The theory of astrophysical masers can be neatly divided into two problems. One is the transfer of radiation through the medium containing the active (maser generating) molecules, and the other is the molecular physics required to calculate the inversion. Most of the complexity of maser theory arises because these problems are coupled. The molecular physics problem appears to be local: we calculate a population inversion at a position in space, which depends on the physical conditions at that point. Unfortunately, these conditions include the radiation field, which requires a solution to the obviously non-local problem of radiation transfer. In turn, radiation transfer requires a knowledge of the populations of the molecular energy levels at all points along a ray and, therefore, of a solution of the molecular physics problem along the path of the same ray.

Non-LTE physics

For gases which are not in local thermal equilibrium (LTE), we must calculate the populations of energy levels by means other than Boltzmann's formula. We do, however, assume that there are sufficient collisions to maintain a Maxwell–Boltzmann distrubution of molecular speeds. For the moment, we also assume that the molecular populations can be accurately calculated from a set of kinetic master equations – often called ‘rate equations’. These equations simply state that, in a steady state, the net population flow into any energy level from all the others is zero.

At present, there are a number of radio astronomy projects in progress which promise to revolutionize our understanding of astrophysical masers and their environments. Masers form only a small part of the impressive science programmes for these new telescopes, so this chapter contains a considerable amount of general material about the instruments and their capabilities.

EVLA

Introduction to the EVLA

The Expanded Very Large Array (EVLA) is a two-phase development of the existing VLA, situated in New Mexico, USA, and operated by the National Radio Astronomy Observatory. The general capabilities of the EVLA can be found at the website http://www.aoc.nrao.edu/evla/. Phase 1 developments are technological, upgrading the existing array to a wide-band system, whilst Phase 2 involves the construction of new antennas, to provide significantly improved spatial resolution, and improved linkage to the VLBA. The technological developments are similar to those of e-MERLIN (see Section 10.2).

Continuum sensitivity improvements range from a factor of 5 at low frequencies to 20 or more above 10 GHz. This is important for maser, and all spectral line, observations, because the spectral line observations rely on continuum sources for calibration. Spectral line sensitivity increases in Phase 1 are modest below 10 GHz, but up to a factor of 3 at higher frequencies. The EVLA will also move to a wide-band receiver system with fibreoptic cabling and digital electronics, allowing complete frequency coverage from 1 to 50 GHz, with a bandwidth of up to 8 GHz per polarization.

Except in very simple cases, it is not possible to solve the coupled molecular physics and radiation transfer problems for masers, or their pumping radiation, analytically. For more realistic problems, we need to resort to numerical solutions. There are many general-purpose radiative transfer codes available. However, a substantial fraction of these require modification to work in situations that may produce masers: inverted populations yield negative absorption coefficients, optical depths and source functions: situations that will cause many codes to fail.

Large velocity gradient approximation

The large velocity gradient (LVG) or Sobolev approximation (Sobolev, 1957) is a means of casting the radiative transfer problem into an entirely local form. In LVG, the integrations that appear in the formal solution of the radiative transfer equation can be carried out, so the line mean intensity can be expressed explicitly as a function of the energy-level populations from the same transition. Elimination of the mean intensities in favour of the population expressions leads to a set of master equations which are non-linear algebraic equations in the populations. The LVG approximation is therefore not really a numerical method, but a clever approximation that allows much simpler numerical methods to be used than suggested by the original problem.

Theory

We begin by selecting the radiation transfer equation for transport along a ray element ds, Eq. (3.78). We do not, at present, assign any particular geometry to the problem, and one of the greatest advantages of the LVG approximation is that it is almost geometry free.

Masers have been detected in a wide variety of astrophysical environments. Perhaps the most astounding feature is the range of scales: the smallest maser environments are objects familiar to us from our own Solar System – comets and planetary atmospheres – whilst the largest masers form in molecular tori around the nuclei of certain galaxies, and may be up to 1 kpc (∼3 × 1019 m) in size. Some of these environments are so violent that, in a naive view, its is difficult to see how the necessary molecules can survive. However, it is the extreme nature of the environments that aids the pumping of masers. Often, we can deduce that gas molecules have motions characteristic of one temperature (a local kinetic temperature) whilst the radiation which is present is characteristic of a different, and usually higher, temperature. Maser molecules cannot attain a distribution of population amongst their energy levels which represents an equilibrium at either temperature, and these nonlocal-thermodynamic-equilibrium (NLTE) conditions allow population inversions to form.

Galactic star-forming regions

The formation of stars from the gravitational collapse of clouds of interstellar gas remains, in its details, one of the great unsolved problems of astrophysics. Our Galaxy, the Milky Way, is a spiral type, which is still forming stars at a significant rate at the current epoch; not all galaxies do. Elliptical and lenticular galaxies have very little interstellar gas compared with spirals, and are forming very few new stars. Within the spiral category, ‘early’ types (Sa, SBa), with large nuclei and tightly wound arms, are comparatively gas-poor compared with ‘late’ types (Sc, SBc), with relatively smaller nuclei and more open spiral arms.

Noise power

The theory of the noise voltage generated by a resistor was developed by Nyquist (1928), following experimental measurements by Johnson (1928). The phenomenon is therefore named after either, or both, of these researchers.

Suppose we have a long coaxial transmission line at temperature T. This line acts as a 1-D cavity for the propagation of electromagnetic waves at a velocity, v, where we will assume v ≃ c. We can then follow the analysis of Section 1.3.4 to obtain the number of available modes. If the transmission line is laid out along the z-axis, the electric and magnetic fields are restricted to the xy-plane, and boundary conditions require the electric field to be zero at the ends of the line, where z = 0 and z = L. Allowed modes along the transmission line are therefore restricted by a z-version of Eq. (1.33) to kz = πmz/L, for integer mz, and the 1-D nature of the problem implies that mz is the only such integer required to define a mode. There are still, however, two independent polarizations allowed (along the x- and y-axes), so we modify the above restriction on modes to k = 2πm/L, where we have dropped the z-subscript.

When we make a set of measurements, it is instinct to try to correlate the observations with other results. One or more motives may be involved in this instinct. For instance we might wish (a) to check that other observers' measurements are reasonable, (b) to check that our measurements are reasonable, (c) to test a hypothesis, perhaps one for which the observations were explicitly made, or (d) in the absence of any hypothesis, any knowledge or anything better to do with the data, to find if they are correlated with other results in the hope of discovering some new and universal truth.

The fishing trip

Take the last point first. Suppose that we have plotted something against something, on a fishing expedition of this type. There are grave dangers on this expedition, and we must ask ourselves the following questions.

1. Does the eye see much correlation? If not, calculation of a formal correlation statistic is probably a waste of time.

2. Could the apparent correlation be due to selection effects? Consider, for instance, the beautiful correlation in Figure 4.1, in which Sandage (1972) plotted radio luminosities of sources in the 3CR catalogue as a function of distance modulus. […]

The distribution of objects on the celestial sphere, or on an imaged patch of this sphere, has ever been a major preoccupation of astronomers. Avoiding here the science of image processing, province of thousands of books and papers, we consider some of the common statistical approaches used to quantify sky distributions in order to permit contact with theory. Before we turn to the adopted statistical weaponry of galaxy distribution, we discuss some general statistics applicable to the spherical surface.

Statistics on a spherical surface

The distribution of objects on the celestial sphere is the distribution of directions of a set of unit vectors. Many other 3D spaces face similar issues of distribution, such as the Poincaré sphere with unit vectors indicating the state of polarization of radiation. Geophysical topics (orientation of paeleomagnetism, for instance) motivate much analysis.

Thus, this is a thriving sub-field of statistics and there is an excellent handbook (Fisher et al., 1987). The emphasis is on statistical modelling and a variety of distributions is available.

Whether He does or not, the concepts of probability are important in astronomy for two reasons.

1. Astronomical measurements are subject to random measurement error, perhaps more so than most physical sciences because of our inability to re-run experiments and our perpetual wish to observe at the extreme limit of instrumental capability. We have to express these errors as precisely and usefully as we can. Thus, when we say ‘an interval of 10-6 units, centred on the measured mass of the Moon, has a 95 per cent chance of containing the true value’, it is a much more quantitative statement than ‘the mass of the Moon is 1 ± 10-6 units’. The second statement really only means anything because of some unspoken assumption about the distribution of errors. Knowing the error distribution allows us to assign a probability, or measure of confidence, to the answer.

2. The inability to do experiments on our subject matter leads us to draw conclusions by contrasting properties of controlled samples. These samples are often small and subject to uncertainty in the same way that a Gallup poll is subject to ‘sampling error’. In astronomy we draw conclusions such as: ‘the distributions of luminosity in X-ray-selected Type I and Type II objects differ at the 95 per cent level of significance.’ Very often the strength of this conclusion is dominated by the number of objects in the sample and is virtually unaffected by observational error.

The study of astrophysical masers is a very young branch of science, with a history extending back no further than the mid 1960s. Even so, the subject has advanced rapidly to the point where masers can be used as tools to investigate problems as diverse as the chemistry of comets and the measurement of the Hubble constant. Arecent (2007) international conference on astrophysical masers had over 120 delegates: hopefully this shows that the subject is as attractive to young astronomers today as it was to the pioneers who first detected these incredibly bright, and at the time mysterious, radio sources just 45 years ago.

The observational side of astrophysical maser research has always been a branch of radio astronomy, and developments in radio techniques continue to govern advances in the knowledge of masers. The inclusion of Chapter 4 is intended to provide the reader with sufficient knowledge of radio techniques to understand modern observing procedures. As most astrophysical masers originate from molecules, there is also a chapter (Chapter 5) devoted to molecular spectroscopy.

Intended readership

The book is aimed at senior undergraduates, postgraduate students and research workers in astronomy, but the first two chapters can be easily read by the non-specialist, as they contain little mathematics and technical detail. The same is true of Chapter 6, which takes a modern view of the main astrophysical environments in which masers form. Chapters 9 and 10, though more specialized, are also accessible to the reader who does not wish to delve into too much mathematical detail.

Of the vast literature, we point to some works which we have found useful, enlightening or just plain entertaining. We bin these into six types (somewhat arbitrarily as there is much overlap): popular, the basic text, the rigorous text, the data analysis manual, the texts considering statistical packages, and the statistics treatments of specialist interest to astronomers.

1. The classic popular books have legendary titles: How to Lie with Statistics (Huff, 1973), Facts from Figures (Moroney, 1965), Statistics in Action (Sprent, 1977) and Statistics without Tears (Rowntree, 1981). They are all fun. To this list we can now add The Lady Tasting Tea: How Statistics Revolutionized Science in the Twentieth Century (Salsburg, 2002), an entertaining exposition of the development of modern statistics; Struck by Lightning: the Curious World of Probabilities (Rosenthal, 2006); Making Sense of Statistics: A Non-mathematical Approach (Wood, 2003), and Dicing with Death: Chance, Risk and Health (Senn, 2003). This latter is a devastatingly blunt, funny and erudite exposition of the importance and application of statistics in decision processes which may affect the lives of millions. As a popular book it is heavy-going in parts; but for scientists, budding or mature, it is a rewarding read.

2. Textbooks come in types (a) and (b), both of which cover similar material for the first two-thirds of each book. They start with descriptive or summarizing statistics (mean, standard deviation), the distributions of these statistics, and move to the concept of probability and hence statistical inference and hypothesis testing, including correlation of two variables. […]

Masers and lasers

The words ‘maser’ and ‘laser’ were originally acronyms: MASER standing for microwave amplification by stimulated emission of radiation, and LASER for the very similar phrase with ‘light’ substituted for ‘microwave’. The important point is that masers and lasers are both derived from the stimulated emission process, and the only difference between them is a rather arbitrary distinction, based on the frequency of radiation they emit. Masers, as laboratory instruments, in fact pre-dated lasers by several years, and both had been completed as practical instruments before the discovery of astrophysical maser sources.

Although this book is about masers, most people are probably more familiar with lasers, so keeping in mind that the two things are very similar, we will begin by considering lasers. Most people probably own several lasers: lasers are used to interpret the information stored on CD and DVD discs; they are also used in many computer printers. Even if they have only a vague idea about how they work, and view lasers as some sort of ‘black box’, tube, or chip that emits light, most people will probably be aware that this light is in some way ‘special’ – that is, it has properties that make it different from the light emitted by, say, a filament electric light bulb. What are these important characteristics? Given time to ponder on this question, most people would proably come up with a list something like this to summarize the important properties of laser light:

• The laser light is extremely bright, or intense.

• The beam is tightly focussed. […]