2 Principles of microthermal measurement of seeing

The pairs of microthermal sensors used in this experiment measure the temperature structure function associated with the turbulence:

 

$$D_{\rm T}(\mbox{\boldmath$\rho$}, h)=\langle\left(T({\bf r}, h) - T({\bf r}+\mbox{\boldmath$\rho$}, h)\right)^{2}\rangle$$ (1)

where $\rho$ is the separation of the sensors and h is the altitude. Assuming fully developed turbulence according to the theory of Kolmogorov (Tatarski 1961), with $\vert\mbox{\boldmath$\rho$}\vert$between the inner and outer scales of the turbulent motion (the "inertial sub- range''), we can obtain the corresponding refractive index structure parameter using (Roddier 1981):

 

$$C_{\rm N}^{2}(h)=\left(80.10^{-6} \frac{P(h)}{T(h)^{2}}\right)^{2} \rho^{-2/3} D_{\rm T}(\mbox{\boldmath$\rho$}, h)$$ (2)

where P(h) is the pressure and T(h) the temperature.

The seeing quality is commonly described in terms of the "Fried parameter'' (Fried 1966):

 

$$r_{0}=\left(0.423k^{2}\sec\gamma\int_{h_{0}}^{\infty}C_{\rm N}^{2}(h){\rm d}h\right)^{-3/5}$$ (3)

where k is the wavenumber, $\gamma$ the zenith angle and h₀ the height of the telescope.

The Fried parameter is interpreted as the spatial coherence scale of the atmosphere. Atmospheric turbulence reduces the image resolution from O($\lambda/D$), where D is the telescope diameter, to O($\lambda/r_{0}$). The exact expression for the full width at half maximum of the "seeing disk'' is (Roddier 1981; Dierickx 1992):

 

$$\varepsilon_{\rm fwhm}=0.98\frac{\lambda}{r_{0}}\cdot$$ (4)