68. Energy in the form of radiant heat and light is continually flowing from the surface of a star into space. The surface layers of material cannot continue to provide this energy for long unless their heat is replenished from below. We are thus led to consider the process of transfer of energy from the interior to the surface.

There are two modes of transfer of heat in material in static equilibrium, viz. conduction and radiation. In both the net flow is in the direction of the temperature gradient from high to low temperature. In both this flow is the resultant of streams of energy in both directions; the stream from the high-temperature region is rather more intense than the stream from the low-temperature region, and the difference constitutes the net flow. In conduction molecules of the hotter region transmit their energy by diffusion and collision to surrounding regions; in radiation the hot material emits aether waves which are absorbed in the surrounding regions. In both cases this transmission is largely neutralised by a similar transmission from the surrounding regions, and the resultant transfer depends on the slight preponderance of the flow from the hotter region.

A third mode of transfer is possible if the limitation to static equilibrium is abandoned. There may be a system of ascending and descending currents in the star by which the material is kept stirred. Heat-energy is then carried from one region to another by actual movement of the matter carrying it—as in the lower part of our own atmosphere.

The study of the mechanical and physical conditions in the deep interior of the stars is undertaken primarily in the hope that an understanding of the internal mechanism will throw light on the external phenomena accessible to observation. More than fifty years have gone by since the general mode of attack was first developed; and the scope of the inquiry has grown so that it now involves much of the recently won knowledge of atoms and radiation, and makes evident the ties which unite pure physics with astrophysics. It would be hard to say whether the star or the electron is the hero of our epic.

The reader will judge for himself whether solid progress has been made. He may, like Shakespeare, take a view less optimistic than my own—

but I hope he will not be so unkind as to continue the quotation—

Re-reading this work I find passages where I have been betrayed into too confident assertion. It is only too true that the most patent clues may mislead, and observational tests of the rough kind here possible sometimes flatter to deceive. But the subject is a fair field for the struggle to gain knowledge by scientific reasoning; and, win or lose, we find the joy of contest.

146. Results reached in the present Chapter have been used in anticipation from § 89 onwards. We must therefore return and take up the problem of the absorption coefficient as it presented itself in § 88. At that stage we were occupied with our first astronomical result of importance, viz. that for the series of giant stars from type M to type A the opacity is nearly constant although the internal temperature increases twelvefold between the beginning and end of the series. This suggested (but, as we now see, wrongly) that the opacity might tend to a constant value at high temperatures and so be the same for all stars. Actually, however, the constancy of the opacity was a statistical result applying to groups of stars presumed to be of the same average mass, and there was no test whether the constancy continued for stars of a different mass.

The radiation in the main interior of a star consists of X rays, and comparison is invited with measurements of absorption of X rays made in the laboratory. In § 105 we have found the absorption coefficient at the centre of Capella to be 49 c.g.s. units. This is of the general order of magnitude of the measured coefficients of most elements for hard X rays; for example, it agrees with the coefficient for iron for wave-length about 0·8 Å. It must, however, be noted that the radiation at the centre of Capella is of much greater wave-length, the maximum intensity being at 3·2 Å.

CEPHEID VARIABLES

123. Although variable stars of the Cepheid type show a periodic change of radial velocity it is improbable that they are binary systems. The theory which now seems most plausible attributes their variation to the pulsation of a single star; and accordingly the varying radial velocity measures the approach and recession of the surface presented towards the observer as the star swells and contracts. If this explanation is correct we have an opportunity of extending the study of the internal state of a star from static to disturbed conditions.

The leading facts about these variables ascertained by observational study are as follows—

About 170 galactic Cepheids are known with periods ranging from a few hours to about 50 days; so-called “orbits” have been determined for 20 of these from measurements of radial velocity. In addition large numbers of Cepheids have been found in some globular clusters; among these periods less than 12 hours are especially prevalent. Cepheids have also been found in the Andromeda nebula.

Relatively few periods are between 0·7 and 3 days, so that the Cepheids may be subdivided into two groups with periods above and below this gap.

The light-range rarely exceeds 1m·2 visual; the photographic range is greater than the visual. The spectral type changes during the period, corresponding to a higher temperature at maximum than at minimum.

The light-curve and the velocity-curve are closely similar; the correspondence is the more marked because both curves are usually unsymmetrical.

1. At first sight it would seem that the deep interior of the sun and stars is less accessible to scientific investigation than any other region of the universe. Our telescopes may probe farther and farther into the depths of space; but how can we ever obtain certain knowledge of that which is hidden behind substantial barriers? What appliance can pierce through the outer layers of a star and test the conditions within?

The problem does not appear so hopeless when misleading metaphor is discarded. It is not our task actively to “probe”; we learn what we do learn by awaiting and interpreting the messages dispatched to us by the objects of nature. And the interior of a star is not wholly cut off from such communication. A gravitational field emanates from it, which substantial barriers cannot appreciably modify; further, radiant energy from the hot interior after many deflections and transformations manages to struggle to the surface and begin its journey across space. From these two clues alone a chain of deduction can start, which is perhaps the more trustworthy because it is only possible to employ in it the most universal rules of nature—the conservation of energy and momentum, the laws of chance and averages, the second law of thermodynamics, the fundamental properties of the atom, and so on. There is no more essential uncertainty in the knowledge so reached than there is in most scientific inferences.

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.