largest instruments. The corresponding light ratio is more than 100,000,000 to I ; and it is an important and not very easy problem to subdivide this range accurately. For this purpose a set of 96 standard stars has been chosen near the North Pole, called the Harvard Polar Sequence; their magnitudes stretch at short inter- vals from the first to the twenty-first, and when once these have been accurately fixed on the absolute scale, it is comparatively easy to determine the magnitudes of any other stars by differential com- parisons. There is some systematic difference between the standard magnitudes of the sequence adopted at Harvard and Mount Wilson respectively for part of the range, which is still being inquired into; but good progress has been made in establishing an accurate and absolute basis for magnitude determinations. Separate standards are needed for visual and photographic magnitudes; their relation has been fixed by international convention so that visual and photo- graphic magnitudes agree for stars of type Ao between 5 m- 5 and 6 m '5. Photographic magnitudes have been determined at numerous observatories, one of the most valuable pioneer investigations being K. Schwarzschild's Gottingen Aktinometrie of the brighter stars. Most of our data of visual magnitudes are due to Harvard (where the late E. C. Pickering alone made a million and a half photometric measures) and to Potsdam observatories. It is now becoming usual to determine "photo-visual" as equivalent to visual magnitudes, i.e. to use a photographic plate of colour-sensitivity corresponding to that of the eye.
Since the photographic plate is most sensitive to blue light and the eye to yellow light, the difference, photographic minus visual magni- tude, gives a quantitative measure of the colour of the star. This is called the " colour-index." As might be expected, it is very approxi- mately a function of the spectral type, so that the spectral type may generally be inferred from the colour-index and vice versa. This affords a very useful method of classifying stars too faint to permit of spectroscopic examination. The colour-index ranges from about o m -5 for the bluest (type B) stars to +l m -9 for the reddest stars (type M). The Draper notation has almost displaced Secchi's and other early nomenclatures of spectral types. The principal stages from the hottest to the coolest are denoted by the letters B, A, F, G, K, M; and intermediate stages are estimated in tenths, e.g. "65" means halfway from Go to Ko. Types B and A correspond to Sec- chi's type I.; F, G, K to type II.; and M to type III. Typical stars are B, Rigel; A, Sirius; F, Procyon; G, the Sun; K, Arcturus; M, Antares. In addition, the somewhat rare Wolf-Rayet stars form type O preceding and hotter than type B; and type N (Secchi's type IV.) appears to form an alternative branch succeeding K and parallel with M, the bifurcation perhaps depending on whether the star has an oxidizing or reducing atmosphere. More recently a type R, probably intermediate between K and N, has been added. In types M and N the temperature is low enough for the spectra of chemical compounds to appear prominently; type M is character- ized especially by titanium oxide, and type N by compounds of car- bon. A catalogue of the spectral types of 230,000 stars classified by Miss A. J. Cannon is in course of publication by the Harvard Observatory; about half of it has already appeared.
Giant and Dwarf Stars. It will be realized that this great gain in quantity and quality of the material available for dis- cussion has permitted of considerable advance in our knowledge of the structure of the stellar universe, since 1910. The most far-reaching of the recent discoveries is the detection of the two classes of " giant " and " dwarf " stars.
To understand this distinction we must go back to Homer Lane's theory of the evolution of gaseous masses (see 25.788). Starting with a very diffuse globe of gas held together by its own gravitational attraction, the conditions of equilibrium require that its temperature must rise when it contracts through radiation of heat. This rise of temperature continues so long as the material is rare enough to follow the laws of a gas; but as the density approaches that of a liquid the changed conditions limit the rise, and ultimately the temperature begins to fall again; the fall continues until the star finally becomes extinct. It follows that any particular temperature is passed through twice, once ascending in a comparatively early stage of evolution, and once descending in a later stage. Now the Draper and other standard classifications of stellar spectra are prac- tically temperature classifications of stars; that is to say, tempera- ture is the primary condition which determines the appearance of the lines and bands distinguishing the spectral types. So in any type of spectrum we have two groups of stars which agree in tem- perature but are wide apart in all other respects; more particularly they differ in diffuseness and stage of evolution. For example, the present effective temperature of the sun is 6,oooC. ; it has a density greater than water and is accordingly in the dense descending stage ; but at an earlier epoch it must have passed through the same temperature ascending. It was then a diffuse globe of about 10 times its present diameter and too times its present surface; the temperature of the surface being the same, it then gave 100 times as much light as now. These two stages are called the dwarf and giant stages respectively, and the most conspicuous outward characteris- tic is the great difference of luminosity, due to the larger surface area in the giant stage.
Instead of having a single sequence of evolution B, A, F, G, K, M we see that a star must start as a giant of type M, ascend the series towards type B, and then descend as a dwarf to type M again. It depends on the mass how far up the series it gets, and probably a star must be three or four times as massive as the sun in order to reach the high temperature of type B. Smaller stars will turn at A, F, or even lower. As Russell has put it, a star of small mass is a poor self-heating affair. The division of giants and dwarfs is most conspicuous for the lower temperatures, G, K, M, since the corre- sponding stages are then furthest apart in the evolutionary sequence; for types A and F the two groups begin to merge into one another, and the division is less easy to recognize.
These conclusions were put forward independently and simul- taneously by H. N. Russell (8) and E. Hertzsprung. The observa- tional evidence drawn from many sources is now overwhelmingly fav- ourable. For stars of known parallax the absolute luminosity can be calculated directly; and when these are grouped according to spec- tral type the bifurcation of the luminosities is evident. The lumi- nosities of the giant stars depend very little on the spectral type (since the rising temperature compensates for the decreasing surface area), and their absolute magnitudes cluster very closely about the value +l m -o. 1 For the dwarfs the decreasing temperature and decreasing surface cause a rapid fall of brightness through the suc- cessive types, and the absolute magnitude fades to about +io m -o for type M. By the new spectroscopic method of determining stellar distances, Adams and Joy (9) have been able to give striking evidence of the two groups; of 58 stars of type M examined they found that 48 were giants with magnitudes between i m -o and +3 m -4, and 10 were dwarfs between +9 m -8 and + io m -7; there was thus a clear gap of six magnitudes separating the groups. Ascending to types K and G the groups draw closer together and begin to commingle, but even in type F the frequency curve shows the two distinct maxima. Further evidence is obtained from the study of eclipsing variable stars (10), since the average densities of these stars may be determined from the period and the light curve. For types B and A the densities are fairly uniform, averaging about one-tenth the den- sity of water; but for lower temperatures they clearly bifurcate, the one branch corresponding to dense stars like the sun and the other to rarefied stars with densities often below that of our atmosphere. W. Crucis, R. Z. Ophiuchi and S. X. Cassiopeiae are examples of stars with densities less than o-ooi, yet giving spectra classed as similar to that of the sun (density 1-38).
Finally all doubt as to the existence of these giant stars is set at rest by Pease and Anderson's direct measurement of the angular diameter of Betelgeuse made with a 2O-ft. interferometer at Mount Wilson in December 1920. The angular diameter was found to be o"-O45. Unfortunately the parallax is too small to be measured with much certainty; but it may be taken as proved that it is less than o"'O5. This makes the linear diameter of Betelgeuse not less than 140 million km. or loo times the sun's diameter. This is an example of a type M giant at the very beginning of the evolutionary sequence.
Spectroscopic Parallaxes. Although giant and dwarf stars of the same temperature have, broadly speaking, the same spectrum, a detailed examination of particular lines reveals distinctive differences. It was early shown by E. Hertzsprung that those spectra marked by Miss Maury as having the " e-characteristic " belonged exclusively to giant stars. More precise criteria were found by W. S. Adams and A. Kohlschiitter in 1914; and the method has been developed by Adams into a means not only of distinguishing the two classes but of determining quantitatively the absolute luminosities of stars. For example, the "enhanced lines" of strontium 4077 and 4215 are relatively strong in stars of high luminosity and weak in those of low luminosity; whereas the " furnace lines " of strontium 4607 and calcium 4455 behave in the reverse manner. Thus measures of the relative intensities of these lines give an indication of the luminosity of the star. In a general way we can understand the reason; en- hanced lines come from ionized atoms, so that they appear when the conditions are favourable to ionization. Other conditions being equal, low density increases the ionization so that the enhanced lines are likely to be strengthened in stars of low density, i.e. the giants as turns out to be the case. Considerable progress in the theory of ionization in stellar atmospheres has been made by M. N. Saha (n), the results being in good agreement with the observed conditions of emission of the corresponding spectral lines. But Adams's spectroscopic method of determining absolute luminosities (and hence parallaxes) is at present entirely empirical ; that is to say, the curve connecting absolute magnitude with the differential inten- sity of the selected lines is first deduced from and tested by stars of known trigonometrical parallax; it is then applied to deduce the luminosities of other stars. Parallaxes determined by this method for 1,650 stars have already been announced (9).
Red Dwarf Stars. Two very feebly luminous stars have been dis- covered which are of special interest owing to their closeness to us. In 1916 E. E. Barnard detected a star of visual magnitude 9 m -7 in
1 The absolute magnitude is the magnitude at a distance of 10 parsecs. The parsec, or distance corresponding to a parallax of I*, is 19-2 Xio 18 miles. The absolute magnitude of the sun is very nearly 5 m -o; thus the zero of absolute magnitude is a star loojtimes as bright as the sun.
