Dust formation in primordial Type II supernovae

Abstract

We have investigated the formation of dust in the ejecta of Type II supernovae (SNe), mostly of primordial composition, to answer the question of where the first solid particles are formed in the Universe. However, we have also considered non-zero progenitor metallicity values up to Z= Z_⊙. The calculations are based on standard nucleation theory, and the scheme has been tested for the first time on the well-studied case of SN1987A, yielding results that are in agreement with the available data. We find that: (i) the first dust grains are predominantly made of silicates, amorphous carbon (AC), magnetite and corundum; and (ii) the largest grains are the AC ones, with sizes around 300 Å, whereas the other grain types have smaller radii, around 10–20 Å. The grain size distribution depends somewhat on the thermodynamics of the ejecta expansion, and variations in the results by a factor ≈2 might occur within reasonable estimates of the relevant parameters. Also, and for the same reason, the grain size distribution is essentially unaffected by metallicity changes. The predictions on the amount of dust formed are very robust: for Z=0, we find that SNe with masses in the range (12–35) M_⊙ produce about 0.08 M_⊙≲M_d≲0.3 M_⊙ of dust per supernova. The above range increases by roughly three times as the metallicity is increased to solar values. We discuss the implications and the cosmological consequences of the results.

stars: formation, supernovae: general, supernovae: individual: SN1987A, dust, extinction, supernova remnants

1 Introduction

Our understanding of galaxy formation is currently making tremendous advances, and recent investigations have also focused on the formation of the first luminous sources (often referred to as Population III objects). Several difficult questions arise when one deals with these peculiarly small collapsed objects (for a discussion see Ferrara 2000), primarily concerning the properties of their first stars and initial mass function (IMF) (Tegmark et al. 1997; Abel et al. 1998; Omukai & Nishi 1999; Bromm, Coppi & Larson 2000; Susa & Umemura 2000; Ripamonti et al. 2000), their response to the energy injection of supernovae (SNe) (MacLow & Ferrara 1999; Ciardi et al. 2000a), their ability to form and preserve enough H₂ to provide the cooling for collapse (Ciardi, Ferrara & Abel 2000b; Haiman, Abel & Rees 2000; Machacek, Bryan & Abel 2001), and their contribution to the reionization (Gnedin & Ostriker 1997; Gnedin 2000; Ciardi et al. 2000a, 2001) and metal enrichment (Ferrara, Pettini & Shchekinov 2000) of the intergalactic medium (IGM).

In spite of this flourishing activity, little attention has been given to the role of dust in these early epochs. At lower redshifts the dramatic effects of dust have been appreciated when estimates of the cosmic star formation rate (SFR) were attempted via ultraviolet/visible surveys of distant galaxies. It was soon realized that approaches based on the ‘dropout’ technique are poorly sensitive to dust-obscured galaxies. Hence, the SFR deduced in this way could represent a severe underestimate of the actual one, if even a rather modest amount of dust is present in the interstellar medium of the star-forming galaxy. Also, some galaxies could be so heavily extinguished that they could be completely missed from the UV/visible census (Cimatti et al. 1997; Ferrara et al. 1999a,b).

Direct indications of the existence of dust at high redshift come from the reddening of background quasars; indirect evidence of dust in damped Lyα systems has been obtained from the relative gas-phase abundances of Zn and Cr (Pettini et al. 1997). Fall, Charlot & Pei (1996) have calculated the cosmic infrared background from dust in damped Lyα systems, and found good agreement with far-infrared (FIR) background deduced from data, which also seem to imply the presence of dust. Recent detection of heavy elements, such as carbon and silicon (Lu et al. 1998; Cowie & Songaila 1998; Ellison et al. 1999), in very low column density Lyα clouds at redshift , can potentially indicate that dust exists also in the Lyα forest: it is quite natural to assume that dust grains are associated with heavy elements. Dust in the forest clouds would be relevant to the understanding of their origin and association with Population III objects, the heavy-element enrichment pattern of the intergalactic medium, and the thermal history of Lyα clouds (Ricotti, Gnedin & Shull 2000; Schaye et al. 2000).

The questions that we pose here are the following. When was dust first formed? Is grain formation possible starting from a metal-free environment? What are the dust properties and amount produced? How are these quantities affected by metallicity changes?

Dwek & Scalo (1980) have shown that dust injection in the interstellar medium (ISM) of the Galaxy from supernovae dominates other sources, if indeed grains can form and survive in the ejecta. This has become clear after the SN1987A event, in which dust has been unambiguously detected (Moseley et al. 1989; Kozasa, Hasegawa & Nomoto 1989). Indeed, the bulk of the refractory elements (characterized by higher melting temperatures, such as Si, Mg, Fe, Ca, Ti, Al, etc.) is injected into the ISM by supernovae (McKee 1989). At high redshift, the contribution to dust production due to evolved stars (M and carbon stars, Wolf–Rayet stars, red giants and supergiants, novae) is even more negligible or absent. The reason is that the typical evolutionary time-scale of these stars (≳1 Gyr) is longer than the age of the Universe, in an EdS cosmology, if (adopting . Thus, it seems clear that, if high-redshift dust exists, it must have been produced by Type II SNe, to which we then devote the rest of this study.

2 Dust formation model

2.1 Dust nucleation and accretion

The formation of solid materials from the gas phase can occur only from a vapour in a supersaturated state. Because of the existence of a well-defined condensation barrier, expressed by a corresponding ‘critical cluster’ size, the formation of solid particles in a gaseous medium is described as a two-step process: (i) the formation of critical clusters; (ii) the growth of these clusters into macroscopic dust grains. The classical theory of nucleation (Feder et al. 1966) gives an expression for the nucleation current, J, i.e. the number of clusters of critical size formed per unit volume and unit time in the gas:

(1)

where with a_o the radius of molecules (or atoms, depending on chemical species) in the condensed phase, σ the specific surface energy (corresponding to surface tension in liquids), k_B the Boltzmann constant and T the gas temperature; m₁ and c₁ are the mass and the concentration of the monomers in the gas phase, respectively; is the volume of a single molecule in the condensed phase; α is the sticking coefficient; and S is the supersaturation ratio, defined below. The subsequent growth of the clusters occurs by accretion and is described by

(2)

with the condition

(3)

where v₁ is the mean velocity of monomers, r(t) is the cluster radius at time t, and r∗ is the cluster critical radius. Equations (1) and (2) describe nucleation and growth of solid particles in a gas composed of a single chemical species, i.e. reactions of the type

However, there are some compounds (like forsterite, Mg₂SiO₄) whose nominal molecule does not exist in the gas phase. These compounds form directly in the solid phase by means of a chemical reaction with the reactants in the gas phase. We need to extend the theory described above to this situation. Following Kozasa & Hasegawa (1987; see also Hasegawa & Kozasa 1988) we consider a vapour in a supersaturated state. In this vapour, grains condense homo-logously via the reaction

(4)

where the A_i represent the chemical species of reactants and products in the gas phase and the ν_i are stoichiometric coefficients, which are positive for reactants and negative for products respectively. We make the following assumptions: (i) the rates of nucleation and grain growth are controlled by a single chemical species, referred to as a key species; (ii) the key species corresponds to the reactant with the least collisional frequency on to a target cluster. In this case, equations (1) and (2) become

(5)

and

(6)

where m_1k, c_1k and v_1k are the mass, concentration and mean velocity of monomers of key species, respectively. In this case the supersaturation ratio is expressed by

(7)

where P_i is the partial pressure of the ith species, R is the gas constant and ΔG_r is the Gibbs free energy for the reaction (4).

We investigate the formation of the following solid compounds: Al₂O₃ (corundum), iron, Fe₃O₄ (magnetite), MgSiO₃ (enstatite), Mg₂SiO₄ (forsterite) and amorphous carbon (AC) grains. These compounds are constituted by the most abundant heavy elements in the ejecta. It is important to note that, as we will see, AC grains can also form if the ejecta composition is richer in oxygen than in carbon as Clayton, Liu & Dalgarno (1999) have shown. Carbon dust can be produced because energetic electrons produced by SN radioactivity are able to dissociate CO molecules, which would otherwise strongly deplete the available carbon. The rate of growth of carbon grains due to free carbon atom association is then faster than their destruction rate due to oxidation. It is not yet clear if these grains will be eventually graphitized by grain temperature and electron bombardment, although evidence from meteoritic analysis seems to indicate that this might be the case (Clayton, Amari & Zinner 1997). The different grain atomic structures however are not particularly relevant for our study.

The numerical constants used in our calculations are summarized in Table 1. The value of the sticking coefficient α is set equal to unity for all reactions; we have checked that the final results are insensitive to a different choice in the plausible range .

Table 1.

Chemical reactions and numerical constants used in dust formation calculations.

2.2 Supernova model

We now describe the adopted model for the SN ejecta. Before the explosion, the progenitor develops the standard ‘onion skin’ stratified structure, with a hydrogen-rich envelope, a helium layer and several thinner heavy-element layers up to an Fe–Ni core. During the explosion, a shock wave propagates through the layers, reheats the gas and triggers the explosive nucleosynthesis phase. This phase lasts for a few hours, then expansion cools the gas and the thermonuclear reactions turn off. After the explosion, the SN starts to expand homologously, with velocity , where R is the distance from the centre. During the first weeks, Rayleigh–Taylor instabilities cause the mixing of the internal layers (Fryxell, Müller & Arnett 1991). The early emergence of X-rays and γ-rays observed in SN1987A (Itoh et al. 1987; Kumagai et al. 1988) can be explained if radioactive ⁵⁶Co is mixed from the internal regions of the star into the external ones; more precisely, observations suggest mixing of the materials in the ejecta at least up to the outer edge of the helium layer. Dust grains are formed by heavy elements, so we focus on the volume containing them, i.e. the sphere of radius R, defined as the radius of the outer edge of the He-rich layer. It is thought that mixing forms clumps of heavy elements embedded in the He-rich layer. As a first approximation, we assume that mixing is complete, and that the gas has uniform density and temperature in the considered volume at any given time. Stated differently, we assume that all the chemical species are mixed at the molecular level in the ejecta.

Photometric observations have shown that a SN emits typically 10⁴⁹ erg in electromagnetic energy, but current theoretical models predict kinetic energies . The expansion velocity v is then given by , where M_tot is the total mass ejected by the SN. We take the chemical composition of the expelled gas from the results of Woosley & Weaver (1995, hereafter WW95), apart from the specific case of SN1987A, see below. They determine the nucleosynthetic yields of isotopes lighter than (zinc) for a grid of stellar masses and metallicities including stars in the mass range and metallicities , 10⁻⁴, 0.01, 0.1, 1. They also give the values for E_kin and M_tot for all the SN models considered. The range is the most relevant mass range for the production of heavy elements. In fact, stars with mass between 8 and 11 M_⊙ are characterized by very thin heavy-element layers, whereas stars heavier than 40 M_⊙ might be rare and give rise to a black hole partially swallowing the nucleosynthetic products (Maeder 1992).

Expansion of the ejecta leads to cooling of the gas. We have already mentioned that the radiation losses are only a few per cent of the total internal energy; their contribution to cooling is even smaller in the first week after the explosion because of the high opacity, which prevents photons from escaping from the inner regions. Therefore, it is a good approximation to assume that the expansion is adiabatic, although this hypothesis becomes less correct in the advanced evolutionary stages. Radiation losses are also partly balanced by the heating provided by radioactive decay (especially of . We neglect these complications here and assume that the expansion is adiabatic. In this case (for an ideal gas) the temperature evolution is given by

where γ is the adiabatic index, T_i and R_i are the temperature and the radius at the beginning of the computation, and t is the time elapsed from this initial epoch. At the temperatures of interest the gas density is ≈10⁸ atom cm⁻³ so the use of the ideal gas law is well justified. We set the values of R_i, T_i and γ as follows. From photometric observations of SN1987A (Catchpole et al. 1987) it is deduced that the photosphere and the outer edge of the He-rich layer overlap ≈70 d after the explosion, when the photospheric temperature is 5400 K and the radius . For the adiabatic index we take as in Kozasa et al. (1989). We generally use these fiducial values in our models, but we will consider the effects of varying the values of R_i and γ when discussing the results.

2.3 Molecule formation

In the ejecta of SN1987A, molecules of CO were detected for the first time in a SN (Meikle et al. 1989, 1993; Bouchet & Danziger 1993) together with SiO (Aitken et al. 1988; Bouchet, Danziger & Lucy 1991). These two molecules are very important for our study because carbon atoms bound in CO are not available to form AC, whereas SiO molecules take part in the chemical pathway leading to the formation of MgSiO₃ and Mg₂SiO₄. We investigate the process of molecule formation under the assumption of chemical equilibrium. In the absence of grains, molecular formation in the gas phase must have been initiated by radiative processes. We assume that formation of CO is dominated by radiative association (Lepp, Dalgarno & McCray 1990; Liu, Dalgarno & Lepp 1992),

with a rate coefficient K_ra(CO). The main destruction process of CO is impact with the energetic electrons produced by the radioactive decay of ⁵⁶Co (Liu & Dalgarno 1995),

with a rate coefficient K_rd(CO). In the steady state, the abundance of CO is given by

(8)

Analogously, radiative association is the most important mechanism for the formation of SiO (Liu & Dalgarno 1996) via the reaction

with a rate coefficient K_ra(SiO). Silicon monoxide is mainly destroyed by impact with energetic electrons,

with a rate coefficient K_rd(SiO), and by charge transfer with positive ions of Ne (Liu & Dalgarno 1996),

with a rate coefficient K_ct(SiO). The abundance of SiO is given by

(9)

We assume for the rate coefficients the following values:

(Dalgarno, Du & You 1990; Gearhart, Wheeler & Swartz 1999), where

(Andreazza, Singh & Sanzovo 1995; Liu & Dalgarno 1996); and

(Liu & Dalgarno 1996). The ejecta are only moderately ionized with fractional ionization ≈10⁻²; hence, we assume .

To calculate the rate coefficient K_rd(CO), we assume the same rate of destruction by energetic electron impact for CO as for SiO, i.e. . High-energy X-rays and γ-rays produced by the chain of radioactive decay ²⁸₅₆Ni → ⁵⁶₂₇Co → ⁵⁶₂₆Fe interact by Compton scattering with the electrons in the ejecta. The average energy deposition rate per particle in the ejecta is (Woosley, Pinto & Hartmann 1989)

Here N_i(⁵⁶Co) is the total number of atoms of ⁵⁶Co in the ejecta; N_tot is the total number of particles in the gas; is the mean γ-ray energy released by each decay; and is the e-folding time of ⁵⁶Co. The deposition function, ƒ, is proportional to the fraction of trapped γ photons,

with the mass column density of the ejecta at some fiducial time t_o, and K₅₆ an average opacity for ⁵⁶Co decay γ-rays. We assume at and . These values are appropriate for SN1987A but, lacking more detailed information, we extrapolate them to all our models. Finally, the estimated destruction rate of CO and SiO by energetic electron impact is

with W_d being the mean energy per dissociation, defined as the energy of primary electrons divided by the number of molecule dissociations (Liu & Victor 1994). For a fractional ionization of the gas ≈10⁻², .

3 A test case: SN1987A

To test and calibrate our dust formation model, we first apply it to SN1987A, a case for which firm evidence of dust formation has been collected. In view of this test, we briefly summarize the relevant observational results.

3.1 Observational results

The most relevant evidence of newly formed dust grains in the ejecta of SN1987A is the blueshift of the line profiles. Spectroscopic observations detected this change between 1988 August and 1989 March (Lucy et al. 1989, 1991). This effect is likely to be caused by the larger attenuation suffered by radiation received from receding matter as a result of dust grains distributed in the ejecta. The condensation efficiency, i.e. the dust mass expressed as a fraction of the maximum value permitted by the elemental abundances in the ejecta, derived by the authors of these observations from their data is ⩽10⁻³. A further interesting observation is the stronger fading of the [Si i] 1.65 μm line flux relative to the continuum after day 530. This can be interpreted as depletion of Si from the gas phase as a consequence of the formation of silicate grains. If the line fading is due solely to depletion, then the condensation efficiency rises to > 50 per cent.

Further evidence of the formation of dust is an IR continuum excess over that expected from a Planck spectrum fitted to the SN emission at optical wavelengths (Roche et al. 1989; Wooden et al. 1993). Dust grains extinguish UV-visible radiation from the central energy source, re-emitting in the IR bands. This process might be responsible for the observed increase in the 10 and 20 μm fluxes around day (Roche et al. 1989; Bouchet & Danziger 1993; Meikle et al. 1993).

3.2 Model results

We now turn to the main results from our nucleation numerical computations obtained by solving the above equations. The chemical composition of the SN1987A ejecta is taken from Nomoto et al. (1991). The value of the expansion velocity of the gas is set to . This is the minimum expansion velocity determined from the H i Pα absorption trough (McGregor 1988; Nulsen et al. 1990), and is thought to represent the expansion velocity of the inner edge of the hydrogen envelope. Fig. 1 shows the mass of dust formed in the ejecta as a function of the time elapsed since explosion for the different solid compounds found to be present. We note that there are two episodes of dust formation.

Figure 1.

Dust formation as a function of time elapsed since explosion in SN1987A.

In the first one, AC (formation time and Al₂O₃ grains are formed. This process is likely to be responsible for the IR excess observed at that time. Essentially all the carbons available end up in AC grains rather than in CO molecules. As a result we find ≈10⁻³ M_⊙ of CO at since the explosion. This is three times less than the value derived observationally by Liu & Dalgarno (1995). Since this fraction is very small, the exact value of CO mass does not matter for our purposes.

In the second episode, magnetite and silicate grains are formed, at about . Two points are worth noting. The first one is that this epoch corresponds to the formation of the predominant fraction of dust mass of the SN (0.57 M_⊙ corresponding to about 84 per cent of the total amount); this dust formation episode might be responsible for the blueshift of line profiles, observed only after day 530, when AC and Al₂O₃ formed. The second point is that the formation of MgSiO₃ and Mg₂SiO₄ might be related to the fading of the Si and Mg lines. The hypothesis that these elements form enstatite and forsterite would also be suggested by the behaviour of the SiO molecule. The silicon monoxide rovibrational line , emission was detected after 160 d (Aitken et al. 1988) and it remained clearly detectable until 519 d (Bouchet et al. 1991) but is no longer detected at 578 d (Roche et al. 1989). The time behaviour of SiO emission can be understood by inspecting Fig. 2, where we show the predicted SiO mass versus the observed mass as a function of time. Note the rapid fall at 660 d, 70 d after the beginning of silicate formation, probably due to the depletion of Si atoms; also our model seems slightly to overpredict (by a factor 3.1) the amount of SiO produced at early times. This could be due to our simplified treatment of the chemical network for this molecule or to inaccuracies in the rate coefficients. The dust formation efficiency deduced here is 95 per cent, consistent with that deduced from the Si line fading.

Figure 2.

Evolution of the SiO mass (solid line) as a function of time compared with the observational data (points) taken from Liu & Dalgarno (1996).

All together, we look on the above results as a satisfactory success of our model in reproducing, at least qualitatively, the principal features of dust formation in SN1987A.

4 Dust in primordial supernovae

In this section we present the general results concerning dust formation in primordial SNe. We take the chemical compositions of the gas from tables 16A and 16B of WW95, and the relevant results are reported in Table 2. There, the chemical composition of the ejecta is given at after the explosion, when strong and electromagnetic reactions have ceased, but many nuclei have not yet decayed into their most stable form. Because dust formation occurs at after explosion, it is necessary to take into account the radioactive decay of such nuclei. In the SN models of WW95, the energy of explosion can be adjusted to give the desired kinetic energy of the ejecta, typically 10⁵¹ erg. Following WW95, we explore the effects of E_kin variation by considering a low , case A) and a high , case B) value for this quantity. We discuss the two cases separately in the following.

Table 2.

Adopted chemical composition of the supernova ejecta after explosion) for metallicity . Data are taken from WW95 and corrected to take into account radioactive decay.

4.1 Low kinetic energy (case A)

As the kinetic energy of the model is relatively low, this is not sufficient for complete expulsion of the heavy elements external to the Ni–Fe core of the most massive SNe, and a variable amount of material falls back on to the core, probably forming a neutron star or a black hole. The fallback will mostly affect the inner layers, containing the heaviest elements; as a result, progenitors with masses larger than ≈20 M_⊙ will be prevented from forming dust. For essentially the same reason, above , only AC grains are formed. Figs 3 and 4 show the amount of dust formed as a function of progenitor mass, and the grain composition. AC grains are typically the first solid particles to condense, depending on the models. The formation of these grains is quite fast with respect to the cooling time-scale of the ejecta: most of the AC dust mass forms in a narrow range of around . Subsequently, at a temperature of ≈1600 K, Al₂O₃ starts to condense, followed by Fe₃O₄, MgSiO₃ and Mg₂SiO₄ at . Clearly this sequence is governed by the condensation temperature (higher for carbon than for silicates) of a given material. Because of this, AC grains form when the density is still high; as a result the accretion rate proceeds rapidly until complete carbon depletion. Under these conditions, few, large grains are formed. Silicate grains, instead, form later and the accretion rate is lower and comparable to the nucleation rate; this leads to the formation of a large number of small grains. Silicate grain growth is also inhibited by the relative paucity of the key species SiO molecule in SN ejecta.

Figure 3.

Dust mass formed as a function of the SN mass (in the range for initial metallicity and kinetic energy of the explosion (case A). Also shown is the grain composition.

Figure 4.

Same as Fig. 3, but for the SN mass range .

The typical size of AC dust grains is , whereas Fe₃O₄ grains have typically , and Al₂O₃, MgSiO₃ and Mg₂SiO₄ grains are even smaller (≈10 Å). In spite of the high condensation temperature, Al₂O₃ grains do not grow to sizes comparable to those of AC, as their growth is limited by the low abundance of Al. In Fig. 5 the grain sizes of the most abundant compounds that form in supernova ejecta are shown for four values of Z (non-zero metallicity cases are discussed later on). The two silicates (enstatite and forsterite) start to condense almost simultaneously; however, Mg₂SiO₄ enters the supersaturation regime earlier than MgSiO₃. For this reason, forsterite grains grow quickly, strongly depleting the Si (or Mg) available. Figs 3 and 4 clearly show that, even starting from a primordial composition, early SNe can contribute a significant amount of dust: for case A, about of dust per SN are produced. Intermediate-mass progenitors are the most efficient sources, being able to convert up to 2 per cent of their mass into solid particles. The reason is that they synthesize a considerable amount of heavy elements without suffering too much from the fallback process mentioned above.

Figure 5.

Grain number density as a function of size for AC grains (open squares), Fe₃O₄ grains (solid triangles) and Mg₂SiO₄ grains (open circles) for the SN model; results are given for four different metallicities of the progenitor. The number density is calculated when the ejecta volume is , i.e. when formation and accretion processes have ceased in all models.

4.2 High kinetic energy (case B)

In this case the kinetic energy of the explosion for the progenitor masses 25, 30, 35 and 40 M_⊙ is chosen equal to for lower masses the energy of explosion is the same as in case A. This energy is sufficient to eject also the inner layers, which now can provide the elements to form grains of various chemical composition, i.e. not only AC as in case A. In Fig. 6 we show the dust mass yield for case B. Now SN up to masses ≈35 M_⊙ are able to form dust. In addition, a SN of 30 M_⊙ is able to produce about 1.3 M_⊙ of dust (4.3 per cent of its mass). The formation sequence and the grain size distribution are very similar to those discussed for case A.

Figure 6.

Same as Fig. 3, but for kinetic energy of the explosion (case B); the SN mass range is . For lower masses case B gives the same results as case A.

4.3 Effects of R_i and γ variations

Up to now we have used the values of R_i and γ deduced from observations on SN1987A. These values represent a reasonable approximation but they might well depend on the specific properties of the SN under examination. Therefore, as a check on the soundness of this approach, we investigate the dependence of our results on different choices for these parameters. As a benchmark, we focus on the SN model (case A) and increase or decrease the standard value of by a factor 2.15; this corresponds to a variation of about 100 times in the initial volume of the ejecta. In Fig. 7 we compare the dust formation evolution and the final size of Fe₃O₄ grains for three values of R_i. The final masses of AC, Al₂O₃ and Fe₃O₄ grains are almost unchanged in the three cases because the gas density remains high enough for the collisional time-scale (regulating the formation/accretion processes) of these materials to remain shorter than the expansion one. However, the behaviour of silicates depends on the choice of R_i. As a general rule, Mg₂SiO₄ grains form first and grow faster than MgSiO₃ ones, thus using up the condensable materials efficiently and ending up with a larger final total mass. However, for larger values of R_i (i.e. larger volume, lower gas density) this process is limited by the fact that the collisional scale becomes longer, thus stopping the accretion at earlier times. The grain size distribution shifts by about a factor 2 as R_i is varied by a factor 2.15. Thus the determination of the grain size distribution is relatively uncertain.

Figure 7.

Dust formation (left, line types as in Fig. 1) and Fe₃O₄ grain size distribution (right) for the SN model; the three cases refer to different values of R_i: (a) , (b) (standard value) and (c) .

The adiabatic index γ gives a measure of the ability of the gas to cool: γ greater than the standard value of 1.25 causes the gas to reach the supersaturation state when the volume of the ejecta is smaller. Therefore, a larger γ case gives results similar to those obtained for the low R_i case discussed above. Fig. 8 shows the dust formation evolution and final grain size distribution of Fe₃O₄ grains for the model (case A) with . The similarity with the previous case with and is evident. It has to be noted though that the variation range for γ (15 per cent) is smaller than that for R_i (50 per cent), which might indicate that the dust formation process is more sensitive to changes in the adiabatic index than in the initial radius.

Figure 8.

Dust formation (top) and Fe₃O₄ grain size distribution (bottom) for the SN with .

5 Extension to higher metallicities

We finally extend our results to non-primordial compositions by exploring the results for an additional three metallicity values , 10⁻² and 1. The chemical composition for these models is also taken from WW95. We start by analysing the dependence of the grain size distribution on metallicity. This is shown in the four panels of Fig. 5. Somewhat surprisingly, the dependence is almost absent, with grain radii ranging from 5 Å to 0.1 μm for all values of Z. Also, the same material segregation is seen, with smaller grains being predominantly composed of silicate and magnetite and the larger ones comprising amorphous carbon. This behaviour can be explained by the fact that the final grain size is governed by the thermodynamics of the ejecta expansion (and therefore sensitive to R_i and γ, as already pointed out before), but poorly affected by the ejecta composition.

The latter, instead, plays a more important role in determining the total amount of dust formed, as seen in Figs 9 and 10 for cases A and B, respectively. Moving from to higher metallicities we observe that a large number of SNe contribute to dust production: a clear example of this is the behaviour of SNe with mass , which increase their dust yield from < 0.1 M_⊙ up to for . The enhancement of dust formation is due to the fact that the density of heavy elements in the ejecta becomes large enough to allow the state of supersaturation to be reached more easily. This trend results in a steady increase of the total amount of the dust produced in the four cases (obviously, when applying these results one has to weigh the appropriate IMF); they are 2.06, 3.6, 4.5 and 5.9 M_⊙ for , 10⁻⁴, 10⁻² and 1, respectively, for case A. However, it is interesting to note that, at each metallicity, the maximum amount of dust produced by a single SN varies little, and it is never higher than approximately 1 M_⊙. Finally, the differences between cases A and B are found to be minor.

Figure 9.

Total dust mass produced as a function of SN mass and different metallicity of the progenitor (case A).

Figure 10.

Same as Fig. 9, but for case B.

6 Summary and discussion

We have investigated the formation of dust in Type II supernovae mostly with primordial abundances, a property characterizing these events in the early Universe; however, we have also considered non-zero metallicity values up to . The calculations are based on standard nucleation theory and the scheme has been tested for the first time on the well-studied case of SN1987A, yielding results that are in satisfactory agreement with the available data (see Section 3). The main results of the paper are the following:

• (i)

  The first solid particles in the Universe are formed by Type II SNe. The dust grains are made of silicates (predominantly Mg₂SiO₄), amorphous carbon (AC), magnetite (Fe₃O₄) and corundum (Al₂O₃), and form about after explosion.

• (ii)

  The largest grains are the AC ones, with sizes around 300 Å, whereas other grain types have smaller radii, around . The grain size distribution depends considerably on the thermodynamics of the ejecta expansion (characterized by their initial radius and adiabatic index) and variations in the results by a factor ≈2 might occur within the estimated range of R_i and γ. Also, and for the same reason, the grain size distribution is essentially unaffected by metallicity changes.

• (iii)

  The amount of dust formed is instead very robust to variations in R_i and γ. For , we find that SNe with masses in the range produce about of dust per supernova in the low kinetic energy explosion case; slightly higher final yields are obtained in the high kinetic energy case. The above range increases by roughly three times as the metallicity is increased to solar values.

The previous results clearly show that it is likely that dust has been present in the Universe since immediately after the first stars appeared. This has a large number of consequences that it will be necessary to study in detail. Among possible effects, the most outstanding ones concern the opacity of the Universe at high z, spectral distortions in the cosmic microwave background (CMB) radiation caused by dust re-emission of absorbed UV-optical light, and catalysation of H₂ molecular hydrogen formation and heavy-element depletion in the interstellar medium of pristine galaxies and in the intergalactic medium.

The first two issues have already been discussed in detail by Loeb & Haiman (1997) and Ferrara et al. (1999a), and we defer the interested reader to those works for details. In short, the expected IGM opacity contributed by dust around the observed wavelength is ∼0.13 and it rapidly increases to ≈0.35 at . The expected CMB spectral distortions due to high-z dust is only ∼1.25–10 times smaller than the current COBE upper limit, but these numbers might depend crucially on the formation epoch and abundance of dust.

In addition to the above effects, Type II SNe can also initiate molecular hydrogen formation on dust grain surfaces rather than in the gas phase, the second process being the only one viable in a dust-free environment. As already mentioned, at high redshift, Type II SNe are the only possible sources of dust, as a result of the short age of the Universe and the long evolutionary time-scales characterizing more conventional dust sources, as for example evolved stars. Thanks to the above results we can now answer the following question: What is the minimum amount of dust required in order for the molecular hydrogen formation on grains to become competitive with the gas-phase one? An order-of-magnitude answer can be obtained by comparing the two formation rates (Fig. 11). At the low densities relevant here, H₂ is formed in the gas phase mainly via the channel

at rate k₈ (the rate coefficient k₈ is given in Abel et al. 1997); formation via the channel, when included, is found to be negligible in our case. Therefore the formation rate in the gas phase is . The formation rate on grain surfaces is instead given by , where γ is the sticking coefficient, c_s is the sound speed in the gas, and σ is the grain cross-section. The equality between the two rates can be cast into the following form: , where 풟 is the dust-to-gas ratio normalized to its Galactic value, T is the gas temperature and x_e the gas ionization fraction. For typical parameters of the Population III objects (Ciardi et al. 2000a), H₂ production on dust grains becomes dominant once 풟 is larger than 5 per cent of the local value. With the dust yields calculated above, we then conclude that only about 50 SN are required to enrich in dust to this level a primordial object. Clearly, early dust formation might play a role in the formation of the first generation of objects.

Figure 11.

Comparison between molecular hydrogen formation rates in the gas phase and on dust grain surfaces as a function of the dust-to-gas ratio and gas temperature. Above the two curves, indicating equality between the rates for two different values of the electron fraction , H₂ formation on grains dominates.

Even in larger galaxies, which will form later on when the overall metallicity and dust levels in the Universe have increased, dust will be at least as important. For example, on the scale of the molecular clouds in a galaxy, it will provide the opacity to stop the infall on forming protostars, hence possibly changing the properties of the IMF, and to allow the cloud to self-shield from damaging H₂ photodissociating UV radiation. Finally, dust photoelectric heating is known to be the major heating source for the diffuse ISM, and hence participating to the onset of its observed multiphase structure (Ricotti, Ferrara & Miniati 1997; Spaans & Norman 1997).

The fate of the dust that we predict from Type II SNe has yet to be determined. What fraction of the grains will be able to survive the passage through the reverse shocks at which the ejecta are thermalized? Grains in a hot gas are essentially destroyed via thermal sputtering, i.e. collisions with ions or electrons with Maxwellian velocity distribution. However, in spite of the still poorly understood underlying physics, it seems unlikely that the efficiency of grain destruction in an adiabatic shock can be higher that 10 per cent (McKee 1989). Grains are more likely to be destroyed behind radiative shocks, by the combined effects of a greatly enhanced gas density and betatron acceleration that increases the grain Larmor frequency. However, if, as expected, the magnetic field is weak at high z (Gnedin, Ferrara & Zweibel 2000), the efficiency cannot be very high. The grains will then follow the fate of the gas and will likely be expelled into the IGM, as Population III objects suffer complete blowaway of their gas (Ciardi et al. 2000a). Once in intergalactic space, dust might have a strong influence, for example, on the determination of cosmological parameters via the observation of high-z (Type I) SN (Aguirre 1999; Croft et al. 2000).

As a final remark, we have seen that SNe with mass above predominantly form amorphous carbon grains (the ejecta of these stars have higher C/O ratios); in doing so, they use up virtually all the available carbon yield in the ejecta. This implies that carbon in the IGM at high redshift will be strongly depleted above redshifts at which only Type II SNe contribute to the metal enrichment of the Universe. It will be then possible to test this prediction once a sample of target sources at becomes available for absorption-line studies.

Acknowledgments

This work was completed while one of us (AF) was a Visiting Professor at the Center for Computational Physics, Tsukuba University, whose support is gratefully acknowledged.