MIXING IN ZERO- AND SOLAR-METALLICITY SUPERNOVAE

Initial models were taken from the surveys of Heger & Woosley (2008) and Woosley & Heger (2007). Both of these papers used the KEPLER code (Weaver et al. 1978; Woosley et al. 2002) to evolve stars through all stable stages of nuclear burning, until their iron cores became unstable to collapse. At this point, pistons located at or near the base of the oxygen shell were used to explode the stars. Heger & Woosley (2008) simulated the evolution and explosion of 10–100 M_☉ stars with zero initial metallicity. Explosion energies ranged from 0.3 to 10 B, where 1 Bethe = 1 B = 10⁵¹ ergs. Woosley & Heger (2007) examined solar-metallicity stars from 12 to 100 M_☉, which were exploded by pistons similar to the other survey, but for a more limited set of masses and energies. Both surveys were limited to nonrotating progenitors. The zero-metallicity stars were assumed to have no mass loss, while the solar-metallicity models took mass loss into account.

Here only two representative stars from each survey are studied: Models z15D and z25D from Heger & Woosley (2008) and Models s15A and s25A from Woosley & Heger (2007). The letter "z" indicates zero initial metallicity, while "s" indicates solar metallicity. The numbers in the models correspond to the initial mass of the star in M_☉, and the final letter is the explosion energy, 1.2 B in each case. The piston was located at the place in the star where the entropy was equal to 4.0k_B/baryon. This corresponded to the base of the oxygen shell. Series sA and zD are thus directly comparable in all respects save their initial metallicity. 15 and 25 M_☉ represent the "canonical" supernova cases with the most commonly employed explosion energy and piston location. 15 and 25 M_☉ stars are also in the same mass range as previous studies of Rayleigh–Taylor mixing in supernovae, allowing for easy comparison.

The one-dimensional progenitor models used in this study lead to spherically symmetric explosions. Mapping the models from one to two dimensions, after the explosion has taken place, has the effect of suppressing low-order departures from spherical symmetry. This work only addresses spherically symmetric explosions with asymmetries of significantly higher mode than l = 1 or 2.

The FLASH code (Fryxell et al. 2000) was used to follow shock propagation and mixing in the models in two dimensions. This code has been extensively verified and tested (Calder et al. 2002; Weirs et al. 2005). FLASH is an adaptive-mesh-refinement code based on an Eulerian implementation of the piecewise parabolic method (PPM) of Colella & Woodward (1984). The code can be configured in a number of different ways. We used the Harten–Lax–van Leer–Einfeldt (HLLE) Riemann solver to resolve shocks. KEPLER uses 19 different isotopes to evolve stellar models through stable stages of nuclear burning. We use the "aprox19" composition module included with the FLASH2.5 distribution to map these isotopes directly to FLASH. The abundances of 19 different isotopes, from ¹H to ⁵⁶Ni, are stored for each grid cell. Explosive nuclear burning could have been followed in FLASH using this network, but the burning was over by the time the models were mapped into FLASH. Mapping the star in at earlier times to follow the explosive burning in two dimensions would have had little effect, as no departure from spherical symmetry is expected until the formation of the reverse shock, which occurs much later in the calculation. FLASH was configured to use axisymmetric coordinates. The gravitational potential was computed using a multipole method to solve Poisson's equation. The mass distribution in this simulation had only slight deviations from spherical symmetry, so the additional gravitational force from overdensities in the simulations was small. The gravitational potential from the point mass at the origin of the grid was added to the potential computed by the multipole solver at each time step. The hydrodynamic equations were solved using an explicit, dimensionally-split approach: the hydrodynamic equations were solved first along one coordinate grid direction, then the other at each time step.

An equation of state was employed that assumed full ionization and included contributions from radiation and ideal gas pressure:

$$\begin{equation} P= \frac{1}{3} a T^4 + \frac{k_{B} T \rho }{m_p \mu } \end{equation} \tag{ 1 }$$

$$\begin{equation} E = \frac{a T^4}{\rho } + 1.5\frac{k_{B} T}{m_p \mu }, \end{equation} \tag{ 2 }$$

where P is the pressure, a is the radiation constant, k_B is Boltzmann's constant, T is the temperature, ρ is the density, m_p is the proton mass, μ is the mean molecular weight, and E is the energy. Although the outer regions of the remnant may not always be fully ionized, the regions where Rayleigh–Taylor mixing takes place are. Exploratory simulations performed in one dimension with FLASH using a Helmholtz equation of state were identical to one-dimensional simulations performed with the perfect gas and radiation equation of state used in the simulations presented in this paper. These one-dimensional simulations were only run to 2 × 10⁴ s. At later times, the density was low enough that the Helmholtz equation of state no longer applied.

The code was configured to use pressure, density, ⁴He, and ¹⁶O as its refinement variables. An error estimate for a block was computed using the second derivative of the chosen refinement variables. If the estimated normalized error in one or more of these variables was greater than a given value, regions were refined until a normalized error is reached that is less than the acceptable value or the maximum level of refinement is reached. Regions of the simulations with steep gradients in one or more of these variables were likely to be refined. If the normalized error estimate was below a certain value, i.e., the absolute value of the second derivative of one or more variables was small, then the region was "de-refined."

The FLASH2.5 release was customized to include a module that inserts a roughly circular zero-gradient inner boundary around the origin. This prevented infalling matter near the origin in the simulations from backing up and affecting the outer regions. Matter was allowed to fall through the zero-gradient, quasi-circular boundary at the center of the model and accumulate on the point mass at the origin. If the radius for the inner boundary passed through one of the inner zones, the zone's interior to that zone was set to be duplicate of the zone on the boundary. While the Cartesian nature of the axisymmetric coordinate system meant that the inner boundary was only as close to circular as one can reproduce with square components, it did not introduce a significant amount of error into the calculation. The inner boundaries were chosen to be within the sonic radius, ensuring that small numerical errors at the interior boundary did not accumulate and affect the flow of fluid upstream from the boundary. The sonic point moved outward, not inward, as the simulation time progressed, so the inner boundary was always within the sonic radius. The falling temperature caused the sound speed to decline with time, while the velocity near the inner boundary increased with time. The gravitational potential resulting from this point mass was updated and added to the gravitational forces computed by the multipole solver at each time step.

A module was also included that locally deposited energy from radioactive decay of ⁵⁶Ni to ⁵⁶Fe. The amount of energy

$$\begin{equation} dE_{{^{56}\mathrm{Ni}}} = \lambda _{{^{56}\mathrm{Ni}}} X_{{^{56}\mathrm{Ni}}} e^{-\lambda _{{^{56}\mathrm{Ni}}}*t}*q({^{56}\mathrm{Ni}}). \end{equation} \tag{ 3 }$$

The decay rate of ⁵⁶Ni, $\lambda _{{^{56}{\rm Ni}}}$, is 1.315 × 10⁻⁶ s⁻¹, and the amount of energy released per gram of decaying ⁵⁶Ni is q(⁵⁶Ni), for which we took the value 2.96 × 10¹⁶ erg g⁻¹. $X_{{^{56}{\rm Ni}}}$ is the fraction of ⁵⁶Ni in the block. The amount of ⁵⁶Co at a given time could be found as a function of the amount of initial ⁵⁶Ni by

$$\begin{equation} X_{{^{56}\mathrm{Co}}} = \frac{\lambda _{{^{56}\mathrm{Ni}}}}{\lambda _{{^{56}\mathrm{Co}}}-\lambda _{{^{56}\mathrm{Ni}}}} X_{{^{56}\mathrm{Ni}}} (e^{-\lambda _{{^{56}\mathrm{Ni}}}t}-e^{-\lambda _{{^{56}\mathrm{Co}}}*t}), \end{equation} \tag{ 4 }$$

so that the energy deposition rate from ⁵⁶Co as a function of time was given by

$$\begin{equation} dE_{{^{56}\mathrm{Co}}} = \frac{\lambda _{{^{56}\mathrm{Ni}}}}{\lambda _{{^{56}\mathrm{Co}}}-\lambda _{{^{56}\mathrm{Ni}}}} X_{{^{56}\mathrm{Ni}}} (e^{-\lambda _{{^{56}\mathrm{Ni}}} t}-e^{-\lambda _{{^{56}\mathrm{Co}}} t})) \lambda _{{^{56}\mathrm{Co}}}*q({^{56}\mathrm{Co}}). \end{equation} \tag{ 5 }$$

We assumed a decay rate for ⁵⁶Co, $\lambda _{{^{56}{\rm Co}}}$, of 1.042 × 10⁻⁷ s⁻¹, and an energy per gram of decaying ⁵⁶Co, q(⁵⁶Co), for which we took the value of 6.4 × 10¹⁶ erg g⁻¹.

The supernova models were evolved with KEPLER to the point where all explosive nuclear burning had ceased and the reverse shock had just begun to form. This occurred at 10³ s and 10⁴ s for 15 and 25 M_☉ stars of solar composition, and at 25 s and 100 s for 15 and 25 M_☉ stars of primordial composition, respectively. At these times, the one-dimensional models from KEPLER were mapped onto a two-dimensional axisymmetric grid and evolved forward in time with the FLASH code. A similar simulation of Model s25A was also performed using a progenitor evolved to 10³ s with KEPLER before being mapped to FLASH. No significant difference in the later evolution of s25A models evolved to 10³ s and 10⁴ s was observed. Only one quadrant of the star was carried in the calculation, enforcing symmetry about the left y- and bottom x-axes, while allowing material to leave the grid through a zero-gradient boundary at the right y- and top x-axes.

An enhanced flow was observed along the x- and y-axes. This flow was not large in comparison with the rest of the simulation, but it was present, as can be seen in Figures 6–7. This is a well-documented artifact of the dimensionally-split approach to solving the hydrodynamic equations. It did not substantially influence the evolution of the simulations presented here.

Perturbations arising from a Cartesian grid are also inevitable. In order to quantify these grid effects, we performed simulations of all stars with a random perturbation in velocity with a maximum amplitude of 0.5% and 2.0%. We also performed simulations with no additional perturbation. We found that perturbations of 2.0% in velocity had a clear effect on the initial scale of the Rayleigh–Taylor instability, increasing the amplitude and scale of the first instabilities to form. This effect is shown for s15A in Figure 1. The case of 0.5% random perturbations to the velocity results in a scale for the initial Rayleigh–Taylor instabilities that is roughly equivalent to the case where the only perturbations were those arising from the Cartesian grid itself. Other models had similar resolution, such that grid perturbations were roughly equivalent to velocity perturbations of 0.5%, and velocity perturbation of 2.0% had a noticeable effect on the initial scale of the instability. Perturbations arising from the grid were no larger than 2%, well within the regime of perturbations expected to arise from convection.

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Because the zD-series models were more compact than sA-series models, the reverse shock reached the centers of zD models faster than the sA models. This had the effect of shutting off mixing in the zD-series before the Rayleigh–Taylor instability had time to become fully nonlinear (see also Herant & Woosley 1994). The initial perturbations had a greater effect on the final state of the zD simulations, while the initial perturbation scale and spectrum are washed out in the sA models as a result of their longer mixing times. A simulation with random velocity perturbations of 5% was performed for Model z25D.

Models were initially mapped onto the two-dimensional grid such that their inner iron cores were resolved with at least four blocks of 16 zones each. This was sufficient to ensure that the rest of the star was accurately resolved. As the simulation progressed and the model stars expanded, the maximum refinement level of the simulation was turned down, so that the model star was always resolved at about the same percentage of the radius of its inner core of ⁴He and heavier isotopes. This is the region where Rayleigh–Taylor mixing takes place. The star expands homologously, ensuring that all regions will be adequately resolved.

The solar composition models were mapped onto a grid 5 × 10¹⁴ cm on a side. The zero-metallicity stars were mapped onto a grid 1.4 × 10¹⁴ cm on a side. The portion of the grid outside the original KEPLER model was initialized with a density proportional to r⁻². Simulations with an outer density proportional to r^−3.1 was also performed for Models s15A, s25A, and z25D. These density profiles span the realistic range of smooth density distributions outside real stars. No difference in the final profiles for density, temperature, pressure, or composition was found between models with different outer density profiles. The density profile of the surrounding material therefore has no effect on the amount of Rayleigh–Taylor mixing that goes on inside the star, provided very little mass as a proportion of the original mass of the star is swept up in the first days of the explosion. The amount of mass added to the grid from ambient density was <2% for all models.

Calculations were run at least until the Rayleigh–Taylor fingers had ceased to move with respect to the mass coordinate of the star. This happened 2 hr after core bounce for Model z15D, 4 hr after core bounce for z25D, and 7 days after core bounce for the s15A and s25A models. All models were followed to 10⁶ s, long after Rayleigh–Taylor mixing had frozen out in the zero-metallicity stars, but long enough that infall though the inner boundary had reached an asymptotic stage and the final remnant mass from these two-dimensional simulations could be determined.

Infall is as important as mixing for determining the final yield of a supernova. For a freely-expanding SNR, the rate at which mass accretes onto the black hole or neutron star at the center of the explosion is given by $\dot{M} \propto t^{-5/3}$ (Chevalier 1989). It takes supernova explosions on the order of 10⁶ s to reach this asymptotic, freely-expanding stage, at which time it is possible to determine the final mass of the stellar remnant by extrapolating from the asymptotic infall rate. Our final remnant masses are compared with those found with one-dimensional calculations carried out by Zhang et al. (2008) with the PANGU code in Figure 13.

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PANGU is a one-dimensional hydrodynamics code based on the second-order semi-discrete finite difference central scheme of Kurganov & Tadmor (2000). Time evolution was carried out by a third-order total variation diminishing Runge–Kutta method (Shu & Osher 1989). Zhang et al. (2008) simulated the explosions of stars in the Woosley & Heger (2007) and Heger & Woosley (2008) surveys. Using this one-dimensional code, the authors followed the evolution of the SNR out to 10⁶ s, to the time at which the accretion rate onto the central remnant had reached an asymptotic dependence on time and the final remnant mass could be determined.

Our final remnant masses showed good agreement with the PANGU results for all models except Model z25D, as shown in Figure 13. The shape of the infall curves for our stars, as shown in Figure 13, matched the PANGU results well, though the remnant masses from FLASH are larger except in the case of s25A. Model s25A was mapped to FLASH after 1 × 10⁴ s of evolution in KEPLER. Although this does not alter the final mixed state of the star, it does alter the amount of mass that accumulates at the inner boundary, since most of the infall occurs during the first 10⁴ s, when the star was still being evolved forward with the Lagrangian code KEPLER. Both 15 solar mass models are in good agreement with the PANGU results. z25D shows the most deviation, most likely because this model experienced the most fallback of any of the models studied here. We were unable to replicate the fine resolution employed by a one-dimensional code, and that is almost certainly the reason for the enhanced infall mass. The initial remnant mass in 2D was sufficiently larger than the 1D remnant mass that the additional force it exerted on the surrounding material was large enough to cause additional infall. This led to a larger remnant, causing a mild runaway effect. While the remnant mass obtained with FLASH for Model z25D is likely inaccurate, Rayleigh–Taylor mixing in this star has stopped by 2 × 10⁴ s, at which point the remnant mass in the FLASH simulation was not significantly larger than that in the PANGU simulation. It is unlikely that increased infall has had a significant impact on the evolution of the Rayleigh–Taylor instability, so the mixing results remain sound.

Random perturbations of amplitude 0.5% and 2% of the original velocity profile were added to the initial model for the solar composition models, as described in Section 2.2. Random perturbations of 2% were added to the zero-metal stars, and an additional simulation with an initial perturbation of 5% was performed for Model z25D. Perturbations of 0.5% produce an original size scale for the Rayleigh–Taylor instabilities that is about the same size as those arising for the nonperturbed case, while the 2% perturbations result in a larger initial scale for the instability, implying that grid perturbations for these models are between 0.5% and 2%, as shown in Figure 1. The Rayleigh–Taylor instabilities can grow for at least as long as the reverse shock takes to reach the origin. The instabilities in the solar-metallicity stars can grow for many e-folding times, a long enough time to wash out the initial scale and spectrum of the instabilities. The final states of the perturbed models appear essentially identical to the unperturbed models. The distribution of the isotopes in both velocity (Figures 14 and 15) and mass space (Figures 16 and 17) has no systematic correlation with the magnitude of the initial perturbation for solar-metallicity stars. The scale of the perturbations one would expect in a real star is set by the magnitude of the convective velocities, which are on the order of 0.5% of the total velocity.

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Perturbations for the zero-metallicity case have a greater effect on the final amount of mixing in these stars. Larger perturbations lead to a larger size scale for the initial Rayleigh–Taylor fingers, which allows them to grow more quickly before the reverse shock passes them, and the pressure gradient is no longer opposite to the density gradient. In these models, the Rayleigh–Taylor instabilities cannot grow for many e-folding times, and so the initial scale of the perturbations matters. This can be seen in Figures 18 and 19, which show the final distribution of isotopes as a function of enclosed mass for different amounts of perturbations. The final amount of mixing is set by the size scale of the initial perturbation. The amount of mixing determines the distribution of isotopes with velocity, as shown in Figures 20 and 21. ¹⁶O and ⁴He are the most affected. The distance in mass space over which these isotopes are mixed varied by around 1 M_☉ for the perturbed models.

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The velocity distribution of chemical isotopes is shown in Figure 22. ⁴⁴Ti was mixed to relatively high velocities in our solar-metallicity stars. The solar stars show ¹H and ⁴He at high velocities of 4 ×  10³ km s⁻¹. These isotopes are also mixed all the way to the core of the solar-metallicity stars, and some fraction of them reaches negative velocities of less than −0.5 × 10³ km s⁻¹. In Model s15A, where the most mixing takes place, we see ¹⁶O and ¹²C mixed out to 2 × 10³ km s⁻¹ in velocity space. Ni does not reach the high velocities observed in 1987A. Model s15A shows a peak in the ⁵⁶Ni velocity at ≈0 km s⁻¹, with a tail extending to 0.7 × 10³ km s⁻¹. Model s25A shows slightly higher velocities, with the Ni distribution peaking at around 0.4 × 10³ km s⁻¹ and reaching out to 0.9 × 10³ km s⁻¹. These higher velocities for Ni are probably due to smaller amounts of fallback in this star.

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The zero-metallicity stars show lower velocities overall. At the time they exploded as supernovae, these stars were more compact than solar composition stars. The supernova shock ran into a higher fraction of the total mass of the star at an earlier time. In neither zero-metal model were the heavier isotopes mixed out to higher velocities than 1 × 10³ km s⁻¹, but in Model z15D the velocities of ⁵⁶Ni and ⁴⁴Ti had slight positive components, whereas in Model z25D the velocity distribution of these isotopes was almost entirely negative. No ⁵⁶Ni or ⁴⁴Ti escaped from Model z25D.

Kifonidis et al. (2006) in their 1987A-type models, which had a metallicity and mass intermediate to the stars studied in this paper, saw a well-mixed heavy element core. The velocity profiles for isotopes from ¹⁶O to the iron group were very similar. Because we did not see the iron core of our solar-metallicity stars mixed to the degree seen in Kifonidis et al. (2006), our ⁵⁶Ni had a slower velocity distribution, while lighter isotopes, from ⁴⁴Ti and ²⁸Si and lighter, were skewed toward higher velocities.

The ⁵⁶Ni velocities observed in the zero-metallicity models were lower than those produced in earlier attempts at modeling SN 1987A, as well. The maximum iron-peak velocities obtained was on the order of 1300 km s⁻¹ (Arnett et al. 1989; Hachisu et al. 1990; Fryxell et al. 1991; Müller et al. 1991; Herant & Benz 1991). In particular, Herant & Benz (1991) found that following the radioactive decay of ⁵⁶Ni and ⁵⁶Co increased the velocity of iron-peak elements slightly, but not enough to match observations of SN 1987A. Our iron-peak velocities were lower still. This was partly due to the large amount of fallback experienced by the primordial composition models. Most of the Ni fell through the inner boundary of our zD-series simulations before it had time to power a nickel bubble like those seen in smoothed particle hydrodynamics (SPH) simulations of SN 1987A (Herant & Benz 1991). Our progenitors were also more compact, with smaller helium cores, than the progenitor models for 1987A used in the above simulations.

⁵⁶Ni was not mixed out to the high velocities seen in SN 1987A. Utrobin (2004) reported that ⁵⁶Ni was mixed out to at least 2.5 × 10³ km s⁻¹. In no model did we see nickel at these high velocities. Significant low-order asymmetry in the explosion is probably necessary to reach ⁵⁶Ni velocities high enough to match observations.

In their 1994 paper on mixing in supernovae with red-supergiant progenitors, which employed progenitor models similar to the s15A and s25A progenitor models in this survey, Herant and Woosley found a similar distribution of isotopes in velocity space to the results presented here. They found a peak in the velocity distribution of ¹⁶O at around 1.4 × 10³ km s⁻¹ for their 25 M_☉ star, we see one at around 1.1 × 10³ km s⁻¹. We see oxygen mixed out to slightly higher velocities in our model s15A than are given in Herant & Woosley (1994). Their models show no negative velocities, which is probably a result of the SPH technique employed, which did not allow for accretion onto a sink particle at the center of the simulation.

The yields of our stars are greatly affected by the amount of fallback and degree of mixing induced by Rayleigh–Taylor instabilities. For models that experienced a great deal of fallback, as in case of the zero-metallicity models, mixing is of crucial importance in determining the final yields. Figure 4 shows the original distribution of isotopes in the four stars studied in this paper. The distribution of isotopes after all mixing has ceased is shown in Figure 3. As Figures 3, 9 and 10 show, layers interior to oxygen in the zero-metallicity stars have not mixed appreciably. A far larger portion of the solar-metallicity stars has been mixed together. The Rayleigh–Taylor instability has penetrated as far as the ⁵⁶Ni core of model s15A, completely disrupting the exterior shells. This can be seen in Figure 11. Figure 12 also shows that Model s25A experienced more mixing than Model z25D (Figure 10), but not quite as much as Model s15A. The lack of mixing into the interior layers of the zero-metallicity models means that very little of the silicon shell, and virtually none of the elements interior to this shell are mixed out. This, coupled with a large amount of infall, means that these Population III stars expel almost no ⁵⁶Ni.

To eliminate the possibility that the absence of Rayleigh–Taylor induced mixing at the edge of the ⁵⁶Ni shell in Models z15D and z25D could have been due to mapping to two dimensions at too late a time we performed four simulations of the interior of Model z25D with an inner boundary of 1 × 10⁹ cm, an order of magnitude smaller than that used to simulate the star in the main simulations presented in this paper. These simulations were run at ten times the resolution of the simulations presented in Section 2.4. KEPLER models were mapped in 10 and 40 s after bounce. To maintain consistency with the simulations that used a mapping from KEPLER at 100 s after bounce, we initially used a reflecting condition at the inner boundary, which did not allow infall. At 100 s after bounce for all of these high-resolution models, the inner boundary condition was changed to be zero gradient. No Rayleigh–Taylor instabilities are seen to develop by 1 × 10³ s of simulation time, the point when the instabilities are clearly defined and growing along the C–O/He interface. Perhaps the perturbations from the Cartesian grid at this resolution are too small to effectively seed the instability. We perturbed the velocity on the grid with an (l = 8)-order Legendre polynomial with an amplitude of 10% of the initial velocity. While this is probably unphysical, it represents a limiting case. The absence of Rayleigh–Taylor-induced mixing with a perturbation this extreme implies that its absence in the larger, lower-resolution simulations is reasonable. The first 1 × 10³ s of the simulation were run at this high resolution. By this point, the Rayleigh–Taylor instabilities are clearly visible at the C–O/He boundary, but there is no hint of the development of this instability at the interface between the ⁵⁶Ni core and the overlying layers. Kifonidis et al. (2006) speculated that the reason they saw mixing at the Si/O boundary when earlier studies did not was the "ad hoc" initialization of these explosions with a piston or thermal bomb, insufficient resolution, or differences in structure between their progenitor models and those employed in previous studies. Insufficient resolution does not appear to be the case here, and our extreme perturbations imply that seeding perturbations through neutrino-driven convection by starting the simulation earlier would have little effect. We can only conclude that this is a valid result arising from the structure of our progenitor star, or a result of the piston explosion mechanism.

Because very little mixing occurs in the layers interior to the mass cut of these zero-metallicity stars, their yields were very sensitive to the amount of fallback that occurs. Figure 23 shows this sensitivity. When the mass of infalling material was increased by 10% of the fiducial value taken from Zhang et al. (2008), almost none of the elements interior to the C–O shell escaped. When the mass of infalling material was decreased by 10% of this fiducial value, more of the elements interior to oxygen escaped and went on to enrich the surrounding gas. Figure 23 shows a comparison of the yields of our zero-metallicity stars with observations of the three most metal-poor stars yet found in the halo of the Milky Way. [X/H] values for all elements were obtained by diluting the yields of our model stars with enough pristine gas to fit observations of [C/H] and [O/H] to individual halo stars. The dotted and dashed lines show the yields resulting from increasing or decreasing the amount of fallback by 10% of its fiducial value, respectively. Model z15D is more sensitive than Model z25D to decreasing the mass cut. The model yields appear much closer to fitting the values of elements heavier than oxygen observed in the metal-poor halo stars.

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The yields of our Population III core collapse supernovae reproduce the extreme overabundance of [C,O/Fe] to the point where Fe is underproduced even when compared to the already low observations of [C,O/Fe] in HMP stars. Carbon is slightly underproduced, and nitrogen is underproduced by 2–3 dex relative to oxygen, as is the iron peak. The iron-peak elements are underproduced because they do not mix sufficiently with the lighter elements in the star and cannot escape falling back onto the remnant at the center. A reduction in the infall mass brings z15D closer to reproducing observations. Two-dimensional versions of the one-dimensional simulations presented in the higher energy explosions of Heger & Woosley (2008) and Nomoto et al. (2007) might eject more of the Ni produced in the more energetic explosions. Rotating models (Hirschi et al. 2008) create more primary nitrogen, which can lead to an increase in the rate of CNO burning at the base of the hydrogen shell, causing some models die as larger red supergiants rather than a compact blue supergiants. In this case, Rayleigh–Taylor mixing would play out in a similar way to the solar models presented in this paper, and more iron would be ejected by these stars. Recent simulations (Scheck et al. 2004, 2006; Burrows et al. 2007a, 2007b, 2007c) point out that the supernova explosion mechanism is probably inherently multidimensional and asymmetric. Asymmetry in the explosion, whether in the form of a jet or a perturbation described by Legendre polynomials of order of l = 1 or l = 2, might also mix more of the nickel core out of the star, bringing the models closer to reproducing observations. Venn & Lambert (2008) have suggested that HMP stars may be "chemically peculiar" stars, in which low iron abundance is caused by separation of gas and dust beyond the stellar surface, followed by accretion of dust-depleted gas. If this is the case—and the authors note that a definitive answer requires additional information—the stars' true metallicity is closer to [X/H] ≈-2 rather than −5.

The supernova light curve is affected by the amount of ⁵⁶Ni in the center of the star that falls back onto the black hole at the center of the explosion. The models for the first supernovae presented in this work are intrinsically dimmer than corresponding supernovae arising from the stars of solar composition, provided they explode with the same amount of energy. In our models of primordial composition supernovae, all or nearly all of the Ni synthesized in the supernova falls back onto the remnant left behind at the center of the explosion. Energy from the radioactive decay of Ni powers the tail of core-collapse supernova light curves. When the energy released in its radioactive decay to ⁵⁶Fe is no longer observable, the supernova light curves will lose their radioactive tails, making them briefer and dimmer than ordinary core-collapse supernova light curves.