Ionization degree and magnetic diffusivity in star-forming clouds with different metallicities
aa r X i v : . [ a s t r o - ph . GA ] J a n MNRAS , 1–23 (2021) Preprint 27 January 2021 Compiled using MNRAS L A TEX style file v3.0
Ionization degree and magnetic diffusivity in star-forming clouds withdifferent metallicities
Daisuke Nakauchi ★ , Kazuyuki Omukai † , and Hajime Susa ‡ Astronomical Institute, Graduate School of Science, Tohoku University, Aoba, Sendai 980-8578, Japan Department of Physics, Faculty of Science, Konan University, Higashi-Nada, Kobe 658-0072, Japan
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
Magnetic fields play such essential roles in star formation as transporting angular momentum and driving outflows from a star-forming cloud, thereby controlling the formation efficiency of a circumstellar disc and also multiple stellar systems. The couplingof magnetic fields to the gas depends on its ionization degree. We calculate the temperature evolution and ionization degreeof a cloud for various metallicities of 𝑍 / Z ⊙ = − , − , − , − , − , − , and 1. We update the chemical network byreversing all the gas-phase processes and by considering grain-surface chemistry, including grain evaporation, thermal ionizationof alkali metals, and thermionic emission from grains. The ionization degree at 𝑛 H ∼ -10 cm − becomes up to eight ordersof magnitude higher than that obtained in the previous model, owing to the thermionic emission and thermal ionization of Kand Na, which have been neglected so far. Although magnetic fields dissipate owing to ambipolar diffusion or Ohmic loss at 𝑛 H < cm − , the fields recover strong coupling to the gas at 𝑛 H ∼ cm − , which is lower by a few orders of magnitudecompared to the previous work. We develop a reduced chemical network by choosing processes relevant to major coolantsand charged species. The reduced network consists of 104 (161) reactions among 28 (38) species in the absence (presence,respectively) of ionization sources. The reduced model includes H and HD formation on grain surfaces as well as the depletionof O, C, OH, CO, and H O on grain surfaces.
Key words: stars: formation, stars: Population III, stars: Population II
Recent development of gravitational wave detectors, such as ad-vanced LIGO, Virgo and KAGRA, revealed the presence of bi-nary black holes (BBHs) that merge within the Hubble time (e.g.,Abbott et al. 2016, 2019). Most of the BHs are as massive as ∼
30 M ⊙ , and their origin is currently debated. Some authors pro-posed that they are originated from isolated binaries formed in low-metallicity star-forming clouds, including Population III (Pop III)stars with the primordial composition (e.g., Kinugawa et al. 2014,2016; Belczynski et al. 2016; Giacobbo et al. 2018). Others pro-posed that these binary systems are formed via gravitational inter-action in low-metallicity dense star clusters (e.g., Rodriguez et al.2016; Antonini & Rasio 2016; Kumamoto et al. 2020; Mapelli 2016;Mapelli et al. 2020; Liu et al. 2020; Tanikawa et al. 2020). In anycase, the merger rate of massive BBHs depends on the formationefficiency and nature of low-metallicity binaries.Without metals and dust grains that work as efficient coolants, aPop III star-forming cloud maintains a warm environment of several100 K because of inefficient H cooling, which makes Pop III starstypically massive (Omukai & Nishi 1998; Bromm et al. 1999, 2002;Omukai & Palla 2001, 2003; Abel et al. 2002; Yoshida et al. 2006). ★ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected]
Recent multi-dimensional (D) radiation hydrodynamical (RHD) sim-ulations have shown that the mass of a Pop III star distributes in a widerange of 10-1000 M ⊙ (Hosokawa et al. 2011, 2016; Stacy et al. 2012,2016; Hirano et al. 2014; Susa et al. 2014), and that Pop III stars areformed in massive binary or multiple stellar systems as a result of thecircumstellar disc formation and its fragmentation (Sugimura et al.2020, see also Machida et al. 2008; Stacy et al. 2010; Clark et al.2011; Greif et al. 2012; Susa 2019; Chiaki & Yoshida 2020). Pop IIIstars may also be born as rapid rotators (Stacy et al. 2011, 2013;Takahashi & Omukai 2017).Massive star formation in a slightly metal-enriched cloud has notbeen studied extensively so far, and is still very uncertain. From2D RHD calculations, Fukushima et al. (2020) studied the protostarformation and evolution within a metal-enriched halo obtained fromthe cosmological simulation of Chiaki et al. (2016). They showedthat despite the strong radiation feedback from a protostar, starsas massive as a few hundred solar masses can be formed. Withlittle mass-loss by radiation-driven winds (Kudritzki 2002), a low-metallicity massive star can be a progenitor of a massive BH.However, the above mentioned works did not account for magneticfields that could change those pictures. When magnetic fields stronglycouple to the gas, they slow down the cloud rotation by transferringthe angular momentum from the center to outwards (so-called mag-netic braking), which suppresses the formation of a circumstellar discand a multiple system via disc fragmentation (e.g., Gillis et al. 1974;Tomisaka 2002; Tsukamoto 2016; Hennebelle & Inutsuka 2019). © D. Nakauchi, K. Omukai, and H. Susa
Magnetic fields also drive outflows which eject a part of the cloud ma-terials back into the interstellar space, decreasing the star-formationefficiency (e.g., Matzner & McKee 2000; Nakamura & Li 2007;Wang et al. 2010; Cunningham et al. 2011; Machida & Hosokawa2013; Federrath et al. 2014a). 3D magnetohydrodynamical (MHD)calculations of low-metallicity star formation have shown that mag-netic braking extracts too much angular momentum from the cloudcenter to form a multiple star system, if the field is stronger than 𝐵 ∼ − G (at ∼ − ) (Machida & Doi 2013; Peters et al.2014; Sharda et al. 2020). About 10% of the mass is removed fromthe cloud into the interstellar space by MHD outflows, if the fieldis stronger than 𝐵 ∼ − G (at ∼ − ) (Machida et al. 2006;Machida & Doi 2013; Tanaka et al. 2018; Higuchi et al. 2019).The magnetic field strength in the interstellar medium (ISM) in ayoung galaxy is still uncertain. While various scenarios are suggestedfor the generation of primordial magnetic fields (e.g., Ando et al.2010; Widrow et al. 2012; Subramanian 2016; McKee et al. 2020),primordial fields are believed to be many orders of magnitude weakerthan the Galactic ISM ( ∼ 𝜇 G; Crutcher et al. 2010). If the ISMin a young galaxy is highly turbulent, weak magnetic fields canbe amplified by the small-scale-dynamo action up to the equiparti-tion ∼ 𝜇 G at ∼ − (Schleicher et al. 2010; Sur et al. 2010,2012; Federrath et al. 2011; Schober et al. 2012; Turk et al. 2012).The presence of micro-Gauss magnetic fields is indicated from theobservation of high- 𝑧 galaxies (Bernet et al. 2008; Mao et al. 2017).The coupling of the magnetic fields to the gas is controlledby its ionization degree. In the present-day star-formation, vari-ous authors have studied the ionization degree in a cloud by con-sidering a simple equilibrium chemistry involving representativeions, electrons and charged grains (Oppenheimer & Dalgarno 1974;Umebayashi & Nakano 1980, 1990; Nakano & Umebayashi 1986;Nakano et al. 2002). They set the abundances of neutral atoms andmolecules constant and treat their depletion fractions on grain sur-faces as model parameters. Recent studies have accounted for a moreelaborated gas-phase chemistry among H, He, C, and O compounds,as well as grain-surface chemistry consisting of the freeze-out of gas-phase species, the desorption of surface species, and molecule forma-tion (Ilgner & Nelson 2006; Furuya et al. 2012; Dzyurkevich et al.2017; Zhao et al. 2018). Marchand et al. (2016) also extended theclassical model by considering grain vaporization, thermal ionizationof alkali metals, and electron ejection from heated grains (so-calledthermionic emission), which elevate the ionization degree at hightemperatures of &
500 K.The ionization degree in a low-metallicity cloud has also beenstudied by Maki & Susa (2004, 2007), and Susa et al. (2015). It wasshown that in a primordial cloud, with its high temperature and lackof dust grains, Li + maintains the ionization degree high enough formagnetic fields to couple to the gas. In a slightly metal-enriched cloudof 𝑍 / Z ⊙ & − , with their large recombination cross section, dustgrains capture electrons and ions, and decrease the ionization degreeso low as to decouple the magnetic fields from the gas. Magneticfields recover the coupling after the dust grains evaporate and hydro-gen ionization raises the ionization degree at 𝑛 H ∼ -10 cm − .However, the previous studies have some flaws in their chemicalmodelling: (i) only a part of the reactions are reversed, and at acertain density, the abundances are switched artificially to the chem-ical equilibrium values of the H/He gas, (ii) thermionic emissionand thermal ionization of alkali metals are not properly considered,and (iii) grain-surface chemistry is not included. In the primor-dial cloud, where only the problem (i) is relevant, Nakauchi et al.(2019) (hereafter Paper I) calculated the ionization degree by revers-ing all the chemical processes. They found that the ionization degree at 𝑛 H ∼ -10 cm − is enhanced by a few orders of magnitude,which couples the magnetic fields to the gas more strongly at thesedensities.In this paper, we update the previous chemical network byaccounting for the processes (i)-(iii), and compute the temper-ature evolution, ionization degree, and resistivity coefficients ina star-forming cloud for a wide range of metallicity, 𝑍 / Z ⊙ = − , − , − , − , − , − , and 1. We find that the ioniza-tion degree at 𝑛 H ∼ -10 cm − becomes up to eight orders ofmagnitude higher than that obtained in the previous model. This isdue to the thermionic emission and thermal ionization of vaporizedK and Na, which are neglected so far. As a result, magnetic fieldsrecover the strong coupling to the gas at much earlier stages (by a feworders of magnitude in density), compared to the previous work. Wealso develop a reduced chemical network by extracting the processesrelevant to major coolants and charged species from the full network.Among various molecule formation processes on grain surfaces, onlyH and HD formation is included in the reduced model by using sim-ple formulae. The reduced model also includes the depletion of O,C, OH, CO, and H O on grain surfaces.This paper consists of the following sections. In Section 2, wedescribe the method and chemical network used to calculate thetemperature evolution and ionization degree in a star-forming cloud.More details about the dust-surface chemistry is summarized in Ap-pendix. In Section 3, we show the results for both cases without andwith ionization sources in Section 3.1 and 3.2, respectively. From theionization degree obtained in Section 3, we discuss the conditionsof magnetic dissipation for both global and turbulent magnetic fieldsin Section 4. After briefly summarizing the results, we discuss theuncertainties and implications of our work in Section 5.
The gravitational contraction of a spherically symmetric cloud iscalculated by way of a one-zone model neglecting rotation, turbu-lence and magnetic fields as in Omukai (2000, 2001, 2012) andOmukai et al. (2005). Owing to higher density, the cloud core col-lapses at a shorter timescale with about the local free-fall time 𝑡 ff = s 𝜋 𝐺 𝜌 , (1)leaving behind a lower density envelope (so-called runaway col-lapse; Larson 1969; Penston 1969). This runaway collapse proceedsin a self-similar way such that the density in the cloud core distributesuniformly across the Jeans length scale 𝜆 J = s 𝜋𝑘 B 𝑇𝐺 𝜇𝑚 H 𝜌 , (2)and the density in the envelope declines with radius following a powerlaw.In our model, the physical quantities in the cloud core are calcu-lated. The density increases in the free-fall time as 𝑑𝜌𝑑𝑡 = 𝜌𝑡 ff , (3)and the temperature 𝑇 is determined by the energy equation: 𝑑𝑒𝑑𝑡 = − 𝑃 𝑑𝑑𝑡 (cid:18) 𝜌 (cid:19) − Λ net , (4)where 𝑃 is the pressure, 𝑒 the internal energy per unit mass, and MNRAS , 1–23 (2021) onization degree in clouds with different metallicities Λ net the net cooling rate per unit mass. The following five processescontribute to the net cooling rate: Λ net = Λ line + Λ chem + Λ grain + Λ cont − Γ ion , (5)where Λ line includes cooling by H Ly 𝛼 emission, molecular lineemissions of H , HD, CO, OH, and H O, and fine-structure lineemissions of CII, CI, and OI, Λ cont the cooling by thermal emissionsfrom gas (e.g, H collision-induced emission; CIE) and dust grains, Λ chem cooling/heating associated with the chemical reactions, and Γ ion the ionization heating by cosmic-ray (CR) injection and decay ofradioactive elements (RE). The formulations of H Ly 𝛼 , H , and HDcooling are referred from Paper I, those of CO, OH, and H O coolingfrom Omukai et al. (2010), and those of fine-structure line cooling, Λ cont and Λ chem from Omukai (2000) (with some updates for CIand OI cooling by Nakauchi et al. 2018). When the cloud becomesopaque, the radiative cooling rates are damped depending on thehydrogen column density of the cloud: 𝑁 H = 𝑛 H 𝜆 J . The ionizationheating rate Γ ion is estimated by assuming that the gas obtains 3.4eV of heat per ionization (Spitzer & Scott 1969).In the gas phase, 1184 chemical reactions are considered amongthe following 63 species: H, H , e − , H + , H + , H + , H − , He, He + ,He + , HeH + , D, HD, D + , HD + , D − , C, C , CH, CH , CH , CH ,C + , C + , CH + , CH + , CH + , CH + , CH + , O, O , OH, CO, H O, HCO,O H, CO , H CO, H O , O + , O + , OH + , CO + , H O + , HCO + , O H + ,H O + , H CO + , HCO + , H CO + , Li, LiH, Li + , Li − , LiH + , Li + , Li + ,K, K + , Na, Na + , Mg, Mg + . The primordial-gas chemistry consistsof 214 reactions, i.e., 107 forward and reverse pairs, which are listedwith their references in Table 1 of Paper I. The ionization processesof Li, Na, and K via H collision and their inverses:H + Li ⇋ H + Li + + 𝑒 H + K ⇋ H + K + + 𝑒 H + Na ⇋ H + Na + + 𝑒 (6)are referred from Ashton & Hayhurst (1973). The other 964 reac-tions (464 forward-reverse pairs and 36 CR-induced processes) arereferred from the UMIST database (McElroy et al. 2013). From thelarge number of reactions listed in the database, we choose those re-actions which contain the following species as reactants or products:H, H , e − , H + , H + , H + , H − , He, He + , HeH + , C, C , CH, CH , CH ,CH , C + , C + , CH + , CH + , CH + , CH + , CH + , O, O , OH, CO, H O,HCO, O H, CO , H CO, H O , O + , O + , OH + , CO + , H O + , HCO + ,O H + , H O + , H CO + , HCO + , H CO + , K, K + , Na, Na + , Mg, Mg + ,but we remove those reactions overlapping with the primordial-gaschemistry. In case a forward-reverse pair is found in the database, weregard the reaction with a positive value for the heat of reaction Δ 𝐸 as forward and calculate its reverse rate coefficient by the methodexplained below.All the gas-phase reactions are reversed so that the fractional abun-dance is calculated correctly both in the non-equilibrium and equilib-rium cases. The rate coefficients for the forward and reverse reactionsare related to each other through the detailed balance principle (e.g.,Draine 2011): 𝑘 rev = 𝑘 fwd 𝐾 eq ( 𝑇 ) , (7)where 𝐾 eq ( 𝑇 ) is the equilibrium constant. For a reaction where 𝑀 reactants R , R , ..., R 𝑀 change into 𝑁 products P , P , ..., P 𝑁 , 𝐾 eq ( 𝑇 ) is calculated from 𝐾 eq ( 𝑇 ) = 𝜋𝑘 B 𝑇ℎ ! ( 𝑀 − 𝑁 ) (cid:18) 𝑚 R ...𝑚 R 𝑀 𝑚 P ...𝑚 P 𝑁 (cid:19) × (cid:18) 𝑧 ( R ) ...𝑧 ( R 𝑀 ) 𝑧 ( P ) ...𝑧 ( P 𝑁 ) (cid:19) 𝑒 − Δ 𝐸 / 𝑘 B 𝑇 , (8)where 𝑚 ( 𝑖 ) and 𝑧 ( 𝑖 ) are the mass and partition function of each atomor molecule, and Δ 𝐸 the heat of reaction, whose values are adoptedfrom the references summarized in Appendix A. In a sufficientlyopaque cloud, atoms (or molecules) are ionized (or dissociated) bythermal radiation trapped in the cloud. In this case, the radiation fieldhas the black-body spectrum ( 𝐽 𝜈 = 𝐵 𝜈 ( 𝑇 ) ), and the rate coefficientfor a radiative-dissociation reaction 𝑘 dissoc is calculated by that ofits reverse reaction, i.e., radiative association 𝑘 assoc through Eq. (8).When the cloud is still optically thin, since 𝑘 dissoc scales linearlywith the radiation intensity 𝐽 𝜈 = ( − 𝑒 − 𝜏 cont ) 𝐵 𝜈 ( 𝑇 ) ( 𝜏 cont is thecontinuum optical depth), 𝑘 dissoc is calculated from (Paper I) 𝑘 dissoc = ( − 𝑒 − 𝜏 cont ) 𝑘 assoc 𝐾 eq ( 𝑇 ) . (9)We adopt the dust model of Pollack et al. (1994), where the dustgrains are assumed to be composed of water ice, organics, troilite,metallic iron, and silicate (olivine and orthopyroxene). Below thevaporization temperature of water ice (100-200 K), the dust-to-gasmass ratio is 9 . × − . When the grain temperature 𝑇 gr exceeds thevaporization temperature of each constituent, the dust-to-gas massratio is decreased by the abundance of each constituent. The graintemperature 𝑇 gr is determined by the energy-balance equation of thedust grains:4 𝜎 SB 𝜅 gr 𝑇 𝛽 esc = Γ gas − dust + 𝜎 SB 𝜅 gr (cid:16) 𝑇 ( − 𝑒 − 𝜏 cont ) + 𝑇 𝑒 − 𝜏 cont (cid:17) , (10)where 𝜅 gr is the Planck mean opacity of dust grains calculated bySemenov et al. (2003), 𝛽 esc = min ( , 𝜏 − ) a factor representing theradiative diffusion effect (Masunaga et al. 1998), Γ gas − dust the energytransfer rate via gas-grain collision (Hollenbach & McKee 1979), and 𝑇 CMB = .
725 K the CMB temperature. The grain size distributionis assumed to follow the Mathis, Rumpl & Nordsieck (MRN)-typepower law (Mathis et al. 1977; Pollack et al. 1985): 𝑦 gr ( 𝑎 gr ) = 𝐶 gr ((cid:0) 𝑎 gr / 𝑎 mid (cid:1) − 𝜆 𝑎 min ≤ 𝑎 gr ≤ 𝑎 mid , (cid:0) 𝑎 gr / 𝑎 mid (cid:1) − 𝜆 𝑎 mid ≤ 𝑎 gr ≤ 𝑎 max , (11)where 𝜆 = . , 𝜆 = . , 𝑎 min = . 𝜇 m , 𝑎 mid = 𝜇 m, and 𝑎 max = 𝜇 m. When we take an integration or average of a physicalquantity over the grain size distribution, the distribution is dividedequally in log 𝑎 gr into 15 bins for 𝑎 min ≤ 𝑎 gr ≤ 𝑎 mid and 5 bins for 𝑎 mid ≤ 𝑎 gr ≤ 𝑎 max . For simplicity, we neglect the grain growth bythe accretion of gas-phase species and by grain-grain collisions.In addition to the gas-phase chemistry, we consider the grain-surface chemistry, the details of which are described in AppendixB. The grain-surface chemistry is divided into three categories: (i)the adsorption of a gas-phase species onto the grain surface, (ii)the desorption of a grain-surface species into the gas-phase, and(iii) molecule formation (e.g., Hasegawa et al. 1992). Dust grainsalso obtain electric charges when gas-phase ions or electrons re-combine with grain-surface species. These charge transfer reactionsfrom gas to grain and grain to grain are taken into account followingDraine & Sutin (1987). Dust grains are assumed to hold up to twocharges, i.e., gr , gr ± , and gr ± , since the abundance of more thantriply charged grains is negligibly small (e.g., Nakano et al. 2002). MNRAS , 1–23 (2021)
D. Nakauchi, K. Omukai, and H. Susa
In a sufficiently warm cloud of &
500 K, the ejection of thermalelectrons from grain surfaces (so-called thermionic emission) is alsotaken into account following Desch & Turner (2015).The fractional abundance of He, D, and Li nuclei relative toH nuclei is set to 𝑦 He = . × − , 𝑦 D = . × − , and 𝑦 Li = . × − , which are derived from the standard Big Bangnucleosynthesis (BBN) theory by using the baryon-to-photon ra-tio of the Planck observation (Cyburt et al. 2016). In the solarmetallicity case ( 𝑍 = Z ⊙ ), the abundance of C, O, Na, Mg, andK nuclei is adopted from the photospheric values of the Sun: 𝑦 C , ⊙ = . × − , 𝑦 O , ⊙ = . × − , 𝑦 Na , ⊙ = . × − , 𝑦 Mg , ⊙ = . × − , and 𝑦 K , ⊙ = . × − (Asplund et al. 2009). Ina metal-poor cloud, the abundance of a heavy element is decreasedin proportion to the metallicity. Below the water ice vaporizationtemperature, 72% of C, 46% of O, 98% of Mg, and, 100% of Na andK are depleted into dust grains. Above the vaporization temperatureof the silicate dust (1000-1500 K for olivine and orthopyroxene), thedepleted metals are released into the gas phase, and their gas-phaseabundance is increased in proportion to the decreased fraction of thesilicate dust (Finocchi & Gail 1997).The calculations are started from the density of 𝑛 H , = − andthe temperature of 𝑇 =
100 K. For the light elements, H, D, He, andLi, the initial abundances are set to be same as the intergalactic valuesin the post-recombination era (Galli & Palla 2013): 𝑦 ( H + ) = − , 𝑦 ( H ) = × − , 𝑦 ( HD ) = × − , and 𝑦 ( Li + ) = 𝑦 Li , and theremaining H, D, and He nuclei are in the neutral atomic state. Forheavy elements, with their high ionization potentials of 11.3 eV forcarbon and 13.6 eV for oxygen, all the C and O nuclei are assumedto exist as neutral atoms, while with its low ionization energy of 7.6eV, all Mg is assumed to present as Mg + .The injection of CR particles and the decay of RE are importantionization sources. The CR ionization rate is calculated with theshielding effect as (Nakano & Umebayashi 1986): 𝜁 CR = 𝜁 ion exp (cid:18) − 𝑁 H . × cm − (cid:19) s − . (12)The CR intensity in the Galactic ISM was once estimated as 𝜁 ion ∼ − s − (Spitzer & Scott 1969), but the recent observationssuggest a much larger value of 2 . × − s − (Neufeld & Wolfire2017). The CR intensity in the ISM in the first galaxies is evenmore uncertain. Since CR particles are generated by the shock-acceleration in a supernova (SN) remnant, the CR intensity would bestronger in a cloud closer to a star-forming galaxy (Stacy & Bromm2007; Nakauchi et al. 2014). This motivates us to consider a widerange for the CR intensity encompassing the Galactic ISM value: 𝜁 ion = , − , − , and 10 − s − . On the other hand, REs aresynthesized in a SN explosion. They are classified into two types de-pending on the decay time (Umebayashi & Nakano 2009; Susa et al.2015). Long-lived REs represented by K have a decay time of > 𝑧 >
6, and theseREs accumulate in the ISM in a galaxy like ordinary metals. Weassume that the ionization rate is given by the Galactic value in thesolar metallicity case and the rate scales linearly with the metallicity: 𝜁 longRE = . × − 𝑍 Z ⊙ s − . (13)On the other hand, short-lived REs represented by Al have a decaytime of ∼ 𝜁 ion = − s − , and the - M o • M o • M o • M o • M o • T ( K ) log n H (cm -3 ) ζ ion =0 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • Figure 1.
Temperature evolution of star-forming clouds without ionizationsources. Individual colored curves correspond to the results with variousmetallicities indicated in the legend. The oblique dashed lines indicate theloci of constant Jeans mass. rate scales linearly with the CR intensity: 𝜁 shortRE = . × − (cid:18) 𝜁 ion − s − (cid:19) s − . (14)The total ionization rate is given by the sum of the contribution fromthe three sources: 𝜁 total = 𝜁 CR + 𝜁 longRE + 𝜁 shortRE . (15) In this subsection, we describe the results without ionizationsources ( 𝜁 ion = Figure 1 shows the temperature evolution of star-forming clouds withvarious metallicities indicated in the legend. The oblique dashed linesindicate the loci of constant Jeans mass. Figure 2 shows the coolingand heating processes in the clouds with 𝑍 / Z ⊙ = (a) 10 − , (b) 10 − ,(c) 10 − , and (d) 1. In Figure 3, we show the fractional abundancesof major coolants, i.e., H , HD (left panel), and H O (right panel). Inthe right panel, solid and dashed curves indicate the H O abundancesin the gas and ice phases, respectively.In almost primordial cases ( 𝑍 / Z ⊙ . − ; Figure 2a), af-ter the short adiabatic contraction phase without enough coolants,H formation proceeds up to 𝑦 ( H ) ∼ − via the H − chan-nel (Peebles & Dicke 1968; Hirasawa et al. 1969):H + 𝑒 → H − + 𝛾 H + H − → H + 𝑒, (16) MNRAS , 1–23 (2021) onization degree in clouds with different metallicities -4-2024 0 5 10 15 l og Λ , Γ ( e r g s - g - ) log n H (cm -3 )(a) ζ ion =0, Z=10 -6 Z o • H lineH formH dissgraingas contcompr -4-2024 0 5 10 15 l og Λ , Γ ( e r g s - g - ) log n H (cm -3 )(b) ζ ion =0, Z=10 -4 Z o • H , HD lineH O lineH formgraincompr -8-6-4-202 0 5 10 15 l og Λ , Γ ( e r g s - g - ) log n H (cm -3 )(c) ζ ion =0, Z=10 -2 Z o • OI, CI lineH O lineH formgraincompr -8-6-4-202 0 5 10 15 l og Λ , Γ ( e r g s - g - ) log n H (cm -3 )(d) ζ ion =0, Z=1Z o • CI lineCO lineH formgraincompr Figure 2.
Contribution to the cooling and heating rates by individual processes for the clouds without ionization sources. Different panels correspond to theresults with 𝑍 / Z ⊙ = (a) 10 − , (b) 10 − , (c) 10 − , and (d) 1, respectively. -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 ) ζ ion =0 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • -10-8-6-4 0 5 10H O (gas) H O (ice) l og y ( H O ) log n H (cm -3 ) ζ ion =0 Figure 3.
Fractional abundances of H , HD (left panel), and H O (right panel) for the cases shown in Figure 1. In the right panel, the solid and dashed curvesindicate the H O abundances in the gas and ice phases, respectively. and H cooling lowers the temperature down to the local mini-mum of ∼
200 K. At 𝑛 H & cm − , H rotational levels reachthe local thermodynamic equilibrium (LTE), which decreases thecooling efficiency and raises the temperature again gradually. At 𝑛 H & cm − , the hydrogen becomes fully molecular via thethree-body reactions (Palla et al. 1983)H + H + H → H + H H + H + H → . (17)The elevated H fraction by these reactions raises the cooling ef-ficiency, but interrupts the temperature increase only temporarilyat 𝑛 H ∼ cm − , because the H lines become optically thickand H formation also contributes to gas heating. The temperature dip at 𝑛 H ∼ cm − is caused by the H collision-induced emis-sion (CIE; Omukai & Nishi 1998). The cloud soon becomes opticallythick to the collision-induced absorption. The very brief adiabatictemperature increase at this moment is followed by gradual temper-ature increase due to the H dissociation cooling.In extremely metal-poor cases ( 𝑍 / Z ⊙ = − -10 − ; Figure 2b),H formation proceeds on grain surfaces, in addition to the H − chan-nel, so that the cloud contracts with lower temperatures via the en-hanced H cooling. Once the temperature decreases below ∼
150 K,HD formation proceeds via the exothermic reactionsH + + D ⇋ H + D + H + D + ⇋ H + + HD , (18) MNRAS , 1–23 (2021)
D. Nakauchi, K. Omukai, and H. Susa and the HD cooling lowers the temperature further until the HD ro-tational levels reach the LTE and its cooling efficiency is reduced. At 𝑛 H ∼ , , and 10 cm − for 𝑍 / Z ⊙ = − , − , and 10 − , re-spectively, OH and H O are formed via the following reactions (Fig-ure 3 right)H + O → OH + 𝛾 H + OH → H + H OH + OH → H O + 𝛾. (19)While H O cooling becomes dominant, it is counteracted by theH formation heating on grain surfaces, and the temperature keepsincreasing gradually. Note that H O formation proceeds mainly viathe gas-phase reactions (Eq. 19), and the pathway via grain-surfacereactions has only a minor effect. After H formation is over at 𝑛 H ∼ , , and 10 cm − (for 𝑍 / Z ⊙ = − , − , and 10 − ),the cloud cools efficiently via dust thermal emission up to the secondlocal minimum. Once the cloud becomes optically thick to the ab-sorption of the dust thermal radiation at 𝑛 H ∼ -10 cm − , thetemperature begins to increase adiabatically with contraction.In the metal-enriched cases of 𝑍 / Z ⊙ & − (Figure 2c, d), OI,CI, and CO line cooling becomes effective, and the cloud cools downto lower temperatures than in more metal-poor clouds. As H for-mation proceeds via the grain-surface reaction and its heating raterises, the temperature starts to increase. After the H formation isover, the temperature remains low ( .
10 K) via dust cooling untilthe cloud becomes opaque to the thermal radiation. In the case of 𝑍 / Z ⊙ & − , HD formation proceeds via the grain-surface pro-cesses rather than via the gas-phase reactions (Eq. 18). However, HDcooling remains a minor process throughout the evolution in thesemetallicities. Therefore, among the molecule formation processeson grain surfaces, only H formation plays an important role in thetemperature evolution.Once the cloud center becomes opaque to the thermal radiation,the temperature increases adiabatically as 𝑇 ∝ 𝑛 𝛾 eff − , with the effec-tive adiabatic index ( 𝛾 eff ) larger than the critical value 𝛾 crit = / 𝑍 / Z ⊙ & − , theadiabatic phase lasts long enough time to stop dynamical contraction,forming a first hydrostatic core (e.g., Larson 1969; Masunaga et al.1998; Omukai et al. 2010). However, the cloud center contracts in aquasi-static way by accreting the surrounding gas and increasing itsmass. As the temperature approaches ∼ occurs and works as the effective cooling. This makesthe temperature increase shallower with 𝛾 eff < /
3, enabling thedynamical contraction of the cloud again. After almost all the H isdissociated into hydrogen atoms, 𝛾 eff is again elevated to 𝛾 eff = / In Figure 4, we show the fractional abundances of the major chargedspecies as a function of density, for the same cases with Figure 2.In the beginning of the collapse, 𝑒 and H + are the major chargedspecies, and the ionization degree decreases slowly via H + radiativerecombination. At 𝑛 H ∼ , , , and 1 cm − (for 𝑍 / Z ⊙ = − , − , − , and 1), along with the H O formation via Eq.(19) (Figure 3 right), molecular ions such as H O + and H O + are produced viaH + + H O → H + H O + H + H O + → H + H O + . (20)They are disrupted immediately via the dissociative recombination: 𝑒 + H O + → H + H O 𝑒 + H O + → + OH 𝑒 + H O + → H + OH , (21)and the ionization degree drops rapidly. Afterwards, the evolution ofionization degree differs between the cases with different metallici-ties.In the case of 𝑍 / Z ⊙ = − (Figure 4a), Li + takes over the majorcation species at 𝑛 H ∼ cm − . The ionization degree continuesdecreasing as Li + and 𝑒 recombine on grain surfaces:Li + + gr −− → Li + gr − 𝑒 + gr − → gr −− . (22)The grain-surface recombination proceeds more rapidly with increas-ing density, and at 𝑛 H ∼ cm − , charged dust grains gr ± becomethe dominant charge carriers, instead of Li + and 𝑒 . However, thisoccurs only temporarily, since the temperature soon becomes highenough ( 𝑇 ∼ 𝑛 H ∼ cm − .Along with grain vaporization, the ionization degree rises rapidlyuntil Li is ionized completely by the thermal radiation trapped in theopaque cloud and by collision with H :Li + 𝛾 → 𝑒 + Li + H + Li → H + 𝑒 + Li + . (23)After Li ionization is completed, hydrogen ionization begins suc-cessively and the ionization degree increases monotonically withincreasing temperature.In the cases of 𝑍 / Z ⊙ & − (Figure 4b-d), after the ionizationdegree drops rapidly via the H O + dissociative recombination (Eq.21), Mg + takes over the major cation species. The abundances ofMg + and 𝑒 are lowered via the grain-surface recombination:Mg + + gr − → Mg + gr 𝑒 + gr → gr − , (24)and finally Mg + and 𝑒 are superseded by charged dust grains gr ± at 𝑛 H ∼ , , and 10 cm − (for 𝑍 / Z ⊙ = − , − , and 1). Theionization degree decreases further as charged dust grains neutralizeeach other:gr + + gr − → . (25)When the temperature reaches ∼ 𝑛 H ∼ cm − ,thermionic emission from neutral grain surfaces becomes important.The ejected thermal electrons are soon absorbed by other neutralgrains:gr → gr + + 𝑒𝑒 + gr → gr − . (26)Through these two processes, the abundance of charged grains gr ± increases rapidly. When the temperature exceeds ∼ MNRAS , 1–23 (2021) onization degree in clouds with different metallicities -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(a) ζ ion =0, Z=10 -6 Z o • eH + H O + Li + Mg + gr - gr -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(b) ζ ion =0, Z=10 -4 Z o • eH + H O + Li + K + Na + Mg + gr + gr - gr -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(c) ζ ion =0, Z=10 -2 Z o • eH + H O + K + Na + Mg + gr + gr - -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(d) ζ ion =0, Z=1Z o • eH + H O + K + Na + Mg + gr + gr - Figure 4.
Fractional abundances of the major charged species as a function of density, for the same cases with Figure 2.
10 15 20solid: our networkdashed: previous network T ( K ) log n H (cm -3 ) ζ ion =0 o • -1 Z o • -2 Z o • -3 Z o • -4 Z o • -18-16-14-12-10-8-6-4 10 15 20solid: our networkdashed: previous network l og ( y ( e ) + y ( g r - )) log n H (cm -3 ) ζ ion =0 o • -1 Z o • -2 Z o • -3 Z o • -4 Z o • Figure 5.
Comparison for the temperature evolution (left panel) and the ionization degree (right panel) calculated by our (solid) and previous (dashed curves)chemical networks. ionization degree is raised dominantly by H ionization until the endof the calculation.Finally, we discuss the differences in the evolution of the tem-perature and ionization degree obtained by our and previous mod-els (Susa et al. 2015). As mentioned in Section 1, previous chemicalmodel has the following problems: (i) thermionic emission and ther-mal ionization of vaporized alkali metals are neglected, (ii) only apart of the reactions are reversed, and the abundances are switchedartificially to the equilibrium values of the gas composed of H, H ,e, H + , He, He + , and He ++ . In Figure 5, we compare the temperatureevolution (left panel) and the ionization degree (right panel) calcu-lated by our (solid) and previous (dashed curves) chemical networks,for the models with 𝑍 / Z ⊙ = − , − , − , − , and 1. In theleft panel, the temperature evolves along the qualitatively similar track in both models, although in our model the dust cooling worksfrom the lower densities and the cloud evolves with lower temper-atures at 𝑛 H < cm − (or 10 cm − ) for 𝑍 / Z ⊙ = − (or10 − , respectively). In the right panel, the ionization degree in bothmodels decreases slowly following almost the same track up to 𝑛 H ∼ cm − . At 𝑛 H ∼ cm − , the ionization degree inour model jumps up via thermionic emission and thermal ionizationof vaporized K and Na. At higher densities, the ionization degreeincreases slowly, following the chemical equilibrium abundances be-tween all the 63 gas-phase species. On the other hand, due to thelack of these processes, the ionization degree in the previous modelkeeps decreasing until the dust grains evaporate completely. Afterthat, the ionization degree increases following the equilibrium valueof the H/He gas. From Figure 5, we find the ionization degree at MNRAS , 1–23 (2021)
D. Nakauchi, K. Omukai, and H. Susa 𝑛 H ∼ -10 cm − higher by up to eight orders of magnitudethan that predicted in the previous model. Next, we describe the effect of ionization sources on the temperatureevolution and ionization degree. CR particles propagating in a cloudlose their energy via the gas ionization, and are thus attenuated whenthe density becomes higher than 𝑛 H ∼ cm − . Therefore, CRionization dominates only at low densities and is superseded byradioactive ionization at 𝑛 H ∼ cm − . In Figure 6, we show the effects of ionization sources on the temper-ature evolution (left column) and the H and HD abundances (rightcolumn) for the cases with 𝜁 ion = (a) 10 − , (b) 10 − , and (c)10 − s − , respectively. Ionization sources not only heat the gas di-rectly, but also enhance the H -cooling efficiency by activating H formation via the electron-catalyzed H − channel (Eq. 16). Both ef-fects are observed at low densities of 𝑛 H . cm − , where theCR ionization dominates over radioactive ionization.The direct heating by CR ionization is observed in all the metal-licity cases. As the CR intensity becomes stronger, the cloud evolveswith higher temperatures at 𝑛 H ∼ cm − . On the other hand,the effect of enhanced H cooling appears only in the low-metallicitycases of 𝑍 / Z ⊙ . − . In these cases, the H abundance at eachdensity becomes higher with increasing 𝜁 ion (Figure 6 right). Oncethe cloud cools via the enhanced H cooling up to 𝑇 .
150 K,HD formation kicks in. Due to the efficient HD cooling, these low-metallicity clouds ( 𝑍 / Z ⊙ . − ) evolve along a similar track at 𝑛 H . cm − . At higher densities, the temperature evolution iscontrolled by H formation heating and dust cooling, both of whichare independent of the ionization degree. Therefore, the temperatureevolution afterwards does not depend on the presence of ionizationsources. In Figure 7, the evolution of the major charged species is plot-ted for the same metallicity cases with Figure 4, but for the dif-ferent ionization rates of 𝜁 ion = − s − (left column) and 𝜁 ion = − s − (right column), respectively. According to Fig-ure 7, the ionization degree takes a higher value at each density withincreasing ionization rate. Gas-phase ions and electrons remain asmajor charge carriers over a wider range of densities compared tothe case without ionization. Charged dust grains never become dom-inant in the very low-metallicity cases of 𝑍 / Z ⊙ . − (or 10 − ) for 𝜁 ion = − s − (or 𝜁 ion = − s − , respectively). With ionizationsources, a positive charge is carried successively by various speciesH + , H + , H O + , HCO + , Mg + , and gr + , whereas a negative charge iscarried only by 𝑒 and gr − . The transition in the major charged speciesis represented by a schematic diagram in Figure 8, and is describedextensively below.In the beginning of the collapse, electrons are produced via the CRionization of hydrogen atoms and removed via the radiative recom-bination of H + . As a result of the balance between these processes,the ionization degree decreases slowly as 𝑦 ( 𝑒 ) ≃ 𝑦 ( H + ) ≃ (cid:18) 𝜁 ion 𝑘 H , rec 𝑛 H (cid:19) / . (27) CR particles also ionize H , produced via the H − channel (Eq. 16)or grain-surface reactions:H + CR → 𝑒 + H + . (28)The resultant H + immediately reacts with H and produces H + via (Oppenheimer & Dalgarno 1974):H + H + → H + H + . (29)As the formation of H O and CO proceeds, positive charges arepassed from H + to other molecular ions, H O + and HCO + , via thefollowing ion-molecule reactions:H + + H O → H + H O + , H + + CO → H + HCO + , H O + HCO + → CO + H O + . (30)Positive charges are also transferred to metallic ion Mg + via thefollowing reactions:H + + Mg → H + H + Mg + , HCO + + Mg → HCO + Mg + . (31)Through these processes, positive charges are carried by molecularions m + (= H + , H O + , HCO + ) and metallic ion Mg + , instead of H + .At the densities where molecular ions are dominant cation species,electrons are removed by the dissociative recombination of m + (e.g.,Eq. 21), whereas they are produced by CR or radioactive ionizationof H . Through the balance between these processes, the ionizationdegree decreases with contraction as 𝑦 ( 𝑒 ) ≃ 𝑦 ( m + ) ≃ (cid:18) 𝜁 ion 𝑘 m , rec 𝑛 H (cid:19) / . (32)Dust grains obtain a negative charge by capturing electrons pro-vided via CR or radioactive ionization of H . Grains also ob-tain a positive charge by colliding with molecular ions or metallicions. Grains can become dominant charge carriers, if the ioniza-tion degree determined by Eq. (32) decreases below the total grainfraction 𝑦 gr = . × − 𝑍 / Z ⊙ before the grain evaporation at 𝑛 H ∼ cm − . This condition can be expressed with respect tometallicity as 𝑍 / Z ⊙ ≃ − (cid:18) 𝜁 ion − s − (cid:19) / , (33)where 𝑘 m , rec = × − cm s − is adopted. This result is consistentwith the numerical calculation. When the metallicity is higher thanEq. (33), dust grains become the dominant population of chargedspecies (Figures 7 c1-d2).When charged grains are prevalent, negative grains are producedvia electron capture of neutral grains and are destroyed via mutualneutralization: 𝑒 + gr → gr − gr + + gr − → . (34)These processes balance each other, controlling their abundances as 𝑦 gr − ≃ 𝑦 gr + ≃ (cid:18) 𝑘 𝑒 − gr 𝑘 gr + − gr − 𝑦 ( gr ) 𝑦 ( 𝑒 ) (cid:19) / . (35)Since the electron fraction is determined by the balance between theradioactive ionization of H and electron capture of neutral grains,Eq. (35) is rewritten as 𝑦 gr − ≃ 𝑦 gr + ≃ (cid:18) 𝜁 ion 𝑘 gr + − gr − 𝑛 H (cid:19) / . (36) MNRAS , 1–23 (2021) onization degree in clouds with different metallicities - M o • M o • M o • M o • M o • T ( K ) log n H (cm -3 )(a) ζ ion =10 -19 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(a) ζ ion =10 -19 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • - M o • M o • M o • M o • M o • T ( K ) log n H (cm -3 )(b) ζ ion =10 -17 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(b) ζ ion =10 -17 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • - M o • M o • M o • M o • M o • T ( K ) log n H (cm -3 )(c) ζ ion =10 -15 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(c) ζ ion =10 -15 Z=010 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • Figure 6.
Effects of ionization sources on the temperature evolution (left column) and H and HD abundances (right column) in the clouds with differentmetallicities. Individual panels correspond to the results with 𝜁 ion = (a) 10 − , (b) 10 − , and (c) 10 − s − , respectively. Charged grain abundances decrease with increasing density, follow-ing Eq. (36).Once the temperature exceeds 𝑇 ≃ 𝑛 H ∼ -10 cm − , dust grains evaporate, and the ionization degreeincreases rapidly via thermal ionization of vaporized alkali metals.The thermal ionization proceeds much more rapidly than radioactiveionization. Therefore, the presence of the ionization sources has noeffect on the ionization degree afterwards.Finally, we discuss the differences of the above results from thoseof the previous models (Susa et al. 2015). In Figure 9, we compare thefractional ionization calculated by our (solid) and previous (dashedcurves) chemical networks for the cases with 𝜁 ion = − s − (leftpanel) and with 𝜁 ion = − s − (right panel). For 𝑛 H > cm − ,CR or radioactive ionization works as minor heating process, so thatthe temperature evolution in both models is the same as in the casewith 𝜁 ion = 𝑛 H ∼ cm − , the fractional ionization in both models decreasesslowly in a qualitatively similar way. At the densities where ions and electrons are the dominant charge carriers, the ionization degree inour model takes higher values, owing to the updated rate coefficientsfor the gas-phase process. Along with the grain evaporation at 𝑛 H ∼ cm − , the ionization degree in our model jumps up via thermalionization of K and Na, while that in the previous model dropsimmediately below < − . Afterwards, the ionization degree doesnot depend on the radioactive ionization rate and evolves along thesame track as in the case with 𝜁 ion = The full chemical network developed above involves a large numberof species (89) and reactions (1482). It requires great computationalresources to implement the full network into multi-D (M)HD cal-culations. To save the computational time, we develop a reduced
MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(a1) ζ ion =10 -17 , Z=10 -6 Z o • eH + H Li + Mg + gr -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(a2) ζ ion =10 -15 , Z=10 -6 Z o • eH + H Li + Mg + gr -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(b1) ζ ion =10 -17 , Z=10 -4 Z o • eH + H H O + K + Na + Mg + gr + gr - -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(b2) ζ ion =10 -15 , Z=10 -4 Z o • eH + H H O + K + Na + Mg + gr -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(c1) ζ ion =10 -17 , Z=10 -2 Z o • eH + H H O + K + Na + Mg + gr + gr - -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(c2) ζ ion =10 -15 , Z=10 -2 Z o • eH + H H O + K + Na + Mg + gr + gr - -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(d1) ζ ion =10 -17 , Z=1Z o • eH + H H O + K + Na + Mg + gr + gr - -15-10-5 0 5 10 15 20 l og y ( i ) log n H (cm -3 )(d2) ζ ion =10 -15 , Z=1Z o • eH + H H O + K + Na + Mg + gr + gr - Figure 7.
Same as Figure 4, but for the models with 𝜁 ion = − s − (left column) and 𝜁 ion = − s − (right column). network to reproduce the fractional abundances of the main coolantsand charged species, by identifying the important processes asso-ciated with these species. The list of the reactions included in thereduced network is shown in Table 1. H and HD formation ongrain surfaces (reactions R6 and R11) is implemented by using the simple formulae of Eqs. (B23) and (B28). In Figure 10, we com-pare the (a) temperature evolution and the abundances of (b) H and HD, (c) 𝑒 , and (d) gr − , which are calculated by using the re-duced (solid) and full (dashed curves) chemical networks, for the MNRAS , 1–23 (2021) onization degree in clouds with different metallicities Table 1.
Reduced chemical network.Number Reaction Number Reaction Number ReactionR1 𝑒 + H + ⇋ H + 𝛾 R37 ∗ H + H + ⇋ H + + 𝛾 R67 𝑒 + gr ⇋ gr − R2 H + 𝑒 ⇋ H − + 𝛾 R38 ∗ H + H + ⇋ H + H + R68 𝑒 + gr + ⇋ grR3 H + H − ⇋ H + 𝑒 R39 ∗ H + H + ⇋ H + H + R69 𝑒 + gr − ⇋ gr −− R4 3H ⇋ H + H R40 ∗ 𝑒 + H + ⇋ H + H R70 H + + gr → H + gr + R5 2H ⇋ + H R41 ∗ 𝑒 + H + ⇋
3H R71 ∗ H + + gr → H + H + gr + R6 2H + grain → H R42 ∗ H + + O ⇋ H + OH + R72 ∗ C + + gr → C + gr + R7 H + HD ⇋ + D R43 ∗ H + + OH ⇋ H + H O + R73 H O + + gr → H + OH + gr + R8 H + + D ⇋ H + D + R44 ∗ H + + H O ⇋ H + H O + R74 Li + + gr → Li + gr + R9 H + D ⇋ H + HD R45 ∗ H + + CO ⇋ H + HCO + R75 Mg + + gr → Mg + gr + R10 H + D + ⇋ H + + HD R46 ∗ H + + O ⇋ H + H O + R76 H + + gr − → H + grR11 H + D + grain → HD R47 ∗ 𝑒 + He + ⇋ He + 𝛾 R77 ∗ H + + gr − → H + H + grR12 H + OH ⇋ O + H R48 ∗ H + He + ⇋ H + H + + He R78 ∗ He + + gr − → He + grR13 H + OH ⇋ H + H O R49 ∗ H + He + ⇋ H + + He R79 ∗ C + + gr − → C + grR14 C + OH ⇋ H + CO R50 ∗ H + He + ⇋ H + + He R80 H O + + gr − → H + OH + grR15 H + O ⇋ OH + 𝛾 R51 ∗ He + + H O ⇋ H + He + OH + R81 Li + + gr − → Li + grR16 H + OH ⇋ H O + 𝛾 R52 ∗ He + + CO ⇋ He + C + + O R82 Mg + + gr − → Mg + grR17 H + + OH ⇋ H + OH + R53 ∗ 𝑒 + C + ⇋ C + 𝛾 R83 H + + gr −− → H + gr − R18 H + + H O ⇋ H + H O + R54 ∗ C + O ⇋ O + CO R84 ∗ H + + gr −− → H + H + gr − R19 H + O + ⇋ H + + O R55 ∗ O + OH ⇋ H + O R85 ∗ C + + gr −− → C + gr − R20 H + O + ⇋ H + OH + R56 ∗ H + + Mg ⇋ H + H + Mg + R86 H O + + gr −− → H + OH + gr − R21 H + OH + ⇋ H + H O + R57 ∗ C + + Mg ⇋ C + Mg + R87 Li + + gr −− → Li + gr − R22 H + H O + ⇋ H + H O + R58 ∗ HCO + + Mg ⇋ HCO + Mg + R88 Mg + + gr −− → Mg + gr − R23 H O + HCO + ⇋ CO + H O + R59 ∗ H + HCO ⇋ H + CO R89 gr + + gr − → gr + grR24 𝑒 + OH + ⇋ H + O R60 † H + CH ⇋ H + C R90 gr + + gr −− → gr + gr − R25 𝑒 + H O + ⇋ H + OH R61 † H + CH ⇋ H + CH R91 ∗ O ⇋ O ( p ) R26 𝑒 + H O + ⇋ H + O R62 † H + C ⇋ CH + 𝛾 R92 ∗ C ⇋ C ( p ) R27 𝑒 + H O + ⇋ + O R63 † H + C ⇋ CH + 𝛾 R93 ∗ OH ⇋ OH ( p ) R28 𝑒 + H O + ⇋ H + H O R64 † CH + O ⇋ H + CO R94 ∗ CO ⇋ CO ( p ) R29 𝑒 + H O + ⇋ H + OH R65 † CH + O ⇋ 𝑒 + HCO + R95 ∗ H O ⇋ H O ( p ) R30 𝑒 + H O + ⇋ + OH R66 † C + H O + ⇋ H + HCO + CR1 ∗ H + CR → 𝑒 + H + R31 𝑒 + HCO + ⇋ H + CO CR2 ∗ He + CR → 𝑒 + He + R32 𝑒 + Mg + ⇋ Mg + 𝛾 CR3 ∗ H + CR → H + 𝑒 + H + R33 H + + Mg ⇋ H + Mg + CR4 ∗ H + CR → 𝑒 + H + R34 𝑒 + Li + ⇋ Li + 𝛾 CR5 ∗ H + CRph → 𝑒 + H + R35 H + + Li ⇋ H + Li + + 𝛾 CR6 ∗ He + CRph → 𝑒 + He + R36 H + Li ⇋ H + 𝑒 + Li + CR7 ∗ C + CRph → 𝑒 + C + CR8 ∗ O + CRph → Notes.
Reactions with asterisks (or daggers) are needed only in the presence (or absence, respectively) of ionization sources. cases with 𝜁 ion = 𝜁 ion = − s − (middle panels),and 𝜁 ion = − s − (right panels).Without ionization sources ( 𝜁 ion = , 𝑒 , H + , H − , D, D + , HD, C,CH, CH , O, OH, CO, H O, O + , OH + , H O + , HCO + , H O + , Li, Li + ,Mg, Mg + , gr , gr + , gr − , gr −− . The following six species, H − , D + ,O + , OH + , H O + , and HCO + , appear as the intermediate products ofthe major chemical species, and their abundances are replaced by thechemical equilibrium values.According to the left panels in Figure 10, the temperature evo-lution and the abundances of H and HD are reproduced almostcompletely by the reduced network. The abundances of 𝑒 and gr − inthe reduced model follows those in the full model over most of therange of densities except at 𝑛 H ∼ -10 cm − (or 10 -10 cm − )for 𝑍 / Z ⊙ = − (or 10 − , respectively), where 𝑦 ( 𝑒 ) and 𝑦 ( gr − ) areoverestimated by up to an order of magnitude and by a factor of afew, respectively. These deviations can be reduced, if the freeze-outof C and O on grain surfaces (reactions R91 and R92) is taken intoaccount. However, these deviations are observed only at a limitedrange of density and metallicity, so that the freeze-out of C and O is omitted in the reduced model with 𝜁 ion =
0. At 𝑛 H > cm − ,the electron fraction is underestimated in the reduced model by up tothree orders of magnitude, owing to the lack of the collisional ion-ization of K and Na (Eq. 6). Even in this case, the ionization degreeis high enough to couple magnetic fields coherent over the cloud sizeto the gas, so that these reactions are negligible, as long as focusingon ordered magnetic fields.With ionization sources, the reduced network consists of 161 re-actions (reactions with no symbols and asterisks in Table 1) amongthe following 38 species: H, H , 𝑒 , H + , H + , H + , H − , He, He + D, D + ,HD, C, C + , O, O , OH, CO, H O, HCO, O + , OH + , H O + , HCO + ,H O + , Li, Li + , Mg, Mg + , gr , gr + , gr − , gr −− , C(p), O(p), OH(p),CO(p), H O(p). For the following eight species, H + , H − , D + , HCO,O + , OH + , H O + , and HCO + , the abundances can be given by theequilibrium values.According to the middle and right panels in Figure 10, the tem-perature evolution as well as the abundances of H , HD, and gr − arereproduced almost completely by the reduced model for both caseswith 𝜁 ion = − s − and 𝜁 ion = − s − . The electron fractionis also reproduced over most of the range of densities. Deviationsfrom the full model appear only a limited range of densities and MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa H C CRph + HCO + Mg + H gr → −− → Mggr gr + COHCO → −− → Mggr gr ⇋ ∗ H OH CR + gr + H O + HCO H C CRph ⇋ ∗ H OH CR + H + He →→ + H + → e C + + gr + + gr + gr + H O + H O + H + H ⇋ ∗ H OH CR !" e H + gr − + gr ⇋ + CRHe CR + CRHe CR /"0.&%1 + CRHe CR H C CRph HH , H C CRph HH , ∗ ⇋ OHCO
Figure 8.
Schematic diagram representing the transition of the major chargedspecies (species enclosed by large circles) with increasing density. Speciesenclosed by small circles indicate the intermediate products of the majorcharged species. are regulated below an order of magnitude, in each metallicity case.With ionization sources, molecular ions, H O + and HCO + , whichare formed via the reactions R44 and R45, respectively, can be thedominant cations at some densities (Figure 7). The abundances ofH O + and HCO + depend on those of H O and CO, which also de-pend on those of O, C, and OH via the reactions R13-R16. Mostof these neutral atoms and molecules are depleted on the grain sur-face at high densities. Therefore, the freeze-out of O, C, OH, CO,and H O (reaction R91-R95) is included in the reduced model withionization sources.
As we have discussed in the previous section, the ionization degree ina star-forming cloud decreases monotonically with increasing densityuntil dust grains evaporate at 𝑛 H ∼ -10 cm − . In a cloudthreaded by magnetic fields, the ionization degree can be low enoughto weaken the coupling between the gas and magnetic fields via suchresistive MHD effects as ambipolar diffusion and Ohmic loss. Inthis section, we discuss the conditions of magnetic dissipation in astar-forming cloud with various metallicities.The diffusion coefficients for ambipolar dissipation and Ohmicloss, 𝜂 ambi and 𝜂 Ohm , are calculated from the following equa-tions (e.g., Wardle & Ng 1999): 𝜂 ambi = 𝑐 𝜋 𝜎 P 𝜎 + 𝜎 − 𝜂 Ohm ,𝜂 Ohm = 𝑐 𝜋𝜎 O , (37)where 𝜎 P , 𝜎 H , and 𝜎 O are the Pedersen, Hall, and Ohmic conductiv- ities, respectively: 𝜎 P = (cid:16) 𝑐𝐵 (cid:17) Õ 𝜈 𝜌 𝜈 𝜏 𝜈 𝜔 𝜈 + 𝜏 𝜈 𝜔 𝜈 𝜎 H = (cid:16) 𝑐𝐵 (cid:17) Õ 𝜈 𝑞 𝜈 | 𝑞 𝜈 | 𝜌 𝜈 𝜔 𝜈 + 𝜏 𝜈 𝜔 𝜈 ,𝜎 O = (cid:16) 𝑐𝐵 (cid:17) Õ 𝜈 𝜌 𝜈 𝜏 𝜈 𝜔 𝜈 . (38)Here the subscript ‘ 𝜈 ’ indicates a charged species with electric charge 𝑞 𝜈 , mass density 𝜌 𝜈 = 𝑚 𝜈 𝑦 ( 𝜈 ) 𝑛 H , and cyclotron frequency 𝜔 𝜈 = 𝑒 | 𝑞 𝜈 | 𝐵 / 𝑚 𝜈 𝑐 . 𝜏 𝜈 is the collision timescale between the charged andneutral particles (Nakano & Umebayashi 1986): 𝜏 − 𝜈 = Õ n 𝜏 − 𝜈, n = Õ n 𝜇 𝜈, n 𝑦 ( 𝜈 ) 𝑦 ( n ) 𝑛 h 𝜎 v i 𝜈, n 𝜌 𝜈 , (39)where the subscript ‘n’ stands for the major neutral species (H, H ,and He), 𝜇 𝜈, n the reduced mass, and h 𝜎 v i 𝜈, n the collision rate coef-ficient. For the collisions between ( 𝑒 , H + , gr ± , and gr ± )-(H, H , andHe), H + -H , C + -H, and HCO + -H , the values for h 𝜎 v i 𝜈, n are takenfrom Pinto & Galli (2008), and for the other cases from Osterbrock(1961).The dependence of 𝜂 ambi and 𝜂 Ohm on the chemical compostionand magnetic field strength can be seen from Eq. (37), by extractingthe dominant terms in the summation of Eq. (38): 𝜂 ambi ≃ 𝜋 (cid:18) 𝐵𝑛 H (cid:19) (cid:0) 𝜇 𝜈, n h 𝜎 v i 𝜈, n 𝑦 ( 𝜈 ) 𝑦 ( n ) (cid:1) − ,𝜂 Ohm ≃ 𝑐 𝜋𝑒 𝜇 𝜈, n h 𝜎 v i 𝜈, n 𝑦 ( n ) 𝑦 ( 𝜈 ) − . (40)Eq. (40) indicates that 𝜂 ambi scales quadratically with the magneticfield strength, whereas 𝜂 Ohm does not depend on the field strength.Moreover, both coefficients are inversely proportional to the ioniza-tion degree.The condition of magnetic dissipation is discussed by calculatingthe magnetic Reynolds number Rm defined below:Rm ( 𝐿 B ) ≡ v n 𝐿 B 𝜂 ambi + 𝜂 Ohm , (41)where v n is the fluid velocity, 𝐿 B the coherent length of the field.Magnetic Reynolds number is expressed as a ratio of the magneticdissipation timescale 𝑡 dis ( 𝐿 B ) = 𝐿 /( 𝜂 ambi + 𝜂 Ohm ) to the fluiddynamical timescale 𝑡 dyn ( 𝐿 B ) = 𝐿 B / v n :Rm ( 𝐿 B ) = 𝑡 dis ( 𝐿 B ) 𝑡 dyn ( 𝐿 B ) . (42)When inequality Rm ( 𝐿 B ) < n ∼ v ff ≡ 𝜆 J /( 𝑡 ff ) and coherent length 𝐿 B ∼ 𝜆 J into Eq. (41). In Figure 11, we showthe contour map of the magnetic Reynolds number Rm ( 𝜆 J ) , for thecases with 𝜁 ion = 𝜁 ion = − s − (middle panels),and 𝜁 ion = − s − (right panels). The thick and thin black curvesrepresent the contours of log Rm = = , , ,
8, and10. In the red-shaded regions, Rm ( 𝜆 J ) < MNRAS , 1–23 (2021) onization degree in clouds with different metallicities -18-16-14-12-10-8-6-4 10 15 20solid: our networkdashed: previous network l og ( y ( e ) + y ( g r - )) log n H (cm -3 ) ζ ion =10 -17 o • -1 Z o • -2 Z o • -3 Z o • -4 Z o • -18-16-14-12-10-8-6-4 10 15 20solid: our networkdashed: previous network l og ( y ( e ) + y ( g r - )) log n H (cm -3 ) ζ ion =10 -15 o • -1 Z o • -2 Z o • -3 Z o • -4 Z o • Figure 9.
Comparison of the fractional ionization calculated by our (solid) and previous (dashed curves) chemical networks for the cases with 𝜁 ion = − s − (leftpanel) and with 𝜁 ion = − s − (right panel). In both models, the temperature evolution is completely same as the case with 𝜁 ion = 𝑛 H > cm − (Figure5 left panel) and is not shown. field strength 𝐵 crit 𝐵 crit = (cid:18) 𝜋𝐺 𝑀 J 𝜌𝜆 J (cid:19) / , (43)above which magnetic pressure suppresses the cloud contraction.Now that a star-forming cloud is concerned, we consider magneticfields weaker than 𝐵 crit (corresponding to the region below the whitecurves in Figure 11).Without ionization sources ( 𝜁 ion =
0; Figure 11 left panels), in-equality Rm ( 𝜆 J ) > 𝐵 ∼ 𝐵 crit , magnetic field lines slip away from the contractingcloud via ambipolar diffusion at 𝑛 H & , , and 10 cm − for 𝑍 / Z ⊙ = − , − , and 1. When charged dust grains becomethe dominant charge carrier, the resistivity is elevated because theionization degree is decreased to an extremely low value and dustgrains have large inertia and collision cross section. Then the mag-netic field is dissipated irrespective of its strength via Ohmic loss at 𝑛 H & , , , and 10 cm − for 𝑍 / Z ⊙ = − , − , − ,and 1 (red-shaded regions). This continues until dust grains evap-orate at 𝑛 H ∼ -10 cm − , where the ionization degree jumpsup via the thermionic emission and thermal ionization of alkali met-als. We find that the magnetic field recovers strong coupling withthe cloud at much earlier stages (by a few orders of magnitudein density), compared to the previous work which showed by ne-glecting the above ionization processes that the recoupling occurs at 𝑛 H ∼ -10 cm − (see Figure 7 of Susa et al. 2015).With increasing ionization rate, the ionization degree at a givendensity becomes higher, elevating the value of Rm ( 𝜆 J ) (c.f., Figure11 middle and right panels). As a result, the resistive MHD effectsare weakened, and the regions of magnetic dissipation (red-shadedregions) shrink, compared to the cases without ionization sources.Especially, in such metal-poor clouds as 𝑍 / Z ⊙ . − (or 10 − ) for 𝜁 ion = − s − (or 𝜁 ion = − s − , respectively), the magneticfield remains coupled with cloud contraction throughout the evolu-tion. In higher metallicity cases, the magnetic field restores strongcoupling with the cloud at 𝑛 H ∼ -10 cm − , which is the sameas in the case with 𝜁 ion =
0. This is because grain evaporation andthermal ionization of alkali metals proceed much more rapidly thanradioactive ionization. Compared to the previous work, the recou-pling occurs at lower densities by a few orders of magnitude (c.f.Figure 7 of Susa et al. 2015).Next, we consider that turbulent motions in a cloud twist and stretch magnetic field lines, generating random fields fluctuating at 𝐿 B = 𝑙 < 𝜆 J (Brandenburg & Subramanian 2005). Such small scalefields are more subject to dissipation. The dissipation condition isjudged at each scale 𝑙 by the magnetic Reynolds numberRm ( 𝑙 ) = v eddy ( 𝑙 ) 𝑙𝜂 ambi + 𝜂 Ohm . (44)In Eq. (44), the velocity spectrum of the turbulent motion v eddy ( 𝑙 ) isassumed to follow a power lawv eddy ( 𝑙 ) = v ff (cid:18) 𝑙𝜆 J (cid:19) 𝜗 ( 𝑙 vis ≤ 𝑙 ≤ 𝜆 J ) , (45)where the viscous scale 𝑙 vis is calculated by using the kinematic vis-cosity 𝜈 vis = 𝑐 𝑠 /( 𝑛 H 𝜎 nn ) and collision cross-section among neutralparticles 𝜎 nn as (Subramanian 1998) 𝑙 vis 𝜆 J = (cid:18) 𝜈 vis v ff 𝜆 J (cid:19) /( 𝜗 + ) . (46)The power-law index takes 𝜗 = / 𝜗 = / ( 𝑙 ) <
1) can be rewritten in terms of the field strength bysubstituting the approximate form of 𝜂 ambi (Eq. 40) into Eq. (44) as: 𝐵 > 𝐵 diss ( 𝑙 ) ∼ (cid:16) 𝜋𝑛 𝜇 𝜈, n h 𝜎 v i 𝜈, n 𝑦 ( n ) 𝑦 ( 𝜈 ) v ff 𝜆 J (cid:17) / × "(cid:18) 𝑙𝜆 J (cid:19) ( 𝜗 + ) − (cid:18) 𝑙 Ohm 𝜆 J (cid:19) ( 𝜗 + ) / , (47)where 𝑙 Ohm is the Ohmic dissipation scale defined by 𝑙 Ohm ≡ 𝜆 J (cid:18) 𝜂 Ohm 𝑢 ff 𝜆 J (cid:19) /( 𝜗 + ) . (48)Fluctuations at scales smaller than 𝑙 Ohm is damped, whereas atscales larger than 𝑙 Ohm , the fluctuating field strength is limited below 𝐵 diss ( 𝑙 ) via ambipolar diffusion.On the basis of the above results, we discuss the maximum fieldstrength achieved if the small-scale dynamo action continues longenough time. Amplification proceeds faster at smaller scales in theeddy timescale of 𝑡 eddy ( 𝑙 ) = 𝑙 / v eddy ( 𝑙 ) = 𝑡 ff ( 𝑙 / 𝜆 J ) − 𝜗 . Without MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa T ( K ) log n H (cm -3 )(a1) ζ ion =0 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • T ( K ) log n H (cm -3 )(a2) ζ ion =10 -17 -6 Z o • -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • T ( K ) log n H (cm -3 )(a3) ζ ion =10 -15 -5 Z o • -4 Z o • -3 Z o • -2 Z o • -1 Z o • o • -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(b1) ζ ion =0 -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(b2) ζ ion =10 -17 -8-6-4-20 0 5 10H HD l og y ( H , HD ) log n H (cm -3 )(b3) ζ ion =10 -15 -14-12-10-8-6-4 0 5 10 15 l og y ( e ) log n H (cm -3 )(c1) ζ ion =0 -14-12-10-8-6-4 0 5 10 15 l og y ( e ) log n H (cm -3 )(c2) ζ ion =10 -17 -14-12-10-8-6-4 0 5 10 15 l og y ( e ) log n H (cm -3 )(c3) ζ ion =10 -15 -16-14-12-10-8 0 5 10 15 l og y ( g r - ) log n H (cm -3 )(d1) ζ ion =0 -16-14-12-10-8 0 5 10 15 l og y ( g r - ) log n H (cm -3 )(d2) ζ ion =10 -17 -16-14-12-10-8 0 5 10 15 l og y ( g r - ) log n H (cm -3 )(d3) ζ ion =10 -15 Figure 10.
Comparison of the (a) temperature evolution and the abundances of (b) H and HD, (c) 𝑒 , and (d) gr − , which are calculated by the reduced (solid)and full (dashed curves) networks, for the cases with 𝜁 ion = 𝜁 ion = − s − (middle panels), and 𝜁 ion = − s − (right panels). dissipation, the amplification continues until the magnetic energyreaches a fraction 𝑓 sat of the total turbulent kinetic energy, i.e., 𝐵 /( 𝜋𝜌 ) ∼ 𝑓 sat v eddy ( 𝜆 J ) / 𝐵 sat ∼ 𝑓 / 𝐵 crit . (49)According to the numerical simulations, the saturation level of theturbulent dynamo changes in the range of 10 − . 𝑓 sat .
1, de-pending strongly on the properties of the turbulent flow, in partic-ular on the sonic Mach number and on the driving mode of theturbulence (Federrath et al. 2011), and on the magnetic Prandtl num- ber, defined as the ratio of kinematic viscosity to magnetic diffu-sivity (Federrath et al. 2014b; Federrath 2016). Amplification abovethe saturation level is prohibited by the backreaction from the field.With dissipation considered, any small-scale fluctuation is damped at 𝑙 < 𝑙
Ohm by Ohmic loss, whereas at 𝑙 ≥ 𝑙 Ohm , the field strength is lim-ited below 𝐵 diss ( 𝑙 ) via ambipolar dissipation. Therefore, magneticfield reaches the saturation level only at scales where the inequal-ity 𝐵 diss ( 𝑙 ) ≥ 𝐵 sat holds. In summary, the maximum field strength MNRAS , 1–23 (2021) onization degree in clouds with different metallicities log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(a1) ζ ion =0, Z=10 -6 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(a2) ζ ion =10 -17 , Z=10 -6 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(a3) ζ ion =10 -15 , Z=10 -6 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(b1) ζ ion =0, Z=10 -4 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(b2) ζ ion =10 -17 , Z=10 -4 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(b3) ζ ion =10 -15 , Z=10 -4 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(c1) ζ ion =0, Z=10 -2 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(c2) ζ ion =10 -17 , Z=10 -2 Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(c3) ζ ion =10 -15 , Z=10 -2 Z o • Rm=1 B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(d1) ζ ion =0, Z=1Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(d2) ζ ion =10 -17 , Z=1Z o • Rm=1B=B crit log Rm 0 5 10 15 20log n H (cm -3 )-10-5 0 5 10 l og B ( G ) -2 0 2 4 6 8 10(d3) ζ ion =10 -15 , Z=1Z o • Rm=1 B=B crit
Figure 11.
The contour map of the magnetic Reynolds number, for the cases with 𝜁 ion = 𝜁 ion = − s − (middle panels), and 𝜁 ion = − s − (right panels). The thick and thin black curves represent the contours of log Rm = = , , ,
8, and 10. In the red-shaded regions, Rm < 𝐵 crit above which magnetic pressure suppresses the gravitationalcontraction of a star-forming cloud. MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (a1) ζ ion =0,Z=10 -6 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (a2) ζ ion =10 -17 ,Z=10 -6 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (a3) ζ ion =10 -15 ,Z=10 -6 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (b1) ζ ion =0,Z=10 -4 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (b2) ζ ion =10 -17 ,Z=10 -4 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (b3) ζ ion =10 -15 ,Z=10 -4 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (c1) ζ ion =0,Z=10 -2 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (c2) ζ ion =10 -17 ,Z=10 -2 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (c3) ζ ion =10 -15 ,Z=10 -2 Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (d1) ζ ion =0,Z=1Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (d2) ζ ion =10 -17 ,Z=1Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 -6-4-20 0 5 10 15 20 l og l / λ J log n H (cm -3 ) (d3) ζ ion =10 -15 ,Z=1Z o • -6-4-20 0 5 10 15 20 l ambi l Ohm l vis B max =B sat B max =B diss B max =0 Figure 12.
The maximum magnetic field strength achievable by the small scale dynamo action at each coherent length as a function of 𝑛 H , for the cases with 𝜁 ion = 𝜁 ion = − s − (middle panels), and 𝜁 ion = − s − (right panels). Here, we set 𝜗 = / 𝑓 sat = − . The colored curves representthe scales of viscous ( 𝑙 vis ; green), Ohmic ( 𝑙 Ohm ; blue), and ambipolar ( 𝑙 ambi ; red) dissipation. In the scales smaller than 𝑙 vis and 𝑙 Ohm , the turbulent field is dampedcompletely ( 𝐵 max =
0; gray-shaded). In the scales smaller than 𝑙 ambi , 𝐵 max is limited to 𝐵 diss by the ambipolar dissipation ( 𝐵 max = 𝐵 diss ; yellow-shaded). Inthe scales larger than 𝑙 ambi , the amplification up to the saturation level is permitted ( 𝐵 max = 𝐵 sat ; non-filled). achieved by dynamo action is: 𝐵 max ( 𝑙 ) = ( 𝑙 ≤ max [ 𝑙 vis , 𝑙 Ohm ]) ,𝐵 diss ( 𝑙 ) ( max [ 𝑙 vis , 𝑙 Ohm ] < 𝑙 ≤ 𝑙 ambi ) ,𝐵 sat ( 𝑙 ambi < 𝑙 ) , (50)where 𝑙 ambi corresponds to the scale where the equality 𝐵 diss ( 𝑙 ambi ) = 𝐵 sat holds.In Figure 12, we show the maximum magnetic field strength achievable by the small scale dynamo action at each coherent lengthas a function of 𝑛 H , for the cases with 𝜁 ion = 𝜁 ion = − s − (middle panels), and 𝜁 ion = − s − (right pan-els). Referring to the numerical simulations of supersonic turbulence,we adopt 𝜗 = / 𝑓 sat = − as fiducial values (Federrath et al.2011, 2014b; Federrath 2016). The colored curves represent thescales of viscous ( 𝑙 vis ; green), Ohmic ( 𝑙 Ohm ; blue), and ambipo-lar ( 𝑙 ambi ; red) dissipation. In the scales smaller than 𝑙 vis and 𝑙 Ohm ,the turbulent field is damped completely ( 𝐵 max =
0; gray-shaded).
MNRAS , 1–23 (2021) onization degree in clouds with different metallicities In the scales smaller than 𝑙 ambi , 𝐵 max is limited to 𝐵 diss by theambipolar dissipation ( 𝐵 max = 𝐵 diss ; yellow-shaded). In the scaleslarger than 𝑙 ambi , the amplification up to the saturation level is al-lowed ( 𝐵 max = 𝐵 sat ; non-filled).Without ionization sources ( 𝜁 ion =
0; left panels), dissipationis negligible in the beginning of collapse at scales larger than 𝑙 / 𝜆 J ∼ − -10 − (non-filled regions), where magnetic fields areamplified to the saturation level successively from the smaller scales.With increasing density and decreasing ionization fraction, ambipo-lar dissipation becomes effective over a wider range of scales (yellow-shaded regions), where the amplification is limited below 𝐵 diss ( 𝑙 ) .Dissipation works from lower densities with increasing metallicity.As charged grains become more abundant by capturing electronsand ions, Ohmic loss works at larger and larger scales, finally damp-ing turbulent magnetic fields at all scales (gray-shaded regions).This is the case until grain evaporation at 𝑛 H & -10 cm − ,where both ambipolar and Ohmic dissipation scales drop rapidly.Afterwards, amplification to the saturation level becomes possiblefrom very small scales of 𝑙 / 𝜆 J > − -10 − to the largest scales of 𝑙 / 𝜆 J ∼
1, irrespective of metallicity.With increasing ionization rate and ionization fraction, turbulentmagnetic fields couple to the gas more strongly at each density, andboth ambipolar dissipation and Ohmic loss works only at smaller andsmaller scales (c.f. Figure 12 middle and right panels). At the largestscales of 𝑙 / 𝜆 J ∼
1, magnetic fields can be amplified to the saturationlevel in the timescale of 𝑡 ff , over a wider range of densities, comparedto the case without ionization sources. In the very metal-poor casesof 𝑍 / Z ⊙ ∼ − (or 10 − ) for 𝜁 ion = − s − (or 𝜁 ion = − s − ,respectively), such efficient amplification continues throughout theevolution. In this paper, we have calculated the temperature and ionization-degree evolution in a star-forming cloud for various metallicities 𝑍 / Z ⊙ = − , − , − , − , − , − , and 1 by using updatedchemical network reversing all the gas-phase processes and account-ing for grain-surface chemistry, including grain evaporation, thermalionization of alkali metals, and thermionic emission. We have alsodiscussed the dissipation conditions of magnetic fields that are or-dered over the cloud scale, as well as that fluctuating at smaller scales.Below, we briefly summarize the results of our work: • At low densities, the ionization degree decreases as the ma-jor ions, such as H + , H + , H O + , and HCO + , recombine with elec-trons (Figures 4 and 7). With increasing density, dust grains captureelectrons and ions with their large recombination cross section, andbecome the major charge carriers. Charged grains also neutralizeeach other via collision. When the temperature exceeds 𝑇 ∼ 𝑛 H ∼ -10 cm − , dust grains evaporate and the ionizationdegree turns to rise via thermionic emission, and thermal ionizationof alkali metals. The ionization degree is elevated continuously byhydrogen ionization afterwards. • With increasing ionization rate, the ionization fraction becomeshigher at a given density (Figure 7). Ions and electrons remain asmajor charge carriers over a wider range of densities compared tothe case without ionization sources. In very low-metallicity casesof 𝑍 / Z ⊙ . − (10 − ) for 𝜁 ion = − s − ( 𝜁 ion = − s − ,respectively), charged grains never become dominant populations. • The ionization degree at 𝑛 H ∼ -10 cm − became up toeight orders of magnitude higher than that obtained in the previousmodel (Figure 5). This is due to the thermionic emission and thermalionization of vaporized alkali metals, which are not included so far.As a result, magnetic fields recover strong coupling to the gas at muchearlier stages (by a few orders of magnitude in density), compared tothe previous work. • We have developed a reduced chemical network that repro-duces the chemical abundances of the major coolants and chargedspecies (Figure 10). The reduced network consists of 104 (or 161)reactions among 28 (38) species in the absence (presence, respec-tively) of ionization sources (Table 1). The reduced model includesH and HD formation on grain surfaces by using simple formulaeobtained in Appendix B. With ionization sources, the reduced modelincludes the depletion of O, C, OH, CO, and H O on grain surfacesto reproduce the abundances of molecular ions, H O + and HCO + ,which become dominant cations at some density ranges. • The coupling of ordered magnetic fields to the gas becomesweak as the ionization degree decreases with increasing density (Fig-ure 11). If magnetic fields are so strong as to suppress the con-traction, magnetic field lines gradually drift out of the cloud viaambipolar diffusion. Once charged grains become the dominant pop-ulation, the ionization degree drops to a value as low as to dissipatemagnetic fields irrespective of their strength via Ohmic loss. Mag-netic fields remain decoupled until the ionization degree jumps up at 𝑛 H ∼ -10 cm − due to grain evaporation. • Magnetic fields couple with the gas more strongly for increasingionization rate. Parameter space of magnetic dissipation shrinks,compared to the cases without ionization sources (Figure 11). Inthe cases of 𝑍 / Z ⊙ . − (or 10 − ) for 𝜁 ion = − s − (or 𝜁 ion = − s − , respectively), magnetic fields remain coupled withthe gas throughout the evolution. • Dissipation of turbulent magnetic fields is negligible in the be-ginning of collapse at scales larger than 𝑙 / 𝜆 J ∼ − -10 − (Figure12). Dissipation is also negligible at 𝑛 H > -10 cm − at verysmall scales of 𝑙 / 𝜆 J ∼ − -10 − . In these cases, magnetic fieldscan be amplified by dynamo action until their energy becomes afraction of the turbulent kinetic energy. At intermediate densities,with increasing density and decreasing ionization fraction, ambipo-lar diffusion or Ohmic loss operates over a wider range of scales.Ambipolar diffusion regulates the magnetic field amplification be-low 𝐵 diss ( 𝑙 ) (Eq. 47), whereas Ohmic loss damps turbulent magneticfields almost completely.So far, we have assumed that the grain-size distribution followsthe MRN-type power law (Mathis et al. 1977) throughout the evolu-tion. Dust grains grow in their size by accreting gas-phase speciesand by colliding with other grains (e.g., Flower et al. 2005). Thisoccurs more frequently with increasing density, so that the size dis-tribution would deviate from the MRN. If large grains become moreabundant, gas-phase ions and electrons could recombine on grainsurfaces more frequently. Then, charged dust grains may becomethe dominant charge carrier from lower densities, extending the do-main of Ohmic dissipation. We have also assumed that dust grainsevaporate immediately maintaining the MRN distribution above thevaporization temperature, whereas grains evaporate by reducing theirsize (e.g., Lenzuni et al. 1995). This assumption remains valid, sincegrain evaporation proceeds with strong temperature dependence.Our calculation is based on the simple one-zone model by as-suming spherical symmetry and by neglecting the backreaction ofmagnetic fields on cloud contraction. Multi-dimensional effects, suchas rotation and turbulence, are prevalent in a star-forming cloud, so MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa that multi-dimensional resistive MHD calculations are required fordiscussing magnetic field effects. Higuchi et al. (2018, 2019) per-formed 3D resistive MHD calculations of low-metallicity clouds byusing the lookup tables of the barotropic EOS and resistivity coeffi-cients calculated by Susa et al. (2015). We have found that the mag-netic fields recover the coupling with the cloud at earlier stages (by afew orders of magnitude in density), compared to Susa et al. (2015).If resistive MHD calculations were performed by using our resis-tivity coefficients, such magnetic effects as magnetically-driven out-flows and magnetic braking would work at earlier stages, comparedto Higuchi et al. (2018, 2019). Since these effects lower the star-formation efficiency and suppress the formation of massive accre-tion disc around a protostar, the formation efficiency of binary andmultiple stellar systems could be lowered.Higuchi et al. (2018, 2019) also assumed that the temperatureevolves following the barotropic EOS, without solving the energyequation consistently with the MHD equations. In the presence ofstrong magnetic fields, the magnetic pressure support delays thecloud contraction compared to the free-fall rate, weakening the com-pressional heating. In addition, the magnetic energy dissipated byambipolar diffusion and Ohmic loss can heat the gas. From theseeffects, temperature evolution could deviate from the track given bythe barotropic EOS. In the primordial clouds, these effects are foundto be small both by analytical (Paper I) and 3D numerical (Sadanariet al. in prep.) calculations. In metal-enriched clouds, where resistiveMHD effects work more strongly, the heating via magnetic energydissipation could be more important. To clarify this point, multi-dimensional resistive MHD calculations by consistently solving theenergy equation with the MHD equations are needed.
ACKNOWLEDGMENTS
The authors wish to express their cordial thanks to Prof. ToyoharuUmebayashi for his continual interests and kind suggestions. Wealso thank Drs. Motomichi Tashiro, Attila G. Császár, and KenjiFuruya for fruitful discussions. Numerical calculations are per-formed by the computer cluster,
Draco , supported by the FrontierResearch Institute for Interdisciplinary Sciences in Tohoku Uni-versity. This work is supported in part by the Grant-in-Aid fromthe Ministry of Education, Culture, Sports, Science and Technol-ogy (MEXT) of Japan (DN:17H06360, 16J02951, KO:17H06360,17H01102, HS:17H02869, 17H01101).
DATA AVAILABILITY
The data underlying this article will be shared on reasonable requestto the corresponding author.
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APPENDIX A: PARTITION FUNCTION AND HEAT OFREACTION
The partition functions are referred from Popovas & Jørgensen(2016) for H , from Vidler & Tennyson (2000) for H O, from theExoMol database (Tennyson & Yurchenko 2012) for HD + , H + ,and LiH + , from the HITRAN database (Gamache et al. 2017) for HD, CH , CH , CO , H O , O H, and H CO, and fromBarklem & Collet (2016) for the remaining atoms and diatomicmolecules. The remaining polyatomic molecules are regarded as rigidrotators, and their partition functions are calculated by consideringthe rotation degree of freedom alone (e.g., Irwin 1988). The val-ues of the rotational constants are cited from the Cologne Databasefor Molecular Spectroscopy for CH , HCO, HCO + , H O + , H O + ,HCO + , and H CO + , from the NIST Computational Chemistry Com-parison and Benchmark Database for H CO + , O H + , CH + , CH + ,and CH + , and from Thompson et al. (2005) for CH + .The heat of reaction Δ 𝐸 is calculated from the ionization (or http://exomol.com/data/molecules/ http://hitran.org/docs/iso-meta/ http://cccbdb.nist.gov/ MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa dissociation) energy of atoms (or molecules) involved in the reaction.The values of ionization (or dissociation) energy are referred from theKIDA database (Wakelam et al. 2012) for the atoms and moleculescomposed of H, He, C, O, Na, Mg, and K nuclei, from the ActiveThermochemical Table for those containing D nuclei, and fromthe NIST-JANAF Thermochemical Table (Chase 1998) for thosecontaining Li nuclei, except for LiH + , whose value is taken fromStancil et al. (1996). APPENDIX B: GRAIN SURFACE CHEMISTRY
The grain-surface chemistry is considered by calculating kineticrate equations in a similar form to the gas-phase chemistry (e.g.,Hasegawa et al. 1992). The grain-surface chemistry is divided intothree categories: (i) the adsorption of a gas-phase species onto thegrain surface, (ii) the desorption of a grain-surface species into thegas-phase, and (iii) molecule formation. For the grain-surface chem-istry, 148 reactions are considered and are listed in Table B1. Dustgrains also exchange an electric charge with gas-phase ions, elec-trons, and other grains. For grain charging, 150 reactions are con-sidered and they are summarized in Table B2. Below, we derive therate coefficient of each reaction in the unit of cm − s − , follow-ing Draine & Sutin (1987), Hocuk et al. (2016), and Esplugues et al.(2016, 2019). B1 Molecule formation on dust grains
B1.1 Adsorption (reactions 1-23)
Atoms and molecules stick on grain surfaces via two types of in-teractions: they are bound weakly via van der Waals interaction andstrongly via covalent bond. The former (so-called physisorption) isconsidered for all the neutral species, whereas the latter (so-calledchemisorption) is considered only for H and D, following Hocuk et al.(2016). For a gas-phase species X, its physisorbed and chemisorbedcounterparts are denoted as X(p) and X(c), respectively.A gas-phase species X sticks at the physisorption site on a grainsurface at the rate of 𝑘 ads ( X ) = 𝑦 gr h 𝜎 gr i v ( X ) 𝑆 ( 𝑇, 𝑇 gr ) , (B1)where 𝑦 gr = 𝑛 gr / 𝑛 H is the total grain fraction relative to hydrogennuclei, h 𝜎 gr i the geometrical cross section of a dust grain averagedover the MRN distribution, v ( X ) = ( 𝑘 B 𝑇 / 𝜋𝑚 X ) / the thermalvelocity of X, and 𝑆 ( 𝑇, 𝑇 gr ) = + . (cid:18) 𝑇 + 𝑇 gr
100 K (cid:19) . + . 𝑇
100 K + . (cid:18) 𝑇
100 K (cid:19) ! − , (B2)the sticking probability of X derived by Hollenbach & McKee(1979).A gas-phase H (or D) sticks directly at a chemisorption site, ifit can cross the energy barrier 𝐸 act between a physisorption andchemisorption site. The reaction barrier is overcome via thermal http://kida.obs.u-bordeaux1.fr/ https://atct.anl.gov/ThermochemicalData/ https://janaf.nist.gov/ hopping or quantum tunneling, so that the transmission probabilityis given by (Esplugues et al. 2016) 𝑇 chem = exp (cid:18) − 𝐸 act 𝑇 gr (cid:19) + exp − Δ r 𝑚 red 𝑘 B 𝐸 act ℏ ! , (B3)where Δ ∼ 𝑘 gc ( X ) = 𝑘 ads ( X ) 𝑇 chem ( − 𝑓 chem ) . (B4)In Eq. (B4), 𝑓 chem is the fraction of the filled chemisorption sitesrepresented by 𝑓 chem = 𝑛 [ H ( c )] + 𝑛 [ D ( c )] 𝑛 gr 𝑁 site , (B5)where 𝑁 site is the total number of adsorption sites on a grain calcu-lated as 𝑁 site = h 𝜎 gr i/ 𝑑 , by assuming that adsorption sites arelocated at the average distance of 𝑑 pp ∼ 𝑘 pc ( H ) = 𝛼 pc ( H )( − 𝑓 chem ) . (B6)In Eq. (B6), the mobility 𝛼 pc ( H ) is calculated from 𝛼 pc ( X ) = p 𝜋𝑇 gr 𝜈 p 𝐸 bind ( H c ) − 𝐸 bind ( H p ) 𝐸 bind ( H c ) − 𝐸 s × exp − Δ p 𝑚 H 𝑘 B ( 𝐸 bind ( H p ) − 𝐸 s ) ℏ ! + 𝜈 s 𝐸 bind ( H p ) − 𝐸 s 𝐸 bind ( H c ) − 𝐸 s exp (cid:18) − 𝐸 bind ( H p ) − 𝐸 s 𝑇 gr (cid:19) , (B7)where 𝜈 = s − is the oscillation frequency, 𝐸 s =
200 K, and 𝐸 bind ( H p ) and 𝐸 bind ( H c ) the binding energies of H(p) and H(c),whose values are referred from Cazaux & Tielens (2010). B1.2 Thermal desorption and CR-induced desorption (reactions24-44)
When the grain is heated, grain-surface species X obtains enoughthermal energy to escape the binding, and X is released into the gasphase. The thermal desorption rate depends on the binding energy ofX, which changes depending on whether the adsorption site is coveredby H O ice or not. The fraction of the adsorption sites covered byH O ice is 𝑓 ice = min (cid:26) 𝑛 [ H O ( p )] 𝑛 gr 𝑁 site , (cid:27) , (B8)whereas that of the bare sites is 𝑓 bare = − 𝑓 ice . Then the ratecoefficient for thermal desorption is calculated from 𝑘 des ( X ) = 𝜈 (cid:20) 𝑓 bare exp (cid:18) − 𝐸 bare , X 𝑇 gr (cid:19) + 𝑓 ice exp (cid:18) − 𝐸 ice , X 𝑇 gr (cid:19) (cid:21) , (B9)where 𝜈 = s − is the oscillation frequency, and 𝐸 bare , X (and 𝐸 ice , X ) the binding energy of X on bare (and icy, respectively) sur-faces, whose values are referred from Esplugues et al. (2019) TableA3. When the grain temperature is so low as <
10 K, despite itslow binding energy, H is depleted on grain surface and forms H -iced layers. Once the grain surface is covered by H ice, the bind-ing on grain surface is weakened, and the binding energy is set to 𝐸 H − H =
100 K (Nakano 1971).
MNRAS , 1–23 (2021) onization degree in clouds with different metallicities CR particles directly hit and heat the grain surfaces, leading tothe desorption of the grain-surface species. CR particles also excitethe gas-phase H , and the UV photons emitted with H de-excitationenable the desorption of surface species. These two effects are consid-ered following Hasegawa & Herbst (1993) and Hocuk et al. (2016),respectively. B1.3 Molecule formation (reactions 45-127)
Grain-surface species move around the grain surface via thermaldiffusion and meet other species at one adsorption site, producinga new molecule. The rate coefficients of the two-body reactions arecalculated following Esplugues et al. (2016).On the bare substrate, two physisorbed species X and Y meet viathermal diffusion in the frequency of R bare = 𝜈 𝑃 bare , (B10)where 𝑃 bare = 𝑓 bare (cid:20) exp (cid:18) − 𝐸 bare , X 𝑇 gr (cid:19) + exp (cid:18) − 𝐸 bare , Y 𝑇 gr (cid:19)(cid:21) . (B11)On the icy substrate, the frequency R ice is calculated by changing thesubscript ‘bare’ into ‘ice’. The energy barrier between two adjacentsites of physisorption is assumed to be 2 / 𝑘 ( X , Y ) = (R bare 𝛿 bare + R ice 𝛿 ice )/( 𝑛 gr 𝑁 site ) , (B12)where 𝛿 bare (and 𝛿 ice ) indicates the desorption probability of theproducts on the bare (and icy, respectively) substrate. The values of 𝛿 bare and 𝛿 ice are referred from Table A4 in Esplugues et al. (2019).Inversely, for reactions with a high activation barrier ( 𝐸 act ), theirrates are determined by the competition between the probabilityof reactants’ encounter ( 𝑃 bare or 𝑃 ice ) and that of overcoming thereaction barrier via thermal crossing or quantum tunneling ( 𝑃 cross ).Therefore, the rate coefficient is decreased from Eq. (B12) by thereaction probability given by (Garrod & Pauly 2011; Esplugues et al.2016): 𝑃 react = 𝑃 cross 𝑃 cross + 𝑃 bare + 𝑃 ice . (B13)The reactants overcome the reaction barrier via thermal crossing withthe probability of 𝑃 therm = exp (cid:18) − 𝐸 act 𝑇 gr (cid:19) , (B14)and via quantum tunneling with 𝑃 tunnel = exp − Δ r 𝑚 red 𝑘 B 𝐸 act ℏ ! , (B15)where Δ ∼ 𝑚 red the reduced massof the reactants. Then the crossing probability is represented by themaximum of the two as 𝑃 cross = max { 𝑃 therm , 𝑃 tunnel } . (B16)The value of 𝐸 act is also referred from Table A4 in Esplugues et al.(2019).There are three additional pathways of H (or HD) formationon grain surfaces. When a gas-phase H hits a physisorbed and chemisorbed H directly (reactions 119-124), H formation proceedsat the rate of 𝑘 H , gp = 𝑘 ads ( H )/( 𝑛 gr 𝑁 site ) , (B17)and 𝑘 H , gc = 𝑘 ads ( H ) 𝑇 chem /( 𝑛 gr 𝑁 site ) , (B18)respectively. When a physisorbed H moves to an adjacent chemisorp-tion site filled by another H (reactions 125-127), H is produced atthe rate of 𝑘 H , pc = 𝛼 pc ( H )/( 𝑛 gr 𝑁 site ) . (B19) B2 Simple formulae for the rate coefficients of H and HDformation on dust grains Here, we derive simple formulae for the rate coefficients of H andHD formation via grain-surface reactions. Without relying on theabundances of grain-surface H and D, these formulae are representedas the fraction of H (and D) atoms that are adsorbed on the grainsurface and return into the gas phase as H (and HD, respectively).First, H is produced dominantly via the reaction between ph-ysisorbed and chemisorbed H atoms (reaction 125), so that the abun-dances of H(p) and H(c) are relevant to the formation efficiency. TheH(p) abundance is determined by the balance among the reactions 1,22, 24, and 125 as 𝑦 [ H p ] = 𝑘 ads ( H ) 𝑦 ( H ) 𝑛 H 𝛼 pc ( H ) + 𝑘 des ( H p ) , (B20)whereas the H(c) abundance is calculated by the balance between thereactions 22 and 125 as 𝑦 [ H c ] = 𝑁 site 𝑛 gr 𝑛 H . (B21)Eq. (B21) indicates that half the chemisorption sites are occupied byH atoms in the steady state. By using Eqs. (B20) and (B21), the H formation rate can be summarized as 𝑑𝑦 ( H ) 𝑑𝑡 = 𝑘 H2 , pc 𝑦 [ H p ] 𝑦 [ H c ] 𝑛 H = 𝑘 gr ( H ) 𝑦 ( H ) 𝑛 H (B22)where 𝑘 gr ( H ) is the rate coefficient given by 𝑘 gr ( H ) = 𝑘 ads ( H ) 𝑓 gr ( H ) (B23)and 𝑓 gr ( H ) = (cid:18) + 𝑘 des ( H p ) 𝛼 pc ( H ) (cid:19) − . (B24) 𝑓 gr ( H ) represents the fraction of H atoms that stick to grain surfaceand return into the gas phase as H . This rate coefficient Eq. (B23)is consistent with that in Cazaux et al. (2008).Next, on the grain surface, HD is produced efficiently via thereactions between physisorbed H (or D) and chemisorbed D (or H,respectively; reactions 126 and 127), so that the abundances of D(p)and D(c), in addition to those of H(p) and H(c), are relevant to theHD formation rate. The balance among the reactions 3, 23, 26, and127 determines the D(p) abundance as 𝑦 [ D p ] = 𝑘 ads ( D ) 𝑦 ( D ) 𝑛 H 𝛼 pc ( D ) + 𝑘 des ( D p ) , (B25)whereas that between the reactions 23 and 126 determines the D(c)abundance as 𝑦 [ D c ] = 𝛼 pc ( D ) 𝛼 pc ( H ) 𝑦 [ D p ] 𝑦 [ H p ] 𝑁 site 𝑛 gr 𝑛 H . (B26) MNRAS , 1–23 (2021) D. Nakauchi, K. Omukai, and H. Susa
Table B1.
Grain surface chemistry.Number Reaction Reference1 H(g) → H(p) 12 H (g) → H (p) 13 D(g) → D(p) 24 HD(g) → HD(p) 25 O(g) → O(p) 16 O (g) → O (p) 17 OH(g) → OH(p) 18 CO(g) → CO(p) 19 CO (g) → CO (p) 110 H O(g) → H O(p) 111 HO (g) → HO (p) 112 H O (g) → H O (p) 113 HCO(g) → HCO(p) 114 H CO(g) → H CO(p) 115 C(g) → C(p) 116 CH(g) → CH(p) 117 CH (g) → CH (p) 118 CH (g) → CH (p) 119 CH (g) → CH (p) 120 H(g) → H(c) 121 D(g) → D(c) 222 H(p) → H(c) 323 D(p) → D(c) 224 H(p) → H(g) 125 H (p) → H (g) 126 D(p) → D(g) 227 HD(p) → HD(g) 228 O(p) → O(g) 129 O (p) → O (g) 130 OH(p) → OH(g) 131 CO(p) → CO(g) 132 CO (p) → CO (g) 133 H O(p) → H O(g) 134 HO (p) → O(g) + OH(g) 135 H O (p) → H O (g) 136 HCO(p) → HCO(g) 137 H CO(p) → H CO(g) 138 C(p) → C(g) 139 CH(p) → CH(g) 140 CH (p) → CH (g) 141 CH (p) → CH (g) 142 CH (p) → CH (g) 143 H(c) → H(g) 144 D(c) → D(g) 245 H(p) + H(p) → H (p) 146 H(p) + D(p) → HD(p) 247 H(p) + O(p) → OH(p) 148 H(p) + OH(p) → H O(p) 149 H(p) + O (p) → HO (p) 150 H(p) + CO(p) → HCO(p) 151 H(p) + HO (p) → H O (p) 152 H(p) + HCO(p) → H CO(p) 153 H(p) + C(p) → CH(p) 154 H(p) + CH(p) → CH (p) 155 H(p) + CH (p) → CH (p) 156 H(p) + CH (p) → CH (p) 157 O(p) + O(p) → O (p) 158 O(p) + C(p) → CO(p) 159 O(p) + CO(p) → CO (p) 160 OH(p) + OH(p) → H O (p) 1 Table B1 – continued Number Reaction Reference61 H(p) + H(p) → H (g) 162 H(p) + D(p) → HD(g) 263 H(p) + O(p) → OH(g) 164 H(p) + OH(p) → H O(g) 165 H(p) + O (p) → HO (g) 166 H(p) + CO(p) → HCO(g) 167 H(p) + HO (p) → H O (g) 168 H(p) + HCO(p) → H CO(g) 169 H(p) + C(p) → CH(g) 170 H(p) + CH(p) → CH (g) 171 H(p) + CH (p) → CH (g) 172 H(p) + CH (p) → CH (g) 173 O(p) + O(p) → O (g) 174 O(p) + C(p) → CO(g) 175 O(p) + CO(p) → CO (g) 176 OH(p) + OH(p) → H O (g) 177 H(p) + H O(p) → H (p) + OH(p) 178 H(p) + HO (p) → OH(p) + OH(p) 179 H(p) + HO (p) → OH(g) + OH(g) 180 H(p) + H O (p) → OH(p) + H O(p) 181 H(p) + H O (p) → OH(g) + H O(p) 182 H(p) + H O (p) → OH(g) + H O(g) 183 H(p) + HCO(p) → H (p) + CO(p) 184 H(p) + HCO(p) → H (g) + CO(p) 185 H(p) + HCO(p) → H (g) + CO(g) 186 H(p) + H CO(p) → H (p) + HCO(p) 187 H(p) + H CO(p) → H (g) + HCO(p) 188 H(p) + H CO(p) → H (g) + HCO(g) 189 H(p) + CO (p) → OH(p) + CO(p) 190 H(p) + CH(p) → H (p) + C(p) 191 H(p) + CH(p) → H (g) + C(p) 192 H(p) + CH(p) → H (g) + C(g) 193 H(p) + CH (p) → H (p) + CH(p) 194 H(p) + CH (p) → H (g) + CH(p) 195 H(p) + CH (p) → H (g) + CH(g) 196 H(p) + CH (p) → H (p) + CH (p) 197 H(p) + CH (p) → H (p) + CH (p) 198 O(p) + OH(p) → H(p) + O (p) 199 O(p) + OH(p) → H(g) + O (p) 1100 O(p) + OH(p) → H(g) + O (g) 1101 O(p) + HO (p) → O (p) + OH(p) 1102 O(p) + HO (p) → O (g) + OH(p) 1103 O(p) + HO (p) → O (g) + OH(g) 1104 O(p) + HCO(p) → H(p) + CO (p) 1105 O(p) + HCO(p) → H(g) + CO (p) 1106 O(p) + HCO(p) → H(g) + CO (g) 1107 O(p) + H CO(p) → H (p) + CO (p) 1108 O(p) + H CO(p) → H (g) + CO (p) 1109 O(p) + H CO(p) → H (g) + CO (g) 1110 H (p) + OH(p) → H(p) + H O(p) 1111 H (p) + OH(p) → H(g) + H O(p) 1112 OH(p) + CO(p) → H(p) + CO (p) 1113 OH(p) + CO(p) → H(g) + CO (p) 1114 OH(p) + CO(p) → H(g) + CO (g) 1115 OH(p) + HCO(p) → H (p) + CO (p) 1116 OH(p) + HCO(p) → H (g) + CO (p) 1117 OH(p) + HCO(p) → H (g) + CO (g) 1118 H (p) + HO (p) → H(p) + H O (p) 1MNRAS , 1–23 (2021) onization degree in clouds with different metallicities Table B1 – continued Number Reaction Reference119 H(g) + H(p) → H (g) 2120 H(g) + D(p) → HD(g) 2121 D(g) + H(p) → HD(g) 2122 H(g) + H(c) → H (g) 3123 H(g) + D(c) → HD(g) 2124 D(g) + H(c) → HD(g) 2125 H(p) + H(c) → H (g) 3126 H(p) + D(c) → HD(g) 2127 D(p) + H(c) → HD(g) 2
References.
1) Esplugues et al. (2019). 2) Thi et al. (2018). 3) Hocuk et al.(2016).
By using Eqs. (B20), (B21), (B25) and (B26), the HD formation ratecan be summarized as 𝑑𝑦 ( HD ) 𝑑𝑡 = 𝑘 gr ( HD ) 𝑦 ( D ) 𝑛 H (B27)where 𝑘 gr ( HD ) = 𝑘 ads ( D ) 𝑓 gr ( HD ) (B28)and 𝑓 gr ( HD ) = (cid:18) + 𝑘 des ( D p ) 𝛼 pc ( D ) (cid:19) − . (B29)These formulae are also consistent with those in Cazaux & Spaans(2009). B3 Collisional charging of dust grains
Dust grains obtain an electric charge, when gas-phase ions and elec-trons recombine with grain-surface species. Charged grains transferits electric charge to other grains via collision with each other. Thelist of the reactions for collisional charging of dust grains is summa-rized in Table B2. Dust grains are assumed to have five charge states:gr , gr ± , and gr ± , since the abundances of more than triply chargedgrains are negligibly small (e.g., Nakano et al. 2002).The rate coefficients for the grain charging via gas-grain and grain-grain collisions are calculated following Draine & Sutin (1987). Agas-phase ion (or an electron) with an electric charge 𝑞 x 𝑒 hits acharged grain with a charge 𝑞 gr 𝑒 (reactions G1-G11) in the frequencyof 𝑘 ( 𝑞 x , 𝑞 gr ) = 𝑆 ( 𝑇, 𝑇 gr ) v ( X ) 𝜋𝑎 ˜ 𝐽 ( 𝜏, 𝜈 ) , (B30)where 𝜏 = 𝑎 gr 𝑘 B 𝑇 ( 𝑞 x 𝑒 ) , 𝜈 = 𝑞 gr 𝑞 x ,˜ 𝐽 ( 𝜏, 𝜈 = ) = + (cid:16) 𝜋 𝜏 (cid:17) / , ˜ 𝐽 ( 𝜏, 𝜈 < ) = h − 𝜈𝜏 i " + (cid:18) 𝜏 − 𝜈 (cid:19) / , (B31)˜ 𝐽 ( 𝜏, 𝜈 > ) = h + ( 𝜏 + 𝜈 ) − / i exp (cid:18) − 𝜃 𝜈 𝜏 (cid:19) , and 𝜃 𝜈 = 𝜈 + 𝜈 − / . The mutual neutralization via grain-grain colli-sions (reactions G12-G15) occurs at the rate of 𝑘 ( 𝑞 , 𝑞 ) = 𝜋 ( 𝑎 + 𝑎 ) v ( gr )( ˜ 𝐽 ( 𝜏 , 𝜈 ) + ˜ 𝐽 ( 𝜏 , 𝜈 ))/ , (B32)between two grains with electric charges 𝑞 and 𝑞 , and radii of 𝑎 and 𝑎 . Table B2.
Collisional charging of dust grains.Number Reaction ReferenceG1 X + + gr → gr + + X 1G2 X + + gr → gr + + X 1G3 X − + gr → gr − + X 1G4 X + + gr + → gr + + X 1G5 X − + gr + → gr + X 1G6 X − + gr + → gr + + X 1G7 X + + gr − → gr + X 1G8 X + + gr − → gr + + X 1G9 X − + gr − → gr − + X 1G10 X + + gr − → gr − + X 1G11 X + + gr − → gr + X 1G12 gr + + gr − → gr + gr 1G13 gr + + gr − → gr + gr − + + gr − → gr + gr 1G15 gr + + gr − → gr + gr + − → gr − + 𝑒 − − → gr + 𝑒 − → gr + + 𝑒 − + → gr + + 𝑒 − References.
1) Draine & Sutin (1987). 2) Desch & Turner (2015).
When a grain is heated above the temperature of ∼
500 K, anelectron bound on the grain surface obtains enough thermal energyto escape into the gas phase (thermionic emission; reactions G16-G19). The rate coefficient of thermionic emission is given by theRichardson law (Desch & Turner 2015) 𝑘 TE ( 𝑞 gr ) = 𝜋𝑎 𝜆 R 𝜋𝑚 𝑒 ( 𝑘 B 𝑇 gr ) ℎ exp − 𝑊 + 𝑞 gr 𝑒 / 𝑎 gr 𝑘 B 𝑇 gr ! , (B33)where 𝑊 = 𝜆 R = / MNRAS000