2 The parameters of atmospheric turbulence

The effects of atmospheric turbulence on astronomical telescopes are commonly characterized as the following atmospheric parameters: the turbulence coherence length, namely Fried's parameter r₀, the Greenwood frequency $f_{\rm G}$, the Tyler frequency $f_{\rm T}$, and the anisoplanatic angle $\theta_0$. These parameters are usually specified at the visible wavelength of 0.55 $\mu$m (Roddier 1981):

$$r_0(\lambda_0) \left[0.423k_0^2~sec(\xi)\mu_0\right]^{-3/5},$$ (1)

$$\theta_0(\lambda_0) \left[2.91k_0^2~sec^{8/3}(\xi)\mu_{5/3}\right]^{-3/5},$$ (2)

$$f_{\rm G}(\lambda_0) 2.31\lambda_0^{-6/5}~sec^{3/5}(\xi)\nu_{5/3}^{3/5},$$ (3)

$$f_{\rm T}(\lambda_0) 0.368D^{-1/6}\lambda_0^{-1}~sec^{1/2}(\xi)\nu_2^{1/2},$$ (4)

where $k_0=2\pi/\lambda_0$ $(\lambda_0=0.55~\mu$m, $\mu_n=\int{\rm d}hC_n^2(h)h^n$, $\nu_n=\int{\rm d}hC_n^2(h)\nu^n(h)$. $\xi$ is the zenith angle. C_n²(h) and $\nu(h)$ are the index-of-refractive model and wind speed profile respectively. Note that formula (4) is only effective for G-tilt, namely for an average gradient of the pupil.