Definition of the rotation system of a mount

Relation between celestial coordinates and the encoders of the mount

The enc (encoder) system is defined by motor encoders of a telescope mount. The usual unit is inc (increment).

The cel (celestial) system is defined by local apparent celestial coordinates. The usual unit is deg (degrees).

The choice of the local apparent celestial coordinates type (ha, dec) or (az, elev) depends on the type of the mount (equatorial, altaz).

Note that astronomical coordinates astro (ra, dec, equinoxe) of celestial object catalogs are not locals but the conversion into cel system is univoque using celestial mechanic relations.

Many difficulties exist to convert systems enc to cel:

• The conversion enc to cel is not bijective because a pointing direction in the sky cel can be realized by two different enc obtained by a back flip of the mount. For example, (ha,dec) = (45, 50) corresponds to the same direction than (ha,dec) = (225,130).
• The zero points of enc may not coorespond to zero points of cel.
• The increasing angles of enc may not correspond to the increasing sense of cel.

We must simplfy the conversion between enc and cel introducting an intermediate universal system which allows to decouple the problems of senses and zero points of encoders with the back flips of the mount.

The conversion of enc to cel will be realized using the rot (rotation) system.

Definition of the angles rotb, rotp

The system of rotb,rotp angles is linked to the machanics of the mount axes. The goal is to have a local spherical coordinate system which have no ambiguity and easily recognizable on any type of mounts.

rotb,rotp are spherical coordinates according the trihedron rx,ry,rz:

• The rz axis is directed toward the mechanical roation pole which is above horizon.
• The (rx,0,ry) plan turns around the rz axis according the rotb (basis angle).
• The rx axis is perpendicular to the rz axis and directed toward the hihest elevation (meridian).
• The ry axis is defined by the direct trihedron with rx and rz.
• The (rx,0,rz) turns around the ry axis according the angle rotp (polar angle).

The origin of rotb,rotp:

• The angle rotp=0 is at the visible pole from the observation site (rx,ry,rz) = (0,0,1).
• The angle rotb=0 is at the meridian with (rx,ry,rz) = (1,0,0).

The sense of rotb,rotp:

• The increasing sense of rotp is direct in the (ry,0,rz) plane observed from the end of the ry axis.
• The increasing sense of rotb is direct in the (rx,0,ry) plane observed from the end of the rz axis.

Attention, the pointed direction (rotb, rotp) may not correspond to (ha, dec) because the optical tube could be offseted by an angle according the zero points of rotb, rotp.

The back flip is univokely identified by the sign of rotp. It is the main advantage of the rot system:

• side = 1 if rotp>=0
• side = -1 if rotp<0

Relations between (rotb, rotp) and (celb, celp)

Northern hemisphere

Relations:

• Hour angle = celb = -rotb
• Declination = celp = 90-abs(rotp)
• Side = sign of rotp

Southern hemisphere

Relations:

• Hour angle = celb = rotb
• Declination = celp = abs(rotp)-90
• Side = sign of rotp