3 The database

The database is organized as a series of figures, one for each of the systems with known transmission curves (179 in all). Given their large number, these figures are available in electronic form only, with Fig. 1 given here as an example. Each figure plots in a compressed, normalized format the band transmission profiles (for a finer view and a tabular version see Paper I or the ADPS web site), and reports for each band the computed data following the scheme outlined in Fig. 4 and discussed in detail later in this section. Table 1 lists the photometric systems considered in this paper and gives the number of the corresponding figure. The photometric systems are grouped in Table 1 in four distinct categories (ultraviolet, optical, infrared, and mixed systems) because of their slightly different figure layout scheme and input spectra (see detailed description in Sects. 3.1 and 3.2 below). Within each category the chronological order is followed. The names of the photometric systems are the same as used in Paper I. For reader's convenience, the number of the corresponding figure in Paper I is given too.

Actual transmission profiles were presented in Paper I for slightly more than half of the censed systems (105 out of 201). This number is expanded here (179 out of 218) in several ways, and Table 1 summarizes the source for band transmission profiles for all the systems included in this paper. First, transmission profiles have been located in literature for some more systems. Second, transmission profiles have been reconstructed for other systems by combining filter, atmospheric and telescope transmissions with detector response. Details and tabular versions of these new transmission profiles are available via the web interface to ADPS. Third, bands that were reported by the authors as obtained with interference filters, but for which the transmission profiles have never been published, are assumed to have a symmetric Gaussian profile. Symmetric Gaussians are generally found to give a satisfactory representation of the actual profiles of interference filters. For example, the v, b, y bands of the uvbyH$\beta$ - Strömgren and Crawford - 1956 system (see Fig. 16) are realized with interference filters. If the band and reddening parameters obtained using their actual profiles are compared with those derived using a Gaussian approximation, a mean difference of $\sim$5.1 Å ($\sim$0.1%) is found for the various types of wavelengths and 3.9 Å ($\sim$2.0%) for the equivalent widths. Similar figures are obtained for interference filters of other photometric systems. It may be than assumed that approximating with symmetric Gaussians the unknown transmission profiles of bands realized with interference filters is an acceptable procedure, suitable to provide results accurate to better than 5% for band wavelengths and better than 10% for band widths. For those interference filters that are particularly wide, the accuracy could be lower.

The data contained in Fig. 3 (and those of the other ones available in electronic form only) are described in the subsections below, following the scheme numbered in Fig. 4. The band parameters (areas     and ) are discussed first, followed by the reddening parameters (areas       and ). In this paper $F(\lambda)$is the transmission profile of the band, $f(\lambda$) the same normalized to 1.0 at its maximum, $S(\lambda)$ the energy distribution of a source spectrum and ${\cal B}(T,\lambda)$ the Planck function for the temperature T. In the figures, quantities for ultraviolet and optical systems are expressed in Å, in $\mu$m for the infrared ones.

3.1 Data scheme for band parameters

3.1.1 Area

The information contained within this area pertains to the pure band transmission profile, without convolution with a source spectrum. Figure 5 provides a graphical representation for some of the quantities here considered.

  

Table 1: List of the 179 photometric systems considered in this paper (all those with known transmission curves among the 218 so far censed by the ADPS project). The names of the systems are the same as in Paper I. Columns 4-6 give respectively the number of the electronic figure devoted to the given system, the number of photometric bands considered in this paper and the type of source for their transmission profiles, while Col. 7 lists the number of the corresponding figure in Paper I ( bn: systems only briefly noted in Paper I, Sect. 3). The sources for the transmission profiles are coded as follow. Tabular: as tabulated by the author(s); graph: derived by us from plots of the transmission profiles published by the author(s); rectangl: rectangular profiles as prescribed by the author(s); reconstr: reconstructed for this paper by combining the detector sensitivity with filter and atmospheric transmission; Gaussian: a Gaussian profile is assumed for the narrow bands obtained with interference filters and for which no other information is available.

 

Table 1: continued.

 

Table 1: continued.

Figure 3: The figure relative to the Sloan DSS - Fukugita et al. - 1996 photometric system is presented as an example of Figs. 9-187 only available in electronic form at the CDS.

Figure 4: Scheme for the organization of numerical data in the figures devoted to the documentation of the photometric systems (see also Fig. 3). The areas in which data are organized are labelled in the same order as they are described in the text in Sects. 3.1.1 to 3.2.5.

Figure 5: Graphical representation of some of the wavelength and width quantities described in Sects. 3.1.1 and 3.1.2. The band is $K_{\rm s}$ from the DENIS - Epchtein et al. - 1994 system (cf. Fig. 172). The shadowed area is the band equivalent width $W_\circ$.

 is the wavelength halfway between the points where the band transmission profile reaches half of the maximum value. In some cases the bands have complicated shapes, with the band transmission profile crossing several times the 50%-line, like the N band of the NQ - Low and Rieke - 1974 system in Fig. 159. In all such cases the most external 50% points are taken in computing $\lambda_{\rm c}$.

 is the mean wavelength of the band:

$$\lambda_\circ = \frac{\int \lambda~F(\lambda)~{\rm d}\lambda}{\int F(\lambda)~{\rm d}\lambda}\cdot$$ (1)

 is the wavelength at which the band transmission profile reaches its maximum.

 is the central wavelength of the approximating Gaussian with an area equal to that of the actual band transmission profile. If both the band and the Gaussian are normalized to 1.0 at peak response, and ${\cal A}$ is the area of the band, the equal-area fitting Gaussian is then:

$$g(\lambda) = \exp{\left[-\pi \frac{(\lambda-\lambda_{\rm gauss})^2}{{\cal A}^2}\right]}\cdot$$ (2)

The value of $\lambda_{\rm gauss}$ has been computed using the Levenberg-Marquardt method (Press et al. 1988).

 is the the full wavelength span between the points where the band transmission profile reaches half of the maximum value. As for $\lambda_{\rm c}$, in the case of complicated band profiles the most external 50% points are taken when computing WHM.

 is the wavelength span between the points where the band transmission profile reaches 10% of the maximum value. In the case of complicated band profiles the most external 10% points are taken in computing W10%. It is worth noticing that for a Gaussian profile of area ${\cal A}$, dispersion $\sigma$ and 1.00 peak transmission it is:

$$W10\% 2{\cal A}\sqrt{(\ln 10)/\pi}\ =\ 1.712\ {\cal A}$$ = (3)

$$2\sigma\sqrt{2\ln 10}\ =\ 4.292\ \sigma.$$ =

 is the wavelength span between the points where the band transmission profile reaches 80% of the maximum value. Again, in the case of complicated band profiles, the most external 80% points are taken in computing W80%. It is worth noticing that for a Gaussian profile of area ${\cal A}$, dispersion $\sigma$ and 1.00 peak transmission it is:

$$W80\% 2{\cal A}\sqrt{(\ln 1.25)/\pi}\ =\ 0.533\ {\cal A}$$ = (4)

$$2\sigma\sqrt{2\ln 1.25}\ =\ 1.336\ \sigma.$$ =

 is the full width at half maximum of the approximating Gaussian with an area equal to that of the actual band transmission profile. If both the band and the Gaussian are normalized to 1.0 at peak response, and ${\cal A}$ is the area of the band, then:

$$2 {\cal A} \sqrt{(\ln 2)/\pi}\ =\ 0.939\ {\cal A}$$                         FWHM = (5)

$$2\sigma\sqrt{2\ln 2}\ =\ 2.355\ \sigma.$$ =

 is the equivalent width of the band transmission profile:

$$W_{\circ}\ =\ \int f(\lambda)~{\rm d}\lambda\ =\ \frac{1}{C} \int F(\lambda)~{\rm d}\lambda$$ (6)

where $C= [ F(\lambda)/f(\lambda)]$ is the normalization factor. $W_\circ$ is the area shadowed in Fig. 5.

 is the 2nd order momentum of the band transmission profile (for a discussion see Golay 1974, pp. 41-43), i.e. the square root of the expression:

$$\mu^2\ =\ \frac{\int (\lambda-\lambda_\circ)^2 ~F(\lambda)~{\rm d}\lambda} {\int F(\lambda)~{\rm d}\lambda}\cdot$$ (7)

For a Gaussian profile it is $\mu\ =\ \sigma$, while for a rectangular band of width W it is $\mu\ =\ \sqrt{3}/6\ W$.

 is the skewness index and it closely resembles the 3rd order moment (the only difference lies in the presence of $\mu^3$):

$$I_{\rm asym}\ =\ \frac{\int (\lambda-\lambda_\circ)^3 ~F(\lambda)~{\rm d}\lambda} {\mu^3 \int F(\lambda)~{\rm d}\lambda}$$ (8)

with $\mu$ being given by Eq. (7). An $I_{\rm asym}<0.0$ pertains to a band with an extended blue wing (as the band i of the ri - Argue - 1967 system in Fig. 45 that has $I_{\rm asym} = -1.32$), and $I_{\rm asym}>0.0$ to a band with an extended red wing (as the band R of the WBVR - Straizys - #Straiz<#494 system in Fig. 81 that has $I_{\rm asym} = +0.66$). Symmetric profiles are characterized by $I_{\rm asym}= 0.0$ (as the band 35 of the system DDO - McClure and Van den Bergh - 1968 in Fig. 51a for which it is $I_{\rm asym}= 0.06$).

 is the kurtosis index and it closely resembles the 4rd order moment (the only difference lies in the presence of $\mu^4$. For the normalization constant -3 see below):

$$I_{\rm kurt}\ =\ \frac{\int (\lambda-\lambda_0)^4 ~F(\lambda)~{\rm d}\lambda} {\mu^4 \int F(\lambda)~{\rm d}\lambda} - 3.$$ (9)

The kurtosis index gives an indication of the balance between the core and the wings of a profile. The -3 is inserted to have $I_{\rm kurt}~=~0.0$ for a Gaussian profile. A $I_{\rm kurt}>0.0$ indicates a band transmission profile more sharply peaked than a Gaussian, i.e. with more wings than core (as the band Pfor the KLMNPQR - Borgman - 1960 system in Fig. 21a that has $I_{\rm kurt} =+1.66$). A $I_{\rm kurt}<0.0$ pertains instead to a band with more core than wings (as the band T₂ for the photoelectric version of the Washington - Canterna - 1976 system in Fig. 94a that has $I_{\rm kurt} = -1.08$). The kurtosis index for a rectangular band is $I_{\rm kurt}=-6/5$, for an equilateral triangular band it is $I_{\rm kurt}=-3/5$, and for an ${\rm e}^{-\vert x\vert}$ profile it is $I_{\rm kurt}=3$.

Figure 6: Values of the 2nd order moment ($\mu$), the skewness and the kurtosis indices for some sample profiles.

Figure 6 provides a graphical representation of $\mu$, $I_{\rm asym}$ and  $I_{\rm kurt}$ for some reference profiles.

Several systems contain rectangular or Gaussian bands. In the corresponding figures only non-redundant data are provided. It is therefore appropriate to summarize here for these two types of profiles the values of the above described parameters. For a rectangular band with 1.00 peak transmission and width W it is:

$$\lambda_c \equiv \lambda_\circ \equiv \lambda_{\rm peak} \equiv \lambda_{\rm gauss}$$ (10)

$$WHM \equiv W10\% \equiv W80\% \equiv W \equiv W_\circ$$ (11)

$$FWHM = 2 W_\circ \sqrt{(\ln{2})/\pi}\ =\ 0.939\ W_\circ$$ (12)

$$\mu = \sqrt{3}/6\ W_\circ\ =\ 0.289\ W_\circ$$ (13)

$$I_{{\rm asym}} = 0$$ (14)

$$I_{{\rm kurt}} = - \frac{6}{5}$$ (15)

and for a Gaussian profiles of peak transmission 1.00 it is:

$$\lambda_c \equiv \lambda_\circ \equiv \lambda_{\rm peak} \equiv \lambda_{\rm gauss}$$ (16)

$$W_\circ = \frac{1}{2} \sqrt{\pi/\ln 2}\ FWHM\ =\ 1.064\ FWHM$$ (17)

$$W10\% = \sqrt{\frac{\ln~ 10 }{\ln~ 2 }} FWHM\ =\ 1.823\ FWHM$$ (18)

$$W80\% = \sqrt{\frac{\ln~ 1.25 }{\ln~ 2 }} FWHM\ =\ 0.567\ FWHM$$ (19)

FWHM = WHM (20)

$$\mu = \sigma \ =\ 0.425\ FWHM$$ (21)

$$I_{{\rm asym}} = 0$$ (22)

$$I_{\rm kurt} = 0.$$ (23)

3.1.2 Area

This area documents how effective wavelengths (upper rows) and widths (in square bracketts) change with the source spectra  $S(\lambda)$ described in Sect. 2. The effective wavelength is:

$$\lambda_{\rm eff}\ =\ \frac{\int \lambda~F(\lambda)~S(\lambda... ... d}\lambda} {\int F(\lambda)~S(\lambda)~{\rm d}\lambda}\cdot$$ (24)

The effective width $W_{\rm eff}$ is the width of a rectangular bandpass of height 1.0, centered at $\lambda_{\rm eff}$, that collects from a source $S(\lambda)$ the same amount of energy going through the filter $f(\lambda)$:

$$\int f(\lambda)~S(\lambda)~{\rm d}\lambda\ =\ \int_{\lambda... ...{\lambda_{\rm eff} + W_{\rm eff}/2} S(\lambda)~{\rm d}\lambda$$ (25)

$W_{\rm eff}$ is the amount by which to multiply the flux per unit wavelength (normally expressed, for example, is erg cm² s^-1 Å^-1 at $\lambda_{\rm eff}$) to obtain the flux through the whole band (normally expressed, for example, in erg cm² s^-1).

Different data schemes are adopted for optical, ultraviolet and infrared systems:

optical systems. All normal and peculiar spectra discussed in Sect. 2 are considered. For normal spectra, both the effective wavelength and width are given, while for peculiar spectra only the effective wavelength is listed;

ultraviolet systems. For UV systems the effective wavelength and width are computed for the Sun ( $T_{\rm eff}=5770$ K), Vega ( $T_{\rm eff}=9550$ K) and $\iota$ Her ( $T_{\rm eff}=17~100$ K), and for three Kurucz's models completing the sequence toward higher temperatures: $T_{\rm eff}= 20~000$, 30 000 and 40 000 with $[Z/Z_\odot]=0.0$ and $\lg g = 5.0$;

near-IR systems. For bands between 1 and 8 $\mu$m the effective wavelength and width are computed for the same set of normal stars used with optical photometric systems;

far-IR systems. For bands longward of 8 $\mu$m the effective wavelength and effective width are computed for blackbody energy distributions characterized by the temperatures $T_{\rm eff}=3000$, 1500, 800, 400, 200 and 100 K.

3.1.3 Area

The effective wavelength of a blackbody energy distribution is:

$$\lambda_{\rm eff}\ =\ \frac{\int \lambda~F(\lambda)~{\cal B}(... ...lambda} {\int F(\lambda)~{\cal B}(T,\lambda)~{\rm d}\lambda}$$ (26)

where ${\cal B}(T,\lambda)$ is the Planck function. In this area it is reported a third order polynomial fit to the behaviour of $\lambda_{\rm eff}$ with blackbody temperature:

$$\lambda_{\rm eff}(T)\ =\ a\ +\ b T_{\rm eff} +\ c T_{\rm eff}^2\ +\ d T_{\rm eff}^3 .$$

To limit the number of significant digits of the polynomial coefficients to three (for a compact writing of the expressions), the temperature must be parametrized. For ultraviolet and optical systems, the temperature is expressed as:

$$\theta\ =\ \frac{2500}{T {\rm (K)}}$$ (27)

and the polynomial fit

$$\lambda_{\rm eff}(T)\ =\ a_\theta\ +\ b_\theta \theta\ +\ c_\theta \theta^2\ +\ {\rm d}_\theta \theta^3$$ (28)

is computed over the interval $50~000\leq T \leq 2500$ K. Within this range of temperatures the polynomial fits provide $\lambda_{\rm eff}(T)$ accurate to better than 3 Å for essentially all systems. For near-IR systems, the temperature is expressed as :

$$\Phi\ =\ \frac{500}{T {\rm (K)}}$$ (29)

and the polynomial fit

$$\lambda_{\rm eff}(T)\ =\ a_\Phi\ +\ b_\Phi \Phi\ +\ c_\Phi \Phi^2\ +\ d_\Phi \Phi^3$$ (30)

extends over the range $10~000\leq T \leq500$ K. Within it, the fit provides $\lambda_{\rm eff}(T)$ accurate to better than 0.01 $\mu$m for most systems. Finally, for far-IR systems, the temperature is expressed as:

$$\Psi\ =\ \frac{100}{T {\rm (K)}}$$ (31)

and the polynomial fit

$$\lambda_{\rm eff}(T)\ =\ a_\Psi\ +\ b_\Psi \Psi\ +\ c_\Psi \Psi^2\ +\ d_\Psi \Psi^3$$ (32)

is computed over the interval $3000\leq T \leq100$ K, where it provides results accurate to 0.02 $\mu$m for nearly all systems.

The coefficients in the polynomial fits are given explicitely even when null: for example, writing 0.00 for $c_\Psi$ means that the actual coefficient of $\Psi^2$ is $\vert c_\Psi\vert < 0.005$ (0.005 being rounded to 0.01). The same applies to all other coefficients used in this paper (Sects. 3.1.4, 3.2.4 and 3.2.5).

3.1.4 Area

The effective width $W_{\rm eff}$, in the case of a blackbody energy distribution, is the width of a rectangular bandpass of height 1.0, centered at $\lambda_{\rm eff}$, that collects from a blackbody ${\cal B}(T,\lambda)$ the same amount of energy going through the filter $f(\lambda)$:

$$\int f(\lambda)~{\cal B}(T,\lambda)~{\rm d}\lambda\ =\ \int... ...\rm eff} + W_{\rm eff}/2} {\cal B}(T,\lambda)~{\rm d}\lambda.$$ (33)

Here it is provided a third order polynomial fit to the behaviour of $W_{\rm eff}$ with the blackbody temperature. The same parametrization of temperature ($\theta$, $\Phi$ and $\Psi$) and ranges of applicability adopted for $\lambda_{\rm eff}(T)$ are mantained for $W_{\rm eff}(T)$ too.

The third order polynomial fits to $W_{\rm eff}(T)$ tipically provide results accurate to 5 Å for UV and optical systems, 0.01 $\mu$m for near-IR systems and 0.02 $\mu$m for far-IR ones.

3.2 Data scheme for reddening parameters

The behaviour with different reddening laws and amount of extinction is investigated in the remaining areas numbered in Fig. 4 for all photometric bands at $\lambda \leq 8~\mu$m. The amount of extinction in a given band is :

$$A(\lambda)\ =\ -2.5 \lg \frac{ \int F(\lambda)~S(\lambda)~\ta... ...a)~{\rm d}\lambda}{ \int F(\lambda)~S(\lambda)~{\rm d}\lambda}$$ (34)

where $\tau(\lambda)$ is the transmission coefficient of the interstellar medium and $\eta$ in the relative mass of the medium. The latter is taken so that a unit amount of medium ($\eta=1$) causes a reddening E_B-V = 1.0 for O-type stars (Straizys #Straiz<#649, p. 10, 100 and 136). In such a case the corresponding hydrogen column density is $N_{\rm H}({\rm HI}+{\rm H}_2)=5.8 \times 10^{21}$atoms cm^-2 (Savage & Mathis 1979) for the average interstellar medium characterized by the standard R_V=3.1 law.

Three extinction laws are considered, labelled according to their R_V = A(V)/E_B-V ratio, as representative of the continuum of extinction laws encountered in Nature (from Fitzpatrick 1999). Their shapes are compared in Fig. 7. Their tabular version can be downloaded via anonymous ftp from astro2.astro.vill.edu (directory pub/fitz/Extinction/FMRCURVE.pro) or from the ADPS web site.

Figure 7: The three extinction laws considered (from Fitzpatrick 1999). The dot marks the V band from the UBV - Johnson and Morgan - 1953 system.

The quantities normally employed in characterizing the effects of reddening and extinction in a given photometric system are color excess ratios $E(\lambda_4 -\lambda_3)/E(\lambda^{\rm ref}_{2} -\lambda^{\rm ref}_{1})$, ratios of total to selective extinction $A(\lambda)/E(\lambda^{\rm ref}_{2}-\lambda^{\rm ref}_{1})$, and absolute extinction ratios $A(\lambda)/A(\lambda^{\rm ref})$. Historically, the adopted $\lambda^{\rm ref}_{i}$ are the B and V bands of the UBV - Johnson and Morgan - 1953 system. Such a choice may not be the optimal one because these bands are rather wide, they embrace a lot of fine structure features of the interstellar extinction and they are not on the R_V-independent part of the extinction curve at $\lambda \geq 8500$ Å. However, we will adopt B and V as $\lambda^{\rm ref}_{i}$ in this paper both for historical reasons and commonality with existing literature. In all computations we explicitely integrate over the whole and exact B and V profiles, and we do not limit ourselves to simply compute at the characteristic wavelength of the band.

Before to proceede, the B and V reference bands must be accurately defined. Throughout this paper we adopt for B and V the so called Vilnius reconstruction (Azusienis & Straizys 1969) of the original UBV - Johnson and Morgan - 1953 system (hereafter VILNIUS-REC-UBV; see Azusienis & Straizys 1969 for details on which Bband to use in combination with U and V bands). The effect of choosing one or another band profile for the reference B and V bands has non-negligible effects. Table 2 (built from data in Figs. 12, 61 and 131, for the R_V=3.1 extinction law and the B3 spectral type) shows the differences between the USA, Vilnius, photographic, Buser (1978), Bessell (1990) and Landolt (1983) versions of the same UBV - Johnson and Morgan - 1953 system. It is evident how the differences cannot be ignored when an accurate analysis is required.

  

Table 2: Values of the reddening parameters $A(\lambda )/A(V)$ and $A(\lambda )/E_{B-V}$ for different reconstructions of the V band of the UBV - Johnson and Morgan - 1953 photometric system. The reference B and V bands are those of the Vilnius reconstruction. The reconstruction by Buser (1978) adopts the same reconstruction as Vilnius for the V band, which explains the identical numbers for the two.

It has to be noted that the reddening expressions depend on the spectral type of the star and the amount of reddening because both change the band effective wavelengths. An example illustrates the effect of the stellar spectral type. For VILNIUS-REC-UBV in Fig. 12b, R_V=3.1 law and $\eta=1$ in Eq. (34), the effective wavelengths of the A(V)/E_B-Vratio for a B3 star are:

$$\frac{A(V)}{E_{B-V}}\ \longrightarrow\ \frac{A (5429\ {\rm\AA})}{E(4336{-} 5429\ {\rm\AA})}$$

while for a M2 star they become:

$$\frac{A(V)}{E_{B-V}}\ \longrightarrow\ \frac{A (5595\ {\rm\AA})}{E(4666{-} 5595\ {\rm\AA})}\cdot$$

Going from spectral type B3 to M2, the $\lambda_{\rm eff}$ of the Bband changes by 330 Å while for the V band the change is only 166 Å. The lever arm (i.e. the distance in $\lambda_{\rm eff}$ between B and V bands) reduces toward redder spectral types, therefore requiring a higher A(V) extinction to match the E_B-V = 1.0 condition. From data in Fig. 12b it is in fact:

$$\begin{eqnarray*}\frac{A(V)}{E_{B-V}} & = & 3.13\ {\rm for\ early\ type\ stars} ... ...\rm for\ Sun-like\ stars} \\ & = & 3.69\ {\rm for\ M\ stars}. \end{eqnarray*}$$

The effect of the reddening is also quite strong. Again from Fig. 12b for VILNIUS-REC-UBV, a $\bigtriangleup E_{B-V} = 1.0$ changes the effective wavelength of the B band by 133 Å, and that of the V band by 113 Å.

The net effect is that the reddening does not translate rigidly the main-sequence over the color-magnitude diagram: the shape of the main-sequence modifies according to the amount of reddening (with obvious implications for classical reddening estimates of clusters). How much the main-sequence shape modifies for different amounts of reddening is grafically represented in Fig. 8 for the UBVRI - Landolt - 1983 system (the curves are available in electronic form from the ADPS web site).

Figure 8: Modification of the main-sequence on the MV, $(B-V)_\circ$ plane of the UBVRI - Landolt - 1983 system for the RV=3.1 extinction law and different reddenings (EB-V=0,1,2) . The curves are shifted so that the solar positions for them coincide (dot). Nearly identical modifications apply for the RV=5.0 and RV=2.1 cases. The unreddened main-sequence is from Drilling & Landolt (2000, their Table 15.7).

3.2.1 Area

The $A(\lambda )/A(V)$ absolute extinction ratio

$$\frac{A(\lambda)}{A(V)}\left\vert \begin{array}{l}\\ R_V \end{array} = X.XX \begin{array}{c}Q_1 \\ Q_2 \end{array} \right.$$ (35)

is reported in this area for the pure band transmission profile, without covolution with a source spectrum and for $\eta=1$ (cf. Eq. (34)). The ratio is computed for the three different extinction laws labelled by their R_V values.

The covolution with a source spectrum modifies however the value of the ratio: the extremes reached over the sources considered in area are listed as values Q₁ and Q₂ in the equation scheme above. For optical systems only spectra of normal stars (cf. Sect. 2) are considered.

3.2.2 Area

The shape of the extinction law $A(\lambda )/A(V)$ can be conveniently and accurately parametrized in term of R_V as :

$$\frac{A(\lambda)}{A(V)}\ =\ a(x)\ +\ \frac{b(x)}{R_V}$$ (36)

where $x=1/\lambda$ (in $\mu$m^-1). The analytical expressions of a(x) and b(x) coefficients are given by Cardelli et al. (1989). They offer a powerful way to derive the extinction at any wavelength for any extinction law parametrized by R_V.

The a(x) and b(x) coefficients can be profitably used to parametrize in terms of R_V the reddening relations normally used :

$$\frac{E(\lambda_2-\lambda_1)}{A(V)} = (a_2 - a_1) + \frac{(b_2-b_1)}{R_V}$$ (37)

$$\frac{A(\lambda)}{E_{B-V}} = \frac{A(\lambda)}{A(V)} R_V\ =\ a R_V + b$$ (38)

$$\frac{E(\lambda-V)}{E_{B-V}} = \frac{A(\lambda)}{E_{B-V}} - R_V\ =\ (a-1) R_V + b$$ (39)

$$\frac{E(\lambda_2-\lambda_1)}{E_{B-V}} = (a_2 - a_1) R_V + (b_2 - b_1).$$ (40)

These relations show that expressing the extinction in terms of $A(\lambda )/A(V)$ or $E(\lambda-V)/E_{B-V}$ is equivalent, provided that R_V is known. If the extinction law is unknown, $A(\lambda )/A(V)$ and $E(\lambda-V)/E_{B-V}$ cease to be equivalent ways to describe the extinction.

The values of a(x) and b(x) depend on the source spectrum and the amount of reddening because both change the $\lambda_{\rm eff}$ at which the coefficients are computed. For example, for the R band of the UBVRI - Landolt - 1983 system, it is $\bigtriangleup \lambda_{\rm eff} =124$ Å when moving from B3 to Sun spectral type, and $\bigtriangleup \lambda_{\rm eff} =264$ Å when changing from E_B-V = 0.0 to E_B-V = 1.0. Such variations are too large to be ignored, and therefore we have computed the a(x) and b(x) coefficients for the B3 and Sun spectral types (representative of the hotter and cooler regions of the HR diagram, respectively) for both E_B-V = 0.0 and E_B-V = 1.0 conditions. They are given in area  in the following format:

$$\begin{tabular}{r\vert rl\vert} \multicolumn{3}{c}{}\\ \cli... ...sl Sun}\\ \cline{2-3} \multicolumn{3}{c}{}\\ \end{tabular}$$

The a(x) and b(x) coefficients are useful also in deriving Q or reddening-free parameters for any extinction law characterized by R_V. A reddening-free parameter is a combination of colors that is independent from interstellar reddening. Given four photometric bands A, B, C and D, the reddening-free parameter and its expression in terms of a(x) and b(x) coefficients are:

$$(A\ -\ B)\ -\ \frac{E_{A-B}}{E_{C-D}}(C\ -\ D)$$                                  Q_ABCD = (41)

$$(A\ -\ B)\ -\ \frac{(a_A - a_B)R_V + (b_A - b_B)}{(a_C - a_D)R_V + (b_C - b_D)}(C\ -\ D) .$$ =

Because a(x) and b(x) depend upon the effective wavelength, the Q_ABCD so derived will be most accurate only in the neighborhood of a given spectral type and for a given amount of reddening.

The best known reddening-free parameter is perhaps that for the UBV - Johnson and Morgan - 1953 system, given by Hiltner & Johnson (1956) as $Q_{UBV}=(U-B) - \Upsilon (B-V)$ where $\Upsilon = 0.72 + 0.05E_{B-V}$ for O-type stars suffering from a standard R_V=3.1 extinction law. Using the values of a(x) and b(x) from Fig. 12b ( VILNIUS-REC-UBV) for the B3 spectral type, Eq. (49) provides $\Upsilon = 0.71$ for E_B-V = 0.0 and $\Upsilon = 0.78$ for E_B-V = 1.0, in excellent agreement with Hiltner and Johson's values. The corresponding figures for a Sun-like source spectrum would be instead $\Upsilon = 0.88$ for E_B-V = 0.0 and $\Upsilon = 1.06$for E_B-V = 1.0.

3.2.3 Area

In this area it is given the second order polynomial fit to extinction versus reddening for the R_V=3.1 extinction law and three different spectral types according to the wavelength region: B3, Sun and M2 for optical and infrared systems, and $\iota$ Her (B3), Sun and a Kurucz's 40 000 K, $[Z/Z_\odot]=0.0$, $\lg g = 5.0$ spectrum for ultraviolet systems. The coefficients are given in the form:

$$\frac{A(\lambda)}{E_{B-V}}\ =\ \alpha\ +\ \beta~E_{B-V}\ =\ (\alpha,\ \beta) % ^r% _{\rm spectrum}.%$$ (42)

The expression provides an exact solution, in the sense that the absorption in the three bands B, V and $\lambda$ is computed for each incremental value of x in Eq. (34). The regression coefficient rgives as an indication of the accuracy of the fit (generally pretty high with |r| close to 1.0).

The $\alpha$ and $\beta$ coefficients can be used to derive the reddening-free parameter Q_ABCD in a direct way (for the R_V=3.1 law, while for others it is necessary to use Eq. (49) above):

$$(A\ -\ B)\ -\ \frac{E_{A-B}}{E_{C-D}}(C\ -\ D)$$                                   Q_ABCD = (43)

$$(A\ -\ B)\ -\ \frac{\alpha_A - \alpha_B + (\beta_A - \beta_B)E_{B-V}}{\alpha_C - \alpha_D + (\beta_C - \beta_D)E_{B-V}}(C\ -\ D).$$ =

From the $\alpha$ and $\beta$ coefficients in Fig. 12b in the VILNIUS-REC-UBV case and B3 spectral type, it is found $\Upsilon = 0.70 + 0.05E_{B-V}$ for $Q_{UBV}=(U-B) - \Upsilon (B-V)$, again in excellent agreement with Hiltner & Johnson (1956) value for O-type stars.

3.2.4 Area

In this area it is reported the first order polynomial fit to the behaviour of $\lambda_{\rm eff}$ with E_B-V for the pure band transmission profile and the R_V=3.1 extinction law:

$$\lambda_{\rm eff}\ =\ \delta\ +\ \gamma\ E_{B-V}.$$ (44)

The regression coefficient r is given as an indication of the accuracy of the fit.

3.2.5 Area

Similarly, a first order polynomial fit to the behaviour of $W_{\rm eff}$ with E_B-V for the pure band transmission profile and the R_V=3.1 extinction law is given:

$$W_{\rm eff}\ =\ \varepsilon\ +\ \zeta\ E_{B-V}.$$ (45)

Again, the regression coefficient r is given as an indication of the accuracy of the fit.