We assume that the observed astronomical target is high-order corrected with
the LGS itself. In this way its size shrinks to a diffraction limited core.
Following Olivier & Gavel (1994), but using a circular
entrance pupil of diameter D,
the final resolution experienced in presence of a residual
jitter 
rms exceeds the diffraction
limit by the factor F given from:
Accepting a degradation of the resolution about 30%
more than the diffraction limit one can find from Eq. (1),
using F=1.3, a maximum residual jitter of 
,
a value comparable to other figures given in the literature (Olivier
et al. 1993).
Defined an aperture diameter of the auxiliary telescopes 
and an
integration time 
a single exposure will collect N* photons
given, for an A0V star of mag V, by (Zombeck 1990):
where 
is expressed in nm and q is the overall quantum
efficiency. Assuming a Poissonian noise the SNR of the observed star
is just 
and, following Tyler & Fried (1982), one
gets:
The additional 
factor has been introduced by Olivier & Gavel
(1994) and it is related to the ratio between the sampling and the
control loop frequency. When such a ratio is 10 as they suggest, some
is obtained.
Combining Eqs. (1)-(3) and assuming the auxiliary telescopes
as two
reflection mirrors and high Q.E. CCD, an overall q=0.8 can be figured out;
a bandwidth 
and a sampling time
s are used as a baseline for the following computation.
In this way one get the following relationship for the limiting magnitude V:
where the argument of the logarithm should be expressed in meters.
Using Bahcall & Soneira (1981) it is easy to retrieve the area in the sky where the probability to get one star is close to 100%.
However the right-handed parameters in Eq. (4) have to be fixed. In the
rest of the paper we assume a Fried parameter of 
in the
visible, corresponding to a median seeing of most of the telescope sites
where large telescopes do exist or are planned in the near term.
The ratio 
between
the diameters of the main and auxiliary telescope is a
more complicated issue. In fact the smaller is the ratio the smaller movements
will be requested to the auxiliary telescope. On the other hand it is to be
pointed out that it is easier to move around smaller telescopes rather
than larger ones.
Ratios smaller than the unity are, obviously, unefficents.
We adopted in the following a figure of 
. This translates into
very small and light, 
, telescopes for a 
class telescope and into 
telescopes for 
large telescope class.
When and a limiting magnitude V=12.8 is
obtained equal to a density of
stars per square degree at the North Galactic Pole (the worst
case) or, in other words,
to a mean search area of 
squared arcmin for a single
star.
However, as pointed out by REM95 two
auxiliary observers are required, forming an angle 
with respect to the main observatory.
In the most general case the angle 
between the two tilt reference stars
with respect the target star will differ from the ideal situation of
and the error associated to tilt measurement could increase.
The amount of this effect can be easily calculated:
one can align one axis (x) with one reference star
(see Fig. 1 (click here)) measuring tilt error along it then one can retrieve
the error relative to the perpendicular axis (y) projecting onto it the
tilt error of the second reference star measured on the oblique axis
(
).
Figure 1: The tilt error increasing is due to non-perpendicular
configuration of
auxiliary telescopes. The indetermination becomes larger
as decreases
The error along the perpendicular axis 
will be given by:
Assuming the errors in determining 
and 
are the same, and equal to , an error 
for
the orthogonal axis y can be worked out:
for departures from 
as much as 
,
an acceptable further degradation of 
% is obtained.