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The Project Gutenberg EBook of Lectures on Stellar Statistics, by Carl Vilhelm Ludvig CharlierThis eBook is for the use of anyone anywhere at no cost and withalmost no restrictions whatsoever.  You may copy it, give it away orre-use it under the terms of the Project Gutenberg License includedwith this eBook or online at www.gutenberg.orgTitle: Lectures on Stellar StatisticsAuthor: Carl Vilhelm Ludvig CharlierRelease Date: July 27, 2007 [EBook #22157]Language: EnglishCharacter set encoding: ISO-8859-1*** START OF THIS PROJECT GUTENBERG EBOOK LECTURES ON STELLAR STATISTICS ***Produced by Susan Skinner and the Online DistributedProofreading Team at http://www.pgdp.net (The originalcopy of this book was generously made available for scanningby the Department of Mathematics at the University ofGlasgow.)

LECTURES
ON STELLAR STATISTICS

BY

C. V. L. CHARLIER

SCIENTIA PUBLISHER
LUND 1921

HAMBURG 1921
PRINTED BYLÜTCKE & WULFF[Pg 3]

CHAPTER I.

APPARENT ATTRIBUTES OF THE STARS.

1.

Our knowledge of the stars is based on theirapparent attributes,obtained from the astronomical observations. The object of astronomyis to deduce herefrom the real orabsolute 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 theirradiation. The characteristics of this radiation may be described indifferent ways, according as the nature of the light is defined. (Undulatorytheory, Emission theory.)

From the statistical point of view it will be convenient to considerthe radiation 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. Theseattributes may be, for instance, the diameter and form of the particles,their mode of rotation, &c. By these attributes the optical and electricalproperties of the radiation are to be explained. I shall not here attemptany such explanation, but shall confine myself to the property whichthe particles have of possessing a different mode of deviating from therectilinear path as they pass from one medium to another. This deviationdepends in some way on one or more attributes of the particles. Letus suppose that it depends on a single attribute, which, with a terminologyderived from the undulatory theory ofHuyghens, may be called thewave-length (λ) of the particle.

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

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

(2) themean wave-length (λ₀) of the radiation, also called (or nearlyidentical with) theeffective wave-length or the colour;[Pg 4]

(3)the dispersion of the wave-length. This characteristic of the radiationmay be determined from thespectrum, which also gives the variationof 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 apparentplace on the sky.

The intensity, the mean wave-length, and the dispersion of thewave-length are in a simple manner connected with thetemperature (T)of the star. According to the radiation laws ofStephan andWien wefind, indeed (compare L. M. 41[1]) that the intensity is proportional to thefourth power ofT, whereas the mean wave-length and the dispersionof the wave-length are both inversely proportional toT. It follows thatwith increasing temperature the mean wave-length diminishes—the colourchanging into violet—and simultaneously the dispersion of the wave-lengthand also even the total length of the spectrum are reduced(decrease).

2.

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

(αδ) = (174435).

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

(αδ) = (174435),

so that the last two figures are in italics.

This notation has been introduced byPickering for variablestars and is used by him everywhere in the Annals of the Harvard[Pg 5]Observatory, but it is also well suited to all stars. This notation gives,simultaneously, the characteristicnumero of the stars. It is true thattwo or more stars may in this manner obtain the same characteristicnumero. They are, however, easily distinguishable from each otherthrough other attributes.

Thegalactic coordinatesl andb are referred to the Milky Way(the Galaxy) as plane of reference. The pole of the Milky Way hasaccording toHouzeau andGould the position (αδ) = (124527). Fromthe distribution of the stars of the spectral type B I have in L. M. II, 14[2]found a somewhat different position. But having ascertained later thatthe real position of the galactic plane requires a greater number of starsfor an accurate determination of its value, I have preferred to employthe position used byPickering in the Harvard catalogues, namely(αδ) = (124028), or

α = 12^h 40^m = 190°, δ = +28°,

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

The galactic longitude (l) is reckoned from the ascending node ofthe Milky Way on the equator, which is situated in the constellationAquila. The galactic latitude (b) gives the angular distance of the starfrom the Galaxy. Onplate I, at the end of these lectures, will be founda fairly detailed diagram from which the conversion of α and δ of a starintol andb may be easily performed. All stars having an apparentmagnitude brighter than 4^m are directly drawn.

Instead of giving the galactic longitude and latitude of a star wemay content ourselves with giving the galacticsquare in which the staris situated. For this purpose we assume the sky to be divided into48 squares, all having the same surface. Two of these squares lie atthe northern pole of the Galaxy and are designated GA₁ and GA₂. Twelvelie north of the galactic plane, between 0° and 30° galactic latitude, andare designated GC₁, GC₂, ..., GC₁₂. The corresponding squares southof the galactic equator (the plane of the Galaxy) are called GD₁, GD₂, ...,GD₁₂. The two polar squares at the south pole are called GF₁ and GF₂.Finally we have 10 B-squares, between the A- and C-squares and10 corresponding E-squares in the southern hemisphere.

[Pg 6]

The distribution of the squares in the heavens is here graphicallyrepresented in the projection ofFlamsteed, which has the advantageof giving areas proportional to the corresponding spherical areas, anarrangement necessary, or at least highly desirable, for all stellar statisticalresearches. It has also the advantage of affording a continuous representationof the whole sky.

The correspondence between squares and stellar constellations isseen fromplate II. Arranging the constellations according to theirgalactic longitude we find north of the galactic equator (in the C-squares)the constellations:—

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

mentioning only one constellation for each square.

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

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

3.

Changes in the position of a star. From the positions of a staron two or more occasions we obtain its apparent motion, also called theproper motion of the star. We may distinguish between asecular partof this motion and aperiodical part. In both cases the motion maybe either a reflex of the motion of the observer, and is then calledparallactic 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 givestheannual parallax (π). In order to obtain available annual parallaxesof a star it is usually necessary for the star to be nearer to us than5 siriometers, corresponding to a parallax greater than 0″.04. More seldomwe may in this manner obtain trustworthy values for a distance amounting[Pg 7]to 10 siriometers (π = 0″.02), or even still greater values. For suchlarge distances thesecular parallax, which is caused by the progressivemotion of the sun in space, may give better results, especially if themean distance of a group of stars is simultaneously determined. Sucha value of the secular parallax is also called, byKapteyn, thesystematicparallax 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 ofNewton,have always used the mean distance of the earth from the sun asuniversal unit of distance. Regarding the distances in the stellar systemthe astronomers have had a varying practice. German astronomers,Seeliger and others, have long used a stellar unit of distance correspondingto an annual parallax of 0″.2, which has been called a “Siriusweite”.To this name it may be justly objected that it has no internationaluse, a great desideratum in science. Against the theoretical definitionof this unit 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 theharmonic mean distance ofthe star and not to its arithmetic mean distance. The same objectionmay be made to the unit “parsec.” proposed in 1912 byTurner.

For my part I have, since 1911, proposed a stellar unit which, bothin name and definition, nearly coincides with the proposition ofSeeliger,and which will be exclusively used in these lectures. Asiriometer isput equal to 10⁶ times the planetary unit of distance, corresponding toa parallax of 0″.206265 (in practice sufficiently exactly 0″.2).

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

5.

In regard totime also, the terrestrial units (second, day, year)are too small for stellar wants. As being consistent with the unit of[Pg 8]distance, I have proposed for the stellar unit of time astellar year (st.),corresponding to 10⁶ years. We thus obtain the same relation betweenthe stellar and the planetary units of length and time, which has theadvantage that avelocity of a star expressed in siriometers per stellar yearis expressed with the same numerals in planetary units of length per year.

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

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 chapterI give 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 platesof the 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 inmagnitudes (m). The reason is partlythat we should otherwise necessarily have to deal with very large numbers,if they were to be proportional to the intensity, and partly that it is provedthat the human eye apprehends quantities of light as proportional tom.

This depends upon a general law in psycho-physics, known asFechner'slaw, which says that changes of the apparent impressionof light are proportional not to the changes of the intensity but to thesechanges divided by the primitive intensity. A similar law is valid forall sensations. A conversation is inaudible in the vicinity of a waterfall.[Pg 9]An increase of a load in the hand from nine to ten hectograms makesno great difference in the feeling, whereas an increase from one to twohectograms is easily appreciable. A match lighted in the day-timemakes no increase in the illumination, and so on.

A mathematical analysis shows that from the law ofFechner itfollows that the impression increases inarithmetical progression (1, 2,3, 4, ...) simultaneously with an increase of the intensity ingeometricalprogression (I,I²,I³,I⁴, ...). It is with the sight the same as withthe hearing. It is well known that the numbers of vibrations of thenotes of a harmonic scale follow each other in a geometrical progressionthough, for the ear, the intervals between the notes are apprehended asequal. The magnitudes play the same rôle in relation to the quantitiesof light as do the logarithms to the corresponding numbers. If a staris considered to have a brightness intermediate between two other starsit is not thedifference but theratio of the quantities of light that isequal in each case.

The branch of astronomy (or physics) which deals with intensitiesof radiation is calledphotometry. In order to determine a certain scalefor the magnitudes we must choose, in a certain manner, thezero-pointof the scale and thescale-ratio.

Both may be chosen arbitrarily. Thezero-point is now almostunanimously chosen by astronomers in accordance with that used bythe Harvard Observatory. No rigorous definition of the Harvardzero-point, as far as I can see, has yet been given (compare howeverH. A. 50[3]), but considering that the Pole-star (α Ursæ Minoris) is usedat Harvard as a fundamental star of comparison for the brighter stars,and that, according to the observations at Harvard and those ofHertzsprung (A. N. 4518 [1911]), the light of the Pole-star is verynearly invariable, we may say thatthe zero-point of the photometricscale is chosen in such a manner that for the Pole-star m = 2.12. Ifthe magnitudes are given in another scale than the Harvard-scale (H. S.),it is necessary to apply the zero-point correction. This amounts, forthe Potsdam catalogue, to -0^m.16.

It is further necessary to determine thescale-ratio. Our magnitudesfor the stars emanate fromPtolemy. It was found that the scale-ratio[Pg 10]—givingthe ratio of the light-intensities of two consecutive classes of magnitudes—accordingto the older values of the magnitudes, was approximatelyequal to 2½. When exact photometry began (with instruments for measuringthe magnitudes) in the middle of last century, the scale-ratio wastherefore put equal to 2.5. Later it was found more convenient to chooseit equal to 2.512, the logarithm of which number has the value 0.4. Themagnitudes being themselves logarithms of a kind, it is evidently moreconvenient to use a simple value of the logarithm of the ratio of intensitythan to use this ratio itself. This scale-ratio is often called thePogson-scale(used byPogson in his “Catalogue of 53 known variable stars”, Astr.Obs. of the Radcliffe Observatory, 1856), and is now exclusively used.

It follows from the definition of the scale-ratio that two stars forwhich the light intensities are in the ratio 100:1 differ by exactly5 magnitudes. A star of the 6^th magnitude is 100 times fainter thana star of the first magnitude, a star of the 11^th magnitude 10000 times,of the 16^th magnitude a million times, and a star of the 21^st magnitude100 million 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 0^m.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.

A consequence of the definition ofm is thatwe also have to do withnegative magnitudes (aswell as with negative logarithms). Thus, for example,forSiriusm = -1.58. The magnitudes of thegreater planets, as well as those of the moon andthe sun, are also negative, as will be seen fromthe adjoining table, where the values are taken from“Die Photometrie der Gestirne” byG. Müller.

The apparent magnitude of the sun is givenbyZöllner (1864). The other values are all found in Potsdam, andallude generally to the maximum value of the apparent magnitude ofthe moon and the planets.

The brightest star isSirius, which has the magnitudem = -1.58.The magnitude of the faintest visible star evidently depends on thepenetrating power of the instrument used. The telescope of[Pg 11]WilliamHerschel, used by him and his son in their star-gauges and otherstellar researches, allowed of the discerning of stars down to the14^th magnitude. The large instruments of our time hardly reach muchfarther, for visual observations. When, however, photographic platesare used, it is easily possible to get impressions of fainter stars, evenwith rather modest instruments. The large 100-inch mirror of theWilson Observatory renders possible the photographic observations ofstars of the 20^th apparent magnitude, and even fainter.

The observations of visual magnitudes are performed almost exclusivelywith the photometer ofZöllner in a more or less improvedform.

7.

Absolute magnitude. The apparent magnitude of a star is changedas the star changes its distance from the observer, the intensity increasingindirectly as the square of the distance of the star. In order to makethe magnitudes of the stars comparable with each other it is convenientto reduce them to their value at a certain unit of distance. As suchwe choose one siriometer. The corresponding magnitude will be calledtheabsolute magnitude and is denoted byM.[4] We easily find fromthe table given in the preceding paragraph that the absolute magnitude ofthe sun, according toZöllner's value ofm, amounts to +3.4, of themoon to +31.2. For Jupiter we findM = +24.6, for VenusM = +25.3.The other planets have approximatelyM = +30.

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

8.

Photographic magnitudes. The magnitudes which have beenmentioned in the preceding paragraphs all refer to observations taken withthe eye, and are calledvisual magnitudes. The total intensity of a star is,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. The[Pg 12]difference in the results obtained with these instruments is due to thecircumstance that different parts of the radiation are taken into account.

The usual photographic plates, which have their principal sensibilityin the violet parts of the spectrum, give us thephotographic 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 yetbeen sufficiently studied, seems however to be rather inconsiderable,and must be neglected in the following.

The photographic magnitude of a star will in these lectures bedenoted bym′, corresponding to a visual magnitudem.

In practical astronomy use is also made of plates which, as the resultof a certain preparation (in colour baths or in other ways), have acquireda distributive function nearly corresponding to that of the eye, andespecially have a maximum point at the same wave-lengths. Suchmagnitudes are calledphoto-visual (compare the memoir ofParkhurstin A. 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 determinem′. The diameterof a star image increases with the time of exposure. This increaseis due in part to the diffraction of the telescope, to imperfect achromatismor spherical 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.

Thezero-point of the photographic magnitudes is so determinedthat this magnitude coincides with the visual magnitude for such starsas belong to the spectral type A0 and havem = 6.0, according to theproposal of the international solar conference at Bonn, 1911.

Determinations of the photographic or photo-visual magnitudes maynow be carried out with great accuracy. The methods for this aremany and are well summarised in the Report of the Council of theR. A. S. of the year 1913. The most effective and far-reaching methodseems to be that proposed bySchwarzschild, called the half-gratingmethod, by which two exposures are taken of the same part of thesky, while at one of the exposures a certain grating is used thatreduces the magnitudes by a constant degree.[Pg 13]

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) thetotal radiation or intensity (I), (2) themeanwave-length (λ₀), (3) thedispersion of the wave-length (σ). In thepreceding paragraphs we have treated of the total radiation of the starsas this is expressed through their magnitudes. The mean wave-lengthis pretty closely defined by thecolour, whereas the dispersion of thewave-length is found from thespectrum 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 theeffective wave-lengthas it is most usually called, or from thecolour-index. We shall revertlater to this question. There are, however, a great many direct eye-estimatesof the colour of the stars.

Colour corresponding to a given spectrum.

Spectrum corresponding to a given colour.

The signs + and - indicate intermediate shades of colour.[Pg 14]

The preceding table drawn up by Dr.Malmquist from the colourobservations ofMüller andKempf 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 7^m.5.

We find from these tables that there is a well-pronouncedregressionin 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, butthe K-stars have upon an average a shade of white in the yellow colour.The coefficient of correlation (r) is not easy to compute in this case,because one of the attributes, the colour, is not strictly graduated (i.e. itis not expressed in numbers defining the colour).[5] Using the coefficientof contingency ofPearson, it is, however, possible to find a fairlyreliable value of the coefficient of correlation, andMalmquist has inthis way foundr = +0.85, a rather high value.

In order to facilitate the discussion of the relation between colourand spectrum it is convenient to deal here with the question of thespectra of the stars.

10.

Spectra of the stars. In order to introduce the discussionI first give a list of the wave-lengths of theFrauenhofer lines in thespectrum, and the corresponding chemical elements.

[Pg 15]

The first column gives theFrauenhofer denomination of each line.Moreover the hydrogen lines α, β, γ, δ, ε are denoted. The secondcolumn gives the name of the corresponding element, to which eachline is to be attributed. The third column gives the wave-length expressedin millionths of a millimeter as unit (μμ).

Onplate 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 theFrauenhofer lines A to H (λ = 760 to 400 μμ), whereasthe photographic spectrum generally lies between F and K (λ = 500to 400 μμ).

In the earliest days of spectroscopy the spectra of the stars wereclassified according to their visual spectra. This classification wasintroduced bySecchi and was later more precisely defined byVogel.The three classes I, II, III ofVogel correspond approximately to thecolour classification into white, yellow, and red stars. Photography hasnow almost entirely taken the place of visual observations of spectra,so thatSecchi's andVogel's definitions of the stellar spectra are nolonger applicable. The terminology now used was introduced byPickering and MissCannon and embraces a great many types, ofwhich we here describe the principal forms as they are defined in Part. IIof Vol. XXVIII of the Annals of the Harvard Observatory. It may beremarked thatPickering first arranged the types in alphabetical order A,B, C, &c., supposing that order to correspond to the temperature ofthe stars. Later this was found to be partly wrong, and in particularit was found that the B-stars may be hotter than those of type A. Thefollowing is the temperature-order of the spectra according to the opinionof the Harvard astronomers.

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

This type embraces mainly stars of relatively small apparent brightness.The brightest is γ Velorum withm = 2.22. We shall find thatthe absolute magnitude of these stars nearly coincides with that of thestars of type B.[Pg 16]

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

Among interesting stars of this type is ζ Puppis (Od), in the spectrumof whichPickering discovered a previously unknown series of heliumlines. They were at first attributed (byRydberg) to hydrogen andwere called “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 manyother bright stars.

Onplate III ε Orionis is taken as representative of this type.

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

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

Type F (Calcium type). The hydrogen lines still rather prominentbut not so broad as in the preceding type. The two calcium lines Hand 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 bright stars.

Type K. The hydrogen lines still fainter. The K-line attains itsmaximum intensity (is not especially pronounced in the figure ofplate III).

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

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

Type M. The spectrum is banded and belongs toSecchi's thirdtype. The flutings are due to titanium oxide.[Pg 17]

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

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

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

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

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

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

11.

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

Let us use the termspectral 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 spectralindexs = 0.0, F0 tos = +1.0, G0 tos = +2.0, K0 tos = +3.0M0 tos = +4.0 and B0 tos = -1.0. Further, let A1, A2, A3, &c., have[Pg 18]the spectral indices +0.1, +0.2, +0.3, &c., and in like manner withthe other intermediate sub-classes. Then it is evident that to all spectralclasses between B0 and M there corresponds a certain spectral indexs.The extreme types O and N are not here included. Their spectralindices may however be determined, as will be seen later.

Though the spectral indices, defined in this manner, are directlyknown for every spectral type, it is nevertheless not obvious that theseries of spectral indices corresponds to a continuous series of valuesof some attribute of the stars. This may be seen to be possible froma comparison 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 spectrait will finally be necessary to change to some extent the definitions of thespectral types, a change which, however, has not yet been accomplished.

12.

We have found in§9 that the light-radiation of a star isdescribed by means of the total intensity (I), the mean wave-length (λ₀)and the dispersion of the wave-length (σ_λ). λ₀ and σ_λ may be deducedfrom the spectral observations. It must here be observed that theobservations give, not the intensities at different wave-lengths but, thevalues of these intensities as they are apprehended by the instrumentsemployed—the eye or the photographic plate. For the derivation ofthe true curve of intensity we must know the distributive function ofthe instrument (L. M. 67). As to the eye, we have reason to believe,from the bolometric observations ofLangley (1888), that the meanwave-length of the visual curve of intensity nearly coincides with thatof the true intensity-curve, a conclusion easily understood fromDarwin'sprinciples of evolution, which demand that the human eye in the courseof time shall be developed in such a way that the mean wave-lengthof the visual intensity curve does coincide with that of the true curve(λ = 530 μμ), when the greatest visual energy is obtained (L. M. 67).As to the dispersion, this is always greater in the true intensity-curvethan in the visual curve, for which, according to§10, it amounts toapproximately 60 μμ. We found indeed that the visual intensity curve isextended, approximately, from 400 μμ to 760 μμ, a sixth part of whichinterval, approximately, corresponds to the dispersion σ of the visual curve.

In the case of the photographic intensity-curve the circumstances[Pg 19]are different. 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 mean wave-lengthfollows the formula ofWien, which says that this wave-length variesinversely as the temperature. The total intensity, according to the lawofStephan, varies directly as the fourth power of the temperature.Even the dispersion is dependent on the variation of the temperature—directlyas the mean wave-length, inversely as the temperature of thestar (L. M. 41)—so that the mean wave-length, as well as the dispersionof the wave-length, is smaller for the hot stars O and B than for thecooler ones (K and M types). It is in this manner possible to determinethe temperature of a star from a determination of its mean wave-length(λ₀) or from the dispersion in λ. Such determinations (from λ₀)have been made byScheiner andWilsing in Potsdam, byRosenbergand others, though these researches still have to be developed to a greaterdegree of accuracy.

14.

Effective wave-length. The mean wave-length of a spectrum,or, as it is often called by the astronomers, theeffective wave-length,is generally determined in the following way. On account of therefraction in the air the image of a star is, without the use of a spectroscope,really a spectrum. After some time of exposure we get a somewhatround image, the position of which is determined precisely by the meanwave-length. This method is especially used with a so-calledobjective-grating,which consists of a series of metallic threads, stretched parallelto each other at equal intervals. On account of the diffraction of thelight we now get in the focal plane of the objective, with the use ofthese gratings, not only a fainter image of the star at the place whereit would have arisen without grating, but also at both sides of thisimage secondary images, the distances of which from the central star arecertain theoretically known multiples of the effective wave-lengths.In this simple manner it is possible to determine the effective wave-length,and this being a tolerably well-known function of the spectral-index,the latter can also be found. This method was first proposed[Pg 20]byHertzsprung and has been extensively used byBergstrand,Lundmark andLindblad at the observatory of Upsala and by others.

15.

Colour-index. We have already pointed out in§9 that thecolour may be identified with the mean wave-length (λ₀). As furtherλ₀ is closely connected with the spectral index (s), we may use thespectral index to represent the colour. Instead ofs there may also beused another expression for the colour, called the colour-index. Thisexpression was first introduced bySchwarzschild, and is definedin the following way.

We have seen that the zero-point of the photographic scale is chosenin such a manner that the visual magnitudem and the photographicmagnitudem′ coincide for stars of spectral index 0.0 (A0). The photographicmagnitudes 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 thanthe visual, and the contrary for the other types. The difference isgreatest for the M-type (still greater for the N-stars, though here forthe present only a few determinations are known), for which stars ifamounts to nearly two magnitudes. So much fainter is a red star ona photographic plate than when observed with the eye.

The difference between the photographic and the visual magnitudesis called the colour-index (c). The correlation between this indexand the spectral-index is found to be rather high (r = +0.96). InL. 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 GIVENSPECTRAL TYPE OR SPECTRAL INDEX.

[Pg 21]

TABLE 1*.

GIVING THE MEAN SPECTRAL INDEX CORRESPONDINGTO A GIVEN COLOUR-INDEX.

From each catalogue of visual magnitudes of the stars we may obtaintheir photographic magnitude through adding the colour-index. Thismay be considered as known (taking into account the high coefficientof correlation betweens andc) as soon as we know the spectral typeof the star. We may conclude directly that the number of stars havinga photographic magnitude brighter than 6.0 is considerably smaller thanthe number of stars visually brighter than this magnitude. There are,indeed, 4701 stars for whichm < 6.0 and 2874 stars havingm′ < 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 theDoppler principle, throughmeasuring the displacement of the lines in the spectrum, this displacementbeing towards the red or the violet according as the star is recedingfrom or approaching the observer.

The velocityW 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 attributesof the stars 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 equatorial coordinates(α and δ) are given in an abridged notation (αδ), where the first fournumbers give the right ascension in hours and minutes and the last[Pg 22]two numbers give the declination in degrees, the latter being printedin 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 perpendicularto the line of sight. These components will be expressed in onecomponent (u₀) parallel to the galactic plane, and one component (v₀)perpendicular to it. If the distance (r) is known we are able to convertthese components into components of the linear velocity perpendicularto the line of sight (U andV).

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

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

The difference between these magnitudes is called thecolour-index(c =m′-m).

IV.The characteristics of the stellar radiation are the mean wave-length(λ₀) and the dispersion (σ) in the wave-length.The mean wave-lengthmay be either directly determined (perhaps aseffective wave-length)or found from the spectral type (spectral index) or from thecolour-index.

There are in all eight attributes of the stars which may be foundfrom the observations:—the spherical position of the star (l,b), itsdistance (r), proper motion (u₀ andv₀), radial velocity (W), apparentmagnitude (m orm′), absolute magnitude (M), spectral type (Sp) orspectral index (s), and colour-index (c). Of these the colour-index,the spectral type, the absolute magnitude and also (to a certain degree)the radial velocity may be considered as independent of the place ofthe observer and may therefore be considered not as only apparent butalso asabsolute attributes of the stars.

Between three of these attributes (m,M andr) a mathematicalrelation exists so that one of them is known as soon as the other twohave been found from observations.[Pg 23]

CHAPTER II.

SOURCES OF OUR PRESENT KNOWLEDGEOF THE STARS.

18.

In this chapter I shall give a short account of the publicationsin which the most complete information on the attributes of the starsmay be obtained, with short notices of the contents and genesis of thesepublications. It is, however, not my intention to give a history of theseresearches. We shall consider more particularly the questions relatingto the position of the stars, their motion, magnitude, and spectra.

19.

Place of the stars.Durchmusterungs. The most completedata on the position of the stars are obtained from the star cataloguesknown as “Durchmusterungs”. There are two such catalogues, whichtogether cover the whole sky, one—visual—performed in Bonn andcalled theBonner Durchmusterung (B. D.), the other—photographic—performedin CapeThe Cape Photographic Durchmusterung (C. P. D.).As the first of these catalogues has long been—and is to some extenteven now—our principal source for the study of the sky and is moreoverthe first enterprise of this kind, I shall give a somewhat detailedaccount of its origin and contents, as related byArgelander in theintroduction to the B. D.

B. D. was planned and performed by the Swedish-FinnArgelander(born in Memel 1799). A scholar ofBessel he was first called asdirector in Åbo, then in Hälsingfors, and from there went in 1836 toBonn, where in the years 1852 to 1856 he performed this greatDurchmusterung.As instrument he used aFrauenhofer comet-seeker withan aperture of 76 mm, a focal length of 650 mm, and 10 times magnifyingpower. The field of sight 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[Pg 24]a narrow dark line perceptible at star illumination. Perpendicular tothis diameter (the “hour-line”) were 10 lines, at each side of a middleline, drawn at a distance of 7′. These lines were drawn with blackoil colour on the glass.

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

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

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

The catalogue was published byArgelander in three parts in theyears 1859, 1861 and 1862[7] and embraces all stars between the poleand 2° south of the equator brighter than 9^m.5, according to the scaleofArgelander (his aim was to register all stars up to the 9^th magnitude).To this scale we return later. The catalogue is arranged inaccordance with the declination-degrees, and for each degree accordingto the right ascension. Quotations from B. D. have the form B. D. 23°.174,which signifies: Zone +23°, star No. 174.

Argelander's work was continued for stars between δ = -2° andδ = -23° bySchönfeld, according to much the same plan, but witha larger instrument (aperture 159 mm, focal length 1930 mm, magnifyingpower 26 times). The observations were made in the years 1876 to 1881and include 133659 stars.[8]

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

[Pg 25]

20.

The Cape Photographic Durchmusterung[9] (C. P. D.). Thisembraces the whole southern sky from -18° to the south pole. PlannedbyGill, the photographs were taken at the Cape Observatory withaDallmeyer lens with 15 cm. aperture and a focal-length of 135 cm.Plates of 30 × 30 cm. give the coordinates for a surface of 5 × 5 squaredegrees. The photographs were taken in the years 1885 to 1890. Themeasurements of the plates were made byKapteyn in Groningen witha “parallactic” measuring-apparatus specially constructed for this purpose,which permits 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 to 1900.

The positions have the same accuracy as in B. D. The wholenumber of stars is 454875.Kapteyn considers the catalogue completeto at least the magnitude 9^m.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 10^m.

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

All these Durchmusterungs are ultimately based on star cataloguesof smaller extent and of great precision. Of these catalogues we shallnot here speak (Compare, however,§23).

A great “Durchmusterung”, that will include all stars to the11^th magnitude, 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 the workfinished in an appreciable time.

21.

Star charts. For the present we possess two great photographicstar charts, embracing the whole heaven:—TheHarvard Map (H. M.)and theFranklin-Adams Charts (F. A. C.).

The Harvard Map, of which a copy (or more correctly two copies)on glass has kindly been placed at the disposal of the Lund Observatory[Pg 26]by Mr.Pickering, embraces all stars down to the 11^th magnitude.It consists of 55 plates, each embracing more than 900 square degreesof the sky. The photographs were taken with a small lens of only2.5 cms. aperture and about 32.5 cms. focal-length. The time of exposurewas one hour. These plates have been counted at the Lund Observatoryby HansHenie. We return later to these counts.

TheFranklin-Adams Charts were made by an amateur astronomerFranklin-Adams, partly at his own observatory (Mervel Hill) inEngland, partly in Cape and Johannesburg, Transvaal, in the years1905-1912. The photographs were taken with aTaylor lens with 25 cm.aperture and a focal-length of 114 cm., which gives rather good imageson a field of 15 × 15 square degrees.

The whole sky is here reproduced on, in all, 206 plates. Eachplate was exposed for 2 hours and 20 minutes and gives images of thestars down to the 17^th magnitude. The original plates are now at theobservatory in Greenwich. Some copies on paper have been made,of which the Lund Observatory possesses one. It shows stars downto the 14^th-15^th magnitudes and gives a splendid survey of the wholesky more complete, indeed, than can be obtained, even for the northsky, by direct observation of the heavens with any telescope at presentaccessible in Sweden.

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 14^th magnitude has been obtained. Weshall later 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. ThisCarte du Ciel (C. d. C.)is still unfinished, but there seems to be a possibility that we shallone day see this work carried to completion. It will embrace stars downto the 14^th magnitude and thus does not reach so far as the F. A. C.,but on the other hand is carried out on a considerably greater scaleand gives better images than F. A. C. and will therefore be of a greatvalue in the future, especially for the study of the proper motionsof the stars.

22.

Distance of the stars. As the determination, from the annualparallax, of the distances of the stars is very precarious if the distance[Pg 27]exceeds 5 sir. (π = 0″.04), it is only natural that the catalogues of star-distancesshould be but few in number. The most complete cataloguesare those ofBigourdan in the Bulletin astronomique XXVI (1909),ofKapteyn andWeersma in the publications of Groningen Nr. 24(1910), embracing 365 stars, and ofWalkey in the “Journal of theBritish Astronomical Association XXVII” (1917), embracing 625 stars.Through the spectroscopic method ofAdams it will be possible toenlarge this number considerably, so that the distance of all stars, forwhich the spectrum is well known, may be determined with fair accuracy.Adams has up to now published 1646 parallax stars.

23.

Proper motions. An excellent catalogue of the proper motionsof the stars isLewis Boss's “Preliminary General Catalogue of 6188stars” (1910) (B. P. C.). It contains the proper motions of all starsdown to the sixth magnitude (with few exceptions) and moreover somefainter stars. The catalogue is considered by the editor only as apreliminary to a greater catalogue, which is to embrace some 25000stars and is now nearly completed.

24.

Visual magnitudes. The Harvard observatory has, under thedirection ofPickering, made its principal aim to study the magnitudesof the stars, and the history of this observatory is at the same timethe history of the treatment of this problem.Pickering, in thegenuine American manner, is not satisfied with the three thirds of thesky visible from the Harvard observatory, but has also founded a daughterobservatory in South America, at Arequipa in Peru. It is thereforepossible for him to publish catalogues embracing the whole heaven frompole to pole. The last complete catalogue (1908) of the magnitudesof the stars is found in the “Annals of the Harvard ObservatoryT. 50” (H. 50). It contains 9110 stars and can be considered as completeto the magnitude 6^m.5. To this catalogue are generally referred themagnitudes which have been adopted at the Observatory of Lund, andwhich 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.) byMüller andKempf, which was published simultaneouslywith H. 50. It contains the magnitude of 14199 stars and embraces[Pg 28]all stars on the northern hemisphere brighter than 7^m.5 (according toB. D.). We have already seen that the zero-point of H. 50 and P. G. C.is somewhat different and that the magnitudes in P. G. C. must beincreased by -0^m.16 if they are to be reduced to the Harvard scale. Thedifference between the two catalogues however is due to some extent tothe colour of the stars, as has been shown by Messrs.Müller andKempf.

25.

Photographic magnitudes. Our knowledge of this subject isstill rather incomplete. The most comprehensive catalogue is the“Actinometrie” bySchwarzschild (1912), containing the photographicmagnitudes of all stars in B. D. down to the magnitude 7^m.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 starsnearest to the pole inParkhurst's “Yerkes actinometry” (1912),[11]which contains all stars in B. D. brighter than 7^m.5 between the poleand 73° northern declination. The total number of stars is 672.

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

A discussion of thecolour-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, the photographicmagnitude may be obtained, with a generally sufficient accuracy,by adding the colour-index according to thetable 1 in§15 above.

26.

Stellar spectra. Here too we find the Harvard Observatoryto be the 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 there included,i.e., for all stars to 6^m.5. MissCannon, at the Harvard Observatory,deserves the principal credit for this great work. Not content withthis result she is now publishing a still greater work embracing morethan 200000 stars. The first four volumes of this work are now[Pg 29]published and contain the first twelve hours of right ascension, so thathalf the work is now printed.[13]

27.

Radial velocity. In this matter, again, we find America to bethe leading nation, though, this time, it is not the Harvard or theMount Wilson but the Lick Observatory to which we have to give thehonour. The eminent director of this observatory,W. W. Campbell,has in a high degree developed the accuracy in the determination ofradial velocities and has moreover carried out such determinations ina large scale. The “Bulletin” No. 229 (1913) of the Lick Observatorycontains the radial velocity of 915 stars. At the observatory of Lund,where as far as possible card catalogues of the attributes of the starsare collected,Gyllenberg has made a catalogue of this kind for theradial velocities. The total number of stars in this catalogue nowamounts to 1640.[14]

28.

Finally I shall briefly mention some comprehensive works onmore special questions regarding the stellar system.

Onvariable stars there is published every year byHartwig inthe “Vierteljahrschrift der astronomischen Gesellschaft” a catalogue ofall known variable stars with needful information about their minima &c.This is the completest and most reliable of such catalogues, and is alwaysup to date. A complete historical catalogue of the variables is given in“Geschichte und Literatur des Lichtwechsels der bis Ende 1915 als sicherveränderlich anerkannten Sterne nebst einem Katalog der Elemente ihresLichtwechsels” vonG. Müller undE. Hartwig. Leipzig 1918, 1920.

Onnebulae we have the excellent catalogues ofDreyer, the“New General Catalogue” (N. G. C.) of 1890 in the “Memoirs ofthe Astronomical Society” vol. 49, the “Index catalogues” (I. C.) in thesame memoirs, vols. 51 and 59 (1895 and 1908). These cataloguescontain all together 13226 objects.

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

[Pg 30]

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 very small,and certainly does not exceed one hundred. These cases refer principallyto such stars as are characterized either by great brilliancy or by agreat proper motion. The principal reason why these stars are betterknown than others is that they lie rather near our solar system. Beforepassing on to consider the stars from more general statistical points ofview, it may therefore be of interest first to make ourselves familiarwith these well-known stars, strongly emphasizing, however, the exceptionalcharacter of these stars, and carefully avoiding any generalizationfrom the attributes we shall here find.

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 1^m.5.

The first column gives the current number, the second the name,the third the equatorial designation (αδ). It should be remembered thatthe first four figures give the hour and minutes in right ascension, thelast two the declination, italics showing negative declination. The fourthcolumn gives the galactic square, the fifth and sixth columns the galacticlongitude and latitude. The seventh and eighth columns give the annualparallax and the corresponding distance expressed in siriometers. Theninth column gives the proper motion (μ), the tenth the radial velocityW expressed in sir./st. (To get km./sec. we may multiply by 4.7375).The eleventh column gives the apparent visual magnitude, the twelfthcolumn the absolute magnitude (M), computed fromm with the helpofr. The 13^th column gives the type of spectrum (Sp), and the lastcolumn the photographic magnitude (m′). The difference betweenm′ andm gives the colour-index (c).[Pg 31]

TABLE 2.

THE APPARENTLY BRIGHTEST STARS.

[Pg 32]

The values of (αδ),m,Sp are taken from H. 50. The values ofl,b are computed from (αδ) with the help of tables in preparation atthe Lund Observatory, or from the original toplate I at the end, allowingthe conversion of the equatorial coordinates into galactic ones. Thevalues of π are generally taken from the table ofKapteyn andWeersmamentioned in the previous chapter. The values of μ are obtained fromB. 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 1^m.5. The brightest of them isSirius, 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 equalto 0.0 (observations in Harvard givec = +0.06). Its apparent magnitudeis -1^m.6, nearly the same as that of Mars in his opposition. Its absolutemagnitude is -0^m.3,i.e., fainter than the apparent magnitude, fromwhich we may conclude that it has a distance from us smaller than onesiriometer. We find, indeed, from the eighth column thatr = 0.5 sir.The proper motion of Sirius is 1″.32 per year, which is rather largebut still not among the largest proper motions as will be seen below.From the 11^th column we find that Sirius is moving towards us witha velocity of 1.6 sir./st. (= 7.6 km./sec.), a rather small velocity. Thethird column shows that its right ascension is 6^h 40^m and its declination-16°. It lies in the square GD₇ and its galactic coordinates are seenin the 5^th and 6^th columns.

The next brightest star isCanopus or α Carinæ at the south sky.If we might place absolute confidence in the value ofM (= -8.2) inthe 12^th column this star would be, in reality, a much more imposingapparition than Sirius itself. Remembering that the apparent magnitudeof the moon, according to§6, amounts to -11.6, we should find thatCanopus, if placed 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 thatof the moon. It is not altogether astonishing that a fanciful astronomershould have thought Canopus to be actually the central star in thewhole stellar 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 ofM[Pg 33]is intimately dependent on that ofr we must consider speculations basedon this value to be very vague. Another reason for a doubt abouta great 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,as will be seen, is seldom found among giant stars. A better supportfor a large distance could on the other hand be found from the smallproper motion of this star. Sirius and Canopus are the only stars inthe sky having a negative value of the apparent visual magnitude.

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

In the fourth column is the galactic square. We call to mind thatall these squares have the same area, and that there is therefore thesame probabilitya priori of finding a star in one of the squares as inanother. The squares GC and GD lie along the galactic equator (theMilky Way). We find now from column 4 that of the 20 stars here consideredthere are no less than 15 in the galactic equator squares andonly 5 outside, instead of 10 in the galactic squares and 10 outside, aswould have been expected. The number of objects is, indeed, too smallto allow us to draw any cosmological conclusions from this distribution,but we shall find in the following many similar instances regardingobjects that are principally accumulated along the Milky Way and arescanty at the galactic poles. We shall find that in these cases we maygenerally conclude from such a partition that we then have to do withobjectssituated 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 smallmaterial here under consideration, we should hence have reason tobelieve that the bright stars lie relatively far from us. In other wordswe should conclude that the bright stars seem to be bright to us notbecause of their proximity but because of their large intrinsic luminosity.Column 8 really tends in this direction. Certainly the distances arenot in this case colossal, but they are nevertheless sufficient to show,in some degree, this uneven partition of the bright stars on the sky.The mean distance of these stars is as large as 7.5 sir. Only α Centauri,[Pg 34]Sirius, Procyon and Altair lie at a distance smaller than one siriometer.Of the other stars there are two that lie as far as 30 siriometers fromour system. These are the two giants Canopus and Rigel. Even if,as has already been said, the distances of these stars may be consideredas rather uncertain, we must regard them as being rather large.

As column 8 shows that these stars are rather far from us, so wefind from column 12, that their absolute luminosity is rather large.The mean absolute magnitude is, indeed, -2^m.1. We shall find thatonly the greatest and most luminous stars in the stellar system havea negative value of the absolute magnitude.

The mean value of the proper motions of the bright stars amountsto 0″.56 per year and may be considered as rather great. We shall,indeed, find that the mean proper motion of the stars down to the6^th magnitude scarcely amounts to a tenth part of this value. On theother hand we find from the table that the high value of this mean ischiefly due to the influence of four of the stars which have a largeproper motion, namely Sirius, Arcturus, α Centauri and Procyon. Theother stars have a proper motion smaller than 1″ per year and for halfthe number of stars the proper motion amounts to approximately 0″.05,indicating their relatively great distance.

That the absolute velocity of these stars is, indeed, rather smallmay be found from column 10, giving their radial velocity, which inthe mean amounts to only three siriometers per stellar year. From thediscussion below of the radial velocities of the stars we shall find thatthis is a rather small figure. This fact is intimately bound up withthe general law in statistical mechanics, to which we return later, thatstars with large masses generally have a small velocity. We thusfind in the radial velocities fresh evidence, independent of the distance,that these bright stars are giants among the stars in our stellarsystem.

We find all the principal spectral types represented among thebright stars. 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) Canopusand Procyon. To the sun type (G) Capella and α Centauri. To theK-type belong Arcturus, Aldebaran and Pollux and to the M-type thetwo red stars Betelgeuze and Antares. Using the spectral indices as[Pg 35]an expression for the spectral types we find that the mean spectralindex of these stars is +1.1 corresponding to the spectral type F1.

31.

Stars with the greatest proper motion. Intable 3 I havecollected the stars having a proper motion greater than 3″ per year.The designations 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 star1830 Groombridge designed as that which possesses the greatest knownproper motion. It is now distanced by two other stars C. P. D. 5^h.243discovered in the year 1897 byKapteyn andInnes on the plates takenfor the Cape Photographic Durchmusterung, andBarnard's star inOphiuchus, discovered 1916. The last-mentioned star, which possessesthe greatest proper motion now known, is very faint, being only ofthe 10^th magnitude, and lies at a distance of 0.40 sir. from our sun andis hence, as will be found fromtable 5 the third nearest star for whichwe know the distance. Its linear velocity is also very great, as wefind from column 10, and amounts to 19 sir./st. (= 90 km./sec.) in thedirection towards the sun. The absolute magnitude of this star is 11^m.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., the propermotion) is so great that the star in 1000 years moves 3°, or as muchas 6 times the diameter of the moon. For this star, as well as for itsnearest neighbours in the table, observations differing only by a year aresufficient for an approximate determination of the value of the propermotion, for which in other cases many tens of years are required.

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 squaresand 12 in the other squares. We shall not build up any conclusionon this irregularity in the distribution, but supported by the generalthesis of the preceding paragraph we conclude only that these starsmust be relatively near us. This follows, indeed, directly from column 8,as not less than eleven of these stars lie within one siriometer from oursun. Their mean distance is 0.87 sir.[Pg 36]

TABLE 3.

STARS WITH THE GREATEST PROPER MOTION.

[Pg 37]

That the great proper motion does not depend alone on theproximity of these stars is seen from column 10, giving the radialvelocities. For some of the stars (4) the radial velocity is for the presentunknown, but the others have, with few exceptions, a rather greatvelocity amounting in the mean to 18 sir./st. (= 85 km./sec.), if noregard is taken to the sign, a value nearly five times as great as theabsolute velocity of the sun. As this is only the component along theline of sight, the absolute velocity is still greater, approximately equalto the component velocity multiplied by √2. We conclude that thegreat proper motions depend partly on the proximity, partly on thegreat linear velocities of the stars. That both these attributes herereally cooperate 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 7^m. This is a consequenceof the fact that the mean distance of these stars is equal to one siriometer,at which distancem andM, indeed, do coincide. We find that thesestars have a small luminosity and may be considered asdwarf stars.According to the general law of statistical mechanics already mentionedsmall bodies upon an average have a great absolute velocity, as we have,indeed, already found from the observed radial velocities of these stars.

As to the spectral type, the stars with great proper motions areall yellow or red stars. The mean spectral index is +2.8, correspondingto the type G8. If the stars of different types are put together we getthe table

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 kindsof nebulae 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 is[Pg 38]possessed by the star (040822) of the eighth magnitude in the constellationPerseus, which retires from us with a velocity of 72 sir./st. or341 km./sec. The nearest velocity is that of the star (010361) whichapproaches us with approximately the same velocity. The followingtable contains all stars with a radial velocity greater than 20 sir./st.(= 94.8 km./sec.). It is based on the catalogue ofVoute mentioned 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 ata distance smaller than one siriometer and 2 at a distance smallerthan 4 siriometers. Among the nearer ones we find the star (050744),identical with C. P. D. 5^h.243, which was the “second” star with greatproper motion. These stars have simultaneously the greatest propermotion and very great linear velocity. Generally we find fromcolumn 9 that these stars with large radial velocity possess alsoa large proper motion. The mean value of the proper motions amountsto 1″.34, a very high value.

In the table we find no star with great apparent luminosity. Thebrightest is the 10^th star in the table which has the magnitude 5.1.The mean apparent magnitude is 7.7. As to the absolute magnitude (M)we see that most of these speedy stars, as well as the stars with greatproper motions intable 3, have a rather greatpositive 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 belongto the yellow or red types (G, K, M), but there are 6 F-stars and,curiously enough, two A-stars. After the designation of their type(A2 and A3) is the letterp (= peculiar), indicating that the spectrumin some respect differs from the usual appearance of the spectrum ofthis type. In the present case the peculiarity consists in the fact thata line of the wave-length 448.1, which emanates from magnesium andwhich we may find onplate III in the spectrum of Sirius, does not occurin the spectrum of these stars, though the spectrum has otherwise thesame appearance as in the case of the Sirius stars. There is reasonto suppose that the absence of this line indicates a low power of radiation(low temperature) in these stars (compareAdams).[Pg 39]

TABLE 4.

STARS WITH THE GREATEST RADIAL VELOCITY.

[Pg 40]

33.

The nearest stars. The star α in Centaurus was long consideredas the nearest of all stars. It has a parallax of 0″.75, correspondingto a distance of 0.27 siriometers (= 4.26 light years). This distance isobtained from the annual parallax with great accuracy, and the resultis moreover confirmed in another way (from the study of the orbit ofthe companion of α Centauri). In the year 1916Innes discovered atthe observatory of Johannesburg in the Transvaal a star of the 10^th magnitude,which seems to follow α Centauri in its path in the heavens, andwhich, in any case, lies at the same distance from the earth, or somewhatnearer. It is not possible at present to decide with accuracy whetherProxima Centauri—as the star is called byInnes—or α Centauri is ournearest neighbour. Then comesBarnard's star (175204), whose largeproper motion we have already mentioned. As No. 5 we find Sirius,as No. 8 Procyon, as No. 21 Altair. The others are of the third magnitudeor fainter. No. 10—61 Cygni—is especially interesting, being thefirst star for which the astronomers, after long and painful endeavoursin vain, have succeeded in determining the distance with the help ofthe annual parallax (Bessel 1841).

From column 4 we find that the distribution of these stars on thesky is tolerably uniform, as might have been predicted. All thesestars have a large proper motion, this being in the mean 3″.42 per year.This was a priori to be expected from their great proximity. The radialvelocity is, numerically, greater than could have been supposed. Thisfact is probably associated with the generally small mass of these stars.

Their apparent magnitude is upon an average 6.3. The brightestof the near stars is Sirius (m = -1.6), the faintest Proxima Centauri(m = 11). Through the systematic researches of the astronomers wemay be sure that no bright stars exist at a distance smaller than one siriometer,for which the distance is not already known and well determined.The following table contains without doubt—we may call them briefly allnear stars—all stars within one siriometer from us with an apparentmagnitude brighter than 6^m (the table has 8 such stars), and probablyalso all near stars brighter than 7^m (10 stars), or even all brighter thanthe eighth magnitude (the table has 13 such stars and two near the limit).Regarding the stars of the eighth magnitude or fainter no systematicinvestigations of the annual parallax have been made and among thesestars we may get from time to time a new star belonging to the sirio[Pg 41]metersphere in the neighbourhood of the sun. To determine thetotal number of stars within this sphere is one of the fundamental problemsin stellar statistics, and to this question I shall return immediately.

TABLE 5.

THE NEAREST STARS.

[Pg 42]The mean absolute magnitude of the near stars is distributed in thefollowing way:—

What is the absolute magnitude of the near stars that are not containedin table? Evidently they must principally be faint stars. Wemay go further and answer thatall stars with an absolute magnitudebrighter than 6^m must be contained in this list. For ifM is equal to 6or brighter,m must be brighter than 6^m, if the star is nearer than onesiriometer. But we have assumed that all stars apparently brighterthan 6^m are known and are contained in the list. Hence also all starsabsolutely brighter than 6^m must be found intable 5. We conclude thatthe number of stars having an absolute magnitude brighter than 6^mamounts 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 in thefollowing way:—

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

We find that the number of stars increases with the spectral index.The unknown stars in the siriometer sphere belong probably, in themain, to the red types.

If we now seek to form a conception of thetotal number in thissphere we may proceed in different ways.Eddington, in his “Stellarmovements”, to which I refer the reader, has used the proper motionsas a scale of calculation, and has found that we may expect to find inall 32 stars in this sphere, confining ourselves to stars apparently brighterthan the magnitude 9^m.5. This makes 8 stars per cub. sir.

We may attack the problem in other ways. A very rough methodwhich, however, is not without importance, is the following. Let ussuppose that the Galaxy in the direction of the Milky Way has anextension of 1000 siriometers and in the direction of the poles of the[Pg 43]Milky Way an extension of 50 sir. We have later to return to thefuller discussion of this extension. For the present it is sufficient toassume these values. The whole system of the Galaxy then hasa volume of 200 million cubic siriometers. Suppose further that thetotal number of stars in the Galaxy would amount to 1000 millions,a value to which we shall also return in a following chapter. Thenwe conclude that the average number of stars per cubic siriometerwould amount to 5. This supposes that the density of the stars in eachpart of the Galaxy is the same. But the sun lies rather near the centreof the system, where the density is (considerably) greater than theaverage density. A calculation, which will be found in the mathematicalpart of these lectures, shows that the density in the centre amountsto approximately 16 times the average density, giving 80 stars per cubicsiriometer in the neighbourhood of the sun (and of the centre). A spherehaving a radius of one siriometer has a volume of 4 cubic siriometers,so that we obtain in this way 320 stars in all, within a sphere witha radius of one siriometer. For different reasons it is probable thatthis number is rather too great than too small, and we may perhapsestimate the total number to be something like 200 stars, of whichmore than a tenth is now known to the astronomers.

We may also arrive at an evaluation of this number by proceedingfrom the number of stars of different apparent or absolute magnitudes.This latter way is the most simple. We shall find in a later paragraphthat the absolute magnitudes which are now known differ between -8and +13. But from mathematical statistics it is proved that the totalrange of a statistical series amounts upon an average to approximately6 times the dispersion of the series. Hence we conclude that thedispersion (σ) of the absolute magnitudes of the stars has approximatelythe value 3 (we should obtain σ = [13 + 8] : 6 = 3.5, but for largenumbers of individuals 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 starsper cubic siriometer brighter than 6^m), we get a relation between thetotal number of stars per cubic siriometer (D₀) and the mean absolutemagnitude (M₀) of the stars, so thatD₀ can be obtained, as soon asM₀ is known. The computation ofM₀ is rather difficult, and is discussed[Pg 44]in a following chapter. Supposing, for the moment,M₀ = 10 we getforD₀ the value 22, corresponding to a number of 90 stars withina distance of one siriometer from the sun. We should then knowa fifth part of these stars.

34.

Parallax stars. In§22 I have paid attention to the nowavailable catalogues of stars with known annual parallax. The mostextensive of these catalogues is that ofWalkey, containing measuredparallaxes of 625 stars. For a great many of these stars the value ofthe parallax measured must however be considered as rather uncertain,and I have pointed out that only for such stars as have a parallaxgreater than 0″.04 (or a distance smaller than 5 siriometers) may themeasured parallax be considered as reliable, as least generally speaking.The effective number of parallax stars is therefore essentially reduced.Indirectly it is nevertheless possible to get a relatively large catalogueof parallax stars with the help of the ingenious spectroscopic methodofAdams, which permits us to determine the absolute magnitude, andtherefore also the distance, of even farther stars through an examinationof the relative intensity of certain lines in the stellar spectra. It maybe that the method is not yet as firmly based as it should be,[15] butthere is every reason to believe that the course taken is the right oneand that the catalogue published byAdams of 500 parallax stars inContrib. from Mount Wilson, 142, already gives a more complete materialthan the catalogues of directly measured parallaxes. I give here a shortresumé of the attributes of the parallax stars in this catalogue.

The catalogue ofAdams embraces stars of the spectral types F, G, Kand M. In order to complete this material by parallaxes of blue starsI add from the catalogue ofWalkey those stars in his catalogue thatbelong to the spectral types B and A, confining myself to stars forwhich the parallax may be considered as rather reliable. There arein all 61 such stars, so that a sum of 561 stars with known distanceis to be discussed.

For all these stars we knowm andM and for the great part ofthem also the proper motion μ. We can therefore for each spectraltype compute the mean values and the dispersion of these attributes.We thus get the following table, in which I confine myself to the meanvalues of the attributes.

[Pg 45]

TABLE 6.

MEAN VALUES OF m, M AND THE PROPER MOTIONS (μ) OF PARALLAX STARSOF DIFFERENT SPECTRAL TYPES.

We shall later consider all parallax stars taken together. We findfromtable 6 that the apparent magnitude, as well as the absolutemagnitude, is approximately the same for all yellow and red stars andeven for the stars of type F, the apparent magnitude being approximatelyequal to +6^m and the absolute magnitude equal to +2^m. For type Bwe find the mean value of M to be -1^m.7 and for type A we findM = +0^m.6. The proper motion also varies in the same way, beingfor F, G, K, M approximately 0″.5 and for B and A 0″.1. As to the meanvalues ofM and μ we cannot draw distinct conclusions from this material,because the parallax stars are selected in a certain way which essentiallyinfluences these mean values, as will be more fully discussed below.The most interesting conclusion to be drawn from the parallax stars isobtained from their distribution over different values ofM. In thememoir referred to,Adams has obtained the following table (somewhatdifferently arranged from the table ofAdams),[16] which gives the numberof parallax stars for different values of the absolute magnitude fordifferent spectral types.

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 typeswe may distinguish between a bright group and a faint group. With[Pg 46]a terminology proposed byHertzsprung the former group is said toconsist ofgiant stars, the latter group ofdwarf 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 TYPESOVER DIFFERENT ABSOLUTE MAGNITUDES.

In the distribution of all the parallax stars we once more finda similar 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 the parallaxstars are arranged according to their apparent proper motion, or evenaccording to their absolute proper motion, a similar bipartition is revealedin their frequency distribution.

Nevertheless the bipartition of the stars into two such distinct classesmust be considered as vague and doubtful. Such anapparent bipartition[Pg 47]is, indeed, necessary in all statistics as soon as individuals are selectedfrom a given population in such a manner as the parallax stars areselected from the stars in space. Let us consider three attributes, sayA,BandC, of the individuals of a population and suppose that the attributeCispositively correlated to the attributesA andB, so that to great orsmall values ofA orB correspond respectively great or small valuesofC. Now if the individuals in the population are statistically selectedin such a way that we choose out individuals having great values ofthe attributesA and small values of the attributeB, then we get a statisticalseries regarding the attributeC, 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 the following.The annual parallax can only be determined for near stars, nearer than,say, 5 siriometers. The direct picking out of these stars is not possible.The astronomers have therefore attacked the problem in the followingway. The near stars must, on account of their proximity, be relativelybrighter than other stars and secondly possess greater proper motionsthan those. Therefore parallax observations are essentially limited to(1) bright stars, (2) stars with great proper motions. Hence the selectedattributes of the stars arem and μ. Butm and μ are both positivelycorrelated toM. By the selection of stars with smallm and great μ weget a series of stars which regarding the attributeM seem to be dividedinto two distinct classes.

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 atall of classes of giant and dwarf stars, not any more than a classificationof this kind can be made regarding the height of the men in a population.What may be statistically concluded from the distribution of the absolutemagnitudes of the parallax stars is only that thedispersion inM isincreased at the transition from blue to yellow or red stars. The fillingup of the gap between the “dwarfs” and the “giants” will probably beperformed according as our knowledge of the distance of the stars isextended, where, however, not the annual parallax but other methodsof measuring the distance must be employed.[Pg 48]

TABLE 8.

THE ABSOLUTELY FAINTEST STARS.

[Pg 49]Regarding the absolute brightness of the stars we may draw someconclusions of interest. We find fromtable 7 that the absolute magnitudeof the parallax stars varies between -4 and +11, the extreme starsbeing of type M. The absolutely brightest stars have a rather greatdistance from us and their absolute magnitude is badly determined.The brightest star in the table is Antares withM = -4.6, which valueis based on the parallax 0″.014 found byAdams. 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 ofAdams. It is probable from a discussion oftheB-stars, to which we return in a later chapter, that the absolutelybrightest stars have a magnitude of the order -5^m or -6^m. If theparallaxes smaller than 0″.01 were taken into account we should findthat Canopus would represent the absolutely brightest star, havingM = -8.17, and next to it we should findRigel, havingM = -6.97,but both these values are based on an annual parallax equal to 0″.007,which is too small to allow of an estimation of the real value of theabsolute magnitude.

If on the contrary theabsolutely faintest stars be considered, theparallax stars give more trustworthy results. Here we have only todo with 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 9^m.

There are in all 19 such stars. The faintest of all known stars isInnes'star “Proxima Centauri” withM = 13.9. The third star isBarnard'sstar withM = 11.7, both being, together with α Centauri, also the nearestof all known stars. The mean distance of all the faint stars 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 withM > 13^m and the mean value ofM,for all stars in the Galaxy, is probably not far from the absolute valueof the faint parallax stars in this table. This problem will be discussedin a later part of these lectures.

PLATE I.

CONVERSION OF EQUATORIAL COORDINATES INTO GALACTIC COORDINATES.

PLATE II.

Squares and Constellations.

PLATE III.

The Harvard Classification of Stellar Spectra.

Plate IV.

Distribution of the parallax stars overdifferent absolute magnitudes.

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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 inM"

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

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