3 Colour transformations

Colour transformations between a physical and a standard filter set always are a source of photometric errors (see e.g. Sterken & Manfroid 1992 or Royer & Manfroid 1996 and references therein). In the particular case of the XMM-OM optical filters, the important discrepancies between the standard and non-standard U, B and V filters renders the choice of adequate transformations even more critical. In order to avoid additional errors introduced by subsequent colour computations, we established colour transformations for both single filters and colour indices.

   

Table 2: Summary of the colour transformations established in this paper. Capital letters refer to Johnson U, B and V filters. A one, two or three digit code i₁,...,i_n characterizes the transformation: n is the number of temperature domains that have to be considered; i_k is the order of the transformation in the $k^{\rm th}$ domain (0 means no transformation is possible), e.g. 102 indicates that one has to divide the temperature domain in 3 distinct parts in which respectively linear, no and second order transformations are possible. The numbering of the domains is going from the hottest to the coolest stars

	U-u	B-b	V-v	U-B	B-V	U-V
u-b	202	102		102
uvw1-u	222
b-v		211 / 3	111 / 2		1
uvw1-b				102
u-v						101 / 1
uvw1-v						102 / 1

We restricted ourselves to a limited set of physically meaningful transformations, i.e. between equivalent or neighbouring filters in both filter sets. We nevertheless introduced transformations to the (U-V) index in order to get colour transformations for colour indices that avoid the b filter, which is the one for which the XMM-OM detector could be the most easily affected by saturation (XMM Users' Handbook, Eds. Dahlem & Shartel 1999). We also established various transformations in which the uvw1 filter supplants the ufilter. Indeed, contrary to the u and U filters, the uvw1filter is defined on the short wavelength side of the Balmer jump. This property renders the transformations based on uvw1 far better than those based on the neighbouring u filter, i.e. they are more linear and essentially possible on a wider temperature range. As expected (Sect. 2.1), the uvw1 band can be used to correct the (U-u) index in the critical domain where the Balmer jump is important (see Sect. 3.1.2).

The colour transformations presented below being established on synthetic spectra, it is obvious that they should be refined through actual observations of suitable standard fields with the satellite and from the ground. The main interest in the transformations given here is that they inform us about the kind of relations that are possible, and their validity domain.

The main characteristics of the considered colour transformations are summarized in Table 2. Two of them are illustrated in Figs. 2 and 3. The analytical forms of the transformations are given below, sorted by categories. Though some formal error bars are smaller, we did not indicate uncertainties smaller than 0.001 mag in the relations given below.

Figure 2: The (V-v) vs. (b-v) diagram. Filled circles are MS stars, open ones are giants and crosses represent metal poor stars. The lowest temperature is 4000 K (bottom right) and the highest is 35000 K (upper left). The relation is monotonic and therefore the transformation is possible over the entire range

3.1 Solar composition main sequence stars

3.1.1 (U-u) vs. (u-b)

4000 K  $\leq~T_{\rm eff}~\leq~5500$ K (K7V - G8V)

(u-b)  $\in~[0.20;1.50]$
$U-u = 0.049 + 0.002 (u-b) - 0.097 (u-b)^2 \pm 10^{-3}$

5500 K  $\leq~T_{\rm eff}~\leq~10500$ K (G8V - B9V)

(u-b)  $\in~[-0.15;0.20].$
No possible transformation.

10500 K  $\leq~T_{\rm eff}~\leq~35000$ K (B9V - 08V)

(u-b)  $\in~[-1.40;-0.15]$
$U-u = -0.016 - 0.268 (u-b) - 0.043 (u-b)^2 \pm 10^{-3}.$

3.1.2 (U-u) vs. (uvw1-u)

3500 K  $\leq~T_{\rm eff}~\leq~7250$ K (M3V - F0V)

(uvw1-u)  $\in~[0.35;1.75]$
$U-u = -0.009 + 0.212 (uvw1-u) - 0.183 (uvw1-u)^2 \pm 4~10^{-3}$

7500 K  $\leq~T_{\rm eff}~\leq~10500$ K (A8V - B9V)

(uvw1-u)  $\in~[-0.10;0.30]$
$U-u = 0.001 - 0.158 (uvw1-u) + 0.860 (uvw1-u)^2\ \pm\break 2~10^{-3}$

10500 K  $\leq~T_{\rm eff}~\leq~35000$ K (B9V - O8V)

(uvw1-u)  $\in~[-0.75;-0.10]$
$U-u = -0.019 - 0.485 (uvw1-u) - 0.128 (uvw1-u)^2 \pm 2~10^{-3}.$

3.1.3 (B-b) vs. (b-v)

4000 K  $\leq~T_{\rm eff}~\leq~4750$ K (K7V - K3V)

(b-v)  $\in~[1.00;1.30]$
$B-b = -0.074 - 0.010 (b-v) \pm 10^{-3}$

4750 K  $\leq~T_{\rm eff}~\leq~10500$ K (K3V - B9V)

(b-v)  $\in~[-0.05;1.00]$
$B-b = -0.085 (b-v) \pm 10^{-3}$

10500 K  $\leq~T_{\rm eff}~\leq~35000$ K (B9V - O8V)

(b-v)  $\in~[-0.30;-0.05]$
$B-b = -0.073 (b-v) -0.065 (b-v)^2 \pm 10^{-3}.$

A third order fit can be applied to the whole interval:

4750 K  $\leq~T_{\rm eff}~\leq~35000$ K (K3V - O8V)

(b-v)  $\in~[-0.30;1.00]$
$B-b = -0.070 (b-v) - 0.040 (b-v)^2 + 0.026 (b-v)^3\ \pm\break 10^{-3}.$

Figure 3: a) (B-b) vs. (u-b) diagram. Symbols have the same meaning as in Fig. 2. Stars have effective temperatures between 3500 K and 35000 K. This is a good illustration of a case where no transformation is possible in the middle part of the temperature domain. b) Same diagram for FRV MS and giant stars (open circles). Kurucz MS stars (filled circles) are given for comparison. Stars below 4000 K have been excluded from this plot

3.1.4 (B-b) vs. (u-b)

4250 K  $\leq~T_{\rm eff}~\leq~5500$ K (K6V - G8V)

(u-b)  $\in~[0.20;1.30]$
$B-b = -0.056 - 0.060 (u-b) + 0.028 (u-b)^2 \pm 10^{-3}$

5500 K  $\leq~T_{\rm eff}~\leq~10500$ K (G8V - B9V)

(u-b)  $\in~[-0.15;0.20].$
No possible transformation.

10500 K  $\leq~T_{\rm eff}~\leq~35000$ K (B9V - O8V)

(u-b)  $\in~[-1.40;-0.15]$
$B-b = 0.003 - 0.009 (u-b) \pm 10^{-3}.$

3.1.5 (V-v) vs. (b-v)

4000 K  $\leq~T_{\rm eff}~\leq~5250$ K (K7V - K0V)

(b-v)  $\in~[0.85;1.30]$
$V-v = 0.061 - 0.136 (b-v) \pm 10^{-3}$

5250 K  $\leq~T_{\rm eff}~\leq~8750$ K (K0V - A3V)

(b-v)  $\in~[0.07;0.85]$
$V-v = 0.0025 - 0.068 (b-v) \pm 10^{-3}$

8750 K  $\leq~T_{\rm eff}~\leq~30000$ K (A3V - B0V)

(b-v)  $\in~[-0.30;0.07]$
$V-v = -0.001 - 0.033 (b-v) \pm 10^{-3}.$

A second order fit can be applied to the whole interval:

4000 K  $\leq~T_{\rm eff}~\leq~30000$ K (K7V - B0V)

(b-v)  $\in~[-0.30;1.30]$
$V-v = -0.001 - 0.041 (b-v) - 0.035 (b-v)^2 \pm 2~10^{-3}.$

3.1.6 (U-B) vs. (u-b)

4250 K  $\leq~T_{\rm eff}~\leq~5500$ K (K6V - G8V)

(u-b)  $\in~[0.20;1.35]$
$U-B = 0.120 + 1.087 (u-b) - 0.135 (u-b)^2 \pm 2~10^{-3}$

5500 K  $\leq~T_{\rm eff}~\leq~10500$ K (G8V - B9V)

(u-b)  $\in~[-0.15;0.20].$
No possible transformation.

10500 K  $\leq~T-{\rm eff}~\leq~35000$ K (B9V - O8V)

(u-b)  $\in~[-1.4;-0.15]$
$U-B = 0.003 + 0.819 (u-b) \pm 5~10^{-3}.$

3.1.7 (U-B) vs. (uvw1-b)

4000 K  $\leq~T_{\rm eff}~\leq~6500$ K (K7V - F5V)

(uvw1-b)  $\in~[0.20;3.20]$
$U-B = -0.292 + 0.710 (uvw1-b) - 0.051 (uvw1-b)^2 \pm 1.2\,10^{-2}$

6500 K  $\leq~T_{\rm eff}~\leq~8750$ K (F5V - A3V)

(uvw1-b)  $\in~[0.10;0.20].$
No possible transformation.

8750 K  $\leq~T_{\rm eff}~\leq~35000$ K (A3V - O8V)

(uvw1-b)  $\in~[-2.20;0.10]$
$U-B = 0.002 + 0.528 (uvw1-b) \pm 8~10^{-3}.$

3.1.8 (B-V) vs. (b-v)

3500 K  $\leq~T_{\rm eff}~\leq~35000$ K (M3V - O8V)

(b-v)  $\in~[-0.30;1.40]$
$B-V = 0.002 + 0.997 (b-v) \pm 9~10^{-3}.$

3.1.9 (U-V) vs. (u-v)

4000 K  $\leq~T_{\rm eff}~\leq~6500$ K (K7V - F5V)

(u-v)  $\in~[0.40;2.80]$
$U-V = 0.133 + 0.947 (u-v) \pm 2.3\, 10^{-2}$

6500 K  $\leq~T_{\rm eff}~\leq~8500$ K (F5V - A4V)

(u-v)  $\in~[0.20;0.40].$
No possible transformation, though the deviation from the general shape of the sequence is very slight (see below for further comments).

8500 K  $\leq~T_{\rm eff}~\leq~35000$ K (A4V - O8V)

(u-v)  $\in~[-1.70;0.20]$
$U-V = 0.008 + 0.843 (u-v) \pm 8~10^{-3}.$

The general shape of our synthetic main sequence is pretty close to a straight line in the (U-V) vs. (u-v) diagram, so that, if precision is not critical (at most 0.1 mag), one can also use the following relation:

4000 K  $\leq~T_{\rm eff}~\leq~35000$ K (K7V - O8V)

(u-v)  $\in~[-1.70;2.80]$
$U-V = 0.100 + 0.930 (u-v) \pm 5.7\,10^{-2}.$

3.1.10 (U-V) vs. (uvw1-v)

4000 K  $\leq~T_{\rm eff}~\leq~7000$ K (K7V - F1V)

(uvw1-v)  $\in~[0.65;4.50]$
$U-V = -0.201 + 0.786 (uvw1-v) - 0.029 (uvw1-v)^2 \pm 1.9\,10^{-2}$

7000 K  $\leq~T_{\rm eff}~\leq~7750$ K (F1V - A7V)

(uvw1-v)  $\in~[0.50;0.65].$
No possible transformation. The problem is however very slight and seems to be due to the Kurucz synthetic spectra rather than to the filters. The best here is to use the general relation proposed below for the whole temperature range.

7750 K  $\leq~T_{\rm eff}~\leq~35000$ K (A7V - O8V)

(uvw1-v)  $\in~[-2.45;0.50]$
$U-V = 0.004 + 0.577 (uvw1-v) \pm 9~10^{-3}.$

The same kind of considerations as for the (U-V) vs. (u-v) case holds concerning a general relation for the whole set of temperatures:

4000 K  $\leq~T_{\rm eff}~\leq~35000$ K (K7V - O8V)

(uvw1-v)  $\in~[-2.45;4.50]$
$U-V = 0.023 + 0.596 (uvw1-v) \pm 4~10^{-2}.$

3.2 Giants

Whenever a colour transformation is possible for MS stars, it is generally also satisfactorily obeyed by giant stars. The difference essentially lies in the validity range of the colour transformations. Each time a forbidden zone appears in the temperature domain (i.e. each time there is a zero in Table 2), the transformation relative to the hottest stars remains valid for giant stars down to $\sim 250$ K cooler than the lower temperature bound defined for MS stars. Symmetrically, the transformation relative to the coolest giant stars is only valid from stars $\sim 250$ K cooler than the upper bound of the temperature interval defined for the corresponding MS star colour transformation (see Fig. 3a).

When colour transformations are possible for MS stars on the whole temperature domain, they are generally valid for giant stars as well (Fig. 2).

3.3 Metal poor stars

In this case, the situation is worse. Stars still obey the same relations as MS stars when colour transformations exist on the whole temperature domain. Even in other cases, the hottest stars still obey the same relations, but the range on which these transformations remain valid is now considerably diminished: the "hot'' relations now hold only for $T_{\rm eff}\geq 11500$ K and the "cool'' ones for 4750 K  $\geq T_{\rm eff}\geq 4000$ K. Cool stars sometimes require different relations. This is true for the (B-b) vs. (u-b) as well as for both (U-u) colour transformations (Fig. 3a).

3.4 Comparison with observed spectra

As a validity check, all colour transformations presented above were also derived on the basis of the mainly-observed FRV spectra. When excluding stars cooler than 4000 K, the comparison between transformations established through Kurucz models and through FRV MS and giant star spectra is excellent, as illustrated in Fig. 3b. The latter ones are of course slightly more dispersed, but the dispersion is fully comparable to what could be expected on the basis of the transformations established in Sect. 3.2. Very small differences occur in some particular colour indices and are quite marginal: the (B-b) colour index discriminates between the FRV MS and giant stars, but this only happens below 5000 K; (V-v) is decreased by $\sim$0.01 mag when calculated on the FRV spectra rather than on the Kurucz spectra; the Kurucz and FRV (U-B) and (U-V) colour indices are slightly discrepant below 5000 K, as are the (U-u) and the (uvw1-u) colour indices below 7000 K.

As a conclusion, before the full inflight calibration on standard fields is performed and reduced, we recommend to use the relations given above to analyse the first data provided by the XMM-OM.

Figure 4: The (u-uvw1) vs. (b-v) colour diagram. Filled circles are MS stars, open ones are giants. Each "temperature track'' (almost vertical) runs from 9000 K (bottom) to 35000 K (top). The reddening runs from AV=0. (leftmost track) to AV=6.(rightmost track) by steps of 0.5

Figure 5: a) The $ic_{\rm red}$ vs. ic0 colour diagram. Filled circles are MS stars, open ones are giants. Each temperature track runs from 9000 K (left) to 35000 K (right). The reddening runs from AV=0. (bottom track) to AV=4.0 (upper track). b) Dependency of the $A_V=A_V(ic_0,ic_{\rm red})$ function on the reddening law. This diagram shows a section at ic0=0.6 in Fig. 5a (RV=3.1, full line) and in each of the equivalent diagrams established for RV=2.6 (dashed line), RV=5.5 (dash-dotted line) and for the SMC reddening law (dotted line)