Emergent Nucleosynthesis from a 1.2 s Long Simulation of a Black Hole Accretion Disk

We simulate a black hole accretion disk system with full-transport general relativistic neutrino radiation magnetohydrodynamics for 1.2 s. This system is likely to form after the merger of two compact objects and is thought to be a robust site of r-process nucleosynthesis. We consider the case of a black hole accretion disk arising from the merger of two neutron stars. Our simulation time coincides with the nucleosynthesis timescale of the r-process (∼1 s). Because these simulations are time-consuming, it is common practice to run for a “short” duration of approximately 0.1–0.3 s. We analyze the nucleosynthetic outflow from this system and compare the results of stopping at 0.12 and 1.2 s. We find that the addition of mass ejected in the longer simulation as well as more favorable thermodynamic conditions from emergent viscous ejecta greatly impacts the nucleosynthetic outcome. We quantify the error in nucleosynthetic outcomes between short and long cuts.

The behavior of accretion disks is sensitive to a number of physical effects including postmerger magnetic field configurations (Rüdiger & Shalybkov 2002;Christie et al. 2019), the nuclear equation of state (Steiner et al. 2013), and neutrino physics (McLaughlin & Surman 2005;Surman et al. 2008).In neutron star mergers, disk ejecta may be accompanied by dynamical ejecta (Dietrich & Ujevic 2017;Radice et al. 2018) that is also sensitive to neutrino physics (Foucart et al. 2023).Accretion disks from the merger of a neutron star-black hole binary are also found to be favorable sites of the r-process (Siegel & Metzger 2017;De & Siegel 2021;Murguia-Berthier et al. 2021;Curtis et al. 2023).
Long-term evolution of accretion disks is consequential for electromagnetic counterparts (Christie et al. 2019;Fernández et al. 2019), as well as the nucleosynthesis that ensues in the aftermath of these cataclysmic events.Recently, significant effort has been devoted to simulations that capture the longlived remnant (Hayashi et al. 2022(Hayashi et al. , 2023;;Kiuchi et al. 2023).To our knowledge, however, no late-time models to date perform detailed radiation transport and nucleosynthesis calculations.Previous work (Miller et al. 2019b(Miller et al. , 2020) ) indicates that at early times, higher-fidelity transport is required to accurately capture the electron fraction of the outflow and thus the nucleosynthetic yields.To date, it is unclear if this result translates to late times during active nucleosynthesis.
In this work, we help resolve this uncertainty.We model a black hole accretion disk system that may arise after the merger of two neutron stars and evolve it for 1.2 s.This duration of time is long enough to explore active nucleosynthesis in the rprocess.We analyze mass ejection, entropy, and electron fraction, which all have a strong influence on the nucleosynthetic outcomes.To analyze the error in the present model calculations arising from computational limitations, we compare these results to the same simulation stopped at 0.12 s.We end with a discussion of the uncertainty that arises in simulated nucleosynthesis yields when using short-duration simulations.

Simulation Details
We extend the full transport general relativistic neutrino radiation magnetohydrodynamics (MHD) simulation of a black hole accretion disk wind system performed in Miller et al. (2019b) using the νbhlight code (Miller et al. 2019a(Miller et al. , 2019b(Miller et al. , 2020) ) to a full 1.2 s.We chose to extend this model, rather than start a new simulation, to make a better apples-to-apples touchpoint with previous results.This calculation took approximately 7 months of walltime, or roughly 1.5 million CPU hr on 33 nodes.
The original model, which we extend, was selected to match one possible outcome of the 2017 observation (Abbott et al. 2019) and uses system parameters informed by Shibata et al. (2017).In particular, we use a stationary Kerr (1963) black hole spacetime for a black hole of mass M BH = 2.58 M e and dimensionless spin a = 0.69.The initial conditions are shown in Figure 1.We approximate the leftover gravitationally bound material via a torus in hydrostatic equilibrium (Fishbone & Moncrief 1976) of constant specific angular momentum, constant entropy of s = 4k b baryon -1 , constant electron fraction of Y e = 0.1, and total mass of M d = 0.12 M e .This torus has an inner radius of r in = 5GM/c 2 and a radius of maximum pressure of = r GM c 10.46 Pmax 2 and starts with a single poloidal magnetic field loop with a minimum ratio of gas to magnetic pressure, β, of 100.
We note that in practice, a neutron star merger produces a continuum of material around the central remnant, not just a bound equatorial flow of constant Y e and entropy.However, the emphasis here is on the accretion flow, so we do not include the other material.We also note that constant, low Y e and entropy is not entirely accurate but is approximately true at the time of "handoff" in this equatorial region.See, for example, Section 3.2 of Radice et al. (2018).We further note that the magnetic field configuration in numerical relativity simulations is a major uncertainty and can have a strong impact on the results.We do not study that phenomenon here, but see, for example, Lund et al. (2023) for a recent discussion of the impact of magnetic fields.
We solve the equations of general relativistic ideal MHD, closed with the SFHo equation of state, described in Steiner et al. (2013) and tabulated in O' Connor & Ott (2010).Neutrinos are evolved with a Monte Carlo method and can interact with matter via emission, absorption, or scattering.For emission and absorption, we use the charged and neutral current interactions as tabulated in Skinner et al. (2019) and summarized in Burrows et al. (2006).Neutrino scattering is implemented as described in Miller et al. (2019a).The Monte Carlo and finite volume methods are coupled via first-order operator splitting.
We use a radially logarithmic, quasi-spherical grid in horizon-penetrating coordinates with N r × N θ × N f = 192 × 168 × 66 grid points with approximately 3.8 × 10 7 Monte Carlo packets.For details on the resolution requirements of the model and why we chose this resolution, see Miller et al. (2019b).
After about 400 ms of runtime, the neutrino opacity in the disk is sufficiently low that neutrinos are essentially free-streaming.At this point, we turn off transport and switch to an optically thin cooling prescription.Essentially, Monte Carlo particles are emitted at the proper rate but are then immediately deleted and not transported or absorbed.
Although our code is Eulerian, we track approximately 1.5 × 10 6 Lagrangian fluid packets, or "tracer particles."Each tracer particle is assigned a mass, representing the statistical weight of the particle.Following Bovard & Rezzolla (2017), we initialize tracer particles uniformly distributed in the volume containing a nontrivial density of gas at the initial time.At each time step, tracer particles are advected with the fluid flow via the equation for fluid four-velocity u μ , three-velocity v i , lapse α, and shift β i .Latin indices range from 1 to 3 and represent spatial directions.Greek indices range from 0 to 3 and represent space and time.Fluid and microphysical data, such as fluid density and temperature, electron fraction, and neutrino reaction rates, are interpolated to tracer positions and recorded per tracer.

Engine Physics
As the system evolves, the magnetorotational instability (MRI ;Velikhov 1959;Balbus & Hawley 1991) self-consistently drives the disk to a turbulent state, which provides the turbulent viscosity necessary for the disk to accrete (Shakura & Sunyaev 1973).This mechanism drives a long-lived accretion flow, which starts as powerful as >1 M e s −1 but sweeps down in accretion rate as the disk expands and cools.Figure 2 shows this behavior.Analytic models of the turbulent viscosity predict that the accretion rate follows a t −5/3 power law before eventually transitioning to exponential decay (Dolence 2011;Tanaka 2011).We include a t −5/3 line to guide the eye.Material undergoing r-process nucleosynthesis is ejected  primarily during the downward-sloping phase of this curve, after approximately 2 × 10 −2 s.Over time, the density drops, causing the accretion rate to drop as the disk drains.The density of the disk for three different times is shown in Figure 3.The electron fraction in the disk and the outflow is set by the relative timescale of fluid motion relative to the timescale on which weak processes are occurring.Following Miller et al. (2020), we compute the weak timescale as and the timescale for fluid motion as for a characteristic disk opening angle and mass-averaged lepton advection velocity The top right panel of each row in Figure 4 shows the ratio of τ + to τ a , the bottom right panel shows the ratio of τ − to τ a , and the left panel shows the ratio of τ + to τ − .In the right panels, a small ratio implies that weak processes dominate.As the ratio grows, weak processes become less important in setting Y e compared to fluid motion, and the electron fraction freezes out.The top row shows the disk at 0.13 s, the middle at 0.51 s, and the bottom at 1.27 s.The left panel shows that τ + is smaller than τ − , indicating that weak processes are driving the  electron fraction up.However, as the disk cools, these weak processes become inefficient compared to fluid motion, and the electron fraction in the disk freezes out.
Outflows begin to be launched early in the lifetime of the disk, although they travel at different speeds and thus become gravitationally unbound at different times.Figure 5 sketches these different components out.The magnetic field powers a jet via the Blandford & Znajek (1977) mechanism; turbulent heating drives a hot, fast disk wind in an hourglass shape out the poles of the disk; and turbulent viscosity drives a slowermoving equatorial outflow.The viscous mechanism eventually unbinds the most mass.In contrast, the jet is the fastest mechanism but unbinds the least mass.While we describe these three outflow mechanisms as separate here, in reality, these mechanisms are difficult to disentangle and thus uniquely quantify.

From Tracer to Trajectory
Once the simulation has completed, we down select tracers that are unbound to study the nucleosynthesis.This filter involves the calculations of two physical constraints.The first is that the tracer is 250 gravitational radii (GM BH /c 2 ) away from the central black hole.The second is that the Bernoulli parameter is B e > 0. The Bernoulli parameter originates in modeling of hydrostatic flows.B e = 0 implies hydrostatic equilibrium, B e < 0 implies a flow infalling into a gravitational potential, and B e > 0 implies a gravitationally unbound flow (Narayan & Yi 1995).
This selection criterion results in 79,556 "short" tracers at 0.12 s and 461,690 "long" tracers at 1.2 s.The difference between these two subsets comes only from running the simulation for an extended duration.Over the course of this additional second of simulation time, the ejected mass increases by a factor of 18.5, with the electron fraction decreasing by 0.1 on average.The temperature and density also show sizable changes in favor of the production of heavy elements.A summary of the difference between short and long evolution is provided in Table 1.
If a tracer is found to be unbound in the short case, it is also unbound in the long case (by the definition of being unbound using the above two constraints).The bulk of the tracers, 382,134 = 461,690-79,556, become unbound on timescales greater than 0.1 s, owing to the dynamics of the central engine.MHD disk models typically drive an early, fast outflow powered by heat and magnetic forces (Siegel & Metzger 2017;Christie et al. 2019) and a late, slow outflow powered by turbulent viscosity (Shakura & Sunyaev 1973).The latter outflow is enhanced by nuclear recombination incorporated into the nuclear statistical equilibrium finite temperature equation of state (Fernández et al. 2019;Fahlman & Fernández 2022;Just et al. 2022;Haddadi et al. 2023).Our disk is no exception, and the more massive late-time outflow is from the slower viscous mechanism.
The total amount of mass unbound in the short tracers is significantly smaller than in the long.At 0.12 s, when the short tracers are extracted, the disk has accreted roughly 9.57 × 10 31 g of mass.The mass in the short tracers accounts for about 3.7% of that accreted mass.At 1.2 s, when the long tracers are extracted, the disk has accreted 9.81 × 10 31 g of mass, only a small fraction more (this is due to the power-law decay shown in Figure 2).However, the total mass in the long tracers is 6.46 × 10 31 g, or 65% of the accreted mass and about 27% of the total mass of the disk.Other late-time models, such as Siegel & Metzger (2018), Christie et al. (2019), andFernández et al. (2019), indicate that late-time outflow can be as much as 40% of the disk mass.Our result, as well as the other literature, indicates that extrapolating the total mass in the outflow at late times based on early-time mass flux will introduce inaccuracies of about an order of magnitude.This is likely due to the different velocities of the outflow, as the fast-moving outflow is less massive than the slower-moving outflow.
We take the set of traces and convert them into a "trajectory" for use in postprocessing nucleosynthesis.A trajectory extends the temperature and density profiles contained in each tracer by assuming a homologous expansion.The simulation of nucleosynthesis for a given trajectory, however, does not start at the point of homologous expansion.Instead, the starting point of our nucleosynthesis calculations begins at the last time the temperature drops below T = 10 (GK).
A homologous expansion is implemented as follows.The velocity is assumed to be constant, yielding an increment of the Cartesian coordinates after a duration of time, dt, dx i = v i × dt.The density is extrapolated as a power law, ρ ∼ ρ e /t 3 , where ρ e is the density at the time of extrapolation.The temperature is extrapolated from the density assuming an ideal gas with Γ = 5/3.6As a consequence of these assumptions, the final time points associated with the trajectories are independent from one another and do not interact hydrodynamically (unlike a tracer).
The endpoints of the tracers (starting points of the homologous expansion) vary drastically.This situation arises naturally from the simulation and thus means the conditions under which heavy element synthesis proceeds will also show large variation.Our results thus highlight the need for future nuclear sensitivity studies to cover a wide range of conditions, as shown in the recent work of Li et al. (2022).
The additional impact of radioactive heating from nuclear processes can be substantial and result in a change in the temperature evolution of the trajectory relative to a homologous expansion.Nevertheless, it is expected to be a larger effect for dynamical ejecta than in disk ejecta (Lippuner & Roberts 2015).For this reason, it will be considered in subsequent work.

Nuclear Inputs
We use Portable Routines for Integrated nucleoSynthesis Modeling (PRISM) to model r-process nucleosynthesis (Sprouse et al. 2021).The nuclear input to PRISM is based on the 2012 version of the Finite Range Liquid Droplet Model (FRDM; Möller et al. 2012Möller et al. , 2016)).Neutron-induced reactions, including radiative capture and fission, are calculated with the CoH 3 statistical Hauser-Feshbach code Kawano (2019Kawano ( , 2021aKawano ( , 2021b)).Rates of β-decay, β-delayed fission, and the associated probabilities of emitting neutrons are calculated assuming a statistical de-excitation from excited states (Mumpower et al. 2016a(Mumpower et al. , 2018)).The REACLIB database is used for secondary reaction rates (Cyburt et al. 2010).Conditions suitable for a robust fission recycling r-process are not found in this work.Therefore, a symmetric 50/50 split is used for fission products in order to increase the computational efficiency of PRISM without impact on the resultant nucleosynthesis or any of our conclusions.

Results
First, we describe the differences in key astrophysical quantities that influence the nucleosynthetic outcomes by running for longer times.We then analyze the nucleosynthesis itself.
In Figure 6, we present the difference in the entropy distribution of the unbound tracers between the long and short runs.The short-duration run has overall less mass ejected, which can be seen from the lower maximum value on the y-axis for traced mass.In addition, the long-duration simulation has a lower average entropy (15.49k B baryon -1 as compared to 19.67 k B baryon -1 in the short case).The shift to lower entropy values as the simulation runs longer arises due to the different ejection mechanisms-the early-time fast outflow is more thermally as opposed to viscously driven and may contain a component of material entrained in the jet.
Also crucial to the resultant nucleosynthesis is the value of the electron fraction at the end of the tracer.Figure 7 compares the distributions of electron fraction between the two cases.We find that the longer-duration simulation has significantly lower Y e than the short-duration simulation due largely to viscous material that became unbound later in the simulation.This strong shift to lower Y e is a harbinger of subsequent heavy element formation.
The additional low entropy and low Y e tracers that are captured in the long-duration run will have slightly different typical nucleosynthetic evolutions as compared with the short tracers.First, a lower electron fraction, with all else being equal, means more neutrons available for capture on seed nuclei and a more robust r-process.This effect is enhanced by lower entropy, which means that material will fall out of equilibrium sooner and experience a more robust r-process.Finally, not only does a lower entropy produce a more robust rprocess, it also changes the shape and position of the peaks in Notes.Averages are indicated with a † and computed by weighting via the mass of each tracer.At the end of a tracer, a homologous expansion is employed.the distribution (Mumpower et al. 2012a(Mumpower et al. , 2012b;;Orford et al. 2018;Vassh et al. 2020Vassh et al. , 2021)).We now turn to the assumption of homologous expansion and contrast the results between early and late times.The evolution of the temperature and density profiles is critical in the first few seconds, as the resultant nucleosynthesis occurs almost entirely in this timescale (Kajino et al. 2019;Sprouse et al. 2022).
In Figure 8, we highlight two individual trajectories.The top panel shows a case where the temperature and density evolution are both altered.In this panel, the long cut (solid) maintains a higher temperature and density for longer than the short cut (dotted).The longer time spent in the 3-1 GK region means the r-process is "hotter," spending more time in (n, γ)⟺(γ, n) equilibrium.In addition, the long-duration tracer spends more time at higher density but by happenstance lands on the homologous expansion curve derived from the short tracer.
The bottom panel of Figure 8 shows a case where the density is orders of magnitude more diffuse in the long run as compared to the short, although the temperature drops off similarly as one would expect from homologous expansion using the short cut.In this case, due to the drop in density, the long-cut nucleosynthesis is less robust than the shortduration cut.
In general, we find that the longer cuts behave as a combination of the temperature and density profiles shown in the two panels of Figure 8.On average, the material in the long cut experiences higher densities at later times with a marginally higher temperature evolution as compared with the short cut.The final two columns of Table 1 highlight these differences where the temperatures are roughly comparable but the density is a factor of 4 larger.
Nuclear reactions scale as the square of the density (Rauscher & Thielemann 2000), so that reaction rates in the long cut are ∼16 times faster than in the short cut.Furthermore, the higher densities are occurring at later times, when reaction rates are more likely to be out of equilibrium, thus substantially favoring more neutron-rich nucleosynthesis (Mumpower et al. 2012c).We find the increase in density at late times to be the primary driver of the differences in the nucleosynthetic outcome between the short and long cuts.
The final abundances for the total mass ejected in each case are shown in Figure 9.The associated elemental abundances are shown in Figure 10.We find that a more robust r-process ensues as emergent viscous material emanates from the disk.The short scenario has a first peak where elements like strontium reside with a reduced third peak production.In contrast, the long simulation shows a complete r-process through the actinides, albeit with a reduced first peak.While the actinides are produced in substantial quantity, we do not find   evidence of fission recycling.Instead, material just makes it to superheavy nuclei (A ∼ 280), which ultimately decay to populate the longer-lived actinides (Holmbeck et al. 2023a).In this simulation, superheavy elements (Z > 103) are not found in sufficient quantity to impact a kilonova signal (Holmbeck et al. 2023b).
The elemental pattern, in particular, shows abundance regions that are clearly simulation uncertainty-dominated (large variation between short and long cuts).This spans nearly the entire pattern from the weak r-process peak (A ∼ 80), to the lanthanides, the third peak (A = 195), and the actinides, while the second peak (A = 130) remains relatively unaltered.
There are also points that cannot be readily explained by simulation uncertainties, since the results from the different cuts of the simulation cannot account for the remaining discrepancy from the solar residuals.In particular, the lighter elements of a "weak" r-process component below Z = 50, as well as the transition nuclei that reside between the second rprocess peak and the lighter lanthanides (50  Z  60), have larger errors from nuclear physics uncertainties than seen from the simulation.Additionally, nuclear physics models like FRDM2012 have a closed N = 126 shell far from stability, which in this simulation results in an overproduction of this peak relative to the solar residuals.Relevant nuclear physics uncertainties for r-process nucleosynthesis have been studied extensively in the works of Mumpower et al. (2016b), Vassh et al. (2019), Misch et al. (2021), andMumpower et al. (2022).
We now quantify the error between the 0.12 and 1.2 s cuts by calculating the percent error, , where X j are the respective final mass fractions.Figure 11 shows this value as a function of proton number (top panel) and mass number (bottom panel).The average percent error for both functions is between 450% and 500%, as indicated by the dashed gray lines.Discrepancies can be found throughout the pattern but are especially astounding for lighter nuclei.A useful rule of thumb derived from this calculation is that for nuclei Z 50, the error in population is roughly a factor of 6, while for Z > 50, the error in population is roughly a factor of 2.
We now address the question of whether or not the stopping point of our simulation at 1.2 s (the long cut) is complete.By complete, we mean that ejecta has stopped impinging on the extraction surface in sufficient amounts such that the electron fraction and other relevant distributions would begin to asymptote, leaving the nucleosynthesis unchanged.To gauge this behavior, we plot in Figure 12 the cumulative mass ejected (gray curve read from the left Y-axis) as a function of time at the extraction surface.The derivative of this quantity, or rate of unbound mass ejection (dashed blue curve), is also shown and can be read from the right Y-axis.While the cumulative mass ejection looks to be slowing down, it is important to note that the salient feature of this curve is its log scale.The bulk of the unbound material arrives at the extraction surface at later times, and the derivative has yet to approach zero.We conclude that simulations must be run for longer times to fully capture the extent of unbound material.

Conclusion
We have simulated a black hole accretion disk system resulting from a binary neutron star merger for 1.2 s using full transport neutrino radiation MHD.We have analyzed the resultant nucleosynthesis, which is greatly impacted as compared with the same simulation cut at 0.12 s.While we find that the total amount of unbound ejecta has yet to completely asymptote (Figure 12), our results provide the first insights of running nucleosynthesis with a long-duration simulation.In particular, we find that emergent viscous material in the plane of the disk is primarily responsible for the vastly different nucleosynthetic outcomes between the short-and long-duration cuts.
Our work shows that by running simulations to later times, lanthanides are produced in similar proportion to the first peak (weak) r-process.To obtain conditions favorable for lighter element production that is in line with the solar pattern, one needs additional processing via neutrinos (that is not found in our simulation) or some other physical mechanism.Monte Carlo transport in νbhlight is only performed in regions of the engine where weak processes are subdominant compared to  fluid motion, i.e., when Y e has frozen out.However, on the timescale of the longer simulation (1 s), these slower processes may matter, and we may be undercounting them.This is one possible source unaccounted for in neutrino processing.
We note that late-time lanthanide-rich outflow from this postmerger disk does not change the fact that the fast-moving lanthanide-poor ejecta may produce an early blue component to a kilonova (Miller et al. 2019b).Moreover, these results cannot be straightforwardly extended to the collapsar case, where the disk is fed and the thermodynamic conditions vary as a power law with time (Miller et al. 2020).
In the near future, longer-duration high-fidelity simulations will become commonplace.We have shown that late-time modeling is required to fully capture the richness of phenomenology in the nucleosynthesis and neutrino sector, and we look forward to continual developments in the community to uncover the details regarding the origin of the heavy elements.
the Monte Carlo quality factor, rad ⎛ ⎝ ⎞ ⎠ minimized over the simulation domain Ω.Here N is the number of emitted Monte Carlo packets and ∂N/∂t is the instantaneous derivative.u is the gas internal energy density by volume, and J is the total frequency-and angle-integrated neutrino emissivity.Equation (A3) roughly encodes how well resolved the radiation field is, with Q rad = 1 a marginal value.
Q rad typically rises with time; as the disk cools, the timescale for weak processes rises until weak freeze-out.In our simulation, we find Q rad ≈ 100 at early times (roughly 20 ms), which is the most difficult regime to capture, and it rises as time goes on.As discussed in Section 2.1, we eventually turn off neutrino absorption when the optical depth becomes too small.As discussed in the conclusion, we subsequently do not include neutrino reprocessing in the nucleosynthesis, which may impact the final yields.

A.3. Artificial Atmosphere Treatment
As discussed in Miller et al. (2019aMiller et al. ( , 2019b)), νbhlight cannot treat true vacuum due to the fact that it models a fluid in the Eulerian frame.Instead, we impose an artificial atmosphere or floor such that where ρ 0 = 10 −5 is a unitless, simulation-dependent parameter; ⌊ρ⌋ = 1.1 × 10 13 g cm −3 is the code unit for density; and r = 1.6 10 min 2 g cm −3 is the minimum density in our tabulated equation of state.The atmosphere is set to approximately virial temperature to prevent it from falling back onto the disk.
Since these long simulations are concerned primarily with outflow, one may be concerned that the artificial atmosphere interferes with the disk wind and may prevent the wind from becoming unbound.To quantify this, we track our artificial atmosphere with a passive scalar to ensure it does not contribute to any reported quantities, such as outflow mass and electron fraction.Tracer particles that end up in mostly atmosphere regions are automatically deleted and not included in the final nucleosynthesis analysis or reported masses.
As the disk wind system evolves, outflows displace the artificial atmosphere by pushing it through the outer boundary of the domain.After about 63 ms, there is no artificial atmosphere remaining.The total amount of atmosphere displaced this way is roughly 10 −5 M e , or ∼10 −4 of the mass of the disk and ≈3 × 10 −4 of the mass in the outflow.Further, we find that the radial momentum flux in atmosphere regions is always more than 3 orders of magnitude less than in wind regions.In total, this gives us confidence that artificial atmosphere is not interfering with dynamics in the outflow.

Figure 1 .
Figure 1.The initial density of our model, as first presented in Miller et al. (2019b).Magnetic field loops are overplotted in black.Figure2.Accretion rate of the disk over the lifetime of the calculation.The blue line segment shows the accretion rate over the duration of the "short" cut; the red line segment shows the extended part of the calculation, referred to as the "long" cut.

Figure 2 .
Figure 1.The initial density of our model, as first presented in Miller et al. (2019b).Magnetic field loops are overplotted in black.Figure2.Accretion rate of the disk over the lifetime of the calculation.The blue line segment shows the accretion rate over the duration of the "short" cut; the red line segment shows the extended part of the calculation, referred to as the "long" cut.
the fluid density, Y e is the electron fraction, and + G Y e and -G Y e are the fluid-neutrino interaction rate for weak processes that increase and decrease the electron fraction, respectively.The times t f and t i bound the time average used to compute τ, θ is the angle off the equator so that θ = 0 is the equator and θ = π/2 is the north pole, -g is the square root of the determinant of the spacetime metric, and u 2 is the theta-component of the fourvelocity of the fluid.SeeMiller et al. (2019a) for a more detailed description of G Y eand Miller et al. (2020) for more details on this timescale analysis procedure.

Figure 3 .
Figure 3.The density of the disk for three different times, 0.13 s (top), 0.51 s (middle), and 1.27 s (bottom), showing the disk drain with time.

Figure 4 .
Figure 4.The electron fraction increasing (τ + ) and decreasing (τ − ) timescales relative to the fluid advection timescale (τ a ) at the same three snapshots in the simulation as in Figure 3.

Figure 5 .
Figure 5. Schematic of the outflow components of the disk.For illustrative purposes, this figure uses a zoomed-out snapshot of the electron fraction Y e of the disk at t = 30 ms and a contrasting color map.

Figure 6 .
Figure 6.Comparison of entropy distributions of tracers between the short and long runs.The short cut (0.12 s) is shown in blue and the long cut (1.2 s) in red.Intermediate snapshots of the entropy distribution are shown between these two snapshots.

Figure 7 .
Figure 7.Comparison of the electron fraction distributions for the short cut (0.12 s; blue) and long cut (1.2 s; red).Intermediate snapshots of the Y e distribution are shown between these two snapshots.

Figure 8 .
Figure 8. Differences between the homologous expansion assumption for short and long trajectories.The dotted lines indicate the short run, while solid lines indicate the long run.The top panel shows a case where both the T 9 and ρ evolution are greatly impacted.The bottom panel shows a case where ρ is greatly impacted.

Figure 9 .
Figure 9. Mean final isotopic abundances at 1 Gyr from the complete ejecta of a neutron star-black hole accretion disk.Solar data in black.

Figure 10 .
Figure 10.Mean final elemental abundances at 1 Gyr from the complete ejecta of a neutron star-black hole accretion disk.Solar data in black.

Figure 11 .
Figure 11.Percent error in the mean final mass fractions as a function of Z or A when using the short-duration simulation.Average values of these functions are represented by the dashed gray lines.

Figure 12 .
Figure 12.The accumulation of unbound mass (solid gray; left Y-axis) and the rate of unbound mass (dashed blue; right Y-axis).

Table 1 A
Summary of the Ending Values for Short and Long Tracer Information