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C.2 Computation of Colour Correction Factors

C.2.1 Derivation of flux density values from in-band power values

In order to derive flux density values (monochromatic fluxes) from measured in-band powers the following corrections and calibrations have to be done:

correction for effective telescope area
correction for reflection losses on all optical elements in the beam from the primary mirror to the detector
correction for the beam profile related to the selected aperture or array
correction for the bandpass transmission, determined by the filter and field lens transmission and the relative detector response. The product of these factors is called the relative system response.

The latter correction contains the determination of the relative contribution of the flux density at the central wavelength of the band to the total flux in the band, depending on the assumed spectral shape of the source. Doing this calculation for a variety of source spectra and computing the ratios provides the colour correction factors between the various SEDs.

The flux conversion can be written in the following form:

IBP = T_{\rm refl} \cdot A \cdot \int f_{\rm beam}(\lambda) \cdot F(\lambda)
\cdot R(\lambda)\,d\lambda
\end{displaymath} (C.2)


  $IBP$ = in-band power (unit: W)

$T_{\rm refl}$ = reflection losses on optical elements
$A$ = primary mirror area
$f_{\rm beam}(\lambda)$ = wavelength dependent beam profile function
$F(\lambda)$ = flux density of source at specific wavelength $\lambda$
$R(\lambda)$ = system response at specific wavelength $\lambda$

Currently the in-band power calculation contains some simplification, in particular related to the beam profiles.

For point source fluxes the following calculation is done:

IBP = T_{\rm refl} \cdot A \cdot f_{\rm psf}(\lambda_{\rm ref}) \cdot
\int F(\lambda) \cdot R(\lambda)\,d\lambda
\end{displaymath} (C.3)


  $f_{\rm psf}(\lambda_{\rm ref})$ = fraction of the point spread function at  the reference wavelength $\lambda_{\rm ref}$

   inside the aperture or pixel $\rightarrow$ Cal-G file PPPSF

For extended source fluxes the corresponding calculation is done:

IBP = T_{\rm refl} \cdot A \cdot (1-\varepsilon^{2}) \cdot \Omega \cdot
\int F(\lambda) \cdot R(\lambda)\,d\lambda
\end{displaymath} (C.4)


  $\varepsilon$ = central obscuration factor by the secondary mirror

$\Omega$ = solid angle of selected aperture or pixel

In order to derive the flux density for the central wavelength, or more general a reference wavelength, of the bandpass the spectral energy distribution function is normalised with respect to the flux density at this wavelength:

F(\lambda) = F(\lambda_{\rm ref}) \cdot \frac{SED(\lambda)}{SED(\lambda_{\rm ref})}
\end{displaymath} (C.5)


  $F(\lambda)$ = flux density at any wavelength $\lambda$

$F(\lambda_{\rm ref})$ = flux density at reference wavelength $\lambda_{\rm ref}$
$SED(\lambda)$ = functional dependence of spectral energy distribution of source with wavelength

Furthermore, for conversion between frequency range and wavelength range the following convention is used:

\lambda \cdot F(\lambda) = \nu \cdot F(\nu)
\end{displaymath} (C.6)


  $F(\lambda)$ = flux density in wavelength range (unit:   Wm$^{-2}$$\mu $m$^{-1}$)

$F(\nu)$ = flux density in frequency range (unit: Wm$^{-2}$Hz$^{-1}$ or Jy)

Solving this equation for $F(\nu)$ yields:

F(\nu) = \frac{\lambda^{2}}{c_{\rm light}} \cdot F(\lambda)
\end{displaymath} (C.7)


  $c_{\rm light}$ = speed of light  

Using the normalisation with respect to the flux density value at the reference wavelength and the convention for conversion of $F(\lambda)$ into $F(\nu)$ the above formula of the in-band power can be solved for $F(\nu)$ for a general source energy distribution $SED(\lambda)$:

F(\nu_{\rm ref}) = \frac{1}{b \cdot c_{\rm light}} \cdot \l...
\frac{IBP}{\int SED(\lambda) \cdot R(\lambda)\,d\lambda}
\end{displaymath} (C.8)


  $b$ = $A \cdot T_{\rm refl} \cdot f_{\rm psf}(\lambda_{\rm ref})$

C.2.2 Flux density conversion factors for different spectral energy distributions

In the formula for the flux density $F(\nu)$ the term, which is dependent on the shape of the spectral energy distribution of the source, is called the flux density conversion factor $a(SED)$

a(SED) = \frac{\lambda_{\rm ref}^{2} \cdot SED(\lambda_{\rm...
...rm light} \cdot \int SED(\lambda) \cdot R(\lambda)\,d\lambda}
\end{displaymath} (C.9)

In the following the conversion factors for specific SEDs are listed:

flux density $\propto \nu^{\alpha}, SED(\nu) = \nu^{\alpha}$ $\rightarrow SED(\lambda) = \frac{c_{\rm light}}{\lambda^{2}} \cdot
...lpha}}{\lambda^{\alpha}} =
\frac{c_{\rm light}^{1+\alpha}}{\lambda^{2+\alpha}}$
a(\nu^{\alpha}) = \frac{1}{c_{\rm light} \cdot
...\cdot \int
\end{displaymath} (C.10)

flux density is a modified black body function with emissivity $\propto \nu^{\beta},
SED(\nu) = \nu^{\beta} \cdot B(\nu,T)$
$\rightarrow SED(\lambda) =
\frac{c_{\rm light}^{\beta}}{\lambda^{\beta}} \cdot
B(\lambda,T)$ ($\beta = 0$: pure black body)
a(\nu^{\beta} \cdot B(\nu,T)) = \frac{B(\lambda_{\rm ref},T...{B(\lambda,T)}{\lambda^{\beta}} \cdot R(\lambda)\,d\lambda}
\end{displaymath} (C.11)

Note that:

B(\lambda,T) = \frac{2 \cdot h \cdot c_{\rm light}^{2}}{\la...
...^{\frac{h \cdot c_{\rm light}}
{k \cdot \lambda \cdot T}}-1}
\end{displaymath} (C.12)


  $h$ = Planck constant

$k$ = Boltzmann constant
B($\lambda$,T) contains the $\frac{c_{\rm light}} {\lambda^{2}}$ correction factor from the frequency to wavelength conversion !

C.2.3 Derivation of colour correction factors from flux density conversion factors

From the general formula of the colour correction in Appendix C.1 it follows that

K_{\rm cc}(SED) = F_{\rm\nu_{\rm ref}}(\nu^{-1})~/~F_{\rm\nu_{\rm ref}}
\end{displaymath} (C.13)

and with the formulae from Appendix C.2.1 for $F(\nu_{\rm ref})$

K_{\rm cc}(SED) = a(\nu^{-1})~/~a(SED)
\end{displaymath} (C.14)

the colour correction factor is the ratio of the flux density conversion factors.

Inserting the formulae from Appendix C.2.2 for both flux density conversion factors this yields

K_{\rm cc}(SED) = \frac{1}{\lambda_{\rm ref} \cdot SED(\lam...
{\int \frac{R(\lambda)}{\lambda}\,d\lambda}
\end{displaymath} (C.15)

In order to calculate the colour correction factors this needs a numerical solution of the two integrals using the tabulated relative system response $R(\lambda_{\rm i})$.

next up previous contents index
Next: C.3 Colour Correction Values Up: C. Colour Corrections Previous: C.1 Application of Tabulated
ISO Handbook Volume IV (PHT), Version 2.0.1, SAI/1999-069/Dc