3 Calculation results

As it follows from (2, 3), one needs to apply some model of C_n² profile in order to calculate the PSF of interest. A convenient 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 in r₀ parametrization (r₀ is the Fried parameter), this profile is expressed as (Kouznetsov et al. 1997):  

$$C_n^2(z) = C_0 r_0^{-5/3}k^{-2} \\ \quad\times \left[ \left... ...}{z_1} \right\} +\exp \left\{ -\frac{z}{z_2} \right\} \right] ,$$ (5)

where $C_0=1.027\ 10^{-3}$ m^-1, $z_0=4.632\ 10^3$ m, z₁=10³ m, $z_2=1.5\ 10^3$ m.

The calculation results with Eqs. (3-5) are presented in Figs. 1-4.

Figure 1 presents some examples of PSF's calculated for different star separations. The graphs at the left-side show the images while the corresponding isophotes of PSF are presented in the right-side graphs. One can see that the isophotes have a non-circular form that indicates clearly an anisotropy in PSF. Note that the PSF's are wider in the guide star direction (horizontal direction) than in the transversal one (vertical direction).

  

Figure 1: Examples of anisotropic PSF's. The PSF's are elongated towards to the direction of the guide star. The PSF's have been calculated for the following conditions: the Fried parameter r0=0.2 m, the telescope diameter D=2 m, the star separations $\gamma =6,$ 7.7 and 10 arcsec (from the top to the bottom)

In what follows we characterize the degree of anisotropy of PSF by its elongation towards the direction of the guide star. This elongation is calculated as the ratio of PSF's maximum width to the minimum one at the 0.5 level of PSF.

Figures 2-4 shows the elongation versus the star separation, Fried parameter and telescope diameter, respectively.

  

Figure 2: Elongation in PSF versus star separation. Note that the elongation decreases with the increase of Fried parameter. In other words, the longer is the wavelength, the smaller is the elongation

  

Figure 3: Elongation in PSF versus Fried parameter

  

Figure 4: Elongation in PSF versus telescope diameter

We show in the graphs above the elongation as a function of the Fried parameter r₀. However if one is interested in wavelength dependence, it can be calculated using the following formula

$$r_0=0.185\lambda ^{6/5}\left[ \int_0^L {\rm d}z C_n^2\left( z\right) \right]^{-3/5}, \nonumber$$

where $\lambda$ denotes the wavelength.

It was recently reported by Close (1998) that "Off-axis guide stars produce an elongation in the science object's PSF towards the direction of the guide star (usually about $0.15\hbox{$^{\prime\prime}$}$ of elongation at 12$\hbox{$^{\prime\prime}$}$ radial distances in J band at CFHT)''. Unfortunately, the information provided in this paper is not sufficient to perform a direct quantitative comparison with our theoretical results. Among other factors, the length difference between the major and minor axes of the observed star depends on the PSF level where the PSF size is measured, and this information is not available in the quoted work. We can, nevertheless, give some data about the expected axes length difference as computed from Eq. (4). Let us consider the case of a 4-m class telescope working at the J-band (1.25 microns) with r₀=0.4 m, C_n² profile specified by Eq. (5) and a separation $\gamma=12\hbox{$^{\prime\prime}$}$ between the guide and observed stars. With the assumed parameters, the major and minor axes of the observed star image (measured at the PSF normalized isophote 0.25) will be $0.27\hbox{$^{\prime\prime}$}$ and $0.14\hbox{$^{\prime\prime}$}$, respectively, showing an excess length of $0.13\hbox{$^{\prime\prime}$}$ towards the direction of the guide star. This excess length will depend on the PSF level chosen to measure it, as it has been already mentioned, and in this example it ranges from $0.04\hbox{$^{\prime\prime}$}$ at PSF level 0.5 to $0.20\hbox{$^{\prime\prime}$}$ at PSF level 0.1.