic outflowsand intermittent mildly magnetized jets is the characteristic vari-ability, which may affect the resulting emission via e.g., internalshocks. In continuous hydrodynamic models the variability is setby the physics of the instabilities that grow on the jet boundary andmix heavy material into the jet. This leads to density fluctuationsof short wavelengths in the jet and to variability timescales of or- Except for the front part of the outflow (of size ∼ R (cid:63) ) that broke out first. der of ∼
10 ms. In intermittent mildly magnetized jets the bound-ary instabilities are quenched due to the presence of magnetic fieldand the variability is governed by the timescales of the jet modula-tions, which are determined by the physics of the launching mecha-nism. This sets a strict constraint on the engine variability timescalesin mildly magnetized GRB jets, and possibly in highly magnetizedjets as well. The engine must be intermittent on timescales that areequivalent to the variability time of the prompt emission, an order of ∼
10 ms. Simulating such short time scales requires high grid res-olutions which are beyond our computational capabilities. Thus wewere limited to simulating jets with intermittent time (cid:38) σ longGRB jets dissipate their magnetic energy in narrow collimation noz-zles that form close to the jet base, resulting in plasma β ∼ σ = . σ jets may besimilar to what we find in mildly magnetized jets.Finally, we conclude this series of papers that explore thestructure and the emission from continuous/intermittent hydrody-namic/magnetized jets by comparing our main findings (see Table2). Continuous hydrodynamic jets maintain ultra-relativistic η s withwhich they generate efficient photospheric emission. Boundary in-stabilities lead to mixing which accounts for the observed light curvevariability and triggers internal shocks. The question whether theseshocks can broaden the spectrum to the observed frequencies re-mains unclear. Magnetic fields inhibit the growth of boundary insta-bilities and the resulting mixing. As a result, the light curve of con-tinuous weakly magnetized jets lacks the variability that is requiredby observations. Intermittent jet launching may generate strong in-ternal shocks that may set the hard tail of the spectrum. However,intermittent hydrodynamic jets were found to be prone to intensemixing that takes place at the head of each high-power jet episode,and reduces the radiative efficiency to be essentially zero. In this pa-per we showed that a magnetization of σ (cid:38) − can stabilize notonly boundary instabilities, but also the head-like mixing of intermit-tent jets. Consequently, intermittent magnetized jets can power effi-cient and variable emission (set by the engine variability), that maybe broaden by strong sub-photospheric internal shocks that emergefrom the modulations of the engine. We therefore conclude that outof all models that we tested, an intermittent mildly magnetized jetis the most plausible one to be consistent with all observables, andthus is the leading candidate as the source of GRBs. ACKNOWLEDGEMENTS
This research is partially supported by an ERC grant (JetNS) (OGand EN). AL and EN acknowledge support by the Israel ScienceFoundation Grant No. 1114/ 17. OB was funded by an ISF grant1657/18 and by an ISF (Icore) grant 1829/12.
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RBs from intermittent mildly magnetized jets Jet type Continuous hydrodynamic Continuous magnetized Intermittent hydrodynamic Intermittent magnetizedMixing source (head/boundary) both; boundary dominates stable head headVariability source (mixing/engine) mixing none both engineVariability timescales ∼ ms none set by engine and mixing set by engineEfficiency high high (cid:46)
1% highSpectrum broadening by shocks unlikely no no possibly
Table 2.
Summary of our 3D simulation results of continuous/intermittent hydrodynamic/magnetized jets, based on this paper and on Gottlieb et al. (2019);Gottlieb et al. (2020a); Gottlieb et al. (2020b); Gottlieb et al. (2021). Shown are the instability and variability sources of the jets, and the observables: variability,efficiency and possibility of hardening the spectrum by shocks. Red color text indicates results that are in tension with observations.
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
APPENDIX A: CONVERGENCE TEST
The baryon entrainment from the cocoon into the jet and the JCIis primarily taking place in the lateral direction. Therefore, our 3DRMHD simulations were chosen to maintain a very high resolutionon the ˆ x − ˆ y plane inside the inner patch where the jet and the JCIreside. In comparison, this resolution is higher than all previous 3Dsimulations of magnetized jets or intermittent jets in a star (Lopez-Camara et al. 2016; Gottlieb et al. 2020a; Gottlieb et al. 2020b). Inthe parallel direction to the jet axis we keep the same cells height ∆ z of that in previous 3D simulations of magnetized jets in a star(Gottlieb et al. 2020a). The cells height is not expected to be affectedby the intermittency of the jets as long as it sustains ∆ z (cid:28) T c . Thiscriterion is held in all of our simulations.We verify that our main conclusions are independent of the reso-lution by conducting a convergence test. We perform an additionalsimulation with an identical setup to that of W , since low σ jets donot deviate from the inner high resolution patch, such that it is easierto study the resolution effects. In the test simulation we double thecells resolution on the ˆ x and ˆ y dimensions, so that we have in total1600 cells on each of the ˆ x and ˆ y axes. Since this simulation is verydemanding, the ˆ z -axis stretches only up to R (cid:63) , and we compare thetwo simulations inside the stellar boundaries.Figure A1 depicts the energy distribution per logarithmic space of η s upon jet breakout from the star, indicating the jet stability insidethe star. The agreement between the original simulation (blue) andthe test one (red) is remarkable at η s (cid:38)
3, which manifests the jetand the JCI regions (Gottlieb et al. 2021) that are inside the highresolution patch. However, the similarity between the cocoons ( η s (cid:46) -1 Figure A1.
A comparison of the energy distribution of the original grid reso-lution (blue) and the higher resolution (red) as a function of η s . The distribu-tions are taken when jet head reaches the stellar surface (11 s in the originalsimulation and 13.5 s in the simulation with the higher resolution.). REFERENCES
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