Almost every issue of the leading astronomical journals includes some polarimetry, either directly or indirectly. Polarimetry as a working tool has clearly come of age. Optical and radio techniques are most advanced, but infrared, sub-millimetre and ultraviolet are following on rapidly, while X-ray techniques are being developed also. There is no technical reason why astronomers should not use polarimetry when it suits their astronomical purposes; polarimetry often yields information that other methods of observation cannot give, and this is the main reason why all astronomers, and today's students in particular, should understand the basic ideas behind polarimetry.

Within the astronomical context, the degree of polarization is often low; a few per cent is typical, though both higher and (much) lower values occur. A polarimetric measurement is basically that of the ratio of the small difference between two signals to their sum. Difference and ratio methods have been devised to measure this small difference without systematic bias or drift errors, but photometric noise (detector noise or photon noise of the signal itself) is always present. To reduce this noise to the low level required for sufficiently accurate polarimetry, considerable observing time on a large telescope is generally needed. Polarimetry should therefore not be used indiscriminately, but only when it provides insight which other methods cannot give. Such judgment also requires a grasp of polarimetric basics.

This book aims to create an awareness of what polarimetry can do and at what price (in observing time, in complexity of equipment and of procedures).

In het land der blinden is Eénoog koning. This saying in my mother tongue contains a sufficient number of Germanic roots for English speakers to guess that the situation depicted is only marginally better than ‘the blind leading the blind’. It aptly describes the current situation in astronomical polarimetry and provides the justification for my attempt to write a primer for students and other polarimetric novices. If we can take today's students straight from polarimetric fundamentals to what is best in modern research practice, then five years from now we shall have a polarization community with both eyes wide open and firmly fixed beyond present-day horizons. That is what this book is about.

Polarimetry, performed mainly by optical or radio specialists, has already made a considerable impact on astronomy, and it deserves to be a standard observational technique, to be used whenever it is best for the job in hand. Accordingly, all astronomers should acquire polarimetric basics. My aim is to allow the reader, starting at first principles, to make use of the very latest literature. To preserve readibility, I have omitted most of the historical development. The References section at the end of the book reflects this attitude; interested readers can always trace the history backwards from modern papers.

I have tried to resist any tendency to write a comprehensive monograph.

In this chapter, the main concepts of polarized radiation will be introduced and discussed. These concepts apply at all wavelengths. Electromagnetic radiation will be treated as a continuous travelling-wave phenomenon. Quantum considerations can be postponed until the moment the radiation strikes a detector and is converted into an electrical signal. Ideal detectors are not sensitive to polarization, and, to the extent that a real-life detector can be seen as an ideal one preceded by polarization optics, quantum and polarization considerations can live side by side without the one influencing the arguments concerning the other. Of the electromagnetic wave, only the electric vector will be considered; the corresponding magnetic vector follows from Maxwell's equations.

Astronomical signals are noise-like. These noise-like variations of electric field strength (of the electromagnetic wave) may be passed through a narrow-band filter, so that a ‘quasi-monochromatic’ wave remains. Such a wave contains a very narrow band of frequencies and may be seen as a sinusoidal carrier wave at signal frequency, modulated both in amplitude and phase by noise-like variations. The highest frequencies (the fastest variations) in the modulating noise determine the width of the sidebands around the carrier wave in the frequency spectrum. Any wide-band (‘polychromatic’) signal may be seen as the sum of many quasi-monochromatic signals, all with different carrier frequencies and generally each with its own amplitude and phase modulation.

In this chapter I shall discuss the scientific reasons for measuring the polarization of astronomical signals. The central question is: ‘What does nature express as polarization rather than as some other property of the signal?’. This, of course, is the scientific point of departure for all astronomical polarimetry, but the basic concepts of polarization and (un)polarized radiation needed clarification before scientific necessity could be discussed properly. This chapter will be only a brief overview of the relevant astronomy; a number of recent reviews are available to help the reader become familiar with the astronomical applications. The subject of this book is polarimetry, the desirability of measuring the polarization will be taken for granted.

The light of most stars is itself unpolarized. In fact, whenever one needs an optical ‘zero-polarization’ reference source, one is generally pushed to use stars rather than lamps. The reason for the low polarization is the great distance (point source) and the spherical symmetry of most stars: any linear polarization there might be is averaged out over the star's visible disc. In the radio domain, antenna properties are highly polarization-dependent, and without specialized techniques large spurious apparent polarization is generated within the instrument. Thus, circumstances conspired to make astronomical polarimetry a late arrival. Even in the spectral regions of greatest instrumental sophistication, polarimetry remained a specialist technique; solar physics has been the notable exception. As a corollary of this lack of attention to polarimetry, awareness of polarization-induced photometric errors within telescopes and instruments has been minimal.

This chapter will focus on those aspects of polarimetric instruments that are peculiar to certain wavelength regions. The concepts discussed in previous chapters will be used freely. Non-polarimetric wavelength-peculiar concepts will generally be taken for granted, but a few are essential and must be recapitulated briefly.

Optical/infrared systems

Optical polarimetric instrumentation has a long history of development. Early polarimeters had errors at the level of a few tenths of a per cent at best, and polarization signals were small, so that polarimetry was very much a specialist craft. B. Lyot was the first to obtain very high accuracy by devising a modulator and using it on the Sun. For stars, the signals were generally so small that photon shot noise was appreciable, and there was little incentive to design sophisticated systems of unavoidably smaller throughput.

The situation has changed drastically within the last decade or two. Larger telescopes are available, CCD detectors now offer thousands of parallel channels of potentially very good accuracy, and improved modulators of high transmission have been devised. The higher signal levels have meant that greater resolution (spectral, temporal, spatial) can be used, and this has had the effect of increasing the degree of polarization provided by nature (less smearing of polarizations from neighbouring resolution elements); the end result is that (i) many more situations within astronomy can usefully be tackled by polarimetry without exceptional cost in telescope time and (ii) ‘common-user’ polarimetry is becoming available in the optical/near-infrared wavelength region (the ‘CCD domain’).

This chapter introduces the tools used by astronomers and instrument designers in describing the action of a medium on the polarization of the radiation passing through it. In the majority of situations encountered in astronomy, the phase of the wave is unimportant, and we need a way to describe the transformation of Stokes parameters, i.e. the changing polarization forms which support the flow of radiant energy. For cases where the phase of the polarized radiation is important (e.g. polarization effects within an optical interferometer, the focusing of a plane wave by a radio telescope, or the amplification of polarized radiation within an astronomical maser), an alternative formulation will be introduced (in section 4.2) that describes the transformation (including phase) of the electric field vibrations of two orthogonal 100% polarized waves (usually linear polarization). In this formulation, partial polarization cannot be handled, and we must make separate calculations for two orthogonal polarizations of the incident radiation, constructing the incoherent sum at the end. Shurcliff (1962, sections 8.6, 8.7, 8.9) details the early history of these two calculi and compares their fields of use; a concise statement of the relationship between the two calculi may be found in Stenflo (1994, section 2.6).

Mueller matrices

As discussed in chapter 2, the four Stokes parameters denote the flow of radiant energy in specific vibrations of the electromagnetic field, and all four are expressed in the same units.

Today, the study of The Structure of the Sun is one of the most exciting and rapidly evolving fields in physics. Helioseismology has provided us with a new tool to measure the physical state of the interior of a star, our Sun. This technique is successful to a depth of 0.7 R⊙ (i.e. 0.3 R⊙ from the centre). Deeper than this, observational data has been scarce. However, data are now becoming available from Earth-bound helioseismic networks (GONG, TON, IRIS, BISON,…) and from experiments on board SOHO (GOLF, MDI, VIRGO). These should allow the spectrum of gravity modes for the Sun to be determined, and thus the physical state of the solar core.

This book provides an up-to-date and comprehensive review of our current understanding of the Sun. Each chapter is written by a world expert. They are based on lectures given at the VIth Canary Islands Winter School on Astrophysics. This timely conference brought together leading scientists in the field, postgraduates and recent postdocs students. The aim was to take stock of the new understanding of the Sun and to focus on avenues for fruitful future research. Eight lecturers, around 60 students, and staff from the IAC met in the Hotel Gran Tinerfe in Playa de las Américas (Adeje, Tenerife) from the 5th to the 16th of December, 1994. It was a fortnight of intense and enjoyable scientific work.

INTRODUCTION

This is almost an impossible task, to summarize the subject of Global Changes in the Sun so I must apologize in advance for limiting the scope of these lectures to issues that have been choosen, in part, because of personal interests. I hope that the references provide the reader with footpoints from which to explore a larger set of questions which bear on this subject.

Here we will not discuss the very long, evolutionary, timescales over which the sun changes, nor will we explore the fast changes associated with flares and other transient phenomena. While these discussions depend on some results from MHD models of the solar magnetic cycle, we will not be concerned with the MHD mechanism. These lectures will not address the questions needed to understand local physical models that describe, for example, granulation. On the other hand we will describe some of the physical problems that connect the small-scale behavior of the sun to its global properties. By “global property” I mean an observable that is connected by physically important timescales to the entire sun: limb shape and brightness, largescale magnetic field, oscillation frequencies, solar luminosity, and solar irradiance are all examples of global properties.

Here we are interested in understanding the deviations of the sun from some standard one-dimensional static stellar model. This is a subject we can hardly approach for other stars, and for the sun it is difficult because the physics of magnetic fields and convection are linked over a wide range of spatial and temporal scales.

ABSTRACT

The X-ray Solar Physics Satellite Yohkoh has provided us with a number of new findings about the high temperature and high energy processes occurring in solar flares, in active regions, and in the background corona. According to these new findings, hot and dense corona above active regions seem to be maintained, at least in part, with the injections of already heated mass along the magnetic loops from the footpoint below. The outermost loops of the magnetic structures of these active regions are expanding away almost continuously in the case of “active” active regions. These give us quite a different and lively picture about the active region corona compared with a previous static picture with steady heating that we had based on the previous low cadence observations. New clues to the mechanism of flares, which were hidden thus-far in the yet fainter and relatively short stages before the start of flares, have been revealed by the wide-dynamic range, high cadence observations with the scientific instruments aboard Yohkoh. Those preflare signatures and their changes containing essential information about the mechanism of flares, now allow us to pursue truer understanding about the flare mechanism. The same merits of Yohkoh (wide-dynamic range and high-cadence observations) have shown us for the first time in its full form the highly dynamical behavior of the faint background corona, together with the influence of the changes in active regions sometimes exerting overwhelming effects on the surrounding corona.

FOREWORD

Broadly speaking, the inverse problem is the inverse of the forward problem. In the case of contemporary helioseismology, the forward problem is usually posed as that of determining the eigenfrequencies of free oscillation of a theoretical model of the sun. That problem is discussed by Christensen-Dalsgaard in this volume. I call inverting that problem the ‘main’ inverse problem. It is the one that I shall be discussing almost exclusively in this chapter. But also included in the forward problem must be the theoretical modelling of the oscillations as they really occur in the sun, forced, we believe, predominantly by the turbulence in the convection zone, and modulated by their nonlinear interactions with other modes of oscillation and by the perturbations they induce to the very convection that drives them, through variations in the turbulent fluxes of heat and momentum. The inverse of that problem is to derive from the fluid motion of the visible layers in the atmosphere of the sun, which I presume to be ‘observed’, estimates of the frequencies that the modes would have had had they not been disturbed by the other forms of motion. The outcome of that prior inversion provides the data for the main inverse problem.

This chapter is entitled: Testing solar models …. By ‘solar models’ is meant any theoretical description of the sun that we might have in mind.

INTRODUCTION

The present chapter addresses the forward problem, i.e., the relation between the structure of a solar model and the corresponding frequencies. As important, however, is the extent to which the frequencies reflect the physics and other assumptions underlying the model calculation. Thus in Section 2 I consider some aspects of solar model computation. In addition, the understanding of the diagnostic potential of the frequencies requires information about the properties of the oscillations, which is provided in Section 3. Section 4 investigates the relation between the properties of solar structure and the oscillations by considering several examples of modifications to the solar models and their effects on the frequencies, while Section 5 considers further analyses of the observed frequencies. Finally, the prospects of extending this type of work to other stars are addressed in Section 6.

A more detailed background on the theory of solar oscillations was given, for example, by Christensen-Dalsgaard & Berthomieu (1991), Gough (1993), and Christensen-Dalsgaard (1994). For other general presentations of the properties of solar and stellar oscillations see, e.g., Unno et al. (1989) and Gough & Toomre (1991).

A little history

The realization that observed frequencies of solar oscillation might provide information about the solar interior goes back at least two decades. Observations of fluctuations in the solar limb intensity (Hill & Stebbins 1975; Hill, Stebbins & Brown 1976), and the claimed detection of a Doppler velocity oscillation with a period close to 160 minutes (Brookes, Isaak & van der Raay 1976; Severny, Kotov & Tsap 1976) provided early indications that global solar oscillations might be detectable and led to the first comparisons of the reported frequencies with those of solar models (e.g. Scuflaire et al. 1975; Christensen-Dalsgaard & Gough 1976; Iben & Mahaffy 1976; Rouse 1977).

ANALYSIS TOOLS AND THE SOLAR NOISE BACKGROUND

When we observe solar oscillations, we are concerned with measuring perturbations on the Sun that are almost periodic in space and time. The periodic waves that interest us are, however, embedded in a background of broadband noise from convection and other solar processes, which tend to obscure and confuse the information we want. Also (and worse), the “almost-periodic” nature of the waves leads to problems in the interpretation of the time series that we measure. Much of the subject of observational helioseismology is thus concerned with ways to minimize these difficulties.

Fourier Transforms and Statistics

A common thread runs through all of the analysis tricks that one plays when looking at solar oscillations data, and indeed through many of the purely instrumental concerns as well: this thread is the Fourier transform. The reason for this commonality is, of course, that we are dealing with (almost) periodic phenomena – either the acoustic-gravity waves themselves, or the light waves that bring us news of them. Since many of the same notions will recur repeatedly, it is worth taking a little time (and boring the cognoscente) to review some of the most useful properties of Fourier transforms and power spectra. In what follows, I shall simply state results and indicate some of the more useful consequences. We shall see below that even when the Big Theorems of Fourier transforms do not apply, (as with Legendre transforms, for instance), analogous things happen, so that the Fourier example is a helpful guide to the kind of problems we may have.