Lectures on Stellar Statistics | Page 8

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 ([lambda]0) and the dispersion of the wave-length ([sigma][lambda]). [lambda]0 and [sigma][lambda] 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 ([lambda] = 530 [mu][mu]), 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 [mu][mu]. We found indeed that the visual intensity curve is extended, approximately, from 400 [mu][mu] to 760 [mu][mu], a sixth part of which interval, approximately, corresponds to the dispersion [sigma] of the visual curve.
In the case of the photographic intensity-curve the circumstances are different. The mean wave-length of the photographic curve is, approximately, 450 [mu][mu], with a dispersion of 16 [mu][mu], 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 ([lambda]0) or from the dispersion in [lambda]. Such determinations (from [lambda]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 ([lambda]0). As further [lambda]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