Our knowledge of the stars is based on their apparent attributes, obtained from the astronomical observations. The object of astronomy is to deduce herefrom the real or absolute attributes of the stars, which are their position in space, their movement, and their physical nature.

The apparent attributes of the stars are studied by the aid of their radiation. The characteristics of this radiation may be described in different ways, according as the nature of the light is defined. (Undulatory theory, Emission theory.)

From the statistical point of view it will be convenient to consider the radiation as consisting of an emanation of small particles from the radiating body (the star). These particles are characterized by certain attributes, which may differ in degree from one particle to another. These attributes may be, for instance, the diameter and form of the particles, their mode of rotation, &c. By these attributes the optical and electrical properties of the radiation are to be explained. I shall not here attempt any such explanation, but shall confine myself to the property which the particles have of possessing a different mode of deviating from the rectilinear path as they pass from one medium to another. This deviation depends in some way on one or more attributes of the particles. Let us suppose that it depends on a single attribute, which, with a terminology derived from the undulatory theory of Huyghens, may be called the wave-length (?) of the particle.

The statistical characteristics of the radiation are then in the first place:—

(1) the total number of particles or the intensity of the radiation;

(2) the mean wave-length (?₀) of the radiation, also called (or nearly identical with) the effective wave-length or the colour;

(3) the dispersion of the wave-length. This characteristic of the radiation may be determined from the spectrum, which also gives the variation of the radiation with ?, and hence may also determine the mean wave-length of the radiation.

Moreover we may find from the radiation of a star its apparent place on the sky.

The intensity, the mean wave-length, and the dispersion of the wave-length are in a simple manner connected with the temperature (T) of the star. According to the radiation laws of Stephan and Wien we find, indeed (compare L. M. 41[1]) that the intensity is proportional to the fourth power of T, whereas the mean wave-length and the dispersion of the wave-length are both inversely proportional to T. It follows that with increasing temperature the mean wave-length diminishes—the colour changing into violet—and simultaneously the dispersion of the wave-length and also even the total length of the spectrum are reduced (decrease).

2.

The apparent position of a star is generally denoted by its right ascension (a) and its declination (d). Taking into account the apparent distribution of the stars in space, it is, however, more practical to characterize the position of a star by its galactic longitude (l) and its galactic latitude (b). Before defining these coordinates, which will be generally used in the following pages, it should be pointed out that we shall also generally give the coordinates a and d of the stars in a particular manner. We shall therefore use an abridged notation, so that if for instance a = 17^h 44^m.7 and d = +35°.84, we shall write

(ad) = (174435).

If d is negative, for instance d = -35°.84, we write

(ad) = (174435),

so that the last two figures are in italics.

This notation has been introduced by Pickering for variable stars and is used by him everywhere in the Annals of the Harvard Observatory, but it is also well suited to all stars. This notation gives, simultaneously, the characteristic numero of the stars. It is true that two or more stars may in this manner obtain the same characteristic numero. They are, however, easily distinguishable from each other through other attributes.

The galactic coordinates l and b are referred to the Milky Way (the Galaxy) as plane of reference. The pole of the Milky Way has according to Houzeau and Gould the position (ad) = (124527). From the distribution of the stars of the spectral type B I have in L. M. II, 14[2] found a somewhat different position. But having ascertained later that the real position of the galactic plane requires a greater number of stars for an accurate determination of its value, I have preferred to employ the position used by Pickering in the Harvard catalogues, namely (ad) = (124028), or

a = 12^h 40^m = 190°, d = +28°,

which position is now exclusively used in the stellar statistical investigations at the Observatory of Lund and is also used in these lectures.

The galactic longitude (l) is reckoned from the ascending node of the Milky Way on the equator, which is situated in the constellation Aquila. The galactic latitude (b) gives the angular distance of the star from the Galaxy. On plate I, at the end of these lectures, will be found a fairly detailed diagram from which the conversion of a and d of a star into l and b may be easily performed. All stars having an apparent magnitude brighter than 4^m are directly drawn.

Instead of giving the galactic longitude and latitude of a star we may content ourselves with giving the galactic square in which the star is situated. For this purpose we assume the sky to be divided into 48 squares, all having the same surface. Two of these squares lie at the northern pole of the Galaxy and are designated GA₁ and GA₂. Twelve lie north of the galactic plane, between 0° and 30° galactic latitude, and are designated GC₁, GC₂, ..., GC₁₂. The corresponding squares south of the galactic equator (the plane of the Galaxy) are called GD₁, GD₂, ..., GD₁₂. The two polar squares at the south pole are called GF₁ and GF₂. Finally we have 10 B-squares, between the A- and C-squares and 10 corresponding E-squares in the southern hemisphere.

The distribution of the squares in the heavens is here graphically represented in the projection of Flamsteed, which has the advantage of giving areas proportional to the corresponding spherical areas, an arrangement necessary, or at least highly desirable, for all stellar statistical researches. It has also the advantage of affording a continuous representation of the whole sky.

The correspondence between squares and stellar constellations is seen from plate II. Arranging the constellations according to their galactic longitude we find north of the galactic equator (in the C-squares) the constellations:—

and south of this equator (in the D-squares):—

mentioning only one constellation for each square.

At the north galactic pole (in the two A-squares) we have:—

and at the south galactic pole (in the two F-squares):—

3.

Changes in the position of a star. From the positions of a star on two or more occasions we obtain its apparent motion, also called the proper motion of the star. We may distinguish between a secular part of this motion and a periodical part. In both cases the motion may be either a reflex of the motion of the observer, and is then called parallactic motion, or it may be caused by a real motion of the star. From the parallactic motion of the star it is possible to deduce its distance from the sun, or its parallax. The periodic parallactic proper motion is caused by the motion of the earth around the sun, and gives the annual parallax (p). In order to obtain available annual parallaxes of a star it is usually necessary for the star to be nearer to us than 5 siriometers, corresponding to a parallax greater than 0.04. More seldom we may in this manner obtain trustworthy values for a distance amounting to 10 siriometers (p = 0.02), or even still greater values. For such large distances the secular parallax, which is caused by the progressive motion of the sun in space, may give better results, especially if the mean distance of a group of stars is simultaneously determined. Such a value of the secular parallax is also called, by Kapteyn, the systematic parallax of the stars.

When we speak of the proper motion of a star, without further specification, we mean always the secular proper motion.

4.

Terrestrial distances are now, at least in scientific researches, universally expressed in kilometres. A kilometre is, however, an inappropriate unit for celestial distances. When dealing with distances in our planetary system, the astronomers, since the time of Newton, have always used the mean distance of the earth from the sun as universal unit of distance. Regarding the distances in the stellar system the astronomers have had a varying practice. German astronomers, Seeliger and others, have long used a stellar unit of distance corresponding to an annual parallax of 0.2, which has been called a “Siriusweite”. To this name it may be justly objected that it has no international use, a great desideratum in science. Against the theoretical definition of this unit it may also be said that a distance is suitably to be defined through another distance and not through an angle—an angle which corresponds moreover, in this case, to the harmonic mean distance of the star and not to its arithmetic mean distance. The same objection may be made to the unit “parsec.” proposed in 1912 by Turner.

For my part I have, since 1911, proposed a stellar unit which, both in name and definition, nearly coincides with the proposition of Seeliger, and which will be exclusively used in these lectures. A siriometer is put equal to 10⁶ times the planetary unit of distance, corresponding to a parallax of 0.206265 (in practice sufficiently exactly 0.2).

In popular writings, another unit: a light-year, has for a very long time been employed. The relation between these units is

5.

In regard to time also, the terrestrial units (second, day, year) are too small for stellar wants. As being consistent with the unit of distance, I have proposed for the stellar unit of time a stellar year (st.), corresponding to 10⁶ years. We thus obtain the same relation between the stellar and the planetary units of length and time, which has the advantage that a velocity of a star expressed in siriometers per stellar year is expressed with the same numerals in planetary units of length per year.

Spectroscopic determinations of the velocities, through the Doppler-principle, are generally expressed in km. per second. The relation with the stellar unit is the following:

Thus the velocity of the sun is 20 km/sec. or 4.22 sir/st. (= 4.22 earth distances from the sun per year).

Of the numerical value of the stellar velocity we shall have opportunity to speak in the following. For the present it may suffice to mention that most stars have a velocity of the same degree as that of the sun (in the mean somewhat greater), and that the highest observed velocity of a star amounts to 72 sir/st. (= 340 km/sec.). In the next chapter I give a table containing the most speedy stars. The least value of the stellar velocity is evidently equal to zero.

6.

Intensity of the radiation. This varies within wide limits. The faintest star which can give an impression on the photographic plates of the greatest instrument of the Mount Wilson observatory (100 inch reflector) is nearly 100 million times fainter than Sirius, a star which is itself more than 10000 million times fainter than the sun—speaking of apparent radiation.

The intensity is expressed in magnitudes (m). The reason is partly that we should otherwise necessarily have to deal with very large numbers, if they were to be proportional to the intensity, and partly that it is proved that the human eye apprehends quantities of light as proportional to m.

This depends upon a general law in psycho-physics, known as Fechner's law, which says that changes of the apparent impression of light are proportional not to the changes of the intensity but to these changes divided by the primitive intensity. A similar law is valid for all sensations. A conversation is inaudible in the vicinity of a waterfall. An increase of a load in the hand from nine to ten hectograms makes no great difference in the feeling, whereas an increase from one to two hectograms is easily appreciable. A match lighted in the day-time makes no increase in the illumination, and so on.

A mathematical analysis shows that from the law of Fechner it follows that the impression increases in arithmetical progression (1, 2, 3, 4, ...) simultaneously with an increase of the intensity in geometrical progression (I, I², I³, I⁴, ...). It is with the sight the same as with the hearing. It is well known that the numbers of vibrations of the notes of a harmonic scale follow each other in a geometrical progression though, for the ear, the intervals between the notes are apprehended as equal. The magnitudes play the same rÔle in relation to the quantities of light as do the logarithms to the corresponding numbers. If a star is considered to have a brightness intermediate between two other stars it is not the difference but the ratio of the quantities of light that is equal in each case.

The branch of astronomy (or physics) which deals with intensities of radiation is called photometry. In order to determine a certain scale for the magnitudes we must choose, in a certain manner, the zero-point of the scale and the scale-ratio.

Both may be chosen arbitrarily. The zero-point is now almost unanimously chosen by astronomers in accordance with that used by the Harvard Observatory. No rigorous definition of the Harvard zero-point, as far as I can see, has yet been given (compare however H. A. 50[3]), but considering that the Pole-star (a UrsÆ Minoris) is used at Harvard as a fundamental star of comparison for the brighter stars, and that, according to the observations at Harvard and those of Hertzsprung (A. N. 4518 [1911]), the light of the Pole-star is very nearly invariable, we may say that the zero-point of the photometric scale is chosen in such a manner that for the Pole-star m = 2.12. If the magnitudes are given in another scale than the Harvard-scale (H. S.), it is necessary to apply the zero-point correction. This amounts, for the Potsdam catalogue, to -0^m.16.

It is further necessary to determine the scale-ratio. Our magnitudes for the stars emanate from Ptolemy. It was found that the scale-ratio—giving the ratio of the light-intensities of two consecutive classes of magnitudes—according to the older values of the magnitudes, was approximately equal to 2½. When exact photometry began (with instruments for measuring the magnitudes) in the middle of last century, the scale-ratio was therefore put equal to 2.5. Later it was found more convenient to choose it equal to 2.512, the logarithm of which number has the value 0.4. The magnitudes being themselves logarithms of a kind, it is evidently more convenient to use a simple value of the logarithm of the ratio of intensity than to use this ratio itself. This scale-ratio is often called the Pogson-scale (used by Pogson in his “Catalogue of 53 known variable stars”, Astr. Obs. of the Radcliffe Observatory, 1856), and is now exclusively used.

It follows from the definition of the scale-ratio that two stars for which the light intensities are in the ratio 100:1 differ by exactly 5 magnitudes. A star of the 6^th magnitude is 100 times fainter than a star of the first magnitude, a star of the 11^th magnitude 10000 times, of the 16^th magnitude a million times, and a star of the 21^st magnitude 100 million times fainter than a star of the first magnitude. The star magnitudes are now, with a certain reservation for systematic errors, determined with an accuracy of 0^m.1, and closer. Evidently, however, there will correspond to an error of 0.1 in the magnitude a considerable uncertainty in the light ratios, when these differ considerably from each other.

A consequence of the definition of m is that we also have to do with negative magnitudes (as well as with negative logarithms). Thus, for example, for Sirius m = -1.58. The magnitudes of the greater planets, as well as those of the moon and the sun, are also negative, as will be seen from the adjoining table, where the values are taken from “Die Photometrie der Gestirne” by G. MÜller.

The apparent magnitude of the sun is given by ZÖllner (1864). The other values are all found in Potsdam, and allude generally to the maximum value of the apparent magnitude of the moon and the planets.

The brightest star is Sirius, which has the magnitude m = -1.58. The magnitude of the faintest visible star evidently depends on the penetrating power of the instrument used. The telescope of William Herschel, used by him and his son in their star-gauges and other stellar researches, allowed of the discerning of stars down to the 14^th magnitude. The large instruments of our time hardly reach much farther, for visual observations. When, however, photographic plates are used, it is easily possible to get impressions of fainter stars, even with rather modest instruments. The large 100-inch mirror of the Wilson Observatory renders possible the photographic observations of stars of the 20^th apparent magnitude, and even fainter.

The observations of visual magnitudes are performed almost exclusively with the photometer of ZÖllner in a more or less improved form.

7.

Absolute magnitude. The apparent magnitude of a star is changed as the star changes its distance from the observer, the intensity increasing indirectly as the square of the distance of the star. In order to make the magnitudes of the stars comparable with each other it is convenient to reduce them to their value at a certain unit of distance. As such we choose one siriometer. The corresponding magnitude will be called the absolute magnitude and is denoted by M.[4] We easily find from the table given in the preceding paragraph that the absolute magnitude of the sun, according to ZÖllner's value of m, amounts to +3.4, of the moon to +31.2. For Jupiter we find M = +24.6, for Venus M = +25.3. The other planets have approximately M = +30.

For the absolute magnitudes of those stars for which it has hitherto been possible to carry out a determination, we find a value of M between -8 and +13. We shall give in the third chapter short tables of the absolutely brightest and faintest stars now known.

8.

Photographic magnitudes. The magnitudes which have been mentioned in the preceding paragraphs all refer to observations taken with the eye, and are called visual magnitudes. The total intensity of a star is, however, essentially dependent on the instrument used in measuring the intensity. Besides the eye, the astronomers use a photographic plate, bolometer, a photo-electric cell, and other instruments. The difference in the results obtained with these instruments is due to the circumstance that different parts of the radiation are taken into account.

The usual photographic plates, which have their principal sensibility in the violet parts of the spectrum, give us the photographic magnitudes of the stars. It is, however, to be remarked that these magnitudes may vary from one plate to another, according to the distributive function of the plate (compare L. M. 67). This variation, which has not yet been sufficiently studied, seems however to be rather inconsiderable, and must be neglected in the following.

The photographic magnitude of a star will in these lectures be denoted by m', corresponding to a visual magnitude m.

In practical astronomy use is also made of plates which, as the result of a certain preparation (in colour baths or in other ways), have acquired a distributive function nearly corresponding to that of the eye, and especially have a maximum point at the same wave-lengths. Such magnitudes are called photo-visual (compare the memoir of Parkhurst in A. J. 36 [1912]).

The photographic magnitude of a star is generally determined from measurements of the diameter of the star on the plate. A simple mathematical relation then permits us to determine m'. The diameter of a star image increases with the time of exposure. This increase is due in part to the diffraction of the telescope, to imperfect achromatism or spherical aberration of the objective, to irregular grinding of the glass, and especially to variations in the refraction of the air, which produce an oscillation of the image around a mean position.

The zero-point of the photographic magnitudes is so determined that this magnitude coincides with the visual magnitude for such stars as belong to the spectral type A0 and have m = 6.0, according to the proposal of the international solar conference at Bonn, 1911.

Determinations of the photographic or photo-visual magnitudes may now be carried out with great accuracy. The methods for this are many and are well summarised in the Report of the Council of the R. A. S. of the year 1913. The most effective and far-reaching method seems to be that proposed by Schwarzschild, called the half-grating method, by which two exposures are taken of the same part of the sky, while at one of the exposures a certain grating is used that reduces the magnitudes by a constant degree.

9.

Colour of the stars. The radiation of a star is different for different wave-lengths (?). As regarding other mass phenomena we may therefore mention:—(1) the total radiation or intensity (I), (2) the mean wave-length (?₀), (3) the dispersion of the wave-length (s). In the preceding paragraphs we have treated of the total radiation of the stars as this is expressed through their magnitudes. The mean wave-length is pretty closely defined by the colour, whereas the dispersion of the wave-length is found from the spectrum of the stars.

There are blue (B), white (W), yellow (Y) and red (R) stars, and intermediate colours. The exact method is to define the colour through the mean wave-length (and not conversely) or the effective wave-length as it is most usually called, or from the colour-index. We shall revert later to this question. There are, however, a great many direct eye-estimates of the colour of the stars.

Colour corresponding to a given spectrum.

Spectrum corresponding to a given colour.

The signs + and - indicate intermediate shades of colour.

The preceding table drawn up by Dr. Malmquist from the colour observations of MÜller and Kempf in Potsdam, shows the connection between the colours of the stars and their spectra.

The Potsdam observations contain all stars north of the celestial equator having an apparent magnitude brighter than 7^m.5.

We find from these tables that there is a well-pronounced regression in the correlation between the spectra and the colours of the stars. Taking together all white stars we find the corresponding mean spectral type to be A0, but to A0 corresponds, upon an average, the colour yellow-white. The yellow stars belong in the mean to the K-type, but the K-stars have upon an average a shade of white in the yellow colour. The coefficient of correlation (r) is not easy to compute in this case, because one of the attributes, the colour, is not strictly graduated (i.e. it is not expressed in numbers defining the colour).[5] Using the coefficient of contingency of Pearson, it is, however, possible to find a fairly reliable value of the coefficient of correlation, and Malmquist has in this way found r = +0.85, a rather high value.

In order to facilitate the discussion of the relation between colour and spectrum it is convenient to deal here with the question of the spectra of the stars.

10.

Spectra of the stars. In order to introduce the discussion I first give a list of the wave-lengths of the Frauenhofer lines in the spectrum, and the corresponding chemical elements.

The first column gives the Frauenhofer denomination of each line. Moreover the hydrogen lines a, , ?, d, e are denoted. The second column gives the name of the corresponding element, to which each line is to be attributed. The third column gives the wave-length expressed in millionths of a millimeter as unit ().

On plate III, where the classification of the stellar spectra according to the Harvard system is reproduced, will be found also the wave-lengths of the principal H and He lines.

By the visual spectrum is usually understood the part of the radiation between the Frauenhofer lines A to H (? = 760 to 400 ), whereas the photographic spectrum generally lies between F and K (? = 500 to 400 ).

In the earliest days of spectroscopy the spectra of the stars were classified according to their visual spectra. This classification was introduced by Secchi and was later more precisely defined by Vogel. The three classes I, II, III of Vogel correspond approximately to the colour classification into white, yellow, and red stars. Photography has now almost entirely taken the place of visual observations of spectra, so that Secchi's and Vogel's definitions of the stellar spectra are no longer applicable. The terminology now used was introduced by Pickering and Miss Cannon and embraces a great many types, of which we here describe the principal forms as they are defined in Part. II of Vol. XXVIII of the Annals of the Harvard Observatory. It may be remarked that Pickering first arranged the types in alphabetical order A, B, C, &c., supposing that order to correspond to the temperature of the stars. Later this was found to be partly wrong, and in particular it was found that the B-stars may be hotter than those of type A. The following is the temperature-order of the spectra according to the opinion of the Harvard astronomers.

Type O (Wolf-Rayet stars). The spectra of these stars consist mainly of bright lines. They are characterized by the bright bands at wave-lengths 463 and 469 , and the line at 501 characteristic of gaseous nebulae is sometimes present.

This type embraces mainly stars of relatively small apparent brightness. The brightest is ? Velorum with m = 2.22. We shall find that the absolute magnitude of these stars nearly coincides with that of the stars of type B.

The type is grouped into five subdivisions represented by the letters Oa, Ob, Oc, Od and Oe. These subdivisions are conditioned by the varying intensities of the bright bands named above. The due sequence of these sub-types is for the present an open question.

Among interesting stars of this type is ? Puppis (Od), in the spectrum of which Pickering discovered a previously unknown series of helium lines. They were at first attributed (by Rydberg) to hydrogen and were called “additional lines of hydrogen”.

Type B (Orion type, Helium stars). All lines are here dark. Besides the hydrogen series we here find the He-lines (396, 403, 412, 414, 447, 471, 493 ).

To this type belong all the bright stars (, ?, d, e, ?, ? and others) in Orion with the exception of Betelgeuze. Further, Spica and many other bright stars.

On plate III e Orionis is taken as representative of this type.

Type A (Sirius type) is characterized by the great intensity of the hydrogen lines (compare plate III). The helium lines have vanished. Other lines visible but faintly.

The greater part of the stars visible to the naked eye are found here. There are 1251 stars brighter than the 6^th magnitude which belong to this type. Sirius, Vega, Castor, Altair, Deneb and others are all A-stars.

Type F (Calcium type). The hydrogen lines still rather prominent but not so broad as in the preceding type. The two calcium lines H and K (396.9, 393.4 ) strongly pronounced.

Among the stars of this type are found a great many bright stars (compare the third chapter), such as Polaris, Canopus, Procyon.

Type G (Sun type). Numerous metallic lines together with relatively faint hydrogen lines.

To this class belong the sun, Capella, a Centauri and other bright stars.

Type K. The hydrogen lines still fainter. The K-line attains its maximum intensity (is not especially pronounced in the figure of plate III).

This is, next to the A-type, the most numerous type (1142 stars) among the bright stars.

We find here ? AndromedÆ, AquilÆ, Arcturus, a CassiopeiÆ, Pollux and Aldebaran, which last forms a transition to the next type.

Type M. The spectrum is banded and belongs to Secchi's third type. The flutings are due to titanium oxide.

Only 190 of the stars visible to the naked eye belong to this type. Generally they are rather faint, but we here find Betelgeuze, a Herculis, Pegasi, a Scorpii (Antares) and most variables of long period, which form a special sub-type Md, characterized by bright hydrogen lines together with the flutings.

Type M has two other sub-types Ma and Mb.

Type N (Secchi's fourth type). Banded spectra. The flutings are due to compounds of carbon.

Here are found only faint stars. The total number is 241. All are red. 27 stars having this spectrum are variables of long period of the same type as Md.

The spectral types may be summed up in the following way:—

The Harvard astronomers do not confine themselves to the types mentioned above, but fill up the intervals between the types with sub-types which are designated by the name of the type followed by a numeral 0, 1, 2, ..., 9. Thus the sub-types between A and F have the designations A0, A1, A2, ..., A9, F0, &c. Exceptions are made as already indicated, for the extreme types O and M.

11.

Spectral index. It may be gathered from the above description that the definition of the types implies many vague moments. Especially in regard to the G-type are very different definitions indeed accepted, even at Harvard.[6] It is also a defect that the definitions do not directly give quantitative characteristics of the spectra. None the less it is possible to substitute for the spectral classes a continuous scale expressing the spectral character of a star. Such a scale is indeed implicit in the Harvard classification of the spectra.

Let us use the term spectral index (s) to define a number expressing the spectral character of a star. Then we may conveniently define this conception in the following way. Let A0 correspond to the spectral index s = 0.0, F0 to s = +1.0, G0 to s = +2.0, K0 to s = +3.0 M0 to s = +4.0 and B0 to s = -1.0. Further, let A1, A2, A3, &c., have the spectral indices +0.1, +0.2, +0.3, &c., and in like manner with the other intermediate sub-classes. Then it is evident that to all spectral classes between B0 and M there corresponds a certain spectral index s. The extreme types O and N are not here included. Their spectral indices may however be determined, as will be seen later.

Though the spectral indices, defined in this manner, are directly known for every spectral type, it is nevertheless not obvious that the series of spectral indices corresponds to a continuous series of values of some attribute of the stars. This may be seen to be possible from a comparison with another attribute which may be rather markedly graduated, namely the colour of the stars. We shall discuss this point in another paragraph. To obtain a well graduated scale of the spectra it will finally be necessary to change to some extent the definitions of the spectral types, a change which, however, has not yet been accomplished.

12.

We have found in §9 that the light-radiation of a star is described by means of the total intensity (I), the mean wave-length (?₀) and the dispersion of the wave-length (s_?). ?₀ and s_? may be deduced from the spectral observations. It must here be observed that the observations give, not the intensities at different wave-lengths but, the values of these intensities as they are apprehended by the instruments employed—the eye or the photographic plate. For the derivation of the true curve of intensity we must know the distributive function of the instrument (L. M. 67). As to the eye, we have reason to believe, from the bolometric observations of Langley (1888), that the mean wave-length of the visual curve of intensity nearly coincides with that of the true intensity-curve, a conclusion easily understood from Darwin's principles of evolution, which demand that the human eye in the course of time shall be developed in such a way that the mean wave-length of the visual intensity curve does coincide with that of the true curve (? = 530 ), when the greatest visual energy is obtained (L. M. 67). As to the dispersion, this is always greater in the true intensity-curve than in the visual curve, for which, according to §10, it amounts to approximately 60 . We found indeed that the visual intensity curve is extended, approximately, from 400 to 760 , a sixth part of which interval, approximately, corresponds to the dispersion s of the visual curve.

In the case of the photographic intensity-curve the circumstances are different. The mean wave-length of the photographic curve is, approximately, 450 , with a dispersion of 16 , which is considerably smaller than in the visual curve.

13.

Both the visual and the photographic curves of intensity differ according to the temperature of the radiating body and are therefore different for stars of different spectral types. Here the mean wave-length follows the formula of Wien, which says that this wave-length varies inversely as the temperature. The total intensity, according to the law of Stephan, varies directly as the fourth power of the temperature. Even the dispersion is dependent on the variation of the temperature—directly as the mean wave-length, inversely as the temperature of the star (L. M. 41)—so that the mean wave-length, as well as the dispersion of the wave-length, is smaller for the hot stars O and B than for the cooler ones (K and M types). It is in this manner possible to determine the temperature of a star from a determination of its mean wave-length (?₀) or from the dispersion in ?. Such determinations (from ?₀) have been made by Scheiner and Wilsing in Potsdam, by Rosenberg and others, though these researches still have to be developed to a greater degree of accuracy.

14.

Effective wave-length. The mean wave-length of a spectrum, or, as it is often called by the astronomers, the effective wave-length, is generally determined in the following way. On account of the refraction in the air the image of a star is, without the use of a spectroscope, really a spectrum. After some time of exposure we get a somewhat round image, the position of which is determined precisely by the mean wave-length. This method is especially used with a so-called objective-grating, which consists of a series of metallic threads, stretched parallel to each other at equal intervals. On account of the diffraction of the light we now get in the focal plane of the objective, with the use of these gratings, not only a fainter image of the star at the place where it would have arisen without grating, but also at both sides of this image secondary images, the distances of which from the central star are certain theoretically known multiples of the effective wave-lengths. In this simple manner it is possible to determine the effective wave-length, and this being a tolerably well-known function of the spectral-index, the latter can also be found. This method was first proposed by Hertzsprung and has been extensively used by Bergstrand, Lundmark and Lindblad at the observatory of Upsala and by others.

15.

Colour-index. We have already pointed out in §9 that the colour may be identified with the mean wave-length (?₀). As further ?₀ is closely connected with the spectral index (s), we may use the spectral index to represent the colour. Instead of s there may also be used another expression for the colour, called the colour-index. This expression was first introduced by Schwarzschild, and is defined in the following way.

We have seen that the zero-point of the photographic scale is chosen in such a manner that the visual magnitude m and the photographic magnitude m' coincide for stars of spectral index 0.0 (A0). The photographic magnitudes are then unequivocally determined. It is found that their values systematically differ from the visual magnitudes, so that for type B (and O) the photographic magnitudes are smaller than the visual, and the contrary for the other types. The difference is greatest for the M-type (still greater for the N-stars, though here for the present only a few determinations are known), for which stars if amounts to nearly two magnitudes. So much fainter is a red star on a photographic plate than when observed with the eye.

The difference between the photographic and the visual magnitudes is called the colour-index (c). The correlation between this index and the spectral-index is found to be rather high (r = +0.96). In L. M. II, 19 I have deduced the following tables giving the spectral-type corresponding to a given colour-index, and inversely.

TABLE 1.
GIVING THE MEAN COLOUR-INDEX CORRESPONDING TO A GIVEN SPECTRAL TYPE OR SPECTRAL INDEX.

TABLE 1*.
GIVING THE MEAN SPECTRAL INDEX CORRESPONDING TO A GIVEN COLOUR-INDEX.

From each catalogue of visual magnitudes of the stars we may obtain their photographic magnitude through adding the colour-index. This may be considered as known (taking into account the high coefficient of correlation between s and c) as soon as we know the spectral type of the star. We may conclude directly that the number of stars having a photographic magnitude brighter than 6.0 is considerably smaller than the number of stars visually brighter than this magnitude. There are, indeed, 4701 stars for which m < 6.0 and 2874 stars having m' < 6.0.

16.

Radial velocity of the stars. From the values of a and d at different times we obtain the components of the proper motions of the stars perpendicular to the line of sight. The third component (W), in the radial direction, is found by the Doppler principle, through measuring the displacement of the lines in the spectrum, this displacement being towards the red or the violet according as the star is receding from or approaching the observer.

The velocity W will be expressed in siriometers per stellar year (sir/st.) and alternately also in km/sec. The rate of conversion of these units is given in §5.

17.

Summing up the remarks here given on the apparent attributes of the stars we find them referred to the following principal groups:—

I. The position of the stars is here generally given in galactic longitude (l) and latitude (b). Moreover their equatorial coordinates (a and d) are given in an abridged notation (ad), where the first four numbers give the right ascension in hours and minutes and the last two numbers give the declination in degrees, the latter being printed in italics if the declination is negative.

Eventually the position is given in galactic squares, as defined in §2.

II. The apparent motion of the stars will be given in radial components (W) expressed in sir/st. and their motion perpendicular to the line of sight. These components will be expressed in one component (u₀) parallel to the galactic plane, and one component (v₀) perpendicular to it. If the distance (r) is known we are able to convert these components into components of the linear velocity perpendicular to the line of sight (U and V).

III. The intensity of the light of the stars is expressed in magnitudes. We may distinguish between the apparent magnitude (m) and the absolute magnitude (M), the latter being equal to the value of the apparent magnitude supposing the star to be situated at a distance of one siriometer.

The apparent magnitude may be either the photographic magnitude (m'), obtained from a photographic plate, or the visual magnitude (m) obtained with the eye.

The difference between these magnitudes is called the colour-index (c = m'-m).

IV. The characteristics of the stellar radiation are the mean wave-length (?₀) and the dispersion (s) in the wave-length. The mean wave-length may be either directly determined (perhaps as effective wave-length) or found from the spectral type (spectral index) or from the colour-index.

There are in all eight attributes of the stars which may be found from the observations:—the spherical position of the star (l, b), its distance (r), proper motion (u₀ and v₀), radial velocity (W), apparent magnitude (m or m'), absolute magnitude (M), spectral type (Sp) or spectral index (s), and colour-index (c). Of these the colour-index, the spectral type, the absolute magnitude and also (to a certain degree) the radial velocity may be considered as independent of the place of the observer and may therefore be considered not as only apparent but also as absolute attributes of the stars.

Between three of these attributes (m, M and r) a mathematical relation exists so that one of them is known as soon as the other two have been found from observations.