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The metal and dust yields of the first massive stars
Stefania Marassi1, Raffaella Schneider1, Marco Limongi1,3, Alessandro Chieffi2, Marco Bocchio1,4, Simone Bianchi4
1INAF/Osservatorio Astronomico di Roma, Via di Frascati 33, 00040 Monteporzio, Italy
2INAF/IASF, Via Fosso del Cavaliere 100, 00133 Roma, Italy
3Kavli Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study,
The University of Tokyo, Kashiwa 277-8583, Japan
4INAF/Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze, Italy E-mail: stefania.marassi@oa-roma.inaf.it
We quantify the role of Population (Pop) III core-collapse supernovae (SNe) as the first cosmic dust polluters. Starting from a homogeneous set of stellar progenitors with masses in the range [13−80]M⊙, we find that the mass and composition of newly formed dust depend on the mixing efficiency of the ejecta and the degree of fallback experienced during the explosion.
For standard Pop III SNe, whose explosions are calibrated to reproduce the average elemental abundances of Galactic halo stars with [Fe/H]<−2.5, between 0.18 and 3.1 M⊙ (0.39−1.76M⊙) of dust can form in uniformly mixed (unmixed) ejecta, and the dominant grain species are silicates. We also investigate dust formation in the ejecta of faint Pop III SN, where the ejecta experience a strong fallback. By examining a set of models, tailored to minimize the scatter with the abundances of carbon-enhanced Galactic halo stars with [Fe/H]<−4, we find that amorphous carbon is the only grain species that forms, with masses in the range 2.7×10−3−0.27M⊙ (7.5×10−4−0.11M⊙) for uniformly mixed (unmixed) ejecta models.
Finally, for all the models we estimate the amount and composition of dust that survives the passage of the reverse shock, and find that, depending on circumstellar medium densities, between 3 and 50% (10 - 80%) of dust produced by standard (faint) Pop III SNe can contribute to early dust enrichment.
stars:low-mass, supernovae: general, ISM: cloud, dust, Galaxy: halo, galaxies:evolution
††pagerange: The metal and dust yields of the first massive stars–Appendix: Final dust grids and tables††pubyear: 2015
Dust grains play a fundamental role in the evolution of stellar populations at high redshift. Population III stars (hereafter Pop III), so far undetected, are responsible of the first chemical enrichment of the high-redshift interstellar medium (ISM). Given the current theoretical limits on the mass of the first stars (Hosokawa et al. 2011; Hosokawa et al. 2012), early metal enrichment is likely to be mostly driven by the first core-collapse supernovae (SNe), with a possible contribution from more massive pair-instability supernovae (although this may strongly depend on the poorly constrained tail of the stellar initial mass function, Hirano et al. 2014, 2015; Susa et al. 2014).
Yet, the amount and properties of grains that can be injected in the high-redshift ISM and contribute to the enrichment depend on the dust condensation efficiencies in SN ejecta and on the destruction suffered by thermal and non-thermal sputtering during the passage of the reverse shock of the SN, on timescales of 104yr (Bianchi & Schneider 2007; Nozawa et al. 2007). Depending on the density of the circumstellar medium where the explosion takes place, the mass fraction of newly formed dust that is able to survive ranges between 2% and 20% for circumstellar medium densities in the range 10−25<ρ/(gr/cm3)<10−23, which corresponds to number densities in the range 0.06<nISM/(cm−3)<6 (Bianchi & Schneider 2007; Nozawa et al. 2007; Silvia et al. 2010; Silvia et al. 2012; Marassi et al. 2014). In addition, the passage of the reverse shock significantly alters the grain size distribution and modifies the grain cross section, changing the dust cooling efficiency (Schneider & Omukai 2010).
Dust formation in SN ejecta has been investigated following two different methods: classical nucleation theory (CNT) and a chemical kinetic approach (Cherchneff & Dwek 2009; Cherchneff & Dwek 2010; Sarangi & Cherchneff 2013). Theoretical models, developed in the framework of CNT, have shown that dust formation can take place in SN ejecta a few hundred days after the explosions and provide predictions on the mass, composition and size distribution of the newly formed grains (Kozasa et al. 1989, 1991; Todini & Ferrara 2001; Nozawa et al. 2003, 2008, 2010, 2011; Schneider et al. 2004; Bianchi & Schneider 2007). These models predict dust masses ≈[0.2−0.6]M⊙ for SN progenitors with masses [12−40]M⊙ and metallicities [0.1−1]Z⊙ (Schneider et al. 2014), in agreement with the dust mass inferred from recent Herschel and ALMA observations of SN 1987A (Matsuura et al. 2011; Indebetouw et al. 2014) and Cas A (Barlow et al. 2010). Although in the past the applicability of CNT in astrophysical context has been questioned (Donn & Nuth 1985; Cherchneff & Dwek 2009), mainly due to the assumption of chemical equilibrium at nucleation, recently Paquette & Nuth (2011) showed that this assumption has a lower impact on the grain mass and size distribution than previously thought. Similarly, Nozawa & Kozasa (2013) demonstrated that CNT is a good approximation in SN ejecta, at least until the collisional timescales of the key molecule is much smaller than the timescale with which the supersaturation ratio increases.
The goal of the present study is to investigate the role of Pop III core-collapse SN as dust polluters, adopting a homogeneous set of metal-free progenitors with masses in the range [13−80]M⊙. Under the assumption that the observed metal poor stars likely formed from gas clouds enriched by Pop III SNe, we use properly calibrated SN explosion models (Chieffi & Limongi 2002) to calculate the formation of dust in the ejecta, applying CNT and accounting for the process of grain growth. We quantify the impact of ejecta mixing on the final dust masses and grain size distributions, analyzing both uniformly mixed models within the helium core and unmixed/stratified cases. To obtain a more realistic estimate of the dust mass that is able to enrich the ISM, we also consider the effects of the reverse shock (Bianchi & Schneider 2007).
Motivated by the observed surface elemental abundances of Carbon Enhanced Metal Poor (CEMP) stars, i.e. stars with an observed overabundance of light elements compared to Fe and - in particular - with [C/Fe]>1 (Beers & Christlieb 2005), we also investigate dust formation in the ejecta of faint Pop III SN, where the ejecta experience mixing and fallback (Umeda & Nomoto 2002; Umeda & Nomoto 2003). Recently, Marassi et al. (2014) showed that dust can be produced in faint Pop III SN ejecta (see Kochanek 2014 for dust formation in solar metallicity faint SNe.)
Here, we extend this previous analysis and we simulate a large set of faint Pop III SN models, searching for a combination of mixing and fallback that provides the best-fit to the observed abundance pattern of all currently known C-enhanced hyper-iron-poor stars (Beers & Christlieb 2005; Yong et al. 2013; Keller et al. 2014; Ishigaki et al. 2014). For these faint SN models we also explore the impact, on the final dust masses, of ejecta mixing and reverse shock.
The plan of the paper is as follows. In Section 2 we briefly summarize the Bianchi & Schneider (2007) dust nucleation model and we describe the upgrated molecular network. In Section 3 we illustrate the main features of Pop III SN progenitors that are modeled using FRANEC stellar evolutionary code (Limongi & Chieffi 2006; Limongi & Chieffi 2012) and their calibration. In Section 4 we present the resulting dust yields for Pop III core-collapse SNe. In Section 5 we describe how we construct, using the mixing and fallback procedure (Umeda & Nomoto 2002), faint Pop III ejecta calibrated with the same procedure described in Marassi et al. (2014). In Section 6 we show the results on dust yields obtained in Pop III faint SN ejecta. In Section 7 we discuss all the results and their dependence on the fallback, ejecta mixing, and reverse shock and we draw our conclusions. In the Appendix we present our final dust data grids that will be available to the community.111The resulting dust and metal yields will be available in electronic format for interested researchers upon request.
2 Dust formation model
To model dust formation in SN ejecta we follow the Bianchi & Schneider (2007) model, where classical nucleation theory in steady state conditions was applied (see Nozawa et al. 2003; Bianchi & Schneider 2007 and references therein). For our calculation, we use a previously developed code that has been applied to core-collapse (Todini & Ferrara 2001; Bianchi & Schneider 2007) and pair-instability SNe (Schneider et al. 2004). This theoretical model has allowed to successfully reproduce the dust masses observed in SNe and young SN remnants (Schneider et al. 2014; Valiante & Schneider 2014). In Bianchi & Schneider (2007) the chemistry of molecular formation is implemented following Todini & Ferrara (2001), but relaxing the assumption of steady state. In the gas-phase, the formation of carbon oxide (CO) and silicon oxide (SiO) was assumed to be driven by radiative association reactions and destroyed by Compton electrons coming from radioactive decay of 56Co. Seven different grain species are formed in SN ejecta: amorphous carbon (AC), iron, corundum (Al2O3), magnetite (Fe3O4), enstatite (MgSiO3), forsterite (Mg2SiO4) and quarz (SiO2). In this work, we follow the formation of the above grain species assuming that seed clusters are formed by a minimum of two monomers, which subsequently grow by accretion of other monomers. The accretion process is regulated by the collisional rate of the key species and depends on the sticking coefficient (defined as the probability that an atom colliding with a grain will stick to it) which we have assumed equal to 1 for all grain species. A discussion of the dependence of the dust yields on these two parameters can be found in Bianchi & Schneider (2007), where is also possible to find a description of the adopted properties of SN dust species to which we refer for more details.
RATE COEFFICIENT
C+O→CO+hν kRA1=1.58×10−17(T/300)0.3exp(−1297.4/T) Dalgarno et al. (1990)
Si+O→SiO+hν kRA2=5.52×10−18(T/300)0.3 UMIST
C+C→C2+hν kRA3=4.36×10−18(T/300)0.3exp(−161.3/T) Andreazza & Sigh (1987)
O+O→O2+hν see text Babb & Dalgarno (1994)
C+O2→CO+O kNN1=5.56×10−11(T/300)0.4exp(26.90/T) UMIST
O+C2→CO+C kNN2=2.0×10−10(T/300)−0.1 UMIST
C+CO→C2+O kNN3=2.94×10−11(T/300)0.5exp(−58025/T) UMIST
O+CO→O2+C see text -
Si+CO→SiO+C kNN6=1.30×10−9exp(−34513/T) UMIST
C+SiO→CO+Si kNN8=1.00×10−16 UMIST
Table 1: Molecular processes considered in this work: the rate coefficients are taken from the UMIST database for astrochemistry 2012; where the origin of the reaction rates is different the corresponding reference is indicated.
Wi [eV]
CO+e−→C+O+e− Wi,CO=125
SiO+e−→Si+O+e− Wi,SiO=110
O2+e−→O+O+e− Wi,O2=125
C2+e−→C+C+e− Wi,C2=125
Table 2: Compton Electron destruction reactions and the corresponding mean energy per ion pair Wi.
2.1 Upgrated Molecular Network
It is well known that CO and SiO molecules play a fundamental role in the dust formation process: CO formation subtracts C-atoms and limits the formation of AC grains. SiO is required to form silicates, such as Mg2SiO4 and MgSiO3. Due to the important role of molecules in the dust formation pathway, we have enlarged our molecular network taking into account other formation/destruction processes involving CO/SiO molecules and their interactions with O2 and C2. These molecular processes are crucial in subtracting gas-phase elements from the ejecta (in particular oxygen and carbon, which are very abundant, see left panel of Fig. 2), that otherwise are free to form dust. We follow the evolution of CO, SiO, O2 and C2 which form through different channels (reaction rates), respectively: (i) radiative association reactions where the formation of molecules takes place through the emission of a photon which carries off the excess energy released during the formation process; (ii) bimolecular, neutral-neutral reactions that involve molecules and atoms. For these two-body reactions, the formation rates of molecules k(T) are given by the usual Arrhenius-type expression,
k(T)=α[T300K]βexp[−γT]cm3s−1 (1)
where T is the temperature of the ejecta gas in K and γ the activation energy in K. In Table 1 we report the rates expressed in the Arrhenius form according to the UMIST database for astrochemistry 2012222http://udfa.ajmarkwick.net. This database is a compilation of molecular rates that have different origin, some are theoretically calculated, others are directly measured in the laboratory. Clearly, there is a degree of uncertainty related to the rate calculations: in some cases we decided to refer to other rate estimates present in the literature that are more robust. This is the case for the radiative association rate coefficient of O2 (RA4), for which we have chosen to fit the expression of the rate coefficient, as a function of temperature, obtained from theoretical calculations in Babb & Dalgarno (1994). In addition, we found that the neutral backward reaction NN4 is negligible (K. Omukai, private communication).
In standard nucleation theory, dust condensation is described in terms of a nucleation current that depends on the abundance of the key species. Hence, contrary to other studies which adopt a chemical kinetic approach (Cherchneff & Dwek 2009, 2010; Sarangi & Cherchneff 2015), we do not follow the formation of carbon chains as a pathway to solid carbon clusters. Thermal fragmentation of the chains through collisions with gas particles, in addition to oxydation reactions similar to NN2, may limit the formation of carbon chains. However, if all C2 were to contribute to carbon dust formation, the estimated carbon dust mass would need to be corrected upward by MC2.
We take into account the destruction due to Compton electrons coming from the 56Ni decay in the ejecta. As observed in SN 1987A, destruction by Compton electrons has a deep impact on the ejecta chemistry. As shown by Woosley et al. (1989), the explosion of SN 1987A has produced 56Ni which decays in ∼6 days into 56Co; the subsequent decay of 56Co into 56Fe deposits energy as γ rays in the SN ejecta, powering the observed light curve. In SN 1987A, the emitted light curve is very well reproduced if 0.075M⊙ mass of 56Co was ejected during the SN explosion.
Here we assume molecules to be destroyed by the impact with energetic electrons produced by the radioactive decay of 56Co, with a rate coefficient kd that depends also on the mean energy per ion pair Wi. According to Woosley et al. (1989), this rate coefficient can be estimated as follows: the thermalized γ-ray energy input rate for a given 56Co mass is given by,
Lγ=9.54×1041(M56Co0.075M⊙)fγ(k56)e−t/τ56erg/s, (2)
where 0.075M⊙ was the adopted mass of 56Co produced in the original Woosley et al. (1989) model, τ56=111.26 days is the e-folding time of 56Co decay, and the function fγ is the deposition function and it is given by one minus the fraction of energy that escapes in photons in the X-ray and γ-ray bands,
fγ(k56)=1−e[−k56ϕ0(t0/t)2], (3)
where ϕ0=ϕ(t0) represents the column depth of the SN at some fiducial time t0 (for t0=106s ϕ0=7×104gr/cm2) and k56=0.033 cm2 gr−1 is an average opacity to γ rays from 56Co decay. The above quoted values of k56 and ϕ0 have been derived by Woosley et al. (1989) for SN 1987A and we will assume these to hold for all the explored SN progenitors. In particular, we assume that once a γ-ray Compton scatters with electrons, it is completely absorbed. Thus, the Lγ can be considered as the electron energy input and the energy transferred to a single gas particle per unit time can be computed as,
Le=Lγ/Npart (4)
where Npart is the number of gas particles in the ejecta. To compute the destruction rate, kd, it is necessary to divide Le by the mean energy per dissociation, Wi. For example, to compute the destruction rate of neutral CO we divide Le by the mean energy per dissociation Wi,CO, obtaining
kd(CO)=Le/Wis−1. (5)
Finally, we assume that all the radioactive energy is deposited uniformly in the ejecta. As it will be clear in what follows, the efficiency of molecule formation/destruction processes depends on the chemical composition and on the thermodynamics of the ejecta. In Section 4, we give a detailed description of the relevant processes for some selected SN models.
Figure 1: Pre-supernova chemical structure for three selected models: 15M⊙ (left panel), 30M⊙ (central panel), 50M⊙ (right panel). The shaded region extends up to the mass coordinate of the He core. Different shaded regions indicate the layers that we will consider in the unmixed SN models (layers A, B, and C from left to right). Colour versions of the figures are available online.
3 PopIII SN progenitors models
Figure 2: Left panel: initial metal abundances in the ejecta of the adopted SN models as a function of the progenitor stellar mass (the abundance of Al has been multiplied by 102). Right panel: mass of dust grains, before the passage of the reverse shock, as a function of the progenitor mass. SIL is the total mass in silicates, including Mg2SiO4, MgSiO3 and SiO2.
Figure 3: Temperature (left panel) and number density (right panel) evolution for 15M⊙ (solid), 30M⊙ (dotted), 50M⊙ (dashed) ejecta models.
The presupernova models adopted in this paper are the ones presented and discussed in details in Limongi & Chieffi (2012). These models span a range of mass between 13 and 80 M⊙ and have a pristine Big Bang initial composition. The evolution of these models has been followed from the pre-main sequence up to the onset of the iron core collapse by means of the FRANEC stellar evolutionary code (Limongi & Chieffi 2006). The explosion of the mantle and the consequent explosive nucleosynthesis have been computed in the framework of the "artificially induced explosion". Then, for each model, the mass cut (Mcut), i.e. the mass coordinate which separates the final remnant from the ejected portion of the mantle, has been fixed by requiring a best fit to the element abundance pattern of the Cayrel average star, as extensively described in Limongi & Chieffi (2012). In Table 3 we summarize the main properties of the SN explosion models.
We construct the Pop III ejecta models, requiring that the thermal, dynamical and chemical evolution of the ejecta evolve consistently with the explosive nucleosynthesis simulation (see next section for details). We have performed dust formation calculations assuming uniform mixing of the elemental abundances in the He core. However, due to the uncertainties related to the efficiency of mixing in metal-free SNe (Joggerst et al. 2009, Heger & Woosley 2010), for some models we also present the results obtained assuming unmixed ejecta. In Fig. 1 we show the chemical structure (elemental mass fraction) as a function of the mass coordinate for three selected pre-supernova models, 15M⊙,30M⊙,and50M⊙ (for a detailed description of the differences emerging in the convective zones we refer the reader to Limongi & Chieffi 2012). The shaded region extends up to the mass-coordinate of the He core, different colours indicate the layers that we will consider in the unmixed SN models, in Section 4. The left boundary of the innermost shaded region corresponds to the adopted mass-cut coordinate. In the left panel of Fig. 2 we show, for each SN model, the initial abundance of metal species which participate to molecules and dust formation; for all the SN models, the ejecta are very rich in carbon and oxygen and the total mass of metals is an increasing function of the progenitor mass. The ejecta mass Meje, as expected, is an increasing function of the progenitor mass and explosion energy (see Table 3).
4 Results: PopIII SN dust yields
Figure 4: Time evolution of molecular (dashed lines) and dust masses (solid lines) for three metal-free SN progenitors: 15 M⊙ (left panel), 30 M⊙ (middle panel), 50 M⊙ (right panel).
This section presents the calculated dust yields that we obtain for Pop III core-collapse SN models. As stated in the previous section, we construct the ejecta models using as initial conditions the thermo-dynamical properties obtained by the SN explosion simulation outputs (Limongi & Chieffi 2012). The ejecta follow an adiabatic expansion and the temperature evolution is given by,
T=T0[1+vejeR0(t−tini)]3(1−γ)whereveje=√10Eexpl3Meje (6)
is the ejecta expansion velocity, γ=1.41 is the adiabatic index, and T0 and R0 are the temperature and radius of the He core at the initial time t=tini. This initial time tini is fixed by requiring that the gas temperature at the radius of the He core, RHecore reaches a temperature of T0=104 K. For all uniformly mixed SN ejecta models (labelled with the progenitor mass) Table 3 reports the thermodynamical properties, the metal yields of the key elements in the nucleation process, the total amount of dust, the mass of molecules and the mass of dust in each grain species. In what follows, we discuss these results in details.
Dust formation in uniformly mixed ejecta
Figure 5: Time evolution of molecular (dashed lines) and dust (solid lines) masses for the 30M⊙ unmixed model: layer A (left panel), layer B (middle panel), layer C (right panel). Colour versions of the figures are available online.
In the right panel of Fig. 2 we plot the mass of dust for different grain species as a function of the progenitor mass. We find that: (i) the total dust mass produced increases with the progenitor mass, ranging between [0.18−3.1] M⊙, and it is dominated by silicates; (ii) the second most abundant dust species is magnetite, due to the initially high iron abundance present in the ejecta (see left panel of Fig. 2) plus the iron produced by 56Ni decay; (iii) amorphous carbon forms only in less massive models, with progenitor masses ≤30M⊙. This is due to the larger mean ejecta density of more massive progenitors, which increases the rates of the major processes leading to the formation of CO molecules, locking carbon atoms and preventing the formation of AC grains (we will return to this point later); (iv) alumina is the less abundant dust species, reflecting the lower abundance of Al in the ejecta.
Using three representative SN progenitors with masses 1530 and 50M⊙, Fig. 3 shows the evolution of the ejecta temperature and number density: when the He core of the three SN progenitors reaches T0=104K, we fixed the initial time tini, which varies between (1.78 - 3.96)×106s (see Table 3). At these initial times, the three ejecta models have radii of 2.44×1014(15 M⊙), 4.68×1014(30 M⊙), and 2.51×1014(50 M⊙) cm, and start the adiabatic expansion with velocities in the range (2960-3086)km s−1 (see eq. 6), and the temperature decreases. As shown in the right panel of Fig. 3, the initial number densities, n0= 7.18×1011(15 M⊙), 7.39×1011(30 M⊙), 1.24×1013(50 M⊙) cm−3, also decreases, showing a step-decrease corresponding to the condensation of the gas-phase elements.
In standard nucleation theory, dust grains condense when the gas becomes supersaturated. For each grain species, this depends on the temperature, density and abundances of the corresponding key element in the ejecta. Fig. 4 shows that grains form between 45 and 120 days after the explosions. Although these timescales are smaller than the values found by Nozawa et al. (2003) for core-collapse Pop III SN with a 25 M⊙ progenitor, due to the different thermal evolution of the ejecta, the resulting dust masses and grain size distribution are consistent because grain condensation occurs in similar physical conditions.
Molecules form according to the formation/destruction rates described in Section 2.1 and are also partially destroyed by the interaction with Compton electrons produced by destruction reactions rates D1−D4 (see Table 2), up to the point where the temperature reaches the condensation window and dust grains start to form. The ejecta density is a strong function of the initial progenitor mass: when the ejecta temperature enters the condensation regime, at around T ∼2000 K, the corresponding density can vary by one order of magnitude, although the dependence on the progenitor mass is not monotonic. Clearly, the chemical evolution of the ejecta depends on the temperature and density at each given time: in Fig. 4 we show, for the same set of SN models, the time dependence of the mass of CO, SiO and of the newly formed dust species.
We start analizing the 15M⊙ model plotted in the left panel of Fig. 4: CO and SiO start forming at around 27 days after the explosion, when T∼5400K. At this time, the CO formation rate is dominated by the neutral process NN2 that rapidly depletes all the available C2. SiO molecules form through the neutral channel NN5 and radiative association process RA2. At around 44 days, when T∼2000K, the impact with Compton electrons (D1) dominates over the other rates and partially destroys CO molecules. The free carbon atoms form AC, which is the first dust species to form, due to its higher condensation temperature. When the temperature decreases to ∼1764K, alumina grains form, which rapidly deplete aluminum from the ejecta. When T drops to ∼1530 K, Mg2SiO4 starts forming - at the same time of MgSiO3- but Mg2SiO4reaches supersaturation first (because it has the largest nucleation current), rapidly consuming Mg and SiO and inhibiting the growth of MgSiO3. Finally, when T∼1400 K, Fe3O4 and SiO2 forms, depleting completely the remaining SiO and iron from the ejecta.
The central panel of Fig. 4 shows the evolution of molecules and dust grains for the 30M⊙ model: at ∼57 days, when 5400≤T≤4250, the dominant process is radiative association of C2 which - due to the large oxygen abundance - is then rapidly depleted in CO through the reaction O+C2→CO+C. When T decreases, at around 66 days, the dominant CO formation process becomes radiative association RA1. Around 93 days, the temperature reaches ∼2000K, nucleation starts, D1 efficiently destroys the CO molecule, leaving C atoms free to form AC and - as in the 15M⊙ model - the CO formation processes are not equally efficient in re-forming the molecule. As a result, AC grains form.
For the more massive model the evolution changes, as shown in the right panel of Fig. 4. The interplay between the larger ejecta density and the richer oxygen content (the mass of oxygen is 18% of the mass of the ejecta) leads to the efficient formation of O2 molecules through radiative association RA4 and also activates efficiently the bimolecular process NN1, which exceeds the destruction rate D1, and becomes the leading process in CO formation. This depletes all the C atoms, inhibiting AC grains condensation. As for the two previous models, at 5400≤T≤5000 the main CO formation channel is O+C2→CO+C. When T decreases, it becomes C+O2→CO+O and it remains so for the subsequent evolution. At 64 days, the nucleation process starts, the grain condensation proceeeds as described for the two previous models, because the condensation sequence of the grain species reflects the corresponding condensation temperatures (Todini & Ferrara 2001; Schneider et al. 2004).
We summarize our results on mixed ejecta models in Table 3. In general, the total mass of CO molecules ranges between [9.60×10−3−5.66]M⊙, increasing with the mass of the ejecta due to the initial abundances of oxygen and carbon. SiO molecules form efficiently through NN5 (at early times) and through RA2, but it is completely depleted in silicates at the end of the nucleation process. C2 molecules are consumed at very early time in the formation process of CO, through the channel NN2. For models less massive than 30M⊙, O2 is under-produced (and consumed rapidly in CO formation process), with respect to more massive models. When M>30M⊙ the higher mean ejecta density and oxygen abundance cause radiative association rate RA4 to become the dominant process. This enables a very efficient NN1 reaction rate which overcomes the destruction rate D1, locking all carbon atoms in CO molecules.
Dust formation in umixed ejecta
The mixing efficiency in SN events is still debated, due to complexity of physical processes associated to core-collapse, such as asphericity and growth of Rayleigh-Taylor instabilities which accompany shock propagation. Besides, these effects are multi-dimensional and this adds an extra source of uncertainty related to emerging differences in 1D-3D numerical simulations (see Joggerst et al. 2009, Heger & Woosley 2010 and references therein). In particular, the occurence and the growth of Rayleigh-Taylor instabilities depends on stellar mass and metallicity. It has been shown that the more compact is the star, the more rapid is the reverse shock propagation, giving less time to these hydrodynamical instabilities to grow (Joggerst et al. 2009). For this reason, we have decided to investigate the impact of ejecta mixing on dust formation and we have considered stratified models, as the limiting cases when Rayleigh-Taylor instabilities do not grow. The unmixed case is based on the hypothesis that, due to inefficient mixing, the elemental abundances reflects the pre-supernova stratified composition: this means that more internal elements, such as magnesium and silicon are not present or present in very little amount in the more external layers of the star, reducing the probability to form silicates.
Figure 6: Mass of dust grains, before the passage of the reverse shock, as a function of the progenitor mass for the three selected unmixed models. SIL is the total mass in silicates, including Mg2SiO4, MgSiO3 and SiO2.
In Fig. 1, for each progenitor model, the shaded regions represent the layers that we consider in the stratified models, which extend from the mass cut coordinate to the outer radius of the He core (RHecore). The number and extent of the layers have been selected based on the criterium of constant abudances of the relevant elemental species within each layer. Hence, the outer radius of layer A corresponds to the mass coordinate where the Si abudance rapidly drops, and the outer radius of layer B is where the abundance of C (Ne) rapidly increases (decreases).
For the 15,30 and 50M⊙ models, we report in Tables 4, 5 and 6 the thermodynamical properties, the initial elemental composition, the masses of molecules and dust for all the layers. The time evolution of the molecular and dust masses for each of the three layers of the 30M⊙ progenitor is shown in Fig. 5. In the stratified ejecta, due to the different temperature evolution of the layers, CO starts to form earlier in the evolution with respect to the corresponding mixed case, and the same is true for SiO, where present. It is worth to note that, compared to the mixed case where O2 is completely depleted in CO, in layers A and B the O2 molecule forms, while AC forms only in layer C. Silicates and alumina form in the two internal layers A and B where Al, Si and Mg are present. Magnetite forms only in layer A where iron is present. The total mass of dust for the 30M⊙ progenitor is 1.2M⊙, about 90% of the mass formed in the mixed case.
Fig. 6 summarizes the results obtained for unmixed ejecta models. The mass of dust for different grain species is shown as a function of the progenitor mass. We find that the total dust mass ranges between [0.39−1.76] M⊙, and it is dominated by silicates. Contrary to the fully mixed cases, for the 15M⊙ and 50M⊙ SN progenitor models, solid iron is able to form due to the initially high iron abundance present in the most internal layer of the ejecta plus the iron produced by 56Ni decay. As a result, in these models the total dust mass formed is ∼3−10% larger than in the mixed case. The CO mass, differently from the mixed case, is not increasing with the progenitor mass, but varies depending on rate formation efficiency which is a function of temperature, mean density and composition of each layer.
Dust destruction by the reverse shock
Figure 7: Left panel: the mass of dust at the end of nucleation, and after the passage of a reverse shock of increasing intensity for all the mixed SN models considered in the present study. From top to bottom: no reverse shock models (no-rev), models with a circumstellar medium density of nISM=0.06 (rev1), 0.6,(rev2), and 6cm−3 (rev3). Right panel: size distribution function of the grains for a 30M⊙ progenitor model before (no-rev, left) and after the passage of the reverse shock (rev1, right).
To describe the impact of the reverse shock on the dust grains formed inside the expanding ejecta and to estimate the surviving dust mass, we follow the approach described in Bianchi & Schneider (2007) for which we summarize the key points: (i) the dynamics of the reverse shock is treated using Truelove & McKee analytic approximations (Truelove & Mckee 1999) which follows the forward and reverse shock evolution as a function of the main ejecta parameters, such as kinetic energy, Eexpl, ejecta mass, Meje, and density of the ISM, ρISM; (ii) the distribution of dust grains in the ejecta is uniform and the size distribution is the same everywhere; (iii) to quantify the role of ρISM we analize three different cases with nISM=0.06(rev1),0.6(rev2),and6cm−3(rev3). As previously shown in Todini & Ferrara (2001) and Nozawa & Kozasa (2003), the nucleation and accretion processes lead to a typical lognormal grain size distribution: in the right panel of Fig. 7 we show the grain size distribution of the 30M⊙ SN model with mixed ejecta, before and after the passage of the reverse shock. The grain sizes range between [10−3−0.5]μm, depending on the grain species and on the SN ejecta models. Since AC is the first grain species to condense, it has sizes larger than the other grains because grain accretion is more efficient at larger densities. For the same reason, alumina grains have the smallest sizes due to the low initial abundance of Al in the ejecta. As a result, the latter grain species is almost completely destroyed by the reverse shock, while the other grain species suffer a partial destruction, with the grain size distribution flattening towards smaller grain sizes.
The dust mass which survives the impact of the reverse shock is reported for all mixed SN models in Table 7, where we have also specified the dust mass of each grain species. In the left panel of Fig. 7 we show, for all Pop III mixed SN models, from top to bottom, the total mass of dust at the end of nucleation and after the impact of the reverse shock for the three increasing values of nISM. In the worst scenario, when the SNe explode in the densest ISM, the reverse shock travels faster and encounters a higher density gas inside the ejecta, increasing the sputtering and causing that only ∼3% of the newly formed dust mass survives. Clearly, the percentage of surviving dust depends on the ejecta model and ranges from 48% to 3%. In Fig. 9 the same histograms are reproduced for 15,30 and 50M⊙ mixed models, showing the impact of the reverse shock on the different grain species. We can see that alumina grains are completely destroyed. Depending on nISM, after the passage of the reverse shock, we see that: (i) in the 15M⊙ model the dominant species becomes AC, differently from the no-rev case, where AC and silicates have comparable masses; (ii) in the 30M⊙ model silicates always dominate the mass of dust; however, the smaller destruction suffered by AC grains, compared to other grain species, alters the original dust composition and after the passage of the reverse shock the mass of AC and silicates are comparable and AC grains are more abundant than Fe3O4 grains; (iii) in the 50M⊙ model, silicates dominate all the grain species, as in the no-rev case.
Fig. 8 illustrates the dust mass which survives to the impact of the reverse shock for the three selected unmixed models (see also Table 8 for the amount of grains of different species in the three layers). After the passage of the reverse shock, depending on the density of the ISM and on the progenitor mass, the percentage of surviving dust ranges from 16% to 100%. Also, compared to the fully mixed models, we note that in each case dust is destroyed to a minor degree. In fact, in all the three unmixed models, most of the dust mass is produced in the innermost layer (layer A). At the time the reverse shock reaches this region of the ejecta, density is low and the effect of the reverse shock is less destructive. In Table 8 we note that, while layer B and C are highly affected by the passage of the reverse shock, dust in layer A survives in large quantities and represents almost the totality of the survived dust mass. These calculations have been performed with an updated version of the code by Bianchi & Schneider (2007) which considers also stratified ejecta, taking into account the gas (density, temperature, elemental abundances) and dust (composition, size distribution) properties in each shell depending on the position within the ejecta.
These results suggest that the effect of the reverse shock must be taken into account in order to have a reliable estimate of the final dust mass and composition.
Figure 8: Mass of dust grains, after the passage of the reverse shock, as a function of the progenitor mass for the three selected unmixed models. From top to bottom: no reverse shock models (no-rev), models with a circumstellar medium density of nISM=0.06(rev1),0.6(rev2)and6cm−3 (rev3).
5 PopIII faint SN progenitors
We know from observations that a fraction of ∼10%−20% of iron-poor stars observed in the Galactic Halo with [Fe/H]<−2 are carbon-rich (Yong et al. 2013; Placco et al. 2014 and references therein). Most notably, the observed frequency of CEMP stars that do not show overabundances of neutron-capture elements, the so-called CEMP-no stars, increases to 80% at [Fe/H]<−4, with interesting implications for chemical evolution models and the formation pathway of these hyper-iron poor stars (de Bennassuti et al. 2014).
Different scenarios have been proposed to explain the observed elemental abundances of CEMP-no stars (see Norris et al. 2013 and references therein). Among these, stellar winds of fast-rotating massive stars (Maeder, Meynet & Chiappini 2015), a single Pop III supernova, experiencing mixing and fallback after the explosion (Umeda & Nomoto 2002; Iwamoto et al. 2005; Tominaga et al. 2007; Tominaga 2009; Heger & Woosley 2010; Tominaga et al. 2014; Yong et al. 2013; Ishigaki et al. 2014; Marassi et al. 2014), or in terms of an almost failed Pop III SN (with a large fallback) exploding in an environment pre-enriched by one or more normal Pop III supernovae (Limongi et al. 2003).
In Marassi et al. (2014) we intepreted the surface element abundances of SMSS J031300, the recently discovered CEMP-no stars with [Fe/H]<−7.1 (Keller et al. 2014), in the framework of the mixing-fallback scenario. In particular, in that paper dust formation in the ejecta of Pop III faint SNe was investigated for the first time, showing that - depending on the extent of mixing experienced by the ejecta and on the partial destruction by the SN reverse shock - between 0.025 and 2.25M⊙ of carbon dust forms. These dust masses are large enough to activate dust-driven fragmentation (Schneider et al. 2012) in the parent star-forming cloud of SMSS J031300, even accounting for the dilution and mixing of the SN ejecta with the surrounding pristine gas (Marassi et al. 2014).
In this section, we use the same procedure applied in Marassi et al. (2014) to estimate the mass and composition of dust that forms in the ejecta of faint Pop III SNe. The SN models have been selected by comparing the predicted elemental yields with the observed surface elemental abundances of the four C-enhanced, hyper iron-poor stars currently known, namely: HE1327-2326 (Frebel et al. 2005), HE0107-5240 (Christlieb et al. 2002), HE0557-4840 (Norris et al. 2007) and SMSS J031300 (Keller et al. 2014). For each of these, we vary the mixing and fallback efficiencies so as to minimize the scatter between the observed abundance pattern and the elemental yields predicted by Pop III SN models (Limongi & Chieffi 2012)333In Marassi et al. (2014) we estimate that in less than 1 Myr the ejecta material can be well mixed and diluted with pristine gas, enriching the gas cloud out of which second generation C-enhanced stars form with a [Fe/H] consistent with the observed values.
The surface chemical abundances of HE1327-2326, HE0107-5240, and HE0557-4840 have been taken from the recent compilation by Norris et al. (2013) (see their Table 4; here we normalize to the solar abundances of Asplund et al. 2009), which are based on high-resolution data, but are determined using 1D, LTE model-atmosphere analysis. The upper limit on silicon for HE1327-2326, HE0107-5240 and HE0557-4840 is taken from Yong et al. (2013). For consistency, we consider the observed elemental abundances of SMSS J031300 as derived from 1D model atmosphere (Marassi et al. 2014). In Fig. 10 we show the comparison between the observed elemental abundances and the best fit models for each of the four stars. We find that the data are reproduced by Pop III SNe with progenitor masses in the range [20−80]M⊙ that experience strong fallback. For all the models we have identified the mass-coordinates that defines the extent of mixing and fallback that better reproduce the observed abundaces, with particular attention on [C/Ca], [Mg/Ca] and [O/Ca] - that are extremely important for dust formation - without exceeding the upper limits on [Fe/Ca]. In the following, we discuss the properties of the faint Pop III SN progenitors inferred from the observed abundances of each star, which we also report in Table 9.
Figure 9: Histograms showing the mass of dust in the different compounds at the end of nucleation, and after the passage of a reverse shock of increasing intensity for three mixed SN models of 15M⊙, 30M⊙ and 50M⊙. From top to bottom: no reverse shock models (no-rev), models with a circumstellar medium density of nISM=0.06(rev1),0.6(rev2)and6cm−3 (rev3).
Figure 10: Comparison between the observed elemental abundance ratios of the CEMP-no stars and the chemical yields of Pop III faint SN with progenitor masses of 30,35,20,and80 M⊙. Mixing and fallback are chosen so as to minimize the scatter with the observations (black points). Dots with arrows show upper limits and filled points with errorbars indicate the detections.
For HE1327-2326, nitrogen has been detected and the observed abundance shows comparable value of [N/Ca] and [C/Ca]. Since a substantial amount of N is produced only by Pop III SN models with progenitor masses in the range [25−35]M⊙ (see Limongi & Chieffi 2012 for more details), we have limited the exploration of the mixing and fallback procedure to this mass range. The best agreement for HE1327-2326 is provided by the 30M⊙ progenitor model with a mass-cut of 5M⊙ and an ejected mass of M(56Ni) of 3.68×10−7M⊙. Note that a [Si/Ca] ratio close to the observed upper limit implies a substantial under-production of [(Na,Mg,Al)/Ca] ratios. Since the amount of Si in the ejecta plays a crucial role in the dust formation process, we chose to fit the [Si/Ca] ratio, even if the abundances of Na, Mg and Al are underproduced.
The same procedure has been applied to HE0107-5240 finding the best agreement for a SN progenitor model with 35M⊙ a mass-cut of 8M⊙ and a M(56Ni) mass of 9.12×10−7M⊙, which reproduces the observed [C/Ca] and [O/Ca] and predicts a [Na/Ca] ratio close to the observed one.
For HE0557-4840 we have searched for progenitors which favors low [N/Ca] due to the low upper-limit inferred from observations. We select a 20M⊙ progenitor with Mcut=5M⊙ and M(56Ni)=9.19×10−7M⊙, that reproduces also the observed [C/Ca] and [Mg/Ca] abundances. We note that none of the models is able to reproduce, at the same time, the observed [O/Ca] and [Mg/Ca] abundances (see also Limongi & Chieffi 2012 and Ishigaki et al. 2014), particularly the low-mass progenitors, which have lower value of [Mg/O] in the ejecta.
Figure 11: Left panel: the mass of dust at the end of nucleation for mixed and unmixed ejecta for the four faint SN progenitors. Central panel: the mass of dust at the end of nucleation and after the passage of a reverse shock of increasing intensity for the four faint SN progenitors with mixed ejecta considered in this study. Right panel: the same as in the central panel but for faint SN models with stratified ejecta. From top to bottom: no reverse shock models (norev), models with nISM=0.06(rev1),0.6(rev2)and6cm−3 (rev3)
Finally, for SMSS J031300 we have reported only the best-fit obtained for the 80M⊙ model (Mcut=24M⊙ and M(56Ni)=1.43×10−7M⊙) selected by Marassi et al. (2014), to which we refer the reader for a detailed discussion.
6 Dust yields from faint Pop III SN
This section presents the results of the dust formation calculation for Pop III faint SN progenitors. To enable a comparison with the results by Marassi et al. (2014), we first discuss the results for uniformly mixed ejecta, which are presented in Table 9 in the Appendix, where we give - for each of the four selected faint SN progenitors - the explosion parameters, the thermodynamical properties, the metal yields of the key elements in the nucleation process, the total amount of dust and the masses of molecules. Similarly to Marassi et al. (2014) we find that for all the progenitor models investigated, the only grain species that is able to condense and grow is amorphous carbon. This is a consequence of the extensive fallback, which determine an ejecta composition dominated by carbon atoms, with an amount of Mg, Si and Al which is too low to enable the condensation of other grain species (Marassi et al. 2014). We find that the molecular reactions involving SiO are negligible. The mass of AC is in the range [2.7×10−3−0.269]M⊙, depending on the initial carbon abundance and on the mean ejecta density and temperature evolution. Due to the interplay of these quantities it is not possible to establish a direct correspondence between progenitor mass, ejected mass and the dust mass produced. As shown in Marassi et al. (2014), AC grains do not forms if the formation channels of carbon monoxide are efficient in subtracting carbon atoms from the ejecta, and this efficiency is greater when the mean ejecta density is larger. As a consequence, we note that HE0557-4840 have the greater condensation efficiency but the smaller AC mass produced. Equally important is the temperature evolution which, in the case of HE0107-5240, enables the formation of C2, subtracting C atoms from the ejecta.
In faint SNe, mixing occurs due to Rayleigh-Taylor instabilities up to a mass-coordinate that is very close to the mass-cut (Umeda & Nomoto 2002); as a consequence, the material beyond the mass-cut is likely to be stratified, and dust nucleation in unmixed ejecta gives a more reliable estimates of the total mass of dust produced. Table 10 shows the results for the unmixed ejecta model. For three of the four progenitor models, the stratified ejecta is composed of two layers A and B that have different average temperature, density and chemical composition. In particular, in all three unmixed models analyzed, the internal layers A have mean ejecta density larger compared to the external layers B, enabling a very efficient formation of CO which inihibits the formation of AC.
The consequence of stratification is a reduction of the total mass of dust produced, that ranges between [7.5×10−4−0.11]M⊙. In the left panel of Fig. 11, we plot the total mass of dust before the passage of the reverse shock for mixed and unmixed models. Similarly to what done in Section 4, we have studied the impact of the reverse-shock on the above dust masses. The grain sizes originally follow a log-normal distribution in the range [0.03−0.22]μm. As expected, after the passage of the shock the distribution flattens and shifts to lower grain sizes, due to the erosion of the larger grains caused by sputtering. In the central and right panels of Fig. 11, we show the mass of dust that survives in uniformly mixed and stratified ejecta models, respectively. We find that, depending on the progenitor model, dust can be totally destroyed (as in the case of HE0557−4840) or that between 10 to 80% of the original mass can survive. The passage of reverse shock in the unmixed case is less destructive than in the case of the fully mixed model. As discussed in Section 4, dust in external layers is highly affected by the reverse shock. On the contrary dust produced in internal layers, as in the case of HE0107−5240, is able to survive in large quantities, finding itself in milder conditions compared to the external layers.
Figure 12: Mass of the SN progenitor, of the remnant, of metals and dust mass as a function of the progenitor mass for standard (left panel) and faint (right panel) Pop III SNe. Note that the mass of metals and dust have been multiplied by Meje/(Mmet+Mdust) (see text).
In this study we have estimated the dust mass produced in the early Universe by Pop III standard and faint core-collapse SNe. For standard core-collapse events, which do not experience a strong fallback, the metal-free progenitors are calibrated to reproduce the average elemental abundances of Galactic halo stars with [Fe/H]<−2.5. We have enlarged our study to include Pop III faint SNe, which are fine-tuned to reproduce the abundance patterns observed on the surface of the sub-sample of carbon-enhanced hyper iron poor stars.
To estimate the dust mass produced that survives the subsequent impact of the reverse shock, we have updated the previously developed code of Bianchi & Schneider (2007) to include formation/destruction processes of CO, SiO, C2 and O2 molecules that are crucial in subtracting gas-phase elements from the ejecta. We have investigated how our poor knowledge of mixing can affect the resulting dust yields by analyzing both uniformly mixed and stratified ejecta. Finally, to have a more realistic estimate of the dust mass that will be injected into the ISM, we have analized the effect of the passage of the reverse-shock. Our main results are that:
Standard core-collapse Pop III SNe are efficient dust producers when the ejecta are uniformly mixed; the newly formed dust mass ranges between [0.18 - 3] M⊙ and is dominated by silicates. Depending on the density of the interstellar medium, the impact of the reverse shock on the newly formed dust is to reduce the original dust mass by a factor which ranges between 3% to 48%, changing the relative abundance of different dust species. The mass of dust produced depends on the degree of mixing experienced by the ejecta. In general, silicates, magnetities and alumina grains are formed in the internal layers, while amorphous carbon forms only in the external layers that are carbon and oxygen dominated. In stratified ejecta dust does not form at the same time in each layer, with silicates forming in the inner layers before AC in the external ones. For the same SN progenitor model, the mass of dust formed in unmixed ejecta can be smaller or higher than in the corresponding mixed case, with variations which amount to 3−10%.
We confirm the results found by Marassi et al. (2014) that dust can form in the ejecta of faint Pop III SNe. As a consequence of the larger fallback experienced with respect to standard core-collapse Pop III SNe, the ejecta is lacking in silicate, magnesium and alumina and the only dust grain that forms is amorphous carbon. The total amount of carbon dust produced in the uniformly mixed case varies between [0.2−0.27]M⊙, and in the stratified case between [7.4×10−4−0.11]M⊙ depending on the model. After the passage of the reverse shock, the dust that survives ranges between 10% to 80%.
The mass and composition of dust formed in the ejecta of Pop III SNe depends on the degree of fallback. In standard Pop III SNe, the mass-cut corresponds to a mass coordinate in the range [1.48−2.44]M⊙. As a result, almost all the mantle is ejected with a greater amount of internal elements, such as silicon and magnesium, favoring the condensation of silicates. The observed elemental abundances of CEMP stars with [Fe/H]<−4 require Pop III SN progenitors with masses in the range [20−80]M⊙ which experience a strong fallback, with mass-cut which correspond to mass coordinates in the range [5−24]M⊙. These faint SNe leave a much larger remnant behind and eject a smaller amount of metals, with mantles dominated by light elements. These properties of the SNe largely affect the mass and composition of dust which condense in the ejecta, as shown by Fig. 12, where we summarize our results showing the mass of metals and of the dominant grain species produced by standard (left panel) and faint (right panel) Pop III SNe. Note that for each SN model, we have rescaled the mass of metals and dust as M∗met=MejeMmet/(Mmet+Mdust) and M∗dust=MejeMdust/(Mmet+Mdust), to clearly display these physical quantities.
Finally we have constructed, starting from our set of metal-free progenitors, a dust data grid for Pop III SNe which provides the initial conditions required to simulate the properties of the first metal-enriched star forming regions at high-redshift. In addition, we plan to extend this analysis to higher metallicity to incorporate these new mass- and metallicity-dependent yields in chemical evolution models with dust.
We thank the anonymous Referee for her/his careful reading of the manuscript and useful comments. The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 306476. We aknowledge financial support from PRIN MIUR 2010-2011, project "The Chemical and dynamical Evolution of the Milky Way and Local Group Galaxies", prot. 2010LY5N2T
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Appendix: Final dust grids and tables
In this appendix we report all the useful tables: for all progenitors we report chemical composition (yields), mass of molecules, mass of dust etc.
Pop III SN - Fully Mixed Ejecta Models
13M⊙ 15M⊙ 20M⊙ 25M⊙ 30M⊙ 35M⊙ 50M⊙ 80M⊙
Eexp
veje
Meje
MHecore
2.81 3.40 5.18 5.37 7.22 11.13 18.11 31.94
2.65×1014 2.44×1014 1.47×1014 3.75×1014 4.68×1014 1.98×1014 2.51×1014 4.78×1014
2.11×106 1.78×106 1.58×106 3.72×106 3.96×106 3.11×106 3.12×106 4.14×106
M(56Ni)
2.20×10−2 3.85×10−2 7.12×10−2 9.54×10−2 2.10×10−1 1.81×10−1 2.86×10−1 4.86×10−1
7.52×10−2 1.56×10−1 2.99×10−1 4.48×10−1 7.02×10−1 6.21×10−1 1.44 2.43
1.38×10−1 2.91×10−1 9.31×10−1 1.96 2.96 4.35 8.60 18.22
9.60×10−3 3.43×10−2 0.65 0.896 1.25 1.45 3.36 5.66
MSiO
- - - - - 0.77 1.41 1.64
7.10×10−2 0.14 1.80×10−2 6.36×10−2 0.16 - - -
MMgSiO3
2.17×10−4 5.10×10−5 - 3.86×10−4 5.62×10−3 - 3.23×10−6 5.76×10−4
MMg2SiO4
5.28×10−2 0.104 0.29 0.27 0.38 0.62 0.59 1.06
MFe3O4
3.00×10−2 5.65×10−2 0.11 0.14 0.30 0.263 0.41 0.70
MSiO2
MAl2O3
0.176 0.35 0.54 0.68 1.32 0.99 1.71 3.07
Table 3: Properties of the Pop III SNe, including the explosion energy [1051 erg], the ejecta velocity [km s−1], the mass of the ejecta, the mass cut and the mass of the helium core [M⊙], the gas number density [cm−3] and the radius of He core, R0 [cm] at t=tini [sec] when adiabatic expansion starts (see text); the initial masses of C, O, Mg, Si, Al, Fe, and 56Ni which decays in 56Co fueling γ-luminosity (see eq. 2), the mass of molecules, CO, SiO, O2, C2 and the grains formed in the expanding ejecta [M⊙]. Each model name identifies the progenitor mass.
15M⊙ Unmixed Ejecta Model
Eexp =0.7
MHecore = 3.40 veje = 2960 Meje = 13.38
zone A (1.62−1.98)M⊙ zone B (1.98−2.22)M⊙ zone C (2.22−3.40)M⊙
3.50×1014 1.96×1014 2.22×1014
3.19×105 1.77×106 1.78×106
3.85×10−2 - -
1.88×10−3 9.48×10−2 5.95×10−2
0.17 0.12 7.50×10−3
3.52×10−2 8.87×10−4 -
4.38×10−3 0.18 1.32×10−2
- - 1.34×10−2
- 1.69×10−2 4.03×10−2
0.10 1.77×10−3 -
0.33 1.88×10−2 4.03×10−2
Table 4: Properties of the 15M⊙ Pop III SN unmixed ejecta model. In unmixed models R0 and n0 are the mean radius and density of the layers.
zone A (1.86−3.99)M⊙ zone B (3.99−5.20)M⊙ zone C (5.20−7.22)
1.26×10−2 0.23 0.45
1.97×10−3 0.46 -
Table 5: Properties of the 30M⊙ Pop III SN unmixed ejecta model.
MHecore = 18.11 veje = 3017 Meje = 47.84
zone A (2.16−4.31)M⊙ zone B (4.31−11.51)M⊙ zone C (11.51−18.11)M⊙
9.56×102 0.106 6.60×10−4
- 8.3×10−4 -
- - 0.12
- 1.58×10−3 -
1.61 2.87×10−2 0.12
Pop III SN - Reverse Shock - Fully Mixed Ejecta Models
MAC 4.78×10−2 9.82×10−2 6.71×10−3 3.59×10−2 0.10 - - -
MMgSiO3 3.45×10−5 1.19×10−5 - 1.20×10−4 1.58×10−3 - 1.44×10−6 1.68×10−4
MMg2SiO4 7.72×10−3 2.13×10−2 0.17 7.45×10−2 9.41×10−2 0.39 0.24 0.27
MFe3O4 1.25×10−3 3.49×10−3 2.35×10−2 1.21×10−2 2.85×10−2 5.54×10−2 7.06×10−2 5.77×10−2
MSiO2 2.22×10−3 8.59×10−3 4.76×10−2 6.33×10−2 0.15 3.37×10−2 0.37 0.47
MAl2O3 7.29×10−5 1.34×10−4 6.82×10−4 2.75×10−4 2.89×10−4 6.33×10−4 1.52×10−4 2.24×10−4
Mdust 5.9×10−2 0.13 0.25 0.19 0.38 0.48 0.68 0.80
MAC 2.43×10−2 5.53×10−2 2.83×10−3 1.71×10−2 5.31×10−2 - - -
MMg2SiO4 2.48×10−3 6.76×10−3 7.82×10−2 2.49×10−2 3.01×10−2 0.16 8.96×10−2 8.66×10−2
MSiO2 7.72×10−4 2.72×10−3 1.87×10−2 2.22×10−2 5.61×10−2 1.17×10−2 0.16 0.17
MAl2O3 3.73


text:
Q: Need Transfer Value PHP I am creating a new website. For that website I will need to transfer a quantity and amount value from one if condition statement into another if condition statement. Both if statements are accessed by separate submit buttons named "checkamt" & "buy".
I need to transfer the "quantity",  "value",  and "net" values to and from the checkamt if statement to the buy if statement.
Here's my code:
<form action="index.php" method="post">
<input type="submit" name="checkamt" value="Check Amount"/>
<input type="submit" name="buy" value="Buy"/>
</form>

   <?php
      if(isset($_POST[checkamt]))
      {
          $qun=1;
          $val=5000;
          $total=$qun*$val;
      }

      if(isset($_POST[buy]))
      {
          echo $qun;
          echo $val;
          echo $total;
      }
   ?>

A: I think that the problem you're having is that variables don't persist on page change. If you want that, you'll need to use a session. First, you must call session_start before anything, including HTML, is sent to the user. Then, you can use the $_SESSION variable.
<?php
    session_start();
?>

<form action="index.php" method="post">
<input type="submit" name="checkamt" value="Check Amount"/>
<input type="submit" name="buy" value="Buy"/>
</form>

   <?php
      if(isset($_POST[checkamt]))
      {
          $_SESSION['qun']=1;
          $_SESSION['val']=5000;
          $_SESSION['total']=$qun*$val;
      }

      if(isset($_POST[buy]))
      {
          echo $_SESSION['qun'];
          echo $_SESSION['val'];
          echo $_SESSION['total'];
      }
   ?>

A: Improve your English! Not sure if this is what you want, but if you want to share the values of your variables between the two ifs? You have to declare them at a higher scope than your if:
<?php
  $qun = 0;
  $val = 0;
  $total = 0;

  if(isset($_POST[checkamt]))
  {
      $qun=1;
      $val=5000;
      $total=$qun*$val;
  }

  if(isset($_POST[buy]))
  {
      echo $qun;
      echo $val;
      echo $total;
  }
?>

A: Not sure if I'm getting the question right, but why not do something like this :
  if(isset($_POST[checkamt]) || isset($_POST[buy]))
  {
      $qun=1;
      $val=5000;
      $total=$qun*$val;

      echo $qun;
      echo $val;
      echo $total;
  }

A: You need to track your variables between the different forms. You can use SESSION like Xeon06 suggested, or do the following. I'm only showing for $qun:
<?php
if(isset($_POST['checkamt'])) {
    $qun=1;
}

if(isset($_POST['buy'])) {
    echo $qun;
}
?>

<form action="index.php" method="post">
<input type="hidden" name="qun" value="<?php echo $qun; ?>" />
<input type="submit" name="checkamt" value="Check Amount"/>
<input type="submit" name="buy" value="Buy"/>
</form>

redpajama_set_name: RedPajamaStackExchange


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text:
Posts from the 'YPLL' Category
Georgia blacks make strong gains in premature death rates; rural white females losing ground
As we've noted in various previous posts, Georgia's premature death rate (known formally as Years of Potential Life Lost before age 75, or YPLL 75) has been improving fairly steadily over the 20 years that the state's Department of Public Health (DPH) has been compiling pertinent data.[1] Between 1994 and 2013, the state's YPLL 75 rate improved from 9,195.6 to 7,104.7, a gain of 19.4 percent. The national median, as reported the Robert W. Johnson Foundation in its latest County Health Rankings, was 7,681, so Georgia is doing a little better than the nation as a whole.
But, as we've noted in past posts, Georgia's improvement has been far from even; we've focused in particular on regional differences and the dramatic gap in YPLL 75 performance between Metro Atlanta and the rest of the state. Until now, however, we haven't looked at racial or gender comparisons, and that produces a couple of interesting headlines. One is that the vast majority of gains in premature death rates between 1994 and 2013 have been made in the black population. The other is that rural white females are losing ground. Read more
A Tale of Two Regions
Our recent research on premature death rates in Georgia produced a couple of unexpected revelations, and we decided to loop back for a closer look. The revelations involve two of our five regions, North Georgia and Middle Georgia. (For a quick primer on that earlier work, click here, here and here.)
For our purposes, North Georgia is made up of 41 counties that lie outside Metro Atlanta and above the gnat line (see map below and click on the map for a larger view). As a region, it has clearly benefited from its proximity to Metro Atlanta; in recent decades, it has posted far and away the second-strongest population and economic growth in the state, outpacing not only Middle and South Georgia, but the Coastal region as well. Read more
Peachcare and Young YPLL
In a recent post, we began to explore premature death rates within Georgia's working-age population, men and women between the ages of 18 and 65. We were initially surprised to learn that improvements in the so-called YPLL 75 Rate for this segment of the state's population lagged gains for the population as a whole. That led us to drill down a bit and look at premature death trends in the younger and older age groups – specifically, Georgians under the age of 18 and between the ages of 65 and 75.
Both groups saw significantly stronger gains in their premature death rates than did working-age Georgians. The question was why; what factors were driving premature death gains for younger and older Georgians that were somehow not impacting working-age Georgians?
Working Age YPLL
Throughout the Partner Up! for Public Health campaign, when we were conducting the research and analysis that enabled us to "connect the dots" between community health and economic vitality at a local level, one of the key health metrics we relied upon was premature death – better known as Years of Potential Life Lost before Age 75, or YPLL 75. YPLL 75 is generally regarded as the Dow Jones Industrial Average of a community's (or a state's, or a nation's) health. If you want to look at one metric and get a sense of a community's health, look at its YPLL 75 rate.
YPLL 75 is part of the formula the Robert Wood Johnson Foundation uses to calculate a county's overall Health Outcomes Rankings, and it's easy enough to pull from the State of Georgia's excellent online public health data system, OASIS (for Online Analytical Statistical Information System). So it was a natural data point to work with. Read more
Chicken and the Egg
In the four years we've been talking about the relationship between health and economic vitality at a community level, the one question we've gotten more than any other is the chicken-and-egg question. Which comes first? Which is the driver? Does health make wealth? Or vice versa?
My answer is that the relationship between health and economic productivity is pretty clear first and foremost at a personal level. We all know we're more productive when we're in good health than when we're down with a cold or the flu, let alone something more serious. We also know that when we're sick, we inevitably have to divert some of our income to medical care.
The same thing plays out at a community level. Recently we updated an analysis of the Georgia counties that came in at the top and bottom of our 2013 Partner Up! for Public Health Power Rankings – Oconee County at the top and side-by-side South Georgia neighbors Crisp and Wilcox counties tied for last.
YPLL: A Primer
Today's subject is YPLL. YPLL is the acronym for Years of Potential Life Lost, which is probably the most common way of gauging premature death and is more or less the Dow Jones Industrial Average of public health. If you want a quick, one-number look at the health status of a county or state or nation, you look at YPLL and whether it's getting better or worse.
Basically, YPLL is calculated by taking the number of people who die in a community in a given year and subtracting their ages at death from 75, which is the most commonly used projected "end point age." I'm 64. If I die this year, I'll contribute 11 years to Fulton County's bucket of Years of Potential Life Lost. An infant who dies at the age of six months would contribute 74.5 years to that same YPLL bucket. Read more

redpajama_set_name: RedPajamaCommonCrawl


text:
InterCure Announces Exclusive International Partnership Agreement with Brand "Binske"
(NASDAQ: INCR) (TSX: INCR.U) (TASE: INCR) (dba Canndoc), announced that it has entered into a definitive licensing agreement with Praetorian Global, Inc. Pursuant to the Agreement, Praetorian Global, Inc., based in Florida, and the parent company of the award-winning cannabis brand, "Binske", agreed to grant InterCure an exclusive multi-year right to cultivate, manufacture, market, and distribute Binske-branded products in major global pharmaceutical markets including Israel, Germany, Australia, UK and others.
InterCure is of the view that Binske has pioneered the premium medical and recreational markets in the United States through its meticulous focus on standards, quality, and consistency. Lauded for its proprietary strains, craft ingredients, full product suite of nearly 200 offerings, and best-in-class packaging, "Binske" offers luxury, artisan-quality products using purposefully sourced ingredients that have earned widespread recognition, making it one of the largest and most recognizable brands in the American market. The "Binske" brand has won numerous cannabis related awards including Leafly's Best Overall Brand, Best Edibles and Best Concentrates. The Company considers Binske's sophisticated product offerings, coupled with their innovative brand identity, to set them apart from the rest of the marketplace. The Agreement brings these award winning products exclusively to Intercure's Israeli hub to manufacture and distribute all "Binske" Branded Products under EU-GMP standards, exclusively to the international pharmaceutical space.
"As the international markets emerge, we feel it is our duty and responsibility to partner with global-leaders like InterCure who possess the same level of commitment to providing patients with the most sophisticated brands and best-in-class product offerings," said Praetorian Global's Founder, Chairman and CEO, Jacob Pasternack. "We have put significant time and resources into developing a nucleus of consumer patronage throughout North America and will now strategically take that same level of passion and dedication towards consumer experience abroad."
As part of the Agreement, Binske will provide the Company access to its intellectual property, including genetics, formulations and know-how for cultivation and manufacturing of Binske-branded cannabis products at the Company's facilities. In addition, "Binske" will provide InterCure with IP relating to extraction formulations and the production of downstream products developed by "Binske", and will support InterCure's team with training of manufacturing and cultivation methods that are tailored for Binske's exacting standards. Under the terms of the Agreement, InterCure will produce and distribute the Branded Products leveraging its international supply chain using the Company's medical cannabis-dedicated pharmacy chain.
During the 7.5-year term of the Agreement, the same lines of core "Binske" Branded Products launched in North America will be manufactured and distributed exclusively by InterCure in the territories stipulated in the Agreement.
This partnership will bring together InterCure's expertise in manufacturing at the EU-GMP pharmaceutical standard with Binske's expertise in crafting a diverse range of flower and non-flower products, including live rosin, live resin, wax, shatter, gummies, chocolates, olive oil and medical cannabis vaping devices. The endeavor will attempt to address the specific needs for medical patients in the agreed territories by replicating the success of Binske's core product offerings in the U.S. While the regulatory framework in the agreed territories is different than the U.S., the Company expects to utilize its GMP supply chain to make the necessary adjustments to comply with all regulatory requirements involved with the manufacturing and distribution of the products in the target markets.
"Binske is already one of the most recognized cannabis brands in the U.S.," said InterCure's Chief Executive Officer, Alexander Rabinovitch. "Partnering with world-class leaders was always a significant part of our strategy and it enabled us to develop our industry leading portfolio of premium branded products. We are excited to partner with Binske and bring their high-quality flower and non-flower products into the pharmaceutical cannabis markets, improving quality of life for patient communities globally."
Forward‐Looking Statements
This press release may contain forward-looking statements. Forward-looking statements may include, but are not limited to, statements relating to InterCure's objectives plans and strategies, the success of the Agreement and overall relationship with Praetorian Global, Inc., as well as statements, other than historical facts, that address activities, events or developments that InterCure intends, expects, projects, believes or anticipates will or may occur in the future. These statements are often characterized by terminology such as "believes," "hopes," "may," "anticipates," "should," "intends," "plans," "will," "expects," "estimates," "projects," "positioned," "strategy" and similar expressions and are based on assumptions and assessments made in light of management's experience and perception of historical trends, current conditions, expected future developments and other factors believed to be appropriate.. Forward-looking statements are not guarantees of future performance and are subject to risks and uncertainties that could cause actual results to differ materially from those expressed or implied in such statements. Many factors could cause InterCure's actual activities or results to differ materially from the activities and results anticipated in forward-looking statements, including, but not limited to, the following: the Company's success of its global expansion plans, its continued growth, the expected operations, financial results business strategy, competitive strengths, goals and expansion and growth plans, expansion strategy to major markets worldwide, the impact of the COVID-19 pandemic, the impact of the war in Ukraine and the conditions of the markets generally. Forward-looking information is based on a number of assumptions and is subject to a number of risks and uncertainties, many of which are beyond InterCure's control, which could cause actual results and events to differ materially from those that are disclosed in or implied by such forward-looking information. Such risks and uncertainties include, but are not limited to: changes in general economic, business and political conditions, changes in applicable laws, the U.S. and Canadian regulatory landscapes and enforcement related to cannabis, changes in public opinion and perception of the cannabis industry, reliance on the expertise and judgment of senior management, as well as the factors discussed under the heading "Risk Factors" in InterCure's Annual Information Form dated April 5, 2022, which is available on SEDAR at www.sedar.com, and under the heading "Risk Factors" and "Cautionary Note Regarding Forward-Looking Statements" in the registration statement on Form 20-F, filed with the Securities Exchange Commission on July 14, 2021, as amended August 3, 2021 and August 18, 2021. InterCure undertakes no obligation to update such forward-looking information, whether as a result of new information, future events or otherwise, except as expressly required by applicable law.
(This information is primarily sourced from InterCure Ltd. Highly Capitalized has neither approved nor disapproved the contents of this news release. Read our Disclaimer here).


For several years Sharesight has partnered with the New Zealand Shareholders Association, and on Saturday 19, August they're holding their annual Investor Conference and AGM in Wellington.
This is a terrific opportunity to network with fellow investors and hear from a great lineup of speakers including Simon Mouter, Managing Director of Spark Ltd and Mark Peterson, CEO of the NZX.
The event is free for members (with a $20 donation) and $50 for the public.
For 16 years the NZSA has been improving outcomes for New Zealand investors through their growing membership ranks. They aim to improve investor knowledge and risk taking, while also holding company directors to account.
The theme for this year's conference is "A World of Change in a Changing World." Given we've been on an 8 year bull market run and seen unprecedented geopolitical events, this topic is perfect given the challenges that investors are facing.
We at Sharesight encourage all Kiwi investors to attend this event!
Register for the August 19 event using this link – 2017 NZSA Investor Conference.


You are here: Home » Uncategorized » Simple tips to plan the spending plan and cut costs whilst in an university?
Simple tips to plan the spending plan and cut buy cialis online5mg. costs whilst in an university?
Some would state that this might be close to impractical to discover ways to plan purchases and save the spending plan. Specially when it comes down to pupils. Shortage of cash may be the problem students experience very often. Let us acknowledge the known undeniable fact that the majority of teenagers are terrible with money saving and strong at money wasting.
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This informative article is mostly about deriving a formula for the economy mode.
Think about everything you have actually and the thing you need, and work out a brief range of your future acquisitions. Then you are not likely to throw money away if you have a goal, for instance, to buy a car. Parties every single day, unneeded acquisitions will disappear to the back ground. So that the point is the fact that something many desirable will prevent you from wasting.
After making a listing of things you need, focus on it. At first put purchases to be made just as feasible after which what's less necessary. It can help you visually understand exactly what your needs that are primary.
So what does it mean to own a spending plan? The facts related to pupils in university? In fact, it is crucial to own a definite understanding of just how money that is much have actually for a second. Whenever every you take some money and do not really know how much is left can eventually end up with a budget hole day. Hence, the notion of having a spending plan (better yet if it is in a written form) act as a way to prevent from wasting additional money than prepared.
This approach evidently won't allow you to be more economical. However it represents quite an effective way to|way that is effective improve your economic habits.
To begin all, agree totally that economic freedom involves those people conserve money. , there are pretty many practices which subscribe to the integrity wallet. Tomorrow your savings generate possibilities of. Now we are going to offer several tips prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs, prednisone dosage for dogs. that are useful allow you to reconsider your attitude towards cash preserving and provide some clues how be kept with absolutely nothing because of the end regarding the month.
It indicates to save lots of 5-10% of one's salary, put it someplace safe and do not touch it. Off the top. Out of this %, you will have developed a fund of accumulations in a run that is long.
Many of us are animals of a practice. It will get much easier to do the same in the future if we get accustomed to saving from 5-10% from each earning. You simply have to do this for a basis that is regular then it's going to work. A lot of people succeeded in enabling financially separate. Therefore are you going to.
Money papers write saving experts are pretty certain that benefiting from tax-deferred cost savings helps accumulate money in a long term. It is really because of multiple income tax prices. Therefore the way as soon as your earning is conserved without having to be taxed. Therefore, money will accumulate even more quickly.
Old college: make a moneybox.
For those who are familiar with a traditional method of money that is saving. Each cent visits a moneybox. Safely locked. A moneybox can distract you against wasting. Needless to say, based on just what banknote you will be investing in, your total amount will develop. In the event Medrol order that you placing 100$ banknotes every month, then, in the long run, you will end up with a decent preserving in your pocket.
Ditch some small purchases that are dailymorning cappuccino or croissant).
5$ per day makes 150$ each month and 1825$ per 12 months. Only a small walk. As opposed to buying it, consider the solution to make it yourself. 100% it's gonna be cheaper.
Never refuse from pupil freebies while offering.
There are lots of possibilities given to students. Pose a question to your peers or pupil communities what occasions to go to, so that it must be cost efficient. For example, there are lots of free museums. It can't be missed – see works of the most extremely remarkable designers. The impression of these masterpieces can not be bought for cash.
As you care able to see, saving and accumulation money is far from being easy. But let us remember about numerous opportunities to get some good freebies, or even to start work in an even more affordable means, get financially separate.
Life in a university is much more diverse while you think. Relish it at full, but never forget exactly what priorities you've got right now.


An advantage that we have here at Stonehenge Advisor Group is that we are an independent agency which means we have access to many different providers. Because of that you'll have wide array of possible life insurance options available to you when you work with us, all at the most competitive rates.
When you talk about life insurance it basically gets broken down to two major categories. One being Term Life Insurance and Permanent Life Insurance, with each having its benefits.
Term life insurance is generally the simplest and cheap for of life insurance. There are different terms associated with this kind of policy, for example there are 10, 20 or 30 years of coverage. That means in that time period if in the unfortunate situation that you'd pass your beneficiaries would be covered. However if the term expires you will need to purchase a new term life insurance policy.
Then with permanent life insurance the major difference between it and term is that it does not expire. Both are designed to give your beneficiaries financial backing when you pass, however with a permanent life insurance policy you won't have to worry about the term. It is more expensive because of this reason and that the money goes into a saving account. However a financial advantage is the premiums never go up, where they might with term life insurance. Also the longer you have a permanent life insurance policy the more money it will accumulate.
There are also a few sub categories when it comes to life insurance which can start getting complicated. We find it best to talk about them in person since each person's needs and concerns are so unique.


text:
/*
 * Jacob Bieker
 * CIS 314 Fall 2015 Lab 1
 * Assigned project
 *
 * This program reads a sorted array from a file and finds a requested number
 *    using recursive or iterative binary search. The array is read from a file
 *    defined by FILE_NAME, which should be written as the number of elements
 *    followed by the elements themselses. each number can be deliniated with
 *    any whitepace character. Also, the maximum size of the array is defined
 *    as MAX_SIZE.
 *
 * NOTE: The array must be sorted!!
 *
 */

#include <stdio.h>
#include <stdlib.h>
#include <time.h>

#define MAX_SIZE 150
#define FILE_NAME "array.dat" //must include quotes

// Implement the rest of the program

int main(void) {

   FILE *in_file;
   int array[MAX_SIZE];
   int size;
   char in_file_name[] = FILE_NAME;

   printf("\n\n=== CIS314 Fall 2014 - Lab 1: Part 2: Program 2 ===\n\n");

   printf("\n\n");

   int desired_number = 0;
   printf("Enter what number to find: ");
   scanf("%d", &desired_number);

   readInput(in_file, in_file_name, array, desired_number);

   return 0;
}

int readInput(FILE *input_file, char file_name[], int values[], int desired_number) {
	int count = 0;
	int value = 0;
	int array_size;
	input_file = fopen(file_name, "r");
	if (feof(input_file) != 0)
	{
		printf("Error: can't open file\n");
		return -1;
	}
	else
	{
		printf("File opened successfully\n");
	}

	//Get the first line of the file, which gives the size of the array
	while (fscanf(input_file, "%d", &value) != EOF) {
		if (count == 0)//The first value is being read
		{
			array_size = value - 1;//So that the later ones start at the correct index
		}
		else {
			values[count - 1] = value;
		}
		count++;
	}
	fclose(input_file);

	int delta;
	clock_t t1, t2;
	t1 = clock();
	int recursive_result = recursiveBinarySearch(values, desired_number, 0, array_size);
	t2 = clock();
	delta = t2 - t1;

	int delta1;
	clock_t t3, t4;
	t3 = clock();
	int iterative_result = iterativeBinarySearch(values, desired_number, array_size);
	t4 = clock();
	delta1 = t3 - t4;

	displayOutput(values, array_size, recursive_result, delta, delta1, iterative_result);

	return 0;
}
int recursiveBinarySearch(int list_of_numbers[], int desired_number, int low_number, int high_number) {
	int mid = (low_number + high_number) / 2;
	if (low_number > high_number)
	{
		return -1;
	}
	else if (list_of_numbers[mid] == desired_number)
	{
		return mid;
	}
	else if (list_of_numbers[mid] < desired_number)
	{
		return recursiveBinarySearch(list_of_numbers, desired_number, mid + 1, high_number);
	}
	else
	{
		return recursiveBinarySearch(list_of_numbers, desired_number, low_number, mid - 1);
	}

}

int iterativeBinarySearch(int list_of_numbers[], int desired_number, int array_size) {
	int low = 0;
	int high = array_size;

	while (low <= high)
	{
		int mid = (high + low) / 2;
		if (mid == desired_number)
		{
			return mid;
		}
		if (list_of_numbers[mid] < desired_number) {
			low = mid + 1;
		}
		else {
			high = mid - 1;
		}
	}

	return -1;
}

int displayOutput(int data_array[], int array_size, int recursive_result, int recursive_time, int iterative_time, int iterative_result) {
	int j = 0;
	printf("The input array is: ");
	for (j = array_size; j >= 0; j--) {
		printf("%d ", data_array[j]);
	}
	if (recursive_result == -1 || iterative_result == -1) {
		printf("\nRecursive result: Did not Find, Time taken: %d \n", recursive_time);
		printf("\nIterative result: Did not Find, Time taken: %d \n", recursive_time);
	}
	else {
		printf("\nRecursive result: %d, Time taken: %d \n", recursive_result, recursive_time);
		printf("\nIterative result: %d, Time taken: %d \n", iterative_result, recursive_time);
	}

	return 0;
}

redpajama_set_name: RedPajamaGithub


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