Introduction

In this chapter, we extend the treatment of rotational excitation, presented in Chapter 4, to encompass vibrational transitions. In practice, the discussion will be limited to linear molecules for which calculations are feasible and quantitative results have been obtained. As compared with rotational transitions, which may be excited at kinetic temperatures T ≈ 10 K, in the case of heavy molecules, or T ≈ 100 K, in the case of light (hydrogen-bearing) molecules, vibrational excitation generally requires T ≈ 1000 K. There are exceptions: the inversion motion (of the nitrogen in the plane of the hydrogen atoms) in NH3 is a form of vibrational motion, and the energy involved is small, of the order of 1 K. Similarly, the torsional motion (of the CH3 relative to the OH group) in CH3OH may be viewed as another form of vibrational motion and involves energies of the order of 100 K. These phenomena have been discussed in Chapter 4. In the present chapter, we shall be concerned with the stretching of chemical bonds within molecules, a process that involves higher energies, on the order of 1000 K.

Vibrational transitions are important in a number of astrophysical contexts, including cool stellar atmospheres, shocked regions of molecular clouds, and planetary nebulae.

Introduction

The interstellar medium has a structure and composition which have been modified by Galactic and stellar evolution over the lifetime of the Universe. While the gas remains rich in primordial H and He, it also contains small but significant amounts of C, N, O and other heavy elements, notably of the Fe-group, which have been produced by nucleosynthesis and then returned, in more or less violent events, to the interstellar medium. In addition to the gas, there is dust in this medium, which contributes only about 1% to the mass but which has important effects on the chemistry and the thermal balance (cf. Chapter 1). The primordial gas, on the other hand, contained no dust and was composed only of those elements that were produced in the primeval fireball — essentially hydrogen and helium, with trace amounts of deuterium, lithium and beryllium. Under these conditions, it is perhaps surprising that molecules existed and even more surprising that they should have played an important role in the evolution of the Universe. Nonetheless, this is believed to have been the case and, in this chapter, we explain why.

The governing equations

The cosmic background radiation field that we observe today has a black-body temperature of 2.73 K; it is a remnant of the ‘big bang’, which is believed to have occurred at the origin of the Universe.

Introduction

Conditions of thermodynamic equilibrium are the exception, rather than the rule, in the interstellar medium. In order to interpret the observed intensities of molecular emission lines, it is usually necessary to know the relevant collisional and radiative transition rates. If the lines are optically thin, they do not undergo significant reabsorption within the region emitting the radiation, and the emitted flux is obtained as the line-of-sight integral of the rate of emission per unit volume of gas. However, it is often the case that strong emission lines are optically thick or, at least, have a significant optical depth (i.e. an optical depth of the order of 1) at their centres. Under these circumstances, it is necessary to solve the equation of radiative transfer in order to predict the emitted line fluxes to a reasonable degree of accuracy.

Solving radiative line transfer problems is no mean task. Both analytical and stochastic (Monte-Carlo) approaches are followed, with the latter being more readily applicable when the geometry or the density distribution does not admit simple treatments; this is likely to be always the case of interstellar molecular clouds. Unfortunately, it is also the case that the geometry and the density distribution are generally poorly known or unknown.

Introduction

An important cooling process in interstellar clouds is the excitation of fine structure transitions in abundant atoms and ions, followed by radiative decay. The relevant transitions are those between the fine structure components of ground terms, such as C0 2p2 3P, O0 2p4 3P and C+ 2p 2Po. By term is meant the LS-coupling state denoted by 2S+1L, where L is the total electronic orbital angular momentum quantum number and S is the total electronic spin angular momentum quantum number. Only the outer (valence) electrons (e.g. 2p2) need to be listed, as the inner shells and sub-shells are closed and have zero resultant angular momenta. Spectrosopic notation is used to denote the orbital angular momentum: ‘s’ for l = 0, ‘p’ for l = 1, ‘d’ for l = 2, …, with upper case letters indicating resultant angular momenta (vector sums of the contributions of the individual valence electrons). Departures from LS-coupling, owing to the spin—orbit interaction, result in the states with different values of J, the total electronic angular momentum (the vector sum of the orbital and spin angular momenta) having slightly different energies; these fine structure states are denoted 2S+1LJ. Thus the ground 3P term of C0 and O0 is a triplet, comprising the three fine structure components with J = 0, 1, 2, and the 2Po ground term of C+ is a doublet with J = 1/2, 3/2.

Introduction

The quantum theory of molecular collisions has been extensively developed over the last three decades. As in many branches of theoretical science, the growth of this subject has been closely linked with the advances in computer technology. Powerful numerical techniques have been developed for solving Schrödinger's equation, which are well adapted to low energy, molecular collision problems, at various levels of approximation. A basic reference text in this context is Atomic, Molecular and Optical Physics Handbook [43]. The complexity of the problems that can be tackled, and the accuracy of the results that can be obtained, continue to be determined by the available computing power.

Any proper discussion of molecular collision processes involves the concept of the potential energy curve or surface. This concept drives from the Born—Oppenheimer approximation, to which we first turn.

The Born—Oppenheimer approximation

For the sake of simplicity, when discussing the basic concepts, we consider the collision between a one-electron atom, A, and a fully-stripped ion, B. The theory which pertains to this illustrative case can be generalized to collisions between many-electron atoms or to collisions between molecules.

When studying a collision problem, we are interested in the relative motions of the particles involved, and not in the motion of the centre of mass (barycentre) of the colliding system.

Stated most simply, fluids are ‘things that flow’. This definition distinguishes between liquids and gases (both fluids) and solids, where the atoms are held more or less rigidly in some form of lattice. Of course, it is always possible to think of substances whose status is ambiguous in this regard, such as those, normally regarded as solids, which exhibit ‘creep’ over sufficiently long timescales (glass would fall into this category). Such borderline cases do not undermine the fact that the vast majority of substances can be readily classified as fluid or not. If they are fluids, then it is important to understand the general problem of how they flow, and under what circumstances they attain equilibrium (i.e. do not flow). These issues, in an astronomical context, form the subject of this book.

There is also a more subtle point about the sorts of systems that can be described as fluids. Although fluids are always in practice composed of particles at a microscopic level, the equations of hydrodynamics treat the fluid as a continuous medium with well-defined macroscopic properties (e.g. pressure or density) at each point. Such a description therefore presupposes that we are dealing with such large numbers of particles locally that it is meaningful to average their properties rather than following individual particle trajectories.

The material in this book is based on lecture notes of a course on astrophysical fluid dynamics which has been given for several years to third-year students at the University of Cambridge. There are several excellent books which cover fluid dynamics from a terrestrial standpoint, but very few provide a full introduction to the concepts and methods used to deal with the highly compressible flows which arise in astrophysical contexts. Our aim with this book is to provide just such an introduction, and we hope that it will also serve as a reference volume for advanced undergraduate and graduate students.

Several people have provided input at various stages of the preparation of this book. In particular we thank Jim Pringle, Donald Lynden-Bell and Giuseppe Lodato for their help. We are also grateful to the students who have taken the course at Cambridge for correcting typographical errors in the lecture notes, drawing our attention to parts where the description was less clear than it should have been, and helping us to develop the exercises.