The last chapter dealt with the interaction of radiation with matter, mostly in the gaseous, but also in the liquid and solid phases. Absorption coefficients of infrared active gases, emission and scattering properties of surfaces, and single scattering albedos and phase functions of aerosols were considered. Applications of these concepts, along with the principles of radiative transfer discussed in Chapter 2, enable us to calculate the emerging radiation field of a planet or satellite.

In this chapter we examine how the physical state of an emitting medium gives rise to the general spectral characteristics of the outgoing radiation field. We begin in Section 4.1 by considering the behavior of a single spectral line in an isothermal atmosphere, both with and without scattering. In Section 4.2 we introduce nonscattering atmospheric models having a more complicated thermal structure, but still consider only a single line. Finally, in Section 4.3 we conclude our investigation of nonscattering models using realistic molecular parameters. Our aim in this chapter is to illustrate the principles behind the analysis of remotely sensed data, especially with regard to how scattering, atmospheric abundances, and thermal structures affect the appearance of the observed spectrum. Later, in Chapter 6, we apply these principles to the descriptions of real planetary spectra.

Models with one isothermal layer

Without scattering

The first model considered consists of a nonscattering gas layer at constant pressure and temperature adjacent to a solid surface of unit emissivity. This model is illustrated in Fig. 4.1.1.

So far we have avoided detailed discussion about two physical phenomena that are crucial in the context of this book and for any understanding of the interaction between matter and radiation in general. These two phenomena are absorption and dispersion, that is, the removal of energy from the electromagnetic field by matter and the dephasing of the electric field components as light streams through the medium. Although we have barely mentioned the existence of these effects, we shall need a deeper insight into both of them. We shall see that retardance, birefringence, and absorption properties of polarization systems, assumed in the preceding sections, are based on these phenomena, whose wavelength dependence is understood in terms of the wavelength dependence of the dielectric permittivity and, hence, of the refractive index of the medium. By studying absorption and dispersion we are producing the necessary bricks with which to build a theory of radiative energy transfer which will be discussed in following chapters. We shall continue to assume unit isotropic magnetic permeability of μ = 1 for the medium.

Certainly, a full account of absorption and dispersion processes can be carried out only within the framework of quantum mechanics.

Spectropolarimetry, as the name suggests, is the measurement of light that has been analyzed both spectroscopically and polarimetrically. In other words, both the wavelength distribution of energy and the vector properties of electromagnetic radiation are measured with the highest possible resolution and accuracy. Thus, spectropolarimetry embraces a number of techniques used in order to characterize light in the most exhaustive way. Such techniques are ultimately based on a theory that, from its beginnings in the closing years of the nineteenth century, finally grew to maturity in the 1990s. Therefore, under the heading of spectropolarimetry we will find several disciplines, which, despite being interrelated or rather, although our aim is to stress their interrelatedness, may be considered independent.

A historical perspective is always helpful for grasping the importance of physical phenomena and their corresponding explanations. The main objective of this chapter is to give a brief description of the salient events and findings in history related to some of the independent disciplines covered in this book.

Various physical processes modify a radiation field as it propagates through an atmosphere. The rate at which the atmosphere emits depends on its composition and thermal structure, while its absorption and scattering properties are defined by the prevailing molecular opacity and cloud structure.

Independently of whether the radiation field is generated internally or is imposed externally, the study of how it interacts with the atmosphere is embodied in the theory of radiative transfer. Many authors have dealt with this theory in various contexts. Monographs include those by Kourganoff (1952), Woolley & Stibbs (1953), Goody (1964), and Goody & Yung (1989). A standard text is by Chandrasekhar (1950), which treats the subject as a branch of mathematical physics. The emphasis is on scattered sunlight in planetary atmospheres and on various problems of astrophysical interest.

Our own approach is somewhat different and emphasizes spectra produced by thermal emission from planetary atmospheres, especially as observed from space platforms. In order to demonstrate the connection between the thermal radiation giving rise to these spectra and the physical state of the atmosphere under consideration, it is necessary to examine how the transport of this radiation is effected. Only then is it possible to have a clear understanding of how the structure of an atmosphere leads to its spectral appearance, a topic considered at length in Chapter 4.

All planets from Mercury to Neptune and most of their satellites have been observed from Earth-based telescopes and at least once, some repeatedly, from spacecraft. Therefore, sufficient information was available to emphasize the physical principles in the discussions in Chapter 6. Trans-Neptunian objects and asteroids have been explored to a much lesser degree. Their small sizes, for many their large heliocentric distances, and their low surface temperatures prevented detailed exploration. Until recently, only a few samples of an enormous amount of objects have been investigated. Therefore, the treatment of these objects, grouped in this chapter, is primarily a summary of presently known properties. Section 7.1 discusses Pluto and its satellite Charon; Section 7.2 is devoted to comets; and Section 7.3 to asteroids.

Pluto and Charon

In 1930, Tombaugh discovered Pluto, the outermost known planet (Reaves, 1997; Marcialis, 1997). Several authors have derived the radius of Pluto with very small uncertainties; unfortunately, the derived values do not overlap. Consequently, only a broad range can be quoted (1145 to 1200 km) within which the true radius of Pluto may fall (Tholen & Buie, 1997). Pluto is by far the smallest planet of our Solar System; it is even smaller than many planetary satellites. Pluto's orbit is highly eccentric and inclined by more than 17° to the ecliptic plane (Malhotra & Williams, 1997). At perihelion (29.7 AU), Pluto is closer to the Sun than Neptune (30.1 AU), and at aphelion it reaches a heliocentric distance of almost 50 AU.

So far, the polarization properties of the simplest conceivable electromagnetic radiation have been described. However, building a polarization theory that is useful in the real world necessarily requires the consideration of light whose spectrum contains a continuous distribution of monochromatic plane waves within a finite width of frequencies. Heisenberg's uncertainty principle implies infinite time intervals for detecting purely monochromatic light (in other words, we can simply say that monochromatic light does not exist in reality). In this section we shall see that the concept of polarization is also applicable to polychromatic light. As a matter of fact, polychromatic light may share the properties of totally polarized radiation and hence be indistinguishable from monochromatic light in so far as polarimetric measurements are concerned. The coherency matrix and the Stokes parameters can also be defined for a polychromatic light beam, although the binding conditions (2.18) for C and (2.22) for I, Q, U, and V will be slightly modified and the new concepts of partial polarization and degree of polarization will naturally come into play.

Polychromatic light as a statistical superposition of monochromatic light

Under the hypotheses of linearity, stationarity, and continuity, one can assume any polychromatic light beam to be the superposition of monochromatic, time-harmonic plane waves of different frequencies within an interval of width Δv around a central frequency v0.

In Chapter 4 we constructed examples of planetary spectra by applying solutions of the radiative transfer equation to model atmospheres of assumed composition and temperature structure. Before we can compare the results of such calculations with measured spectra we have to understand the modifications the emerging radiation field experiences in the recording process performed by radiometric instruments. A full comprehension of the detailed functioning of instruments is also necessary for the planning and the design of remote sensing investigations. Therefore, in Chapter 5 we discuss the principles of infrared instrumentation. We concentrate on instruments for space use, but the physical principles are equally applicable to ground-based astronomical sensors. On several occasions we refer to such Earth-based devices. It is neither possible nor useful to mention all infrared instruments ever flown in space or ever used for planetary work with ground-based telescopes. Instead, we analyze the physical concepts of different design approaches. To illustrate these concepts we occasionally show diagrams of specific instruments as well as samples of results obtained with them.

Radiometric devices have certain common characteristics. For example, most radiometric instruments contain optical elements to channel planetary radiation onto a detector. Telescopes are often essential parts of these designs. Following a brief introduction in Section 5.1, the subject of telescopes is discussed in Section 5.2. In the process of imaging a planetary surface element onto the detector, fundamental limits in spatial resolution are encountered. These limits, set by diffraction, are discussed in Section 5.3.

So far we have been dealing with the propagation of light through media whose refractive indices have been assumed to be constant with position (the assumption of homogeneity). We are now able to embark on the study of the propagation of light through media whose refractive indices – and hence absorptive and dispersive properties – may vary along the ray path; a differential treatment is then in order. More specifically, we shall deal with stratified media whose material properties are constant in planes perpendicular to a given direction. Moreover, our study will not only include passive systems but emission properties of the medium will also be considered (although in the most simplified way).

There are three main hypotheses we should add to proceed with the development that follows:

1. We shall assume that the absorptive, dispersive, and emissive properties of the medium are independent of the light-beam Stokes vector. This is in fact a linear approximation that holds in many astrophysical applications, where, even though the medium may be dependent on the whole radiation field, the angular width of the beam (indeed within the realm of geometrical optics) is so small that its contribution to the physical conditions of the medium can be neglected (e.g., Landi Degl' Innocenti and Landi Degl' Innocenti 1981).

2. […]

The preceding chapter demonstrates how the basic thermal, compositional, and cloud structures of planetary atmospheres can be inferred from infrared measurements. Some information on surface properties is also available. So far, however, there has been no discussion of how underlying physical processes cause these structures to develop and evolve. That is the purpose of this chapter.

We divide the discussion into four topics. In Section 9.1 we are concerned with the one-dimensional thermal equilibrium configuration of an atmosphere in the absence of internal motion. In Section 9.2 we expand the temperature field to three dimensions and investigate the dynamical properties of atmospheres. In Section 9.3 we address the question of how determinations of chemical composition imply the evolution of planets and the Solar System as a whole. Finally, in Section 9.4 we review measurements of the excess heat emitted by the planets, and discuss the importance of these measurements for determining the status of planetary evolution in the present epoch.

Radiative equilibrium

The absorption of solar radiation leads to heating within the atmosphere, while cooling is achieved by the emission of infrared radiation. Thermal gradients are established, and the magnitudes and directions of these gradients, coupled with the forces of gravity and planetary rotation, give rise to imbalances in local pressure fields that lead to atmospheric motions. These internal motions are responsible for additional energy transport, and it is the balance of the dynamical and radiative heating and cooling rates that determines the ultimate thermal structure of the atmosphere.

With the radiative transfer equation for polarized light to hand, we shall proceed to find solutions and to exploit them both, the equation and its solutions, in order to obtain information about the medium. This chapter is devoted to solutions of the RTE and to the first and simplest diagnostics one can obtain from the observed Stokes profiles. The main emphasis is on concepts rather than numerical details. The latter may be found in the literature (some of the most recent papers are recommended in the bibliography) and in fact are still in continuous evolution and debate. Most of the concepts we describe in this chapter, however, may be said to be well founded nowadays and will help the reader in understanding the topic.

This chapter is devoted to recalling a number of results of importance for development in later chapters. Most of these concepts are assumed to be already known to the reader, and those derivations that are missing will be found in textbooks on optics and electromagnetism. The main aim here is to provide a summary of the polarization properties of the simplest electromagnetic wave one can conceive: the monochromatic, time-harmonic, plane wave.

The terms light and electromagnetic wave will be understood as synonymous throughout the text. More specifically, we will be referring to the visible part of the spectrum and its two nearest neighbors, the ultraviolet and the infrared. Many of the topics discussed are also applicable to other wavelength regions. In particular, it is worth noting that radio observations use most of the concepts we shall be developing here for the optical region, although they are not in principle necessary for that wavelength range.

Polarimetric accuracy is one of the most important goals of modern astronomy. The definition itself of polarimetric accuracy, however, is difficult since we mostly measure polarization differences and are uncertain in establishing the zero level, which is often set by convention. Hence, by “accuracy” we shall understand the sensitivity to variations of the polarization level. Besides the greatest polarimetric accuracy, every astronomical observation should ideally pursue the highest spectral, spatial, and temporal resolution with the widest spatial and spectral coverage. However, all these goals are hard to accomplish at the same time and one always needs to compromise depending on the specific objectives a given observation is aimed at. The amount of available photons from the Sun is never sufficient. In fact, it is equal per resolution element to that from a scarcely resolved star of the same effective temperature. This observational fact is easy to understand (e.g., Mihalas, 1978) if one takes into account the invariance with distance of the specific intensity (energy per unit normal surface, per unit time, per unit frequency interval, per unit solid angle) and its proportionality to the photon distribution function (number of photons per unit volume, per unit frequency interval, per unit solid angle).

Solar polarimetry is, of course, a part of the game and has several limiting factors that govern the final accuracy of the measurements.

The discussions of the equation of transfer and the solution of this equation in Chapter 2 rest entirely on concepts of classical physics. Such treatment was possible because we considered a large number of photons interacting with a volume element that, although it was assumed to be small, was still of sufficient size to contain a large number of individual molecules. But with the assumption of many photons acting on many molecules we have only postponed the need to introduce quantum theory. Single photons do interact with individual atoms and molecules. The optical depth, τ(υ), depends on the absorption coefficients of the matter present, which must fully reflect quantum mechanical concepts. The role of quantum physics in the derivation of the Planck function has already been discussed in Section 1.7. Both the optical depth and the Planck function appear in the radiative transfer equation (2.1.47).

The interaction of radiation with matter can take many forms. The photoelectric effect, the Compton effect, and pair generation–annihilation are processes that occur at wavelengths shorter than those encountered in the infrared. Infrared photons can excite rotational and vibrational modes of molecules, but they are insufficiently energetic to excite electronic transitions in atoms, which occur mostly in the visible and ultraviolet. Therefore, a discussion of the interaction of infrared radiation with matter in the gaseous phase needs to consider only rotational and vibrational transitions, while in the solid phase lattice vibrations in crystals must be included.