Theoretical calculation of radiant power emitted by atmospheric water vapour in the 20$\mu$m band above Mauna Kea

• Atmospheric models show that the 20$\mu$m flux from H₂O (assuming 1mm pwv) is $\sim$1 W m^-2 sr ^-1. Fig 3 shows a theoretical curve of growth (radiance vs mm pwv) for the atmosphere above Mauna Kea, fitted with a 13 order Chebyshev.
• Telescope diameter $\phi$ = 125 mm. Area=1.23 x 10^-2 m²
• Field of view: defined by 10 m patch of sky at an altitude of 1 km, $\theta$ = 10^-2 rad, $\Omega$ = 7.85 x 10^-5 sr
• Total efficiency, $\eta \sim$10% (optics, filters, chopping)
• Power on detector = 1 x 1.23 x 10^-2 x 7.85 x 10^-5 x 0.1 = 9.7 x 10^-8 W
• Detector sensitivity, D^*, = 2 x 10⁹ cm $\sqrt{\rm {Hz}}$/W
• NEP = $\frac{\sqrt{\rm {Area}}}{\rm {D}^*}$ = 5 x 10^-11 W/ $\sqrt{\rm {Hz}}$
• S/N = $\frac{9.7 \rm {x} 10^{-8}}{5 \rm {x} 10^{-11}} \approx$ 2000
• Atmospheric models of flux vs pwv, well described by Chebyshev polynomials, show that this S/N translates to a $\Delta$pwv of $\sim$ 1$\mu$m (1$\sigma$) detection in a 1 second integration.