[Transcriber's Note: This text uses utf-8 (unicode) file encoding. Ifyou don't see 8 characters in the next line:α β γ ε λ ο ς ″make sure your text reader's "character set" or "file encoding" is setto Unicode (UTF-8). You may also need to change the default font. As alast resort, use the latin-1 version of the file instead. 

In the original text, the units h and m, and ordinals th and st wereprinted as superscripts. For readability, they have not been representedas such in this file. Similarly for the + and - signs when used todescribe intermediate stellar colours. 

Other superscripts are indicated by the carat symbol, ^, and subscriptsby an underline, _. ]

LECTURESON STELLAR STATISTICS

BY

C. V. L. CHARLIER

SCIENTIA PUBLISHERLUND 1921

HAMBURG 1921PRINTED BY _LÜTCKE & WULFF_

CHAPTER I. 

APPARENT ATTRIBUTES OF THE STARS. 

1. Our knowledge of the stars is based on their _apparent_ attributes, obtained from the astronomical observations. The object of astronomy isto deduce herefrom the real or _absolute_ attributes of the stars, whichare 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 indifferent 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 theradiation as consisting of an emanation of small particles from theradiating body (the star). These particles are characterized by certainattributes, which may differ in degree from one particle to another. These attributes may be, for instance, the diameter and form of theparticles, their mode of rotation, &c. By these attributes the opticaland electrical properties of the radiation are to be explained. I shallnot here attempt any such explanation, but shall confine myself to theproperty which the particles have of possessing a different mode ofdeviating from the rectilinear path as they pass from one medium toanother. This deviation depends in some way on one or more attributes ofthe particles. Let us suppose that it depends on a single attribute, which, with a terminology derived from the undulatory theory ofHUYGHENS, may be called the _wave-length_ (λ) of the particle. 

The statistical characteristics of the radiation are then in the firstplace:--

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

(2) the _mean wave-length_ (λ_0) of the radiation, also called (ornearly identical with) the _effective_ wave-length or the colour;

(3) _the dispersion of the wave-length_. This characteristic of theradiation may be determined from the _spectrum_, which also gives thevariation of the radiation with λ, and hence may also determine the meanwave-length of the radiation. 

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

The intensity, the mean wave-length, and the dispersion of thewave-length are in a simple manner connected with the _temperature_(_T_) of the star. According to the radiation laws of STEPHAN and WIENwe find, indeed (compare L. M. 41[1]) that the intensity is proportionalto the fourth power of _T_, whereas the mean wave-length and thedispersion of the wave-length are both inversely proportional to _T_. Itfollows that with increasing temperature the mean wave-lengthdiminishes--the colour changing into violet--and simultaneously thedispersion of the wave-length and also even the total length of thespectrum are reduced (decrease). 

2. _The apparent position of a star_ is generally denoted by its rightascension (α) and its declination (δ). Taking into account the apparentdistribution of the stars in space, it is, however, more practical tocharacterize the position of a star by its galactic longitude (_l_) andits galactic latitude (_b_). Before defining these coordinates, whichwill be generally used in the following pages, it should be pointed outthat we shall also generally give the coordinates α and δ of the starsin a particular manner. We shall therefore use an abridged notation, sothat if for instance α = 17h 44m. 7 and δ = +35°. 84, we shall write

 (αδ) = (174435). 

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

 (αδ) = (1744{35}), so that the last two figures are in italics. 

[Transcriber's Note: In this version of the text, the last two figuresare enclosed in braces to represent the italics. ]

This notation has been introduced by PICKERING for variable stars and isused by him everywhere in the Annals of the Harvard Observatory, but itis also well suited to all stars. This notation gives, simultaneously, the characteristic _numero_ of the stars. It is true that two or morestars may in this manner obtain the same characteristic _numero_. Theyare, however, easily distinguishable from each other through otherattributes. 

The _galactic_ coordinates _l_ and _b_ are referred to the Milky Way(the Galaxy) as plane of reference. The pole of the Milky Way hasaccording to HOUZEAU and GOULD the position (αδ) = (124527). From thedistribution of the stars of the spectral type B I have in L. M. II, 14[2] found a somewhat different position. But having ascertained laterthat the real position of the galactic plane requires a greater numberof stars for an accurate determination of its value, I have preferred toemploy the position used by PICKERING in the Harvard catalogues, namely(αδ) = (124028), or

 α = 12h 40m = 190°, δ = +28°, which position is now exclusively used in the stellar statisticalinvestigations at the Observatory of Lund and is also used in theselectures. 

The galactic longitude (_l_) is reckoned from the ascending node of theMilky Way on the equator, which is situated in the constellation_Aquila_. The galactic latitude (_b_) gives the angular distance of thestar from the Galaxy. On plate I, at the end of these lectures, will befound a fairly detailed diagram from which the conversion of α and δ ofa star into _l_ and _b_ may be easily performed. All stars having anapparent magnitude brighter than 4m are directly drawn. 

Instead of giving the galactic longitude and latitude of a star we maycontent ourselves with giving the galactic _square_ in which the star issituated. For this purpose we assume the sky to be divided into 48squares, all having the same surface. Two of these squares lie at thenorthern pole of the Galaxy and are designated GA_1 and GA_2. Twelve lienorth of the galactic plane, between 0° and 30° galactic latitude, andare designated GC_1, GC_2, . .. , GC_12. The corresponding squares southof the galactic equator (the plane of the Galaxy) are called GD_1, GD_2, . .. , GD_12. The two polar squares at the south pole are called GF_1 andGF_2. Finally we have 10 B-squares, between the A- and C-squares and 10corresponding E-squares in the southern hemisphere. 

The distribution of the squares in the heavens is here graphicallyrepresented in the projection of FLAMSTEED, which has the advantage ofgiving areas proportional to the corresponding spherical areas, anarrangement necessary, or at least highly desirable, for all stellarstatistical researches. It has also the advantage of affording acontinuous representation of the whole sky. 

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

 Hercules, Cygnus, Cepheus, Cassiopæa, Auriga, Gemini, Canis Minor, Pyxis, Vela, Centaurus, Scorpius, Ophiuchus, and south of this equator (in the D-squares):--

 Aquila, Cygnus, Lacerta, Andromeda, Perseus, Orion, Canis Major, Puppis, Carina, Circinus, Corona australis, Sagittarius, mentioning only one constellation for each square. 

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

 Canes Venatici and Coma Berenices, and at the south galactic pole (in the two F-squares):--

 Cetus and Sculptor. 

3. _Changes in the position of a star. _ From the positions of a star ontwo or more occasions we obtain its apparent motion, also called the_proper motion_ of the star. We may distinguish between a _secular_ partof this motion and a _periodical_ part. In both cases the motion may beeither 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 itsdistance from the sun, or its parallax. The periodic parallactic propermotion is caused by the motion of the earth around the sun, and givesthe _annual parallax_ (π). In order to obtain available annualparallaxes of a star it is usually necessary for the star to be nearerto us than 5 siriometers, corresponding to a parallax greater than0″. 04. More seldom we may in this manner obtain trustworthy values for adistance amounting to 10 siriometers (π = 0″. 02), or even still greatervalues. For such large distances the _secular_ parallax, which is causedby the progressive motion of the sun in space, may give better results, especially if the mean distance of a group of stars is simultaneouslydetermined. Such a value of the secular parallax is also called, byKAPTEYN, the _systematic_ parallax of the stars. 

When we speak of the proper motion of a star, without furtherspecification, 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, aninappropriate unit for celestial distances. When dealing with distancesin our planetary system, the astronomers, since the time of NEWTON, havealways used the mean distance of the earth from the sun as universalunit of distance. Regarding the distances in the stellar system theastronomers have had a varying practice. German astronomers, SEELIGERand others, have long used a stellar unit of distance corresponding toan annual parallax of 0″. 2, which has been called a “Siriusweite”. Tothis name it may be justly objected that it has no international use, agreat desideratum in science. Against the theoretical definition of thisunit it may also be said that a distance is suitably to be definedthrough another distance and not through an angle--an angle whichcorresponds moreover, in this case, to the _harmonic_ mean distance ofthe star and not to its arithmetic mean distance. The same objection maybe made to the unit “parsec. ” proposed in 1912 by TURNER. 

For my part I have, since 1911, proposed a stellar unit which, both inname and definition, nearly coincides with the proposition of SEELIGER, and which will be exclusively used in these lectures. A _siriometer_ isput equal to 10^6 times the planetary unit of distance, corresponding toa parallax of 0″. 206265 (in practice sufficiently exactly 0″. 2). 

In popular writings, another unit: a _light-year_, has for a very longtime been employed. The relation between these units is

 1 siriometer = 15. 79 light-years, 1 light-year = 0. 0633 siriometers. 

5. In regard to _time_ also, the terrestrial units (second, day, year)are too small for stellar wants. As being consistent with the unit ofdistance, I have proposed for the stellar unit of time a _stellar year_(st. ), corresponding to 10^6 years. We thus obtain the same relationbetween the stellar and the planetary units of length and time, whichhas the advantage that a _velocity_ of a star expressed in siriometersper stellar year is expressed with the same numerals in planetary unitsof length per year. 

Spectroscopic determinations of the velocities, through theDOPPLER-principle, are generally expressed in km. Per second. Therelation with the stellar unit is the following:

 1 km. /sec. = 0. 2111 sir. /st. , = 0. 2111 planetary units per year, 1 sir. /st. = 4. 7375 km. /sec. 

Thus the velocity of the sun is 20 km. /sec. Or 4. 22 sir. /st. (= 4. 22earth distances from the sun per year). 

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

6. _Intensity of the radiation. _ This varies within wide limits. Thefaintest star which can give an impression on the photographic plates ofthe greatest instrument of the Mount Wilson observatory (100 inchreflector) is nearly 100 million times fainter than Sirius, a star whichis itself more than 10000 million times fainter than the sun--speakingof apparent radiation. 

The intensity is expressed in _magnitudes_ (_m_). The reason is partlythat we should otherwise necessarily have to deal with very largenumbers, if they were to be proportional to the intensity, and partlythat it is proved that the human eye apprehends quantities of light asproportional 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 areproportional not to the changes of the intensity but to these changesdivided by the primitive intensity. A similar law is valid for allsensations. A conversation is inaudible in the vicinity of a waterfall. An increase of a load in the hand from nine to ten hectograms makes nogreat difference in the feeling, whereas an increase from one to twohectograms is easily appreciable. A match lighted in the day-time makesno increase in the illumination, and so on. 

A mathematical analysis shows that from the law of FECHNER it followsthat the impression increases in _arithmetical_ progression (1, 2, 3, 4, . .. ) simultaneously with an increase of the intensity in _geometrical_progression (_I_, _I_^2, _I_^3, _I_^4, . .. ). It is with the sight thesame as with the hearing. It is well known that the numbers ofvibrations of the notes of a harmonic scale follow each other in ageometrical progression though, for the ear, the intervals between thenotes are apprehended as equal. The magnitudes play the same rôle inrelation to the quantities of light as do the logarithms to thecorresponding numbers. If a star is considered to have a brightnessintermediate 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 ofradiation is called _photometry_. In order to determine a certain scalefor 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 almostunanimously chosen by astronomers in accordance with that used by theHarvard 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 (α Ursæ Minoris) is used at Harvardas 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, wemay say that _the zero-point of the photometric scale is chosen in sucha manner that for the Pole-star _m_ = 2. 12_. If the magnitudes are givenin another scale than the Harvard-scale (H. S. ), it is necessary toapply the zero-point correction. This amounts, for the Potsdamcatalogue, to -0m. 16. 

It is further necessary to determine the _scale-ratio_. Our magnitudesfor the stars emanate from PTOLEMY. It was found that thescale-ratio--giving the ratio of the light-intensities of twoconsecutive classes of magnitudes--according to the older values of themagnitudes, was approximately equal to 2½. When exact photometry began(with instruments for measuring the magnitudes) in the middle of lastcentury, the scale-ratio was therefore put equal to 2. 5. Later it wasfound more convenient to choose it equal to 2. 512, the logarithm ofwhich number has the value 0. 4. The magnitudes being themselveslogarithms of a kind, it is evidently more convenient to use a simplevalue of the logarithm of the ratio of intensity than to use this ratioitself. This scale-ratio is often called the POGSON-scale (used byPOGSON in his “Catalogue of 53 known variable stars”, Astr. Obs. Of theRadcliffe Observatory, 1856), and is now exclusively used. 

It follows from the definition of the scale-ratio that two stars forwhich the light intensities are in the ratio 100:1 differ by exactly 5magnitudes. A star of the 6th magnitude is 100 times fainter than a starof the first magnitude, a star of the 11th magnitude 10000 times, of the16th magnitude a million times, and a star of the 21st magnitude 100million times fainter than a star of the first magnitude. The starmagnitudes are now, with a certain reservation for systematic errors, determined with an accuracy of 0m. 1, and closer. Evidently, however, there will correspond to an error of 0. 1 in the magnitude a considerableuncertainty in the light ratios, when these differ considerably fromeach other. 

 Sun -26m. 60 Full moon -11m. 77 Venus - 4m. 28 Jupiter - 2m. 35 Mars - 1m. 79 Mercury - 0m. 90 Saturn + 0m. 88 Uranus + 5m. 86 Neptune + 7m. 66

A consequence of the definition of _m_ is that we also have to do with_negative_ magnitudes (as well as with negative logarithms). Thus, forexample, for _Sirius_ _m_ = -1. 58. The magnitudes of the greaterplanets, as well as those of the moon and the sun, are also negative, aswill 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 othervalues are all found in Potsdam, and allude generally to the maximumvalue of the apparent magnitude of the moon and the planets. 

The brightest star is _Sirius_, which has the magnitude _m_ = -1. 58. Themagnitude of the faintest visible star evidently depends on thepenetrating power of the instrument used. The telescope of WILLIAMHERSCHEL, used by him and his son in their star-gauges and other stellarresearches, allowed of the discerning of stars down to the 14thmagnitude. The large instruments of our time hardly reach much farther, for visual observations. When, however, photographic plates are used, itis easily possible to get impressions of fainter stars, even with rathermodest instruments. The large 100-inch mirror of the Wilson Observatoryrenders possible the photographic observations of stars of the 20thapparent magnitude, and even fainter. 

The observations of visual magnitudes are performed almost exclusivelywith the photometer of ZÖLLNER in a more or less improved form. 

7. _Absolute magnitude. _ The apparent magnitude of a star is changed asthe star changes its distance from the observer, the intensityincreasing indirectly as the square of the distance of the star. Inorder to make the magnitudes of the stars comparable with each other itis convenient to reduce them to their value at a certain unit ofdistance. As such we choose one siriometer. The corresponding magnitudewill be called the _absolute_ magnitude and is denoted by _M_. [4] Weeasily find from the table given in the preceding paragraph that theabsolute 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 hithertobeen 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 ofthe absolutely brightest and faintest stars now known. 

8. _Photographic magnitudes. _ The magnitudes which have been mentionedin the preceding paragraphs all refer to observations taken with theeye, and are called _visual_ magnitudes. The total intensity of a staris, however, essentially dependent on the instrument used in measuringthe intensity. Besides the eye, the astronomers use a photographicplate, bolometer, a photo-electric cell, and other instruments. Thedifference in the results obtained with these instruments is due to thecircumstance that different parts of the radiation are taken intoaccount. 

The usual photographic plates, which have their principal sensibility inthe violet parts of the spectrum, give us the _photographic_ magnitudesof the stars. It is, however, to be remarked that these magnitudes mayvary from one plate to another, according to the distributive functionof the plate (compare L. M. 67). This variation, which has not yet beensufficiently studied, seems however to be rather inconsiderable, andmust be neglected in the following. 

The photographic magnitude of a star will in these lectures be denotedby _m′_, corresponding to a visual magnitude _m_. 

In practical astronomy use is also made of plates which, as the resultof a certain preparation (in colour baths or in other ways), haveacquired a distributive function nearly corresponding to that of theeye, and especially have a maximum point at the same wave-lengths. Suchmagnitudes are called _photo-visual_ (compare the memoir of PARKHURST inA. J. 36 [1912]). 

The photographic magnitude of a star is generally determined frommeasurements of the diameter of the star on the plate. A simplemathematical relation then permits us to determine _m′_. The diameter ofa star image increases with the time of exposure. This increase is duein part to the diffraction of the telescope, to imperfect achromatism orspherical aberration of the objective, to irregular grinding of theglass, and especially to variations in the refraction of the air, whichproduce an oscillation of the image around a mean position. 

The _zero-point_ of the photographic magnitudes is so determined thatthis magnitude coincides with the visual magnitude for such stars asbelong to the spectral type A0 and have _m_ = 6. 0, according to theproposal of the international solar conference at Bonn, 1911. 

Determinations of the photographic or photo-visual magnitudes may now becarried out with great accuracy. The methods for this are many and arewell summarised in the Report of the Council of the R. A. S. Of the year1913. The most effective and far-reaching method seems to be thatproposed by SCHWARZSCHILD, called the half-grating method, by which twoexposures are taken of the same part of the sky, while at one of theexposures a certain grating is used that reduces the magnitudes by aconstant degree. 

9. _Colour of the stars. _ The radiation of a star is different fordifferent wave-lengths (λ). As regarding other mass phenomena we maytherefore mention:--(1) the _total radiation_ or intensity (_I_), (2)the _mean wave-length_ (λ_0), (3) the _dispersion of the wave-length_(σ). In the preceding paragraphs we have treated of the total radiationof the stars as this is expressed through their magnitudes. The meanwave-length is pretty closely defined by the _colour_, whereas thedispersion of the wave-length is found from the _spectrum_ of the stars. 

There are blue (B), white (W), yellow (Y) and red (R) stars, andintermediate colours. The exact method is to define the colour throughthe mean wave-length (and not conversely) or the _effective_ wave-lengthas it is most usually called, or from the _colour-index_. We shallrevert later to this question. There are, however, a great many directeye-estimates of the colour of the stars. 

_Colour corresponding to a given spectrum. _

 _Sp. _ _Colour_ _Number_ B3 YW- 161 A0 YW- 788 A5 YW 115 F5 YW, WY- 295 G5 WY 216 K5 WY+, Y- 552 M Y, Y+ 95 ----------------------------- Sum . .. 2222

_Spectrum corresponding to a given colour. _

 _Colour_ _Sp. _ _Number_ W, W+ A0 281 YW- A0 356 YW A5 482 YW+, YW- F3 211 WY G4 264 WY+, Y- K1 289 Y, Y+ K4 254 RY-, RY K5 85 -------------------------------- Sum . .. 2222

The signs + and - indicate intermediate shades of colour. 

The preceding table drawn up by Dr. MALMQUIST from the colourobservations of MÜLLER and KEMPF in Potsdam, shows the connectionbetween the colours of the stars and their spectra. 

The Potsdam observations contain all stars north of the celestialequator having an apparent magnitude brighter than 7m. 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 spectraltype to be A0, but to A0 corresponds, upon an average, the colouryellow-white. The yellow stars belong in the mean to the K-type, but theK-stars have upon an average a shade of white in the yellow colour. Thecoefficient 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] Usingthe coefficient of contingency of PEARSON, it is, however, possible tofind a fairly reliable value of the coefficient of correlation, andMALMQUIST has in this way found _r_ = +0. 85, a rather high value. 

In order to facilitate the discussion of the relation between colour andspectrum it is convenient to deal here with the question of the spectraof the stars. 

10. _Spectra of the stars. _ In order to introduce the discussion I firstgive a list of the wave-lengths of the FRAUENHOFER lines in thespectrum, and the corresponding chemical elements. 

 _FRAUENHOFER line_ _Element_ λ A 759. 4 B 686. 8 C(α) H (hydrogen) 656. 3 D_1 Na (sodium) 589. 6 D_3 He 587. 6 E Fe (iron) 527. 0 F(β) H 486. 2 (γ) H 434. 1 G Ca (calcium) 430. 8 h(δ) H 410. 2 H(ε) Ca(H) 396. 9 K Ca 393. 4

The first column gives the FRAUENHOFER denomination of each line. Moreover the hydrogen lines α, β, γ, δ, ε are denoted. The second columngives the name of the corresponding element, to which each line is to beattributed. The third column gives the wave-length expressed inmillionths of a millimeter as unit (μμ). 

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

By the visual spectrum is usually understood the part of the radiationbetween the FRAUENHOFER lines A to H (λ = 760 to 400 μμ), whereas thephotographic spectrum generally lies between F and K (λ = 500 to 400μμ). 

In the earliest days of spectroscopy the spectra of the stars wereclassified according to their visual spectra. This classification wasintroduced by SECCHI and was later more precisely defined by VOGEL. Thethree classes I, II, III of VOGEL correspond approximately to the colourclassification into white, yellow, and red stars. Photography has nowalmost entirely taken the place of visual observations of spectra, sothat SECCHI's and VOGEL's definitions of the stellar spectra are nolonger applicable. The terminology now used was introduced by PICKERINGand Miss CANNON and embraces a great many types, of which we heredescribe the principal forms as they are defined in Part. II of Vol. XXVIII of the Annals of the Harvard Observatory. It may be remarked thatPICKERING 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 foundthat the B-stars may be hotter than those of type A. The following isthe temperature-order of the spectra according to the opinion of theHarvard astronomers. 

_Type O_ (WOLF-RAYET stars). The spectra of these stars consist mainlyof bright lines. They are characterized by the bright bands atwave-lengths 463 μμ and 469 μμ, and the line at 501 μμ characteristic ofgaseous nebulae is sometimes present. 

This type embraces mainly stars of relatively small apparent brightness. The brightest is γ Velorum with _m_ = 2. 22. We shall find that theabsolute magnitude of these stars nearly coincides with that of thestars of type B. 

The type is grouped into five subdivisions represented by the lettersOa, Ob, Oc, Od and Oe. These subdivisions are conditioned by the varyingintensities of the bright bands named above. The due sequence of thesesub-types is for the present an open question. 

Among interesting stars of this type is ζ Puppis (Od), in the spectrumof which PICKERING discovered a previously unknown series of heliumlines. They were at first attributed (by RYDBERG) to hydrogen and werecalled “additional lines of hydrogen”. 

_Type B_ (Orion type, Helium stars). All lines are here dark. Besidesthe hydrogen series we here find the He-lines (396, 403, 412, 414, 447, 471, 493 μμ). 

To this type belong all the bright stars (β, γ, δ, ε, ζ, η and others)in Orion with the exception of Betelgeuze. Further, Spica and many otherbright stars. 

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

_Type A_ (Sirius type) is characterized by the great intensity of thehydrogen 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 6th magnitude which belong tothis type. Sirius, Vega, Castor, Altair, Deneb and others are allA-stars. 

_Type F_ (Calcium type). The hydrogen lines still rather prominent butnot so broad as in the preceding type. The two calcium lines H and K(396. 9, 393. 4 μμ) 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 relativelyfaint hydrogen lines. 

To this class belong the sun, Capella, α Centauri and other brightstars. 

_Type K. _ The hydrogen lines still fainter. The K-line attains itsmaximum intensity (is not especially pronounced in the figure of plateIII). 

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

We find here γ Andromedæ, β Aquilæ, Arcturus, α Cassiopeiæ, Pollux andAldebaran, which last forms a transition to the next type. 

_Type M. _ The spectrum is banded and belongs to SECCHI's third type. Theflutings 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, αHerculis, β Pegasi, α Scorpii (Antares) and most variables of longperiod, which form a special sub-type _Md_, characterized by brighthydrogen 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 tocompounds 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 sametype as Md. 

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

 White stars:--SECCHI's type I:--Harvard B and A, Yellow " :-- " " II:-- " F, G and K, Red " :-- " " III:-- " M, " " :-- " " IV:-- " N. 

The Harvard astronomers do not confine themselves to the types mentionedabove, but fill up the intervals between the types with sub-types whichare 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 theextreme types O and M. 

11. _Spectral index. _ It may be gathered from the above description thatthe definition of the types implies many vague moments. Especially inregard to the G-type are very different definitions indeed accepted, even at Harvard. [6] It is also a defect that the definitions do notdirectly give _quantitative_ characteristics of the spectra. None theless it is possible to substitute for the spectral classes a continuousscale expressing the spectral character of a star. Such a scale isindeed implicit in the Harvard classification of the spectra. 

Let us use the term _spectral index_ (_s_) to define a number expressingthe spectral character of a star. Then we may conveniently define thisconception 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 thespectral indices +0. 1, +0. 2, +0. 3, &c. , and in like manner with theother intermediate sub-classes. Then it is evident that to all spectralclasses between B0 and M there corresponds a certain spectral index _s_. The extreme types O and N are not here included. Their spectral indicesmay however be determined, as will be seen later. 

Though the spectral indices, defined in this manner, are directly knownfor every spectral type, it is nevertheless not obvious that the seriesof spectral indices corresponds to a continuous series of values of someattribute of the stars. This may be seen to be possible from acomparison with another attribute which may be rather markedlygraduated, namely the colour of the stars. We shall discuss this pointin another paragraph. To obtain a well graduated scale of the spectra itwill finally be necessary to change to some extent the definitions ofthe spectral types, a change which, however, has not yet beenaccomplished. 

12. We have found in §9 that the light-radiation of a star is describedby means of the total intensity (_I_), the mean wave-length (λ_0) andthe dispersion of the wave-length (σ_λ). λ_0 and σ_λ may be deduced fromthe spectral observations. It must here be observed that theobservations give, not the intensities at different wave-lengths but, the values of these intensities as they are apprehended by theinstruments employed--the eye or the photographic plate. For thederivation of the true curve of intensity we must know the distributivefunction of the instrument (L. M. 67). As to the eye, we have reason tobelieve, from the bolometric observations of LANGLEY (1888), that themean wave-length of the visual curve of intensity nearly coincides withthat of the true intensity-curve, a conclusion easily understood fromDARWIN's principles of evolution, which demand that the human eye in thecourse of time shall be developed in such a way that the meanwave-length of the visual intensity curve does coincide with that of thetrue curve (λ = 530 μμ), when the greatest visual energy is obtained (L. M. 67). As to the dispersion, this is always greater in the trueintensity-curve than in the visual curve, for which, according to §10, it amounts to approximately 60 μμ. We found indeed that the visualintensity curve is extended, approximately, from 400 μμ to 760 μμ, asixth part of which interval, approximately, corresponds to thedispersion σ of the visual curve. 

In the case of the photographic intensity-curve the circumstances aredifferent. The mean wave-length of the photographic curve is, approximately, 450 μμ, with a dispersion of 16 μμ, which is considerablysmaller than in the visual curve. 

13. Both the visual and the photographic curves of intensity differaccording to the temperature of the radiating body and are thereforedifferent for stars of different spectral types. Here the meanwave-length follows the formula of WIEN, which says that thiswave-length varies inversely as the temperature. The total intensity, according to the law of STEPHAN, varies directly as the fourth power ofthe temperature. Even the dispersion is dependent on the variation ofthe temperature--directly as the mean wave-length, inversely as thetemperature of the star (L. M. 41)--so that the mean wave-length, aswell as the dispersion of the wave-length, is smaller for the hot starsO and B than for the cooler ones (K and M types). It is in this mannerpossible to determine the temperature of a star from a determination ofits mean wave-length (λ_0) or from the dispersion in λ. Suchdeterminations (from λ_0) have been made by SCHEINER and WILSING inPotsdam, by ROSENBERG and others, though these researches still have tobe developed to a greater degree of accuracy. 

14. _Effective wave-length. _ The mean wave-length of a spectrum, or, asit is often called by the astronomers, the _effective_ wave-length, isgenerally determined in the following way. On account of the refractionin 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 roundimage, the position of which is determined precisely by the meanwave-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 thediffraction 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 atthe place where it would have arisen without grating, but also at bothsides of this image secondary images, the distances of which from thecentral star are certain theoretically known multiples of the effectivewave-lengths. In this simple manner it is possible to determine theeffective wave-length, and this being a tolerably well-known function ofthe spectral-index, the latter can also be found. This method was firstproposed 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 colourmay be identified with the mean wave-length (λ_0). As further λ_0 isclosely connected with the spectral index (_s_), we may use the spectralindex to represent the colour. Instead of _s_ there may also be usedanother expression for the colour, called the colour-index. Thisexpression was first introduced by SCHWARZSCHILD, and is defined in thefollowing way. 

We have seen that the zero-point of the photographic scale is chosen insuch a manner that the visual magnitude _m_ and the photographicmagnitude _m′_ coincide for stars of spectral index 0. 0 (A0). Thephotographic magnitudes are then unequivocally determined. It is foundthat their values systematically differ from the visual magnitudes, sothat for type B (and O) the photographic magnitudes are smaller than thevisual, and the contrary for the other types. The difference is greatestfor the M-type (still greater for the N-stars, though here for thepresent only a few determinations are known), for which stars if amountsto nearly two magnitudes. So much fainter is a red star on aphotographic plate than when observed with the eye. 

_The difference between the photographic and the visual magnitudes iscalled the colour-index (_c_). _ The correlation between this index andthe 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-typecorresponding to a given colour-index, and inversely. 

TABLE 1. 

_GIVING THE MEAN COLOUR-INDEX CORRESPONDING TO A GIVEN SPECTRAL TYPE ORSPECTRAL INDEX. _

+-------------------+----------------+| Spectral | Colour-index || type | index | |+-------+-----------+----------------+| B0 | -1. 0 | -0. 46 || B5 | -0. 5 | -0. 23 || A0 | 0. 0 | 0. 00 || A5 | +0. 5 | +0. 23 || F0 | +1. 0 | +0. 46 || F5 | +1. 5 | +0. 69 || G0 | +2. 0 | +0. 92 || G5 | +2. 5 | +1. 15 || K0 | +3. 0 | +1. 38 || K5 | +3. 5 | +1. 61 || M0 | +4. 0 | +1. 84 |+-------+-----------+----------------+

TABLE 1*. 

_GIVING THE MEAN SPECTRAL INDEX CORRESPONDING TO A GIVEN COLOUR-INDEX. _

+----------------+-------------------+| Colour-index | Spectral || | index | type |+----------------+---------+---------+| | | || -0. 4 | -0. 70 | B3 || -0. 2 | -0. 80 | B7 || 0. 0 | +0. 10 | A1 || +0. 2 | +0. 50 | A5 || +0. 4 | +0. 90 | A9 || +0. 6 | +1. 30 | F3 || +0. 8 | +1. 70 | F7 || +1. 0 | +2. 10 | G1 || +1. 2 | +2. 50 | G5 || +1. 4 | +2. 90 | G9 || +1. 6 | +3. 30 | K3 || +1. 8 | +3. 70 | K7 || +2. 0 | +4. 10 | M1 |+----------------+---------+---------+

From each catalogue of visual magnitudes of the stars we may obtaintheir photographic magnitude through adding the colour-index. This maybe considered as known (taking into account the high coefficient ofcorrelation between _s_ and _c_) as soon as we know the spectral type ofthe star. We may conclude directly that the number of stars having aphotographic magnitude brighter than 6. 0 is considerably smaller thanthe 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 α and δ atdifferent times we obtain the components of the proper motions of thestars perpendicular to the line of sight. The third component (_W_), inthe radial direction, is found by the DOPPLER principle, throughmeasuring the displacement of the lines in the spectrum, thisdisplacement being towards the red or the violet according as the staris 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 ofthese units is given in §5. 

17. Summing up the remarks here given on the apparent attributes of thestars we find them referred to the following principal groups:--

I. _The position of the stars_ is here generally given in galacticlongitude (_l_) and latitude (_b_). Moreover their equatorialcoordinates (α and δ) are given in an abridged notation (αδ), where thefirst four numbers give the right ascension in hours and minutes and thelast two numbers give the declination in degrees, the latter beingprinted 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 radialcomponents (_W_) expressed in sir. /st. And their motion perpendicular tothe line of sight. These components will be expressed in one component(_u_0_) parallel to the galactic plane, and one component (_v_0_)perpendicular to it. If the distance (_r_) is known we are able toconvert these components into components of the linear velocityperpendicular to the line of sight (_U_ and _V_). 

III. _The intensity of the light_ of the stars is expressed inmagnitudes. We may distinguish between the _apparent_ magnitude (_m_)and the _absolute_ magnitude (_M_), the latter being equal to the valueof the apparent magnitude supposing the star to be situated at adistance 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 meanwave-length (λ_0) and the dispersion (σ) in the wave-length. _The meanwave-length_ may be either directly determined (perhaps as _effective_wave-length) or found from the spectral type (spectral index) or fromthe colour-index. 

There are in all eight attributes of the stars which may be found fromthe observations:--the spherical position of the star (_l_, _b_), itsdistance (_r_), proper motion (_u_0_ and _v_0_), radial velocity (_W_), apparent magnitude (_m_ or _m′_), absolute magnitude (_M_), spectraltype (_Sp_) or spectral index (_s_), and colour-index (_c_). Of thesethe colour-index, the spectral type, the absolute magnitude and also (toa certain degree) the radial velocity may be considered as independentof the place of the observer and may therefore be considered not as onlyapparent but also as _absolute_ attributes of the stars. 

Between three of these attributes (_m_, _M_ and _r_) a mathematicalrelation exists so that one of them is known as soon as the other twohave been found from observations. 

FOOTNOTES:

[Footnote 1: Meddelanden från Lunds Observatorium, No. 41. ]

[Footnote 2: Meddelanden från Lunds Astronomiska Observatorium, SerieII, No. 14. ]

[Footnote 3: Annals of the Harvard Observatory, vol. 50. ]

[Footnote 4: In order to deduce from _M_ the apparent magnitude at adistance corresponding to a parallax of 1″ we may subtract 3m. 48. Toobtain the magnitude corresponding to a parallax of 0″. 1 we may add1. 57. The latter distance is chosen by some writers on stellarstatistics. ]

[Footnote 5: The best colour-scale of the latter sort seems to be thatof OSTHOFF. ]

[Footnote 6: Compare H. A. 50 and H. A. 56 and the remarks in L. M. II, 19. ]

CHAPTER II. 

SOURCES OF OUR PRESENT KNOWLEDGE OF THE STARS. 

18. In this chapter I shall give a short account of the publications inwhich the most complete information on the attributes of the stars maybe obtained, with short notices of the contents and genesis of thesepublications. It is, however, not my intention to give a history ofthese researches. We shall consider more particularly the questionsrelating to the position of the stars, their motion, magnitude, andspectra. 

19. _Place of the stars. _ _Durchmusterungs. _ The most complete data onthe position of the stars are obtained from the star catalogues known as“Durchmusterungs”. There are two such catalogues, which together coverthe whole sky, one--visual--performed in Bonn and called the _BonnerDurchmusterung_ (B. D. ), the other--photographic--performed in Cape _TheCape Photographic Durchmusterung_ (C. P. D. ). As the first of thesecatalogues has long been--and is to some extent even now--our principalsource for the study of the sky and is moreover the first enterprise ofthis kind, I shall give a somewhat detailed account of its origin andcontents, as related by ARGELANDER in the introduction to the B. D. 

B. D. Was planned and performed by the Swedish-Finn ARGELANDER (born inMemel 1799). A scholar of BESSEL he was first called as director in Åbo, then in Hälsingfors, and from there went in 1836 to Bonn, where in theyears 1852 to 1856 he performed this great _Durchmusterung_. Asinstrument he used a FRAUENHOFER comet-seeker with an aperture of 76 mm, a focal length of 650 mm, and 10 times magnifying power. The field ofsight had an extension of 6°. 

In the focus of the objective was a semicircular piece of thin glass, with the edge (= the diameter of the semicircle) parallel to the circleof declination. This edge was sharply ground, so that it formed anarrow dark line perceptible at star illumination. Perpendicular to thisdiameter (the “hour-line”) were 10 lines, at each side of a middle line, drawn at a distance of 7′. These lines were drawn with black oil colouron the glass. 

The observations are performed by the observer A and his assistant B. Ais in a dark room, lies on a chair having the eye at the ocular and caneasily look over 2° in declination. The assistant sits in the roombelow, separated by a board floor, at the _Thiede_ clock. 

From the beginning of the observations the declination circle is fixedat a certain declination (whole degrees). All stars passing the field ata distance smaller than one degree from the middle line are observed. Hence the name “Durchmusterung”. When a star passes the “hour line” themagnitude is called out by A, and noted by B together with the time ofthe clock. Simultaneously the declination is noted by A in the darkness. On some occasions 30 stars may be observed in a minute. 

The first observation was made on Febr. 25, 1852, the last on March 27, 1859. In all there were 625 observation nights with 1841 “zones”. Thetotal number of stars was 324198. 

The catalogue was published by ARGELANDER in three parts in the years1859, 1861 and 1862[7] and embraces all stars between the pole and 2°south of the equator brighter than 9m. 5, according to the scale ofARGELANDER (his aim was to register all stars up to the 9th magnitude). To this scale we return later. The catalogue is arranged in accordancewith the declination-degrees, and for each degree according to the rightascension. Quotations from B. D. Have the form B. D. 23°. 174, whichsignifies: Zone +23°, star No. 174. 

ARGELANDER's work was continued for stars between δ = -2° and δ = -23°by SCHÖNFELD, according to much the same plan, but with a largerinstrument (aperture 159 mm, focal length 1930 mm, magnifying power 26times). The observations were made in the years 1876 to 1881 and include133659 stars. [8]

The positions in B. D. Are given in tenths of a second in rightascension and in tenths of a minute in declination. 

20. _The Cape Photographic Durchmusterung_[9] (C. P. D. ). This embracesthe whole southern sky from -18° to the south pole. Planned by GILL, thephotographs were taken at the Cape Observatory with a DALLMEYER lenswith 15 cm. Aperture and a focal-length of 135 cm. Plates of 30 × 30 cm. Give the coordinates for a surface of 5 × 5 square degrees. Thephotographs were taken in the years 1885 to 1890. The measurements ofthe plates were made by KAPTEYN in Groningen with a “parallactic”measuring-apparatus specially constructed for this purpose, whichpermits of the direct obtaining of the right ascension and thedeclination of the stars. The measurements were made in the years 1886to 1898. The catalogue was published in three parts in the years 1896 to1900. 

The positions have the same accuracy as in B. D. The whole number ofstars is 454875. KAPTEYN considers the catalogue complete to at leastthe magnitude 9m. 2. 

In the two great catalogues B. D. And C. P. D. We have all starsregistered down to the magnitude 9. 0 (visually) and a good way belowthis limit. Probably as far as to 10m. 

A third great Durchmusterung has for some time been in preparation atCordoba in Argentina. [10] It continues the southern zones of SCHÖNFELDand is for the present completed up to 62° southern declination. 

All these Durchmusterungs are ultimately based on star catalogues ofsmaller extent and of great precision. Of these catalogues we shall nothere speak (Compare, however, §23). 

A great “Durchmusterung”, that will include all stars to the 11thmagnitude, has for the last thirty years been in progress at differentobservatories proposed by the congress in Paris, 1888. The observationsproceed very irregularly, and there is little prospect of getting thework finished in an appreciable time. 

21. _Star charts. _ For the present we possess two great photographicstar charts, embracing the whole heaven:--The _Harvard Map_ (H. M. ) andthe _FRANKLIN-ADAMS Charts_ (F. A. C. ). 

_The Harvard Map_, of which a copy (or more correctly two copies) onglass has kindly been placed at the disposal of the Lund Observatory byMr. PICKERING, embraces all stars down to the 11th magnitude. Itconsists of 55 plates, each embracing more than 900 square degrees ofthe sky. The photographs were taken with a small lens of only 2. 5 cms. Aperture and about 32. 5 cms. Focal-length. The time of exposure was onehour. These plates have been counted at the Lund Observatory by HansHENIE. We return later to these counts. 

The _FRANKLIN-ADAMS Charts_ were made by an amateur astronomerFRANKLIN-ADAMS, partly at his own observatory (Mervel Hill) in England, partly in Cape and Johannesburg, Transvaal, in the years 1905-1912. Thephotographs were taken with a _Taylor_ lens with 25 cm. Aperture and afocal-length of 114 cm. , which gives rather good images on a field of 15× 15 square degrees. 

The whole sky is here reproduced on, in all, 206 plates. Each plate wasexposed for 2 hours and 20 minutes and gives images of the stars down tothe 17th magnitude. The original plates are now at the observatory inGreenwich. Some copies on paper have been made, of which the LundObservatory possesses one. It shows stars down to the 14th-15thmagnitudes and gives a splendid survey of the whole sky more complete, indeed, than can be obtained, even for the north sky, by directobservation of the heavens with any telescope at present accessible inSweden. 

The F. A. C. Have been counted by the astronomers of the LundObservatory, so that thus a complete count of the number of stars forthe whole heaven down to the 14th magnitude has been obtained. We shalllater have an opportunity of discussing the results of these counts. 

A great star map is planned in connection with the Paris cataloguementioned in the preceding paragraph. This _Carte du Ciel_ (C. D. C. ) isstill unfinished, but there seems to be a possibility that we shall oneday see this work carried to completion. It will embrace stars down tothe 14th magnitude and thus does not reach so far as the F. A. C. , buton the other hand is carried out on a considerably greater scale andgives better images than F. A. C. And will therefore be of a great valuein the future, especially for the study of the proper motions of thestars. 

22. _Distance of the stars. _ As the determination, from the annualparallax, of the distances of the stars is very precarious if thedistance exceeds 5 sir. (π = 0″. 04), it is only natural that thecatalogues of star-distances should be but few in number. The mostcomplete catalogues are those of BIGOURDAN in the Bulletin astronomiqueXXVI (1909), of KAPTEYN and WEERSMA in the publications of Groningen Nr. 24 (1910), embracing 365 stars, and of WALKEY in the “Journal of theBritish Astronomical Association XXVII” (1917), embracing 625 stars. Through the spectroscopic method of ADAMS it will be possible to enlargethis number considerably, so that the distance of all stars, for whichthe spectrum is well known, may be determined with fair accuracy. ADAMShas up to now published 1646 parallax stars. 

23. _Proper motions. _ An excellent catalogue of the proper motions ofthe stars is LEWIS BOSS's “Preliminary General Catalogue of 6188 stars”(1910) (B. P. C. ). It contains the proper motions of all stars down tothe sixth magnitude (with few exceptions) and moreover some fainterstars. The catalogue is considered by the editor only as a preliminaryto a greater catalogue, which is to embrace some 25000 stars and is nownearly completed. 

24. _Visual magnitudes. _ The Harvard observatory has, under thedirection of PICKERING, made its principal aim to study themagnitudes of the stars, and the history of this observatory is at thesame time the history of the treatment of this problem. PICKERING, inthe genuine American manner, is not satisfied with the three thirds ofthe sky visible from the Harvard observatory, but has also founded adaughter observatory in South America, at Arequipa in Peru. It istherefore possible for him to publish catalogues embracing the wholeheaven from pole to pole. The last complete catalogue (1908) of themagnitudes of the stars is found in the “Annals of the HarvardObservatory T. 50” (H. 50). It contains 9110 stars and can be consideredas complete to the magnitude 6m. 5. To this catalogue are generallyreferred the magnitudes which have been adopted at the Observatory ofLund, and which are treated in these lectures. 

A very important, and in one respect even still more comprehensive, catalogue of visual magnitudes is the “Potsdam General Catalogue” (P. G. C. ) by MÜLLER and KEMPF, which was published simultaneously with H. 50. It contains the magnitude of 14199 stars and embraces all stars on thenorthern hemisphere brighter than 7m. 5 (according to B. D. ). We havealready seen that the zero-point of H. 50 and P. G. C. Is somewhatdifferent and that the magnitudes in P. G. C. Must be increased by-0m. 16 if they are to be reduced to the Harvard scale. The differencebetween the two catalogues however is due to some extent to the colourof the stars, as has been shown by Messrs. MÜLLER and KEMPF. 

25. _Photographic magnitudes. _ Our knowledge of this subject is stillrather incomplete. The most comprehensive catalogue is the“Actinometrie” by SCHWARZSCHILD (1912), containing the photographicmagnitudes of all stars in B. D. Down to the magnitude 7m. 5 between theequator and a declination of +20°. In all, 3522 stars. The photographicmagnitudes are however not reduced for the zero-point (compare §6). 

These is also a photometric photographic catalogue of the stars nearestto the pole in PARKHURST's “Yerkes actinometry” (1912), [11] whichcontains all stars in B. D. Brighter than 7m. 5 between the pole and 73°northern declination. The total number of stars is 672. 

During the last few years the astronomers of Harvard and Mount Wilsonhave produced a collection of “standard photographic magnitudes” forfaint stars. These stars, which are called the _polar sequence_, [12] alllie in the immediate neighbourhood of the pole. The list is extendeddown to the 20th magnitude. Moreover similar standard photographicmagnitudes are given in H. A. 71, 85 and 101. 

A discussion of the _colour-index_ (_i. E. _, the difference between thephotographic and the visual magnitudes) will be found in L. M. II, 19. When the visual magnitude and the type of spectrum are known, thephotographic magnitude may be obtained, with a generally sufficientaccuracy, by adding the colour-index according to the table 1 in §15above. 

26. _Stellar spectra. _ Here too we find the Harvard Observatory to bethe leading one. The same volume of the Annals of the HarvardObservatory (H. 50) that contains the most complete catalogue of visualmagnitudes, also gives the spectral types for all the stars thereincluded, _i. E. _, for all stars to 6m. 5. Miss CANNON, at the HarvardObservatory, deserves the principal credit for this great work. Notcontent with this result she is now publishing a still greater workembracing more than 200000 stars. The first four volumes of this workare now published and contain the first twelve hours of rightascension, so that half the work is now printed. [13]

27. _Radial velocity. _ In this matter, again, we find America to be theleading nation, though, this time, it is not the Harvard or the MountWilson but the Lick Observatory to which we have to give the honour. Theeminent director of this observatory, W. W. CAMPBELL, has in a highdegree developed the accuracy in the determination of radial velocitiesand has moreover carried out such determinations in a large scale. The“Bulletin” No. 229 (1913) of the Lick Observatory contains the radialvelocity of 915 stars. At the observatory of Lund, where as far aspossible card catalogues of the attributes of the stars are collected, GYLLENBERG has made a catalogue of this kind for the radial velocities. The total number of stars in this catalogue now amounts to 1640. [14]

28. Finally I shall briefly mention some comprehensive works on morespecial questions regarding the stellar system. 

On _variable stars_ there is published every year by HARTWIG in the“Vierteljahrschrift der astronomischen Gesellschaft” a catalogue of allknown variable stars with needful information about their minima &c. This is the completest and most reliable of such catalogues, and isalways up to date. A complete historical catalogue of the variables isgiven in “Geschichte und Literatur des Lichtwechsels der bis Ende 1915als sicher veränderlich anerkannten Sterne nebst einem Katalog derElemente ihres Lichtwechsels” von G. MÜLLER und E. HARTWIG. Leipzig1918, 1920. 

On _nebulae_ we have the excellent catalogues of DREYER, the “NewGeneral Catalogue” (N. G. C. ) of 1890 in the “Memoirs of theAstronomical Society” vol. 49, the “Index catalogues” (I. C. ) in thesame memoirs, vols. 51 and 59 (1895 and 1908). These catalogues containall together 13226 objects. 

Regarding other special attributes I refer in the first place to theimportant Annals of the Harvard Observatory. Other references will begiven in the following, as need arises. 

FOOTNOTES:

[Footnote 7: “Bonner Sternverzeichnis” in den AstronomischenBeobachtungen auf der Sternwarte zu Bonn, Dritter bis Fünfter Band. Bonn1859-62. ]

[Footnote 8: “Bonner Durchmusterung”, Vierte Sektion. Achter Band derAstronomischen Beobachtungen zu Bonn, 1886. ]

[Footnote 9: “The Cape Photographic Durchmusterung” by DAVID GILL and J. C. KAPTEYN, Annals of the Cape Observatory, vol. III-V (1896-1900). ]

[Footnote 10: “Cordoba Durchmusterung” by J. THOME. Results of theNational Argentine Observatory, vol. 16, 17, 18, 21 (1894-1914). ]

[Footnote 11: Aph. J. , vol. 36. ]

[Footnote 12: H. A. , vol. 71. ]

[Footnote 13: H. A. , vol. 91, 92, 93, 94. ]

[Footnote 14: A catalogue of radial velocities has this year beenpublished by J. VOUTE, embracing 2071 stars. “First catalogue of radialvelocities”, by J. VOUTE. Weltevreden, 1920. ]

CHAPTER III. 

SOME GROUPS OF KNOWN STARS. 

29. The number of cases in which all the eight attributes of the starsdiscussed in the first chapter are well known for one star is verysmall, and certainly does not exceed one hundred. These cases referprincipally to such stars as are characterized either by greatbrilliancy or by a great proper motion. The principal reason why thesestars are better known than others is that they lie rather near oursolar system. Before passing on to consider the stars from more generalstatistical points of view, it may therefore be of interest first tomake ourselves familiar with these well-known stars, stronglyemphasizing, however, the exceptional character of these stars, andcarefully avoiding any generalization from the attributes we shall herefind. 

30. _The apparently brightest stars. _ We begin with these objects sowell known to every lover of the stellar sky. The following tablecontains all stars the apparent visual magnitude of which is brighterthan 1m. 5. 

The first column gives the current number, the second the name, thethird the equatorial designation (αδ). It should be remembered that thefirst four figures give the hour and minutes in right ascension, thelast two the declination, italics showing negative declination. Thefourth column gives the galactic square, the fifth and sixth columns thegalactic longitude and latitude. The seventh and eighth columns give theannual parallax and the corresponding distance expressed in siriometers. The ninth column gives the proper motion (μ), the tenth the radialvelocity _W_ expressed in sir. /st. (To get km. /sec. We may multiply by4. 7375). The eleventh column gives the apparent visual magnitude, thetwelfth column the absolute magnitude (_M_), computed from _m_ with thehelp of _r_. The 13th column gives the type of spectrum (_Sp_), and thelast column the photographic magnitude (_m′_). The difference between_m′_ and _m_ gives the colour-index (_c_). 

TABLE 2. 

_THE APPARENTLY BRIGHTEST STARS. _

+--+---------------------+----------+--------+-----+-------+-------+-------+| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | Position | Distance || | _Name_ |----------+--------+-----+-------+-------+-------+| | | (αδ) | Square | _l_ | _b_ | π | _r_ |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || 1|Sirius |(0640{16})| GD_7 | 195°| - 8° |0″. 876 | 0. 5 || 2|Canopus |(0621{52})| GD_8 | 229 | -24 | 0. 007 | 29. 5 || 3|Vega |(183338) | GC_2 | 30 | +17 | 0. 094 | 2. 2 || 4|Capella |(050945) | GC_5 | 131 | + 5 | 0. 066 | 3. 1 || 5|Arcturus |(141119) | GA_2 | 344 | +68 | 0. 075 | 2. 7 || 6|α Centauri |(1432{60})| GD_10 | 284 | - 2 | 0. 759 | 0. 3 || 7|Rigel |(0509{08})| GD_6 | 176 | -24 | 0. 007 | 29. 5 || 8|Procyon |(073405) | GC_7 | 182 | +14 | 0. 324 | 0. 6 || 9|Achernar |(0134{57})| GE_8 | 256 | -59 | 0. 051 | 4. 0 ||10|β Centauri |(1356{59})| GC_10 | 280 | + 2 | 0. 037 | 5. 6 ||11|Altair |(194508) | GD_1 | 15 | -10 | 0. 238 | 0. 9 ||12|Betelgeuze |(054907) | GD_6 | 168 | - 8 | 0. 030 | 6. 9 ||13|Aldebaran |(043016) | GD_5 | 149 | -19 | 0. 078 | 2. 8 ||14|Pollux |(073928) | GC_6 | 160 | +25 | 0. 064 | 3. 2 ||15|Spica |(1319{10})| GB_8 | 286 | +51 | . . | . . ||16|Antares |(1623{26})| GC_11 | 320 | +14 | 0. 029 | 7. 1 ||17|Fomalhaut |(2252{30})| GE_10 | 348 | -66 | 0. 138 | 1. 5 ||18|Deneb |(203844) | GC_2 | 51 | + 1 | . . | . . ||19|Regulus |(100312) | GB_6 | 196 | +50 | 0. 033 | 6. 3 ||20|β Crucis |(1241{59})| GC_10 | 270 | + 3 | 0. 008 | 25. 8 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || | Mean | . . | . . | . . | 23°. 5|0″. 134 | 7. 3 |+--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+| 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 |+--+---------------------+------+--------+---------+---------+----+------+| | | Motion | Magnitude | Spectrum || | _Name_ +------+--------+---------+---------+----+------+| | | μ | _W_ | _m_ | _M_ |_Sp_| _m′_ |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || 1|Sirius | 1″. 32| - 1. 56| -1m. 58 | -0m. 3 |A |-1. 58 || 2|Canopus | 0. 02| + 4. 39| -0. 86 | -8. 2 |F |-0. 40 || 3|Vega | 0. 35| - 2. 91| 0. 14 | -1. 6 |A | 0. 14 || 4|Capella | 0. 44| + 6. 38| 0. 21 | -2. 8 |G | 1. 13 || 5|Arcturus | 2. 28| - 0. 82| 0. 24 | -1. 9 |K | 1. 62 || 6|α Centauri | 3. 68| - 4. 69| 0. 33 | +3. 2 |G | 1. 25 || 7|Rigel | 0. 00| + 4. 77| 0. 34 | -7. 0 |B8p | 0. 25 || 8|Procyon | 1. 24| - 0. 74| 0. 48 | +1. 5 |F5 | 1. 17 || 9|Achernar | 0. 09| . . | 0. 60 | -2. 4 |B5 | 0. 87 ||10|β Centauri | 0. 04| + 2. 53| 0. 86 | -2. 9 |B1 | 0. 45 ||11|Altair | 0. 66| - 6. 97| 0. 89 | +1. 2 |A5 | 1. 12 ||12|Betelgeuze | 0. 03| + 4. 43| 0. 92 | -3. 3 |Ma | 2. 76 ||13|Aldebaran | 0. 20| +11. 63| 1. 06 | -1. 2 |K5 | 2. 67 ||14|Pollux | 0. 07| + 0. 82| 1. 21 | -1. 3 |K | 2. 59 ||15|Spica | 0. 06| + 0. 34| 1. 21 | . . |B2 | 0. 84 ||16|Antares | 0. 03| - 0. 63| 1. 22 | -3. 0 |Map | 3. 06 ||17|Fomalhaut | 0. 37| + 1. 41| 1. 29 | +0. 4 |A3 | 1. 43 ||18|Deneb | 0. 00| - 0. 84| 1. 33 | . . |A2 | 1. 42 ||19|Regulus | 0. 25| . . | 1. 34 | -2. 7 |B8 | 1. 25 ||20|β Crucis | 0. 06| + 2. 74| 1. 50 | -5. 6 |B1 | 1. 09 |+--+---------------------+------+--------+---------+---------+----+------+| | | | | | | | _m′_ || | Mean | 0″. 56| 3. 26| +0m. 64 | -2m. 1 |F1 |+1. 13 |+--+---------------------+------+--------+---------+---------+----+------+

The values of (αδ), _m_, _Sp_ are taken from H. 50. The values of _l_, _b_ are computed from (αδ) with the help of tables in preparation at theLund Observatory, or from the original to plate I at the end, allowingthe conversion of the equatorial coordinates into galactic ones. Thevalues of π are generally taken from the table of KAPTEYN and WEERSMAmentioned in the previous chapter. The values of μ are obtained from B. P. C. , those of the radial velocity (_W_) from the card catalogue inLund already described. 

There are in all, in the sky, 20 stars having an apparent magnitudebrighter than 1m. 5. The brightest of them is _Sirius_, which, owing toits brilliancy and position, is visible to the whole civilized world. Ithas a spectrum of the type A0 and hence a colour-index nearly equal to0. 0 (observations in Harvard give _c_ = +0. 06). Its apparent magnitudeis -1m. 6, nearly the same as that of Mars in his opposition. Itsabsolute magnitude is -0m. 3, _i. E. _, fainter than the apparentmagnitude, from which we may conclude that it has a distance from ussmaller than one siriometer. We find, indeed, from the eighth columnthat _r_ = 0. 5 sir. The proper motion of Sirius is 1″. 32 per year, whichis rather large but still not among the largest proper motions as willbe seen below. From the 11th column we find that Sirius is movingtowards us with a velocity of 1. 6 sir. /st. (= 7. 6 km. /sec. ), a rathersmall velocity. The third column shows that its right ascension is 6h40m and its declination -16°. It lies in the square GD_7 and itsgalactic coordinates are seen in the 5th and 6th columns. 

The next brightest star is _Canopus_ or α Carinæ at the south sky. If wemight place absolute confidence in the value of _M_ (= -8. 2) in the 12thcolumn this star would be, in reality, a much more imposing apparitionthan Sirius itself. Remembering that the apparent magnitude of the moon, according to §6, amounts to -11. 6, we should find that Canopus, ifplaced at a distance from us equal to that of Sirius (_r_ = 0. 5 sir. ), would shine with a lustre equal to no less than a quarter of that of themoon. It is not altogether astonishing that a fanciful astronomer shouldhave thought Canopus to be actually the central star in the wholestellar system. We find, however, from column 8 that its supposeddistance is not less that 30 sir. We have already pointed out thatdistances greater than 4 sir. , when computed from annual parallaxes, must generally be considered as rather uncertain. As the value of _M_is intimately dependent on that of _r_ we must consider speculationsbased on this value to be very vague. Another reason for a doubt about agreat value for the real luminosity of this star is found from its typeof spectrum which, according to the last column, is F0, a type which, aswill be seen, is seldom found among giant stars. A better support for alarge distance could on the other hand be found from the small propermotion of this star. Sirius and Canopus are the only stars in the skyhaving a negative value of the apparent visual magnitude. 

Space will not permit us to go through this list star for star. We maybe satisfied with some general remarks. 

In the fourth column is the galactic square. We call to mind that allthese squares have the same area, and that there is therefore the sameprobability _a priori_ of finding a star in one of the squares as inanother. The squares GC and GD lie along the galactic equator (the MilkyWay). We find now from column 4 that of the 20 stars here consideredthere are no less than 15 in the galactic equator squares and only 5outside, instead of 10 in the galactic squares and 10 outside, as wouldhave been expected. The number of objects is, indeed, too small to allowus to draw any cosmological conclusions from this distribution, but weshall find in the following many similar instances regarding objectsthat are principally accumulated along the Milky Way and are scanty atthe galactic poles. We shall find that in these cases we may _generally_conclude from such a partition that we then have to do with objects_situated far from the sun_, while objects that are uniformlydistributed on the sky lie relatively near us. It is easy to understandthat this conclusion is a consequence of the supposition, confirmed byall star counts, that the stellar system extends much farther into spacealong the Milky Way than in the direction of its poles. 

If we could permit ourselves to draw conclusions from the small materialhere under consideration, we should hence have reason to believe thatthe bright stars lie relatively far from us. In other words we shouldconclude that the bright stars seem to be bright to us not because oftheir proximity but because of their large intrinsic luminosity. Column8 really tends in this direction. Certainly the distances are not inthis case colossal, but they are nevertheless sufficient to show, insome degree, this uneven partition of the bright stars on the sky. Themean distance of these stars is as large as 7. 5 sir. Only α Centauri, Sirius, Procyon and Altair lie at a distance smaller than onesiriometer. Of the other stars there are two that lie as far as 30siriometers from our system. These are the two giants Canopus and Rigel. Even if, as has already been said, the distances of these stars may beconsidered as rather uncertain, we must regard them as being ratherlarge. 

As column 8 shows that these stars are rather far from us, so we findfrom column 12, that their absolute luminosity is rather large. The meanabsolute magnitude is, indeed, -2m. 1. We shall find that only thegreatest and most luminous stars in the stellar system have a negativevalue of the absolute magnitude. 

The mean value of the proper motions of the bright stars amounts to0″. 56 per year and may be considered as rather great. We shall, indeed, find that the mean proper motion of the stars down to the 6th magnitudescarcely amounts to a tenth part of this value. On the other hand wefind from the table that the high value of this mean is chiefly due tothe influence of four of the stars which have a large proper motion, namely Sirius, Arcturus, α Centauri and Procyon. The other stars have aproper motion smaller than 1″ per year and for half the number of starsthe proper motion amounts to approximately 0″. 05, indicating theirrelatively great distance. 

That the absolute velocity of these stars is, indeed, rather small maybe found from column 10, giving their radial velocity, which in the meanamounts to only three siriometers per stellar year. From the discussionbelow of the radial velocities of the stars we shall find that this is arather small figure. This fact is intimately bound up with the generallaw in statistical mechanics, to which we return later, that stars withlarge masses generally have a small velocity. We thus find in the radialvelocities fresh evidence, independent of the distance, that thesebright stars are giants among the stars in our stellar system. 

We find all the principal spectral types represented among the brightstars. To the helium stars (B) belong Rigel, Achernar, β Centauri, Spica, Regulus and β Crucis. To the Sirius type (A) belong Sirius, Vega, Altair, Fomalhaut and Deneb. To the Calcium type (F) Canopus andProcyon. To the sun type (G) Capella and α Centauri. To the K-typebelong Arcturus, Aldebaran and Pollux and to the M-type the two redstars Betelgeuze and Antares. Using the spectral indices as anexpression for the spectral types we find that the mean spectral indexof these stars is +1. 1 corresponding to the spectral type F1. 

31. _Stars with the greatest proper motion. _ In table 3 I have collectedthe stars having a proper motion greater than 3″ per year. Thedesignations are the same as in the preceding table, except that thenames of the stars are here taken from different catalogues. 

In the astronomical literature of the last century we find the star 1830Groombridge designed as that which possesses the greatest known propermotion. It is now distanced by two other stars C. P. D. 5h. 243discovered in the year 1897 by KAPTEYN and INNES on the plates taken forthe Cape Photographic Durchmusterung, and BARNARD's star in Ophiuchus, discovered 1916. The last-mentioned star, which possesses the greatestproper motion now known, is very faint, being only of the 10thmagnitude, and lies at a distance of 0. 40 sir. From our sun and ishence, as will be found from table 5 the third nearest star for which weknow the distance. Its linear velocity is also very great, as we findfrom column 10, and amounts to 19 sir. /st. (= 90 km. /sec. ) in thedirection towards the sun. The absolute magnitude of this star is 11m. 7and it is, with the exception of one other, the very faintest star nowknown. Its spectral type is Mb, a fact worth fixing in our memory, asdifferent reasons favour the belief that it is precisely the M-type thatcontains the very faintest stars. Its apparent velocity (_i. E. _, theproper motion) is so great that the star in 1000 years moves 3°, or asmuch as 6 times the diameter of the moon. For this star, as well as forits nearest neighbours in the table, observations differing only by ayear are sufficient for an approximate determination of the value of theproper motion, for which in other cases many tens of years arerequired. 

Regarding the distribution of these stars in the sky we find that, unlike the brightest stars, they are not concentrated along the MilkyWay. On the contrary we find only 6 in the galactic equator squares and12 in the other squares. We shall not build up any conclusion on thisirregularity in the distribution, but supported by the general thesis ofthe preceding paragraph we conclude only that these stars must berelatively near us. This follows, indeed, directly from column 8, as notless than eleven of these stars lie within one siriometer from our sun. Their mean distance is 0. 87 sir. 

TABLE 3. 

_STARS WITH THE GREATEST PROPER MOTION. _

+--+---------------------+----------+--------+-----+-------+-------+-------+| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | Position | Distance || | _Name_ |----------+--------+-----+-------+-------+-------+| | | (αδ) | Square | _l_ | _b_ | π | _r_ |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || 1|Barnards star |(175204) | GC_12 | 358°| +12° |0″. 515 | 0. 40 || 2|C. Z. 5h. 243 |(0507{44})| GE_7 | 218 | -35 | 0. 319 | 0. 65 || 3|Groom. 1830 |(114738) | GA_1 | 135 | +75 | 0. 102 | 2. 02 || 4|Lac. 9352 |(2259{36})| GE_10 | 333 | -66 | 0. 292 | 0. 71 || 5|C. G. A. 32416 |(2359{37})| GF_2 | 308 | -75 | 0. 230 | 0. 89 || 6|61 Cygni |(210238) | GD_2 | 50 | - 7 | 0. 311 | 0. 66 || 7|Lal. 21185 |(105736) | GB_5 | 153 | +66 | 0. 403 | 0. 51 || 8|ε Indi |(2155{57})| GE_9 | 304 | -47 | 0. 284 | 0. 73 || 9|Lal. 21258 |(110044) | GB_4 | 135 | +64 | 0. 203 | 1. 02 ||10|O^2 Eridani |(0410{07})| GE_5 | 168 | -36 | 0. 174 | 1. 19 ||11|Proxima Centauri |(1422{62})| GD_10 | 281 | - 2 | 0. 780 | 0. 26 ||12|Oe. A. 14320 |(1504{15})| GB_9 | 314 | +35 | 0. 035 | 5. 90 ||13|μ Cassiopeiæ |(010154) | GD_4 | 93 | - 8 | 0. 112 | 1. 84 ||14|α Centauri |(1432{60})| GD_10 | 284 | - 2 | 0. 759 | 0. 27 ||15|Lac. 8760 |(2111{39})| GE_10 | 332 | -44 | 0. 248 | 0. 83 ||16|Lac. 1060 |(0315{43})| GE_7 | 216 | -55 | 0. 162 | 1. 27 ||17|Oe. A. 11677 |(111466) | GB_8 | 103 | +50 | 0. 198 | 1. 04 ||18|Van Maanens star |(004304) | GD_8 | 92 | -58 | 0. 246 | 0. 84 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || | Mean | . . | . . | | 41° |0″. 298 | 0. 87 |+--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+| 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 |+--+---------------------+------+--------+---------+---------+----+------+| | | Motion | Magnitude | Spectrum || | _Name_ +------+--------+---------+---------+----+------+| | | μ | _W_ | _m_ | _M_ |_Sp_| _m′_ |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || 1|Barnards star |10″. 29| -19 | 9m. 7 | +11m. 7|Mb |11. 5 || 2|C. Z. 5h. 243 | 8. 75| +51 | 9. 2 | +10. 1 |K2 |10. 6 || 3|Groom. 1830 | 7. 06| -20 | 6. 5 | +5. 0 |G5 | 7. 6 || 4|Lac. 9352 | 6. 90| +2 | 7. 5 | +8. 2 |K | 8. 9 || 5|C. G. A. 32416 | 6. 11| +5 | 8. 2 | +8. 5 |G | 9. 1 || 6|61 Cygni | 5. 27| -13 | 5. 6 | +6. 5 |K5 | 7. 2 || 7|Lal. 21185 | 4. 77| -18 | 7. 6 | +9. 1 |Mb | 8. 9 || 8|ε Indi | 4. 70| -8 | 4. 7 | +5. 4 |K5 | 6. 3 || 9|Lal. 21258 | 4. 47| +14 | 8. 5 | +8. 5 |Ma |10. 3 ||10|O^2 Eridani | 4. 11| -9 | 4. 7 | +4. 3 |G5 | 5. 8 ||11|Proxima Centauri | 3. 85| . . | 11. 0 | +13. 9 |. . |13. 5 ||12|Oe. A. 14320 | 3. 75| +61 | 9. 0 | +5. 1 |G0 | 9. 9 ||13|μ Cassiopeiæ | 3. 73| -21 | 5. 7 | +4. 4 |G3 | 6. 8 ||14|α Centauri | 3. 68| -5 | 0. 3 | +3. 2 |G | 1. 2 ||15|Lac. 8760 | 3. 53| +3 | 6. 6 | +7. 0 |G | 7. 5 ||16|Lac. 1060 | 3. 05| +18 | 5. 6 | +5. 1 |G5 | 6. 7 ||17|Oe. A. 11677 | 3. 03| . . | 9. 2 | +9. 1 |Ma |11. 0 ||18|Van Maanens star | 3. 01| . . | 12. 3 | +12. 7 |F0 |12. 9 |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || | Mean | 5″. 00| 17. 8 | 7m. 3 | +7m. 6|G8 | 8. 7 |+--+---------------------+------+--------+---------+---------+----+------+

That the great proper motion does not depend alone on the proximity ofthese stars is seen from column 10, giving the radial velocities. Forsome of the stars (4) the radial velocity is for the present unknown, but the others have, with few exceptions, a rather great velocityamounting in the mean to 18 sir. /st. (= 85 km. /sec. ), if no regard istaken to the sign, a value nearly five times as great as the absolutevelocity of the sun. As this is only the component along the line ofsight, the absolute velocity is still greater, approximately equal tothe component velocity multiplied by √2. We conclude that the greatproper motions depend partly on the proximity, partly on the greatlinear velocities of the stars. That both these attributes here reallycooperate may be seen from the absolute magnitudes (_M_). 

The apparent and the absolute magnitudes are for these stars nearlyequal, the means for both been approximately 7m. This is a consequenceof the fact that the mean distance of these stars is equal to onesiriometer, at which distance _m_ and _M_, indeed, do coincide. We findthat these stars have a small luminosity and may be considered as_dwarf_ stars. According to the general law of statistical mechanicsalready mentioned small bodies upon an average have a great absolutevelocity, as we have, indeed, already found from the observed radialvelocities of these stars. 

As to the spectral type, the stars with great proper motions are allyellow or red stars. The mean spectral index is +2. 8, corresponding tothe type G8. If the stars of different types are put together we get thetable

 _Type_ _Number_ _Mean value of M_ G 8 5. 3 K 4 7. 5 M 4 9. 6

We conclude that, at least for these stars, the mean value of theabsolute magnitude increases with the spectral index. This conclusion, however, is not generally valid. 

32. _Stars with the greatest radial velocities. _ There are some kinds ofnebulae for which very large values of the radial velocities have beenfound. With these we shall not for the present deal, but shall confineourselves to the stars. The greatest radial velocity hitherto found ispossessed by the star (040822) of the eighth magnitude in theconstellation Perseus, which retires from us with a velocity of 72sir. /st. Or 341 km. /sec. The nearest velocity is that of the star(010361) which approaches us with approximately the same velocity. Thefollowing table contains all stars with a radial velocity greater than20 sir. /st. (= 94. 8 km. /sec. ). It is based on the catalogue of VOUTEmentioned above. 

Regarding their distribution in the sky we find 11 in the galacticequator squares and 7 outside. A large radial velocity seems thereforeto be a galactic phenomenon and to be correlated to a great distancefrom us. Of the 18 stars in consideration there is only one at adistance smaller than one siriometer and 2 at a distance smaller than 4siriometers. Among the nearer ones we find the star (050744), identicalwith C. P. D. 5h. 243, which was the “second” star with great propermotion. These stars have simultaneously the greatest proper motion andvery great linear velocity. Generally we find from column 9 that thesestars with large radial velocity possess also a large proper motion. Themean value of the proper motions amounts to 1″. 34, a very high value. 

In the table we find no star with great apparent luminosity. Thebrightest is the 10th star in the table which has the magnitude 5. 1. Themean apparent magnitude is 7. 7. As to the absolute magnitude (_M_) wesee that most of these speedy stars, as well as the stars with greatproper motions in table 3, have a rather great _positive_ magnitude andthus are absolutely faint stars, though they perhaps may not be directlyconsidered as dwarf stars. Their mean absolute magnitude is +3. 0. 

Regarding the spectrum we find that these stars generally belong to theyellow or red types (G, K, M), but there are 6 F-stars and, curiouslyenough, two A-stars. After the designation of their type (A2 and A3) isthe letter _p_ (= peculiar), indicating that the spectrum in somerespect differs from the usual appearance of the spectrum of this type. In the present case the peculiarity consists in the fact that a line ofthe wave-length 448. 1, which emanates from magnesium and which we mayfind on plate III in the spectrum of Sirius, does not occur in thespectrum of these stars, though the spectrum has otherwise the sameappearance as in the case of the Sirius stars. There is reason tosuppose that the absence of this line indicates a low power of radiation(low temperature) in these stars (compare ADAMS). 

TABLE 4. 

_STARS WITH THE GREATEST RADIAL VELOCITY. _

+--+---------------------+----------+--------+-----+-------+-------+-------+| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | Position | Distance || | _Name_ |----------+--------+-----+-------+-------+-------+| | | (αδ) | Square | _l_ | _b_ | π | _r_ |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || 1|A. G. Berlin 1366 |(040822) | GD_5 | 141°| -20° |0″. 007 | 30. 8 || 2|Lal. 1966 |(010361) | GD_4 | 93 | - 2 | 0. 016 | 12. 9 || 3|A. Oe. 14320 |(1504{15})| GB_9 | 314 | +35 | 0. 035 | 5. 9 || 4|C. Z. 5h. 243 |(0507{44})| GE_7 | 218 | -35 | 0. 319 | 0. 6 || 5|Lal. 15290 |(074730) | GC_6 | 158 | +26 | 0. 023 | 9. 0 || 6|53 Cassiop. |(015563) | GC_4 | 98 | + 2 | . . | . . || 7|A. G. Berlin 1866 |(055719) | GD_6 | 159 | - 2 | 0. 021 | 9. 8 || 8|W Lyræ |(181136) | GC_2 | 31 | +21 | . . | . . || 9|Boss 1511 |(0559{26})| GD_7 | 200 | -20 | 0. 012 | 17. 0 ||10|ω Pavonis |(1849{60})| GD_11 | 304 | -24 | . . | . . ||11|A. Oe. 20452 |(2017{21})| GE_10 | 351 | -31 | 0. 015 | 13. 5 ||12|Lal. 28607 |(1537{10})| GB_10 | 325 | +34 | 0. 033 | 6. 2 ||13|A. G. Leiden 5734 |(161132) | GB_1 | 21 | +45 | 0. 002 | 89. 2 ||14|Lal. 37120 |(192932) | GC_2 | 33 | + 6 | 0. 050 | 4. 1 ||15|Lal. 27274 |(1454{21})| GB_9 | 308 | +34 | 0. 013 | 16. 2 ||16|Lal. 5761 |(030225) | GD_5 | 126 | -28 | 0. 039 | 5. 1 ||17|W. B. 17h. 517 |(172906) | GC_12 | 358 | +20 | 0. 014 | 14. 1 ||18|Lal. 23995 |(1247{17})| GB_8 | 271 | +46 | 0. 012 | 17. 0 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || | Mean | . . | . . | | 23°. 9|0″. 041 | 16. 7 |+--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+| 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 |+--+---------------------+------+--------+---------+---------+----+------+| | | Motion | Magnitude | Spectrum || | _Name_ +------+--------+---------+---------+----+------+| | | μ | _W_ | _m_ | _M_ |_Sp_| _m′_ |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || 1|A. G. Berlin 1366 | 0″. 54| +72 | 8m. 9 | +1m. 4 |F0 | 9. 4 || 2|Lal. 1966 | 0. 64| -69 | 7. 9 | +2. 3 |F3 | 8. 5 || 3|A. Oe. 14320 | 3. 75| +61 | 9. 0 | +5. 1 |G0 | 9. 9 || 4|C. Z. 5h. 243 | 8. 75| +51 | 9. 2 | +10. 1 |K2 |10. 6 || 5|Lal. 15290 | 1. 96| -51 | 8. 2 | +3. 4 |G0 | 9. 1 || 6|53 Cassiop. | 0. 01| -44 | 5. 6 | . . |B8 | 5. 5 || 7|A. G. Berlin 1866 | 0. 76| -40 | 9. 0 | +4. 0 |F0 | 9. 9 || 8|W Lyræ | . . | -39 | var. | . . |Md | var. || 9|Boss 1511 | 0. 10| +39 | 5. 2 | -1. 0 |G5 | 6. 4 ||10|ω Pavonis | 0. 14| +38 | 5. 1 | . . |K | 6. 5 ||11|A. Oe. 20452 | 1. 18| -38 | 8. 1 | +2. 4 |G8p | 9. 4 ||12|Lal. 28607 | 1. 18| -36 | 7. 3 | +3. 3 |A2p | 7. 4 ||13|A. G. Leiden 5734 | 0. 04| -35 | 8. 3 | -1. 5 |K4 | 9. 9 ||14|Lal. 37120 | 0. 52| -34 | 6. 6 | +3. 5 |G2 | 7. 6 ||15|Lal. 27274 | 0. 79| +34 | 8. 3 | +2. 2 |F4 | 8. 9 ||16|Lal. 5761 | 0. 86| -32 | 8. 0 | +4. 4 |A3p | 8. 1 ||17|W. B. 17h. 517 | 0. 63| -31 | 8. 6 | +2. 8 |F1 | 9. 1 ||18|Lal. 23995 | 0. 88| +30 | 8. 2 | +2. 0 |F3 | 8. 8 |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || | Mean | 1″. 34| 16. 7 | 7m. 7 | +3m. 0 |F9 | 8. 5 |+--+---------------------+------+--------+---------+---------+----+------+

33. _The nearest stars. _ The star α in Centaurus was long considered asthe nearest of all stars. It has a parallax of 0″. 75, corresponding to adistance of 0. 27 siriometers (= 4. 26 light years). This distance isobtained from the annual parallax with great accuracy, and the result ismoreover confirmed in another way (from the study of the orbit of thecompanion of α Centauri). In the year 1916 INNES discovered at theobservatory of Johannesburg in the Transvaal a star of the 10thmagnitude, which seems to follow α Centauri in its path in the heavens, and which, in any case, lies at the same distance from the earth, orsomewhat nearer. It is not possible at present to decide with accuracywhether _Proxima Centauri_--as the star is called by INNES--or αCentauri is our nearest neighbour. Then comes BARNARD's star (175204), whose large proper motion we have already mentioned. As No. 5 we findSirius, as No. 8 Procyon, as No. 21 Altair. The others are of the thirdmagnitude or fainter. No. 10--61 Cygni--is especially interesting, beingthe first star for which the astronomers, after long and painfulendeavours in vain, have succeeded in determining the distance with thehelp of the annual parallax (BESSEL 1841). 

From column 4 we find that the distribution of these stars on the sky istolerably uniform, as might have been predicted. All these stars have alarge proper motion, this being in the mean 3″. 42 per year. This was apriori to be expected from their great proximity. The radial velocityis, numerically, greater than could have been supposed. This fact isprobably associated with the generally small mass of these stars. 

Their apparent magnitude is upon an average 6. 3. The brightest of thenear stars is Sirius (_m_ = -1. 6), the faintest Proxima Centauri (_m_ =11). Through the systematic researches of the astronomers we may be surethat no bright stars exist at a distance smaller than one siriometer, for which the distance is not already known and well determined. Thefollowing table contains without doubt--we may call them briefly all_near_ stars--all stars within one siriometer from us with an apparentmagnitude brighter than 6m (the table has 8 such stars), and probablyalso all near stars brighter than 7m (10 stars), or even all brighterthan the eighth magnitude (the table has 13 such stars and two near thelimit). Regarding the stars of the eighth magnitude or fainter nosystematic investigations of the annual parallax have been made andamong these stars we may get from time to time a new star belonging tothe siriometer sphere in the neighbourhood of the sun. To determinethe total number of stars within this sphere is one of the fundamentalproblems in stellar statistics, and to this question I shall returnimmediately. 

TABLE 5. 

_THE NEAREST STARS. _

+--+----------------------+----------+--------+-----+-------+-------+-------+| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 |+--+----------------------+----------+--------+-----+-------+-------+-------+| | | Position | Distance || | _Name_ |----------+--------+-----+-------+-------+-------+| | | (αδ) | Square | _l_ | _b_ | π | _r_ |+--+----------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || 1|Proxima Centauri |(1422{62})| GD_10 | 281°| - 2° |0″. 780 | 0. 26 || 2|α Centauri |(1432{60})| GD_10 | 284 | - 2 | 0. 759 | 0. 27 || 3|Barnards p. M. Star |(175204) | GC_12 | 358 | +12 | 0. 515 | 0. 40 || 4|Lal. 21185 |(105736) | GB_5 | 153 | +66 | 0. 403 | 0. 51 || 5|Sirius |(0640{16})| GD_7 | 195 | - 8 | 0. 376 | 0. 55 || 6| . . |(1113{57})| GC_6 | 158 | + 3 | 0. 337 | 0. 60 || 7|τ Ceti |(0139{16})| GF_1 | 144 | -74 | 0. 334 | 0. 62 || 8|Procyon |(073405) | GC_7 | 182 | +14 | 0. 324 | 0. 64 || 9|C. Z. 5h. 243 |(0507{44})| GE_7 | 218 | -35 | 0. 319 | 0. 65 ||10|61 Cygni |(210238) | GD_2 | 50 | - 7 | 0. 311 | 0. 66 ||11|Lal. 26481 |(1425{15})| GB_9 | 124 | -40 | 0. 311 | 0. 66 ||12|ε Eridani |(0328{09})| GE_5 | 153 | -42 | 0. 295 | 0. 70 ||13|Lac. 9352 |(2259{36})| GE_10 | 333 | -66 | 0. 292 | 0. 71 ||14|Pos. Med. 2164 |(184159) | GC_2 | 56 | +24 | 0. 292 | 0. 71 ||15|ε Indi |(215557) | GE_9 | 304 | -47 | 0. 284 | 0. 73 ||16|Groom. 34 |(001243) | GD_3 | 84 | -20 | 0. 281 | 0. 73 ||17|Oe. A. 17415 |(173768) | GC_8 | 65 | +32 | 0. 268 | 0. 77 ||18|Krüger 60 |(222457) | GC_3 | 72 | 0 | 0. 256 | 0. 81 ||19|Lac. 8760 |(2111{39})| GE_10 | 332 | -44 | 0. 248 | 0. 88 ||20|van Maanens p. M. Star|(004304) | GE_3 | 92 | -58 | 0. 246 | 0. 84 ||21|Altair |(194508) | GD_1 | 15 | -10 | 0. 238 | 0. 87 ||22|C. G. A. 32416 |(2359{37})| GF_2 | 308 | -75 | 0. 230 | 0. 89 ||23|Bradley 1584 |(1129{32})| GC_6 | 252 | +28 | 0. 216 | 0. 95 |+--+----------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || | Mean | . . | . . | . . | 30°. 8|0″. 344 | 0. 67 |+--+----------------------+----------+--------+-----+-------+-------+-------+

+--+-----------------------+------+--------+---------+---------+----+------+| 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 |+--+-----------------------+------+--------+---------+---------+----+------+| | | Motion | Magnitude | Spectrum || | _Name_ +------+--------+---------+---------+----+------+| | | μ | _W_ | _m_ | _M_ |_Sp_| _m′_ |+--+-----------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || 1|Proxima Centauri | 3″. 85| . . | 11m. 0 | +13m. 9 |. . |13. 5 || 2|α Centauri | 3. 68| - 5 | 0. 33 | + 3. 2 |G | 1. 25 || 3|Barnards p. M. Star | 10. 29| -19 | 9. 7 | +11. 7 |Mb |11. 5 || 4|Lal. 21185 | 4. 77| -18 | 7. 6 | + 9. 1 |Mb | 8. 9 || 5|Sirius | 1. 32| - 2 | -1. 58 | - 0. 3 |A |-1. 58 || 6| . . | 2. 72| . . | . . | . . |. . |12. 5 || 7|τ Ceti | 1. 92| - 3 | 3. 6 | + 4. 6 |K0 | 4. 6 || 8|Procyon | 1. 24| - 1 | 0. 48 | + 1. 5 |F5 | 0. 90 || 9|C. Z. 5h. 243 | 8. 75| +51 | 9. 2 | +10. 1 |K2 |10. 6 ||10|61 Cygni | 5. 27| -13 | 5. 6 | + 6. 5 |K5 | 7. 2 ||11|Lal. 26481 | 0. 47| . . | 7. 8 | + 8. 7 |G5 | 8. 9 ||12|ε Eridani | 0. 97| + 3 | 3. 8 | + 4. 6 |K0 | 4. 8 ||13|Lac. 9352 | 6. 90| + 2 | 7. 5 | + 8. 2 |K | 8. 9 ||14|Pos. Med. 2164 | 2. 28| . . | 8. 9 | + 9. 6 |K |10. 3 ||15|ε Indi | 4. 70| - 8 | 4. 7 | + 5. 4 |K5 | 6. 3 ||16|Groom. 34 | 2. 89| + 1 | 8. 1 | + 8. 8 |Ma | 9. 5 ||17|Oe. A. 17415 | 1. 30| . . | 9. 1 | + 9. 7 |K |10. 5 ||18|Krüger 60 | 0. 94| . . | 9. 2 | + 9. 6 |K5 |10. 8 ||19|Lac. 8760 | 3. 53| + 3 | 6. 6 | + 7. 0 |G | 7. 5 ||20|van Maanens p. M. Star | 3. 01| . . | 12. 3 | +12. 7 |F0 |12. 9 ||21|Altair | 0. 66| - 7 | 0. 9 | + 1. 2 |A5 | 1. 12 ||22|C. G. A. 32416 | 6. 11| + 5 | 8. 2 | + 8. 5 |G | 9. 1 ||23|Bradley 1584 | 1. 06| - 5 | 6. 1 | + 6. 2 |G | 6. 9 |+--+-----------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || | Mean | 3″. 42| 9. 1 | 6m. 3 | +7m. 3 |G6 | 7. 5 |+--+-----------------------+------+--------+---------+---------+----+------+

The mean absolute magnitude of the near stars is distributed in thefollowing way:--

 _M_ 0 1 3 4 5 6 7 8 9 10 11 12 13 Number 1 2 1 2 1 2 1 4 4 1 1 1 1. 

What is the absolute magnitude of the near stars that are not containedin table? Evidently they must principally be faint stars. We may gofurther and answer that _all stars with an absolute magnitude brighterthan 6m_ must be contained in this list. For if _M_ is equal to 6 orbrighter, _m_ must be brighter than 6m, if the star is nearer than onesiriometer. But we have assumed that all stars apparently brighter than6m are known and are contained in the list. Hence also all stars_absolutely_ brighter than 6m must be found in table 5. We conclude thatthe number of stars having an absolute magnitude brighter than 6mamounts to 8. 

If, finally, the spectral type of the near stars is considered, we findfrom the last column of the table that these stars are distributed inthe following way:--

 Spectral type B A F G K M Number 0 2 2 5 9 3. 

For two of the stars the spectrum is for the present unknown. 

We find that the number of stars increases with the spectral index. Theunknown stars in the siriometer sphere belong probably, in the main, tothe red types. 

If we now seek to form a conception of the _total_ number in this spherewe may proceed in different ways. EDDINGTON, in his “Stellar movements”, to which I refer the reader, has used the proper motions as a scale ofcalculation, and has found that we may expect to find in all 32 stars inthis sphere, confining ourselves to stars apparently brighter than themagnitude 9m. 5. This makes 8 stars per cub. Sir. 

We may attack the problem in other ways. A very rough method which, however, is not without importance, is the following. Let us supposethat the Galaxy in the direction of the Milky Way has an extension of1000 siriometers and in the direction of the poles of the Milky Way anextension of 50 sir. We have later to return to the fuller discussion ofthis extension. For the present it is sufficient to assume these values. The whole system of the Galaxy then has a volume of 200 million cubicsiriometers. Suppose further that the total number of stars in theGalaxy would amount to 1000 millions, a value to which we shall alsoreturn in a following chapter. Then we conclude that the average numberof stars per cubic siriometer would amount to 5. This supposes that thedensity of the stars in each part of the Galaxy is the same. But the sunlies rather near the centre of the system, where the density is(considerably) greater than the average density. A calculation, whichwill be found in the mathematical part of these lectures, shows that thedensity in the centre amounts to approximately 16 times the averagedensity, giving 80 stars per cubic siriometer in the neighbourhood ofthe sun (and of the centre). A sphere having a radius of one siriometerhas a volume of 4 cubic siriometers, so that we obtain in this way 320stars in all, within a sphere with a radius of one siriometer. Fordifferent reasons it is probable that this number is rather too greatthan too small, and we may perhaps estimate the total number to besomething like 200 stars, of which more than a tenth is now known to theastronomers. 

We may also arrive at an evaluation of this number by proceeding fromthe number of stars of different apparent or absolute magnitudes. Thislatter way is the most simple. We shall find in a later paragraph thatthe absolute magnitudes which are now known differ between -8 and +13. But from mathematical statistics it is proved that the total range of astatistical series amounts upon an average to approximately 6 times thedispersion of the series. Hence we conclude that the dispersion (σ) ofthe absolute magnitudes of the stars has approximately the value 3 (weshould obtain σ = [13 + 8] : 6 = 3. 5, but for large numbers ofindividuals the total range may amount to more than 6 σ). 

As, further, the number of stars per cubic siriometer with an absolutemagnitude brighter than 6 is known (we have obtained 8 : 4 = 2 stars percubic siriometer brighter than 6m), we get a relation between the totalnumber of stars per cubic siriometer (_D_0_) and the mean absolutemagnitude (_M_0_) of the stars, so that _D_0_ can be obtained, as soonas _M_0_ is known. The computation of _M_0_ is rather difficult, and isdiscussed in a following chapter. Supposing, for the moment, _M_0_ = 10we get for _D_0_ the value 22, corresponding to a number of 90 starswithin a distance of one siriometer from the sun. We should then know afifth part of these stars. 

34. _Parallax stars. _ In §22 I have paid attention to the now availablecatalogues of stars with known annual parallax. The most extensive ofthese catalogues is that of WALKEY, containing measured parallaxes of625 stars. For a great many of these stars the value of the parallaxmeasured must however be considered as rather uncertain, and I havepointed out that only for such stars as have a parallax greater than0″. 04 (or a distance smaller than 5 siriometers) may the measuredparallax be considered as reliable, as least generally speaking. Theeffective number of parallax stars is therefore essentially reduced. Indirectly it is nevertheless possible to get a relatively largecatalogue of parallax stars with the help of the ingenious spectroscopicmethod of ADAMS, which permits us to determine the absolute magnitude, and therefore also the distance, of even farther stars through anexamination of the relative intensity of certain lines in the stellarspectra. It may be that the method is not yet as firmly based as itshould be, [15] but there is every reason to believe that the coursetaken is the right one and that the catalogue published by ADAMS of 500parallax stars in Contrib. From Mount Wilson, 142, already gives a morecomplete material than the catalogues of directly measured parallaxes. Igive here a short resumé of the attributes of the parallax stars in thiscatalogue. 

The catalogue of ADAMS embraces stars of the spectral types F, G, K andM. In order to complete this material by parallaxes of blue stars I addfrom the catalogue of WALKEY those stars in his catalogue that belong tothe spectral types B and A, confining myself to stars for which theparallax may be considered as rather reliable. There are in all 61 suchstars, so that a sum of 561 stars with known distance is to bediscussed. 

For all these stars we know _m_ and _M_ and for the great part of themalso the proper motion μ. We can therefore for each spectral typecompute the mean values and the dispersion of these attributes. We thusget the following table, in which I confine myself to the mean values ofthe attributes. 

TABLE 6. 

_MEAN VALUES OF _m_, _M_ AND THE PROPER MOTIONS (μ) OF PARALLAX STARS OFDIFFERENT SPECTRAL TYPES. _

+---+-------+-------+-------+-------+|Sp. |Number | _m_ | _M_ | μ |+---+-------+-------+-------+-------+| B | 15 | +2. 03 | -1. 67 | 0″. 05 || A | 46 | +3. 40 | +0. 64 | 0. 21 || F | 125 | +5. 60 | +2. 10 | 0. 40 || G | 179 | +5. 77 | +1. 68 | 0. 51 || K | 184 | +6. 17 | +2. 31 | 0. 53 || M | 42 | +6. 02 | +2. 30 | 0. 82 |+---+-------+-------+-------+-------+

We shall later consider all parallax stars taken together. We find fromtable 6 that the apparent magnitude, as well as the absolute magnitude, is approximately the same for all yellow and red stars and even for thestars of type F, the apparent magnitude being approximately equal to +6mand the absolute magnitude equal to +2m. For type B we find the meanvalue of M to be -1m. 7 and for type A we find M = +0m. 6. The propermotion also varies in the same way, being for F, G, K, M approximately0″. 5 and for B and A 0″. 1. As to the mean values of _M_ and μ we cannotdraw distinct conclusions from this material, because the parallax starsare selected in a certain way which essentially influences these meanvalues, as will be more fully discussed below. The most interestingconclusion to be drawn from the parallax stars is obtained from theirdistribution over different values of _M_. In the memoir referred to, ADAMS has obtained the following table (somewhat differently arrangedfrom the table of ADAMS), [16] which gives the number of parallax starsfor different values of the absolute magnitude for different spectraltypes. 

A glance at this table is sufficient to indicate a singular and wellpronounced property in these frequency distributions. We find, indeed, that in the types G, K and M the frequency curves are evidentlyresolvable into two simple curves of distribution. In all these types wemay distinguish between a bright group and a faint group. With aterminology proposed by HERTZSPRUNG the former group is said to consistof _giant_ stars, the latter group of _dwarf_ stars. Even in the starsof type F this division may be suggested. This distinction is still morepronounced in the graphical representation given in figures (plate IV). 

TABLE 7. 

_DISTRIBUTION OF THE PARALLAX STARS OF DIFFERENT SPECTRAL TYPES OVERDIFFERENT ABSOLUTE MAGNITUDES. _

+-------------------------------------------------+| | | | | | | || || M | B | A | F | G | K | M || All ||-------+----+----+-----+-----+-----+-----++------|| | | | | | | || || - 4 | . . | . . | . . | . . | . . | 1 || . . || - 3 | . . | . . | . . | . . | . . | . . || . . || - 2 | 1 | 4 | 1 | 7 | . . | 2 || 15 || - 1 | 2 | 7 | 7 | 28 | 15 | 4 || 63 || - 0 | 3 | 10 | 6 | 32 | 40 | 10 || 91 || + 0 | 1 | 11 | 6 | 7 | 14 | 11 || 50 || + 1 | 1 | 3 | 20 | 9 | 4 | 1 || 38 || + 2 | . . | 5 | 48 | 26 | . . | 1 || 80 || + 3 | . . | 1 | 32 | 36 | 2 | . . || 71 || + 4 | . . | 1 | 5 | 25 | 25 | . . || 56 || + 5 | . . | 1 | . . | 6 | 25 | . . || 32 || + 6 | . . | 2 | . . | 3 | 10 | . . || 15 || + 7 | . . | 1 | . . | . . | 14 | . . || 15 || + 8 | . . | . . | . . | . . | 3 | 7 || 10 || + 9 | . . | . . | . . | . . | 2 | 4 || 6 || +10 | . . | . . | . . | . . | . . | . . || . . || +11 | . . | . . | . . | . . | . . | 1 || 1 ||-------+----+----+-----+-----+-----+-----++------|| Total | 8 | 46 | 125 | 179 | 154 | 42 || 554 |+-------------------------------------------------+

In the distribution of all the parallax stars we once more find asimilar bipartition of the stars. Arguing from these statistics someastronomers have put forward the theory that the stars in space aredivided into two classes, which are not in reality closely related. Theone class consists of intensely luminous stars and the other of feeblestars, with little or no transition between the two classes. If theparallax stars are arranged according to their apparent proper motion, or even according to their absolute proper motion, a similar bipartitionis revealed in their frequency distribution. 

Nevertheless the bipartition of the stars into two such distinct classesmust be considered as vague and doubtful. Such an _apparent_bipartition is, indeed, necessary in all statistics as soon asindividuals are selected from a given population in such a manner as theparallax stars are selected from the stars in space. Let us considerthree attributes, say _A_, _B_ and _C_, of the individuals of apopulation and suppose that the attribute _C_ is _positively_ correlatedto the attributes _A_ and _B_, so that to great or small values of _A_or _B_ correspond respectively great or small values of _C_. Now if theindividuals in the population are statistically selected in such a waythat we choose out individuals having great values of the attributes _A_and small values of the attribute _B_, then we get a statistical seriesregarding the attribute _C_, which consists of two seemingly distinctnormal frequency distributions. It is in like manner, however, that theparallax stars are selected. The reason for this selection is thefollowing. The annual parallax can only be determined for near stars, nearer than, say, 5 siriometers. The direct picking out of these starsis not possible. The astronomers have therefore attacked the problem inthe following way. The near stars must, on account of their proximity, be relatively brighter than other stars and secondly possess greaterproper motions than those. Therefore parallax observations areessentially limited to (1) bright stars, (2) stars with great propermotions. Hence the selected attributes of the stars are _m_ and μ. But_m_ and μ are both positively correlated to _M_. By the selection ofstars with small _m_ and great μ we get a series of stars whichregarding the attribute _M_ seem to be divided into two distinctclasses. 

The distribution of the parallax stars gives us no reason to believethat the stars of the types K and M are divided into the two supposedclasses. There is on the whole no reason to suppose the existence at allof classes of giant and dwarf stars, not any more than a classificationof this kind can be made regarding the height of the men in apopulation. What may be statistically concluded from the distribution ofthe absolute magnitudes of the parallax stars is only that the_dispersion_ in _M_ is increased at the transition from blue to yellowor red stars. The filling up of the gap between the “dwarfs” and the“giants” will probably be performed according as our knowledge of thedistance of the stars is extended, where, however, not the annualparallax but other methods of measuring the distance must be employed. 

TABLE 8. 

_THE ABSOLUTELY FAINTEST STARS. _

+--+---------------------+----------+--------+-----+-------+-------+-------+| 1| 2 | 3 | 4 | 5 | 6 | 7 | 8 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | Position | Distance || | _Name_ |----------+--------+-----+-------+-------+-------+| | | (αδ) | Square | _l_ | _b_ | π | _r_ |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || 1|Proxima Centauri |(1422{62})| GD_10 | 281°| - 2° |0″. 780 | 0. 26 || 2|van Maanens star |(004304) | GE_8 | 92 | -58 | 0. 246 | 0. 84 || 3|Barnards star |(175204) | GC_12 | 358 | +12 | 0. 515 | 0. 40 || 4|17 Lyræ C |(190332) | GC_2 | 31 | +10 | 0. 128 | 1. 60 || 5|C. Z. 5h. 243 |(0507{44})| GE_7 | 218 | -35 | 0. 319 | 0. 65 || 6|Gron. 19 VIII 234 |(161839) | GB_1 | 29 | +44 | 0. 162 | 1. 27 || 7|Oe. A. 17415 |(173768) | GB_8 | 65 | +32 | 0. 268 | 0. 77 || 8|Gron. 19 VII 20 |(162148) | GB_2 | 41 | +43 | 0. 133 | 1. 55 || 9|Pos. Med. 2164 |(184159) | GC_2 | 56 | +24 | 0. 292 | 0. 71 ||10|Krüger 60 |(222457) | GC_8 | 72 | 0 | 0. 256 | 0. 81 ||11|B. D. +56°532 |(021256) | GD_8 | 103 | - 4 | 0. 195 | 1. 06 ||12|B. D. +55°581 |(021356) | GD_8 | 103 | - 4 | 0. 185 | 1. 12 ||13|Gron. 19 VIII 48 |(160438) | GB_1 | 27 | +46 | 0. 091 | 2. 27 ||14|Lal. 21185 |(105736) | GB_5 | 153 | +66 | 0. 403 | 0. 51 ||15|Oe. A. 11677 |(111466) | GB_3 | 103 | +50 | 0. 198 | 1. 04 ||16|Walkey 653 |(155359) | GB_2 | 57 | +45 | 0. 175 | 1. 18 ||17|Yerkes parallax star |(021243) | GD_8 | 107 | -16 | 0. 045 | 4. 58 ||18|B. D. +56°537 |(021256) | GD_8 | 103 | - 4 | 0. 175 | 1. 18 ||19|Gron. 19 VI 266 |(062084) | GC_3 | 97 | +27 | 0. 071 | 2. 80 |+--+---------------------+----------+--------+-----+-------+-------+-------+| | | | | | | | sir. || | Mean | . . | . . | . . | 27°. 5|0″. 244 | 0. 99 |+--+---------------------+----------+--------+-----+-------+-------+-------+

+--+---------------------+------+--------+---------+---------+----+------+| 1| 2 | 9 | 10 | 11 | 12 | 13 | 14 |+--+---------------------+------+--------+---------+---------+----+------+| | | Motion | Magnitude | Spectrum || | _Name_ +------+--------+---------+---------+----+------+| | | μ | _W_ | _m_ | _M_ |_Sp_| _m′_ |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || 1|Proxima Centauri | 3″. 85| . . | 11m. 0 |+13m. 9 |. . |13. 5 || 2|van Maanens star | 3. 01| . . | 12. 3 | +12. 7 |F0 |12. 95 || 3|Barnards star | 10. 29| -19 | 9. 7 | +11. 7 |Mb | 8. 9 || 4|17 Lyræ C | 1. 75| . . | 11. 3 | +10. 3 |. . |12. 5 || 5|C. Z. 5h. 243 | 8. 75| +51 | 9. 2 | +10. 1 |K2 |10. 68 || 6|Gron. 19 VIII 234 | 0. 12| . . | 10. 3 | + 9. 8 |. . | . . || 7|Oe. A. 17415 | 1. 30| . . | 9. 1 | + 9. 7 |K |10. 5 || 8|Gron. 19 VII 20 | 1. 22| . . | 10. 5 | + 9. 6 |. . | . . 0 || 9|Pos. Med. 2164 | 2. 28| . . | 8. 9 | + 9. 6 |K |10. 3 ||10|Krüger 60 | 0. 94| . . | 9. 2 | + 9. 6 |K5 |10. 8 ||11|B. D. +56°532 | . . | . . | 9. 5 | + 9. 4 |. . | . . ||12|B. D. +55°581 | . . | . . | 9. 4 | + 9. 2 |G5 |10. 2 ||13|Gron. 19 VIII 48 | 0. 12| . . | 11. 1 | + 9. 3 |. . | . . ||14|Lal. 21185 | 4. 77| -18 | 7. 6 | + 9. 1 |Mb | 8. 9 ||15|Oe. A. 11677 | 3. 03| . . | 9. 2 | + 9. 1 |Ma |11. 0 ||16|Walkey 653 | . . | . . | 9. 5 | + 9. 1 |. . | . . ||17|Yerkes parallax star | . . | . . | 12. 4 | + 9. 1 |. . | . . ||18|B. D. +56°537 | . . | . . | 9. 4 | + 9. 0 |. . | . . ||19|Gron. 19 VI 266 | 0. 09| . . | 11. 3 | + 9. 0 |. . | . . |+--+---------------------+------+--------+---------+---------+----+------+| | | |sir. /st. | | | | _m′_ || | Mean | 2″. 96| 29. 3 | 10m. 0 | +9m. 9 |K1 |10. 9 |+--+---------------------+------+--------+---------+---------+----+------+

Regarding the absolute brightness of the stars we may draw someconclusions of interest. We find from table 7 that the absolutemagnitude of the parallax stars varies between -4 and +11, the extremestars being of type M. The absolutely brightest stars have a rathergreat distance from us and their absolute magnitude is badly determined. The brightest star in the table is Antares with _M_ = -4. 6, which valueis based on the parallax 0″. 014 found by ADAMS. So small a parallaxvalue is of little reliability when it is directly computed from annualparallax observations, but is more trustworthy when derived with thespectroscopic method of ADAMS. It is probable from a discussion of the_B_-stars, to which we return in a later chapter, that the absolutelybrightest stars have a magnitude of the order -5m or -6m. If theparallaxes smaller than 0″. 01 were taken into account we should find thatCanopus would represent the absolutely brightest star, having _M_ =-8. 17, and next to it we should find RIGEL, having _M_ = -6. 97, but boththese values are based on an annual parallax equal to 0″. 007, which istoo small to allow of an estimation of the real value of the absolutemagnitude. 

If on the contrary the _absolutely faintest_ stars be considered, theparallax stars give more trustworthy results. Here we have only to dowith near stars for which the annual parallax is well determined. Intable 8 I give a list of those parallax stars that have an absolutemagnitude greater than 9m. 

There are in all 19 such stars. The faintest of all known stars isINNES' star “Proxima Centauri” with _M_ = 13. 9. The third star isBARNARD's star with _M_ = 11. 7, both being, together with α Centauri, also the nearest of all known stars. The mean distance of all the faintstars is 1. 0 sir. 

There is no reason to believe that the limit of the absolute magnitudeof the faint stars is found from these faint parallax stars:--Certainlythere are many stars in space with _M_ > 13m and the mean value of _M_, for all stars in the Galaxy, is probably not far from the absolute valueof the faint parallax stars in this table. This problem will bediscussed in a later part of these lectures. 

FOOTNOTES:

[Footnote 15: Compare ADAMS' memoirs in the Contributions from MountWilson. ]

[Footnote 16: The first line gives the stars of an absolute magnitudebetween -4. 9 and -4. 0, the second those between -3. 9 and -3. 0, &c. Thestars of type B and A are from WALKEY's catalogue. ]

[Illustration: PLATE I. 

_CONVERSION OF EQUATORIAL COORDINATES INTO GALACTIC COORDINATES. _]

[Illustration: PLATE II. 

_Squares and Constellations. _]

[Illustration: PLATE III. 

_The Harvard Classification of Stellar Spectra. _]

[Illustration: Plate IV. 

_Distribution of the parallax stars over different absolutemagnitudes. _]

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[Transcriber's Note: The following corrections have been made to theoriginal text. 

Page 4: "Terrestial distances" changed to "Terrestrial distances"

Page 9: "we must chose, " changed to "we must choose, "

Page 12: "acromasie" changed to "achromatism"

Page 15: "inparticular" changed to "in particular"

"supposing, that" changed to "supposing that"

Page 16: "393. 4 mm" changed to "393. 4 μμ"

Page 20: "for which stars if" changed to "for which stars it"

Page 22: "sphaerical" changed to "spherical"

Page 23: "principal scource" changed to "principal source"

Page 25: "lense with 15 cm" changed to "lens with 15 cm"

Page 27: "Through the spectroscopie method" changed to "Through thespectroscopic method"

"made to its principal" changed to "made its principal"

"american manner" changed to "American manner"

Page 35: "many tenths of a year" changed to "many tens of years"

Page 38: "same appearence" changed to "same appearance"

Page 47: "red stears" changed to "red stars"

"_dispersion_ in M" changed to "_dispersion_ in _M_"

Page 49: "smaller the" changed to "smaller than"]