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 (λ0) or from the dispersion in λ. Such determinations (from λ0) 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 (λ0). As further λ0 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 |