The analysis was performed separately for each photometric region.
The method was applied several times using as variables the observed
Strömgren-Crawford indices ((b-y), m1, c1 and 
) and the
bracket, reddening free, indices ([m1], [c1] and ). So, each
analysis yielded different definitions of the p coordinate.
From a photometric point of view, a star belonging to the early
region is a star located in a specific zone of the diagrams 
and 
(Strömgren 1966). For normal stars this corresponds
approximately to spectral types B0-B9. However, since peculiar stars
are bluer than normal stars, some A0-A2 peculiar stars are also classified
into the early region.
In general, a linear combination of the 
colours will not be
reddening free.
Although standard calibrations to compute reddening are not suitable
for CP2 stars, we dereddened them as if they were normal stars
since the uncertainties are of about 0.010 mag
(Maitzen 1980)
and we will come back to this problem in Sect. 6 (click here).
The number of stars as a function of the computed reddening is shown in
Fig. 1 (click here). 59 (13%) CP2 stars and 277 (21%) normal
stars have a colour excess greater than 0.10 mag. To avoid the effect of
the reddening, we built a subsample
of almost unreddened stars, removing all those with E(b-y) > 0.10 mag,
and we performed the discriminant analysis.
The MDA yielded the following value for p:
Figure 1: Histogram of colour excess for normal (dashed line) and CP2
(solid line) stars classified into the early region
the zero point being introduced in order that 
for
normal stars.
To estimate the order of the errors of the coefficients,
we built different subsamples of normal and CP2 stars by randomly reducing
the whole samples in 10% of stars. The MDA applied to these subsamples
shows that the coefficients vary by about 2-3%.
A polynomial was fitted to the normal stars:
and a 
was defined. So, p0 gives a standard
relation for normal stars (
) and 
, for a given
star, is the difference between its p coordinate and the p coordinate of
a normal star with the same [u-b]. The polynomial was fitted as a
function of [u-b], in order to avoid the reddening in the x-axis,
following Maitzen & Vogt (1983). Figure 2 (click here) shows the p coordinate
for the normal and CP2 samples and the standard relation of normal stars.
As a threshold value for peculiarity, we defined 
above which
50% of the CP2 stars are located.
The percentage of normal stars with respect to
the total number of stars with 
shows the associated "contamination''.
This contamination is an indicator of the capability of to
separate CP2 from normal stars. In the present case, a threshold of
= 1.50 yields a contamination of about 2%.
The percentage of contamination increases quickly as the threshold decreases.
Figure 2: p vs. [u-b] for the stars of the early region with
mag. The ordinate p
resulted from the application of the Multiple Discriminant Analysis
using (b-y), m1, c1 and as variables describing the
behaviour of both populations: peculiar and normal stars.
Dots represent normal stars and full circles CP2 stars.
The solid line is the adopted standard relation for normal stars and
the dashed line indicates the threshold above which 50% of the
CP2 stars are placed
Table 6 (click here) shows the efficiency of
as a function of the [u-b] (i.e. temperature). The
best segregation
is obtained for high values of [u-b], whereas
for the hottest stars
([u-b]<0.7) it is poor. This is due to the decreasing sensitivity of
the v-band to indicate higher opacities among hotter
B-type stars.
A fraction of the contamination can be explained by the presence
of undetected binaries. The binary nature can increase the value of p
of the primary component up to 0.1 for a B5-type star and by up to
0.6 for an A0-type star. The maximum increase is obtained when the
secondary is an A3-A5 type star.
So, in some combinations of spectral types, normal binary stars could
be misclassified as CP2 stars.
On the other hand, p is not reddening free: following Eq. (1) and applying
the common extinction curve we obtain E(p)= -4.7E(b-y), so
reddening decreases p and , and we cannot differentiate a very
reddened peculiar star from an unreddened normal star.
For the total sample, including reddened stars, the value of the contamination is about 3%, while the detection remains around 50%.
In order to analyze the effect of the peculiarity in each filter, it
is useful to rewrite the Eq. (1) as follows:
The contribution of y-band to p is due to the feature at
, whereas v is modified by the 
feature.
The coefficient of u shows that this band does not play any significant
role in p. The negative sign of the coefficient is due to the
fact that the index for CP2 stars is lower than for normal stars.
The contribution of b, in spite of its coefficient, is small, because
it lies in a region only slightly affected by the peculiarity.
The use of bracket, reddening free, indices to characterize the populations in the MDA yielded to a contamination of 13%. The effectiveness is smaller than with observed indices, probably because we are removing part of the information contained in the (b-y) colour.
As mentioned above and since the peculiar stars are bluer than the normal stars, some CP2 stars with spectral types A0-A2 are photometrically classified into the early region. We also carried out a discriminant analysis with a sample limited by spectral type from B0 to B9, i.e. instead of classifying the stars into the early region by their photometry we classified the stars by their spectral type. The contamination indicator is higher than before (15%), very probably because we excluded stars (see Fig. 2 (click here)) which exhibit the largest p values from our sample.
Since inferred from Eqs. (1) and (2) yield the lowest level of
contamination, this peculiarity parameter is the one adopted from now on.
| [u-b] | CP2 | normal | % | % |
| range | stars | stars | detection | contamination |
| 0.5-0.7 | 35 | 232 | 9 | 0 |
| 0.7-0.9 | 129 | 217 | 49 | 2 |
| 0.9-1.1 | 87 | 204 | 52 | 4 |
| 1.1-1.3 | 92 | 267 | 62 | 0 |
| 1.3-1.5 | 70 | 104 | 41 | 2 |
| 0.5-1.5 | 413 | 1024 | 50 | 2 |
The parameter defined by Eqs. (1) and (2) is only able to
detect CP2 stars. A sample of 71 CP3 stars extracted from
Renson et al. catalogue showed a mean value of equal
to 0.16 (s.d. 0.41). This is because is mainly sensitive to the
depression, not present in the CP3 stars.
Figure 3: The same as Fig. 2, but for the intermediate region
Table 7
lists 60 stars extracted from HM catalogue
that we consider likely to be peculiar (i.e. with 
).
To build the table we considered all early stars with complete photometry
in the HM catalogue and removed
the stars belonging to Renson et al. catalogue, all those having
or Geneva photometry (which are
more effective to detect CP2 stars than system, see Sect. 5)
and the stars having a quoted peculiar spectra in SIMBAD data base.
A similar analysis was applied to the intermediate and late regions,
with subsamples of stars with mag.
Unfortunately, the 's obtained are less efficient
(Figs. 3 (click here) and 4 (click here)) than in the early
region.
The calculated values of p and the standard relation for normal stars are, for the intermediate region:
and for the late region:
Figure 4: The same as Fig. 2, but for the late region
50% of detection is at = 1.25 and 2.00 mag, with
contamination levels of 29% and 17%, for intermediate and late regions,
respectively.
Due to the small size of the samples, the
errors of the coefficients are larger than for the early region: around 6%
for the intermediate region and 4% for the late region.
Several authors have pointed out the similarity between cool CP2 stars,
mainly Sr
stars, and metallic stars, also named CP1 stars. To test this point,
we applied the MDA to a sample of metallic stars
extracted from The General Catalogue of Ap and Am stars by
Renson et al. (1991). The sample contains 494 stars
classified into the late photometric region according to the same algorithm
used above. Considering two populations, normal
stars and metallic stars, the MDA yielded:
The efficiency of this p
is shown in Fig. 5 (click here).
With
the contamination is equal to 1.1%.
The dependence of p on each colour does not differ excessively from the
p obtained for cool CP2 stars. So, the MDA confirms the
mentioned similarity between CP1 and cool CP2 stars.
Cameron (1967) already noticed that cool CP2
stars are placed on a c1 vs. m1 diagram at the same location
as Am stars. All attempts at separating both types using
these indices were unsuccessful.
Figure 5: The same as Fig. 2, but for Am stars