The two lines of investigation which are brought together in the present theory of the equilibrium of a star originate in two classical papers—

1. J. Homer Lane. On the Theoretical Temperature of the Sun. Amer. Journ. of Sci. and Arts, Series 2, Vol. 4, p. 57 (1870).

2. K. Schwarzschild. Ueber das Gleichgewicht der Sonnenatmosphäre. Göttingen Nachrichten, 1906, p. 41.

The latter paper develops the theory of radiative equilibrium in a form appropriate to the outer layers of a star.

Investigations up to the year 1907 are brought together in

1. 3. R. Emden. Gaskugeln: Anwendungen der Mechanischen Wärmetheorie. (B. G. Teubner, Leipzig and Berlin, 1907.)

which contains important developments by Emden himself. The most relevant portions are here summarised in §§ 54–63. Schwarzschild's work, which had newly appeared, is described by Emden, p. 330, but the book is in the main a study of convective equilibrium.

Two further references of historic interest may be added—

1. 4. R. A. Sampson. On the Rotation and Mechanical State of the Sun. Memoirs R.A.S. 51, p. 123 (1894).

2. 5. I. Bialobjesky. Sur l'Équilibre Thermodynamique d'une SphÈre Gazeuse Libre. Bull. Acad. Sci. Cracovie, May, 1913.

The first definitely postulates radiative equilibrium rather than convective equilibrium in the sun's interior. The second takes account of radiation pressure and demonstrates its importance in investigations of the internal equilibrium of a star.

For other early papers the references in Emden's Gaskugeln should be consulted.

My own investigations originated in an attempt to discuss a problem of Cepheid variation.

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.

White dwarfs

Objects lying five or more magnitudes below the main sequence are called ‘white dwarfs’.

The name was attributed to them because the spectrum of the first star analysed, 40 Eri B (Adams 1914), was of type A and the star white, whereas dwarfs were thought to be all of type M and therefore red. So the term calls attention to the inconsistency between the color or spectral type and the luminosity (or absolute magnitude) of the star. Since, however, the stars are neither white nor dwarfs, it would be more logical to call them ‘degenerates’ (D) because of the equation of state of the matter composing them. We shall use the latter term.

Detection methods. Degenerates, being intrinsically faint, are difficult to single out among the large number of faint stars. (Let us remember that there are about 2 × 107 stars brighter than B = 15.) Detection methods are thus of fundamental importance. Essentially three methods have been used to isolate candidates. The first is to search for nearby stars through their proper motion, and retain those which are blue. The second is to search spectroscopically for blue stars where no intrinsically bright stars are expected. The third is to search for faint companions to brighter nearby stars.

The first method was proposed and exploited by Luyten, from 1931 on. From an examination of about 108 star images on plates taken at different epochs, Luyten selected those stars whose proper motion is larger than a given amount (for instance 17″).

Normal stars

An O-type spectrum is characterized by the simultaneous presence of lines of neutral and ionized helium. If ionized helium is absent, the spectrum corresponds to a B-type. The lines of ionized helium strengthen toward earlier stars, whereas neutral helium lines and hydrogen lines decline toward earlier types. Table 8.1, taken from Conti (1973), illustrates this by way of the equivalent widths of some of the stronger lines.

Because of this antagonistic behavior of He ii and He i, we can use them for classification. The MK system uses the ratio He ii λ4541/He i λ4471 as the criterion for spectral type for the interval 03–09 (for an illustration, see figure 8.1). The He ii lines are grouped in the so-called Pickering series (analogous to the Bracket series of hydrogen) with lines at λλ5414,4850,4542,4339,4200,4100, 4025 and 3968,…, and the Fowler series (analogous to the Paschen series of hydrogen) with lines at λλ4656,3204,…

Besides helium, some other elements are also present in O-type spectra. Among these we have Si iv(λ4089) and C iii(λλ4068,4647 and 4651) in late O-type stars. N iii(λλ4634,4640) if present is only seen in emission.

In these stars emission lines are very often present and can appear with varying degrees of intensity, from barely perceptible to rather strong, although they are never as broad and intense as in Wolf–Rayet stars. They can appear in different objects as a weak central emission on a wider absorption, or as a complete fill-in, or as a moderately strong emission.

Normal stars

The spectra of stars of class M are characterized by strong absorption bands of TiO, and by the great number and strength of metallic lines which practically block the spectrum for λ < 4000 Å.

In the Harvard system four subdivisions of M-type stars were kept: Ma, Mb, Mc and Md. The Mt Wilson observers replaced these types by the decimal subdivisions M0, 1,…, 6 in which Ma corresponds to M0–M2, Mb to M3–M5, and Mc to M5 and later. The last group, Md, is characterized by variable emission lines, so that any Md object should be denoted Mxe, x standing for the decimal subtype and e for emission. The Mt Wilson system uses mainly the intensities of the TiO bands for classification and this is also done in the Yerkes system. The reason is that the bands strengthen so rapidly with advancing type that it becomes difficult to find stretches of continuum where atomic lines can be compared reliably, except for the earlier subtypes. The most important TiO bands are listed in table 14.1.

The first bands which appear in the yellow and red are λ5167 and λ7054, and λ4954 in the blue at M0. After M2 many others can be seen; the ones especially useful are marked with an asterisk in table 14.1.