Collapsar Gamma-Ray Bursts Grind Their Black Hole Spins to a Halt

The spin of a newly formed black hole (BH) at the center of a massive star evolves from its natal value due to two competing processes: accretion of gas angular momentum that increases the spin and extraction of BH angular momentum by outflows that decreases the spin. Ultimately, the final, equilibrium spin is set by a balance between both processes. In order for the BH to launch relativistic jets and power a γ-ray burst (GRB), the BH magnetic field needs to be dynamically important. Thus, we consider the case of a magnetically arrested disk (MAD) driving the spin evolution of the BH. By applying the semianalytic MAD BH spin evolution model of Lowell et al. to collapsars, we show that if the BH accretes ∼20% of its initial mass, its dimensionless spin inevitably reaches small values, a ≲ 0.2. For such spins, and for mass accretion rates inferred from collapsar simulations, we show that our semianalytic model reproduces the energetics of typical GRB jets, L jet ∼ 1050 erg s−1. We show that our semianalytic model reproduces the nearly constant power of typical GRB jets. If the MAD onset is delayed, this allows powerful jets at the high end of the GRB luminosity distribution, L jet ∼ 1052 erg s−1, but the final spin remains low, a ≲ 0.3. These results are consistent with the low spins inferred from gravitational wave detections of binary BH mergers. In a companion paper by Gottlieb et al., we use GRB observations to constrain the natal BH spin to be a ≃ 0.2.


INTRODUCTION
Black holes (BHs) are the product of a massive star corecollapse at the end of its life (collapsar; Woosley 1993).Before the formation of the BH, the stellar core can undergo an intermediate stage during which it collapses into a protoneutron star (PNS).The large mass reservoir in the stellar core leads to a high mass accretion rate onto the PNS.Once the PNS accretes mass above M NS,max 2.2 M (Margalit & Metzger 2017;Aloy & Obergaulinger 2021;Obergaulinger & Aloy 2022), it collapses to a BH.Observationally, the least massive observed BHs are M min 2M NS,max , suggesting the presence of a mass gap between M NS,max and M min (Bailyn et al. 1998;Özel et al. 2010;Farr et al. 2011;Mandel et al. 2017).Such gap implies that after their formation and while the stellar collapse is ongoing, BHs continue to accrete mass that is at least comparable to their natal mass, M NS,max (Belczynski et al. 2012;Kovetz et al. 2017).
As it accretes gas, the BH gains mass and angular momentum, so that its spin can either increase by accretion, or decrease by generating collimated Poynting-flux dominated outflows (jets) that extract BH rotational energy (Penrose & Floyd 1971).Numerical simulations of rotating collapsars have shown that in the absence of collimated outflows or jets (Shapiro & Shibata 2002;Shibata & Shapiro 2002; Fujonatan.jacqueminide@northwestern.edu jibayashi et al. 2020, 2022), or if the jets are powered hydrodynamically, rather than by the rotational energy of the BH (MacFadyen & Woosley 1999;Janiuk et al. 2008), the BH spins up by the end of the explosion process to a dimensionless spin a ≈ 1 (see however, Chan et al. 2018).We are unaware of numerical studies that consider both spin-up by accretion and spin-down by jet launching.
Several observational techniques have been used over the years to constrain the spin of BHs via electromagnetic (EM) emission of the BH accretion disk, from x-ray reflection spectroscopy (e.g., García et al. 2014) to thermal continuum fitting (e.g., Zhang et al. 1997;McClintock et al. 2014;Zhu et al. 2019).Although these methods suggest that at least some of the BHs are rapidly spinning, these measurements may depend on the poorly understood accretion physics of BHs (see Middleton 2016;Reynolds 2021, for reviews).A relatively new and more robust technique to infer the BH spin is through gravitational wave detections of binary BH mergers by LIGO/Virgo/KAGRA (LVK).Such studies consistently indicate that pre-merger BHs are slowly spinning (Farr et al. 2017;Tiwari et al. 2018;Roulet & Zaldarriaga 2019;Abbott et al. 2020;Hoy et al. 2022).
Some of the massive progenitors of BHs are stripped envelope stars (e.g., Gal-Yam et al. 2022).These stars are associated with the detection of γ-ray bursts (GRBs), powered by relativistic jets launched from the BH.The enormous energy of those jets indicates that they are powered electromagnet-ically (e.g., Lyutikov & Blandford 2003;Leng & Giannios 2014;Liu et al. 2015) via the extraction of BH rotational energy by the magnetic fields threading the BH (BZ; Blandford & Znajek 1977).Therefore, jetted explosions link the birth of BHs and their spin with the formation of relativistic jets in the stellar core, providing a unique opportunity to study BHs through the observables of GRBs.In a companion paper, Gottlieb et al. (2023), we argue that GRB observables favor BHs with low natal spins.Here, we analyze the interplay between the BH spin and the jet to study the BH spin evolution, and the final spin at the end of the stellar collapse.
A spinning BH exchanges angular momentum with its disk-jet accretion system, which results in both hydrodynamic torques through accretion, and magnetic torques through jet launching acting on the BH.This continues until the BH reaches equilibrium spin.In order for the jets to be launched, the BH needs to possess dynamically-important magnetic fields (Komissarov & Barkov 2009).BHs in this state are in or near the magnetically-arrested disk (MAD) state (Bisnovatyi-Kogan & Ruzmaikin 1974, 1976;Narayan et al. 2003;Tchekhovskoy et al. 2011). Recently, Lowell et al. (2023) used simulations of Tchekhovskoy et al. (2011Tchekhovskoy et al. ( , 2012) ) to compute the torques applied by a MAD system to a spinning BH.They constructed a semi-analytic model that could reproduce the behavior of magnetohydrodynamic torques of MADs on BHs.They found that MADs spin down BHs to a relatively low equilibrium spin, a eq 0.07, more efficiently than the spin-up by a standard thin disk (Bardeen 1970).For example, an initially maximally spinning BH of a 0 = 1 can reach the equilibrium spin, a eq 0.07, by accreting only 50% of its initial mass during the MAD state.By contrast, for a standard thin disk, a minimally spinning BH, a 0 = 0, needs to accrete ∼ 200% of its initial mass to reach the equilibrium spin, a = 1.
In this Letter, we build on the model of Lowell et al. (2023) to show that the final spin of collapsar BHs associated with Poynting-flux dominated jets is almost inevitably small.In §2 we outline the reasoning of why BHs end up being slowly spinning at the end of stellar collapse.In §3 we present the semi-analytic model of BH spin evolution.In §4 we compare the model with GRB observables to show that for any reasonable stellar collapse scenario, BHs spin down/up to an equilibrium spin of a ≈ 0.1.We summarize and conclude in §5.

LOW FINAL BH SPIN IN A NUTSHELL
The angular momentum exchange between the BH, the disk, and the jets leads to magnetohydrodynamic torques on the BH.On one hand, the angular momentum of the matter falling onto the BH drives an accelerating hydrodynamic torque.On the other, the accreting matter advects magnetic fields onto the BH: the magnetic flux threading the BH pow-ers relativistic jets that apply decelerating magnetic torques on the BH.
In the MAD state, the jet power and EM torque are linked to the accretion power by the jet efficiency, which in turn depends on the BH spin (Lowell et al. 2023).Hence, the torques acting on the BH depend only on the accretion rate and the BH spin (Gottlieb et al. 2023).Consequently, the final BH spin depends solely on the initial BH spin, a 0 and the total accreted mass, m f .We normalize the total accreted mass by the BH initial mass, M 0 ≡ M(t = 0), to define where we consider a mass accretion rate ṁ to compute the total accreted mass, We stress that χ does not represent the mass growth of the BH, but the total accreted mass on the BH.The increment in BH mass is lower than the accreted mass because some of the accreted energy is deposited into launching BH-powered relativistic jets, thus χ = M(t→∞) M 0 .For a sufficiently high accretion rate, e.g.χ 1 2 , the final BH spin reaches equilibrium spin, a f a eq = 0.07 (Lowell et al. 2023).Ultimately, the accreted mass is likely related to the stellar mass M .Thus, we define One might naively expect most of the stellar envelope to fall onto the BH.However, the powerful jets and disk outflows will unbind a considerable fraction of the stellar envelope.The percentage of the stellar envelope that is accreted by the BH was estimated by Gottlieb et al. (2022b) by measuring the percentage of mass that remains bound at the end of the simulation.They found that ∼ 20 s after the collapse, λ reaches an asymptotic value of λ ≈ 0.1, e.g.10% of the stellar mass will be accreted onto the BH.For such accretion fraction, a stellar envelope of 20 M and an initial BH mass of 2.5 M result in χ = 0.8, as roughly needed for explaining the mass gap, well above the critical value for reaching equilibrium spin.
To compute the BH spin evolution in time, we model the BH spin by coupling the spin evolution equations to an imposed accretion rate (Moderski & Sikora 1996;Lowell et al. 2023) Final spin of the BH as a function of the inverse of the accreted mass, 1/χ (Eq.1), and the initial BH spin a 0 .All BHs that accrete more than half of their original mass, i.e. 1/χ < 2 showed by the white dashed line, spin down to equilibrium spin, a eq 0.07.The simulation of Gottlieb et al. (2022b) with initial spin a 0 = 0.8 and χ 0.3 is marked by the red star.Most BHs spin down to a ≤ 0.3 as long as they accrete 15% of their initial mass, irrespective of their initial spin.
where the spin-up parameter can be written as where M is the mass of the BH, dm = ṁ dt is the accreted mass, η EM (a) is the jet launching efficiency, e HD and l HD are the hydrodynamic energy and angular momentum fluxes, and k(a) = Ω F /Ω H is the angular frequency of the magnetic field lines over the angular frequency of the event horizon.The numerical values of e HD and l HD , and the functions η EM (a) and k(a) are taken from Lowell et al. (2023) 1 .
Figure 1 depicts the final BH spin from multiple spin-down solutions computed with different initial BH spins a 0 , and total accreted mass χ.All solutions with 1/χ < 2 reach equilibrium spin, a eq = 0.07, as was found by Lowell et al. (2023).The spin-down is also efficient for lower values of accreted mass.For example, even for a small total accreted mass of 0.2M 0 , i.e. 1/χ = 5, the final BH spin is a f 0.2.The simulation of Gottlieb et al. (2022b) with an initial spin a 0 = 0.8 and a χ 0.3, marked by the red star in Fig. 1, should reach a f 0.1.This demonstrates that even if the natal BH spin is high and χ is below the critical value of χ = 0.5, the spindown is substantial.We conclude that for any reasonable accreted mass, e.g.m f ∼ M , collapsar BHs inevitably spin down to low spins of a f ≈ 0.1, independent of the mass accretion rate.
We stress that the spin evolution model of Lowell et al. (2023) is only valid for an engine that has reached the MAD state and is radiatively inefficient.We verify that the spin evolution model of Lowell et al. (2023) is valid for collapsars by showing that it is compatible with the spin-up parameter in collapsar simulations with a = 0.8 and a = 0.1 (see Appendix A).In collapsar simulations the system reaches the MAD state relatively fast, t < 1 s (see Appendix A).However, the system could take longer to reach the MAD state with different initial conditions, as discussed in §4.2.

BH SPIN EVOLUTION
The magnitude and time dependence of the accretion onto the BH depend on the stellar mass and density profile, respectively.1D core-collapse simulations find that density profiles from stellar evolution models, ρ(r) ∝ r −2.5 , flatten prior to the BH formation to ρ(r) ∝ r −1.5 (Halevi et al. 2022).For freefall of a typical stellar envelope mass, numerical and analytic results show that this power-law leads to a steady BH accretion of ṁ 10 −2 M s −1 (Gottlieb et al. 2022b(Gottlieb et al. , 2023)).If we were to extrapolate that rate to typical GRB durations of a few dozens of seconds, the BH would accrete m f ≈ M .For M 0 M NS,max , this corresponds to a 50% increase in the BH mass and final spin a f a eq .
For a roughly constant accretion rate until time t f , we adopt the following time-dependency of ṁ, where is the characteristic accretion duration, the time where the mass reservoir, m f = M λ, has been exhausted.For simplicity we adopt a constant ṁ.However, we note that a constant accretion rate is not strictly consistent with the simulations of Gottlieb et al. (2023), which feature a small decrease of ṁ with time.Furthermore, the asymptotic behavior of ṁ is not constrained, as discussed in §5.
Figure 2 demonstrates the evolution of the BH mass (a) and spin (b) for different initial spins, where t = 0 is the MAD state activation time, assuming λ = 0.2, χ = 1.2.The BH mass saturates once the accretion stops, denoted by t f , whereas the BH spin saturates when a = a eq , before t f .All solutions converge to equilibrium spin, with lower initial spins reaching a f = a eq faster.The final BH mass is practically independent of the initial spin.

OBSERVABLES OF THE SPIN-DOWN MODEL
We examine the compatibility of our semi-analytic solution with the observed GRB characteristics: duration t f , average luminosity L jet (Eq.12), and time evolution (Eq.13).The jet power is dictated by the mass accretion rate ṁ and jet launching efficiency η EM , which in turn depends solely on the BH spin (Lowell et al. 2023) To compute the GRB characteristics, we model the jet propagation inside the star with the semi-analytic model of Harrison et al. (2018), which relies on the formalism of hydrodynamic jets (Bromberg et al. 2011, see caveats in §5).We adopt the following mass density profile in the star where r H is the radius of the BH and R is the radius of the star.The density ρ 0 is normalized using the total mass of the star, M .Using the jet power from our engine evolution model and the above stellar profile, we compute the jet propagation within the stellar envelope and the breakout time t b .
The jet power at the jet head is computed using the retarded time, t − z h /c, where z h is the position of the jet head.The cumulative energy carried by the jet is approximated to be the jet energy that does not cross the reverse shock before breakout We then define t 1 and t 2 which are the times when 5% and 95% of the jet energy has been released.Thus, 90% of the jet energy is released between in a time scale T 90 = t 2 −t 1 .To represent the characteristic jet power of every solution, we define the average jet power To quantify the evolution of the jet power, we define the ratio We impose R L jet ≤ 2.5 so that the jet power remains roughly constant during the GRBs duty cycle (e.g., McBreen et al. 2002).

BH spin evolution of typical GRBs
To obtain the characteristic GRB jet power, one needs to consider the highly uncertain γ-ray radiative efficiency γ .We choose a fiducial value of γ = 0.5, so the jet power and energy are L jet = L jet,obs / γ , and E jet = E jet,obs / γ .The typical range of the GRB jet luminosity and energy are 6×10 49 erg s −1 ≤ L jet ≤ 4×10 51 erg s −1 and 3×10 50 erg ≤ E jet ≤ 6 × 10 51 erg, respectively (Goldstein et al. 2016).Figure (3(a)) delineates the evolution of the jet power and energy for different initial BH spins.We assume ṁ = 3 × 10 −2 M s −1 , which is consistent with our choice of stellar profile and the values measured by Gottlieb et al. (2023).While higher initial spins lead to higher power and more energetic jets, all BHs reach an equilibrium spin within the typical long GRB duration.Consequently, all jets also converge to the same value, L jet = η EM (a = a eq ) × ṁc 2 8 × 10 49 erg s −1 , consistent with the typical GRB jet power.The unshocked jet element breakout time from the star, t b − R c , marked by a filled circle, represents the time from which the time evolution in the jet power can be observed.If the BH spin is still evolving considerably at t > t b , there will be visible variations in the observed jet power, in tension with observations.Fig. 3(b) shows the jet energy, as calculated in Eq. ( 11).The energy of all jets launched from BHs with initial spins of a 0 0.1 is dominated by the energy released when the BH reaches equilibrium spin, L jet (a = a eq ), and converges to E jet ≈ 5 × 10 51 erg by the end of the GRB, at t = t f .This jet energy is within 1σ of the observed jet energy distribution (Goldstein et al. 2016).Figure 4 displays the dependence of the average jet luminosity on the mass accretion rate and initial spin.The average jet luminosity is governed by the more luminous phase of the jet.Thus, for low initial spins, a 0 < a eq , the average jet power does not depend on the initial spin, and depends only on the accretion rate.This is consistent with Fig. 3(a), which shows that low spins quickly reach equilibrium spin, thus L jet ≈ η(a eq ) ṁc 2 .This trend is reversed for high initial spins, a 0 > a eq , where the contour lines are primarily vertical, i.e. the average jet power is dictated by the jet luminosity when the BH spin is a 0 .
Most BH spins and mass accretion rates are consistent within 1σ with the observed jet power, outlined by the red lines in Fig. 4. The jet power obtained in the solutions under the white line varies by less than a factor of 2.5, as shown in Eq. ( 13).Only solutions with low initial spins, a 0 0.1, The area under the white contour constrains the emerging jets with minimum time evolution in their emission, characterized by R L jet ≤ 2.5.The red lines represent the observational 1σ spread around the average jet power (Goldstein et al. 2016).Only solutions with weak initial spins, a 0 ≤ 0.1, and accretion rate of 2 × 10 −2 M s −1 ṁ 1.5 × 10 −1 M s −1 are consistent with both constraints.
reach equilibrium spin at t < t 1 , thus exhibit a flat jet power curve that could reproduce the observational data (Fig. 3a).

BH spin of powerful GRB jets
Fig. 4 shows that powerful GRBs with L jet / γ ≥ 5 × 10 51 erg s −1 are excluded from the variation constrain.Here we show that the most powerful GRB jets can emerge by delaying the activation of the MAD state (the jet launching).To investigate the effects of delaying the onset of the MAD state, we introduce the function Φ(t) that represents the disk state with respect to MAD where t MAD is the characteristic time for the disk to become MAD.When t t MAD , the disk acts as a standard viscously accreting hydrodynamic disk, and its torques on the BH are modeled with the standard theory of Bardeen (1970).We write a modified set of spin evolution equations that follow the spin evolution in Bardeen (1970) up until the system becomes MAD 1 and where e Ba and s Ba are taken from Bardeen (1970), and s MAD is defined in Eq. ( 6).We reintroduce the jet power in Eq. ( 9) with the magnetic flux saturation parameter as When Φ(t) = 1, the magnetic field of the BH and inner disk have saturated and we recover Eqs. ( 4) and ( 5).When the t t MAD , there is no magnetic jet torque breaking the BH.Furthermore, the Bardeen (1970) accelerating hydrodynamic torque on the BH is larger than for a MAD (Lowell et al. 2023).This leads to a far greater equilibrium spin, a eq = 1.Thus, by having a long t MAD , the BH reaches a higher final spin.Solutions with t MAD ≥ t f are excluded from the parameter space, since they would reach the MAD state after the mass reservoir is exhausted.We note that Φ(t) is a continuous function of t, and thus the magnetic flux on the engine and the jet power gradually increases until Φ(t) saturates.See Appendix B for the temporal evolution of solutions with high t MAD .
First-principles numerical simulations of collapsars (e.g., Gottlieb et al. 2022a,b) show that the disk reaches a MAD state soon after the core collapse.If the disk does not become MAD early on, it energizes an expanding accretion shock that hampers magnetic flux on the disk such that the disk cannot become MAD at later times, disfavoring long t MAD .However, those simulations explored only a limited range of magnetic field profiles.It is possible that a low net magnetic flux within the star, or a magnetic flux profile that is concentrated far away from the core, would take a long time to saturate the central engine, leading to a long t MAD .Hence, t MAD will depend on the initial magnetic field profile and the magnetic flux transport with the stellar envelope and the disk.
Figure 5 depicts the average jet power (panel a) and final BH spin (panel b) as a function of ṁ and a 0 for t MAD = 70 s and χ = 1.5.Fixing χ in Fig. 5 leads to t f being anticorrelated with ṁ, so high accretion rates lead to fast reservoir depletion.Thus, we exclude the solution with ṁ 5 × 10 −2 M s −1 as this entails t f t MAD .
In Fig. 5(a), the white line delineates R L jet = 2.5, above which are shown solutions with a low variation, R L jet < 2.5.The red contours represent the observational 1σ spread around average jet power (Goldstein et al. 2016).High jet power, L jet ≥ 5 × 10 51 erg s −1 , solutions with low variability are obtained above both contours, and are weakly dependent on a 0 .However, very high initial spins, a 0 0.4, are excluded since they do not satisfy the variability constraint.
In Figure 5(b), the maximum final spin a f 0.35 is obtained at t f t MAD , since the MAD state does not have 49.4 50.0 50.6 51.1 51.7 Figure 5.Effect of long t MAD = 70 s on the average jet power, L jet (panel a) and final spin, a f (panel b) as functions of the accretion rate ṁ and the initial spin, a 0 .We assume χ = 1.5 (M = 25 M , M 0 = 2.5 M , λ = 0.15, R = 1 R ).The right axis displays the GRB duration t f , using the accretion rate and χ from Eq. ( 8).We do not plot solutions that have t MAD > t f (white area).Above the white contour lie the least variable jets, characterized by R L jet < 2.5.The red lines constrain the observational 1σ spread around the average GRB jet power, assuming γ = 0.5.A large t MAD allows for strong jets, L jet ≥ 5 × 10 51 erg, while maintaining small variation, R L jet , and small final spins, a f ≤ 0.3 enough time to spin down the BH.At longer t f (lower ṁ), the BH can have a substantial spin-down, and reach closer to a = a eq .Although the final spin is a f 0.35 or lower, it peaks at a 0.6 before spinning down.This peak is due to the Bardeen (1970) accelerating torque acting on the BH before Φ(t) saturates (t < t MAD ).The peak in spin leads to a peak in jet power, so the jet reaches high energies while maintaining a low variation (see Appendix B).We conclude that delaying the jet activation to ∼ 70 s yields jets that are compatible with the most energetic GRBs.

Summary
In this Letter we show that BH spin evolution to low spins is unavoidable in magnetically arrested collapsars.The final BH spin only weakly depends on the initial spin: it is primarily sensitive to the ratio between the total accreted mass and the BH mass at the onset of the MAD state.For physically motivated values of accreted mass, this results in a low BH spin.Achieving a high final spin is challenging even for conservative values of BH accretion of 20% its initial mass, for which the final spin reaches a f 0.1.This will lead to a statistical final spin distribution centered around the equilibrium spin, a eq = 0.07.This is consistent with Bayesian estimation of the BH spin distribution from gravitational wave measurements by LVK (Abbott et al. 2019;García-Bellido et al. 2021;Edelman et al. 2022), which constrains the spin distribution of merging BHs to be centered around a 0.15.Furthermore, the spin distributions show that highly spinning BHs, a > 0.7, at least those that end up in merging binaries, should be rare or nonexistent.However, it is unclear if BH growth through consecutive mergers or different formation channels would lead to a similar spin distribution.
For consistency, we also check if our BH-powered jets are representative of typical GRBs and do not display any trends in their time evolution.In Table 1, we summarize the required mass accretion and MAD activation times for reproducing the variety of GRB powers.We find that most types of GRBs favor low initial spins, a 0 ≤ 0.1, thanks to their low variation, with the exception of strong jets, L jet,obs / γ 5×10 51 erg s −1 , that can display little variation up to a higher initial spin, a 0 0.35.
In a companion paper (Gottlieb et al. 2023), we show that slowly spinning BH, with a 0 0.2, generate jets with typical GRB powers.GRBs with higher initial spins would exhibit too much variation.Weak GRBs can be explained by ṁ 2 × 10 −2 M s −1 , at a low initial spin, a 0 < 0.1 (see Fig. 4).In order to obtain low variation for powerful GRB jets, we introduce a delayed jet activation time, t MAD ∼ t f .This allows the BH to spin up during the first stage of accretion in the absence of jet activity.The rapidly spinning BH then powers a strong jet, and ultimately spins down to a f 0.3.A long t MAD could be the consequence of a star with a weak magnetic field or a magnetic flux that has a steep radial profile, see §4.2.

Limitations
We discuss the limitations of our semi-analytic model, and how to improve those in future work.
• In §4 we use a semi-analytic hydrodynamic model to solve the jet propagation in the star.On the one hand, Gottlieb et al. (2020); Gottlieb & Nakar (2022) showed that weakly magnetized jets do not develop local hydrodynamical instabilities, and they propagate ∼ 2 times faster than their hydrodynamic counterparts.
On the other hand, we consider strongly magnetized jets, which are subject current-driven instabilities that may slow the jets down.In the absence of numerical modeling of the propagation of such jets, we adopted the numerical solution for hydrodynamic jets by Harrison et al. (2018).Nevertheless, in the companion paper Gottlieb et al. (2023), we find that firstprinciples, strongly magnetized, jets with a typical GRB power fail to retain their relativistic nature upon breakout, due to strong interactions with the star.This is in contrast to hydrodynamic and weakly magnetized models, which do not consider the tilt of the jet launching and kink instabilities.It is thus of utmost importance to generalize jet propagation models based on first-principles simulations.
• We delay the activation of the jet by introducing the timescale by which the formation of the MAD is delayed post core-collapse, t MAD .However, we did not consider the engine deactivation time, which we set at t f , i.e. when the mass reservoir is exhausted.Magnetic field transport is responsible for the emergence or end of the MAD state (Tchekhovskoy et al. 2011).
Once the magnetic flux saturates on the BH, the inner disk reaches the MAD state.The engine deactivation time then depends on the magnetic field structure of the star and the magnetic field transport after and during the stellar collapse.Jacquemin-Ide et al. ( 2021) measured how the magnetic field transport depends on the strength of the initial magnetic field in accretion disks.However, how this results can be generalized to collapsars or to longer timescales is unclear, as the magnetic field transport mechanism remains poorly understood.Thus, the jet could turn off before t f if the magnetic reservoir is exhausted or the magnetic structure reorganizes before t f (Gottlieb et al. 2022a).Long dura-tion global collapsar simulations are needed to better constrain the engine evolution on long timescales.
• Our model is built on 3D GRMHD simulations in which the disk is not cooled: they are meant to represent highly super-Eddington accretion systems (Lowell et al. 2023).This is also the case for the simulations of Gottlieb et al. (2022bGottlieb et al. ( , 2023)).In reality, for the high accretion regimes of GRBs, neutrinos take on as the dominant cooling agent (e.g., Chevalier 1989;Batta & Lee 2014;Siegel et al. 2019).Cooling reduces the disk geometric thickness, which may modify the magnetic and hydrodynamic spin-down torques that in turn could shift the equilibrium spin and the timescale on which the BH reaches that spin.Lowell et al. (2023) argued that thin accretion could lead to higher equilibrium spins, a eq ∼ 0.3.Even though this value is larger than the one we consider by a factor of ∼ 3, it still is a relatively low spin.The biggest uncertainty that thin MADs introduce is the change in the efficiency of the spin-down torques.It is uncertain if a thin MAD requires a higher accreted mass to reach the equilibrium spin.Finally, Gottlieb et al. (2023) find an accretion rate that is not violent enough for neutrino cooling to be very strong for the mass accretion rates involved, so neutrino cooling might not considerably change our results.
• The simulations of Gottlieb et al. (2023) show that the accretion rate, although slowly varying, is not entirely constant with time.Numerically, due to the limited simulation runtime it is very hard to constrain the accretion rate on very long timescales.Furthermore, it is unclear how the spin-down might affect the non-linear feedback of jets on the accretion rate (Gottlieb et al. 2022a).Because of this, we do not include any temporal variation in the mass accretion rate in our model.
In fact, when we include a slowly varying ṁ, we find that this makes it hard to satisfy the lack of variation in jet power suggested by observations (McBreen et al. 2002).
• First-principles collapsar simulations feature wobbly jets (Gottlieb et al. 2022b), owing to the spontaneous tilt of the accretion disk.Such a wobble would alter the inferred jet energy from the one used in this paper.For example, if the tilt jet angle is 0.2 rad and its opening angle is 0.1 rad, then only 10% of the jet energy is observed for a given line of sight (Gottlieb et al. 2022c).Thus, the presence of the wobble increases the total GRB jet energy by about an order of magnitude.Such an increase would favor solutions on the border of the parameter space shown in Fig. 4, ṁ 10 −1 M s −1 , and long t MAD values.

Figure 3 .
Figure 3. Evolution of L jet (panel a) and E jet (panel b) as functions of time and different initial spins, a 0 ∈ [5 × 10 −3 , 1.0].The circles in panel (a) mark t b − R c , which is the time after which unshocked jet elements can emerge from the star.The left and right triangles in panel (b) represent t 1 and t 2 , respectively.The vertical dotted line marks the time at which accretion stops t f .All solutions are computed using ṁ = 3 × 10 −2 M s −1 , λ = 0.2, M = 15 M , R = R , M 0 = 2.5 M , and χ = 1.2.

Figure 4 .
Figure 4. Jet power as a function of ṁ and the BH initial spin.The area under the white contour constrains the emerging jets with minimum time evolution in their emission, characterized by R L jet ≤ 2.5.The red lines represent the observational 1σ spread around the average jet power(Goldstein et al. 2016).Only solutions with weak initial spins, a 0 ≤ 0.1, and accretion rate of 2 × 10 −2 M s −1 ṁ 1.5 × 10 −1 M s −1 are consistent with both constraints.

Figure 6 .
Figure 6.Spin-up parameter vs. time for collapsar simulations with BH spin values of a = 0.1 and a = 0.8.The pink and dark blue dashed lines show moving averages of s for a = 0.8 and a = 0.1, respectively.The horizontal orange and light blue lines show the values calculated from the MAD model for a = 0.8 and a = 0.1, respectively.The time where the system reaches the MAD state in the a = 0.8 simulation, t MAD , is shown by the vertical black line.