Because of Earth rotation it is necessary to move auxiliary telescopes around the main one in order to maintain the perspective superposition between LGS and tilt reference star.
The tracking path shape depends on several parameters:
target star declination 
, its angular separation 
from the tilt reference star and position angle PA of the
reference with respect to the target.
The geometry of the problem is shown in Fig. 2 (click here).
In the following an height 
(Happer et al.
1995) for the mesospheric Sodium layer is assumed.
Figure 2: Geometrical layout of the LGS and tilt reference star with
respect to the observatory. The observatory plane coordinates
are South-East
oriented while the coordinates of the plane tangent to celestial
sphere
in the LGS are "altazimuthal-like'' oriented with North
along 
axis but with opposite versus
To obtain tilt reference star position in the Sodium Layer plane coordinate
system, placed at an altitude H and centered on the LGS, two rotations
are needed: the first around the
axis by 
degrees, in order to make parallel
the plane tangent to the celestial sphere in the LGS and the Sodium Layer
plane, the second around the axis passing through the LGS and
perpendicular to the Sodium Layer plane, by the azimuth angle A,
in order to align 
axes to xy axes:
where the azimuth A, the height h and the projected separation 
on the Sodium Layer plane are given by the following set of relationships:
being 
the observatory latitude and HA the target hour angle.
The angle m is related to the parallactic angle q by the following
pair of equations:
One can finally retrieve the auxiliary telescope position taking into
account (see Fig. 3 (click here)) that its coordinates will be identical to
those of the tilt reference star on the Sodium Layer plane but with inverted
sign:
Figure 3: Position of the auxiliary telescope with respect to
observatory. Displacing the small telescope of an amount equal
to the projection of the tilt reference star on Sodium Layer
plane but with opposite versus, allows the auxiliary telescope
to see LGS superimposed on the reference star
Some results obtained for the observatory latitude
N
(corresponding to the site of the National Telescope Galileo, Barbieri
1996)
are shown in Fig. 4 (click here). The tracking
is performed along four hours of HA across meridian (
) and the filled circles are placed at steps of 
.
In Fig. 4 (click here)a are drawn different paths
calculated for values of interested by a zenithal
distance of 
with a 
step,
and 
; in this particular case
the telescope movement is southward.
Figure 4: Paths of auxilary telescope for different values of ,
PA and . In all cases the observatory latitude is
N. In a) is shown an example
with different declinations and fixed position angle
and separation. In b) also the separation is
varying while in c) this last is kept fixed and
position angle varies. For further details see the text
In Fig. 4 (click here)b are
shown paths with fixed but different
values of : note that, as expected, the shapes are identical
but for a scale factor.
In Fig. 4 (click here)c the paths are calculated for different PA values
and fixed 
. The isolated filled dot
identifies observatory position (located, in all cases, at the origin of
the coordinate axis).