2 Principle of seeing measurements

The correlation between the astronomical seeing and the atmospheric turbulence has been investigated by various authors (Barletti et al. 1974, 1976; Ken Knight et al. 1977; Marks et al. 1996). They have measured the small scale temperature structure functions in order to evaluate the refractive index structure constants for atmospheric layers. These refractive index structures have widely been used for calculating the astronomical seeing quality. Their investigations reveal that the parameter which gives a measure of the optical turbulence intensity related to the refractive index inhomogeneities is the refractive index structure coefficient $C_{\rm N}^{2}$ (Coulman 1969; Vernin & Muñoz-Tuñón 1994), which is a measure of the average variability of the refractive index of light in the atmosphere (Erasmus & Thompson 1986). The parameter $C_{\rm N}^{2}$ is connected with the temperature structure coefficient $C_{\rm T}^{2}$ of the microthermal field variations, which produce fluctuations in the refractive index at optical wavelengths (Coulman 1969; Barletti et al. 1974). The relationship between refractive index fluctuations ($C_{\rm N}^{2}$) and thermal irregularities ($C_{\rm T}^{2}$) at height h is given by

$$C_{\rm N}^{2}(h) = \left( \frac{80 \ 10^{-6} \times P(h)}{T^{2}(h)} \right)^{2} C_{\rm T}^{2}(h)$$ (1)

where P(h) and T(h) are the pressure in millibar (mb) and the absolute temperature in Kelvin (K) respectively at height h in metre. Hence the knowledge of $C_{\rm T}^{2}(h)$ as a function of altitude is prerequisite for the estimation of the astronomical seeing quality of any place. Following Barletti (1974) and Marks et al. (1996), the process for evaluating $C_{\rm T}^{2}(h)$ involves the measurement of the temperature function $D_{\rm T}(r,h)$ at points P₁ and P₂ at same level h, but horizontally separated by a distance r given by

$$D_{\rm T}(r,h) = \ < \left[T(P_{1}) - T(P_{2})\right]^{2} \gt$$ (2)

where, T(P) is the temperature at point P and angle brackets denote the ensemble average. As defined by Obukhov (1949) this is related to $C_{\rm T}^{2}(h)$ by

$$D_{\rm T}(r,h) = C_{\rm T}^{2}(h) r^{2/3}$$ (3)

In our case, measuring the temperature differences at two points which are horizontally separated by one metre apart, the value of $D_{\rm T}(r,h)$ is numerically equal to the $C_{\rm T}^{2}(h)$ value in ${\rm K}^{2}\, {\rm m}^{-2/3}$ units. The relationship between r₀, the Fried's parameter and $C_{\rm N}^{2}$ as a function of height h through the atmosphere has been given by Fried (1966) as

$$r_{0} = \left( 16.7 \lambda^{-2} \int_{0}^{\infty}C_{\rm N}^{2}(h){\rm d}h\right)^{-3/5}$$ (4)

where $\lambda$ is the wavelength and $C_{\rm N}^{2}$ is the refractive index structure constant, which gives a measure of the optical turbulence intensity related to the refractive index inhomogeneities in the atmosphere at height h. Vernin & Muñoz-Tuñón (1992) have mentioned that each turbulent layer at its altitude contributes to the degradation of the image according to the intensity of the turbulence. Thus $C_{\rm N}^{2}$ represents the sum of the contributions from all turbulent layers in the atmosphere. From the theory of wave propagation in turbulent media given by Tatarski (1961) and its relevant application to the astronomical seeing quality (Roddier 1981; Coulman 1985) the relationship between seeing ($\epsilon_{\rm fwhm}$) and r₀ is given by Dierickx (1992) as

$$\epsilon_{\rm fwhm} = 0.98 \frac{\lambda}{r_{0}}$$ (5)

where r₀ represents the diameter of the telescope aperture for which diffraction limited image resolution is equal to the full width at half maximum (fwhm) of the seeing limited image. Thus, r₀ takes into account all the different turbulent layers encountered by the light beam before reaching the ground (Vernin & Muñoz-Tuñón 1992). Using expressions (4) and (5), it is possible to write seeing as a function of $C_{\rm N}^{2}(h)$ as

$$\epsilon_{\rm fwhm} = 5.25 \lambda^{-1/5}\left(\int_{0}^{\infty}C_{\rm N}^{2}(h) {\rm d}h\right)^{3/5}.$$ (6)

The refractive index structure constant in this case represents the sum of the contribution from all turbulent layers in the atmosphere (Marks et al. 1996). Vernin & Muñoz-Tuñón (1992) have mentioned that, in order to assess the quality of an astronomical site, it is not sufficient to measure only the optical turbulence integrated over the whole atmosphere but also to evaluate the relative contribution to the smearing of the image from each intervening slab. This information is valuable for deciding height of the telescope location above the ground level in order to obtain better angular resolution images. The turbulence contributions to seeing ($\epsilon_{\rm fwhm}$) originating from different layers is given by

$$\epsilon_{\rm fwhm}({\rm total}) = \left(\sum_{i} \epsilon_{i}^{5/3}\right)^{3/5}$$ (7)

where $\epsilon_{i}$ is the seeing contributed by $i^{\rm th}$ turbulent layer.