Introduction

Presentation and scope

This chapter contains the proceedings of the course on analysis of 3D spectrographic data given as part of the XVII Canary Island Winter School of Astrophysics. It provides an overview of some basic and generic analysis techniques for 3D spectrographic data.

It includes a description of an arbitrary selection of tasks with, whenever possible, examples on real data and a lot of discussion about noise and errors. To illustrate the examples, we will make a heavy use of tools that are part of the XOasis software developed at the Centre de Recherche Astrophysique de Lyon (CRAL) and of 3D datasets obtained using the TIGER and OASIS instruments.

This course is not limited to pure 3D analysis techniques as the core of the analysis of 3D datasets is either identical or similar to what is done for regular spectrographic data.

It has some obvious caveats and limitations:

• It is not exhaustive but contains a rather arbitrary selection of tasks and tools that we have considered as unavoidable.

• The methods and examples are biased toward extragalactic astronomy.

• It is limited to the data analysis techniques used in visible and near-infrared (NIR) astronomy.

• It does not address those used in the radio and X-ray communities (long-time users of 3D spectrography).

Data analysis

Before starting, we need to define better what data analysis is and, in particular, where it starts and stops in the process leading from raw data to ready-to-publish information (see Figure 4.1).

Introduction

We found in the previous chapter that, if a massive star runs out of nuclear fuel, it would lose its equilibrium and begin to shrink. Even when nuclear fuel is available to the star, it may be insufficient to meet the demands for the star's equilibrium. In the early 1930s the young astrophysicist Subrahmanyan Chandrasekhar had encountered a somewhat similar situation when discussing the state of stars like the Sun, after they run out of their nuclear fuel. He found that the star can still sustain equilibrium if its internal matter can attain the degenerate state. Degeneracy can arise if the density of matter is so high that all available energy levels of atoms are filled up, up to some low energy. In such a situation further compression of matter is not possible and gravity is held at bay. This is an excellent example of a macroscopic effect of quantum mechanics: a star as massive as the Sun feels an effect whose origin is in quantum mechanics. We cannot describe it in detail since that would take us farther away from our present interest.

The early work on degenerate matter by R. H. Fowler had shown that every star on sufficient compression attains degeneracy, thereby ensuring that the star would rest in peace in a state of very high density and small radius. It was felt that white dwarf stars are precisely the stars which are in this state.

Introduction

Integral field spectroscopy (IFS) is a technique to obtain both spatial (x,y) and spectral (λ) information of a more or less continuous area of the sky simultaneously on the detector. Only a few instrumental concepts allow 3D information on 2D detectors to be obtained, and all of these are based on field splitters such as fibre bundles, lens array, or image slicers (see Figure 7.1) to sample the field of view. Each sampled element is then dispersed using a classic spectrograph and produces a spectrum on the detector. Depending on the field splitter used, the geometry of the spectra on the detectors may be very different. This diversity leads to the creation of very specific reduction techniques and/or packages, i.e. one per instrument built (e.g. P3d, Becker, 2001). Combined with the inherent complexity of 3D techniques, such software diversity has reduced the use of IFS for decades to a handful of specialists, mainly those involved in the teams building such instruments.

Conscious that this would be a handicap IFS specialists Walsh and Roth (2002) have started to standardize techniques and tools for integral field units (IFU). Recently, the Euro3D Research Training Network (RTN), whose aim was to promote 3D spectroscopy all over Europe (Walsh and Roth, 2002), made a great effort to create a standard data format (Kissler-Patig et al., 2004) for storing and exchanging 3D data, developing an application programming interface, API (Pécontal-Rousset et al., 2004), to ease the use of such a data format and creating a visualization tool (Sánchez, 2004) usable by any existing IFU.

Introduction

In the previous chapter we described how the dynamics of the scalar and vector fields can be described in a manner consistent with the special theory of relativity. Given the fact that Newtonian gravity is described by a gravitational potential φN(t, x), which satisfies the Poisson equation ∇2φN = 4πGρm (where ρm is the mass density), it might seem that one could construct a theory for gravity consistent with special relativity by suitably generalizing the Poisson equation for the gravitational potential. It turns out, however, that this is not so straightforward. The natural description of the gravitational field happens to be completely different and is intimately linked with the geometrical properties of the spacetime. We will be concentrating on such a description from Chapter 4 onwards in this book.

The key purpose of the present chapter is to explain in physical terms why such a geometrical description for gravity is almost inevitable. We shall first describe several difficulties that arise in any attempt to provide a purely field theoretic description of gravity in flat spacetime. We will then give a series of simple thought experiments that illustrate an intimate connection between gravitational fields and spacetime geometry. None of this can be thought of as a mathematically rigorous proof that gravity must be described as spacetime geometry; however, it goes a long way in showing that such a description is most natural and, of course, consistent with all known facts about gravity.

Introduction

We begin our study of general relativity and curved spacetime in this chapter. Chapters 4 and 5 will develop the necessary mathematical apparatus to deal with curved spacetime. As in the case of electromagnetism, the study of gravity can be divided into two separate – but interconnected – aspects. In this chapter and the next, we will study the influence of gravity on other physical systems (like particles, photons, ideal fluids, fields, etc.) without worrying about how a given gravitational field is generated – which will be discussed in Chapter 6.

All the topics introduced in this chapter will be required in the subsequent chapters and form core material for general relativity. In particular, we will start introducing index-free vector notation more liberally in the coming chapters and familiarity with the ideas and notation developed in Section 4.6.1 will be crucial. We will use units with c = 1 unless otherwise indicated.

Metric tensor and gravity

The arguments presented in the previous chapter suggest that a weak gravitational field cannot be distinguished from a modified spacetime interval as far as mechanical phenomena are concerned. We shall now generalize this result by postulating that all aspects of gravitational fields allow a geometrical description. We thus extend the tentative conclusion of the previous chapter to include arbitrarily strong gravitational fields and all physical phenomena. This leads to Einstein's theory of gravitation, which is the most beautiful of all existing physical theories.

Introduction

In Euclidean geometry or in the pseudo-Euclidean spacetime of special relativity, the geometrical properties are invariant under translations and rotations. The same is not necessarily true of the non-Euclidean spacetimes of general relativity. As we shall see in Chapter 8, the spacetime geometry is intimately related to the distribution of gravitating matter (and energy). A completely general spacetime arising from an arbitrary distribution of gravitating objects will not have any symmetries at all. Such cases are difficult to solve as solutions of Einstein's gravitational equations. It is, however, easier to solve problems where mass distributions have certain symmetries. For example, a point mass in an otherwise empty space is expected to generate a solution that has spherical symmetry about that point. Cases like these may be looked upon as approximations to reality. A similar approach is adopted in Newtonian gravitation. For example, as a first approximation the gravitating masses in the Solar System (the Sun and the planets) are treated as spherical distributions. In this chapter we will look at certain symmetric spacetimes that will be of use in solving specific problems in general relativity. The main question that we shall begin with is that of how to identify a symmetry in a given spacetime. How do we discover an intrinsic property like symmetry, when given the spacetime metric?

We will have occasion to use symmetric and antisymmetric tensors.

We have come to the end of our account of the theories of relativity: special and general. While the former was briefly reviewed in the first chapter, we spent 16 chapters presenting the general theory from scratch. After preparing the background of vectors and tensors in the curved spacetime, we introduced the notions of parallel propagation, covariant differentiation, spacetime curvature and symmetries of motion. We then introduced physics through the notions of the action principle and energy-momentum tensors.

This was the appropriate stage to introduce the basics of general relativity: the principle of equivalence, Einstein's field equations and their Newtonian limit. Following these notions, we introduced the Schwarzschild solution and the various tests of general relativity, largely within the Solar System. We also discussed the budding field of gravitational radiation and the attempts to detect it coming from cosmic sources. Our next topic was relativistic astrophysics, which deals with compact massive objects such as supermassive stars and black holes. We also briefly touched upon the very interesting topic of gravitational lensing. This was followed by a discussion of some highlights of relativistic cosmology.

This presentation is indicative of the scope of general relativity. While it has created a niche for itself in theoretical physics as a remarkable intellectual exercise, it has also justified its status as the most effective physical theory of gravitation by explaining and predicting several gravitational phenomena.

Preface

The topic of the XVII IAC Winter School is ‘3D Spectroscopy’: a powerful astronomical observing technique, which has been in use since the early stages of the first prototype instruments about a quarter of a century ago. However, this technique is still not considered a standard common user tool among most present-day astronomers.

3D Spectroscopy (hereafter ‘3D’) is also called ‘integral field spectroscopy’ (IFS), sometimes ‘two-dimensional’ or even ‘area’ spectroscopy, and commonly also ‘three-dimensional’ spectroscopy; in other areas outside astronomy it is called ‘hyperspectral imaging’, and so forth. It is already this diversity in the nomenclature that perhaps reflects the level of confusion. For practical reasons, the organizers of this Winter School and the Euro3D network (which will be introduced below) have adopted the terminology ‘3D’, which is intuitively descriptive, but, as a caveat early on, is conceptually misleading if we restrict our imagination to the popular picture of the ‘datacube’ (Figure 1.1). Although this term will commonly be used throughout this book, we need to point out for the reasons given later in the first chapter that the idealized picture of an orthogonal cube with two spatial, and one wavelength, coordinate(s) is inappropriate in the most general case.

Whatever the terminology, it is the aim of this Winter School to help alleviate the apparent lack of insight into 3D instrumentation, its use for astronomical observations, the complex problems of data reduction and analysis, and to spread knowledge among a significant number of international young researchers at the beginning of their career.