Astron. Astrophys. 350, 985-996 (1999)

4. A main sequence (V-I)₀ versus M_V relation from nearby Hipparcos stars

The confinements of a color - magnitude diagram like Fig. 2 may be interpreted by means of a minimum and maximum reddening caused by the diffuse ISM in the galactic disk as found in Jonch-Sorensen & Knude (1994), Fig. 5. If the reddening extremes, of the diffuse atomic part of the interstellar medium, are known and a color - magnitude relation is available for the relevant photometric bands the distances to the less and most reddened features may be found by shifting the color - magnitude relation, much like main sequence fitting for clusters.

The Hipparcos and Tycho Catalogues (ESA 1997) offer the opportunity to derive a (V-I)₀ - M_V relation. For this purpose we have extracted stars with   20 mas and with luminosity classification V to define our main sequence. There is about 1800 such stars in the Catalogues and the Hertzsprung - Russell diagram shows that the luminosity classification (LC here after) is reliable, only very few, if any, class IV and III stars appears to have been included (see Fig. 4b and c). Since the upper distance limit for this sample is only about 50 pc we may assume that reddening can only have a minor influence on the M_V determination. From the compilation of uvby photometry and Hipparcos parallaxes Vergely et al. (1998) have demonstrated that the average color excess is negligible within 60 - 80 pc and that the standard dispersion is less than 0.020 within the same distances. Diffuse clouds are, however, present at very small distances as shown by the EUVE shadowing experiment, Berghöfer et al. (1998). These shadowing clouds are at distances 50 pc and have 0.050.

The (V-I) colors from the Hipparcos and Tycho Catalogues are only measured for 3000 Cataloque entries. For the remaining stars this color is estimated from 19 different methods, only 12 of these are used for the 20 mas sample. For details see The Hipparcos and Tycho Catalogues Vol. 1, Sect. 1.3, Appendix 5. Stars included in the 20 mas sample has V in the range from 4 to 12. A (B-V) - (V-I) diagram shows that a few (V-I) colors between 1.0 and 2.0 probably are estimated wrongly, some of these are seen in the comparison of Fig. 4b to Fig. 4c, their estimated (V-I) value is too blue for their luminosity by 03 or alternatively their absolute magnitudes are too faint by +02 relative to the observed color sample, see discussion later in this section. In Fig. 3a we display how the Cataloque value of (V-I) has been estimated for our stars with 20 mas. Only 4 methods contribute larger numbers. About 300 values are seen to have been actually measured. Fig. 3b shows that the measured sample covers the complete (V-I)₀ range we discuss. Our final relation, see Fig. 4a, shows a kink at (V-I)₀ 0.5. According to Fig. 3b this color is estimated from method A, H and L respectively; not by just one method. No color is seen to depend on a single estimate. Finally we show in Fig. 3c how the relative accuracy / varies with (V-I)₀. Apart from two of the reddest bins the median relative accuracy is better than 10, the two worst cases pertain to two of the faintest stars in the Hipparcos sample. The 10 accuracy together with the photometric error in our observed V, (V-I) CCD data determine the error in distances from the main sequence fitting. In Fig. 4f we show the relative error of the observed and estimated sample separately.

Fig. 3. a  Hipparcos determination of (V-I) for luminosity class V stars with 20 mas. Capital letters refer to to the method as described in Sect. 1.3 Appendix 5 in Volume 1 of the Hipparcos and Tycho Catalogues. A is a direct observation; for three most frequent calibrations H, I and L please consult Vol. 1. b  Color range covered by the various methods estimating (V-I). c  Relative parallax error as a function of (V-I). The solid curve displays the median calculated for 0.1 color bins

Fig. 4a-f. LC V stars with 20 mas. a  The thick solid curve is the average MV in 0.1 (V-I) bins. The three thinner curves are the (V-I) - MV relations from Hawley et al. (1999) for [Fe/H] = -0.3, 0.0, +0.2, the latter is the most luminous for a given color. b  Estimated (V-I) values only. c  Observed (V-I) values only. d   is MV = MV(estimated) - MV(observed) for 0.1 (V-I) bins. depicts the error of the mean for the same 0.1 (V-I) bins computed for the complete sample. + are (MV(V-I) - MV(V-I+0.1))/m.e. e  + points with estimated (V-I) values, points with measured (V-I). f  As Fig. 3c but separate symbols for measured () and estimated (+) data

If the color distribution is homogeneous the formal color standard deviation in one bin is expected to be 0.03 which is the value we find. The intrinsic standard deviation of M_V in one color bin is in the range magnitude, the luminosity effects of varying metallicity is included in this intrinsic scatter as is the evolution across the main sequence band.

The (V-I)₀ color is sorted in 0.1 mag bins. In Fig. 4b and c the complete sample is shown separated in data with estimated (V-I) and observed (V-I) respectively the two samples are seen to follow similar M_V trends. The only major difference is the presence of 20-30 estimated (V-I) values in the color range from 1.0 to 2.0 that are less luminous than expected from the data with observed (V-I) in the same range. As the discussion below shows the location of the deviant points may not exclusively be due to errors in the estimated (V-I) but rather caused by a relatively large value of compared to for the observed sample, see Fig. 4f. We do, however, prefer the combined sample to the observed one regardless of a slightly increase of the standard deviation in M_V from 0.3 to 0.4-0.5. But the inclusion of the points with estimated color does not change the error of the mean, see Fig. 4d for values of the latter. As seen in Fig. 4c the observed sample with (V-I) 0.4 could be too sparse for the M_V estimates and in this color range the estimated data follow the observed ones quite closely and Fig. 4f shows that there is no difference in the range for the two subsamples. For these colors the width of the main sequence band is influenced by fast stellar evolution, the combined sample helps keeping down the formal error.

It is of course an issue whether the calibrating sample is a fair match to the state of evolution sampled in the field under investigation.

In Fig. 4d we depict the difference M_V M_V(V-I)_est - M_V(V-I)_obs defined for each 0.1 color bin as a function of (V-I). The general agreement is good for (V-I) 1.5 although the absolute magnitude calculated from the estimated data set seem systematically fainter than those from the observed sample. For (V-I) 1.5 = 0140.14. The diamonds display the formal mean error of for each (V-I) bin pertaining to the combined sample. We note that of all the colors the agreement between M_V(est) and M_V(obs) seems best around (V-I) = 05 just where the (V-I)₀ - M_V relation, see Fig. 4a, has one of its kinks and that apart for the most extreme red and blue bins the mean error is smaller than 01.

Why is M_V for the estimated sample slightly fainter than for the observed sample? As Fig. 4d shows this effect is largest for bins redder than (V-I) = 1.4-1.5. One would expect, because of their low luminosity, that the reddest stars would generally be apparently fainter than the bluer ones and that the chance to have an observed (V-I) increases with the apparent brightness of the star. In Fig. 4e we plot V vs. (V-I); indicate an observed point and + points with estimated (V-I). We note that for the bluest stars almost all are measured and for the reddest stars the faintest all have estimated color, beyond V-I 1.5 the two samples are almost discrepant, but Fig. 4e does not imply that the estimated V-I values necessarily have been estimated too blue regardless of the immediate impression that for stars fainter than V 10 the estimated colors all are bluer than the observed ones.

Since the stars with estimated colors are the faintest one might expect that they could have parallaxes with inferior measurements than the brighter ones with observed V-I. Fig. 4f is an enlargement of Fig. 3c with the two subsamples indicated. We notice that the points bluer than 04 have 0.05 independent of how (V-I) was procured. For the points redder than 1.4-1.5 the same is the case for stars with observed V-I whereas the estimated points have 0.10, a marked difference that may influence the M_V determination.

The different relative accuracy in the parallax data for the observed and estimated color samples introduces different biases in the two subsets. According to Lutz & Kelker (1973) a correction M_V = M_V(true) - M_V(observed) depending on is required. An impression of the size of this correction may be obtained from Lutz & Kelker (1973) with = 0.05 representative for the observed color sample and = 0.12 typical for the redder part of the estimated color sample corrections become -0.02 and -0.18 respectively. The difference between the two observed absolute magnitudes, for a given color V-I, is accordingly expected as M_V(est) - M_V(obs) = 0.18-0.02 02, which should be compared to the differences shown as filled circles in Fig. 4d, the sign of the predicted difference is correct and so is the size of the difference, recall that for (V-I) 1.5 = 0140.14.

The resulting (V-I)₀ - M_V relation is shown as the thick solid curve in Fig. 4a we note the wavy shape of the relation, if it had been featureless the adoption to the observed data would have been more difficult. The M_V values plotted in Fig. 4d and the not completely negligible Lutz-Kelker correction 02 possibly required for the estimated color sample would make the M_V values of the (V-I)₀ - M_V relation brighter by 02. Such a correction would increase the distances derived in the following section with only 10 so we have decided not to apply this correction, but its presence should be born in mind.

In Fig. 4d we also plot (+) the dimensionless measure (M_V(V-I)-M_V(V-I+0.1))/m.e., in the figure. The values are scaled down by a factor of 100 so we may use the scale of the figure's ordinate axis unchanged. The (V-I)₀ - M_V relation is determined with such small mean errors that the difference between neighboring points relative to the mean error is found in the range from 3 to 25. Only two ratios are smaller than 3, 2.9 and 2.3 for the (V-I) bins [0.10.2] and [1.71.8] respectively. Apparently the detailed shape of the (V-I)₀ - M_V relation is significant at a rather high level.

We have found it needless to correct the estimated absolute magnitudes for any Malmquist bias. This conclusion results from an investigation of the size of the Malmquist correction. For the luminous stars we expect only negligible corrections but for the less luminous ones even the low upper distance limit of 50 pc may require some correction. We proceeded as follows. The 1800 stars were sorted in V05 magnitude bins and the ln(N(V)) versus V diagram was produced, N(V) is the number of values in the V05 bin. The slope d(ln(N(V)))/dV was found to be constant in the three magnitude intervals: [-6.5], [6.59.0] and [9.012.5] confining all stars we extracted from the Hipparcos and Tycho Catalogues. For each star a correction according to its V magnitude was assigned, the correction computed for any (V-I)₀ interval is the plain average of these corrections. We further assume that the classification error is = 03/ (see p. 455 in Vol. 1 of the Hipparcos and Tycho Catalogues) where n is the number of stars in the (V-I)₀ bin in question. The classification error is the error of the mean and not the error of an individual M_V determination; is so small that is has no significance for the classification error. The final corrections range from being less than a mmag to 023 valid for the two reddest intervals included (V-I)₀ [1.81.9] and [1.92.0]. We find only a few stars with 20 mas, LC V and (V -I)₀ 2.0.

The large correction for the two reddest 01 (V-I) bins is mainly due to the small number of stars in these color intervals and within 50 pc, only 12 stars in each. We have therefore chosen not to use the Malmquist corrected (V-I)₀ - M_V relation since the bias is negligible for the complete (V -I)₀ range apart from the two very reddest intervals. The main thing is that the Malmquist bias correction will not change the wavy appearance of the (V - I)₀ - M_V relation.

As may be seen in Fig. 4a the (V-I)₀ - M_V relation covers a range of almost 10 magnitudes. It includes stars from +1.0^m to about +10^m, that is from early A0 to about M2 dwarfs.

The reliability of the (V-I)₀ - M_V relation may be judged from Fig. 4a where we have included three main sequences relations with [Fe/H] = -0.3, 0.0 and +0.2 respectively from Hawley et al. (1999). These relations are a combination of an empirical sequence valid for very nearby stars and theoretical relations. The only possible deviation for the common color range is noticed at (V-I)₀ 0.5 where we suggest a slightly more luminous M_V value than Hawley et al. As the data points in Fig. 4b and c indicate we may have included a few stars that should have been classified LC IV or even LC III. One could think of a criterion that would rid the sample of these possibly evolved stars but we do not want to introduce such an ad hoc criterion. And the shape of our relation is probably right for the main sequence as such and not only for a blue envelope. This conclusion is corroborated by Fig. 3.5.4 in the Hipparcos and Tycho Catalogues, Vol. 1. a diagram valid for stars with / 10 which also pertain to our sample, Fig. 3c. The blue envelope of the main sequence in Fig. 3.5.4 confirms the general shape of the relation we suggest.

© European Southern Observatory (ESO) 1999

Online publication: October 14, 1999
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