Skydips and determination of the forward efficiency

In a skydip, the atmospheric emission, as seen by the receiver ( \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}), is measured at equally spaced airmass ( \ensuremath{a}). \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} is a combination of the atmospheric emission \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{tot}{}}{}{^\ensuremath{\mathrm{tot}}}} and of losses \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}

  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{...
...\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}.
$ (7)
And the atmospheric emission is the sum of the atmospheric emission in both receiver bands
  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{tot}{}}...
...}}{}{^\ensuremath{\mathrm{ima}}}}}{1+\ensuremath{G_\ensuremath{\mathrm{im}}}},
$ (8)
each contribution computed from an equivalent atmospheric temperature \ensuremath{T_\ensuremath{\mathrm{atm}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} and opacity $\tau$
  $\displaystyle \ensuremath{T_\ensuremath{\mathrm{emi}}\ifthenelse{\equal{sig}{}}...
...\ifthenelse{\equal{ima}{}}{}{_\ensuremath{\mathrm{ima}}}} \right) } \right\}}.
$ (9)
The zenith opacity can be written as a combination of a dry and wet components
  $\displaystyle \tau = \ensuremath{\tau\ifthenelse{\equal{dry}{}}{}{_\ensuremath{...
...}}+ \ensuremath{\tau\ifthenelse{\equal{wet}{}}{}{_\ensuremath{\mathrm{wet}}}},
$ (10)
where \ensuremath{\tau\ifthenelse{\equal{dry}{}}{}{_\ensuremath{\mathrm{dry}}}} is the opacity due to the permanent components of the atmosphere (mainly oxygen) and \ensuremath{\tau\ifthenelse{\equal{wet}{}}{}{_\ensuremath{\mathrm{wet}}}} is proportional to the varying amount of water vapor amount ( \ensuremath{\mathrm{w_{H_2O}}}) in the atmosphere: $\ensuremath{\tau\ifthenelse{\equal{wet}{}}{}{_\ensuremath{\mathrm{wet}}}}=
\ensuremath{\mathrm{\alpha_{H_2O}}}\,\ensuremath{\mathrm{w_{H_2O}}}$

Assuming that \ensuremath{F_\ensuremath{\mathrm{eff}}} and \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} are independent of the elevation and that \ensuremath{\tau\ifthenelse{\equal{dry}{}}{}{_\ensuremath{\mathrm{dry}}}} and \ensuremath{\mathrm{\alpha_{H_2O}}} are correctly modeled by an atmospheric model, we obtain that \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} is a function of \ensuremath{F_\ensuremath{\mathrm{eff}}}, \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}, \ensuremath{G_\ensuremath{\mathrm{im}}}, \ensuremath{\mathrm{w_{H_2O}}} and the airmass ( \ensuremath{a}). If \ensuremath{G_\ensuremath{\mathrm{im}}} and \ensuremath{T_\ensuremath{\mathrm{loss}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}} are assumed to be measured by other means, then \ensuremath{F_\ensuremath{\mathrm{eff}}} and \ensuremath{\mathrm{w_{H_2O}}} can be fitted through the measured couples ( \ensuremath{T_\ensuremath{\mathrm{sky}}\ifthenelse{\equal{}{}}{}{^\ensuremath{\mathrm{}}}}, \ensuremath{a}).