Astrochemical Modeling: Practical Aspects of Microphysics in Numerical Simulations

Astrochemical Modelling: Practical Aspects of Microphysics in Numerical Simulations is a comprehensive and detailed guide to dealing with the standard problems that students and researchers face when they need to take into account astrochemistry in their models, including building chemical networks, determining the relevant processes, and understanding the theoretical challenges and the numerical limitations. The book provides chapters covering the theoretical background on the predominant areas of astrochemistry, with each chapter following theoretical background with information on existing databases, step-by-step computational examples with solutions to recurrent problems, and an overview of the different processes and their numerical implementation.

Furthermore, a section on case studies provides concrete examples of computational modelling usage for real-world applications and cases where the techniques can be applied is also included.

• Provides theoretical background on topics that is followed by computational examples and tailored tutorials to allow for full understanding and replication of techniques
• Written by theoreticians and authors with direct experience on the computational implementation to provide a realistic and pragmatic approach to common problems
• Details up-to-date information on available databases, tools and benchmarks for practical usage, forming a good starting point for introductory readers and a reference for actual implementation for more advanced researchers

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 23, 2023
• ISBN: 9780323972574

Book preview

Preface

Stefano Bovino; Tommaso Grassi     

All models are wrong, some are useful

George Box

Modeling is a crucial component of understanding astrophysical objects. Although, by definition, models are a limited representation of reality, they are yet one of the most effective elements in the toolbox of the astrophysical community. Through the years, models improved by including more and more physical processes but, at the same time, they became increasingly computationally demanding. In fact, their computational footprint has become a crucial limitation in applying models to interpret observations, the latter having reached a remarkable level of detail in recent years, therefore requiring more powerful and accurate models. For this reason, one of the pivotal challenges for the astrophysical community is to reduce the computational cost of models by simplifying (or removing) some of the less impacting processes or improving specific numerical techniques. However, the details and the methods employed are rarely well described in literature, distributed over countless papers, or handed down in some obscure way by more experienced modelers, and, in many cases, the beginner should reinvent the wheel.

To cope with this problem, we decided to gather in this book several authors from different fields with recognized experience in numerical modeling or in the theoretical aspects that are crucial to modeling. This approach introduces a point of view and some specific hands-on techniques that only experienced modelers can provide. Therefore, we identified four main broad areas, such as chemistry, that include not only standard gas and dust chemistry but also practical suggestions on how to design chemical networks and evolve them in time numerically. Then we illustrated less local aspects, such as the interaction of gas and dust with radiation and cosmic rays, to move then into the third part that describes the importance of microphysics in simulations. Finally, we included the coupling with hydrodynamics and synthetic observations, and we concluded the book with a set of case studies, where the reader can find how to model specific astrophysical environments. The latter should be considered a starting point for developing more complex models.

Although this book is not a comprehensive guide on how to model thermochemistry from a numerical point of view, we aim to present a relatively large sample of the most commonly used techniques, also introducing for the first time in a single work, some specific topics, as for example an essential guide on how to integrating differential equations within the astrochemical context.

This book aims to give the reader a general overview of the most compelling problems in the numerical modeling of thermochemistry and microphysics and to help beginner or advanced modelers to understand what are the most relevant processes to include, or, more importantly, what are the unnecessary processes. In practice, this book will help students and researchers understand how to worry selectively about processes that have different importance in the global picture, and that can be modeled with various degrees of complexity and accuracy.

As Editors, we structured the book with coherent contents and interconnected chapters, hoping to create a valuable tool for instructors, undergraduate studies, and not only for researchers in the advanced stage of their careers. For this reason, we also provide an illustrative computational framework guiding the readers through a more practical approach.

Part 1: Chemistry

Outline

Chapter 1. Gas-phase chemistry

Chapter 2. Designing a gas-phase chemical network

Chapter 3. Time-dependent integration of chemical networks

Chapter 4. Chemistry on interstellar dust grains

Chapter 1: Gas-phase chemistry

James Babb    Center for Astrophysics | Harvard & Smithsonian, Cambridge, MA, United States

Abstract

Gas-phase chemistry and closely-related atomic and molecular processes are at the foundation of large-scale astrochemical modeling and describe energy, mass, and photon interactions down to the quantum scale and must be considered in formulating chemical networks. We provide a brief introduction.

Keywords

Atomic and molecular processes; ions; electrons; molecules; chemical reactions

1.1 Overview

Astrochemical modeling involves the joining of many components and the chapters of this book are designed to provide practical approaches at all stages. The descriptions of relevant gas phase chemistry from an atomic and molecular physics perspective deserves discussion and will be the main topic of the chapter. Subsequent chapters will revisit in more detail some of the topics discussed here and some of the particular applications of specific atomic and molecular processes; these will be cross-referenced accordingly.

Many parameters need to be specified in formulating an astrochemical model and these parameters affect what chemical processes and reactions need to be treated, for example, abundances of elements and temperatures in the environments being modeled. The descriptions in this chapter will be kept as general as possible within the realm of what is relevant for astrochemistry. Sources of data listed here should be considered to be representative rather than exhaustive.

The chapter is organized into three sections. A brief discussion of sources of basic atomic and molecular data is given in Sect. 1.2. Then the interplay between electrons, ions, atoms, molecules, and photons is organized into Sect. 1.3 on Atomic and Molecular Processes and Sect. 1.4 on Chemical Reactions. The division into Processes and Reactions is mainly organizational: Processes will include electrons and photons, while Reactions will involve solely atoms, ions, and molecules.

A familiarity with atomic and molecular spectroscopic notations and conventions such as atomic units ( , , and , respectively the reduced Planck constant, the electron rest mass, and the elementary charge) will be assumed. Generally speaking, the processes will be considered nonrelativistically.

1.2 Sources of basic atomic and molecular data

Ionization potentials are available at NIST for atoms and their ions using the ionization energies forms at the Atomic Spectra Database (ASD)¹ and for molecules at the Computational Chemistry Comparison and Benchmark DataBase (CCCBDB).² One classic source for diatomic molecule data, including dissociation energies, vibrational-rotational energies and parameters, dipole moments, reduced masses (μ), and more, is the book by Huber and Herzberg (1979).³ Electron affinities (the energy released in forming an anion from an atom) can be found in Ning and Lu (2022) and references therein. Proton affinities are also available at the CCCBDB.⁴ Electric dipole polarizabilities can be found for neutral atoms in Schwerdtfeger and Nagle (2019) and for neutral molecules in Hohm (2013). The reader should note that the value of depends on the quantum states of the atom or molecule and might have a tensor component (anisotropy); see the book by Bonin and Kresin (1997) for further details on experimental and theoretical aspects of polarizabilities. Atomic and molecular partition functions are available from various sources, a recent tutorial paper concerning atoms and ions provides some instructive background and contains useful references (Alimohamadi and Ferland, 2022); some additional considerations concerning molecular data sources are given by Tennyson et al. (2020). Keep in mind that diverse collections of critically reviewed and compiled data can be found in standard reference books in any local scientific library. In addition, comprehensive references to basic and applied physics focusing on the mutual interactions of atoms, molecules, ions, electrons, and photons are available (Smirnov, 2018; Drake, 2023). Additional resources are discussed in Sect. 5.4.

1.3 Atomic and molecular processes

In formulating astrochemical models, quantitative descriptions of various atomic and molecular processes are required. These data might describe collisions, photo-processes (absorption, emission), ionization, etc., and are often measured or calculated as cross-sections and expressed as rate coefficients or rates that enter into models.

When a cross-section is available, it will describe a specific process that can most likely be expressed in plain language. Formally, labels such as initial and final states and collisional energies will be given, though cross-sections for two-body processes are usually simply written as , where E is the relative kinetic energy and state labels are suppressed but carried contextually. The rate coefficient can be obtained by averaging over a Maxwellian velocity distribution at thermal equilibrium temperature T, which, when expressed in terms of E, yields

(1.1)

where μ is the reduced mass, is the Boltzmann constant, and with the standard units of cm³ s−1 for the two-body rate coefficient . From a practical viewpoint, will not be available from a single calculation or experiment over the energy range indicated in Eq. (1.1). Depending on the temperature, the exponential in the integrand could diminish the relative contributions of the high-energy values of . Values of can be sourced, as available, from different references and then smoothly joined to span a sufficient energy range to ensure an accurate evaluation of the integral for ; usually, cubic splines are sufficient to interpolate between data points. Often, the low-energy and high-energy trends of can be set using power laws or other physical guidelines in which case the energy range can be extended continuously. If resonances appear in cross-sections, some consideration of their potential contributions in evaluating Eq. (1.1) is warranted (a Lorentzian fit to the resonance might be sufficient). Note that a partial cross-section usually refers to a specific dissociation channel, such as in the context of molecular photodissociation (e.g., Sec. 3.1 of Heays et al., 2017), though the term may also be used to a quantum-mechanical cross-section for a specific partial wave, e.g., , where l is the angular momentum of the partial wave. Once the terminology is understood, desired cross-sections can be obtained through the standard procedure of averaging over initial states and summing over final states (Friedrich, 1991, p. 108). Rate coefficients can also be obtained directly, such as through kinetics, see Sect. 2.1.1. Cross-sections for photo-processes, if available as functions of frequency, can be expressed as rates (s−1); see, for example, Eq. (5.24) for photoionization or Eq. (11.56) for photodissociation.

1.3.1 Specific processes

A simple list identifying atomic and molecular processes written in terms of reactants and products is instructive and is given in Table 1.1. Some of these processes are not discussed in later chapters, but may be present in tables distributed with chemical networks to be discussed later. Charge transfer is included in Table 1.1; while it does not strictly meet the process definition of involving electrons or photons, its presence will be explained in Sect. 1.3.1.9.

Table 1.1

1.3.1.1 Radiative recombination

Radiative recombination is the capture of an electron by an ion into the ground or excited states of a neutral atom (if the initial ion has charge one, otherwise resulting in an ion) with the emission of a photon yielding a continuum (since this is a free-bound transition). After the capture, the states will decay (cascade) towards the ground state yielding spectral lines. State-selected rates can be tabulated for each recombination into a specific state and these can be summed to obtain total rates. Using detailed balance (Light et al., 1969) or the Milne relation, Eq. (11.57), each state-selective rate can be related to the photoionization cross-section from the state, which is amenable to calculation (see Sect. 1.3.1.2). A recent paper (Kotelnikov and Milstein, 2019) gives a good brief review of the major earlier works and the calculation of radiative recombination rates for hydrogen from photoionization cross-sections. Multielectron atoms and ions are treated theoretically through large-scale calculations of photoionization cross-sections; see, for example, Nahar and Pradhan (1995). While such calculations require significant resources, advances in computing have made many-level calculations feasible and successful, for example, see Seager et al. (2000) (mentioned in Chap. 14) for an application to the recombination of , , and + in the early Universe. Further details can be found on theoretical aspects in Bates and Dalgarno (1962), Flannery (1976), and Hahn (1997), and on computational aspects in Badnell et al. (2003), where discussions can be found of processes important for high electron densities – dielectronic recombination and three-body recombination of two electrons and an ion yielding an electron and an ion or neutral – and the influence of external electric fields, which are not discussed here.

Sources of data include the CHIANTI database (mentioned in Sect. 13.3.1 and see Del Zanna and Young 2020 for an overview) and the NORAD-Atomic-Data site.⁵

1.3.1.2 Photoionization

In addition to CHIANTI and NORAD (see above), for atoms and ions ( to 26), a collection of calculated values can be found at the TOPbase database.⁶ For photoionization of atoms and molecules, see the PHoto Ionization/Dissociation RATES (phidrates) database⁷ (Huebner and Mukherjee, 2015). The theoretical treatment of atomic systems is reviewed by, for example, Burke (1976). See Sects. 5.2, 5.3, and 5.4 (databases) for more detailed discussion.

1.3.1.3 Radiative association

The radiative association process may occur between two atoms or molecules or between an atom or a molecule and an ion. The available rate coefficients for two atoms or for an atom and an ion are primarily theoretical, calculated either semiclassically or obtained from quantum-mechanically calculated cross-sections using theoretical techniques that are fairly mature. Nyman et al. (2015) reviewed the theoretical approaches and gives a bibliographic listing of calculations for the formation of many diatomic systems and a few polyatomic systems. Quantum-mechanical calculations of polyatomic molecule formation by radiative association are scarce, see Stoecklin et al. (2018) for a recent example. In addition, theoretical calculations of radiative association rate coefficients involving the formation of molecules larger than diatomics were carried out using a phase space theory (Herbst, 1982); see, for example, a recent application by Wakelam et al. (2009). Additional discussion is given in Chap. 2. Some experimental results for polyatomic ion formation are available for the radiative association of ions and neutrals, see Gerlich (1995).

In particular for radiative association processes involving two atoms or an atom and an ion, it is straightforward to obtain radiative association cross-sections from photodissociation cross-sections using microscopic reversibility, see Eq. (7) in Babb (2015) for an application to where state-resolved cross-sections were utilized. If photodissociation cross-sections from each level of the initial molecule are not accounted for, the calculated radiative association process obtained by summation may not be complete. In such cases, a direct quantum-mechanical calculation of the radiative association process may be preferable. Similarly, rate coefficients for radiative association can be obtained from photodissociation rates by application of the Milne relation, see Eq. (11.57).

1.3.1.4 Photodissociation

The review by Heays et al. (2017) is another source of data for astrophysical applications and additional databases are listed there. See Sects. 6.2.2 and 6.6 for more detailed discussion.

1.3.1.5 Radiative attachment

Rate coefficients for radiative attachment can be obtained using detailed balance from photodetachment (Sect. 1.3.1.6) rates. An example applied to can be found in McLaughlin et al. (2017). Likewise, cross-sections for radiative attachment can be obtained using microscopic reversibility from cross-sections for photodetachment; for an example, see the treatment of in Douguet et al. (2013), their Eq. (11). The introductory section of Yurtsever et al. (2020) provides a succinct summary of the astrophysical motivation to determine radiative attachment rate coefficients for more complex molecules to understand their formation, stemming from the identification of (McCarthy et al., 2006) in astronomical sources, see also Walsh et al. (2009). For complex molecules, the phase space theory as developed by Herbst (1982) for radiative association can be applied to calculate radiative attachment rate coefficients (Herbst and Osamura, 2008). A recent detection of highlights the need for more work on the rate coefficients for the radiative attachment mechanism applied to nitriles of the form (Cernicharo et al., 2023).

1.3.1.6 Photodetachment

Photodetachment is the inverse of radiative attachment, as mentioned in the previous section. Photodetachment can be a removal process for anions and thus rates are usually available in astrochemical databases for common atoms and molecules. An example of a systematic calculation of photodetachment rates for linear carbon chains, including and for , for , and , can be found in Lara-Moreno et al. (2019); this paper also illustrates the use of detailed balance to calculate radiative attachment rate coefficients. Photodetachment from grains, in the context of cosmic dust, is discussed in Chap. 9.

1.3.1.7 Dissociative recombination

Some general properties of dissociative recombination (also known as dissociative electron recombination) processes are discussed in Florescu-Mitchell and Mitchell (2006) and in Chap. 2. There has been steady progress in obtaining reliable rate coefficients, both experimentally and theoretically. On the experimental side, the advent of cryogenic storage rings facilitated measurements for molecular ions in their lowest vibrational and rotational states, which are highly relevant to astrochemical applications; for a recent example involving , see Paul et al. (2022). Similarly, in Novotný et al. (2019) results from cryogenic storage ring experiments on dissociative recombination of are summarized and applied to primordial gas evolution. Theoretically, calculations using multichannel quantum defect theory (MQDT) are becoming more prevalent, see Jiang et al. (2021) and references therein. The MQDT method can yield branching ratios for the production of atoms in various states by dissociative recombination. For example, in a recent paper (Boffelli et al., 2023), the yields for production of and in dissociative recombination of (a removal process for the molecular ion) are calculated for applications to sulfur astrochemistry. The dissociative recombination process is important in atmospheric chemistry and plasma physics leading to data production that is also sometimes of value for astrochemistry, e.g., for (Djuissi et al., 2022), which is seen in industrial processes and tokamaks, as well as in the diffuse interstellar medium (Schilke et al., 2014).

1.3.1.8 Associative detachment

This process, sometimes called associative electron detachment, can be important in dust-free environments. The process for and − is introduced in Sect. 10.1.6.2; recent measurements of the rate coefficients were carried out, see Kreckel et al. 2010; Miller et al. 2011; Gerlich et al. 2012, and some theoretical developments were given by Grozdanov and McCarroll (2019). Other chemistries are possible, for example, theoretical calculations on the destruction of by associative detachment with are given by Jerosimić et al. (2018). Note that the removal of anions by photodetachment (Sect. 1.3.1.6) can work against the formation of molecules by associative detachment.

1.3.1.9 Charge transfer

The charge transfer process is included in this section (rather than in the next section Chemical Reactions), given its long and fundamental history in atomic and molecular collision physics. A recent review provides a nice summary of theoretical and experimental cross-section data with their relevant energy ranges (Gu and Shah, 2023, Tables 1.1 and 1.2). See also Sect. 5.4 (databases) and the short review by Stancil (2001).

An important type of charge transfer process is dissociative charge transfer. For example,

is an efficient mechanism destroying in environments such as, for example, supernovae ejecta (Lepp et al., 1990), while the reaction is a source of in dense photon-dominated regions (Sternberg and Dalgarno, 1995, reaction R222). Recently, new methods were developed allowing laboratory measurements of the rate coefficient for this reaction in the range of temperatures between 0 and 25 K (Martins et al., 2022).

1.3.2 Other processes

Some additional atomic and molecular processes are introduced and discussed in detail in Sect. 8.4 and shown in Table 8.3, where, in particular, momentum transfer and collisions involving protons are introduced, e.g., (proton) dissociative excitation and (proton) dissociative ionization.

1.4 Chemical reactions

The amount and types of data concerning chemical reactions are vast, but fortunately, specialized databases are available for astrochemical applications, such as the Kinetic Database for Astrochemistry (KIDA),⁸ and these databases are discussed further in Chaps. 2 and 10. Databases compiled for other purposes can also be helpful, such as for combustion (see Sect. 2.2.1). Similarly, there are available databases compiled for applications to exoplanetary atmosphere modeling, such as Stand2015 (Rimmer and Helling, 2016), which extensively covers , , , and chemistry with nine other elements and contains thousands of reactions. However, even though evaluation and curation is performed in assembling these databases, diligence may be required in selecting particular reactions. For example, the RATE06 (Woodall et al., 2007) database contained data on more than 4573 reactions involving 420 chemical species: Röllig (2011) reevaluated tabulated fits to the low-temperature behavior of several dozen rate coefficients; the revised fits were implemented subsequently in the latest version of the database, RATE2012 (McElroy et al., 2013).

Though it feels speculative, it seems reasonable to expect that in future years machine-learning methods will be applied extensively to the problem of reaction rate constants for astrochemistry; certainly, there is activity in chemistry-at-large (Komp et al., 2022). See also Koner et al. (2019) for an application of a machine learning process to the prediction of cross-section data.

1.4.1 Ion–neutral reactions

Of the listed reactions in astrochemical reaction databases, a majority are those between an ion and a neutral atom or molecule; their relative importance arises from the attractive interaction with the neutral collider arising from the charge of the ion. For example, in one of the first general compilations intended for astrochemical applications Prasad and Huntress (1980) list 1060 positive ion–molecule reactions out of over 1400 in their library of reactions; moreover, of those 1060, only one-quarter had been studied in the lab at the time. The close connection between ion–molecule reactions and experimental laboratory spectroscopy of molecular ions is emphasized in a recent review (McGuire et al., 2020). See Sect. 2.2.4.1 for further discussion.

1.4.2 Neutral–neutral reactions

See Chap. 2 for further discussion of estimates and databases. Some of the considerations in adopting experimental and theoretical data on neutral–neutral reactions for astrochemical purposes are discussed in Smith et al. (2004). Some practical notes on using chemical kinetics methods for reversing rate coefficients can be found in Rimmer and Helling (2016, App. B). Recently tunneling at low temperatures has become a topic of renewed interest partly stemming from the finding that the rate coefficient for the astrochemically-interesting reaction of is two orders larger at 65 K than at 200 K (Shannon et al., 2013; Nguyen et al., 2019). A recent representative publication is the experimental and theoretical work on the low temperature kinetics of (West et al., 2023).

1.4.3 Mutual neutralization

Schematically written as

, the reaction is an important anion removal process in negative ion chemistry, see Dalgarno and McCray (1973) and Lepp and Dalgarno (1988) for some considerations and example applications. Representative recent calculations include those for (Belyaev and Voronov, 2020) and for (Mezei et al., 2011).

References

Alimohamadi and Ferland, 2022 P. Alimohamadi, G.J. Ferland, Publ. Astron. Soc. Pac. 2022;134, 073001.

Babb, 2015 J.F. Babb, Astrophys. J. Suppl. 2015;216:21.

Badnell et al., 2003 N.R. Badnell, M.G. O'Mullane, H.P. Summers, et al., Astron. Astrophys. 2003;406:1151.

Bates and Dalgarno, 1962 D.R. Bates, A. Dalgarno, D.R. Bates, ed. Atomic and Molecular Processes. New York: Academic Press; 1962:245–271.

Belyaev and Voronov, 2020 A.K. Belyaev, Y.V. Voronov, Astrophys. J. 2020;893:59.

Boffelli et al., 2023 J. Boffelli, F. Gauchet, D.O. Kashinski, et al., Mon. Not. R. Astron. Soc. 2023;522:2259.

Bonin and Kresin, 1997 K.D. Bonin, V.V. Kresin, Electric-Dipole Polarizabilities of Atoms, Molecules, and Clusters. Singapore: World Scientific; 1997.

Burke, 1976 P.G. Burke, P.G. Burke, B.L. Moiseiwitsch, eds. Atomic Processes and Applications. Amsterdam: North-Holland; 1976:199–248.

Cernicharo et al., 2023 J. Cernicharo, J.R. Pardo, C. Cabezas, et al., Astron. Astrophys. 2023;670:L19.

Dalgarno and McCray, 1973 A. Dalgarno, R.A. McCray, Astrophys. J. 1973;181:95.

Del Zanna and Young, 2020 G. Del Zanna, P.R. Young, Atoms 2020;8:46.

Djuissi et al., 2022 E. Djuissi, A. Bultel, J. Tennyson, I.F. Schneider, V. Laporta, Plasma Sources Sci. Technol. 2022;31, 114012.

Douguet et al., 2013 N. Douguet, S. Fonseca dos Santos, M. Raoult, et al., Phys. Rev. A 2013;88, 052710.

Drake, 2023 Drake, G.W.F. (Ed.), 2023. Springer Handbook of Atomic, Molecular, and Optical Physics. 2nd edn. Springer Nature Switzerland, Cham.

Flannery, 1976 M.R. Flannery, P.G. Burke, B.L. Moiseiwitsch, eds. Atomic Processes and Applications. Amsterdam: North-Holland; 1976:407–466.

Florescu-Mitchell and Mitchell, 2006 A.I. Florescu-Mitchell, J.B.A. Mitchell, Phys. Rep. 2006;430:277.

Friedrich, 1991 H. Friedrich, Theoretical Atomic Physics. Berlin, New York: Springer; 1991.

Gerlich, 1995 D. Gerlich, Phys. Scr. T 1995;59:256.

Gerlich et al., 2012 D. Gerlich, P. Jusko, Š. Roučka, et al., Astrophys. J. 2012;749:22.

Grozdanov and McCarroll, 2019 T.P. Grozdanov, R. McCarroll, J. Phys. Chem. A 2019;123:3090.

Gu and Shah, 2023 L. Gu, C. Shah, arXiv e-prints arXiv:2301.11335; 2023.

Hahn, 1997 Y. Hahn, Rep. Prog. Phys. 1997;60:691.

Heays et al., 2017 A.N. Heays, A.D. Bosman, E.F. van Dishoeck, Astron. Astrophys. 2017;602:A105.

Herbst, 1982 E. Herbst, Chem. Phys. 1982;65:185.

Herbst and Osamura, 2008 E. Herbst, Y. Osamura, Astrophys. J. 2008;679:1670.

Hohm, 2013 U. Hohm, J. Mol. Struct. 2013;1054:282.

Huber and Herzberg, 1979 K.P. Huber, G. Herzberg, Molecular Spectra and Molecular Structure. New York: Van Nostrand Reinhold; 1979.

Huebner and Mukherjee, 2015 W.F. Huebner, J. Mukherjee, Planet. Space Sci. 2015;106:11.

Jerosimić et al., 2018 S.V. Jerosimić, F.A. Gianturco, R. Wester, Phys. Chem. Chem. Phys. 2018;20:5490.

Jiang et al., 2021 X. Jiang, J. Forer, C.H. Yuen, M. Ayouz, V. Kokoouline, Phys. Rev. A 2021;104, 042801.

Komp et al., 2022 E. Komp, N. Janulaitis, S. Valleau, Phys. Chem. Chem. Phys. 2022;24:2692.

Koner et al., 2019 D. Koner, O.T. Unke, K. Boe, R.J. Bemish, M. Meuwly, J. Chem. Phys. 2019;150, 211101.

Kotelnikov and Milstein, 2019 I.A. Kotelnikov, A.I. Milstein, Phys. Scr. 2019;94, 055403.

Kreckel et al., 2010 H. Kreckel, H. Bruhns, M. Čížek, et al., Science 2010;329:69.

Lara-Moreno et al., 2019 M. Lara-Moreno, T. Stoecklin, P. Halvick, ACS Earth Space Chem. 2019;3:1556.

Lepp and Dalgarno, 1988 S. Lepp, A. Dalgarno, Astrophys. J. 1988;324:553.

Lepp et al., 1990 S. Lepp, A. Dalgarno, R. McCray, Astrophys. J. 1990;358:262.

Light et al., 1969 J.C. Light, J. Ross, K.E. Shuler, A.R. Hochstim, ed. Kinetic Processes in Gases and Plasmas. New York: Academic Press; 1969:281.

Martins et al., 2022 F.B.V. Martins, V. Zhelyazkova, F. Merkt, New J. Phys. 2022;24, 113003.

McCarthy et al., 2006 M.C. McCarthy, C.A. Gottlieb, H. Gupta, P. Thaddeus, Astrophys. J. Lett. 2006;652:L141.

McElroy et al., 2013 D. McElroy, C. Walsh, A.J. Markwick, et al., Astron. Astrophys. 2013;550:A36.

McGuire et al., 2020 B.A. McGuire, O. Asvany, S. Brünken, S. Schlemmer, Nat. Rev. Phys. 2020;2:402.

McLaughlin et al., 2017 B.M. McLaughlin, P.C. Stancil, H.R. Sadeghpour, R.C. Forrey, J. Phys. B, At. Mol. Phys. 2017;50, 114001.

Mezei et al., 2011 J.Z. Mezei, J.B. Roos, K. Shilyaeva, N. Elander, Å. Larson, Phys. Rev. A 2011;84, 012703.

Miller et al., 2011 K.A. Miller, H. Bruhns, J. Eliášek, et al., Phys. Rev. A 2011;84, 052709.

Nahar and Pradhan, 1995 S.N. Nahar, A.K. Pradhan, Astrophys. J. 1995;447:966.

Nguyen et al., 2019 T.L. Nguyen, B. Ruscic, J.F. Stanton, J. Chem. Phys. 2019;150, 084105.

Ning and Lu, 2022 C. Ning, Y. Lu, J. Phys. Chem. Ref. Data 2022;51, 021502.

Novotný et al., 2019 O. Novotný, P. Wilhelm, D. Paul, et al., Science 2019;365:676.

Nyman et al., 2015 G. Nyman, M. Gustafsson, S.V. Antipov, Int. Rev. Phys. Chem. 2015;34:385.

Paul et al., 2022 D. Paul, M. Grieser, F. Grussie, et al., Astrophys. J. 2022;939:122.

Prasad and Huntress, 1980 S.S. Prasad, W.T. Huntress Jr., Astrophys. J. Suppl. 1980;43:1.

Rimmer and Helling, 2016 P.B. Rimmer, C. Helling, Astrophys. J. Suppl. 2016;224:9.

Röllig, 2011 M. Röllig, Astron. Astrophys. 2011;530:A9.

Schilke et al., 2014 P. Schilke, D.A. Neufeld, H.S.P. Müller, et al., Astron. Astrophys. 2014;566:A29.

Schwerdtfeger and Nagle, 2019 P. Schwerdtfeger, J.K. Nagle, Mol. Phys. 2019;117:1200.

Seager et al., 2000 S. Seager, D.D. Sasselov, D. Scott, Astrophys. J. Suppl. 2000;128:407.

Shannon et al., 2013 R.J. Shannon, M.A. Blitz, A. Goddard, D.E. Heard, Nat. Chem. 2013;5:745.

Smirnov, 2018 B.M. Smirnov, Atomic Particles and Atom Systems: Data for Properties of Atomic Objects and Processes. 2nd edn. Springer Series on Atomic, Optical, and Plasma Physics. Cham: Springer; 2018;vol. 51,.

Smith et al., 2004 I.W.M. Smith, E. Herbst, Q. Chang, Mon. Not. R. Astron. Soc. 2004;350:323.

Stancil, 2001 P.C. Stancil, G. Ferland, D.W. Savin, eds. Spectroscopic Challenges of Photoionized Plasmas. Astronomical Society of the Pacific Conference Series. 2001;vol. 247:3.

Sternberg and Dalgarno, 1995 A. Sternberg, A. Dalgarno, Astrophys. J. Suppl. 1995;99:565.

Stoecklin et al., 2018 T. Stoecklin, P. Halvick, H.-G. Yu, G. Nyman, Y. Ellinger, Mon. Not. R. Astron. Soc. 2018;475:2545.

Tennyson et al., 2020 J. Tennyson, S.N. Yurchenko, A.F. Al-Refaie, et al., J. Quant. Spectrosc. Radiat. Transf. 2020;255, 107228.

Wakelam et al., 2009 V. Wakelam, J.C. Loison, E. Herbst, et al., Astron. Astrophys. 2009;495:513.

Walsh et al., 2009 C. Walsh, N. Harada, E. Herbst, T.J. Millar, Astrophys. J. 2009;700:752.

West et al., 2023 N.A. West, L.H.D. Li, T.J. Millar, et al., Phys. Chem. Chem. Phys. 2023;25:7719.

Woodall et al., 2007 J. Woodall, M. Agúndez, A.J. Markwick-Kemper, T.J. Millar, Astron. Astrophys. 2007;466:1197.

Yurtsever et al., 2020 E. Yurtsever, M. Satta, R. Wester, F.A. Gianturco, J. Phys. Chem. A 2020;124:5098.

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¹  https://www.nist.gov/pml/atomic-spectra-database.

²  http://cccbdb.nist.gov/ie1x.asp.

³  https://rios.mp.fhi.mpg.de/index.php.

⁴  http://cccbdb.nist.gov/palistx.asp.

⁵  https://norad.astronomy.osu.edu/.

⁶  http://cds.u-strasbg.fr/topbase/xsections.html.

⁷  https://phidrates.space.swri.edu/.

⁸  https://kida.astrochem-tools.org.

Chapter 2: Designing a gas-phase chemical network

Olli Sipiläa; Maxime Ruaudb,c    aCenter for Astrochemical Studies, Max-Planck-Institut für Extraterrestrische Physik, Garching, Germany

bCarl Sagan Center, SETI Institute, Mountain View, CA, United States

cNASA Ames Research Center, Mountain View, CA, United States

Abstract

In this chapter, we give an overview of designing a custom gas-phase chemical network for one's purposes. We describe some fundamental aspects of the problem, such as which units to use, how the reaction rates and rate coefficients are defined, which types of reactions are typically considered in simulations, and how one connects a network of reactions on the mathematical level. We describe the various potential sources of input data for the new network. Practical hints for building a network are given, emphasizing how to avoid potential pitfalls during network development; for example, ensuring that the total number of atomic nuclei is conserved during the simulation. References to benchmarks are given. The chapter ends with a brief introduction to isotopologue and spin-state chemistry.

Keywords

Chemical reactions; databases; isotopes; molecules; numerical methods

Astrophysical models that include chemistry aim at understanding how the available elements (i.e., C, O, N, S, P, Fe, …) combine to form molecules we observe in space. More specifically, they aim to compute the evolution of the concentration of the different chemical compounds present in the environment as a function of time.

These models consider complex chemical systems that may contain hundreds of chemical compounds, involved in thousands of chemical reactions. Solving this kind of system relies on the availability of chemical and microphysical data. Collecting these data and building a chemical network represent a complex task, not only due to the large number of reactions involved, but also because many of these reactions are not well known.

This chapter will discuss the caveats related to one of the key questions in astrochemistry, i.e., to design a chemical network to include chemistry in, e.g., hydrodynamical models.

2.1 General considerations

This section gives a brief introduction to reaction kinetics and, in particular, the notion of rate equation, reaction rates, and coefficients. It also introduces the notion of chemical networks.

Note on units. To obtain the rate of formation or destruction of a given species in a chemical reaction (see Sect. 2.1.1), astrochemical codes express the concentrations of the participating species as number densities, i.e., counts of molecules in unit volume, in units of . The concentration of species i is most often expressed as (or in some cases ), which is the convention that is also followed here. However, in the literature, results are often expressed in terms of abundances, which is meant the concentration of a species with respect to another quantity, usually the volume density. The abundance is typically denoted as x, and can denote either or , where is the total hydrogen number density, .

In many applications, for example, in hydrodynamical models, it may be beneficial, or even necessary, to express concentrations as mass densities instead of number densities. In the conversion between the mass density and the number density, one must consider the mean atomic or molecular weight. As an example of the interconversion between mass and number density, let us consider deriving the number density of dust grains. Starting from the dust-to-gas mass ratio , and writing where is the molecular weight per hydrogen atom, is the mass of the hydrogen atom, and is the total gas number density, with , where is the mass of one dust grain,¹ we obtain

(2.1)

Here the conversion from mass density to number density is mediated by , whose value is ∼1.4 for a medium composed of ∼90% hydrogen and ∼10% helium (by abundance). In other applications, one may need to consider the average molecular weight per free particle ( instead; Kauffmann et al. 2008).

2.1.1 Reaction rates and coefficients

For a set of chemical compounds X with concentration , the chemical evolution of the system is controlled by the set of differential equations

(2.2)

In the following, we will assume that formation and destruction rates are only induced by chemistry, but they could also contain transport terms (i.e., advection and diffusion of chemical species, see Chap. 12).

The rate of gas-phase processes can be written in terms of the molecular concentration of the different reactants and a rate coefficient k. The rate coefficient defines the intrinsic velocity of the chemical reaction, and its unit depends on the type of reaction. There are three types of elementary reactions: unimolecular, bimolecular, and termolecular reactions. Bimolecular reactions, which involve the collision of two chemical compounds, are the most common type of reactions in interstellar gas.

Suppose we have a bimolecular reaction A + B → C. The rate of the reaction is

(2.3)

where k has units of cm³ s−1 in this case,² and the negative signs indicate the species destruction. The rate coefficient k is usually a function of the temperature and can be expressed by the Arrhenius equation

(2.4)

where is the activation energy (in K), A the preexponential term, and n a dimensionless factor that gives the temperature dependence of the rate coefficient. This reaction is called a second-order reaction because the rate of the reaction depends on the square of the concentration of the reactants. For unimolecular reactions (and more generally, all first-order reactions), the rate of the reaction linearly depends on the concentration of the reactant. The rate of the unimolecular reaction A → B is

(2.5)

For this type of reaction, the rate coefficient has units of s−1. This is the case of gas-phase photo-processes, as well as processes involving cosmic rays, and for which the different techniques used to compute the rate coefficient k will be discussed in Chap. 6 and 8.

At high densities (i.e., typically larger than cm−3), termolecular reactions (or three-body reactions), in the form of

, can become important. In most cases, these reactions are association reactions where the third body (M) acts as a stabilizer of the excited product formed during the collision of the pair of reactants A and B. The third body (M) is any inert atom or molecule that can remove the excess of energy from the excited product and leaves C in a bound state. An important reaction of this type is , which is thought to have played an important role in the condensation of the primordial gas which led to the formation of the first generation of stars (see, e.g., Flower and Harris, 2007). The rate of this type of reaction is

(2.6)

with k in cm⁶ s−1. Since the concentration of M is not affected by the reaction, it does not need to be evolved, and the rate of the reaction only depends on its concentration.³

2.1.2 Chemical networks

A chemical reaction network connects a set of chemical compounds to a set of chemical reactions. Most chemical reaction networks used to study the chemistry in astrophysical environments, take into account hundreds of chemical compounds coupled through thousands of elementary chemical reactions. Because of the presence of nonlinear terms, finding an analytical solution of this kind of system is rarely feasible, and the problem must be solved numerically (see Chap. 3). In this section, we introduce the mathematical representation of such a network.

Suppose we have a network of chemical species involved in M unimolecular and bimolecular reactions. Such a network can be formalized mathematically as a system of coupled ordinary differential equations (ODEs) similar to Eq. (2.2). For each species i, we have

(2.7)

where is the rate coefficient for the bimolecular reaction between species Xi and Xj and the rate coefficient for the unimolecular reaction involving species j. The first two terms of the right-hand side include all reactions that lead to the formation of species Xi, while the last two terms account for its destruction.

In some environments, the chemical timescales can be relatively short compared to the dynamical ones, and the chemistry can approach steady state. This is the case of interstellar photodissociation regions, for instance (although high-resolution observations show that this may not be true at small scales where dynamical effects become important), where the chemistry is dominated by rapid photoionization and photodissociation by stellar radiation. In the case where the steady-state approximation can be employed, we have a system of N dependent