A Comparison of Spectroscopic and Trigonometric Parallaxes

The calibration of spectroscofric absolute magnitudes has been made by selecting groups of stars with similar intensities for pairs of "sensitive" lines and adjusting the calibration-curves so that the absolute magnitudes which these lines give for each group shall agree with those determined for the same stars from other data (trigo- nometric parallaxes, proper motions, etc.). Since the spectroscopic data are affected by accidental errors, this practically unavoidable process gives a regressioii-curve which inevitably underestimates the deviations of the individual absolute magnitudes from the general mean for the spectral class concerned, while giving an accurate value I or this mean. The magnitude of this effect can be determined by the comparison of the spectro- scopic and trigonometric parallaxes of the same stars, provided that the accidental errors of the latter are known. Formulae are developed for the determination of the constants involved in this problem, and of the statistical errors of the constants derived from samples of finite size. The spectroscopic parallaxes of Mo~unt Wilson Contribution No. 511 are then com- pared with the trigonometric parallaxes given in Schlesinger's General Catalogue (Yale, 1936). For the 1140 stars of the main sequence which are available, the real dispersion in absolute magnitude about the mean for the subclass of the Draper classification corresponds to a standard deviation of ±34 per cent in the parallax, while the errors of the spectroscopic determination (including the effects of real differences between stars whose sensitive lines appear the same) give a standard deviation of ±38 per cent. The means for all the stars of a given subclass are accurate within 5 per cent, except for a few faint M dwarfs. The differences between the individual spectroscopic absolute magnitudes and the mean for the subclass should be multiplied by 1.19 to give an im- proved calibration for groups selected according to the spectroscopic criteria, but by 2.6 in order to give values which wifi be correct in the mean for groups of stars selected by criteria independent of the spectroscopic data. When these stars are divided into seven groups according to spectral type, the results for all are similar, although with some- what larger fluctuations than would be anticipated from errors of sampling. Comparison of the mean for groups of stars selected by spectroscopic absolute magnitude and by proper motion fully confirms the anticipated differences. For the giant stars, 732 in number, the parallaxes are smaller; but good mean results can be obtained. The standard deviation in parallax, corresponding to real dispersion in absolute magnitude, is ± 52 per cent (increased by the inclusion of supergiants and subgiants), while that arising from the spectroscoJ~ic errors is ± 38 per cent. The general means of the spectroscopic and trigonometric parallaxes agree perfectly. The differences from the mean require no correction for grouping according to spectroscopic criteria, but a factor of 1.6 when the criteria are independent. A special discussion of ~7 stars of great absolute brightness shows that the probable errors assigned in "Schlesinger's Catalogue" represent correctly the true accidental errors of the determinations. Formulae and tables are given (sec. 13) for the rapid application of the corrections to the spectroscopic parallaxes. These are important for only a small fraction of the whole number of stars. Their applicabifity to supergiants is doubtful. * Contributions from the Mount Wilson Observatory, Carnegie Institution of Wash- ington, No. 589. `Research Associate, Mount Wilson Observatory. 38