Seeding the second star – II. CEMP star formation enriched from faint supernovae

ABSTRACT

Carbon-enhanced metal-poor (CEMP) stars are the living fossils holding records of chemical enrichment from early generations of stars. In this work, we perform a set of numerical simulations of the enrichment from a supernova (SN) of a first generation of metal-free (Pop III) star and the gravitational collapse of the enriched cloud, considering all relevant cooling/heating processes and chemical reactions as well as the growth of dust grains. We adopt faint SN models for the first time with progenitor masses M_PopIII = 13–|$80 \ {\rm M_{\bigodot }}$|⁠, which yield C-enhanced abundance patterns ([C/Fe] = 4.57–4.75) through mixing and fallback of innermost layers of the ejecta. This model also considers the formation and destruction of dust grains. We find that the metals ejected by the SN can be partly re-accreted by the same dark matter minihalo, and carbon abundance of the enriched cloud A(C) = 3.80–5.06 is lower than the abundance range of observed CEMP stars (A(C) ≳ 6) because the mass of the metals ejected by faint SNe is smaller than normal core-collapse SNe due to extensive fallback. We also find that cloud fragmentation is induced by gas cooling from carbonaceous grains for |$M_{\rm Pop III}= 13 \ {\rm M_{\bigodot }}$| even with the lowest iron abundance [Fe/H] ∼ −9. This leads to the formation of low-mass stars, and these ‘giga metal-poor’ stars can survive until the present-day Universe and may be found by future observations.

stars: formation, stars: low-mass, stars: Population II, stars: Population III, ISM: abundances, galaxies: evolution

1 INTRODUCTION

Metal-poor stars are important astronomical objects that hold the signatures of chemical abundance in the early phase of metal enrichment in the Universe. In particular, extremely metal-poor (EMP) stars with metallicities [Fe/H] < −3 are considered to form in gas clouds enriched by only a few progenitors (Audouze & Silk 1995; Ryan, Norris & Beers 1996; Cayrel et al. 2004).¹ Inversely, we can indirectly see the nucleosynthesis of the progenitors from the metallicities and elemental abundances of EMP stars. This approach to chemical evolution in the early Universe is called galactic archaeology (Salvadori, Schneider & Ferrara 2007; de Bennassuti et al. 2014, 2017; Graziani et al. 2015; Hartwig et al. 2018; Komiya et al. 2020). Also, for the ancient stars to be observed in the present day, they should be of low mass (⁠|${\lt}0.8 \ {\rm M_{\bigodot }}$|⁠). The metallicity distribution of EMP stars gives a constraint on their initial mass functions (IMFs) with different metallicities.

Another important aim of this study is to search for the first generation of metal-free (Population III or Pop III) stars that first modify cosmic structure formation through their radiative and supernova (SN) feedback. So far, metal-free stars have not been observed although a large number (∼5000) of metal-poor stars have been identified in the Milky Way halo and Local Group dwarf galaxies in large survey campaigns and follow-up spectroscopic observations. This indicates that Pop III stars are predominantly massive. Numerical studies (Bromm, Coppi & Larson 1999; Abel, Bryan & Norman 2002; Yoshida et al. 2003) also predict that massive Pop III stars with M_PopIII ∼ 10–|$1000 \ {\rm M_{\bigodot }}$| form in the metal-free gas clouds hosted by low-mass (⁠|${\sim}10^6 \ {\rm M_{\bigodot }}$|⁠) dark matter (DM) haloes (minihaloes, MHs) at redshift z ≳ 6. The fragmentation of the pristine clouds is significantly reduced due to the lack of efficient gas coolants.

On the other hand, the survival of stars with non-zero metallicities indicates that the additional gas cooling due to heavy elements induces the fragmentation of their parent clouds. In particular, thermal emission cooling of dust grains becomes dominant at high densities n_H ∼ 10¹²–10¹⁴ cm⁻³, corresponding to small Jeans masses M_J ∼ 0.01–|$0.1 \ {\rm M_{\bigodot }}$|⁠, which may be indicative of low-mass star formation (Omukai 2000; Omukai et al. 2005; Schneider et al. 2002, 2003, 2006, 2012a). Several authors have studied dust-induced fragmentation, but they assume that the metal-poor clouds have the solar elemental abundance ratio and the same composition and size distribution of grains as in the local interstellar medium (ISM; Omukai 2000; Ritter et al. 2015; Smith et al. 2015; Safranek-Shrader et al. 2016; see, however, Schneider et al. 2006).

Following the discovery of the EMP star |$\rm SDSS \ J1029{+}1729$| by Caffau et al. (2011), with a total metallicity of only |${\it Z} = 4.5\times 10^{-5} \ {\rm {\it Z}_{\bigodot }}$|⁠, Schneider et al. (2012b) showed that its surface elemental abundances suggest that its birth cloud was enriched by the metal and dust yields of normal core-collapse SNe (CCSNe) and that the gas cooled through silicate dust cooling and fragmented. The same method was then applied by Marassi et al. (2015) to investigate the origin of |$\rm SMSS \ J0313{-}6708$|⁠, a carbon-enhanced star with an upper limit on its surface iron abundance of only [Fe/H] < −7.1 (Keller et al. 2014). It was suggested that, similarly to other carbon enhanced metal-poor (CEMP) stars with [Fe/H] ≲ −3 (so-called Group II and III of CEMP stars; Yoon et al. 2016),² this star could originate from dust-cooling and fragmentation of a collapsing gas cloud previously enriched by the yields of faint SNe. In faint SNe, mixing and fall back of the innermost layers of the ejecta into the central compact remnant can yield the C-enhanced abundance pattern of ejected materials. Since ⁵⁶Ni, the main source of γ-ray photons through its radioactive decay, is also depleted, this type of SNe is called faint SNe (Umeda & Nomoto 2003). In addition, Marassi et al. (2014, 2015) show that the carbonaceous grains are produced from faint SNe and could be responsible for the formation of |$\rm SMSS \ J0313{-}6708$|⁠. Hence, it was suggested that the distinctive elemental abundances between C-normal EMP (CN-EMP) and C-enhanced EMP stars point to a different type of SN explosion as the main formation sites of their surface elemental abundance, but to a common formation pathway based on either silicates or carbon dust cooling (Marassi et al. 2015). This scenario was further investigated by Chiaki, Tominaga & Nozawa (2017), who showed that the observed lower limits of carbon abundance A_cr(C) ∼ 6 for CEMP (for reference, the solar abundance is |$A_{\bigodot }({\rm {C}}) = 8.43$|⁠; Asplund et al. 2009) and of iron abundance of [Fe/H]_cr ∼ −5 for CN-EMP stars confirm that gas cooling by dust-driven cooling and fragmentation in their birth clouds is required, and that the dominant grain species are carbon and silicates, respectively.

Although these models successfully reproduce the relative element to element abundances of observed stars, the studies for the absolute abundance of metals/grains in enriched clouds have so far been limited to normal CCSNe. In our previous study (Chiaki & Wise 2019, hereafter Paper I), we followed the metal enrichment from a normal CCSN with [C/Fe] = 0.18. We found that only a fraction f_ret ∼ 0.4 per cent of metals return to the MH because the SN shell interacts with dense cosmological filaments and loses its energy through radiative cooling. Still, the metallicity of an enriched cloud [Fe/H] = −3.62 is consistent with the metallicity range of observed C-normal stars.³ Mainly, silicate grains ejected from the SN induce gas cooling in the enriched cloud and fragmentation occurs. Hence, the scenario proposed in Paper I could explain the origin of C-normal stars from a single CCSN.

For faint SNe, the ejected metal mass is smaller than for normal CCSNe because the ejecta partially falls back. Also, faint SNe exploding in a spherically symmetric manner have smaller explosion energies than normal CCSNe with the same progenitor mass. This can affect the return fraction of metals f_ret. Therefore, it is necessary to investigate whether the enriched clouds have C abundances consistent with the observed level (A(C) ≳ 6).

In this study, we follow the metal enrichment from a faint SN and the gravitational collapse of an enriched cloud with a set of cosmological simulations. To quantify the absolute metal and dust abundances in the enriched cloud, for the first time, we employ a nucleosynthesis/nucleation model of Pop III faint SNe produced by Marassi et al. (2014, hereafter M14). The metal mass and explosion energy are consistently derived so that the model reproduces the elemental abundance of the CEMP star |$\rm SMSS \ J0313{-}6708$| (Keller et al. 2014) for given progenitor masses. We then follow the gravitational collapse of the enriched cloud with all relevant cooling/heating processes including carbon grain cooling and chemical reactions as well as growth of dust grains (Chiaki et al. 2015) to determine whether the gas cloud is able to fragment, allowing the formation of low-mass and long-lived stars, such as |$\rm SMSS \ J0313{-}6708$|⁠.

The structure of this paper is as follows: In Section 2, we describe our numerical methods. Then, the results are presented in Section 3. We discuss the observability of stars forming in the simulated clouds enriched by faint SNe and other issues in Section 4. Finally, this paper is concluded in Section 5. Throughout the simulations, we adopt the cosmological parameters Ω_m = 0.3089, Ω_CDM = 0.2603, Ω_Λ = 0.6911, and H₀ = 67.74 km s⁻¹ Mpc⁻¹ (Planck Collaboration XIII 2016). We run the simulations in comoving coordinates but we describe physical quantities in proper coordinates throughout this paper, unless otherwise specified. All the figures in this paper are created with the yt toolkit (Turk et al. 2011).⁴

2 NUMERICAL MODELS

Since the numerical method is nearly the same as Paper I, we briefly describe it in Sections 2.1 and 2.2. We detail a faint SN model that is, for the first time, included in three-dimensional simulations of this work in Section 2.3.

2.1 Basic setup

We perform cosmological simulations with the adaptive mesh refinement (AMR) hydrodynamics code enzo (Bryan et al. 2014).⁵ The dynamics of DM is followed with an N-body particle-mesh solver (Efstathiou et al. 1985; Bryan & Norman 1997). The hydrodynamics equations are solved with the piecewise parabolic method (PPM) in an Eulerian frame (Woodward & Colella 1984; Bryan et al. 1995) and a Harten-Lax-van Leer-Contact (HLLC) Riemann solver, which accurately captures hydrodynamical shocks and computes advection of chemical species across contact discontinuities. We set the controlling parameters for the flux calculation and interpolation of field values between computational grids as |${\tt FluxCorrection} = 2$| and |${\tt ConservativeInterpolation} = 0$| to minimize numerical errors. Although errors for the total metal mass accumulates up to ∼7 per cent at ∼500 Myr after an SN explosion even with the above setting, this is below the typical observational error of stellar abundances (∼0.1 dex).

Computational cells are progressively refined by a factor of 2 in space when satisfying the following criteria:

• The baryon mass in a cell exceeds 3m_b,0 × 2^−0.2l on a refinement level l, where m_b,0 is the mean baryon mass on the root grid.

• The DM particle mass contained by a cell exceeds 3m_dm,0, where m_dm,0 is the mean DM mass on the root grid.

• The local Jeans length λ_J is resolved less than 64 cells.

The negative coefficient −0.2 in the spectral index of the criterion means (i) the super-Lagrangian refinement criterion for the gas component while the criterion (ii), the Lagrangian, for the DM. When the baryon density starts to increase in the runaway collapse phase, cells are refined mostly on the criterion (iii). This criterion warranties that the local Jeans length is resolved sufficiently to prevent spurious fragmentation (Truelove et al. 1997; Turk et al. 2012).

We generate the initial conditions of the simulations in a periodic box with a side of 300 kpc (comoving) with music (Hahn & Abel 2011). We initially run a DM-only simulation with a base resolution 64³ and identify the most massive halo with a mass |$3.8\times 10^{6} \ {\rm M_{\bigodot }}$| at redshift z = 10 with a Friends-of-Friends (FOF) algorithm. By initially refining the halo region with two additional AMR levels, i.e. with higher spatial resolution by a factor of 4, we restart the simulation adding the baryon component. With this zoom-in strategy, the effective resolution is 256³ and the minimum DM particle mass is |$53.4 \ {\rm M_{\bigodot }}$|⁠.

2.2 Simulations of CEMP star formation

In this section, we describe the numerical methods that follow the formation of a Pop III star (Section 2.2.1), its radiative and SN feedback (Section 2.2.2), and recollapse of an enriched cloud in the MH (Section 2.2.3).

2.2.1 Pop III star formation

The main coolant of a primordial cloud is hydrogen molecules (H₂) and, in some cases, hydrogen deuteride (HD) molecules. To calculate the fractions and cooling rates of H₂ and HD, we solve the non-equilibrium chemistry with a modified version of a chemistry/cooling library grackle (Smith et al. 2017, Paper I).⁶ We solve a chemical network of 49 reactions for 15 primordial species, e⁻, H⁺, H, H⁻, H|$_2^+$|⁠, H₂, D⁺, D, D⁻, HD⁺, HD, He, He⁺, He^{2 +}, and HeH⁺. This chemical network includes the collisional ionization/recombination of H/He and formation/dissociation of H₂/HD molecules. We compute the rates of radiative cooling: inverse Compton, bremsstrahlung, H/He line transition, H₂ ro-vibrational transition, and HD rotational transition cooling. When H₂ molecules form, the binding energy (4.48 eV per molecule) is converted into the thermal energy (see Omukai 2000). The continuum opacity κ_p of the primordial component is taken from Mayer & Duschl (2005). Throughout this paper, the mass fraction of hydrogen nuclei is X_H = 0.76 and number fraction of deuterium relative to hydrogen nuclei is y_D = 3.4 × 10⁻⁵.

We then simulate the formation of a Pop III star. In reality, a star forms from a hydrostatic core with a central density n_H ∼ 10¹⁹ cm⁻³, where gas cooling from the endothermic reaction of hydrogen molecular dissociation becomes ineffective (Larson 1969; Penston 1969). In this work, to save the computational cost, we put a Pop III star particle, representing a single star, with the following criteria:

• gas density exceeds 10⁶ cm⁻³;

• gas flow is convergent (⁠|$\nabla \cdot {{\boldsymbol v}} \lt 0$|⁠);

• the cooling time is less than the dynamical time;

• H₂ fraction exceeds a critical value (10⁻³).

Fig. 1 shows the snapshot of the MH just before the time t_form of Pop III star formation. At this time (z = 12.1), the DM and baryon mass is |$M_{\rm halo, dm}= 1.53\times 10^{6}$||${\rm M}_{\bigodot}$| and |$M_{\rm halo, b}= 2.57\times 10^{5} \ {\rm M_{\bigodot }}$|⁠, respectively, within the virial radius R_halo = 288 pc. The H₂ fraction reaches ∼10⁻³ through the H⁻ process (Omukai 2000) and the temperature at the centre decreases because of H₂ cooling. In this region, the above criteria for Pop III star formation are satisfied.

Figure 1.

Density-weighted projection of density, temperature, and H₂ abundance of the minihalo just before Pop III star formation (redshift z = 12.1) in a box with a side 1 kpc centred on the density maximum. The horizontal and vertical axes are, respectively, parallel to the major and minor axes of the momentum of inertia of the region with densities above 0.1 cm⁻³.

From the time t_form, we run three simulations, in each of which we put a single Pop III star with masses 13.5, 50, and |$80 \ {\rm M_{\bigodot }}$| in the MH (hereafter called the run F13, F50, and F80, respectively).

2.2.2 Radiation and SN feedback from a Pop III star

During the main sequence of the Pop III star, we solve the radiative transfer with a module moray (Wise & Abel 2011). The ionization photon emission rate Q(H) and the lifetime of the stars t_life are taken from Schaerer (2002) and reported in Table 1. We calculate the H, He, and He⁺ ionization rates by integrating the number of photons consumed by the atoms from the location of the star particle to each computational cell with the cross-section of Verner et al. (1996). The H₂ dissociation rate is computed from the column density of H₂ with a self-shielding function of Draine & Bertoldi (1996). With these rates, the photon reactions are consistently calculated, coupled with the non-radiative reactions.

Table 1.

Pop III models.

1Run	2MPopIII	3tlife	4Q(H)
	|$({\rm M_{\bigodot }})$|	(Myr)	(1048 s−1)
F13	13.5	11.8	1.09
F50	50	3.49	34.6
F80	80	3.03	84.9

Notes. (1) ID of runs; (2–4) mass, lifetime, and ionizing photon emission rate of our Pop III model stars.

After the time t_life, we uniformly add explosion energy and metals within a sphere with a radius R_sh = 10 pc centred on the star particle, mimicking the end of the free-expansion phase. We assume that the explosion energy has been completely converted into the thermal energy. We also assume that the ejecta has been completely mixed during the free-expansion phase and its elemental composition is uniform. The explosion energy E_SN and mass of ejected metals/grains of our models are described in Section 2.3 and listed in Table 2.

Table 2.

SN models.

1Run	2ESN	3Mcut	4Mmet	5MC	6MO	7MFe	8[C/Fe]	9MAC	|$^{10} {\cal D}_{\rm AC}$|	11rAC,cool
	(1051 erg)	|$({\rm M_{\bigodot }})$|	|$({\rm M_{\bigodot }})$|	|$({\rm M_{\bigodot }})$|	|$({\rm M_{\bigodot }})$|	|$({\rm M_{\bigodot }})$|		|$({\rm M_{\bigodot }})$|		(μm)
F13	0.5	1.7	0.119	0.079	0.039	1.06 × 10−6	4.62	1.70 × 10−2	1.49 × 10−8	0.096
F50	2.6	11	3.54	0.989	2.551	1.47 × 10−5	4.57	3.89 × 10−4	4.11 × 10−10	0.012
F80	5.2	22.5	4.31	1.089	3.213	1.05 × 10−5	4.75	3.74 × 10−2	2.51 × 10−8	0.064

Notes. (1) ID of runs; (2) explosion energy; (3) mass cut; (4–7) metal, carbon, oxygen, and iron mass in ejecta; (8) carbon-to-iron abundance ratio; (9) mass of amorphous carbon (AC); (10) mass ratio of AC to gas; (11) characteristic radius of AC grains (see the text).

2.2.3 Collapse of the enriched clouds

To include the chemical/thermal evolution of clouds enriched by faint SNe, we solve 40 reactions of 19 metal species: C⁺, C, CH, CH₂, CO⁺, CO, CO₂, O⁺, O, OH⁺, OH, H₂O⁺, H₂O, H₃O⁺, O|$_2^+$| , O₂, Si, SiO, and SiO₂. In our cooling model, we include C ii, C i, and O i fine-structure transition line cooling and CO, OH, and H₂O molecular rotational transition line cooling. The optical depth in each transition line is calculated with the Sobolev length approximation. To reduce the computational cost, we estimate the local velocity gradient to be |$|\nabla \cdot {{\boldsymbol v}}| = 1/3 t_{\rm ff}$|⁠, where t_ff = (3π/32Gρ)^1/2 is the free-fall time for a density ρ (Omukai 2000).

We model the chemistry and cooling of dust grains most carefully. In this work, we consider seven grain species: alumina (Al₂O₃), metallic iron (Fe), magnetite (Fe₃O₄), enstatite (MgSiO₃), forsterite (Mg₂SiO₄), amorphous carbon (AC), and silica (SiO₂) with arbitrary grain size distributions. Further, we include grain growth (accretion of gas-phase metals on to grains) as chemical reactions with a reaction rate given by the geometrical cross-section of the grains multiplied by the thermal velocity of metal molecules (Kozasa & Hasegawa 1987). The cooling rates, H₂ formation rates on grain surfaces, continuum opacity, and grain growth rates are calculated for each species. We calculate the continuum optical depth as τ_cont = (κ_pρ + ∑_iκ_iρ_i)λ_J with an opacity κ_i (Nozawa et al. 2008) and density ρ_i of a grain species i.

We terminate the simulations at a density n_H ∼ 10¹⁶ cm⁻³ above which further cloud fragmentation does not occur. In this optically thick region, radiative cooling becomes ineffective, and thus stable hydrostatic cores (first cores) form (Larson 1969).

We run the simulations on the PACE cluster in Georgia Institute of Technology on three nodes, using 28 cores/node. The maximum AMR level eventually reaches 32–34, corresponding to a spatial size of ∼0.01 au. Even protostars with a size ∼1 au can be resolved. The runs F13, F50, and F80 take 69, 47, and 41 d with total computational costs of 140 000, 94 000, and 83 000 core-hours, respectively.

2.3 SN models

We adopt faint SN models calculated by M14. In these models, explosion energy E_SN, ⁵⁶Ni mass, and initial ejection velocity are chosen so that the abundance ratios of heavy elements are fitted with the elemental abundance of the most iron-poor CEMP star, |$\rm SMSS \ J0313{-}6708$| (Keller et al. 2014), for each progenitor mass. The mass cut, M_cut, depends on the initial ejection velocity. The total mass of synthesized metals, M_met, is 0.119–|$4.31 \ {\rm M_{\bigodot }}$|⁠. Because of the fall back of the inner layers of the ejecta, the metal mass |$0.119 \ {\rm M_{\bigodot }}$| for F13 is smaller than that of normal CCSNe with the same progenitor mass (0.3–|$1 \ {\rm M_{\bigodot }}$|⁠). The masses of C, O, and Fe range between 0.079–|$1.089$|⁠, 0.039–|$3.213$|⁠, and 1.06 × 10⁻⁶–|$1.05\times 10^{-5} \ {\rm M_{\bigodot }}$|⁠, respectively. This gives a C-enhanced abundance ratio of [C/Fe] = 4.57–4.75. In these spherical explosion models, the explosion energy for F13 is smaller (0.5 × 10⁵¹ erg) than the normal CCSN model with the same progenitor mass used in Paper I by a factor of 2.

M14 follow the formation/destruction of dust grains in faint SNe. First, dust grains form in the expanding ejecta at the time ∼100 d after the SN explosion. At the time when gas temperature decreases below the sublimation temperature of grains (∼2000 K), seed clusters form through aggregation of monomers and grow by accreting the gas-phase species. In C-rich ejecta, AC is the dominant species while the mass fraction of other species is at most 10⁻⁵. The newly formed grains have an approximately lognormal size distribution function with a mean radius that depends on the C density and temperature at the grain formation time. Then, M14 follow the destruction of dust grains in SN remnants. When the ejecta sweeps up the circumstellar gas with a mass comparable to the ejecta mass, reverse shocks propagate to the region where grains have formed. Through the collision of high-energy ions, a fraction of the atoms on grain surfaces returns to the gas phase. Dust destruction occurs at ∼1000 years after the SNe. This should occur in the three-dimensional simulations but we do not include the dust destruction to save the numerical cost. Instead, we use the dust model of M14 as an initial condition.

The efficiency of grain destruction depends on the density ρ_amb of ambient gas around an SN. M14 calculate the grain properties for ρ_amb = 10⁻²⁵–10⁻²³ g cm⁻³ for each progenitor mass. We employ the model for ρ_amb = 10⁻²⁴ g cm⁻³. Table 2 shows the mass M_AC of AC grains produced by faint SNe after the formation/destruction processes. While the fraction of C locked up into AC grains is 4 and 20 per cent for F13 and M80, respectively, it is only 0.04 per cent for F50. The ejecta density is larger for F50 than for the other progenitor masses, and thus C atoms are oxidized to form CO molecules more rapidly than condensed into AC grains. We can expect that the low abundance of AC prevents the fragmentation of a cloud enriched by the SN. The dashed curves of Fig. 2 show the size distribution of AC grains after the destruction. When grains are sputtered, their size decreases. A tail appears on the smaller end of the initial lognormal size distribution. Following Chiaki et al. (2015, 2017), we also estimate a characteristic grain radius given by r_AC,cool = 〈r³〉_AC/〈r²〉_AC = 0.012–0.096 μm from the grain properties in our SN models, where 〈rⁿ〉_AC = ∫rⁿφ_AC(r)dr is the nth moment of size distribution function φ_AC(r) at a grain radius r normalized to unity.

Figure 2.

Distribution of the number fraction of AC grains relative to hydrogen nuclei against grain radii in the clouds enriched by Pop III SNe with progenitor masses |$13$| (orange), |$50 $| (purple), and |$80 \ {\rm M_{\bigodot }}$| (green). The dashed and solid curves denote the size distribution of grains in envelope and centre of the clouds before and after grain growth occurs, respectively.

3 RESULTS

3.1 Enrichment from faint SNe

The evolution of SN shells is affected by the density structure of the H ii region created by the ultraviolet (UV) radiation from the progenitor star. Panel (a) of Figs 3 and 4 shows the density, temperature, and carbon abundance, A(C), just after the SN. We do not show the plot for F50 because it is essentially the same as for F80. From these figures, we can see the structure of outer part of H ii region which has not been affected by the SN shock. For F13, the H ii region formed around the Pop III star has a radius ∼100 pc defined by a D-type shock front. In the region, the density and temperature become ∼0.01 cm⁻³ and 10⁴ K, respectively, and hydrogen atoms are fully ionized. For F50 and F80, the ionization front is decoupled from the D-type shock at a radius ∼100 pc and reaches ∼ kpc beyond the virial radius because of the large ionizing photon emission rates Q(H) = (3–8) × 10⁴⁹ s⁻¹.

Figure 3.

Slice of density, temperature, and C abundance for a progenitor model with a mass |$M_{{\rm pr}}= 13 \ {\rm M_{\bigodot }}$| at the time t_SN = 0.41 Myr (panel a), 15.22 Myr (panel b), and 29.16 Myr (panel c) after the SN explosion in a box with a side 2 kpc centred on the centroid of the MH. The axes of the windows are the same as Fig. 1.

Figure 4.

Same as Fig. 3 but for a progenitor mass |$80 \ {\rm M_{\bigodot }}$| at the time t_SN = 0.77 (panel a), 19.36 (panel b), and 79.77 Myr (panel c) from the SN explosion.

After the SN explosion, a hot bubble (∼10⁷ K) forms around the Pop III remnant. Consistent with analytical blastwave models, the temperature declines adiabatically until the time-scale for radiative cooling becomes smaller than the dynamical time of the shell (∼0.1–1 Myr). Then, the shell loses its energy through radiative cooling and its expansion is driven by the hot inner bubble in the pressure-driven snowplough (PDS) phase. The inertial force from the decelerated ejecta to the outer swept-up pristine materials drives Rayleigh–Taylor (RT) instabilities and metal mixing between the two regions.

After the inner hot bubble loses its thermal energy, the shell expansion is driven under the conservation of momentum (momentum-conserving snowplough, MCS phase). Figs 3(b) and 4(b) show that the temperature of the inner region begins to decline at a time t_SN = 10–20 Myr after the SN explosion. The RT fingers develop toward less dense regions in the SN shell. An anticorrelation between density and metal abundance can be seen. Then, the shell collides with dense clumps at distances of ∼100 pc on the filaments and loses its kinetic energy.

Finally, the clumps return to the MH as shown in Figs 3(c) and 4(c). The collision of the clumps induces the mixing of the metals that remain in the halo centre. For all models, the internal enrichment of the MH occurs and the enriched cloud begins to collapse gravitationally. We estimate the time t_ret when the density of the enriched cloud increases up to 10⁶ cm⁻³ to be 86.6, 304, and 564 Myr for F13, F50, and F80, respectively (see Table 3). For F50 and F80 where the H ii region expands beyond the virial radius of the MH, it is expected that the SN shell entirely expands into the void region (Kitayama & Yoshida 2005; Chiaki, Susa & Hirano 2018). We find that the gas infall continues along the cosmic filaments and a part of the shell eventually returns to the MH, although it takes longer time than for F13. For F80, the time t_ret = 564 Myr corresponds to redshift z_ret = 6.01. Because we do not consider a UV background, our results are only applicable to volumes that are not ionized by external sources. However, the same sequence can be generalized and is just as likely to occur slightly earlier (i.e. z ∼ 8) in a different large-scale environment.

Fig. 5 shows the temporal evolution of mass within the virial radius from t_form to t_ret. While the DM mass (solid curves) monotonically increases up to (2.72–|$7.47)\times 10^{6} \ {\rm M_{\bigodot }}$| through the accretion of DM and a halo merger (at z ∼ 8), the baryon mass (dashed curves) decreases from the time of SN explosions (represented by vertical dotted lines) because the gas is expelled from the MH. At ∼30 Myr after the explosion, the baryon mass turns to increase up to (3.00–|$2.73)\times 10^{5} \ {\rm M_{\bigodot }}$| because the pristine gas continues to be accreted into the MH. The accretion of pristine gas leads to the dilution of metals.

Figure 5.

Temporal evolution of DM and baryon mass within the virial radius of the MH (solid and dashed curves, respectively) from the time of Pop III star formation. The vertical dotted lines show the time of SN explosions for progenitor masses 13 (orange), 50 (purple), and |$80 \ {\rm M_{\bigodot }}$| (green).

Fig. 6 shows the distribution of A(C) as a function of distance from the centre of the enriched cloud. In the outskirt of the enriched cloud, the metal fraction as well as C abundance shows a large scatter. While the anticorrelation between density and metallicity is still present in some regions, these two quantities are correlated in the other regions where the metal-rich SN shell infalls into the MH (Fig. 4c). The metallicity converges within a radius ∼0.1 (for F13) and ∼10 pc (for F50 and F80) from the cloud centre, indicating the minimum length-scale of the fluctuation of metallicity. C and Fe abundances in the uniform metallicity region are ([Fe/H], A(C)) = (−9.25, 3.80), (−7.94, 5.06), and (−8.28, 4.90) for F13, F50, and F80, respectively. The dust-to-gas mass ratio is |${\cal D}_{\rm AC} = 1.49\times 10^{-8}$|⁠, 4.11 × 10⁻¹⁰, and 2.51 × 10⁻⁸ (Table 2). The comparison between the dust cooling rate and the gas compressional heating rate suggests that clouds with dust-to-gas mass ratios above |${\cal D}_{{\rm cr}}= (2.6$|–6.3) × 10⁻⁹ are likely to fragment into low-mass clumps (Schneider et al. 2012a). We now describe the succeeding runaway collapse phase of the enriched clouds, focusing on their thermal evolution and fragmentation properties.

Figure 6.

Distribution of C abundance A(C) as a function of distance from the density maximum of the enriched clouds for progenitor masses (a) 13, (b) 50, and (c) |$80 \ {\rm M_{\bigodot }}$| at the time when we terminate the simulations.

3.2 Recollapse of enriched clouds

In this section, we investigate the chemothermal evolution of the cloud enriched by faint SNe, comparing the results for the three progenitor models. Fig. 7 shows the thermal evolution of the clouds. In the earlier stage of the evolution where the density of cloud centre is below ∼10¹² cm⁻³, molecular cooling is dominant. Since metal molecules can contribute to gas cooling for C abundances A(C) ≳ 5, in Section 3.2.1, we discuss the thermal evolution in the early collapse stage, focusing on the difference between the two cases with

• lower C abundance (A(C) ∼ 4 for F13),

• higher C abundance (A(C) ∼ 5 for F50 and F80).

Figure 7.

Temporal evolution of temperature of the cloud cores enriched by Pop III SNe with progenitor masses 13 (orange), 50 (purple), and |$80 \ {\rm M_{\bigodot }}$| (green). We output snapshots at every time when the highest AMR level is refined and plot the average density and temperature in the region with densities >n_H,max/3, where n_H,max is the maximum density.

In the late collapse stage (n_H ≳ 10¹² cm⁻³), the thermal evolution mainly depends on the amount of AC grains through dust thermal emission cooling. In Section 3.2.2, we compare the two cases with

• lower dust amount (⁠|${\cal D}_{\rm AC} \sim 10^{-10}$| for F50),

• higher dust amount (⁠|${\cal D}_{\rm AC} \sim 10^{-8}$| for F13 and F80).

3.2.1 Early stage of collapse: n_H ≲ 10¹² cm⁻³

Figs 8 and 9 show the evolution of the abundances of electrons, ions, molecules, and grains and of cooling/heating rates, respectively, as a function of the central density of the cloud core. At densities n_H ∼ 1–10² cm⁻³, hydrogen molecules are the dominant coolant. H₂ molecules form mainly through the H⁻-process:

$$\begin{eqnarray*} {\rm H} + {\rm e^-} &\rightarrow & {\rm H^-} + \gamma , \nonumber \\ {\rm H^-} + {\rm H} &\rightarrow & {\rm H_2} + {\rm e^-}. \nonumber \end{eqnarray*}$$

The grain surface reaction also contributes to molecular formation. The H₂ fraction relative to hydrogen nuclei becomes ∼10⁻³ for all models. When the temperature drops below 150 K, HD molecules form through the reactions

$$\begin{eqnarray*} {\rm H_2} + {\rm D} &\rightarrow & {\rm HD} + {\rm H} , \nonumber \\ {\rm H_2} + {\rm D^+} &\rightarrow & {\rm HD} + {\rm H^+}. \nonumber \end{eqnarray*}$$

HD cooling becomes dominant at n_H ∼ 10²–10⁴ cm⁻³. Panels (a) of Figs 10 and 11 show the morphology of the cloud core for F13 and F80 models, respectively. We do not show the plot for F50 because the cloud morphology is similar to that for F80 in the early collapse stage. For all models, HD cooling leads to a bar-mode instability on a spherically symmetric core (Hanawa & Matsumoto 2000; Lai 2000). This cloud deformation is the precursor of fragmentation (Tsuribe & Omukai 2006). However, low-mass stars that can survive until the present day are unlikely to form through molecular cooling because the corresponding Jeans mass is still relatively large (M_J ∼ 10²–|$10^4 \ {\rm M_{\bigodot }}$|⁠) as indicated by the black dotted lines in Fig. 7.

Figure 8.

Evolution of the abundance of electrons, atoms, ions, molecules, and grains in the cloud cores enriched by the Pop III SNe with progenitor masses 13 (panel a), 50 (panel b) and |$80 \ {\rm M_{\bigodot }}$| (panel c) as a function of density of the cloud cores.

Figure 9.

Evolution of cooling and heating rates of the cloud cores enriched by Pop III SNe with progenitor masses (a) 13, (b) 50, and (c) |$80 \ {\rm M_{\bigodot }}$| as a function of core density. We plot the heating rates of adiabatic gas compression (‘adia’; red solid) and H₂ formation (‘form’; purple solid) and radiative cooling rates of H₂ (blue solid), HD (green solid), C i (black dotted), O i (orange dotted), CO (orange dashed), OH (green dashed), H₂O (cyan dashed), dust (grey solid), and CIE (blue dotted).

Figure 10.

Density-weighted projection of density and temperature in a box centred on the density maximum of the enriched clouds for a progenitor mass |$13 \ {\rm M_{\bigodot }}$|⁠. We plot the snapshots at the time when the maximum density n_H,max reaches ∼10⁷ (panel a), 10¹² (panel b), and 10¹⁶ cm⁻³ (panel c). The circle and plus symbols are plotted on the density maxima of a hydrostatic core and clumps forming through fragmentation induced by dust thermal emission cooling, respectively. The x- and z-axes of the windows are, respectively, parallel to the major and minor axes of the momentum of inertia of the region with densities >n_H,max/3.

Figure 11.

Same as Fig. 10 but for a progenitor mass |$80 \ {\rm M_{\bigodot }}$|⁠. In this run, fragmentation does not occur and thus only a hydrostatic core is shown by the circle.

At densities n_H ∼ 10⁴–10⁸ cm⁻³, the temperature is smaller (30 K) for F50 and F80 than for F13 (150 K) because OH cooling becomes dominant in the former case. OH and O₂ molecules form mainly through the reactions

$$\begin{eqnarray*} {\rm O} + {\rm H} \rightarrow {\rm OH} + \gamma . \nonumber \end{eqnarray*}$$

at a density ∼10⁴ cm⁻³. For F13, OH cooling is less efficient because the O abundance ([O/H] = −5.33) is smaller than for the other models ([O/H] = −3.34 and −3.44 for F50 and F80, respectively). Further, O is rapidly depleted into CO molecules because C has a larger abundance with respect over O ([C/O] = +0.691) compared to the other models ([C/O] = −0.0269 and −0.0855 for F50 and F80).

At densities n_H ∼ 10⁸–10¹² cm⁻³, the temperature increases rapidly due to gas heating along with H₂ formation through three-body reactions

$$\begin{eqnarray*} {\rm H} + {\rm H} + {\rm H} &\rightarrow & {\rm H_2} + {\rm H} , \nonumber \\ {\rm H} + {\rm H} + {\rm H_2} &\rightarrow & {\rm H_2} + {\rm H_2} . \nonumber \end{eqnarray*}$$

The temperature increase is more significant in models F50 and F80 than in model F13 because the temperature just before H₂ formation heating becomes dominant is smaller (∼100 K) owing to cooling from OH as well as H₂O molecules forming through the reaction

$$\begin{eqnarray*} {\rm OH} + {\rm H_2} \rightarrow {\rm H_2O} + {\rm H}. \nonumber \end{eqnarray*}$$

This rapid gas heating stabilizes the cloud against the deformation of the cloud core. A hot spherical hydrostatic core defined by accretion shocks forms at a radius ∼10⁻² pc for F80 (Fig. 11b), whereas density fluctuations remain within this radius scale through the moderate gas heating for F13 (Fig. 10b). Fig. 12 shows the radial profile of density, temperature, and radial velocity. The velocity increases inwards by ∼1 km s⁻¹ at a radius ∼10⁻² pc where temperature rapidly increases for F50 and F80 while the velocity smoothly decreases for F13.

Figure 12.

Radial profile of density, temperature, and radial velocity at the time when we terminate our simulations for progenitor masses 13 (orange), 50 (purple), and |$80 \ {\rm M_{\bigodot }}$| (green).

3.2.2 Late stage of collapse: n_H ≳ 10¹² cm⁻³

For F13, AC grains grow in size through accretion of gas-phase C atoms. Fig. 2 shows the size distribution of AC grains before (dashed curves) and after (solid curves) grain growth. The size distribution shifts toward the larger radii by Δr = 0.02 μm. Fig. 8(a) shows that the abundance of C atoms locked up into AC grains (black dotted curve) slightly increases at a density n_H ∼ 10¹³ cm⁻³. The dust-to-gas mass ratio increases from 1.49 × 10⁻⁸ to 3.57 × 10⁻⁸ (Tables 2 and 3) and the grain cooling rate is enhanced. For the other models, grain growth is not effective because C atoms are mostly oxidized into CO molecules at densities n_H ∼ 10⁵ cm⁻³ due to the larger O/C abundance ratio.

Table 3.

Abundance in the enriched clouds and fragmentation properties.

1Run	2tret	3zret	4Mhalo,dm	5Mhalo,b	6[Fe/H]	7A(C)	|$^{8} {\cal D}_{\rm AC}$|	9rAC,cool	10Nfrag	11Mps	12Rps
	(Myr)		(⁠|${\rm M_{\bigodot }}$|⁠)	(⁠|${\rm M_{\bigodot }}$|⁠)				(μm)		(⁠|${\rm M_{\bigodot }}$|⁠)	(au)
F13	86.6	10.2	2.72 × 106	3.00 × 105	−9.25	3.80	3.57 × 10−8	0.119	5	0.0421	1.17
F50	304	7.73	3.41 × 106	1.85 × 105	−7.94	5.06	4.52 × 10−10	0.012	1	0.0294	1.62
F80	564	6.01	7.47 × 106	2.73 × 105	−8.28	4.90	2.70 × 10−8	0.065	1	0.0138	0.868

Notes. (1) ID of runs; (2–3) time and redshift when SN shells return to original MHs; (4–5) dark matter and baryon mass of haloes at t_ret; (6–7) carbon and iron abundance in recollapsing clouds; (8–9) dust-to-gas mass ratio and characteristic grain radius of AC after grain growth; (10) number of fragments; (11–12) mass and radius of the first hydrostatic core.

At densities n_H = 10¹²–10¹⁴ cm⁻³, while the temperature evolves nearly isothermally for the model with low dust content (F50), the temperature decreases due to cooling by dust thermal emission for the models with high dust content (F13 and F80). The condition where the dust cooling rate exceeds the adiabatic compressional heating (the main heating source of the clouds) at a given density n_H and temperature T is

$$\begin{eqnarray*} \frac{{\cal D}_{\rm AC}}{r_{\rm AC, cool}} &\gt & 1.86\times 10^{-7} \ {\rm \mu m}^{-1} \left(\frac{T}{500 \ {\rm K}} \right) ^{-1/2} \nonumber \\ &&\times \left(\frac{n_{{\rm H}}}{10^{13} \ {\rm cm^{-3}}} \right) ^{-1/2} \end{eqnarray*}$$

(1)

(Schneider et al. 2012a; Chiaki et al. 2015). Assuming the values |${\cal D}_{\rm AC}$| and r_AC,cool achieved after grain growth (Table 3), we find that, for models F13 and F80, |${\cal D}_{\rm AC}$| / r_AC,cool = 3.00 × 10⁻⁷ and 4.15 × 10⁻⁷ μm⁻¹, respectively, above the critical value while, for F50 3.77 × 10⁻⁸ μm⁻¹, below the critical value.

Finally, in the region with densities >10¹³ cm⁻³, since the grains become thermally coupled with the gas, the temperature starts to increase again. Then, the gas becomes optically thick to continuum radiation. The cloud core evolves nearly adiabatically and a hydrostatic core, which can not contract further, forms as we can see at a radius ∼1 au (∼10⁻⁶ pc) in Fig. 12 (Larson 1969; Penston 1969).

3.3 Fragmentation of enriched clouds

Whether fragmentation occurs in the later stage of collapse or not depends on the different thermal evolution among the three progenitor models. For F50, cloud fragmentation does not occur because of inefficient dust cooling (equation 1). Fig. 13 shows the snapshot of the cloud core in the later collapse stage for F50. A disc structure forms around a central hydrostatic core (circle) as for primordial clouds (Clark et al. 2011; Greif et al. 2012; Hirano & Bromm 2017) because the temperature evolution in the later stage is similar to the primordial clouds as H₂ cooling is dominant (Fig. 9b). In Fig. 13, we show the disc from the face-on and edge-on views. The disc is gravitationally unstable and dense spiral arms form. The arms might fragment in the further evolution of the accretion disc as we discuss in Section 4.

Figure 13.

From the left- to right-hand side: density-weighted projection of density, temperature, and the Toomre Q parameter (see Section 4.2.3) in a box with a side 50 au centred on the density maximum of the enriched clouds for a progenitor masses |$50 \ {\rm M_{\bigodot }}$|⁠. The top and bottom panels of each column show the disc from the face-on and edge-on view, respectively. The circle denotes the radius of a hydrostatic core forming in the central optically thick region.

For higher dust content, although dust cooling becomes effective in both runs, the cloud core fragments into isolated filamentary structures for F13 (Fig. 10c) while fragmentation does not occur for F80 (Fig. 11c). In the latter case, only short filaments connected with the central core appear. For the filaments to fragment, their aspect ratio should grow up to 20–30 (Tsuribe & Omukai 2006; Chiaki, Yoshida & Hirano 2016). In this model, rapid gas heating along with H₂ molecular formation has made the cloud stable against the deformation at densities n_H = 10⁸–10¹² cm⁻³ before dust cooling occurs. Therefore, the time-scale for deformation of the spherical cloud core to sufficiently thin filaments is longer than the dynamical time-scale of the cloud, while the opposite is true for F13 model. As seen in Section 3.2.1, this difference of heating rate is due to the different cooling efficiency of metal molecules (OH/H₂O) before the onset of H₂ formation heating.

Our detailed chemistry/cooling model shows that the fragmentation properties of the clouds enriched by faint SNe depend on multiple cooling/heating processes (1) dust thermal emission cooling, (2) H₂ formation heating, and (3) OH/H₂O molecular cooling (see also Chiaki et al. 2016).

More quantitatively, we count the number and mass of clumps within 100 au from the maximum density with a clump-finding algorithm. This identifies isolated clumps by using iso-density contours (Smith et al. 2009; Turk et al. 2011). If a candidate clump does not meet the user-defined validators, it is merged with another clump or discarded. In this work, we impose the following three criteria:

• the minimum number of cells in a clump is 20;

• the minimum density of the cell in a clump is n_H,th = 10¹³ cm⁻³;

• the minimum mass of a clump is |$M_{\rm {th}} = 10^{-3} \ {\rm M_{\bigodot }}$| (∼1 Jupiter mass).

In the criteria (ii), we choose the threshold density n_H,th = 10¹³ cm⁻³, where dust cooling becomes dominant, to select clumps formed through dust-induced fragmentation. Here we do not require the clumps to be gravitationally bound. For the most massive clump, which has developed a hydrostatic core, we estimate the mass M_ps and radius R_ps above which the gravitational energy |$3 G M_{\rm ps}^2 / 5R_{\rm ps}$| exceeds the total thermal energy. We hereafter call the primary clump as ‘protostar’. Since the computational time is set by the short dynamical time of the dense protostar, when we terminate the simulation the secondary clumps are still in the early phase of their gravitational growth. Therefore, in this work, we just show their mass derived from the clump-finding algorithm. We hereafter call the secondary clumps as just ‘clumps’.

For F13, the number N_frag of identified clumps is 5 (Table 3). The circle and plus symbols in Fig. 10(c) depict the distribution of the primary protostar and the secondary clumps, respectively. The mass and radius of the protostar are (M_ps, R_ps) = (⁠|$0.0421 \ {\rm M_{\bigodot }}$|⁠, 1.17 au). The masses of the four clumps are 1.06 × 10⁻², 2.35 × 10⁻³, 1.55 × 10⁻³, and |$1.21\times 10^{-3} \ {\rm M_{\bigodot }}$|⁠. The separation of clumps (∼10 au) and mass of the protostar are comparable to the local Jeans length and mass for n_H = 10¹⁴ cm⁻³ and T = 1000 K, which indicates that the fragmentation is induced by dust cooling. For F50 and F80, the clump-finding algorithm can not identify any secondary clumps (N_frag = 1). The mass and radius of the protostar are |$0.0294 \ {\rm M_{\bigodot }}$|⁠, 1.62 au and |$0.0138 \ {\rm M_{\bigodot }}$|⁠, 0.868 au for F50 and F80, respectively.

Although these protostars and clumps will gain mass by accreting the ambient gas in the subsequent evolution until they reach the main-sequence as discussed in Section 4, dust-induced low-mass fragments can form in model F13.

4 DISCUSSION

4.1 Prediction of observations

We have so far presented the results of our simulation of metal enrichment from faint SNe and the fragmentation properties of the enriched clouds. If the protostars in the clouds grow to low-mass stars and the MH is accreted into the Milky Way halo or Local Group dwarf galaxies through DM halo mergers, they can be observed as EMP stars. In this section, we discuss their observability. The colored symbols in Fig. 14 depict the distribution of elemental abundances of the enriched clouds on the A(C)–[Fe/H] diagram used in Yoon et al. (2016). We also plot the abundances of protostars forming in the cloud enriched from a normal CCSN with an |$M_{\rm Pop III}= 13 \ {\rm M_{\bigodot }}$| (black square) in the simulation of our previous study (Paper I), hereafter called the run C13. The filled and open symbols indicate the runs where fragmentation occurs and not, respectively.

Figure 14.

Elemental abundances of clouds enriched by faint SNe for progenitor masses |$M_{\rm Pop III}= 13$| (orange diamond), 50 (purple triangle), and |$80 \ {\rm M_{\bigodot }}$| (green circle) on the A(C)–[Fe/H] plane. The black square shows the C and Fe abundances of the clouds enriched by a normal CCSN with an |$M_{\rm Pop III}= 13 \ {\rm M_{\bigodot }}$| from our previous simulation (Paper I). The filled and open symbols indicate the runs where cloud fragmentation occurs or not, respectively. The dots show the abundances of observed stars retrieved from the SAGA data base (Suda et al. 2008): Blue dots show C-normal stars ([C/Fe] < 0.7) or stars whose C abundances are not constrained, green dots show CEMP-s stars ([C/Fe] > 0.7 and [Ba/Fe] > 0.5), and red dots show CEMP-no stars ([C/Fe] > 0.7 and [Ba/Fe] < 0.5) or CEMP stars whose Ba abundances are not constrained. We also plot the criterion for CEMP stars ([C/Fe] = 0.7; Aoki et al. 2007). The black solid line shows the critical condition defining the carbon grain dominant region (red shaded) and silicate dominant region (blue shaded). In the grey shaded region below the black solid curve, no stars have been identified, which may suggest that dust cooling is ineffective in their parent clouds and low-mass stars hardly form (C17).

For comparison, we plot the abundances of observed EMP stars from the SAGA data base (Suda et al. 2008). The blue dots represent the C-normal stars with [C/Fe] < 0.7 or stars whose C abundances are not constrained. The other colors represent C-enhanced stars with [C/Fe] > 0.7, following the definition of Aoki et al. (2007). The green and grey dots represent CEMP-s with [Ba/Fe] > 0.5 and CEMP-no with [Ba/Fe] < 0.5 or stars whose Ba abundances are not constrained, respectively. As C17 found, no stars have been observed in the grey shaded area bounded by the black solid curve.

From a comparison of our simulation results with the observational data, we can see that the abundances for C13 ([Fe/H], A(C)) = (−3.62, 4.99) are in the range of observed C-normal stars ([Fe/H] > −5) and can explain the formation of C-normal stars, whereas the abundances for F13, F50, and F80 ([Fe/H] = [−9.25: −7.94] and A(C) = [3.80: 5.06]) are below the range of observed CEMP Group III stars (A(C) ≳ 6), although the abundance ratio [C/Fe] = 4.57–4.75 is consistent with the observed values.

We now discuss the reason why the elemental abundances for the faint SN models are systematically smaller than those for the normal CCSN model. The elemental abundances of recollapsing clouds depend on the following three parameters (Ritter et al. 2015; Paper I). First, the mass of an element M ejected from SNe, M_M, is an important value. Secondly, the fraction f_ret of metals that return to the MH determines the elemental abundances of the enriched clouds. We can estimate this as the fraction of solid angles where metal dispersion is interrupted on the contact surfaces between the SN shell and cosmic filaments or interstellar gas clumps. Finally, the dilution efficiency of the metals is important, i.e. the mass of pristine gas with which the metals are mixed. The simplest estimate is to use the pristine gas mass within the virial radius, similar to semi-analytic works (e.g. de Bennassuti et al. 2017; Hartwig et al. 2018). Since the distribution of elements is not uniform within the virial radius of the MH, we consider only the dense central cloud where the elemental abundances are uniform (Fig. 6). In this work, we define the pristine gas mass as the Jeans mass M_cloud, at the density 10³ cm⁻³ (see the black dotted lines in Fig. 7). With these three parameters (M_M, f_ret, and M_cloud), we can estimate C and Fe abundances in the enriched cloud to be

$$\begin{eqnarray*} A ({\rm C}) &=& 12 + \log \left(\frac{f_{\rm ret}M_{\rm {C}} / \mu _{\rm {C}}}{X_{{\rm H}}M_{\rm {cloud}}} \right), \end{eqnarray*}$$

(2)

$$\begin{eqnarray*} {\rm [Fe/H]} &=& 12 + \log \left(\frac{f_{\rm ret}M_{\rm {Fe}} / \mu _{\rm {Fe}}}{X_{{\rm H}}M_{\rm {cloud}}} \right) - A_{\bigodot }({\rm {Fe}}), \end{eqnarray*}$$

(3)

where μ_C = 12 and μ_Fe = 56 are the molecular masses of C and Fe, respectively.

For C13, the ejected mass of C and Fe is (M_C, M_Fe) = (0.5 |${\rm M_{\bigodot }}$|⁠, |$0.2 \ {\rm M_{\bigodot }}$|⁠). The metal dispersion is interrupted by a clump with a distance D_cl = 50 pc and radius R_cl = 6 pc. Therefore, |$f_{\rm ret}= \pi R_{\rm cl}^2 / 4\pi D_{\rm cl}^2 \sim 0.004$|⁠. With a cloud mass |$M_{\rm {cloud}} = 2000 \ {\rm M_{\bigodot }}$|⁠, we can estimate the abundances as ([Fe/H], A(C)) = (−3.6, 5.0), consistent with the simulation result.⁷ Even with the small f_ret, we can explain the formation of C-normal EMP stars. For F13, the mass of ejected metals are (M_C, M_Fe) = (⁠|$0.08 \ {\rm M_{\bigodot }}$|⁠, |$10^{-6} \ {\rm M_{\bigodot }}$|⁠). Since the density structure of ISM is similar to that for C13, the return fraction is f_ret ∼ 0.004. With the cloud mass |$M_{\rm {cloud}} = 4000 \ {\rm M_{\bigodot }}$|⁠, ([Fe/H], A(C)) = (−9.2, 3.9). Compared to C13, the ejected C mass is smaller by a factor of 6. Also, the cloud mass is larger because the gas temperature is higher due to the smaller metallicity. Consequently, the solid angle where metal dispersion is interrupted becomes smaller.

For F50 and F80, the metal yield is (M_C, M_Fe) = (⁠|$1 \ {\rm M_{\bigodot }}$|⁠, |$10^{-5} \ {\rm M_{\bigodot }}$|⁠). The radius of the interacting region of the shell with the ambient clump is larger D_cl = 100 pc than for F13 because filaments and clumps within R_cl are photoevaporated by the strong UV emission. It takes longer time until the SN shells return and the clump has gravitationally grown to R_cl = 7 pc. With |$f_{\rm ret}= \pi R_{\rm cl}^2 / 4\pi D_{\rm cl} \sim 0.001$| and |$M_{\rm {cloud}} = 1000 \ {\rm M_{\bigodot }}$|⁠, ([Fe/H], A(C)) = (−8.0, 5.1), which is consistent with our simulation results. Compared to C13, the ejected C mass is larger by a factor of 3. However, f_ret is smaller by a factor of 3 because the radius of the region within which metals are mixed is larger with larger explosion energies (E_SN = (2.6–5.2) × 10⁵¹ erg). Consequently, A(C) in the enriched clouds becomes comparable to that for C13. The value is smaller than A(C) of the observed CEMP Group III stars.

For the MH and Pop III faint SNe investigated here, the enrichment from a single faint SNe can hardly explain the formation of CEMP stars with the observed metallicity range. We discuss their formation paths in general cases in Section 4.2.1.

We also find that cloud fragmentation occurs through dust cooling for F13 even with the lowest metallicity ([Fe/H] ∼ −9). This is because the condensation efficiency of C into AC grains is large (f_AC,C = 0.262) and the cloud avoids rapid H₂ formation heating (Section 3.2.1). This indicates that low-mass stars with a metallicity [Fe/H] ∼ −9 can form but the abundance is below the region where CEMP stars have been observed so far (grey shaded region in Fig. 14).

C17 estimated the effective grain radius of AC to be r_AC,cool/f_AC,C ∼ 10 μm at which the dust cooling rate overcomes the compressional heating rate (equation 1) above the critical C abundance A_cr(C) ∼ 6. The dust model of M14 used in this work suggests the formation of smaller grains with r_AC,cool/f_AC,C = 0.454 μm for F13 in faint SNe. Grains smaller than 10 μm can induce cloud fragmentation for C abundances lower than A_cr(C) ∼ 6.

A semi-analytic study of galactic chemical evolution including the contributions from faint SNe (Komiya et al. 2020) gives a consistent result with our simulations. They point out that faint SNe might overproduce stars with significantly small metallicities ([Fe/H] < −7) compared to the metallicity distribution function (MDF) of observed stars because these SNe produce smaller mass of metals than normal CCSNe (see also de Bennassuti et al. 2014, 2017). Unfortunately, the MDF in the low metallicity end still has a large uncertainty because the number of observed stars with [Fe/H] < −4.5 is small (∼10). Future surveys and spectroscopic studies with Subaru/PFS (Takada et al. 2014) and the Thirty-Meter Telescope (TMT; Skidmore et al. 2015) will probe the chemical enrichment process in the early Universe.

4.2 Caveats

4.2.1 General initial conditions

In this work, we have investigated the metal enrichment from faint SNe with progenitor masses M_PopIII = 13, 50, and |$80 \ {\rm M_{\bigodot }}$| in an MH with an |$M_{\rm halo}= 2.1\times 10^{6} \ {\rm M_{\bigodot }}$|⁠. The MH is relatively high-mass, compared to the mass range of MHs derived from large-volume cosmological simulations, 2 × 10⁵–|$3\times 10^{6} \ {\rm M_{\bigodot }}$| (Susa, Hasegawa & Tominaga 2014; Hirano et al. 2014, 2015). Chiaki et al. (2018) investigated CEMP star formation for different initial conditions with M_halo = 3 × 10⁵–|$3\times 10^{6}$| and M_PopIII = 20–|$30 \ {\rm M_{\bigodot }}$|⁠, and found that the elemental abundances in recollapsing clouds are larger for lower mass MHs. For a high-mass MH (⁠|$3\times 10^{6} \ {\rm M_{\bigodot }}$|⁠), the SN shell becomes gravitationally unstable during its expansion because the H ii region does not expand outside the virial radius, and the shell rapidly loses its thermal energy once it encounters the neutral medium. The SN shell enriches a nearby cloud and collapses away from the explosion site, offset from the halo centre. A smaller fraction of metals mixes into the cloud that has already begun to collapse. Consequently, the elemental abundances are smaller than the low-mass case (⁠|$3\times 10^{5} \ {\rm M_{\bigodot }}$|⁠), where the explosion disrupts the MH and recollapses into the halo centre. For low-mass MHs, the iron abundance can be estimated to be in the range [Fe/H] = [ − 8.63: −4.34], and thus the formation of CEMP stars can be explained by the enrichment from a single faint SN in several cases. In Chiaki et al. (2018), the resolution was limited (∼1 pc) compared to this work, and the formation of dust grains was not considered. We will investigate CEMP star formation for a wide range of MH and Pop III stellar masses with a resolution similar to the present study in large cosmological simulations in future works.

4.2.2 Multiple source of metals

In this work, we restrict our focus on Pop III star formation in a single MH. There are several MHs around the central MH within 1 kpc (Fig. 1) and they might contribute to their mutual enrichment (Smith et al. 2015). For F80, it takes 500 Myr until the metals from the central MH return and, during this time interval, the metals ejected by other MHs may possibly enrich the MH. The effect of multienrichment will be to create a superposition of elemental abundances having multiple SN progenitors (Salvadori et al. 2007, 2010; de Bennassuti et al. 2015, 2017; Hartwig et al. 2018). However, in this work, we consider the formation of a single Pop III star in the MH. Numerical simulations of primordial clouds suggest that the stellar cluster formation through disc fragmentation may occur (Turk, Abel & O’Shea 2009; Clark et al. 2011; Greif et al. 2012). A cosmological simulation has also indicated that multiple Pop III stars form in each MH (Skinner & Wise 2020). In the case of multiple SNe in an MH, Ritter et al. (2015) found that the gas in the MH is evacuated by the first SN blastwave but a non-negligible fraction (∼0.3) of metals ejected from the secondary SNe contribute to the enrichment of the MH.

In galactic archaeology, the number of progenitors of EMP stars is important value as the mass and explosion energy of the Pop III stars can be directly traced back from the elemental abundances of singly enriched stars (Hartwig et al. 2018; Ishigaki et al. 2018; Choplin et al. 2019). In their semi-analytic studies, de Bennassuti et al. (2017) predict that stars with metallicities [Fe/H] ≲ −5 can be the very second generation of stars, which is the metallicity range that we have investigated. In order to study the formation of stars with larger metallicities, we will investigate multiple star formation in each MH and mutual metal enrichment between close MHs in cosmological simulations to see the frequency of multienrichment in forthcoming papers.

4.2.3 Growth of protostars

When we terminate the simulations, protostars have formed in the optically thick region. Their mass is |${\sim}0.01 \ {\rm M_{\bigodot }}$|⁠, comparable to the Jeans mass with density ∼10¹⁴ cm⁻³ and temperature ∼1000 K. We should note that, until the protostars reach the zero-age main sequence, their masses grow through the accretion of the ambient materials (Hosokawa et al. 2011; Hirano et al. 2014). For F13, if the protostars remain low-mass, with a mass less than |$0.8 \ {\rm M_{\bigodot }}$| despite competitive accretion among the fragments, the stars will be observed in the present day. For F50 and F80, gas accretion is partly halted by ionization feedback from the protostar but the protostar can grow up to |${\sim}100 \ {\rm M_{\bigodot }}$| with a sufficiently large accretion rate |${\sim}10^{-3} \ {\rm M_{\bigodot }}\,{\rm yr}^{-1}$| (Fukushima et al., 2020). The massive star will explode as an SN and contribute to the enrichment of the succeeding generation of stars.

We should also note that the accretion discs forming around the protostars are in most cases gravitationally unstable to fragment (Turk et al. 2009; Clark et al. 2011; Greif et al. 2012). This disc fragmentation can occur even in the primordial clouds. This fragmentation mode is distinctive from cloud fragmentation induced by radiative cooling (Chiaki et al. 2016). Although the critical conditions for disc fragmentation are still debated (Chon & Hosokawa 2019; Liao, Turk & Schive 2019; Inoue & Yoshida 2020), Takahashi, Tsukamoto & Inutsuka (2016) find that a spiral arm structure developing in a disc can fragment when its minimum Toomre Q = c_sΩ_epi/πGΣ becomes less than 0.6, where Ω_epi is the epicyclic frequency and Σ is the column density in the perpendicular direction of the disc. For F50, spiral arms developed (Fig. 13). We estimate the Toomre Q parameter by approximating Ω_epi = 2Ω, where Ω is the gas angular momentum, and we find that Q in the spiral arms is below 0.6 (blue coloured region in the upper right-hand panel of Fig. 13), which suggests that the arms will fragment. However, since the gas accretion rate on to the protostar is higher (⁠|${\sim}c_{\rm s}^3 /G \sim 3\times 10^{-3} \ {\rm M_{\bigodot }}\,{\rm yr}^{-1}$|⁠) than in present-day star-forming regions (⁠|${\sim}10^{-5} \ {\rm M_{\bigodot }}\,{\rm yr}^{-1}$|⁠), disc fragmentation might result in the formation of a massive star cluster.

It is numerically challenging to follow the entire accretion process (∼10⁵ yr) because the computational time-step becomes extremely short (≲ yr) to meet the Courant condition of the dynamics of dense protostellar cores. In order to follow the late phases of accretion and disc fragmentation, in a future study, we will continue the simulations by using additional numerical techniques such as sink particles, with which the dense region is masked and replaced with Lagrangian particles.

4.2.4 Formation of EMP stars with peculiar abundances

In this series of papers, we have investigated the formation processes of C-normal and CEMP-no stars. For the formation of other classes of EMP stars, additional nucleosynthesis mechanisms are required to be considered. First, s-process elements enhancement of CEMP-s (CEMP Group I) stars can be explained by the binary mass transfer from companion stars in the asymptotic giant branch (AGB) phase (Suda et al. 2004; Komiya et al. 2020). Since the progenitors are low-mass (1–|$7 \ {\rm M_{\bigodot }}$|⁠), the companion stars begin to contribute to the enrichment in more massive and metal-rich haloes ([Fe/H] ≳ −3; Fig. 14) than MHs. Therefore, to see the evolution of low-mass stars and effect of binary transfer, it is required to extend our simulations spatially and temporarily.

Also, rotating stars (‘spinstars’) can produce lighter elements (C–Al) and trans-iron elements in their envelopes (Meynet et al. 2006; Choplin & Hirschi 2020). The envelopes are blown away by SN explosions and contribute to C and s-process element enhancement (Choplin et al. 2019). Since rotating stars also produce nitrogen, the formation of N-enhanced stars such as CS 22952−015 and BS 16550−087 could be explained with the spinstar models. Some spinstars are considered to be associated with jet-like SNe (Tominaga 2009). This aspherical explosion model can reproduce Fe-deficient abundance patterns even with larger explosion energies than spherical explosion models because Fe can fall back on to the compact remnants from the equatorial plane. As a result, the elemental abundances of the ejecta depend on the direction from the jet axis. Ezzeddine et al. (2019) speculate that Zn-enhancement of the star HE 1327−2326 could be explained by the external enrichment of a cloud in a specific angle range from the jet axis of an SN. We will explore the effect of the aspherical structure of ejecta on the metal enrichment in forthcoming papers.

5 CONCLUSION

CEMP stars, the most iron-poor class of stars, hold fossil records of the chemical enrichment in the early Universe (de Bennassuti et al. 2017), and their origin is still under debate. In this work, we follow the metal enrichment from faint SNe and gravitational collapse of the enriched clouds with high-resolution numerical simulations. We find that C abundance of the clouds is A(C) = 3.80, 5.06, and 4.90 for faint SN models with progenitor masses M_PopIII = 13, 50, and |$80 \ {\rm M_{\bigodot }}$|⁠, respectively. These are smaller than the abundance range of observed CEMP stars (A(C) > 6), while the abundance of a cloud enriched by a normal CCSN is consistent with the metallicity range of observed C-normal stars (Paper I). This behaviour occurs because the mass of metals produced by faint SNe and the fraction of metals incorporated into the clouds are smaller than in normal CCSN model. We also find that our detailed chemistry/cooling model enables us to follow the entire process of fragmentation. AC grain cooling induces fragmentation of the enriched cloud for model F13 even with the lowest C abundance A(C) = 3.80. This result indicates that low-mass stars with metallicities [Fe/H] ∼ −9 may form through cooling by carbon dust grains produced by faint SNe and re-accreted on to the same MH. Such ‘giga metal-poor’ stars may be observed in the future as a larger number of samples of EMP stars will be collected.

So far, ∼5000 metal-poor stars have been observed. With future observational facilities, Subaru/PFS (Takada et al. 2014) and TMT (Skidmore et al. 2015), samples will increase. To prepare for future surveys, it is urgent to study the origin of metal-poor stars from the theoretical side. We have focused on the formation process of C-normal and CEMP-no stars in this series of papers. The other classes of stars, which show neutron-capture element enhancement and peculiar elemental abundances, are believed to form by multiple types of progenitors. We will extend our numerical model to enable comprehensive studies of their formation processes, while being constrained by large samples of observed metal-poor stars. Such a study should reveal elusive details of the chemical enrichment and star formation in the early Universe.

ACKNOWLEDGEMENTS

GC is supported by Overseas Research Fellowships of the Japan Society for the Promotion of Science (JSPS). JHW is supported by National Science Foundation grants AST-1614333 and OAC-1835213, and NASA grants NNX17AG23G and 80NSSC20K0520. The simulation was performed with NSF’s XSEDE allocation AST-120046 on the Comet and Stampede2 resources and also on the Georgia Tech PACE compute system. The figures in this paper are constructed with the plotting library matplotlib (Hunter 2007).

DATA AVAILABILITY

The versions of enzo, grackle, and yt used in this work are available at https://github.com/genchiaki/enzo-dev/tree/metal-dust, https://github.com/genchiaki/grackle/tree/metal-dust, and https://github.com/genchiaki/yt/tree/metal-dust, respectively. The script for yt used in this work is available at https://github.com/genchiaki/Analysis_CEMP. The simulation data will be shared on reasonable request to the authors.

Footnotes

1

The logarithmic number abundance ratio between elements A and B relative to solar one |${\rm [A/B]} = \log (n_{\rm A}/n_{\rm B}) - \log (n_{\rm A}/n_{\rm B})_{\bigodot }$| is often used to measure the metal content and peculiarity of elemental abundance ratio of a star. We also use the logarithmic abundance A(M) = log ϵ(M) = 12 + log (n_M/n_H) of an element M. We hereafter use the solar abundance of Asplund et al. (2009).

2

Instead, Group I stars are distributed mostly at intermediate metallicities, with [Fe/H] ∼ −2, and show also s-process element enhancement. Because of this, their carbon enhancement is believed to originate from mass transfer from an AGB companion in a binary system (Suda et al. 2004). Alternatively, Group I stars could form in regions that have been previously enriched by the explosion of rotating massive stars, which could produce C/N and s-process elements in their envelopes (Meynet et al. 2006; Choplin, Tominaga & Ishigaki 2019).

3

In Paper I, we presented the iron abundance of the recollapsing cloud as [Fe/H] = −3.42. In the simulation, we used the solar metallicity |${\it Z}_{\bigodot } = 0.012\,95$| as in this work, and the metallicity of the recollapsing cloud was |${\it Z} = 3.4\times 10^{-6} = 2.6\times 10^{-4}\,{\rm {\it Z}_{\bigodot }}$|⁠. However, when we converted Z from |${\rm [{Fe}/H]} = 12 + \log (X({\rm {Fe}}) Z / \mu _{\rm Fe} X_{{\rm H}}) - A_{\bigodot }({\rm {Fe}})$|⁠, we used |${\rm {\it Z}_{\bigodot }}= 0.02$|⁠. If we use |${\rm {\it Z}_{\bigodot }}= 0.012\,95$|⁠, [Fe/H] is consistently estimated to be −3.62.

4

https://yt-project.org/.

5

https://enzo-project.org/.

6

https://grackle.readthedocs.io/.

7

Compared to the estimation in Paper I, we improve the approximation accuracy, and thus the estimation is slightly changed.