Calibrations for B-type stars in the Geneva photometric system1

New Astronomy Reviews 43 (1999) 343–387 www.elsevier.nl / locate / newar

Calibrations for B-type stars in the Geneva photometric system 1 N. Cramer 2 Geneva Observatory, CH-1290 Sauverny, Switzerland

Abstract The Geneva seven-colour photometric system has been applied for almost four decades. Its data bank presently contains 345 000 measurements of 47 600 stars of all types. About 40% of the latter are of types O, B and first A. The light of these bright and massive stars is practically without exception extinguished by interstellar dust. Here, we describe various calibrations made in a three-dimensional orthogonal reddening-free representation of the system, optimised for B-stars, and that separates temperature and gravity effects. Central to all calibrations lies the empirical determination of the intrinsic colours, described here in some detail. Comparisons with synthetic photometry are made. Calibrations for effective temperature, bolometric correction, absolute magnitude are discussed as well as correlations with MK type, standard U,B,V ¨ photometry and the Stromgren b index. The unavoidable limitations imposed upon photometry by undetected causes of scatter, such as in binary systems, are evaluated. Open questions and future prospects are mentioned.  1999 Elsevier Science B.V. All rights reserved. PACS: 95.75.De; 95.85.Kr; 97.10.Ri; 98.38.–j Keywords: Techniques: photometric; Stars: early-type; Stars: fundamental parameters; Dust: extinction

1. Introduction The Geneva seven-colour photometric system was defined in 1960 and has been used without interruption in both hemispheres up to the present date. The current data base contains measurements of 47 600 stars of all spectral types, most of them measured at least three times, and some much more often due to their use as standards or to their being the subjects of specific studies of variability. The system may be considered as ‘‘closed’’ (i.e. restricted duplication of instrumentation, unique reduction procedure; see Rufener, 1985) and the data are therefore extremely homogeneous and accurate. This implies that cali1

Based on data acquired at the La Silla (ESO, Chile), Jungfraujoch and Gornergrat (HFSJG International Foundation, Switzerland), and Haute-Provence (OHP, France) observatories. 2 E-mail address: [email protected] (N. Cramer)

brations, or any other procedures set up to extract information from the measurements, are unique. They can be applied to all the available data, even to those acquired over several decades, without any adjustments being necessary. A general review of the system was presented earlier by Golay in Vistas in Astronomy (Golay, 1980). About 40% of these 47 600 stars can be loosely qualified as ‘‘B-type’’ (or, more precisely O, B and first A-types), and the properties of our photometry regarding them will be the subject of the following discussion. The hot, highly ionised, radiative atmospheres of these massive stars account for a relatively simple spectral energy distribution in the visible, marked by the salient features of the Balmer jump and the hydrogen lines. Their energy distribution, which is the dominant feature perceived by multicolour photometry, is less affected by variations in chemical composition than that of their cooler coun-

1387-6473 / 99 / $ – see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S1387-6473( 99 )00041-X

344

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

terparts. Moreover, these young stars are (in our galactic neighbourhood, at any rate) not highly dispersed in initial chemical abundances. These factors reduce the colorimetric dispersion and contribute to simplify calibration processes designed to determine basic quantities such as effective temperature and absolute magnitude, for example. B-type stars are, however, much less numerous than cooler types, and their greater distances unavoidably link the study of their population by multicolour photometry to the effects of interstellar extinction by dust. Fortunately, for effective temperatures higher than about 10 4 K, the vector of interstellar reddening in various colour index representations stands well separated from those related to the variations of other major astrophysical quantities. This allows us to de-redden such stars with confidence and, conversely, to study the distribution of interstellar dust efficiently with their aid. Calibrations for B-type stars are therefore best done in reddening-free representations. Such will be the nature of those discussed below. Calibrations are essentially made by correlating in an optimum manner the observable photometric quantities with given physical quantities that have been determined otherwise. The strategy that is followed will depend on the properties of the system – i.e. the way its different passbands sample the energy distribution – and on the quantity to be estimated. Quite generally, optimisation includes the search for simplicity, which is often also that of the minimal set of assumptions regarding the calibration data. Thus, one gives greatest weight to ‘‘primary’’ physical data, whereas data of the same nature but derived via a more elaborate series of processes will be given less weight, if not discarded altogether. Even primary data do, however, ‘‘evolve’’ with time as instrumental techniques get better – e.g. trigonometric parallaxes – and some types of calibrations will be more ‘‘robust’’ than others depending on the nature of those data. The advent of good theoretical stellar atmosphere models has decisively aided the interpretation of multicolour photometry during these last twenty years. Synthetic photometry done by filtering theoretical fluxes through sets of passbands links the basic physical quantities to the observed fluxes and helps to determine the regions where colour-colour

diagrams perform best, thus revealing the proper shape of a possible calibration relation. Such a method will not, however, provide a workable calibration by itself. Synthetic photometry accumulates the uncertainties over the models with those affecting the true shapes of the passbands. All the more so, regarding the latter, in an ‘‘open’’ system such as the standard UBV system which is widely reproduced throughout the astronomical community. The use of empirical – i.e. ‘‘minimally assuming’’ data can, however, be applied profitably to adjust synthetic relations in most homogeneous sets of ¨ photometric data (see Kunzli et al., 1997). As mentioned above, the photometric analysis of B stars has to be done in a reddening-free context. The Geneva system is particularly well suited to that end due to the variety of reddening-free ‘‘Q-type’’ parameters that can be constructed with the five intermediate bands U, B1, B2, V1, G (see Fig. 1). This has led us to establish a set of three orthogonal parameters named X, Y and Z (Cramer & Maeder, 1979) which are optimised for the treatment of B stars. The three-dimensional classification scheme makes full use of the five intermediate bands and all the calibrations that will be presented here have been made in this reddening-free photometric space. The following discussion will first present a short overview of the Geneva system and its historical background and will then pass on to a more detailed review of the calibrations. The presentation will strive to be as clear as possible with the object of providing the reader who would like to use the Geneva system with a well defined ‘‘toolbox’’. Much of the discussion will rely on tables and figures, which are far more eloquent than words. Specific virtues or difficulties will be brought up within context. Many of the techniques and calibrations discussed can be improved upon. Ways to do so will be mentioned when conceivable.

2. The Geneva system

2.1. Historical context The Geneva photometric system was created by M. Golay in the late fifties when photomultiplier tubes began to become easily available. The pass-

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

bands (Fig. 1, Rufener & Nicolet, 1988; Nicolet, 1996) were chosen with the object of reproducing the general properties of Johnson’s U,B,V system with the help of an almost identical V band but with slightly different U and B bands presenting less overlap at the Balmer jump. The other four intermediate bands, B1, B2, V1 and G, were added in view of approximating the stellar classification properties of the Barbier-Chalonge-Divan (BCD) low resolution photographic spectrophotometry that was based on a measurement of the Balmer jump, an estimate of the hydrogen line strengths and on the determination of gradients over the Balmer and Paschen continua. The first systematic observations were carried out by Golay and Rufener in February 1960 at the Sphinx Observatory (3600 m) of the Jungfraujoch High Altitude Research Station with the aid of a first photometer named ‘‘P2’’. Together with that instrumentation, Rufener also conceived a new measurement and reduction technique based on a twin Bouguer method that makes use of two contravarying extinction stars, and called by him the ‘‘M and D’’ technique (Rufener, 1964; Rufener, 1985). That technique allows the determination of ‘‘instantaneous’’ extinction lines at given moments (usually 4 to 6 per night), and allows an accurate determination of atmospheric extinction and of its evolution throughout the duration of the observations.

345

Only three other photometers have since been equipped with Geneva filters. The latest one, called ‘‘P7’’ (Burnet & Rufener, 1979), is a double-channel instrument that ensures the quasi-simultaneous measurement through the 7 filters of star and star plus sky background by means of a single photomultiplier, fast revolving filter wheel and beam selecting chopper. Most of the Geneva photometry in the southern hemisphere has been carried out with P7, which is also the most accurate of the four instruments due to its rapid sampling technique (Bartholdi et al., 1984).

2.2. Homogeneity The conservation of the system has been the primary concern of Geneva photometry. The limited reproduction of the system and long-term occupation of a small number of observing sites have greatly contributed to the stability of measurement conditions. Instrumental stability is achieved by using filters coming from the same batch of Schott glass and by thermostatically controlling the filters and detectors. The photometers are sealed so as to prevent dust and the condensation of water vapour from altering the instrumental response. The long term dedication of each telescope to a given photometer enhances instrumental stability. The constancy of the reduction techniques (Rufener, 1964; Rufener, 1985) ultimately ensures

Fig. 1. The passbands of the Geneva photometric system.

346

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

homogeneity. The whole data bank is periodically recalculated, and mean values of colours and magnitudes incorporating the new measurements are computed (see Cramer 1989). Concurrently, a variety of stability tests based on the inter-comparison of batches of old and new measurements are carried out to ascertain the conservation of the system (see Rufener, 1981; Rufener, 1988). These tests have consistently shown that the homogeneity of the colours exceeds the 10 23 magnitude level over the whole extent of the catalogue. Studies of microvariability have demonstrated that the homogeneity of the V-magnitudes is also close to the millimagnitude level (see Waelkens, 1991; Burki, 1999). The latter reference concerns a study of slow variability in the millimagnitude range. Ground-based observations, however carefully they are carried out, do not guarantee the absence of small seasonally dependent drifts of scale. These are dependent on the quality of the global definition of the standard and affect more obviously the mag-

nitudes than the indices, which are relative quantities. Various strategies have been successfully used by photometrists to eliminate such periodic drifts within small limits that are of the order of one percent. To achieve notably better homogeneity the observations need the continuous whole-sky conditions that can only be encountered in space. The most homogeneous and accurate system of magnitudes presently available is that acquired by the Hipparcos satellite. The possibility of improving the scale of Geneva photometry by classical means, but in confrontation with that data, is presently being examined by Bartholdi, Burki and Nicolet. A more general review of the homogeneity of the Geneva system was presented elsewhere by the author (Cramer, 1994a).

2.3. The data The Geneva photometric data bank currently contains about 345 000 individual measurements

Fig. 2. Distribution of the stars measured in the Geneva system in galactic co-ordinates. The selection effects due to various specific observational projects are apparent, as explained in the text.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

with the corresponding averaged colours and magnitudes of 47 600 stars. These are displayed in Fig. 2 in galactic co-ordinates. Some selection effects over the data may be seen. They reflect the various observing programs. Local concentrations correspond to star clusters or to surveys done in selected areas. The right hand side of the figure corresponds to the southern hemisphere, where the faster P7 photometer and better atmospheric conditions in the Atacama desert have boosted the rate of data acquisition. The photometric horizon of the ESO La Silla observatory may be seen as an arch extending into positive latitudes from the southern galactic plane. A survey in the southern galactic polar region appears as a band south of latitude 2408. The restriction of the data to the early-type stars alone provides us with a still respectable sample of 19 500 stars, as is displayed in Fig. 3. As expected, the great majority of these are concentrated towards the galactic plane. The photometric catalogue lists the average normalised colours U, V, B1, B2, V1, G and the V-

347

magnitude m V of each object (see its last ‘‘paper’’ edition: Rufener (1988), for details). These colours are in reality the colour indices [U2B], [V2B], [B12B], [B22B], [V12B] and [G2B] normalised to B. Thus, within the context of Geneva [U,B,V] diagrams discussed here, we shall sometimes use the normalised index [V2B]. This only affects the sign of the axis graduation and will be clearly indicated on the figures when necessary. Square brackets are used throughout by convention to distinguish the Geneva indices from those of other systems. The following relation converts the indices listed in the catalogue to magnitudes: m i 5 m V 2 [V2 B] 1 [i 2 B], where i is one of the 7 passbands. The reddening-free parameters mentioned above are derived from the three traditional reddening-free ‘‘Q-type’’ parameters d, D and g of the Geneva system. They can be expressed in terms of linear combinations of the normalised colours, and are defined as follows:

Fig. 3. Distribution of the O, B and first A-type stars of the current Geneva data base, in galactic coordinates.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

348

X 5 0.3797 1 1.3764U 2 1.2162B1 2 0.8498B2 2 0.1554V1 1 0.8449G Y 5 2 0.8288 1 0.3235U 2 2.3228B1 1 2.3363B2 1 0.7495V1 2 1.0865G Z 5 2 0.4572 1 0.0255U 2 0.1740B1 1 0.4696B2 2 1.1205V1 1 0.7994G • The X parameter has properties that are similar to those of the U2B index by measuring the Balmer jump, but benefits of a larger range of variation and has the important advantage of being reddening-free. It is the optimal temperature indicator for B stars in the Geneva system (note that the constant term has been corrected for a small error of less than a millimagnitude that propagated through all earlier publications). • Y is much less sensitive to the Balmer discontinuity, but maximises the sensitivity to the hydrogen lines via the B22B1 index embedded in the relation. This makes it the best gravity indicator for B stars in the Geneva system. • Z is virtually independent of temperature and gravity effects. It optimises the detection of variations of gradient over the range covered by | 0.01 mag) the V band. Very weakly dispersed (s 5 for normal B stars, It detects peculiarity (Ap, Bp stars) and has been shown to be correlated, in certain cases, with the surface magnetic field ˚ absorption feature intensity via the 5200 A (Cramer & Maeder, 1980). The following table (Table 1) gives the standard deviations of the six normalised indices, visual magnitude and the parameters calculated according to the procedure defined by Rufener (1988) and expressed in units of 10 23 mag. The weight P is a

value synonymous to the number of good measurements. The photometric colour-classification of all the data is presented in the four following figures. The first, Fig. 4a, is the classical [U,B,V] (Geneva) diagram. One may note the large number of reddened B stars which appear to fan out from the main sequence in the upper part of the diagram, and the separation of the K to M giants (below) from the main sequence which ultimately tends to turn up again. The three figures 4b, 4c, 4d, display the general classification properties of the three-dimensional X,Y,Z reddening-free representation. The optimisation for B stars is clearly shown in Fig. 4b by the almost rectilinear sequence developing in the X| 0. A marked luminosity effect direction at Y 5 displaces the evolved early types from that sequence towards the upper part of the diagram. The separation of the K and M giants from the dwarfs is rather better distinguished than in the U,B,V diagram with a turn-up of the giants sequence close to M0. The X,Z projection of Fig. 4c shows the continuing rectilinearity of the sequence of B stars in the other dimension, and the even clearer separation of the K–M stars. The Y,Z projection of Fig. 4d sights along the sequence of the B stars and has the singular property of gathering almost 20 000 stars, i.e. about 40% of the whole catalogue, into a narrow region only a few hundredths of a magnitude wide. The early-type supergiants stand out to the right of the clump with an amplitude reaching 0.3 mag. The vertical ‘‘beak’’ detaching itself downwards contains the magnetic Ap stars. The late type giants are again well separated from the dwarfs. One notes that the M giants sequence, containing the Mira-type variables, arcs well above the plot and does not interfere with the B types. The X,Y,Z parameter space is indeed an optimal representation that separates the B stars from all later types.

Table 1 Standard deviations of indices, visual magnitude and parameters (s 3 10 3 ) P

U

B

V

B1

B2

V1

G

mV

X

Y

Z

1 2 3 5 10

7.7 5.5 4.5 3.5 2.4

4.1 2.9 2.4 1.8 1.3

4.6 3.2 2.6 2.0 1.4

4.6 3.2 2.6 2.0 1.4

4.5 3.2 2.6 2.0 1.4

4.5 3.2 2.6 2.0 1.4

5.7 4.1 3.3 2.6 1.8

6.2 4.4 3.6 2.8 1.9

14.0 9.9 8.1 6.3 4.4

16.0 11.3 9.2 7.1 5.0

6.1 4.3 3.5 2.7 1.9

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

349

Fig. 4. (a) The classical U,B,V diagram in the Geneva system. (b) The reddening-free representation of the Geneva X,Y,Z parameters. The X,Y diagram shown here demonstrates the optimisation for the early-type stars which occupy an almost rectilinear sequence along the X direction. (c) The X,Z projection shows the remarkably rectilinear sequence of the B-stars. The excellent separation of K and M-type dwarfs and giants is also apparent. (d) The Y,Z projection has the uncommon property of gathering almost 20 000 O and B-type stars in the small clump to the right of the figure at Y 5 0. Those detaching themselves towards the extreme right are the early-type supergiants. Those descending and forming a sort of ‘‘beak’’ are the magnetic Ap and Bp stars.

3. Calibrations for B stars

3.1. Properties of the restricted X,Y-diagram The portion of the X,Y,Z-space occupied by the O,

B and early A-stars is easily separated from the rest of the data by cutting off at small negative values of Y (typically Y , 2 0.08) and by restricting Z. The thus isolated region is shown in the X,Y-plane of Fig. 5a where the supergiants and giants stand out in the

350

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 5. (a) The X,Y-diagram restricted to the O, B and first A-types. The Y parameter is sensitive to gravity in that region, and the stars that stand out in the upper part of the diagram are the bright giants and supergiants. (b) The restricted X,Z-diagram showing the very narrow dispersion in Z of the majority of the 20 000 stars contained therein. The ‘‘tail’’ rising above X 5 0 with negative values is due to the uncorrected residual reddening of highly extinguished O stars. The negative dispersion at larger X is due to the magnetic Ap and Bp stars ˚ An observational sampling bias favouring B8 stars relatively to B9 explains the concentration which present an absorption feature at 5200 A. at X 5 0.9.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

upper portion of the sequence. The range of the X parameter is quite important, 1.8 mag starting from the O stars, on the left, up to the B9–A0 main sequence stars and giants, on the right. The separation of gravity develops concurrently with the importance of the hydrogen lines as X increases. Practically indiscernible at the highest temperatures, it reaches its maximum extent of about 0.35 mag at the location of the earliest A stars. Looking perpendicularly in the X,Z projection of Fig. 5b, we note the rectilinear nature and extremely narrow dispersion (about 0.01 mag r.m.s.) of the sequence. The stars that stand out with negative values of Z are almost exclusively Ap and Bp stars (see Cramer & Maeder, 1980; North & Cramer, 1984). Z is therefore a parameter that is temperatureand gravity-independent in this temperature range. It will thus be irrelevant in any calibrations involving those quantities. Apart from some exceptional deviations presented by Wolf-Rayet stars, the dispersion seen at the extreme left of the figure is essentially

351

due to residual effects of extinction affecting highly reddened O stars, as will be discussed later. The basic properties of the X,Y-plane are best illustrated in Fig. 6 by the solar composition models (Kurucz, 1993) filtered by Nicolet using the last published definition of the passbands (Rufener & Nicolet, 1988). This figure, which could be perceived as the first step towards a synthetic calibration of the diagram, serves us here to delimit the capabilities of the photometry: • Photometric sensitivity to temperature and gravity is very low for the O- and first B-stars with T eff . 3 3 10 4 K. Spectroscopic information is far superior. • Definition improves very satisfactorily at lower temperatures, down to about 9 3 10 3 K. • Photometric discrimination is impeded by regions of ambiguity where overlapping of the grid occurs. That is the case for some supergiants with the lowest surface gravities, and is one of the

Fig. 6. Synthetic grid of the stellar atmosphere models computed by Kurucz (1993) in the X,Y-diagram. Such theoretical projections of physical parameters into representations that are observationally determined offer precious guidelines to their interpretation.

O6 O7 O8 O9V O9.5V B0V B0.5V B1V B1.5V B2V B2.5V B3V B4V B5V B6V B7V B8V B8.5V B9V B9.5V A0V A1V A2V A3V A4V A5V A6V A7V A8V

Y 0.010 0.011 0.012 0.015 0.018 0.020 0.018 0.015 0.010 0.002 2 0.001 0.000 0.001 0.005 0.015 0.020 0.044 0.050 0.046 0.034 0.011 2 0.017 2 0.049 2 0.071 2 0.090 2 0.110 2 0.125 2 0.139 2 0.157

X

2 0.009 2 0.006 2 0.003 0.034 0.072 0.103 0.193 0.260 0.324 0.468 0.546 0.609 0.689 0.756 0.878 0.930 1.129 1.217 1.404 1.493 1.557 1.595 1.615 1.609 1.593 1.535 1.490 1.430 1.339

Class V

Table 2 MK type versus X,Y,Z relation

Z 2 0.020 2 0.017 2 0.012 2 0.007 2 0.004 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.003 2 0.002 2 0.002 2 0.001 2 0.002 2 0.004 2 0.008 2 0.012 2 0.015 2 0.01 2 0.022 2 0.025 2 0.027 O9IV O9.5IV B0IV B0.5IV B1IV B1.5IV B2IV B2.5IV B3IV B4IV B5IV B6IV B7IV B8IV B9IV B9.5IV A0IV A1IV A2IV A3IV

X 2 0.001 0.055 0.092 0.137 0.229 0.280 0.356 0.520 0.656 0.710 0.750 0.899 0.959 1.062 1.315 1.480 1.565 1.640 1.621 1.600

Y 0.010 0.011 0.012 0.012 0.011 0.010 0.010 0.011 0.017 0.020 0.025 0.040 0.048 0.056 0.065 0.054 0.036 2 0.026 2 0.066 2 0.113

Class IV Z 2 0.012 2 0.010 2 0.007 2 0.005 2 0.002 2 0.001 2 0.001 2 0.001 2 0.001 2 0.001 2 0.001 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.003 2 0.007 2 0.011 2 0.014 O9III O9.5III B0III B0.5III B1III B1.5III B2III B3III B4III B5III B6III B7III B8III B9III B9.5III A0III A1III A2III A3III A5III A6III

X 0.022 0.040 0.044 0.126 0.201 0.284 0.324 0.598 0.694 0.837 0.912 0.941 1.057 1.265 1.375 1.550 1.663 1.653 1.612 1.532 1.450

Y 0.015 0.016 0.017 0.022 0.022 0.021 0.020 0.019 0.019 0.032 0.042 0.048 0.068 0.093 0.100 0.095 0.043 2 0.025 2 0.074 2 0.107 2 0.160

Class III Z 2 0.007 2 0.005 2 0.003 2 0.003 2 0.003 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.003 2 0.003 2 0.006 2 0.011 2 0.018 2 0.023 2 0.028 O9II O9.5II B0II B0.5II B1II B1.5II B2II B3II B4II B5II B7II B8II B9II A0II A5II A7II

X 0.017 0.020 0.045 0.108 0.154 0.220 0.253 0.377 0.580 0.723 0.880 0.942 1.004 1.685 2.161 2.031

Y 0.014 0.015 0.016 0.018 0.020 0.022 0.023 0.025 0.037 0.045 0.054 0.060 0.070 0.259 0.267 0.142

Class II Z 2 0.010 2 0.008 2 0.007 2 0.005 2 0.003 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.002 2 0.004 2 0.005 2 0.007 2 0.011 2 0.019

352 N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

B0Ia B0.5Ia B1Ia B1.5Ia B2Ia B2.5Ia B3Ia B4Ia B5Ia B6Ia B8Ia B9Ia A0Ia A1Ia A2Ia A3Ia A5Ia F0Ia F2Ia F5Ia

0.017 0.053 0.058 0.128 0.160 0.220 0.300 0.340 0.397 0.480 0.530 0.666 0.810 0.980 1.200 1.429 1.650 1.787 1.889 1.795

X

Table 2. Continued Class Ia

0.015 0.017 0.018 0.023 0.030 0.035 0.047 0.055 0.065 0.083 0.095 0.120 0.150 0.180 0.225 0.270 0.300 0.302 0.250 0.200

Y 0.020 0.039 0.132 0.271 0.387 0.530 0.620 0.743 0.830 0.926 1.080 1.370 1.550 1.805 2.100 2.150 2.100 1.599

2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.002 0.005 B0Iab B0.5Iab B1Iab B2Iab B3Iab B5Iab B6Iab B7Iab B8Iab B9Iab A0Iab A2Iab A3Iab A7Iab F0Iab F2Iab F3Iab F5Iab

X

Z

Class Iab

0.015 0.017 0.023 0.035 0.050 0.068 0.081 0.100 0.120 0.140 0.170 0.240 0.280 0.290 0.260 0.190 0.104 2 0.108

Y 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.005 2 0.006 2 0.008 2 0.010 2 0.012

Z B0Ib B0.5Ib B1Ib B1.5Ib B2Ib B2.5Ib B3Ib B5Ib B6Ib B7Ib B8Ib B9Ib A0Ib A1Ib A2Ib A3Ib A4Ib A5Ib A7Ib F0Ib

0.058 0.082 0.140 0.196 0.260 0.320 0.465 0.600 0.670 0.810 0.890 1.100 1.422 1.650 1.720 1.900 2.080 2.106 2.090 2.055

X

Class Ib

0.020 0.021 0.024 0.030 0.039 0.043 0.050 0.065 0.072 0.090 0.100 0.145 0.229 0.280 0.290 0.300 0.290 0.270 0.232 0.145

Y

2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004 2 0.004

Z

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387 353

354

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

reasons why the following calibrations will be restricted to classes V to III. But the onset of the most insidious region is at T eff , 9 3 10 3 K, i.e. the early A-type stars, where the effects of temperature and gravity (in X and Y, respectively) begin to cross over (into Y and X, respectively). This shortcoming is not exclusive to the Geneva system. It besets most multicolour photometric systems applied in the visible. • More generally, uniqueness of X and Y does not always imply uniqueness of other related quantities (such as absolute magnitude) or of partially qualitative notions (such as MK class). This is most relevant here in the region of the supergiants, and such cases will appear at several places in the following discussion. The first remark this figure brings to mind is that the photometry is very well suited to do stellar classification within a quite wide domain of B-type stars. The second is that spectroscopic information is mandatory and that, indeed, the two techniques are to be considered as being largely complementary to eachother in all applications.

3.2. Correlation with MK spectral types The resolving of T eff and log g in the X,Y-diagram bears within itself the basis of the MK two-dimensional spectral classification. Straightforward correlations can be established for both parameters with B stars classified in the MK system. A first calibration of that sort was presented by Cramer & Maeder (1979). A more elaborate correlation was presented by Cramer (1994b). A slightly improved version of the latter is presented in Fig. 7, and is listed in Table 2. The calibration is based on about 4000 stars, some 80% of which have MK types taken from the various editions of the Michigan catalogue (Houk & Cowley (1975) and subsequent volumes). The absence of discrimination among O stars as well as the abovementioned regions of ambiguity are readily noted in Fig. 7. One also notices a region of ambiguity regarding MK type for B4 to B7 stars of types IV and III. Houk has pointed out (private communication, 1994) that this does indeed correspond to a difficulty in the spectral classification procedure. The error bars in the figure reflect the dispersion (s N 21 / 2 ) of each correlation in the extreme cases of

the class V and Ia sequences, and have been included here for the purpose of illustration. A question arises regarding the relevance of such a calibration. This question has been asked the author at several occasions, notably by spectroscopists. The first part of an answer lies in the fact that a good correlation between photometry, which is highly quantitative, and spectral classification, which relies somewhat more on the appreciation of the practitioner, helps both communities to ‘‘speak the same language’’ regarding physical parameters. Another, and important, aspect is that a good calibration in terms of a homogeneous photometric system not only allows the two partners of the initial correlation to agree with each-other, but also enables us to examine, via the photometric homogeneity, that of other sources of spectroscopic data. In other words, the consistency of MK types derived by various authors may be directly compared even if their samples of stars do not overlap, provided photometry is available for all the data

3.3. The intrinsic colours problem The knowledge of the intrinsic colours of a stellar population in a given photometric system is of fundamental importance. It is a necessary condition for all calibrations of physical parameters in the system. De-reddening of stellar groups or of individual stars can only be properly carried out if the corresponding intrinsic colours can be estimated. Comparisons of synthetic photometry of stellar models with observations must be done relatively to an accurate, but empirically consistent sequence. In the particular case of the inevitably distant O and B stars, reddening is always intimately connected with the observations. It is not possible to define a nonreddened sequence in colour-colour or colour-magnitude diagrams by simply taking the observed blue envelope, as may be legitimately done in many cases for cooler stars. The blue envelope will unavoidably be reddened initially and will favour peculiar objects. A pertinent process of extrapolation applied to normal stars of the population has to be utilised for the calibration. The process will be all the more accurate the closer one is able to approach the conditions of ‘‘minimal assumption’’ regarding the gathering of the calibration data. Synthetic photometry of stellar atmosphere models

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

355

Fig. 7. Correlation of the MK classification with the X,Y parameters. The photometric discrimination is very good, except for the O-B1 stars and in some regions occupied by the late A- and first F-type supergiants and later A-types. The error bars shown here for the two extreme cases of the class V and Ia stars illustrate the probable errors of the correlation in each case.

can be seen at first glance as a possible aid in the process. It is, however, heavily burdened with assumptions concerning the true knowledge of the passbands as well as regarding the reality of the wavelength dependence of the model fluxes. It can nevertheless be of precious assistance through its prediction of the expected shape of the relation or, even better, by its confirmation of a completely empirical calibration. The latter case is substantially true for the empirically derived intrinsic colours of O and B stars of classes V to III in the Geneva system (Cramer, 1982; Cramer, 1993), and that we shall now discuss.

3.3.1. The calibration techniques 3.3.1.1. The six basic indices The strategy applied here to the empirical determination of intrinsic colours makes use of the classification properties described above. Namely,

the equivalence between a given location in the X,Y-diagram and a specific MK type, thus implying also the corresponding absolute magnitude and intrinsic colours. The calibration process first leading to the intrinsic indices [U2B] 0 and [B2V] 0 of O and B class V to III stars in the Geneva system is described in detail by Cramer (1982). We summarise here the basic assumptions: 1. Each small neighbourhood (outside zones of ambiguity) in the X,Y plain only contains groups of stars with virtually identical intrinsic colours. 2. To each one of those stars can be assigned an absolute magnitude, i.e. also a distance in the form of an apparent distance modulus m V 2 MV . No de-reddening is required yet. Only an absolute magnitude calibration in terms of X,Y is necessary. That given by Cramer & Maeder (1979) was used initially. 3. Interstellar extinction in the solar neighbourhood

356

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

has, on the whole, two components: a diffuse, large scale and more or less homogeneous component, to which local concentrations of interstellar dust add random contributions according to the line of sight. 4. The diffuse extinction will then define the position of the blue envelope of the plot for each neighbourhood in a m V 2 MV versus [B2V] diagram (a minimum absorption gradient of 3 3 10 24 mag / pc was found to be representative of the galactic plane and the corresponding reddening line was used as an extrapolating envelope). Contrarily to a simple blue limit usually determined by one or two stars, this envelope rests on a much greater number of stars, and is therefore better defined. Its extrapolation to the zero distance, i.e. its asymptotic value, then determines the intrinsic value [B2V] 0 of the small neighbourhood. Each ‘‘blue envelope’’ value was then intentionally reddened by 0.01 mag to account for observational scatter and reduce blue biasing. A sample of 3000 well measured stars distributed over 107 cells in the X,Y-diagram defines an equal number of empirical [B2V] 0 points in this manner. The slope of the reddening line, d [U2B] /d [B2V] 5 0.64 for B stars, allows us to determine the corresponding [U2B] 0 by adjusting to the local [B2V] zero-point (see Cramer, 1982). The two sets of intrinsic points are then smoothed globally by least squares with a third degree polynomial in X and Y, giving the two intrinsic colour estimators [B2 V] 0 (X,Y) and [U2B] 0 (X,Y) (see Table 3). The method has the advantage of being fully empirical and reduces the recourse to external data to the absolute minimum (the initial absolute magnitude calibration data). It relies on the homogeneity of the photometry. Even if it may at first awake the suspicion of being a ‘‘bootstrap method’’, it is in reality very robust. The absolute magnitude determination does not have to be very precise. The blue envelope of each cell is essentially determined by the closest and least reddened stars, where the enveloping reddening line is steepest and its location in terms of [B2V] best defined. A substantial advantage of these estimators is their ability to estimate intrinsic indices of individual stars regardless of prior

knowledge of their evolutionary stage (within the class V to III limits). This is of great value for studies of field stars, but can also be used to good effect in the cases of clusters masked by patchy foreground reddening (see Raboud et al., 1997) or even in studies involving different clusters disposed along the same line of sight (see Carrier et al., 1999). The overall accuracy of the E[B2V] estimate is reckoned to be better than 0.02 mag (this can be verified in Figs. 9a–9d). The internal error depends on the accuracy of the measured [B2V] index, on that of the X,Y parameters and on the local slope of the calibration relation. The map of the mean (i.e. all photometric weights considered) internal error is shown over the background of 12 800 well measured stars lying within the validity range of the calibration, in Fig. 8. The large scale behaviour of reddening determined in this manner is illustrated in Figs. 9a–9c with the E[B2V] versus (m 2 M) 0 diagram for three domains of galactic latitude (note that m V0 5 m V 2 2.75E[B2V] for B stars in the Geneva system and E(B2V ) Johnson 5 0.842E[B2V] Geneva ). The two high latitude plots are not very different in their apparent properties, except for the better populated southern hemisphere sample. They both show the first significant onset of extinction at about 120 pc. The plot centred on the galactic plane is, as expected, more absorbed. The minimum gradient is seen there to be close to 2 3 10 24 mag / pc, with the first strongly absorbed regions appearing at about 250 pc. The Fig. 9d is a cut-out of the least reddened stars of the whole data. It serves to show how the estimated colour excesses tend to ‘‘funnel down’’ the reddening lines and justifies the extrapolating process used to fix the zero-points. The other four intrinsic indices [B12B] 0 , [B22B] 0 , [V12B] 0 and [G2B] 0 were estimated with a sample of almost 10 000 stars distributed over 157 X,Y cells (see Cramer, 1993) by first correlating [B22B] with [B2V] and [U2B2] with [U2B] for the stars in each cell. The corresponding zero-points that are [B2 V] 0 (X,Y) and [U2B] 0 (X,Y) then define two sets of values for [B22B] 0 of which the average was taken. The three remaining indices [B12B] 0 , [V12B] 0 and [G2B] 0 were then derived directly by solving the system of three equations established by the definition of the X,Y,Z parameters. Z, however, was

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

357

Fig. 8. Mean internal standard deviation of the E[B2V] colour excess estimate for the 12 800 stars plotted in the background, all weights considered. The error is smaller than 0.01 mag over the whole domain.

not systematically set to zero. It exhibits a weak residual sensitivity to reddening, as illustrated in Fig. 10 for the albeit extreme case of the well measured O stars. The insensitivity of Z to temperature and gravity allows us to correct these residual effects by simple linear extrapolation. The de-reddened value Z0 was estimated for each cell by linear regression and used in the corresponding set of equations. Polynomial smoothing then defines the four new estimators [B12B] 0 (X,Y), [B22B] 0 (X,Y), [V12

B] 0 (X,Y) and [G2B] 0 (X,Y). The details of this second step in the determination of the intrinsic colours are given by Cramer (1993). The definitions of the six estimators are presented in tabular form in Table 3. The validity range of these estimators is reflected by the distribution of the background stars plotted in Fig. 8. They apply to classes V to III. The MK calibration of Table 1 is easily translated via these estimators into the six normalised indices as well as

Table 3 Coefficients of the intrinsic colour estimators [k 2 B] 0 (X,Y): [k 2 B] 0 (X,Y) 5 a 0 1 a 1 X 1 a 2 Y 1 a 3 XY 1 a 4 X 2 1 a 5 Y 2 1 a 6 XY 2 1 a 7 X 2 Y 1 a 8 X 3 1 a 9 Y 3 , k 5 U, V, B1, B2, V1, G Index

a0

a1

a2

a3

a4

a5

a6

a7

a8

a9

[U2B] [V2B] [B12B] [B22B] [V12B] [G2B]

0.0380 1.3431 0.7413 1.6408 2.0062 2.5758

0.9057 2 0.3227 0.0937 2 0.0878 2 0.2480 2 0.2907

2 0.0625 0.2400 2 0.2001 0.2403 0.0974 0.0734

2 0.2409 2 0.3371 2 0.0480 0.0721 0.4180 0.4734

2 0.0518 0.1582 2 0.0152 0.0061 0.0752 0.0823

4.8627 1.6294 1.0014 2 2.3129 2 6.2176 2 9.9578

2 2.6551 2 1.1422 2 0.5845 1.2373 3.1873 5.3204

2 0.0340 0.4078 2 0.0249 2 0.0151 2 0.0139 0.0013

0.0240 2 0.0638 0.0067 2 0.0039 2 0.0350 2 0.0398

1.9123 2 5.4104 1.6738 1.6564 0.8075 1.1033

358

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 9. (a) Estimated colour excess versus distance within a 68 wide strip centred on the galactic equator for stars in the validity range of the calibrations. Several reddening lines corresponding to different values of constant extinction are plotted. A gradient of 2 3 10 24 mag / pc gives a good account of extinction in the most transparent regions within the first kiloparsec. The distances are derived here with the MV (X,Y) relation calibrated with Hipparcos satellite parallaxes. (b) Estimated colour excess versus distance north of b 5 38 showing the greater transparency of the medium, as expected. (c) Estimated colour excess versus distance south of b 5 2 38. The more densely populated plot is the result of the greater acquisition of data in the southern hemisphere. (d) Estimated colour excess versus distance restricted to E[B2V] , 0.2 mag for all latitudes for stars in the validity range of the calibrations. The need to apply an extrapolating process for the determination of intrinsic colours is demonstrated by the rarity of B- and first A stars within the first 100 pc. The data are seen to converge convincingly along the reddening lines towards ‘‘zero distance’’. Scatter is well accounted for, giving negative excesses of the order of 10 22 mag, as should be the case for a well adjusted zero-point. The 5 closest stars in the figure are respectively a LYR, a CRB, b PER, b UMA, g UMA.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

four other indices often used in Geneva photometry, and is given in Table 4.

3.3.1.2. The colour excess ratios of the system The knowledge of the intrinsic colours of all normalised indices allows us to compute the corresponding colour excesses, and thereby the colour excess ratios of all other combinations of indices. These ratios are essential for the consistent analysis of given stellar samples in various colour-colour diagrams. Particularly so in the case of cluster sequences. The estimators defined above reduce the process to a set of straightforward correlations. We have chosen here to correlate all the two-by-two combinations of Geneva colours with the [B2V] colour excess, since that index is universally used to express interstellar reddening. A sample of 559 well measured and carefully selected mid-B type stars (0.65 , X , 0.95

359

and 2 0.02 , Y , 0.60) is used to that end. One of the correlations is shown in Fig. 11, and the whole set of ratios relative to [B2V] is presented in Table 5. Other ratios, not involving the [B2V] index, can easily be derived from those values. The visual absorption A V can also be readily estimated by all indices with the help of these relations (A V 5 2.75E[B2V] for B stars in the Geneva system). The calibrations of intrinsic colours and colour excess ratios presented here for the O, B and earliest A stars of classes V to III are specific to the observational data of the Geneva system. They have been derived through a totally empirical process and with minimal recourse to assumptions and to external data. They are arguably the most robust that can be achieved within their limited range. The zero-points rely most heavily on the observed colours of the closest stars. These have all been measured and the accumulation of new data will not have any signifi-

Fig. 10. The small residual effect of reddening on the Z parameter illustrated by means of the O stars. The insensitivity of Z to temperature and gravity makes this straightforward representation possible. It was corrected throughout the process leading to the intrinsic colour estimators, so as to improve the accuracy of the latter. Z essentially measures variations of gradient in the spectral region spanned by the V ˚ absorption feature of the magnetic filter, and it works like a de-reddened [V12G] 2 [V12G] standard colour difference. It detects the 5200 A Ap and Bp stars.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

360

Table 4. MK type versus intrinsic colour correlation [U2B] 0

[B2V] 0

[B12B] 0

[B22B] 0

[V12B] 0

[G2B] 0

[B12B2] 0

[U2B2] 0

[B22V1] 0

[V12G] 0

Class V O6 O7 O8 O9V O9.5V B0V B0.5V B1V B1.5V B2V B2.5V B3V B4V B5V B6V B7V B8V B8.5V B9V B9.5V A0V A1V A2V A3V

0.029 0.032 0.035 0.069 0.103 0.131 0.210 0.269 0.325 0.452 0.521 0.576 0.645 0.702 0.805 0.849 1.016 1.091 1.256 1.339 1.407 1.457 0.494 1.501

2 1.349 2 1.348 2 1.347 2 1.336 2 1.325 2 1.316 2 1.290 2 1.272 2 1.255 2 1.221 2 1.203 2 1.191 2 1.175 2 1.163 2 1.143 2 1.134 2 1.106 2 1.091 2 1.051 2 1.023 2 0.991 2 0.960 2 0.929 2 0.915

0.739 0.739 0.739 0.742 0.745 0.747 0.755 0.762 0.768 0.782 0.789 0.794 0.801 0.805 0.813 0.816 0.826 0.831 0.848 0.859 0.872 0.885 0.898 0.905

1.644 1.644 1.644 1.641 1.638 1.636 1.628 1.622 1.615 1.601 1.594 1.589 1.582 1.578 1.570 1.567 1.556 1.549 1.532 1.521 1.508 1.495 1.482 1.475

2.009 2.008 2.007 1.998 1.989 1.982 1.962 1.948 1.934 1.904 1.887 1.875 1.860 1.849 1.829 1.821 1.791 1.776 1.736 1.709 1.678 1.648 1.616 1.599

2.578 2.577 2.576 2.565 2.554 2.545 2.523 2.506 2.490 2.454 2.435 2.420 2.402 2.388 2.363 2.353 2.315 2.296 2.249 2.217 2.182 2.146 2.109 2.089

2 0.905 2 0.905 2 0.905 2 0.899 2 0.894 2 0.889 2 0.873 2 0.860 2 0.847 2 0.819 2 0.804 2 0.794 2 0.782 2 0.773 2 0.757 2 0.751 2 0.730 2 0.718 2 0.684 2 0.662 2 0.636 2 0.610 2 0.584 2 0.570

2 1.614 2 1.611 2 1.609 2 1.572 2 1.535 2 1.505 2 1.418 2 1.353 2 1.290 2 1.149 2 1.072 2 1.013 2 0.937 2 0.876 2 0.765 2 0.718 2 0.540 2 0.459 2 0.276 2 0.182 2 0.101 2 0.038 0.012 0.027

2 0.365 2 0.364 2 0.364 2 0.357 2 0.351 2 0.346 2 0.334 2 0.326 2 0.319 2 0.302 2 0.293 2 0.287 2 0.278 2 0.271 2 0.259 2 0.254 2 0.236 2 0.227 2 0.204 2 0.188 2 0.171 2 0.153 2 0.134 2 0.124

2 0.569 2 0.569 2 0.569 2 0.567 2 0.565 2 0.563 2 0.560 2 0.558 2 0.556 2 0.551 2 0.548 2 0.545 2 0.542 2 0.539 2 0.534 2 0.532 2 0.524 2 0.520 2 0.513 2 0.508 2 0.503 2 0.498 2 0.493 2 0.490

Class IV O9IV O9.5IV B0IV B0.5IV B1IV B1.5IV B2IV B2.5IV B3IV B4IV B5IV B6IV B7IV B8IV B9IV B9.5IV A0IV A1IV A2IV

0.037 0.087 0.121 0.161 0.242 0.287 0.354 0.496 0.613 0.659 0.694 0.820 0.870 0.957 1.172 1.319 1.403 1.505 1.509

2 1.346 2 1.328 2 1.317 2 1.304 2 1.279 2 1.266 2 1.247 2 1.211 2 1.185 2 1.175 2 1.169 2 1.146 2 1.138 2 1.122 2 1.079 2 1.038 2 1.006 2 0.937 2 0.914

0.739 0.744 0.748 0.752 0.760 0.764 0.771 0.785 0.794 0.798 0.800 0.808 0.811 0.818 0.835 0.852 0.864 0.893 0.904

1.643 1.638 1.635 1.632 1.624 1.619 1.613 1.599 1.589 1.585 1.583 1.574 1.571 1.564 1.545 1.527 1.515 1.488 1.475

2.007 1.993 1.985 1.975 1.954 1.943 1.928 1.896 1.871 1.862 1.855 1.832 1.823 1.806 1.763 1.723 1.693 1.627 1.600

2.576 2.560 2.550 2.538 2.514 2.501 2.483 2.444 2.414 2.403 2.394 2.364 2.352 2.331 2.279 2.233 2.198 2.122 2.090

2 0.904 2 0.894 2 0.888 2 0.880 2 0.864 2 0.855 2 0.842 2 0.814 2 0.795 2 0.788 2 0.783 2 0.765 2 0.759 2 0.746 2 0.709 2 0.676 2 0.650 2 0.595 2 0.571

2 1.606 2 1.551 2 1.515 2 1.471 2 1.381 2 1.332 2 1.258 2 1.102 2 0.976 2 0.926 2 0.889 2 0.754 2 0.701 2 0.607 2 0.373 2 0.208 2 0.112 0.017 0.034

2 0.364 2 0.355 2 0.349 2 0.343 2 0.331 2 0.324 2 0.315 2 0.297 2 0.282 2 0.277 2 0.272 2 0.258 2 0.252 2 0.243 2 0.218 2 0.196 2 0.178 2 0.140 2 0.124

2 0.569 2 0.567 2 0.565 2 0.563 2 0.560 2 0.558 2 0.555 2 0.548 2 0.543 2 0.541 2 0.539 2 0.532 2 0.529 2 0.525 2 0.516 2 0.510 2 0.505 2 0.495 2 0.490

Class III O9III O9.5III B0III B0.5III B1III B1.5III B2III B3III

0.058 0.075 0.078 0.152 0.218 0.290 0.326 0.563

2 1.340 2 1.334 2 1.334 2 1.310 2 1.289 2 1.267 2 1.257 2 1.197

0.741 0.742 0.742 0.749 0.755 0.763 0.766 0.789

1.642 1.641 1.640 1.634 1.628 1.621 1.617 1.594

2.001 1.997 1.996 1.977 1.960 1.943 1.935 1.882

2.569 2.563 2.562 2.539 2.520 2.500 2.490 2.427

2 0.901 2 0.898 2 0.898 2 0.885 2 0.873 2 0.858 2 0.851 2 0.805

2 1.584 2 1.566 2 1.563 2 1.483 2 1.410 2 1.330 2 1.292 2 1.031

2 0.359 2 0.356 2 0.355 2 0.342 2 0.332 2 0.322 2 0.318 2 0.288

2 0.567 2 0.567 2 0.566 2 0.562 2 0.559 2 0.556 2 0.555 2 0.545

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

361

Table 4. Continued

B4III B5III B6III B7III B8III B9III B9.5III A0III A1III A2III A3III

[U2B] 0

[B2V] 0

[B12B] 0

[B22B] 0

[V12B] 0

[G2B] 0

[B12B2] 0

[U2B2] 0

[B22V1] 0

[V12G] 0

0.646 0.767 0.830 0.855 0.952 1.124 1.216 1.366 1.489 1.516 1.506

2 1.178 2 1.155 2 1.144 2 1.141 2 1.127 2 1.100 2 1.083 2 1.046 2 0.984 2 0.933 2 0.912

0.797 0.805 0.809 0.810 0.815 0.825 0.831 0.846 0.871 0.894 0.906

1.586 1.577 1.573 1.572 1.566 1.555 1.548 1.533 1.507 1.486 1.473

1.864 1.841 1.830 1.826 1.810 1.780 1.764 1.729 1.672 1.624 1.595

2.406 2.376 2.361 2.356 2.333 2.295 2.276 2.237 2.172 2.118 2.085

2 0.789 2 0.772 2 0.764 2 0.762 2 0.752 2 0.730 2 0.717 2 0.687 2 0.636 2 0.593 2 0.567

2 0.940 2 0.810 2 0.743 2 0.717 2 0.615 2 0.431 2 0.332 2 0.166 2 0.018 0.030 0.033

2 0.278 2 0.264 2 0.257 2 0.254 2 0.243 2 0.225 2 0.215 2 0.196 2 0.164 2 0.138 2 0.122

2 0.541 2 0.535 2 0.531 2 0.530 2 0.524 2 0.515 2 0.512 2 0.508 2 0.501 2 0.494 2 0.490

cant incidence regarding the possibility of improving their accuracy. Their exclusively empirical nature renders them ideally suitable for comparisons with more elaborately derived quantities such as those proposed by synthetic photometry, for example. The colour excess ratios derived here also bear the potential to improve the ‘‘Q-type’’ parameters (d, D and g) that were defined during the early stages of

the photometric system, when the data for reddened stars were much less abundant. The improvement would thus also operate on the X,Y,Z parameters by changing the constants in the linear expressions. All the calibrations would have to be readjusted within the new parameter system. Another approach would be to re-scale the new parameters so as to reproduce the same values for the least reddened stars, thus preserving the extant calibrations. This was attempt-

Fig. 11. Example of the correlation defining the colour excess ratio E[B22V1] /E[B2V] .

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

362 Table 5 Colour excess ratios relative to [B2V] CI

E[CI] /E[B2V]

s

U2B B12B B22B V12B G2B U2B1 B12B2 B22V1 V12G U2B2 B22G U2V1 U2G U2V B12V1 B12G B12V V12V G2V B22V

0.654 0.158 2 0.195 2 0.943 2 1.208 0.497 0.351 0.750 0.266 0.848 1.014 1.597 1.861 1.654 1.100 1.364 1.157 0.062 2 0.210 0.807

0.003 0.002 0.002 0.002 0.003 0.001 0.001 0.002 0.003 0.003 0.003 0.004 0.005 0.003 0.003 0.004 0.002 0.002 0.003 0.002

ed by the author, and a re-adjusted set of new X,Y,Z parameters does indeed slightly improve the reddening correction. The gain was, however, not deemed to be significant enough to justify a revision of former work in regard to the inevitable risk of confusion when two sets of similar parameters are available. But, the way does lie open for such an iteration in future work.

3.3.2. A comparison with synthetic indices In the ideal situation of total consistency the empirically derived intrinsic colours should coincide with the synthetically computed positions of the stellar atmosphere models. As mentioned earlier, there is some reason to expect a certain measure of discrepancy. The question that arises is how much. A good way of illustrating the typical magnitude of the discrepancy is to compare the predictions of our intrinsic colour estimators with the corresponding synthetic values of the Kurucz (1993) models filtered by Nicolet using the Rufener & Nicolet (1988) Geneva passbands. A first approach may be taken by computing the X,Y,Z values of each (solar abundances) model by means of the synthetic normalised indices. The thus derived X,Y values can then be used to restore the

normalised indices with the aid of the intrinsic colour estimators. The process is somewhat circular but, if everything were perfect, the two sets of normalised indices should be identical. The result is shown in Fig. 12, where the differences are plotted as a function of X for the log g 5 4.5 sequence of models only, so as to avoid cluttering up the figure. The overall conclusion is that the synthetic values appear to be redder than the empirical ones: the deviations for normalised indices using a filter bluer than B are positive, and negative otherwise. The deviation behaves proportionally to the range of the index, i.e. to wavelength. The possible slight red excess of the synthetic colours was indeed suggested by Rufener & Nicolet (1988). A more direct comparison can be made by using real observations. For the sake of clarity, the comparison is made in the familiar [U,B,V] diagram. The stars selected have measurements weighted P $ 3, and are chosen among the least reddened ones of classes V to III (E[B2V] # 0.08 for X $ 0.6 and E[B2V] # 0.12 for X , 0.6). The result is shown in Figs. 13a–13c: • In Fig. 13a, the overall aspect of the selected stars superposed on the synthetic lines of the models with log g ranging from 5 to 3.5 does not, at first glance, give an impression of significant discord. A few stars do transgress the log g 5 3.5 line and may appear to be a bit too blue. • If the individual stars are de-reddened by the estimators, as in Fig. 13b, the observed sequence narrows down and becomes decidedly bluer that those of the models. The cut-off at the bottom right of the stellar sequence reflects the cut-off that was made at Y 5 2 0.06 to discard types later than A2. • The differences in Fig. 12 suggest that the indices of the models taken relatively to the empirical values are about 0.035 mag too large for [U2B] 0 and about 0.050 mag too small for [V2B] 0 . If we apply inverse corrections to the models (20.035 and 10.050, respectively) we get the very satisfactory situation seen in Fig. 13c. The upper envelope (higher g, class V) of the empirical sequence follows the log g 5 4.5 line throughout its length. The top of the sequence favours a somewhat higher gravity whereas the bottom

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

363

Fig. 12. Difference between the synthetic normalised indices and those derived by the intrinsic colour estimators, as a function of X. One notes a colour dependent behaviour of the amplitude of the deviation.

tends to drift towards lower log g. This is in good agreement with predictions of stellar models. The shape of the lower envelope (class III) is also well fitted by the log g 5 4 and 3.5 lines. Here, we have also included the predictions for lower gravities, and which serve to hint at the difficulty of de-reddening supergiants. Now, the absolute ratio of the two corrective shifts d [U2B] /d [B2V] 5 0.035 / 0.050 5 0.70, does lies suspiciously close to the slope of the reddening line (0.64 for B stars in the Geneva system). One could object that the empirical intrinsic colours may be too strongly de-reddened. However, Such discrepancies, particularly that of 0.05 mag in [V2B], are difficult to explain by a zero-point error of the empirical calibration alone. A shift of that point by more than 0.01 mag, either blueward or redward, would stand out quite conspicuously in Figs. 9a–9c and particularly so in Fig. 9d. The main body of the discrepancy must reside in the combination of the small uncertainties over the model flux distributions

with those inherent to the definition of the passbands. The latter is achieved through a difficult and elaborate process that depends on the quality of external spectrophotometric data. A revision of the determination of the Geneva passbands is planned by Nicolet (see Nicolet, 1996) for the near future. The main object of this comparison was to illustrate the possible dangers inherent to the uncritical application of synthetic photometry in analyses that require high accuracy. The other was to show that small and properly made adjustments can locally virtually restore the initial aims of a good synthetic photometry.

3.3.3. Inevitable noise Maybe the most ubiquitous source of uncertainty in photometrically derived quantities is undetected multiplicity. Various estimates of the frequency of binaries among B stars encountered in the literature lie within the range of 40% to 50%, i.e. more than half the whole population of individual stars of that type are members of a system. If in a binary system

364

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 13. (a) Synthetic Geneva U,B,V colours of the models with log g 5 3.5, 4.0, 4.5, 5.0 compared with the least reddened well measured O, B and first A stars of classes V to III in the Geneva catalogue. (b) Same comments as for (a), but with the stars de-reddened by the intrinsic colour estimators. The estimated intrinsic sequence is definitely bluer than the synthetic locus. (c) Synthetic colours shifted by 20.035 mag in [U2B] and 0.050 mag in [V2B] as suggested in Fig. 12. The fit is excellent throughout the whole sequence. Note that a shift of the zero-point of [V2B] 0 by that amount would be unacceptable in Fig. 9d, thus implying that the greater part of the discrepancy resides in the synthetic photometry. The shifted lines of lower gravity have been added in this figure.

a secondary is detected and the relative contributions of the components can be estimated, standard calibrations can be applied after the corresponding

corrections have been made to the observed colours. Otherwise, and actually in the majority of cases, the results derived by photometric calibrations will be

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

affected by unseen companion stars. Indeed, the establishment of a calibration itself may be governed by two different points of view. The first is to assume that it is impossible to put together a perfectly ‘‘clean’’ sample of single reference stars large enough to carry out a ‘‘pure’’ calibration. In any case, when applied randomly and on a large scale to the real population of stars, such a calibration would systematically be biased. So, for statistical investigations, a calibration based on a non-selective sample of data would appear to be most suitable. The second approach is to strive to establish a calibration that is as pure as possible. The major difficulty lies in the choice of the data, some of which will also have to be corrected for multiplicity. Empirical calibrations voluntarily biased in favour of single stars will have to be used carefully in statistical studies. They are, however, best suited for comparisons with synthetic predictions or for the

365

interpretation of theoretical models which most often concern single stars. In practice, neither of these two objectives are fully attained. For the advocates of the first, the temptation to repress peculiar data during the calibration process is difficult to resist. For the second, the sheer volume of potentially valuable data that have to be discarded causes a painful dilemma. So, whatever may be the character of a given calibration, multiplicity is a major source of noise, and the accurate assessment of its incidences difficult. Here, we illustrate the effects of binarity by their incidence in the X,Y parameter diagram. They are shown with the help of the ‘‘binarity loops’’ of Fig. 14 distributed along the sequences of class V and III stars. The basis of the class V sequence is a set of 34 points of an empirical ZAMS made with the help of cluster sequences and harmonised over the six normalised indices with the aid of the colour excess ratios of Table 5 (Burki, Cramer, Mermilliod; un-

Fig. 14. Effects of binarity in the X,Y-diagram. The loops are computed by adding a string of less luminous stars from a ZAMS sequence to a variety of primaries of classes V and III.

366

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

published). Each loop is computed at each step of the ZAMS by adding the fainter portion of the sequence to the primary, one-by-one. The same operation is carried out along the class III sequence, but with primaries having MV estimated by a calibration of the X,Y-plane and with intrinsic normalised indices computed by the relations above. The less luminous ZAMS stars are then added as before. The figure shows that the behaviour is quite complex, and that the effects are not negligible even in this case of ‘‘simple’’ binarity. The shapes of the loops in photometric diagrams always reflect the general shape of the lower sequence. The more contorted is the lower part of the sequence, the more so are the loops. A portion of the figure is enlarged in Fig. 15, so as to better show the effect of the magnitude difference d MV between the two components. We see that the maximum deviation occurs at different d MV for different MV of the primary. This is best illustrated in Fig. 16 where the maximum colour deviation expressed as an ‘‘Euclidian distance’’ is plotted as a

function of the MV of the primary (the two cases at MV 5 0.82 and 1.00 are bimodal). Determinations of photometric sensitivity of this kind can be of some use within the context of colour-dependent bias estimation in the selection of ‘‘pure’’ calibration samples. As spectroscopic detection of binaries loses in efficiency with d MV . 2, the more sensitive the photometry is locally regarding binarity – i.e. the greater is the d MV at maximum photometric distance – the more vulnerable the calibration data become to undetected multiplicity. Regarding the incidence of binarity on the determination of colour excesses, it is clear that in a two-colour representation, such as the U,B,V diagram, nothing can be done for unknown binaries. The effect will quite generally be to overestimate the excess (up to a few 10 22 mag) as the presumed zero-point will be taken by default at the standard sequence. The consequence is to overestimate total absorption A V by up to almost 0.1 mag for late B binaries. But, since binarity affects the X,Y parameters in the sense of the composite, one could hope

Fig. 15. Enlargement of four loops of Fig. 14 illustrating the influence of the difference in magnitudes d MV between the two components.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

367

Fig. 16. Dependence of the maximum colour deviation, expressed as a distance in the X,Y-plane, as a function of the brightness of the primary (class V). The difference in magnitudes d MV between components at maximum colour deviation is given for each point.

that the intrinsic colours then estimated by these parameters will reflect the effect operating on the true indices. That this is partially realised is shown in Fig. 17 which compares the estimated sequence (shifted by 0.3 mag in [V2B] for clarity) with the true sequence in [V2B]. The binarity loops are reproduced, but at a reduced scale, and still lead to a, nevertheless, attenuated overestimate of colour excess. Rotation, which is an important feature particularly among early-type stars, is also a significant cause of dispersion in photometric diagrams. The relative effects of rotation are more difficult to assess since they are model dependent. An early study giving Geneva synthetic colours of these effects was made by Maeder & Peytremann (1970) for a variety of masses, rotation velocities and apparent inclinations. One model (5M( ) lies within the validity region of our calibrations and has been used in earlier publications (Cramer & Maeder, 1979; Cramer, 1984b; Cramer, 1994b). The general effect

is to displace the colours in the sense of decreasing gravity and temperature with increasing rotation velocity. The greatest displacement is in the X parameter with a maximum deviation d X 5 0.231 mag. The intrinsic colour estimators are, however, not greatly induced into error. The synthetic index [U2B], for example, which is best related to X, is reproduced within an absolute deviation , 0.06 mag in all cases.

3.4. Effective temperatures The determination of effective temperature is basically associated with multicolour photometry. It is one of the ‘‘first order’’ effects to which it responds by its colorimetric nature. The literature is well provided in calibrations and estimates of T eff either photometric or spectroscopic. Fundamental empirical data of that sort are still, however, not very common and one has to go back to the original

368

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 17. Binarity loops in the Geneva U,B,V diagram, and their partial reproduction by the intrinsic colour estimators. The estimated sequence is shifted by 0.3 mag in [V2B] to facilitate comparison. The colour excesses of unknown binaries are systematically overestimated by reduction to the standard sequence in U,B,V photometry. The estimators are also affected, but to a lesser extent.

interferometrically based work of Code et al. (1976) for the most direct estimate of T eff for a few nearby stars. Pending future apparent angular diameter measurements of a larger number of stars by new instruments doted with an interferometric capability, such as the VLT, and their proper interpretation via atmosphere models, the T eff determined with the aid of the Narrabri interferometer still have to be given highest priority in calibrations. Empirical calibrations of effective temperatures and bolometric corrections based on the results of Code et al. (1976) and using the X parameter alone have been presented elsewhere (Cramer & Maeder, 1979; Cramer, 1984a). The latter publication gives estimators of these quantities in the form of fourth degree polynomials in X, valid for B stars of classes V to III. In that spectral range, effective temperature is almost independent of the Y parameter. Here, we may go a step further by using the shape

of the T eff versus X relation predicted by synthetic photometry to fit the data. The result is expected to take better account of the physics that characterise the form of the sequence than does a polynomial fit. This is achieved in a straightforward manner as shown in Fig. 18. The empirical data displayed correspond to the class V to III O-B stars of Code et al.’s list, where the stars that the authors corrected for binarity have also been adjusted here to reproduce the colours of the primary. A small shift (0.035 mag) of the X values of the (Kurucz, 1993) solar models was made to optimise the fit. The bright giants and supergiants (empty dots), which were not considered for calibration, are also shown for the purpose of comparison. The present calibration scheme has the advantage of involving the Y parameter through its discrimination of gravity (no correction of scale was attempted for that parameter). The log g 5 3.5, 4, 4.5 and 5

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

369

Fig. 18. Kurucz’s (1993) models adjusted to fit a log T eff versus X relation through the empirical data of Code et al. for O, B and first A stars of classes V to III. The lines of lower gravity as well as the bright giants and supergiants have been included for comparison.

lines are in good agreement with the polynomial calibration (Cramer, 1984a), though slightly cooler for 0.7 , X , 1.2 (B4V to B8.5V) but well within the error bars of the empirical data. A quite instructive comparison can be made with standard MK-type calibrations of effective temperature by using the MK correlation of Table 2. This is ¨ shown in Fig. 19, where the calibrations of BohmVitense (1981) and Schmidt-Kaler (1982) were used. In spite of the 5% errors on T eff , the sequences diverge significantly for 0.4 , X , 0.7 (B2V to B5V) and somewhat also for 0.9 , X , 1.1 (B7V2B8V). Neither do the uncertainties over the mean X values (horizontal error bars) help to account for the deviations. As mentioned earlier, our spectral type correlation rests essentially on the MK-types of the Michigan catalogue (Houk & Cowley, 1975) which have been systematically used in our data base. Maybe these deviations reflect some of the dispersion, or biasing, that occurs throughout the various

sources of spectral types published in the general literature. The corresponding bolometric corrections of the models are given in Fig. 20 (which also includes the supergiants for comparison). They agree well with those of Code et al, though they may seem to be a trifle too low. Not enough, however, to motivate a further attempt of adjustment. In the present context, the earlier discussion regarding possible small errors of the synthetic indices (Figs. 12 to 13c) could suggest that one could ‘‘correct’’ the synthetic parameters by modifying the normalised indices accordingly. The resulting changes to the parameters (5 3 10 24 in X and 15 3 10 24 in Y) are not, however, significant enough to do so. The two new calibrations are presented in Table 6 for log g 5 3.5, 4, 4.5 and 5, and can be read in that form with steps of 0.05 in X (except for a few extreme points of sequences). The corresponding

370

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

¨ Fig. 19. The relation of Fig. 18 compared with the MK type T eff calibrations of Bohm-Vitense and Schmidt-Kaler. The X value is determined by our MK type calibration. The vertical error bars are the estimated 5% errors on T eff and the horizontal ones the dispersion in X.

values of Y enable the interpolation among the sequences. An interpolation over Y does allow an estimate of log g to be made. That was not, however, the object of this calibration, and such values should not be taken at face value and used as precise log g determinations. A more elaborate fitting procedure of ¨ synthetic photometry was carried out by Kunzli et al. (1997) to that end and applies to stars down to G-types. The present, more simple and restricted calibration is quite comparable regarding the T eff determination, but tends relatively to that work to slightly underestimate gravity with a small difference in Y of the order of 0.02 mag. Finally, one must bear in mind that, however sophisticated it may be, a calibration can perform no better than the data it is based upon. Regarding the determination of effective temperature and bolometric correction, these calibrations (including the 1984 polynomial forms) all lie within the uncertainties of the fundamental data.

We must therefore wait for a significant improvement of the latter to be able to honestly affirm that a given calibration truly supersedes the others.

3.5. Absolute magnitudes 3.5.1. Three calibrations The determination of absolute magnitudes is a long standing fundamental issue that has involved all facets of astronomy and astrophysics (see for instance Rowan-Robinson, 1985). The basic distance scale is determined through trigonometric parallaxes and in some cases that were traditionally beyond the reach of the latter, by the moving cluster method (Sco-Cen, Hyades). Photometric and spectroscopic calibrations using cluster sequences have been scaled to that fundamental, but restricted, sample of nearby stars. The recently published results of the Hipparcos astrometric satellite have now dramatically improved our knowledge regarding the trigonometric distance

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

371

Fig. 20. Synthetic relations for the bolometric correction compared with those of Code et al. Same comments as for Fig. 18.

scale. While greatly simplifying the calibration of indirect methods, such as photometry, the new parallax determinations have nevertheless brought up some new questions. As several investigations have shown, the colour-absolute magnitude relationship, even when metallicity is accounted for, does not always seem to be as straightforward as formerly believed (see Van Leeuwen & Hansen Ruiz (1997); Mermilliod et al. (1997a), regarding the Pleiades cluster). One of the first aims following the definition of the X,Y,Z parameters was to establish an absolute magnitude calibration which would allow field stars to be integrated into studies of stellar evolution. The first paper devoted to that photometric representation (Cramer & Maeder, 1979) thus also discussed an absolute magnitude calibration based on the X,Y parameters. Virtually no primary measurements of distance (apart from the Scorpio-Centaurus association) were available for B stars at that time, and the calibration relied essentially on cluster distance modulus determinations published in the literature.

The calibration consisted of a succession of segments along X for which specific quadratic relations were fitted locally. The reference data were voluntarily biased against binarity, when possible, and included a few bright giants and supergiants. The global standard deviation over the residuals amounted to 0.38 mag for the calibration sample of 199 stars. Subsequently, several (unpublished) calibrations were made with the object of improving the estimate and expanding the range of validity. The last of those was published in the Appendix of Raboud et al. (1997) and was based on the study of 30 solar composition clusters by Meynet et al. (1993). A third degree polynomial form in X and Y similar to those of the intrinsic colour estimators discussed above was used. There too, the biasing toward single stars was carried out by an iterative process. The brightness of binaries is systematically underestimated in an absolute magnitude versus colour relation, shifting those belonging to clusters apparently into the foreground. Such stars were removed from the basic sample until the distributions around the mean

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

372

Table 6 Relation between T eff , B.C., X and Y for classes V to III X

log T eff (3.5)

T eff (3.5)

BC (3.5)

Y (3.5)

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350 1.400 1.450 1.500 1.550 1.600 1.650 1.700 1.750 1.800 1.850

4.499 4.472 4.447 4.419 4.391 4.367 4.348 4.328 4.310 4.290 4.272 4.252 4.234 4.216 4.199 4.184 4.168 4.154 4.141 4.129 4.116 4.106 4.097 4.088 4.079 4.070 4.062 4.055 4.046 4.038 4.030 4.022 4.015 4.005 3.995 3.986 3.970 3.944

31 29 27 26 24 23 22 21 20 19 18 17 17 16 15 15 14 14 13 13 13 12 12 12 11 11 11 11 11 10 10 10 10 10 9 9 9 8

2 3.078 2 2.939 2 2.817 2 2.678 2 2.539 2 2.406 2 2.299 2 2.196 2 2.093 2 1.981 2 1.878 2 1.770 2 1.663 2 1.560 2 1.457 2 1.370 2 1.273 2 1.196 2 1.119 2 1.053 2 0.981 2 0.924 2 0.868 2 0.817 2 0.770 2 0.724 2 0.678 2 0.642 2 0.606 2 0.565 2 0.524 2 0.483 2 0.447 2 0.406 2 0.360 2 0.319 2 0.257 2 0.160

0.082 0.082 0.081 0.079 0.077 0.076 0.076 0.077 0.078 0.079 0.081 0.083 0.085 0.087 0.089 0.093 0.096 0.100 0.104 0.108 0.113 0.117 0.122 0.127 0.131 0.135 0.140 0.145 0.149 0.151 0.152 0.152 0.151 0.150 0.148 0.136 0.116 0.065

log T eff (4.5)

T eff (4.5)

4.602 4.573 4.521 4.484 4.449 4.417 4.389 4.366 4.343 4.318

39 37 33 30 28 26 24 23 22 20

X 0.012 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400

530 645 990 225 605 300 260 300 415 490 725 875 145 455 800 260 725 260 840 455 070 760 490 260 990 760 530 340 105 915 725 530 340 105 875 685 340 800

987 373 220 453 093 093 500 220 040 807

BC (4.5) 2 3.765 2 3.555 2 3.263 2 3.088 2 2.899 2 2.739 2 2.586 2 2.457 2 2.324 2 2.181

Y (4.5) 0.081 0.073 0.058 0.046 0.039 0.032 0.026 0.021 0.017 0.013

X 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350 1.400 1.450 1.500 1.550 1.600 1.650 1.687 2 2 2 X 2 0.040 2 0.025 0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

log T eff (4.0)

T eff (4.0)

BC (4.0)

Y (4.0)

4.546 4.505 4.473 4.441 4.410 4.384 4.362 4.339 4.318 4.297 4.277 4.255 4.236 4.217 4.200 4.184 4.168 4.154 4.141 4.129 4.118 4.106 4.097 4.086 4.079 4.069 4.060 4.050 4.044 4.034 4.023 4.011 3.999 3.984 3.955 2 2 2

35 31 29 27 25 24 23 21 20 19 18 17 17 16 15 15 14 14 13 13 13 12 12 12 11 11 11 11 11 10 10 10 9 9 9 – – –

2 3.391 2 3.145 2 2.991 2 2.832 2 2.668 2 2.535 2 2.411 2 2.288 2 2.165 2 2.053 2 1.929 2 1.806 2 1.693 2 1.581 2 1.473 2 1.375 2 1.283 2 1.201 2 1.119 2 1.053 2 0.991 2 0.924 2 0.873 2 0.821 2 0.770 2 0.724 2 0.683 2 0.637 2 0.591 2 0.539 2 0.493 2 0.432 2 0.381 2 0.303 2 0.170 2 2 2

0.082 0.074 0.068 0.063 0.058 0.055 0.053 0.052 0.050 0.049 0.049 0.048 0.048 0.049 0.050 0.051 0.053 0.055 0.058 0.061 0.064 0.067 0.070 0.073 0.075 0.077 0.079 0.080 0.080 0.076 0.067 0.058 0.047 0.019 2 0.047 2 2 2

167 987 733 633 680 193 013 833 807 833 913 987 220 500 833 273 707 247 833 473 120 760 500 193 987 733 473 220 067 807 553 247 987 633 013

log T eff (5.0)

T eff (5.0)

4.699 4.639 4.574 4.523 4.482 4.447 4.417 4.388 4.362 4.335

49 43 37 33 30 27 26 24 22 21

986 563 481 313 306 983 138 430 995 628

BC (5.0) 2 4.469 2 3.988 2 3.585 2 3.296 2 3.104 2 2.912 2 2.745 2 2.598 2 2.443 2 2.283

Y (5.0) 0.083 0.073 0.056 0.034 0.019 0.008 2 0.003 2 0.011 2 0.019 2 0.027

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

373

Table 6. Continued X

log T eff (4.5)

T eff (4.5)

BC (4.5)

Y (4.5)

X

log T eff (5.0)

T eff (5.0)

BC (5.0)

Y (5.0)

0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.350 1.400 1.450 1.500 1.500

4.296 4.273 4.251 4.231 4.212 4.195 4.178 4.163 4.150 4.138 4.126 4.114 4.104 4.093 4.083 4.071 4.062 4.054 4.040 4.027 4.013 3.988 3.955

19 18 17 17 16 15 15 14 14 13 13 13 12 12 12 11 11 11 10 10 10 9 9

2 2.047 2 1.914 2 1.781 2 1.663 2 1.550 2 1.447 2 1.345 2 1.253 2 1.175 2 1.103 2 1.037 2 0.970 2 0.909 2 0.853 2 0.796 2 0.750 2 0.693 2 0.632 2 0.575 2 0.509 2 0.437 2 0.324 2 0.191

0.010 0.006 0.004 0.001 0.000 2 0.000 2 0.000 2 0.000 0.000 0.001 0.002 0.003 0.003 0.003 0.003 0.002 2 0.000 2 0.006 2 0.014 2 0.034 2 0.060 2 0.105 2 0.163

0.400 0.450 0.500 0.550 0.600 0.650 0.700 0.750 0.800 0.850 0.900 0.950 1.000 1.050 1.100 1.150 1.200 1.250 1.300 1.325 1.300 2 2

4.310 4.286 4.262 4.240 4.221 4.201 4.186 4.170 4.156 4.141 4.128 4.115 4.105 4.091 4.084 4.069 4.056 4.040 4.024 3.991 3.954 2 2

20 19 18 17 16 15 15 14 14 13 13 13 12 12 12 11 11 10 10 9 8 – –

2 2.136 2 1.983 2 1.841 2 1.707 2 1.591 2 1.476 2 1.379 2 1.290 2 1.207 2 1.129 2 1.059 2 0.988 2 0.924 2 0.854 2 0.796 2 0.726 2 0.655 2 0.572 2 0.463 2 0.322 2 0.200 2 2

2 0.034 2 0.041 2 0.046 2 0.051 2 0.055 2 0.059 2 0.061 2 0.064 2 0.066 2 0.068 2 0.069 2 0.072 2 0.075 2 0.079 2 0.085 2 0.094 2 0.108 2 0.130 2 0.179 2 0.226 2 0.275 2 2

787 760 833 013 300 680 067 553 140 733 373 013 707 400 093 787 527 320 967 653 300 733 013

estimated distances became symmetrical. For that calibration also, the standard deviation over the residuals was found to be 0.40 mag over the 483 reference stars. The most objective and precise distance data were finally provided for early-type stars in the solar neighbourhood by the Hipparcos satellite late in 1996. This enabled a totally independent calibration to be made on the basis of 6044 field B- to A stars and 134 members of the Scorpio-Centaurus association. After selection over well defined parallaxes (p /s (p ) $ 5), their location in the X,Y-diagram (Y $ 2 0.06) and discrimination against known binaries (including those detected by Hipparcos), 1580 stars of classes V to III remained for the calibration. These were then de-reddened by means of the intrinsic colour estimators, thus providing the reference sample of MV . The resulting calibration is shown in Fig. 21 where it is mapped in the X,Y-plane with the calibration stars. The following polynomial form, of fourth degree in X and quadratic in Y, defines the new MV (X,Y) relation:

398 305 280 391 640 888 341 795 316 838 428 018 745 335 130 720 378 968 558 806 986

MV (X,Y) 5 a 0 1 a 1 Y 1 a 2 Y 2 1 a 3 X 1 a 4 XY 1 a 5 XY 2 1 a 6 X 2 1 a 7 X 2 Y 1 a 8 X 2 Y 2 1 a 9 X 3 1 a 10 X 3 Y 1 a 11 X 4 with a 0 5 2 3.7528, a 1 5 2 21.0189, a 2 5 2 161.4130, a 3 5 5.2076, a 4 5 2 2.5839, a 5 5 138.1530, a 6 5 2 2.8463, a 7 5 13.2480, a 8 5 2 29.8913, a 9 5 2.9365, a 10 5 2 4.0510, a 11 5 2 1.2997. Another view of the relation is shown in Fig. 22, projected along the X direction, and in which the standard errors over the Hipparcos parallaxes are transformed into absolute magnitude. The distribution in Y of the data is seen to justify the quadratic term in that parameter. A more detailed discussion of the calibration is given by Cramer (1997). Here too, the standard deviation over the residuals (after subtraction of the component inherent to

374

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 21. Absolute magnitude calibration of the X,Y-diagram based on the Hipparcos satellite parallaxes. The calibration stars are plotted in the figure.

Fig. 22. The MV (X,Y) relation with the calibration stars. The error bars correspond to the standard deviations of the Hipparcos parallaxes.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

375

Fig. 23. Histogram of the residuals over the calibration data in terms of distance. The fitted distribution has a hwhm of 60 pc for the sample extending out to about 300 pc.

Hipparcos) amounts to 0.44 mag, and is comparable to those of the earlier calibrations. Translated into terms of distance, we observe the mean dispersion of 60 pc (hwhm) shown in Fig. 23 (which includes the Hipparcos uncertainty) within the approximately 300 pc range of the calibration data. A sizeable part of these ubiquitous four tenths of a magnitude of uncertainty arises from the gradient of the calibration operating on the uncertainties over the parameters themselves. The mean internal standard deviation of MV (X,Y) is mapped into the X,Y-diagram of Fig. 24 with the same 12 800 stars as in Fig. 8, and shows that an uncertainty of the order of 0.20 mag is unavoidable for purely metrological reasons. Among the underlying astrophysical causes of dispersion, rotation can be expected to contribute to dispersion by about 0.1 mag in the average (see Cramer & Maeder, 1979). However, binarity still remains the major perturbing agent of an MV calibration. The importance, and complexity of the binarity loops for this calibration are given in Fig. 25 for main sequence stars with indication of the d MV of the components. As already mentioned, the effects of binarity in a photometric diagram depend on the shape of the related sequence. For stars of types later than B8V (X $ 1.1), the variation of colour due to a bend in the lower sequence generates an error on the estimate that may even exceed the 0.753 mag that

would be caused by an identical twin. Hence, the dispersion of the residuals for the various attempts to calibrate MV should not be considered as a cause of surprise, but rather as an inevitable limitation.

3.5.2. Some related comments The absolute magnitude calibrations used above, particularly the polynomial forms, are global relations which cannot presume to perfectly account for the true physical connection with MV at all points of the validity range. Separate calibrations adjusted over smaller ranges where the colour variation gradient is more important, or where interference due to overlapping sequences can occur, may reduce the overall zero-point error (O2B1 stars, see for example Appendix of Raboud et al. (1997), or specifically A0 to A2 stars). Nevertheless, examination of the X,Y distribution of the residuals of our last calibration does not display any clear colour-dependent bias, which implies that the gain would not be very important in regard to the other causes of dispersion which are much larger. The ultimate local calibration is achieved in a ‘‘photometric box’’. This technique, which is specific to Geneva photometry and depends critically on its homogeneity, was introduced by Golay et al. (1969) and developed by Nicolet (Nicolet, 1981a; Nicolet, 1981b; Nicolet, 1993; Nicolet, 1994). The object is

376

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 24. Mean internal standard deviation of the MV (X,Y) calibration in the X,Y-diagram, all weights considered, with 12 800 stars plotted in the background. An error ranging from 0.1 mag to about 0.4 mag, depending on the local declivity of the relation, can be expected for metrological reasons alone. Uncertainties of that order are common to all photometric estimates of MV .

to extract the maximum amount of information by simultaneously using the values measured through the whole set of filters. This is done by considering small neighbourhoods in the six, or less, dimensional space of the normalised colour indices, or else in a three-dimensional parameter space (d, D, g or X, Y, Z) if reddening is present. The neighbourhood is centred on the colours of a given star. The distance to the border of the neighbourhood is defined by an appropriate metric which can be adjusted to take into account the accuracy of the individual colour indices or parameters. The assumption made in this ‘‘topological’’ approach to multicolour photometry is that the properties of any number of stars, or calibration points, lying within the given neighbourhood can be equated with each- other, provided that the radius of the ‘‘photometric box’’ is small enough. Typically, the radius of a box is of the order of 10 22 to 2 3 10 22 mag. The box technique has been success-

fully used to inter-relate the distances of open clusters (Nicolet, 1981b). A proper application to the Hipparcos parallax data would be expected to decisively improve the distance scale out to even very remote clusters possessing stars which can be photometrically equated with Hipparcos stars, and Nicolet is presently working on that project. Some exceptions to the photometric discrimination of the technique have been shown however to exist. When the Pleiades cluster star Pleione went through its last shell phase, it became for some time photometrically indistinguishable in the Geneva system from a supergiant, even when the box method was used (see Cramer et al., 1995). A different comment may be made in regard to the correction of interstellar extinction. The Hipparcos parallaxes have unquestionably revealed a number of problems affecting the determination of cluster distances by using photometric sequences, notably for

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

377

Fig. 25. The effect of undetected binarity on the absolute magnitude estimate for a sequence of class V stars distributed along X. The result is to always underestimate the brightness of the pair. The magnitude difference d MV between the two components is indicated along the loops. Close to the bend of the sequence in the X,Y-diagram, the effect can even exceed the 0.753 mag of identical twins.

the Pleiades cluster, as mentioned above. The distance estimated for the cluster by the present calibration (which did not initially have access to Pleiades data) reproduces the classical value of 130 pc instead of the Hipparcos determination of 116 pc (Mermilliod et al., 1997a). Adjustments of helium content to explain the apparently lower luminosity of its members do not seem to work out satisfactorily (Van Leeuwen & Hansen Ruiz, 1997). A very peculiar reddening law with a total to selective absorption ratio R of more than twice the standard value would go in the sense of solving the discrepancy, but is unlikely to have gone undetected in view of the extent to which the cluster has been studied at all wavelengths. Maybe the simplest explanation could be provided by the presence of a small neutral, or almost achromatic extinction in that direction. The cluster is known to be located at the fringe of a large molecular cloud complex. The properties of extinction in the visible are not well known in such media. Indeed, to the author’s knowl-

edge, the absence of a weak grey component which could sometimes be present in interstellar extinction has not been definitely proven yet. Its presence on a very large scale could even have cosmological consequences, by increasing the distance estimates of ‘‘standard candles’’ and thus affecting the accuracy of determinations of the Hubble constant. But – to return to the more mundane environment of our own galactic neighbourhood – the correction of interstellar extinction through its spectrally selective nature alone has proven to perform quite satisfactory up to now, though only within current observational limits, and the discussion of neutral absorption has been neglected for lack of observational means. The question remains open. Indeed, if one were to choose among the various manners of handling the problem of neutral absorption, a method involving the very accurate knowledge of distances of reddened stars and clusters would be favoured as being one of the most direct, and that may just be what we are beginning to see with the Hipparcos parallaxes.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

378

4. Comparisons with other systems Having established a number of calibrations in the Geneva system, particularly the most basic one of all regarding the intrinsic colours of the reddened B stars, it becomes important to establish accurate relations with other systems. The first reason to do so is to be able to transfer data acquired in one system into the framework of the other. In other words, and as was done with the MK classification, but in a much more narrow sense – since the basic natures of the data are the same – so as to be able to ‘‘speak the same dialect’’. The consequence of this, which is also the main reason for seeking compatibility, is that calibrations, or even synthetic photometry done in each system can be directly inter-compared. If all were perfect, everything would agree. If disagreements do occur, then the objects of contention have to be re-examined to the general benefit of both parties. Here, we will briefly discuss two different types of comparisons for B stars with the help of two examples. The first, in a de-reddened context, with the classical U,B,V system. The second in the redden¨ ing-free context of the Stromgren b index. The two original calibrations were presented in Cramer (1984a); Cramer (1984b).

4.1. Correlating with the classical U,B,V system The correlation for B stars (Cramer, 1984a) was achieved in a manner similar to the second step of the intrinsic colours determination. The contents of small zones in the X,Y-diagram were correlated with each other regarding the U2B and B2V indices of both systems. The reference U,B,V data were taken from Nicolet’s (1978) homogeneous compilation. Interstellar extinction generated the correlation line in each case, and the zero-points were adjusted to those given by the [U,B,V] 0 (X,Y) relation. Cubic polynomial expressions in X and Y were again used to globally smooth the individual estimates. The two estimators are: (k 2 l) 0 (X,Y) 5 a 0 1 a 1 X 1 a 2 Y 1 a 3 XY 1 a 4 X 2 1 a 5 Y 2 1 a 6 XY 2 1 a 7 X 2 Y 1 a 8 X 3 1 a9Y 3

where k 2 l 5 U2B, B2V. For (U2B) a 0 5 2 1.1333, a 1 5 0.7745, a 2 5 2 0.3593, a 3 5 2 0.2738, a 4 5 2 0.0813, a 5 5 2.7456, a 6 5 2 2.1926,

a 7 5 2 0.0117,

a 8 5 0.0211,

a 9 5 4.6680; and for (B2V) a 0 5 2 0.3115, a 1 5 0.2556, a 2 5 2 0.0692, a 3 5 0.1316, a 4 5 2 0.1281, a 5 5 2 1.6989, a 6 5 1.0061, a 7 5 2 0.2832, a 8 5 0.0530, a 9 5 5.2059. Their validity is limited to the same range in X,Y as the other calibrations, i.e. to the stars of classes V to III, with Y . 2 0.06. The (U2B) estimator is slightly gravity-dependent whereas the relation for (B2V) is practically independent of log g for classes V to III over the same validity range. A detailed discussion of the estimators is to be found in the original paper. These calibrations were initially conceived with the object of examining the consistency of synthetic photometry done in both systems and, more particularly, to compare effective temperature calibrations. The comparison of synthetic colours proved at first to be somewhat problematic, but the consistency of the transformed effective temperature scales was shown to be good. The deviations noted in Fig. 19 regarding the MK type versus T eff relation did, however, persist. We may now examine some aspects of the transformation with the help of more recent synthetic photometry and observations. With the publication of his 1993 models, Kurucz also gave synthetic standard U,B,V indices for them. It is then interesting to see how the same models, filtered with the Geneva passbands, perform regarding those indices after having been transformed by our estimators. The transformation was based on the synthetic X,Y values, without any attempt to correct the latter for the reasons stated above in the context of effective temperatures. The resulting comparison is given in Fig. 26 within the limits of the range of the estimators. For the four sets of models with log g encompassing MK classes V to III, one notes a colour dependent discrepancy in (U2B) varying from about 0.06 mag

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

379

Fig. 26. Comparison of the synthetic U,B,V indices of Kurucz’s (1993) models having log g 5 3.5, 4.0, 4.5, 5.0 with their counterparts in the Geneva system transformed into the standard U,B,V system. The estimated values are bluer. A colour related behaviour, i.e. varying with X, is apparent for (U2B).

at O2B1 stars, to zero at B92A0, and a more or less constant shift of (B2V) of about 0.03 mag. Looking in the U,B,V diagram of Fig. 27 at the models with their transformed counterparts (slashed lines), we see that the estimated positions of the (Geneva) models are slightly bluer. The zero-points of the estimators have, of course, been adjusted to those of the intrinsic colour calibration in the Geneva system. There too the synthetic [U2B] 0 and [B2V] 0 indices appeared too red, though by a larger amount regarding [B2V] than in this case. Here, the value of the d (U2B) /d (B2V) ratio is far from that of the slope of the reddening line, weakening the case for excessively de-reddened zero-points. We may also recall the argument to that end based on Figs. 9a–9d. The present figure also contains 50 very well measured (partly from the standard) and little reddened stars of types O- to A0 and of classes V to III. Their Geneva intrinsic colours are transformed into the standard U,B,V diagram (crosses) and re-reddened (open circles) as follows:

E( B2V)John 5 0.842E[B2V]Gen , E( U2B)John 5 0.73E( B2V)John , where 0.842 is the very well defined ratio (Cramer, 1984a) between the B2V excesses of both systems. Being the slope of a correlation, it is derived in a zero-point independent manner. The corresponding data measured in the standard system and compiled by Mermilliod et al. (1997b) are then plotted (full circles) into the figure. One notices that, apart from very few discrepancies, the true standard U,B,V values are well reproduced by the transformations, particularly so for (B2V). These comparisons serve to restate some of the difficulties and uncertainties which are inherent to the applications of synthetic photometry. Particularly so when the object is to achieve predictions with an accuracy equal to that of the photometric measurements themselves. The effective temperature and Bolometric Correction relations of Table 6 are easily transformed into the U,B,V system via these estimators restricted to

380

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 27. Synthetic U,B,V diagram of Kurucz’s (1993) models with log g 5 3.5, 4.0, 4.5, 5.0 (full lines) and their transformed Geneva counterparts (dashed lines). The latter, the estimators of which are adjusted to the zero-point of the Geneva intrinsic colours, are slightly bluer. A sample of 50 well measured stars in the Geneva system is transformed into the U,B,V diagram (crosses) and re-reddened on the basis of their scaled Geneva E[B2V] (open circles). They are compared with the indices measured in the standard system (full circles).

their validity range (Y . 2 0.06). They are given in Table 7. The transformed relations for log g 5 3.5, 4.0, 4.5, 5.0 are compared in Figs. 28 and 29 with the classical relations of Schmidt-Kaler (1982). They do not present any fundamental disagreements with each-other. We do, however, notice the steeping slope of the log g 5 5 line of the semi-empirical relation as it rises above 38 000 K. We also recognise the slightly higher temperatures of the classical relation in the region where the MK class versus T eff calibration (Fig. 19) also shows that trend. Much new data have been acquired in the Geneva system since 1984. Equally so in the standard U,B,V system which has also benefited from compilations with homogenisation (Mermilliod et al., 1997b). A revision and more thorough discussion of the correlation

in the future could prove to be profitable, though it is not expected to significantly modify the estimators.

¨ 4.2. Correlating with the Stromgren Hb index The most straightforward type of correlation of the X,Y parameters is obviously made with other reddening-free photometric quantities. That with the ¨ Stromgren Hb index is particularly interesting because of the extremely different natures of the two photometries. The X,Y parameters essentially reflect the behaviour of the continuum over the visible spectral range, whereas the b index specifically measures the strength of the Hb line. Moreover, the Geneva passbands only marginally include that line. An excellent correlation for normal stars is obtained

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

381

Table 7 Geneva T eff calibration transformed to standard U,B,V U2B 2 1.15 2 1.10 2 1.05 2 1.00 2 0.95 2 0.90 2 0.85 2 0.80 2 0.75 2 0.70 2 0.65 2 0.60 2 0.55 2 0.50 2 0.45 2 0.40 2 0.35 2 0.30 2 0.25 2 0.20 2 0.15 2 0.10 2 0.05 0.00 0.05 0.10 U2B 2 1.15 2 1.10 2 1.05 2 1.00 2 0.95 2 0.90 2 0.85 2 0.80 2 0.75 2 0.70 2 0.65 2 0.60 2 0.55 2 0.50 2 0.45 2 0.40 2 0.35 2 0.30 2 0.25 2 0.20 2 0.15 2 0.10 2 0.05

B2V (3.5) 2 0.329 2 0.309 2 0.293 2 0.277 2 0.261 2 0.247 2 0.234 2 0.222 2 0.211 2 0.200 2 0.189 2 0.180 2 0.171 2 0.161 2 0.152 2 0.143 2 0.133 2 0.123 2 0.113 2 0.100 2 0.087 2 0.074 2 0.056 2 0.038 2 0.015 0.011 B2V (4.5) 2 0.327 2 0.304 2 0.286 2 0.270 2 0.256 2 0.242 2 0.229 2 0.217 2 0.206 2 0.193 2 0.183 2 0.172 2 0.161 2 0.149 2 0.138 2 0.127 2 0.114 2 0.101 2 0.088 2 0.072 2 0.054 2 0.034 2 0.011

log T eff (3.5) 4.503 4.467 4.432 4.393 4.359 4.332 4.304 4.276 4.247 4.220 4.193 4.169 4.147 4.125 4.106 4.090 4.074 4.058 4.044 4.029 4.016 4.001 3.988 3.973 3.957 3.943 log T eff (4.5) 4.589 4.514 4.466 4.424 4.386 4.356 4.323 4.294 4.263 4.233 4.206 4.182 4.160 4.142 4.125 4.108 4.093 4.077 4.062 4.049 4.034 4.020 4.004

BC (3.5) 2 3.111 2 2.909 2 2.737 2 2.553 2 2.363 2 2.214 2 2.053 2 1.909 2 1.743 2 1.576 2 1.421 2 1.277 2 1.151 2 1.030 2 0.927 2 0.835 2 0.743 2 0.668 2 0.593 2 0.519 2 0.461 2 0.387 2 0.329 2 0.271 2 0.210 2 0.153 BC (4.5) 2 3.599 2 3.225 2 2.984 2 2.760 2 2.565 2 2.392 2 2.197 2 2.025 2 1.852 2 1.674 2 1.519 2 1.375 2 1.243 2 1.128 2 1.025 2 0.933 2 0.841 2 0.766 2 0.685 2 0.611 2 0.541 2 0.479 2 0.398

B2V (4.0) 2 0.328 2 0.307 2 0.289 2 0.273 2 0.258 2 0.244 2 0.230 2 0.219 2 0.207 2 0.196 2 0.185 2 0.175 2 0.164 2 0.155 2 0.145 2 0.135 2 0.125 2 0.114 2 0.104 2 0.091 2 0.075 2 0.058 2 0.038 2 0.012 0.018 0.060 B2V (5.0) 2 0.322 2 0.301 2 0.284 2 0.270 2 0.257 2 0.244 2 0.233 2 0.222 2 0.211 2 0.200 2 0.189 2 0.178 2 0.166 2 2 2 2 2 2 2 2 2 2

log T eff (4.0) 4.555 4.500 4.454 4.411 4.376 4.346 4.316 4.285 4.253 4.227 4.200 4.176 4.153 4.133 4.115 4.097 4.083 4.068 4.051 4.041 4.026 4.010 3.997 3.984 3.967 3.947 log T eff (5.0) 4.592 4.518 4.470 4.431 4.395 4.362 4.329 4.298 4.267 4.241 4.214 4.192 4.171 2 2 2 2 2 2 2 2 2 2

BC (4.0) 2 3.295 2 3.111 2 2.898 2 2.685 2 2.495 2 2.323 2 2.145 2 1.984 2 1.806 2 1.633 2 1.473 2 1.335 2 1.197 2 1.082 2 0.973 2 0.881 2 0.795 2 0.720 2 0.645 2 0.576 2 0.513 2 0.438 2 0.375 2 0.312 2 0.227 2 0.135 BC (5.0) 2 3.702 2 3.278 2 3.041 2 2.818 2 2.631 2 2.444 2 2.250 2 2.063 2 1.877 2 1.698 2 1.553 2 1.417 2 1.281 2 2 2 2 2 2 2 2 2 2

382

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 28. Comparison of the T eff relations of Fig. 18 transformed into the (U2B) index, with the classical relations of Schmidt-Kaler.

Fig. 29. Comparison of the BC relations of Fig. 20 transformed into the (U2B) index, with the classical relations of Schmidt-Kaler.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

(Cramer, 1984b) and is expressed by the b (X,Y) estimator:

b (X,Y) 5 a 0 1 a 1 X 1 a 2 Y 1 a 3 XY 1 a 4 X 2 1 a 5 Y 2 1 a 6 XY 2 1 a 7 X 2 Y 1 a 8 X 3 1 a 9 Y 3 where a 0 5 2.5909, a 1 5 0.0667, a 2 5 2 0.6801, a 3 5 2 0.2559, a 4 5 0.1748, a 5 5 2 2.4676, a 6 5 0.1448, a 7 5 0.2582, a 8 5 2 0.0612, a 9 5 0.4418. The relation, defined for O, B and first A type stars or classes V to III, does nevertheless extrapolate correctly to bright giants and supergiants, as far as has been verified for a few dozen cases. The details of the calibration are discussed in the original paper. It is mapped into the X,Y-plane of Fig. 30 which also contains almost 12 800 stars as a background within the validity range. The standard

383

deviation over the residuals for the 950 calibration stars is only 0.016 mag. Such a good correlation encourages us to form what we may call a ‘‘xenoindex’’, i.e. an index composed of quantities issuing from two different photometric systems. Here, we consider the db 5 b (X,Y) 2 b ‘‘xenoindex’’. Due to the specific nature of the b index, db is very sensitive to any departures from the norm that may occur in the Hb line. That is particularly evident in the case of emission in the hydrogen lines and makes db a powerful detector of Be stars in their emission phase. The X,Y parameters are indeed virtually unaffected for moderate emission, and much less so than b when strong hydrogen emission lines are present. Fig. 31 (taken from Cramer, 1984b) shows the large amplitude of the variation for Be stars and its dependence on temperature and rotational velocity. In the same paper, it was suggested that rotation was less susceptible to affect the b index than was formerly believed; this was later confirmed by Garrison & Gray (1991). The db index was also shown to have the capacity of

Fig. 30. The b (X,Y) relation overlaid with 12 800 stars within its validity range.

384

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

Fig. 31. The db 5 b (X,Y) 2 b ‘‘xenoindex’’ applied to Be stars. The latter are detected with high efficiency at the higher temperatures. Very fast rotators stand out conspicuously in that population. To be most effective, measurements should be carried out simultaneously in both systems. However, the b (X,Y) estimate being much less affected by emission, knowledge of the epoch of the true b index measurement alone is most important. An estimate of the contribution of the emission feature’s equivalent width in the Hb line is given by d W . 34 3 db ˚ units (Figures taken from Cramer (1984b)). in A

estimating the equivalent width of the emission feature in Hb ; a feat usually achieved only by medium to high resolution spectroscopy. The b (X,Y) estimator equally provoked an early brush with synthetic photometry. The (1979) Kurucz models (Kurucz, 1979), as seen in Geneva photometry, then became directly comparable via the estimator with their synthetic counterparts expressed in terms of the b index by Schmidt (1979). A rather large line-width dependent discrepancy (see Cramer, 1984b) was seen, that led to the dilemma: either the width of the Hb line of the models was overestimated, or the calibration of the synthetic photometry was in error. That the latter contention was true was subsequently suggested by Moon & Dworetsky (1985) and corrected by Lester et al. (1986). The two correlations with other systems that we have just briefly reviewed show that it is relatively easy to establish estimators based on the X,Y reddening-free parameters within a restricted domain of

early-type stars. These correlations can be obtained either for reddening-free indices or for those subject to extinction. The latter, however, require further steps also involving correlation to determine their intrinsic colours, the consequence then being that the zero-points are adjusted to those of our intrinsic colour calibrations.

5. Concluding remarks

5.1. Retrospective We have given here a presentation of the types of calibrations that can be established for the massive early-type stars in the Geneva system. Their relatively sparse spatial distribution restricted to the vicinity of the galactic plane makes that distant population universally subject to extinction by interstellar dust. The analysis by multicolour photometry

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

must be undertaken within a setting that is as far as possible unaffected by extinction. The three reddening-free orthogonal parameters called X, Y and Z span the whole visible spectrum by combining the five U, B1, B2, V1 and G intermediate width bands. They are optimised for the analysis of O and B stars and constitute the best possible framework for studying the latter in the Geneva system. Their original determination was done by ‘‘fitting by hand’’, though later re-determinations by factorial analysis came up with exactly the same definitions. To make proper use of a reddening-free representation, it is of central importance to know the intrinsic colours of the various indices that contribute to establish it. The necessary search for the zeropoints is best done within a context of ‘‘minimal assumption’’, i.e. as empirically as possible. That was basically the approach that led us to the definitions given above, and which are probably also the most ‘‘robust’’ of the calibrations presented here. The knowledge of the colours of synthetic photometry and of the corresponding empirically determined reddening-free values then enables us to examine their general consistency and accuracy. This is particularly important when comparing the necessarily non-reddened predictions of synthetic indices with observations that are virtually all reddened. We have seen that empirical calibrations can suggest small corrections to good synthetic photometry which then allow the latter to achieve its full potential. Generally speaking, synthetic photometry, if it is to be applied within a context requiring high accuracy, has to be used with care. Less robust are calibrations based on data that are subject to ‘‘evolution’’ as astronomical and astrophysical techniques get better. That is true for effective temperatures and absolute magnitudes, for example; though the latter have decisively benefited recently from the results of the Hipparcos astrometric satellite. The former will also do so as the next generation of high angular resolution instruments become operational. Quite generally, the fitting of a third degree polynomial has proven to be a good option for calibrating in the X,Y-diagram. It has the advantage of having a compact form that one can integrate into software that is not well adapted to interpolation in

385

tables. Higher degrees are to be avoided as they may easily diverge rapidly at extrapolation, though a fourth degree term in X does improve the absolute magnitude calibration of the brighter part of the main sequence. The general strategy during calibration should be governed by the search for the simplest relation giving a non-biased account of the original data. Biasing is most usually colour dependent in global calibrations and has to be carefully investigated. Some relationships do not yield, however, to a manageable polynomial fitting. Such is the case for the semi-empirical T eff calibration that is given in tabular form in Tables 6 and 7. The most restricted, but not least effective calibration that is embodied by a ‘‘photometric box’’, has to be used here within a reddening-free reference system. The incidence of undetected astrophysical causes of scatter such as rotation and binarity, for example, has to be constantly borne in mind. The latter is particularly troublesome and affects photometric parallaxes in no trivial fashion. Such effects are unavoidable and, unless one has access to different data such as those obtained by spectroscopy to enable corrections to be made, they set a natural limit to the analytical capability of photometry. The properties of the three-dimensional reddeningfree representation used here enable a very consistent treatment of B stars to be undertaken in a manner that is only reproduced with difficulty for the later types. The photometric effects of varying metallicity rapidly increase at temperatures lower than 10 4 K while those longer lived stellar populations also become more mixed in chemical composition. The interstellar reddening vector tends to merge with those of varying metallicity or gravity in certain diagrams at lower temperatures, impeding reddeningfree photometric analysis. Different strategies have to be followed for optimising calibrations in that domain. These options lie outside the scope of the present discussion, but have been presented elsewhere (Grenon, 1981; Grenon, 1993; Mennessier & Grenon, 1985).

5.2. Prospective We can mention a few more directions in which the photometric ‘‘toolmaker’’ can progress regarding the types of calibrations discussed above: Correla-

386

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387

tions with further popular photometric systems, such as the uvby system, can be established. These may not necessarily have to be confined to the visible alone. For instance, our intrinsic colour calibration and its correlation with the standard U,B,V system served to calibrate the intrinsic colours of the ESO IR system (see Dougherty et al., 1993). Regarding the U,B,V system, Golay (unpublished) has shown that good global correlations can be established with the corresponding Geneva indices even within a reddened context. A future calibration of that type is planned by Cramer and Golay. More accurate synthetic photometry will enable us to calibrate the X,Y-diagram with greater confidence in terms of age and mass by means of stellar evolution models, thus allowing us to easily estimate those parameters for isolated field stars. A first calibration in that sense was done by North & Cramer (1981). Better synthetic colours will also allow us to devise more consistent strategies for de-reddening supergiants, a subject that has been neglected here for lack of reliability of extant methods. The reproduction of the Geneva system has been attempted with a CCD detector. A first application of the X,Y,Z parameters measured that way ‘‘in the field’’ (see Raboud et al., 1997) proved to be acceptable, but not as satisfactory as classical photoelectric aperture photometry. The situation will get better with time, and the sampling of B stars will extend well beyond the present range of a couple of kiloparsecs. It could even extend to the Magellanic Clouds. The result will then be that variations of the reddening law will become more pronounced and, in particular, very different values of metallicity will be encountered. Strategies will have to be developed to detect and estimate these variations and the calibrations will have to be adapted accordingly. Finally, the multicolour photometric systems carried aloft by future astronomical satellites would profit by their capacity to define reddening-free parameter spaces akin to the X,Y,Z-space. Interstellar extinction will be a major component of the data acquired for distant objects of all types. The spectral sampling would have the virtue of being able to extend from the ultraviolet to the infrared. This implies the choice of a sufficient number of well located bands. With enough bands, several such photometric spaces could

even be defined within the same system, each one optimised by principal component analysis to detect a specific physical quantity. But that is the subject of a whole new discussion.

References Bartholdi, P., Burnet, M., & Rufener, F., 1984, A&A, 134, 290. ¨ Bohm-Vitense, E., 1981, ARAA, 19, 295. Burki, G., 1999, A&A, in press. Burnet, M. & Rufener, F., 1979, A&A, 74, 54. Carrier, F., Burki, G., & Richard, C., 1999, A&A, 341, 469. Code, A.D., Davis, J., Bless, R.C., & Hanbury Brown, R., 1976, ApJ, 203, 417. Cramer, N., 1982, A&A, 112, 330. Cramer, N., 1984a, A&A, 132, 283. Cramer, N., 1984b, A&A, 141, 215. Cramer, N., 1993, A&A, 269, 457. Cramer, N., 1994a, in: C. Sterken, M. De Groot (Eds.), The impact of long term monitoring on variable star research, NATO ASI, Kluwer Acad. Publ., Dordrecht, p. 405. Cramer, N., 1994b, in: C. Corbally, R.O. Gray, R.F. Garrison (Eds.), The MK Process at 50 Years, ASPC, 60, 172. Cramer, N., 1997, in: Hipparcos Venice ’97, ESA SP-402, 311. Cramer, N. & Maeder, A., 1979, A&A, 78, 305. Cramer, N. & Maeder, A., 1980, A&A, 88, 135. Cramer, N., Doazan, V., Nicolet, B., de la Fuente, A., & Barylak, M., 1995, A&A, 301, 811. Dougherty, S.M., Cramer, N., van Kerkwijk, M.H., Taylor, A.R., & Waters, L.B.F.M., 1993, A&A, 273, 503. Garrison, R.F. & Gray, R.O., 1991, in: A.G. Davis Philip et al., (Eds.) Precision Photometry: Astrophysics of the Galaxy, L. Davis Press, Schenectady, p. 173. Golay, M., 1980, in: Vistas in Astronomy 24, part 2, A. Beer, K. Pounds, P. Beer (Eds.), Pergamon Press, Oxford. Golay, M., Peytremann, E., & Mader, A., 1969, Publ. Obs. ` ´ A, No 76, p. 44. Geneve, Ser. Grenon, M., 1981, in: Astrophysical Parameters for Globular Clusters, IAU Coll. 68. Grenon, M., 1993, in: W.W. Weiss, A. Baglin (Eds.), Inside the Stars, IAU Coll. 137, ASPC, Vol. 40. Houk, N. & Cowley, A.P., 1975, University of Michigan Catalogue of two-dimensional Spectral Types for the HD Stars, Vol. 1, Univ. of Michigan, Ann Arbor. ¨ Kunzli, M., North, P., Kurucz, R.L., & Nicolet, B., 1997, A&AS, 122, 51. Kurucz, R.L., 1979, ApJS, 40, 1. Kurucz, R.L., 1993, Kurucz CD-ROM, 13, ATLAS9 stellar atmosphere program and 2 km / s grid. Lester, J.B., Gray, R.O., & Kurucz, R.L., 1986, ApJS, 61, 509. Maeder, A. & Peytremann, E., 1970, A&A, 7, 120. Mennessier, M.O. & Grenon, M., 1985, in: D.S. Hayes et al., (Eds.), Calibration of fundamental Stellar quantities, IAU Coll. 111, Reidel Publ. Co., Dordrecht.

N. Cramer / New Astronomy Reviews 43 (1999) 343 – 387 Mermilliod, J.C., Turon, C., Robichon, N., Arenou, F., & Lebreton, Y., 1997a, in: Hipparcos Venice ’97, ESO SP-402, 643. Mermilliod, J.-C., Hauck, B., & Mermilliod, M., 1997b, A&AS, 124, 349. Meynet, G., Mermilliod, J.-C., & Maeder, A., 1993, A&AS, 98, 477. Moon, T.T. & Dworetsky, M.M., 1985, MNRAS, 217, 305. Nicolet, B., 1981a, A&A, 97, 85. Nicolet, B., 1981b, A&A, 104, 185. Nicolet, B., 1993, in: Poster Papers on Stellar Photometry, IAU Coll. 136, Dublin Institute for Advanced Studies. Nicolet, B., 1994, in: C. Corbally, R.O. Gray, R.F. Garrison, (Eds.), The MK Process at 50 Years, ASPC, 60, 182. Nicolet, B., 1996, BaltA, 5, 417. North, P. & Cramer, N., 1981, A&AS, 43, 395. North, P. & Cramer, N., 1984, A&AS, 58, 387. Raboud, D., Cramer, N., & Bernasconi, P.A., 1997, A&A, 325, 167.

387

Rowan-Robinson, M., 1985, The cosmological distance ladder, W.H. Freeman and Company, New York. ` A, 66, 413. Rufener, F., 1964, Publ. Obs. Geneve Rufener, F., 1981, A&AS, 45, 207. Rufener, F., 1985, in: D.S. Hayes et al., (Eds.), Calibration of fundamental Stellar quantities, IAU Coll. 111, Reidel Publ. Co., Dordrecht, p. 253. Rufener, F., 1988, Catalogue of Stars measured in the Geneva Observatory Photometric system (fourth edition), Observatoire ` de Geneve, Sauverny. Rufener, F. & Nicolet, B., 1988, A&A, 206, 357. Schmidt, E.G., 1979, AJ, 84, 1739. Schmidt-Kaler, Th., 1982, in: Landolt Bornstein (New Series) IV/ 2b. Van Leeuwen, F. & Hansen Ruiz, C.S., 1997, in: Hipparcos Venice ’97, ESA SP-402, 689. Waelkens, C., 1991, A&A, 246, 453.