Cygnus A (Figure 13.1) is one of the strongest radio sources in the sky. It was discovered by Hey in his early survey of the radio sky (Hey et al. 1946a, 1946b); its trace can even be discerned on Reber's 1944 map (see Appendix 3 for a brief history). It is, however, an inconspicuous object optically, and it was not identified until its position was known to an accuracy of 1 arcmin (Smith 1951; Baade and Minkowski 1954). Its optical counterpart was found to be an eighteenth-magnitude galaxy with a recession velocity of 17 000 km s-1, that is, with redshift z= 0.06, indicating a distance of almost 1000 million light years. The source was shown to be double, with an overall size of more than a minute of arc, through the pioneering interferometry observations of Jennison and das Gupta (1953). Furthermore, the large angular size and double-lobed shape of Cygnus A were such distinctive features that similar radio sources might well be recognizable at very much greater distances; this was confirmed in 1960 when the radio source 3C 295 was identified by Minkowski with another similar galaxy, this time with a redshift of 0.45. Another galaxy, NGC 5128, had already been identified as a radio source known as Centaurus A (Bolton et al.1949); this again showed the characteristic double-lobed shape, but with an angular diameter of 4° it was obviously much closer.

In astrophysical contexts, the propagation of radio waves is governed, as for other parts of the electromagnetic spectrum, by the laws of radiative transfer and refraction. In radio astronomy, however, there is an emphasis on classical (non-quantized) radiative and refractive processes. Synchrotron radiation is the dominant radiation process at the longer wavelengths; spectral-line emission is observed mainly at shorter wavelengths. Maser action, the microwave equivalent of lasers, is encountered in several astrophysical contexts: this is due to the low energy of radio photons which can be significantly amplified by small population inversions in rotational and vibrational energy levels. Refraction is important in astrophysical plasmas; even though these are usually electrically neutral, protons have a negligible effect and the electron gas can have a significant effect on the velocity of radio waves. In the presence of a magnetic field, birefringence can lead to Faraday rotation of the plane of polarization.

In this chapter we set out the basic theories of radiative transfer, and outline the processes of radiation that are of particular importance in radio astronomy: free-free emission, line emission (and particularly maser emission) in dilute gas and synchrotron radiation. Free- free emission, or bremsstrahlung, is the main source in ionized hydrogen clouds, whereas synchrotron radiation is responsible for the background radiation in our Galaxy (Chapter 8) and is also practically universal in discrete radio sources from supernova remnants to quasars.

Objects satisfying universal mapping properties are in a sense trivial if you look at them from one side, but not trivial if you look at them from the other side. For example, maps from an object to the terminal object 1 are trivial; but if, after establishing that 1 is a terminal object, one counts the maps whose domain is 1, 1 → X, the answer gives us valuable information about X. A similar remark is valid about products. Mapping into a product B1 × B2 is trivial in the sense that the maps X → B1 × B2 are precisely determined by the pairs of maps X → B1, X → B2 which we could study without having the product. However, specifying a map B1 × B2 → X usually cannot be reduced to anything happening on B1 and B2 separately, since each of its values results from a specific ‘interaction’ of the two factors.

Binary operations and actions

In this session we will study two important cases of mapping a product to an object. The first case is that in which the three objects are the same, i.e. maps B × B → B. Such a map is called a binary operation on the object B. The word ‘binary’ in this definition refers to the fact that an input of the map consists of two elements of B.

Much mathematical struggle (e.g. ‘solving equations’) aims to partially invert a given transformation. In particular, in the case of a functor Φ from one category to another, Kan (1958) noticed that there is sometimes a uniquely determined functor in the opposite direction that, while not actually inverting Φ, is the ‘best approximate inverse’ in either a left- or a right-handed sense. The given functor is typically so obvious that one might not have mentioned it, whereas its resulting adjoint functor is a construction bristling with content that moves mathematics forward.

The uniqueness theorem for adjoints permits taking chosen cases of their existence as axioms. This unification guides the advance of homotopy theory, homological algebra and axiomatic set theory, as well as logic, informatics, and dynamics.

Roughly speaking these reverse functors may adjoin more action, as in the free iteration Φ!(X) of initial data on X, or the chaotic observation Φ*(X) of quantities in X, where Φ strips some of the ‘activity’ from objects Y in its domain. When Φ is instead the full inclusion of constant attributes into variable ones, then the reverse functors effect an ‘averaging’, as in the existential quantification Φ!(X) and the universal quantification Φ*(X) of a predicate X. Exponentiation of spaces satisfies the exponential law that Φ*()L is right adjoint to Φ=()×L. Similarly, implication L ⇒ () of predicates is right adjoint to ()&L.