Ionisation.

173. The determination of the degree of ionisation of the atoms under the conditions of temperature and density found in the stars is important in connection with the following applications—

(a) We derive from it the molecular weight μ which is required for nearly all numerical calculations. Accuracy is important since μ is often raised to a rather high power in the formulae. We have to find—

(1) What is the most probable value of μ for the stars in general? (The standard value adopted by us is 2·1.)

(2) What is the magnitude of the differential effects (more particularly as affecting the mass-luminosity relation) caused by differences of μ between different stars?

(3) What is the change of μ between the centre and the outer parts of a star?

(b) A knowledge of the ionisation is required in connection with theories of absorption, since each ionisation destroys an absorbing mechanism; in particular, it determines the “guillotine” correction to the opacity on Kramers' theory.

(c) It determines the energy of ionisation of a star and hence the ratio of specific heats γ, which is important in the study of the pulsations of Cepheids.

Another subject appropriately treated in connection with ionisation is the determination of the deviation of stellar material from the laws of a perfect gas.

The results generally depend appreciably on the chemical constitution of a star.

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.

What are spherical data? By a spherical measurement we simply mean the orientation of a straight line in space. In some scientific contexts we would wish to regard the line as directed, in others as undirected; in the first case we call it a vector and in the second case an axis.

Spherical data arise in many areas of scientific experimentation and observation. As examples of vectorial data from various fields, we instance from Astrophysics the arrival directions of showers of cosmic rays; from Structural Geology the facing directions of conically folded planes; from Palaeomagnetism the measurements of magnetic remanence in rocks; from Meteorology the observed wind directions at a given place; and from Physical Oceanography the measurements of ocean current directions. Examples of axial data from various fields include, from Crystallography the directions of the optic axes of quartz crystals in a sample of quartzite pebbles; from Astronomy the normals to the orbital planes of a number of comets; from Structural Geology the measurements of poles to joint planes or to axial plane cleavage surfaces; and from Animal Physiology the orientations of the dendritic fields at different sites in the retina of a cat's eye, in response to stimulus by polarised light.

Again, observations which are not in any way orientations can sometimes be usefully re-expressed in the form of orientations and analysed as spherical data; in Social Science, for example, it has been the practice to analyse data on occupational judgments by individuals as unit vectors.

Introduction

The contents of this chapter constitute a tool-kit for use in the subsequent chapters on data analysis. §3.2 deals with some basic mathematical methods for vectors and matrices; §3.3 and §3.4 are concerned with methods of data display, and qualitative (or descriptive) features of spherical data sets. In particular, the basic method we have adopted for displaying vectorial data, which may cover both hemispheres, is explained in §3.3.1. §3.5 describes some standard statistical methods for deciding whether a given random sample of observations is adequately fitted by some specified probability distribution, and whether two independent samples have been drawn from the same (unspecified) distribution; §3.6 describes the use of simulation as an aid in complicated analyses; §3.7 describes jackknife procedures and permutation tests; and §3.8 is a brief discourse on problems of data collection.

The mathematical results presented in §3.2 are purely for reference purposes, and no derivations are given; most, if not all, of the results are available in standard texts.

Mathematical methods for unit vectors and axes in three dimensions

Mean direction, resultant length and centre of mass

Consider a collection of points P1, …,Pn on the surface of the unit sphere centred at O, with Pi corresponding to a unit vector with polar coordinates (θi, φi) and direction cosines xi, = sin θi, cos φi, yi = sin θi, sin φi, zi, = cos θi, i = 1,…, n.