Definitions

Define the following variables:

F

The focal length (mm).

$\psi$

The sky position angle in radians.

$\psi$₀

The offset between the coordinate system of the guide probe and that of the instrument rotator, in radians.

$\epsilon$

The handedness of the guide-probe coordinate system ($\pm$1).

x y

Guide probe coordinates, in encoder units.

x$$_min x$$_max y$$_min y$$_max

xy limits on the area within which guide stars may be selected, in encoder units.

r $\theta$

Polar coordinates of the guide probe (encoder units; r is positive by definition).

$\theta$_min θ_max r$$_min r$$_max

Limits on the area within which guide stars may be selected, in polar coordinates; encoder units).

e$$_r e$$_θ

The linear and angular encoder scales in mm/encoder unit and rad/encoder unit, respectively.

x$$₀ y$$₀

The position of the centre of rotation in encoder units in the guide probe coordinate system (used for xy systems only).

r$$₀

The radial offset for guide probe coordinates, in encoder units (used for r$\theta$ systems only).

All of the existing A&G boxes on the La Palma telescopes use xy stages for their guide probes, with the exception of the WHT Cassegrain system, which has an r$\theta$ arrangement. A general description of the search areas may be given as follows (note that not all configurations use every constraint):

1. $x$_min < x < x_max and $y$_min < y < y_max (within a rectangle).
2. $[(x-x$₀)² + (y-y₀)²]^1/2 ≤r_max or $r\; \le r$_max (within a circle: this may be either a result of vignetting or a restriction on the mechanical travel of the probe).
3. $(x-x$₀)²/l² + (y-y₀)²/m² > 1 (inner ellipse excluded). A minimum radial travel for an r$\theta$ system is a special case. A similar condition, $[(x-x$₀)² + (y-y₀)²]^1/2 > r_min or $r\; >\; r$_min is used for the ROTATOR ANY option, where the available area is always a circular annulus.
4. - or $x\; >\; x$+ or - or $y\; >\; y$+ (i.e. the area $x$- < x < x+ and $y$- < y < y+ is excluded).
5. The probe azimuth is in the range $\theta$_min < θ< θ_max.