Modeling primordial gas in numerical cosmology

NEW ASTRONOMY New Astronomy 2 (1997) 181-207

ELSEVIER

Modeling primordial gas in numerical cosmology Tom Abel’, Peter Anninos*, Yu Zhang3, Michael L. Norman4 Laboratov

for Computational

Astrophysics. National Champaign.

Center for Supercomputing Applications,

405 N. Mathews Ave., Urbana,

University of Illinois

at Urbana.

IL 61801, USA

Received 9 August 1996; accepted 18 March 1997 Communicated by Edmund Bertschinger

Abstract We have reviewed the chemistry and cooling behavior of low-density (n 5 lo4 cm-j ) primordial gas and devised a model which involves 19 collisional and 9 radiative processes and is applicable for temperatures in the range 1 K < T < 10’ K. In a companion paper (Anninos et al., 1997)[NewA, 2, 2091 numerical methods are presented that unify the modeling of non-equilibrium primordial gas chemistry and cooling dicussed here with cosmological hydrodynamics. We derived new fits of rate coefficients for the photo-attachment of neutral hydrogen, the formation of molecular hydrogen via H-, charge exchange between H, and H’, electron detachment of H- by neutral hydrogen, dissociative recombination of Hl with slow of H,. Furthermore it was found that the molecular hydrogen electrons, photodissociation of Hl, and photodissociation produced through the gas-phase processes, Hl + H + H, + H’, and H- + H +H, + e-, is likely to be converted into its para configuration on a faster time scale than the formation time. We have tested the model extensively and shown it to agree well with former studies. We further studied the chemical kinetics in great detail and devised a minimal model which is substantially simpler than the full reaction network but predicts correct abundances. This minimal model shows convincingly that 12 collisional processes are sufficient to model the H, He, H’, H-, He’, He++, and H2 abundances in low density primordial gas for applications with no radiation fields. 0 1997 Elsevier Science B.V. PACS: 98.80; 95.30.Dr; 95.30.Ft; 82.20.P Keywords: Atomic processes; Molecular processes:

Plasmas; ISM: molecules;

1. Introduction Since

Saslaw

Radiation mechanisms:

+hv,

HSe--+H& Zipoy

(1967)

tance of the gas phase reactions primordial gas

realized

the impor-

for H2 formation

in

[email protected] [email protected] [email protected] [email protected]

(2)

and

H;

(3)

+hu,

+H+H,+H+,

and Peebles

1384-1076/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved. S1384-1076(97)00010-9

PII

(1)

H-+H+H,+e-,

H+ +H+H; ‘E-mail: ‘E-mail: ‘E-mail: ‘E-mail:

thermal

& Dicke

(4) (1968)

formulated

their theory

182

T. Abel et al. / New Astronomy

for globular cluster formation based on H, cooling of collapsing primordial density fluctuations, many models have been put forward employing this effect to explain the origin of structures on a vast range of mass scales. For example, in the cosmic string model of structure formation (see Moessner & Brandenberger (1996), and references therein), molecular hydrogen is believed to trigger the fragmentation of late cosmic string wakes. This is because, for the weak accretion shocks encountered during the formation of late cosmic string wakes, thermal instabilities due to hydrogen line, or bremsstrahlung cooling are not accessible (Rees, 1986). Kashlinsky & Rees (1983) discussed fragmentation of primordial gas that cools via H,. Couchman & Rees (1986) envisaged star cluster sized objects collapsing via H, cooling at high redshift which then reionize the intergalactic medium. More recently, two models for the origin of halo globular clusters and spheroid stars based on a cooling instability triggered by H, formation have been put forward by Vietri & Pesce (1995) and Padoan et al. (1997). Furthermore, in studies of primordial star formation, molecular hydrogen cooling plays a central role (see Stahler (1986) for review). All these applications are of fundamental importance for our understanding of the origin of structure in the universe and its subsequent evolution. All of the above problems are inherently multidimensional and a method that computes the hydrodynamics along with the chemistry is desirable. In order to form H, efficiently through the above gas phase reactions, electrons and protons have to be abundant at relatively low temperatures (T < lo4 K) allowing the formed molecules to survive. This situation arises naturally in shock heated gas, where the gas recombines slower than it can cool, and an enhanced ionized fraction (over the equilibrium value) is reached, despite the low temperatures. This has been convincingly shown by many previous investigations (Hollenbach & McKee, 1979; Mac Low & Shull, 1986; Shapiro & Kang, 1987; Anninos & Norman, 1996). Furthermore, at the recombination epoch, the universe expands so fast that recombination will not be complete and the gas is left with a

2 (1997) 181-207

residual abundance of free electrons (Peebles, 1993). This residual ionized fraction can be enough to form a substantial amount of H, in structures that only have very weak virialization shocks (Couchman & Rees, 1986; Tegmark et al., 1997). However, the chemistry has to be solved self-consistently with the structure formation equations, which is computationally difficult due to the stiff nature of the reaction network. This has forced former studies to use steady state shock approximations (e.g. Shapiro & Kang, 1987) or to assume homogeneous spherical free falling spheres (e.g. Palla et al., 1983). Recently Haiman et al. (1996) presented a study of collapsing small scale structure that incorporated the time dependent chemistry in an one-dimensional hydrodynamics code. In a companion paper (Anninos et al., 1997), we discuss a numerical method that unifies the time dependent chemistry with multidimensional cosmological hydrodynamics. Together with the chemistry model which accurately predicts abundances, and the cooling model for primordial gas which is valid over temperatures 1< T < -10’ K and densities below - lo4 cm-3, this introduces a powerful tool for investigating many aspects of structure formation in three-dimensional hydrodynamical simulations. This paper is organized in the following way: In the next section we review some general properties of primordial gas such as nucleosynthesis constraints and the coronal limit. In Section 2.2 we discuss general arguments on how to select the dominant reactions. We then give an elaborate presentation of the collisional and radiative processes which we find to be important, and the rate coefficients we have adopted. Section 5 gives an overview of the processes found to be negligible for the range of applications we are interested in. We also review the cooling mechanisms needed for our model and discuss the molecular cooling and heating functions in the low-density limit. Finally we present an extensive discussion of the performance of our model for an application to a strong shock wave arising from the collapse of a single pancake or cosmological sheet. The complete chemical model is summarized in Appendix A. A discussion of why HZ

T. Abel et al. I New Astron0m.v

is most likely to be in its para configuration in Appendix B.

2. General properties

of primordial

is given

gas

What are the major species determining the physics in primordial gas? Here we briefly summarize what we include in our cooling model and defer the detailed discussion to the following sections. We know from nucleosynthesis and observational constraints that 7Li - lo-“, D - 10e5, 3He - 10p5, and 0.236 I 4He i 0.254. Here D, He, Li denote the mass fractions in the primordial gas of deuterium, helium and lithium, respectively. Hence D, 3He, and Li have, compared to neutral hydrogen, very low abundances. For the line cooling of D, 3He, and Li to contribute significantly to the overall cooling of the gas, their excitation rates would have to be many orders of magnitude higher than the corresponding processes of neutral hydrogen. However, this is not the case and those species, at least in their atomic and ionic forms, will have negligible influence on the hydrodynamics of the gas. The work by Lepp & Shull (1984) indicates that molecular hydrogen was by far the most abundant molecule in the universe at redshifts between recombination and before the first stars formed. It forms by radiative association of the negative hydrogen ion with neutral hydrogen, and charge exchange between H and Hl. The lowest energy rotational or vibrational state defines a minimal temperature and hence a minimal Jeans mass. In the case of molecular hydrogen, this minimum temperature is - 10’ K (see also Mac Low & Shull, 1986; Shapiro & Kang, 1987) which translates to a Jeans mass of

312

(n/cm

-3

)

-l/2

,

2 (1997)

181-207

183

rough estimate of the minimum mass scale one has to resolve in numerical simulations. If one includes also HD and LiH which will be excited only at temperatures above 112 K and 21 K, respectively, one has to be able to resolve masses of an order of magnitude smaller. A study by Puy et al. (1993) was able to give fairly good estimates on the primordial HD abundance after recombination. They found that HD was the second most abundant molecule with nHD - 10-3’5n,;. We know, therefore, that molecular hydrogen will always be the dominant molecular coolant for temperatures above - 100 K. Hence choosing not to include molecules other than H, effectively means that we restrict our attention to The lithium masses 2 lo4 M,(nllOO cm-3). chemistry has been rigorously discussed recently by Stancil et al. (1996), who found that much less primordial LiH is formed than was previously expected. They also suggested that a significant fraction of LiH might only be formed by three body reactions. Since our model focuses on number densities 5 lo4 cme3 and three-body reactions only become important at higher densities, we conclude that, in the density regime we are interested in, LiH will not be important. The papers mentioned above did not include H: which, due to its low abundance, is not believed to be a significant coolant. From the calculations by Lenzuni et al. (1991) we know that, in fact, the Hl abundance can be of the same order as the Hl abundance. However, their results also show that it has a negligible effect on the molecular hydrogen abundance. We also did not include either He molecular ions nor hydrogen ion clusters (Hl with n 2 4) because their expected abundances are even smaller than the ones of the molecules mentioned above. He-(4P) is a metastable ion that is formed from excited states of He only. Since only the ground states are populated at the densities we are interested in (see the following discussion of the coronal limit) it is clear that it is negligible.

(5)

indicating that for applications such as proto-galaxies, where virial number densities are 5 100 cmm3, we expect a Jeans mass of Z lo4 M,. This is a

2.1. The coronal limit Atoms and molecules

have very complex intrinsic

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T. Abel et al. I New Astronomy

properties, and their chemical behavior varies, sometimes drastically, with the specific quantum-mechanical state they occupy. For atoms and ions at moderate or low densities like in the solar corona (electron number density n, - 108-lo9 cmm3), the following features of thermodynamical equilibrium do not hold (Sobelman, 1979): Boltzmann distribution of atoms over excited states. Saha distribution of atoms over degrees of ionization. Principle of detailed balance. The velocity distribution of the free electrons is, however, as a rule nearly always Maxwellian. In this low-density limit we know that the level distributions are given by

where Nf denotes the number density of the species i in its level k, vaj, is the rate coefficient for collisional excitation from level j up to level k, and A, is the total probability for spontaneous transition from all higher levels down to k. This approximation is applicable if

4

ne e (VCQ . One important assumption here is that collisional excitations outweigh radiative excitations, which is always true as long as there are only moderate external radiation fluxes. For the s-p excitation in helium one finds that the coronal limit holds up to electron densities of - lOI cme3. (Here we used (~(5~)-6 X 10e9 cm3 s-l at 400 K from Janev et al. (1987), 2.3.1, and A, = 1.8 X lo9 s-i (p. 297 of Sobelman, 1979)). Thus the coronal limit is valid in the intended cosmological applications concerning structure formation. The important conclusion is that nearly all atoms are in their ground states and it is therefore not necessary to treat multilevel atoms. Comparing the time scales for collisional excita-

2 (1997) 181-207

tion, collisional de-excitation and also the transition probabilities of molecular hydrogen for radiative decay, one finds that for low densities (nn < lo4 cme3) the population of excited rotational and vibrational states of H, is orders of magnitude smaller than the ground state population. This fact manifests itself in the work of Lepp & Shull (1983) and Dove et al. (1987) who studied the dissociation of molecular para-hydrogen by collisions with helium and hydrogen atoms. They found that at low densities the dissociation rate coefficient approaches a constant value, which corresponds to the direct collisional dissociation out of the (v = 0, J = 0) level of the ground electronic state. A strong radiation field in the UV is capable of populating excited (electronic, rotational, and vibrational) states and hence changing the chemical behavior of H,. However, a detailed study (Shull, 1978) shows that only for fluxes greater than -5 X lo-l5 erg cm-’ s-’ Hz-’ sr-’ does UV pumping become effective. Hence considering all species to be in their ground-states is justified by the coronal limit for the atoms and ions and what one could call a “cosmological limit” for the molecules. 2.2. Selection of reactions and rate coeficients Although primordial gas is a simple mixture of hydrogen and helium, the ongoing physical reactions in it are numerous. For example, in Janev et al. (1987) one finds more than 70 reactions involving only the ground states of our species. It is obvious that in order to construct a computationally feasible model that some selections have to be made. The quest is to make them as good as possible i.e. without neglecting any important physics. To neglect a collisional processes one has to make sure that for all reactants and products the rate will, for any temperature and density, never contribute more than 1% to the right hand side of the rate equations. the reaction enthalpy is, at all temperatures and densities, less than one percent of the total cooling (heating) rates of the gas.

T. Abel et al. I New Astronomy

We used these criteria to construct network out of hundreds of reactions literature as well as in databases.

3. Collisional

our reaction found in the

processes

In Appendix A, we present all included processes, their rate coefficients and the corresponding reference. In this section we discuss the reliability of our adopted rate coefficients and their sources. Further information on atomic rates can be found at the Data for 1996, Atomic Dima Verners Astrophysics* page. 3.1. Ionizing processes For the cosmological problems we are interested in, we have seen above that spontaneous decay is much faster than collisional excitation (coronal limit). This immediately suggests the non-LTE character of our problem. Qualitatively speaking, the radiation field is in our case much smaller than in the LTE case, since there are not enough collisional excitations to build up a Planckian spectrum. Therefore, if we compare in the following collisional and radiative processes, we can use arguments derived for strict LTE to make qualitative statements for non-LTE conditions by assuming that the radiation temperature is much less than the matter temperature. This is always true in the matter dominated phase of the universe, when radiation fields other than the CBR are negligible. We adopt here the estimate of Mihalas (1978), p. 123, of the ratio between the total number of photoionizations Ri, and the total number of collisional ionizations Clk, R. rkz C,, ---

’

2 (1997) 181-207

185

where Ej is the ionization energy out of the level i, k, is the Boltzmann constant, m is the electron mass, T, the electron temperature, e the electron charge, c the speed of light, and W is defined by J, = WB,(T,), where J, is the specific intensity at the frequency v, and B,(T,) denotes the Planck spectrum of a black body with temperature TR. .This gives for the ionization of hydrogen out of the ground state (Ei = 13.6 eV)

In thermal equilibrium with T, = TR - lo4 and n, 1 cmm3, this ratio is of the order 1015, and therefore radiative ionization dominates by far. However, for Tc-lOx TR - lo’, the ratio +?$ is of the order 1O-47 and we see that, for all of our intended applications, collisional ionization is the dominating process. It is important, however, to include photoionization to study the influence of external radiation fields on the chemistry and cooling of the structure-forming gas, such as Lyman Alpha clouds in the vicinity of quasars. The collisional ionization processes of hydrogen and helium atoms and ions (procs. (l)-(3)) are dominated by free electron collisions since their thermal velocities are (Am, lm,)“2 = 43A”2 times higher than for an ion of A times the proton mass. (1) Collisional ionization of hydrogen. We use the fit given in 2.1.5 of Janev et al. (1987). For temperatures below 1000 K - 0.086 eV we take it to be zero since it is smaller than 10e6’ cm3 s-l. Since we do not expect to encounter temperatures above lo5 eV in our simulations, no relativistic corrections have to be made. The ionization threshold is 13.6 eV. (3) Collisional ionization of helium. We use the fit given in 2.3.9 by Janev et al. (1987). For temperatures below 4000 K we take it to be zero since it is smaller than 1O-38 cm3 s-‘. The threshold lies at 24.6 eV (5) Collisional ionization of He*. The rate is taken from the Aladdin Database (1989) which is part of the Atomic and Molecular

186

Data Information national Atomic

T. Abel et al. I New Astronomy

System6 (AMDIS) of the InterEnergy Agency7 (IAEA). The

energy threshold is 54.4 eV Again we only have to consider ionization out of the ground state and no relativistic effects.

3.2. Radiative recombination Radiative recombination is the inverse reaction of photoionization. With a similar argument as the one we used to discuss the relative importance of collisional to radiative ionization, one finds (Mihalas, 1978) that radiative recombination always outweighs the collisional one. This is especially valid in the low-density limit since collisional recombination is, as are all three-body processes, typically negligible for densities 5 lo8 cme3. (2) Radiative recombination to hydrogen. In the coronal limit we assume that recombination can happen into any quantum state which, if it is an excited state, will spontaneously decay to the ground state. Therefore, the total rate,coefficient is the sum of the rate coefficients a;, for all n = 1, .... TV.Ferland et al. (1992) computed hydrogenic rate coefficients which are, in principle, exact. We have fitted their data (the sum of all rate coefficients for n = 1, .... 1000) to a form similar to what Janev et al. (1987) used in their compendium. We made sure that the fit is accurate for temperatures from 1 K to lo9 K to better than one percent. (4), (6) Recombination to helium. He+ is the only species in our model subject to di-electronic recombination, which dominates at high temperatures (T > 6 X lo4 K). Since radiative and di-electronic recombination rates are independent of density, their sum gives the overall recombination rate. For the radiative recombination process, we employ the rate coefficient given by Cen (1992). For the di-electronic recombination, we adopt the one given by Aldrovandi & Pequignot (1973).

6http://www.iaea.or.at/programslri/nds/amdisin~o.htm ‘htttxllwww.iaea.or.at/

2 (1997) 181-207

Since He’+ is a hydrogenic ion, one can obtain its recombination rate by scaling the hydrogen rate as follows: k, = 2k,(T/4) (Osterbrock, 1989). (7) Photo-attachment of H and e to H-. The rate coefficient from Hutchins (1976) is stated to be accurate to 10% in the temperature range 100 K ( = 0.0086 eV) I T 5 2500 K ( = 0.254 eV). The cross section for the inverse reaction has been calculated by Wishart (1979) to within 1% around the threshold (2 X lOI Hz < v < 2 X lOI Hz). From that data alone one can compute the rate for photoattachment from 2000 K to 10000 K, using the principle of detailed balance or simply the Saha equation. To cover a greater temperature range, however, we use the cross section given in De Jong (1972) for all frequencies outside of the interval given by Wishart (1979). With that we are able to compute and fit this rate in the temperature range from 1 K to lOa K, with an accuracy to within a few percent for 1 K < T < 100 K, about one percent for 100 K < T < IO4 K, and better than 10% for T > lo4 K. At LTE, the Saha equation has to be valid and the rate at non-LTE will be naturally the same as in LTE since it is an atomic property. The SahaBoltzmann ionization equation reads, k k at,

_=-det

N,,, u,+, 2(2nmkT)3’2 N, Ne = u, h3

where N, denotes the free electron number density, N,,, the number density of neutral hydrogen atoms, N,. the number density of H- ions, kdet the rate coefficient for photo-detachment, k,,, the rate coefficient for photo-attachment, u,, u,+, the respective partition functions, Z, the threshold for photo-detachment of H- (0.755 eV), m the electron mass, and k the Boltzmann constant. The single bound state of H- is ‘S and the ground state of H is ‘S. Hence u r+, = 2 and u, = 1. The rate coefficient, kdet, for photo-attachment in LTE is derived by integrating the cross section over a Plan&an black body spectrum. The fit, which is given in the appendix, equals the one given by Hutchins (1976) to within a few percent in the range where the latter is applicable.

187

T. Abel et al. I New Astronomy2 (1997) 181-207 (8) Formation

of molecular

hydrogen

via H-.

The rate coefficient has been taken from Janev et al. (1987), 7.3.2.b, and is based on theoretical cross sections of Browne & Dalgamo (1969) and normalized to the experimental results of Hummer et al. (1960) in the region of 500 eV to 4 X lo4 eV. We averaged their energy dependent rate coefficient over a Maxwellian velocity distribution of the incident particle. The data are only reliable for temperatures above 0.1 eV. However, we, know that the rate coefficient is constant for small temperatures and we also see that the energy dependent rate coefficient is already the same for 0.1 eV and 1 eV of the incident particle for a temperature of 0.1 eV. Therefore we take the rate coefficient at 0.01 eV to be the same as at 0.1 eV, fit it from 0.1 eV to lo4 eV, and set it to be constant for lower values. This leads to a value of the rate coefficient for low temperatures about 10% higher than the constant values given by Bieniek (1980) and De Jong (1972). Launay et al. (1991) computed the rate coefficient of this reaction including its dependence on the ro-vibrational states of the produced H, molecules. Using the potential derived by Senekowitsch et al. (1984) they found a thermal rate at 300 K of 1.49 X 1O-9 which lies 4% above the rate we derived from the data of Janev et al. (1987). (9) Formation of Hi. The rate coefficient for this radiative association reaction has been calculated by Ramaker & Peek (1976) and fitted by Shapiro & Kang (1987) as well as by Rawlings et al. (1993). We prefer the fit given by Shapiro and Kang, because it covers a greater temperature interval. (10) Hz formation via Hl. The rate constant for this charge exchange reaction between neutral hydrogen and Hi was measured by Karpas et al. (1979) to an accuracy better than 20%. This rate is constant at low energies because the distribution of final states in phase space is determined by the energy released in the reaction, which is almost independent of the incident energy (Peebles, 1993). By assuming it to be constant at high temperatures as well, we potentially introduce an error in the H, abundance because the dominant

destruction processes will be balanced by an rect formation rate. However, this error will significant because the formation of hydrogen cules is dominated by the H- formation path. summarizes the rate coefficients for formation and Hl as well as the ones for HZ. (II) Charge exchange between molecular

incornot be moleFig. 1 of H hydro-

gen and H’.

This is the inverse reaction of the above. The rate coefficient was derived by Donahue and Shull using detailed balance and the Karpas et al. (1979) data to

k,, =6.4X

lo-“exp(

-y)

cm3 s-l.

That rate differs drastically from the one we have computed using the data provided by Janev et al. (1987), which is accurate also for high temperatures. The data of Janev et al. has to be considered more accurate than the simple detailed balance argument of Donahue & Shull (1991). However, comparing results of our model with the one of Shapiro & Kang (1987) we find that drastic differences in this rate do not change the final abundances significantly. (12) Dissociation of molecular hydrogen by electrons.

We employ the rate coefficient that Donahue & Shull ( 1991) derived from the cross section given by Corrigan ( 1965). (13) Dissociation of molecular hydrogen by neutral hydrogen.

Dove and Mandy used fully 3D quasi-classical trajectory calculations, using the Siegbahn-LiuTruhlar-Horowitz potential energy surface, to calculate the rate coefficient in the low density limit. They found significant differences to the former “standard” values given by Lepp & Shull (1983). We use the results for the total rate coefficient of Dove & Mandy (1986), which assumes that all the quasibound states will dissociate. Although their rate coefficient is not accurate at temperatures well below 500 K, this does not lead to a significant error in the H, abundance as can clearly be seen from the discussion in Section 7.1

188

T. Abel et al. I New Astronomy

2 (1997) 181-207

temperature

in K

Fig. 1. The rate coefficients for photo-attachment to H- (dashed line), Hl formation (dotted line) and the two H, forming reactions, procs. (8) and (10) (solid lines).

(14) Collisional detachment of H- by electrons. We take the rate coefficient from Janev et al. (1987), 7.1.1, which is based on experimental results. The net energy loss is the threshold energy of 0.755 eV. (IS) Collisional electron detachment of H- by neutral hydrogen. The formation of an auto-dissociating state of molecular hydrogen (Eq. (16) in Section 5.6) seems to be far more effective in destroying H- than the direct collisional detachment H-+H+H+H+e-. The rate coefficient for the auto-dissociating state below 0.1 eV is very uncertain. We have linearly extrapolated the data given by Janev et al. (1987) to lower temperatures. This overestimates the rate coefficient in this range. However, it does not introduce any significant error in the abundances since collisions of H- with H in this temperature range are more likely to form a stable H, molecule

than to destroy H-. In fact, we expect this reaction to be important only in a very small range around 1 eV, since for higher temperatures, the dissociation by free electrons is far more effective than the one by H. Nevertheless a more accurate rate coefficient for this reaction would be desirable. (16) Mutual neutralization between H- and H+. The rate coefficient given by Dalgamo & Lepp (1987) is consistent with the measurements by Szucs et al. (1984) and Peart et al. (1985). See the latter reference for a review on the experimental difficulties and uncertainties of ion-ion collisions. (17) Hi formation in H- and H+ collision. Shapiro & Kang (1987) derived the rate coefficient from the cross section measured by Poulaert et al. (1978) in the range from 0.001 eV to 3 eV, and found it to have a E-o.9 dependence with a value of 1.5 X IO-l4 cm* at 0.003 eV The fit given in Shapiro & Kang (1987), however, is discontinuous at lo4 K. We modified it by about 4%, which surely lies within the uncertainties of the experimental data, to make it continuous. The result is given in the appendix.

T. AM

et al. / New Astronomy

(18) Dissociative recombination of Hl with slow electrons. The rate for this reaction was calculated using multichannel quantum defect theory (MQDT) by Nakashima et al. (1987) in the energy range from 0.02 to - 1 eV Their result is much smaller at low temperatures than previously assumed (see, e.g., Shapiro & Kang, 1987; Mitchell & Deveau, 1983). More recently, Schneider et al. (1994) recalculated this rate coefficient also using MQDT. Their tabulated rate coefficient for dissociative recombination of the Hi ground state via the 2: state is a factor of two to four times smaller than the rough fit given by Nakashima at temperatures above 1000 K. Our fit of the Schneider et al. (1994) data neglects the fact that the rate is increasing for T < 50 K. This is reasonable since one does not expect this reaction to be important for temperatures smaller than 100 K due the small abundance of free electrons and Ht. (19) Neutralization between Hi and Hp. This mutual neutralization reaction takes place at low energies ( < 1 eV) where Hl and H- coexist. Unfortunately, the only three measurements we are aware of, Aberth et al. in Moseley et al. (1975), Szucs et al. (1983), and Dolder & Peart (1985), give data only for energies above 3 eV We use the rate coefficient given by Dalgarno & Lepp (1987) despite its uncertainty. This equals the coefficient given by Prasad & Huntress (1980) to within 25%. The uncertainty in this rate is not important, as we will see in Section 7, since this process is negligible in cosmological applications.

4. Photoionization

and dissociation

processes

In general, we will only consider photoionization (dissociation) out of the ground state since we assume the abundance of excited states to be negligible. The rate of a particular photoionization (dissociation) reaction is given by:

2 (1997)

181-207

189

where k denotes either H, He+, H-, Hl or H,, i(v) is the intensity of the radiation field, q,, the threshold energy for which photoionization (dissociation) is possible, and ak, is the frequency dependent photoionization (dissociation) cross section of species k. (20)-(22) Photoionization of H, He and He l. These cross sections have been studied in great detail, and since they are relatively simple to calculate, the available data are very accurate. We use the typical expression given in Osterbrock ( 1989). The threshold frequencies are hv = 13.6, 24.6, and 54.4 eV, for H, He, and He+, respectively. (23) Photo-detachment of the H- ion. The best data available are given by Wishart (1979) with a stated accuracy within 1%. The fitting formula for the cross section given in Shapiro & Kang (1987) is accurate to - 10% at the high energy end. This is a normal characteristic of such fitting formulas. Although it is adequate for our present purposes, we recommend using the tabulated values of Wishart ( 1979) where they are available, since using the fitting formula in the integral of Eq. (6) leads to an overestimation of the corresponding rate. The data of Wishart (1979) is available from Tom’s Primordial Gas Chemistry Page’ in electronic form. H- is destroyed efficiently by the CBR for redshifts L 110. A simple fit for k,, depending on the radiation temperature that is accurate to better than 10% can be found in Tegmark et al. (1997). (24) Photoionization of molecular hydrogen. The high threshold of 15.42 eV assures that the CBR will not be able to photoionize H, in the post-recombination universe. The rate is taken from Shapiro & Kang (1987). (25), (26) Photodissociation of H_T. There is a conceptual difficulty in the treatment of this process since for strong radiation fields and high densities, the excited states of Hl will be populated. The photodissociation cross sections for the excited states are much higher than for the ground state. However, the levels might not be in their thermal equilibrium distribution. To treat this exactly, one

190

T. Abel et al. I New Astronomy

would have to include all Hi levels explicitly, which for our applications would not be worth the effort since most of the H, is formed via H-. By using only the cross section for the ground state, we will overestimate the Hl abundance. Using the LTE rate one can derive a lower limit and estimate the resulting error. The rate for photodissociation of Hl by a Planckian radiation field with temperature TR is derived from the equilibrium constant given in Sauval & Tatum (1984), 1.8

k;y(TR)

= 6.00 x lo-l9 X exp(45 KIT,&,,

K LTE

=

lOdex(c,

+ c,log,,x

+ c~(log,,x)3

SC’ ,

(7)

+ c2(log,,#

+ c4(log,gx)4 -x

d) (7&J’

2 (1997) 181-207

C’I&(u’), which then decay to the continuum of the ground state. This is called the two-step Solomon process (cf. Stecher & Williams, 1967). Allison & Dalgarno (1970) computed the band oscillator strengths, and Dalgarno & Stephens (1970) derived the probability that these states decay into the continuum of the ground state. The dominant photodissociation paths are through absorption in the Lyman Band to the vibrational states 6 < u’ < 20. This means that H, dissociation happens mostly in a very narrow frequency range 12.24 eV < hv < 13.5 1 eV Assuming the incident radiation field to be nearly constant around hF = 12.87 eV, which correwe sponds to v ’ = 13, and neglecting self-shielding, can derive a rate constant for photodissociation of H, through

, 63)

co = 9.9835,

c, = - 6.64 x lo-‘,

c2 = - 1.4979, cj = - 1.95 X lo-‘, d = 2.6508,

(9) cd = 0.7486,

- 1.1 x lo8

j(y)

s

ergs Hz-‘s-‘cm-*

-1

,

(10)

where k, is Boltzmann’s constant in erg/K, and x = 5040 K/T,. The most recent data for dissociation out of the ground state as well as the LTE rates can be found in Stancil (1994). We have fitted the cross section for the ground state to better than 2% (see (25) in the appendix). This fit is more accurate, and also does not show the unfortunate divergent character as the one given in Shapiro & Kang (1987) which was drawn from Dunn (1968). The threshold energy is 2.65 eV For photons with energies greater than 30 eV, Hz can be dissociated with two protons as products (proc. (26)). For this we adopted the rate given in Shapiro & Kang (1987). (27) Photodissociation of H2 by pre-dissociation. Photodissociation of the ground state of molecular hydrogen (X12:(u)) happens mostly through absorption in the Lyman-Werner Bands to the electronically and vibrationaly excited states, B ‘2: (u ‘) and

the radiation flux in where j denotes ergs cm-’ Hz-’ at h$ = 12.87 eV, fi’ is the oscillator strength for the transition X1x: (u = 0) to that B ‘L$~(u’) BrZ:(u’), P;; is the probability decays to the continuum, 4,,,.(v) is the line profile, and .Ire’/(mc) = 2.65 X lo-’ cm* is the classical total cross section. (28) Direct photodissociation of H,. Direct photodissociation of the ground state by absorbtion into the continua of the Lyman and Werner systems has been studied by Allison & Dalgarno (1969). We have fitted their cross sections within the stated accuracy of their data with, S -’

%

=‘(~;,“+f$O)+(l-y+1

,:

1)(

Cr;,’ + C$)

(11) where

T. Abel et al. / New Astronomy r;:( = IO-‘” dex(l5.1289-1.05139xkv)cm',

x

1

da-31.41

n T"=

14.675eV
+1.8042X 10~*(ku)'-4.2339X 10-r(kv)')cmL.

(12)

16.820eV
191

2 (1997) 181-207

we first derive the numbers of high energy photoelectrons produced in a typical quasar radiation field. Shapiro & Kang (1987) modeled a quasar radiation flux with: --a

lO_'X Xdex(l3.5311-0.9182618hv)cm".

14.615eV
F,. = 1.0 X lo-“6

17.1 eV.

erg cm-'

s-’

Hz-‘,

(13) = ,()-fX

(/ LY

X

dex(12.0218406-0.819429kv)cm'. dex(16.04@-

l.082438hv)cmz.

14.159eV
(14)

(15)

and y is the ortho- to para-H, ratio. Since they only gave the data for photon excess energies up to 3 eV, one might make quite significant errors in deriving the rate coefficient when integrating over this cross section. The magnitude of the error depends on the shape of the radiation spectrum.

5. Brief summary

of neglected processes

In this section we briefly summarize some processes which we find to be negligible. Although we can almost be certain that this summary is far from complete, we hope to stimulate a discussion which should lead us to a more detailed understanding of the primordial gas chemistry, and enable us to find the minimum set of reactions which describes primordial gas accurately enough for the desired cosmological applications. 5. I. Photoionization ionization

heating and secondary

For all simulations in which strong external radiation fluxes are present, we have to look at the fate of the photoelectrons produced in the photoionization reactions. To estimate the importance of this effect

where vn is the Lyman edge frequency, (Y= 0.7, 0.7 eV < hv 5 12.4 keV, and E is a parameter introduced to adjust the amplitude of the flux (or effectively the distance of the quasar). Although Sargent et al. (1989) argued that cr L 1 might be a better fit, we will derive an upper limit of the produced photoelectrons with the more drastic value of cy = 0.7. We calculate the fraction nF:m/n!$ of electrons produced by photons in the range from hv, to 2hv,, and the ones produced by photons in the range of 2hv, to 12.4 keV:

low

n Y.= high n y. e-

For such a hard spectrum with (Y= 0.7, we obtain n t: _ lnhyl$_ = 10, whereas for a more soft spectrum this number increases (e.g. 13 for (Y= 1.0). Because photoelectrons have an energy of h(v - v,), the ones produced by photons in the range from hv,, to 2hv, will not be capable of collisionally ionizing any species other than H-. Therefore, all they can do is to either go into heat, which means scatter with other electrons or ions and equilibrate to a Maxwellian distribution, or excite other atoms, ions or molecules. The photoelectrons above 2hv, carry enough energy to also ionize (dissociate) other species. The very high energetic ones are capable of ionizing (dissociating) many atoms (molecules) while cooling down. Shull & Van Steenberg (1985) gave fits for the fraction of photoelectrons above 100 eV which go into heat, ionization, and excitation, respectively. We, however, find that the total energy of all photoelectrons above 100 eV is only 1.5% of the total energy of all the photoelectrons produced in the photoionization of neutral hydrogen by a typical quasar flux.

T. Abel et al. I New Astronomy

192

Roughly 30% of all photoelectrons will have energies in the range of hv,, to 100 eV Fig. 3 of Shull & Van Steenberg (1985) shows that even for a mostly neutral gas, a maximum of about 30% of the secondary electrons will cause further ionizations. Hence, by leaving out secondary ionization, the error of the ionized fraction will be S 10%. The estimates above were based on the photoionization cross section of hydrogen, but they hold in general for the following reasons. First, the cross sections for He and He+ scale as ye3 in the high frequency limit, and second, nucleosynthesis predicts Therefore the photoelectrons produced nH - 16n,,. in photoionizations of neutral hydrogen are the dominant ones. Clearly, the above discussion only justifies the approximation of leaving out the effects of photoelectrons for applications where the radiation spectrum is approximated well by a power law as given in Eq. (16). 5.2. Exciting

of electrons

also included tral hydrogen

the photo-attachment (proc. (7)).

5.4. Proton collisions hydrogen

with atoms and ions

Electrons can photo-attach with, ionize, and excite, neutral hydrogen and helium and also ionize, recombine with, and excite their ions. We included all radiative recombination and collisional ionization reactions (procs. (l)-(6), (14)) except the double ionization of helium since it does not significantly influence the ionization balance due to its high threshold. The dominant excitations (see above) are treated through the appropriate cooling functions. We

reaction

of neu-

with neutral helium and

Below temperatures of - 1 eV, the neutral atoms are the most abundant species, whereas free protons and He+ are, even in the non-equilibrium case, more than two orders of magnitude less abundant than H, and He, respectively (Shapiro & Kang, 1987). For the equilibrium case, this difference is far more dramatic. We have checked all the proton collisions listed in Janev et al. (1987) and find that the rate coefficients rapidly decrease for temperatures 5 10 eV. This is especially true for collisional ionization by protons. The exception to the rule are charge exchange reactions such as H(n) + p + p + H(n),

collisions

For our intended low density applications, the coronal limit (see above) holds, and we therefore know that any excitation of H, H-, He, He+, and H, will be followed by a spontaneous decay back into the ground state and ensure that practically every atom or ion will be in its ground state most of the time. An excitation does not change the abundance of our model species and does not enter the reaction network explicitly. However, the kinetic energy loss is accounted for through the use of the appropriate cooling functions. 5.3. Collisions

2 (1997) 181-207

n = 1, 2, 3, .. .. ,

where n denotes the radial quantum number. Since we expect nearly all hydrogen atoms to be in the ground state, the n = 1 case will be the most probable process. Although it does not enter the reaction network, since it “produces what it destrays”, it is an important reaction theoretically, since it ensures tight thermal and spatial coupling between the protons and the neutrals due to its relatively high rate coefficient of - 1O-8 cm3 s- ’ . All rate coefficients for the excitation of neutral hydrogen and helium by protons out of the ground state are by many orders of magnitude smaller than the ones for excitation by electrons. Since charge neutrality requires that there are about as many free electrons as protons, it is clear that excitations by electrons dominate excitations by protons. 5.5. Collisions ions

of hydrogen

with helium and their

An interesting feature of the reaction network presented in this paper is that all species which are built up by hydrogen nuclei do not directly chemical-

T. Abel et al. I New Astronomy

ly interact with the species formed by helium nuclei. This is due to the following reasons: Electrons are much more effective in ionizing than any other species since their mean velocity is (Am,lm,)“* -‘I 43A”* times higher than for an ion with A times the proton mass. Charge exchange between H and He+ has a rate coefficient - 1.9 X IO-l5 cm3 s-’ and is, according to Couchman (1985), negligible. H, H:, and H, are not capable of ionizing helium. They are relatively fragile and have very small abundances at temperatures * 1 eV, where their kinetic energies would allow the ionization of helium and its ion. Although He is a noble gas HeH+, can be formed. Unpublished work by Stancil (1996) shows that the fractional abundance of HeHf in the background universe will be - 10-14. Hence it seems to be negligible. However, the HeH+ chemistry is not well understood yet, and a more detailed study that discusses its chemistry and cooling behavior in collapsing clouds is desirable. The arguments why the reaction network does not include processes where helium or its ions destroy HP, Hl ,or H, are given in the subsequent sections. 5.6. Neglected

dissociating

reactions

of H

2 (1997)

We did not include mutual neutralization of Hwith He’. The available data from AMDIS” of the IAEA12, which are only valid for energies above 42eV, indicates that the cross section of mutual neutralization of H- with He+ is higher than for mutual neutralization with protons. This is what one might naively expect, especially when considering that the Coulomb attraction acts as the driving force for this reaction, competing with the kinetic motion of the reactants. Since He + is heavier than H, its mean velocity will be slower. However, the Coulomb attraction is the same, and therefore mutual neutralization with He+ should occur with a higher rate. We only leave out this reaction due to the lack of reliable data. However, the expected error is small since, at low temperatures, most of the H- will react with neutral hydrogen and form molecular hydrogen (proc. (7)). At higher temperatures ( > 8 000 K), collisional detachment by free electrons (proc. (14)) will dominate the destruction of H-. Neutralization between H- and He++ can be neglected due to the high temperature threshold for He++ formation. Whenever He++ is formed, the temperature is so high that nearly all H- will be destroyed. 5.7. Omitted reactions for H2 formation Molecular hydrogen radiative association

Electron detachment of H- through neutral helium has, according to the AMDIS9 of the IAEA”, a cross section of the order - lo-l6 cm’, and is therefore many orders smaller than the cross section (- 10-9-10m7 cm*; Janev et al. (1987), 7.3.2a) of reaction

193

181-207

formation

by excited

atom

H(n=2)+H(n=l)+H,+hv

Exactly the same argument holds for the electron detachment of H through molecular hydrogen, - 10-‘8_10-‘6 cm* which has a cross section (AMDIS).

has a very small rate coefficient, due to the zero dipole moment of molecular hydrogen (Latter & Black, 1991). Also, for the cosmological applications we are interested in, the n = 2 population is extremely small in all cosmological applications, making this process less significant than the dominant H, formation mechanisms (procs. (8), (10)). We also neglect the formation path where Hl first forms by excited atomic radiative association and then forms molecular hydrogen through the charge

“http://www.iaea.or.at/programs/~/nds/amdisin~o.htm “‘http://www.iaea.or.at/

“http://www.iaea.or.at/programs/ri/nds/amdisintro.htm ‘*http://www.iaea.or.at/

H-+H--+H~*(ZR)+H+H+ee.

(16)

T. Abel et al. / New Astronomy 2 (1997) 181-207

194

exchange with neutral hydrogen (proc. (10)) (Rawlings et al., 1993) H(n=2)+H(n=

l)+Ht

+e-,

dissociation, 4.476 eV are lost. In the following we will discuss dissociating reactions which we find to be negligible compared to the three dissociating reactions we have included in our model and which are illustrated in Fig. 2.

H;+H(n=l)-+H,+H+. 5.8.1. Dissociation of molecular hydrogen by e This process can, in certain circumstances such as dense and hot gas in circumstellar environments, dominate the formation by excited atomic radiative association. In the intended cosmological applications, these processes will never be significant since the population of excited states is always negligible. 5.8. Collisional dissociation of H, and Hf Since hydrogen molecules are the main coolant for gas at temperatures below - 1 eV, we pay special attention to its destruction mechanisms. We follow Shapiro & Kang (1987) and assume that, at every

i) Hf +e-+(H,)*

10-‘h lo4

+H-

+H.

Wadehra & Bardsley (1978) showed that this dissociative attachment reaction depends strongly on the vibrational and rotational states. For low density gas (nu < lo5 cmm3) in which nearly all the hydrogen molecules will be in their ground state, its rate is of the order lo-l5 cm3 s- ’ for an electron temperature of 1 eV. For lower electron temperatures, this rate drops drastically. After comparing this to Fig. 2, it becomes clear that this process will never play an important role for the destruction of H, in our applications. It is, however, a crucial process in situations with either densities above lo4 cmm3 or

lo5

lo6

10’

10s

temperaturein K Fig. 2. Rate coefficients for the dissociation of molecular hydrogen by neutral hydrogen (dot-dashed line), and free electrons (dashed line). The light solid line depicts the rate coefficient for charge exchange between H’ and H,, given by Donahue and Shull (1991). whereas the thick solid line is our numerical integration from the Janev et al. (1987) data.

T. Abel et al. / New Astronomy

intense ultraviolet radiation fluxes because, under such circumstances, the excited states will become populated (for excitation of molecular hydrogen in intense UV fluxes see Shull (1978)). ii) H, +e-+H*

+H”

+e-.

Our process ( 12) is H,+e-+H(ls)+H(ls)+e-, and dominates

(17)

all other reactions

or the type

H?+e-+H*+H*+e-.

+H,

effective due to their higher single temperature fluid.

velocities

in the

Additionally, the known rate coefficients for the ionization of H, and the dissociation of Hl by protons (3.2.5, 3.2.6 of Janev et al., 1987) decrease rapidly for temperatures below 10 eV. Therefore, we are confident that dissociation by protons is negligible.

(18)

Intuitively one would have expected this because, in proc. (18), the incident energy has to be enough to excite the produced H atoms. However, the 2s level already lies 10.2 eV above the ground state, which only a few electrons have at the low temperatures ( < 1 eV) where molecular hydrogen exists. From the data given in Janev et al. (1987), Section 2.25, it is evident that

H,+e

195

2 (1997) 181-207

a3z: i c’n,

+H(ls)+H(ls)+ei

will also be dominated by the above reaction (Eq. (17)). Notice that the rate coefficient of reaction 2.2.6 in Janev et al. (1987) is a factor 10 too high! (Personal communication with Bill Langer, 1995.) 5.8.2. Dissociation of molecular hydrogen by H+ The positive hydrogen ion is able to destroy molecular hydrogen through the charge exchange reaction ( 11). The direct collisional dissociation, H++H+2H+H+, has often been left out by former studies. We also were not able to find a rate coefficient for this reaction. It seems to be negligible because at low temperatures where a significant H, fraction exists there will be nearly no free protons due to the high recombination rate at low temperatures. dissociation by electrons is likely to be more

5.8.3. Dissociation of molecular hydrogen by He Dove & Mandy (1986) found that He in comparison to H is very inefficient in dissociating H,(O,O). They further state that the collision between two closed shell species tends to cause the molecular bond to stiffen, whereas a collision with an atom weakens and loosens the molecular bond. In a follow up paper, Dove et al. (1987) applied their result to interstellar densities and found that the dissociation through neutral helium is negligible and that HZ-He collisions do not excite the vib./rot. populations of HZ in low density gas. Following these results, we do not include the dissociation of H,(O,O) by He.

5.8.4. Dissociation of molecular hydrogen by He + The number fraction of He in primordial gas is about 10%. Due to the high ionization threshold of He+ (54.4 eV) it will not be abundant at low temperatures. Furthermore, from the scaling relations of the rate coefficients given in Janev et al. (1987), we know the dissociation of H, by He+ occurs 0.35 times less than the corresponding proton reaction. Therefore we can safely conclude that this reaction can be omitted.

5.8.5. Dissociation of H2 by H, The dissociation rate coefficient for H, on H, is of the same order and temperature dependence as the one for the dissociation by neutral hydrogen (Lepp & Shull, 1983). However, we do not include this reaction since the H, to H fraction (and hence the rate) is always smaller than - lo-*, and is therefore negligible.

T. Abel et al. I New Astronomy

196

6. The cooling and heating functions optically thin gas

for

To model the cooling behavior of the gas correctly, we incorporate the following cooling and heating mechanisms: 0 Compton cooling l Recombination cooling due to hydrogen helium l Line cooling of hydrogen and helium l Bremsstrahlung cooling l Molecular formation and line cooling l Photoionization heating l Photodissociation heating

and

The cooling functions for Compton, recombination, line, and bremsstrahlung cooling that we have used are given in Anninos et al. (1997). The molecular cooling and heating rates are somewhat controversial in the literature and, therefore, deserve special attention. 6.1. Molecular

cooling

6.1.1. Formation cooling of molecular hydrogen The reaction enthalpy (1.83 eV for process (10) and 3.53 eV for process (8)) is mainly released in ro-vibrational excitations of the molecule and not in kinetic energy of the products. We assume that all the reaction energy goes into excitation and is radiated away through the subsequent spontaneous radiative decay to the ground state (Shapiro & Kang, 1987), with the following cooling formula: A Hz =

formation

nH(3.53n,-k,

+ 1.38n,:)

eV cme3

s-’ .

These reactions, however, are important heat sources at high densities (n 2 lo* cmm3) where collisional de-excitation can transform most of the excitation energy into kinetic (thermal) energy. The density dependent heating rates are given by Hollenbach & McKee (1979).

2 (1997) 181-207

6.1.2. Line cooling of molecular hydrogen Recently Martin et al. (1996) have derived the molecular hydrogen cooling function for H,-H collisions with a complete set of rate coefficients. Their result differs substantially from former cooling functions (e.g., Hollenbach & McKee, 1979; Lepp & Shull, 1983) especially at number densities exceeding 10 cmm3. Lepp and Shull’s expression seems to overestimates (underestimate) the cooling at low (high) temperatures by an order of magnitude. Although the rate coefficients of Martin et al. (1996) are only accurate for temperatures above 600 K, some of them have been found to agree with QCT calculations at 300 K. For lower temperatures, however, the cooling function of Martin et al. (1996) has to be understood as a lower limit (personal communication with Peter Martin, 1996). A FORTRAN77 routineI that computes their fitting formula can be found on our WWW-Site. From Appendix B we know that the number density ni of ortho-H, can be very small for applications where our model is valid. Therefore, if one were to estimate the cooling behavior to high accuracy, a separate treatment of the ortho and para-H, states is required. 6.1.3. Photodissociation heating Our model includes the photodissociation processes of H,, processes (27) and (28). Black & Dalgarno (1977) argue that, for typical radiation the photodissociation by the two-step fluxes, Solomon process (27) yields 0.4 eV per atom pair. Hence the corresponding heating function is I& = 6.4 X 10-‘3nHzkz,

erg cme3

s-’ .

(19)

For direct dissociation into the continua of the Lyman and Werner systems, the situation is similar to photoionization heating. Since the reaction channel leading to the excitation of the produced hydrogen atoms is only accessible for excess photon energies greater than 10.2 eV, we can safely assume that all the excess energy of the dissociating photons

T. Abel et al. I New Astronomy

will be shared as kinetic energies of the produced hydrogen atom pairs. Hence the heating functions can be written,

“lh

where i(v) denotes the radiation intensity and the integral is evaluated for the ortho and para-H, component separately because of their different threshold energies.

7. Application

and discussion

2 (1997)

181-207

197

Very massive pancakes have strong virialization shocks (u,~- 100 km/s), which leads to the destruction of primordial pre-shock hydrogen molecules. The post-shock gas cools faster than it recombines and a significant free electron fraction remains even at low temperatures (T +C 104). This can clearly be seen in Fig. 3 where the fractional abundances of all species in the post shock gas are plotted as a function of temperature. Evidently Hl is always much less abundant than H-. He+ and He++ are also found to have non-equilibrium abundances at low temperatures. These results agree well with the ones given in Shapiro & Kang (1987).

of the model 7.1. Hz chemistry

In this section we will use data from a Id high resolution study of a cosmological sheet with wavelength 4 Mpc to discuss non-equilibrium effects and the chemical dynamics in primordial gas. The details of the simulation parameters can be found in Anninos & Norman (1996)

The right hand side of an individual rate equation is given by a sum of terms kjjn,nj, where kii is the rate coefficient for the collisional processes of species i with j, which is taken positive if it produces the species under consideration, or negative if it

Fig. 3. The fractional abundance of nine species in a collapsing pancake of wavelength 4 Mpc are shown vs. temperature. The data are taken from Anninos and Norman (1996). Clearly the non-equilibrium enhancement of free electrons at low temperatures can be seen. Note also the similar non-equilibrium behavior of the He’. and He++ fraction.

198

T. Abel et al. I New Asrronomy

destroys it. To see what processes dominate the chemistry of species 1, let us consider the rate kjjn,nj divided by n,, which we call the rate per l-atom (molecule). Comparing this to the sum over all 1 producing/destroying processes (the evaluated right hand side), much insight about the ongoing processes can be gained. Actually this quantity equals the inverse reaction time scale and hence provides a measure of the time in which a particular species reaches equilibrium. We show such plots for H,, Hl, and H- (Figs. 4, 6, 5). The complete H, chemistry in our model is illustrated in Fig. 4. All molecular hydrogen producing reactions, (8), (10) and (19), are shown as solid lines as well as their sum (labeled formation). The destroying processes (1 l), (12), and (13) are depicted by dash-dotted lines. It is clear that the molecular hydrogen abundance at temperatures above - 7000 K is a result of the balance between the producing and destroying processes, what one

loJ)

~ -

2 (1997) 181-207

could call a “collisional equilibrium”. It can also be seen that, from 0.5 eV to 1 eV, the charge exchange reaction of H, and free protons is the most efficient molecular hydrogen destruction path. At higher temperatures, hydrogen molecules are destroyed most efficiently by free electrons. Obviously here in the pancake collapse, destruction by neutral hydrogen atoms has a negligible influence on the molecular hydrogen abundance. For the production of H,, we find that proc. (8) and (19) are always dominated by the fast H- formation path (proc. (8)). In summary, we find that the molecular hydrogen fraction is governed dominantly by three processes. Formation via H-, destruction through charge exchange with free protons, and destruction by free electrons. These insights can be used also for analytical estimates of the molecular hydrogen fraction during pancake collapse. Let us illustrate this briefly on the molecular hydrogen abundance for tempera-

“““’ “““’

formation - destruction

10-l’

1000

10000

100000

temperature(K) Fig. 4. The inverse of the reaction time scale in s-’ for all processes in our model that involve hydrogen molecules. The relative rates the producing collisional processes, (8), (lo), and (19), are illustrated by solid lines. The H, destroying processes, (11). (12). and (13). shown with dot-dashed lines. The thick solid (thick dot-dashed) line depicts the inverse of the sum of reaction time scales of all producing (destroying) processes. The divergence of these producing and destroying curves at low temperatures indicates that the molecule abundance is out of equilibrium.

for are H, H,

T. Abel et al. I New Astronomy

7.2. The H- and Hi chemistry

tures 7000 K 5 T 5 lo4 K. Here the H, abundance can obviously be derived through the equilibrium condition dnH21dt = 0 with

%I2

-=k,n,~n, dt

199

2 (1997) 181-207

The H- ion is found to reach chemical equilibrium faster than either the Courant, dynamical or cosmological time scales. Hence the overall destruction and production rates are the same at all temperatures (see Fig. 5). Our model includes one H producing reaction, the radiative attachment (proc. (7)), and six destroying reactions. Fig. 5 clearly illustrates that processes (17) and (19) play no role, processes (15) and (16) little role, and processes (7), (8), and (14) the main role in the H- chemistry. Since the atomic data we used for the dominant reactions (7), (8), and (14) are very accurate at all temperatures, we have ensured that the model will predict the H abundance accurately. An analogous plot for the Ht chemistry is given in Fig. 6. over the Obviously H 2’ is in chemical equilibrium temperature ranges in our pancake study. All of its

-kllnH2nH+ =O,

ks nHk II - nH?=-X, where x denotes the ionized fraction. Looking up the rate coefficients, we can read off the qualitative temperature dependence and find that, in the temperature range 6000 K 5 T 5 2 X lo4 K, the molecular hydrogen abundance increases with decreasing temperature. For lower temperatures, collisional processes are not efficient enough in destroying hydrogen molecules and it will be produced as long as there exists a significant amount of H- ions.

~ formation - - - destruction

/’

/‘\

i 14; I lo-l4

/’ /’ /

lo-l6

100

/

/‘19

/’

/’

‘\

‘1

1’

,I-* ,’ I I

\

I I

, ,, / 1000

1 10000

‘\ 100000

temperature (K) Fig. 5. The inverse of the reaction time scale in s-’ are shown for the collisional processes involving H-, m&ding

the single H producing reaction (7). and all its destruction processes (S), (14), (15), (16), (17), and (19). It is evident that H- reaches its equilibrium abundance within the hydrodynamical and cooling time scale since the sum of the production rates equals the sum of the destruction rates. Here the thick dot-dashed lines illustrate the major H- destroying processes, (8) and (14).

2cnl

T. Abel et al. I New Astronomy

__

2 (1997) 181-207

formation

lo-l4 100

1000

10000

100000

temperature (K) Fig. 6. The inverse of the reaction time scale in s-’ for the collisional processes involving H: are shown. The Hz producing processes (9). ( 11), and (17) are depicted by solid lines and the H: destroying reactions (lo), (18), and (19) correspond to the dot-dashed lines. Analogous to H- (Fig. 5), it is clear that Hl reaches its equilibrium abundance within the hydrodynamical and cooling time scales.

three production mechanisms contribute significantly in different temperature regimes. The radiative association reaction (9) contributes strongly at temperatures below a few thousand Kelvin and dominates at temperatures - 2 X lo4 K. Over the remaining high temperature regime the charge exchange of H, with H+ is the dominant Hl producing reaction. Furthermore, it is clear from Fig. 6 that one can safely neglect the destroying mechanism, proc. (19), because it does not contribute significantly, at any temperature, to the overall destruction of Hc. It is also evident that, at all temperatures below - 8 X lo3 K, nearly every Hl molecule that is produced will be converted to a neutral hydrogen molecule by proc. (10). At higher temperatures, Hl will be destroyed by electrons. The data of Schneider et al. (1994) for the rate coefficient of this process, and consequently our fit, are quite inaccurate at such high temperatures and we do not expect the H: abundance at temperatures above lo4 K to be reliable. We

see that process (19) was never important in determining the abundances of H-, Hi, and of H,, and can therefore be neglected in future applications. 7.3. A minimal model From the preceding section it is clear that not all of the reactions in our model are equally important. We devised a minimal model that incorporates only the essential processes needed to accurately model the formation of molecular hydrogen. From Fig. 4 it is evident that the only reaction involving H: which contributes strongly to the H, chemistry is the charge exchange reaction (11). Its rate is independent of the Hl abundance since it is a Hi producing reaction and we can conclude that, at least for strong shocks, the H, abundance is independent of the Hz chemistry. For a minimal model we can use this to advantage and leave out reactions (9), (lo), (17) and (19) so as to avoid solving an Hl rate equation.

T. Abel et al. I New Astronomy 2 (1997) 181-207

Looking at Fig. 5, we also find that process (15) is negligible and that process (16) is only marginally important. Clearly (13) can be left out as well since it obviously does not contribute to the H, chemistry. Furthermore. we note that it is not necessary to include the contributions from the reaction involving H- and H, to the major species H, He, and their ions. With these simplifications 7 of the 19 reactions are eliminated and the reaction network is reduced to the following: dn H dt

= kgtH+n,

dn,+ = k,n,n, dt dn He dt

klnHne ,

(21)

- k2nH+ne,

(22)

-

= k4nHe+ne - k3nHene,

201

have to be included ( + k,, in the denominator of Eq. (27)), and for even higher redshifts the reactions involving Hl are needed. Of the 12 reactions in the minimal model

seven are highly accurate to within 2 percent, namely processes (l)-(3) and (5)-(8); three are sufficiently accurate with errors 5 5%, namely (4), (II), and (14); however processes (12) and (16) might only be accurate to within a factor of a few.

detailed calculation and/or measurement of the rate coefficient for reaction (16) is especially desirable.

(23)

dn,, + = k3nHene + k6nHe++ne - k4nHe+ne, dt

(24)

8. Conclusions

dnHe++ ___ = k5nHe+ne - k6nHe++ne, dt

(25)

We have derived a reliable time dependent chemistry and cooling model for primordial gas that, in connection with our new numerical method (Anninos et al., 1997), provides a powerful tool to investigate primordial structure formation in multidimensional numerical calculations. The model is designed to be applicable for densities below lo4 cmm3 and temperatures < 10’ K. We have discussed the influence of the ortho-H, to para-H, ratio on the cooling behavior of the gas, derived new fits to atomic and molecular data, and developed a minimal model capable of describing primordial gas in applications where no external radiation fields are considered. The current chemistry model is designed for lowdensity cosmological applications. To extend this model to higher densities will be difficult mostly due to the lack of molecular data that account for the effects of rotational and vibrational excitations on the chemical behavior of the molecules. Without these data, however, will it be impossible to accurately model the transition from the non-LTE low-density regime to LTE which is very important for primordial protostar formation.

dn,,

T=k

8nH-nH

-

nH2(kl,nH+

where the number density equilibrium condition nHm =

+

k,,ne)

of H-

3

is given

(26)

by the

k7nHne

k,n,

+ k,,n,+

+ k,4ne ’

(27)

The derivation of the number density of H- from its equilibrium value is justified by comparing Figs. 4,5. Since they show the inverse of the reaction time scales of the different processes, it is clearly evident that all reactions determining the H- abundance occur on much faster time scales than those responsible for the H, chemistry. We have checked this minimal model extensively and find the H, abundance to generally agree with the full model to within a few percent over the entire temperature range of the particular application. Note that this model is only applicable for redshifts 5 110 when no external radiation fields are present. For redshifts 110’ - 2 5 300, the photo-detachment reaction would

202

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2 (1997) 181-207

9.Related websites

Appendix A.

At our WWW-Site, The LCA Cooling Model14, we present all the atomic and molecular data discussed here in detail. We also provide FORTRAN routines that compute the rate coefficients and solve the rate equations. Tom Abel continues to collect molecular data related to primordial gas at Tom’s Primordial Gas Chemistry Page15. Furthermore, the Dima Verners, Atomic Data for Astrophysics’6 page represents a superb reference for atomic data as does the Atomic and Molecular Data Information System ” of the International Atomic Energy Agency18. Applications of our chemistry model and the numerical methods introduced in the companion paper please see our Cosmic String” and First Structure Formation*’ ?VWW pages.

Here we present all the rate coefficients used in our model. All fits are applicable for temperatures ranging from 1 K to 10’ K. Since we are interested in numerical applications we paid more attention to the accuracy of the fits than to the simplicity of the formulas. A FORTRAN program” that computes these rate coefficients can be obtained at our WWW site. The temperatures are in eV unless stated otherwise.

Acknowledgments We would like to thank Dimitri Mihalas, Mordecai-Mark Mac Low, Zoltan Haiman, Max Tegmark, and Evelyne Roueff for helpful discussion. Furthermore, we are grateful for helpful correspondences with Bill Langer, Uffe Hellsten, Phil Stancil, Jonathan Rawlings, Alex Dalgamo, Peter Martin, and Stephen Lepp. This work was done partly under the auspices of the Grand Challenge Cosmology Consortium funded by NSF grant ASC93 18 185. T.A. happily acknowledges the hospitality of the Max Planck Institut fur Astrophysik and the Insituto Astrofisica de Canarias where parts of this work have been carried out.

Appendix B.

Reactions and rates

Ortho-H, to para-H, ratio

Many of the collisional rates have only been computed for the para configuration and are applicable only if para-H, is is the dominant modification (Flower & Watt, 1984). For our purposes it plays another significant role because the selection rules for the allowed rotational transitions of the two configurations are different. This is crucial in determining the exact cooling behavior at low temperatures, since low density primordial gas will cool mostly in the rotational lines E3 - E, = 844.5 K for the ortho configuration and E, - E, = 509.9 K for para-H,. At formation, the two different modifications will be abundant corresponding to their statistical weights, namely

n(ofiho)_ NJ = 1) _ . nW-a> n(J= 0) 3

(B.1)

Flower & Watt (1984) argue for a single interconversion mechanism at low densities, H,(ortho) + H+ _j H,(para) + H’ + 170.5 K , (B.2) and suggest a temperature independent rate coefficient of k,, = 3 X lo-” cm3 s-i. If the J = 2 state

T. Abel et al. I New Astronomy 2 (1997) 181-207 Table I Collisional

ionization

and radiative

recombination

of hydrogen

203

and helium

(I) H t e- +H’ + 2e- Janev et al., 1987 (2.1.5) k, = exp[-32.71396786 t 13.536556 In(T) - 5.73932875 In(T)* t 1.56315498 In(T)’ - 0.2877056 III(T)~t 3.48255977 X lo-’ X In(T)’ - 2.63197617 X IO-’ X ln(T)‘t 1.11954395 X 10-‘?n(Tf - 2.03914985 X lO~‘ln(T)‘] cm’s_’ (2) H’ t e +H t y Our fit to data of Ferland et al. (1992) k2 = exp[-28.6130338 - 0.72411256 In(T) - 2.02604473 X 10~zln(T)2 - 2.38086188 X IO-‘In(T)’ - 3.21260521 X 10m41n(T) - 1.42150291 X lO~‘ln(T)’ t 4.98910892 X 10-bln(T)6 t 5.75561414 X 10-‘in(T)’ - 1.85676704 X IO-%(T) - 3.07113524 X 10~yln(T)‘] cm’ s-’ (3) He + em -He’ + 2e- Janev et al., 1987 (2.3.9) k, = exp[-44.09864886 + 23.91596563 In(T) - 10.7532302 In(T)’ t 3.05803875 ln(T)j - 0.56851189 ln(T)4 t 6.79539123 X IO-‘In(T) -5.00905610X 10~‘1n(T)6+2.067236l6Xl0~4ln(T)7-3.64916l4lXl0~6ln(T)R] cm’s_’ (4) He’ t em + He t y Cen (1992) and Aldrovandi & Pequignot (1973) Radiative: k,, = 3.925 X 10-‘3T-06353 cm’ s-’ Dielectronic: k,, = 1.544 X 10-YT-“*exp( - -)[0.3 t exp(v)]

cm’ s-’

(5) He’ t em +He” + 2e- AMDIS Database (1989) k, = exp[-68.71040990 t 43.93347633 In(T) - 18.4806699 In(T)* + 4.70162649 In(T)’ - 0.76924663 ln(Ty t 8.113042 X IO-‘ln(Ty - 5.32402063 X IO-‘ln(TT t 1.97570531 X 10-41n(T)7 - 3.16558106 X 10-61n(T)*] cm3 S-I (6) He’+ t e- -+He’ k, = 2 X kz(T/4)

Table 2 The formation

paths of H,

(7) H + e- +Hk, =

t y Scaling k, (see, e.g., Osterbrock 1989)

+ y This work from data by Wishart

(1979).

T in K

-18 ” 7620 0 152310*,#)(7, --3.274xI0-40&iT1 3 -1 1.429X10 T T T cm s for T 5 6WO K 3,802 x 10-‘7T0 19981%,0(T) dex(4.0415 X 10-510g~,(T) 5.447 X IO-?ogf,(T)) cm3 s-’ otherwise. {

(8) H + H- -+ H, + e- Our integration of data from Janev et al. (1987) T >O.l eV: k, = exp[-20.06913897 + 0.22898 In(T) + 3.5998377 X 10-21n(T)2 - 4.55512 X IO-?n(T)’ t 1.0732940 X 10-41n(T)5 - 8.36671960 X 10-61n(T)6 + 2.23830623 X 10-‘In(T)‘] cm3 s-’ T CO.1 eV: k, = 1.428 X 10m9 cm3 s-l (9) H + H+ +Hi k, = {

+ y Shapiro

& Kang (1987)

3.833 X lo-l6 X T’ * cm’ s-‘, 5,81 x lo-l” x (o,20651 T)-“.**91Xlog(0.*0651XTI cmz s-I,

(IO) Ht +H-+H, tH+ Karpas et al. (1979) -1” 3 -I k,,, = (6.4? 1.2) X 10 cm s

for T < 0.577 eV for T 2 0.577 eV

- 3.10511544

X 10m41n(T)4

204

T. Abel et al. 1 New Astronomy

Table 3 Other collisional

processes

involving

H-, Hi,

2 (1997) 181-207

and H,

(11) H, +H++H; +H This work ln(k, ,) = - 24.24914687 + 3.40082444 In(T) - 3.89800396 in(T)* + 2.04558782 In(T)’ - 0.541618285 ln(T)4 + 8.41077503 X 10~21n(T)S - 7.87902615 X 10-?n(T)6 + 4.13839842 X 10-41n(T)7 - 9.36345888 X 10-61n(T)8 cm’ SC’ (12) H, + e- +2H

+ e- Donahue

& Shull (1991). T is in K

k,, = 5.6 X lo-“T”‘exp(-v)

cm3 SC’

(13) Hz + H+3H Dove & Mandy (1986) k,, = 1.067 X 10-‘“T20’2 X exp(-(4.463/T)(l

+ 0.2472 T)“‘*)

cm3 s-’

(14) H- + e- -+H + 2e- Janev et al., 1987 (7.1.1) k,, = exp[-18.01849334 + 2.3608522 In(T) - 0.28274430 In(T)’ + 1.62331664 X 10-Zln(T)3 - 3.36501203 X 10-Zln(T)J + 1.17832978 X 10-*1n(T)5 - 1.65619470 X 10-31n(T)6 + 1.06827520 X 10-41n(T)’ - 2.63128581 X lO?n(T)*) cm3 s-’ (15) H- + H+2H + e- Our modification to the Janev et al. (1987) data T >O.l eV: k,, = exp[-20.37260896 + 1.13944933 In(T) - 0.14210135 In(T)’ + 8.4644554 X 10-31n(T)3 - 1.4327641 X 10-‘ln(T)4 + 2.0122503 X 10-41n(T)5 + 8.6639632 X 10-51n(T)6 - 2.5850097 X 10m51n(T)’ + 2.4555012 X 10m61n(T)* - 8.0683825 X IO-*lt~(T)~] cm’ SC’ T CO.1 eV: k,, = 2.5634 X 10m9 X T’ 78’Shcm’ SC’ (16) H- + H’ --f 2H Dalgamo & Lepp (1987), k,, = 7 X lO-8(&)~“2 cm3 SC’ (17) H- + H+-+Hl

+ e- Our modification 2.291 X 10~‘0T-04

k,, =

to fit of Shapiro

cm3 SC’,

1 8.4258 X 10-‘“T-‘~4exp(-l.301/T)

1.0 X lo-* cm3 SC’, { 1.32 X 10~6T-“7b

(19) H; + H- +H,

cm3 SC’,

+ H Dalgarno

k,, = 5 X LO-‘(E!$A)I/2

for T < 1.719 eV otherwise

et al. (1994) data. T is in K for

T<617

for

T>617K

& Lepp (1987),

K

T is in K

cm’ s-’

would be significantly populated, transformed to ortho by H,(J = 2) + H+ +H,(J

& Kang (1987)

cm3 s-l,

(18) H,: + e- --3 2H Our fit to the Schneider k,, =

T is in K

para-H,

could be

the neutral hydrogen number density, and the ionized fraction, respectively. The time evolution of the ionized fraction is determined from

= 1) + H+ + 170.5 K. dx

However, at the low density limit mostly the ground level will be populated and Eq. (B.2) will determine the ortho to para-H, ratio. Assuming that we have a constant total number density of molecular hydrogen, n2, the rate equation for par-a-H, fti simply becomes

z=

- k2nx2,

+x(t)

=

X0 1 + x,k,nt ’

Using n, = ng + $ and Eq. (B.4), we derive the simple solution for the number density of para-H,

(B.3) where ni, n, x, denote the ortho-H,

number

density,

(B.4)

From this we clearly can see that if

T. Abel et al. I New Astronomy Table 4 Photoionization

and photodissociation

2 (1997) 181-207

205

processes

(20) H+y+H++e(21) He+y+He’+e-

Osterbrock(l974)

or, = 7.42 X 10-‘8(1.66(~)-205

- 0.66(<)-3a5)

cm’,

for v> I+,.

(22) He’ + y + He++ + e Osterbrock ( 1974) az, 22 = 2(<)” ex;9““x~::;;;1~fl, where A = 6.30 X lo-” (23) H-+y+H+e-

E =qw,

hv,, = 13.62’ eV

De Jong (1972)

o?? = 7.928 X lO’(v - v,$“i (24) Hz + y+H:

“3

cm*,

for hv > hv,, = 0.755 eV

+ e- O’Neil & Reinhardt

(1978) for

hv < 15.42 eV

6.2 X IO-“hv

- 9.4 X LO-” cm*,

for 15.42 eV 5

hv < 16.50 eV

1.4 x lo-‘*hv

- 1.48 X lo-”

for 16.50 eV 5

0, Y 0z4 =

cm*

( (25) H; + y +H log,,,(cr;:)

2.5 X lo-‘*(hv)-’

cm*,

” cm*,

for

hv < 17.7 eV hv?

17.7 eV

+ H’ Our fit to Stancil (1994) = - 1.6547717 X lo6 + 1.8660333 X lO?n(v)

- 7.8986431

X lO’ln(v)* + 148.73693 In(v)’ - 1.0513032 ln(v)J

(26) Hz +y+2H’ +e- Shapiro & Kang (1987) log,,,(u>,) = - 16.926 - 4.528 X 10m2hv + 2.238 X 10m4(hv)’ + 4.245 X IO-‘(hv)’ (27)H,+y-+H:-+H+HThiswork Neglecting self-shielding (28) Hz + y +H 0% =+

k,, = 1.1 X lO’j(;)

SC’

where j(i),

is thexadiation

for 30 eV < hv < 90 eV

flux in ergs s-’ cm-* at hi = 12.87 eV

+ H This work

(u;;+ffzwd,)+(l

-&)(a:;

k,, -+z k,, ortho-H, to para-H, within one recombination time.

+o$)

will not change

k, -K kopr ortho-H, to para-H, will change dramatically within one recombination time.

Since k, is well fitted by 1.8 X 10-‘0T-0’65 cm3 s-‘, it convincingly shows that the latter case always applies. Obviously the recombination time scale also sets the formation time scale of H, since the electrons act as catalysts (see also Abel, 1995; Tegmark et al., 1997). Hence when H, is formed by the gas phase reactions (7) through (lo), it might immediately be converted to its para configuration. The subsequent cooling of the gas will therefore happen mostly in the E, - E, line of para-H,. For completely neutral hydrogen gas at the low density and temperature limit, the ortho-H, to paraH, ratio will be given by 9 exp(-170.5 K/T) (Mandy & Martin, 1993). Sun & Dalgamo ( 1994)

computed the rate coefficients for the odd transitions of low lying rotational levels by collisions with atomic hydrogen, in the temperature range 30 K1000 K. For temperatures below 300 K these reactions have rate coefficients -C 10-r’ cm3 s-r, i.e. for such low temperatures in low density gas (nn 1 cme3) the corresponding reaction time scales are z+ 10 Gyrs. Hence, if the gas can cool faster than it recombines below 300 K, the above result is unchanged. However, the ortho-Hi, to para-H, will be different in different environments. Once the complete set of rate coefficients for the collisional excitation and dissociation of H, molecules by H atoms as given by Martin & Mandy (1995) are extended to temperatures below 450 K, and extended to include Hz-H, collisions as well as Hz-H+ reactions, it will be possible to study the ortho-H, to para-H, ratio and its effect on the cooling behavior of primordial gas for the entire range in which our reaction network is applicable.

206

T. Abel et al. I New Astronomy

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