The purpose of the book

The analysis of data in the form of directions in space, or equivalently of positions of points on a spherical surface, is required in many contexts in the Earth Sciences, Astrophysics and other fields. While the contexts vary, the statistical methodology required is common to most of these data situations. Some of the methods date back to the beginning of the century, but the main developments have been from about 1950 onwards. A large body of results and techniques is now disseminated throughout the literature. This book aims to present a unified and up-to-date account of these methods for practical use.

It is directed to several categories of reader:

1. to the working scientist dealing with spherical data;

2. to undergraduate or graduate students whose taught courses or research require an understanding of aspects of spherical data analysis, for whom it would be a useful supporting text or working manual;

3. to statistical research workers, for reference with regard to current solved and unsolved problems in the field.

Because of the range of readership, priority has been given to providing a manual for the working scientist. Statistical notions are spelt out in some detail, whereas the statistical theory underlying the methods is, by and large, not included; for the statistician, references are given to this theory, and to related work. In particular, only one procedure is given for any specific problem. In some cases, the choice of procedure is dictated by certain optimality considerations, whereas in other cases a somewhat arbitrary choice has been made from several essentially equivalent procedures.

Introduction

Many different ways of representing a three-dimensional unit vector or axis have been developed over the centuries, due not only to the requirements of different disciplines (Astronomy, Geodesy, Geology, Geophysics, Mathematics, …) but also to diverse needs within a discipline: in Geology, for example, there appear to be five or six systems in current use. In this book, we shall use either polar coordinates or the corresponding direction cosines for all purposes of statistical analysis. The following sub-section (§2.2) defines several of the coordinate systems and gives the mathematical relationship of each to polar coordinates.

Later chapters of this book, concerned with statistical analysis, abound with words and phrases which have particular meanings in Statistics, and, possibly, rather different meanings in other areas. A good example of this is the word “sample”, which for our purposes is loosely taken to mean a collection of measurements of a particular characteristic, but which has a general scientific meaning of an observational or sampling unit (e.g. a drill-core specimen on which a single measurement may be made). §2.3 gives definitions of a number of such words and phrases.

Spherical coordinate systems

The type of data we shall be dealing with will be either directed lines or undirected lines. For the former, the measurements will be unit vectors, such as the direction of magnetisation of a rock specimen, or the direction of palaeocurrent flow. For the latter, which we shall term axes (cf. §2.3), the line measured might be the normal to a fracture plane, and so have no sense (direction) unless this is ascribed on some other basis.

Normal stars

A B-type star is an object exhibiting neutral helium lines in its spectrum, but no ionized helium lines. The latter are characteristic of O-type stars. Neutral helium lines are invisible in A-type stars. (For illustration, see figure 9.1.)

The maximum strength of the He i lines is reached in early B subclasses, around B2. Hydrogen lines on the other hand have their maximum strength at A2 and therefore along the B-type star sequence hydrogen and helium exhibit an opposite trend. Table 9.1 provides the equivalent widths of the stronger lines, taken from Didelon (1982). All lines of elements other than hydrogen in the region λλ3600–4800 are less intense than 1.3 Å.

Table 9.1 shows that for quantitative classification we can use in principle H and He i line strengths alone. However, the Balmer lines are too intense to use for visual classification and we must look for other, weaker lines in the λλ3600–4800 region. Elements having weaker lines are listed in table 9.2. As can be seen from the table, the number of elements visible diminishes toward later B-types. For stars between B5 and A0 only a few lines are left and so all have to be used for classification. Equivalent widths for most of these lines are given by Didelon (1982).

Normal stars

G-type stars are characterized by weak hydrogen lines which become comparable in strength to the lines of some metals. Metallic lines increase both in number and in intensity toward later spectral subdivisions, and molecular bands of CH and CN become easily visible features.

In order to fix ideas, we quote in table 12.1 the equivalent widths of some strong lines.

We have not given the intensity of the G-band, which is easily observable at classification dispersion, but which breaks down on the low plate factor spectrograms needed to measure equivalent widths.

The spectral type is established by the comparison of hydrogen and metal lines, like Fe λ4143 and Hδ: they are about equally intense at G8 when seen at 80 Å/mm. Instead of this pair of lines, Fe λ4045/H λ4101 or Fe λ4384/H λ4340 and λ4921/H λ4861 may also be used. For types later than G5 the Ca i λ4226 line becomes sensitive to temperature and can be used for determination of spectral type as Ca i λ4226/H λ4101 (see figure 12.1).

If it is suspected that there are composition anomalies, the hydrogen-to-metal ratio should not be used but should be replaced by Cr λ4254/Fe λ4250 and Cr λ4274/Fe/ λ4271 (Keenan and McNeil 1976).

If for instance the star has weak metal lines, the ratio between hydrogen and metallic lines is earlier than it should really be, and only the ratio of two metal features can provide the right spectral type.

We shall examine in this last chapter some issues which are relevant to further progress in the field of classification. We shall group the issues into three sections. The first concerns the incorporation of additional information into the ‘classical’ scheme. The second is about groupings of superior hierarchical order. In the third section, we consider the future of classification.

Incorporation of new information

A question we have considered briefly in various chapters is the incorporation of new data into the classical scheme. To give an example, suppose that a large number of spectra covering the region λλ1200–3000 became available and that we are interested in a particular group of objects, for instance HB stars. In the classical region (λλ3600–4800) this is a homogeneous group (see section 10.4), but in the UV region we discover that half of the stars observed exhibit a feature at λ3040 not present in the other stars. A similar situation arises if some DC stars (i.e. degenerates having a continuous spectrum with no lines) are discovered in the UV to display carbon features. We could imagine these stars being studied in yet another region of the spectrum, for instance the 10–100 µm region, and finding there that an HB or a DC star has an infrared excess, indicating the presence of a circumstellar dust cloud.

Having seen in some detail spectroscopic and photometric classification methods, we shall compare them in this chapter. We shall examine first their ‘problem solving capability’ and ‘information content’. Finally we shall discuss the relation between classification and physical parameters. (For more details, see Jaschek 1982.)

Problem solving capability

In the chapter on spectral classification we have seen that in the Yerkes system there are two parameters, according to which stars can be arranged. If a star cannot be assigned a unique place in the scheme, it is called ‘peculiar’. We have also seen that in some cases abbreviations are needed for stars with varying degrees of rotation and that in some cases magnetic fields can be detected by the inspection of spectrograms. Therefore the list of parameters which can be ascertained from spectrograms is:

1. spectral type

2. luminosity class

3. spectral peculiarity

4. rotation

5. magnetic field.

Without going into details we may say that the spectral type corresponds to stellar surface temperature, luminosity class to stellar luminosity and spectral peculiarity to either abnormal atmospheric structures or anomalies in the abundance of chemical elements.

Our next question is whether these parameters can only be determined spectroscopically, or if photometry is able to do the same or better. This is a crucial question because it will determine the choice of instrumentation to attack a given problem.

Normal stars

According to the Harvard system an F-type star is characterized by strong Ca ii (K and H) lines, which become much stronger than the hydrogen lines of the Balmer series. A multitude of fainter metallic lines accompanies both features. At F0, as already mentioned I(K) = I(H + Hε); at G0, I(K) » I(H). Whereas at 100 Å/mm in A-type stars the Balmer lines are remarkable for their strength, in F-type stars they are no longer conspicuous. Another feature which appears at this dispersion is the G-band (near λ4300), which is due to the molecule CH; this feature appears around F3 and strengthens toward the later subtypes. The feature is constituted by the head of a molecular band and tends to dissolve when observed at lower plate factors.

In order to fix these ideas, table 11.1 provides the equivalent widths of some strong lines.

Besides the strong lines, there exists a host of weak lines, which produce, as we have seen, an increasing blocking. They also become so numerous that the undisturbed continuum is hard to see, except at low plate factors.

The spectral type is obtained from intensity ratios involving medium intensity features mostly from neutral elements. Attention has to be paid to two facts. The first is that because of the large number of lines present, any feature is a blend of several lines, except at very small plate factors.

Normal stars

According to the Harvard system, an A-type star is an object in which strong Balmer lines are accompanied by many faint to moderately strong lines. These metallic lines increase gradually in strength from A0 to A9. A-type stars differ from B-type stars in that in the former there is no He i line. The difference between A- and F-type stars is subtler; in the latter the metallic lines are more numerous and stronger.

The behavior of metallic lines can be illustrated by that of the Ca ii lines H (λ3968) and K (λ3933). On 100 Å/mm plates these lines are very weak at A0. Since H is close to Hε (λ3970), only a faint K-line and a broad H + Hε line are seen. At A5 approximately, K is half as strong as H + Hε, and at F0, I(K) ≃ I(H + Hε). Table 10.1 provides the equivalent widths of some typical lines.

Table 10.1 shows that hydrogen has its maximum at A2, decreasing from thereon; Ca ii and metals increase in strength toward later types. In principle therefore the hydrogen lines and the Ca ii lines, representing the metals, can be used for quantitative spectral type assignment. For visual classification of A-type stars, the lines are however too strong and fainter features have to be used.