3 Sky coverage

In order to fulfile condition 3 the NGSs are to be located within an angle $D/(2h_{\rm max})$ from the optical axes, where $h_{\rm max}$ is the altitude of the highest relevant perturbing layer (here and in the following we assume zenital or nearly zenital observations). The area where the ${\cal N}$ NGSs stars are to be found is given, expressed in square degrees, by the following:

$$S= \pi \left( \frac{180}{\pi} \times \frac{D}{2h_{\rm max}}\right)^2 \approx 7.96 \, 10^{-6} \, D^2.$$ (7)

The numerical estimation is based upon the Roggermann et al. (1995) model, assuming $h_{\rm max} = 1.8 \, 10^4$ m and hence D is hereafter expressed in meters.

In contrast, recall that the usable area for classical NGS-based adaptive optics system is characterized by a circular zone of radius $\theta_0$, the so called isoplanatic patch, of size $0.3 r_0/h_{\rm avg}$.Assuming an average $h_{\rm avg} \approx 1.25 \, 10^3$ m for both the SLC-N and HV-21 models (the two give respectively $1.53 \, 10^3$ m and $9.73 \, 10^2$ m for $h_{\rm avg}$) a numerical estimation can be made also:

$$S_{\rm c}=\pi \left( \frac{180}{\pi} \times \frac{0.3 r_0}{h_{\rm avg}} \right)^2 \approx 3.71 \, 10^{-5}$$ (8)

where again the result is given in square degrees.

It should be pointed out in the latter that a single suitable NGS is to be found. However it is also remarkable that the points of the sky satisfying this last condition are biased by the presence of a relatively bright NGS within a small angle. Because of light scattering (or, at least, to the non negligible extension of the PSF) the sky background will be affected by some light negatively impacting extremely deep imaging. The classical sky coverage, or probability to find out a suitable NGS, is given by:

$$P_{\rm c}=1-\exp(-S_{\rm c}\rho)$$ (9)

regardless of the telescope diameter D. Using a limiting magnitude V₀=13.0 as reported in Sect. 2, sky coverage of $P_{\rm c} \approx 0.2\%$ for the Galactic poles ($b=90^\circ$) and $P_{\rm c}\approx 2\%$ for the Galactic plane ($b=0^\circ$) are retained.

In the tomographic case ${\cal N}$ stars are to be found and the probabilities composed in a multiplicative manner:

$$P = \left[ 1-\exp(- S \rho) \right]^{\cal N}.$$ (10)

  

Figure 2: Sky coverage for various conditions vs. telescope diameter. Continuous line is for V=12, $b=0^\circ$; dotted line is for V=12, $b=90^\circ$;dashed line is for V=14, $b=0^\circ$; dotted and dashed line is for V=14, $b=90^\circ$. Note for comparison the sky coverage obtained by classical NGS-based adaptive optics. The lines indicating 50% and 90% of sky coverage are also reported

Using the numerical estimation given in Eq. (7):

$$P \approx \left[ 1- \exp \left( -7.96 \, 10^{-6} D^2 \rho \right) \right]^{\cal N}$$ (11)

where one can note the dependence both from $\rho$ and D. Inversion of Eq. (11) for D gives the following:

$$D_{\rm P}\approx \sqrt{ \frac{- \ln \left( 1 - P^{1/{\cal N}} \right)}{7.96 \, 10^{-6} \rho} } \cdot$$ (12)

Imposing P=0.50 or P=0.90 one can find the diameter where 50% and 90% of sky coverage is reached:

$$D_{50} \approx \frac{482}{\sqrt{\rho}}; \; \; \; D_{90} \approx \frac{680}{\sqrt{\rho}}\cdot$$ (13)

Solving Eq. (12) for the classical NGS-based probabilities (P=0.02 for $b=0^\circ$ and P=0.002 for $b=90^\circ$) one can find out also the critical diameter $D_{\rm c}$ as defined in the first section. All these results are summarized in Table 2.

  

Table 2: The diameter of the telescopes, expressed in meters, where NGS-based tomographic technique provides sky coverages close to the NGS-based classic adaptive optics ($D_{\rm c}$), where the sky coverage reaches 50% (D₅₀) and 90% (D₉₀) for different limiting magnitude V and galactic latitude b. In the second column the average number of stars per square degree, brighter than V is given

Equation (10) and followings do not impose any particular geometry for the ${\cal N}$ stars. Hence there is some chance that these stars are placed on the sky in a way that avoids properly sensing some portion of the highest layers. This problem is only mentioned here.