The first detections of molecular emissions from species such as CO and HCN in external but relatively nearby galaxies were made in the 1970s. From the 1990s, detections were made of CO emission from high-redshift objects, culminating in the remarkable identification in 2003 of high-excitation CO in a gravitationally lensed quasar at a redshift of 6.4. At that time, this was the most distant quasar known. In standard cosmology a redshift of 6.4 represents material when the Universe was merely a few percent of its present age. The discovery of molecular emission in such a distant object demonstrated that chemistry was occurring very early in the evolution of the Universe, and its products – molecules – must therefore also be widespread. In more recent years, it has been firmly established that chemistry in external galaxies can be complex and well developed. So the question arises: Can we use molecular emissions from distant galaxies to explore the physical conditions in them and their likely evolutionary status, as we can do for various regions of the Milky Way (cf. Chapter 5)?

There are a couple of points to be borne in mind, before simply applying the ideas in Chapter 5 to external galaxies. The most important one is that – apart from the nearest objects – most galaxies will be spatially unresolved; the telescope beam will usually encompass the entire galaxy being observed, so that emissions from many types of region are compounded. However, the detection of, say, CO, SiO, and CH3OH emissions in a spatially unresolved galaxy does not mean that they occur in the same region of that galaxy: the first molecule may indicate the presence of cold tenuous clouds, the second strong shocks, and the third dense star-forming cores. External galaxies will, in general, contain the variety of regions and sources similar to those that we can identify in the Milky Way.

In this chapter we summarise briefly the notation of molecular spectroscopy, with examples of transitions used to identify molecular species in the interstellar medium. We also describe how radiation is transported in the interstellar medium, introducing ideas that will be needed in Chapters 8 and 9. Finally, we discuss the processes that determine level populations of molecules in the interstellar medium.

Molecular Spectroscopy

Whereas most atomic spectra are determined simply by transitions between individual electronic states, molecular spectra are more complex because molecules have additional degrees of freedom associated with vibration and rotation. Each electronic state of a molecule possesses a manifold of vibrational levels, and each of those vibrational levels has a ladder of rotational levels associated with it. Electronic transitions in molecules therefore occur between specific vibrational and specific rotational levels in each electronic state. The equivalent of a single atomic line corresponding to an electronic transition is – for a molecule – replaced by a set of many lines (see Figure 1.2 for part of the H2 electronic transition B–X, between the two lowest electronic states, showing the vibrational and rotational structure). These electronic transitions often lie in the ultraviolet spectrum.

Molecules also possess transitions that have no counterpart in atoms. Molecules may undergo transitions between specific rotational states of vibrational states, that is, ro-vibrational transitions. These transitions usually occur in the near-infrared range. Pure rotational transitions may also occur between rotational states of the same vibrational level. These tend to lie in the far-infrared, or in the millimetre to submillimetre range of the electromagnetic spectrum.

In Chapter 3 we saw that the chemistry producing molecular tracers in interstellar space is driven in a variety of ways: by electromagnetic radiation at UV and X-ray wavelengths; by ionisation caused by cosmic ray particles; by reactions on grains or in their icy mantles; and in chemistry induced by gas dynamics. In many types of region, some or all of these processes may act together; in others, one of them may dominate. The atomic and molecular tracers produced reflect the nature of the chemical drivers that are operating.

However, there is another complication. Each chemical driver does not always operate at the same rate; a driver may vary from place to place in the interstellar medium of a galaxy, or from galaxy to galaxy, or even from time to time. For example, the local UV radiation field in a galaxy with a transient high star-formation rate may be much more intense than the value corresponding to the mean intensity in the Milky Way Galaxy. Similarly, cosmic rays are accelerated by magnetohydrodynamical events and so locations of high dynamical activity may have much greater fluxes of cosmic rays than in more quiescent regions. Also, chemistry on dust grains is obviously affected by the dust:gas ratio and by the nature of the surfaces of the dust; both of these may vary from place to place within the Milky Way Galaxy, and from galaxy to galaxy. Finally, gas dynamical events causing shocks or turbulent mixing can be an important driver of the chemistry.

It is clear that molecular line emissions can yield important information about the physical conditions of the gas and dust in our own as well as in external galaxies. What is not so clear is how to transform observational results into such physically meaningful information. This chapter aims at providing some simple recipes to aid the observer in achieving this goal. In this chapter we describe the relationships between the observable quantities in submillimetre molecular astronomy and the physical information that the observer would like to obtain. Inevitably, several approximations are made in order to derive such information, depending on the spectral and spatial resolution available, as well as the number of observable molecular transitions of any one species.

We start by relating what we measure with submillimetre and radio telescopes, that is, the antenna temperature, Ta, to the fundamental molecular constants and the relevant astronomical parameters. What we would like to know are the column densities of the observed species and the gas temperatures and number densities of the local gas. We shall describe first how to obtain these quantities if the gas is in Local Thermal Equilibrium (LTE), that is, where the level populations are dominated by collisions in a gas at a uniform temperature. The methods of obtaining this information depend on the types of molecule involved. We conclude this chapter by discussing the conversion of column densities into fractional abundances compared to the total hydrogen abundance, quantities that are directly comparable to the predictions of chemical models. In Chapter 9 we describe some approximations routinely used when LTE does not apply.

What Drives Cosmic Chemistry?

It is a relatively straightforward matter to use freely available computer codes and lists of chemical reactions to compute abundances of molecular species for many types of interstellar or circumstellar region. For example, the UDfA, Ohio, and KIDA websites (see Chapter 9) provide lists of relevant chemical reactions and reaction rate data. Codes to integrate time-dependent chemical rate equations incorporating these data are widely available and provide as outputs the chemical abundances as functions of time. For many circumstances, the codes are fast, and the reaction rate data (from laboratory experiments and from theory) have been assessed for accuracy. The required input data define the relevant physical conditions for the region to be investigated.

These codes and databases are immensely useful achievements that are based on decades of research. However, the results from this approach do not readily provide the insight that addresses some of the questions we posed in Chapter 1: What are the useful molecular tracers for observers to use, and how do these tracers respond to changes in the ‘drivers’ of the chemistry? Observers do not need to understand all the details of the chemical networks (which may contain thousands of reactions), but it is important to appreciate how the choice of the tracer molecule may be guided by, and depend on, the physical conditions in the regions they wish to study.

Why Are Molecules Important in Astronomy?

Molecules pervade the cooler, denser parts of the Universe. As a useful rule of thumb, cosmic gases at temperatures of less than a few thousand K and with number densities greater than one hydrogen atom per cm3 are likely to contain some molecules; even the Sun's atmosphere is very slightly molecular in sunspots (where the temperature – at about 3200 K – is lower than the average surface temperature). However, if the gas kinetic temperatures are much lower, say about 100 K or less, and gas number densities much higher, say more than about 1000 hydrogen atoms per cm3, the gas will usually be almost entirely molecular. The Giant Molecular Clouds (GMCs) in the Milky Way and in other spiral galaxies are clear examples of regions that are almost entirely molecular. The denser, cooler components of cosmic gas, such as the GMCs in the Milky Way Galaxy, contain a significant fraction of the nonstellar baryonic matter in the Galaxy. Counterparts of the GMCs in the Milky Way are found in nearby spiral galaxies (see Figure 1.1). Although molecular regions are generally relatively small in volume compared to hot gas in structures such as galactic jets or extended regions of very hot X-ray–emitting gas in interstellar space, their much higher density offsets that disparity, and so compact dense objects may be more massive than large tenuous regions.

This book is about Lagrangians and Hamiltonians. To state it more formally, this book is about the variational approach to analytical mechanics. You may not have been exposed to the calculus of variations, or may have forgotten what you once knew about it, so I am not assuming that you know what I mean by, “the variational approach to analytical mechanics.” But I think that by the time you have worked through the first two chapters, you will have a good grasp of the concept.

We being with a review of introductory concepts and an overview of background material. Some of the concepts presented in this chapter will be familiar from your introductory and intermediate mechanics courses. However, you will also encounter several new concepts that will be useful in developing an understanding of advanced analytical mechanics.

Kinematics

A particle is a material body having mass but no spatial extent. Geometrically, it is a point. The position of a particle is usually specified by the vector r from the origin of a coordinate system to the particle. We can assume the coordinate system is inertial and for the sake of familiarity you may suppose the coordinate system is Cartesian. See Figure 1.1.

Introduction

The calculus of variations is a branch of mathematics which considers extremal problems; it yields techniques for determining when a particular definite integral will be a maximum or a minimum (or, more generally, the conditions for the integral to be “stationary”). The calculus of variations answers questions such as the following.

1. • What is the path that gives the shortest distance between two points in a plane? (A straight line.)

2. • What is the path that gives the shortest distance between two points on a sphere? (A geodesic or “great circle.”)

3. • What is the shape of the curve of given length that encloses the greatest area? (A circle.)

4. • What is the shape of the region of space that encloses the greatest volume for a given surface area? (A sphere.)

The technique of the calculus of variations is to formulate the problem in terms of a definite integral, then to determine the conditions under which the integral will be maximized (or minimized). For example, consider two points (P1 and P2)inthe x–y plane. These can be connected by an infinite number of paths, each described by a function of the form y = y(x). Suppose we wanted to determine the equation y = y(x) for the curve giving the shortest path between P1 and P2.

The purpose of this book is to give the student of physics a basic overview of Lagrangians and Hamiltonians. We will focus on what are called variational techniques in mechanics. The material discussed here includes only topics directly related to the Lagrangian and Hamiltonian techniques. It is not a traditional graduate mechanics text and does not include many topics covered in texts such as those by Goldstein, Fetter and Walecka, or Landau and Lifshitz. To help you to understand the material, I have included a large number of easy exercises and a smaller number of difficult problems. Some of the exercises border on the trivial, and are included only to help you to focus on an equation or a concept. If you work through the exercises, you will better prepared to solve the difficult problems. I have also included a number of worked examples. You may find it helpful to go through them carefully, step by step.