Hydrodynamics and Nucleosynthesis of Jet-driven Supernovae. I. Parameter Study of the Dependence on Jet Energetics

The association of gamma-ray bursts (GRBs) with massive-star explosions has stimulated vast interest in the past decades (for a review see, e.g., Tsuruta et al. 2018). Several well-observed examples include GRB 980425 (SN 1998bw; Galama et al. 1998), and GRB030329 (SN 2003dh; Hjorth et al. 2003; Stanek et al. 2003). They show a much higher peak luminosity than canonical core-collapse supernovae (SNe) such as SN 1987A that occurred in the Large Magellanic Cloud (for a review see, e.g., Nomoto et al. 1994a). These events hint at an explosion energy typically 10 times above ordinary SNe, named hypernovae (Iwamoto et al. 1998). The explosion of a bare CO core (Type Ic SN) can resemble many features observed in these events (see, e.g., Woosley 1993). The removal of H and He envelopes could be a result of tidal interaction from its companion in a binary system (Nomoto et al. 1994b) or efficient stellar wind mass loss. The bifurcation of high-mass-star explosions into hypernovae and faint-SN branches (Moriya et al. 2010; Nomoto et al. 2010) indicates the presence of an inner energy source after the core-collapse event.

The progenitor CO core is likely formed from stars with initial masses of M_prog ∼25–140 M_⊙ (Heger & Woosley 2002). For M_prog ≳ 80 M_⊙, the electron–positron pair instability induces significant pulsation and surface mass ejection before its final explosion (e.g., Ohkubo et al. 2009; Takahashi et al. 2016; Woosley 2017, 2018; Leung et al. 2019a; Woosley 2019). The CO core formed in these stars is so compact that the bounce shock fails to disrupt the entire star and results in the black hole formation (Sukhbold et al. 2016; Powell et al.2021).

The jet-forming accretion disk is suggested to be closely related to metal-poor stars in view of nucleosynthesis and mixing (Nomoto et al. 2010). The suppressed mass loss in a lower-metallicity star allows the star to maintain its angular momentum, so that the remnant black hole becomes rapidly rotating. Furthermore, the angular momentum of the infalling matter supports the formation of an accretion disk. A minimum rotation is necessary for sustaining the accretion disk and launching the jet (MacFadyen & Woosley 1999).

While the jet is an essential component in this model, the exact formation mechanism remains unclear. It is a matter of debate whether the jet is driven by magnetohydrodynamical (MHD) instabilities, by neutrinos, or by radiation. For the neutrino case (Liu et al. 2017), the extremely high temperature at the inner boundary of the accretion disk (∼10–20 MeV) provides the necessary neutrino heat deposition as an energy source for the jet; however, the jet may not be strong enough to sustain until the stellar surface (Wei et al. 2019). For the radiation case, the explosion is similar to the MHD case, but the photons may destroy most of the metal nuclei during its propagation to the surface (Shibata & Tominaga 2015). The jet stability is also closely related to the jet geometry such as the opening angle (Aloy et al. 2000; Zhang et al. 2003, 2004).

Some low-metallicity stars show clear signs of collapsar explosion such as the indicative Zn production (Maeda & Nomoto 2003b; Aoki et al. 2014). The presence of a stellar survey (e.g., SAGA database; Suda et al. 2011) has largely extended the catalog of stellar abundance, especially those with a low metallicity (<10⁻² Z_⊙), which can be enriched by single or a few explosions (Hartwig et al. 2019). The abundance patterns of some carbon-enhanced metal-poor stars provide direct evidence of these asymmetric explosion models (Tominaga 2009), e.g., HE 1327–2326 (Ezzeddine et al. 2019). The explosion morphology of ejected ⁴⁴Ti and ⁵⁶Ni can hint at the explosion history (Magkotsios et al. 2010).

The stellar evolution (see, e.g., Woosley et al. 1993; Ohkubo et al. 2009; Nomoto et al. 2013; Woosley & Heger 2015) and nucleosynthesis (see e.g., Woosley & Weaver 1995; Heger & Woosley 2002; Tominaga et al. 2007b; Limongi & Chieffi 2012; Nomoto et al. 2013; Umeda & Yoshida 2017; Grimmett et al. 2018) of (low-metallicity) massive stars have been systematically studied. We notice that the jet propagation breaks the spherical symmetry assumed by these models. Multidimensional hydrodynamics simulations with nucleosynthesis are necessary to consistently trace the energy deposition of the jet and the associated nuclear reactions (see, e.g., Maeda & Nomoto 2003a, 2003b; Couch et al. 2009; Maeda & Tominaga 2009; Nagataki 2009). The asymmetric energy deposition creates a high-entropy flow within the jet opening angle that synthesizes elements such as Ti, V, Cr, and Zn. These elements are generally absent in a spherically symmetric model. Given the uncertainty of the jet energetics, it becomes interesting to investigate how the chemical composition can serve as an alternative constraint. We therefore carry out a parameter survey on the explosive nucleosynthesis of jet-driven SNe. By comparing with some metal-poor stars, we explore the corresponding parameters of jet energetics that can reproduce the observed chemical abundances.

In Section 2 we present the numerical methods used for modeling the jet-driven SN. In Section 3 we report our characteristic model, which aims at representing typical jet-driven SNe, and we examine its energetics and chemical abundance patterns. In Section 4 we present our parameter survey. We examine the diversity of a jet in terms of the jet duration, energy deposition rate, deposited energy, and jet open angle. In Section 5 we examine the chemical abundance patterns and how they vary with each of the parameters explored. In Section 6 we compare our models with those in the literature. Then, we discuss in detail the possible observables of our explosion models, including the remnant black hole mass and correlations among ejecta mass, energy, velocity, and ⁵⁶Ni mass. After that, we further compare the Sc–Ti–V correlation and the high [Zn/Fe] ratio as observed in metal-poor stars reported recently in the literature. Finally, we give our conclusions.

We use the pre-SN model of the M_prog = 40 M_⊙ zero-metallicity star as the initial model (Tominaga et al. 2007a). Prior to the start of the simulation, the Fe core is removed from the simulation and is assumed to have formed a compact object. In Table 1 we tabulate the important progenitor parameters and setting of the jet energetics. In this work all progenitor models are assumed to be spherically symmetric and nonrotating. We further discuss the implications of our approach in Section 6.9.

Table 1. Massive-star Progenitor Model Used by the Characteristic Model and the Setting of the Jet

Model	Mass (M⊙)	Radius (km)
Progenitor star	40	2.0 × 107
He star mass	15	6.0 × 105
C–O core mass	13.9	6.1 × 104
Si core mass	3.81	9.4 × 103
Jet parameter	Variable	Value
Deposited energy	Edep	1.5 × 1052 erg
Energy deposition rate	${\dot{E}}_{\mathrm{dep}}$	1.2 × 1053 erg s−1
Jet deposition time	tjet	0.125 s
Jet opening angle	θjet	15°

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Here we present the hydrodynamical and thermodynamical history of the characteristic model to outline the important characteristics of the jet-powered explosion. In each simulation, we characterize the jet by the duration of the jet, t_jet,0, and the energy deposition rate, ${\dot{E}}_{\mathrm{dep},0}$.

In Figure 1 we plot the tracer particles of the characteristic model at 3.75, 7.50, 11.25, and 15.00 s. The red and blue marks correspond to the tracers that can escape from the system and are bound by the system, depending on their individual total energy e = ∣v∣²/2 + Φ(r), where v is the velocity and Φ is the local gravitational potential at the end of simulations. Some outer tracer particles are classified as bound because they have a marginally negative energy. However, those tracers are likely to be ejected through shock compression as the density lowers. It takes ∼10 s for the shock to completely reach the envelope of the star (He envelope extends to a much larger radius). The jet angle has increased to about 30°. At t = 15 s, where the simulation ends, the jet-accelerated particles have already broken out of the surface and reach as far as 2.0 × 10⁵ km. The rapidly expanding flow swept away most matter along the cone shape with an opening angle 15°. The ejected matter gradually falls to the inner part of the simulation box, which will be later accreted.

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It is worthwhile to note that the final ejecta has an angular extension of 30°. The same phenomenon appears in other models besides the characteristic model. The expansion is due to the competition between thermal expansion along the angular direction and the radial propagation of the shock. A stronger shock implies a shorter travel time from the inner mass cut to the surface. At the same time, it also implies a stronger thermal expansion. The similarity of our models suggests that these effects approximately cancel each other.

In Figure 2 we plot the total, kinetic, internal, and gravitational energies of the characteristic model taken from the hydrodynamical simulation. Within the first 1 s, the jet has already finished injecting the energy to the system where the total energy increases to ∼4 × 10⁵¹ erg. A small bump in the internal energy can also be seen. They show that the energy deposition from the jet creates a shock that provides significant shock heating by compression. After that, the internal energy drops, showing that the high-velocity jet continues to lose its energy as work done to accelerate the outer matter in the star. Meanwhile, the kinetic and gravitational potential energies grow slowly. At t = 10 s, all energies reach their asymptotic values within ∼10%. Not all energy from the shock can be transferred to the ejecta because part of that is lost when the shock-heated fluid parcels expand and do work on the fluid elements along the angular direction. In addition, near the bottom of the shock-heated fluid parcel, part of the matter falls back. They both dissipate the deposited energy. We also added the total kinetic energy of the tracers as a comparison.

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In Figure 3 we plot the thermodynamics history using the ejected tracer particles. The distribution of particles shows two groups. In the low-temperature branch, the tracers are not directly excited by the shock. They have a density from 10² to 10⁵ g cm⁻³. They preserve the maximum temperature from the progenitor from ∼10⁸ to 2 × 10⁹ K. For tracer particles in the high-temperature branch, they are excited by the jet directly. They follow a steeper ρ − T relation for the density range from 10^5.5 to 10^8.5 g cm⁻³. Some ejected particles close to the mass cut initially have a maximum temperature as high as (10–15) × 10⁹ K. Between ${\mathrm{log}}_{10}{\rho }_{\max }=6\mbox{--}7$, the cluster corresponds to the tracers indirectly excited by the jet. All tracers in the model do not enter nuclear statistical equilibrium (NSE). Instead, they only achieve α-rich freezeout. ⁵

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In Figure 4 we plot the abundance pattern for the characteristic model. We calculate the post-process nucleosynthesis of the tracers until all major exothermic reactions cease. Then, we wait for all short-lived radioactive isotopes to decay. We use two lines to show the abundance at two times and half of the solar values for comparison. Figure 4 shows the following:

• 1.  
  For most of the lower-mass elements from C to Ca, the abundance ratios of α-chain elements to Fe are marginally compatible with the solar composition.
• 2.  
  Elements like Si, S, and the nearby odd number elements are underproduced relative to Fe. ⁶
• 3.  
  On the contrary, elements from Ti onward are comparable to the solar composition.
• 4.  
  The high production of elements like V and Zn is consistent with the high-entropy environment experienced in the shock-heated matter.
• 5.  
  Mn, which is mostly produced by Type Ia SNe, is also underproduced in our model.

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We first present the models studied in this work. For all models we name according to their configurations. For example, for Model S40-0250-4000-15, we use the 40 M_⊙ star model. The energy deposition rate is 0.25 times the characteristic model defined as 1.2 × 10⁵³ erg s⁻¹. The energy deposition time is 4 times the characteristic model defined as 0.125 s. The jet opening angle is 15°. We use this notation to describe all input physics we used to change the configuration of the jet. In Table 2 we tabulate the models studied in this work. A total of 33 models are presented.

Table 2. The Models Presented in This Work

Model	Mprog	Mini	${\dot{E}}_{\mathrm{dep}}$	tdep	Edep	θjet	Mej	M(56Ni)	M(Ti)	M(V)	M(Cr)	M(Zn)
S40-1000-0125-15	40	15	120	0.015625	1.875	15	0.81	0.018	2.18	0.66	0.90	<0.01
S40-2000-0125-15	40	15	240	0.015625	3.750	15	0.57	0.037	2.75	0.33	0.66	0.01
S40-0500-0250-15	40	15	60	0.031250	1.875	15	0.33	0.003	0.03	0.02	0.04	<0.01
S40-1000-0250-15	40	15	120	0.031250	3.750	15	0.25	0.013	0.38	0.25	0.50	<0.01
S40-2000-0250-15	40	15	240	0.031250	7.500	15	1.18	0.07	1.30	0.97	0.71	0.09
S40-0250-0500-15	40	15	30	0.031250	1.875	15	0.50	0.005	0.77	0.22	0.44	0.02
S40-0500-0500-15	40	15	60	0.062500	3.750	15	0.27	0.008	0.18	0.02	0.15	<0.01
S40-1000-0500-15	40	15	120	0.062500	7.500	15	1.17	0.07	2.02	1.56	1.08	0.03
S40-2000-0500-15	40	15	240	0.062500	15.00	15	4.36	0.19	5.61	1.82	2.21	0.17
S40-4000-0500-15	40	15	480	0.062500	30.00	15	9.56	0.57	14.1	2.38	9.05	4.98
S40-8000-0500-15	40	15	960	0.062500	60.00	15	10.22	0.71	21.8	4.36	11.1	2.52
S40-0125-1000-15	40	15	15	0.125000	1.875	15	0.04	0.003	0.11	0.03	0.04	<0.01
S40-0250-1000-15	40	15	30	0.125000	3.750	15	0.06	0.006	0.08	0.06	0.11	<0.01
S40-0500-1000-15	40	15	60	0.125000	7.500	15	1.34	0.076	2.31	1.78	1.23	0.03
S40-1000-1000-7.5	40	15	120	0.125000	15.00	7.5	3.15	0.04	1.33	0.02	0.55	0.07
S40-1000-1000-15*	40	15	120	0.125000	15.00	15	4.13	0.30	7.31	2.57	4.17	4.81
S40-1000-1000-30	40	15	120	0.125000	15.00	30	1.55	0.075	1.89	0.43	1.35	0.20
S40-1000-1000-45	40	15	120	0.125000	15.00	45	0.90	0.005	0.01	0.13	0.12	<0.01
S40-2000-1000-15	40	15	240	0.250000	30.00	15	6.89	0.49	13.9	2.43	7.65	7.14
S40-4000-1000-15	40	15	480	0.250000	60.00	15	10.21	0.65	13.9	2.50	10.3	5.48
S40-8000-1000-15	40	15	960	0.250000	120.0	15	10.74	0.79	16.2	2.73	13.7	40.7
S40-0125-2000-15	40	15	15	0.250000	3.750	15	0.65	0.055	6.51	1.02	1.95	0.02
S40-0250-2000-15	40	15	30	0.250000	7.500	15	1.27	0.081	6.49	0.75	2.38	1.10
S40-0500-2000-15	40	15	60	0.250000	15.00	15	2.39	0.13	9.72	2.38	1.93	3.75
S40-1000-2000-15	40	15	120	0.250000	30.00	15	5.32	0.44	14.5	5.26	8.20	2.18
S40-2000-2000-15	40	15	240	0.250000	60.00	15	8.10	0.59	15.3	2.74	9.35	4.83
S40-4000-2000-15	40	15	480	0.250000	120.0	15	10.33	0.68	13.9	2.45	10.3	5.84
S40-0250-4000-15	40	15	30	0.500000	15.00	15	1.62	0.15	4.08	1.99	2.95	4.52
S40-0500-4000-15	40	15	60	0.500000	30.00	15	1.42	0.065	1.65	0.82	0.71	1.50
S40-1000-4000-15	40	15	120	0.500000	60.00	15	5.59	0.46	1.13	3.40	7.91	4.28
S40-2000-4000-15	40	15	240	0.500000	120.0	15	8.78	0.64	11.9	2.38	9.64	1.68
S40-0500-8000-15	40	15	60	1.000000	60.00	15	3.02	0.21	6.31	2.40	7.96	5.22
S40-1000-8000-15	40	15	120	1.000000	120.0	15	10.74	0.92	36.7	1.52	10.3	2.19

Note. M_prog and M_ini are the zero-age main-sequence progenitor mass and the initial mass of the simulation in units of M_⊙; ${\dot{E}}_{\mathrm{dep}}$ is the energy deposition rate in units of 10⁵¹ erg s⁻¹; t_dep is the total energy deposition time in units of s; θ_jet is the open angle of the jet in units of degrees; E_dep is the total energy deposited by the jet in units of 10⁵¹ erg; M_ej and M(⁵⁶Ni) are the ejecta and included ⁵⁶Ni mass in units of M_⊙; M(Ti), M(V), M(Cr), and M(Zn) are the final stable masses of Ti, V, Cr, and Zn in units of 10⁻⁴, 10⁻⁴, 10⁻³, and 10⁻³ M_⊙, respectively. The model with an asterisk is our characteristic model.

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We plot in Figure 5 the thermodynamics record for tracers for Models S40-1000-0500-15 and S40-1000-2000-15 by their maximum temperature and density, binned by their maximum density. We observe that the jet duration affects mostly the tracer particles near the core with a density ≥10^5.5 g cm⁻³. The maximum temperature is ∼20% higher for the model with a longer jet duration. It also triggers a wider spread in the maximum temperature among particles. The model with a higher jet duration also has tracer particles with higher ${\rho }_{\max }$. The longer energy deposition allows the shock to maintain its strength, which suppresses fallback into the central remnant. All tracers are burnt in either α-rich freezeout or incomplete burning.

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In Figure 6 we plot the thermodynamics trajectories similar to Figure 5, but for Models S40-0500-1000-15 and S40-2000-1000-15. Again, for tracer particles with ${\rho }_{\max }\gt {10}^{5.5}$ g cm⁻³, derivation appears between the behaviors of the two models, driven by the differences in the adopted jet parameters. The model with a high energy deposition again has a higher ${T}_{\max }$ and a wider spread. However, the ${\rho }_{\max }$ range of the two models is comparable.

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In Figure 7 we plot the statistics of the ejected tracer particles for Models S40-1000-1000-7.5, S40-1000-1000-30, and S40-1000-1000-45. The geometry of the shock plays an important role in the thermodynamical evolution of the ejecta. At an angle above 30°, the inner part of the ejecta has a temperature ∼15 × 10⁹ K, where some of the tracers in the Si layer are also ejected. When the jet angle increases, the shock is dispersed by its area that the shock has a larger surface area which makes the shock strength drop. Thus the shock heating is less efficient. A lower temperature maximum of 6 × 10⁹ K is recorded. The shock compression with the maximum density of ∼10⁷ g cm⁻³ is also weaker than the previous case. On the other hand, as θ decreases to 75, the more concentrated energy deposition leads to a stronger shock, with stronger heating and compression for a given ${\rho }_{\max }$. The distribution suggests that the jet energy has the largest effect when θ_jet ≈ 30°. This can be understood as the competition of two factors, the deposited mass and the shock strength. At a small jet angle, the initial shock is strong but the deposited mass is small. Thus, the effect of shock heating is limited. At a large jet angle, the shock is weak but more mass is affected.

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In Figure 8 we plot the statistics of the ejected tracer particles for Models S40-0500-2000-15 and S40-2000-0500-15. The two models assume the same total deposition energy E_jet but with different t_jet and ${\dot{E}}_{\mathrm{jet}}$. Both models exhibit similarities in the statistic of tracers, especially at low ${\rho }_{\max }$. This suggests that the outer envelope is less sensitive to the jet characteristics. On the other hand, the differences in the high ${\rho }_{\max }$ demonstrate the sensitivity of the temperature range in the ejecta on t_jet. A longer energy deposition helps inner tracers to escape from the star.

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It might look contradictory that S40-2000-050-15, which has a higher ${T}_{\max }$, has a lower ejected mass than S40-0500-2000-15. In fact, it is because the total ejected mass depends on two factors: the shock strength and its sustainability. With a shorter t_jet, the early shock dissipates its energy and the matter in the inner region does not have sufficient energy input to maintain its expansion. On the other hand, a longer t_jet means a weaker shock. But the expansion is long enough that the matter from the inner core becomes unbound.

In this section we have considered all three variables E_dep, ${\dot{E}}_{\mathrm{dep}}$, and t_jet as independent variables. However, by definition ${E}_{\mathrm{dep}}={\dot{E}}_{\mathrm{dep}}{t}_{\mathrm{dep}}$, which means that these variables are not fully independent. Changing, for example, t_jet while keeping ${\dot{E}}_{\mathrm{dep}}$ constant still changes E_dep. Note that all three variables are not yet well constrained by observational data. When we fix one of the variables, effectively we are observing the jet dependence of the nucleosynthesis yield on one of the slices of the two-dimensional surface in the parameter space. Further constraints, such as a precise measurement of E_dep in real collapsars, will indicate which parameter "slices" presented in this section are necessary for the comparison.

We first examine how the chemical abundance patterns of the ejecta depend on the jet duration. We compare in Figure 9 the abundance pattern for two contrasting models S40-1000-0500-15 (t_jet = 0.5t_jet,0) and S40-1000-2000-15 (t_jet = 2t_jet,0). We remark that the two models also differ in the total deposited energy, but the early shock, which we will show to largely change the iron-group element (IGE) synthesis, is identical in both models. We also remark that in this section all elements are referred as the ratios to ⁵⁶Fe, instead of the absolute values of the mass fraction.

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The two models share similar abundance patterns for lighter elements, including O, Ne, and Mg. A similar pattern can be observed for intermediate-mass elements (IMEs), including Si, S, Ar, and Ca. But there is a minor enhancement in P and Sc for the model with a longer energy deposition. IGEs from Ti to Ni are also similar, with overproduction in ⁴⁸Ti and ⁵¹V. Substantial differences are found for ⁵⁵Mn and ^64,66,67Zn. The weaker explosion model has lower IGE abundances than the other by an order of magnitude.

In Figure 10 we plot the final abundance of Models S40-0500-1000-15 (${\dot{E}}_{\mathrm{dep}}=0.5{\dot{E}}_{\mathrm{dep},0}$) and S40-2000-1000-15 (${\dot{E}}_{\mathrm{dep}}=2{\dot{E}}_{\mathrm{dep},0}$). The two models differ by the energy deposition rate to be half of and double the characteristic model. The model with a higher energy deposition rate ejects a larger fraction of near-surface material because of a higher energy deposited. As a result, a higher abundance of C, O, and Mg is found. As the global ⁵⁶Ni mass increases, and as more material from the deeper core is ejected, the IME and the IGE abundances are suppressed. On the contrary, for the model with a lower deposition rate, the much smaller production of ⁵⁶Fe (or ⁵⁶Ni before decay) allows the formation of peculiar abundance patterns. This includes a supersolar production of ⁴⁶⁻⁴⁷Ti, ⁵¹V, ⁵⁹Co, and ⁵⁸⁻⁶²Ni. The similar ⁶⁴Zn production suggests that both models have experienced a similar high-entropy phase.

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In Figure 11 we plot [X_i /⁵⁶Fe] for Models S40-1000-1000-30 (θ_jet = 30°) and S40-1000-1000-45 (θ_jet = 45°). The two models differ from each other by the opening angle of the jet, which is two or three times that of the characteristic model. When the jet opening angle increases, the more pronounced envelope ejection is reflected by the enhanced abundance of C, O, Ne, and Mg. The IME production is suppressed. The ratios of Cr, Fe, and Mn are enhanced owing to the lower ⁵⁶Fe production. But elements yielded in typical α-rich freezeout, such as Ni and Zn, are not well produced.

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In Figure 12 we plot [X_i /⁵⁶Fe] for Models S40-2000-0500-15 and S40-0500-2000-15. The two models differ from each other by the jet duration and energy deposition rate, while the total deposited energy is fixed. The abundance patterns of the two models are similar. Elements from C to Ca and from Fe to Ni show very good agreement with each other. In particular, the α-chain isotopes, such as ²⁸Si, ³²S, ³⁶Ar, and ⁴⁰Ca, show almost complete overlap.

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Model S40-0500-2000-15 has a more pronounced Ti, V, Cu, and Zn production. These are the elements that require high entropy for the formation. A long t_jet is important in synthesizing these elements.

There are not many works in the literature that extensively cover the effects of the jet on hydrodynamics and nucleosynthesis presented in this work. Tominaga (2009) presents contrasting models that explicitly study how the energetics of the jet affects the associated nucleosynthesis. Here we compare the input physics used in their work and the numerical results. In Table 3 we compare the numerical algorithms used in their work and in this work.

For the progenitor, we use the same 40 M_⊙ zero-metallicity star at the onset of collapse as the progenitor. While the exact grid and hydrodynamical schemes are different, we choose a compatible choice of mass cut and jet energetics. We refer to their Model A as our Model S40-1000-1000-15. They observe the remnant of mass 9.1 M_⊙, while ours is ∼10.9 M_⊙. The larger remnant mass is because we observe that the very outer envelope could include the very outer He envelope, which is extended and low in binding energy, and the prescription ensures that net energy is deposited to outgoing matter. This adds an extra ∼1 M_⊙ to the ejecta mass. Notice that this does not change the nucleosynthetic pattern because the density and temperature in the He envelope are too low for significant reactions to occur.

Figure 1 shows a very similar distribution of ejecta and remnant structure to their Figure 4(a). The whole He layer and the C+O layer are ejected after the explosion. A disk shape structure that is bound is concentrated near the innermost Si layer and C+O layer.

For nucleosynthesis, our model predicts a higher Fe production, which leads to lower C, O, and Ne ratios and the global abundance pattern, as the abundances are taken as the ratio to ⁵⁶Fe. Both our and their works show an underproduction of Sc and Mn. Their model shows a flat distribution of Ti, V, and Cr, while V in our model is slightly overproduced and is higher than Ti and Cr. At last, for IGEs, our model shows a higher stable Ni, Cu, and Zn, suggesting more contribution from inner ejecta.

Globally, our model consists of more tracers with complete burning, especially α-rich freezeout elements. The difference of nucleosynthesis is closely related to the EOS and the nuclear reaction network available. The exact temperature of the shock front can be sensitive to the EOSs and the numerical shock-capturing scheme. The numerical dissipation may also affect how the shock heating behaves in the outer layer of the stellar envelope.

Recent observations of gravitational wave signals measured by advanced-LIGO and VIRGO demonstrated the existence of black hole–black hole mergers and neutron star–neutron star mergers (Abbott et al. 2019). The third observation run (The LIGO Scientific Collaboration et al. 2021b) has discovered a wide distribution of black hole mass and spins. The black hole statistics (The LIGO Scientific Collaboration et al. 2021a) could be directly related to the mass ejection process of this class of SNe.

In Figure 13 we plot the ejecta mass distribution as a function of the deposited energy E_dep. As E_dep increases, the ejected mass increases significantly and approaches an asymptotic value. Almost the entire envelope is accreted in the low-energy limit. In the high-energy limit, about two-thirds of the envelope is ejected. This corresponds to the black hole mass range from 5 to 15 M_⊙. The asymptotic limit exists because the Si and C+O cores outside the jet cone cannot be directly excited by this mechanism. They always fall back and remain bound. Below ∼2 × 10⁵¹ erg, the jet fails to eject any observable mass.

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Early modeling of the light curve and spectra from SN 1998bw (e.g., Maeda et al. 2002) suggested an explosion energy of ∼10⁵² erg and an ejecta mass of 2–3 M_⊙. This mass range implies that the deposited energy of the jet ranges between 0.5 × 10⁵² erg and 2 × 10⁵² erg. This also constrains the deposition time for this transient to about O(1) s. The final black hole mass will be ∼12–13 M_⊙, depending on the exact details of the jet.

Due to the massive envelope and aspherical deposition of energy, the typical efficiency of energy deposition is low. This means that the energy contained in the ejecta, as the sum of their internal and kinetic energy, can be much lower than the total energy deposited by the jet. Here we examine how the two quantities are related.

To obtain the ejecta energy, we compute the sum of ejecta kinetic energy where the tracers have a positive sum of their kinetic energy and gravitational energy. In Figure 14 we plot the ejecta total energy against deposited energy by the jet. We also plot a line for E_ejecta = 0.1 E_dep. The straight line, which corresponds to an efficiency of 10%, shows the typical trend of the models. Dispersion occurs at both very low and very high E_jet. At the low-energy limit, the statistical fluctuation becomes significant. At the high-energy limit, how the jet deposits energy affects the mass ejection. This demonstrates the nonlinear interaction of how the shocked matter leads to mass ejection.

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The hypernova SN 1998bw associated with GRB 980425 has shown a possibility of triggering a jet from an aspherical explosion. Here we further examine the ejected ⁵⁶Ni mass as a function of the explosion energy. These quantities can be directly constrained by the light-curve shape and the spectral lines. Here we explore the correlation of this pair of quantities.

In Figure 15 we plot the final ejected ⁵⁶Ni mass against the explosion energy. The explosion energy shows an almost monotonic relation with the ⁵⁶Ni mass, which grows from ∼10⁻² M_⊙ to as high as 0.8 M_⊙ in the high-energy limit. Different configurations of the jet produce a dispersion of the ⁵⁶Ni mass of about 0.2–0.4 M_⊙. Similar to the ejected mass, the ejecta ⁵⁶Ni mass also levels off at 4 × 10⁵¹ erg.

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SN 1998bw has a derived ⁵⁶Ni mass of 0.2–0.6 M_⊙. Our models suggest that, in order to achieve the derived ⁵⁶Ni mass, a minimum of explosion energy >10⁵² erg is necessary to reconcile with the lower limit 0.2 M_⊙. On the other hand, in the upper limit, E_dep from 2 × 10⁵² erg to 8 × 10⁵² erg will be necessary.

Another important observable pair of a jet-driven SN is the ⁵⁶Ni mass against typical ejecta velocity. This pair of variables can be derived from the light-curve peak luminosity L_peak against the Si ii velocity inferred from the spectra. Here we estimate the typical ejecta velocity by first calculating the ejecta total energy E_ej and its mass M_ej. The typical ejecta velocity is defined as ${v}_{\mathrm{ej}}=\sqrt{2{E}_{\mathrm{ej}}/{M}_{\mathrm{ej}}}.$

In Figure 16 we plot the ⁵⁶Ni mass against v_ej for the models presented in this work. The distribution of data points is more scattered than the E_ej–E_dep pair. At a low v_ej, the two quantities have a linear relation but with a large dispersion. This trend extends from 4 × 10³ km s⁻¹to 7 × 10³ km s⁻¹. At a high v_ej, the ⁵⁶Ni mass is almost independent of the ejecta velocity.

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The estimation does not fully capture the details of the ejecta velocity because the the outer layer of ejecta can move faster than the bulk of ejecta, after the ejecta develops a homologous expansion profile. The deposition of the radioactive decay energy of ⁵⁶Ni by γ-ray may also create secondary changes to the asymptotic velocity profile (Blinnikov et al. 2006). Thus, the velocity here represents the average velocity of the total ejecta.

To further compare the models, we compare our models with the observational results from superluminous SNe Ib/c (De Cia et al. 2018), SNe Ic-BL (Taddia et al. 2019), and luminous SNe Ib/c (Gomez et al. 2022). The parameters are taken from their fitting results using analytical models. The observational data also show a knee structure as in our models, with low ejecta mass spanning a wide range of ⁵⁶Ni from 10⁻² to 10⁰ M_⊙. Meanwhile, models with a high v_ej correspond to a smaller range but a higher ⁵⁶Ni mass value. Some SNe show a higher v_ej ∼ 2 × 10⁴ km s⁻¹. This is likely to be an aspherical effect where the ejected ⁵⁶Ni has a higher velocity than the bulk of ejecta.

Another observable pair presented is the ⁵⁶Ni mass, M_Ni, against the ejecta mass, M_ej. These two variables can be directly extracted from the shape and peak luminosity of the light curves.

In Figure 17 we plot the ⁵⁶Ni mass against the total ejecta mass for the models in this work. Our data show a strong correlation between this pair of observables. The quantity pair exhibits a power-law scaling. A lower M_ej leads to a larger dispersion in M_Ni, and the dispersion decreases in the high-M_ej limit. We also use the observational data to demonstrate how our models can indicate the explosion history of observed hypernovae.

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SN 1998bw is the first evidence for GRB–hypernova correlation (Galama et al. 1998). The SN is modeled as an explosion of a massive CO star (Iwamoto et al. 1998; Woosley et al. 1999) with a fit of ⁵⁶Ni ∼ 0.5 M_⊙ and M_ej ∼ 10 M_⊙. SN 2006aj is regarded as a hypernova explosion on the dim side. Fitting of the spectra and light-curve provides estimates M_ejecta = 3 M_⊙ and ⁵⁶Ni ∼ 0.2 M_⊙.

Both models favor models with t_jet > t_jet,0 to fit this observable pair. These data points show that the observed hypernovae are diversified in the parameter space of jet energetics. Our models agree with the trend derived from observed hypernovae.

We further compare the statistical trend of Type Ib/c(-BL) SNe that are in the superluminous and luminous branch. Our models show the slope of this variable pair being consistent with the slope and the mass range presented in their derived models, especially the SN Ic/BL models in Taddia et al.(2019).

However, there are a few SNe from Gomez et al. (2022) with M_Ni below ∼10⁻² where our models are persistently higher than theirs. They correspond to a very low energy deposition, for which we expect that the interpretation of ⁵⁶Ni could encounter greater uncertainties due to late-time fallback and low statistics.

Here we study how the Sc–Ti–V correlation depends on our models. This correlation has been observed in metal-poor stars (Sneden et al. 2016). The metal-poor star HD 84937, together with others, has shown that the three element ratios are correlated. The amount of Ti increases with V and Sc. Here we focus on the Ti-V relation.

To obtain the two quantities [Ti/Fe] and [V/Fe], we use the chemical composition of the ejected particles. Then, we compute the corresponding Ti and V masses. In Figure 18 we plot [V/Fe] against [Ti/Fe] for the models presented in this work. In Sneden et al. (2016) this pair of elements shows a correlation of 45° based on the metal-poor stars derived in Roederer et al. (2014). We overlay the stellar abundance data on our models.

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There is an intrinsic scattering in the model prediction due to the nonlinear dependence of the deposited energy on how the ejecta is heated and how much matter is ejected. Some sequences of models, at such a time as t_jet = 0.5t_jet,0, show a clear trend of how [Ti/Fe] scales with [V/Fe], which spans from ∼ −1.0 to 0.5 in [Ti/Fe]. The trend becomes less obvious for other choices of t_jet. The trend of our models is comparable to the trend of observational data. Some of the models lie on the outskirts of the clusters. However, no model in our sequence directly intersects with the cluster. This can be attributed to the progenitor considered. The production of [Ti/Fe] in general peaks around a maximum temperature of ∼3.5 × 10⁹ K. At a higher temperature, the production of IGEs surpasses the production of Ti. As shown in Figure 3, the jet excites the matter to a maximum temperature above 5 × 10⁹ K. Thus, the production of IGEs dominates and suppresses the abundance ratios [Ti/Fe]. This suggests that to reconcile with the observational group, a jet of lower energy or a more compact (i.e., lower-mass) progenitor is necessary.

By comparing with Figure 17, we observe that the high-${\dot{E}}_{\mathrm{dep}}$ models can simultaneously correspond to observed high [Ti/Fe] versus [V/Fe] stars and hypernovae. However, the low-${\dot{E}}_{\mathrm{dep}}$ models are less relevant to the observed abundances in these stars.

In our models, the Sc production is below the solar value, regardless of the use of ν − p process in nuclear reactions. Our model shows some Sc production during the jet propagation, but it is destroyed at a later time. This suggests that the underproduction has a systematic dependence on the progenitor. Further extension to other realistic progenitor will help us understand the diversity of Sc production in this class of explosion.

Recent analysis of a few low-metallicity stars shows an enhanced production of [Zn/Fe] from low-metallicity SNe. These results suggest that the aspherical explosion is due to the local high-entropy environment as in the electron capture supernova and hypernova branch (Nomoto et al. 2010). Early works demonstrated the connection between metal-poor stars and aspherical explosions from the peculiar pattern of some carbon enriched metal-poor stars, including HE 1300+0157, HE 1327–2326, HE 0107–5240, and HE 1424–0241 (e.g., Tominaga 2009). Some of these objects, in particular HE 1300+0157 and HE 1327–2326, have [C/Fe] as high as 1–4. To explain the low Fe abundance, the ejecta restricted within a certain angle is necessary to produce these results.

In Figure 19, we show the ratio [Zn/Fe] taken from our models as a function of the explosion energy. At the low-energy limit, the ratio [Zn/Fe] is suppressed for two reasons. First, the jet deposits insufficient energy, where the heated matter in the core does not have the necessary momentum to break through the heavy envelope. Second, at this low energy, the ejecta does not go through the high-entropy phase. The temperature of the ejecta cannot reach NSE for ⁵⁶Ni production or further. On the other hand, in the high-energy limit, the upper limit of [Zn/Fe] in our models can always reach the values suggested by observational data. Scatter of [Zn/Fe] exists for a given E_dep. It is because the duration of energy deposition can also affect the ejected mass and the production of ⁵⁶Fe, which alter the ratio directly.

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We compare our models with the recent observations of HE 1327–2326. This star shows a significant enhancement of [Zn/Fe] = 0.8 ± 0.25. Such a high ratio indicates that the high-entropy environment typical in jet-driven SNe is necessary. In order to realize the high [Zn/Fe] in HE 1327–2326, we observe that from our models E_dep = 10⁵² erg to 4 × 10⁵² erg is necessary for producing the expected [Zn/Fe]. In Tables 4–5 we present the chemical abundance tables in Tables 4–5 for the stable and radioactive isotopes in the ejecta of our collapsar models where the [Zn/Fe] ratio satisfies the observed value. The results agree that the high-entropy environment for the α-rich freezeout is essential for the high Zn production.

In this work we have considered only the 40 M_⊙ zero-metallicity star as the progenitor. We do not consider models of other masses owing to the expensive computation of these multidimensional models. For lower-mass models, there could be substantial fallback accretion before the black hole forms. The infalling matter will be distinctive from our models. On the other hand, we expect that fewer changes appear for higher-mass models, as the progenitor structure does not vary qualitatively. The Si and C+O cores are in general nondegenerate. To a good approximation, extending this approach to both higher- and lower-mass star models may demonstrate the diversity of jet-driven SNe.

In this work we also assumed the progenitor stellar model to be spherically symmetric and nonrotating. Indeed, the formation of GRBs through a black hole accretion disk requires an initial angular momentum. Since we are not directly modeling how the infall of matter triggers the outburst of energy, the rotation component is less important. Furthermore, the rotation in typical stars is less than ∼10% of the critical rotation. The dynamical effect of rotation on the ejecta is small. One possible effect of the neglect of rotation in the stellar model is the difference in the outer layer, where rotational mixing is important during main-sequence evolution. However, as most IGE synthesis is relevant in the Si and inner C+O layer, rotation plays a less important role. Despite that, the study of how rotation is coupled to stellar evolution and the explosion will be important for future quantitative comparisons.

In this work we have examined the parameter dependence of the hydrodynamics and nucleosynthesis of jet-driven SNe. We use the 40 M_⊙ zero-metallicity star at the onset of Fe core collapse as the progenitor with a mass cut at ∼1.4 M_⊙. We treat the energetics of the jet, including the energy deposition rate, energy deposition time, and jet open angle. We vary the total deposited energy from ∼10⁵⁰ to ∼10⁵² erg.

We observe the following features in the explosion morphology and nucleosynthesis pattern:

• 1.  
  The ejecta mass is sensitive to the energy deposition rate and the energy deposition duration. A large energy deposition can trigger a mass ejection along the jet open angle and the outermost CO and He layers. The ejecta composition features the presence of Ni, Cu, and Zn in general.
• 2.  
  The ejected mass is also sensitive to the jet open angle. A wider jet open angle results in a more dispersed energy deposition. There is less matter along the jet open angle and more matter in the outer layer. The ejecta in a wide angle jet contains more light elements, including C, O, Ne, and IMEs (e.g., Si, S, and Ar). The ejecta contains much fewer IGEs, especially Ni, Cu, and Zn. Observational data of metal-poor stars in Ti, V, and Cr for the light IGEs and Ni, Co, and Zn for the heavy IGEs can constrain directly the explosion energetics.
• 3.  
  The corresponding explosion results in a moderately bright event compared to Type Ic SNe. Typical collapsar explosions produce 0.1–0.3 M_⊙ of ⁵⁶Ni. In terms of the ejecta mass, a weaker explosion can eject matter as low as ∼0.01 M_⊙, while a strong explosion can eject ∼10 M_⊙. The effective energy deposition in the ejecta is ∼10% of the actual deposited energy. The observed energy of the ejecta could therefore be a probe of the initial energy deposition by the jet, which constrains the jet formation environment.
• 4.  
  The ejected ⁵⁶Ni mass is strongly sensitive to the deposited energy. It ranges from ∼10⁻³ M_⊙ in the SN regime to ∼1 M_⊙ in the hypernova regime. The corresponding ⁵⁶Ni mass against v_ej shows a knee pattern. The lower v_ej grows with ⁵⁶Ni mass, while at high v_ej ⁵⁶Ni mass is capped above.
• 5.  
  We examine the relation between [V/Fe] and [Ti/Fe] of our models. We show that our jet-driven SNe can reproduce the 45° slope derived from the observed metal-poor stars (Roederer et al. 2014; Sneden et al. 2016). This connects with the metal-poor star statistics that the observed abundance pattern originates from early jet-driven SNe. However, an exact matching will require the extension to other progenitor models.
• 6.  
  We compare the model abundance patterns with the recently observed Zn-enriched metal-poor star HE 1327+2326. The high Zn ratio coincides with our jet-driven SN models with an explosion energy ∼1 × 10⁵² erg. Future observations and statistics of this abundance ratio from metal-poor stars may provide important constraints on the jet energetics.