3 Calculation results with the Hufnagel and experimental C_n² profiles

A useful model for C_n² profile corresponding to the night-time observation conditions was suggested by Hufnagel (1974). This model is an analytical approximation of the experimental data, and it can be expressed as

$$\begin{eqnarray} C_{n}^{2}\left( z\right) =C_{0}^{2}\left[ \left( \frac{z}{z_{0}... ...}{z_{1}}\right\} +\exp \left\{ -\frac{z}{z_{2}} \right\} \right] ,\end{eqnarray}$$ (7)

where C₀² is the arbitrary scaling factor with dimension of length in the power -2/3, $\begin{array} {c} z_{0}=4.632\ 10^{3}{\ }m\end{array}$, $\begin{array} {c} z_{1}=10^{3}{\ }m\end{array}$, and $\begin{array} {c} z_{2}=1.5\ 10^{3}{\ }m.\end{array}$

In the case of Hufnagel profile, the residual structure function is expressed analytically in terms of generalized hypergeometric functions. The final expression for $D_{\rm S_{R}}$ can be obtained after substitution of Eq. (7) into Eq. (3) evaluating the integrals in the following sequence: over $\varphi ,$ $\theta ,$ z, $\kappa$. The intermediate integrals are evaluated applying the Mellin transformation technique, and the formulas needed for evaluations can be found in (Prudnikov et al. 1988).

The experimental C_n² profile used here is the median of 800 profiles obtained during three nights at the 2.1 m telescope of San Pedro Martir observatory, Baja California Mexico, in April 1997. The relevant profiles in an actual adaptive optics observation are the quasi-instantaneous ones, for the comparison of our results obtained with theoretical and experimental profiles, a median profile is better suited because it has a statistical value. A detailed description of this observing campaign will be presented in Avila et al. (1998). The measurements were accomplished with the generalized scidar (G-SCIDAR) which was suggested by Fuchs (1995) as a generalized version of the scidar technique originally proposed by Rocca et al. (1974). The data reduction consists in computing the spatial autocorrelation function of short exposure-time images of the scintillation pattern which is produced by a double star detected on a virtual plane a few kilometers beneath the pupil. A maximum entropy algorithm is used to retrieve the C_n² profile from the measured autocorrelation function. The experimental setup can be found in Avila et al. (1997), where the first implementation of the G-SCIDAR at a telescope focus is described.

Both the Hufnagel's and experimental profiles are plotted in Fig. 1. The scaling factor C₀² for Hufnagel profile has been chosen in such a way that the integrals $\begin{array} {c} \int_{0}^{L}{\rm d}z{\ }C_{n}^{2}\left( z\right)\end{array}$ calculated for both profiles be equal. From the practical point of view it means that the same seeing conditions are considered for both cases.

  

Figure 1: Theoretical (Hufnagel) and experimental (San Pedro Mártir) Cn2 profiles. The corresponding Fried parameter is equal to 16 cm (for wavelength 0.55 $\mu$m). The scaling factor for Hufnagel model has been chosen in such a way that the Fried parameters associated with two profiles be equal

Figures 2 and 3 plot the Strehl ratio calculated for both profiles versus the angular separation between the two stars. Figure 2 shows the ratio of interest for 2.1 m telescope which operates now in San Pedro Mártir observatory, while Fig. 3 presents the results for 6.5 m telescope which is planned for building in this observatory. Both graphs are plotted for three wavelengths which are of interest in astronomical applications: 0.55, 1.25 and 2.2 $\mu$m. Using these graphs, one can estimate the upper limit of the efficiency of off-axis adaptive correction.

  

Figure 2: Strehl ratio versus the angular separation between the stars for 2.1 m telescope

As one can see from Figs. 2 and 3, despite of the seeing conditions associated with both profiles are equal, there is a significant difference in results to be obtained for off-axis adaptive correction. This difference appears because the profiles differ strongly in structure. The theoretical profile assumes a more or less smooth behavior of C_n² with altitude, while the experimental one has a pronounced layered structure with a great amount of the turbulence strength concentrated near to the ground. Since, as it follows from Eq. (5), the quality of correction is affected by the quantity $\begin{array} {c} \int_{0}^{L}{\rm d}zz^{5/3}C_{n}^{2}\left( z\right) ,\end{array}$ the profile with a turbulence strength concentrated mainly near-to-the ground gives better results for off-axis correction.

  

Figure 3: Strehl ratio versus the angular separation between the stars for 6.5 m telescope

Basing on the results obtained, one may arrive to some conclusions in what concerns to the choice of the telescope location. On the one side, an off-axis adaptive correction is a promising approach which allows to override certain difficulties arising in observations of weak stars. However, the information which can be extracted from the conventional measurements of seeing conditions is not sufficient to predict the quality of such correction. So one needs to carry out some more advanced experiments allowing for the reconstruction of C_n² profile. Such measurements can be performed making use of different experimental techniques. We believe that the G-SCIDAR method is among the best candidates because it combines a good accuracy of profile reconstruction with a relatively low cost of measurements.

Acknowledgements

We are indebted to Dr. J. Vernin for kindly authorizing our use of the G-SCIDAR data in this investigation. This work was supported by Consejo Nacional de Ciencia y Tecnologia (Mexico) project 1020P-E9507 and by Sistema Nacional de Investigadores (Mexico).