- C.2.1 Derivation of flux density values from in-band power values
- C.2.2 Flux density conversion factors for different spectral energy distributions
- C.2.3 Derivation of colour correction factors from flux density conversion factors

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:

- a)
- correction for effective telescope area
- b)
- correction for reflection losses on all optical elements in the beam from the primary mirror to the detector
- c)
- correction for the beam profile related to the selected aperture or array
- d)
- 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:

(C.2) |

where:

= in-band power (unit: W)

= reflection losses on optical elements

= primary mirror area

= wavelength dependent beam profile function

= flux density of source at specific wavelength

= system response at specific wavelength

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:

(C.3) |

where:

= fraction of the point spread function at the reference wavelength

inside the aperture or pixel Cal-G file PPPSF

For extended source fluxes the corresponding calculation is done:

(C.4) |

where:

= central obscuration factor by the secondary mirror

= 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:

(C.5) |

where:

= flux density at any wavelength

= flux density at reference wavelength

= functional dependence of spectral energy distribution of source with wavelength

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

(C.6) |

where:

= flux density in wavelength range (unit: Wmm)

= flux density in frequency range (unit: WmHz or Jy)

Solving this equation for yields:

(C.7) |

where:

= speed of light

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

(C.8) |

where:

=

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

In the formula for the flux density the term, which is dependent on the shape of the spectral energy distribution of the source, is called the flux density conversion factor

(C.9) |

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

- 1)
- flux density

(C.10) - 2)
- flux density is a modified black body function with
emissivity

(: pure black body)

(C.11)

Note that:

(C.12) |

where:

= Planck constant

= Boltzmann constant

B(,T) contains the 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

(C.13) |

and with the formulae from Appendix C.2.1 for

(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

(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 .