4.1. Limited probabilities for selection of associated and foreground couples
The individual probabilities can be used for distinguishing associated couples of OB and WR stars from foreground stars. The distribution of these quantities for 206 couples is shown in Fig. 1. The associated stars of the two stellar populations constitute couples with substantially smaller distances than those of the foreground ones. From Fig. 1 we see that associated couples can be selected when . On the other hand, corresponds to foreground couples. Taking and would make a stronger criterion but would mean the lost of many associated couples. We keep the first criterion as a compromise for our purpose.
4.2. The ratio OB stars to RSGs (OB/R)
Humphreys and Sandage (1980) found that the ratio of the number of blue stars- to-red supergiants (B/R) decreases as a function of the galactocentric distance in M 33. Later on Freedman (1984) has taken into account errors on the ratio (B/R) and concluded the result has a low statistical weight in the centre of M 33. She inferred that the data of Humphreys and Sandage (1980) when plotted with their error bars are in apparent contradiction with their previous result. Freedman has calculated the error using the following relations: where and are respectively the maximum and minimum fluctuations of the ratio (B/R). and are the numbers of blue and red stars respectively in each bin of galactocentric distance. The errors defined in this way are correct if and are independent statistical variables. But OB stars and RSGs correlate as it is the case M 33 then the errors bars on the ratio (OB/R) are
where is the coefficient of correlation between blue and red stars (see Eadie et al. (1971)). Here we take , where R5 is given by Eq. 2. This substitution seems approximately correct. Fig. 2 shows the ratio (OB/R) as a function of the galactocentric distance in the deprojected plane of the galaxy. We used a position angle and an inclination of the plane of the galaxy . The error bars of the ratio (OB/R) in this figure are not crucial for assessing the existence (or non-existence) of a radial gradient of the ratio (OB/R). They just indicate the statistical weight of the ratios (OB/R) in each radial bin. We define the statistical weight in each bin as:
where , where n is the number of bins and are defined by Eq. 5for each radial bin. We obtained a coefficient of correlation for data in Fig. 2 . This correlation is not very strong but is statistically significant. Therefore it appears that the observational evidence for a radial gradient of the ratio (B/R) found by Humphreys and Sandage (1980) is real.
Table 3. The ratios (OB/R), R5 and RN5 between OB stars and RSGs
4.3. Observational evidences of the evolution of massive stars
Table 1 indicates a tight correlation between WC stars and OB stars. We also obtain tight correlation between WC stars and RSGs (). This result suggests that the WC progenitors with higher masses evolve first to RSGs and then loose their envelopes. No other stellar population in M 33 shows a correlation with RSGs as strong as WC stars. If WN stars have a smaller initial mass than WC stars, we should find less WC than WN stars. Indeed in M 33 were found twice more WN (89) than WC stars (41). The small coefficient correlation between WN stars and RSGs (R5 = 0.33) calculated on the basis of MBHS96 data suggests that lower mass stars () rather evolve directly to WN stars.
The gradient in the number of blue to red stars seen by Humphreys & Sandage (1980) and confirmed by the present study can be interpreted by a chemical abundance gradient effect (Henry & Howard, 1995) as explained by MLA.
© European Southern Observatory (ESO) 1998
Online publication: August 6, 1998