The capability of the technique to recover the absolute tilt of the LGS is related to the joint probability to find two stars, whose magnitude is equal or brighter than the limiting magnitude defined in the second section, located at angular distances respectively d₁ and d₂ from the observed object as small as possible and forming with respect to the latter an angle of about

, in order to reduce to a minimum the error on the tilt determination.

Projecting the problem to the Earth surface, we have the joint probability to exploit two tracks for the small telescopes as close as possible to the main observatory. The area in the neighbourhood of the observatory will have different occupation frequency densities for the auxiliary telescopes depending upon the ratio between the main and the auxiliary telescope diameters

, the Fried parameter r₀, the tolerance angle

around the nominal position of

and the coordinates of the object.

Using the values given in Sect. 2, several simulations about the ground occupation for

N have been performed. For a density of

the mean distance of the star nearest to the object <d₁> and of the nearest second star <d₂>, within different tolerance angles (view as a cone of aperture

centered around the

position) have been computed averaging over 5000 simulation runs.

Figure 5: Mean distance from the observed object of the nearest second star versus the tolerance angle

: the filled circles calculated through the simulations are well fitted by the -1/2-law of Eq. (14) and the asymptotic trend gives a value of

The mean distance of the nearest star <d₁> depends only upon the star density n, and its value is equal to:

The computation of the second nearest star is more complicated because of the constraint imposed by the first nearest star. Using the observation that the square of the mean distance is proportional to the available number of stars in a given sky region one can figure out the following relationship:

where the fixed addictive term is due to take into account the finite value of <d₂> when

approaches

. Numerical simulations and fitting of the obtained data lead to an estimation, for our case study, of

and

. The results are plotted in Fig. 5 (click here).

In order to define the tracks distribution around the observatory we performed a MonteCarlo simulation in which we calculated the tracks for the star and its almost perpendicular companion closest to the object for different tolerance angles, using a star field of poissonian mean density equal to

and objects of random zenithal distance

considering an observation time of 4 hours across the meridian (

The results for the case

are shown in Fig. 6 (click here): the simulated data are rappresented with filled circles and a poissonian fitting is superimposed. The figure shows a poissonian-like distance distribution both for the nearest star and for its perpendicular companion. We performed 5000 simulation runs to figure out the described results.

Figure 6: Distribution of the tracks around the observatory: the MonteCarlo data results are well fitted by a Poissonian curve. The upper plot, referred to the nearest star, shows a sharper shape with respect to the case of the perpendicular companion (lower plot)

A simple view of the MonteCarlo simulation with only 200 tracks is shown in Fig. 7 (click here): the tracks of the perpendicular companion are marked at their ends with filled circles. The different distances of the paths for the two reference stars are evident.

Given a circular area of radius R around the observatory it is possible to retrieve the percentage of tracks inside it, and to define the sky coverage of the system. The results are shown in Table 1 (click here) for different tolerance angles where the other parameters are the same used in this section; accepting a further degradation of the measurement error due to the non perpendicularity of the two reference stars (29%, 48% and 73% respectively for

and

), it is possible, within 1 km, to project the LGS through the auxiliary telescopes and track them for approximately all the reference stars found in proximity of the observed object.