Lectures on Stellar Statistics Part 2

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

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

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

_Type A_ (Sirius type) is characterized by the great intensity of the hydrogen lines (compare plate III). The helium lines have vanished.

Other lines visible but faintly.

The greater part of the stars visible to the naked eye are found here.

There are 1251 stars brighter than the 6th magnitude which belong to this type. Sirius, Vega, Castor, Altair, Deneb and others are all A-stars.

_Type F_ (Calcium type). The hydrogen lines still rather prominent but not so broad as in the preceding type. The two calcium lines H and K (396.9, 393.4 ) strongly p.r.o.nounced.

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

_Type G_ (Sun type). Numerous metallic lines together with relatively faint hydrogen lines.

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

_Type K._ The hydrogen lines still fainter. The K-line attains its maximum intensity (is not especially p.r.o.nounced in the figure of plate III).

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

We find here ? Andromedae, Aquilae, Arcturus, a Ca.s.siopeiae, Pollux and Aldebaran, which last forms a transition to the next type.

_Type M._ The spectrum is banded and belongs to SECCHI's third type. The flutings are due to t.i.tanium oxide.

Only 190 of the stars visible to the naked eye belong to this type.

Generally they are rather faint, but we here find Betelgeuze, a Herculis, Pegasi, a Scorpii (Antares) and most variables of long period, which form a special sub-type _Md_, characterized by bright hydrogen lines together with the flutings.

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

_Type N_ (SECCHI's fourth type). Banded spectra. The flutings are due to compounds of carbon.

Here are found only faint stars. The total number is 241. All are red.

27 stars having this spectrum are variables of long period of the same type as Md.

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

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 mentioned above, but fill up the intervals between the types with sub-types which are designated by the name of the type followed by a numeral 0, 1, 2, ..., 9. Thus the sub-types between A and F have the designations A0, A1, A2, ..., A9, F0, &c. Exceptions are made as already indicated, for the extreme types O and M.

11. _Spectral index._ It may be gathered from the above description that the definition of the types implies many vague moments. Especially in regard to the G-type are very different definitions indeed accepted, even at Harvard.[6] It is also a defect that the definitions do not directly give _quant.i.tative_ characteristics of the spectra. None the less it is possible to subst.i.tute for the spectral cla.s.ses a continuous scale expressing the spectral character of a star. Such a scale is indeed implicit in the Harvard cla.s.sification of the spectra.

Let us use the term _spectral index_ (_s_) to define a number expressing the spectral character of a star. Then we may conveniently define this conception in the following way. Let A0 correspond to the spectral index _s_ = 0.0, F0 to _s_ = +1.0, G0 to _s_ = +2.0, K0 to _s_ = +3.0 M0 to _s_ = +4.0 and B0 to _s_ = -1.0. Further, let A1, A2, A3, &c., have the spectral indices +0.1, +0.2, +0.3, &c., and in like manner with the other intermediate sub-cla.s.ses. Then it is evident that to all spectral cla.s.ses between B0 and M there corresponds a certain spectral index _s_.

The extreme types O and N are not here included. Their spectral indices may however be determined, as will be seen later.

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

12. We have found in --9 that the light-radiation of a star is described by means of the total intensity (_I_), the mean wave-length (?_0) and the dispersion of the wave-length (s_?). ?_0 and s_? may be deduced from the spectral observations. It must here be observed that the observations give, not the intensities at different wave-lengths but, the values of these intensities as they are apprehended by the instruments employed--the eye or the photographic plate. For the derivation of the true curve of intensity we must know the distributive function of the instrument (L. M. 67). As to the eye, we have reason to believe, from the bolometric observations of LANGLEY (1888), that the mean wave-length of the visual curve of intensity nearly coincides with that of the true intensity-curve, a conclusion easily understood from DARWIN's principles of evolution, which demand that the human eye in the course of time shall be developed in such a way that the mean wave-length of the visual intensity curve does coincide with that of the true curve (? = 530 ), when the greatest visual energy is obtained (L.

M. 67). As to the dispersion, this is always greater in the true intensity-curve than in the visual curve, for which, according to --10, it amounts to approximately 60 . We found indeed that the visual intensity curve is extended, approximately, from 400 to 760 , a sixth part of which interval, approximately, corresponds to the dispersion s of the visual curve.

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

13. Both the visual and the photographic curves of intensity differ according to the temperature of the radiating body and are therefore different for stars of different spectral types. Here the mean wave-length follows the formula of WIEN, which says that this wave-length varies inversely as the temperature. The total intensity, according to the law of STEPHAN, varies directly as the fourth power of the temperature. Even the dispersion is dependent on the variation of the temperature--directly as the mean wave-length, inversely as the temperature of the star (L. M. 41)--so that the mean wave-length, as well as the dispersion of the wave-length, is smaller for the hot stars O and B than for the cooler ones (K and M types). It is in this manner possible to determine the temperature of a star from a determination of its mean wave-length (?_0) or from the dispersion in ?. Such determinations (from ?_0) have been made by SCHEINER and WILSING in Potsdam, by ROSENBERG and others, though these researches still have to be developed to a greater degree of accuracy.

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

15. _Colour-index._ We have already pointed out in --9 that the colour may be identified with the mean wave-length (?_0). As further ?_0 is closely connected with the spectral index (_s_), we may use the spectral index to represent the colour. Instead of _s_ there may also be used another expression for the colour, called the colour-index. This expression was first introduced by SCHWARZSCHILD, and is defined in the following way.

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

_The difference between the photographic and the visual magnitudes is called the colour-index (_c_)._ The correlation between this index and the spectral-index is found to be rather high (_r_ = +0.96). In L. M.

II, 19 I have deduced the following tables giving the spectral-type corresponding to a given colour-index, and inversely.

TABLE 1.

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

+-------------------+----------------+ | 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 obtain their photographic magnitude through adding the colour-index. This may be considered as known (taking into account the high coefficient of correlation between _s_ and _c_) as soon as we know the spectral type of the star. We may conclude directly that the number of stars having a photographic magnitude brighter than 6.0 is considerably smaller than the number of stars visually brighter than this magnitude. There are, indeed, 4701 stars for which _m_ < 6.0="" and="" 2874="" stars="" having="" _m'_=""><>

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

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

17. Summing up the remarks here given on the apparent attributes of the stars we find them referred to the following princ.i.p.al groups:--

I. _The position of the stars_ is here generally given in galactic longitude (_l_) and lat.i.tude (_b_). Moreover their equatorial coordinates (a and d) are given in an abridged notation (ad), where the first four numbers give the right ascension in hours and minutes and the last two numbers give the declination in degrees, the latter being printed in italics if the declination is negative.

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

II. _The apparent motion of the stars_ will be given in radial components (_W_) expressed in sir./st. and their motion perpendicular to the line of sight. These components will be expressed in one component (_u_0_) parallel to the galactic plane, and one component (_v_0_) perpendicular to it. If the distance (_r_) is known we are able to convert these components into components of the linear velocity perpendicular to the line of sight (_U_ and _V_).